Journal of Functional Analysis 258 (2010) 1–19 www.elsevier.com/locate/jfa
On OL∞ structure of nuclear, quasidiagonal C ∗ -algebras Caleb Eckhardt Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, United States Received 25 June 2008; accepted 2 October 2009
Communicated by Alain Connes
Abstract We continue the study of OL∞ structure of nuclear C ∗ -algebras initiated by Junge, Ozawa and Ruan. In particular, we prove if OL∞ (A) < 1.005, then A has a separating family of irreducible, stably finite representations. As an application we give examples of nuclear, quasidiagonal C ∗ -algebras A with OL∞ (A) > 1. © 2009 Elsevier Inc. All rights reserved. Keywords: Operator spaces; C ∗ -algebras; Quasidiagonal; Nuclear C ∗ -algebras
1. Introduction This paper continues the study of OL∞ -structure of nuclear C ∗ -algebras initiated by Junge, Ozawa and Ruan in [8]. Before describing the contents of this paper, we recall the necessary definitions and results. Let V and W be n-dimensional operator spaces and consider the completely bounded version of Banach–Mazur distance: dcb (V , W ) = inf ϕcb ϕ −1 cb : ϕ : V → W is a linear isomorphism . Let A be a C ∗ -algebra. For λ > 1 we say that OL∞ (A) λ if for every finite-dimensional subspace E ⊂ A, there exist a finite-dimensional C ∗ -algebra B and a subspace E ⊂ F ⊂ A such E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.004
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C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
that dcb (F, B) λ. Then define OL∞ (A) = inf λ: OL∞ (A) λ . A is a rigid OL∞ space if for every > 0 and every x1 , . . . , xn ∈ A there is a finite-dimensional C ∗ -algebra B and a complete isometry ϕ : B → A such that dist(xi , ϕ(B)) < for i = 1, . . . , n. OL∞ is an interesting invariant for C ∗ -algebras, particularly when one considers the interplay between OL∞ and various approximation properties of C ∗ -algebras. It follows easily from the definition, that if OL∞ (A) < ∞, then there is a net of matrix algebras (Mni ) and linear maps αi : A → Mni , βi : Mni → A such that βi αi tends to the identity on A pointwise and supi αi cb βi cb < ∞. Pisier showed [11, Theorem 2.9] that this implies A is nuclear. Conversely, it was shown in [8] if A is nuclear, then OL∞ (A) 6. This estimate was improved in [7] when the authors showed that all nuclear C ∗ -algebras A have OL∞ (A) 3. So, OL∞ is most useful when restricted to nuclear C ∗ -algebras. Another important approximation property is quasidiagonality (QD). We refer the reader to the survey article [5] for information on QD C ∗ -algebras. The following relationships between QD and OL∞ were established in [8]: A is a rigid OL∞ space
(i)
−→
OL∞ (A) = 1
(ii)
−−→
A is nuclear & QD.
Blackadar and Kirchberg showed [3, Proposition 2.5] that all 3 of the above assertions are equivalent if A is either simple or both prime and antiliminal. The main purpose of this paper is to give examples showing that the converse of (ii) does not hold in general. In Section 2 we prove the necessary technical results used throughout the paper. Section 3 contains our first counterexamples to (ii). Section 4 contains some results about permanence properties about OL∞ . In Section 5 we prove the main result that all unital C ∗ -algebras A with OL∞ (A) < 1.005 have a separating family of irreducible, stably finite representations. This provides a larger class of nuclear quasidiagonal C ∗ -algebras A with OL∞ (A) > 1, but also has implications for the converse of (i) which we discuss at the end of the paper. 2. Technical lemmas In this section, we gather some technical lemmas needed for Sections 3 and 4, and fix our notation. Throughout the paper, if H is a Hilbert space, we let B(H ) denote the space of bounded linear operators on H . For H n-dimensional we write 2 (n), and Mn for B(2 (n)). We write ucp and cpc as shorthand for “unital completely positive” and “completely positive contraction” respectively. For linear maps ϕ : V → W between operator spaces we write ϕ (n) for idMn ⊗ ϕ : Mn (V ) → Mn (W ), and ϕcb = supn ϕ (n) . Furthermore if ϕ is injective, we write ϕ −1 for the norm of the map ϕ −1 : ϕ(V ) → V . We write ⊗ for the minimal tensor product of C ∗ algebras. The following lemma is implicit in the proof of [8, Theorem 3.2]. √ Lemma 2.1. Let 0 < δ < 1/ 2, and let A be a unital C ∗ -algebra with OL∞ (A) < 1 + δ 2 /2. Let F ⊂ A be a finite subset. Then there is a finite-dimensional C ∗ -algebra B, a linear map ϕ : B → A with ϕcb < 1 + δ 2 /2 and a ucp map ψ : A → B such that F ⊂ ϕ(B) and ψϕ − idB cb < 1 + δ 2 /2 2 δ 2 + δ 4 /4 .
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In [8, Theorem 3.2] the authors require δ < 1/16. The reason for this is to guarantee that ψ is approximately multiplicative on F . We will not need approximate multiplicativity in this paper, √ which is why we are able to relax this condition to δ < 1/ 2. We need the following slight variation of Lemma 2.1. √
Lemma 2.2. Let λ < ( 1+2 3 )1/2 and A be a unital C ∗ -algebra with OL∞ (A) < λ. Let F ⊂ A be a finite subset. Then there is a finite-dimensional C ∗ -algebra B, a ucp map ψ : A → B and a unital, self-adjoint map ϕ : B → A such that: (i) ϕcb <
√λ
1−λ
(ii) F ⊂ ϕ(B). (iii) ψϕ = idB . (iv) ϕψ|F = idF .
2(λ2 −1)
.
Proof. Without loss of generality suppose F consists of positive elements. We apply Lemma 2.1 with λ = 1 + δ 2 /2 to obtain a finite-dimensional C ∗ -algebra B, a ucp map ψ : A → B, and a linear map ϕ : B → A such that F ⊂ ϕ(B), ϕcb < λ and ψϕ − idB cb < λ 2(λ2 − 1) < 1. Then ψϕ is invertible in the Banach algebra of all completely bounded maps on B. Let ϕ = ϕ(ψϕ)−1 . Then ϕ cb ϕcb (ψϕ)−1 cb λ
1
1 − λ 2(λ2 − 1)
.
Then ϕ satisfies (i)–(iii). Moreover, since ψ is unital and ψϕ = idB , it follows that ϕ is unital. Finally, let ϕ
(x) = 1/2(ϕ (x) + ϕ (x ∗ )∗ ), for x ∈ B. Then ϕ
is unital, self-adjoint and
ϕ cb ϕ cb . Since ψ is positive, it follows that ψϕ
= idB . To see (ii), let b ∈ B such that ϕ (b) ∈ F . Since F consists of positive elements, b = ψϕ (b) 0. Hence, ϕ
(b) = 1/2(ϕ (b) + ϕ (b)∗ ) = ϕ (b) ∈ F . Condition (iv) is a consequence of (ii) and (iii). 2 Lemma 2.3. Let A be a unital C ∗ -algebra and let x ∈ A. Set x1 =
x1 x ∈ M2 ⊗ A. x1 x∗
(2.1)
Then x1 = 2x. Proof. Without loss of generality, assume that x = 1. Clearly x1 2. For the reverse inequality, suppose that A ⊂ B(H ) unitally for some Hilbert space H . By spectral theory there is a sequence of unit vectors (ηk ) ⊂ H such that lim x ∗ xηk − ηk = 0.
k→∞
For each k ∈ N set 1 ξk = √ 2
xηk ηk
∈ H ⊕ H.
(2.2)
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Then ξk 1 and by (2.2), it follows that 1 2xηk = 2. lim x1 ξk = lim √ ∗ x xηk + ηk k→∞ k→∞ 2 Hence, x1 2.
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Lemma 2.4. Let A and B be C ∗ -algebras with A unital, and 1/2 < r 1. Let ϕ : A → B be a cpc such that for every k ∈ N and a ∈ Mk ⊗ A with a 0, ϕ (k) (a) ra. Then ϕ is injective with ϕ −1 cb (2r − 1)−1 . Proof. Let n ∈ N and x ∈ Mn ⊗ A. Let x1 ∈ M2 ⊗ (Mn ⊗ A) be as in Lemma 2.3. Then, x1 0 and x1 = 2x. By assumption, we have 2rx ϕ (2n) (x1 ) xϕ (n) (1) ϕ (n) (x) = ϕ (n) (x)∗ xϕ (n) (1) x1 ϕ (n) (x) (n) ∗ ϕ (x) x1 x + ϕ (n) (x). Hence ϕ (n) (x) (2r − 1)x, from which we conclude that −1 ϕ (2r − 1)−1 . cb
2
We recall the following well-known corollary to Stinespring’s Theorem. Lemma 2.5. Let A and B be unital C ∗ -algebras and ψ : A → B a ucp map. Then for every a ∈ A, we have ψ(a)∗ ψ(a) ψ(a ∗ a). Lemma 2.6. Let L1 and L2 be Hilbert spaces and n ∈ N. Let ϕ : Mn → B(L1 ) ⊕ B(L2 ) be an √ injective cpc with ϕ −1 cb = r −1 < 2/( 6 − 1). Let ϕi : Mn → B(Li ) denote the coordinate maps of ϕ for i = 1, 2. Suppose there is a k ∈ N and a ∈ Mk ⊗ Mn of norm 1 and a 0 such that (k) ϕ (a) = s < r 2 + r − 1 /r. 2
Then ϕ1 is injective and 2 −1 −1 ϕ r − 1 − r . 1 cb 1−s
(2.3)
C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
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Proof. By [13, Theorem 2.10] every bounded map ψ from an operator space into Mn is completely bounded with ψ (n) = ψcb . So, we may assume that k = n. Also by [13, Theorem 2.10], to prove inequality (2.3) it suffices to show that for every x ∈ Mn ⊗ Mn of norm 1, we have 2 (n) ϕ (x) r − 1 − r . 1 1−s
(2.4)
By Wittstock’s extension theorem [9, Theorem 8.2], let : B(L1 ) ⊕ B(L2 ) → Mn ψ cb = ϕ −1 cb = r −1 . Let ψ = r ψ . Then be an extension of ϕ −1 : ϕ(Mn ) → Mn with ψ ψcb = 1 and ψϕ(x) = rx
for all x ∈ Mn .
(2.5)
By the factorization theorem for completely bounded maps [9, Theorem 8.4] there is a unital representation (π, H ) of Mn ⊗ B(L1 ) ⊕ B(L2 ) = B L1 ⊗ 2 (n) ⊕ B L2 ⊗ 2 (n) and isometries S, T : 2 (n) ⊗ 2 (n) → H such that T ∗ π(x)S = ψ (n) (x)
for every x ∈ B L1 ⊗ 2 (n) ⊕ B L2 ⊗ 2 (n) .
(2.6)
Let qL1 = π(1L1 ⊗2 (n) , 0) ∈ B(H ) and qL2 = π(0, 1L2 ⊗2 (n) ) ∈ B(H ). We now show that the ranges of S and T are almost included in qL1 (H ). Let ξ1 ∈ 2 (n) ⊗ 2 (n) be a norm 1 eigenvector for a with eigenvalue 1. Let ω1 ∈ Mn ⊗ Mn (n) be the orthogonal projection onto Cξ1 . Then ω1 a. Since ϕ2 is cp, we have ϕ2 (ω1 ) (n) ϕ2 (a) = s. Extend ξ1 to an orthonormal basis ξ1 , ξ2 , . . . , ξn2 for 2 (n) ⊗ 2 (n). For i = 1, . . . , n2 define the rank 1 operators, ωi (η) = η, ξi ξ1 ,
for η ∈ 2 (n) ⊗ 2 (n).
Then ωi ωj∗ = δi,j ω1 ,
for 1 i, j n2 .
(2.7)
2 2 Let η = ni=1 αi ξi ∈ 2 (n) ⊗ 2 (n) of norm 1 and ωη = ni=1 α i ωi ∈ Mn ⊗ Mn . By (2.7) and Lemma 2.5, it follows that (n) ϕ (ωη ) = ϕ (n) (ωη )ϕ (n) (ωη )∗ 1/2 2
2
n2
i=1
2
1/2
(n) |αi |2 ϕ2 (ω1 )
s 1/2 .
(2.8)
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Combining (2.5) and (2.6), we have rξ1 = rωη (η) = ψ (n) ◦ ϕ (n) (ωη )η = T ∗ π ϕ (n) (ωη ) Sη. Therefore, by (2.8) 2 r 2 π ϕ (n) (ωη ) Sη (n) 2 2 (n) = π ϕ1 (ωη ), 0 qL1 Sη + π 0, ϕ2 (ωη ) qL2 Sη qL1 Sη2 + sqL2 Sη2 .
(2.9)
Combining (2.9) with the fact that S is an isometry, we obtain 1 = qL1 Sη2 + qL2 Sη2 r 2 − sqL2 Sη2 + qL2 Sη2 . Since η ∈ 2 (n) ⊗ 2 (n) was an arbitrary vector of norm 1, it follows that qL2 S
1 − r2 1−s
1/2 (2.10)
.
Define ψ ∗ : B(L1 ) ⊕ B(L2 ) → Mn by ψ ∗ (x) = ψ(x ∗ )∗ . By the complete positivity of ϕ it follows that ψ ∗ ϕ = r · idMn . Moreover note that ∗ (n) (x) = S ∗ π(x)T . ψ So, by replacing ψ with ψ ∗ (and hence S with T ) in the above proof we obtain ∗ T qL = qL T 2 2
1 − r2 1−s
1/2 .
Let x ∈ Mn ⊗ Mn be arbitrary of norm 1. By (2.5), (2.6), then (2.10) and (2.11), we have r = ψ (n) ϕ (n) (x) = T ∗ π ϕ (n) (x) S (n) (n) = T ∗ qL1 π ϕ1 (x), 0 qL1 + qL2 π 0, ϕ2 (x) qL2 S (n) (n) ϕ1 (x) + T ∗ qL2 π 0, ϕ2 (x) qL2 S (n) 1 − r 2 ϕ1 (x) + . 1−s This proves (2.4) and the lemma.
2
(2.11)
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We will be careful with our norm estimates throughout the paper. Thus, the technical nature of Lemma 2.6. Colloquially it states; regardless of the value of n ∈ N, if ϕ is almost a complete isometry, then either ϕ1 or ϕ2 is almost a complete isometry. In particular, we have: Corollary 2.7. Let L1 , L2 , n and ϕ be as in Lemma 2.6, but with ϕ −1 cb = r −1 < 125/124. Then either ϕ1 or ϕ2 is injective, and −1 ϕ 1 + (r − 1)1/3 −1 i cb for either i = 1 or i = 2. Proof. If ϕ2 is injective with ϕ2−1 cb < (1 + (r − 1)1/3 )−1 , we are done. If not, then there is (n) an x ∈ Mn ⊗ Mn of norm 1 such that ϕ2 (x) < 1 + (r − 1)1/3 . Then Lemma 2.4 provides an a ∈ Mn ⊗ Mn of norm 1 with a 0 such that (n) 1 ϕ (a) 1 + ϕ (n) (x) 1 2 + (r − 1)1/3 . 2 2 2 2 (n)
We now apply Lemma 2.6 with s = 12 (1 + ϕ2 (x)) to obtain, 2 −1 −1 −1 ϕ r − 1 − r 1 + (r − 1)1/3 , 1 cb 1−s which holds whenever 124/125 < r 1.
2
Finally, we recall 2 useful perturbation lemmas. Lemma 2.8. (See [16, Proposition 1.19].) Let A be a unital C ∗ -algebra and N an injective von Neumann algebra. Let ϕ : A → N be a unital self-adjoint map with ϕcb 1 + for some > 0. Then there is a ucp map t : A → N such that t − ϕcb . Lemma 2.9. (See [12, Lemma 2.13.2].) Let 0 < < 1 and X be an operator space. Let (xi , xi )ni=1 ∗ be a biorthogonal system with xi ∈ X and xi ∈ X . Let y1 , . . . , yn ∈ X be such that
xi xi − yi < .
Then there is a complete isomorphism w : X → X such that w(yi ) = xi and wcb w −1 cb 1+ 1− . 3. First examples √
For 1 λ < ( 1+2 3 )1/2 , let f (λ) =
λ , √ 1 − λ 2(λ − 1)
(3.1)
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and consider the real polynomial, g(y) = y(1 + y)(y − 1)(2 − y) − 2(2 − y)2 + 1.
(3.2)
Note that f (λ) → 1 as λ → 1, and g(1) = −1. Let λ in the domain of f be such that g f (λ ) < 0.
(3.3)
A calculation shows that any λ < 1.005 satisfies (3.3). Theorem 3.1. Let A be a unital C ∗ -algebra and let λ satisfy (3.3). Suppose that A has a unital faithful representation (π, Hπ ) = (ρ ⊕ σ, Hρ ⊕ Hσ ), such that ker(σ ) = {0}. Furthermore suppose there is a sequence (xn ) in the unit sphere of A such that ρ(xn ) is an isometry for each n, and ρ(xn xn∗ ) → 0 strongly in B(Hρ ). Then OL∞ (A) λ . Proof. Let a ∈ ker(σ ) be positive and norm 1. Choose n large enough so ρ(1 − xn xn∗ )ρ(a)ρ(1 − xn xn∗ ) = 0. Set y = xn , and let −1 b = 1 − yy ∗ a 1 − yy ∗ 1 − yy ∗ a 1 − yy ∗ . Then σ (b) = 0, hence 1 = b = ρ(b). Since ρ(1 − yy ∗ ) is a projection, it follows that ρ(b) ρ(1 − yy ∗ ), hence π(b) π 1 − yy ∗ .
(3.4)
Suppose that OL∞ (A) < λ , and obtain a contradiction. Let F = {b, y, y ∗ }. Let f and g be as in (3.1) and (3.2). We apply Lemma 2.2 to obtain a finite-dimensional C ∗ -algebra B, a ucp map ψ : π(A) → B and a unital, self-adjoint map ϕ : B → π(A) such that ϕcb < f (λ ),
ψϕ = idB ,
and ϕψ|π(F ) = idπ(F ) .
(3.5)
By Lemma 2.8, there is a ucp map t : B → B(Hρ ) ⊕ B(Hσ ) such that t − ϕcb < f (λ ) − 1.
(3.6)
Let n ∈ N and x ∈ Mn ⊗ B. Since ψϕ = idB , it follows that ϕ (n) (x) x. Therefore, (n) (n) (n) t (x) ϕ (x) − ϕ (x) − t (n) (x) x − f (λ ) − 1 x = 2 − f (λ ) x. Hence t is injective with −1 t 2 − f (λ ) −1 . cb
(3.7)
C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
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Let qρ and qσ denote the orthogonal projections of Hρ ⊕ Hσ onto Hρ ⊕ 0 and 0 ⊕ Hσ respectively. By (3.5) and (3.6) we have qσ tψ π(b) qσ ϕψ π(b) + f (λ ) − 1 = σ (b) + f (λ ) − 1 = f (λ ) − 1.
(3.8)
Let p ∈ B be a minimal central projection such that pψ(π(b)) = ψ(π(b)). Then pB ∼ = Mn for some n ∈ N. Using (3.7) and (3.8), we apply Lemma 2.6 with s = qσ t pψ π(b) qσ tψ π(b) f (λ ) − 1 and −1 r −1 = (t|pB )−1 2 − f (λ ) cb
to obtain, (qρ t|pB )−1 cb
2(2 − f (λ ))2 − 1 2 − f (λ )
−1 .
(3.9)
Recall that for any finite C ∗ -algebra C and any contractive x ∈ C, we have 1 − xx ∗ = 1 − x ∗ x .
(3.10)
In particular, (3.10) holds for any finite-dimensional C ∗ -algebra. We will use (3.9) to “isolate” ρ, then the fact that ρ(A) violates (3.10) to arrive at a contradiction: by (3.5) f (λ )−1 ϕ−1 cb ψ π(b) = pψ π(b) p 1 − ψ π(y)π y ∗ by (3.4) p 1 − ψ π(y) ψ π y ∗ (by Lemma 2.5) = p 1 − ψ π y ∗ ψ π(y) by (3.10) (qρ t|pB )−1 cb qρ t p 1 − ψ π y ∗ ψ π(y) (qρ t|pB )−1 cb qρ t 1 − ψ π y ∗ ψ π(y) (qρ t|pB )−1 cb qρ − qρ t ψ π y ∗ t ψ π(y) (by Lemma 2.5) (qρ t|pB )−1 cb qρ − qρ ϕ ψ π y ∗ ϕ ψ π(y) + t − ϕcb 1 + ϕcb by (3.5) = (qρ t|pB )−1 cb ρ 1 − y ∗ y + t − ϕcb 1 + ϕcb 2(2 − f (λ ))2 − 1 −1 f (λ ) − 1 1 + f (λ ) .
2 − f (λ ) The last line follows because ρ(y) is an isometry, by (3.5), (3.6) and (3.9). Hence g(f (λ )) > 0, a contradiction. 2
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In [8] it was asked if there were any nuclear, quasidiagonal C ∗ -algebras A with OL∞ (A) > 1. We give some examples of such algebras. Let λ satisfy (3.3). Example 3.2. Let s ∈ B(2 ) denote the unilateral shift. Then, A = C ∗ (s ⊕ s ∗ ) is nuclear and quasidiagonal. Applying Theorem 3.1 with ρ : A → C ∗ (s), σ : A → C ∗ (s ∗ ) and (xn ) = (s ⊕ s ∗ )n , we have OL∞ (A) > λ . Before the author obtained Theorem 3.1, Narutaka Ozawa outlined for me an alternate proof that OL∞ (C ∗ (s ⊕ s ∗ )) > 1. The proof was based on the observation that for any finitedimensional C ∗ -algebra B and any partial isometry v ∈ B, we have 1 − vv ∗ Murray–von Neumann equivalent to 1 − v ∗ v. But, if we let (eij ) denote matrix units for B(2 ) and T = s ⊕ s ∗ , then T is a partial isometry and 1 − T ∗ T = 0 ⊕ e11 and 1 − T T ∗ = e11 ⊕ 0. So, 1 − T ∗ T and 1 − T T ∗ are not Murray–von Neumann equivalent in C ∗ (s ⊕ s ∗ )
= B(2 ) ⊕ B(2 ). One can use these facts and arguments similar to Lemmas 2.2 and 2.8 to show that OL∞ (C ∗ (s ⊕ s ∗ )) > 1. Example 3.3. (See [6, Example IX.11.2].) Let D1 and D2 be commuting diagonal operators with joint essential spectrum RP2 , the real projective plane. Let s be as in Example 3.2. Set A = C ∗ (s ⊕ D1 , 0 ⊕ D2 ). Then, A is easily seen to be an extension of nuclear C ∗ -algebras and hence is nuclear. As is shown in [6], A is quasidiagonal. Applying Theorem 3.1, with ρ : A → C ∗ (s), σ : A → C ∗ (D1 , D2 ) and (xn ) = (s ⊕ N1 )n , we have OL∞ (A) > λ . 4. Permanence properties We now investigate a couple permanence properties of OL∞ . Let B ⊂ A be nuclear C ∗ -algebras with OL∞ (A) = 1. In general, we do not have OL∞ (B) = 1. Indeed let B = C ∗ (s ⊕ s ∗ ) from Example 3.2. It is easy to see that s ⊕ s ∗ is a compact perturbation of a unitary operator u ∈ B(2 ⊕ 2 ). Let A = C ∗ (u) + K(2 ⊕ 2 ). Then A is nuclear and inner quasidiagonal [3, Definition 2.2]. By [3, Theorem 4.5], A is a strong NF algebra, which is a rigid OL∞ -space by [2, Theorem 6.1.1]. Hence OL∞ (A) = 1, but OL∞ (B) > 1. In contrast to this situation, if B is an ideal we have the following: Theorem 4.1. Let A be a unital C ∗ -algebra and J an ideal of A. If OL∞ (A) = 1, then OL∞ (J ) = 1. Proof. Let > 0 and E ⊂ J a finite-dimensional subspace. Without loss of generality suppose E has a basis of positive elements x1 , . . . , xn ∈ E with xi = 1 for each i = 1, . . . , n. Let x1 , . . . , xn ∈ J ∗ such that xi , xj = δi,j . Set M = xi . Define δ1 (δ) = (1 − δ) − (1 −
√ −1 δ ) 1 − (1 − δ)2
for 0 δ < 1.
Note that δ1 (δ) → 1 as δ → 0. √ Choose δ > 0 small enough so 2 δ /M and (2δ1 (δ) − 1)−1 1 + .
C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
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By Lemma 2.2 we obtain a finite-dimensional C ∗ -algebra B = N i=1 Mni , a ucp map ψ : A → B and a unital, self-adjoint map ϕ : B → A with ϕcb 1 + δ such that ϕψ|E = idE and ψϕ = idB . is almost com ⊂ J ∗∗ such that F We will construct a finite-dimensional subspace E ⊂ F ∗ pletely isometric to a finite-dimensional C -algebra and then apply a key theorem from [8] to obtain a subspace E ⊂ F ⊂ J such that F is almost completely isometric to a finite-dimensional C ∗ -algebra. Since A∗∗ is injective, Lemma 2.8 provides a ucp map t : B → A∗∗ such that t − ϕcb δ. Then t is injective and t −1 cb (1 − δ)−1 . Let p ∈ A∗∗ be the central projection such that pA∗∗ = J ∗∗ . Let t1 : B → J ∗∗ be defined by t1 (x) = pt (x) and t2 : B → A∗∗ by t2 (x) = (1 − p)t (x). Returning to the C ∗ -algebra B, let q1 , . . . , qN ∈ B be the minimal central projections such that qi B ∼ = Mni . Let √ I = 1 i N : sup qi ψ(xj ) δ . 1j n
(4.1)
Set q = i ∈/ I qi and C = qB. −1 We now show that t1 : C → J ∗∗ is injective with t1 |−1 C cb (2δ1 (δ) − 1) . We first show t1 restricted to each summand of C is almost a complete isometry. √ To this end, let I c = {1, . . . , N} \ I and j ∈ I c . Then there is an xi such that qj ψ(xi ) > δ. Since xi ∈ J ∩ E, we have (1 − p)ϕψ(xi ) = (1 − p)xi = 0.
(4.2)
Since ψ is ucp, qj ψ(xi )−1 qj ψ(xi ) ∈ qj B ∼ = Mn
j
(4.3)
is norm 1 and positive. Combining (4.2) and (4.3) we obtain t2 qj ψ(xi )−1 qj ψ(xi ) qj ψ(xi )−1 t2 ψ(xi ) −1 = qj ψ(xi ) (1 − p)tψ(xi ) −1 qj ψ(xi ) (1 − p)ϕψ(xi ) + δ √ δ. We apply Lemma 2.6 to t1 : qj B → J ∗∗ with −1 √ s = t2 qj ψ(xi ) qj ψ(xi ) δ and −1 −1 r −1 = t|−1 qj B cb t|B cb (1 − δ) to obtain −1 t1 | δ1 (δ)−1 qj B cb
for all j ∈ I c .
(4.4)
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Now, let k ∈ N be arbitrary and a = pletely positive, by (4.4),
j∈ / I (1k
⊗ qj )a ∈ Mk ⊗ C be positive. Since t1 is com-
(k) t (a) supt (k) (1k ⊗ qj )a δ1 (δ)a. 1
j∈ /I
1
By Lemma 2.4, t1 : C → J ∗∗ is injective with t1−1 cb (2δ1 (δ) − 1)−1 . t1 (C) does not necessarily contain x1 , . . . , xn . We fix this with a perturbation. Since pxi = xi = ϕψ(xi ) for i = 1, . . . , n, it follows from (4.1) that √ xi − t1 ψ(xi )q xi − t1 ψ(xi ) + δ √ = xi − ptψ(xi ) + δ √ xi − pϕψ(xi ) + δ + δ √ = δ + δ. Set yi = t1 (ψ(xi )q) ∈ J ∗∗ for i = 1, . . . , n. Then, n
√ xi xi − yi M(2 δ ) .
i=1
By Lemma 2.9 there is a complete isomorphism w : J ∗∗ → J ∗∗ such that w(yi ) = xi for i = 1, . . . , n and wcb w −1 cb (1 + )/(1 − ). = wt1 (C) ⊂ J ∗∗ . Then E ⊂ F and Let F 2 , C) (1 + ) 2δ1 (δ) − 1 −1 < (1 + ) . dcb (F (1 − ) 1−
By [8, Theorem 4.3] there is a subspace F ⊂ J such that E ⊂ F and dcb (F, C) < (1 + )2 (1 − )−1 . Since > 0 was arbitrary, it follows that OL∞ (J ) = 1. 2 Remark 4.2. It is not known if Theorem 4.1 holds in general, i.e. if J is an ideal of A do we always have OL∞ (J ) OL∞ (A)? Remark 4.3. Blackadar and Kirchberg have shown [2, Proposition 6.1.7] that every hereditary subalgebra of a rigid OL∞ space is also a rigid OL∞ space. It is not known if Theorem 4.1 can be extended to include hereditary sub C ∗ -algebras. Finally, we need the following Proposition for Section 5. For C ∗ -algebras A and B, let A B denote the algebraic tensor product of A and B. Proposition 4.4. Let A1 and A2 be nuclear C ∗ -algebras. Then OL∞ (A1 ⊗ A2 ) OL∞ (A1 )OL∞ (A2 ).
C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
13
Proof. Let E ⊂ A1 A2 be a finite-dimensional subspace and > 0. For i = 1, 2 choose finitedimensional subspaces Fi ⊂ Ai , finite-dimensional C ∗ -algebras Bi and linear isomorphisms ϕi : Fi → Bi , such that ϕi cb ϕi−1 cb OL∞ (Ai ) +
and E ⊂ F1 F2 .
Let ⊗min denote the minimal operator space tensor product. Recall that for C ∗ -algebras the minimal operator space tensor product coincides with the minimal C ∗ -tensor product (see [12, p. 228]). Furthermore by [12, 2.1.3], ϕ1 ⊗ ϕ2 : F1 ⊗min F2 → B1 ⊗ B2 cb ϕ1 cb ϕ2 cb . We have a similar inequality for (ϕ1 ⊗ ϕ2 )−1 = ϕ1−1 ⊗ ϕ2−1 . Since A1 A2 is dense in A1 ⊗ A2 , it follows that OL∞ (A1 ⊗ A2 ) inf OL∞ (A1 ) + OL∞ (A2 ) + = OL∞ (A1 )OL∞ (A2 ). >0
2
5. Irreducible representations and OL∞ This section contains the main theorem (Theorem 5.4). We first recall the necessary definitions and prove some preliminary lemmas. Definition 5.1. Let A be a unital C ∗ -algebra. Recall that x ∈ A is an isometry if x ∗ x = 1. An isometry is called proper, if xx ∗ = 1. A is called finite if it contains no proper isometries. A is called stably finite if Mn ⊗ A is finite for every n ∈ N. We will call a representation π of A finite (resp. stably finite) if A/ ker(π) is finite (resp. stably finite). Lemma 5.2. Let H be a separable Hilbert space and x ∈ B(H ) be a proper isometry. Then there is a unitary u ∈ B(H ) such that (ux)n (ux)∗n → 0 strongly. Proof. It is well known (see [6, Theorem V.2.1]) that there is a closed subspace K ⊂ H such that relative to the decomposition H = K ⊕ K ⊥ , we have x = s ⊕ w where s ∈ B(K) is unitarily equivalent to sα , the unilateral shift of order α (for some α = 1, 2, . . . , ∞), and w is a unitary in B(K ⊥ ). In particular s n s ∗n → 0 strongly in B(K). Without loss of generality, assume that w = idK ⊥ . Suppose first that K ⊥ is infinite-dimensional. Since x|K is a proper isometry, K is also infinite-dimensional. Since H is separable, K ∼ = K ⊥ . Under this identification and relative to the decomposition H = K ⊕ K, let
0 1 u= ∈ B(H ). 1 0 Then for n ∈ N we have, 2n
∗2n
(ux) (ux)
s n s ∗n = 0
0 n s s ∗n
14
C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
and 2n+1
(ux)
∗(2n+1)
(ux)
s n s ∗n = 0
0 . s (n+1) s ∗(n+1)
Hence, (ux)n (ux)∗n → 0 strongly. Suppose now that dim(K ⊥ ) = n < ∞. Let {f1 , . . . , fn } be an orthonormal basis for K ⊥ . Since s is unitarily equivalent to a shift, let e1 , . . . , en ∈ K be an orthonormal set such that sei = ei+1
for i = 1, . . . , n − 1 and e1 ⊥ x(H ).
Define u ∈ B(H ) by u(ei ) = fi and u(fi ) = ei for i = 1, . . . , n and u(η) = η for η ⊥ span{e1 , . . . , en , f1 , . . . , fn }. Then u is unitary and (ux)2n (H ) ⊥ span{e1 , . . . , en , f1 , . . . , fn }. Hence for every k 2n we have (ux)2n+k = x k (ux)2n . Therefore, ∗ (ux)2n+k (ux)∗(2n+k) s k ⊕ 0K ⊥ s k ⊕ 0K ⊥ → 0 strongly.
2
We recall the following definitions (see [10, Section 4.1]). Let A be a C ∗ -algebra. An ideal J of A is called primitive if J is the kernel of some (nonzero) irreducible representation of A. Let Prim(A) denote the set of all primitive ideals of A. For a subset X ⊂ Prim(A), and an ideal J of A let ker(X) =
I
and
hull(J ) = I ∈ Prim(A): J ⊂ I .
I ∈X
Then Prim(A) is a topological space with closure operation X → hull(ker(X)) (see [10, Theorem 4.1.3]). The following is an easy consequence of [10, Theorem 4.1.3]. Lemma 5.3. Let A be a C ∗ -algebra and X ⊂ Prim(A). Then X is dense if and only if ker(X) = {0}. Theorem 5.4. Let A be a separable unital C ∗ -algebra with OL∞ (A) < λ , where λ satisfies (3.3). Then A has a separating family of irreducible, stably finite representations. Proof. We first show that A has a separating family of irreducible, finite representations. We assume that A does not have a separating family of irreducible, finite representations and prove that OL∞ (A) > λ . Let Y = y ∈ A: ∃J ∈ Prim(A) (y + J ∈ A/J is a proper isometry) . Then Y is not empty. For each y ∈ Y, let O(y) = J ∈ Prim(A): 1 − y ∗ y + J < 1/4 and 1 − yy ∗ + J > 3/4 ,
(5.1)
C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
15
CO(y) = Prim(A) \ hull ker O(y) . We will now prove the following statement: (∃y ∈ Y) CO(y) is not dense in Prim(A) .
(5.2)
(If Prim(A) is Hausdorff, then (5.2) is immediate by [10, Proposition 4.4.5]. But Prim(A) is not Hausdorff in general.) Since A is separable, let (yn ) ⊂ Y be a dense sequence. Suppose that (5.2) does not hold. Then CO(yn ) is a dense, open subset of Prim(A) for each n ∈ N. Since Prim(A) is a Baire space, (see [10, Theorem 4.3.5]) the following set is dense in Prim(A): X=
∞
CO(yn ).
n=1
If there is a J ∈ X such that A/J is not finite, then there is a y ∈ Y such that y + J is a proper isometry. Then there is an n ∈ N such that yn y ∗ − yy ∗ + y ∗ yn − y ∗ y < 1/8. n
n
But this implies that J ∈ O(yn ) ∩ X ⊂ hull ker O(yn ) ∩ X = ∅. Hence for every J ∈ X, A/J is finite. Since X is dense, ker(X) = {0} by Lemma 5.3. Then A has a separating family of irreducible finite representations, a contradiction. This completes the proof of (5.2). We now build representations ρ and σ that satisfy Theorem 3.1. Let y ∈ Y satisfy (5.2). For each J ∈ CO(y) let σJ be an irreducible representation of A such that ker(σJ ) = J . Let σ = J ∈CO(y) σJ . Since CO(y) is not dense, we have ker(σ ) =
J = ker CO(y) = {0}.
(5.3)
J ∈CO(y)
Let {Ji }i∈I ⊂ O(y) be an at most countable subset such that ker {Ji }i∈I = ker O(y) .
(5.4)
For i ∈ I , let ρi be an irreducible representation of A such that ker(ρi ) = Ji . Let ρ = By (5.3) and (5.4) we have ker(ρ ⊕ σ ) = ker O(y) ∩
J ∈CO(y)
J
=
J ∈Prim(A)
J = {0}.
i∈I
ρi .
(5.5)
16
C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
By definition (5.1), for every i ∈ I , we have 1 − ρi (y ∗ y) < 1/4. Hence ρi (y) is left invertible and ρi (y ∗ y) is invertible. We note that ρi (y) is not right invertible. Indeed, if ρi (y) is right invertible, then there is a unitary u ∈ ρi (A) such that ρi (y) = u|ρi (y)|. Then by (5.1) we have 3/4 < 1 − ρi yy ∗ = 1 − uρi y ∗ y u∗ = 1 − ρi y ∗ y < 1/4, a contradiction. For each i ∈ I , let −1/2 zi = ρi (y) ρi y ∗ y . Then zi∗ zi = 1, but zi zi∗ = 1 because ρi (y) is not right invertible. Hence, zi ∈ ρi (A) is a proper isometry for each i ∈ I . Define the continuous function f : R+ → R+ by f (t) =
8 √ t 3 3 t −1/2
if 0 t 3/4, if t > 3/4.
Let x = yf (y ∗ y) ∈ A. Since sp(ρi (y ∗ y)) ⊂ [3/4, 1], it follows that ρi ( x ) = zi for each i ∈ I . Let x ∈ A be norm 1 such that ρ(x) = ρ( x ) (such a lifting is always possible, see [16, Remark 8.6]). Let Hi denote the Hilbert space associated with ρi . For each i ∈ I , Lemma 5.2 provides a unitary ui ∈ B(Hi ) such that (ui zi )n (ui zi )∗n → 0 strongly in B(Hi ), as n → ∞.
(5.6)
Since each ρi has a different kernel, they are mutually inequivalent. So, by [10, Theorem 3.8.11] ρ(A)
=
ρi (A)
=
i∈I
Set u =
i∈I
B(Hi ).
i∈I
ui . Since ρi (x) = zi , by (5.6) we have n ∗n uρ(x) uρ(x) → 0 strongly in B(Hi ). i∈I
By Kaplansky’s density theorem (see [14, Theorem II.4.11]) there is a sequence (uk ) of unitaries from ρ(A) such that uk → u in the strong* topology. From this we obtain sequences (kr ) and (nr ) such that n ∗n ukr ρ(x) r ukr ρ(x) r → 0 strongly as r → ∞.
(5.7)
For each r ∈ N let xr ∈ A be norm 1 such that ρ(xr ) = ukr . By (5.3) and (5.5) we apply Theo
rem 3.1 with the sequence (xr x)∞ r=1 and deduce that OL∞ (A) > λ .
We now return to the general case. Suppose that OL∞ (A) < λ . Let H be a separable, infinite-dimensional Hilbert space. Let K denote the compact operators on H and K 1 be the unitization of K. Since K 1 is an AF algebra, OL∞ (K 1 ) = 1. By Proposition 4.4, OL∞ (A ⊗ K 1 ) OL∞ (A) < λ .
C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
17
By the above proof there is a subset X ⊂ Prim(A ⊗ K 1 ) with ker(X) = {0} and (A ⊗ K 1 )/J finite for each J ∈ X. By [1, IV.3.4.23], Prim A ⊗ K 1 = J ⊗ K 1 + A ⊗ I : J ∈ Prim(A), I = {0}, K . So, without loss of generality we may assume X = {Ji ⊗ K 1 }i∈I with Ji ∈ Prim(A). Since is exact, (A ⊗ K 1 )/(Ji ⊗ K 1 ) = (A/Ji ) ⊗ K 1 , so A/Ji is stably finite. Furthermore, by the exactness of K 1 , we have K1
{0} =
Ji ⊗ K 1 = Ji ⊗ K 1 . i∈I
So, ker({Ji }i∈I ) = {0}.
i∈I
2
We are now in a position to give a new class of examples of nuclear, quasidiagonal C ∗ algebras A with OL∞ (A) > 1. Example 5.5. Let A be a unital nuclear C ∗ -algebra without a separating family of irreducible stably finite representations (in particular any non-finite nuclear, C ∗ -algebra). Let C(A)1 = (C0 (0, 1] ⊗ A)1 be the unitization of the cone of A. Since A is nuclear, so is C(A)1 . By [15, Proposition 3] C(A)1 is quasidiagonal. For t ∈ (0, 1], let It = {f ∈ C0 (0, 1]: f (t) = 0}. By [1, IV.3.4.23] every non-essential primitive ideal of C(A)1 is of the form It ⊗ A + C0 (0, 1] ⊗ J for some J ∈ Prim(A) and 0 < t 1. Furthermore, by [1, IV.3.4.22], C0 (0, 1] ⊗ A / It ⊗ A + C0 (0, 1] ⊗ J ∼ = A/J. From this we deduce that C(A)1 cannot have a separating family of irreducible, stably finite representations, hence OL∞ (C(A)1 ) > λ by Theorem 5.4. 6. Questions and remarks Recall from the Introduction: Question 6.1. (See [8, Question 6.1].) If OL∞ (A) = 1, is A a rigid OL∞ space? Blackadar and Kirchberg showed [3, Theorem 4.5] that a C ∗ -algebra A is nuclear and inner quasidiagonal if and only if A is a strong NF algebra (see [2, Definition 5.2.1]). In [8] it was shown that A is a strong NF algebra if and only if A is a rigid OL∞ space. Furthermore, by [3, Proposition 2.4] any C ∗ -algebra with a separating family of irreducible quasidiagonal representations is inner quasidiagonal. Therefore if there is a C ∗ -algebra A with OL∞ (A) = 1, but which is not a rigid OL∞ space, then A cannot have a separating family of irreducible, quasidiagonal representations, but A must
18
C. Eckhardt / Journal of Functional Analysis 258 (2010) 1–19
have a separating family of irreducible stably finite representations by Theorem 5.4. Let A ⊂ B(H ) from Example 3.3. Then A + K(H ) is stably finite and prime, hence has a faithful stably finite representation. On the other hand by [4], the unique irreducible representation of A+K(H ) is not quasidiagonal. Hence, A + K(H ) is a possible counterexample to Question 6.1. Finally, recall the question raised by Blackadar and Kirchberg: Question 6.2. (See [2, Question 7.4].) Is every nuclear stably finite C ∗ -algebra quasidiagonal? There are some interesting relationships between Question 6.2 and OL∞ structure. Proposition 6.3. Let A be either simple or both prime and antiliminal. If 1 < OL∞ (A) <
√ 1 + 5 1/2 2
then A is (nuclear) stably finite, but not quasidiagonal. Proof. This follows from [3, Corollary 2.6] and [8, Theorem 3.4].
2
In light of Theorem 5.4, we have the following similar relationship: Proposition 6.4. Let A be a C ∗ -algebra such that every primitive quotient is antiliminal. If 1 < OL∞ (A) < 1.005 then some quotient of A is (nuclear) stably finite, but not quasidiagonal. Proof. This follows from [3, Corollary 2.6] and Theorem 5.4.
2
Acknowledgments A portion of the work for this paper was completed while the author took part in the Thematic Program on Operator Algebras at the Fields Institute in Toronto, ON in the Fall of 2007. I would like to thank Narutaka Ozawa for a helpful discussion about this work and my advisor Zhong-Jin Ruan for all his support. References [1] B. Blackadar, Operator algebras, in: Theory of C ∗ -Algebras and von Neumann Algebras, Operator Algebras and Non-Commutative Geometry, III, in: Encyclopaedia Math. Sci., vol. 122, Springer-Verlag, Berlin, 2006. [2] Bruce Blackadar, Eberhard Kirchberg, Generalized inductive limits of finite-dimensional C*-algebras, Math. Ann. 307 (3) (1997) 343–380. [3] Bruce Blackadar, Eberhard Kirchberg, Inner quasidiagonality and strong NF algebras, Pacific J. Math. 198 (2) (2001) 307–329. [4] L.G. Brown, The universal coefficient theorem for Ext and quasidiagonality, in: Operator Algebras and Group Representations, vol. I, Neptun, 1980, in: Math. Stud. Monogr. Ser., vol. 17, Pitman, Boston, MA, 1984, pp. 60–64. [5] Nathanial P. Brown, On quasidiagonal C ∗ -algebras, in: Operator Algebras and Applications, in: Adv. Stud. Pure Math., vol. 38, Math. Soc. Japan, Tokyo, 2004, pp. 19–64.
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[6] Kenneth R. Davidson, C ∗ -Algebras by Example, Fields Inst. Monogr., vol. 6, American Mathematical Society, Providence, RI, 1996. [7] M. Junge, N.J. Nielsen, Zhong-Jin Ruan, Q. Xu, COLp spaces—the local structure of non-commutative Lp spaces, Adv. Math. 187 (2) (2004) 257–319. [8] Marius Junge, Narutaka Ozawa, Zhong-Jin Ruan, On OL∞ structures of nuclear C ∗ -algebras, Math. Ann. 325 (3) (2003) 449–483. [9] Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Stud. Adv. Math., vol. 78, Cambridge University Press, Cambridge, 2002. [10] Gert K. Pedersen, C ∗ -Algebras and Their Automorphism Groups, London Math. Soc. Monogr. Ser., vol. 14, Academic Press Inc., London, 1979. [11] Gilles Pisier, The operator Hilbert space OH, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (585) (1996). [12] Gilles Pisier, Introduction to Operator Space Theory, London Math. Soc. Lecture Note Ser., vol. 294, Cambridge University Press, Cambridge, 2003. [13] R.R. Smith, Completely bounded maps between C ∗ -algebras, J. London Math. Soc. (2) 27 (1) (1983) 157–166. [14] M. Takesaki, Theory of Operator Algebras, I, Encyclopaedia Math. Sci., vol. 124, Springer-Verlag, Berlin, 2002. Reprint of the first edition, 1979, Operator Algebras and Non-Commutative Geometry, vol. 5. [15] Dan Voiculescu, A note on quasi-diagonal C ∗ -algebras and homotopy, Duke Math. J. 62 (2) (1991) 267–271. [16] Simon Wassermann, Exact C ∗ -Algebras and Related Topics, Lect. Notes Ser., vol. 19, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994.
Journal of Functional Analysis 258 (2010) 20–49 www.elsevier.com/locate/jfa
Groupoid normalizers of tensor products Junsheng Fang a , Roger R. Smith a,∗,1 , Stuart A. White b , Alan D. Wiggins c a Department of Mathematics, Texas A&M University, College Station, TX 77843, USA b Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK c Department of Mathematics and Statistics, University of Michigan–Dearborn, Dearborn, MI 48128, USA
Received 15 September 2008; accepted 7 October 2009
Communicated by N. Kalton
Abstract We consider an inclusion B ⊆ M of finite von Neumann algebras satisfying B ∩ M ⊆ B. A partial isometry v ∈ M is called a groupoid normalizer if vBv ∗ , v ∗ Bv ⊆ B. Given two such inclusions Bi ⊆ Mi , i = 1, 2, we find approximations to the groupoid normalizers of B1 ⊗ B2 in M1 ⊗ M2 , from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis Bi ∩ Mi ⊆ Bi , i = 1, 2. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries v ∈ M satisfying vBv ∗ ⊆ B and v ∗ v, vv ∗ ∈ B. © 2009 Elsevier Inc. All rights reserved. Keywords: Groupoid normalizer; Tensor product; von Neumann algebra; Finite factor
1. Introduction The focus of this paper is an inclusion B ⊆ M of finite von Neumann algebras. Such inclusions have a rich diverse history, first being studied by Dixmier [3] in the context of maximal abelian subalgebras (masas) of II1 factors. These inclusions provided the basic building blocks for the * Corresponding author.
E-mail addresses:
[email protected] (J. Fang),
[email protected] (R.R. Smith),
[email protected] (S.A. White),
[email protected] (A.D. Wiggins). 1 Partially supported by a grant from the National Science Foundation. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.005
J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
21
theory of subfactors developed by Jones in [9] and today they are a key component in the study of structural properties of II1 factors using the deformation-rigidity techniques introduced by Popa in [14]. In [3], Dixmier introduced a classification of masas in II1 factors using normalizers, defining NM (B) = {u a unitary in M: uBu∗ = B}. A masa B ⊂ M is Cartan or regular if these normalizers generate M and singular if NM (B) ⊂ B. Feldman and Moore demonstrated the importance of Cartan masas, and hence normalizers, in the study of II1 factors, showing that inclusions of Cartan masas arise from measurable equivalence relations and that, up to orbit equivalence, these relations determine the resulting inclusion [7,8]. Given two inclusions Bi ⊂ Mi of masas in II1 factors, it is immediate that an elementary tensor u1 ⊗ u2 of unitaries ui ∈ Mi normalizes the tensor product inclusion B = B1 ⊗ B2 ⊂ M = M1 ⊗ M2 if and only if each ui normalizes Bi . As a simple consequence, the tensor product of Cartan masas is again Cartan. More generally, the operation of passing to the von Neumann algebra generated by the normalizers was shown to commute with the tensor product operation for masas inside II1 factors, in the sense that the equality NM1 (B1 ) ⊗ NM2 (B2 ) = NM1 ⊗M2 (B1 ⊗ B2 )
(1.1)
holds. This was proved when both masas are singular in [19] and the general case was established by Chifan in [1]. Since the containment from left to right in (1.1) is immediate, the problem in both cases is to eliminate the possibility that some unexpected unitary in the tensor product normalizes B1 ⊗ B2 . This difficulty was overcome in [19] and [1] by employing techniques of Popa [15] to analyse the basic construction algebra M, eB of Jones [9]. Beyond the masa setting, (1.1) holds when each Bi satisfies Bi ∩ Mi = C1, the defining property of irreducible subfactors. When each Bi has finite Jones index in Mi , the identity (1.1) can be deduced from results of [13]. The infinite index case was established in [20], where every normalizing unitary of such a tensor product of irreducible subfactors was shown to be of the form w(v1 ⊗ v2 ), where w is a unitary in B1 ⊗ B2 and each vi ∈ NMi (Bi ). Some other situations where (1.1) holds are discussed in [6]. For general inclusions Bi ⊆ Mi of finite von Neumann algebras, the commutation identity (1.1) can fail. Indeed, taking each Mi to be a copy of the 3 × 3 matrices and each Bi ∼ = C ⊕ M2 (C), one obtains inclusions with NMi (Bi ) ⊂ Bi , yet there are non-trivial normalizers of B1 ⊗ B2 inside M1 ⊗ M2 . This is due to the presence of partial isometries v in Mi \ Bi with v ∗ Bi v ⊆ Bi , as the non-trivial unitary normalizers of B1 ⊗ B2 can all be writvBi v ∗ ⊆ Bi and ten in the form j xj (v1,j ⊗ v2,j ), where xj lie in B1 ⊗ B2 and the vi,j are partial isometries ∗ ⊆ B and v ∗ B v with vi,j Bi vi,j i,j i i,j ⊆ Bi . Defining the groupoid normalizers of a unital inclusion B ⊂ M to be the set GN M (B) = {v a partial isometry in M: vBv ∗ ⊆ B, v ∗ Bv ⊆ B}, the example discussed above satisfies the commutation identity GN M1 (B1 ) ⊗ GN M2 (B2 ) = GN M1 ⊗M2 (B1 ⊗ B2 ) .
(1.2)
In this paper we examine groupoid normalizers of tensor product algebras, establishing (Corollary 5.6) the identity (1.2) whenever Bi ⊆ Mi are inclusions of finite von Neumann algebras with separable preduals satisfying Bi ∩ Mi ⊆ Bi for each i. In [4] Dye shows that every groupoid normalizer v of a masa B in M is of the form v = ue for some projection e = v ∗ v ∈ B and some unitary normalizer u of B in M, see also [18, Lemma 6.2.3]. The same result holds by a direct computation when B is an irreducible subfactor of M, so that in these two cases
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J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
NM (B) = GN M (B) and (1.2) directly generalizes (1.1) established in [1] and [20] respectively. The following example shows why the hypothesis Bi ∩ Mi ⊆ Bi (which is satisfied by both masas and irreducible subfactors) is necessary in this result. Example 1.1. Consider the subalgebra B=
0 α 0
α 0 0
0 0 β
: α, β ∈ C ⊆ M3 ,
and note that B ∩ M3 strictly contains B. A direct computation shows that GN (B) = M2 ⊕ C, and so GN M3 (B) ⊗ GN M3 (B) ∼ = M4 ⊕ M2 ⊕ M2 ⊕ C. However, B ⊗ B is isomorphic to CI4 ⊕ CI2 ⊕ CI2 ⊕ C inside M9 , and GN M3 ⊗M3 (B ⊗ B) is M4 ⊕ M4 ⊕ C. A new feature of [20] was the notion of one-sided normalizers of an irreducible inclusion B ⊂ M of II1 factors, namely those unitaries u ∈ M with uBu∗ B. These cannot arise for finite index inclusions by index considerations, or in the case when B ⊂ M is a masa. To establish (1.1) for irreducible subfactors, it was necessary to first establish the general form of a one-sided normalizer of a tensor product of irreducible subfactors and then deduce the normalizer result from this. The same procedure is necessary here, so we introduce the notion of an intertwiner to study groupoid normalizers in a one-sided situation. Definition 1.2. Given an inclusion B ⊆ M of von Neumann algebras satisfying B ∩ M ⊆ B, (1) define the collection GN M (B) of intertwiners of B in M by (1) GN M (B) = v a partial isometry in M: vBv ∗ ⊆ B, v ∗ v ∈ B . (1)
We will write GN (1) (B) for GN M (B) when there is no confusion about the underlying algebra M. We use the superscript (1) to indicate that our intertwiners are one-sided, namely that although vBv ∗ ⊆ B, we are not guaranteed to have a containment v ∗ Bv ⊆ B. Note that (1) v ∈ GN M (B) if, and only if, both v and v ∗ lie in GN M (B). Note too that while the groupoid normalizers form a groupoid, the intertwiners do not. Finally, the terminology intertwiner comes from the fact that, under the hypothesis B ∩ M ⊆ B, these are exactly the partial isometries that witness the embeddability of a corner of B into itself inside M in the sense of Popa’s intertwining procedure for subalgebras from [14,15]. We obtain a similar commutation result to (1.2) for intertwiners. In fact our main theorem, stated below, obtains more as it gives approximate forms for intertwiners and groupoid normalizers of tensor products. Theorem 1.3. Let Bi ⊂ Mi be inclusions of finite von Neumann algebras with separable preduals and with fixed faithful normal traces τi on Mi . Moreover, suppose that Bi ∩ Mi ⊆ Bi for i = 1, 2. For v ∈ GN (1) (B1 ⊗ B2 ) and ε > 0, there exist k ∈ N and operators x1 , . . . , xk ∈ B1 ⊗ B2 , M1 ⊗M2 intertwiners w1,1 , . . . , w1,k of B1 in M1 and intertwiners w2,1 , . . . , w2,k of B2 in M2 such that
J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
k
xj (w1,j ⊗ w2,j ) < ε, v − j =1
23
(1.3)
2
where the · 2 -norm arises from the trace τ1 ⊗ τ2 on M1 ⊗ M2 . If in addition v is a groupoid normalizer, then each wi,j can be taken to be a groupoid normalizer rather than just an intertwiner. The intertwiner form of Theorem 1.3 is established as Theorem 4.7 and additional analysis in Section 5 enables us to deduce the groupoid normalizer form of Theorem 1.3 as Theorem 5.5. For the remainder of the introduction we give a summary of the main steps used to establish these results and where they can be found in the paper. Given inclusions Bi ⊆ Mi of finite von Neumann algebras with Bi ∩ Mi ⊆ Bi , write B ⊂ M (1) for the tensor product inclusion B1 ⊗B2 ⊆ M1 ⊗M2 . Let v ∈ GN M (B). Then the element v ∗ eB v is a projection in the basic construction algebra M, eB , the properties of which are recalled in Section 2. Section 3 discusses the properties of these projections in the basic construction arising from intertwiners. In particular, we show that the projection v ∗ eB v is central in the cut-down (B ∩ M, eB )v ∗ v (Lemma 3.2) and construct an explicit projection Pv ∈ Z(B ∩ M, eB ) with Pv v ∗ v = v ∗ eB v. We need to construct this projection explicitly rather than appeal to general theory, as its properties (established in Lemma 3.8) are crucial subsequently. Since the basic construction factorizes as a tensor product M, eB ∼ = M1 , eB1 ⊗ M2 , eB2 , Tomita’s commutation theorem gives Z B ∩ M, eB ∼ = Z B1 ∩ M1 , eB1 ⊗ Z B2 ∩ M2 , eB2 .
(1.4)
For each i = 1, 2, let Qi denote the supremum of all projections in Z(Bi ∩ Mi , eBi ) of the form ∗ (1) ∗ j wi,j eBi wi,j , where the wi,j lie in GN Mi (Bi ) and satisfy wi,j wi,k = 0 when j = k. If we can show that Pv Q1 ⊗ Q2 ,
(1.5)
then it will follow that we can approximate Pv in L2 ( M, eB ) by projections of the form (1) ∗ j (w1,j ⊗ w2,j ) eB (w1,j ⊗ w2,j ) for intertwiners wi,j ∈ GN Mi (Bi ). To do this, we use the fact that projections in the tensor product (1.4) of abelian von Neumann algebras can be approximated by sums of elementary tensors of projections, and so it is crucial that the original projection v ∗ eB v be central in (B ∩ M, eB )v ∗ v, for which the hypothesis B ∩ M ⊆ B is necessary. Finally, we push the approximation for Pv down to M and obtain the required approximation for v in M (see Theorem 4.7). Most of Section 4 is taken up with establishing (1.5). We give a technical result (Theorem 4.1), which in particular characterizes when a projection in the basic construction arises from an intertwiner. By applying Theorem 4.1 to Pv and the inclusion Z B1 ∩ M1 , eB1 ⊗ B2 ⊆ Z B1 ∩ M1 , eB1 ⊗ M2 , regarded as a direct integral of inclusions of finite von Neumann algebras, we are able to establish Pv 1 ⊗ Q2 in Lemma 4.6 and so (1.5) follows by symmetry. It should be noted that the introduction of the projections Qi is essential in order to make use of measure theory, particularly the
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J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
uniqueness of product measures on σ -finite spaces [17, p. 312]. The canonical trace on the basic construction need not be a semifinite weight on Z(Bi ∩ Mi , eBi ) but does have this property on the compression Z(Bi ∩ Mi , eBi )Qi where it can be treated as a measure (see Lemma 2.6 and the discussion preceding Definition 4.4). The remaining difficulty is to check that the projection Pv satisfies the hypotheses of Theorem 4.1, for which we require certain order properties of the pull down map on the basic construction. These are described in the next section, in which we also set out our notation, review the properties of the basic construction, and establish some technical lemmas. Finally, the paper ends with Section 5, which handles the additional details required to deduce the groupoid normalizer result (Theorem 5.5) from our earlier work. 2. Notation and preliminaries Throughout the paper, all von Neumann algebras are assumed to have separable preduals. The basic object of study in this paper is an inclusion B ⊆ M of finite von Neumann algebras, where M is equipped with a faithful normal trace τ satisfying τ (1) = 1. We always assume that M is standardly represented on the Hilbert space L2 (M, τ ), or simply L2 (M). The letter ξ is reserved for the image of 1 ∈ M in this Hilbert space, and J will denote the isometric conjugate linear operator on L2 (M) defined on Mξ by J (xξ ) = x ∗ ξ , x ∈ M, and extended by continuity to L2 (M) from this dense subspace. Then L2 (B) is a closed subspace of L2 (M), and eB denotes the projection of L2 (M) onto L2 (B), called the Jones projection. The von Neumann algebra generated by M and eB is called the basic construction and is denoted by M, eB [2,9]. Let EB denote the unique trace preserving conditional expectation of M onto B. In the next proposition we collect together standard properties of eB , EB and M, eB from [9,13,10,18]. Proposition 2.1. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi)
eB (xξ ) = EB (x)ξ , x ∈ M. eB xeB = EB (x)eB = eB EB (x), x ∈ M. M ∩ {eB } = B. M, eB = J BJ , Z( M, eB ) = J Z(B)J . eB has central support 1 in M, eB . Span{xeB y: x, y ∈ M} generates a ∗-strongly dense subalgebra, denoted MeB M, of M, eB . x → eB x and x → xeB are injective maps for x ∈ M. MeB and eB M are ∗-strongly dense in M, eB eB and eB M, eB respectively. eB M, eB eB = BeB = eB B. (MeB M) M, eB (MeB M) ⊆ MeB M. There is a unique faithful normal semifinite trace Tr on M, eB satisfying Tr(xeB y) = τ (xy),
x, y ∈ M.
(2.1)
This trace is given by the formula Tr(t) =
∞
tJ vi∗ ξ, J vi∗ ξ ,
t ∈ M, eB + ,
(2.2)
i=1
where the vi ’s are partial isometries in M, eB satisfying
∞
∗ i=1 vi eB vi
= 1.
J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
25
(xii) The algebra MeB M is · 2,Tr -dense in L2 ( M, eB , Tr) and · 1,Tr -dense in L1 ( M, eB , Tr). (xiii) Given inclusions Bi ⊂ Mi of finite von Neumann algebras for i = 1, 2, the basic construction M1 ⊗ M2 , eB1 ⊗B2 is isomorphic to M1 , eB1 ⊗ M2 , eB2 . Under this isomorphism, the canonical trace Tr on M1 ⊗ M2 , eB1 ⊗B2 is given by Tr1 ⊗ Tr2 , where Tri is the canonical trace on Mi , eBi . (xiv) There is a well-defined map Ψ : MeB M → M, given by Ψ (xeB y) = xy,
x, y ∈ M.
(2.3)
This is the pull down map of [13], where it was shown to extend to a contraction from L1 ( M, eB , Tr) to L1 (M, τ ). Using part (xii) of the previous proposition, the equation Tr (xeB y)z = τ (xy)z = τ Ψ (xeB y)z ,
x, y, z ∈ M,
(2.4)
shows that Ψ is the pre-adjoint of the identity embedding M → M, eB and is, in particular, positive. The basic properties of Ψ are set out in [13], but we will need more detailed information on this map than is currently available in the literature. We devote much of this section to obtaining further properties of Ψ , the main objective being to apply them in Lemma 4.5. In the next three lemmas, the inclusion B ⊂ M is always of arbitrary finite von Neumann algebras with a fixed faithful normalized normal trace τ on M, inducing the trace Tr on M, eB . Lemma 2.2. Let x ∈ L1 ( M, eB )+ ∩ M, eB . If Ψ (x) ∈ L1 (M) ∩ M, then Ψ (x) x. Proof. It suffices to show that
Ψ (x)yξ, yξ xyξ, yξ ,
y ∈ M.
(2.5)
The maximality argument, preceding [18, Lemma 4.3.4], to establish part (xi) of Proposition 2.1 can be easily modified to incorporate the requirement that v1 = 1. Thus there are vectors ξi = J vi∗ ξ ∈ L2 (M) so that (2.2) becomes Tr(t) =
∞
tξi , ξi ,
t ∈ M, eB + ,
(2.6)
i=1
where ξ1 = ξ . Now, for y ∈ M, we may use the M-modularity of Ψ to write
Ψ (x)yξ, yξ = Ψ y ∗ xy ξ, ξ = τ Ψ y ∗ xy = Tr y ∗ xy .
(2.7)
It follows from (2.6) and (2.7) that ∞
Ψ (x)yξ, yξ = xyξ, yξ + xyξi , yξi xyξ, yξ ,
i=2
establishing that Ψ (x) x.
2
y ∈ M,
(2.8)
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J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
We now extend this result to tensor products. Let N be a semifinite von Neumann algebra with a specified faithful normal semifinite trace TR. In [5], Effros and Ruan identified the predual of a tensor product of von Neumann algebras X and Y by (X ⊗ Y )∗ = X∗ ⊗op Y∗ , the operator space projective tensor product of the preduals. In the presence of traces, this identifies L1 (X ⊗ Y ) with L1 (X) ⊗op L1 (Y ), so I ⊗ Ψ is well defined, positive, and bounded from L1 (N ⊗ M, eB , TR ⊗ Tr) to L1 (N ⊗ M, TR ⊗ τ ), being the pre-adjoint of the identity embedding N ⊗ M → N ⊗ M, eB . Following [21, Chapter IX], we will always assume that N is faithfully represented on L2 (N, TR), for which span{y ∈ N : TR(y ∗ y) < ∞} is a dense subspace. Lemma 2.3. Let x ∈ L1 (N ⊗ M, eB )+ ∩ (N ⊗ M, eB ). If (I ⊗ Ψ )(x) ∈ L1 (N ⊗ M) ∩ (N ⊗ M), then (I ⊗ Ψ )(x) x. Proof. Suppose that the result is not true. Then we may find a finite projection p ∈ N , elements yi ∈ pNp and zi ∈ M, 1 i k, so that
k k
yi ⊗ zi ξ, yi ⊗ zi ξ > 0, x − (I ⊗ Ψ )(x) i=1
since such sums
k
i=1 yi
(2.9)
i=1
⊗ zi ξ are dense in L2 (N, TR) ⊗2 L2 (M, τ ). Then the inequality
(I ⊗ Ψ ) (p ⊗ 1)x(p ⊗ 1) (p ⊗ 1)x(p ⊗ 1)
(2.10)
fails. The element on the left of (2.10) is (p ⊗ 1)((I ⊗ Ψ )(x))(p ⊗ 1), and so is bounded by hypothesis. The restriction of I ⊗ Ψ to L1 (pNp ⊗ M, eB ) is the pull down map for the inclusion pNp ⊗ B ⊆ pNp ⊗ M of finite von Neumann algebras with basic construction pNp ⊗ M, eB . The failure of (2.10) then contradicts Lemma 2.2 applied to this inclusion, establishing that (1 ⊗ Ψ )(x) x. 2 The next lemma completes our investigation of the order properties of pull down maps. Lemma 2.4. If x ∈ L1 (N ⊗ M, eB )+ is unbounded, then so also is (1 ⊗ Ψ )(x). Proof. Suppose that (1 ⊗ Ψ )(x) is bounded. Following [21, Section IX.2], we may regard x as a self-adjoint positive densely defined operator on L2 (N ⊗ M, eB ). For n 1, let pn ∈ N be the spectral projection of x for the interval [0, n]. Then pn x x, so (I ⊗ Ψ (pn x)) (I ⊗ Ψ )(x), since I ⊗ Ψ is the pre-adjoint of a positive map. In particular, I ⊗ Ψ (pn x) is bounded. By Lemma 2.3 applied to pn x, (I ⊗ Ψ )(x) (I ⊗ Ψ )(pn x) pn x.
(2.11)
Since n 1 was arbitrary, we conclude from (2.11) that x is bounded, a contradiction which completes the proof. 2 We note for future reference that these results are equally valid for pull down maps of the form Ψ ⊗ I , due to symmetry. These lemmas will be used in Section 4 to derive an important inequality. The next lemma formulates exactly what will be needed.
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27
Lemma 2.5. Let Bi ⊆ Mi , i = 1, 2, be inclusions of finite von Neumann algebras with pull down maps Ψi . Let B ⊆ M be the inclusion B1 ⊗ B2 ⊆ M1 ⊗ M2 . If x ∈ L1 ( M, eB )+ ∩ M, eB is such that (Ψ1 ⊗ Ψ2 )(x) ∈ L1 (M) ∩ M and the inequality (Ψ1 ⊗ Ψ2 )(x) 1
(2.12)
is satisfied, then (I ⊗Ψ2 )(x) ∈ L1 ( M1 , eB1 ⊗M2 )+ ∩( M1 , eB1 ⊗M2 ) and (I ⊗Ψ2 )(x) 1. Proof. Using the isomorphism of Proposition 2.1(xiii), Ψ1 ⊗ Ψ2 is the pull down map for M, eB . Since (Ψ1 ⊗ I )((I ⊗ Ψ2 )(x)) = (Ψ1 ⊗ Ψ2 )(x) is a bounded operator by hypothesis, it follows from Lemma 2.4 that (I ⊗ Ψ2 )(x) is also bounded in M1 , eB1 ⊗ M2 . Thus the three operators x, (I ⊗ Ψ2 )(x) and (Ψ1 ⊗ Ψ2 )(x) are all bounded, and so we may apply Lemma 2.3 twice to the pull down maps I ⊗ Ψ2 and Ψ1 ⊗ I to obtain (Ψ1 ⊗ Ψ2 )(x) (I ⊗ Ψ2 )(x) x. The result then follows from (2.13) and the hypothesis (2.12).
(2.13)
2
In the proof of Lemma 4.6, we will need the following fact regarding inclusions of finite von Neumann algebras B ⊂ M with B ∩ M ⊆ B. Here, and elsewhere in the paper, we consider inclusions induced by cut-downs. Recall that if Q ⊆ N is an inclusion of von Neumann algebras and q is a projection in Q, then Q ∩ N q = (qQq) ∩ (qN q),
Z Q ∩ N q = Z (qQq) ∩ (qN q) ,
(2.14)
see, for example, [18, Section 5.4]. Lemma 2.6. Let B ⊆ M be a containment of finite von Neumann algebras such that B ∩ M ⊆ B. If p ∈ M is a nonzero projection, then there exists a nonzero projection q ∈ B which is equivalent to a subprojection of p. Observe that if M is a finite factor, then Lemma 2.6 is immediate. Our proof of Lemma 2.6 is classical, proceeding by analysing the center-valued trace on P . Alternatively one can establish the lemma by taking a direct integral over the center. Since we have been unable to find this fact in the literature we give the details for completeness. Proof of Lemma 2.6. Let denote the center-valued trace on M. We will make use of two properties of from [12, Theorem 8.4.3]. The first is that p1 p2 if and only if (p1 ) (p2 ), and the second is that p1 ∼ p2 if and only if (p1 ) = (p2 ). The hypothesis B ∩ M ⊆ B implies that B ∩ M = Z(B) and, in particular, that Z(M) ⊆ Z(B). For some sufficiently small c > 0, the spectral projection z of (p) for the interval [c, 1] is nonzero, and (pz) cz. Since Bz ⊆ Mz also satisfies the relative commutant hypothesis, it suffices to prove the result under the additional restriction (p) c1 for some constant c > 0. Let n c−1 be any integer. Suppose that it is possible to find a nonzero projection q ∈ B and an orthogonal set {q, p2 , . . . , pn } of equivalent projections in M. The sum of these projections has central trace equal to n (q) and is also bounded by 1, so that (q) n−1 1 c1. But then q p and we are done. Thus we may assume that there is an absolute bound on the length of
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J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
any such set, and we may then choose one, {q1 , p2 , . . . , pn }, of maximal length. By cutting by the central support of q1 , we may assume that this central support is 1. Now consider the inclusion q1 Bq1 ⊆ q1 Mq1 , and note that (q1 Bq1 ) ∩ q1 Mq1 = q1 B ∩ M = q1 Z(B) = Z(q1 Bq1 )
(2.15)
from (2.14). Let f1 and f2 be nonzero orthogonal projections in q1 Bq1 and q1 Mq1 respectively. By the comparison theory of projections, there exists a projection z ∈ Z(q1 Mq1 ) ⊆ Z(q1 Bq1 ) so that zf1 zf2 ,
(1 − z)f2 (1 − z)f1 ,
(2.16)
the equivalence being taken in q1 Mq1 . Now zf1 ∈ q1 Bq1 and is equivalent to a subprojection p0 of zf2 q1 . Then the pair zf1 , p0 is equivalent to orthogonal pairs below each pi , 2 i n, which will contradict the maximal length of {q1 , p2 , . . . , pn } unless zf1 = 0. Similarly (1 − z)f2 = 0. Thus f1 and f2 have orthogonal central supports in q1 Bq1 and so [18, Lemma 5.5.3] shows that q1 Bq1 is abelian. Eq. (2.15) then shows that q1 Bq1 is a masa in q1 Mq1 , and so another application of [18, Lemma 5.5.3] shows that q1 Mq1 is also abelian. Thus q1 Bq1 = q1 Mq1 . Now the projection 1 − q1 − p2 − · · · − pn must be 0, otherwise it would have a nonzero subprojection equivalent to a nonzero projection q˜1 ∈ q1 Mq1 = q1 Bq1 , since q1 has central support 1, and q˜1 would lie in a set of n + 1 equivalent orthogonal projections. Thus q1 , p2 , . . . , pn are abelian projections in M with sum 1, so M is isomorphic to L∞ (Ω) ⊗ Mn for some measure space Ω. Identify p and q1 with measurable Mn -valued functions. Since q1 is abelian, the rank of q1 (ω) is 1 almost everywhere, and the rank of p(ω) is at least 1 almost everywhere since
(p) c1. Then q1 is equivalent to a subprojection of p since (q1 ) (p). This completes the proof. 2 We conclude this section with a brief explanation of an averaging technique in finite von Neumann algebras which we will use subsequently. It has its origins in [2], but is also used extensively in [14]. If η ∈ L2 (M) and U is a group of unitaries in M then the vector can be averaged over U . This is normally associated with amenable groups, but can be made to work in this setting without this assumption. Form the · 2 -norm closure K of K = conv uηu∗ : u ∈ U . There is a unique vector η˜ ∈ K of minimal norm, and uniqueness of η˜ implies that uηu ˜ ∗ = η˜ for all u ∈ U . We refer to η˜ as the result of averaging η over U , and many variations of this are possible. We give an example of this technique by establishing a technical result which will be needed in the proof of Theorem 4.7. Recall (Proposition 2.1(xii)) that MeB M is · 2,Tr -dense in L2 ( M, eB , Tr). Consider a ∗∞ subalgebra A which is strongly dense in M. If x, y ∈ M, then fix sequences {xn }∞ n=1 , {yn }n=1 from A converging strongly to x and y, respectively. Then (x − xn )eB 2 = Tr eB (x − xn )∗ (x − xn )eB 2,Tr = τ (x − xn )∗ (x − xn ) = (x − xn )ξ, (x − xn )ξ ,
(2.17)
J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
29
so xn eB → xeB in · 2,Tr -norm. Thus xn eB y → xeB y so, given ε > 0, we may choose n0 so large that xn0 eB y − xeB y 2,Tr < ε/2. The same argument on the right allows us to choose n1 so large that xn0 eB y − xn0 eB yn1 2,Tr < ε/2, whereupon xeB y − xn0 eB yn1 2,Tr < ε. The conclusion reached is that the algebra AeB A = { ni=1 xi eB yi : xi , yi ∈ A} is · 2,Tr -norm dense in L2 ( M, eB , Tr). In the next lemma, we will use this when M is a tensor product M1 ⊗ M2 where we take A to be the algebraic tensor product M1 ⊗ M2 . Lemma 2.7. Let B1 , B2 be von Neumann subalgebras of finite von Neumann algebras M1 , M2 and let B = B1 ⊗ B2 , M = M1 ⊗ M2 . Then L2 Z B ∩ M, eB , Tr = L2 Z B1 ∩ M1 , eB1 , Tr1 ⊗2 L2 Z B2 ∩ M2 , eB2 , Tr2 . Note that, although Tri is a semifinite trace on Mi , eBi , it need not be semifinite on Z(Bi ∩ Mi , eBi ). This is why the lemma cannot be obtained immediately from the uniqueness of product measures on σ -finite measure spaces. Proof of Lemma 2.7. If zi ∈ Z(Bi ∩ Mi , eBi ), i = 1, 2, then z1 ⊗ z2 ∈ Z(B ∩ M, eB ) and
z1 ⊗ z2 2,Tr = z1 2,Tr1 z2 2,Tr2 . This shows the containment from right to left. Suppose that z ∈ Z(B ∩ M, eB ) with Tr(z∗ z) < ∞. Then z lies in L2 ( M, eB , Tr) so can be approximated in · 2,Tr -norm by sums of the form ki=1 xi eB yi with xi , yi ∈ M1 ⊗ M2 . The preceding remarks then allow us to assume that xi and yi lie in the algebraic tensor product M1 ⊗ M2 . Thus, given ε > 0, we may find elements ai , ci ∈ M1 , bi , di ∈ M2 so that n
(ai ⊗ bi )(eB1 ⊗ eB2 )(ci ⊗ di ) z − i=1
ε.
(2.18)
2,Tr
This may be rewritten as n
(ai eB1 ci ) ⊗ (bi eB2 di ) z − i=1
ε,
(2.19)
2,Tr
and then as n
f i ⊗ gi z − i=1
ε,
(2.20)
2,Tr
where fi ∈ M1 eB1 M1 and gi ∈ M2 eB2 M2 . We may further suppose that the set {g1 , . . . , gn } is linearly independent. For j = 1, 2, let Nj be the von Neumann algebra generated by Bj and Bj ∩ Mj , eBj , and note that z commutes with N1 ⊗ N2 . Let K = conv
w
n
i=1
∗
ufi u ⊗ gi : u ∈ U(N1 ) ,
Ki = convw ufi u∗ : u ∈ U(N1 )
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J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
for 1 i n. Then K ⊆ ni=1 Ki ⊗ gi . By [18, Lemma 9.2.1] K and each Ki are closed in their respective
· 2 -norms. If k ∈ K is the element of minimal · 2 -norm in K then it may be written as k = ni=1 ki ⊗ gi with ki ∈ Ki . Since k is invariant for the action of U(N1 ⊗ 1), we see that n
uki u∗ − ki ⊗ gi = 0,
u ∈ U(N1 ).
(2.21)
i=1
The linear independence of the gi ’s allows us to conclude that uki u∗ = ki for 1 i n and u ∈ U(N1 ). Thus ki ∈ N1 ∩ M1 , eB1 = Z(B1 ∩ M1 , eB1 ). The inequality (2.20) is preserved by averaging in this manner over U(N1 ⊗ 1) so, replacing each fi by ki if necessary, we may assume that fi ∈ Z(B1 ∩ M1 , eB1 ) for 1 i n. Now repeat this argument on the right, averaging over U(1 ⊗ N2 ), to replace the gi ’s by elements of Z(B2 ∩ M2 , eB2 ). With these changes, (2.20) now approximates z by a sum from L2 Z B1 ∩ M1 , eB1 , Tr1 ⊗2 L2 Z B2 ∩ M2 , eB2 , Tr2 which proves the containment from left to right and establishes equality.
2
3. Projections in the basic construction In this section, we relate intertwiners of a subalgebra to certain projections in the basic construction. We consider a finite von Neumann algebra M and a von Neumann subalgebra B whose unit will always coincide with that of M. For the most part, we will be interested in the condition B ∩ M ⊆ B (equivalent to B ∩ M = Z(B)), but we will make this requirement explicit when it is needed. Lemma 3.1. Let B be a von Neumann subalgebra of a finite von Neumann algebra M and let v ∈ GN (1) (B). 1. Then v ∗ eB v is a projection in (B ∩ M, eB )v ∗ v. 2. Suppose q is a projection in B. Then v ∗ eB v lies in (B ∩ M, eB )q if, and only if, v ∗ v ∈ Z(B)q = Z(qBq). Proof. 1. The element v ∗ eB v is positive in M, eB . Since vv ∗ ∈ B and so commutes with eB , the following calculation establishes that v ∗ eB v is a projection: ∗ 2 v eB v = v ∗ eB vv ∗ eB v = v ∗ vv ∗ eB v = v ∗ eB v.
(3.1)
For an arbitrary b ∈ v ∗ vBv ∗ v, ∗ v eB v b = v ∗ eB vbv ∗ v = v ∗ vbv ∗ eB v = v ∗ vbv ∗ vv ∗ eB v = b v ∗ eB v ,
(3.2)
where the second equality uses vbv ∗ ∈ B to commute this element with eB . Thus (3.2) establishes that v ∗ eB v ∈ (v ∗ vBv ∗ v) ∩ v ∗ v M, eB v ∗ v which is (B ∩ M, eB )v ∗ v (see (2.14)).
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2. Suppose now that v ∗ v ∈ Z(B)q = Z(qBq). It is immediate that Z(qBq) ⊆ Z((B ∩ M, eB )q), so we have a decomposition B ∩ M, eB q = B ∩ M, eB v ∗ v ⊕ B ∩ M, eB q − v ∗ v .
(3.3)
We have already shown that v ∗ eB v is in the first summand of (3.3) so must lie in (B ∩ M, eB )q. Conversely, the hypothesis on v ∗ eB v implies that v ∗ eB v = v ∗ eB vq = qv ∗ eB v, so the pull down map gives v ∗ v = v ∗ vq = qv ∗ v, showing that v ∗ v ∈ qBq. For each b ∈ B, v ∗ eB vqbq = qbqv ∗ eB v. Applying the pull down map gives v ∗ vqbq = qbqv ∗ v and hence v ∗ v ∈ Z(qBq). 2 We now strengthen this lemma under the additional hypothesis that B ∩ M ⊆ B. Recall from part (iv) of Proposition 2.1 that J Z(B)J = Z( M, eB ). When B ⊂ M is a finite index inclusion of irreducible subfactors, Lemma 3.2 is contained in [13, Propositions 1.7(2) and 1.9]. The proof follows the extension to infinite index inclusions of irreducible subfactors in [20, Lemma 3.3]. Lemma 3.2. Let B be a von Neumann subalgebra of a finite von Neumann algebra M and suppose that B ∩ M ⊆ B. Let v ∈ GN (1) (B). Then the projection v ∗ eB v is central in (B ∩ M, eB )v ∗ v. Proof. Define two projections p, q ∈ B by p = v ∗ v and q = vv ∗ . Now consider an arbitrary x ∈ (B ∩ M, eB )p = (pBp) ∩ p M, eB p. Then, for each b ∈ B, vxv ∗ vbv ∗ = vxpbpv ∗ = vpbpxv ∗ = vbv ∗ vxv ∗ ,
(3.4)
showing that vxv ∗ ∈ (vBv ∗ ) ∩ (q M, eB q). We next prove that qeB is central in (vBv ∗ ) ∩ (q M, eB q). It lies in this algebra by the previous calculation and Lemma 3.1, as qeB = v(v ∗ eB v)v ∗ . Take t ∈ (vBv ∗ ) ∩ (q M, eB q) to be self-adjoint and let η be tξ ∈ L2 (M). Now take a sequence {xn }∞ n=1 from M converging in · 2 -norm to tξ . Since t = qt = tq, we may assume ∞ that the sequence {xn }∞ n=1 lies in qMq, otherwise replace it by {qxn q}n=1 . ∗ For each u in the unitary group U(pBp), t commutes with vuv and so J vuv ∗ J vuv ∗ η = J vuv ∗ J vuv ∗ tξ = J vuv ∗ J tvuv ∗ ξ = tJ vuv ∗ J vuv ∗ ξ = tvuv ∗ vu∗ v ∗ ξ = tqξ = tξ = η,
(3.5)
where the third equality holds because vuv ∗ ∈ B so that J vuv ∗ J ∈ ( M, eB ) . For each n 1 and each u ∈ U(pBp), J vuv ∗ J vuv ∗ xn ξ − η = J vuv ∗ J vuv ∗ (xn ξ − η) 2 2 xn ξ − η 2 ,
(3.6)
from (3.5). If we let yn be the element of qMq obtained by averaging xn over the unitary group vU(pBp)v ∗ ⊆ qBq, then (3.6) gives
yn ξ − η 2 xn ξ − η 2 ,
n 1,
(3.7)
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while yn ∈ (vBv ∗ ) ∩ qMq. Since vBv ∗ ∩ qMq = vBv ∗ ∩ vMv ∗ = v B ∩ M v ∗ = vZ(B)v ∗ ⊆ qBq,
(3.8)
we see that yn ∈ qBq for n 1. From (3.7) it follows that η ∈ L2 (qBq). For b ∈ B, tqbξ = tJ b∗ qJ ξ = J b∗ qJ tξ = lim J b∗ qJyn ξ = lim yn qbξ, n→∞
(3.9)
n→∞
showing that tqbξ ∈ L2 (qB). Thus L2 (qB) is an invariant subspace for t. The projection onto it is qeB , so tqeB = qeB tqeB . Since t is self-adjoint, we obtain that qeB commutes with (vBv ∗ ) ∩ (q M, eB q), establishing centrality. It was established in Eq. (3.4) that vxv ∗ ∈ (vBv ∗ ) ∩ (q M, eB q) whenever x ∈ (B ∩ M, eB )p, and so each such vxv ∗ commutes with qeB . Thus v ∗ eB vx = v ∗ qeB vxv ∗ v = v ∗ vxv ∗ qeB v = xv ∗ eB v for x ∈ (B ∩ M, eB )p, showing that v ∗ eB v is central in (B ∩ M, eB )p.
(3.10) 2
Since the centrality of the projections v ∗ eB v in (B ∩ M, eB )v ∗ v is crucial to our subsequent arguments, the following corollary highlights why the hypothesis B ∩ M ⊆ B is essential. Corollary 3.3. Let B be a von Neumann subalgebra of a finite von Neumann algebra M. Then eB is central in B ∩ M, eB if and only if B ∩ M ⊆ B. Proof. If B ∩ M ⊆ B, then centrality of eB is a special case of Lemma 3.2 with v = 1. Conversely, suppose that eB is central in B ∩ M, eB and consider x ∈ B ∩ M. Then x commutes with eB so x ∈ B by Proposition 2.1(iii). Thus B ∩ M ⊆ B. 2 Lemma 3.4. Let B be a von Neumann subalgebra of a finite von Neumann algebra M, suppose that B ∩ M ⊆ B, and let v ∈ GN (1) (B). Then any subprojection of v ∗ eB v in (B ∩ M, eB )v ∗ v has the form pv ∗ eB v where p is a central projection in B. Proof. By Lemma 3.1(i), v ∗ eB v ∈ (B ∩ M, eB )v ∗ v. Suppose that a projection q ∈ (B ∩ M, eB )v ∗ v lies below v ∗ eB v. Then (vqv ∗ )2 = vqv ∗ vqv ∗ = vqv ∗ , so vqv ∗ is a projection. The relation vqv ∗ vbv ∗ = vq v ∗ vbv ∗ v v ∗ = v v ∗ vbv ∗ v qv ∗ = vbv ∗ vqv ∗ ,
b ∈ B,
(3.11)
shows that vqv ∗ ∈ (vBv ∗ ) ∩ vv ∗ M, eB vv ∗ . Moreover, vqv ∗ vv ∗ eB vv ∗ = eB vv ∗ , so there exists a projection f ∈ vv ∗ Bvv ∗ such that vqv ∗ = f eB . For b ∈ B, f vbv ∗ eB = f eB vbv ∗ = vqv ∗ vbv ∗ = vbv ∗ vqv ∗ = vbv ∗ f eB ,
(3.12)
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and so f vbv ∗ = vbv ∗ f . Thus f ∈ vBv ∗ ∩ vv ∗ Bvv ∗ .
(3.13)
If b0 ∈ B is such that vv ∗ b0 vv ∗ commutes with vBv ∗ , then vv ∗ b0 vv ∗ vbv ∗ = vbv ∗ vv ∗ b0 vv ∗ ,
b ∈ B.
(3.14)
b ∈ B.
(3.15)
Multiply on the left by v ∗ and on the right by v to obtain v ∗ b0 vv ∗ vbv ∗ v = v ∗ vbv ∗ vv ∗ b0 v, Thus v ∗ b0 v ∈ v ∗ vBv ∗ v ∩ v ∗ vMv ∗ v = v ∗ v B ∩ M v ∗ v = v ∗ vZ(B)v ∗ v.
(3.16)
Consequently, vv ∗ b0 vv ∗ ∈ vZ(B)v ∗ . It follows that (vBv ∗ ) ∩ vv ∗ Bvv ∗ ⊆ vZ(B)v ∗ , and so there is a central projection p ∈ Z(B) so that f = vpv ∗ . We now have q = v ∗ f eB v = v ∗ vpv ∗ eB v = pv ∗ eB v, as required.
(3.17)
2
In Section 4, we wish to use the projection v ∗ eB v to investigate an intertwiner v of a tensor product B = B1 ⊗ B2 ⊂ M1 ⊗ M2 = M, where each Bi ∩ Mi ⊆ Bi . In conjunction with Proposition 2.1(xiii), Tomita’s commutation theorem gives B ∩ M, eB ∼ = B1 ∩ M1 , eB1 ⊗ B2 ∩ M2 , eB2 .
(3.18)
By Lemma 3.2, such an intertwiner gives rise to a central projection v ∗ eB v in (B ∩ M, eB )v ∗ v. Unfortunately, in general the projection v ∗ v will not factorize as an elementary tensor of projections b1 ⊗ b2 , with bi ∈ Bi , and so the algebra (B ∩ M, eB )v ∗ v will not decompose as a tensor product. This prevents us from applying tensor product techniques to the projection v ∗ eB v directly. However, standard von Neumann algebra theory (see, for example, [11]) gives a central projection P ∈ B ∩ M, eB such that v ∗ eB v = P v ∗ v. Since we need to ensure that the projection P fully reflects the properties of v, we cannot just appeal to the general theory to obtain P , so we give an explicit construction in Definition 3.5 below. The subsequent lemmas set out the properties of P that we require later. Definition 3.5. Let B ⊂ M be an inclusion of finite von Neumann algebras with B ∩ M ⊆ B (1) and let v ∈ GN M (B). Let z ∈ Z(B) be the central support of v ∗ v. Define p0 to be v ∗ v, and let {p0 , p1 , . . .} be a family of nonzero pairwise orthogonal projections in B which is maximal with respect to the requirements that pn z and each pn is equivalent in B to a subprojection in B of p0 . Since two projections in a von Neumann algebra with non-orthogonal central supports have equivalent nonzero subprojections, maximality gives n0 pn = z. For n 1, choose partial isometries wn ∈ B so that wn∗ wn = qn p0 and wn wn∗ = pn . Then define
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vn = vwn∗ ∈ GN M (B). Lemma 3.1 shows that vn∗ eB vn ∈ (B ∩ M, eB )vn∗ vn and this space is ∗ (B ∩ M, eB )pn since vn∗ vn = wn v ∗ vwn∗ = pn . In particular, {vn eB vn }n0 is a set of pairwise orthogonal projections so we may define a projection Pv = n0 vn∗ eB vn in M, eB . (1)
Lemma 3.6. With the notation of Definition 3.5, the projection Pv is central in B ∩ M, eB and satisfies Pv v ∗ v = v ∗ eB v. Proof. This projection is Pv = n0 wn v ∗ eB vwn∗ . By Lemma 3.2 there exists t ∈ Z(B ∩ M, eB ) so that v ∗ eB v = tv ∗ v, and so Pv becomes Pv =
wn tv ∗ vwn∗ =
n0
twn v ∗ vwn∗ =
n0
twn p0 wn∗ = tz.
(3.19)
n0
Thus Pv ∈ B ∩ M, eB and Pv v ∗ v = tv ∗ v = v ∗ eB v since z is the central support of v ∗ v. Since z ∈ Z(B) ⊂ Z(B ∩ M, eB ), it follows that Pv = tz also lies in Z(B ∩ M, eB ). 2 Remark 3.7. The proof of Lemma 3.6 shows that Pv is the minimal projection in B ∩ M, eB
with Pv v ∗ v = v ∗ eB v. This gives a canonical description of Pv which is independent of the choices made in Definition 3.5. The explicit formulation of the definition is useful in transferring properties from v ∗ eB v to Pv . We now identify the subprojections of Pv . This will be accomplished by the next lemma, be a sequence which considers a wider class of projections needed subsequently. Let {vi }∞ i=1 ∗ from GN (1) (B) satisfying vi vj∗ = 0 for i = j , let p ∈ B be the projection ∞ i=1 vi vi and let ∞ ∗ P ∈ M, eB be the projection i=1 vi eB vi . In particular, the projection Pv of Definition 3.5 is of this form. Let N(P ) denote the von Neumann algebra N (P ) = {x ∈ pMp: xP = P x} ⊆ M. Lemma 3.8. Let P =
∞
∗ i=1 vi eB vi
be as above, and let p =
∞
∗ i=1 vi vi
(3.20) ∈ B.
(i) If x ∈ pMp satisfies xP = 0, then x = 0; (ii) The map x → xP is a ∗-isomorphism of N (P ) into M, eB ; (iii) A projection Q ∈ M, eB satisfies Q P if and only if there exists a projection f ∈ N (P ) such that Q = f P . Moreover, if P and Q lie in B ∩ M, eB , then f ∈ Z(B) and Q has the same form as P . Proof. (i) Suppose that x ∈ pMp and xP = 0. Then x
∞
vi∗ eB vi = 0.
(3.21)
i=1
Multiply on the right in (3.21) by vk∗ vk to obtain xvk∗ eB vk = 0 for k 1. The pull down map gives xvk∗ vk = 0. Summing over k shows that xp = 0 and the result follows since x = xp. (ii) Since P ∈ N(P ) , the map x → xP is a ∗-homomorphism on N (P ). It has trivial kernel, by (i), so is a ∗-isomorphism.
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(iii) If f ∈ N(P ), then it is clear that f P is a projection below P , since f commutes with P . Conversely, consider a projection Q P with Q ∈ M, eB . The introduction of partial sums below is to circumvent some questions of convergence. Define Pk = ki=1 vi∗ eB vi . Then limk→∞ Pk = P strongly, so Pk QPk converges strongly to P QP = Q. For m, n 1, let bm,n ∈ B be the element such that bm,n eB = eB vm Qvn∗ eB . Then Pk QPk =
k
∗ vm eB vm Qvn∗ eB vn =
m,n=1
=
k
k
∗ vm bm,n eB vn
m,n=1 ∗ vm bm,n eB vn vn∗ vn =
m,n=1
k
∗ vm bm,n vn vn∗ eB vn .
(3.22)
m,n=1
∗ ∗b Now define xk ∈ pW ∗ (GN (1) (B))p by xk = km,n=1 vm m,n vn . The relations vi vj = 0 for i = j allow us to verify that xk P = Pk QPk , and consequently xk P = P xk∗ ,
k 1,
(3.23)
since Pk QPk is self-adjoint. Thus ∞
xk vi∗ eB vi =
i=1
∞
vi∗ eB vi xk∗ .
(3.24)
i=1
The sums in (3.24) converge in · 1 -norm, so we may apply the pull down map to obtain xk p = pxk∗ . Since xk = xk p, we conclude that xk is self-adjoint. Thus, from (3.23), xk commutes with P , and so lies in N (P ). From above, xk P 0 and xk P 1, so xk 0 and xk 1 by (ii). Let f be a σ -weak accumulation point of the sequence {xk }∞ k=1 . Then f 0, f 1, and f ∈ N(P ). Since xk P = Pk QPk , we conclude that f P = Q. It now follows from (ii) that f is a projection in N (P ). If P ∈ B ∩ M, eB , then the pull down map gives bp = pb for b ∈ B, so p ∈ Z(B). If also Q ∈ B ∩ M, eB , then Q = f P and f p. If b ∈ Bp then commutation with Q gives (bf − f b)P = 0, so f ∈ B ∩ M = Z(B), by (i). Finally Q = f P = i (vi f )∗ eB (vi f ), so is of the same form as P . 2 4. Intertwiners of tensor products In this section we will prove one of our main results, the equality of W ∗ (GN (1) (B1 )) ⊗ ∗ W (GN (1) (B2 )) and W ∗ (GN (1) (B1 ⊗ B2 )), where Bi ⊆ Mi , i = 1, 2, are inclusions of finite von Neumann algebras satisfying Bi ∩ Mi ⊆ Bi . The key theorem for achieving this is the following one, which enables us to detect those central projections in corners of the relative commutant of the basic construction which arise from intertwiners. It is inspired by [1, Proposition 2.7], although is not a direct generalization of that result. For comparison, [1, Proposition 2.7] shows that, in the case of a masa A, a projection P ∈ A ∩ M, eA which is subequivalent to eA dominates an operator v ∗ eB v for some v ∈ GN (A). Example 4.3 below will show that such a result will not hold in general without additional hypotheses.
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Theorem 4.1. Let A be an abelian von Neumann algebra with a fixed faithful normal semifinite weight Φ. Let B be a von Neumann subalgebra of a finite von Neumann algebra M with a faithful normal trace τ satisfying B ∩ M ⊆ B. Fix a projection q ∈ A ⊗ B and suppose that P ∈ (A ⊗ (B ∩ M, eB ))q is a nonzero projection such that P (1 ⊗ eB ) in A ⊗ M, eB , and satisfies (Φ ⊗ Tr)(P r) (Φ ⊗ τ )(qr)
(4.1) (1)
for all projections r ∈ Z(q(A ⊗ B)q). Then there exists an element v ∈ GN A⊗M (A ⊗ B) such that P = v ∗ (1 ⊗ eB )v. Before embarking on the proof, let us recall that, for a finite von Neumann algebra M with a faithful normal trace τ , we regard the Hilbert space L2 (M) as the completion of M in the norm x 2,τ = τ (x ∗ x)1/2 and L1 (M) as the completion of M in the norm x 1,τ = τ (|x|). The Cauchy–Schwarz inequality gives xy ∗ 1,τ x 2,τ y 2,τ for x, y ∈ M and so this inequality allows us to define ζ η∗ ∈ L1 (M) for ζ, η ∈ L2 (M). In particular, if (yn ) is a sequence in M converging to η ∈ L2 (M), then yn∗ yn → η∗ η in L1 (M). Recall too that we can regard elements of L2 (M) as unbounded operators on L2 (M) affiliated to M. The only fact we need about these unbounded operators is that if η ∈ L2 (M) satisfies η∗ η ∈ M (regarded as a subset of L1 (M)), then in fact η ∈ M. This follows as η has a polar decomposition v(η∗ η)1/2 , where v is a partial isometry in M and (η∗ η)1/2 is an element of L2 (M), which lies in M if η∗ η does. Proof of Theorem 4.1. The first case that we will consider is where A = C and Φ is the identity map. Then the hypothesis becomes Tr(P r) τ (qr)
(4.2)
for all projections r ∈ Z(B)q. Since P eB , there exists a partial isometry V ∈ M, eB such that P = V ∗ V and V V ∗ eB . Define the map θ : qBq → BeB by θ (qbq) = V qbqV ∗ = eB V qbqV ∗ eB ,
b ∈ B.
(4.3)
Then θ is a ∗-homomorphism since V ∗ V commutes with qBq, and so there is a ∗-homomorphism φ : qBq → B so that θ (qBq) = φ(qbq)eB for qbq ∈ qBq. Thus qbqV ∗ = qbqV ∗ V V ∗ = V ∗ V qbqV ∗ = V ∗ eB φ(qbq) = V ∗ φ(qbq)
(4.4)
for qbq ∈ qBq. Now define η ∈ L2 (M) by η = J V ∗ ξ , and observe that η = 0 since V J η = V V ∗ ξ = b0 ξ , where V V ∗ = b0 eB for some b0 ∈ B. If we apply (4.4) to ξ , then the result is qbqJ η = V ∗ φ(qbq)ξ = V ∗ J φ qb∗ q J ξ = J φ qb∗ q J V ∗ ξ = J φ qb∗ q η, qbq ∈ qBq,
(4.5)
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37
where we have used M, eB = (J BJ ) to commute J φ(qb∗ q)J with V ∗ . Multiply (4.10) on the left by J and replace b by b∗ to obtain ηqbq = φ(qbq)η,
qbq ∈ qBq.
(4.6)
Taking b = 1, (4.5) becomes J qJ η = φ(q)η,
(4.7)
J qV ∗ ξ = φ(q)η.
(4.8)
φ(q)ξ = V J φ(q)η,
(4.9)
so
Multiply on the left by V J to obtain
showing that φ(q)η = 0. From (4.7), φ(q)ηq = φ(q)η = 0, and this allows us to assume in (4.6) that the vector η is nonzero and satisfies ηq = η, φ(q)η = η by replacing η with φ(q)ηq if necessary. For unitaries u ∈ qBq, (4.6) becomes φ u∗ ηu = φ u∗ φ(u)η = φ(q)η = η.
(4.10)
Choose a sequence {xn }∞ n=1 from M such that xn ξ → η in · 2,τ -norm. Since φ(q)ηq = η, we may assume that φ(q)xn q = xn for n 1. Let yn be the element of minimal · 2,τ -norm in convw {φ(u∗ )xn u: u ∈ U(qbq)}. Since ∗ φ u xn uξ − η = φ u∗ (xn ξ − η)u xn ξ − η 2,τ 2,τ 2,τ
(4.11)
for all u ∈ U(qBq), we see that yn ξ − η 2,τ xn ξ − η 2,τ , so yn ξ → η in · 2,τ -norm and φ(u∗ )yn u = yn for n 1 by the choice of yn . Then yn u = φ(u)yn for u ∈ U(qBq), so yn qbq = φ(qbq)yn ,
n 1, qbq ∈ qBq.
(4.12)
Thus yn∗ yn qbq = yn∗ φ(qbq)yn ,
n 1, qbq ∈ qBq,
(4.13)
and this implies that yn∗ yn ∈ (qBq) ∩ qMq = (B ∩ M)q = Z(B)q for each n 1. The discussion preceding the proof ensures that yn∗ yn → η∗ η in L1 (M), so we see that η∗ η ∈ L1 (Z(B)q). For each z ∈ Z(B)q, ∗ τ η ηzq = lim τ y ∗ yn qz = lim yn qzξ, yn ξ n n→∞ n→∞ ∗ ∗ = J z qJ η, η = J z qJ J V ∗ ξ, J V ∗ ξ = z∗ qV ∗ ξ, V ∗ ξ = V z∗ qV ∗ ξ, ξ = φ z∗ q ξ, ξ = τ φ(zq) .
(4.14)
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J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
Now Tr(P zq) = Tr V ∗ V zq = Tr V zqV ∗ = Tr φ(zq)eB = τ φ(zq) .
(4.15)
Thus, from (4.14), (4.15) and the hypothesis (4.2), ∗ τ η ηr = Tr(P r) τ (r)
(4.16)
for all projections r ∈ Z(B)q. Since Z(B)q is abelian, simple measure theory allows us to conclude from (4.16) that η∗ η ∈ Z(B)q (rather than just L1 (Z(B)q)) and so η ∈ M, by the discussion prior to the start of the proof. Moreover, (4.16) also gives η 1, by taking r to be the spectral projection of η∗ η for the interval (c, ∞) where c > 1 is arbitrary. Since η ∈ L2 (M) has been proved to lie in M, we rename this nonzero operator as x ∈ M. From above, x 1 and x ∗ x ∈ Z(B)q. Since J V ∗ ξ = η = xξ = xJ ξ , for y ∈ M and b ∈ B, (V − x)yξ, bξ = yξ, V ∗ J b∗ J ξ − xyξ, bξ
= yξ, J b∗ J V ∗ ξ − xyξ, bξ
= yξ, J b∗ xJ ξ − xyξ, bξ
= yξ, x ∗ bξ − xyξ, bξ = 0,
(4.17)
and so eB V = eB x, implying that V = eB x. Thus P = V ∗ V = x ∗ eB x. Since P 2 = P , x ∗ eB x = x ∗ eB xx ∗ eB x = x ∗ EB xx ∗ eB x,
(4.18)
so the pull down map gives x ∗ x = x ∗ EB xx ∗ x.
(4.19)
This equation is x ∗ (1 − EB (xx ∗ ))x = 0, so (1 − EB (xx ∗ ))1/2 x = 0. Thus EB (xx ∗ )x = x. If we multiply on the right by x ∗ and apply EB , then we conclude that EB (xx ∗ ) is a projection. Moreover, EB xx ∗ xx ∗ = xx ∗ ,
(4.20)
so EB (xx ∗ ) xx ∗ since x 1. The trace then gives equality, and so x is a partial isometry with x ∗ x, xx ∗ ∈ B. Since x = xq and x ∗ eB x = P , which commutes with qBq, we obtain xbx ∗ eB = xqbqx ∗ xx ∗ eB = xqbqx ∗ eB xx ∗ = xx ∗ eB xqbqx ∗ = eB xbx ∗ ,
b ∈ B, (4.21)
showing that xBx ∗ ⊆ B. Thus x ∈ GN (1) (B) and P = x ∗ eB x. This completes the proof when A = C and Φ is the identity map.
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39
The second case is when A is an arbitrary abelian von Neumann algebra, and Φ is bounded, so we may assume that Φ is a state on A since (4.1) is unaffected by scaling. The result now follows from the first case by replacing the inclusion B ⊆ M by A ⊗ B ⊆ A ⊗ M. The trace on A ⊗ M is Φ ⊗ τ and eA⊗B is 1 ⊗ eB , so the canonical trace on A ⊗ M, eA⊗B is Φ ⊗ Tr. The last case is where Φ is a faithful normal semifinite weight. Then there is a family {fλ }λ∈Λ of orthogonal projections in A with sum 1 such that Φ(fλ ) < ∞ for each λ ∈ Λ. If Φλ denotes the restriction of Φ to Afλ , then Φλ is bounded. If we replace P , A, Φ and q by respectively P (fλ ⊗ 1), Afλ , Φλ and q(fλ ⊗ 1), then we are in the second case. Thus there exists, for each (1) λ ∈ Λ, a partial isometry vλ ∈ GN Af ⊗M (Afλ ⊗ B) so that P (fλ ⊗ 1) = vλ∗ (eAfλ ⊗B )vλ . The λ (1) central support of vλ lies below fλ ⊗ 1 so we may define v ∈ GN A⊗M (A ⊗ B) by v = λ∈Λ vλ , and it is routine to check that P = v ∗ eB v. 2 Theorem 4.1 characterizes those projections in the basic construction which arise from intertwiners. Corollary 4.2. Given an inclusion B ⊆ M of finite von Neumann algebras with B ∩ M ⊆ B and a projection q ∈ B, a projection P ∈ (B ∩ M, eB )q is of the form v ∗ eB v for some intertwiner v ∈ GN (1) (B) if and only if P eB in M, eB and Tr(P r) τ (qr) for all projections r ∈ Z(B)q. Furthermore in this case the domain projection v ∗ v must lie in Z(B)q. Proof. Taking A = C and Φ to be the identity in Theorem 4.1 shows that any projection P satisfying the conditions of the corollary is of the form v ∗ eB v for some intertwiner v ∈ GN (1) (B). Lemma 3.1(ii) then shows that v ∗ v ∈ Z(B)q. Conversely, given an intertwiner v ∈ GN (1) (B), Lemma 3.1(ii) shows that v ∗ eB v ∈ (B ∩ M, eB )q precisely when v ∗ v ∈ Z(B)q. The other two conditions of the corollary follow as v ∗ eB v ∼ vv ∗ eB eB in M, eB , and for a projection r ∈ Z(B)q, Tr(v ∗ eB vr) = τ (v ∗ vr) τ (qr). 2 The tracial hypothesis (4.1) of Theorem 4.1 is an extra ingredient in this theorem as compared with [1, Proposition 2.7]. The following example shows that Theorem 4.1 can fail without this hypothesis. Example 4.3. Let R be the hyperfinite II1 factor and fix an outer automorphism θ of period two. Let M = M3 ⊗ R and let B=
a 0 0
0 b θ (c)
0 : a, b, c ∈ R ⊆ M. c θ (b)
Note that B ∼ = Z(B) =√Ce11 ⊕ = R ⊕ (R θ Z2 ). It is straightforward to verify that B ∩ M√ C(e22 + e33 ) where {ei,j }3i,j =1 are the matrix units. Let P ∈ M, eB be ( 2 e12 )eB ( 2 e21 ). Since EB (e22 ) = (e22 + e33 )/2, P is a projection, and it is routine to verify that P commutes with B. If there is a nonzero intertwiner v ∈ M such that v ∗ eB v P , then v = ve11 . Direct calculation shows that v would then have the form we11 for some partial isometry w ∈ R, so v ∗ eB v would be qe11 eB for some projection q ∈ R. However, this nonzero projection is orthogonal to P and so cannot lie under it. Thus the conclusion of Theorem 4.1 fails in this case. Note that Tr(P e11 ) = τ (2e11 ) = 2/3, while τ (e11 ) = 1/3, so the tracial hypothesis of Theorem 4.1 is not satisfied.
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It is worth noting that B ∩ M, eB can be explicitly calculated in this case. This algebra is abelian and √ five-dimensional with minimal projections e11 eB , (1 − e11 )eB , (1 − e11 )(1 − eB ), √ ( 2 e12 )eB ( 2 e21 ), and e21 eB e12 + e31 eB e13 . The corresponding B-bimodules in L2 (M) are generated respectively by the vectors e11 , e22 + e33 , e22 − e33 , e12 , and e21 . For the remainder of the section we fix inclusions Bi ⊆ Mi of finite von Neumann algebras satisfying Bi ∩ Mi ⊆ Bi for i = 1, 2, and we denote the inclusion B1 ⊗ B2 ⊆ M1 ⊗ M2 by B ⊆ M. For i = 1, 2, let Si = sup Pj ∈ Z Bi ∩ Mi , eBi : Tri (Pj ) < ∞ . Note that Si acts as the identity on L2 (Z(Bi ∩ Mi , eBi ), Tri ). Given v ∈ GN M (B), the projection Pv of Definition 3.5 satisfies Tr(Pv ) 1, and Pv ∈ Z(B ∩ M, eB ) by Lemma 3.6. It follows from Lemma 2.7 that Pv S1 ⊗ S2 . Although Tri restricted to Z(Bi ∩ Mi , eBi ) might not be semifinite, it does have this property on the abelian von Neumann algebra Ai = Z(Bi ∩ Mi , eBi )Si by the choice of Si . Moreover, each Pv is an element of A1 ⊗ M2 , eB2 . We need two further projections which we define below. (1)
Definition 4.4. For i = 1, 2, let Pi denote the collection of projections R ∈ Z(Bi ∩ Mi , eBi ) which are expressible as R=
vn∗ eBi vn ,
vn ∈ GN (1) (Bi )
n1
where {vn∗ vn }n1 is an orthogonal set of projections in Bi . Such a projection satisfies Tri (R) 1. Let Qi be the supremum of the projections in Pi , so Qi Si . Our next objective is to show that each projection Pv arising from an intertwiner lies below (1) Q1 ⊗ Q2 . For the next two lemmas, the ∗ let v ∈ GNM (B) be fixed. We continue to∗ employ notation Pv for the projection n vn eB vn ∈ Z(B ∩ M, eB ) which satisfies Pv v v = v ∗ eB v. Let A1 be the abelian von Neumann algebra Z(B1 ∩ M1 , eB1 )S1 on which Φ, the restriction of Tr1 , is semifinite. Lemma 4.5. If r is a projection in A1 ⊗ Z(B2 ), then (Φ ⊗ Tr2 )(Pv r) (Φ ⊗ τ2 )(r).
(4.22)
Proof. There is a measure space (Ω, Σ, μ) so that A1 corresponds to L∞ (Ω) while Φ is given by integration with respect to the σ -finite measure μ. Then Pv is viewed as a projection-valued function Pv (ω), with the same representation r(ω) for r. For i = 1, 2, let Ψi be the pull down map for Mi , eBi . Then (Ψ1 ⊗ Ψ2 )(Pv ) = n vn∗ vn which is a projection, so has norm 1. By Lemma 2.5, (I ⊗ Ψ2 )(Pv ) 1, and so this element of A1 ⊗ M2 can be represented as a function f (ω) with f (ω) 1 almost everywhere. It follows that
J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
(Φ ⊗ τ2 )(r) =
τ2 r(ω) dμ(ω)
Ω
41
τ2 r(ω)f (ω) dμ(ω)
Ω
= (Φ ⊗ τ2 ) (I ⊗ Ψ2 )(Pv )r = (Φ ⊗ τ2 ) (I ⊗ Ψ2 )(Pv r) = (Φ ⊗ Tr2 )(Pv r),
(4.23)
where the penultimate equality is valid because r ∈ A1 ⊗ Z(B2 ) and the M2 -bimodular property of Ψ2 applies. 2 Lemma 4.6. For v ∈ GN (1) (B), the associated projection Pv ∈ Z(B ∩ M, eB ) satisfies Pv Q1 ⊗ Q2 . Proof. The remarks preceding Definition 4.4 show that Pv S1 ⊗ S2 . We will prove the stronger inequality Pv S1 ⊗ Q2 , which is sufficient to establish the result since we will also have Pv Q1 ⊗ S2 by a symmetric argument. As before, let A1 denote Z(B1 ∩ M1 , eB1 )S1 and let Φ be the restriction of Tr1 to A1 . Then, as noted earlier, A1 is an abelian von Neumann algebra and Φ is a faithful normal semifinite weight on A1 . Let {qn }∞ n=1 be a maximal family of nonzero orthogonal projections in Z(B1 ) ⊗ B2 so that Pv qn = wn∗ (1 ⊗ eB2 )wn for partial isometries wn ∈ A1 ⊗ M2 which are intertwiners of A1 ⊗ B2 . Let q = ∞ n=1 qn , defining q to be 0 if no such projections exist. We will first show that Pv q, so suppose that (1 − q)Pv = 0. The central support of 1 ⊗ eB2 in A1 ⊗ M2 , eB2 is 1, so there is a nonzero subprojection
with Q 1 ⊗ eB2 in this algebra. The projection Pv has the Q of (1 − q)P v in A1 ⊗ M2 , eB2 form Pv = n0 vn∗ eB vn where n0 vn∗ vn is the central support z ∈ Z(B) of v ∗ v ∈ B. With this notation, (3.20) becomes N (Pv ) = {x ∈ zMz: xPv = Pv x}. By Lemma 3.8(iii), there is a projection f ∈ N (Pv ) so that Q = f Pv . Both Q and Pv commute with B1 ⊗ 1, so the relation (b1 ⊗ 1)Q − Q(b1 ⊗ 1) = 0 for b1 ∈ B1 becomes ((b1 ⊗ 1)f − f (b1 ⊗ 1))Pv = 0. The element (b1 ⊗ 1)f − f (b1 ⊗ 1) lies in N (Pv ) so, by Lemma 3.8(i), (b1 ⊗ 1)f = f (b1 ⊗ 1) for all b1 ∈ B1 . This shows that f ∈ B1 ∩ N (Pv ). Moreover, f = (1 − q)f follows from the equation 0 = qQ = qf Pv ,
(4.24)
which implies that qf = 0 since qf ∈ N (Pv ). Note that (1 − q)z = 0, otherwise zf = 0. Now B1 ∩ N (Pv ) ⊆ B1 ∩ M = Z(B1 ) ⊗ M2
(4.25)
Z(B1 ) ⊗ B2 ∩ Z(B1 ) ⊗ M2 = Z(B1 ) ⊗ Z(B2 ) ⊆ Z(B1 ) ⊗ B2 .
(4.26)
and
Thus the inclusion (1 − q)z Z(B1 ) ⊗ B2 z(1 − q) ⊆ (1 − q)z B1 ∩ N (Pv ) z(1 − q)
(4.27)
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J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
has the property that the first algebra contains its relative commutant in the second algebra, which is the hypothesis of Lemma 2.6. Thus we may choose a nonzero projection b ∈ (1 − q)z(Z(B1 ) ⊗ B2 )z(1 − q) with b f in (1 − q)z(B1 ∩ N (Pv ))z(1 − q). Let w be a partial isometry in this algebra with w ∗ w = b and ww ∗ f , and note that w commutes with Pv by definition of N (Pv ). Then bPv = w ∗ wPv = Pv w ∗ wPv ∼ wPv w ∗ = ww ∗ Pv f Pv
(4.28)
in A1 ⊗ M2 , eB2 . Since b z, Lemma 3.8(i) ensures that bPv = 0. Moreover, bPv 1 ⊗ eB2 in A1 ⊗ M2 , eB2 since f Pv has this property. Consider now a projection r ∈ (A1 ⊗ Z(B2 ))b. The inequality Φ ⊗ Tr2 (Pv r) Φ ⊗ τ (r)
(4.29)
is valid by Lemma 4.5. Thus the hypotheses of Theorem 4.1 are satisfied with P replaced by bPv . (1) We conclude that there is an element w ∈ GN A ⊗M (A1 ⊗ B2 ) so that bPv = w ∗ (1 ⊗ eB2 )w. 1 2 Since b lies under 1 − q, this contradicts maximality of the qi ’s, proving that Pv q = Pv . Thus Pv =
∞
wn∗ (1 ⊗ eB2 )wn ,
(4.30)
n=1
∗ which we also write as Pv = ∞ n=1 Wn Wn where Wn is defined to be (1 ⊗ eB2 )wn ∈ A1 ⊗ M2 , eB2 . As in Lemma 4.5, we regard (A1 , Tr1 ) as L∞ (Ω) for a σ -finite measure space (Ω, μ). We can then identify A1 ⊗ M2 , eB2 with L∞ (Ω, M2 , eB2 ), and we write elements of this tensor product as uniformly bounded measurable functions on Ω with values in M2 , eB2 . Then Tr(Pv ) =
∞
Tr2 Wn (ω)∗ Wn (ω) dμ(ω).
(4.31)
n=1
Since Tr(Pv ) < ∞, we may neglect a countable number of null sets to conclude that ∞
∗ Wn (ω) Wn (ω) < ∞, Tr2 Pv (ω) = Tr2
ω ∈ Ω,
(4.32)
n=1
from which it follows that Pv (ω) Q2 for ω ∈ Ω. Thus Pv 1 ⊗ Q2 which gives Pv S1 ⊗ Q2 , since the inequality Pv S1 ⊗ S2 has already been established. 2 We are now in a position to approximate an intertwiner in a tensor product. Theorem 4.7. Let Bi ⊆ Mi , i = 1, 2, be inclusions of finite von Neumann algebras satisfying (1) Bi ∩ Mi ⊆ Bi . Given v ∈ GN M ⊗M (B1 ⊗ B2 ) and ε > 0, there exist x1 , . . . , xk ∈ B1 ⊗ B2 , (1)
1
2
(1)
w1,1 , . . . , w1,k ∈ GN M1 (B1 ) and w2,1 , . . . , w2,k ∈ GN M2 (B2 ) such that:
J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
1. xj 1 for each j ; 2.
k
xj (w1,j ⊗ w2,j ) v − j =1
43
(4.33)
< ε. 2,τ (1)
Proof. Write B ⊆ M for the inclusion B1 ⊗ B2 ⊆ M1 ⊗ M2 and fix v ∈ GN M (B). Recall from Lemmas 3.6 and 4.6 that there is a projection Pv ∈ Z(B ∩ M, eB ) satisfying Pv v ∗ v = v ∗ eB v, of the set Pi of projections in Z(Bi ∩ Mi , eBi ) and Pv Q1 ⊗ Q2 where Qi is the supremum specified in Definition 4.4. Thus Qi = k Ri,k for some countable sum of orthogonal projections Ri,k ∈ Pi . By Lemma 3.8(iii), any subprojection of Ri,k in Z(Bi ∩ Mi , eBi )is also in Pi , so for some it follows that every subprojection of Qi in B ∩ Mi , eBi is also of the form k Ri,k ∈P . countable sum of orthogonal projections Ri,k i The restriction Φi of Tri to Z(Bi ∩ Mi , eBi )Qi is a normal semifinite weight on this abelian von Neumann algebra Ai , and (Ai , Tri ) can be identified with L∞ (Ωi ) for a σ -finite measure space (Ωi , Σi , μi ). Since Pv Q1 ⊗ Q2 , this operator can be viewed as an element of L∞ (Ω1 × Ω2 , μ1 × μ2 ), and it also lies in the corresponding L2 -space since Tr(Pv ) 1. By Lemma 2.7 and the previous paragraph, Pv can be approximated in · 2,Tr -norm by finite sums of orthogonal projections of the form R1 ⊗ R2 , each lying in Pi . These elementary tensors correspond to measurable rectangles in Ω1 × Ω2 . By the definition of Pi , each Ri is close in (1) ∗ e w , with w
· 2,Tri -norm to a finite sum kj =1 wi,j Bi i,j i,j ∈ GN Mi (Bi ). This allows us to make (1)
the following approximation: given ε > 0, there exist finite sets {wi,j }kj =1 ∈ GN Mi (Bi ), i = 1, 2, such that k
∗ ∗ w1,j eB1 w1,j ⊗ w2,j eB2 w2,j < ε. (4.34) Pv − j =1
2,Tr
If we multiply on the right in (4.34) by v ∗ eB = v ∗ eB vv ∗ eB , then the result is k
∗ ∗ ∗ ∗ w1,j eB1 w1,j ⊗ w2,j eB2 w2,j v eB v eB − j =1
< ε,
(4.35)
2,Tr
using the fact that Pv v ∗ v = v ∗ eB v. A typical element of the sum in (4.35) is ∗ ∗ w1j ⊗ w2,j (eB1 ⊗ eB2 )(w1,j ⊗ w2,j )v ∗ (eB1 ⊗ eB2 ) ∗ ⊗ w ∗ )x ∗ e , where x ∗ = E ((w ∗ ∗ which has the form (w1,j B 1,j ⊗ w2,j )v ) ∈ B has xj j 2,j j B
(w1,j ⊗ w2,j )v ∗ 1. Thus (4.35) becomes
k ∗
∗ ∗ (w1,j ⊗ w2,j ) xj eB v − j =1
< ε.
(4.36)
2,Tr
For each y ∈ M,
yeB 22,Tr = Tr eB y ∗ yeB = τ y ∗ y = y 22,τ ,
(4.37)
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J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
and so (4.36) implies that k
xj (w1,j ⊗ w2,j ) v − j =1
as required.
< ε,
2,τ
2
Corollary 4.8. Let Bi ⊆ Mi , i = 1, 2, be inclusions of finite von Neumann algebras satisfying Bi ∩ Mi ⊆ Bi . Then W ∗ GN (1) (B1 ⊗ B2 ) = W ∗ GN (1) (B1 ) ⊗ W ∗ GN (1) (B2 ) .
(4.38)
There are two extreme cases where the hypothesis B ∩ M ⊆ B is satisfied. The first is when B is an irreducible subfactor where the result of Theorem 4.7 can be deduced from the stronger results of [20]. The second is when B is a masa in M where Theorem 4.7 is already known [1]. The following example explains the preference given to intertwiners over unitary normalizers in intermediate cases, even in a simple setting. Example 4.9. Let M be a II1 factor, let p ∈ M be projection whose trace lies in (0, 1/2), and let B = pMp + (1 − p)M(1 − p). This subalgebra has no non-trivial unitary normalizers, essentially because τ (p) = τ (1 − p). However, the tensor product B ⊗ B ⊆ M ⊗ M does have such normalizers because the compressions by p ⊗ (1 − p) and (1 − p) ⊗ p, which have equal traces, are conjugate by a unitary normalizer u which is certainly outside B ⊗ B. According to Theorem 4.7, u can be obtained as the limit of finite sums of elementary tensors from W ∗ (GN (1) (B)) ⊗ W ∗ (GN (1) (B)). 5. Groupoid normalizers of tensor products In this section we return to the groupoid normalizers GN M (B), namely those v ∈ M such (1) that v, v ∗ ∈ GN M (B). Our objective in this section is to establish a corresponding version of Theorem 4.7 for GN (B), and consequently we will assume throughout that any inclusion B ⊆ M satisfies the relative commutant condition B ∩ M ⊆ B. We will need to draw a sharp distinction between those intertwiners v that are groupoid normalizers and those that are not, and so we introduce the following definition. Definition 5.1. Say that v ∈ GN (1) (B) is strictly one-sided if the only projection p ∈ Z(Bv ∗ v) = Z(B)v ∗ v for which vp ∈ GN (B) is p = 0. When B is an irreducible subfactor of M then any unitary u ∈ M satisfying uBu∗ B is a strictly one-sided intertwiner (see [20, Example 5.4] for examples of such unitaries). Given v ∈ GN (B), recall from Section 3 that there is a projection Pv ∈ Z(B ∩ M, eB ) such that Pv v ∗ v = v ∗ eB v, and Pv has the form Pv =
n0
vn∗ eB vn ,
(5.1)
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45
∗ w = 0 for where there exist partial isometries wn ∈ B so that vn = vwn∗ ∈ GN (1) (B), and wm n m = n. Letting pn denote the projection wn wn∗ ∈ B, it also holds that vn∗ vn = pn . We will employ this notation below.
Lemma 5.2. Let v ∈ GN (B), and let u ∈ GN (1) (B) be strictly one-sided. Then Pv u∗ eB u = 0. Proof. By Lemmas 3.2(i) and 3.6, u∗ eB u ∈ Z(B ∩ M, eB )u∗ u and Pv ∈ Z(B ∩ M, eB ), showing that Pv and u∗ eB u are commuting projections. Let Q denote the projection Pv u∗ eB u in Z(B ∩ M, eB )u∗ u which lies below both Pv and u∗ eB u. From Lemmas 3.4 and 3.8 we may find projections z ∈ Z(B) and f ∈ M ∩ {Pv } such that Q = zu∗ eB u = f Pv .
(5.2)
From (5.1), write Pv as the strongly convergent sum Pv =
wn v ∗ eB vwn∗ ,
(5.3)
n0
so that (5.2) becomes
f wn v ∗ eB vwn∗ = zu∗ eB u.
(5.4)
n0
If we multiply (5.4) on the right by pj = wj wj∗ and on the left by eB u, noting that uzu∗ ∈ B, then the result is eB bj vwj∗ = eB uzu∗ upj = eB uu∗ uzpj = eB uzpj
(5.5)
for each j 0, where bj = EB (uf wj v ∗ ) ∈ B. Thus bj vwj∗ = uzpj ,
j 0.
(5.6)
If we sum (5.6) over j 0, then the right-hand side will converge strongly, implying strong convergence of j 0 bj vwj∗ . If we return to (5.4) and multiply on the left by eB u, then we obtain uz =
bn vwn∗
(5.7)
n0
with strong convergence of this sum. It follows that, for each b ∈ B, zu∗ buz = lim
k→∞
∗ wn v ∗ bn∗ bbm vwm
(5.8)
m,nk
strongly, and thus zu∗ buz ∈ B since v ∗ bn∗ bbm v ∈ B. Thus the projection p = zu∗ u ∈ Z(Bu∗ u) satisfies pu∗ Bup ⊆ B. Since u is strictly one-sided, we conclude that zu∗ u = 0, showing that
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J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
Q = zu∗ eB u = 0 from (5.2). This proves the result.
(5.9)
2
We can use the preceding lemma to show that a strictly one-sided intertwiner v ∈ GN (1) (B) has the property that the only projection p ∈ Bv ∗ v for which vp ∈ GN (B) is p = 0. Indeed, take such a projection p for which w = vp ∈ GN (B). Then Pv w ∗ w = w ∗ eB w so that Pv Pw by Remark 3.7. Lemma 5.2 then gives Pw v ∗ eB v = 0. Thus Pw w ∗ eB w = w ∗ eB w = 0 and so w = 0. Lemma 5.3. Let v ∈ GN (1) (B). Then there exist orthogonal families of orthogonal projections en , fn ∈ v ∗ vBv ∗ v such that (en + fn ) = v ∗ v, each ven is strictly one-sided, and each vfn lies in GN (B). Proof. Let {en } be a maximal orthogonal family of projections in Z(B)v ∗ v such that ven is of orthogonal projections strictly one-sided, and set e = en . Then choosea maximal family ∗ v then the result is proved, If e + f = v {fn } ∈ Z(B)(v ∗ v − e) such that vfn ∈GN (B). n n so consider the projection g = v ∗ v − en − fn ∈ Z(B)v ∗ v and suppose that g = 0. Then vg cannot be strictly one-sided otherwise the maximality of {en } would be contradicted. Thus there exists z ∈ Z(B) such that vgz is a nonzero element of GN (B). But this contradicts maximality of {fn }, proving the result. 2 We now return to considering two containments Bi ⊆ Mi satisfying Bi ∩ Mi ⊆ Bi , and the tensor product containment B = B1 ⊗ B2 ⊆ M1 ⊗ M2 = M. The next lemma is the key step required to obtain a version of Theorem 4.7 for groupoid normalizers. We need a result from the perturbation theory of finite von Neumann algebras. For any containment A ⊆ N , where N has a specified trace τ , recall that N ⊂δ,τ A means that sup x − EA (x) 2 : x ∈ N, x 1 δ where · 2 is defined using the given trace τ . If τ is scaled by a constant λ, then ⊂δ,λτ is the same as ⊂δ/√λ,τ . Then [16, Theorem 3.5] (see also [18, Theorem 10.3.5]), stated for normalized traces, has the following general interpretation: if A ⊆ N and N ⊂δ,τ A for some δ < (τ (1)/23)1/2 , then there exists a nonzero projection p ∈ Z(A ∩ N ) such that Ap = pNp. Lemma 5.4. Let v ∈ GN (1) (B1 ) and w ∈ GN (1) (B2 ). If v (or w) is strictly one-sided then v ⊗ w ∈ GN (1) (B) is strictly one-sided. Proof. Without loss of generality, suppose that v is strictly one-sided. Fix a nonzero projection p ∈ Z(B)(v ∗ v ⊗ w ∗ w), and let τi be the faithful normalized normal trace on Mi ⊃ Bi . Given ε > 0, we may choose projections pi ∈ Z(B1 )v ∗ v and qi ∈ Z(B2 )w ∗ w, 1 i k, with p − k i=1 pi ⊗ qi 2,τ < ε and the pi ’s orthogonal since these projections lie in abelian von Neumann algebras. Here, · 2,τ is with respect to the normalized trace τ = τ1 ⊗ τ2 on M. Now pi B1 pi = pi v ∗ vB1 v ∗ vpi ⊆ pi v ∗ B1 vpi
(5.10)
since vB1 v ∗ ⊆ B1 . If it were true that pi v ∗ B1 vpi ⊂δ,τ1 pi B1 pi for some δ < (τ (pi )/23)1/2 , then it would follow from [16, Theorem 3.5] that there exists a nonzero projection pi ∈ (pi B1 pi ) ∩
J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
47
pi v ∗ B1 vpi ⊆ Z(B1 )pi such that pi pi B1 pi = pi pi v ∗ B1 vpi pi , contradicting the hypothesis that v is strictly one-sided. Thus there exists bi ∈ B1 satisfying ∗ pi v bi vpi 1,
1/2 d2,τ1 pi v ∗ bi vpi , pi B1 pi τ1 (pi )/23 ,
(5.11)
where d2,τ1 (x, A) = inf{ x − a 2,τ1 : a ∈ A} for any von Neumann algebra A. Since each pi lies under v ∗ v, the projections vpi v ∗ lie in B1 and are orthogonal. We may then define an element b ∈ B by b=
vpi v ∗ bi vpi v ∗ ⊗ wqi w ∗ ,
(5.12)
i
and the orthogonality gives b 1. Moreover, (v ⊗ w)∗ b(v ⊗ w) =
pi v ∗ bi vpi ⊗ qi ,
(5.13)
i
and 2 d2,τ (v ⊗ w)∗ b(v ⊗ w), B
τ1 (pi )τ2 (qi ) /23
i
> τ (p) − ε /23.
(5.14)
The right-hand side of (5.13) is unchanged by pre- and post-multiplication by the projection i pi ⊗ qi , and this is close to p. This leads to the estimate
p(v ⊗ w)∗ b(v ⊗ w)p − (v ⊗ w)∗ b(v ⊗ w) 2 p − pi ⊗ q i 2,τ i
< 2ε. (5.15)
2,τ
From (5.14) and (5.15), 1/2 d2,τ p(v ⊗ w)∗ b(v ⊗ w)p, B > τ (p) − ε /23 − 2ε.
(5.16)
A sufficiently small choice of ε then shows that p(v ⊗ w)∗ b(v ⊗ w)p ∈ / B, and thus v ⊗ w is strictly one-sided. 2 We can now give the two-sided counterpart of Theorem 4.7. Theorem 5.5. Let Bi ⊆ Mi , i = 1, 2, be inclusions of finite von Neumann algebras satisfying Bi ∩ Mi ⊆ Bi . Given v ∈ GN M1 ⊗M2 (B1 ⊗ B2 ) and ε > 0, there exist x1 , . . . , xk ∈ B1 ⊗ B2 , w1,1 , . . . , w1,k ∈ GN M1 (B1 ) and w2,1 , . . . , w2,k ∈ GN M2 (B2 ) such that: 1. xj 1 for each j ; 2.
k
xj (w1,j ⊗ w2,j ) v − j =1
< ε. 2,τ
(5.17)
48
J. Fang et al. / Journal of Functional Analysis 258 (2010) 20–49
Proof. Consider v ∈ GN M1 ⊗M2 (B1 ⊗ B2 ). Following the proof of Theorem 4.7, given ε > 0, there exist elements vi,j ∈ GN (1) (Bi ), 1 j k, so that k
∗ (v1,j ⊗ v2,j ) eB (v1,j ⊗ v2,j ) Pv − j =1
<ε
(5.18)
2,Tr
as in (4.34). Using Lemma 5.3, we may replace this sum with one of the form
(w1,j ⊗ w2,j )∗ eB (w1,j ⊗ w2,j ) + (x1,j ⊗ x2,j )∗ eB (x1,j ⊗ x2,j ) where the wi,j ’s are two-sided and, for each j , at least one of x1,j , x2,j is strictly one-sided. By Lemma 5.4, each x1,j ⊗ x2,j is strictly one-sided, so (x1,j ⊗ x2,j )∗ eB (x1,j ⊗ x2,j )Pv = 0 by Lemma 5.2. If we multiply on the right by Pv , then
∗ Pv − (w1,j ⊗ w2,j ) eB (w1,j ⊗ w2,j )Pv j
< ε.
(5.19)
2,τ
Simple approximation allows us to obtain the same estimate for some finite subcollection of the w1,j and w2,j , say w1,1 , . . . , w1,k and w2,1 , . . . , w2,k . We now continue to follow the proof of Theorem 4.7 from (4.34) to obtain the required xj . 2 Just as in Section 4, the next corollary follows immediately. Corollary 5.6. Let Bi ⊆ Mi , i = 1, 2, be inclusions of finite von Neumann algebras satisfying Bi ∩ Mi ⊆ Bi . Then GN M1 (B1 ) ⊗ GN M2 (B2 ) = GN M1 ⊗M2 (B1 ⊗ B2 ) . Acknowledgments The work in this paper originated during the Workshop in Analysis and Probability, held at Texas A&M University during Summer 2007. It is a pleasure to express our thanks to both the organizers of the workshop and to the NSF for providing financial support to the workshop. References [1] [2] [3] [4] [5] [6] [7]
I. Chifan, On the normalizing algebra of a masa in a II1 factor, arXiv:math.OA/0606225, 2006. E. Christensen, Subalgebras of a finite algebra, Math. Ann. 243 (1979) 17–29. J. Dixmier, Sous-anneaux abéliens maximaux dans les facteurs de type fini, Ann. of Math. (2) 59 (1954) 279–286. H.A. Dye, On groups of measure preserving transformation, I, Amer. J. Math. 81 (1959) 119–159. E.G. Effros, Z.-J. Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (2) (1990) 163–187. J. Fang, On completely singular von Neumann subalgebras, Proc. Edinb. Math. Soc. (2) 52 (2009) 607–618. J. Feldman, C.C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, I, Trans. Amer. Math. Soc. 234 (2) (1977) 289–324. [8] J. Feldman, C.C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, II, Trans. Amer. Math. Soc. 234 (2) (1977) 325–359. [9] V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1) (1983) 1–25.
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[10] V. Jones, V.S. Sunder, Introduction to Subfactors, London Math. Soc. Lecture Note Ser., vol. 234, Cambridge University Press, Cambridge, 1997. [11] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. I, Grad. Stud. Math., vol. 15, American Mathematical Society, Providence, RI, 1997, Elementary theory, reprint of the 1983 original. [12] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II, Grad. Stud. Math., vol. 16, American Mathematical Society, Providence, RI, 1997, Advanced theory, corrected reprint of the 1986 original. [13] M. Pimsner, S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1) (1986) 57–106. [14] S. Popa, On a class of type II1 factors with Betti numbers invariants, Ann. of Math. (2) 163 (3) (2006) 809–899. [15] S. Popa, Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I, Invent. Math. 165 (2006) 369–408. [16] S. Popa, A.M. Sinclair, R.R. Smith, Perturbations of subalgebras of type II1 factors, J. Funct. Anal. 213 (2) (2004) 346–379. [17] H.L. Royden, Real Analysis, third ed., Macmillan Publishing Company, New York, 1988. [18] A.M. Sinclair, R.R. Smith, Finite von Neumann Algebras and Masas, London Math. Soc. Lecture Note Ser., vol. 351, Cambridge University Press, Cambridge, 2008. [19] A.M. Sinclair, R.R. Smith, S.A. White, A. Wiggins, Strong singularity of singular masas in II1 factors, Illinois J. Math. 51 (4) (2007) 1077–1083. [20] R.R. Smith, S.A. White, A. Wiggins, Normalizers of irreducible subfactors, J. Math. Anal. Appl. 352 (2) (2009) 684–695. [21] M. Takesaki, Theory of Operator Algebras, II, Operator Algebras and Non-commutative Geometry, vol. VI, Encyclopaedia Math. Sci., vol. 125, Springer-Verlag, Berlin, 2003.
Journal of Functional Analysis 258 (2010) 50–66 www.elsevier.com/locate/jfa
Linear maps preserving the minimum and reduced minimum moduli A. Bourhim a,∗,1,2 , M. Burgos b,3 , V.S. Shulman c a Département de Mathématiques et de Statistique, Université Laval, Québec (Québec), Canada G1K 7P4 b Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain c Department of Mathematics, Vologda State Technical University, Vologda, Russia
Received 16 September 2008; accepted 2 October 2009
Communicated by Paul Malliavin
Abstract We describe linear maps from a C ∗ -algebra onto another one preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the reduced minimum modulus. © 2009 Elsevier Inc. All rights reserved. Keywords: C ∗ -algebras; Linear preserver problems; Minimum modulus; Surjectivity modulus; Reduced minimum modulus
1. Introduction In the last few decades, there has been a considerable interest in the so-called linear preserver problems which concern the characterization of linear or additive maps on matrix algebras or operator algebras or more generally on Banach algebras that leave invariant a certain function, a certain subset, or a certain relation; see for instance the survey papers [8,13,21,22,29] and the * Corresponding author.
E-mail addresses:
[email protected],
[email protected] (A. Bourhim),
[email protected] (M. Burgos),
[email protected] (V.S. Shulman). 1 Current address: Department of Mathematics, Syracuse University, 215 Carnegie Building, Syracuse, NY 13244, USA. 2 Supported by an adjunct professorship at Laval university. 3 Partially supported by the Junta de Andalucía PAI project FQM-3737 and the I+D MEC project MTM2007-65959. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.003
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references therein. One of the most important question in this active research area is the famous problem initiated by Kaplansky’s discussion in [20] that asks whether a spectrum-preserving linear map from a semisimple unital Banach algebra A onto another one B is a Jordan isomorphism. This problem remains far from being solved even when A and B are C ∗ -algebras but it has a positive solution when B is commutative or when A and B are either von Neumann algebras or Banach algebras with big socles; see for instance [1,2,6,7,10–12,17,19,23,32,34]. Recently, Mbekhta described unital surjective linear maps on the algebra L(H ) of all bounded linear operators on an infinite dimensional complex Hilbert space H preserving several spectral quantities such as the minimum, the surjectivity and the reduced minimum moduli. In [26], he proved that a unital surjective linear map from L(H ) onto itself preserves either the minimum modulus or the surjectivity modulus if and only if it is a self-adjoint automorphism. While in [25], he showed that a unital surjective linear map on L(H ) preserves the reduced minimum modulus precisely when it is either a self-adjoint automorphism or a self-adjoint anti-automorphism. These results have been extended by the first two authors to the setting of surjective linear maps between C ∗ -algebras of real rank zero preserving different spectral quantities; see [4,5]. For the nonunital case, Mbekhta closed his paper [25] with the following natural conjecture. Conjecture. (See [25].) A surjective linear map Φ : L(H ) → L(H ) preserves the reduced minimum modulus if and only if there are unitary operators U, V ∈ L(H ) such that ϕ takes either the form ϕ(T ) = U T V (T ∈ L(H )) or ϕ(T ) = U T tr V (T ∈ L(H )). In this paper, we unify and extend all the results from [4,5,25,26] by a characterization of (not necessarily unital) surjective linear maps between C ∗ -algebras A and B preserving the minimum, the surjectivity, the maximum, and the reduced minimum moduli. We do not impose any condition on the C ∗ -algebras A and B, and also consider the nonsurjective case. One of the most important steps in the proofs of our results is to show that the maps we are dealing with preserve the invertibility in both directions. Thus, the obtained results not only provide a positive answer to Mbekhta’s conjecture but are also considered as a positive answer to a version of Kaplansky’s problem. The contents of this paper are as follows. In Section 2, we gather some basic definitions and preliminary properties of different spectral quantities which are needed for the proofs of our main results stated in Section 3. In Section 4, we collect some auxiliary lemmas which are the main ingredients for the proofs, presented in Sections 5 and 6, of our results. The last section is devoted to some consequences of the obtained results. 2. Preliminaries Throughout this paper, the term Banach algebra means a unital complex associative Banach algebra, with unit 1, and a C ∗ -algebra means a unital (complex associative) C ∗ -algebra. Let A be a Banach algebra, and Inv(A) be the group of all invertible elements of A. For an element a in A, let σ (a), ∂σ (a) and r(a) denote the spectrum, the boundary of the spectrum and the spectral radius of a, respectively. 2.1. Notation Let A and B be Banach algebras. A linear map Φ : A → B is called unital if Φ(1) = 1, and it is said to be a Jordan homomorphism if Φ(a 2 ) = Φ(a)2 for all a ∈ A. Equivalently, the map
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Φ is a Jordan homomorphism if and only if Φ(ab + ba) = Φ(a)Φ(b) + Φ(b)Φ(a) for all a and b in A. It is called a Jordan isomorphism provided that it is a bijective Jordan homomorphism. Clearly, every homomorphism and every anti-homomorphism is a Jordan homomorphism. It is well known that if Φ : A → B is a Jordan homomorphism, then Φ(aba) = Φ(a)Φ(b)Φ(a)
(2.1)
for all a, b ∈ A. Moreover, if Φ is a Jordan isomorphism, then Φ strongly preserves invertibility, that is Φ a −1 = Φ(a)−1
(2.2)
for every invertible element a in A. We finally recall that if A and B are C ∗ -algebras, then the map Φ is said to be self-adjoint provided that Φ(a ∗ ) = Φ(a)∗ for all a ∈ A, and it is called positive if Φ(a) is positive for every positive element a ∈ A. Self-adjoint Jordan homomorphisms are called Jordan ∗-homomorphisms. 2.2. Minimum, surjectivity and maximum moduli in Banach algebras Let X be a complex Banach space, and let L(X) be the algebra of all bounded linear operators on X. The minimum modulus of an operator T ∈ L(X) is defined by m(T ) := inf{T x: x ∈ X, x = 1}, and the surjectivity modulus of T is defined by q(T ) := sup{ε 0: εBX ⊆ T (BX )}, where as usual BX denotes the closed unit ball of X. Note that m(T ) > 0 if and only if T is injective and has closed range, and that q(T ) > 0 if and only if T is surjective. Moreover, m(T ) = inf{T S: S ∈ L(X), S = 1} and q(T ) = inf{ST : S ∈ L(X), S = 1}; see [28, Theorem II.9.11]. It is also well known that, if X = H is a Hilbert space and T ∈ L(H ) is a bounded linear operator on H , then m(T ) = inf{λ: λ ∈ σ (|T |)} and q(T ) = inf{λ: λ ∈ σ (|T ∗ |)}, where T ∗ is the adjoint of T and |T | := (T ∗ T )1/2 is the absolute value of T . Thus, m(T ) = q(T ∗ ), and m(T ) > 0 (resp. q(T ) > 0) if and only if T is left invertible (resp. right invertible). We now translate these definitions and properties to the setting of Banach algebras. Let A be a Banach algebra, and let a be an element of A. The minimum modulus and the surjectivity modulus of a are defined respectively by m(a) := m(La ) = inf ax: x ∈ A, x = 1 , and q(a) := m(Ra ) = inf xa: x ∈ A, x = 1 , where La and Ra are the left and right multiplication operators by a. The maximum modulus of a is defined by M(a) := max m(a), q(a) . Obviously, m(a) = 0 (respectively q(a) = 0) if and only if a is a left (respectively right) topological divisor of zero. Also M(a) = 0 if and only if a is a topological divisor of zero. The following lemmas summarize some elementary properties of the quantities defined above.
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Lemma 2.1. Let A be a Banach algebra. The following statements hold. (i) If a ∈ A is an invertible element, then −1 M(a) = m(a) = q(a) = a −1 . (ii) For every a, b ∈ A, m(a)m(b) m(ab) am(b) and q(a)q(b) q(ab) q(a)b.
(2.3)
(iii) For every a, b ∈ A, m(a) − m(b) a − b and q(a) − q(b) a − b. Proof. See [28, Section II.9], and [30, Section I.5].
(2.4)
2
Lemma 2.2. If A is a C ∗ -algebra, then the following assertions hold. (i) For every a ∈ A, m(a) = inf{λ: λ ∈ σ (|a|)} and q(a) = inf{λ: λ ∈ σ (|a ∗ |)}, where |a| := (a ∗ a)1/2 is the absolute value of a. (ii) For every a ∈ A, m(a) = q(a ∗ ). (iii) For every a ∈ A, m(a) > 0 (respectively q(a) > 0) if and only if a is left (respectively right) invertible. (iv) If u and v are unitary elements in A, then m(a) = m(uav), q(a) = q(uav), and M(a) = M(uav) for all a ∈ A. Proof. Since any C ∗ -algebra can be considered as an algebra of operators on a Hilbert space, we have m(a) = inf{λ: λ ∈ σ (|a|)} and q(a) = inf{λ: λ ∈ σ (|a ∗ |)} for all a ∈ A, and the first statement is established. The second and the third statements are immediate consequences of the first one, and the last affirmation follows straightforwardly. 2 Notice that for an element a in a Banach algebra A, the approximate point spectrum, σap (.), the surjective spectrum, σs (.), and their intersection, σap,s (.), given by σap (a) := λ ∈ C: m(a − λ) = 0 , σs (a) := λ ∈ C: q(a − λ) = 0 , σap,s (a) := λ ∈ C: M(a − λ) = 0
(2.5) (2.6) (2.7)
are closed subsets of σ (a) containing ∂σ (a). 2.3. Von Neumann regularity in Banach algebras Let A be a Banach algebra. An element a ∈ A is called (von Neumann) regular if it has a generalized inverse, that is, if there exists b ∈ A that satisfies a = aba and b = bab. Obviously, if a is regular, then so are the left and right multiplication operators by a, and thus their ranges
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aA = La (A) and Aa = Ra (A) are both closed. Also observe that if a and b are two elements of A such that a = aba, then a is regular and b = bab is a generalized inverse of a. This shows, in particular, that the generalized inverse of a regular element a is not unique. In fact, if a = aba and b = bab, then for every x ∈ A, the element y = b + x − baxab also satisfies a = aya and thus yay is a generalized inverse of a. Finally, note that if a has a generalized inverse b, then p = ab and q = ba are idempotents in A satisfying aA = pA and Aa = Aq. Recall that the reduced minimum modulus of an operator T ∈ L(X) is given by γ (T ) = inf{T (x): dist(x, Ker(T )) 1} if T = 0 and γ (T ) = ∞ if T = 0. It is positive precisely when T has closed range; see for instance [28, II.10]. The reduced minimum modulus (or the conorm) of an element a in a Banach algebra A, is defined as the reduced minimum modulus of the left multiplication operator by a, γ (a) := γ (La ) =
inf{ax: dist(x, Ker(La )) 1} if a = 0, ∞ if a = 0.
If b is a generalized inverse of a, with a = 0, then b−1 γ (a) baabb−1 ;
(2.8)
see [14, Theorem 2]. Regular elements in C ∗ -algebras are studied by Harte and Mbekhta in [14,15]. They proved that if A is C ∗ -algebra, then a ∈ A is regular if and only if aA is closed, equivalently γ (a) > 0, and that 2 γ (a)2 = γ a ∗ a = inf λ: λ ∈ σ a ∗ a \ {0} = γ a ∗ .
(2.9)
Furthermore, they showed that if a is a regular element, then −1 γ (a) = a † ,
(2.10)
where a † is the Moore–Penrose inverse of a, that is, the unique element b ∈ A for which a = aba, b = bab and the associated idempotents ab and ba are self-adjoint; see [15, Theorem 2]. For an element a in a Banach algebra A, denote by reg(a) the regular set of a, that is, the set of all λ ∈ C such that there exist a neighborhood Uλ of λ and an analytic function b : Uλ → A such that b(μ) is a generalized inverse of a − μ for any μ ∈ Uλ . The generalized spectrum (also called Saphar spectrum) of a is given by σg (a) := C \ reg(a), and the Kato spectrum of a is defined as
(2.11) σK (a) := λ ∈ C: lim γ (a − μ) = 0 . μ→λ
The following properties of the generalized spectrum and the Kato spectrum are well known (see [28, Sections 12, 13], and [24,27]): (1) (2) (3) (4)
0∈ / σg (a) if and only if a is regular and Ker(La ) ⊆ n1 a n A. 0∈ / σK (a) if and only if aA is closed and Ker(La ) ⊆ n1 a n A. ∂σ (a) ⊆ σK (a) ⊆ σg (a) ⊆ σ (a). If A is a C ∗ -algebra, then σg (a ∗ ) = σg (a), and σg (a) = σK (a) for all a ∈ A.
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3. Statement of main results In the sequel let c(.) stand for any of the spectral quantities m(.), q(.), M(.) and γ (.), and let σc (.) denote the spectra described in term of the spectral quantity c(.) by (2.5), (2.6), (2.7) and (2.11), respectively. A linear map Φ : A → B between two Banach algebras A and B is said to preserve the spectral quantity c(.) if c(Φ(a)) = c(a) for all a ∈ A. The description of such maps is the main object of this paper. Assume that H is an infinite dimensional complex Hilbert space, and let T tr denote the transpose of all operators T ∈ L(H ) with respect to an arbitrary but a fixed orthonormal basis in H . Mbekhta’s results, mentioned in the introduction, assert that every unital surjective linear map Φ : L(H ) → L(H ) which preserves the spectral quantity c(.) is an isometry. It is worth recalling that a surjective linear isometry Φ on L(H ) is either Φ(T ) = U T V
T ∈ L(H ) ,
or Φ(T ) = U T tr V
T ∈ L(H ) ,
for some unitary operators U, V ∈ L(H ). More generally, Kadison, in his celebrated paper [18], proved that a surjective linear map between two C ∗ -algebras A and B is an isometry if and only if it is a Jordan ∗-isomorphism multiplied by a unitary element in B. Thus, Mbekhta’s conjecture can be rephrased by saying that a surjective linear map on L(H ) preserves the reduced minimum modulus if and only if it is an isometry. Based on the Mbekhta’s results and conjecture, the following arises in a natural way. Conjecture. Let A and B be two C ∗ -algebras, and let Φ be a linear map from A onto B. If Φ preserves the spectral quantity c(.), then Φ is an isometry. The main results of this paper lead to a positive solution to this conjecture. Furthermore, we describe nonsurjective unital linear maps that preserve the spectral quantity c(.). Theorem 3.1. Let A and B be C ∗ -algebras. If Φ : A → B is a unital linear map such that c(x) = c(Φ(x)) for all x ∈ A, then Φ is an isometric Jordan ∗-homomorphism. It should be noted that not every unital isometric Jordan ∗-homomorphism preserves the minimum or surjectivity moduli. For example, the linear map Φ : T → T tr on L(H ), for an infinite dimensional complex Hilbert space H , is an isometric Jordan ∗-isomorphism and preserves neither the minimum modulus nor the surjectivity modulus. But of course, this map preserves both the maximum and the reduced minimum moduli. On the other hand, the same map preserves all these spectral quantities when H is finite dimensional. In fact, given a unital ∗-antihomomorphism Φ : A → B between C ∗ -algebras A and B, we always have m(Φ(x)) = q(x) and σ (x ∗ x) = σ (Φ(x)Φ(x)∗ ) for all x ∈ A, and Φ preserves both the maximum and the reduced minimum moduli. Moreover, if for either A or B every left invertible element is invertible, then Φ preserves the minimum and the surjectivity moduli as well. Note also that a unital isometric self-adjoint map between C ∗ -algebras can extend spectra and preserve spectral radius but it needs not be a Jordan homomorphism. As an example one can take the map from the algebra L∞ (T) of all bounded measurable functions on the unit circle T into L(H 2 (T)) that takes each function φ to the Toeplitz operator Tφ with symbol φ. Here, H 2 (T) denotes the classical Hardy space on T.
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Finally, we point out that the above theorem extends the main results of [25,26], and does not impose the surjectivity condition on Φ. However, if Φ is surjective then the fact that Φ is unital is not needed, as is shown by our next result which also generalizes the main results of [25,26] and answers positively the above conjecture. Theorem 3.2. Let A be a semisimple Banach algebra and let B be a C ∗ -algebra. If Φ : A → B is a surjective linear map for which c(x) = c(Φ(x)) for all x ∈ A, then A (for its norm and some involution) is a C ∗ -algebra, and Φ is an isometric Jordan ∗-isomorphism multiplied by a unitary element of B. Before closing this section, we shall make a couple of comments. Assume that A and B are two C ∗ -algebras, and that Φ is a Jordan ∗-isomorphism ϕ multiplied on the left by a unitary element u of B. The map Φ can be written as a Jordan ∗-isomorphism multiplied on the right by a unitary element of B. Indeed, we have Φ(.) = uϕ(.) = uϕ(.)u∗ u, and uϕ(.)u∗ is a Jordan ∗-isomorphism. In Theorem 3.2, the role of A or B being a C ∗ -algebra is symmetrical since, from Lemma 4.2, a surjective linear map Φ : A → B between Banach algebras is bijective provided it preserves the spectral quantity c(.). 4. Technical lemmas In this section, we assemble some auxiliary lemmas which will be needed for the proofs of Theorems 3.1 and 3.2. The first one, which is the key tool for our main results, characterizes hermitian elements in a Banach algebra in terms of the minimum, surjectivity, maximum and reduced minimum moduli. Before stating this lemma, it will be convenient to introduce some definition and notation. Recall that an element h of a Banach algebra A is called hermitian if it has real numerical range, that is, if f (h) ∈ R, for all f in the Banach dual space of A with f (1) = 1 = f . We shall denote by H (A) the closed real subspace of A of all hermitian elements of A. It is well known that an element in a C ∗ -algebra is hermitian if and only if it is self-adjoint. By the Vidav– Palmer theorem, a unital Banach algebra A is a C ∗ -algebra if and only if it is generated by its hermitian elements, that is, if A = H (A) + iH (A). In this case, the involution is given by (h + ik)∗ := h − ik
h, k ∈ H (A) ;
(4.12)
see [3, §5, §6]. Lemma 4.1. Let a be an element of a Banach algebra A. The following assertions are equivalent. (i) (ii) (iii) (iv)
a is hermitian. 1 + ita = 1 + ◦(t) as t → 0. c(1 + ita) = 1 + ◦(t) as t → 0. c(1 + ita) 1 + ◦(t) as t → 0.
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Proof. The equivalence (i) ⇔ (ii) is well known; see for instance [3, Lemma 5.2]. The implication (ii) ⇒ (iii) is obvious since c(1 + ita) = (1 + ita)−1 −1 for sufficiently small t. While the implication (iii) ⇒ (iv) is obviously trivial. Now, assume that c(1 + ita) 1 + (t) with (t) = ◦(t) as t → 0. We always have 1 + ◦(t) = 1 + (t) c(1 + ita) 1 + ita.
(4.13)
On the other hand, for sufficiently small t, we have −1 1 + (t) c(1 + ita) = (1 + ita)−1 .
(4.14)
So, (1 + ita)−1 1 + δ(t)
δ(t) = ◦(t), as t → 0 .
(4.15)
Since 1 − ita = (1 + ita)−1 (1 + t 2 a 2 ), we get 1 − ita 1 + δ(t) 1 + t 2 a2 = 1 + η(t)
η(t) = ◦(t), as t → 0 .
(4.16)
Replacing t by −t, we get 1 + ita 1 + η(−t) = 1 + ◦(t),
as t → 0.
(4.17)
It follows from (4.13) and (4.17) that 1 + ita = 1 + ◦(t), as t → 0, and so a is hermitian; as desired. 2 The next lemma shows in particular that linear maps between Banach algebras (not necessarily semisimple) which preserve the spectral quantity c(.) are always injective. Its proof is simple and relies on Lemma 4.1. Lemma 4.2. Assume that d(.) denotes also any of the spectral quantities m(.), q(.), M(.) or γ (.), and let A and B be two Banach algebras. If Φ is a linear map from A into B for which 1 c(Φ(1)) and c(Φ(x)) d(x) for all x ∈ A, then Φ is injective. Proof. Assume that Φ(a0 ) = 0 for some a0 ∈ A, and let us prove that a0 = 0. For every α ∈ C, we have 1 c(Φ(1)) = c(Φ(1 + αa0 )) d(1 + αa0 ). Thus, 1 d(1 + ita0 )
and 1 d(1 − ta0 )
for all t ∈ R. By Lemma 4.1, we see that both a0 and ia0 are hermitian. This shows that a0 = 0, and implies that Φ is injective; as desired. 2 The last lemma is an immediate consequence of [33, Corollary 1.4]. We give here an alternative elementary proof which avoids certain background from the theory of non-associative algebras.
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Lemma 4.3. Every unital injective Jordan ∗-homomorphism Φ : A → B between two C ∗ -algebras A and B is an isometry. Proof. The unital injective Jordan ∗-homomorphism Φ can be decomposed into a sum of a ∗-homomorphism and ∗-anti-homomorphism acting to the universal enveloping von Neumann algebra B of B. More precisely, there is a central projection e ∈ B such that the map x → eΦ(x) is a ∗-homomorphism and x → (1 − e)Φ(x) is a ∗-anti-homomorphism. Replacing the original multiplication in B by x ∗ y := 12 (exy + (1 − e)yx) (x, y ∈ B), the map Φ becomes a unital injective ∗-homomorphism of C ∗ -algebras and hence it is an isometry. 2 5. Proof of Theorem 3.1 Theorem 3.1 is an immediate consequence of the following more general result together with Lemma 4.2. Theorem 5.1. Assume that d(.) denotes also any of the spectral quantities m(.), q(.), M(.) or γ (.). Let A and B be two C ∗ -algebras and let Φ : A → B be a unital linear map such that c(x) d(Φ(x)) for all x ∈ A. Then Φ is a Jordan ∗-homomorphism. Moreover, if Φ is injective, then Φ is isometric. Proof. In view of Lemma 4.3, we only need to establish the first part of the theorem. For every self-adjoint element a ∈ A, we have 1 + ◦(t) c(1 + ita) d Φ(1 + ita) = d 1 + itΦ(a) ,
as t → 0,
and Lemma 4.1 implies that Φ(a) is also self-adjoint. Therefore, given a ∈ A, and h, k ∈ H (A) such that a = h + ik, we get ∗ Φ a ∗ = Φ(h − ik) = Φ(h) − iΦ(k) = Φ(h) + iΦ(k) = Φ(h + ik)∗ = Φ(a)∗ . This implies that Φ is a self-adjoint map. Moreover, as ∂σ Φ(x) ⊆ σd Φ(x) ⊆ σc (x) ⊆ σ (x) for all x ∈ A, and positive elements are self-adjoint elements with nonnegative spectrum, it is clear that Φ is positive. Hence, Φ = 1 by [31, Corollary 1]. Now, let a be a self-adjoint element in A and t sufficiently small such that the unitary element u = eita ∈ A has spectrum strictly contained in the unit circle T. We have ∂σ Φ(u) ⊆ σd Φ(u) ⊆ σc (u) = σ (u) T, and the interior, int(σ (Φ(u))), of σ (Φ(u)) is a subset of the open unit disc D. In fact, it is closed and open in D since int(σ (Φ(u))) = σ (Φ(u)) ∩ D, and thus either int(σ (Φ(u))) = ∅ or int(σ (Φ(u))) = D. The second possibility cannot occur since ∂σ (Φ(u)) T, and Φ(u) is invertible. Moreover, Φ(u) u = 1,
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59
and Φ(u)−1 = d Φ(u) −1 c(u)−1 = u−1 = 1, which shows that Φ(u) is unitary. Since Φ(u) = 1 + itΦ(a) − 12 t 2 Φ(a 2 ) + · · · and Φ(u)∗ = Φ(u∗ ) = 1 − itΦ(a) − 12 t 2 Φ(a 2 ) + · · · , we have 1 = Φ(u)Φ(u)∗ = Φ(u)Φ u∗
1 2 2 1 2 2 = 1 + itΦ(a) − t Φ a + · · · 1 − itΦ(a) − t Φ a + · · · 2 2 whence 2 t Φ(a)2 − Φ a 2 = ◦ t 2 . We deduce that Φ(a)2 = Φ(a 2 ), and hence Φ is a Jordan homomorphism. This completes the proof. 2 Note that the assumption of injectivity of Φ in Theorem 5.1 is necessary. Consider a unital ∗-homomorphism Φ : A → B between C ∗ -algebras A and B and note that σ Φ(x)∗ Φ(x) = σ Φ x ∗ x ⊂ σ x ∗ x for all x ∈ A. From this, we infer that c(x) c(Φ(x)) for all x ∈ A. 6. Proof of Theorem 3.2 Roughly speaking, Theorem 3.2 asserts that a surjective linear map Φ between two C ∗ -algebras A and B preserving the spectral quantity c(.) is an isometric Jordan ∗-isomorphism multiplied by a unitary element via Φ(1). An important step of the proof of this theorem is to show that Φ(1) is invertible. Of course, the one-sided invertibility of Φ(1) is obvious when c(.) represents either m(.) or q(.) or M(.). But this fact is not obviously seen when c(.) coincides with γ (.), and therefore some extra efforts are needed to establish it. So, it is more convenient to split Theorem 3.2 into two results, and prove them separately. The first one deals with the case when c(.) represents either m(.) or q(.) or M(.), and says little bit more than what has been stated in Theorem 3.2. Theorem 6.1. Assume that d(.) denotes also any of the spectral quantities m(.), q(.), or M(.). Let A be a semisimple Banach algebra and let B be a C ∗ -algebra. If Φ : A → B is a surjective linear map for which d(Φ(1)) 1 and c(x) d(Φ(x)) for all x ∈ A, then A (for its norm and some involution) is a C ∗ -algebra, and Φ is an isometric Jordan ∗-isomorphism multiplied by a unitary element of B. Proof. We first assume that d(.) = m(.), and let us prove that b := Φ(1) is invertible. We have 1 m(Φ(1)) = m(b) c(1) = 1, and m(b) = 1. Thus, the ideal bB is closed and b is not a left divisor of zero. To see that b is invertible, it suffices to show that b is not a right divisor of zero.
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So, assume that there is y ∈ B such that b∗ y = 0 = y ∗ b. Since Φ is surjective, there is x ∈ A such that y = Φ(x). For every α ∈ C, we have 2 c(1 + αx)2 m Φ(1) + αΦ(x) = m(b + αy)2 = m (b + y)∗ (b + y) = m b∗ b + |α|2 y ∗ y 1 − |α|2 y2 . The last inequality holds because the spectral function m(.) is contractive; see (2.4). Now, we have 1/2 c(1 + itx) 1 − t 2 y2
1/2 and c(1 − tx) 1 − t 2 y2
for all t ∈ R. By Lemma 4.1, we see that both x and ix are hermitian. This shows that x = 0, and implies that y = Φ(x) = 0 as well. Thus b = Φ(1) is invertible; as desired. Now, consider the unital linear map ϕ : A → B defined by ϕ(x) := b−1 Φ(x) for all x ∈ A, and note that both Φ and ϕ are bijective; see Lemma 4.2. We have b−1 −1 = m(b) = 1, and m ϕ(x) b−1 m Φ(x) = m Φ(x) c(x)
(6.18)
for all x ∈ A; see (2.3). Equivalently, m(y) c ϕ −1 (y)
(6.19)
for all y ∈ B. Thus, c 1 + itϕ −1 (y) = c ϕ −1 (1 + ity) m(1 + ity) for all y ∈ B and t ∈ C. It follows from Lemma 4.1 that ϕ −1 (y) ∈ H (A) whenever y ∈ H (B). As B is a C ∗ -algebra and ϕ −1 (H (B)) ⊆ H (A), we have A = H (A) + iH (A). By the Vidav– Palmer theorem, A is a C ∗ -algebra for its norm and the involution given by (4.12). In view of Theorem 5.1 and (6.19), we see that ϕ is an isometric Jordan ∗-isomorphism. In order to conclude the proof, we need to see that b is unitary. For every x ∈ Inv(A), we have Φ(x)−1 = ϕ(x −1 )b−1 . Moreover, it follows from (6.18) that Φ(x)−1 = x −1 for all x ∈ Inv(A). Since ϕ is an isometry, it follows that −1 −1 ϕ x b = Φ(x)−1 = x −1 = ϕ x −1 for all x ∈ Inv(A). Thus, yb−1 = y for all y ∈ Inv(B), and consequently b is unitary. With little bit effort, the proof runs in a similar way if the spectral quantity d(.) represents either q(.) or M(.). 2
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The second result describes surjective linear maps between C ∗ -algebras that preserve the reduced minimum modulus. Theorem 6.2. Let A be a semisimple Banach algebra and let B be a C ∗ -algebra. If Φ : A → B is a surjective linear map preserving the reduced minimum modulus, then A (for its norm and some involution) is a C ∗ -algebra, and Φ is an isometric Jordan ∗-isomorphism multiplied by a unitary element of B. Proof. We only need to prove that b := Φ(1) is invertible as the rest of the proof goes in the same lines as the one of Theorem 6.1. We first show that b is right invertible. Note that, since γ (b) = γ (Φ(1)) = γ (1) = 1, the element b is regular in B and the ideals bB, Bb are both closed. Let x ∈ A and y ∈ B such that y = Φ(x) and b∗ y = 0 = y ∗ b. For every α ∈ C, we have 2 γ (1 + αx)2 = γ Φ(1) + αΦ(x) = γ (b + αy)2 = γ b∗ b + |α|2 y ∗ y , and −2 lim γ b∗ b + |α|2 y ∗ y = lim γ (1 + αx)2 = lim (1 + αx)−1 = 1 = γ b∗ b .
|α|→0
|α|→0
|α|→0
By [15, Theorem 6], we see that lim|α|→0 (b∗ b + |α|2 y ∗ y)† − (b∗ b)† = 0. Thus, for sufficiently small α ∈ C, we have ∗ b b + |α|2 y ∗ y † b∗ b + |α|2 y ∗ y − b∗ b † b∗ b < 1, and ∗ γ b b + |α|2 y ∗ y − 1 = γ b∗ b + |α|2 y ∗ y − γ b∗ b |α|2 y2 ; see [15, Theorem 5(5.2)]. In particular, for sufficiently small α ∈ C, we have γ (1 + αx)2 = γ b∗ b + |α|2 y ∗ y 1 − |α|2 y2 . We therefore have 1/2 γ (1 + itx) 1 − t 2 y2
1/2 and γ (1 − tx) 1 − t 2 y2
for all sufficiently small t ∈ R. By Lemma 4.1, we see that both x and ix are hermitian. This shows that x = 0, and implies that y = Φ(x) = 0 as well. Thus b = Φ(1) is right invertible. Similar arguments show that b is left invertible as well, and consequently b is an invertible element of B; as desired. 2
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7. Consequences In this section, we exemplify our main results in the case of surjective linear maps on L(H ), for an infinite dimensional complex Hilbert space H , preserving the spectral quantity c(.). We begin with the following result which in particular solves Mbekhta’s conjecture. Theorem 7.1. Assume that H is a Hilbert space. A linear map Φ from L(H ) onto itself preserves the reduced minimum modulus if and only if there are two unitary operators U and V such that either Φ(T ) = U T V (T ∈ L(H )), or Φ(T ) = U T tr V (T ∈ L(H )). Proof. Obviously, the “if part” always holds. While, the “only if part” is a direct consequence of Theorem 6.2. 2 Using results about invertibility preserving linear maps on L(H ), more can be obtained. Theorem 7.2. Assume that H is a Hilbert space. For a linear map Φ from L(H ) onto itself, the following statements are equivalent. (i) There are α, β > 0 such that βγ (T ) γ (Φ(T )) αγ (T ) for all T ∈ L(H ). (ii) There are two invertible operators A and B such that either Φ(T ) = AT B (T ∈ L(H )), or Φ(T ) = AT tr B (T ∈ L(H )). Proof. It is easy to verify that the implication (ii) ⇒ (i) always holds. Assume that there are α, β > 0 such that βγ (T ) γ Φ(T ) αγ (T )
(7.20)
for all T ∈ L(H ), and let us prove that Φ preserves invertibility in both directions. To do that, let us first show that T0 := Φ(1) is one-sided invertible. Note that, since γ (T0 ) β > 0, the operator T0 has a closed range. To see that T0 is one-sided invertible, it suffices to show that either Ker(T0 ) or Ker(T0∗ ) is trivial. Assume for the way of contradiction that both Ker(T0 ) and Ker(T0∗ ) are nontrivial, and pick two unit vectors x ∈ Ker(T0 ) and y ∈ Ker(T0∗ ). Let > 0, and note that x is orthogonal to Ker(T0 + y ⊗ x), where y ⊗ x is the rank one operator defined by z → z, xy. We therefore have γ (T0 + y ⊗ x) (T0 + y ⊗ x)x = . Since Φ is surjective, there is R ∈ L(H ) such that Φ(R) = y ⊗ x. For sufficiently small , we have 1 + R is invertible and −1 β (1 + R)−1 = βγ (1 + R) γ Φ(1 + R) = γ (T0 + y ⊗ x) . Since the left side of the inequality tends to β as goes to 0, we get a contradiction. This shows that T0 = Φ(1) is semi-invertible; as desired. Now, we are ready to show that T0 is invertible. Assume without loss of generality that T0 is left invertible, and let R0 be a left inverse of T0 . We consider the unital linear map ϕ defined by ϕ(T ) := R0 Φ(T ) (T ∈ L(H )), and note that m(ϕ(T )) R0 m(Φ(T )) R0 γ (Φ(T )) αR0 γ (T ) for all T ∈ L(H ). We therefore have σg (T ) ⊆ σap (ϕ(T )) for all T ∈ L(H ), and thus
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r(T ) r(ϕ(T )) for all T ∈ L(H ). From this, it follows that ϕ is injective. Indeed, let R ∈ L(H ) be an operator such that ϕ(R) = 0, and pick T ∈ L(H ). For every λ ∈ C, we have r(λR + T ) r ϕ(λR + T ) = r ϕ(T ) . As λ → r(λR + T ) is a subharmonic function on C, Liouville’s theorem implies that r(R + T ) = r(T ). Because T is an arbitrary operator in L(H ), the spectral characterization of the radical, together with the semisimplicity of L(H ) imply that R = 0, and hence ϕ is injective. Thus Φ is injective as well and is, in fact, a bijective map. Let S ∈ L(H ) such that Φ(S) = 1 − T0 R0 , and note that ϕ(S) = R0 Φ(S) = R0 (1 − T0 R0 ) = 0. It follows that S = 0, and Φ(S) = Φ(0) = 1 − T0 R0 = 0 which shows that T0 is right invertible. Therefore, T0 is invertible; as desired. Note that, since Φ is a bijective map and its inverse Φ −1 satisfies the analogous inequalities to (7.20), we only need to show that Φ preserves the invertibility in one direction. Let T ∈ L(H ) be an invertible operator and let us show that Φ(T ) is invertible as well. Consider the linear surjective map Ψ on L(H ) defined by Ψ (S) := Φ(T S)
S ∈ L(H ) ,
and note that −1 β T −1 γ (S) γ Ψ (S) αT γ (S)
S ∈ L(H ) .
By what has been shown above, it follows that Ψ (1) = Φ(T ) is invertible. Finally, apply the main result of either [7] or [17] to get the desired forms of Φ. This establishes the implication (i) ⇒ (ii), and completes the proof. 2 We also have the following result. Theorem 7.3. Assume that d(.) represents either m(.) or q(.), and that H is an infinite dimensional Hilbert space. For a linear map Φ from L(H ) onto itself, the following statements are equivalent. (i) (ii) (iii) (iv)
1 d(T0 ) for some T0 ∈ Φ −1 ({1}), and d(T ) d(Φ(T )) for all T ∈ L(H ). 1 d(Φ(1)), and d(Φ(T )) d(T ) for all T ∈ L(H ). d(Φ(T )) = d(T ) for all T ∈ L(H ). There are two unitary operators U and V such that Φ(T ) = U T V (T ∈ L(H )).
Proof. In view of Theorem 6.1, we always have (ii) ⇔ (iii) ⇔ (iv). Of course, the implication (iv) ⇒ (i) is always there. Now, assume that 1 d(T0 ) for some T0 ∈ Φ −1 ({1}), and d(T ) d(Φ(T )) for all T ∈ L(H ), and suppose for instance that d(.) represents m(.). We have m(T0 ) = 1 > 0, and T0 has a left
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inverse R0 ; i.e., R0 T0 = 1. Consider the unital surjective map Ψ defined on L(H ) by Ψ (T ) := Φ(T T0 )
T ∈ L(H ) .
We have m Ψ (T ) = m Φ(T T0 ) m(T T0 ) m(T )m(T0 ) = m(T ) for all T ∈ L(H ). By Theorem 5.1, Ψ is a Jordan ∗-homomorphism of L(H ), and the well-known theorem of Herstein [16] tells us that Ψ is either a ∗-homomorphism or a ∗anti-homomorphism. In particular, Ker(Ψ ) is a closed two-sided ideal of L(H ) and either Ker(Ψ ) = {0} or Ker(Ψ ) contains all finite rank operators; see [9]. The last possibility cannot occur since otherwise L(H )/Ker(Ψ ) will be mapped onto L(H ) by either an isomorphism or anti-isomorphism. This is a contradiction since L(H ) has a big socle while L(H )/Ker(Ψ ) has a trivial one. Thus, Ψ is either a ∗-automorphism or a ∗-anti-automorphism. Now, it is easy to see that the injectivity of Φ follows from the injectivity of Ψ . Indeed, let S0 ∈ L(H ) such that Φ(S0 ) = 0 and let us show that S0 = 0. We have 0 = Φ(S0 ) = Ψ (S0 R0 ), and S0 R0 = 0. Thus S0 = S0 R0 T0 = 0, and Φ is injective. In fact, it is bijective and its inverse satisfies m Φ −1 (1) = m(T0 ) = 1 and m Φ −1 (T ) m(T ) for all T ∈ L(H ). Therefore, Theorem 6.1 applied to Φ −1 shows that there are two unitary operators U and V such that either Φ(T ) = U T V (T ∈ L(H )), or Φ(T ) = U T tr V (T ∈ L(H )). It is easy to show that the second possibility cannot occur since the map t → U T tr V maps left invertible operators into right invertible ones. 2 Remark 7.4. Let S ∈ L(H ) be a noninvertible isometry on H , and consider the surjective linear maps defined on L(H ) by Φ : T → T S
and Ψ : T → S ∗ T .
Of course, these maps do not have the form given in Theorem 7.3(iv) and are not even injective. As m(T ) m(Φ(T )) and q(T ) q(Ψ (T )) for all T ∈ L(H ), we see that the condition “1 d(T0 ) for some T0 ∈ Φ −1 ({1})” in the first statement of Theorem 7.3 is necessary. Also note that, since q(Φ(T )) q(T ) and m(Ψ (T )) m(T ) for all T ∈ L(H ), the condition “1 d(Φ(1))” in the second statement of Theorem 7.3 is necessary as well. Similar proof to the one of Theorem 7.3 yields the following result. The details are omitted. Theorem 7.5. Assume that H is a Hilbert space. For a linear map Φ from L(H ) onto itself, the following statements are equivalent. (i) 1 M(T0 ) for some T0 ∈ Φ −1 ({1}), and M(T ) M(Φ(T )) for all T ∈ L(H ). (ii) 1 M(Φ(1)), and M(Φ(T )) M(T ) for all T ∈ L(H ). (iii) M(Φ(T )) = M(T ) for all T ∈ L(H ).
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(iv) There are two unitary operators U and V such that either Φ(T ) = U T V (T ∈ L(H )), or Φ(T ) = U T tr V (T ∈ L(H )). Remark 7.6. Similar results to Theorem 7.2 can be obtained when replacing γ (.) by either m(.) or q(.) or M(.). Also essential versions of the above results in the line of [4,5] can be stated. References [1] B. Aupetit, Spectrum-preserving linear map between Banach algebra or Jordan–Banach algebra, J. London Math. Soc. 62 (2000) 917–924. [2] B. Aupetit, H.T. Mouton, Spectrum preserving linear mappings in Banach algebras, Studia Math. 109 (1) (1994) 91–100. [3] F.F. Bonsall, J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, London Math. Soc. Lecture Note Ser., vol. 2, Cambridge University Press, London, New York, 1971. [4] A. Bourhim, M. Burgos, Linear maps preserving the minimum modulus, Oper. Matrices, in press. [5] A. Bourhim, M. Burgos, Linear maps preserving regularity in C ∗ -algebras, Illinois J. Math., in press. [6] M. Brešar, A. Fošner, P. Šemrl, A note on invertibility preservers on Banach algebras, Proc. Amer. Math. Soc. 131 (2003) 3833–3837. [7] M. Brešar, P. Šemrl, Linear maps preserving the spectral radius, J. Funct. Anal. 142 (2) (1996) 360–368. [8] M. Brešar, P. Šemrl, Linear preservers on B(X), Banach Center Publ. 38 (1997) 49–58. [9] J.W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. Math. 42 (4) (1941) 839–873. [10] M.D. Choi, D. Hadwin, E. Nordgren, H. Radjavi, P. Rosenthal, On positive linear maps preserving invertibility, J. Funct. Anal. 59 (3) (1984) 462–469. [11] J. Dieudonné, Sur une généralisation du groupe orthogonal a quatre variables, Arch. Math. 1 (1949) 282–287. [12] A. Gleason, A characterization of maximal ideals, J. Anal. Math. 19 (1967) 171–172. [13] A. Guterman, C.K. Li, P. Šemrl, Some general techniques on linear preserver problems, Linear Algebra Appl. 315 (2000) 61–81. [14] R. Harte, M. Mbekhta, On generalized inverses in C ∗ -algebras, Studia Math. 103 (1992) 71–77. [15] R. Harte, M. Mbekhta, Generalized inverses in C ∗ -algebras, II, Studia Math. 106 (1993) 129–138. [16] I.N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956) 331–341. [17] A. Jafarian, A.R. Sourour, Spectrum preserving linear maps, J. Funct. Anal. 66 (1986) 255–261. [18] R.V. Kadison, Isometries of operator algebras, Ann. Math. 54 (1951) 325–338. ˙ [19] J.P. Kahane, W. Zelazko, A characterization of maximal ideals in commutative Banach algebras, Studia Math. 29 (1968) 339–343. [20] I. Kaplansky, Algebraic and Analytic Aspect of Operator Algebras, American Mathematical Society, Providence, RI, 1970. [21] C.K. Li, S. Pierce, Linear preserver problems, Amer. Math. Monthly 108 (2001) 591–605. [22] C.K. Li, N.K. Tsing, Linear preserver problems: A brief introduction and some special techniques, Linear Algebra Appl. 162–164 (1992) 217–235. [23] M. Marcus, R. Purves, Linear transformations on algebras of matrices: The invariance of the elementary symmetric functions, Canad. J. Math. 11 (1959) 383–396. [24] M. Mbekhta, Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux, Glasg. Math. J. 29 (1987) 129–175. [25] M. Mbekhta, Linear maps preserving the generalized spectrum, Extracta Math. 22 (2007) 45–54. [26] M. Mbekhta, Linear maps preserving the minimum and surjectivity moduli of operators, preprint. [27] M. Mbekhta, A. Ouahab, Opérateur s-régulier dans un espace de Banach et théorie spectrale, Acta Sci. Math. (Szeged) 59 (1994) 525–543. [28] V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Oper. Theory Adv. Appl., vol. 139, Birkhäuser Verlag, Basel, 2003. [29] S. Pierce, et al., A survey of linear preserver problems, Linear Multilinear Algebra 33 (1992) 1–129. [30] C.E. Rickart, General Theory of Banach Algebras, Kreiger, New York, 1974. [31] B. Russo, H.A. Dye, A note on unitary operators in C ∗ -algebras, Duke Math. J. 33 (1966) 413–416.
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[32] A.R. Sourour, Invertibility preserving linear maps on L(X), Trans. Amer. Math. Soc. 348 (1) (1996) 13–30. [33] J.D.M. Wright, Jordan C ∗ -algebras, Michigan Math. J. 24 (3) (1977) 291–302. ˙ [34] W. Zelazko, Characterization of multiplicative linear functionals in complex Banach algebras, Studia Math. 30 (1968) 83–85.
Journal of Functional Analysis 258 (2010) 67–98 www.elsevier.com/locate/jfa
Resolvent estimates and decomposable extensions of generalized Cesàro operators Alexandru Aleman, Anna-Maria Persson ∗ Department of Mathematics, Lund University, PO Box 118, S-221 00 Lund, Sweden Received 19 December 2008; accepted 9 October 2009 Available online 21 October 2009 Communicated by N. Kalton
Abstract We determine the spectrum of generalized Cesàro operators with essentially rational symbols acting on various spaces of analytic functions, including Hardy spaces, weighted Bergman and Dirichlet spaces. Then we show that in all cases these operators are subdecomposable. © 2009 Elsevier Inc. All rights reserved. Keywords: Cesàro operator; Spaces of analytic functions; Decomposable extension
1. Introduction Let H (D) denote the space of all analytic functions in the unit disc D and consider for g ∈ H (D) the generalized Cesàro operator Cg defined by 1 Cg f (z) = z
z
f (ζ )g (ζ ) dζ,
z ∈ D, f ∈ H (D).
0
The classical Cesàro operator C, which is usually defined on sequence spaces by its action on a sequence x = (x0 , x1 , . . .): * Corresponding author.
E-mail addresses:
[email protected] (A. Aleman),
[email protected] (A.-M. Persson). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.006
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1 xk , n+1 n
(Cx)n =
k=0
1 is obtained for g (z) = 1−z . The boundedness and compactness of such operators defined on various spaces of analytic functions, like Hardy spaces, weighted Bergman and Dirichlet spaces, or even Bloch-type spaces, have attracted a lot of attention (see for example, [6,4,7,12,16,24,30], or the survey papers [29,3] and the references therein). For the sake of completion we include a list of short definitions of those spaces which are relevant for this work. For 0 < p < ∞, the Hardy spaces H p are defined by
H = f ∈ H (D), p
p f p
2π 2π it p dt it p dt f re f e = <∞ , = lim 2π 2π r→1− 0
0
while H ∞ is the space of all bounded analytic functions on the unit disc endowed with the p,α supremum norm. The standard weighted Bergman spaces La , α > −1, 0 < p < ∞ are defined by p,α La
p α p f (z) 1 − |z| dA(z) < ∞ . = f ∈ H (D), f Lp,α = a
D p,0
p
For the unweighted case α = 0 we simply write La = La . The weighted Dirichlet spaces D p,α , p,α α > p − 1, consist of those analytic functions in D whose derivative belongs to La , that is,
p p α p D p,α = f ∈ H (D), f D p,α = f (0) + f (z) 1 − |z| dA(z) < ∞ . D
Finally, we will also consider the well-known growth classes γ A−γ = f ∈ H (D): f −γ = sup 1 − |z| f (z) < ∞ , z∈D
−γ
for γ > 0, as well as the closed subspace A0 , γ > 0 of A−γ defined by γ −γ A0 = f ∈ A−γ : lim sup 1 − |z| f (z) = 0 . |z|→1
The description of the spectrum of the operators considered above appears to be a delicate matter, even for the classical example C. Siskakis (see [26,28]) developed a method based on composition semigroups to obtain the spectrum of C on Hardy spaces and unweighted Bergman spaces, and the result was extended to certain weighted Dirichlet spaces by Galanopoulos in [11]. Dahlner [8] used a different approach essentially based on Hardy-type inequalities and deterp,α mined the fine spectrum of C acting on standard weighted Bergman spaces La , α > −1, 1 < p < ∞. In all cases, it turns out that the spectrum of the Cesàro operator is a closed disc
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which consists of the origin and the points λ ∈ C \ {0} such that exp(g/λ) does not belong to the 1 space in question, where g(z) = log 1−z . Using Fredholm theory in a quite clever way, Young (see [31,32]) was able to show that this result extends to the case when the derivative of the symbol g is the sum of a rational function with poles on the unit circle and a bounded function. The spaces considered by Young are H 2 and L2a , but the result continues to hold for all stanp,α dard weighted Bergman spaces La , for α −1 and 1 < p < ∞ (see [2]). In connection with the spectra of such operators, we should point out that for the closely related, non-normalized version of Cg , defined by z Tg f (z) =
f (ζ )g (ζ ) dζ,
z ∈ D, g, f ∈ H (D),
0
Pommerenke [22] observed that if this operator is bounded on H 2 (or on any other Banach space of analytic functions in the unit disc containing the constants), then its spectrum always contains the set of λ ∈ C \ {0} such that exp(g/λ) does not belong to the space. However, very recently, it has been shown in [5] that for general symbols g the spectrum of Tg on a weighted Bergman p,α space La can be much larger. A further development in this direction has been inspired by the pioneering work of Kriete and Trutt [14] who proved in 1971 that the usual Cesàro operator C is subnormal on H 2 . More recently, using the connection between this operator and certain composition semigroups, Miller, Miller and Smith [20] showed that C is subdecomposable on H p , 1 < p < ∞, that is, it has a decomposable extension, and then Miller and Miller [17,18] extended the result to the case p of unweighted Bergman spaces La , for p 2. Recall that an operator T on the Banach space X is called decomposable if for any pair of open sets U ; V ⊂ C with U ∪ V = C there exist T -invariant subspaces M, N ⊂ X such that σ (T |M) ⊂ U , σ (T |N ) ⊂ V and M + N = X. The semigroup technique used in the above papers fails to work for other values of the parameter p. However, the approach developed by Dahlner [8] yields the same result for C acting on all stanp,α dard weighted Bergman spaces La , with α > −1, and 1 < p < ∞. Finally, the nonreflexive case p = 1 could not be handled with any of the above methods. The question whether C on H 1 (or L1,α a ) is subdecomposable was answered affirmatively by the second author in [21]. Actually, all results mentioned here state that when acting on the spaces in question, C has Bishop’s property (β). The fact that this property characterizes subdecomposability has been proved by Albrecht and Eschmeier [1] (see also [10]). Much less is known about the corresponding questions for generalized Cesàro operators. In 1 [19] Miller, Miller and Neumann proved that Cg has Bishop’s property (β) when g (z) = 1−z 2, n
z g (z) = 1−z , or when (1 − z)g (z) extends analytically in a larger disc and has positive real part in D. Moreover, it is pointed out in [2] that their technique applies to the corresponding operators considered on the unweighted Bergman spaces. In this paper we consider generalized Cesàro operators with essentially rational symbols, that is, g = r + h, where r is a rational function with simple poles on the unit circle and h is a smooth function, h ∈ H ∞ . We give a unified approach to the description of the spectrum of such operators and estimate their resolvent. As a consequence we prove that these generalized Cesàro operators are subdecomposable on all Banach spaces of analytic functions listed in this introduction. Our method relies on certain common properties of these spaces which are briefly discussed below.
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a−z (1) Composition with conformal automorphisms of the disc ψa (z) = 1−az induces bounded operators on these spaces, but they not necessarily uniformly bounded in a ∈ D. However, there exists a number γ > 0 depending only on the space such that the weighted composition operators γ Cψa given by
γ γ Cψa f = ψa f ◦ ψa are uniformly bounded in a ∈ D. (2) Composition operators Cφ f = f ◦ φ induced by polynomials of degree one φ : D → D are bounded on these spaces and satisfy an estimate of the form Cφ X
c , (1 − |φ(0)|)γ
with the same constant γ > 0 as in (1). For Hardy and Bergman spaces as well as for the growth classes considered here, the estimate holds for arbitrary analytic self-maps of the disc. This is no longer true for Dirichlet spaces but the simple maps needed for our purposes do satisfy the estimate. A proof will be given in Section 3. (3) The operator of multiplication by the independent variable Mξ f (z) = zf (z) is bounded and bounded below all spaces considered here and its spectrum equals the closed unit disc. (4) The polynomials are dense in all spaces listed above, except the growth classes A−γ , and all of them contain the functions z → log(1 − bz) for all b ∈ ∂D. The results for A−γ will be −γ then deduced with help of A0 . (5) The norm on these spaces can be localized with help of composition operators. This is a more technical fact which is needed only to deal with the case when the derivative of the symbol has several poles. It can be expressed as follows. Given distinct points b1 , . . . , bn ∈ ∂D, there exists a finite set F contained in the set F of Riemann maps from D onto the interior of C 5 -Jordan curves contained in D which fix the origin, such that: (i) If φ ∈ F then φ(D) contains exactly one of the points bj , 1 j n, and the composition operator Cφ is bounded on X. (ii) If f ∈ H (D) satisfies f ◦ φ ∈ X for all φ ∈ F then f ∈ X. These are the only properties needed in our proofs and for this reason our results hold for any Banach space of analytic functions where (1)–(5) are fulfilled. Therefore we have stated our theorems in the general context. We prove that generalized Cesàro operators with essentially rational symbols are bounded on such a space and describe their fine spectrum in our main theorem, Theorem 5.1. In particular, it turns out that the spectrum of such an operator equals a finite union of closed discs having the origin on their boundary and determined by the poles of the rational part of g and its residua. Moreover, the Fredholm index is defined on the complement of the boundaries of these discs and equals the sum of their characteristic functions. Finally, there is an operator-valued function defined a.e. on C whose value at λ is a left-inverse of λI − Cg and whose operator norm is locally integrable on C \ {0}. This last result easily implies that Cg has Bishop’s property (β), and hence it is subdecomposable.
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The paper is organized as follows. In Section 2 we derive a formal expression for the resolvent of Cg which is then used to obtain information about the range of λI − Cg when g has poles on the boundary. In Section 3 we explore the assumptions (1)–(3) and develop the basic tools needed for our main estimate which could be seen as a Hardy-type inequality for such Banach spaces of analytic functions. This technical result together with its proof are presented in Section 4. In Section 5 we gather the information obtained in the previous sections and give the proof of the main result mentioned above. 2. General form of the resolvent The present section is devoted to some preliminary considerations concerning the resolvent of the generalized Cesàro operator 1 Cg f (z) = z
z
f (ζ )g (ζ ) dζ,
z ∈ D,
(2.1)
0
where g is a fixed analytic function in D with g(0) = 0. We consider first Cg as an operator on the space H (D) of all analytic functions in D and derive a formula for its resolvent operator, that is, the solution f of the resolvent equation λf − Cg f = h,
h ∈ H (D).
(2.2)
For computational purposes it is convenient to introduce the analytic function u in D \ (−1, 0] which satisfies the equality zu (z) = g (z). u(z) A simple computation shows that
u(z) = z
g (0)
z exp
g (ζ ) − g (0) dζ , ζ
z ∈ D \ (−1, 0],
(2.3)
0
where for the possibly non-integer power above we choose the principal branch, that is, 1g (0) = 1. Note that the function uα extends analytically to D if and only if αg (0) is a nonnegative integer. Another direct computation shows that if h = 0 Eq. (2.2) has a nonzero solution if and only if λ = g n(0) , n ∈ Z+ . In this case all solutions have the form 1
f (z) = c
u λ (z) , z
c ∈ C, z ∈ D.
Given an analytic function h in D we shall denote throughout in what follows by m0 (h) the multiplicity of its zero at the origin (m0 (h) = 0 if h(0) = 0). Also, by ξ we denote the identity function on D, that is, ξ(z) = z.
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Proposition 2.1. Let λ ∈ C \ {0}.
(i) If g (0) = 0, or m0 (h) > Re g λ(0) − 1 then Eq. (2.2) has the analytic solution f in D given by z
1
h(z) u(z) λ f (z) = Rλ h(z) = + 2 λ λ z
1
u(ζ )− λ g (ζ )h(ζ ) dζ.
0
(ii) If g (0) = 0 and Re g λ(0) − 1 m0 (h), let m be an arbitrary positive integer with m >
Re g λ(0) − 1 if λ = has the solution
g (0) n ,
f (z) =
n ∈ Z+ , and if λ =
m−1
g (0) n
βk (h, λ)ξ + Rλ h − k
for some n ∈ Z+ , let m = n. Then (2.2)
m−1
k=0
βk (h, λ)(λI − Cg )ξ
k
,
k=0
where the coefficients βk (h, λ) are the uniquely determined solutions of the linear system h(j ) (0) =
m−1
(j ) (λI − Cg )ξ k (0)xk ,
0 j m − 1.
(2.4)
g (0) n
for some n ∈ Z+
k=0
(iii) The solution given above is unique if λ = 1 λ
g (0) n ,
n ∈ Z+ , and if λ =
then all solutions have the form f + cu , c ∈ C. Proof. (i) We multiply by the independent variable and differentiate both sides of (2.2) to obtain λzf (z) + λf (z) − f (z)g (z) =
d zh(z) dz
or equivalently, 1 d 1 g (z) − f (z) = zh(z) . f (z) + z λz λz dz
1
We then use the integrating factor z → zu(z)− λ to get 1 1 d d 1 f (z)zu(z)− λ = u(z)− λ zh(z) dz λ dz
(2.5)
for all z ∈ D \ (−1, 0]. Using our assumption, we find the particular solution 1
u(z) λ f (z) = λ
1 0
u(tz)−1/λ tzh (tz) + h(tz) dt,
(2.6)
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since the integral on the right is convergent in this case. It is also easily seen that the right-hand side defines an analytic function in the whole disc, that is, we have 1
u(z) λ f (z) = λz
z
u(ζ )−1/λ ζ h (ζ ) + h(ζ ) dζ,
z ∈ D,
0
because the nonanalytic powers of z cancel out. Recall that use again the assumption to obtain the equality in (i). (ii) Write
zu (z) u(z)
g (0) Gk (z), (λI − Cg )ξ k (z) = zk λ − k+1
= g (z), integrate by parts and
k 0,
with Gk ∈ H (D), Gk (0) = 1. If m is a nonnegative integer as in the statement, then the matrix with entries (j ) (λI − Cg )ξ k (0),
0 k, j m − 1,
is lower triangular with nonzero diagonal entries k!(λ − the statement has a unique solution
g (0) k+1 ).
Thus the linear system given in
t β0 (h, λ), . . . , βm−1 (h, λ) ∈ Cm and clearly, the function h1 = h −
m−1
βk (h, λ)(λI − Cg )ξ k
k=0
has a zero of order m at the origin. Then part (i) applies to h1 , that is, λRλ h1 − Cg Rλ h1 = h1 , which shows that f is a solution of (2.2). (iii) follows by linearity. 2 We want to estimate the coefficients βk (h, λ) given in part (ii) above. It turns out that this can be done in a fairly general context. As usual, by Banach spaces of analytic functions in D we mean Banach spaces continuously contained in the locally convex space H (D). Note that if X1 , X2 are such Banach spaces with X1 ⊂ X2 , the inclusion map from X1 into X2 is automatically continuous by the closed graph theorem. Lemma 2.1. Let X be a Banach space of analytic functions in D which contains the polynomials and is contained in the growth class A−γ for some γ > 0 and assume that Cg is bounded on X. Given λ ∈ C \ {0}, let m be an arbitrary positive integer if λ = g n(0) , n ∈ Z+ , or m = n if λ = g n(0) for some n ∈ Z+ . Then for h ∈ X, the solutions βk (h, λ), 0 k m − 1 of the linear system (2.4)
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define continuous linear functionals on X and there exists a constant c0 > 0 depending only on X such that max{1, ξ j X } k−1 βk (·, λ) ck |λ| + Cg k (k + 1)! γ . (0) 0 |λ − gk+1 | j =0 Proof. The same argument as in the proof of Proposition 2.1 shows that (2.4) has a unique solution. The values βk (h, λ), k 0 can be computed in the following way. Set gk = (λI −Cg )ξ k , h−1 = h and define hk , k 0 inductively by (k)
hk = hk−1 −
hk−1 (0) gk(k) (0)
(k)
gk = hk−1 −
hk−1 (0) k!(λ −
g . g (0) k k+1 )
Clearly, for m as in the statement hm−1 = h −
m−1 k=0
h(k) k−1 (0) k!(λ −
g , g (0) k k+1 )
has a zero of order m at the origin, hence, (k)
βk (h, λ) =
hk−1 (0) k!(λ −
g (0) k+1 )
,
k 0.
For functions f ∈ A−γ we have the standard Cauchy estimate on the circle centered at the origin 1 , and of radius rk = 1 − k+1 (k) f (0) k!r −k (1 − rk )−γ f −γ ek!(k + 1)γ f −γ . k Using this inequality together with the fact that X is continuously contained in A−γ we obtain βk (h, λ) e(k + 1)γ hk−1 −γ c(k + 1)γ hk−1X , (0) (0) |λ − gk+1 | |λ − gk+1 |
k 0,
(2.7)
where c > 0 is a constant depending only on X. From the definition of hk we have that hk−1 X |λ| + Cg ξ k X g (0) |λ − k+1 | k c(k + 1)γ |λ| + Cg ξ X , = hk−1 X 1 + (0) |λ − gk+1 |
hk X hk−1 X + c(k + 1)γ
and by iterating this inequality we obtain for k 1 hk−1 X hX
k−1 j =0
j c(j + 1)γ |λ| + Cg ξ X . 1+ (0) |λ − gj +1 |
(2.8)
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Finally, note that if g (0) = 0, then inf λ
|λ| + Cg =1 |λ|
and if g (0) = 0,
(j + 1)Cg inf = min 1, min 1, c1 (X)−1 , j,λ |λ − g (0) | |g (0)| j +1 |λ| + Cg
where c1 (X) = sup f (0). f X =1
Then the result follows from (2.7) and (2.8).
2
These general considerations can be applied in order to study the spectrum and resolvent of Cg on such spaces. A preliminary result is given below. The proof follows directly from Proposition 2.1 and the closed graph theorem, so that it will be omitted. Given a space X as above and m ∈ Z+ , we denote by Xm the subspace of X consisting of functions with a zero of order at least m at the origin. Note that Xm is closed in X. Corollary 2.1. Let X be a Banach space of analytic functions in D which contains the polynomials, is contained in A−γ for some γ > 0 and assume that Cg is bounded on X. (i) If g (0) = 0 the point spectrum of Cg |X is void and if g (0) = 0 we have
n g (0) + (0) g : n∈Z , u ∈X . σp (Cg |X) = n
(ii) If Y λ = (λI − Cg )X is closed for some λ = 0, g n(0) , n ∈ Z+ , then for every positive integer
m > Re g λ(0) − 1, the operator Rλ defined in Proposition 2.1(i) is a bounded operator from Ymλ into Xm . If rλ,m denotes its norm, then the left inverse T (λ) : Y λ → X of λI − Cg with m−1 βk (·, λ)ξ k . T (λ) rλ,m + 1 + rλ,m λI − Cg k=0
Our next application concerns symbols g with a special type of singularity on the unit circle. Proposition 2.2. Assume that there exist constants a, b ∈ C \ {0} with |b| = 1 such that g (z) − a(1 − bz)−1 has a finite limit at b. Let X be a Banach space of analytic functions in D which is contained in A−γ for some γ > 0. Then for λ ∈ C \ {0}, m ∈ Z+ with Re
a > γ, λ
m > Re
g (0) − 1, λ
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we have that rb yb,λ (h) = lim
r→1
1
u(ζ )− λ g (ζ )h(ζ ) dζ,
0
defines a continuous linear functional on Xm , which is nonzero if X contains the polynomials. If Cg is bounded on X then (λI − Cg )Xm = (λI − Cg )X ∩ Xm ⊂ ker yb,λ . Proof. It is a simple matter to verify that the assumption on g implies that the function z → (1 − bz)a u(z) is bounded and bounded away from zero near b. From the fact that X is continuously contained in A−γ it follows that on the line segment from the origin to b the integrand 1 u− λ g h satisfies an inequality of the form −1 u λ g h(tb) c(1 − t)Re aλ −γ −1 hX , for some fixed constant c > 0 and all t ∈ [0, 1]. Then by the dominated convergence theorem we obtain that yb,λ is a continuous linear functional on X b yb,λ (h) =
1
u(ζ )− λ g (ζ )h(ζ ) dζ.
0 1
Also, by the above argument, we have that u(tb)− λ g (tb)h(tb) dt is a finite nonzero measure on [0, 1] which cannot annihilate all polynomials, hence yb,λ = 0 if X contains these functions. It remains to verify the equalities in the statement. The first one is trivial. By Proposition 2.1(i) we 1 have that if h ∈ (λI − Cg )X ∩ Xm then Rλ h + cu λ ξ −1 ∈ X for some complex constant c which −γ is nonzero only if λ = g n(0) for some n ∈ Z+ . Since X is continuously contained in A0 this implies that 1 u λ (rb) = 0. lim (1 − r)γ Rλ h(rb) + c r→1 rb −γ
We also have h ∈ A0
which yields
lim (1 − r)
γ
r→1
1
u λ (rb) λ2 rb
rb
1
u λ (rb) g (ζ )h(ζ ) dζ + c rb
− λ1
u(ζ )
= 0.
0
From the fact that z → (1 − bz)a u(z) has a finite nonzero limit at b, and Re aλ > γ , we obtain that c = 0 and rb lim
r→1 0
1
u(ζ )− λ g (ζ )h(ζ ) dζ = 0.
2
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3. Weighted conformal invariance and composition operators In this section we shall consider Banach spaces of analytic functions which satisfy some additional properties related to composition and weighted composition operators. In order to motivate our assumptions, let us recall two common properties of such operators on Hardy and weighted Bergman spaces. (1) These spaces are invariant under composition with the conformal automorphisms of D a−z , z ∈ D, where a ∈ D is fixed, but the norms of the composition operators given by ψa (z) = 1−az induced by these maps Cψa f = f ◦ ψa are not uniformly bounded in a ∈ D. However, for a γ suitable choice of γ > 0 the weighted composition operators Cψa defined by γ γ Cψa f = ψa f ◦ ψa
(3.9)
become uniformly bounded (actually they are isometries) on the spaces in question. Using a change of variable, one can easily verify from the definition that γ = p1 for H p , and γ = α+2 p p,α for La . (2) Given any analytic self-map φ of the unit disc the composition operator Cφ defined by Cφ f = f ◦ φ is bounded on these spaces and its norm is dominated (1 − |φ(0)|)−γ , where γ happens to be the same parameter as in (1) (see for example [25]). Motivated by these facts we consider Banach spaces of analytic functions in D which satisfy the weighted conformal invariance property described in (1), as well as a weaker form of (2). More precisely, our Banach spaces X are continuously contained in the locally convex space H (D) and have the following properties depending on a fixed number γ > 0. γ
(A1) The weighted composition operators Cψa given by (3.9) are uniformly bounded on X. (A2) For every function φ : D → D of the form φ(z) = ρz + λ, λ, ρ ∈ C, the composition operator Cφ is bounded on X and there exists a constant c > 0 such that Cφ
c . (1 − |φ(0)|)γ
Besides the Hardy and weighted Bergman spaces there are other classical Banach spaces of analytic functions in the unit disc which satisfy these two assumptions. −γ The simplest additional examples are the Banach spaces A−γ , A0 , γ > 0 which satisfy (A1) and (A2) with the same parameter γ . The verification of this fact is based on a straightforward application of the Schwarz–Pick lemma. For example, if φ : D → D is analytic and f ∈ A−γ , we have γ φ (z) f φ(z) f −γ
|φ (z)|γ 2γ f −γ , γ (1 − |φ(z)|) (1 − |z|2 )γ
and f φ(z) f −γ f −γ
2γ 2γ (1 − |φ(0)|2 )γ f −γ (1 − |φ(z)|2 )γ (1 − |z|2 )γ |1 − φ(0)φ(z)|2γ 4γ . (1 − |φ(0)|)γ (1 − |z|2 )γ
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It is also easy to show that A0 satisfies (A1) and (A2). In the case of weighted Dirichlet spaces, the situation is more complicated. The proposition below can be deduced from a recent result obtained in [5]. Proposition 3.1. Let p 1, α > −1 with α p − 1. Then the weighted Dirichlet space D p,α p,α satisfies (A1) for some γ > 0 if and only if α > p − 2. In this case, γ = α+2 p − 1 and D satisfies (A2) as well. Proof. Note that γ γ (1 − |a|2 )γ Cψa 1 = ψa = (1 − aξ )2γ and use the standard estimates for weighted Bergman space norms (see [13]) to conclude that γ Cψa 1 are uniformly bounded in D p,α , if and only if γ α+2 p − 1. If this inequality is strict then γ sup Cψa f (0): f ∈ D p,α , f 1 γ = 1 − |a|2 sup f (a): f ∈ D p,α , f 1 → ∞ when |a| → 1, and from the estimate α+2 f (a) = O 1 − |a| p −1 ,
f ∈ D p,α , |a| → 1,
(3.10)
γ
we see that the operators Cψa cannot be uniformly bounded for a ∈ D. Now assume that γ = α+2 p − 1 > 0 and note from above that γ sup Cψa f (0): f ∈ D p,α , f 1, a ∈ D < ∞. p,α
By the weighted conformal invariance of the space La , we have that
γ p C f (z) 1 − |z|2 α dA(z) ψa
D
=
f (z) + γ ψ ψ −2 ψa (z) f (z)p 1 − |z|2 α dA(z) a
a
D
=
f (z) + γ (1 − az)−1 f (z)p 1 − |z|2 α dA(z).
D
Using the rotational invariance of D p,α , it is easy to verify that the last integral on the right stays bounded when a ∈ D and f ∈ D p,α with f 1, if and only if sup f ∈D p,α f 1 D
α |f (z)|p 1 − |z|2 dA(z) < ∞. p |1 − z|
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Now a special case of Theorems 5.1 and 5.2 in [5] yields the inequality D
p |f (z)|p 2 α f (0)p + |f (z)| 1 − |z|2 α+p dA(z) 1 − |z| dA(z) c |1 − z|p |1 − z|p D
2 cf , p
p
which gives the desired result. The verification of (A2) with this value of the parameter γ follows p,α directly from the corresponding result for La together with the pointwise estimate (3.10). If φ : D → D has the form φ(z) = ρz + λ then |ρ| + |λ| 1 and p f ◦ φp = f (λ) + |ρ|p
f (ρz + λ)p 1 − |z|2 α dA(z)
D
c1 (1 − |λ|)
and the proof is complete.
α+2 p −1
f +
c2 |ρ|p (1 − |λ|)
α+2 p
f
c3 (1 − |λ|)
α+2 p −1
f ,
2
Let us list a number of direct consequences of the two assumptions above. γ
Remarks 3.1. (1) If (A1) holds then the weighted composition operators Cψa are also uniformly γ bounded below on X, since (Cψa )2 equals the identity operator on this space. (2) Each of the assumptions (A1) or (A2) implies that X is (continuously) contained in A−γ . If (A1) holds then γ γ γ sup 1 − |a|2 f (a) = supCψa f (0) c supCψa f
a∈D
a∈D
a∈D
for some positive constant c and all f ∈ X. The proof that (A2) implies that X is contained in A−γ is similar and can be found in Lemma 3.1 from [21]. Some deeper properties of such spaces of analytic functions are proved below. Given a function h ∈ H (D) with hX ⊂ X we shall denote throughout by Mh the operator of multiplication by h, that is Mh f = hf , f ∈ X. A direct application of the closed graph theorem shows that Mh is bounded on X. Also recall that ξ denoted the identity function on D. Proposition 3.2. Let X be a Banach space of analytic functions on D which satisfies (A1) and assume, in addition, that the operator Mξ is bounded on X with spectrum σ (Mξ |X) = D. Then: (i) For all η ∈ C with Re η > 0, and all a ∈ D we have (1 − aξ )−η X ⊂ X and there exists a fixed constant c > 0 such that Re η (1 − aξ )−η f (1 + |η|)c f X . X (1 − |a|)Re η
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(ii) There exists c > 0 such that n M c n, ξ
for all n ∈ Z+ . (iii) If h ∈ H (D) is such that h ∈ A−δ for some δ ∈ (0, 1) then hX ⊂ X and there exists cδ > 0 such that Mh
2 (k) h (0) + h
−δ
.
k=0
Proof. (i) The first part of the statement follows by basic functional analysis from the assumption that σ (Mξ |X) = D. In fact, M(1−aξ )−η =
1 2πi
(1 − aλ)−η (λ − Mξ )−1 dλ.
(3.11)
|λ|=2−|a|
For |λ| > 1 let b = 1/λ and use Remarks 3.1(1) to obtain (λ − ξ )−1 f c1 C γ (λ − ξ )−1 f ψb X X for some fixed constant c1 > 0. Note that Cψb (λ − ξ )−1 f = γ
b(1 − bξ ) γ C f, 1 − |b|2 ψb
and use (A1) together with the assumption that Mξ is bounded to obtain (λ − ξ )−1 f c2 |b|(1 + Mξ ) f X . 1 X 1 − |b|2
(3.12)
In particular, since b = λ−1 , (i) holds for η = 1. If Re η > 1 we use (3.11) and (3.12) to obtain (1 − aξ )−η f X
c2 1 − |a|
2π 1 − a 2 − |a| eit − Re η dt 0
cη , (1 − |a|)Re η
by a standard estimate for integrals (see [9]). Finally, if Re η < 1, we let be an arbitrary continuous linear functional on X and f ∈ X. We consider the analytic function η f (a) = (1 − aξ )−η f ,
a ∈ D.
Since η f (a) = −η (1 − aξ )−η−1 f ,
(3.13)
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we have from above η (a) f
|η|c1+Re η X∗ f X , (1 − |a|)1+Re η
a ∈ D.
Using again the well-known estimates for analytic functions (see again [9, Chapter 5]) we obtain (1 − aξ )−η f
Re η
|η|c1 X∗ f X , (1 − |a|)Re η
a ∈ D,
and (i) follows. To see (ii) note that for ∈ X ∗ and f ∈ X 1f (a) =
∞
a n Mξn f .
n=0
As in the proof of Lemma 2.1 we use the coefficient estimates for A−1 to obtain n M f c n1
f −1 ,
ξ
and the result follows from (i). Finally, to prove (iii), note that if h ∈ H (D) satisfies the condition in the statement then (n) h (0) δ−3 n! c1 (n + 1) and thus Mh =
∞ (n) h (0)
n!
n=0
where, by (ii), the sum converges in B(X).
Mξn ,
2
Proposition 3.3. Let X be a Banach space of analytic functions on D which satisfies (A1) and (A2). Given a linear fractional map φ with φ(D) ⊂ D, denote by λφ the center and γ by ρφ the radius of the disc φ(D). Let Cφ be the weighted composition operator defined by γ Cφ f = (φ )γ f ◦ φ. Then there exists a constant c > 0 such that γ C c φ
for all such linear fractional maps φ.
γ
ρφ
(1 − |λφ |)γ
,
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Proof. Write φ(z) = ρφ ψa (z) + λφ for some a ∈ D. Then γ γ φ f ◦ φ = ρφ ψa f ◦ (ρφ ψλ + λφ ), and by (A1) and (A2) we have γ C f c1 ρ γ f ◦ (ρφ ξ + λφ ) c2 φ φ X X for some positive constants c1 , c2 and all f ∈ X.
γ
ρφ
(1 − |λφ |)γ
f X ,
2
When working with the assumptions (A1), (A2) in the general context, we encounter some technical difficulties which in most concrete cases are easily dealt with. These arise from the fact that our assumptions only regard the norms of certain composition and weighted composition operators and do not give any information about the dependence of such operators on their symbols. This is a crucial matter for our further purposes and is addressed in the next lemma, at the cost of an additional assumption. Lemma 3.1. Let X be a Banach space of analytic functions in the unit disc that satisfies (A1), (A2) and such that Mξ is bounded on X with σ (Mξ |X) = D. Assume, in addition, that polynomials are dense in X. For some fixed r ∈ (1, ∞), let (ψn ), (φn ) be convergent sequences in H (rD) with limits ψ, φ ∈ H (rD), such that each φn is a linear fractional map with φn (D) ⊂ D. If φ(D) ⊂ D then Cφ is bounded on X and (Mψn Cφn ) converges to Mψ Cφ in the strong topology on B(X). Proof. Note that under our assumptions we have Mψn = ψn (Mξ ) =
1 2πi
ψn (ζ )(ζ − Mξ )−1 dξ
|ζ |= 1+r 2
which shows that Mψn → Mψ in the operator norm. It will then suffice to prove that (Cφn ) is bounded in B(X). Indeed, if this holds, we can use the fact that for every polynomial f we have Cφn f = f (Mφn )1, hence, by the first part of the proof Cφn f → Cφ f , for all polynomials f , and since polynomials are dense in X, the result follows. To verify the above assertion, note first that if φ is constant then by the previous argument we have that Mφn 1 − δ for some fixed δ > 0 and all sufficiently large n. For such n and f ∈ X we can write Cφn f =
∞ f (k) (0) k=0
k!
φnk .
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For every ε > 0 we have an estimate of the form |f (k) (0)| cε (1 + ε)k f X , k! for some fixed cε > 0 and all k 0, and if we choose ε > 0 such that (1 + ε)Mφn 1 − ε, we obtain Cφn f X
cε f X . ε
If φ is not constant then the centers λn of the discs φn (D) must satisfy |λn | 1 − δ for some fixed δ > 0 and all n. In addition, (φn ) converges uniformly on compact subsets of rD to φ and φ (z) = 0, z ∈ rD, which implies that ((φn )−γ ) converges uniformly on compact subsets of rD. Then γ γ Cφn = M(φn )−γ Cφn M(φn )−γ Cφn , and by the first part of the proof together with Proposition 3.3 we obtain again that (Cφn ) is bounded in B(X). 2 A direct consequence of the lemma is the fact that a space X with the properties in the statement is automatically invariant under composition with rotations. Indeed, by (A2), the operators Ct,μ , 0 < t < 1, |μ| = 1, defined by Ct,μ f (z) = f (tμz), are uniformly bounded on X, and by Lemma 3.1 Ct,μ has a limit in the strong operator topology on B(X). This implies that the operators of composition with rotations, Cμ f (z) = f (μz) are uniformly bounded on X. Note also that these operators are uniformly bounded below, since Cμ Cμ = I . 4. Basic estimates for Cesàro-like operators It turns out that the assumptions considered in the previous section provide a powerful tool for the study of certain generalized Cesàro operators. Our approach follows a known idea based on composition semigroups (see [26–28,21]), but the general context considered here requires a number of additional steps. The more technical part is deferred to the present section, more precisely, the following lemma which will play an essential role for our further developments. Throughout in what follows, for a fixed nonnegative integer N , we will denote by HN∞ the Banach algebra HN∞ = f ∈ H (D): f (N ) ∈ H ∞ , with the natural norm N (n) (N ) f (0) + f . f N,∞ = ∞ n=0
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Lemma 4.1. Let X be a Banach space of analytic functions in the unit disc that satisfies (A1), (A2) for some γ > 0, and such that polynomials are dense in X. Assume in addition that: (A3) Mξ is bounded and bounded below on X with σ (Mξ |X) = D. Given μ, ν ∈ C, and a positive integer m with m > Re μ − 1, define for f ∈ Xm Tμ,ν f (z) = zμ−1 (1 − z)−ν
z
ζ −μ (1 − ζ )ν−1 f (ζ ) dζ.
0
Then there exists a constant c 1 depending only on X such that: (i) If Re ν < γ and m > 2γ + Re(μ − ν) − 1, then Tμ,ν f ∈ Xm whenever f ∈ Xm and Tμ,ν f X
(1 + |Im(ν − μ)|)cm f X . γ − Re ν
(ii) If Re ν > γ and m > |Re μ| + ν + 4, then Tμ,ν f ∈ Xm whenever f ∈ Xm with 1
t −μ (1 − t)ν−1 f (t) dt = 0,
0
and m Tμ,ν f X 1 + Im(ν − μ) eπ(|Im μ|+|Im ν|)
c0m f X . Re ν − γ
Proof. (i) Since polynomials are in X and Mξ is bounded below on this space, we can easily conclude that the backward shift Bf =
f − f (0) ξ
is a bounded linear operator on X. Consider the one-parameter family of functions ϕt (z) =
(e−t
e−t z , − 1)z + 1
z ∈ D.
(4.14)
These functions form a semigroup under composition since ϕt (z) = ϕ −1 e−t ϕ(z) ,
z ∈ D,
where ϕ is the starlike function ϕ(z) = z(1 − z)−1 , z ∈ D. Using also this equality, we can verify that the functions ϕt satisfy
A. Aleman, A.-M. Persson / Journal of Functional Analysis 258 (2010) 67–98
ϕt (0) = z,
lim ϕt (z) = 0,
85
(4.15)
t→∞
d ϕ(ϕt (z)) ϕt (z) = − = −ϕt (z) 1 − ϕt (z) , dt ϕ (ϕt (z))
(4.16)
and ϕt (z) 2 d . ϕt (z) = et dz z
ϕt (z) =
(4.17)
Now in the expression defining Tμ,ν , we integrate along the path t → ϕt (z), t ∈ [0, ∞), use (4.16) and (4.17), and obtain Tμ,ν f (z) = z
μ−1
−ν
z
(1 − z)
ζ −μ+m (1 − ζ )ν−1 B m f (ζ ) dζ
0
= zμ−1 (1 − z)−ν
∞
−μ+m+1
ϕt
ν (z) 1 − ϕt (z) f ϕt (z) dt
0
∞ = zm
eνt
ϕt (z) z
−μ+ν+m+1
f ϕt (z) dt
0
∞ =z
m
e(ν−γ )t
ϕt (z) z
−μ+ν+m+1−2γ
γ ϕt (z) f ϕt (z)
0
∞ =z
m
e
(ν−γ )t
ϕt (z) z
−μ+ν+m+1−2γ
γ Cϕt f (z) dt.
0
Let η = −μ + ν + m + 1 − 2γ . By Lemma 3.1 we have that t → M(ϕt /ξ )η Cϕγt is a strongly continuous B(X)-valued function on [0, ∞). Moreover, if m > 2γ + Re(μ − ν) − 1, i.e. Re η > 0, then by Proposition 3.2(i) we have Re η M(ϕt /ξ )η 1 + |Im η| c1 ,
t 0.
By Proposition 3.3 and the straightforward computation λϕt =
1 − e−t 1 , 2 − e−t 2
ρϕt 1,
we obtain γ C c2 , ϕt
t 0.
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Thus ∞ Tμ,ν f =
e(ν−γ )t M(ϕt /ξ )η Cϕγt f dt, 0
and Re η Tμ,ν f X c2 1 + |Im η| c1 f X
∞
e−t Re(ν−γ ) dt
0
Re η = c2 1 + |Im η| c1 f X
1 . Re(ν − γ )
The result follows then easily from the assumption that m > 2γ + Re(μ − ν) − 1. (ii) Assume that Re ν > γ . For f ∈ X and a ∈ D \ {1}, let a fa = f −
−μ+m (1 − ζ )ν−1 f (ζ ) dζ 1 (ζ − a) a −μ+m (1 − ζ )ν−1 dζ 1 (ζ − a)
Γ (m + 1 − μ + ν) =f − Γ (m + 1 − μ)Γ (ν)
1
t m−μ (1 − t)ν−1 f t + (1 − t)a dt,
0
and note that a (ζ − a)m−μ (1 − ζ )ν−1 fa (ζ ) dζ = 0. 1
Now for a ∈ D \ {1} and f ∈ X let 2γ +1
Fa (z) = (1 − a)
(z − a)
m+μ+3
−ν
z
(1 − z)
(ζ − a)m−μ (1 − ζ )ν−1 fa (ζ ) dζ a
= (1 − a)2γ +1 (z − a)m+μ+3 (1 − z)−ν
z (ζ − a)m−μ (1 − ζ )ν−1 fa (ζ ) dζ.
(4.18)
1
We claim that if f is a polynomial then Fa belongs to X and the X-valued function a → Fa is analytic in D and extends continuously to D, with sup
m Fa X 1 + Im(ν − μ) eπ(|Im μ|+|Im ν|)
a∈∂D\{1}
where c0 1 depends only on X.
c0m f X , Re ν − γ
(4.19)
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If we assume the claim for the moment, we can proceed as follows. Since F0 = Mξm+4 Tμ,ν Mξm f0 , we can apply the maximum principle and (4.19) to obtain that m+4 M Tμ,ν Mξm f0 X = F0 X ξ
sup
Fa X
a∈∂D\{1}
m 1 + Im(ν − μ) eπ(|Im μ|+|Im ν|)
c0m f X , Re ν − γ
for every polynomial f . The same argument as in the proof of Proposition 2.2 (see also the estimate of fa X below) shows that 1 yμ,ν (f ) =
t −μ (1 − t)ν−1 f (t) dt
0
defines a continuous linear functional on Xm . Now if f ∈ ker yμ,ν , and (fn ) is a sequence of polynomials which converges to f in X, it follows directly by the assumptions and that ξ m (B m fn )0 → f in X. From the last inequality above we deduce that (Mξm+4 Tμ,ν Mξm (B m fn )0 ) is a Cauchy sequence in X which converges pointwise to Mξm+4 Tμ,ν f , and (ii) follows. To complete the proof it remains to verify the claim. This part of the argument is more involved. We begin by estimating fa X . From the fact that X is continuously contained in A−γ together with the elementary computation 2 Re(1 − a) − |1 − a|2 = 1 − |a|2 0, we obtain 1 t m−μ (1 − t)ν−1 f t + (1 − t)a dt 0
1 c1 f X 0
t m−Re μ (1 − t)Re ν−1 dt (1 − |1 + (1 − t)(a − 1)|2 )γ −2γ
1
c1 f X |1 − a|
t m−Re μ−γ (1 − t)Re ν−1−γ dt 0
Γ (m + 1 − Re μ − γ )Γ (Re ν − γ ) = c1 |1 − a|−2γ f X . Γ (m + 1 − 2γ + Re(ν − μ)) If ε, δ > 0, then by Stirling’s formula we have for all w with Re w > ε + δ Re w Γ (w) δ π |Im δ|+k2 Re w 1 + |Im w| , Γ (Re w − δ) k1 |w| e 2
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and for all w with Re w > δ Γ (w) − π |Im δ| , Γ (Re w − δ) k3 e 2 where k1 , k2 > 0 depend only on ε and δ and k3 > 0 only on δ. Thus if m > Re μ + γ , then Γ (m + 1 − μ + ν)Γ (m + 1 − Re μ − γ )Γ (Re ν − γ ) Γ (m + 1 + Re(ν − μ) − 2γ )Γ (m + 1 − μ)Γ (ν) Re(m+1−μ+ν) k4 |m + 1 − μ + ν|2γ 1 + Im(ν − μ) π Im(ν − μ) + |Im μ| + |Im ν| + k5 Re(m + 1 − μ + ν) , × exp 2 where k4 , k5 > 0 depend only on γ , that is, only on X. If we now use also the assumption that m > Re(ν − μ) + 1, it follows that there exists c2 1 depending only on X such that the righthand side of the above inequality does not exceed m c2m 1 + Im(ν − μ) eπ(|Im μ|+|Im ν|) . This gives an estimate of the form fa X |1 − a|−2γ c3m exp π |Im μ| + |Im ν| f X ,
(4.20)
where c3 1 depends only on X. It is also clear that the X-valued function a → (1 − a)2γ +1 fa is analytic in D and extends continuously to D. We now turn to the actual proof of the claim. In the first equality in (4.18) we integrate along the line segment from a to z and conclude that Fa is analytic in D for all a ∈ D \ {1}. If we integrate in the second equality in (4.18) along the line segment from 1 to z we obtain Fa (z) = −(1 − a)2γ +1 (z − a)m+μ+3 1 ×
m−μ t ν−1 1 + t (z − 1) − a fa 1 + t (z − 1) dζ.
(4.21)
0 (4)
Since Re ν > γ > 0 and m > Re μ + 4, it follows that if f is a polynomial, Fa (z) is bounded for
a−t (a, z) ∈ D \ {1} × D \ : t <1 1−t
.
But then Fa 4,∞ is bounded. Now Proposition 3.2(iii) implies that Fa ∈ X, but even more than that is true: If 0 < r < 1 and Cr Fa (z) = Fa (rz),
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then limr→1− Cr Fa = Fa in H3∞ , uniformly in a ∈ D. By Proposition 3.2(iii) it follows that limr→1− Cr Fa = Fa in X, uniformly in a ∈ D. It is also easy to verify that a → Cr Fa is analytic in D and continuous in D, hence, so is a → Fa . Then by a repeated application of Proposition 3.2(iii) and Lemma 3.1 we obtain (recall that Re μ < m) M
(z−a)μ+m+3
c4 m3 2m ,
|a| = 1,
and that t → M(1+t (ξ −1)−a)m−μ is a continuous B(X)-valued function on [0, 1] with M(1+t (ξ −1)−a)m−μ c5 m3 2m . Moreover, if φt (z) = 1 + t (z − 1), then t → Cφt is a strongly continuous B(X)-valued function on [0, 1), with Cφt c6 t −γ ,
t ∈ [0, 1),
where this last estimate is obtained directly from (A2). Thus
2γ +1
Fa = −(1 − a)
1 t ν−1 M(1+t (ξ −1)−a)m−μ Cφt fa dt.
M(z−a)m+μ+3 0
Then the claim follows from these estimates, (4.20) and (4.21), and the proof of the lemma is complete. 2 The additional assumption that polynomials are dense in our space X excludes from the list of examples the growth classes A−γ , γ > 0, this condition is fulfilled only by the “little oh” −γ version A0 . There is however a simple way to deal with the larger spaces as well. Given a Banach space X of analytic functions in D, we let X˜ ⊂ H (D) be the space of functions that are pointwise limits of bounded sequences in X. For f ∈ X˜ we set f X˜ = inf lim inf fn X : fn ∈ X, lim fn (z) = f (z), z ∈ D . n→∞
n→∞
˜ · ˜ ) is a Banach space of analytic functions in D, and that It is quite easy to show that (X, X the inclusion map from X into X˜ is contractive. Moreover, if the linear span of functionals of evaluation at points of D is dense in X ∗ then the inclusion map above is isometric, and X˜ is isometrically isomorphic to X ∗∗ . As a concrete example we have −γ A0 = A−γ
for all γ > 0. More generally, the conclusion holds true for any Banach space X of analytic functions in D which satisfies the assumptions (A1)–(A3) and contains the polynomials as a dense subspace.
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Indeed, this follows from the fact that the composition operators Ct , 0 < t < 1, with Ct f (z) = f (tz) satisfy lim Ct f = f,
t→1−
f ∈ X.
Then Ct∗ also converges strongly to the identity on X ∗ when t → 1− , and using Cauchy’s formula, we can verify that for every y ∈ X ∗ , Ct∗ y belongs to the closed linear span of the functionals of evaluation at points of D. This leads to the following result. Corollary 4.1. If X satisfies (A1)–(A3) then the conclusion of Lemma 4.1 continues to hold when ˜ X is replaced by X. Our first application concerns the boundedness of generalized Cesàro operators whose symbol has an essentially rational derivative. Theorem 4.1. Let X be a Banach space of analytic functions in the unit disc that satisfies (A1)– (A3). Assume in addition that: (A4) Polynomials are dense in X and log(1 − bξ ) ∈ X for all b ∈ ∂D. If g ∈ H (D) with g(0) = 0 has a derivative of the form g (z) =
n
ak
1 − bk z k=1
+ h(z),
where a1 , . . . , an ∈ C, b1 , . . . , bn ∈ ∂D, and h ∈ H (D) with hX ⊂ X, then the generalized Cesàro ˜ If a1 = · · · = an = 0 then Cg is compact on any operator Cg is bounded on both spaces X and X. of these spaces. Proof. For 0 t < 1 let φt (z) = tz, z ∈ D. It is a simple matter to show that the composition operators Cφt , 0 < t < 1 are compact on X. For example, this follows immediately from Proposition 3.2(iii). Moreover, by Lemma 3.1 we have that the B(X)-valued function t → Cφt is strongly continuous on [0, 1], and if h˜ is the primitive of h which vanishes at the origin then 1 Ch˜ f =
t Cφt Mh f dt = lim
Cφt Mh f dt,
t→1−
0
0
where the limit is taken in B(X). Clearly, this implies that Ch˜ is compact on X. Finally, by the ˜ any compact operator on X extends to a compact operator from X˜ into X. definition of X, To prove the theorem it remains to show that Cgb defines a bounded operator on X, where gb (z) = b log(1 − bz),
|b| = 1,
since the boundedness of this operator on X˜ is again immediate by definition. Moreover, it will be sufficient to prove that for some fixed nonnegative integer m, the operators Cgb Mξm are bounded
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91
on X. Indeed, using the density of polynomials in X together with the fact that Mξ is bounded below, we conclude that the backward shift Bf =
f − f (0) ξ
is a bounded linear operator on X. Then every f ∈ X can be written as f = Pm (f ) + Mξm B m f, where Pm (f ) is the Taylor polynomial of degree m of f . Clearly, Pm = I − Mξm B m is bounded on X, and by assumption (A3), it follows that Cgb Pm are bounded linear operators on X. Thus Cgb = Cgb Pm + Cgb Mξm B m is then bounded on X. Now this last assertion follows directly from Lemma 4.1. Indeed, as pointed out in the previous section, the operators of composition with rotations are bounded and invertible on X, so that we can restrict our attention to the case when b = 1. But then Cg1 = T0,0 which is bounded on Xm by the result mentioned above. 2 5. Spectra of generalized Cesàro operators For all concrete examples of Banach spaces of analytic functions in D considered in this paper, it turns out that the results concerning the Cesàro operator obtained in the previous section, can be extended with little effort to all generalized Cesàro operators of the form given in Theorem 4.1. This is due to a localization technique which is explained in the proposition below. We denote by F the set of Riemann maps from D onto the interior of C 5 -Jordan curves contained in D, which fix the origin. As it is well known (see [23, p. 49]), if φ ∈ F then φ ∈ H4∞ and φ has a continuous, zero-free extension to D. −γ
Proposition 5.1. If X is one of the spaces H p , La , D p,α , A−γ , A0 positive integer n: p,α
then it satisfies for every
(A5(n)) Given distinct points b1 , . . . , bn ∈ ∂D, there exists a finite set F ⊂ F such that: (i) If φ ∈ F then φ(D) contains exactly one of the points bj , 1 j n, and the composition operator Cφ is bounded on X. (ii) If f ∈ H (D) satisfies f ◦ φ ∈ X for all φ ∈ F then f ∈ X. Proof. Consider any set of open arcs Ij ⊂ ∂D, 1 j n such that Ij ∩ {b1 , . . . , bn } = {bj },
1 j n,
n
Ij = ∂D.
j =1
Construct C 5 -Jordan curves Υj with Ij ⊂ Υj , and Υj \ Ij ⊂ D, and such that the interior of Υj contains the set
1 Sj = z: |z| < ∪ z ∈ D \ {0}: z/|z| ∈ Ij . 2
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Let φj be a conformal map from D onto the interior of Υj , with φj (0) = 0. Then for all z ∈ D we have 1 − |z| 1 − |φj (z)|. If |z| → 1 in φj−1 (Sj ) then |φj (z)| → 1, which immediately implies that 1 − |z| cj 1 − φj (z) , for some constant cj > 0 and all z ∈ φj−1 (Sj ). Now (A5(n)) can be proved with the following p,α argument. For example, if f ∈ La
f ◦ φj p,α cj p
f ◦ φj (z)p φ (z)2 1 − φj (z) α dA(z) j
φj−1 (Sj )
= cj
f (z)p 1 − |z| α dA(z),
Sj
−γ and since j Sj = D, this implies (A5(n)). The cases when X = A−γ , or X = A0 are similar. If X = D p,α we have as above p p f ◦ φj D p,α f (0) + cj
f ◦ φj (z)p φ (z)2 1 − φj (z) α dA(z) j
φj−1 (Sj )
and the result follows with the same argument. Finally, the reasoning can be applied to the Hardy spaces as well, since for f ∈ H p p p f p ∼ f (0) +
f (z)p−2 f (z)2 1 − |z| dA(z).
2
D
Note that (A5(1)) is satisfied by any Banach space of analytic functions with F = {ξ }. Moreover, for any finite family F of conformal maps with the properties in (A5(n)), we have by the closed graph theorem that f → max f X , φ∈F
defines an equivalent norm on X. We can now turn to the main result of this paper, the description of the fine spectrum together with resolvent estimates for generalized Cesàro operators Cg whose symbol g has an essentially rational derivative. More precisely, we focus on symbols g ∈ H (D) with g(0) = 0 and g (z) =
n
ak
k=1
1 − bk z
+ h(z),
(5.22)
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where h ∈ H3∞ . Note that if all coefficients a1 , . . . , an are zero the corresponding generalized Cesàro operator Cg is compact by Theorem 4.1, so that its spectrum consists only of eigenvalues and is essentially described in Section 2. More precisely, in this case we have σ (Cg ) =
g (0) : n ∈ Z+ ∪ {0}. n
For this reason we shall assume that all coefficients a1 , . . . , an are nonzero and that the points b1 , . . . , bn ∈ ∂D are distinct. Theorem 5.1. Let X be a Banach space of analytic functions in the unit disc that satisfies (A1)– (A5(n)) with some constant γ > 0. Let g ∈ H (D) with g(0) = 0 such that g has the form (5.22) with a1 , . . . , an = 0 and b1 , . . . , bn ∈ ∂D distinct. (i) The point spectrum σp (Cg ) is void if g (0) = 0 and if g (0) = 0 then σp (Cg ) =
ka kaj g (0) − j : k ∈ Z+ , Re < γ , (1 − ξ ) g (0) ∈ X, 1 j n . k g (0)
(ii) σ (C) = σp (C) ∪ ( nj=1 {λ ∈ C: |λ − (iii) σe (C) = {λ ∈ C: |λ − given by
aj 2γ
|=
|aj | 2γ }
aj 2γ
|
|aj | 2γ }).
and for λ ∈ C \ σe (Cg ), the Fredholm index of λI − Cg is
ind(λI − Cg ) =
n
χj (λ),
j =1 a
|a |
where j = {ζ ∈ C: |ζ − 2γj | < 2γj }, and χj denotes its characteristic function. (iv) If T (λ) : (λI − Cg )X → X denotes the left inverse of λI − Cg , λ ∈ C \ (σe (Cg ) ∪ σp (Cg )) then λ → T (λ) is locally integrable on C \ {0}. Proof. We will use repeatedly the following simple observations. If F is the family of conformal maps given by (A5(n)) corresponding to the points b1 , . . . , bn , then based on the invariance under composition with rotations, we can assume without loss of generality that every φ ∈ F satisfies φ −1 {b1 , . . . , bn } = {1}. If φ ∈ F , φ(1) = bj then the function satisfies 1 − bj φ(z) = 1−z
1 0
1−bj φ 1−ξ
has a continuous, zero-free extension to D, and
φ 1 + t (z − 1) dt,
arg 1 − bj φ(z) < π. 1−z
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Then any complex power of this function belongs to H3∞ . Thus, 1 − bj φ τ , 1−ξ 3,∞
(1 − bk φ)τ
3,∞
,
1 k n,
(φ/ξ )τ , 3,∞
are locally bounded functions of τ ∈ C. Finally, the function u given by (2.3) is n
u(z) = zg (0) h1 (z)
(1 − bj z)−aj ,
(5.23)
j =1
z where h1 (z) = exp( 0 h(ζ )−h(0) dζ ). By the assumption that h ∈ H3∞ it follows that hτ1 ∈ H3∞ ζ for any complex τ , and that τ → hτ1 3,∞ is locally bounded on C. (i) From (A5(n)) and the above considerations it follows immediately that if g (0) = 0, and k
kaj
−
k ∈ Z+ then u g (0) ∈ X if and only if (1 − ξ ) g (0) ∈ X, 1 j n. Moreover, such an integer kaj < γ , because the assumptions in the theorem imply that X is contained k must satisfy Re g (0) −γ
in A0 . Indeed, since X is continuously contained in A−γ there exists c1 > 0 such that for every f ∈ X and every polynomial q γ γ lim sup 1 − |z| f (z) = lim sup 1 − |z| f (z) − q(z) c1 f − qX , |z|→1
|z|→1
and the claim follows by (A4). The result now follows from Corollary 2.1(i). In order to prove the remaining assertions we are going to relate the operator Rλ , λ = 0, defined in Proposition 2.1, to the Cesàro-like operators considered in the previous section. For m > Re λ1 − 1, f ∈ Xm and φ ∈ F with φ(1) = bj , we have f ◦ φ ∈ Xm and 1
(u ◦ φ) λ λ (Rλ f ) ◦ φ(z) = λf ◦ φ(z) + φ(z) 2
φ(z) 1 u− λ (ζ )g (ζ )f (ζ ) dζ 0
= λf ◦ φ(z) +
(u ◦ φ) φ(z)
1 λ
z
1
(u ◦ φ)− λ (ζ )(g ◦ φ) (ζ )f ◦ φ(ζ ) dζ
0 g (0) λ −1
= λf ◦ φ(z) + z = λf where μ =
(1 − z)
φ φ ◦ φ(z) + vλ (z)Tμ,νj wλ f
g (0) λ , νj
=
aj λ
z vλ (z)
ζ−
g (0) λ
aj
(1 − ζ ) λ −1 wλ (ζ )f ◦ φ(ζ ) dζ
0
◦ φ (z),
aj
(u ◦ φ) λ (1 − z) λ z
aj λ
, and
1
φ vλ (z) =
−
g (0) λ −1
φ(z)
1
,
φ wλ (z) =
aj
(u ◦ φ)− λ (1 − z)− λ z
g (0) λ
(1 − z)(g ◦ φ) (z).
A. Aleman, A.-M. Persson / Journal of Functional Analysis 258 (2010) 67–98
95
By the remarks made at the beginning of the proof, vλ , wλ ∈ H3∞ and vλ 3,∞ , wλ 3,∞ are locally bounded functions of λ ∈ C \ {0}. The last equality above can be then written as φ
φ
φ
λ2 (Rλ f ) ◦ φ = λCφ f + Mv φ Tμ,νj Mwφ Cφ f. λ
(ii)–(iv) If |λ −
aj 2γ
|>
|aj | 2γ
φ
(5.24)
λ
for 1 j n, then Re νj = Re
aj < γ, λ
1 j n,
and by Lemma 4.1(i), Tμ,ν defines a bounded operator on Xm for sufficiently large m. Then by (5.24) and (A5(n)) it follows that Rλ is bounded on Xm . a |a | Assume that |λ− 2γj | < 2γj for j ∈ J with ∅ = J ⊂ {1, . . . , n}, and recall from Proposition 2.2 that (λI − Cg )Xm ⊂
ker ybj ,λ .
j ∈J
If φ ∈ F with φ(1) = bj , j ∈ / J then by the previous argument, (Rλ f ) ◦ φ ∈ X, whenever f ∈ Xm with m sufficiently large. If φ ∈ F with φ(1) = bj , j ∈ J then by and a direct computation based on a change of variable we see that f ∈ ker ybj ,λ if and only if 1
t −μ (1 − t)ν−1 wλ (t)f ◦ φ(t) dt = 0. φ
0
Then Lemma 4.1(ii) shows that for sufficiently large m and f ∈ φ ∈ X for all such φ ∈ F . Thus if m is large enough, then
(λI − Cg )Xm ⊂
j ∈J
ker ybj ,λ we have (Rλ f ) ◦
ker ybj ,λ ,
j ∈J
and by (i) λI − Cg is injective in this case. For such m we conclude that if λ∈ /
n
∂j ,
j =1
then (λI − Cg )|Xm is Fredholm with ind(λI − Cg ) =
n
χj (λ).
j =1
Then by a well-known result (see for example [15, p. 285, Proposition 3.7.1]) the same holds true for λI − Cg , which proves (ii) and (iii).
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Finally, from the assumptions in Lemma 4.1 we see that if λ lies in a compact K ⊂ C \ {0} then we can choose a fixed integer m(K) such that the above considerations hold for all λ ∈ K and m = m(K). Then this lemma combined with (5.24) and (A5(n)) gives an estimate of the form c (K, X) c(K, X) c (K, X) n = , dist(λ, j =1 ∂j ) dist(λ, σe (Cg )) 1j n Re νj − γ
Rλ f max
(5.25)
for all f ∈ (λI − Cg )Xm(K) . We now apply Corollary 2.1(ii) with m = m(K),
rλ,m =
c (K, X) , dist(λ, σe (Cg ))
and Lemma 2.1 to conclude that if λ ∈ C \ σe (Cg ) and λ = T (λ) : (λI − Cg )X → X of λI − Cg satisfies T (λ)
c (K, X) dist(λ, σe (Cg ))
m(K)−1 j =0
g (0) k ,
k ∈ Z+ , then the left inverse
1 |λ −
g (0) j +1 |
.
Now general operator theory shows that T (λ) is continuous on C \ σe (Cg ), and (iv) follows. 2 ˜ The proof is identical to the one above. A similar result holds for the space X. Corollary 5.1. Let X be a Banach space of analytic functions in the unit disc that satisfies (A1)– (A5(n)) with some constant γ > 0. Then the conclusions in Theorem 5.1(ii)–(iv) continue to ˜ Moreover, the point spectrum σp (Cg ) is void if g (0) = 0 and if hold when X is replaced by X. g (0) = 0 then
kaj kaj g (0) − g (0) + ˜ ˜ : k ∈ Z , Re γ , (1 − ξ ) σp (Cg |X) = ∈ X, 1 j n . k g (0) We should also point out that for n = 1 the above results hold for every space X that satisfies (A1)–(A4), since the assumption (A5(1)) is automatically fulfilled. Finally, we can use the resolvent estimate (iv) to show that all generalized Cesàro operators considered in this section have decomposable extensions. This is a consequence of the following general result. Proposition 5.2. Let X be a Banach space S ∈ B(X). Assume that for almost every λ ∈ C there exists T (λ) ∈ B((λI − S)X, X) such that T (·) is locally integrable on C \ {0} and T (λ)(λI − S) = I a.e. Then S has a decomposable extension. Proof. According to the Albrecht–Eschmeier criterion (see [15, p. 138]) it will be sufficient to show that S has Bishop’s property (β), i.e. for every open set U ⊂ C, a sequence (fn ) of
A. Aleman, A.-M. Persson / Journal of Functional Analysis 258 (2010) 67–98
97
X-valued analytic functions in U with the property that z → (S − z)fn (z),
(5.26)
converges to zero uniformly on compact subsets of U , converges itself to zero uniformly on compact subsets of U . A compact subset K of the open set U ⊂ C can be covered by a finite unions of discs Dj , 1 j N , centered at zj ∈ K, of radius rj > 0 such that |z − zj | 3rj ⊂ U,
2rj |z − zj | 3rj ⊂ U \ {0}.
If f : U → X is analytic, then f is subharmonic in U , hence it satisfies for 1 j N sup f (z) z∈Dj
4 5πrj2 4 5πrj2
f (ζ ) dA(ζ )
2rj <|ζ −zj |<3rj
T (ζ )(S − ζ )f (ζ ) dA(ζ )
2rj <|ζ −zj |<3rj
(S − z)f (z) 4 sup 5πrj2 2rj |z−zj |3rj
T (ζ ) dA(ζ ).
2rj <|ζ −zj |<3rj
Thus if (fn ) has the property that the sequence given in (5.26) converges to zero uniformly on K, the above estimate shows that (fn ) converges uniformly to zero on K, and the result follows. 2 Corollary 5.2. If X is a Banach space of analytic functions in the unit disc that satisfies (A1)– (A5(n)), then every generalized Cesàro operator of the form (5.22) is subdecomposable on X ˜ and X. Acknowledgment We would like to thank the referee for useful suggestions and for carefully reading the manuscript. References [1] E. Albrecht, J. Eschmeier, Analytic functional models and local spectral theory, Proc. London Math. Soc. (3) 75 (2) (1997) 323–348. [2] E. Albrecht, T.L. Miller, M.M. Neumann, Spectral properties of generalized Cesàro operators on Hardy and weighted Bergman spaces, Arch. Math. (Basel) 85 (5) (2005) 446–459. [3] A. Aleman, A class of integral operators on spaces of analytic functions, in: Proceedings of the Winter School in Operator Theory and Complex Analysis, Univ. Málaga Secr. Publ., Málaga, 2007, pp. 3–30. [4] A. Aleman, J.A. Cima, An integral operator on H p and Hardy’s inequality, J. Anal. Math. 85 (2001) 157–176. [5] A. Aleman, O. Constantin, Spectra of integration operators on weighted Bergman spaces, J. Anal. Math., in press. [6] A. Aleman, A.G. Siskakis, An integral operator on H p , Complex Var. Theory Appl. 28 (2) (1995) 149–158. [7] A. Aleman, A.G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (2) (1997) 337–356.
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[8] A. Dahlner, Some resolvent estimates in harmonic analysis; Decomposable extension of the Cesàro operator on the weighted Bergman space and Bishop’s property, Doctoral Thesis in Mathematical Sciences, 2003:4, Centre for Mathematical Sciences, Lund University, 2003. [9] P.L. Duren, The Theory of H p Spaces, Academic Press, New York, 1970. [10] J. Eschmeier, M. Putinar, Spectral theory and sheaf theory, III, J. Reine Angew. Math. 354 (1918) 150–163. [11] P. Galanopoulos, The Cesàro operator on Dirichlet spaces, Acta Sci. Math. (Szeged) 67 (1–2) (2001) 411–420. [12] D. Girela, J.A. Pelaez, Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Anal. 241 (1) (2006) 334–358. [13] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Grad. Texts in Math., vol. 199, Springer-Verlag, New York, 2000. [14] T.L. Kriete, D. Trutt, The Cesàro operator in l 2 is subnormal, Amer. J. Math. 93 (1971) 215–225. [15] K.B. Laursen, M.M. Neumann, An Introduction to Local Spectral Theory, London Math. Soc. Monogr. Ser., vol. 20, Clarendon Press, Oxford University Press, New York, 2000, xii+591 pp. [16] S. Li, S. Stevi´c, Riemann–Stieltjes-type integral operators on the unit ball in Cn , Complex Var. Elliptic Equ. 52 (6) (2007) 495–517. [17] V.G. Miller, T.L. Miller, On the approximate point spectrum of the Bergman space Cesàro operator, Houston J. Math. 27 (2) (2001) 479–494. [18] V.G. Miller, T.L. Miller, The Cesàro operator on the Bergman space A2 (D), Arch. Math. (Basel) 78 (5) (2002) 409–416. [19] V.G. Miller, T.L. Miller, M.M. Neumann, Growth conditions, compact perturbations and operator subdecomposability, with applications to generalized Cesàro operators, J. Math. Anal. Appl. 301 (1) (2005) 32–51. [20] V.G. Miller, T.L. Miller, R.C. Smith, Bishop’s property (β) and the Cesàro operator, J. London Math. Soc. (2) 58 (1) (1998) 197–207. [21] A.-M. Persson, On the spectrum of the Cesàro operator on spaces of analytic functions, J. Math. Anal. Appl. 340 (2) (2008) 1180–1203. [22] Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation, Comment. Math. Helv. 52 (4) (1977) 591–602. [23] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren Math. Wiss., vol. 299, Springer-Verlag, Berlin, 1992. [24] J. Rättyä, Integration operator acting on Hardy and weighted Bergman spaces, Bull. Austral. Math. Soc. 75 (3) (2007) 431–446. [25] J.H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics, SpringerVerlag, New York, 1993, xvi+223 pp. [26] A.G. Siskakis, Composition semigroups and the Cesàro operator on H p , J. London Math. Soc. (2) 36 (1) (1987) 153–164. [27] A.G. Siskakis, The Cesàro operator is bounded on H 1 , Proc. Amer. Math. Soc. 110 (2) (1990) 461–462. [28] A.G. Siskakis, On the Bergman space norm of the Cesàro operator, Arch. Math. (Basel) 67 (4) (1996) 312–318. [29] A. Siskakis, Volterra operators on spaces of analytic functions – a survey, in: Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, Univ. Sevilla Secr. Publ., Seville, 2006, pp. 51–68. [30] J. Xiao, Riemann–Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball, J. London Math. Soc. (2) 70 (1) (2004) 199–214. [31] S.W. Young, Generalized Cesàro operators and the Bergman space, J. Operator Theory 52 (2) (2004) 341–351. [32] S.W. Young, Spectral properties of generalized Cesàro operators, Integral Equations Operator Theory 50 (1) (2004) 129–146.
Journal of Functional Analysis 258 (2010) 99–131 www.elsevier.com/locate/jfa
Multiple positive solutions for a class of concave–convex elliptic problems in RN involving sign-changing weight Tsung-fang Wu Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan Received 6 March 2009; accepted 14 August 2009 Available online 21 August 2009 Communicated by H. Brezis
Abstract In this paper, we study the multiplicity of positive solutions for the following concave–convex elliptic equation: ⎧ ⎨ −u + u = fλ (x)uq−1 + gμ (x)up−1 u0 ⎩ u ∈ H 1 RN ,
in RN , in RN ,
∗ where 1 < q < 2 < p < 2∗ (2∗ = N2N −2 if N 3, 2 = ∞ if N = 1, 2) and the parameters λ, μ 0. We assume that fλ (x) = λf+ (x) + f− (x) is sign-changing and gμ (x) = a(x) + μb(x), where the functions f± , a and b satisfy suitable conditions. © 2009 Elsevier Inc. All rights reserved.
Keywords: Semilinear elliptic equations; Sign-changing weight; Multiple positive solutions
1. Introduction In this paper, we consider the multiplicity results for positive solutions of the following concave–convex elliptic equation: E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.08.005
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
⎧ ⎨ −u + u = fλ (x)uq−1 + gμ (x)up−1 u0 ⎩ u ∈ H 1 RN ,
in RN , in RN ,
(Efλ ,gμ )
∗ where 1 < q < 2 < p < 2∗ (2∗ = N2N −2 if N 3, 2 = ∞ if N = 1, 2) and the parameters λ, μ 0. We assume that fλ (x) = λf+ (x) + f− (x) and gμ (x) = a(x) + μb(x) where the functions f± , a and b satisfy the following conditions: ∗
p ) with f± (x) = ± max{±f (x), 0} ≡ 0 and there exists a positive (D1) f ∈ Lq (RN ) (q ∗ = p−q number rf− such that
c exp −rf− |x| f− (x) −
for some c > 0 and for all x ∈ RN ;
(D2) a, b ∈ C(RN ) and there are positive numbers ra , rb with rb < min{rf− , ra , q} such that 1 a(x) 1 − c0 exp −ra |x|
for some c0 < 1 and for all x ∈ RN
and b(x) d0 exp −rb |x|
for some d0 > 0 and for all x ∈ RN ;
(D3) b(x) → 0 and a(x) → 1 as |x| → ∞. Elliptic problems in bounded domains involving concave and convex terms have been studied extensively since Ambrosetti, Brezis and Cerami [5] considered the following equation: ⎧ ⎨ −u = λuq−1 + up−1 in Ω, (Eλ ) u>0 in Ω, ⎩ u ∈ H01 (Ω), where 1 < q < 2 < p 2∗ , λ > 0 and Ω is a bounded domain in RN . They found that there exists λ0 > 0 such that Eq. (Eλ ) admits at least two positive solutions for λ ∈ (0, λ0 ), a positive solution for λ = λ0 and no positive solution exists for λ > λ0 (see also Ambrosetti, Azorero and Peral [4] for more references therein). Actually, Adimurthy, Pacella and Yadava [2], Damascelli, Grossi and Pacella [13], Ouyang and Shi [22] and Tang [25] proved that there exists λ0 > 0 such that there are exactly two positive solutions of Eq. (Eλ ) in the unit ball B N (0; 1) for λ ∈ (0, λ0 ), exactly one positive solution for λ = λ0 and no positive solution exists for λ > λ0 . Generalizations of the result of Eq. (Eλ ) (involving sign-changing weight) were done by Brown and Wu [9,10], de Figueiredo, Gossez and Ubilla [16] and Wu [29,30]. However, little has been done for this type of problem in RN . We are only aware of the works [12,18,21,28] which studied existence of solutions for some related concave–convex elliptic problem in RN (not involving sign-changing weight). Furthermore, we do not know of any results for concave–convex elliptic problems in RN involving sign-changing weight functions. In this paper, we will study this topic. The following theorems are our main results. Theorem 1.1. Suppose that the functions f± , a and b satisfy the conditions (D1)–(D3). Let Sp p−q Λ0 = (2 − q)2−q ( fp−2 )p−2 ( p−q ) , where Sp is a best Sobolev constant for the imbedding + ∗ Lq
of H 1 (RN ) into Lp (RN ). Then
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
101
(i) for each λ > 0 and μ > 0 with λp−2 (1 + μb∞ )2−q < ( q2 )p−2 Λ0 , Eq. (Efλ ,gμ ) has at least two positive solutions; p−2 (ii) there exist positive numbers λ0 , μ0 with λ0 (1 + μ0 b∞ )2−q < ( q2 )p−2 Λ0 such that for λ ∈ (0, λ0 ) and μ ∈ (0, μ0 ), Eq. (Efλ ,gμ ) has at least three positive solutions. Note that the positive numbers λ0 , μ0 are independent of f− . Therefore, if f− Lq ∗ is sufficiently small, we have the following result. Theorem 1.2. If in addition to the conditions (D1)–(D3), we still have (D4) a(x) 1 on RN with a strict inequality on a set of positive measure; (D5) ra > 2, μ0 μ0 and ν0 such that for λ ∈ (0, λ0 ), μ ∈ (0, μ0 ) then there exist positive numbers λ0 λ0 , ∗ and f− Lq < ν0 , Eq. (Efλ ,gμ ) has at least four positive solutions. Among other interesting similar problems, Adachi and Tanaka [1] investigated the following non-homogeneous elliptic equation: ⎧ ⎨ −u + u = a(x)up−1 + h(x) in RN , (E a,h ) u>0 in RN , ⎩ 1 N u∈H R , where h(x) ∈ H −1 (RN )\{0} is nonnegative and a(x) ∈ C(RN ) which satisfy a(x) 1 = lim a(x) |x|→∞
and a(x) 1 − c0 exp −(2 + δ)|x|
for some c0 < 1, δ > 0 and for all x ∈ RN .
Using Eq. (E a,0 ) does not admit any ground state solution and Bahri–Li’s minimax argument [6], they proved that Eq. (E a,h ) has at least four positive solutions under the assumption hH −1 is sufficiently small. In the following sections, we proceed to prove Theorems 1.1, 1.2. We use the variational methods to find positive solutions of Eq. (Efλ ,gμ ). Associated with Eq. (Efλ ,gμ ), we consider the energy functional Jfλ ,gμ in H 1 (RN ) 1 1 Jfλ ,gμ (u) = u2H 1 − 2 q
fλ |u|q dx − RN
1 p
gμ |u|p dx, RN
where uH 1 = ( RN |∇u|2 + u2 dx)1/2 is the standard norm in H 1 (RN ). It is well known that the solutions of Eq. (Efλ ,gμ ) are the critical points of the energy functional Jfλ ,gμ in H 1 (RN ) (see Rabinowitz [23]). This paper is organized as follows. In Section 2, we give some notations and preliminaries. In Section 3, we establish the existence of a local minimum for Jfλ ,gμ . In Section 4, we give an
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
estimate of energy. In Section 5, we discuss some concentration behavior in the Nehari manifold. In Sections 6, 7, we prove Theorems 1.1, 1.2. 2. Notations and preliminaries Throughout this paper, we denote by Sp the best Sobolev constant for the embedding of H 1 (RN ) into Lp (RN ) which is given by Sp =
u2H 1
> 0. u∈H 1 (RN )\{0} ( RN |u|p dx)2/p inf
In particular,
1 |u|p dx
for all u ∈ H 1 RN \{0}.
−1
p
Sp2 uH 1
RN
First, we define the Palais–Smale (simply (PS)) sequences, (PS)-values, and (PS)-conditions in H 1 (RN ) for Jfλ ,gμ as follows. Definition 2.1. (i) For β ∈ R, a sequence {un } is a (PS)β -sequence in H 1 (RN ) for Jfλ ,gμ if Jfλ ,gμ (un ) = β + o(1) and Jf λ ,gμ (un ) = o(1) strongly in H −1 (RN ) as n → ∞. (ii) β ∈ R is a (PS)-value in H 1 (RN ) for Jfλ ,gμ if there exists a (PS)β -sequence in H 1 (RN ) for Jfλ ,gμ . (iii) Jfλ ,gμ satisfies the (PS)β -condition in H 1 (RN ) if every (PS)β -sequence in H 1 (RN ) for Jfλ ,gμ contains a convergent subsequence. As the energy functional Jfλ ,gμ is not bounded below on H 1 (RN ), it is useful to consider the functional on the Nehari manifold
Nfλ ,gμ = u ∈ H 1 RN \{0} Jf λ ,gμ (u), u = 0 . Thus, u ∈ Nfλ ,gμ if and only if u2H 1
− RN
fλ |u| dx − q
gμ |u|p dx = 0.
RN
Note that Nfλ ,gμ contains every non-zero solution of Eq. (Efλ ,gμ ). Furthermore, we have the following results. Lemma 2.2. The energy functional Jfλ ,gμ is coercive and bounded below on Nfλ ,gμ . Proof. If u ∈ Nfλ ,gμ , then, by the Hölder and Sobolev inequalities, Jfλ ,gμ (u) =
1 1 1 1 2 − uH 1 − − (λf+ + f− )|u|q dx 2 p q p RN
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
103
1 1 1 1 − u2H 1 − − λf+ |u|q dx 2 p q p
RN
p−q p−2 −q q u2H 1 − λ f+ Lq ∗ Sp 2 uH 1 . 2p pq
(2.1)
2
Thus, Jfλ ,gμ is coercive and bounded below on Nfλ ,gμ .
The Nehari manifold Nfλ ,gμ is closely linked to the behavior of the function of the form hu : t → Jfλ ,gμ (tu) for t > 0. Such maps are known as fibering maps and were introduced by Drábek and Pohozaev in [14] and are also discussed in Brown and Zhang [11] and Brown and Wu [9,10]. If u ∈ H 1 (RN ), we have
t2 tq hu (t) = u2H 1 − 2 q
tp fλ |u| dx − p
gμ |u|p dx;
q
RN
h u (t) = tu2H 1 − t q−1
RN
fλ |u|q dx − t p−1
RN
RN
h
u (t) = u2H 1 − (q − 1)t q−2
gμ |u|p dx;
fλ |u|q dx − (p − 1)t p−2
RN
gμ |u|p dx.
RN
It is easy to see that th u (t) = tu2H 1
−
fλ |tu| dx −
RN
q
gμ |tu|p dx
RN
and so, for u ∈ H 1 (RN )\{0} and t > 0, h u (t) = 0 if and only if tu ∈ Nfλ ,gμ , i.e., positive critical points of hu correspond to points on the Nehari manifold. In particular, h u (1) = 0 if and only if u ∈ Nfλ ,gμ . Thus, it is natural to split Nfλ ,gμ into three parts corresponding to local minima, local maxima and points of inflection. Accordingly, we define
N+ fλ ,gμ = u ∈ Nfλ ,gμ
N0fλ ,gμ = u ∈ Nfλ ,gμ
N− fλ ,gμ = u ∈ Nfλ ,gμ
h (1) > 0 ; u
h (1) = 0 ; u
h (1) < 0 . u
− 0 We now derive some basic properties of N+ fλ ,gμ , Nfλ ,gμ and Nfλ ,gμ .
/ N0fλ ,gμ . Lemma 2.3. Suppose that u0 is a local minimizer for Jfλ ,gμ on Nfλ ,gμ and that u0 ∈ Then Jf λ ,gμ (u0 ) = 0 in H −1 (RN ).
Proof. The proof is essentially the same as that in Brown and Zhang [11, Theorem 2.3] (or see Binding, Drábek and Huang [8]). 2
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
For each u ∈ Nfλ ,gμ , we have h
u (1) = u2H 1
− (q − 1)
fλ |u| dx − (p − 1)
gμ |u|p dx
q
RN
RN
= (2 − p)u2H 1 − (q − p)
fλ |u|q dx
(2.2)
gμ |u|p dx.
(2.3)
RN
= (2 − q)u2H 1 − (p − q) RN
Then we have the following result. Lemma 2.4.
0 q (i) For any u ∈ N+ fλ ,gμ ∪ Nfλ ,gμ , we have RN fλ |u| dx > 0.
p (ii) For any u ∈ N− fλ ,gμ , we have RN gμ |u| dx > 0. 2
Proof. The result now follows immediately from (2.2) and (2.3). Let Λ0 = (2 − q)2−q
p−2 f+ Lq ∗
p−2
Sp p−q
p−q .
Then we have the following result. Lemma 2.5. For each λ > 0 and μ 0 with λp−2 (1 + μb∞ )2−q < Λ0 , we have N0fλ ,gμ = ∅. Proof. Suppose the contrary. Then there exist λ > 0 and μ 0 with 2−q < Λ0 λp−2 1 + μb∞ such that N0fλ ,gμ = ∅. Then, for u ∈ N0fλ ,gμ , by (2.2) and the Hölder and Sobolev inequalities we have −q p − q p−q q f+ Lq ∗ uH 1 fλ |u|q dx λSp2 u2H 1 = p−2 p−2 RN
and so u2H 1
q q−2
Sp
2 p − q 2−q . λf+ Lq ∗ p−2
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
105
Similarly, using (2.3) and the Sobolev inequality we have −p 2−q p u2H 1 = (a + μb)|u|p dx 1 + μb∞ Sp2 uH 1 , p−q RN
which implies u2H 1
p p−2
Sp
2−q (1 + μb∞ )(p − q)
2 p−2
for all μ 0.
Hence, we must have 2−q λp−2 1 + μb∞ (2 − q)2−q
p−2 f+ Lq ∗
which is a contradiction. This completes the proof.
p−2
Sp p−q
p−q = Λ0
2
In order to get a better understanding of the Nehari manifold and fibering maps, we consider the function mu : R+ → R defined by mu (t) = t 2−q u2H 1 − t p−q gμ |u|p dx for t > 0. (2.4) RN
Clearly tu ∈ Nfλ ,gμ if and only if mu (t) =
RN
m u (t) = (2 − q)t 1−q u2H 1
fλ |u|q dx. Moreover, − (p − q)t
gμ |u|p dx
p−q−1
(2.5)
RN
and so it is easy to see that, if tu ∈ Nfλ ,gμ , then t q−1 m u (t) = h
u (t). Hence, tu ∈ N+ fλ ,gμ (or
N− fλ ,gμ ) if and only if mu (t) > 0 (or < 0).
Suppose u ∈ H 1 (RN )\{0}. Then, by (2.5), mu has a unique critical point at t = tmax,μ (u) where tmax,μ (u) =
(2 − q)u2H 1
(p − q) RN gμ |u|p dx
1 p−2
>0
(2.6)
and clearly mu is strictly increasing on (0, tmax,μ (u)) and strictly decreasing on (tmax,μ (u), ∞) with limt→∞ mu (t) = −∞. Moreover, if λp−2 (1 + μb∞ )2−q < Λ0 , then mu tmax,μ (u) =
2−q p−q
2−q
p−2
2−q − p−q
p−q p−2
2(p−q)
uHp−2 1
2−q
( RN gμ |u|p dx) p−2 2−q 2−q p p−2 uH 1 p−2 2 − q p−2 q
= uH 1 p p−q p−q RN gμ |u| dx
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
p−2 λf+ Lq ∗
Sp p−q
p−q p−2
2−q 1 + μb∞
2−q p−2
fλ |u|q dx
RN
fλ |u|q dx.
> RN
Thus, we have the following lemma. Lemma 2.6. For each u ∈ H 1 (RN )\{0} we have the following.
(i) If RN fλ |u|q dx 0, then there is a unique t − = t − (u) > tmax,μ (u) such that t − u ∈ N− fλ ,gμ and hu is increasing on (0, t − ) and decreasing on (t − , ∞). Moreover, Jfλ ,gμ (t − u) = sup Jfλ ,gμ (tu).
(2.7)
t0
(ii) If
RN
fλ |u|q dx > 0, then there are unique 0 < t + = t + (u) < tmax,μ (u) < t − = t − (u)
− − + + − such that t + u ∈ N+ fλ ,gμ , t u ∈ Nfλ ,gμ , hu is decreasing on (0, t ), increasing on (t , t ) and decreasing on (t − , ∞). Moreover,
Jfλ ,gμ (t + u) =
inf
0ttmax,μ (u)
Jfλ ,gμ (tu);
Jfλ ,gμ (t − u) = sup Jfλ ,gμ (tu). tt +
(2.8)
(iii) t − (u) is a continuous function for u ∈ H 1 (RN )\{0}. 1 u 1 N − (iv) N− fλ ,gμ = {u ∈ H (R ) | u 1 t ( u 1 ) = 1}. H
H
1 N Proof. Fix u ∈ H
(R )\{0}.
(i) Suppose RN fλ |u|q dx 0. Then mu (t) = RN fλ |u|q dx has a unique solution t − > tmax,μ (u) such that m u (t − ) < 0 and h u (t − ) = 0. Hence, by t q−1 m u (t) = h
u (t), hu has a unique critical point at t = t − and h
u (t − ) < 0. Thus, t − u ∈ N− fλ ,gμ and (2.7) holds.
q dx > 0. Since m (t (ii) Suppose f |u| (u)) > RN fλ |u|q dx, the equation mu (t) = u max,μ RN λ
q dx has exactly two solutions t + < t f |u| (u) < t − such that m u (t + ) > 0 and max,μ RN λ
− mu (t ) < 0. Hence, there are exactly two multiples of u lying in Nfλ ,gμ , that is, t + u ∈ N+ fλ ,gμ
q−1 m (t) = h
(t), h has critical points at t = t + and t = t − with and t − u ∈ N− u u u fλ ,gμ . Thus, by t
+
− hu (t ) > 0 and hu (t ) < 0. Thus, hu is decreasing on (0, t + ), increasing on (t + , t − ) and decreasing on (t + , ∞). Therefore, (2.8) must hold. (iii) By the uniqueness of t − (u) and the extremal property of t − (u), we have t − (u) is a continuous function for u ∈ H 1 (RN )\{0}. u − (iv) For u ∈ N− fλ ,gμ . Let v = u 1 . By parts (i), (ii), there is a unique t (v) > 0 such that H
− − u 1 u 1 − − t − (v)v ∈ N− fλ ,gμ or t ( uH 1 ) uH 1 u ∈ Nfλ ,gμ . Since u ∈ Nfλ ,gμ , we have t ( uH 1 ) uH 1 = 1, and this implies
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
N− ⊂ u ∈ H 1 RN fλ ,gμ Conversely, let u ∈ H 1 (RN ) such that t
−
1 u =1 . t− uH 1 uH 1
1 u − uH 1 t ( uH 1 ) = 1.
u uH 1
107
Then
u ∈ N− fλ ,gμ . uH 1
Thus, N− = u ∈ H 1 RN fλ ,gμ This completes the proof.
1 u =1 . t− uH 1 uH 1
2
− Remark 2.1. (i) If λ = 0, then, by Lemma 2.6(i) N+ f0 ,gμ = ∅, and so Nf0 ,gμ = Nf0 ,gμ for all μ 0. (ii) If λp−2 (1 + μb∞ )2−q < Λ0 , then, by (2.2), for each u ∈ N+ fλ ,gμ we have
u2H 1 <
p−q p−2
1/(p−2)
fλ |u|q dx Λ0
−q
Sp2
RN
p−q q f+ Lq ∗ uH 1 , p−2
and so 1/(2−q) −q 1/(p−2) 2 p − q f+ Lq ∗ Sp uH 1 Λ0 p−2
for all u ∈ N+ fλ ,gμ .
(2.9)
3. Existence of a first solution First, we remark that it follows from Lemma 2.5 that − Nfλ ,gμ = N+ fλ ,gμ ∪ Nfλ ,gμ
for all λ > 0 and μ 0 with λp−2 (1 + μb∞ )2−q < Λ0 . Furthermore, by Lemma 2.6 it follows − that N+ fλ ,gμ and Nfλ ,gμ are non-empty and, by Lemma 2.2, we may define αf+λ ,gμ =
inf
u∈N+ fλ ,gμ
Jfλ ,gμ (u)
and αf−λ ,gμ =
inf
u∈N− f
Jfλ ,gμ (u).
λ ,gμ
Then we have the following result. Theorem 3.1. We have the following: (i) αf+λ ,gμ < 0 for all λ > 0 and μ 0 with λp−2 (1 + μb∞ )2−q < Λ0 .
(ii) If λp−2 (1 + μb∞ )2−q < ( q2 )p−2 Λ0 , then αf−λ ,gμ > c0 for some c0 > 0.
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
In particular, for each λ > 0 and μ 0 with λp−2 (1 + μb∞ )2−q < ( q2 )p−2 Λ0 , we have αf+λ ,gμ = infu∈Nfλ ,gμ Jfλ ,gμ (u). Proof. (i) Let u ∈ N+ fλ ,gμ . Then, by (2.2), u2H 1
p−q < p−2
fλ |u|q dx. RN
Hence, by (2.1) and Lemma 2.4, Jfλ ,gμ (u) =
p−2 p−q u2H 1 − 2p pq
<−
(p − q)(2 − q) 2pq
fλ |u|q dx RN
fλ |u|q dx < 0 RN
and so αf+λ ,gμ < 0.
(ii) Let u ∈ N− fλ ,gμ . Then, by (2.3) and the Sobolev inequality, 2−q u2H 1 < p−q
−p p gμ |u|p dx 1 + μb∞ Sp2 uH 1 ,
RN
which implies p 2(p−2)
uH 1 > Sp
2−q (1 + μb∞ )(p − q)
1/(p−2)
for all u ∈ N− fλ ,gμ .
(3.1)
By (2.1) and (3.1), we have
p−2 p−q −q 2−q uH 1 − λ f+ Lq ∗ Sp 2 2p pq q pq p−2 2−q > Sp2(p−2) (1 + μb∞ )(p − q) 2−q p−2 p − 2 p(2−q) p−q 2−q −q 2(p−2) Sp × −λ f+ Lq ∗ Sp 2 . 2p (1 + μb∞ )(p − q) pq q
Jfλ ,gμ (u) uH 1
Thus, if λp−2 (1 + μb∞ )2−q < ( q2 )p−2 Λ0 , then αf−λ ,gμ > c0 This completes the proof.
2
for some c0 > 0.
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
109
Now, we consider the following semilinear elliptic problem:
in RN ,
−u + u = |u|p−2 u u ∈ H 1 RN .
(E ∞ )
Associated with Eq. (E ∞ ), we consider the energy functional J ∞ in H 1 (RN ) 1 1 J (u) = u2H 1 − 2 p ∞
|u|p dx. RN
Consider the minimizing problem: inf J ∞ (u) = α ∞
u∈N∞
where
N∞ = u ∈ H 1 RN \{0} J ∞ (u), u = 0 . It is known that Eq. (E ∞ ) has a unique positive radially solution w0 (x) such that J ∞ (w0 ) = α ∞ (see [7,19]). Then the following proposition provides a precise description for the (PS)-sequence of Jfλ ,gμ . Proposition 3.2. (i) If {un } is a (PS)β -sequence in H 1 (RN ) for Jfλ ,gμ with β < αf+λ ,gμ + α ∞ , then there exist a subsequence {un } and a non-zero u0 in H 1 (RN ) such that un → u0 strongly in H 1 (RN ) and Jfλ ,gμ (u0 ) = β. Moreover, u0 is a solution of Eq. (Efλ ,gμ ). 1 N (ii) If {un } ⊂ N− fλ ,gμ is a (PS)β -sequence in H (R ) for Jfλ ,gμ with αf+λ ,gμ + α ∞ < β < αf−λ ,gμ + α ∞ , then there exist a subsequence {un } and a non-zero u0 in H 1 (RN ) such that un → u0 strongly in H 1 (RN ) and Jfλ ,gμ (u0 ) = β. Moreover, u0 is a solution of Eq. (Efλ ,gμ ). Proof. Similarly to the argument in Wu [28, Proposition 4.6] (or see Adachi and Tanaka [1, Proposition 1.9]). 2 Theorem 3.3. For each λ > 0 and μ 0 with λp−2 (1 + μb∞ )2−q < Λ0 , the functional Jfλ ,gμ + has a minimizer u+ λ,μ in Nfλ ,gμ and it satisfies + (i) Jfλ ,gμ (u+ λ,μ ) = αfλ ,gμ ,
(ii) u+ λ,μ is a positive solution of Eq. (Efλ ,gμ ), (iii) u+ λ,μ H 1 → 0 as λ → 0.
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
Proof. By the Ekeland variational principle [15] (or see Wu [29, Proposition 1]), there ex-sequence for Jfλ ,gμ . Then, by Proposition 3.2, ists {un } ⊂ N+ fλ ,gμ such that it is a (PS)α + fλ ,gμ
+ there exist a subsequence {un } and u+ λ,μ ∈ Nfλ ,gμ a non-zero solution of Eq. (Efλ ,gμ ) such that + + + + 1 N un → u+ λ,μ strongly in H (R ) and Jfλ ,gμ (uλ,μ ) = αfλ ,gμ . Since Jfλ ,gμ (uλ,μ ) = Jfλ ,gμ (|uλ,μ |)
+ + and |u+ λ,μ | ∈ Nfλ ,gμ , by Lemma 2.3 we may assume that uλ,μ is a positive solution of Eq. (Efλ ,gμ ). Finally, by (2.2) and the Hölder and Sobolev inequalities, −q + 2−q u 1 < λ p − q f+ q ∗ Sp2 L λ,μ H p−2
and so u+ λ,μ H 1 → 0 as λ → 0.
2
4. The estimate of energy First, we let w0 (x) be a unique radially symmetric positive solution of Eq. (E ∞ ) such that ∞ 0 ) = α . Then, by the result in Gidas, Ni and Nirenberg [17], for any ε > 0, there exist positive numbers Aε , B0 and Cε such that Aε exp −(1 + ε)|x| w0 (x) B0 exp −|x| (4.1) J ∞ (w
and ∇w0 (x) Cε exp −(1 − ε)|x| .
(4.2)
Let wl (x) = w0 (x + le),
for l ∈ R and e ∈ SN −1 ,
where SN −1 = {x ∈ RN | |x| = 1}. Then we have the following results. Proposition 4.1. For each λ > 0 and μ > 0 with λp−2 (1 + μb∞ )2−q < Λ0 , we have αf−λ ,gμ < αf+λ ,gμ + α ∞ . Proof. Let u+ λ,μ be a positive solution of Eq. (Efλ ,gμ ) as in Theorem 3.3. Then Jfλ ,gμ u+ λ,μ + twl 2 1 1 = u+ λ,μ + twl H 1 − 2 q
q 1 fλ u+ λ,μ + twl dx − p
RN
1 ∞ Jfλ ,gμ u+ λ,μ + J (twl ) + p
RN
− RN
p
t p wl dx −
1 p
p gμ u+ λ,μ + twl dx
RN p
gμ t p wl dx RN
twl + q−1 + q−1 uλ,μ + η dη dx (λf+ + f− ) − uλ,μ 0
(4.3)
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
1 − p
111
+ p p p−1 p − t p wl − p u+ twl dx uλ,μ + twl − u+ λ,μ λ,μ
RN
1 p
αf+λ ,gμ + J ∞ (twl ) +
p
(1 − g0 )t p wl dx RN
μ − p
bt
p
p wl dx
+
RN
−
1 p
twl
|f− |
RN
= αf+λ ,gμ + J ∞ (twl ) + −
1 p
dη dx
0
+ p p p−1 p uλ,μ + twl − u+ − t p wl − p u+ twl dx λ,μ λ,μ
RN
η
q−1
tp p
p
(1 − g0 )wl dx −
μt p p
RN
p
bwl dx +
tq q
RN
q
|f− |wl dx RN
+ p p p−1 p uλ,μ + twl − u+ − t p wl − p u+ twl dx. λ,μ λ,μ
(4.4)
RN
By Brown and Zhang [11] and Willem [27], we know that J ∞ (twl ) α ∞
for all l ∈ R.
(4.5)
Thus, by (4.4) and (4.5), we have Jfλ ,gμ u+ λ,μ + twl αf+λ ,gμ −
1 p
+α
∞
tp + p
p (1 − g0 )wl dx
μt p − p
RN
p bwl dx
tq + q
RN
q
|f− |wl dx RN
+ p p p−1 p uλ,μ + twy − u+ − t p wl − p u+ twl dx. λ,μ λ,μ
(4.6)
RN
Since + + Jfλ ,gμ u+ λ,μ + twl → Jfλ ,gμ uλ,μ = αfλ ,gμ < 0 as t → 0 and Jfλ ,gμ u+ λ,μ + twl → −∞ as t → ∞, we can easily find 0 < t1 < t2 such that + ∞ Jfλ ,gμ u+ λ,μ + twl < αfλ ,gμ + α
for all t ∈ [0, t1 ] ∪ [t2 , ∞).
(4.7)
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
Thus, we only need to show that there exists l0 > 0 such that for l > l0 , + ∞ sup Jfλ ,gμ u+ λ,μ + twl < αfλ ,gμ + α .
(4.8)
t1 tt2
We also remark that (u + v)p − up − v p − pup−1 v 0 for all (u, v) ∈ [0, ∞) × [0, ∞). Thus,
+ p p p−1 p uλ,μ + twl − u+ − t p wl − p u+ twl dx 0. λ,μ λ,μ
(4.9)
RN
From the condition (D2) and (4.1) (1 − g0 )t
p
p wl dx
p c0 B0
RN
exp −ra |x| exp −p|x + le| dx
RN p c0 B0
exp − min{ra , p} |x| + |x + le| dx
|x|
exp − min{ra , p} |x| + |x + le| dx
+ c0 B0
|x|l
p c0 B0 l N
exp − min{ra , p}l |x| + |x + e| dx
|x|<1
p + c0 B0 exp − min{ra , p}l
exp − min{ra , p} |x + le| dx
|x|l
p
p exp − min{ra , p}l dx + C0 B0 exp − min{ra , p}l
c0 B0 l N |x|<1 p C0 B0 l N
exp − min{ra , p}l for l 1
(4.10)
and
p
p
bwl dx = RN
b(x − le)w0 (x) dx
RN
=
p min w0 (x) d0
x∈B N (1)
p min w0 (x)
x∈B N (1)
b(x − le) dx
B N (1)
exp −rb |x| − rb l|e| dx
B N (1)
p min w0 (x) D0 exp(−rb l).
x∈B N (1)
(4.11)
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
113
From the condition (D1) and the same argument of inequality (4.10), we also have
q |f− |wl
dx
RN
q cB0
exp −rf− |x| exp −q|x + le| dx
RN
q cB0 l N exp − min{rf− , q}l for l 1.
(4.12)
Since rb < min{rf− , ra , q} min{rf− , ra , p} and t1 t t2 , by (4.6)–(4.12), we can find l1 max{l0 , 1} such that + ∞ sup Jfλ ,gμ u+ λ,μ + twl < αfλ ,gμ + α
for all l > l1 .
t0
To complete the proof of Proposition 4.1, it remains to show that there exists a positive number t∗ − such that u+ λ,μ + t∗ wl ∈ Nfλ ,gμ . Let 1 u > 1 ∪ {0}; t− uH 1 uH 1 N 1 u 1 − <1 . U2 = u ∈ H R t uH 1 uH 1
U1 = u ∈ H 1 R N
1 N 1 N Then N− fλ ,gμ separates H (R ) into two connected components U1 and U2 , and H (R )\
+ N− fλ ,gμ = U1 ∪ U2 . For each u ∈ Nfλ ,gμ , we have
1 < tmax,μ (u) < t − (u). Since t − (u) =
1 u − uH 1 t ( uH 1 ),
+ then N+ fλ ,gμ ⊂ U1 . In particular, uλ,μ ∈ U1 . We claim that
there exists t0 > 0 such that u+ λ,μ + t0 wl ∈ U2 . First, we find a constant c > 0 such that u+ +twl
0 < t − ( u+λ,μ+tw λ,μ
l H1
) < c for each t 0. Suppose the contrary. Then there exists a sequence {tn } u+ +tn wl
such that tn → ∞ and t − ( u+λ,μ+t
t − (v
− n )vn ∈ Nfλ ,gμ ,
λ,μ
u+ λ,μ +tn wl + uλ,μ +tn wl H 1
by the Lebesgue dominated convergence theorem,
RN
p gμ vn dx
1 = p + uλ,μ + tn wl H 1 =
→ we have
n wl H 1
) → ∞ as n → ∞. Let vn =
1
u+ λ,μ tn
RN
p
p gμ u+ dx λ,μ + tn wl
RN
gμ
+ wl H 1 R N
u+ λ,μ tn
p
gμ wl dx p
wl H 1
as n → ∞,
p + wl
dx
. Since
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
2 [t − (vn )]q 1 − Jfλ ,gμ t − (vn )vn = t (vn ) − 2 q
q
fλ vn dx −
[t − (vn )]p p
RN
→ −∞
p
gμ vn dx RN
as n → ∞,
this contradicts the fact that Jfλ ,gμ is bounded below on Nfλ ,gμ . Let t0 =
2 p − 2 2 c − u+ λ,μ H 1 ∞ 2pα
1 2
+ 1.
Then + u + t0 wl 2 1 = u+ 2 1 + t 2 wl 2 1 + o(1) 0 λ,μ λ,μ H H H 2 + 2 + 2 > uλ,μ H 1 + c − uλ,μ H 1 + o(1) 2 u+ λ,μ + t0 wl 2 − > c + o(1) > t + o(1) u+ λ,μ + t0 wl H 1
as l → ∞.
Thus, there exists l2 l1 such that for l > l2 , u+ 1 λ,μ + t0 wl − <1 t u+ u+ λ,μ + t0 wl H 1 λ,μ + t0 wl H 1 + or u+ λ,μ + t0 wl ∈ U2 . Define a path γl (s) = uλ,μ + st0 wl for s ∈ [0, 1]. Then
γl (0) = u+ λ,μ ∈ U1 , Since
1 u − uH 1 t ( uH 1 )
γl (1) = u+ λ,μ + t0 wl ∈ U2 .
is a continuous function for non-zero u and γl ([0, 1]) is connected, there
− exists sl ∈ (0, 1) such that u+ λ,μ + sl t0 wl ∈ Nfλ ,gμ . This completes the proof.
2
Then we have the following result. Theorem 4.2. For each λ > 0 and μ > 0 with λp−2 (1 + μb∞ )2−q < ( q2 )p−2 Λ0 , Eq. (Efλ ,gμ ) − − − has positive solution u− λ,μ ∈ Nfλ ,gμ such that Jfλ ,gμ (uλ,μ ) = αfλ ,gμ . Proof. Analogous to the proof of Wu [30, Proposition 9], one can show that by the Ekeland variational principle (see [15]), there exist minimizing sequences {un } ⊂ N− fλ ,gμ such that Jfλ ,gμ (un ) = αf−λ ,gμ + o(1) and Jf λ ,gμ (un ) = o(1)
in H −1 RN .
Since αf−λ ,gμ < αf+λ ,gμ + α ∞ , by Theorem 3.1(ii) and Proposition 3.2 there exist a subse-
− quence {un } and u− λ,μ ∈ Nfλ ,gμ a non-zero solution of Eq. (Efλ ,gμ ) such that
un → u− λ,μ
strongly in H 1 RN .
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
115
− − − Since Jfλ ,gμ (u− λ,μ ) = Jfλ ,gμ (|uλ,μ |) and |uλ,μ | ∈ Nfλ ,gμ , by Lemma 2.3, we may assume that
u− λ,μ is a positive solution of Eq. (Efλ ,gμ ).
2
5. Concentration behavior We need the following lemmas. Lemma 5.1. We have inf
u∈Nf0 ,g0
Jf0 ,g0 (u) = inf∞ J ∞ (u) = α ∞ . u∈N
Furthermore, Eq. (Ef0 ,g0 ) does not admit any solution u0 such that Jf0 ,g0 (u0 ) = infu∈Nf0 ,g0 Jf0 ,g0 (u). 2−q 1/(p−2) Proof. Let wl be as in (4.3). Then, by Lemma 2.6, there is a unique t − (wl ) > ( p−q ) such − that t (wl )wl ∈ Nf0 ,g0 for all l > 0, that is
− t (wl )wl 2 1 = H
q f− t − (wl )wl dx +
RN
p g0 t − (wl )wl dx.
RN
Since wl 2H 1
=
|wl |p dx =
RN
for all l 0,
f− |wl |q dx → 0 and RN
2p ∞ α p−2
(1 − g0 )|wl |p dx → 0 as l → ∞, RN
we have t − (wl ) → 1 as l → ∞. Thus, lim Jf0 ,g0 t − (wl )wl = lim J ∞ t − (wl )wl = α ∞ . l→∞
l→∞
Then inf
u∈Nf0 ,g0
Jf0 ,g0 (u) inf∞ J ∞ (u) = α ∞ . u∈N
Let u ∈ Nf0 ,g0 . Then, by Lemma 2.6(i), Jf0 ,g0 (u) = supt0 Jf0 ,g0 (tu). Moreover, there is a unique t ∞ > 0 such that t ∞ u ∈ N∞ . Thus, Jf0 ,g0 (u) Jf0 ,g0 t ∞ u J ∞ t ∞ u α ∞ and so infu∈Nf0 ,g0 Jf0 ,g0 (u) α ∞ . Therefore, inf
u∈Nf0 ,g0
Jf0 ,g0 (u) = inf∞ J ∞ (u) = α ∞ . u∈N
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Next, we will show that Eq. (Ef0 ,g0 ) does not admit any solution u0 such that Jf0 ,g0 (u0 ) = infu∈Nf0 ,g0 Jf0 ,g0 (u). Suppose the contrary. Then we can assume that there exists u0 ∈ Nf0 ,g0 such that Jf0 ,g0 (u0 ) = infu∈Nf0 ,g0 Jf0 ,g0 (u). Then, by Lemma 2.6(i), Jf0 ,g0 (u0 ) = supt0 Jf0 ,g0 (tu0 ). Moreover, there is a unique tu0 > 0 such that tu0 u0 ∈ N∞ . Thus, α∞ =
inf
u∈Nf0 ,g0
Jf0 ,g0 (u) = Jf0 ,g0 (u0 ) Jf0 ,g0 (tu0 u0 )
q
tu J (tu0 u0 ) − 0 q ∞
f− |u0 | dx α q
∞
q
tu − 0 q
RN
f− |u0 |q dx. RN
This implies RN f− |u0 |q dx = 0 and so u0 ≡ 0 in {x ∈ RN | f− (x) = 0} form the condition (D1). Therefore, α ∞ = inf∞ J ∞ (u) = J ∞ (tu0 u0 ). u∈N
By the Lagrange multiplier and the maximum principle, we can assume that tu0 u0 is a positive solution of (E ∞ ). This contradicts
u0 ≡ 0 in x ∈ RN f− (x) = 0 and completes the proof.
2
Lemma 5.2. Suppose that {un } is a minimizing sequence in Nf0 ,g0 for Jf0 ,g0 . Then
(i) RN f− |un |q dx = o(1);
(ii) RN (1 − g0 )|un |p dx = o(1). Furthermore, {un } is a (PS)α ∞ -sequence in H 1 (RN ) for J ∞ . Proof. For each n, there is a unique tn > 0 such that tn un ∈ N∞ , that is p
tn2 un 2H 1 = tn
|un |p dx.
RN
Then, by Lemma 2.6(i), Jf0 ,g0 (un ) Jf0 ,g0 (tn un ) = J ∞ (tn un ) +
α∞ +
p tn
p
tn p
q
(1 − g0 )|un |p dx − RN
(1 − g0 )|un |p dx −
p RN
q tn
tn q
f− |un |q dx RN
f− |un |q dx.
q RN
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
117
Since Jf0 ,g0 (un ) = α ∞ + o(1) from Lemma 5.1, we have
q
tn q
f− |un |q dx = o(1) RN
and
p
tn p
(1 − g0 )|un |p dx = o(1). RN
We will show that there exists c0 > 0 such that tn > c0 for all n. Suppose the contrary. Then we may assume tn → 0 as n → ∞. Since Jf0 ,g0 (un ) = α ∞ + o(1), by Lemma 2.2, un is uniformly bounded and so tn un H 1 → 0 or J ∞ (tn un ) → 0 and this contradicts J ∞ (tn un ) α ∞ > 0. Thus, f− |un |q dx = o(1) RN
and (1 − g0 )|un |p dx = o(1), RN
this implies un 2H 1 =
|un |p dx + o(1)
RN
and J ∞ (un ) = α ∞ + o(1). Moreover, by Wang and Wu [26, Lemma 7], we have {un } is a (PS)α ∞ -sequence in H 1 (RN ) for J ∞ . 2 The following lemma is a key lemma in proving our main result. Lemma 5.3. There exists d0 > 0 such that if u ∈ Nf0 ,g0 and Jf0 ,g0 (u) α ∞ + d0 , then RN
x |∇u|2 + u2 dx = 0. |x|
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
Proof. Suppose the contrary. Then there exists a sequence {un } ⊂ Nf0 ,g0 such that Jf0 ,g0 (u) = α ∞ + o(1) and RN
x |∇un |2 + u2n dx = 0. |x|
Moreover, by Lemma 5.2, we have {un } is a (PS)α ∞ -sequence in H 1 (RN ) for J ∞ . It follows from Lemma 2.2 that there exist a subsequence {un } and u0 ∈ H 1 (RN ) such that un u0 weakly in H 1 (RN ). By the concentration–compactness principle (see Lions [20] or Struwe [24, Theorem 3.1]), there exist a sequence {xn } ⊂ RN , and a positive solution w0 ∈ H 1 (RN ) of Eq. (E ∞ ) such that un (x) − w0 (x − xn )
H1
→ 0 as n → ∞.
(5.1)
Now we will show that |xn | → ∞ as n → ∞. Suppose the contrary. Then we may assume that {xn } is bounded and xn → x0 for some x0 ∈ RN . Thus, by (5.1)
q f− (x)w0 (x − xn ) dx + o(1)
f− |un |q dx = RN
RN
q f− (x + xn )w0 (x) dx + o(1)
= RN
q f− (x + x0 )w0 (x) dx + o(1),
= RN
this contradicts the result of Lemma 5.2: RN f− |un |q dx = o(1). Hence we may assume |xxnn | → e as n → ∞, where e ∈ SN −1 . Then, by the Lebesgue dominated convergence theorem, we have
x |∇un |2 + u2n dx = |x|
0= RN
=
RN
x + xn |∇w0 |2 + w02 dx + o(1) |x + xn |
2p ∞ α e + o(1), p−2
which is a contradiction. This completes the proof.
2
− By (2.3), (2.6) and Lemma 2.6, for each u ∈ N− fλ ,gμ there is a unique t0 (u) > 0 such that
t0− (u)u ∈ Nf0 ,g0
and t0− (u) > tmax,0 (u) =
(2 − q)u2H 1
(p − q) RN g0 |u|p dx
1 p−2
> 0.
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
119
Let 2−q p p−2 (p − q)(1 + μb/a∞ ) p−2 (p − q)(1 + μb/a∞ ) 1 + f− Lq ∗ θμ = . p−q 2−q (2 − q)Sp2−q
Then we have the following results. Lemma 5.4. For each λ > 0 and μ > 0 with λp−2 (1 + μb∞ )2−q < ( q2 )p−2 Λ0 we have the following. + ∞ (i) [t0− (u)]p < θμ for all u ∈ N− fλ ,gμ with Jfλ ,gμ (u) < αfλ ,gμ + α .
pq − p ∞ (ii) RN g0 |u| dx θμ (p−q) α for all u ∈ Nfλ ,gμ with Jfλ ,gμ (u) < αf+λ ,gμ + α ∞ . + ∞ Proof. (i) For u ∈ N− fλ ,gμ with Jfλ ,gμ (u) < αfλ ,gμ + α , we have
u2H 1 −
fλ |u|q dx −
RN
gμ |u|p dx = 0
RN
and (2 − q)u2H 1 < (p − q)
gμ |u|p dx.
RN
We distinguish two cases. Case (A): t0− (u) < 1. Since θμ > 1 for all μ > 0, we have − p t0 (u) < 1 < θμ . Case (B): t0− (u) 1. Since − p t0 (u)
2 q g0 |u| dx = t0− (u) u2H 1 − t0− (u)
f− |u|q dx
p
RN
2 u2H 1 + t0− (u)
RN
|f− ||u| dx , q
RN
we have − p−2 u2H 1 +
t0 (u)
q RN |f− ||u| p RN g0 |u| dx
Moreover, by (2.3) and the Sobolev inequality,
dx
.
(5.2)
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
u2H 1
p−q < 2−q
p−q 1 + μb/a∞ gμ |u| dx 2−q
p
RN
g0 |u|p dx
RN
(p − q)(1 + μb/a∞ ) p 2
(2 − q)Sp
p
uH 1
(5.3)
and so uH 1
p
(2 − q)Sp2 (p − q)(1 + μb/a∞ )
1 p−2
(5.4)
.
Thus, by (5.2)–(5.4) and the Sobolev inequality,
q − p−2 p−q N f− |u| dx t0 (u) 1+ R 1 + μb/a∞ 2−q u2H 1 p−q f− Lq ∗ 1+ q 1 + μb/a∞ 2−q 2−q Sp2 uH 1 2−q (p − q)(1 + μb/a∞ ) p−2 (p − q)(1 + μb/a∞ ) 1 + f− Lq ∗ p−q 2−q (2 − q)Sp2−q or [t − (u)]p θμ . (ii) By Lemma 5.1 and t0− (u)u ∈ Nf0 ,g0 , α ∞ Jf0 ,g0 t0− (u)u 1 1 1 − 2 1 − p 2 t0 (u) uH 1 + t (u) g0 |u|p dx = − − 2 q q p 0 <
1 − p 1 − t0 (u) g0 |u|p dx, q p
RN
RN
and this implies
pq 1 α∞. g0 |u| dx − [t0 (u)]p p − q p
RN
By part (i), we can conclude that g0 |u|p dx RN
pq α∞ θμ (p − q)
+ ∞ for all u ∈ N− fλ ,gμ with Jfλ ,gμ (u) < αfλ ,gμ + α . This completes the proof.
2
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
121
By the proof of Proposition 4.1, there exist positive numbers t∗ and l2 such that u+ λ,μ + t∗ wl ∈ − Nfλ ,gμ and + ∞ Jfλ ,gμ u+ λ,μ + t∗ wl < αfλ ,gμ + α
for all l > l2 .
Furthermore, we have the following result. Lemma 5.5. There exist positive numbers λ0 and μ0 with 2−q p−2 λ0 1 + μ0 b∞
p−2 q < Λ0 2
such that for every λ ∈ (0, λ0 ) and μ ∈ (0, μ0 ), we have x |∇u|2 + u2 dx = 0 |x| RN
+ ∞ for all u ∈ N− fλ ,gμ with Jfλ ,gμ (u) < αfλ ,gμ + α . + − ∞ Proof. For u ∈ N− fλ ,gμ with Jfλ ,gμ (u) < αfλ ,gμ + α , by Lemma 2.6(i) there exists t0 (u) > 0
such that t0− (u)u ∈ Nf0 ,g0 . Moreover,
Jfλ ,gμ (u) = sup Jfλ ,gμ (tu) Jfλ ,gμ t0− (u)u t0
λ[t − (u)]q = Jf0 ,g0 t0− (u)u − 0 q −
μ[t0− (u)]p p
f+ |u|q dx RN
b|u|p dx. RN
Thus, by Lemma 5.4 and the Hölder and Sobolev inequalities, λ[t − (u)]q Jf0 ,g0 t0− (u)u Jfλ ,gμ (u) + 0 q
f+ |u|q dx +
μ[t0− (u)]p p
RN
< αf+λ ,gμ + α ∞ +
q/p λθμ
q
b|u|p dx RN
−q
q
f+ Lq ∗ Sp 2 uH 1 +
μθμ b∞ − p2 p Sp uH 1 . p
Since Jfλ ,gμ (u) < αf+λ ,gμ + α ∞ < α ∞ , by (2.1) in Lemma 2.2, for each λ > 0 and μ > 0 with
λp−2 (1 + μb∞ )2−q < ( q2 )p−2 Λ0 , there exists a positive number c independent of λ, μ such − + ∞ c for all u ∈ Nfλ ,gμ with Jfλ ,gμ (u) < αfλ ,gμ + α . Therefore, that uH 1 q/p λθμ μθμ b∞ − p2 p −q q Jf0 ,g0 t0− (u)u < αf+λ ,gμ + α ∞ + f+ Lq ∗ Sp 2 Sp c + c . q p
122
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131 p−2
Let d0 > 0 be as in Lemma 5.3. Then there exist positive numbers λ0 and μ0 with λ0 μ0 b∞ )2−q < ( q2 )p−2 Λ0 such that for λ ∈ (0, λ0 ) and μ ∈ (0, μ0 ), Jf0 ,g0 t − (u)u < α ∞ + d0 .
(1 +
(5.5)
Since t0− (u)u ∈ Nf0 ,g0 and t0− (u) > 0, by Lemma 5.3 and (5.5) RN
2 2 x − ∇ t0 (u)u + t0− (u)u dx = 0, |x|
and this implies RN
x |∇u|2 + u2 dx = 0 |x|
+ ∞ for all u ∈ N− fλ ,gμ with Jfλ ,gμ (u) < αfλ ,gμ + α .
2
6. Proof of Theorem 1.1 In the following, we use an idea of Adachi and Tanaka [1]. For c ∈ R+ , we denote
[Jfλ ,gμ c] = u ∈ N− fλ ,gμ u 0, Jfλ ,gμ (u) c . We then try to show for a sufficiently small σ > 0 cat Jfλ ,gμ αf+λ ,gμ + α ∞ − σ 2.
(6.1)
To prove (6.1), we need some preliminaries. Recall the definition of Lusternik–Schnirelman category. Definition 6.1. (i) For a topological space X, we say a non-empty, closed subset Y ⊂ X is contractible to a point in X if and only if there exists a continuous mapping ξ : [0, 1] × Y → X such that for some x0 ∈ X ξ(0, x) = x
for all x ∈ Y,
ξ(1, x) = x0
for all x ∈ Y.
and
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
123
(ii) We define
cat(X) = min k ∈ N there exist closed subsets Y1 , . . . , Yk ⊂ X such that
Yj is contractible to a point in X for all j and
k
Yj = X .
j =1
When there do not exist finitely many closed subsets Y1 , . . . , Yk ⊂ X such that Yj is con tractible to a point in X for all j and kj =1 Yj = X, we say cat(X) = ∞. We need the following two lemmas. Lemma 6.2. Suppose that X is a Hilbert manifold and F ∈ C 1 (X, R). Assume that there are c0 ∈ R and k ∈ N, (i) F (x) satisfies the Palais–Smale condition for energy level c c0 ; (ii) cat({x ∈ X | F (x) c0 }) k. Then F (x) has at least k critical points in {x ∈ X; F (x) c0 }. Proof. See Ambrosetti [3, Theorem 2.3].
2
Lemma 6.3. Let X be a topological space. Suppose that there are two continuous maps Φ : SN −1 → X,
Ψ : X → SN −1
such that Ψ ◦ Φ is homotopic to the identity map of SN −1 , that is, there exists a continuous map ζ : [0, 1] × SN −1 → SN −1 such that ζ (0, x) = (Ψ ◦ Φ)(x) ζ (1, x) = x
for each x ∈ SN −1 ,
for each x ∈ SN −1 .
Then cat(X) 2. Proof. See Adachi and Tanaka [1, Lemma 2.5].
2
For l > l2 , we define a map Φfλ ,gμ : SN −1 → H 1 (RN ) by Φfλ ,gμ (e)(x) = u+ λ,μ + sl t0 wl
for e ∈ SN −1 ,
where u+ λ,μ + sl t0 wl is as in the proof of Proposition 4.1. Then we have the following result.
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
Lemma 6.4. There exists a sequence {σl } ⊂ R+ with σl → 0 as l → ∞ such that Φfλ ,gμ SN −1 ⊂ Jfλ ,gμ αf+λ ,gμ + α ∞ − σl . − Proof. By Proposition 4.1, for each l > l2 we have u+ λ,μ + sl t0 wl ∈ Nfλ ,gμ and
+ ∞ sup Jfλ ,gμ u+ λ,μ + twl < αfλ ,gμ + α
uniformly in e ∈ SN −1 .
t0
+ ∞ Since Φfλ ,gμ (SN −1 ) is compact, Jfλ ,gμ (u+ λ,μ +sl t0 wl ) αfλ ,gμ +α −σl , so that the conclusion holds. 2
From Lemma 5.5, we define Ψfλ ,gμ : Jfλ ,gμ < αf+λ ,gμ + α ∞ → SN −1 by
R Ψfλ ,gμ (u) =
| RN N
x 2 |x| (|∇u| x 2 |x| (|∇u|
+ u2 ) dx + u2 ) dx|
.
Then we have the following results. Lemma 6.5. Let λ0 , μ0 be as in Lemma 5.5. Then for each λ ∈ (0, λ0 ) and μ ∈ (0, μ0 ), there exists l∗ l2 such that for l > l∗ , the map Ψfλ ,gμ ◦ Φfλ ,gμ : SN −1 → SN −1 is homotopic to the identity. Proof. Let Σ = {u ∈ H 1 (RN )\{0} |
x 2 RN |x| (|∇u|
+ u2 ) dx = 0}. We define
Ψ fλ ,gμ : Σ → SN −1 by
R Ψ fλ ,gμ (u) =
| RN N
x 2 |x| (|∇u| x 2 |x| (|∇u|
+ u2 ) dx + u2 ) dx|
as an extension of Ψfλ ,gμ . Since wl ∈ Σ for all e ∈ SN −1 and for l sufficiently large, we let γ : [s1 , s2 ] → SN −1 be a regular geodesic between Ψ fλ ,gμ (wl ) and Ψ fλ ,gμ (Φfλ ,gμ (e)) such that γ (s1 ) = Ψ fλ ,gμ (wl ), γ (s2 ) = Ψ fλ ,gμ (Φfλ ,gμ (e)). By an argument similar to that in Lemma 5.3, there exists a positive number l∗ l2 such that for l > l∗ , l e ∈ Σ for all e ∈ SN −1 and θ ∈ [1/2, 1). w0 x + 2(1 − θ )
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
125
We define ζl (θ, e) : [0, 1] × SN −1 → SN −1 by ζl (θ, e) =
γ (2θ (s1 − s2 ) + s2 ) l Ψ fλ ,gμ (w0 (x + 2(1−θ) e)) e
for θ ∈ [0, 1/2); for θ ∈ [1/2, 1); for θ = 1.
Then ζl (0, e) = Ψ fλ ,gμ (Φfλ ,gμ (e)) = Ψfλ ,gμ (Φfλ (e)) and ζl (1, e) = e. By the standard regularN ity, we have u+ λ,μ ∈ C(R ). First, we claim that limθ→1− ζl (θ, e) = e and limθ→ 1 − ζl (θ, e) = 2
Ψ fλ ,gμ (w0 (x + le)). (a) limθ→1− ζl (θ, e) = e: since RN
2 2 l l x ∇ w0 x + dx e + w0 x + e |x| 2(1 − θ ) 2(1 − θ ) = RN
=
x− |x −
l 2 2(1−θ) e ∇ w0 (x) l 2(1−θ) e|
2p α ∞ e + o(1) p−2
2 + w0 (x) dx
as θ → 1− ,
then limθ→1− ζl (θ, e) = e. (b) limθ→ 1 − ζl (θ, e) = Ψ fλ ,gμ (w0 (x + le)): since Ψ fλ ,gμ ∈ C(Σ, SN −1 ), we obtain 2
limθ→ 1 − ζl (θ, l) = Ψ fλ ,gμ (w0 (x + le)). 2
Thus, ζl (θ, e) ∈ C([0, 1] × SN −1 , SN −1 ) and ζl (0, e) = Ψfλ ,gμ Φfλ ,gμ (e) ζl (1, e) = e
for all e ∈ SN −1 ,
for all e ∈ SN −1 ,
provided l > l∗ . This completes the proof.
2
Lemma 6.6. For each λ ∈ (0, λ0 ), μ ∈ (0, μ0 ) and l > l∗ , functional Jfλ ,gμ has at least two critical points in [Jfλ ,gμ < αf+λ ,gμ + α ∞ ]. Proof. Applying Lemmas 6.3 and 6.5, we have for λ ∈ (0, λ0 ), μ ∈ (0, μ0 ) and l > l∗ , cat Jfλ ,gμ αf+λ ,gμ + α ∞ − σl 2. By Proposition 3.2, Lemma 6.2, Jfλ (u) has at least two critical points in [Jfλ ,gμ < αf+λ ,gμ + α ∞ ]. 2
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
We can now complete the proof of Theorem 1.1: (i) by Theorems 3.3, 4.2; (ii) for λ ∈ (0, λ0 ) and μ ∈ (0, μ0 ), from Theorem 3.3 and Lemma 6.6, Eq. (Efλ ,gμ ) has three positive solutions − − + + − − u+ λ,μ , u1 , u2 such that uλ,μ ∈ Nfλ ,gμ and ui ∈ Nfλ ,gμ for i = 1, 2. This completes the proof of Theorem 1.1. 7. Proof of Theorem 1.2 For c > 0, we define J0,cg0 (u) =
1 2
RN
N0,cg0
1 p
|∇u|2 + u2 dx −
cg0 |u|p dx, RN
= u ∈ H 1 RN \{0} J0,cg (u), u = 0 . 0
Recall that for each u ∈ H 1 (RN )\{0} there exist a unique t − (u) > 0 and t0 (u) > 0 such that t − (u)u ∈ N− fλ ,gμ and t0 (u)u ∈ N0,g0 . Let
B = u ∈ H 1 RN \{0} u 0 and uH 1 = 1 . Then we have the following result. Lemma 7.1. For each u ∈ B we have the following. (i) There is a unique t0c = t0c (u) > 0 such that t0c u ∈ N0,cg0 and p−2 sup J0,cg0 (tu) = J0,cg0 t0c u = 2p t0
cg0 |u|p dx
−2 p−2
.
RN
(ii) For ρ ∈ (0, 1),
Jfλ ,gμ
t (u)u
p
(1 − ρ) p−2
−
(1 + μb/a∞ )
2 p−2
q 2 2−q (ρSp ) q−2 λf+ Lq ∗ 2−q J0,g0 t0 (u)u − 2q
and p q 2 2−q (ρSp ) q−2 λf+ Lq ∗ + f− Lq ∗ 2−q . Jfλ ,gμ t − (u)u (1 + ρ) p−2 J0,g0 t0 (u)u + 2q
Proof. (i) For each u ∈ B, let 1 1 K(t) = J0,cg0 (tu) = t 2 − t p 2 p
RN
cg0 |u|p dx,
T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
then K(t) → −∞ as t → ∞, K (t) = t − t p−1
1)t p−2 RN cg0 |u|p dx. Let
RN
t0c = t0c (u) =
127
cg0 |u|p dx and K
(t) = 1 − (p −
cg0 |u|p dx
1 2−p
> 0.
RN
Then K (t0c ) = 0, t0c u ∈ N0,cg0 and K
(t0c ) = 2 − p < 0. Thus, there is a unique t0c = t0c (u) > 0 such that t0c u ∈ N0,cg0 and p−2 sup J0,cg0 (tu) = J0,cg0 t0c u = 2p t0
cg0 |u| dx p
−2 p−2
.
RN
(ii) Let c = (1 + μb/a∞ )/(1 − ρ). Then for each u ∈ B and ρ ∈ (0, 1), we get
−q q q fλ t0c u dx λSp2 f+ Lq ∗ t0c uH 1
RN
2 −q 2−q q q q 2 (ρSp ) 2 λf+ Lq ∗ 2−q + ρ 2 t0c uH 1 q 2 2 q 2 2−q qρ t c u2 1 . = (ρSp ) q−2 λf+ Lq ∗ 2−q + 0 H 2 2
(7.1)
Then, by part (i) and (7.1), sup Jfλ ,gμ (tu) Jfλ ,gμ t0c u t0
q 2 1−ρ t c u2 1 − 2 − q (ρSp ) q−2 λf+ q ∗ 2−q 0 L H 2 2q p (1 + μb/a∞ ) − g0 t0c u dx p
RN q 2−q 2 (ρSp ) q−2 λf+ Lq ∗ 2−q = (1 − ρ)J0,cg0 t0c u − 2q p
q 2 (p − 2)(1 − ρ) p−2 2−q (ρSp ) q−2 λf+ Lq ∗ 2−q = − 2
2q 2p((1 + μb/a∞ ) RN g0 |u|p dx) p−2 p
=
(1 − ρ) p−2 (1 + μb/a∞ )
2 p−2
q 2 2−q (ρSp ) q−2 λf+ Lq ∗ 2−q . J0,g0 t0 (u)u − 2q
By Lemma 2.6 and Theorem 3.1, sup Jfλ ,gμ (tu) = Jfλ ,gμ t − (u)u . t0
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T.F. Wu / Journal of Functional Analysis 258 (2010) 99–131
Thus, Jfλ ,gμ t − (u)u
p
(1 − ρ) p−2 (1 + μb/a∞ )
2 p−2
q 2 2−q (ρSp ) q−2 λf+ Lq ∗ 2−q . J0,g0 t0 (u)u − 2q
Moreover, by the Hölder, Sobolev and Young inequalities,
−q q fλ |tu|q dx λf+ Lq ∗ + f− Lq ∗ Sp 2 tuH 1
RN q 2 2−q qρ (ρSp ) q−2 λf+ Lq ∗ + f− Lq ∗ 2−q + tu2H 1 . 2 2
Therefore, Jfλ ,gμ (tu)
q 2 (1 + ρ) 2 2 − q 1 t + (ρSp ) q−2 λf+ Lq ∗ + f− Lq ∗ 2−q − 2 2q p
g0 |tu|p dx RN
(1 + ρ)
p p−2
2−q (ρSp ) J0,g0 t0 (u)u + 2q
q q−2
2 λf+ Lq ∗ + f− Lq ∗ 2−q
and so p q 2 2−q (ρSp ) q−2 λf+ Lq ∗ + f− Lq ∗ 2−q . Jfλ ,gμ t − (u)u (1 + ρ) p−2 J0,g0 t0 (u)u + 2q
This completes the proof.
2
Since αf−λ ,gμ > 0 for all λ ∈ (0, λ0 ) and μ ∈ (0, μ0 ), we define Ifλ ,gμ (u) = sup Jfλ ,gμ (tu) = Jfλ ,gμ t − (u)u > 0, t0 ∗ where t − (u)u ∈ N− fλ ,gμ . We observe that if λ, μ and f− Lq are sufficiently small, Bahri–Li’s minimax argument [6] also works for Jfλ ,gμ . Let
Γfλ ,gμ = γ ∈ C B N (0, l), B γ |∂B N (0,l) = wl /wl H 1
for large l.
Then we define βfλ ,gμ =
inf
sup Ifλ ,gμ γ (x)
γ ∈Γfλ ,gμ x∈RN
and β0,g0 = inf
sup I0,g0 γ (x) .
γ ∈Γ0,g0 x∈RN
By Lemma 7.1(ii), for 0 < ρ < 1, we have p
βfλ ,gμ
(1 − ρ) p−2 (1 + μb/a∞ )
2 p−2
β0,g0 −
q 2 2−q (ρSp ) q−2 λf+ Lq ∗ 2−q 2q
(7.2)
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and q 2 2−q (ρSp ) q−2 λf+ Lq ∗ + f− Lq ∗ 2−q . 2q
p
βfλ ,gμ (1 + ρ) p−2 β0,g0 +
(7.3)
We need the following results. Lemma 7.2. α ∞ < β0,g0 < 2α ∞ . Proof. Bahri and Li [6] prove that Eq. (E0,g0 ) admits at least one positive solution u0 and J0,g0 (u0 ) = β0,g0 < 2α ∞ . Moreover, by the condition (D4), Eq. (E0,g0 ) does not have a positive ground state solution. Hence, α ∞ < β0,g0 < 2α ∞ . 2 Theorem 7.3. Let λ0 , μ0 be as in Lemma 5.5. Then there exist positive numbers λ 0 λ0 , μ0 μ0 and ν0 such that for λ ∈ (0, λ0 ), μ ∈ (0, μ0 ) and f− Lq ∗ < ν0 , we have αf+λ ,gμ + α ∞ < βfλ ,gμ < αf−λ ,gμ + α ∞ . Furthermore, Eq. (Efλ ,gμ ) has a positive solution vfλ ,gμ such that Jfλ ,gμ (vfλ ,gμ ) = βfλ ,gμ . Proof. By Lemma 7.1(ii), we also have that for 0 < ρ < 1 p
αf−λ ,gμ
(1 − ρ) p−2 (1 + μb/a∞ )
2 p−2
α∞ −
q 2 2−q (ρSp ) q−2 λf+ Lq ∗ 2−q 2q
and p
αf−λ ,gμ (1 + ρ) p−2 α ∞ +
q 2 2−q (ρSp ) q−2 λf+ Lq ∗ + f− Lq ∗ 2−q . 2q
For any ε > 0 there exist positive numbers λ1 λ0 , μ1 μ0 and ν1 such that for λ ∈ (0, λ1 ), ∗ μ ∈ (0, μ1 ) and f− Lq < ν1 , we have α ∞ − ε < αf−λ ,gμ < α ∞ + ε. Thus, 2α ∞ − ε < αf−λ ,gμ + α ∞ < 2α ∞ + ε. μ2 μ0 and ν2 Applying (7.2) and (7.3) for any δ > 0 there exist positive numbers λ2 λ0 , such that for λ ∈ (0, λ2 ), μ ∈ (0, μ2 ) and f− Lq ∗ < ν2 , we have β0,g0 − δ < βfλ ,gμ < β0,g0 + δ.
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Fix a small 0 < ε < (2α ∞ − β0,g0 )/2, since α ∞ < β0,g0 < 2α ∞ , choosing a δ > 0 such that for λ1 , λ2 }, μ < μ0 = min{ μ1 , μ2 } and f− Lq ∗ < ν0 = min{ν1 , ν2 }, we get λ < λ0 = min{ αf+λ ,gμ + α ∞ < α ∞ < βfλ ,gμ < 2α ∞ − ε < αf−λ ,gμ + α ∞ . Therefore, by Proposition 3.2, we obtain that (Efλ ,gμ ) has a positive solution vfλ ,gμ such that Jfλ ,gμ (vfλ ,gμ ) = βfλ ,gμ . 2 μ0 ) and We can now complete the proof of Theorem 1.2: for λ ∈ (0, λ0 ), μ ∈ (0, f− Lq ∗ < ν0 , from Theorems 1.1, 7.3, Eq. (Efλ ,gμ ) has at least four positive solutions. Acknowledgment The author is grateful for the referee’s valuable suggestions and helps. References [1] S. Adachi, K. Tanaka, Four positive solutions for the semilinear elliptic equation: −u + u = a(x)up + f (x) in RN , Calc. Var. Partial Differential Equations 11 (2000) 63–95. [2] Adimurthy, F. Pacella, L. Yadava, On the number of positive solutions of some semilinear Dirichlet problems in a ball, Differential Integral Equations 10 (6) (1997) 1157–1170. [3] A. Ambrosetti, Critical points and nonlinear variational problems, Bull. Soc. Math. France Mémoire 49 (1992). [4] A. Ambrosetti, G.J. Azorero, I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996) 219–242. [5] A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519–543. [6] A. Bahri, Y.Y. Li, On the min–max procedure for the existence of a positive solution for certain scalar field equations in RN , Rev. Mat. Iberoamericana 6 (1990) 1–15. [7] H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I. Existence of ground state, Arch. Ration. Mech. Anal. 82 (1983) 313–345. [8] P.A. Binding, P. Drábek, Y.X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations 5 (1997) 1–11. [9] K.J. Brown, T.F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electron. J. Differential Equations 69 (2007) 1–9. [10] K.J. Brown, T.F. Wu, A fibering map approach to a potential operator equation and its applications, Differential Integral Equations, in press. [11] K.J. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations 193 (2003) 481–499. [12] J. Chabrowski, João Marcos Bezzera do Ó, On semilinear elliptic equations involving concave and convex nonlinearities, Math. Nachr. 233–234 (2002) 55–76. [13] L. Damascelli, M. Grossi, F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 631–652. [14] P. Drábek, S.I. Pohozaev, Positive solutions for the p-Laplacian: Application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997) 703–726. [15] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 17 (1974) 324–353. [16] D.G. de Figueiredo, J.P. Gossez, P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal. 199 (2003) 452–467. [17] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1978) 209–243. [18] J.V. Goncalves, O.H. Miyagaki, Multiple positive solutions for semilinear elliptic equations in RN involving subcritical exponents, Nonlinear Anal. 32 (1998) 41–51. [19] M.K. Kwong, Uniqueness of positive solution of u − u + up = 0 in RN , Arch. Ration. Mech. Anal. 105 (1989) 243–266.
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[20] P.L. Lions, The concentration–compactness principle in the calculus of variations. The local compact case, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) 102–145 and 223–283. [21] Z. Liu, Z.Q. Wang, Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Phys. 56 (2005) 609–629. [22] T. Ouyang, J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem II, J. Differential Equations 158 (1999) 94–151. [23] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., American Mathematical Society, 1986. [24] M. Struwe, Variational Methods, second ed., Springer-Verlag, Berlin, Heidelberg, 1996. [25] M. Tang, Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 705–717. [26] H.C. Wang, T.F. Wu, Symmetry breaking in a bounded symmetry domain, NoDEA Nonlinear Differential Equations Appl. 11 (2004) 361–377. [27] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. [28] T.F. Wu, Multiplicity of positive solutions for semilinear elliptic equations in RN , Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 647–670. [29] T.F. Wu, On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function, Commun. Pure Appl. Anal. 7 (2008) 383–405. [30] T.F. Wu, Multiplicity results for a semilinear elliptic equation involving sign-changing weight function, Rocky Mountain J. Math. 39 (2009) 995–1012.
Journal of Functional Analysis 258 (2010) 132–160 www.elsevier.com/locate/jfa
Remarks on common hypercyclic vectors Stanislav Shkarin Queens’s University Belfast, Department of Pure Mathematics, University road, Belfast, BT7 1NN, UK Received 6 March 2009; accepted 29 June 2009 Available online 18 July 2009 Communicated by N. Kalton
Abstract We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator T on a complex Fréchet space X and a set Λ ⊆ R+ × C which is not of zero three-dimensional Lebesgue measure, the family {aT + bI : (a, b) ∈ Λ} has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. It allows to show that if D = {z ∈ C: |z| < 1} and ϕ ∈ H∞ (D) is non-constant, then the family {zMϕ : b−1 < |z| < a −1 } has a common hypercyclic vector, where Mϕ : H2 (D) → H2 (D), Mϕ f = ϕf , a = inf{|ϕ(z)|: z ∈ D} and b = sup{|ϕ(z)|: |z| ∈ D}, providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family {aTb : a, b ∈ C \ {0}} has a common hypercyclic vector, where Tb f (z) = f (z − b) acts on the Fréchet space H(C) of entire functions on one complex variable. © 2009 Elsevier Inc. All rights reserved. Keywords: Hypercyclic operators; Hypercyclic vectors
1. Introduction All vector spaces in this article are assumed to be over K being either the field C of complex numbers or the field R of real numbers. Throughout this paper all topological spaces and topological vector spaces are assumed to be Hausdorff. As usual, Z+ is the set of non-negative integers, R+ is the set of non-negative real numbers, N is the set of positive integers, K = K \ {0}, E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.032
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D = {z ∈ C: |z| < 1} and T = {z ∈ C: |z| = 1}. By a compact interval of the real line we mean a set of the shape [a, b] with −∞ < a < b < ∞. That is, a singleton is not considered to be an interval. For topological vector spaces X and Y , L(X, Y ) stands for the space of continuous linear operators from X to Y . We write L(X) instead of L(X, X) and X ∗ instead of L(X, K). For T ∈ L(X, Y ), the dual operator T ∗ : Y ∗ → X ∗ acts according to the formula T ∗ f (x) = f (T x). Recall [21] that an F -space is a complete metrizable topological vector space and a Fréchet space is a locally convex F -space. For a subset A of a vector space X, symbol span(A) stands for the linear span of A. Definition 1.1. Let X and Y be topological spaces and F = {Ta : a ∈ A} be a family of continuous maps from X to Y . An element x ∈ X is called universal for F if the orbit {Ta x: a ∈ A} is dense in Y and F is said to be universal if it has a universal element. We denote the set of universal elements for F by the symbol U(F ). A continuous linear operator T acting on a topological vector space X is called hypercyclic if the family of its powers {T n : n ∈ Z+ } is universal. Corresponding universal elements are called hypercyclic vectors for T . The set of hypercyclic vectors for T is denoted by H(T ). That is, H(T ) = U({T n : n ∈ Z+ }). If {Ta : a ∈ A} is a family of continuous linear operators on topological vector space X, we denote H{Ta : a ∈ A} =
H(Ta ).
a∈A
That is, H{Ta : a ∈ A} consists of all vectors x ∈ X that are hypercyclic for each Ta , a ∈ A. Recall that a topological space X is called Baire if the intersection of any countable family of dense open subsets of X is dense. Hypercyclic operators and universal families have been intensely studied during last few decades, see surveys [14,15] and references therein. It is well known [14] that the set of hypercyclic vectors of a hypercyclic operator on a separable metrizable Baire topological vector space is a dense Gδ -set. It immediately follows that any countable family of hypercyclic operators on such a space has a dense Gδ -set of common hypercyclic vectors (= hypercyclic for each member of the family). We are interested in the existence of common hypercyclic vectors for uncountable families of continuous linear operators. First results in this direction were obtained by Abakumov and Gordon [1] and León-Saavedra and Müller [18]. Theorem AG. Let T be the backward shift on 2 . That is, T ∈ L(2 ), T e0 = 0 and T en = en−1 for n ∈ N, where {en }n∈Z+ is the standard orthonormal basis of 2 . Then H{aT : a ∈ K, |a| > 1} is a dense Gδ -set. The following result is of completely different flavor. It is proven in [18] for continuous linear operators on Banach spaces although the proof can be easily adapted [23] for continuous linear operators acting on arbitrary topological vector spaces. Theorem LM. Let X be a complex topological vector space and T ∈ L(X). Then U(F ) = H(zT ) = H(T ) for any z ∈ T, where F = {wT n : w ∈ T, n ∈ Z+ }. In particular, H{zT : z ∈ T} = H(T ). It follows that the family {zT : z ∈ T} has a common hypercyclic vector, whenever T is a hypercyclic operator. A result similar to the above one was recently obtained by Conejero, Müller
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and Peris [9] for operators acting on separable F -spaces (see [23] for a proof in a more general setting). Recall that a family {Tt }t∈R+ of continuous linear operators on a topological vector space is called an operator semigroup if T0 = I and Tt+s = Tt Ts for any t, s ∈ R+ . Theorem CMP. Let X be a topological vector space and {Tt }t∈R+ be an operator semigroup on X. Assume also that the map (t, x) → Tt x from R+ × X to X is continuous. Then H(Tt ) = U(F ) for any t > 0, where F = {Ts : s > 0}. In particular, H{Ts : s > 0} = H(Tt ) for any t > 0. It follows that if {Tt }t∈R+ is an operator semigroup such that the map (t, x) → Tt x is continuous and there exists t > 0 for which Tt is hypercyclic, then the family {Ts : s > 0} has a common hypercyclic vector. Bayart [2] provided families of composition operators on the space of holomorphic functions on D, which have common hypercyclic vectors. Costakis and Sambarino [11], Bayart and Matheron [4], Chan and Sanders [8] and Gallardo-Gutiérrez and Partington [12] proved certain sufficient conditions for a set of families of continuous linear operators to have a common universal vector. In all the mentioned papers the criteria were applied to specific sets of families. For instance, Costakis and Sambarino [11] proved the following theorem. Theorem CS. Let H(C) be the complex Fréchet space of entire functions on one variable, D ∈ L(H(C)) be the differentiation operator Df = f and for each a ∈ C, Ta ∈ L(H(C)) be the translation operator Ta f (z) = f (z − a). Then H{Ta : a ∈ C }, H{aT1 : a ∈ C } and H{aD: a ∈ C } are dense Gδ -sets. The criteria by Bayart and Matheron were applied to various families of operators including families of weighted translations on Lp (R), composition operators on Hardy spaces Hp (D) and backward weighted shifts on p . We would like to mention just one example of the application of the criterion from [4], which is related to our results. Example BM. As in Theorem CS, let Ta be translation operators on H(C). For each s ∈ R+ and z ∈ T, consider the family Fs,z = {ns Tnz : n ∈ Z+ }. Then
U(Fs,z ) is a dense Gδ -subset of H(C).
(s,z)∈R+ ×T
Chan and Sanders [8] found common universal elements of certain sets of families of backward weighted shifts on 2 . Gallardo-Gutiérrez and Partington [12] proved a modification of the Costakis–Sambarino criterion and applied it to obtain common hypercyclic vectors for families of adjoint multipliers and composition operators on Hardy spaces. Finally, we would like to mention the following application by Costakis and Mavroudis [10] of the Bayart–Matheron criterion. Theorem CM. Let D be the differentiation operator on H(C) and p be a non-constant polynomial. Then H{ap(D): a ∈ C } is a dense Gδ -set. Although the most of the mentioned criteria look quite general, they are basically not applicable to finding common hypercyclic vectors of families that are not smoothly labeled by one real parameter. Note that although the families in Theorems AG, CS and CM are formally speaking labeled by a complex parameter a, Theorem LM allows to reduce them to families labeled
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by one real parameter. Example BM is, of course, genuinely two-parametric, but it is not about a common hypercyclic vector. On the other hand, one can artificially produce huge families of operators with a common hypercyclic vector. For example, take all operators for which a given vector is hypercyclic. The following result provides a common hypercyclic vector for a twoparametric family of operators. It strengthens the first part of Theorem CS and kind of improves Example BM. Theorem 1.2. Let Ta for a ∈ C be the translation operator Ta f (z) = f (z − a) acting on the complex Fréchet space H(C) of entire functions on one complex variable. Then H{bTa : a, b ∈ C } is a dense Gδ -set. A common hypercyclic vector from the above theorem is even more monstrous than the holomorphic monsters provided by Theorem CS. Godefroy and Shapiro [13] considered adjoint multiplication operators on function Hilbert spaces. Recall that if U is a connected open subset of Cm , then a function Hilbert space H on U is a Hilbert space consisting of functions f : U → C holomorphic on U such that for any z ∈ U the evaluation functional χz : H → C, χz (f ) = f (z) is continuous. A multiplier for H is a function ϕ : U → C such that ϕf ∈ H for each f ∈ H. It is well known [13] that any multiplier is bounded and holomorphic. Each multiplier gives rise to the multiplication operator Mϕ ∈ L(H), Mϕ f = ϕf (continuity of Mϕ follows from the Banach closed graph theorem). Its Hilbert space adjoint Mϕ is called an adjoint multiplication operator. Godefroy and Shapiro proved that there is f ∈ H, which is cyclic for Mϕ for any non-constant multiplier ϕ for H and demonstrated that if ϕ : U → C is a non-constant multiplier for H and ϕ(U ) ∩ T = ∅, then Mϕ is hypercyclic, see also the related paper by Bourdon and Shapiro [7]. Godefroy and Shapiro also raised the following question [13, p. 263]. Question GS. Let H be a Hilbert function space on a connected open subset U of Cm . Does the family of all hypercyclic adjoint multiplications on H have a common hypercyclic vector? Recall that any T ∈ L(H(C)) such that T is not a scalar multiple of the identity and T D = DT is hypercyclic. The following question was raised by Richard Aron. Question A. Let D be the family of all continuous linear operators on H(C), which are not scalar multiples of the identity and which commute with the differentiation operator D. Is it true that there is a common hypercyclic vector for all operators from the family D? The next result allows us to answer negatively both of the above questions. Theorem 1.3. Let X be a complex topological vector space such that X ∗ = {0}, T ∈ L(X) and Λ be a subset of R+ × C. Assume also that the family {aT + bI : (a, b) ∈ Λ} has a common hypercyclic vector. Then the set Λ has zero three-dimensional Lebesgue measure. Corollary 1.4. The family {aD + bI : a > 0, b ∈ C} of continuous linear operators on H(C) does not have a common hypercyclic vector. Corollary 1.5. Let H be a Hilbert function space on a connected open subset U of Cm and ϕ be a non-constant multiplier for H. Then the family {M : a > 0, b ∈ C, (b + aϕ)(U ) ∩ T = ∅} b+aϕ of hypercyclic operators does not have a common hypercyclic vector.
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Corollaries 1.4 and 1.5 follow from Theorem 1.3 because M = aMϕ + bI and the sets b+aϕ of pairs (a, b) involved in the definition of the families in Corollaries 1.4 and 1.5 are non-empty open subsets of R+ × C and therefore have non-zero 3-dimensional Lebesgue measure. In fact, Theorem 1.3 shows that even relatively small subfamilies of the families from Questions GS and A fail to have common hypercyclic vectors. As usual, H2 (D) is the Hardy space of the unit disk. It is well known that H2 (D) is a function Hilbert space on D and the set of multipliers for H2 (D) is the space H∞ (D) of bounded holomorphic functions f : D → C. Let ϕ ∈ H∞ (D) be non-constant. Using the mentioned criterion by Godefroy and Shapiro for hypercyclicity of adjoint multiplications together with the fact that a contraction or its inverse cannot be hypercyclic, is hypercyclic if and only if b−1 < |z| < a −1 , where a = inf we see that zMϕ = Mzϕ z∈D |ϕ(z)| and b = supz∈D |ϕ(z)|. Probably, expecting the answer to Question GS to be negative, Bayart and Grivaux [3] raised the following question. Question BG. Let ϕ ∈ H∞ (D) be non-constant, a = infz∈D |ϕ(z)| and b = supz∈D |ϕ(z)|. Is it true that the family {zMϕ : b−1 < |z| < a −1 } has common hypercyclic vectors? We prove a sufficient condition on a family of scalar multiples of a given operator to have a common hypercyclic vector and use it to answer Question BG affirmatively. It is worth noting that Gallardo-Gutiérrez and Partington [12] found a partial affirmative answer to the above question. Theorem 1.6. Let X be a separable complex F -space, T ∈ L(X) and 0 a < b ∞. Assume also that there is a map (k, c) → Fk,c sending a pair (k, c) ∈ N × (a, b) to a subset Fk,c of X satisfying the following properties: (1.6.1) (1.6.2) (1.6.3) (1.6.4)
Fk,c ⊆ w∈T ker(T k − wck I ) for each (k, c) ∈ N × (a, b); {c ∈ (a, b): Fk,c ∩ V = ∅} is open in (a, b) for any open subset V of X and k ∈ N; Fc = ∞ k=1 Fk,c is dense in X for any c ∈ (a, b); For any k1 , . . . , kn ∈ N, there is k ∈ N such that nj=1 Fkj ,c ⊆ Fk,c for each c ∈ (a, b).
Then H{zT : b−1 < |z| < a −1 } is a dense Gδ -set. Note that (1.6.1) is satisfied if Fk,c ⊆ ker(T k − ck I ), which is the case in all following applications of Theorem 1.6. If X is a complex locally convex topological vector space and U is a non-empty open subset of Cm , then we say that f : U → X is holomorphic if f is continuous and for each g ∈ X ∗ , g ◦ f : U → C is holomorphic. Theorem 1.7. Let m ∈ N, X be a complex Fréchet space, T ∈ L(X) and U be a connected open subset of Cm . Assume also that there exist holomorphic maps f : U → X and ϕ : U → C such that ϕ is non-constant, Tf (z) = ϕ(z)f (z) for each z ∈ U and span{f (z): z ∈ U } is dense in X. Denote a = infz∈U |ϕ(z)| and b = supz∈U |ϕ(z)|. Then H{zT : b−1 < |z| < a −1 } is a dense Gδ -set. Corollary 1.8. Let m ∈ N, U be connected non-empty open subset of Cm , H be a function Hilbert space on U and ϕ be a non-constant multiplier for H, a = infz∈U |ϕ(z)| and b = supz∈U |ϕ(z)|. Then H{zT : b−1 < |z| < a −1 } is a dense Gδ -set.
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Corollary 1.9. Let T ∈ L(H(C)) be such that T D = DT and T = cI for any c ∈ C. Then H{zT : z ∈ C } is a dense Gδ -set. Corollary 1.10. Let X be a separable Fréchet space, T ∈ L(X), 0 a < b ∞ and T ∈ L(X). Assume also that for any α, β ∈ R such that a < α < β < b, there exist a dense subset E of X and a map S : E → E such that T Sx = x, α −n T n x → 0 and β n S n x → 0 for each x ∈ E. Then H{zT : b−1 < |z| < a −1 } is a dense Gδ -set. Note that Corollary 1.8 gives an affirmative answer to Question BG, Corollary 1.9 contains Theorem CM as a particular case, while Corollary 1.10 may be considered as an analog of the Kitai criterion [17]. The above results on common hypercyclic vectors for scalar multiples of a given operator may lead to an impression that for 0 < a < b < ∞ and a continuous linear operator T on a Fréchet space, hypercyclicity of aT and bT implies the existence of common hypercyclic vectors for the family {cT : a c b}. This impression is utterly false as follows from the next theorem. For a continuous linear operator T on a topological vector space X, we denote MT = {c > 0: cT is hypercyclic}. Theorem 1.11. I. There exists S ∈ L(2 ) such that MS = {1, 2}. II. There exists T ∈ L(2 ) such that MT is an open interval, but any A ⊂ R+ for which the family {cT : c ∈ A} has common hypercyclic vectors is of zero Lebesgue measure. 2. Yet another general criterion Lemma 2.1. Let A be a set and X, Y and Ω be topological spaces such that Ω is compact. For ∈ Ω let Fω = each a ∈ A let (ω, x) → Fa,ω x be a continuous map from Ω × X to Y . For any ω {Fa,ω : a ∈ A} treated as a family of continuous maps from X to Y . Denote U∗ = ω∈Ω U(Fω ). Then GV =
−1 Fa,ω (V ) is open in X for any open subset V of Y.
(2.1)
ω∈Ω a∈A
Moreover, for any base V of topology of Y , U∗ =
GV .
(2.2)
V ∈V
In particular, U∗ is a Gδ -set if Y is second countable. Proof. Let x ∈ GV . Then for any ω ∈ Ω, there exists a(ω) ∈ A such that Fa(ω),ω x ∈ V . Continuity of the map ω → Fa,ω x implies that for each ω ∈ Ω, Wω = {α ∈ Ω: Fa(ω),α x ∈ V } is an open neighborhood of ω in Ω. Since any Hausdorff compact space is regular, for any ω ∈ Ω, we can pick an open neighborhood Wω of ω in Ω such that, Wω ⊆ Wω . Since {Wω :ω ∈ Ω} is an open covering of the compact space Ω, there are ω1 , . . . , ωn ∈ Ω such that Ω = nj=1 Wω j . Continuity of the map (α, z) → Fa,α z and compactness of Wω imply that for any j ∈ {1, . . . , n},
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there is a neighborhood Uj of x in X such that Fa(ωj ),α z ∈ V for any α ∈ Wω j and z ∈ Uj . Let U = nj=1 Uj . Since Ω = nj=1 Wω j , for any z ∈ U and ω ∈ Ω, there exists j ∈ {1, . . . , n} such that Fa(ωj ),ω z ∈ V . Hence U ⊆ GV . Thus any point of GV is interior and therefore GV is open. The equality (2.2) follows immediately from the definition of U∗ . 2 The main tool in the proof of Theorem 1.2 is the following criterion. It is a simultaneous generalization of results by Chan and Sanders [8, Theorem 2.1] and Grosse-Erdmann [14, Theorem 1]. The latter is exactly the next proposition in the case when Ω is a singleton. Proposition 2.2. Let A be a set and X, Y, Ω be topological spaces such that X is Baire, Y is second countable and Ω is compact. For each a ∈ A, let (ω, x) → Fa,ω x be a continuous map from Ω × X to Y . Let Fω = {Fa,ω : a ∈ A} for ω ∈ Ω and U∗ = ω∈Ω U(Fω ). Then U∗ is a Gδ -subset of X. Moreover, the following conditions are equivalent. (2.2.1) U∗ is dense in X. (2.2.2) For any non-empty open set U in X and any non-empty open set V in Y , there exists x ∈ U such that V ∩ {Fa,ω x: a ∈ A} = ∅ for each ω ∈ Ω. Proof. Let V be a countable base of the topology of Y . By Lemma 2.1, U∗ is a Gδ -set. Assume that (2.2.2) is satisfied. For any V ∈ V and n ∈ N, condition (2.2.2) implies that GV defined by (2.1) is dense in X. By Lemma 2.1, each GV is a dense open subset of X. Since X is Baire, (2.2) implies that U∗ is a dense Gδ -subset of X. Hence (2.2.2) implies (2.2.1). Next, assume that (2.2.1) is satisfied and U , V are non-empty open subsets of X and Y respectively. Since U∗ is dense in X, there is x ∈ U∗ ∩ U . Let ω ∈ Ω. Since x ∈ U(Fω ), there is a ∈ A such that Fa,ω x ∈ V . Hence (2.2.2) is satisfied. 2 Using Proposition 2.2 and the fact that in a Baire topological space the class of dense Gδ sets is closed under countable intersections, we immediately obtain the following corollary. Corollary 2.3. Let A be a set and X, Y, Ω be topological spaces such that X is Baire, Y is second countable and Ω is the union of its compact subsets Ωn for n ∈ N. For each a ∈ A, let (ω, x) → F a,ω x be a continuous map from Ω × X to Y . Let Fω = {Fa,ω : a ∈ A} for ω ∈ Ω and U∗ = ω∈Ω U(Fω ). Then U∗ is a Gδ -subset of X. Moreover, the following conditions are equivalent. (2.3.1) U∗ is dense in X. (2.3.2) For each n ∈ N, any non-empty open set U in X and any non-empty open set V in Y , there exists x ∈ U such that V ∩ {Fa,ω x: a ∈ A} = ∅ for each ω ∈ Ωn . Recall that if X is a topological vector space, A is a set and {fn }n∈Z+ is a sequence of maps from A to X, then we say that fn uniformly converges to 0 on A if for any neighborhood W of 0 in X, there is n ∈ Z+ such that fk (a) ∈ W for any a ∈ A and any k n. Definition 2.4. Let X and Y be topological vector spaces, A be a set and Ω be a topological space. We use the symbol LΩ,A (X, Y )
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to denote the set of maps (ω, a, n, x) → Tω,a,n x from Ω × A × Z+ × X to Y such that Tω,a,n ∈ L(X, Y ) for each (ω, a, n) ∈ Ω × A × Z+ and the map (ω, x) → Tω,a,n x from Ω × X to X is continuous for any (a, n) ∈ A × Z+ . If T ∈ LΩ,A (X, Y ) is fixed, Λ ⊆ Z+ , u ∈ X and U is a subset of Y , we denote M(u, Λ, U ) = {ω ∈ Ω: Tω,a,n u ∈ U for some n ∈ Λ and a ∈ A}.
(2.3)
Proposition 2.5. Let A be a set, X be a Baire topological vector space, Y be a separable metrizable topological vector space, Ω be a compact topological space and T ∈ LΩ,A (X, Y ) be such that (2.5.1) E = {x ∈ X: Tω,a,n x → 0 as n → ∞ uniformly on Ω × A} is dense in X; (2.5.2) for any non-empty open subset U of Y , there exist m ∈ N and compact subsets Ω1 , . . . , Ωm of Ω such that Ω = m j =1 Ωj and for any j ∈ {1, . . . , m}, l ∈ Z+ and a neighborhood W of 0 in X, there are a finite set Λ ⊂ Z+ and u ∈ W for which min Λ l and Ωj ⊆ M(u, Λ, U ). Then U∗ =
ω∈Ω
U(Fω ) is a dense Gδ -subset of X, where Fω = {Tω,a,n : a ∈ A, n ∈ Z+ }.
Proof. Let U0 be a non-empty open subset of X and U be a non-empty open subset of Y . Pick y0 ∈ U and a neighborhood W of zero in Y such that y0 + W + W ⊆ U . Then V = y0 + W is a non-empty open subset of Y and V + W ⊆ U . According to (2.5.2), there exist compact subsets Ω1 , . . . , Ωm of Ω such that Ω = m j =1 Ωj and for any j ∈ {1, . . . , m}, l ∈ Z+ and any neighborhood W1 of 0 in X, there are a finite set Λ ⊂ Z+ and u ∈ W1 such that min Λ l and Ωj ⊆ M(u, Λ, V ).
(2.4)
We shall construct inductively u0 , . . . , um ∈ E ∩ U0 and finite sets Λ1 , . . . , Λm ⊂ Z+ such that for 0 j m, Ωp ⊆ M(uj , Λp , U )
for 1 p j.
(2.5)
By (2.5.1), the linear space E is dense in X. Hence we can pick u0 ∈ U0 ∩ E, which will serve as the basis of induction. Assume now that 1 q m and u0 , . . . , uq−1 ∈ E ∩ U0 and finite subsets Λ1 , . . . , Λq−1 of Z+ satisfying (2.5) with 0 j q − 1 are already constructed. We shall construct uq ∈ E ∩ U0 and a finite subset Λq of Z+ satisfying (2.5) with j = q. Consider the set G = u ∈ X: Ωp ⊆ M(u, Λp , U ) for 1 p q − 1 . Since Ωp are compact and U is open, Lemma 2.1 implies that G is open in X. According to (2.5) with j = q − 1, uq−1 ∈ G. Since uq−1 ∈ E, there exists l ∈ Z+ such that Tω,a,n uq−1 ∈ W
for any n l and any (ω, a) ∈ Ω × A.
(2.6)
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Since uq−1 ∈ G ∩ U0 , and G ∩ U0 is open in X, W1 = (G ∩ U0 ) − uq−1 is a neighborhood of 0 in X. According to (2.4), there exists a finite subset Λq of Z+ such that min Λq l
and G1 = u ∈ W1 : Ωq ⊆ M(u, Λq , V ) = ∅.
By Lemma 2.1, G1 is open in X. Since E is dense in X, we can pick u ∈ G1 ∩ E. Denote uq = uq−1 + u. We shall see that uq and Λq satisfy (2.5) with j = q. Since uq−1 , u ∈ E and E is a linear space, we have uq ∈ E. Since u ∈ W1 = (G ∩ U0 ) − uq−1 , we get uq ∈ G ∩ U0 . In particular, uq ∈ U0 ∩ E and uq ∈ G. By definition of G, Ωp ⊆ M(uq , Λp , U ) for 1 p q − 1. Since u ∈ G1 , for any ω ∈ Ωq , there exist nω ∈ Λq and aω ∈ A such that Tω,aω ,nω u ∈ V . Since nω ∈ Λq and min Λq l, we have nω l. According to (2.6), Tω,aω ,nω uq−1 ∈ W . The equality uq = uq−1 + u and linearity of Tω,aω ,nω imply Tω,aω ,nω uq ∈ V + W ⊆ U . Since ω ∈ Ωq is arbitrary, Ωq ⊆ M(uq , Λq , U ). This completes the proof of (2.5) for j = q and the inductive construction of u0 , . . . , um and Λ1 , . . . , Λm satisfying (2.5). Since Ω is the union of Ωj with 1 j m, (2.5) for j = m implies that um ∈ U0 and Ω = M(um , Z+ , U ). That is, for any ω ∈ Ω there are a ∈ A and n ∈ Z+ such that Tω,a,n um ∈ U . Since U0 and U are arbitrary non-empty open subsets of X and Y respectively, condition (2.2.2) is satisfied. By Proposition 2.2, U∗ is a dense Gδ -subset of X. 2 Since for any δ > 0, any compact interval of the real line is the union of finitely many intervals of length δ, we immediately obtain the following corollary. Corollary 2.6. Let A be a set, X be a Baire topological vector space, Y be a separable metrizable topological vector space, Ω be a compact interval of R and T ∈ LΩ,A (X, Y ) be such that (2.5.1) is satisfied and (2.6.2) for any non-empty open subset U of Y , there exists δ > 0 such that for any compact interval J ⊆ Ω of length δ, l ∈ Z+ and a neighborhood W of 0 in X, there exist a finite set Λ ⊂ Z+ and u ∈ W for which min Λ l and J ⊆ M(u, Λ, U ). Then U∗ =
ω∈Ω
U(Fω ) is a dense Gδ -subset of X, where Fω = {Tω,a,n : a ∈ A, n ∈ Z+ }.
3. Operator groups with the Runge property In this section we prove a statement more general than of Theorem 1.2. Definition 3.1. Let X be a locally convex topological vector space and {Tz }z∈C be an operator group. That is, Tz ∈ L(X) for each z ∈ C, T0 = I and Tz+w = Tz Tw for any z, w ∈ C. We say that {Tz }z∈C has the Runge property if for any continuous seminorm p on X there exists c = c(p) > 0 such that for any finite set S of complex numbers satisfying |z − z | c for z, z ∈ S, z = z , any ε > 0 and {xz }z∈S ∈ X S , there is x ∈ X such that p(Tz x − xz ) < ε for each z ∈ S. Lemma 3.2. For each a ∈ C let Ta ∈ L(H(C)) be the translation operator Tf (z) = f (z − a). Then the group {Ta }a∈C has the Runge property. Proof. Let p be a continuous seminorm on H(C). Then there exists a > 0 such that p(f ) q(f ) for each f ∈ H(C), where q(f ) = a max|z|a |f (z)|. Take any c > 2a. We shall show that c
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satisfies the condition from Definition 3.1. Let ε > 0, S be a finite set of complex numbers such S each z ∈ S consider the disk Dz = that |z − z | c for z, z ∈ S, z = z and {fz }z∈S ∈ H(C) . For {w ∈ C: |z + w| a} and let D = z∈S Dz . Since |z − z | c for z, z ∈ S, z = z , the closed disks Dz are pairwise disjoint. It follows that C \ D is connected. By the classical Runge theorem, any function holomorphic in a neighborhood of the compact set D can be with any prescribed accuracy uniformly on D approximated by a polynomial. Thus there is a polynomial f such that supw∈Dz |f (w) − fz (z + w)| < ε/a for any z ∈ S. Equivalently, sup|w|a |f (w − z) − fz (w)| < δ for any z ∈ S. Using the definitions of Tz and q, we obtain p(Tz f − fz ) q(Tz f − fz ) < ε for each z ∈ S. 2 It is also easy to show that the translation group satisfies the Runge property when acting on the Fréchet space C(C) of continuous functions f : C → C with the topology of uniform convergence on compact sets. Recall that an operator semigroup {Tt } is called strongly continuous if the map (t, x) → Tt x is separately continuous. Theorem 3.3. Let X be a separable Fréchet space and {Tz }z∈C be a strongly continuous operator group on X with the Runge property. Then the family {aTb : a ∈ K , b ∈ C } has a dense Gδ -set of common hypercyclic vectors. According to Lemma 3.2, Theorem 1.2 is a particular case of Theorem 3.3. The rest of this section is devoted to the proof of Theorem 3.3. We need a couple of technical lemmas. Lemma 3.4. For each δ, C > 0, there is R > 0 such that for any n ∈ N, there exists a finite set S ⊂ C such that |z| ∈ N and nR + c |z| (n + 1)R − c for any z ∈ S, |z − z | c for any z | < δ/|z|. z, z ∈ S, z = z and for each w ∈ T, there exists z ∈ S such that |w − |z| Proof. Without loss of generality, we may assume that 0 < δ < 1. Pick m ∈ N such that 2m c and h ∈ N such that h (40 · m)/δ. We shall show that R = hm satisfies the desired condition. Pick n ∈ N and consider k = k(n) ∈ N defined by the formula k = [ π(n+1)m 2δn ] + 1, where [t] is the integer part of t ∈ R. For 1 j k let nj = nR + 2j m. Clearly nj are natural numbers and n1 = nR + 2m nR + c. On the other hand, nk = nR + 2mk (n + 1)R − 2m. Indeed, the last inequality is equivalent to 2(k + 1) h, which is an easy consequence of the two inequalities + 2 πm h > (40 · m)/δ and k + 1 π(n+1)m 2δn δ + 2. Thus, nR + c n1 nj nk (n + 1)R − 2m (n + 1)R − c
for 1 j k.
Now we can define a finite set S of complex numbers in the following way: πi(lk + j ) S = {zj,l : 1 j k, 0 l 2nh − 1}, where zj,l = nj exp nhk
(3.1)
(3.2)
and exp(z) stands for ez . Clearly for each zj,l ∈ S, we have |zj,l | = nj ∈ N. Moreover, according to (3.1), nR + c |z| (n + 1)R − c for any z ∈ S. Next, let z, z ∈ S and z = z . Then z = zj,l and z = zp,q for 1 j, p k, 0 l, q 2nh − 1 and (j, l) = (p, q). If j = p, then |z − z | ||z| − |z || = |nj − np | = 2m|j − p| 2m c. If j = p, then l = q and
πil πi πiq
= 2nj sin π . n exp |z − z | = nj exp − exp − 1 j
nh nh
nh 2nh
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The inequality sin x |z − z |
2x π for 0 4πnj 2nj 2nR 2πnh = nh > nh
x π/2, the inequality nj > nR and the equality R = hm
= 2m c. Thus |z − z | c for any z, z ∈ S, z = z . Finally, imply consider the set Σ = {z/|z|: z ∈ S}. Clearly 1 j k, πij πi(lk + j ) = exp : : 1 j 2nhk Σ = exp nhk 0 l 2nh − 1 nhk = z ∈ C: z2nhk = 1 . It immediately follows that
π πi
π πm
= 2 sin = . sup min |w − z| = 1 − exp
2nhk 4nhk 2nhk 2nRk w∈T z∈Σ −1 −1 Since k > π(n+1)m 2δn , we get supw∈T minz∈Σ |w − z| < δ(n + 1) R . That is, for any w ∈ T, z δ z there exists z ∈ S such that |w − |z| | < R(n+1) . Since |z| < R(n + 1), we obtain |w − |z| | < δ/|z|, which completes the proof. 2
Lemma 3.5. Let X be a locally convex topological vector space and {Tz }z∈C be an operator group on X such that the map (u, h) → Th u from X × C to X is continuous. Let also x ∈ X and p be a continuous seminorm on X. Then there exist a continuous seminorm q on X and δ > 0 such that p q and for any a ∈ R, w ∈ T, n ∈ N and y ∈ X satisfying q(x − ean Twn y) < 1, we have p(x − ebn Tzn y) < 1 whenever b ∈ R and z ∈ T are such that |a − b| < δ/n and |w − z| < δ/n. Proof. Since the map (u, h) → Th u from X × C to X is continuous, there is θ > 0 and a continuous seminorm q on X such that p(x − Th x) 1/4 and p(Th u) q(u)/4 for any u ∈ X whenever |h| θ . In particular, p(u) q(u)/4 q(u) for each u ∈ X. Pick r ∈ (0, θ ) and assume that a, b ∈ R, w, z ∈ T, n ∈ N and y ∈ X are such that q(x − ean Twn y) < 1, |a − b| < r/n and |w − z| < r/n. Then p(ean Twn y) q(ean Twn y) q(x) + 1. Since |a − b| < r/n, we have |e(b−a)n − 1| < er − 1. Hence
p ebn Twn y − ean Twn y = e(b−a)n − 1 p ean Twn y er − 1 q(x) + 1 .
(3.3)
Since |nw − nz| < r < θ and p(Th u) q(u)/4 for any u ∈ X whenever |h| θ , we have
p T(z−w)n x − ean Tzn y = p T(z−w)n x − ean Twn y q x − ean Twn y /4 < 1/4. Since |(z − w)n| < r < θ , we get p(x − T(z−w)n x) 1/4. Using this inequality together with the last display and the triangle inequality, we obtain p(x − ean Tzn y) 1/2. The latter together with (3.3) and the triangle inequality gives p(x − ebn Tzn y) < (er − 1)(q(x) + 1) + 1/2. Hence any δ ∈ (0, θ ) satisfying (eδ − 1)(q(x) + 1) < 1/2, satisfies also the desired condition. 2 3.1. Proof of Theorem 3.3 By Theorems LM and CMP, H(bTa ) = H(b Ta ) if |b| = |b | and a/a ∈ R+ . Hence the set of common hypercyclic vectors of the family {aTb : a ∈ K , b ∈ C } coincides with the set G of common hypercyclic vectors for the family {eb Ta : (a, b) ∈ T × R}. Thus it remains to show that
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G is a dense Gδ -subset of X. Fix d > 0. According to Corollary 2.3, it suffices to demonstrate that for any non-empty open subsets U and V of X, there is y ∈ U such that for any a ∈ T and b ∈ [−d, d] there is n ∈ N for which ebn Tan y ∈ V .
(3.4)
Pick a continuous seminorm p on X and u, x ∈ X such that {y ∈ X: p(u − y) < 1} ⊆ U and {y ∈ X: p(x − y) < 1} ⊆ V . By the uniform boundedness principle [21], strong continuity of {Tz }z∈C implies that the map (z, v) → Tz v from C × X to X is continuous. By Lemma 3.5, there is a continuous seminorm q on X and δ > 0 such that p(v) q(v) for any v ∈ X and
for any a, b ∈ R, w, z ∈ T, n ∈ N and y ∈ X satisfying q x − ean Twn y < 1,
|a − b| < δ/n and |w − z| < δ/n, we have p g − ebn Tzn y < 1.
(3.5)
Since {Tz }z∈C has the Runge property, there is c > 0 such that for any finite set S ⊂ C with |z − z | c for z, z ∈ S, z = z , any ε > 0 and any {xz }z∈S ∈ X S , there exists y ∈ X such that q(Tz y − xz ) < ε for any z ∈ S.
(3.6)
Let R > 0 be the number provided by Lemma 3.4 for the just chosen δ and c. By Lemma 3.4, for each n ∈ N there is a finite set Sn ⊂ C such that |z| ∈ N and nR + c |z| (n + 1)R − c for c for any z, z ∈ Sn , z = z and for each w ∈ T, there is z ∈ Sn such that any z ∈ Sn , |z − z | z δ −1 = ∞, we can pick d , . . . , d ∈ [−d, d] for which |w − |z| | < |z| . Since ∞ 1 k n=1 n k δR −1 δR −1 dn − , dn + . [−d, d] ⊆ n+1 n+1
(3.7)
n=1
Let S = kn=1 Sn and Λ = S ∪ {0}. It is straightforward to see that Λ is a finite set, |z| ∈ Z+ for any z ∈ Λ and |z − u| c for any z, u ∈ Λ, z = u. Let N = max{|z|: z ∈ Λ} and ε = d −N . By (3.6), there is y ∈ X such that q(u − y) < ε and q(Tz y − e−cn |z| x) < ε for each z ∈ S. Then p(u − y) q(u − y) < ε < 1 and therefore f ∈ U . By definition of ε, q(x − ecn |z| Tz y) < 1 for each z ∈ S. Let now a ∈ T and b ∈ [−d, d]. According to (3.7), there is n ∈ {1, . . . , k} such −1 that |b − dn | < δR n+1 . By the mentioned property of the set Sn , we can choose z ∈ Sn such that z δ δ |a − |z| | < |z| . Since |z| < R(n + 1), we have |b − dn | < |z| . By (3.5), p(x − eb|z| Ta|z| y) < 1. b|z| Hence e Ta|z| f ∈ V , which completes the proof of (3.4) and that of Theorem 3.3. 4. Scalar multiples of a fixed operator In this section we shall prove Theorems 1.6 and 1.7 as well as Corollaries 1.8, 1.9 and 1.10. Recall that a subset A of a vector space is called balanced if zx ∈ A for any x ∈ A and z ∈ K satisfying |z| 1. It is well known that any topological vector space has a base of open neighborhoods of zero consisting of balanced sets. For two subsets A, B of a vector space X we say that A absorbs B if there exists c > 0 such that B ⊆ zA for any z ∈ K satisfying |z| c. Obviously, if A is balanced, then A absorbs B if and only if there is c > 0 for which B ⊆ cA.
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Lemma 4.1. Let X be a topological vector space and U be a non-empty open subset of X. Then there exist a non-empty open subset V of X and a balanced neighborhood W of zero in X such that V + W ⊆ U and W absorbs V . Proof. Pick u ∈ U and a balanced neighborhood W0 of zero in X such that u + W0 + W0 + W0 ⊆ U . Denote V = u + W0 and W = W0 + W0 . Clearly V is a non-empty open subset of X, W is a balanced neighborhood of 0 in X and V + W = u + W0 + W0 + W0 ⊆ U . Since W0 is a neighborhood of 0 in X, we can pick c 1 such that u ∈ cW0 . Since W0 is balanced and c 1, W0 ⊆ cW0 and therefore V = u + W0 ⊆ cW0 + W0 ⊆ c(W0 + W0 ) = cW . Since W is balanced, W absorbs V . 2 To any continuous linear operator T on a complex topological vector space X there corresponds T ∈ LR,T (X, X) defined by the formula Tt,w,n x = wetn T n x. We will use the symbol M(T , u, Λ, U ) to denote the sets defined in (2.3) for T. In other words, for Λ ⊆ Z+ , t ∈ R, u ∈ X and a subset U of X, we write M(T , u, Λ, U ) = t ∈ R: wetn T n u ∈ U for some n ∈ Λ and w ∈ T . Lemma 4.2. Let X be a complex topological vector space, W be a balanced neighborhood of 0 in X, c > 0, k ∈ N and δ ∈ (0, (2ck)−1 ]. Then for any m ∈ N, any α ∈ [−c, c], any w ∈ T, any neighborhood W0 of zero in X and any x ∈ cW such that T k x = we−αk x, there exist u ∈ W0 and a finite set Λ ⊂ N such that min Λ m and [α + δ, α + 2δ] ⊆ M(T , u, Λ, x + W ). Proof. Let α ∈ [−c, c], w ∈ T and any x ∈ cW be such that T k x = we−αk x. For each p ∈ N consider up = e−2δkp x. Since T k x = we−αk x, we see that for 0 j p, T
(p+j )k
up = e
−α(p+j )k −2δkp
e
w
p+j
2pδ w p+j x. x = exp −(p + j )k α + p+j
That is, wj e(p+j )kθj T (p+j )k up = x
for 1 j p, where θj = α +
2δp and wj = w −p−j ∈ T. p+j (4.1)
Let now 0 l p − 1 and θ ∈ [θl+1 , θl ]. Since e(p+l)kθ T (p+l)k up = e(p+l)k(θ−θl ) e(p+l)kθl T (p+l)k up , using (4.1) with j = l, we obtain
wl e(p+l)kθ T (p+l)k up = e(p+l)k(θ−θl ) x = x + e(p+l)k(θ−θl ) − 1 x. Taking into account that −(θl − θl+1 ) θ − θl 0 and using the inequality 0 1 − e−t t for t 0, we see that |e(p+l)k(θ−θl ) − 1| (p + l)k(θl − θl+1 ). This inequality, the inclusion x ∈ cW the last display and the fact that W is balanced imply that
wl e(p+l)kθ T (p+l)k up ∈ x + c e(p+l)k(θ−θl ) − 1 W ⊆ x + c(p + l)k(θl − θl+1 )W.
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2pδ 2δ Since θl − θl+1 = (p+l)(p+l+1) p+l and δ (2ck)−1 , we have c(p + l)k(θl − θl+1 ) 1. Thus (p+l)kθ T (p+l)k up ∈ x + W whenever θ ∈ [θl+1 , θl ]. It folaccording to the above display, wl e lows that [θl+1 , θl ] ⊆ M(T , up , Λp , x + W ) for 0 l p − 1, where Λp = {(p + j )k: 0 j p}. Since the sequence {θj }0j p decreases, θ0 = α + 2δ and θp = α + δ, we see that p−1 [α + δ, α + 2δ] = l=0 [θl+1 , θl ]. Since [θl+1 , θl ] ⊆ M(T , up , Λp , x + W ) for 0 l p − 1, we have [α + δ, α + 2δ] ⊆ M(T , up , Λp , x + W ) for any p ∈ N. Clearly min Λp = pk → ∞ and up = e−2δkp x → 0 in X as p → ∞. Thus we can pick p ∈ N such that min Λp > m and up ∈ W0 . Then u = up and Λ = Λp for such a p satisfy all desired conditions. 2
We shall prove a statement more general than Theorem 1.6. Theorem 4.3. Let X be a separable complex F -space, T ∈ L(X) and 0 a < b ∞. Assume also that the following condition is satisfied. (4.3.1) For any compact interval J ⊂ (a, b) and any non-empty open subset V of X, there exist k = k(J, V ) ∈ N and a dense subset C = C(J, V ) of J such that V∩
ker T k − wck I = ∅
for each c ∈ C.
w∈T
Then H{zT : b−1 < |z| < a −1 } is a dense Gδ -set. Proof. Let α0 , α, β ∈ R be such that b−1 < eα0 < eα < eβ < a −1 . For each ω ∈ [α, β] consider the family Fω = {zeωn T n : z ∈ T, n ∈ Z+ }. We shall apply Corollary 2.6 with A = T, Tω,a,n = aeωn T n and Ω = [α, β]. First, pick a compact interval J ⊂ (a, e−β ). For each non-empty open subset V0 of X, we can use (4.3.1) to find x ∈ V0 , k ∈ N, r ∈ J and w ∈ T such that T k x = wr k x. The latter equality implies that x is a sum of finitely many eigenvectors of T corresponding to eigenvalues λj with |λj | = r < e−β . Hence eβn T n x → 0 as n → ∞. Since V0 is an arbitrary non-empty open subset of X and x ∈ V0 , we see that the space E = {x ∈ X: eβn T n x → 0} is dense in X. It immediately follows that for any x ∈ E, zeωn T n x → 0 as n → ∞ uniformly for (z, ω) ∈ T × [α, β]. Hence (2.5.1) is satisfied. Let now U be a non-empty open subset of X. By Lemma 4.1, there exist a balanced neighborhood W of zero in X and a non-empty open subset V of X such that V + W ⊆ U and W absorbs V . Since W absorbs V , there is c > 0 such that V ⊆ cW . According to (4.3.2), we can pick k ∈ N and a dense subset R of [α0 , β] for which V∩
ker T k − we−rk I = ∅ for any r ∈ R.
(4.2)
w∈T
Let δ0 = min{(2ck)−1 , α − α0 } and r ∈ R. By (4.2), we can pick wr ∈ T and xr ∈ V ⊆ cW such that T k xr = wr −rk xr . By Lemma 4.2, for any neighborhood W0 of zero in X and any m ∈ N, there exist u ∈ W0 and a finite set Λ ⊂ N satisfying min Λ m and [r + δ0 , r + 2δ0 ] ⊆ M(T , u, Λ, xr + W ). Pick δ ∈ (0, δ0 ). Since R is dense in [α0 , β] and δ0 α − α0 , it is easy to see that each compact interval J ⊆ [α, β] of length at most δ is contained in [r + δ0 , r + 2δ0 ] for
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some r ∈ R. Thus for each compact interval J ⊆ [α, β] of length at most δ, any neighborhood W0 of zero in X and any m ∈ N, there exist r ∈ R, u ∈ W0 and a finite set Λ such that min Λ m and J ⊆ M(T , u, Λ, xr + W ). The latter inclusion means that for each t ∈ J , there exist wt ∈ T and nt ∈ Λ such that wt T nt u ∈ xr + W . Since xr ∈ V and V + W ⊆ U , we get wt T nt u ∈ U . That is, for any compact interval J ⊆ [α, β] of length at most δ, any neighborhood W0 of zero in X and any m ∈ N, there exist u ∈ W0 and a finite set Λ such that min Λ m and J ⊆ M(T , u, Λ, U ). Thus (2.6.2) is also satisfied. By Corollary 2.6, Hα,β =
U(Fω ) is a dense Gδ -subset of X whenever b−1 < eα < eβ < a −1 .
ω∈[α,β]
By Theorem LM, U(Fω ) = H(zeω T ) for any ω ∈ R and z ∈ T. Hence Hα,β = H{zT : eα |z| eβ }. From the above display it now follows that H{zT : b−1 < |z| < a −1 } is a dense Gδ subset of X as the intersection of a countable family of dense Gδ -sets. 2 4.1. Proof of Theorem 1.6 We shall prove Theorem 1.6 by means of applying Theorem 4.3. To do this it suffices to demonstrate that (4.3.1) is satisfied. Let J ⊂ (a, b) be a compact interval and V be a non-empty open subset of X. For any k ∈ N let Ok = {c ∈ (a, b): Fk,c ∩ V = ∅}. By (1.6.2), Ok are open subsets of (a, b). According to (1.6.3), {Ok : k ∈ N} is an open covering of (a, b). Since J is compact, we can pick k1 , . . . , kn ∈ N such that J ⊆ nj=1 Okj . By (1.6.4), there is k ∈ N for which nj=1 Fkj ,c ⊆ Fk,c for any c ∈ (a, b). Hence Ok ⊇ nj=1 Okj ⊇ J . It follows that for any c ∈ J , thereis x ∈ Fk,c ∩ V . According to (1.6.1), there is w ∈ T for which x ∈ ker(T k − wck I ). Thus V ∩ w∈T ker(T k − wck I ) = ∅ for any c ∈ J . That is, (4.3.1) is satisfied with C = J . It remains to apply Theorem 4.3 to conclude the proof of Theorem 1.6. 4.2. Proof of Theorem 1.7 Recall that a map h from a topological space X to a topological space Y is called open if h(U ) is open in Y for any open subset U of X. Recall also that a subset A of a connected open subset U of Cm is called a set of uniqueness if any holomorphic function ϕ : U → C vanishing on A is identically zero. The following lemma contains few classical results that can be found in virtually any book on complex analysis. Lemma 4.4. Let m ∈ N and U be a connected open subset of Cm . Then any non-empty open subset of U is a set of uniqueness and any non-constant holomorphic map ϕ : U → C is open. Moreover, if m = 1, then any subset of U with at least one limit point in U is a set of uniqueness. We need the following generalization of the last statement of Lemma 4.4 to the case m > 1. Although it is probably known, the author was unable to locate a reference. Lemma 4.5. Let m ∈ N, U be a connected open subset of Cm , ϕ : U → C be a non-constant holomorphic map and A be a subset of C with at least one limit point in ϕ(U ). Then ϕ −1 (A) is a set of uniqueness. In particular, if a = infz∈U |ϕ(z)|, b = supz∈U |ϕ(z)|, c ∈ (a, b) and G is a dense subset of T, then ϕ −1 (cG) is a set of uniqueness.
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Proof. Assume the contrary. Then there exists a non-zero holomorphic function f : U → C such that f |ϕ −1 (A) = 0. Let a ∈ ϕ(U ) be a limit point of A and w ∈ U be such that ϕ(w) = a. Pick a convex open subset V of Cm such that w ∈ V ⊆ U . For any complex one-dimensional linear subspace L of Cm , VL = (w + L) ∩ V can be treated as a convex open subset of C. If ϕL = ϕ|VL is non-constant, then by Lemma 4.4, ϕL : VL → C is open. Since a = ϕ(w) is a limit point of A, it follows that w is a limit point of ϕL−1 (A). Using the one-dimensional uniqueness theorem, we see that ϕL−1 (A) is a set of uniqueness in VL . Since f vanishes on ϕ −1 (A) ⊇ ϕL−1 (A), f |VL = 0. On the other hand, if ϕL is constant, then (ϕ − a)|VL = 0. Since L is arbitrary, we have f (ϕ − a)|V = 0. Since V , being a non-empty open subset of U , is a set of uniqueness, we have f · (ϕ − a) = 0. Since f ≡ 0, there is a non-empty open subset W of U such that f (z) = 0 for any z ∈ W . The equality f · (ϕ − a) = 0 implies that ϕ(z) = a for any z ∈ W . Since W is a set of uniqueness, ϕ ≡ a. We have arrived to a contradiction. Thus ϕ −1 (A) is a set of uniqueness. Assume now that a = infz∈U |ϕ(z)|, b = supz∈U |ϕ(z)|, c ∈ (a, b) and G is a dense subset of T. Since U is connected cT ∩ ϕ(U ) = ∅. Since ϕ is open, the set ϕ(U ) is open in C. Thus density of G in T implies that cG ∩ ϕ(U ) is dense in cT ∩ ϕ(U ), which is an open subset of cT. Hence cG has plenty of limit points in ϕ(U ) and it remains to apply the first part of the lemma. 2 We shall prove Theorem 1.7 by means of applying Theorem 1.6. First, note that density of span{f (z): z ∈ U } implies separability of X. Let Fk,c = span f (z): z ∈ U, ϕ(z)k = ck
for k ∈ N and c ∈ (a, b).
In order to apply Theorem 1.6 it suffices to verify that the map (k, c) → Fk,c satisfies conditions (1.6.1)–(1.6.4). First, from the equality Tf (z) = ϕ(z)f (z) it follows that T k x = ck x for any x ∈ Fk,c . Hence (1.6.1) is satisfied. Clearly Fk,c ⊆ Fm,c whenever k is a divisor of m. Hence for any c ∈ (a, b) and any k1 , . . . , kn ∈ N, Fkj ,c ⊆ Fk,c for 1 j n, where k = k1 · · · · · kn . Thus (1.6.4) is satisfied. It is easy to see that Fc =
∞
Fk,c = span f (z): ϕ(z) ∈ cG ,
where G = z ∈ T: zk = 1 for some k ∈ N .
k=1
In order to prove (1.6.3), we have to show that Fc is dense in X. Assume the contrary. Since Fc is a vector space and X is locally convex, we can pick g ∈ X ∗ such that g = 0 and g(x) = 0 for each x ∈ Fc . In particular, g(f (z)) = 0 whenever ϕ(z) ∈ cG. By Lemma 4.5, ϕ −1 (cG) is a set of uniqueness. Since the holomorphic function g ◦ f vanishes on ϕ −1 (cG), it is identically zero. Hence g(f (z)) = 0 for any z ∈ U , which contradicts the density of span{f (z): z ∈ U } in X. This contradiction completes the proof of (1.6.3). It remains to verify (1.6.2). Let k ∈ N, V be a non-empty open subset of X and G = {c ∈ (a, b): Fk,c ∩ V = ∅}. We have to show that G is open in R. Let c ∈ G. there exist z1 , . . . , zn ∈ U and λ1 , . . . , λn ∈ C such that ϕ(zj )k = ck Then n for 1 j n and j =1 λj f (zj ) ∈ V . Since f is continuous, we can pick ε > 0 such that zj + εDm ⊂ U for 1 j n and nj=1 λj f (wj ) ∈ V for any choice of wj ∈ zj + εDm . By Lemma 4.4, ϕ is open and therefore there exists δ > 0 such that ϕ(zj ) + cδD ⊆ ϕ(zj + εDm ) for 1 j n. In particular, since |ϕ(zj )| = c, we see that (1 − δ, 1 + δ)ϕ(zj ) ⊂ ϕ(zj + εD) for 1 that wj ∈ zj + εDm j n. Hence for each s ∈ (1 − δ, 1 + δ), we can pick w1 , . . . , wn ∈ U such k k k k and ϕ(wj ) = sϕ(zj ) for 1 j n. Then ϕ(wj ) = s ϕ(zj ) = (cs) and nj=1 λj f (wj ) ∈ V since wj ∈ zj + εD. Hence cs ∈ G for each s ∈ (1 − δ, 1 + δ) and therefore c is an interior point
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of G. Since c is an arbitrary point of G, G is open. This completes the proof of (1.6.2). It remains to apply Theorem 1.6 to conclude the proof of Theorem 1.7. 4.3. Proof of Corollary 1.8 Note that H∗ with the usual norm is a Banach space. Consider the map f : U → H∗ defined by the formula f (z)(x) = x(z). It is straightforward to verify that f is holomorphic, Mϕ∗ f (z) = ϕ(z)f (z) for each z ∈ U and span{f (z): z ∈ U } is dense in H∗ . The latter is a consequence of the fact that evaluation functionals separate points of H. Using Theorem 1.7, we immediately obtain that G0 = H{zMϕ∗ : b−1 < |z| < a −1 } is a dense Gδ -subset of H∗ . Now consider the map R : H → H∗ , Rx(y) = y, x, where ·,· is the scalar product of the Hilbert space H. According to the Riesz theorem, R is an R-linear isometric isomorphism (it happens to be complex conjugate linear). It is also easy to see that R −1 S ∗ R = S for any S ∈ L(H), where S ∗ is the dual of S and S is the Hilbert space adjoint of S. Hence G = R −1 (G0 ), where G = H{zMϕ : b−1 < |z| < a −1 }. Since R is a homeomorphism from H onto H∗ , G is a dense Gδ -subset of H. 4.4. Proof of Corollary 1.9 Consider the map f : C → H(C) defined by the formula f (w)(z) = ewz . It is easy to see that f is holomorphic, span{f (z): z ∈ C} is dense in H(C) and for each w ∈ C, ker(D − wI ) = span{f (w)}. In particular, Df (w) = wf (w) and using the equality T D = DT , we get wTf (w) = DTf (w) for each w ∈ C. Hence Tf (w) ∈ ker(D − wI ) = span{f (w)} for any w ∈ C. Thus there exists a unique function ϕ : C → C such that Tf (w) = ϕ(w)f (w) for each w ∈ C. Using the fact that f is holomorphic and each f (w) does not take value 0, one can easily verify that ϕ is holomorphic. Moreover, since T is not a scalar multiple of identity, ϕ is non-constant. By the Picard theorem, any non-constant entire function takes all complex values except for maybe one. Hence infw∈C |ϕ(w)| = 0 and supw∈C |ϕ(w)| = ∞. By Theorem 1.7, H{zT : z ∈ C } is a dense Gδ -subset of H(C). 4.5. Proof of Corollary 1.10 First, we consider the case K = C. Let a < α < β < b. By the assumptions, there is a dense subset E of X and a map S : E → E such that T Sx = x, α −n T n x → 0 and β n S n x → 0 for each x ∈ E. Let U = {w ∈ C: α < |w| < β}. Since X is locally convex and complete, the rela∞ −n T n x → 0 and β n S n x → 0 ensure that for each w ∈ U , the series −n n tions α n=1 w T x and ∞ n n n=1 w S x converge in X for any x ∈ E. Thus we can define ∞
−n n ux,w = x + w T x + w n S n x for w ∈ U and x ∈ E. n=1
Using the relations T Sx = x for x ∈ E and T ∈ L(X), one can easily verify that T ux,w = wux,w for each x ∈ E and w ∈ U . Now we consider Fk,c = span ux,w : x ∈ E, w k = ck
for k ∈ N and c ∈ (α, β).
We shall show that Fk,c for k ∈ N and c ∈ (α, β) satisfy conditions (1.6.1)–(1.6.4). First, the equality T ux,w = wux,w implies that T k y = ck y for any y ∈ Fk,c . Hence (1.6.1) is satis-
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fied. Clearly Fk,c ⊆ Fm,c whenever k is a divisor of m. Hence for any c ∈ (α, β) and any k1 , . . . , kn ∈ N, Fkj ,c ⊆ Fk,c for 1 j n, where k = k1 · · · · · kn . Thus (1.6.4) is satisfied. It is easy to see that Fc =
∞
Fk,c = span{ux,w : x ∈ E, w ∈ cG},
where G = z ∈ T: zk = 1 for some k ∈ N .
k=1
In order to prove (1.6.3), we have to show that Fc is dense in X. Assume the contrary. Since Fc is a vector space and X is locally convex, we can pick g ∈ X ∗ such that g = 0 and g(y) = 0 for each y ∈ Fc . Hence for any x ∈ E and w ∈ cG, we have fx (w) = 0, where fx (w) = g(ux,w ). It is easy to verify that for any x ∈ E, the function fx : U → C is holomorphic. Since fx vanishes on cG, the uniqueness theorem implies that each fx is identically zero. On the other hand, the 0th Laurent coefficient of fx is g(x). Hence g(x) = 0 for any x ∈ E. Since E is dense in X, we get g = 0. This contradiction completes the proof of (1.6.3). It remains to verify (1.6.2). Let k ∈ N, V be a non-empty open subset of X and G = {c ∈ (α, β): Fk,c ∩ V = ∅}. We have to show that G is open in R. Let c ∈ G. Then there exist x1 , . . . , xn ∈ E and w1 , . . . , wn , λ1 , . . . , λn ∈ C such that wjk = ck for 1 j n and nj=1 λj uxj ,wj ∈ V . Since for any fixed x ∈ E, the map w → ux,w is continuous, there is δ > 0 such that ys ∈ V if |c − s| < δ, where ys = nj=1 λj uxj ,swj /c . On the other hand, ys ∈ Ek,s for each s and therefore (c − δ, c + δ) ∩ (α, β) ⊆ G. Hence c is an interior point of G. Since c is an arbitrary point of G, G is open. This completes the proof of (1.6.2). By Theorem 1.7, H{zT : β −1 < |z| < α −1 } is a dense Gδ -set whenever a < α < β < b. Hence the set of common hypercyclic vectors of the family {zT : b−1 < |z| < a −1 } is a dense Gδ -subset of X as a countable intersection of dense Gδ -sets. The proof of Corollary 1.10 in the case K = C is complete. Assume now that K = R. Let XC = X ⊕ iX and TC (u + iv) = T u + iT v be complexifications of X and T respectively. It is straightforward to see that TC satisfies the same conditions with EC = E + iE and SC (u + iv) = Su + iSv taken as E and S. Corollary 1.10 in the complex case implies that H0 = H{zTC : z ∈ C, b−1 < |z| < a −1 } is a dense Gδ -subset of XC . Clearly H = H{zT : z ∈ R, b−1 < |z| < a −1 } contains the projection of H0 onto X along iX and therefore in dense in X. The fact that H is a Gδ -subset of X follows from Corollary 2.3. 5. Counterexamples on hypercyclic scalar multiples We find operators, whose existence is assured by Theorem 1.11 in the class of bilateral weighted shifts on 2 (Z). Recall that if w = {wn }n∈Z is a bounded sequence of non-zero scalars, then the unique Tw ∈ L(2 (Z)) such that Tw en = wn en−1 for n ∈ Z, where {en }n∈Z is the canonical orthonormal basis of the Hilbert space 2 (Z), is called the bilateral weighted shift with the weight sequence w. Hypercyclicity of bilateral weighted shifts was characterized by Salas [20], whose necessary and sufficient condition is presented in a more convenient shape in [22]. Theorem S. Let Tw be a bilateral weighted shift on 2 (Z). Then Tw is hypercyclic if and only if for any k ∈ Z+ ,
(k − n + 1, k) + w (k + 1, k + n)−1 = 0, lim w
n→∞
where w (a, b) =
b j =a
|wj | for a, b ∈ Z, a b.
(5.1)
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It is well known and easy to see that a bilateral weighted shift Tw is invertible if and only if infn∈Z |wn | > 0. In this case condition (5.1) can be rewritten in the following simpler form. Theorem S . Let Tw be an invertible bilateral weighted shift on 2 (Z). Then Tw is hypercyclic if and only if
lim w (−n, 0) + w (0, n)−1 = 0.
(5.2)
n→∞
5.1. Proof of Theorem 1.11, Part II First, we prove few elementary lemmas. The following one generalizes the fact that the set of hypercyclic vectors of a hypercyclic operator is dense. Lemma 5.1. Let X be a topological vector space and Abe a family of pairwise commuting continuous linear operators on X. Then the set H(A) = T ∈A H(T ) is either empty or dense in X. Proof. Let x ∈ H(A) and S ∈ A. We have to show that H(A) is dense in X. Since x is a hypercyclic vector for S, O(S, x) = {S n x: n ∈ Z+ } is dense in X and therefore S has dense range. Take any T ∈ A. Since T S = ST , O(T , S m x) = S m (O(T , x)) for each m ∈ Z+ . Since x ∈ H(T ) and S m has dense range, O(T , S m x) is dense in X. Hence S m x ∈ H(T ) for any T ∈ A and m ∈ Z+ . That is, O(S, x) ⊆ H(A). Since O(S, x) is dense in X, so is H(A). 2 Lemma 5.2. Let X be a locally convex topological vector space, T ∈ L(X), A ⊆ (0, ∞) and x ∈ H{cT : c ∈ Λ}. Assume also that there exists a non-empty open subset U of X such that
n−1 < ∞,
where QU = n ∈ N: a n T n x ∈ U for some a ∈ A .
(5.3)
n∈QU
Then A has zero Lebesgue measure. Proof. Clearly we can assume that A = ∅ and therefore Λ = ∅, where Λ= ln(A) = {ln a: a ∈ A}. Since X is Hausdorff and locally convex, we can find a continuous seminorm p on X such that V = U ∩ {u ∈ X: 1 < p(u) < e} is non-empty. It suffices to show that Λ has zero Lebesgue measure. Let α ∈ Λ and m ∈ N. Since x is hypercyclic for eα T and V is open, we can find n m such that eαn T n ∈ V ⊆ U . Then n ∈ QU and p(eαn T n x) ∈ (1, e). Hence α ∈ (αn , βn ),
where αn =
− ln(p(T n x)) 1 − ln(p(T n x)) and βn = . n n
Since α ∈ Λ is arbitrary, we obtain Λ⊆
n∈QU , nm
(αn , βn )
for any m ∈ N.
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On the other hand, (αn , βn ) is an interval of length n−1 . Then (5.3) and the last display imply that Λ can be covered by intervals with arbitrarily small sum of lengths. That is, Λ has zero Lebesgue measure. 2 For k ∈ N, we denote mk = 2
3k 2
,
Ik−
7 9 + Ik = n ∈ N: mk < n mk and = n ∈ N: mk n < mk , 8 8 7 9 Ik = Ik− ∪ Ik+ ∪ {mk } = n ∈ N: mk n mk . (5.4) 8 8
Consider the sequence w = {wn }n∈Z defined by the formula ⎧ ⎨ 28 wn = 2−8 ⎩ 1
if n ∈ Ik− ∪ −Ik+ , k ∈ N, if n ∈ Ik+ ∪ −Ik− , k ∈ N, otherwise.
(5.5)
Clearly w is a sequence of positive numbers and 0 < 2−8 = infn∈Z wn < supn∈Z wn = 28 < ∞. Hence Tw is an invertible bilateral weighted shift. In order to prove Part II of Theorem 1.11 it is enough to verify the following statement. Example 5.3. Let w be the weight sequence defined by (5.5) and T = Tw be the corresponding bilateral weighted shift on 2 (Z). Then MT = (1/2, 2) and any Λ ⊆ (1/2, 2) has Lebesgue measure 0 if the family {aT : a ∈ Λ} has a common hypercyclic vector. Proof. Using the definition (5.5) of the sequence w, it is easy to verify that for any n ∈ N, ⎧ 8n−7m +8 k ⎨2 β(n) = 29mk −8n ⎩ 1
if n ∈ Ik− , k ∈ N, if n ∈ Ik+ , k ∈ N, otherwise,
where β(n) =
n
wj .
(5.6)
j =0
Moreover, wn−1 = w−n for any n ∈ Z. Using this fact and the equality w0 = 1, we get ⎧ ⎨ β(n)β(j − 1)−1 w (j, n) = β(−1 − n)β(−j )−1 ⎩ β(n)β(−j )−1
if j 1, if n −1, if j 0 and n 0
for any j, n ∈ Z, j n,
(5.7)
where the numbers w (j, n) are defined in (5.1). In particular, w (0, n) = β(n) and w (−n, 0) = β(n)−1 for each n ∈ N. This observation together with Theorem S and the fact that aT = Taw for a = 0 imply that for a > 0,
aT is hypercyclic if and only if lim β(n)−1 a n + a −n = 0.
(5.8)
n→∞
By (5.6), 1 β(n) 2n for n ∈ N, which together with (5.8) implies that MT ⊆ (1/2, 2). On the other hand, by (5.6), β(mk ) = 2mk for each k ∈ N. Hence β(mk )−1 (a mk + a −mk ) → 0 as k → ∞ for any a ∈ (1/2, 2). According to (5.8), aT is hypercyclic if 1/2 < a < 2. Hence MT = (1/2, 2).
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Let now Λ be a non-empty subset of (1/2, 2) such that the family {aT : a ∈ Λ} has common hypercyclic vectors. We have to demonstrate that Λ has zero Lebesgue measure. Pick ε > 0 ε < 2−8 . By Lemma 5.1, there is a common hypercyclic vector x of the family such that 1−ε {aT : a ∈ Λ} such that x − e−1 < ε. Let ∞ n n Ik . Q = n ∈ N: a T x − e0 < ε for some a ∈ Λ and J =
k=1
First, we show that Q ⊆ J . Let n ∈ Q. Then there is a ∈ Λ such that a n T n x − e0 < ε. Hence
and a n T n x, e−n−1 < ε.
n n
a T x, e0 > 1 − ε
Using (5.7), we get a n T n x, e0 = a n β(n)xn and a n T n x, e−n−1 = a n β(n)−1 x−1 . Then from the last display it follows that and a n β(n)−1 wn |x−1 | < ε.
a n β(n)|xn | > 1 − ε
Since x − e−1 < ε, |x−1 | > 1 − ε and |xn | < ε. Then according to the last display, β(n) >
1−ε 1−ε max a n , a −n > 28 > 1. ε ε
By (5.6), β(j ) = 1 if j ∈ / J . Hence n ∈ J . Since n is an arbitrary element of Q, we get Q ⊆ J . Next, we show that (Q − Q) ∩ N ⊆ J . Indeed, let m, n ∈ Q be such that m > n. Since m, n ∈ Q, we can pick a, b ∈ Λ such that a n T n x − e0 < ε and bm T m x − e0 < ε. In particular,
n n
a T x, e0 > 1 − ε,
n n
a T x, em−n < ε and
m m
b T x, e0 > 1 − ε,
m m
b T x, en−m < ε.
Using (5.7), we get
a n T n x, e0 = a n β(n)xn , m m b T x, e0 = bm β(m)xm ,
a n T n x, em−n = a n β(m)β(m − n)−1 xm , m m b T x, en−m = bm β(n)β(m − n − 1)−1 xn .
According to the last two displays, β(m − n − 1) >
1 − ε n −m a b ε
and β(m − n) >
1 − ε −n m a b . ε
Since β(m − n) = β(m − n − 1)wm−n 2−8 β(m − n − 1) from the last display it follows that β(m − n) > 2−8
1−ε 1−ε max a n b−m , a −n bm 2−8 > 1. ε ε
Since β(j ) = 1 if j ∈ / J , we have m − n ∈ J . Hence (Q − Q) ∩ N ⊆ J .
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Let now k ∈ N and m, n ∈ Q ∩ Ik be such that m > n. Since (Q − Q) ∩ N ⊆ J , we have k−1 k m − n ∈ J . Since m, n ∈ Ik , we get m − n m4k < 7m = min I . Hence m − n ∈ k j =0 Ij , 8 9mk−1 < 2mk−1 , where m0 = 1. Hence Q ∩ Ik 8 7mk n 8 m2k for any n ∈ Ik and therefore
where I0 = ∅. Then |m − n| elements. On the other hand,
n−1 2mk−1
n∈Q∩Ik
has at most 2mk−1
2 4mk−1 = 2−k , mk mk
where the last inequality follows from the definition of mk . Since Q ⊆ J and J is the union of disjoint sets Ik , we obtain
n−1 =
n∈Q
∞
n−1
k=1 n∈Q∩Ik
∞
2−k = 1 < ∞.
k=1
Using the definition of Q and Lemma 5.2, we now see that Λ has zero Lebesgue measure.
2
5.2. Proof of Theorem 1.11, Part I Consider the sequences {an }n∈Z and {wn }n∈Z defined by the formulae ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 8−1 an = 8 ⎪ ⎪ ⎪ 2−1 ⎪ ⎪ ⎪ −1 ⎪ ⎩4 16
if |n| 5 or −2 · 5k n < −5k , or −5k+1 n < −4 · 5k , k ∈ N, if −3 · 5k n < −2 · 5k , k ∈ N, if −4 · 5k n < −3 · 5k , k ∈ N, if 2 · 5k < n 4 · 5k , k ∈ N, if 5k < n 2 · 5k , k ∈ N, if 4 · 5k < n 5k+1 , k ∈ N;
wn =
1 n(n − 1)−1 an (n + 1)n−1 an
if |n| 1, if n 2, if n −2.
(5.9)
It is easy to see that w is a bounded sequence of positive numbers and infn∈Z wn > 0. Hence the bilateral weighted shift Tw is invertible. In order to prove Part I of Theorem 1.11 it is enough to verify the following statement. Example 5.4. Let w be the weight sequence defined by (5.9) and S = Tw be the corresponding bilateral weighted shift on 2 (Z). Then MS = {1, 2}. Proof. Using (5.9), one can easily verify that ⎧ k n ⎨ 45 −n if 5k < n 2 · 5k , k ∈ N, −n k < n 4 · 5k , k ∈ N, γ+ (n) = 2 aj , (5.10) where γ+ (n) = if 2 · 5 ⎩ n−5k+1 j =0 if 4 · 5k < n 5k+1 , k ∈ N, 16 ⎧ 0 ⎨1 if 5k < n 2 · 5k or 4 · 5k < n 5k+1 , k ∈ N, k aj . γ− (n) = 82·5 −n if 2 · 5k < n 3 · 5k , k ∈ N, where γ− (n) = ⎩ n−4·5k k k j =−n if 3 · 5 < n 4 · 5 , k ∈ N, 8 (5.11)
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For brevity we denote β+ (n) = w (0, n) and β− (n) = w (−n, 0), where w (k, l) are defined in (5.1). By definition of w, β+ (n) = nγ+ (n)
and β− (n) =
γ− (n) n
for any n ∈ N.
(5.12)
According to (5.10) and (5.11), γ+ (5k ) = γ− (5k ) = 1 and γ+ (3 · 5k ) = γ− (3 · 5k ) = 8−5 for k any k ∈ N. Using (5.12), we get β+ (5k )−1 = β− (5k ) = 5−k → 0 and (23·5 β+ (3 · 5k ))−1 = k 23·5 β− (3 · 5k ) = 3−1 5−k → 0 as k → ∞. Applying Theorem S to S = Tw and 2S = T2w , we see that S and 2S are both hypercyclic. Let c > 0 be such that cS = Tcw is hypercyclic. By Theorem S , there exists a strictly increasing sequence {nj }j ∈N of positive integers such that k
−1 + cnj β− (nj ) → 0 as j → ∞. cnj β+ (nj )
(5.13)
Let kj be the integer part of log5 nj . Then nj = bj 5kj , where 1 bj < 5. Passing to a subsequence, if necessary, we can additionally assume that bj → b ∈ [1, 5] as j → ∞. Using (5.10) and (5.11), one can easily verify that convergence of bj to b implies that lim γ+ (nj )1/nj = λ+ (b)
j →∞
and
lim γ− (nj )1/nj = λ− (b),
j →∞
(5.14)
where the continuous positive functions λ+ and λ− on [1, 5] are defined by the formula ⎧ −1 ⎨ 4b −1 λ+ (b) = 1/2 ⎩ 1−5b−1 16
if 1 b < 2, if 2 b 4, if 4 < b 5
and λ− (b) =
1 −1 82b −1 −1 81−4b
if b ∈ [1, 2] ∪ [4, 5], (5.15) if 2 < b 3, if 3 < b < 4.
According to (5.12), lim
n→∞
β+ (n) γ+ (n)
1/n = 1 and
lim
n→∞
β− (n) γ− (n)
1/n = 1.
From (5.14) and the above display it follows that
−1/nj −1 lim cnj β+ (nj )1/nj = cλ+ (b)
j →∞
and
1/nj lim cnj β+ (nj )1/nj = cλ− (b).
j →∞
These equalities together with (5.13) imply that (cλ+ (b))−1 1 and cλ− (b) 1. In particuλ− (b) lar, λλ−+ (b) (b) 1. On the other hand, (5.15) implies that λ+ (b) > 1 for b ∈ (1, 3) ∪ (3, 5). Hence b ∈ {1, 3, 5}. If b ∈ {1, 5}, then λ− (b) = λ+ (b) = 1 and the inequalities (cλ+ (b))−1 1 and cλ− (b) 1 imply that c 1 and c−1 1. That is, c = 1. If b = 3, then λ− (b) = λ+ (b) = 1/2 and the inequalities (cλ+ (b))−1 1 and cλ− (b) 1 imply that c/2 1 and 2/c 1. That is, c = 2. Thus c ∈ {1, 2}. Hence MS = {1, 2}. 2
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155
6. Proof of Theorem 1.3 The main tool in the proof is the following result by Macintyre and Fuchs. The following theorem is a part of Theorem 1 in [19]. Theorem MF. Let d > 0, n ∈ N and z1 , . . . , zn ∈ C. Then there exist n closed disks D1 , . . . , Dn on the complex plane such that their radii r1 , . . . , rn satisfy nj=1 rj2 4d 2 and n
|z − zj |−2 <
j =1
n(1 + ln n) d2
for any z ∈ C \
n
Dj .
(6.1)
j =1
We also need the following elementary lemma. Lemma 6.1. Let X be a topological vector space, T ∈ L(X) and f ∈ X ∗ \ {0}. Assume also that there exists a polynomial p such that p(T ) is hypercyclic. Then the sequence {(T ∗ )n f }n∈Z+ is linearly independent. Proof. Assume that the sequence {(T ∗ )n f }n∈Z+ is linearly dependent. Then we can pick n ∈ N such that (T ∗ )n f ∈ L = span{f, T ∗ f, . . . , (T ∗ )n−1 f }. It follows that L is a non-trivial finite dimensional invariant subspace for T ∗ . Hence L⊥ = {x ∈ X: g(x) = 0 for any g ∈ L} is a closed linear subspace of X of finite positive codimension invariant for T . Clearly L⊥ is also invariant for p(T ). We have obtained a contradiction with a result of Wengenroth [24], according to which hypercyclic operators on topological vector spaces have no closed invariant subspaces of positive finite codimension. 2 We are ready to prove Theorem 1.3. Let X be a complex topological vector space such that X ∗ = {0}, T ∈ L(X) and Λ be a non-empty subset of R × C for which the family A = {ea (T + bI ): (a, b) ∈ Λ} has a common hypercyclic vector. In order to prove Theorem 1.3 it suffices to show that Λ has zero three-dimensional Lebesgue measure. Pick a non-zero f ∈ X ∗ . By Lemma 5.1, the set H(A) of common hypercyclic vectors for operators from A is dense in X. Since H(A) is also closed under multiplications by non-zero scalars, we can pick x ∈ H(A) such that f (x) = 1. For each n ∈ N consider the complex polynomial n
n
∗ n−j n T f (x)bj . pn (b) = f (T + bI ) x = j
(6.2)
j =0
Clearly pn is a polynomial of degree n with coefficient 1 = f (x) in front of bn (such polynomials are usually called monic). Differentiating (6.2) by b, we obtain that pn (b) = nf ((T +bI )n−1 x) = npn−1 (b). That is, pn = npn−1
for each n ∈ N.
(6.3)
Applying (6.3) twice, one can easily verify that
pn /pn = n2
pn−1 2 1 pn−2 − 1− for each n 2. n pn pn
(6.4)
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S. Shkarin / Journal of Functional Analysis 258 (2010) 132–160
The equality (6.4) immediately implies the following inequality:
2
p /pn n2 pn−2 − pn−1
n
2p p
n n
for each n 2.
(6.5)
Lemma 6.2. For any (a, b) ∈ Λ and k ∈ Z+ , the sequence {vn }nk is dense in Ck+1 , where vn = ean (pn (b), pn−1 (b), . . . , pn−k (b)). Proof. Assume the contrary. Then there exist (a, b) ∈ Λ and a non-empty open subset W of / W for each n k. Let S = ea (T + bI ). By definition of pm , for 0 j k, Ck+1 such that vn ∈
k−j n−k
ean pn−j (b) = ean f (T + bI )n−j x = eaj f S n−j x = eaj S ∗ f S x . / W can be rewritten as S n−k x ∈ / R −1 (W ), where the linear operator Thus the relation vn ∈ k+1 is defined by the formula R:X→C
k−l+1 (Ry)l = ea(l−1) S ∗ f (y)
for 1 l k + 1.
By Lemma 6.1, continuous linear functionals f, S ∗ f, . . . , (S ∗ )k f are linearly independent. It follows that R is continuous and surjective. Hence V = R −1 (W ) is a non-empty open subset of X. Thus S n−k x does not meet the non-empty open set V for each n k, which is impossible since x ∈ H(S). 2 By Lemma 6.2 with k = 2, for any (a, b) ∈ Λ, the sequence {vn = ean (pn (b), pn−1 (b), pn−2 (b))}n2 is dense in C3 . Since the map F : C × C2 → C3 , F (u, v, w) = (u, v/u, w/u) is continuous and has dense range, {F (un ): n 2, pn (b) = 0} is dense in C3 . That is, an e pn (b), pn−1 (b)/pn (b), pn−2 (b)/pn (b) : n 2, pn (b) = 0 is dense in C3 . It follows that any (a, b) ∈ Λ is contained in infinitely many sets Cn , where
Cn = (a, b) ∈ R × C: 1 < ean pn (b) < e, pn−1 (b)/pn (b) < 1, pn−2 (b)/pn (b) > 8 . That is, Λ ⊆ Λ∗ =
∞
Cn .
(6.6)
m=1 nm
Clearly, Cn ⊆ R × Bn , where
Bn = b ∈ C: pn−1 (b)/pn (b) < 1, pn−2 (b)/pn (b) > 8 . Applying the inequality (6.5), we see that
Bn ⊆ Bn = b ∈ C: pn (b)/pn (b) 3n2 .
(6.7)
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Since pn is a monic polynomial of degree n, there exist z1 , . . . , zn ∈ C such that pn (b) =
n
(b − zj )
and therefore
n
pn (b)/pn (b) = − (b − zj )−2 .
j =1
j =1
By Theorem MF with d = n−1/3 , there are n closed disks D1 , . . . , Dn on the complex plane such that their radii r1 , . . . , rn satisfy n
rj2 4n−2/3
and
j =1
p (b)/pn (b)
n
n
|b − zj |−2 < n5/3 (1 + ln n) for any b ∈ C \
j =1
n
Dj .
j =1
Since n5/3 (1 + ln n) 3n2 for any n ∈ N, we see that Bn ⊆
n
j =1 Dj .
Hence
n
μ2 (Bn ) μ2 Bn π rj2 4πn−2/3 , j =1
where μk is the k-dimensional Lebesgue measure. For each b ∈ Bn , Ab,n = {a ∈ R: (a, b) ∈ Cn } can be written as
an
− ln |pn (b)| 1 − ln |pn (b)|
Ab,n = a ∈ R: 1 < e pn (b) < e = , , n n which is an interval of length n−1 . Hence μ1 (Ab,n ) = n−1 for each b ∈ Bn . By the Fubini theorem, μ2 (Bn ) 4πn−5/3 . μ3 (Cn ) = μ1 (Ab,n )μ2 (db) = n Bn
According to (6.6) and the above estimate, we obtain ∞
μ3 Λ∗ inf 4π n−5/3 = 0 m∈N
n=m
since
∞
n−5/3 < ∞.
n=1
Thus μ3 (Λ∗ ) = 0 and therefore μ3 (Λ) = 0 since Λ ⊆ Λ∗ . The proof of Theorem 1.3 is complete. 7. Concluding remarks and open problems Lemma 6.1 implies the following easy corollary. Corollary 7.1. Let X be a topological vector space such that 0 < dim X ∗ < ∞. Then X supports no hypercyclic operators.
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Proof. Assume that T ∈ L(X) is hypercyclic and f ∈ X ∗ , f = 0. By Lemma 6.1, the sequence {(T ∗ )n f }n∈Z+ is linearly independent, which contradicts the inequality dim X ∗ < ∞. 2 In particular, F -spaces X = Lp [0, 1] × Kn for 0 < p < 1 and n ∈ N support no hypercyclic operators. Indeed, the dual of X is n-dimensional. On the other hand, each separable infinite dimensional Fréchet space supports a hypercyclic operator [6] and there are separable infinite dimensional F -spaces [16] that support no continuous linear operators except the scalar multiples of I and therefore support no hypercyclic operators. However the following question remains open. Question 7.2. Let X be a separable F -space such that X ∗ is infinite dimensional. Is it true that there exists a hypercyclic operator T ∈ L(X)? Part I of Theorem 1.11 shows that there exists a continuous linear operator S on 2 such that MS = {1, 2}, where MS = {a > 0: aS is hypercyclic}. Using the same basic idea as in the proof of Theorem 1.11, one can construct an invertible bilateral weighted shift S on 2 (Z) such that MS is a dense subset of an interval and has zero Lebesgue measure. In particular, MS and its complement are both dense in this interval. It is also easy to show that for any F -space X and any T ∈ L(X), MT is a Gδ -set. If X is a Banach space, then MT is separated from zero by the number T −1 . These observations naturally lead to the following question. Question 7.3. Characterize subsets A of R+ for which there is S ∈ L(2 ) such that A = MS . In particular, is it true that for any Gδ -subset A of R+ such that inf A > 0, there exists S ∈ L(2 ) for which A = MS ? In the proof of Part II of Theorem 1.11 we constructed an invertible bilateral weighted shift T on 2 (Z) such that MT = (1/2, 2) and any subset A of (1/2, 2) such that the family {aT : a ∈ A} has a common hypercyclic vector must be of zero Lebesgue measure. It is also easy to see that our T enjoys the following extra property. Namely, if E = span{en : n ∈ Z} and x ∈ E, then for 2 1/2 < α < β < 2, we have α −mk T mk x → 0 and β mk T −mk x → 0 with mk = 23k . This shows that the convergence to zero condition in Corollary 1.10 cannot be replaced by convergence to 0 of a subsequence. Note that, according to the hypercyclicity criterion [5], the latter still implies hypercyclicity of all relevant scalar multiples of T . Recall that for 0 < s 1 the Hausdorff outer measure μs on R is defined as μs (A) = limδ↓0 μs,δ (A) with μs,δ (A) = inf (bj − aj )s , where the infimum is taken over all sequences {(aj , bj )} of intervals of length δ, whose union contains A. The number inf{s ∈ (0, 1]: μs (A) = 0} is called the Hausdorff dimension of A. With basically the same proof Lemma 5.2 can be strengthened in the following way. Lemma 7.4. Let X be a locally convex topological vector space, T ∈ L(X), s ∈ (0, 1], A ⊆ (0, ∞) and x be a common hypercyclic vector forthe family {cT : c ∈ Λ}. Assume also that there exists a non-empty open subset U of X such that n∈QU n−s < ∞, where QU is defined in (5.3). Then μs (A) = 0. Using Lemma 7.4 instead of Lemma 5.2, one can easily see that the operator T constructed in the proof of Part II of Theorem 1.11 has a stronger property. Namely, any A ⊂ R+ such that the family {cT : c ∈ A} is hypercyclic has zero Hausdorff dimension.
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159
Theorem CMP guarantees existence of common hypercyclic vectors for all non-identity operators of a universal strongly continuous semigroup {Tt }t0 on an F -space. On the other hand, Theorem CS shows that the non-identity elements of the 2-parametric translation group on H(C) have a common hypercyclic vector. The latter group enjoys the extra property of depending holomorphically on the parameter. Note that Theorem 1.2 strengthens this result. Question 7.5. Let X be a complex Fréchet space and {Tz }z∈C be a holomorphic strongly continuous operator group. Assume also that for each z ∈ C , the operator Tz is hypercyclic. Is it true that the family {Tz : z ∈ C } has a common hypercyclic vector? Question 7.6. Let X be a complex Fréchet space and {Tz }z∈C be a holomorphic strongly continuous operator group. Assume also that for each z, a ∈ C , the operator aTz is hypercyclic. Is it true that the family {aTz : a, z ∈ C } has a common hypercyclic vector? An affirmative answer to the following question would allow to strengthen Theorem 1.7. Question 7.7. Let T be a continuous linear operator on a complex separable Fréchet space X and 0 a < b ∞. Assume also that for any α ∈ (a, b), the sets
Eα = span
|z|<α
ker(T − zI )
and Fα = span
ker(T − zI )
|z|>α
are both dense in X. Is it true that the family {zT : b−1 < |z| < a −1 } has common hypercyclic vectors? It is worth noting that according to the Kitai criterion for T from the above question, zT is hypercyclic for any z ∈ C with b−1 < |z| < a −1 . It also remains unclear whether the natural analog of Theorem 1.3 holds in the case K = R. For instance, the following question is open. Question 7.8. Does there exist a continuous linear operator T on a real Fréchet space such that the family {aT + bI : a > 0, b ∈ R} has a common hypercyclic vector? Acknowledgment The author would like to thank Richard Aron for interest and helpful comments. References [1] E. Abakumov, J. Gordon, Common hypercyclic vectors for multiples of backward shift, J. Funct. Anal. 200 (2003) 494–504. [2] F. Bayart, Common hypercyclic vectors for composition operators, J. Operator Theory 52 (2004) 353–370. [3] F. Bayart, S. Grivaux, Hypercyclicity and unimodular point spectrum, J. Funct. Anal. 226 (2005) 281–300. [4] F. Bayart, É. Matheron, How to get common universal vectors, Indiana Univ. Math. J. 56 (2007) 553–580. [5] J. Bés, A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999) 94–112. [6] J. Bonet, A. Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998) 587–595. [7] P. Bourdon, J. Shapiro, Spectral synthesis and common cyclic vectors, Michigan Math. J. 37 (1990) 71–90. [8] K. Chan, R. Sanders, Common supercyclic vectors for a path of operators, J. Math. Anal. Appl. 337 (2008) 646–658.
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[9] J. Conejero, V. Müller, A. Peris, Hypercyclic behaviour of operators in a hypercyclic C0 -semigroup, J. Funct. Anal. 244 (2007) 342–348. [10] G. Costakis, P. Mavroudis, Common hypercyclic entire functions for multiples of differential operators, Colloq. Math. 111 (2008) 199–203. [11] G. Costakis, M. Sambarino, Genericity of wild holomorphic functions and common hypercyclic vectors, Adv. Math. 182 (2004) 278–306. [12] E. Gallardo-Gutiérrez, J. Partington, Common hypercyclic vectors for families of operators, Proc. Amer. Math. Soc. 136 (2008) 119–126. [13] G. Godefroy, J. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991) 229– 269. [14] K. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999) 345–381. [15] K. Grosse-Erdmann, Recent developments in hypercyclicity, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 97 (2003) 273–286. [16] N. Kalton, N. Peck, J. Roberts, An F -Space Sampler, London Math. Soc. Lecture Note Ser., vol. 89, Cambridge University Press, Cambridge, 1984. [17] C. Kitai, Invariant closed sets for linear operators, Thesis, University of Toronto, 1982. [18] F. León-Saavedra, Vladimir Müller, Rotations of hypercyclic and supercyclic operators, Integral Equations Operator Theory 50 (2004) 385–391. [19] A. Macintyre, W. Fuchs, Inequalities for the logarithmic derivatives of a polynomial, J. London Math. Soc. 15 (1940) 162–168. [20] H. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995) 993–1004. [21] H. Schaefer, Topological Vector Spaces, Springer, Berlin, 1971. [22] S. Shkarin, Non-sequential weak supercyclicity and hypercyclicity, J. Funct. Anal. 242 (2007) 37–77. [23] S. Shkarin, Universal elements for non-linear operators and their applications, J. Math. Anal. Appl. 348 (2008) 193–210. [24] J. Wengenroth, Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc. 131 (2003) 1759–1761.
Journal of Functional Analysis 258 (2010) 161–195 www.elsevier.com/locate/jfa
Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map Mourad Bellassoued a , Mourad Choulli b,∗ a University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerte, 7021 Jarzouna Bizerte, Tunisia b Laboratoire LMAM, UMR 7122, Université Paul Verlaine-Metz et CNRS, Ile du Saulcy, 57045 Metz cedex, France
Received 10 March 2009; accepted 17 June 2009 Available online 4 July 2009 Communicated by J. Coron
Abstract We consider the problem of stability estimate of the inverse problem of determining the magnetic field entering the magnetic Schrödinger equation in a bounded smooth domain of Rn with input Dirichlet data, from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichletto-Neumann map associated to the solutions of the magnetic Schrödinger equation. We prove in dimension n 2 that the knowledge of the Dirichlet-to-Neumann map for the magnetic Schrödinger equation measured on the boundary determines uniquely the magnetic field and we prove a Hölder-type stability in determining the magnetic field induced by the magnetic potential. © 2009 Elsevier Inc. All rights reserved. Keywords: Stability estimate; Schrödinger inverse problem; Magnetic field; Dirichlet-to-Neumann map
1. Introduction In this paper we study an inverse problem for the dynamical Schrödinger equation in the presence of a magnetic potential. Such an equation appears naturally in some mathematical models related to certain quantum dynamical systems. We shall consider the physically important case of a real valued magnetic potential. We will see below that in this case one has conservation of * Corresponding author.
E-mail addresses:
[email protected] (M. Bellassoued),
[email protected] (M. Choulli). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.010
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M. Bellassoued, M. Choulli / Journal of Functional Analysis 258 (2010) 161–195
charge. The dynamical Schrödinger equation plays also an important role in geometry. We refer to [42] and references therein for more details. Throughout this paper we assume that Ω is an open bounded subset of Rn , n 2, with C ∞ boundary Γ . Given T > 0, we consider the following initial boundary value problem (IBVP in short) for the Schrödinger equation with a magnetic potential, where Q = (0, T ) × Ω and Σ = (0, T ) × Γ , (i∂ + )u = 0, in Q, t A u(0, ·) = 0, in Ω, u = f, on Σ,
(1.1)
where A =
n (∂j + iaj )2 = + 2iA · ∇ + i div(A) − |A|2 . j =1
Here A = (aj )1j n ∈ W 1,∞ (Ω; Rn ) is the magnetic potential. We may define the operator ΛA (f ) = (∂ν + iA · ν)u,
f ∈ L2 (Σ),
where ν = ν(x) denotes the unit outward normal to Γ at x. We call ΛA the Dirichlet-to-Neumann map (DN map in short) associated to the IBVP (1.1). We consider the inverse problem to know whether the DN map ΛA determines uniquely the magnetic potential A. First of all, let us observe that there is an obstruction to uniqueness. In fact as it was noted in [19], the DN map is invariant under the gauge transformation of the magnetic potential. Namely, given Ψ ∈ C 1 (Ω) such that Ψ |Γ = 0 one has e−iΨ A eiΨ = A+∇Ψ ,
e−iΨ ΛA eiΨ = ΛA+∇Ψ ,
(1.2)
and ΛA = ΛA+∇Ψ . Therefore, the magnetic potential A cannot be uniquely determined by the DN map ΛA . From a geometric view point this can be seen as follows. The vector field A defines the connection given by the one form αA = nj=1 aj dxj , and the non-uniqueness manifested in (1.2) says that the best we could hope to reconstruct from the DN map ΛA is the 2-form called the magnetic field dαA given by dαA =
n ∂ai i,j =1
∂xj
−
∂aj ∂xi
dxj ∧ dxi .
Physically, our inverse problem consists in determining the magnetic field dαA induced by the magnetic potential A of an inhomogeneous medium by probing it with disturbances generated on the boundary. The data are responses of the medium to these disturbances which are measured on the boundary and the goal is to recover the magnetic field dαA which describes the property of the medium. Here we assume that the medium is quiet initially and f is a disturbance which is used to probe the medium. Roughly speaking, the data is (∂ν + iν · A)u measured on the boundary for different choices of f .
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The uniqueness in the determination of time-dependent electromagnetic potential, appearing in a Schödinger equation in a domain with obstacles, from the DN map was proved by Eskin [21]. The main ingredient in his proof is the construction of geometric optics solutions. In [1], Avdonin et al. use the so-called BC (boundary control) method to prove that the DN map determines the time-independent electrical potential in a one dimensional Schrödinger equation. The problem of stability in determining the time-independent potential in a Schödinger equation from a single boundary measurement was studied by Baudouin and Puel [2]. They establish Lipschitz stability estimate by a method based essentially on an appropriate Carleman inequality. More recently, Mercado, Osses and Rosier [35] improved the results in [2]. In both above mentioned papers, the main assumption is that the part of the boundary where the measurement is made must satisfy a geometric condition (related to geometric optics condition insuring observability). Recently, we showed [6] that this geometric condition can be relaxed provided that the potential is known near the boundary. The key idea was the following: we used an FBI transform to change the Schödinger equation near the boundary into a heat equation for which we have a useful Carleman inequality involving a boundary term and without any geometric condition. In recent years significant progress has been made for the problem of identifying the electrical potential. In [39], Rakesh and Symes prove that the DN map determines uniquely the timeindependent potential in a wave equation. Ramm and Sjöstrand [40] has extended the result in [39] to the case of time-dependent potentials. Isakov [25] has considered the simultaneous determination of a zeroth order coefficient and a damping coefficient. A key ingredient in the existing results is the construction of complex geometric optics solutions of the wave equation, concentrated along a line, and the relationship between the hyperbolic DN map and the X-ray transform play a crucial role. In [5], Bellassoued and Benjoud use complex geometric optics solutions concentrating near lines in any direction to prove that the DN map determines uniquely the magnetic field induced by a magnetic potential in a magnetic wave equation. In this work, the DN map gives an equivalent information to the responses on the whole boundary for all possible input disturbances. Cipolatti and Lopez [16] consider the inverse problem of recovering the time-independent damping coefficient in a wave equation from the DN map. They prove Lipschitz or Hölder stability. Moreover in [16] it is proved that if an unknown coefficient belongs to a given finite dimensional vector space, then the uniqueness follows by a finite number of measurements on the whole boundary. All results mentioned above are concerned with the full data, i.e., measurements are made on the whole boundary. The uniqueness by a local DN map is well solved (e.g., Belishev [3], Eskin [18–20], Eskin and Ralston [22], Katchlov, Kurylev and Lassas [28], Kurylev and Lassas [31]). The stability estimates in the case where the DN map is considered on the whole lateral boundary were established in Cipolatti and Lopez [16], Stefanov and Uhlmann [43], Sun [44]. However the stability by a local DN map is not discussed comprehensively. For it, see Isakov and Sun [27] where a local DN map yields a stability result in determining a coefficient in a subdomain. For the DN map for an elliptic equation, the paper by Calderón [11] is a pioneering work. We also refer to Bukhgeim and Uhlamnn [10], Hech and Wang [23], Salo [41] and Uhlmann [46] as a survey. In [17] Dos Santos Ferreira, Kenig, Sjostrand, Uhlmann prove that the knowledge of the Cauchy data for the Schrödinger equation in the presence of magnetic potential, measured on possibly very small subset of the boundary, determines uniquely the magnetic field. In [45], Tzou proves a log log-type estimate which show that the magnetic field and the electric potential of the magnetic Schrödinger equation depends stably on the DN map even when the boundary measurement is taken only on a subset that is slightly larger than the half of the boundary. In [15], Cheng
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and Yamamoto prove that the stability estimation imply the convergence rate of the Tikhonov regularized solutions. As for results by a finite number of data of DN map, see Bellassoued, Jellali and Yamamoto [7,8], Cheng and Nakamura [14], Cipolatti and Lopez [16], Rakesh [38]. There are many other works on DN maps and our references are far from being complete. See also Cardoso and Mendoza [12], Rachele [37], Uhlmann [46] as related papers. Let us mention that the method using Carleman inequalities was initiated by Bukhgeim and Klibanov [9]. Furthermore, as for applications of Carleman estimates to inverse problems, we can refer to Bellassoued [4], Imanuvilov and Yamamoto [24], Isakov [26], Klibanov [29], Klibanov and Timonov [30,33]. Most of those papers treat the determination of spatially varying functions by a single measurement. As for observability inequalities by means of a Carleman estimate, see [30]. In the present paper, we prove a Hölder-type estimate which shows that a magnetic field induced by a magnetic potential depends stably on the DN map. We organize this paper as follows. We state our main results, Theorem 2.1 and its corollary, in Section 2. We collect in Section 3 all the results on the initial-boundary value problem for the magnetic Schrödinger equation that are needed in the analysis of our inverse problem. In Section 4 we construct the so-called geometric optics solutions to the magnetic Schrödinger equation. These solutions constitute the main ingredient in our proof of stability estimate for the inverse problem. We establish in Section 5 a preliminary technical estimate that we use in Section 6 to prove a stability estimate for the X-ray transform of some functions related to our problem. Finally, we complete the proof of Theorem 2.1 and its corollary in Section 7. 2. Stability estimate We first need to define the notion of a solution in the transposition sense of the IBVP (1.1). To this end, we introduce the following subspace of H 2,1 (Q) = {u ∈ L2 (Q); ∂i u, ∂ij u, ∂t u ∈ L2 (Q)} X (Q) = v ∈ H 2,1 (Q); v(T , ·) = 0, v = 0 on Σ . We define on X (Q) the following bounded antilinear form (v) = f ∂ν v, v ∈ X (Q). Σ
Usually, the transposition solution is defined by duality. We proceed as follows. If u ∈ H 2,1 (Q) is a solution of (1.1) then multiplying the first equation of (1.1) by v ∈ X (Q), we find, by using an integration by parts with respect to the variable t and Green’s formula with respect to the variable x, u(i∂t + A )v = (v), v ∈ X (Q). (2.1) Q
Suppose that we can show that for any ϕ ∈ L2 (Q), there exists v ∈ X (Q) such that (i∂t + A )v = ϕ and | (v)| CϕL2 (Q) . In this case one can extend (by using the Hahn– Banach extension theorem) to a bounded antilinear form on L2 (Q). Therefore, we find (think
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about Riesz’s representation theorem) u ∈ L2 (Q) such that (2.1) holds. Unfortunately, we cannot do this for Schrödinger type equations. In fact to solve Schrödinger type equations we need more than L2 (Q) regularity for the non-homogenous term. To overcome this difficulty, we have to replace (2.1) by a weaker form:
u, (i∂t + A )v
H ,H
= (v)
with v is in some appropriate space, where H is a subspace of L2 (Q) and ·,· is the duality pairing between H and H . In the next section we show that we can choose H = H01 (0, T ; L2 (Ω)). More precisely, we prove that the IBVP (1.1) has a unique transposition solution u = u(f ) ∈ H −1 (0, T ; L2 (Ω)) and, in addition, the mapping f → ∂ν u(f ) define a bounded operator from L2 (Σ) into H −1 (0, T ; H −3/2 (Γ )). We then define the DN map ΛA (f ) = (∂ν + iA · ν)u,
f ∈ L2 (Σ).
Clearly ΛA ∈ B(L2 (Σ), H −1 (0, T ; H −3/2 (Γ ))), where B(L2 (Σ), H −1 (0, T ; H −3/2 (Γ ))) is the Banach space of linear bounded operators from L2 (Σ) into H −1 (0, T ; H −3/2 (Γ )). In the sequel the norm of ΛA in B(L2 (Σ), H −1 (0, T ; H −3/2 (Γ ))) will denoted by ΛA . We now introduce the following subset of W 1,∞ (Ω; Rn ), where ∇A = ∇ + iA,
A = A ∈ W 1,∞ Ω; Rn ; ϕ → |||∇A ϕ|||L2 (Ω) defines an equivalent norm on H01 (Ω) . In Lemma 3.1 below we give a characterization of this set. The main result of this paper can be stated as follows Theorem 2.1. Let M > 0 be a given constant and A1 , A2 ∈ W 3,∞ (Ω; Rn ) ∩ A such that Ai W 3,∞ M, i = 1, 2. We assume that A1 = A2 and ∂j A1 = ∂j A2 on Γ . Then there exists a constant C > 0, depending only on A1 , M, Ω, T and n, such that dαA1 − dαA2 H −1 (Ω) CΛA1 − ΛA2 κ , where κ = ( 32 n + 15)−1 . By Theorem 2.1, we can readily derive the following stability estimate Corollary 2.1. Let A1 , A2 ∈ W 3,∞ (Ω; Rn ) ∩ A such that Ai H s (Ω) M, i = 1, 2, where s > n/2 + 3. We assume that A1 = A2 and ∂j A1 = ∂j A2 on Γ . Then there exists a constant C > 0, depending only on A1 , M, Ω, T and n, such that
dαA1 − dαA2 L∞ (Ω) CΛA1 − ΛA2 κ . Here κ =
κ(s−1−n/2) , 2s
(2.2)
where κ is as in the preceding theorem.
Our proof is inspired by techniques used by Bukhgeim and Uhlmann [10], Dos Santos Ferreira, Kenig, Sjostrand, Uhlmann [17] and Salo [41] for proving an uniqueness theorem related to an inverse elliptic problem. Their idea in turn goes back to the earlier work of Calderón [11]. We note that, contrary to the elliptic case, geometric optics solutions interact with the interior of
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Ω in the dynamical case. The main idea is to probe the medium by real geometric optics solutions of the Schrödinger equation, concentrated along a line, starting on one side of the boundary, and measure the responses of the medium on other side of the boundary. A response gives a line integral of dαA . 3. The magnetic Schödinger equation Even if they are not explicitly listed in literature, some of the results of this section are more or less known. For sake of completeness, we give most of details. Let Ω be as before. That is a bounded C ∞ smooth domain of Rn with boundary Γ . We recall that to A = (aj ) ∈ W 1,∞ (Ω, Rn ) we associated the operator A =
n (∂j + iaj )2 . j =1
We first observe that −A with the domain D(−A ) = H01 (Ω) ∩ H 2 (Ω) is an unbounded self-adjoint operator on L2 (Ω). This can be easily seen from Green’s formula. Indeed, if u, v ∈ D(−A ) then −A uv dx = − Ω
(∂j + iaj )2 uv dx Ω
j
Ω
j
=− (∂j + iaj )u(−∂j + iaj )v dx =−
Ω
=
u(−∂j + iaj )2 v dx
j
u(−A v) dx. Ω
The following lemma gives a characterization of the set A introduced in the previous section. We recall that ∇A = ∇ + iA and we consider V = (v, w) ∈ H01 (Ω)2 ; ∇v = wA and ∇w = −vA . Lemma 3.1. The mapping u → ∇A uL2 (Ω) defines a norm on H01 (Ω), equivalent to the norm ∇uL2 (Ω) if and only if one of following conditions holds. (i) 0 is not an eigenvalue of −A or 0 is an eigenvalue of −A and for each corresponding eigenfunction u, (Re(u), Im(u)) ∈ / V. (ii) 1 is not an eigenvalue of the operator TA =
(−0 )−1 0
0 (−0 )−1
|A|2 div(A)
− div(A) |A|2
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or 1 is an eigenvalue of TA and N (I − TA ) ∩ V = {0}. Here −0 is the operator − with domain H01 (Ω) ∩ H 2 (Ω). Proof. We have |||∇A u|||L2 (Ω) |||∇u|||L2 (Ω) + |||A|||L∞ (Ω) uL2 (Ω) ,
u ∈ H01 (Ω).
In view of Poincaré’s inequality, this estimate implies |||∇A u|||L2 (Ω) C0 |||∇u|||L2 (Ω) , for some positive constant C = C(Ω, |||A|||L∞ (Ω) ). Next, we claim that there exists a positive constant C1 (Ω, A) such that uL2 (Ω) C1 |||∇A u|||L2 (Ω) ,
for all u ∈ H01 (Ω)
(3.1)
if one of conditions (i) and (ii) is satisfied. Otherwise, we find a sequence (uk ) in H01 (Ω) such that uk L2 (Ω) = 1 and |||∇A uk |||L2 (Ω) converges to 0. In particular (|∇uk |) is bounded in L2 (Ω), i.e. (uk ) is bounded in H01 (Ω). Therefore, (uk ) converges strongly in L2 (Ω) to u ∈ L2 (Ω) and weakly in H01 (Ω). Therefore ∇A u = 0. This condition implies: (a) that (Re(u), Im(u)) ∈ V and A u = ∇A · ∇A u = 0; or (b) (v, w) = (Re(u), Im(u)) ∈ V and
v = div(wA) = div(A)w + ∇w · A = div(A)w − |A|2 v, w = div(−vA) = − div(A)w − ∇v · A = − div(A)v − |A|2 w.
v It can be seen that this system is equivalent to (I − TA ) w = 0. Hence, if (i) or (ii) is satisfied then u = 0. But this contradicts the fact that uL2 (Ω) = limk→+∞ uk L2 (Ω) = 1. Combining (3.1) with the following inequality |||∇u|||L2 (Ω) |||∇A u|||L2 (Ω) + AL∞ (Ω) uL2 (Ω) , we find ∇uL2 (Ω) C2 |||∇A u|||L2 (Ω) ,
for all u ∈ H01 (Ω),
where C2 = C2 (Ω, A) is some positive constant. This completes the proof. 2 We have seen above that the unbounded operator −A is self-adjoint. Therefore PA = −iA , with D(PA ) = D(−A ) is skew-adjoint. Since D(PA ) is dense in L2 (Ω) we deduce from a classical result (see for instance [13]) that PA (and also −PA ) generates on L2 (Ω) a one-parameter group of isometries (SA (t)). Lemma 3.2. (i) For each u0 ∈ H01 (Ω), |||∇A SA (t)u0 |||L2 (Ω) = |||∇A u0 |||L2 (Ω) .
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(ii) Let f ∈ L1 (0, T ; H01 (Ω)) and t v(t) =
SA (t − s)f (s) ds. 0
Then v ∈ C([0, T ]; H01 (Ω)) and vC ([0,T ];H 1 (Ω)) f L1 (0,T ;H 1 (Ω)) . 0
0
(iii) Let M > 0 be a given constant, A1 , A2 ∈ W 1,1 (Ω, Rn )∩A such that |||A1 −A2 |||L∞ (Ω) M. Let f ∈ W 1,1 (0, T ; L2 (Ω)) with f (0) = 0. Then v given by t v(t) =
SA2 (t − s)f (s) ds 0
(which belongs to C([0, T ]; D(PA2 )) ∩ C 1 ([0, T ]; L2 (Ω))) satisfies: for all 0 < 1, ∇v(t)
L2 (Ω)
C −1 f (t)L1 (0,T ;L2 (Ω)) + 2 f (t)L1 (0,T ;L2 (Ω)) ,
where the constant C = C(A1 , M) doesn’t depend on . Proof. (i) We start with smooth u0 . We assume that u0 ∈ D(PA2 ). Therefore u(t) = SA (t)u0 ∈ C 1 ([0, +∞[; D(PA )). We set 1 E(t) = 2
∇A u(t)2 dx.
Ω
Then we have 1 E (t) = 2
∇A u(t) · ∇A u (t) + ∇A u (t)∇A u(t) dx
Ω
= Re
∇A u (t) · ∇A u(t) dx
Ω
−A
= Re Ω
u(t)u (t) dx
2 = Re i u (t) dx = 0. Ω
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169
Hence E(t) = E(0) =
1 2
|∇A u0 |2 dx. Ω
Next, let u0 ∈ H01 (Ω) and let (uk0 ) be a sequence in D(PA2 ) converging to u0 in H01 (Ω). If u(t) = SA (t)u0 and uk (t) = SA (t)uk0 then the preceding result leads
∇A uk (t) − ∇A ul (t)2 dx =
Ω
∇A uk − ∇A ul 2 dx. 0
0
Ω
Since we have also
k u (t) − ul (t)2 dx =
Ω
k u − ul 2 dx, 0
0
Ω
we conclude that (uk ) is a Cauchy sequence in C([0, T ]; H01 (Ω)). But we already know that (uk ) converges to u in C([0, T ]; L2 (Ω)). Consequently, u ∈ C([0, T ]; H01 (Ω)). Finally, passing to the limit in the identity ∇A SA (t)uk
0 L2 (Ω)
= ∇A uk0 L2 (Ω) ,
we find ∇A SA (t)u0
L2 (Ω)
= |||∇A u0 |||L2 (Ω) .
(ii) As t ∇A v(t) =
∇A SA (t − s)f (s) ds,
(3.2)
0
we have, for small h > 0, t+h t ∇A v(t + h) − ∇A v(t) = ∇A SA (t + h − s)f (s) ds − ∇A SA (t − s)f (s) ds 0
0
t
t
=
∇A SA (t + h − s)f (s) ds − 0
+
∇A SA (t − s)f (s) ds 0
t+h ∇A SA (t + h − s)f (s) ds t
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t =
∇A SA (t − s) SA (h)f (s) − f (s) ds
0
t+h + ∇A SA (t + h − s)f (s) ds. t
This and (i) imply ∇A v(t + h) − ∇A v(t)
t L2 (Ω)
∇A SA (h)f (s) − ∇A f (s)
L2 (Ω)
0
t+h + ∇A f (s)L2 (Ω) ds. t
Clearly the second term of the right-hand side in the last inequality converges to 0 as h tends to 0. On the other hand, since t → ∇A SA (t)f (s) is continuous for a.e. s ∈ (0, T ), we deduce that ∇A SA (h)f (s) converges to ∇A f (s) for a.e. s ∈ (0, T ), as h tends to 0. Since ∇A SA (h)f (s) − ∇A f (s)
L2 (Ω)
2∇A f (s)L2 (Ω) ,
we can then apply the dominated convergence theorem to conclude that the first term of the right-hand side converges also to 0 as h tends to 0. In the other words, we prove that v ∈ C([0, T ]; H01 (Ω)). Finally, (3.2) and (i) lead vC ([0,T ];H 1 (Ω)) f L1 (0,T ;H 1 (Ω)) . 0
0
(iii) We have t
v (t) =
SA2 (t − s)f (s) ds.
0
Therefore v(t)
L2 (Ω)
f L1 (0,T ;L2 (Ω))
and v (t) 2 f L1 (0,T ;L2 (Ω)) . L (Ω) On the other hand, we have v (t) = iA2 v(t) + f (t).
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An application of Green’s formula leads
v (t)v(t) dx = i
Ω
∇A v(t)2 dx + 2
Ω
=i Ω
f (t)v(t) dx Ω
∇A v(t)2 dx + 2
t
f (s)v(t) dx ds.
Ω 0
Hence ∇A v(t)2 2 v(t)L2 (Ω) v (t)L2 (Ω) + f (t)L1 (0,T ;L2 (Ω)) v(t)L2 (Ω) 2 L (Ω) 2f (t)L1 (0,T ;L2 (Ω)) f (t)L1 (0,T ;L2 (Ω)) . This and the elementary inequalities 2ab 2 a 2 + −2 b2 ,
√ √ √ a + b a + b implies
2 ∇A v(t) 2 f (t)L1 (0,T ;L2 (Ω)) + 2 f (t)L1 (0,T ;L2 (Ω)) . 2 L (Ω) But ∇A v(t) 2 ∇A2 v(t)L2 (Ω) + |||A1 − A2 |||L∞ (Ω) v(t)L2 (Ω) 1 L (Ω) ∇A2 v(t)L2 (Ω) + M v(t)L2 (Ω) . Therefore 2 ∇A v(t) 2 f (t)L1 (0,T ;L2 (Ω)) + 2 f (t)L1 (0,T ;L2 (Ω)) + M f (t)L1 (0,T ;L2 (Ω)) 1 L (Ω) 2+M f (t) 1 2 (Ω)) + 2 f (t) L1 (0,T ;L2 (Ω)) . L (0,T ;L The conclusion follows then from the fact that w → |||∇A1 w(t)|||L2 (Ω) defines an equivalent norm on H01 (Ω). 2 Let us introduce the mapping
S : L2 (Ω) × L1 0, T ; L2 (Ω) → C [0, T ]; L2 (Ω) , t (u0 , f ) → S (u0 , f )(t) = SA (t)u0 +
SA (t − s)f (s) ds. 0
Let us recall that if (u0 , f ) ∈ D(PA ) × C([0, T ], D(PA )) then
S (u0 , f ) ∈ C [0, T ]; D(PA ) ∩ C 1 [0, T ]; L2 (Ω) .
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Therefore ∂ν S (u0 , f ) is well defined as an element of L2 (Σ) (in fact, it belongs to C([0, T ]; H 1/2 (Γ ))). Theorem 3.1. The operator (u0 , f ) ∈ D(PA ) × C([0, T ], D(PA )) → ∂ν S (u0 , f ) ∈ L2 (Σ) can be extended to a bounded operator from H01 (Ω) × L1 (0, T ; H01 (Ω)) into L2 (Σ). Proof. Since Ω is C ∞ -smooth, there exists γ ∈ C ∞ (Ω, Rn ) such that γ = ν on Γ . We fix (u0 , f ) ∈ D(PA ) × C([0, T ], D(PA )) and we set u = S (u0 , f ). We will use the fact that u is the solution of the following initial value problem.
u (t) = PA u(t) + f (t) = iA u(t) + f (t), u(0) = u0 .
t ∈ [0, T ],
From the first equation in the initial value problem above, we obtain u (t)γ · ∇A u(t) dx = iA u(t)γ · ∇A u(t) dx + f (t)γ · ∇A u(t) dx. Ω
Ω
Ω
We apply Green’s formula to the first term of the right-hand side of this identity. After some elementary calculations, we get 2 1 Re i u (t)γ · ∇A u(t) dx = (γ )t ∇A u(t) · ∇A u(t) dx + γ · ∇ ∇A u(t) dx 2 Ω
Ω
−
Ω
∂ν u(t)2 dσ + Re i f (t)γ · ∇A u(t) dx .
Γ
Ω
Here and henceforth, γ = (∂l γk ). But 2 2 2 γ · ∇ ∇A u(t) dx = − div(γ ) ∇A u(t) dx + ∂ν u(t) dσ. Ω
Ω
Γ
This identity follows also from Green’s formula. Therefore 2 1 Re i u (t)γ · ∇A u(t) dx = (γ )t ∇A u(t) · ∇A u(t) dx − div(γ )∇A u(t) dx 2 Ω
Ω
−
1 2
Γ
Hence T Re i u (t)γ · ∇A u(t) dx dt 0 Ω
Ω
∂ν u(t)2 dσ + Re i f (t)γ · ∇A u(t) dx . Ω
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T =
1 (γ ) ∇A u(t) · ∇A u(t) dx dt − 2 t
T
0 Ω
−
1 2
173
2 div(γ ∇A u(t dx dt
0 Ω
T
T ∂ν u(t)2 dσ dt + Re i f (t)γ · ∇A u(t) dx dt .
0 Γ
(3.3)
0 Ω
We have T T i Re i u (t)γ · ∇A u(t) − u (t)γ · ∇A u(t) dx dt. u (t)γ · ∇A u(t) dx dt = 2 0 Ω
0 Ω
But 2 u (t)γ · ∇A u(t) − u (t)γ · ∇A u(t) = u (t)γ · ∇u(t) − u (t)γ · ∇u(t) − i aj u(t) . j
Consequently T T i u (t)γ · ∇u(t) − u (t)γ · ∇u(t) dx dt u (t)γ · ∇A u(t) dx dt = Re i 2 0 Ω
0 Ω
+
1 2
Ω
2 2 aj u(T ) − u(0) .
(3.4)
j
We now make an integration by parts with respect to the variable t. We find T
u (t)γ · ∇A u(t) dx dt
0 Ω
=
u(T )γ · ∇u(T ) − u(0)γ · ∇u(0) dx −
Ω
T
u(t)γ · ∇A u (t) dx dt.
0 Ω
Once again, if we apply Green’s formula to the last term in the right-hand side in the identity above then we get T
u (t)γ · ∇A u(t) dx dt
0 Ω
= Ω
u(T )γ · ∇u(T ) − u(0)γ · ∇u(0) dx −
T 0 Ω
div u(t)γ u (t) dx dt
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M. Bellassoued, M. Choulli / Journal of Functional Analysis 258 (2010) 161–195
u(T )γ · ∇u(T ) − u(0)γ · ∇u(0) dx +
= Ω
T
∇u(t) · γ u (t) dx dt
0 Ω
T
+
div(γ )u(t)u (t) dx dt.
0 Ω
That is T
u (t)γ · ∇A u(t) − ∇A u(t) · γ u (t) dx dt =
0 Ω
T
div(γ )u(t)u (t) dx dt
0 Ω
+
u(T )γ · ∇u(T ) − u(0)γ · ∇u(0) dx.
Ω
This and (3.4) give T i u(T )γ · ∇u(T ) − u(0)γ · ∇u(0) dx Re i u (t)γ · ∇A u(t) dx dt = 2 0 Ω
Ω
+
1 2
Ω
i + 2
2 2 aj u(T ) − u(0) dx
j
T
div(γ )u(t)u (t) dx dt.
(3.5)
0 Ω
In this identity, we replace in the last term of the right-hand side u (t) by iA u(t) + f (t). That is we have T
div(γ )u(t)u (t) dx dt =
0 Ω
T
T div(γ )u(t)iA u(t) dx dt +
0 Ω
div(γ )u(t)f (t) dx dt. 0 Ω
A simple application of Green’s formula leads T
T div(γ )u(t)A u(t) dx dt = −
0 Ω
2 div(γ )∇A u(t) dx dt
0 Ω
T − 0 Ω
u(t)∇A div(γ ) · ∇A u(t) dx dt.
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175
Hence T
T div(γ )u(t)u (t) dx dt
2 div(γ )∇A u(t) dx dt
= −i
0 Ω
0 Ω
T
u(t)∇A div(γ ) · ∇A u(t) dx dt
− 0 Ω
T +
div(γ )u(t)f (t) dx dt. 0 Ω
This identity, combined with (3.5), implies T i Re i u(T )γ · ∇u(T ) − u(0)γ · ∇u(0) dx u (t)γ · ∇A u(t) dx dt = 2 0 Ω
Ω
+
1 2
Ω
1 + 2
2 2 aj u(T ) − u(0)
j
T
2 div(γ )∇A u(t) dx dt
0 Ω
i − 2
T
u(t)∇A div(γ ) · ∇A u(t) dx dt
0 Ω
i + 2
T div(γ )u(t)f (t) dx dt. 0 Ω
We deduce by comparing (3.3) and (3.6) T 0 Γ
∂ν u(t)2 dσ dt = 2
T
(γ )t ∇A u(t) · ∇A u(t) dx dt
0 Ω
T +
u(t)∇A div(γ ) · ∇A u(t) dx dt
0 Ω
−i Ω
u(T )γ · ∇u(T ) − u(0)γ · ∇u(0) dx
(3.6)
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−
2 2 aj u(T ) − u(0) +
j
Ω
T
T 0 Ω
T
−i
u(t)∇ div(γ ) · ∇u(t) dx dt
div(γ )u(t)f (t) dx dt − 2 Re 0 Ω
if (t)γ · ∇A u(t) dx dt .
0 Ω
In view of Lemma 3.2, it follows from this identity that there exists a constant C = C(Ω, A) such that
∂ν uL2 (Σ) C u0 H 1 (Ω) + f L1 (0,T ;H 1 (Ω)) . 0
0
We complete the proof by noting that D(PA ) × C([0, T ], D(PA )) is dense in H01 (Ω) × L1 (0, T ; H01 (Ω)). 2 Our proof of Theorem 3.1 was inspired by that of Machtyngier [34] (see also Baudouin and Puel [2]). The original idea goes back to Lasiecka, Lions and Triggiani [32], where they prove the same result for a wave IBVP. Corollary 3.1. The operator (u0 , f ) ∈ D(PA ) × C 1 ([0, T ]; L2 (Ω)) → ∂ν S (u0 , f ) ∈ L2 (Σ) can extended to a bounded operator from H01 (Ω) × W 1,1 (0, T ; L2 (Ω)) into L2 (Σ). Proof. Let (u0 , f ) ∈ H01 (Ω) × W 1,1 (0, T ; L2 (Ω)) into L2 (Σ). We split u = S (u0 , f ) into two terms u = v + w, where v = S (u0 , 0) and w = S (0, f ). From Theorem 3.1, ∂ν v ∈ L2 (Σ) and ∂ν vL2 (Σ) C0 u0 H 1 (Ω) , 0
(3.7)
for some positive constant C0 = C0 (Ω). On the other hand, we know that
w ∈ C 1 [0, T ]; L2 (Ω) ∩ C [0, T ]; D(PA ) and t
PA w = w (t) − f (t) = SA (t)f (0) +
SA (s)f (t − s) ds − f (t).
0
Therefore A w(t)
L2 (Ω)
C1 f W 1,1 (0,T ;L2 (Ω)) ,
for some positive constant C1 = C1 (T ). Furthermore, if we fix λ not being in the spectrum of A then from the classical elliptic estimate we have
M. Bellassoued, M. Choulli / Journal of Functional Analysis 258 (2010) 161–195
w(t)
H 2 (Ω)
C2 (A − λ)w(t)L2 (Ω)
C2 A w(t)L2 (Ω) + |λ|w(t)L2 (Ω) ,
177
t ∈ [0, T ],
where C2 = C2 (Ω, A) is a positive constant. But, we already proved A w(t)
L2 (Ω)
C1 f W 1,1 (0,T ;L2 (Ω)) .
Since w(t)
L2 (Ω)
f L1 (0,T ;L2 (Ω)) ,
we get w(t)
H 2 (Ω)
C3 f W 1,1 (0,T ;L2 (Ω)) ,
t ∈ [0, T ].
Here C3 = C3 (Ω, A, T ) is a positive constant. This and the fact that the trace operator ϕ ∈ H 2 (Ω) → ∂ν ϕ ∈ L2 (Γ ) is bounded imply ∂ν wL2 (Σ) C4 f W 1,1 (0,T ;L2 (Ω)) ,
(3.8)
for some positive constant C4 = C4 (Ω, A, T ). Finally, a combination of (3.7) and (3.8) leads
∂ν uL2 (Σ) C5 u0 H 1 (Ω) + f W 1,1 (0,T ;L2 (Ω)) , 0
where C5 = C5 (Ω, A, T ) is a positive constant.
2
Theorem 3.1 and Corollary 3.1 are still valid if we replace the mapping S by the following one
S : L2 (Ω) × L1 0, T ; L2 (Ω) → C [0, T ]; L2 (Ω) , (uT , f ) → S(uT , f )(t) = SA (−T + t)uT +
−T +t
SA (−T + t − s)f (s + T ) ds. 0
We note that v(t) = S(uT , f )(t) is nothing more than the weak solution for the backward Schrödinger equation for the magnetic laplacian with Dirichlet boundary condition, or, equivalently, the mild solution of the following Cauchy problem
v (t) = PA v(t) + f (t), v(T ) = uT .
t ∈ (0, T ),
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In the rest of this section we prove the existence and uniqueness of the transposition solution of the IBVP (1.1). We fix f ∈ L2 (Σ). From Corollary 3.1, we deduce, since H01 (0, T ; L2 (Ω)) is a subspace of W 1,1 (0, T ; L2 (Ω)), that the operator
F ∈ H01 0, T ; L2 (Ω) → ∂ν v(F ) ∈ L2 (Σ), is bounded, where v(F ) = S (0, F ) ∈ C 1 ([0, T ]; L2 (Ω)) ∩ C([0, T ]; D(PA )). Hence the linear form
F ∈ H01 0, T ; L2 (Ω) → f ∂ν v(F ) dσ Σ
is bounded. Therefore, there exists u ∈ H −1 (0, T ; L2 (Ω)) = (H01 (0, T ; L2 (Ω))) such that u, F H −1 (0,T ;L2 (Ω)),H 1 (0,T ;L2 (Ω)) = 0
f ∂ν v(F ) dσ,
F ∈ H01 0, T ; L2 (Ω) .
f ∂ν v(F ) dσ,
F ∈ H01 0, T ; L2 (Ω) .
Σ
Or equivalently u, F H −1 (0,T ;L2 (Ω)),H 1 (0,T ;L2 (Ω)) = 0
Σ
Moreover uH −1 (0,T ;L2 (Ω)) = sup Re f ∂ν v(F ) dσ ; F H 1 (0,T ;L2 (Ω)) = 1 f L2 (Σ) . 0
Σ
But any v ∈ Y (Q), where
Y (Q) = w ∈ C 1 [0, T ]; L2 (Ω) ∩ C [0, T ]; D(PA ) ; (i∂t + A )w ∈ H01 0, T ; L2 (Ω) and w(T , ·) = 0 , is such that v = v(F ) with F = (i∂t + A )v. Therefore, we get
u, (i∂t + A )v
H −1 (0,T ;L2 (Ω)),H01 (0,T ;L2 (Ω))
=
f ∂ν v dσ,
for each v ∈ Y (Q).
Σ
The solution u obtained in this way will be called the transposition solution of the IBVP (1.1). Next we show how ∂ν u can be defined as an element of H −1 (0, T ; H −3/2 (Γ )). To do so, for ϕ ∈ H01 (0, T ; H 3/2 (Γ )), let ψ(t) ∈ H 2 (Ω) be the unique solution of the following boundary value problem
ψ(t) = 0 in Ω, ψ = ϕ(t) on Γ.
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179
Using the classical H 2 elliptic regularity, we easily deduce that ψ belongs to H01 (0, T ; H 2 (Ω)). From Green’s formula, if v is a sufficiently smooth function defined over Q then
∂ν vϕ dσ dt =
Σ
vψ dx dt. Q
Inspired by this formula, we can define ∂ν u as an element of H −1 (0, T ; H −3/2 (Γ )) by duality in the following way, where we used that u ∈ H −1 (0, T ; H −2 (Ω)), ∂ν u, ϕ H −1 (0,T ;H −3/2 (Γ )),H 1 (0,T ;H 3/2 (Γ )) = u, ψ H −1 (0,T ;H −2 (Ω)),H 1 (0,T ;H 2 (Ω)) . 0
0
Indeed, according one more time to the H 2 elliptic estimate, there exist a positive constant K = K(Ω) such that ∂ν u, ϕ H −1 (0,T ;H −3/2 (Γ )),H 1 (0,T ;H 3/2 (Γ )) uH −1 (0,T ;H −2 (Ω)) ψH 1 (0,T ;H 2 (Ω)) 0
0
KuH −1 (0,T ;H −2 (Ω)) ϕH 1 (0,T ;H 3/2 (Γ )) . 0
Since uH −1 (0,T ;L2 (Ω)) Cf L2 (Σ) , for some positive constant C, we get uH −1 (0,T ;H −2 (Ω)) uH −1 (0,T ;L2 (Ω)) Cf L2 (Σ) and then ∂ν u, ϕ H −1 (0,T ;H −3/2 (Γ )),H 1 (0,T ;H 3/2 (Γ )) KCf L2 (Σ) ϕH 1 (0,T ;H 3/2 (Γ )) . 0
0
We summarize this in the following theorem. Theorem 3.2. For any f ∈ L2 (Σ), the IBVP (1.1) has a unique transposition solution u ∈ H −1 (0, T ; L2 (Ω)). In addition, the mapping f → ∂ν u defines a bounded operator from L2 (Σ) into H −1 (0, T ; H −3/2 (Γ )). 4. Geometric optics solutions We construct geometric optics solutions for the magnetic Schrödinger equation. Below, we shall make use the following Green’s formula. For A ∈ W 1,∞ (Ω; Rn ), the following identity holds
(A uv − uA v) dx =
Ω
(∂ν + iν · A)uv − u(∂ν + iν · A)v dσ,
Γ
for all u, v ∈ H 1 (Ω) such that u, v ∈ L2 (Ω).
(4.1)
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Let w = w(t, x) ∈ C 1 ([0, T ]; L2 (Ω)) ∩ C([0, T ]; H 2 (Ω)) satisfies w(0, x) = 0,
x∈Ω
and w(t, x) = 0,
(t, x) ∈ Σ.
(4.2)
Let h be defined on Σ be such that h = H|Σ , where H ∈ C 1 ([0, T ]; L2 (Ω)) ∩ C([0, T ]; H 2 (Ω)) is such that (i∂t + A )H ∈ C([0, T ]; D(PA )). From the results in Section 3, we can easily deduce that there exists a unique solution u1 ∈ C 1 ([0, T ]; L2 (Ω)) ∩ C([0, T ]; H 2 (Ω)) of the following backward magnetic Schrödinger IBVP
(i∂t + A )u1 = 0 in Q, in Ω, u1 (T , ·) = 0 on Σ. u1 = h
(4.3)
w and u1 being as above, we have the following identity
(i∂t w + A w)u1 dx dt = Q
w(i∂t u1 + A u1 ) dx dt − Q
Σ
=−
(∂ν + iν · A)wu1 dσ dt
(∂ν + iν · A)wu1 dσ dt.
(4.4)
Σ
In fact, in order to prove (4.4), we use (4.2), (4.3) and we apply Green’s formula (4.1) to the left-hand side of (4.4). Let R > 0 such that Ω ⊂ B(0, R) and we set DR = B(0, R + 1)\B(0, R). Let φ ∈ C0∞ (Rn ) such that supp(φ) ⊂ DR . In the sequel we assume σ > supp φ ∩ Ω = ∅,
2R+1 T .
(4.5)
Then
(supp φ − σ T ω) ∩ Ω = ∅,
∀ω ∈ Sn−1 .
(supp φ + σ T ω) ∩ Ω = ∅1 , (4.6)
We note that the function Φ given by Φ(t, x) = φ(x − tω) solves in R × Rn the transport equation (∂t + ω · ∇)Φ(t, x) = 0. 1 Indeed, if x ∈ (supp φ − σ T ω) then x is of the form x = y − σ T ω with y ∈ supp φ. Therefore x σ T − y > 2R + 1 − y. Or supp φ ⊂ B(0, R + 1). Hence x > R. This implies that x ∈ / Ω because Ω ⊂ B(0, R). Similarly we prove (supp φ + σ T ω) ∩ Ω = ∅.
M. Bellassoued, M. Choulli / Journal of Functional Analysis 258 (2010) 161–195
Finally, for ω ∈ Sn−1 , we set
t
ω · A(x − sω) ds ,
b(t, x) = exp −i 0
where we extended A by 0 outside Ω. We have
ω · ∇b = −ib
t k
0
ωk
ωj ∂j ak (x − sω) ds = ib
j
t ωk
k
d ak (x − sω) ds
0
= iω · A(x − tω)b − iω · Ab = −∂t b − iω · Ab. Therefore b satisfies (∂t + ω · ∇A )b = (∂t + ω · ∇ + iω · A)b = 0. For ω ∈ Sn−1 , we consider the following subspace of H 2 (Rn ) Hω2 (DR ) = φ ∈ H 2 Rn ; ω · ∇φ ∈ H 2 Rn and supp(φ) ⊂ DR . This space is equipped with its natural norm. Namely Nω (φ) = φH 2 (Rn ) + ω · ∇φH 2 (Rn ) . In the rest of this paper we assume that σ 1. We prove in the present section the following lemma. Lemma 4.1. Fix ω ∈ Sn−1 and let A ∈ W 3,∞ (Ω; Rn ) ∩ A. If φ ∈ Hω2 (DR ) then (i∂t + A )u = 0,
(t, x) ∈ Q,
has a solution of the form u(t, x) = Φ(2σ t, x)b(2σ t, x)eiσ (x·ω−σ t) + ψσ (t, x) satisfying
u ∈ C 1 [0, T ]; L2 (Ω) ∩ C [0, T ]; H 2 (Ω) , where ψσ satisfies ψσ = 0,
for all (t, x) ∈ Σ
and ψσ (0, x) = 0,
x ∈ Ω.
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Moreover σ ψσ L2 (Q) + ∇ψσ L2 (Q) CNω (φ)
(4.7)
where C is a constant depending only on Ω, T , n and A. We have a similar result by replacing above ψσ (0, ·) = 0 by ψσ (T , ·) = 0. Proof. For simplicity, we use the following notations in this proof. Eσ (x, t) = eiσ (x·ω−σ t)
and ϕσ (t, x) = Φ(2σ t, x)b(2σ t, x).
Clearly, ψσ must be a solution of the following IBVP (i∂ + )ψ = G, t
A
ψ(0, ·) = 0, ψ = 0,
in Q, in Ω, on Σ,
(4.8)
where G = −(i∂t + A )(Eσ ϕσ ). Since Eσ and ϕσ are the respective solutions of the following two equations
(i∂t + A )E = −2σ ω · A − |A|2 + i div(A) E, (∂t + 2σ · ∇ + 2iσ ω · A)ϕ = 0, we deduce that G = Eσ A ϕσ . By our assumptions G ∈ H01 (0, T ; L2 (Ω)). Therefore, by the results in Section 3, the IBVP (4.8) has a unique solution
ψσ ∈ C 1 [0, T ]; L2 (Ω) ∩ C [0, T ]; H 2 (Ω) ∩ H01 (Ω) and T ψσ L2 (Q) C
G0 (2σ t, ·)
L2 (Ω)
dt
0
C σ
G0 (s, ·)
L2 (Ω)
ds
R
C φH 2 (Rn ) . σ
Here G0 = A ϕσ . Moreover, in view of (iii) of Lemma 3.2, we have for any > 0, ∇ψσ (t)
L2 (Ω)
T Cε
2
σ G0 (2σ t, ·)L2 (Ω) + σ ∂t G0 (2σ t, ·)L2 (Ω) dt
0
+ε
−1
T
G0 (2σ t, ·)
L2 (Ω)
0
dt.
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183
Choosing ε = σ −1 , we obtain ∇ψσ (t) 2 C L (Ω)
G0 (s, ·)
R
L2 (Ω)
ds +
∂t G0 (s, ·)
L2 (Ω)
C φH 2 (Rn ) + ω · ∇φH 2 (Rn ) . The proof is complete.
ds
R
(4.9)
2
5. Preliminary estimate Let ω ∈ Sn−1 and A1 , A2 ∈ W 3,∞ (Ω; Rn ) ∩ A be given such that Aj W 3,∞ M. We set A = A2 − A1
and for ω ∈ S
n−1
,
t
b(t, x) = (b2 b1 )(t, x) = exp −i
ω · A(x − sω) ds . 0
Here
t
bj (t, x) = exp −i
ω · Aj (x − sω) ds ,
j = 1, 2.
0
Since A1 − A2 = 0 on Γ , the zero extension of A outside Ω, still denoted by A, belongs to H 1 (Rn ). Therefore we can consider dαA as a function in L2 (Rn ) supported in Ω. Lemma 5.1. We assume that σ > 2R/T . Then there exists a constant C = C(A1 , M) > 0 such that for any ω ∈ Sn−1 and φ1 , φ2 ∈ Hω2 (DR ) the following estimate holds T σ ω · A(x)(φ2 φ1 )(x − 2σ tω)b(2σ t, x) dx dt 0 Rn
C σ 2 ΛA1 − ΛA2 + σ −1 Nω (φ1 )Nω (φ2 ).
Proof. Let φ2 ∈ Hω2 (DR ). Then Lemma 4.1 guarantees the existence of the geometric optics solution u2 of the equation (i∂t + A2 )u = 0 in Q, of the form u2 (t, x) = Φ2 (2σ t, x)b2 (2σ t, x)eiσ (x·ω−σ t) + ψ2,σ (t, x), where ψ2,σ satisfies
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M. Bellassoued, M. Choulli / Journal of Functional Analysis 258 (2010) 161–195
σ ψ2,σ L2 (Q) + ∇ψ2,σ L2 (Q) CNω (φ2 ), ψ2,σ = 0,
on Σ,
ψ2,σ (0, ·) = 0,
in Ω,
(5.1)
and
u2 ∈ C 1 [0, T ]; L2 (Ω) ∩ C [0, T ]; H 2 (Ω) . We note that according to (iii) of Lemma 3.2, the constant C in (5.1) can be chosen independent on A2 , but can depend on M and A1 . Let fσ,2 (t, x) = Φ2 (2σ t, x)b2 (2σ t, x)eiσ (x·ω−σ t) ,
x ∈ Γ, t ∈ (0, T )
and we denote by v the solution of the following IBVP ⎧ ⎨ (i∂t + A1 )v = 0, in Q, v(0, ·) = 0, in Ω, ⎩ on Σ. v = u2 := fσ,2 , As we have seen in the preceding section
v ∈ C 1 [0, T ]; L2 (Ω) ∩ C [0, T ]; H 2 (Ω) . Defining w = v − u2 , we get
(i∂t + A1 )w = 2iA · ∇u2 + V (x)u2 , inQ, w(0, ·) = 0, in Ω, w = 0, on Σ,
where V = i div (A) − |A2 |2 + |A1 |2 . We observe that
w ∈ C 1 [0, T ]; L2 (Ω) ∩ C [0, T ]; H 2 (Ω) ∩ H01 (Ω) . Next let u1 ∈ C 1 (0, T ; L2 (Ω)) ∩ C(0, T ; H 2 (Ω)) be a solution of the equation (i∂t + A1 )u = 0,
in Q,
having the form u1 (t, x) = Φ1 (2σ t, x)b1 (2σ t, x)eiσ (x·ω−2σ t) + ψ1,σ (t, x), where ψ1,σ satisfies
(5.2)
M. Bellassoued, M. Choulli / Journal of Functional Analysis 258 (2010) 161–195
185
σ ψ1,σ L2 (Q) + ∇ψ1,σ L2 (Q) CNω (φ1 ), ψ1,σ = 0,
on Σ,
ψ1,σ (T , ·) = 0,
in Ω.
(5.3)
Such a solution exists according to Lemma 4.1. It follows from identity (4.4),
(i∂t + A1 )wu1 dx dt =
Q
2iA · ∇u2 u1 dx dt +
Q
=−
V (x)u2 u1 dx dt Q
(∂ν + iA1 · ν)wu1 dσ dt.
(5.4)
Σ
As A = 0 on Γ , a combination of (5.2) and (5.4) gives
2iA · ∇u2 u1 dx dt + Q
V (x)u2 u1 dx dt = −
Q
(ΛA1 − ΛA2 )(fσ,2 )fσ,1 dσ dt Σ
= − (ΛA1 − ΛA2 )(fσ,2 ), fσ,1 ,
(5.5)
where fσ,1 (t, x) = Φ1 (2σ t, x)b1 (2σ t, x)eiσ (x·ω−2σ t) ,
(t, x) ∈ Σ
and, for simplicity, we set ·,· for the duality pairing between H −1 (0, T ; H −3/2 (Γ )) and H01 (0, T ; H 3/2 (Γ )). We have
2iA · ∇u2 u1 dx dt = −
Q
2σ ω · A(x)(Φ2 Φ 1 )(2σ t, x)(b2 b1 )(2σ t, x) dx dt Q
+
2iA · ∇ Φ2 (2σ t, x)b2 (2σ t, x) Φ 1 (2σ t, x)b1 (2σ t, x) dx dt
Q
+
2iA · ∇ Φ2 (2σ t, x)b2 (2σ t, x) eiσ (x·ω−σ t) ψ 1,σ (t, x) dx dt
Q
+
2iA · ∇ψ2,σ (t, x)Φ 1 (2σ t, x)b1 (2σ t, x)e−iσ (x·ω−σ t) dx dt
Q
+
2iA · ∇ψ2,σ (t, x)ψ 1,σ (t, x) dx dt Q
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M. Bellassoued, M. Choulli / Journal of Functional Analysis 258 (2010) 161–195
−
2σ ω · A(x)b2 (2σ t, x)Φ2 (2σ t, x)ψ 1,σ (t, x)eiσ (x·ω−σ t) dx dt Q
2σ ω · A(x)(φ2 φ 1 )(x − 2σ tω)(b2 b1 )(2σ t, x) dx dt + Iσ . (5.6)
=− Q
Using (5.1) and (5.3), we obtain |Iσ | Cσ −1 Nω (φ2 )Nω (φ1 ).
(5.7)
Consequently, from (5.5), (5.6) and (5.7), we obtain σ ω · A(x)(φ2 φ 1 )(x − 2σ tω)(b2 b1 )(2σ t, x) dx dt Q
−1 C V u2 u1 dx dt + (ΛA1 − ΛA2 )(fσ,2 )fσ,1 dσ dt + σ Nω (φ2 )Nω (φ1 ) . (5.8) Σ
Q
On the other hand (5.1) and (5.3) imply V (x)u2 u1 dx dt Cσ −1 Nω (φ2 )Nω (φ1 )
(5.9)
Q
and we have
(ΛA − ΛA )(fσ,2 )fσ,1 dσx dt = (ΛA − ΛA )(fσ,2 ), fσ,1 1 2 1 2 Σ
ΛA1 − ΛA2 fσ,2 L2 (Σ) fσ,1 H 1 (0,T ;H 3/2 (Γ )) Cσ 2 Nω (φ1 )Nω (φ2 )ΛA1 − ΛA2 .
(5.10)
From (5.8), (5.9) and (5.10), we derive T σ ω · A(x)(φ2 φ 1 )(x − 2σ tω)b(2σ t, x) dx dt 0 Rn
C σ 2 Nω (φ1 )Nω (φ2 )ΛA1 − ΛA2 + σ −1 Nω (φ1 )Nω (φ2 ) .
This completes the proof of the lemma.
2
(5.11)
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6. An intermediate estimate We establish a stability estimate for the X-ray transform of a function related to our main result. We shall use the same notations and assumptions as in the previous sections. We recall that the X-ray transform of a function f , defined on Rn , is given by P(f )(ω, x) := f (x + sω) ds, ω ∈ Sn−1 , x ∈ Rn . R
We can see that P(f )(ω, x) represents the integral of f over the straight line passing through x in the direction of ω and we observe that P(f )(ω, x) does not change if x is moved in the direction of ω. Therefore we restrict x to ω⊥ = {θ ∈ Rn ; θ · ω = 0} and we can consider P(f ) as a function on the tangent bundle T = {(ω, x): ω ∈ Sn−1 , x ∈ ω⊥ } (e.g., Natterer [36]). The following so-called projection slice theorem shows how the Fourier transform of the parallel beam radiograph can be obtained from the Fourier transform of the X-ray attenuation density. Hereafter, fˆ denotes the Fourier transform of the function f − n2 ˆ f (x)e−ix.ξ dx. f (ξ ) = (2π) Rn
Lemma 6.1. Let f ∈ L1 (Rn ) and ω ∈ Sn−1 . Then Pf (ω, .) ∈ L1 (ω⊥ ) and √
Pf (ω, .) ˆ(ξ ) := e−ix·ξ P(f )(ω, x) dx = 2π fˆ(ξ ) ω⊥
for all ξ ∈ ω⊥ . Proof. Lemma 6.1 was proved in [36]. For the sake of completeness we recall briefly its proof. Obviously, Pf (ω, x) dx f (x + tω) dt dx = f 1 < +∞. ω⊥ R
ω⊥
For f ∈ L1 (Rn ), the change of variable y = x + tω ∈ ω⊥ ⊕ Rω = Rn yields dy = dx dt and, after noting that ξ ∈ ω⊥ implies x · ξ = x · ξ + tω · ξ = y · ξ ,
n−1 Pf (ω, .) ˆ(ξ ) = (2π)− 2
ω⊥ R
f (x + tω)e−ix.ξ dtdx
√ √ − n2 f (y)e−iy.ξ dy = 2π fˆ(ξ ). = 2π(2π) Rn
This completes the proof.
2
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We introduce the notation ∂ai ∂A (x) = ωi (x), ρj (x) = ω · ∂xj ∂xj n
x ∈ Rn .
(6.1)
i=1
For ω ∈ Sn−1 , let D− R (ω) = {x ∈ DR , x · ω < 0}. Here DR is as in Section 4. That is DR = B(0, R + 1)\B(0, R). We prove in this section the following lemma. Lemma 6.2. Let σ0 = 2R/T . Then there exists a constant C = C(A1 , M) > 0 such that for all 2 ω ∈ Sn−1 and φ ∈ Hω2 (DR ) satisfying supp(φ) ⊂ D− R (ω) and ∂j φ ∈ Hω (DR ), j = 1, . . . , n, the following estimate φ 2 (x)P(ρj )(ω, x) exp −i ω · A(x + sω) ds dx Rn
R
1 2 C σ ΛA1 − ΛA2 + Nω (φ)Nω (∂j φ) σ
holds for any σ > σ0 and j ∈ {1, . . . , n}. Proof. Let φ1 , φ2 ∈ Hω2 (DR ) such that supp(φj ) ⊂ D− R (ω), we have T σ ω · A(x)(φ2 φ 1 )(x − 2σ tω)b(2σ t, x) dx dt 0 Rn
T =
2σ t σ ω · A(x)(φ2 φ 1 )(x − 2σ tω) exp −i ω · A(x − sω) ds dx dt
0 Rn
0
T =
σ ω · A(x + 2σ tω)(φ2 φ 1 )(x)b(2σ t, x + 2σ tω) dx dt 0
Rn
=
T (φ2 φ 1 )(x)
Rn
i = 2 i = 2
2σ t σ ω · A(x + 2σ tω) exp −i ω · A(x + sω) ds dx dt
0
0
T
(φ2 φ 1 )(x) Rn
d exp −i dt
0
ω · A(x + sω) ds dt dx 0
2σ T
(φ2 φ 1 )(x) exp −i Rn
2σ t
ω · A(x + sω) ds − 1 dx. 0
(6.2)
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We choose φ1 and φ2 such that φ2 = φ, φ1 = ∂j φ, j = 1, . . . , n. By an application of Green’s formula, (6.2) yields T σ ω · A(x)(φ2 φ 1 )(x − 2σ tω)b(2σ t, x) dx dt 0
Rn
i =− 2 1 =− 2
2σ T ∂ φ (x) ω · A(x + sω) ds dx exp −i ∂xj 2
Rn
0
∂ φ (x) ∂xj
Rn
2σ T
0
2σ T
ω · A(x + sω) ds exp −i
2
ω · A(x + sω) ds dx. 0
Since the support of A is contained in B(0, R), we have
2σ T
ω · A(x + sω) ds =
ω · A(x + sω) ds
(6.3)
R
0
/ B(0, R), for when x ∈ D− R (ω). In fact, for s 2σ T and x ∈ DR it is easy to see that (x + sω) ∈ any σ > σ0 . So 2σ T
∞ ω · A(x + sω) ds =
0
ω · A(x + sω) ds. 0
2 2 2 2 On the other hand, if s 0 and x ∈ D− R , we get |x + sω| = |x| + s + 2sx · ω R and then A(x + sω) = 0. This way, (6.3) is obtained. Substituting (6.3) into Eq. (6.2), we obtain
T σ ω · A(x)(φ2 φ1 )(x − 2σ tω)b(2σ t, x) dx dt 0
Rn
1 =− 2 1 =− 2
∂ φ (x) ∂xj
2
Rn
R
ω · A(x + sω) ds exp −i ω · A(x + sω) ds dx
φ 2 (x)P(ρj )(ω, x) exp −i Rn
R
ω · A(x + sω) ds dx,
(6.4)
R
where ρj is given by (6.1). By (6.4) and the estimate from Lemma 5.1, we conclude that for any σ > σ0 ,
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φ 2 (x)P(ρj )(ω, x) exp −i ω · A(x + sω) ds dx Rn
R
C σ 2 ΛA1 − ΛA2 + Nω (φ)Nω (∂j φ) . σ
The proof is then complete.
2
7. Estimate for the magnetic potential In this last section we use the estimate of the preceding section to prove Theorem 2.1 and its corollary. We shall use the following notations. For x0 ∈ ω⊥ ∩B(0, R +1/2) we have B(x0 , 1/2)∩ω⊥ ⊂ B(0, R + 1) ∩ ω⊥ . Let rx0 =
3 2 R+ − |x0 |2 , 4
and x1 = x0 − rx0 ω. n It is not difficult to check that B(x1 , 1/4) ⊂ D− R (ω) = {x ∈ R ; x · ω < 0}. We start with an estimate for the Fourier transform of βij ,
∂aj ∂ai (x) − (x), ∂xj ∂xi
βij (x) =
i, j = 1, . . . , n.
Lemma 7.1. Let σ0 = 2R/T . Then there exists a constant C = C(A1 , M) > 0 such that the following estimate 1 β ij (ξ ) C σ 2 ΛA1 − ΛA2 + ξ 5 σ holds for all σ > σ0 and ξ ∈ Rn , where ξ =
1 + |ξ |2 .
Proof. We fix x0 ∈ ω⊥ ∩ B(0, R + 1/2). Let h ∈ C0∞ ((0, 1/8)) such that h2 (t) dt = 1 R
and let φ0 ∈ C0∞ (ω⊥ ∩ B(x0 , 1/8)) and φ0 0. We put i
i y − (y · ω)ω exp ω · A(y + sω) ds . 2
1/2
φ(y) = h(y · ω + rx0 )e− 2 y·ξ φ0
R
(7.1)
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Then we have supp(φ) ⊂ B(x1 , 1/4) ⊂ D− R (ω). We observe that ω·∇ R
∂A ω· (y + sω) ds = 0 ∂xj
(7.2)
and since A ∈ W 3,∞ (Ω; Rn ), we conclude that φ and ∂j φ ∈ Hω2 (DR ) for any j = 1, . . . , n. The change of variable y = x + tω ∈ ω⊥ ⊕ Rω, dy = dx dt yields, after noting that ξ ∈ ω⊥ ,
φ 2 (x)P(ρj )(ω, x) exp −i ω · A(x + sω) ds dx
Rn
=
R
φ (x + tω)P(ρj )(ω, x + tω) exp −i ω · A(x + sω) ds dx dt 2
R ω⊥
=
R
h2 (t)e−ix·ξ φ0 (x)P(ρj )(ω, x) dx dt
R ω⊥
=
e−ix·ξ φ0 (x)P(ρj )(ω, x) dx.
(7.3)
ω⊥
Then from (7.3) and the estimate in Lemma 6.2 we deduce e−ix·ξ φ0 (x)P(ρj )(ω, x) dx C σ 2 ΛA − ΛA + 1 Nω (φ)Nω (∂j φ). 1 2 σ ω⊥
As φ is given by (7.1) and Nω (φ) = φH 2 (Rn ) + ω · ∇φH 2 (Rn ) , an elementary calculation gives, for any ξ ∈ ω⊥ , Nω (φ)Nω (∂j φ) C ξ 5 , i
where used (7.2). We note that the term ξ 5 comes from the exponential factor e− 2 y·ξ in formula (7.1). From the last two inequalities we derive that, for any ξ ∈ ω⊥ , e−ix·ξ P(ρj )(ω, x) dx C σ 2 ΛA − ΛA + 1 ξ 5 . (7.4) 1 2 σ ω⊥
Consequently, it follows from (7.4) and the identity in Lemma 6.1 that, for any ξ ∈ ω⊥ , 1 ρ j (ξ ) C σ 2 ΛA1 − ΛA2 + ξ 5 . σ
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Since ρ j (ξ ) =
n
i (ξ ) = ω i ξj A
i=1
n
n
ij (ξ ) ωi ξj Ai (ξ ) − ξi Aj (ξ ) = ωi β
i=1
for all ξ ∈ ω⊥ ,
i=1
and ω ∈ Sn−1 is arbitrary, we get, for any ξ ∈ Rn , 1 β ij (ξ ) C σ 2 ΛA1 − ΛA2 + ξ 5 . σ This completes the proof of Lemma 7.1.
(7.5)
2
We can now terminate the proof of Theorem 2.1. In the sequel C denotes a generic constant that can depend only on A1 , M, Ω, T and n. Using (7.5) we get |ξ |γ
1 2 β ij (ξ )2 ξ −2 dξ C σ 2 ΛA1 − ΛA2 + ξ 8 dξ σ |ξ |γ
1 2 n+8 C σ 2 ΛA1 − ΛA2 + γ . σ
On the other hand, we assumed that Aj W 3,∞ (Rn ) M, j = 1, 2. Hence ij L2 (Rn ) = βij L2 (Ω) C. β Therefore βij 2H −1 (Rn )
=
β ij (ξ )2 ξ −2 dξ +
β ij (ξ )2 ξ −2 dξ
|ξ |>γ
|ξ |γ
C σ γ
4 n+8
γ n+8 1 ΛA1 − ΛA2 + 2 + 2 . σ γ 2
Choosing σ 2 = γ n+10 ,
(7.6)
we obtain 1 βij H −1 (Rd ) C γ k ΛA1 − ΛA2 + γ where k = 32 n+14. The argument above is valid if σ > σ0 . By (7.6) we need to take γ sufficiently 1
large. So there exists a m > 0 such that if ΛA1 − ΛA2 < m and γ = ΛA1 − ΛA2 − k+1 we have σ σ0 and by (7.6) we obtain βij H −1 (Rn ) CΛA1 − ΛA2 κ ,
(7.7)
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where κ = 1/(k + 1). Now if ΛA1 − ΛA2 m. Then we have βij H −1 (Rn ) βij L∞ (Rn ) 2AW 3,∞ (Rn )
4M m1/(k+1)
m1/(k+1)
4M ΛA1 − ΛA2 κ , mμ
(7.8)
where we used the a priori bound AW 3,∞ (Rn ) A1 W 3,∞ (Rn ) + A2 W 3,∞ (Rn ) 2M. Therefore, (7.7) also holds. Thus it follows from (7.8) that dαA1 − dαA2 H −1 (Ω)
βij H −1 (Rn ) CΛA1 − ΛA2 κ .
i,j
This ends the proof of Theorem 2.1. Corollary (2.1) is now an easy consequence of an interpolation inequality and Theorem 2.1. Let δ > 0 such that s − 1 = n2 + 2δ. By Sobolev’s embedding theorem, since Aj ∈ W 3,∞ (Ω, Rn ), we obtain dαA1 − dαA2 L∞ (Ω) CdαA1 − dαA2
H
n +δ 2 (Rn )
1−β
β
CdαA1 − dαA2 H −1 (Rn ) dαA1 − dαA2 H s−1 (Rn )
CΛA1 − ΛA2 μ , which yields the desired estimate with κ =
κ(s−1−n/2) . 2s
Acknowledgment The authors thank the referee for his valuable remarks which have improved the presentation of this paper. References [1] S. Avdonin, S. Lenhart, V. Protopopescu, Determining the potential in the Schrödinger equation from the Dirichlet to Neumann map by the Boundary Control method, J. Inverse Ill-Posed Probl. 13 (5) (2005) 317–330. [2] L. Baudouin, J.-P. Puel, Uniqueness and stability in an inverse problem for the Schröndinger equation, Inverse Problems 18 (2002) 1537–1554. [3] M. Belishev, Boundary control in reconstruction of manifolds and metrics (BC method), Inverse Problems 13 (1997) R1–R45. [4] M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients, Appl. Anal. 83 (2004) 983–1014. [5] M. Bellassoued, H. Benjoud, Stability estimate for an inverse problem for the wave equation in a magnetic field, Appl. Anal. 87 (3) (2008) 277–292. [6] M. Bellassoued, M. Choulli, Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by an arbitrary boundary observation, J. Math. Pures Appl. 91 (3) (2009) 233–255. [7] M. Bellassoued, D. Jellali, M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal. 85 (2006) 1219–1243. [8] M. Bellassoued, D. Jellali, M. Yamamoto, Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, preprint.
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Journal of Functional Analysis 258 (2010) 196–207 www.elsevier.com/locate/jfa
Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations ✩ Jing-huai Liu ∗ , Xiao-qiu Song College of Science, China University of Mining and Technology, Xuzhou, 221008, China Received 13 March 2009; accepted 4 June 2009 Available online 24 June 2009 Communicated by J. Coron
Abstract In this paper, applying the theory of semigroups of operators to evolution family and Banach fixed point theorem, we prove the existence and uniqueness of an (a) almost automorphic (weighted pseudo almost automorphic) mild solution of the semilinear evolution equation x (t) = A(t)x(t) + f (t, x(t)) in Banach space under conditions. © 2009 Elsevier Inc. All rights reserved. Keywords: Almost automorphic; Weighted pseudo almost automorphic; Semilinear evolution equation; Evolution family; Exponentially stable
1. Introduction Almost automorphic functions are more general than almost periodic functions and they were introduced by Bochner. For more details about this topics we refer to the recent book [17] where the author gave an important overview about the theory of almost automorphic functions and their applications to differential equations. Almost automorphic solutions in the context of differential equations had been studied by several authors [10–12,16]. N’Guérékata [9] and Xiao [13,19,20] ✩
Supported by the Science and Technology Foundation of China University of Mining and Technology (No. OK060156). * Corresponding author. E-mail addresses:
[email protected] (J.-h. Liu),
[email protected] (X.-q. Song). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.007
J.-h. Liu, X.-q. Song / Journal of Functional Analysis 258 (2010) 196–207
197
established existence and uniqueness theorems of pseudo almost automorphic solutions to some semilinear abstract differential equations. Recently, N’Guérékata [3] introduced the concept of weighted pseudo almost automorphic, which generalizes the one of weighted pseudo almost periodicity [4–7,21], and the author proved some interesting properties of the space of weighted pseudo almost automorphic functions like the completeness and the composition theorem, which have many applications in the context of differential equations. The purpose of this paper is to deal with the existence and uniqueness of an (a) almost automorphic (weighted pseudo almost automorphic) mild solution of the following semilinear evolution equation in a Banach space X x (t) = A(t)x(t) + f t, x(t) ,
t ∈ R, x ∈ J,
(1)
where J ∈ {AA(X), WPAA(R, ρ)}, AA(X)(WPAA(R, ρ)) is the set of all almost automorphic (weighted pseudo almost automorphic) functions from R to X and the family {A(t), t ∈ R} of operators in X generates an exponentially stable evolution family {U (t, s), t s}. 2. Preliminaries In this section we recall some definitions and fix notations which will be used in the sequel. We assume that X is a Banach space endowed with the norm · . N, R and C stand for the sets of positive integer, real and complex numbers. We denote by B(X) the Banach space of all bounded linear operators from X to itself. Cb (R, X)(Cb (R × X, X)) is the space of all bounded continuous functions from R to X (R ×X to X). U stands for the collection of functions (weights) ρ : R → (0, ∞), which are locally integrable over R with ρ > 0 (a.e.). For a given r > 0 and each ρ ∈ U, set r m(r, ρ) =
ρ(x) dx.
−r
Let U∞ denote the set of all ρ ∈ U with limr→∞ m(r, ρ) = ∞ and Ub denote the set of all ρ ∈ U∞ such that ρ is bounded with infx∈R ρ(x) > 0. {U (t, s), t s} is an exponentially stable evolution family, if there exists M 1 and δ > 0 such that U (t, s) Me−δ(t−s) ,
for t s.
Definition 2.1. A continuous function f : R → X is said to be almost automorphic if for every sequence of real numbers {sn }n∈N , there exists a subsequence {τn }n∈N such that limm→∞ limn→∞ f (t + τn − τm ) = f (t), for each t ∈ R. This limit means that g(t) = limn→∞ f (t + τn ) is well defined for each t ∈ R, and limn→∞ g(t − τn ) = f (t), for each t ∈ R. Denote by AA(X) the set of all such functions. Definition 2.2. A continuous function f : R × X → X is said to be almost automorphic if f (t, x) is almost automorphic for each t ∈ R uniformly for all x ∈ K, where K is any bounded subset of X. That is to say, for every sequence of real numbers {sn }n∈N , we can extract a subsequence {τn }n∈N such that g(t, x) = limn→∞ f (t + τn , x) is well defined for each t ∈ R for all x ∈ K, and limn→∞ g(t − τn , x) = f (t, x) for each t ∈ R and x ∈ K. Denote by AA(R × X, X) the set of all such functions.
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Lemma 2.3. (See [15].) (AA(X), · AA(X) ) is a Banach space with the supremum norm given by f AA(X) = supf (t). t∈R
Lemma 2.4. (See [15].) Assume that f : R → X is almost automorphic, then f is bounded. Lemma 2.5. (See [8].) Let f : R × X → X be almost automorphic in t ∈ R, x ∈ X and assume that f (t, x) satisfies a Lipschitz condition in x uniformly in t ∈ R. Then x(t) ∈ AA(X) implies f (t, x(t)) ∈ AA(X). Set
1 PAA0 (X) = ϕ(t) ∈ Cb (R, X): lim r→∞ 2r
PAA0 (R × X, X) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
r
ϕ(t) dt = 0 ;
−r
1 ϕ(t, x) ∈ Cb (R × X, X): lim r→∞ 2r
r
⎫ ⎪ ⎬ ϕ(t, x) dt = 0 ⎪
−r
uniformly for x in any bounded subset of X
⎪ ⎪ ⎭
.
Definition 2.6. (See [14].) A continuous function f : R → X(R × X → X) is said to be pseudo almost automorphic if it can be decomposed as f = g + ϕ, where g ∈ AA(X)(AA(R × X, X)) and ϕ ∈ PAA0 (X)(PAA0 (R × X, X)). Denote by PAA(X)(PAA(R × X, X)) the set of all such functions. Now for ρ ∈ U∞ define
1 PAA0 (R, ρ) = ϕ(t) ∈ Cb (R, X): lim r→∞ m(r, ρ)
r
ϕ(t)ρ(t) dt = 0 ;
−r
⎫ ⎧ r ⎪ ⎪ ⎪ ⎪ 1 ⎨ ϕ(t, x) ∈ C (R × X, X): lim ϕ(t, x)ρ(t) dt = 0 ⎬ b r→∞ m(r, ρ) . PAA0 (R × X, ρ) = ⎪ ⎪ −r ⎪ ⎪ ⎭ ⎩ uniformly for x ∈ X Definition 2.7. (See [3].) A bounded continuous function f : R → X(R × X → X) is said to be weighted pseudo almost automorphic if it can be decomposed as f = g + ϕ, where g ∈ AA(X)(AA(R × X, X)) and ϕ ∈ PAA0 (R, ρ)(PAA0 (R × X, ρ)). Denote by WPAA(R, ρ)(WPAA(R × X, ρ)) the set of all such functions.
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Remark 2.8. (See [3].) When ρ = 1, we obtain the standard spaces PAA(X) and PAA(R × X, X). Lemma 2.9. (See [3].) The decomposition of a weighted pseudo almost automorphic function is unique for any ρ ∈ Ub . Lemma 2.10. (See [3].) If ρ ∈ Ub , (WPAA(R, ρ), · WPAA(R,ρ) ) is a Banach space with the supremum norm given by f WPAA(R,ρ) = supf (t). t∈R
Lemma 2.11. (See [3].) Let f = g + ϕ ∈ WPAA(R, ρ) where ρ ∈ U∞ , g ∈ AA(R × X, X) and ϕ ∈ PAA0 (R × X, ρ). Assume both f and g are Lipschitzian in x ∈ X uniformly in t ∈ R. Then x(t) ∈ WPAA(R, ρ) implies f (t, x(t)) ∈ WPAA(R, ρ). From [1, Theorem 2.3] and [2], we have the following lemma: Lemma 2.12. Let Σθ = {λ ∈ C: |arg λ| θ } ∪ {0} ⊂ ρ(A(t)), θ ∈ ( π2 , π) if there exist a constant k0 and a set of real numbers α1 , α2 , . . . , αk , β1 , . . . , βk with 0 βi < αi 2, i = 1, 2, . . . , k such that k A(t) λ − A(t) −1 A(t)−1 − A(s)−1 K0 (t − s)αi |λ|βi −1 , i=1
for t, s ∈ R, λ ∈ Σθ \ {0} and there exists a constant M 0 such that λ − A(t) −1
M , 1 + |λ|
λ ∈ Σθ .
Then there exists a unique evolution family {U (t, s), t s > −∞}. Definition 2.13. A mild solution to Eq. (1) is a continuous function x(t) : R → X satisfying x(t) = U (t, s) x(s) +
t
U (t, r)f r, x(r) dr,
(2)
s
for t s and all s ∈ R. Remark 2.14. If {U (t, s), t s > −∞} is exponentially stable, then there exist M 1 and δ > 0 such that U (t, s) Me−δ(t−s) , for t s. When s → −∞, Eq. (2) can be replaced by t x(t) = −∞
U (t, s)f s, x(s) ds,
t ∈ R.
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3. Almost automorphic mild solutions In this section we consider the following conditions: (A1 ) The function f (t, x) is almost automorphic; (A2 ) The function f (t, x) is Lipschitz continuous, that is, there exists a positive number Lf such that f (t, x) − f (t, y) Lf x − yAA(X) , for all t ∈ R and x, y ∈ AA(X); (A3 ) U (t, s), t s, is an exponentially stable evolution family on X; (A4 ) For every sequence of real numbers {sn }n∈N , there exists a subsequence {τn }n∈N and for any fixed s ∈ R, > 0, there exists an N ∈ N such that, for all n > N , it follows that U (t + τn , s + τn ) − U (t, s) e− 2δ (t−s) , for all t s ∈ R; Moreover U (t − τn , s − τn ) − U (t, s) e− 2δ (t−s) , for all t s ∈ R. We define a mapping F by t (F x)(t) =
U (t, s)f s, x(s) ds,
t ∈ R.
−∞
Lemma 3.1. If x(s) is almost automorphic, then function F x is almost automorphic. Proof. First we observe that F is bounded. By condition (A1 ) and Lemma 2.4, f (t, x(t)) is almost automorphic and hence it is bounded, we assume that there exists M1 > 0, such that f (t, x(t))AA(X) M1 . So (F x)(t)
t
U (t, s)f s, x(s) ds
−∞
t
Me−δ(t−s) M1 ds
−∞
M M1 < ∞. δ Hence F x is bounded. Now we show that (F x)(t) is almost automorphic with respect to t ∈ R. By condition (A1 ) the function f (t, x(t)) is almost automorphic, that is f (t, x(t)) ∈ AA(R × X, X) by condition (A2 ) and Lemma 2.5, f (t, x(t)) ∈ AA(X), we define f (t, x(t)) = H (t). Now let {sn } ∈ N be an arbitrary sequence of real numbers. Since H (t) ∈ AA(X), there exists a subsequence {τn }n∈N such that (H1 ) g(t) = limn→∞ H (t + τn ) is well defined for each t ∈ R; (H2 ) limn→∞ g(t − τn ) = H (t) for each t ∈ R.
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Now consider t+τ n
(F x)(t + τn ) =
U (t + τn , s)H (s) ds −∞
t =
U (t + τn , s + τn )H (s + τn ) ds.
−∞
So we have (F x)(t + τn )
t
U (t + τn , s + τn )H (s + τn ) ds
−∞
t
Me−δ(t−s) H (s + τn ) ds
−∞
M1
M , δ
for all n = 1, 2, . . . .
For (H1 ), for any fixed s ∈ R and > 0, there exists an N1 ∈ N such that, for all n > N1 , it follows that H (s + τn ) − g(s) < . In addition, by condition (A4 ), for s and above, there exists an N2 ∈ N such that, for all n > N2 , it follows that U (t + τn , s + τn ) − U (t, s) < e− 2δ (t−s) . Thus, taking N = max{N1 , N2 }, for all n > N , U (t + τn , s + τn )H (s + τn ) − U (t, s)g(s) U (t + τn , s + τn ) − U (t, s)H (s + τn ) + U (t, s)H (s + τn ) − g(s) δ
M1 e− 2 (t−s) + M e−δ(t−s) . As n → ∞, we have U (t + τn , s + τn )H (s + τn ) → U (t, s)g(s) for each s ∈ R fixed and any t s. Notice that U (t + τn , s + τn )H (s + τn ) Me−δ(t−s) H ,
for t s.
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So, by the Lebesgue’s dominated convergent theorem, we get (F x)(t + τn ) → (Gx)(t)
as n → ∞,
t where (Gx)(t) = −∞ U (t, s)g(s) ds, t ∈ R. We can show in a similar way that (Gx)(t − τn ) → (F x)(t)
as n → ∞,
for each t ∈ R. This shows that F x ∈ AA(X). This completes the proof.
2
Theorem 3.2. Let f (t, x) and U (t, s) satisfy all the conditions of (A1 )–(A4 ). Then Eq. (1) has a unique almost automorphic mild solution whenever Lf < Mδ . Proof. By Lemma 3.1, we have F x ∈ AA(X). For x, y ∈ AA(X), it is easy to check that t t (F x)(t) − (Fy)(t) = U (t, s)f s, x(s) ds − U (t, s)f s, y(s) ds −∞
t
−∞
U (t, s)f s, x(s) − f s, y(s) ds
−∞
t Lf x − yAA(X)
Me−δ(t−s) ds
−∞
=
M Lf x − yAA(X) . δ
So we have (F x)(t) − (Fy)(t)
AA(X)
M Lf x − yAA(X) . δ
For 0 < Mδ Lf < 1 and by Banach contraction principle, F has a unique fixed point x ∈ AA(X), such that F x = x. Fixing s ∈ R, we have t
U (t, r)f r, x(r) dr.
x(t) = −∞
Since U (t, s) = U (t, r)U (r, s), for t r s (see [18, Chapter 5, Theorem 5.2]), let x(a) = a −∞ U (a, s)f (s, x(s)) ds. So a U (t, a)x(a) = −∞
U (t, s)f s, x(s) ds.
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For t a, t
t
U (t, s)f s, x(s) ds =
U (t, s)f s, x(s) ds −
−∞
a
a
U (t, s)f s, x(s) ds
−∞
= x(t) − U (t, a)x(a). So that t x(t) = U (t, s)x(a) +
U (t, s)f s, x(s) ds.
a
It follows that x(t) satisfies Eq. (2). Hence x(t) is a mild solution of Eq. (1). In conclusion, x(t) is the unique mild solution to Eq. (1), this completes the proof.
2
Remark 3.3. Using arguments similar to those in this paper we can deal with the existence and uniqueness of an almost automorphic solution for x (t) = Ax(t) + f t, x(t) ,
t ∈ R, x ∈ AA(X),
where A is the infinitesimal generator of a C0 -semigroup {T (t)}t0 . Here our evolution operator becomes U (t, s) = T (t − s). In this case we have the mild solution given by t x(t) =
T (t − s)f s, x(s) ds,
for t ∈ R.
−∞
4. Weighted pseudo almost automorphic mild solutions In this section we consider the following conditions (A3 ), (A4 ) and: (B1 ) The function f (t, x) is weighted pseudo almost automorphic; (B2 ) The function f (t, x) is Lipschitz continuous, that is, there exists a positive number Lf such that f (t, x) − f (t, y) Lf x − yWPAA(R,ρ) , for all t ∈ R and x, y ∈ WPAA(R, ρ), ρ ∈ U∞ . We also define a mapping V by t (V x)(t) = −∞
U (t, s)f s, x(s) ds,
t ∈ R.
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Lemma 4.1. If x(s) is weighted pseudo almost automorphic, then function V x is weighted pseudo almost automorphic. Proof. By condition (B1 ), the function f (t, x(t)) is weighted pseudo almost automorphic, that is f (t, x(t)) ∈ WPAA(R × X, ρ), by condition (B2 ) and Lemma 2.11, f (t, x(t)) ∈ WPAA(R, ρ). Let f t, x(t) = g(t) + ϕ(t), where g ∈ AA(X) and ϕ ∈ PAA0 (R, ρ). Then t (V x)(t) =
U (t, s)f s, x(s) ds
−∞
t =
t U (t, s)g(s) ds +
−∞
U (t, s)ϕ(s) ds.
−∞
Let t v(t) =
t U (t, s)g(s) ds
and w(t) =
−∞
U (t, s)ϕ(s) ds.
−∞
Similar as in the proof of Lemma 3.1, it follows that v ∈ AA(X). So v is almost automorphic. In order to show that V x ∈ WPAA(R, ρ), we need to show that w(t) ∈ PAA0 (R, ρ), that is, we need to show that 1 lim r→∞ m(r, ρ)
r
w(t)ρ(t) dt = 0.
−r
We have 1 0 lim r→∞ m(r, ρ)
r −r
w(t)ρ(t) dt lim
1 r→∞ m(r, ρ)
1 = lim r→∞ m(r, ρ)
r t
Me−δ(t−s) ϕ(s)ρ(s) ds dt
−r −∞
r
−r dt
−r
1 + lim r→∞ m(r, ρ)
Me−δ(t−s) ϕ(s)ρ(s) ds
−∞
r
t dt
−r
−r
Me−δ(t−s) ϕ(s)ρ(s) ds.
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205
Now let 1 I1 = lim r→∞ m(r, ρ)
r
−r dt
−r
Me−δ(t−s) ϕ(s)ρ(s) ds.
−∞
By using the Fubini theorem, one has 1 I1 = lim r→∞ m(r, ρ)
−r e
ϕ(s) ds
δs
−∞
r
Me−δt ρ(t) dt
−r
−r M −δr 1 δr sup ϕ(t) ρL1 (R) e −e eδs ds lim Loc r→∞ m(r, ρ) t∈R δ −∞
lim
r→∞
M 1 supϕ(t)ρL1 (R) 2 e−2δr Loc m(r, ρ) t∈R δ
−I .
Since ϕ(t) is bounded and limr→∞ m(r, ρ) = ∞, then I1 = 0. Also let 1 I2 = lim r→∞ m(r, ρ)
r −r
ϕ(t)ρ(t) dt
t
Me−δ(t−s) ds
−r
r M −δ(t−s) t 1 ϕ(t)ρ(t) dt e = lim r→∞ m(r, ρ) δ −r −r
r M 1 −δ(t+r) ϕ(t)ρ(t) dt. = lim I −e r→∞ m(r, ρ) δ −r
Since −r t r and δ > 0, then Mδ (I − e−δ(t+r) ) is bounded. Furthermore, ϕ ∈ PAA0 (R, ρ), then I2 = 0. This completes the proof. 2 Theorem 4.2. Let f (t, x) and U (t, s) satisfy all the conditions of (B1 ), (B2 ), (A3 ), (A4 ) and 0 < Mδ Lf < 1. Then Eq. (1) has a unique weighted pseudo almost automorphic mild solution. Proof. By Lemma 4.1, we can see V maps WPAA(R, ρ) into WPAA(R, ρ). Let x, y ∈ WPAA(R, ρ), and observe t t (V x)(t) − (V y)(t) = U (t, s)f s, x(s) ds − U (t, s)f s, y(s) ds −∞
t −∞
−∞
U (t, s)f s, x(s) − f s, y(s) ds
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t Lf x − yWPAA(R,ρ)
Me−δ(t−s) ds
−∞
M Lf x − yWPAA(R,ρ) . δ
So we have M (V x)(t) − (V y)(t) Lf x − yWPAA(R,ρ) . WPAA(R,ρ) δ For 0 < Mδ Lf < 1, V is a contractive mapping. By Lemma 2.10, WPAA(R, ρ) is complete. Therefore, by the Banach fixed point theorem, V has a unique fixed point x ∈ WPAA(R, ρ) such that V x = x. Fixing s ∈ R, we have t x(t) =
U (t, r)f r, x(r) dr.
−∞
Since U (t, s) = U (t, r)U (r, s), for t r s (see [18, Chapter 5, Theorem 5.2]), similar as in the proof of Theorem 3.2, it follows that x(t) satisfies Eq. (2). Hence x(t) is the unique mild solution of Eq. (1). 2 Remark 4.3. When ρ = 1, we can deal with the existence and uniqueness of a pseudo almost automorphic solution for x (t) = A(t)x(t) + f (t, x(t)), t ∈ R, x ∈ PAA(X). In this case we have t the mild solution given by x(t) = −∞ U (t, s)f (s, x(s)) ds, for t ∈ R. Remark 4.4. When U (t, s) = T (t − s), we can deal with the existence and uniqueness of a weighted pseudo almost automorphic solution for x (t) = Ax(t) + f (t, x(t)), t ∈ R, x ∈ WPAA(R, ρ), where A is the infinitesimal generator of a C0 -semigroup {T (t)}t0 . In this case t we have the mild solution given by x(t) = −∞ T (t − s)f (s, x(s)) ds, for t ∈ R. Remark 4.5. When ρ = 1, U (t, s) = T (t − s), we can also deal with the existence and uniqueness of a pseudo almost automorphic solution for x (t) = t Ax(t) + f (t, x(t)), t ∈ R, x ∈ PAA(X). In this case we have the mild solution given by x(t) = −∞ T (t − s)f (s, x(s)) ds, for t ∈ R. Acknowledgment The authors are grateful to the referee for the valuable comments and suggestions. References [1] P. Acquistapace, Evolution operators and strong solution of abstract linear parabolic equations, Differential Integral Equations 1 (1998) 433–457. [2] P. Acquistapace, B. Terreni, A unified approach to abstract linear parabolic equations, Rend. Sem. Mat. Univ. Padova 78 (1987) 47–107.
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[3] J. Blot, G.M. Mophou, G.M. N’Guérékata, D. Pennequin, Weighted pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Anal. 71 (2009) 903–909. [4] N. Boukli-Hacene, K. Ezzinbi, Weighted pseudo almost periodic solutions for some partial functional differential equations, Nonlinear Anal. 71 (2009) 3612–3621. [5] T. Diagana, Weighted pseudo almost periodic functions and applications, C. R. Acad. Sci. Paris Ser. I 343 (10) (2006) 643–646. [6] T. Diagana, Weighted pseudo almost periodic solutions to some differential equations, Nonlinear Anal. 68 (2008) 2250–2260. [7] T. Diagana, Existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations, J. Math. Anal. Appl. 350 (2009) 18–28. [8] T. Diagana, H.R. Henriquez, E.M. Hernández, Almost automorphic mild solutions to some partial neutral functional–differential equations and applications, Nonlinear Anal. 69 (2008) 1485–1493. [9] K. Ezzinbi, S. Fatajou, G.M. N’Guérékata, Pseudo almost automorphic solutions to some neutral partial functional differential equations in Banach spaces, Nonlinear Anal. 70 (2009) 1641–1647. [10] K. Ezzinbi, G.M. N’Guérékata, Massera type theorem for almost automorphic solutions of functional differential equations of neutral type, J. Math. Anal. Appl. 316 (2006) 707–721. [11] K. Ezzinbi, G.M. N’Guérékata, Almost automorphic solutions for some partial functional differential equations, J. Math. Anal. Appl. 328 (1) (2007) 344–358. [12] J.A. Goldstein, G.M. N’Guérékata, Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc. 133 (2005) 2401–2408. [13] J. Liang, G.M. N’Guérékata, Ti-Jun Xiao, Jun Zhang, Some properties of pseudo-almost automorphic functions and applications to abstract differential equations, Nonlinear Anal. 70 (2009) 2731–2735. [14] J. Liang, J. Zhang, T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2008) 1493–1499. [15] G.M. N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic Publishers, New York, London, Moscow, 2001. [16] G.M. N’Guérékata, Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, Semigroup Forum 69 (2004) 80–86. [17] G.M. N’Guérékata, Topics in Almost Automorphy, Springer-Verlag, New York, 2005. [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. [19] T.-J. Xiao, J. Liang, J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum 76 (3) (2008) 518–524. [20] T.-J. Xiao, X.-X. Zhu, J. Liang, Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear Anal. 70 (2009) 4079–4085. [21] L. Zhang, Y. Xu, Weighted pseudo-almost periodic solutions of a class of abstract differential equations, Nonlinear Anal. 71 (2009) 3705–3714 .
Journal of Functional Analysis 258 (2010) 208–240 www.elsevier.com/locate/jfa
Riesz bases for p-subordinate perturbations of normal operators Christian Wyss University of Bern, Mathematical Institute, Sidlerstrasse 5, 3012 Bern, Switzerland Received 16 March 2009; accepted 1 September 2009 Available online 12 September 2009 Communicated by D. Voiculescu
Abstract For a class of unbounded perturbations of unbounded normal operators, the change of the spectrum is studied and spectral criteria for the existence of a Riesz basis with parentheses of root vectors are established. A Riesz basis without parentheses is obtained under an additional a priori assumption on the spectrum of the perturbed operator. The results are applied to two classes of block operator matrices. © 2009 Elsevier Inc. All rights reserved. Keywords: Perturbation theory; Subordinate perturbation; Riesz basis; Eigenvector expansion; Spectrum
1. Introduction Since for non-normal operators there is no analogue of the spectral theorem, the existence of a Riesz basis (possibly with parentheses) of root vectors is an important property: it allows e.g. the construction of non-trivial invariant subspaces and yields spectral criteria related to semigroup generation. For a class of non-normal perturbations of normal operators we establish different conditions in terms of the spectrum which imply the existence of such Riesz bases. Our assumptions on the multiplicities of the eigenvalues are weaker than in classical perturbation theorems. We consider an operator T = G + S on a Hilbert space where G is unbounded, normal, has a compact resolvent, and the (generally also unbounded) perturbation S is p-subordinate to G, i.e. Su bu1−p Gup
for all u ∈ D(G)
E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.001
C. Wyss / Journal of Functional Analysis 258 (2010) 208–240
209
where p ∈ [0, 1[ and b 0; this implies that S is relatively bounded to G with G-bound 0. In Theorem 6.1 we prove that T admits a Riesz basis with parentheses of root vectors if the eigenvalue multiplicities of G satisfy a certain asymptotic growth condition. This growth condition is weaker than the one in a similar result by Markus and Matsaev ([18], [17, Theorem 6.12]). In Theorem 6.2 we obtain a Riesz basis with parentheses under a spectral condition of different type: we impose no restriction on the multiplicities and instead assume that the eigenvalues of G lie on sufficiently separated line segments, see Fig. 4. If we know a priori that the eigenvalues of the perturbed operator T are uniformly separated, then Theorem 6.2 even yields a Riesz basis without parentheses. An example for such a situation may be found in Theorem 7.3. In contrast to our result, classical perturbation theorems for Riesz bases without parentheses such as Kato [12, Theorem V.4.15a], Dunford and Schwartz [8, Theorem XIX.2.7] and Clark [5] require that almost all eigenvalues of G are simple. Apart from the above mentioned theorems, a wide range of existence results for Riesz bases of root vectors may be found in the literature, both for abstract operator settings and for concrete applications. Dissipative operators, for example, were considered by several authors; references and some results may be found in [10]. For generators of C0 -semigroups, a Riesz basis of eigenvectors implies the so-called spectrum determined growth assumption, see [6, Theorem 2.3.5]. Zwart [29] obtained Riesz bases for generators of C0 -groups, while Xu and Yung [28] constructed Riesz bases with parentheses for semigroup generators. Riesz basis properties of root vectors are also investigated for operator pencils, see e.g. [3,26]. Pencils coming from concrete physical problems were studied in [1,19]. Finally there are simple examples of non-normal operators whose eigenvectors are complete but do not form a Riesz basis, see e.g. [7]. In this paper, we follow ideas due to Markus and Matsaev [17, Chapter 1] to prove the existence of Riesz bases of root vectors. In Section 2 we start by deriving a completeness theorem for the system of root vectors of an operator with compact resolvent. Unlike the classical Keldysh theorem on completeness ([13], [17, §4]), where the resolvent belongs to a von Neumann– Schatten class, we assume here that it is uniformly bounded on an appropriate sequence of curves. In Section 3 we then recall the notion of a Riesz basis consisting of subspaces and provide a sufficient condition for its existence in terms of projections. Although a Riesz basis consisting of (finite-dimensional) subspaces is equivalent to a Riesz basis with parentheses, we use the basis of subspaces notion in the formulation of our theorems, since it is more convenient. In Section 4 we study in detail the change of the spectrum of a normal operator under a p-subordinate perturbation. The basic observation here is that if the spectrum of G lies on rays from the origin, then the spectrum of T lies inside parabolas around these rays. Based on the localization of the spectrum, several estimates for Riesz projections of T are obtained in Section 5. In Section 6 we derive our main existence results for Riesz bases of root vectors. In fact, these results also hold in the more general setting where G is a possibly non-normal operator with compact resolvent, a Riesz basis of root vectors and an appropriate spectrum, see Proposition 6.6 and Remark 6.7. Section 7 is devoted to applications of our theory. As a simple example, we deduce the existence of Riesz bases of root vectors for ordinary differential operators. This result is well known, see e.g. Clark [5], Dunford and Schwartz [8, Theorem XIX.4.16], and Shkalikov [21,22]. Next, we consider diagonally dominant block operator matrices. Riesz bases of root vectors were obtained by Jacob, Trunk and Winklmeier [11] for operator matrices associated with damped vibrations of thin beams and by Kuiper and Zwart [15] for a class of Hamiltonian operator matrices from control theory. In Theorem 7.2 we consider operator matrices whose entries may all be
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C. Wyss / Journal of Functional Analysis 258 (2010) 208–240
unbounded, whereas in [11,15] some of the entries were always bounded. Theorem 7.3 applies to a class of Hamiltonians which is different from the one in [15]. While the eigenvalues of the diagonal part of the Hamiltonian in [15] are simple, we consider the case of double eigenvalues. Finally, we briefly illustrate how a Riesz basis of root vectors of a Hamiltonian operator matrix allows the construction of infinitely many solutions of a corresponding operator Riccati equation. This will be the subject of a forthcoming paper, see also [27]. 2. Completeness of the system of root subspaces We derive a completeness theorem for the system of root subspaces of an operator with compact resolvent, which applies to a different situation than the classical theorem of Keldysh ([13], [17, §4]). Let T be an operator on a Banach space with a compact isolated part σ ⊂ σ (T ) of the spectrum. Let Γ be a simply closed, positively oriented integration contour with σ in its interior and σ (T ) \ σ in its exterior. Then i (T − z)−1 dz (1) P= 2π Γ
defines a projection such that R(P ) and ker P are T -invariant, R(P ) ⊂ D(T ), and σ (T |R(P ) ) = σ,
σ (T |ker P ) = σ (T ) \ σ.
P does not depend on the particular choice of Γ and is called the Riesz projection associated with σ (or Γ ); R(P ) is the corresponding spectral subspace. For a proof see [9, Theorem XV.2.1] or [12, Theorem III.6.17]. For an eigenvalue λ of T we call L(λ) =
ker(T − λ)k
k∈N
the root subspace of T corresponding to λ; the non-zero elements of L(λ) are the root vectors. A sequence of root vectors x1 , . . . , xn ∈ L(λ) is called a Jordan chain if (T − λ)xk = xk−1 for k 2 and (T − λ)x1 = 0. In the case that T has a compact resolvent, its spectrum consists of isolated eigenvalues only; so for every eigenvalue λ there is the associated Riesz projection Pλ , which satisfies R(Pλ ) = L(λ). Recall that for an operator T with compact resolvent on a Hilbert space its adjoint T ∗ also has a compact resolvent. Lemma 2.1. Let T be an operator with compact resolvent on a Hilbert space and M the subspace generated by all root subspaces of T , i.e., the set of all finite linear combinations of root vectors of T . If P is the Riesz projection of T ∗ corresponding to an eigenvalue λ ∈ σ (T ∗ ), then M ⊥ ⊂ ker P . Moreover, M ⊥ is T ∗ -invariant and (T ∗ −z)−1 -invariant for every z ∈ (T ∗ ); in particular (T ∗ ) ⊂ (T ∗ |M ⊥ ). Proof. We have λ ∈ σ (T ∗ ) if and only if λ ∈ σ (T ). Observe that if P is the Riesz projection of T ∗ corresponding to λ, then P ∗ is the Riesz projection of T corresponding to λ.
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Since R(P ∗ ) ⊂ M we find M ⊥ ⊂ R(P ∗ )⊥ = ker P . Now let v ∈ M and z ∈ (T ∗ ). Then T v, (T − z¯ )−1 v ∈ M and we have u ∈ M ⊥ ∩ D(T ∗ )
⇒
u ∈ M⊥
⇒
(T ∗ u|v) = (u|T v) = 0, ∗ (T − z)−1 uv = u(T − z¯ )−1 v = 0.
Therefore M ⊥ is T ∗ - and (T ∗ − z)−1 -invariant, and this in turn implies the inclusion (T ∗ ) ⊂ (T ∗ |M ⊥ ). 2 Corollary 2.2. Let T and M be as above. Then (T ∗ |M ⊥ ) = C. Proof. Since T has a compact resolvent, the same holds for T ∗ and T ∗ |M ⊥ . Consequently if λ ∈ σ (T ∗ |M ⊥ ), then λ is an eigenvalue of T ∗ |M ⊥ , i.e., T ∗ u = λu for some u ∈ M ⊥ \ {0}. In particular λ is an eigenvalue of T ∗ and we have u ∈ R(P ) where P is the Riesz projection of T ∗ corresponding to λ. Now the previous lemma implies u ∈ M ⊥ ⊂ ker P and hence u = 0, which is a contradiction. Therefore σ (T ∗ |M ⊥ ) = ∅. 2 Theorem 2.3. Let T be an operator with compact resolvent on a Hilbert space H with scalar product (·|·). If there exists a sequence of bounded regions (Uk )k∈N such that C = k∈N Uk , ∂Uk ⊂ (T ) for all k, and there is a constant C 0 with (T − z)−1 C
for z ∈ ∂Uk , k ∈ N,
then the system of root subspaces of T is complete.1 Proof. Let M be as before. For u, v ∈ M ⊥ we consider the holomorphic function defined by f (z) = (T ∗ |M ⊥ − z)−1 uv . From the previous corollary we know that its domain of definition is C. Since ∗ (T | ⊥ − z)−1 (T ∗ − z)−1 = (T − z¯ )−1 M
for z¯ ∈ (T ),
we see that |f (z)| Cuv holds for z¯ ∈ ∂Uk . Using the maximum principle, we find that |f (z)| Cuv for every z ∈ C; by Liouville’s theorem f is constant. Since u and v have been arbitrary, the mapping z → (T ∗ |M ⊥ − z)−1 is also constant. For u ∈ M ⊥ we obtain (T ∗ |M ⊥ )−1 u = (T ∗ |M ⊥ − I )−1 u
⇒
(T ∗ |M ⊥ − I )(T ∗ |M ⊥ )−1 u = u
(T ∗ |M ⊥ )−1 u = 0
⇒
u = 0.
⇒
Hence M ⊥ = {0}, i.e., M ⊂ H is dense.
2
1 A system of subspaces in H is called complete if the subspace generated by the system is dense in H .
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Corollary 2.4. Let T be an operator with compact resolvent on a Hilbert space. Suppose that almost all eigenvalues of T lie in a finite number of pairwise disjoint sectors
π with 0 < ψj , j = 1, . . . , n. Ωj = z ∈ C | arg z − θj | < ψj 4 If there are constants C, r0 0 such that (T − z)−1 C for z ∈ / Ω1 ∪ · · · ∪ Ωn , |z| r0 and for each sector Ωj there is a sequence (xk )k∈N with xk → ∞ and (T − z)−1 C for z ∈ Ωj , Re e−iθj z = xk , k ∈ N, then the system of root subspaces of T is complete. 3. Riesz bases of subspaces We recall the closely related concepts of Riesz bases, Riesz bases with parentheses, and Riesz bases of subspaces, see [26, §1], [10, Chapter VI], [23, §15] and [27, §2] for more details. Definition 3.1. Let H be a separable Hilbert space. (i) A sequence (vk )k∈N in H is called a Riesz basis of H if there is an isomorphism J : H → H such that (J vk )k∈N is an orthonormal basis of H . (ii) A sequence of closed subspaces (Vk )k∈N of H is called a Riesz basis of subspaces of H if there is an isomorphism J : H → H such that (J (Vk ))k∈N is a complete system of pairwise orthogonal subspaces. Other notions for Riesz bases of subspaces are “unconditional basis of subspaces” [26] or “l 2 -decomposition” [23]. The sequence (vk )k∈N is a Riesz basis if and only if inf vk > 0, sup vk < ∞, and every x ∈ H has a unique representation x=
∞
αk vk ,
αk ∈ C,
k=0
where the convergence of the series is unconditional. There is a similar characterization for Riesz bases of subspaces, see [27, §2.2] and [10, §VI.5] for a proof: Proposition 3.2. For a sequence (Vk )k∈N of closed subspaces of H the following assertions are equivalent: (i) (Vk )k∈N is a Riesz basis of subspaces for H . (ii) The sequence (Vk )k∈N is complete and there exists c 1 such that c
−1
2 xk xk xk 2 c 2
k∈F
for all finite subsets F ⊂ N and xk ∈ Vk .
k∈F
k∈F
(2)
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(iii) Every x ∈ H has a unique representation x = of the series is unconditional.
∞
k=0 xk
213
with xk ∈ Vk , where the convergence
To refer to the constant in (2), we shall also speak of a Riesz basis of subspaces with constant c. A sequence (vk )k∈N in a Hilbert space H is called a Riesz basis with parentheses if there exists a Riesz basis of subspaces (Vk )k∈N of H and a subsequence (nk )k of N with n0 = 0 such that (vnk , . . . , vnk+1 −1 ) is a basis of Vk . In this case every x ∈ H has a unique representation x=
nk+1 −1 ∞ k=0
αj vj ,
αj ∈ C,
j =nk
where the series over k converges unconditionally. The definition of a Riesz basis of subspaces generalizes naturally to a family of closed subspaces (Vk )k∈Λ , where the index set Λ is either finite or countably infinite; Proposition 3.2 continues to hold in this context. In particular, a finite family (V1 , . . . , Vn ) of closed subspaces is a Riesz basis of H if and only if the subspaces form a direct sum H = V1 ⊕ · · · ⊕ Vn . Despite this equivalence, the Riesz basis notion is convenient even for finite families to specify the constant c in (2). An example is the next lemma, which is used in the proof of Theorem 6.2 to show that the root subspaces of an operator form a Riesz basis. Lemma 3.3. Let (Wk )k∈Λ be a Riesz basis of subspaces of H with constant c0 . Let (Vkj )j ∈Jk be Riesz bases of subspaces of Wk for all k ∈ Λ with common constant c1 . Then the family (Vkj )k∈Λ, j ∈Jk is a Riesz basis of subspaces of H with constant c0 c1 . Proof. Since (Wk )k∈Λ is complete in H and (Vkj )j ∈Jk is complete in Wk for every k ∈ Λ, the family (Vkj )k∈Λ, j ∈Jk is complete in H . Consider F ⊂ Λ finite, Fk ⊂ Jk finite for each k ∈ F , and xkj ∈ Vkj . Then, using (2), we obtain 2 2 xkj c0 xkj c1 xkj 2 = c0 c1 xkj 2 c0 k∈F j ∈Fk
k∈F j ∈Fk
and similarly
k∈F, j ∈Fk
xkj 2 c0−1 c1−1
k∈F
k∈F, j ∈Fk
j ∈Fk
xkj 2 .
k∈F j ∈Fk
2
Note that the existence of the common constant c1 is guaranteed if only finitely many Jk consist of more than one element. Our next aim is to derive a sufficient condition for a sequence of projections to generate a Riesz basis of subspaces. Lemma 3.4. Let (xk )k∈N be a sequence in a Banach space. If there exists C 0 such that for bij every reordering φ : N −→ N and every n ∈ N we have nk=0 xφ(k) C, then n sup εk xk 2C. n∈N, εk =±1 k=0
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Proof. Let ε0 , . . . , εn ∈ {−1, 1} and consider reorderings φ1 and φ2 that move all +1 and all −1 in the sequence (ε0 , . . . , εn ), respectively, to its beginning. Then, with n1 , n2 appropriate, we obtain n n n n n 1 2 2 εk xk xk + xk = xφ1 (k) + xφ2 (k) 2C. k=0
k=0 εk =+1
k=0 εk =−1
k=0
k=0
Lemma 3.5. Let H be a Hilbert space, x0 , . . . , xn ∈ H , and
E = (ε0 , . . . , εn ) εk = ±1 . Then 2n+1
n
xk 2 =
ε0 x0 + · · · + εn xn 2 .
ε∈E
k=0
Proof. We use induction on n. The statement is true for the case n = 0 since 2x0 2 = x0 2 + − x0 2 . Now suppose the statement holds for some n 0; let
= (ε0 , . . . , εn+1 ) εk = ±1 E and write xε = ε0 x0 + · · · + εn xn . Then
ε0 x0 + · · · + εn+1 xn+1 2 =
ε∈E
xε + xn+1 2 + xε − xn+1 2 ε∈E
2xε 2 + 2xn+1 2 = 2 = xε 2 + 2 · 2n+1 xn+1 2 ε∈E
= 2n+2
n
ε∈E
xk 2 + xn+1 2 .
2
k=0
Lemma 3.6. Let P0 , . . . , Pn be projections in a Hilbert space H with Pj Pk = 0 for j = k. Then C −2
n k=0
where C = max{
n
n 2 n 2 2 Pk x Pk x C Pk x2
k=0 εk Pk | εk
k=0
for all x ∈ H
k=0
= ±1}.
Proof. We write xk = Pk x and use the last lemma considering that ε ∈ E for which ε0 x0 + · · · + εn xn becomes maximal. Then we obtain n k=0
n n 2 n 2 2 Pk x ε0 x0 + · · · + εn xn = εk Pk xk C Pk x . 2
2
k=0
k=0
k=0
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On the other hand, if we choose ε ∈ E such that ε0 x0 + · · · + εn xn is minimal, we find n 2 n n 2 Pk x = εk Pk εk xk k=0
k=0
k=0
C 2 ε0 x0 + · · · + εn xn 2 C 2
n
Pk x2 .
2
k=0
The following statement is a slight modification of a result2 in the book of Markus [17, Lemma 6.2]. Proposition 3.7. Let H be a Hilbert space and (Pk )k∈N a sequence of projections in H satisfying Pj Pk = 0 for j = k. Suppose that the family (R(Pk ))k∈N is complete in H and that ∞ (Pk x|y) Cxy for all x, y ∈ H
(3)
k=0
with some constant C 0. Then (R(Pk ))k∈N is a Riesz basis of subspaces of H with constant c = 4C 2 . Proof. From n n (Pk x|y) Cxy y P x k k=0
k=0
n
we conclude that k=0 Pk C for all n ∈ N. This assertion remains valid after an arbitrary rearrangement of the sequence (Pk )k∈N since (3) still holds for the rearranged sequence. An application of Lemmas 3.4, 3.6 and Proposition 3.2 now completes the proof. 2 We end this section with a remark on the connection between Riesz bases of finite-dimensional invariant subspaces of an operator and Riesz bases with parentheses of root vectors, see also [27, §2.3]. Remark 3.8. Let T be an operator on a Hilbert space. Since every finite-dimensional T -invariant subspace3 admits a basis consisting of Jordan chains, it is immediate that a Riesz basis of finitedimensional T -invariant subspaces is equivalent to a Riesz basis with parentheses of Jordan chains such that each Jordan chain lies inside some parenthesis. As a consequence of Lemma 3.3, a Riesz basis of finite-dimensional invariant subspaces where almost all subspaces are onedimensional is equivalent to a Riesz basis of eigenvectors and finitely many Jordan chains. 2 Under the weaker assumption ∞ |(P x|y)| < ∞ for all x, y ∈ H , the existence of the Riesz basis of subspaces is k k=0
proved, but without obtaining an estimate for the constant c. 3 In general, a subspace U is called T -invariant if T (U ∩ D(T )) ⊂ U . If we speak of a finite-dimensional T -invariant subspace U , we additionally assume that dim U < ∞ and U ⊂ D(T ).
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4. Spectral enclosures for p-subordinate perturbations The concept of p-subordination is in a certain sense an interpolation between the notions of boundedness and relative boundedness. We start with a result for relatively bounded perturbations. Lemma 4.1. Let G and S be operators on a Banach space with D(G) ⊂ D(S) and T = G + S. If 0 < ε < 1 and z ∈ (G) such that S(G − z)−1 ε,
(4)
then z ∈ (T ) and (T − z)−1
S(T − z)−1
1 (G − z)−1 , 1−ε
ε . 1−ε
Moreover if Γ ⊂ (G) is a simply closed, positively oriented integration contour and (4) holds for all z ∈ Γ , then Γ ⊂ (T ) and for the Riesz projections Q and P of G and T associated with Γ there are isomorphisms R(Q) ∼ = R(P ),
ker Q ∼ = ker P .
Proof. (4) implies the convergence of the Neumann series ∞ k −1 −1 −S(G − z)−1 = I + S(G − z) k=0
with I + S(G − z)−1 −1
1 1 . 1 − S(G − z)−1 1 − ε
Since T − z = I + S(G − z)−1 (G − z), we conclude that z ∈ (T ) with (T − z)−1 (G − z)−1 I + S(G − z)−1 −1
1 (G − z)−1 . 1−ε
The identity S(T − z)−1 = S(G − z)−1 (I + S(G − z)−1 )−1 yields S(T − z)−1 ε(1 − ε)−1 . To prove the assertion about the Riesz projections, consider the operators Tr = G + rS for r ∈ [0, 1]. We have the power series expansion ∞ −1 k I + rS(G − z)−1 = r k −S(G − z)−1 , k=0
r ∈ [0, 1],
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which converges uniformly in z ∈ Γ . Consequently Γ ⊂ (Tr ), and −1 (Tr − z)−1 = (G − z)−1 I + rS(G − z)−1 is continuous in r uniformly for z ∈ Γ . Hence the Riesz projections Pr of Tr associated with Γ also depend continuously on r. Now if Pr − Ps < 1, then there are isomorphisms R(Pr ) ∼ = R(Ps ),
ker Pr ∼ = ker Ps ,
see [12, §I.4.6]. Since r ranges over a compact interval, the proof is complete.
2
In the following we consider p-subordinate perturbations of unbounded normal operators. Recall that a closed, densely defined operator G on a Hilbert space is said to be normal if GG∗ = G∗ G,
(5)
i.e., D(GG∗ ) = D(G∗ G) and GG∗ u = G∗ Gu for all u ∈ D(GG∗ ), see e.g. [12]. In particular, every selfadjoint operator is normal. If z ∈ (G), then G is normal if and only if the resolvent (G − z)−1 is a normal operator. Moreover, (5) implies that Gu = G∗ u for u ∈ D(GG∗ ). The concept of p-subordinate perturbations was studied by Krein [14, §I.7.1] and Markus [17, §5], see also [27, §3.2]. Definition 4.2. Let G, S be operators on some Banach space and p ∈ [0, 1]. Then S is said to be p-subordinate to G if D(G) ⊂ D(S) and there exists b 0 such that Su bu1−p Gup
for all u ∈ D(G).
(6)
In this case there is a minimal constant b 0 such that (6) holds, which is called the psubordination bound of S to G. If S is p-subordinate to G with p < 1, then S is relatively bounded to G with relative bound 0; if also 0 ∈ (G) and q > p, then S is q-subordinate to G. An example for p-subordination are differential operators, see Example 7.1. Remark 4.3. In the case that G and S are operators on a Hilbert space and that G is normal with compact resolvent and 0 ∈ (G), the following can be shown [17, §5]: If SG−p is bounded with 0 p 1, then S is p-subordinate to G. If S is p-subordinate to G with 0 p < 1, then SG−q is bounded for all q > p; in particular, S is relatively compact to G. Now we investigate the change of the spectrum of G under the perturbation S for the case that σ (G) lies on rays from the origin. We denote sectors in the complex plane by
Ω(ϕ− , ϕ+ ) = reiϕ r 0, ϕ− < ϕ < ϕ+
and Ω(ϕ) = Ω(−ϕ, ϕ).
In the next lemma the strip 3 corresponds to large gaps of σ (G) on the positive real axis, compare Fig. 1. Sufficient conditions for the existence of such gaps may be found in Proposition 5.8, Theorem 6.2 and Lemma 6.5.
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Fig. 1. The situation of Lemma 4.4.
Lemma 4.4. Let G be a normal operator on a Hilbert space H such that σ (G) ∩ Ω(2ϕ− , 2ϕ+ ) ⊂ R0 with −π ϕ− < 0 < ϕ+ π . Let S be p-subordinate to G with bound b, 0 p < 1, and T = G + S. Then for α > b, b/α < ε < 1, and 0 < ψ < min{−ϕ− , ϕ+ , π/2} there exists r0 > 0 such that the sets
1 = z ∈ Ω(ϕ− , ϕ+ ) |z| r0 , z ∈ / Ω(ψ) ,
2 = z = x + iy ∈ Ω(ψ) |z| r0 , |y| αx p ,
3 = z = x + iy ∈ Ω(ψ) |z| r0 , |y| αx p dist z, σ (G) satisfy 1 ∪ 2 ∪ 3 ⊂ (T ), and for z ∈ 1 ∪ 2 ∪ 3 we have S(G − z)−1 ε,
(T − z)−1
(1 − ε)−1 , dist(z, σ (G))
S(T − z)−1
ε . 1−ε
Furthermore there is a constant M > 0 such that (T − z)−1 M
for all z ∈ 1 ∪ 2 ∪ 3 .
Proof. We write d = dist(z, σ (G)) and use a consequence of the spectral theorem for normal operators, see [12, §V.3.8]: (G − z)−1 = sup
1 1 = , |λ − z| d λ∈σ (G)
G(G − z)−1 = I + z(G − z)−1 1 + |z| . d
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219
With the definition of p-subordination this yields p 1 S(G − z)−1 u bG(G − z)−1 up (G − z)−1 u1−p b 1 + |z| u d d 1−p for every u ∈ H . In order to apply Lemma 4.1, we thus have to show that |z| p 1 ε. C =b 1+ d d 1−p
(7)
First we analyze the geometry of the situation: For z = x + iy we have the implications π π or arg z ϕ+ ϕ− arg z − 2 2 π π arg z min ϕ+ , max ϕ− , − 2 2
⇒
d |z|,
(8)
⇒
d |y|,
(9)
as well as π 2 |arg z| ψ
ψ |arg z|
⇒
|y| |z| sin ψ,
(10)
⇒
x |z| cos ψ.
(11)
Now let z ∈ 1 . If ϕ− arg z −π/2 or π/2 arg z ϕ+ , then (8) yields C 2p b|z|p−1 ε, provided r0 is large enough. If ψ | arg z| π/2, then (9) and (10) imply d |z| sin ψ and hence p 1 1 ε C b 1+ sin ψ (|z| sin ψ)1−p for r0 sufficiently large. For z ∈ 2 , the implications (9) and (11) apply and with |y| αx p we find d αx p . For p > 0 we use the Minkowski inequality to get the estimate |z| p x + |y| p x p + |y|p α −1 d + d p 1 1+ 1+ 1+ 1+ = 2 + d 1−p , p p d d d d α i.e., C 2bd p−1 +b/α. Since b/α < ε and d α(|z| cos ψ)p , we obtain C ε for r0 sufficiently large. On the other hand, if p = 0 then d α and C = b/d b/α < ε. In the case z ∈ 3 , (9) and (11) apply, and we have d αx p by definition of the set 3 . In the same manner as for z ∈ 2 , we conclude that C ε if r0 is large enough. Finally, to prove that (T − z)−1 is uniformly bounded, we need to show that d −1 is bounded independently of z. For z ∈ 1 we have either d |z| r0 > 0 or
d |z| sin ψ r0 sin ψ > 0.
For z ∈ 2 ∪ 3 we obtain p d α |z| cos ψ α(r0 cos ψ)p > 0.
2
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Fig. 2. The spectrum after a p-subordinate perturbation.
Theorem 4.5. Let G be a normal operator whose spectrum lies on finitely many rays eiθj R0 with 0 θj < 2π , j = 1, . . . , n. Let T = G + S where S is p-subordinate to G with bound b and 0 p < 1. Then for every α > b there exists r0 > 0 such that σ (T ) ⊂ Br0 (0) ∪
n iθ
e j (x + iy) x 0, |y| αx p ,
(12)
j =1
cf. Fig. 2. If G has a compact resolvent, then so has T . Proof. Without loss of generality, we assume θ1 < θ2 < · · · < θn and set θ0 = θn − 2π , θn+1 = θ0 + 2π . Then we may, after a rotation by θj , apply Lemma 4.4 to each sector Ω(θj −1 , θj +1 ). More precisely, we apply the lemma to the operators e−iθj G, e−iθj S, e−iθj T with ϕ+ = (θj +1 − θj )/2, ϕ− = (θj −1 − θj )/2, and some suitable ε. This yields the implication θj + θj +1 θj −1 + θj arg z , |z| r0 2 2
z ∈ eiθj (x + iy) x 0, |y| αx p
z ∈ σ (T ), ⇒
with some r0 0 for each j = 1, . . . , n. If G has compact resolvent, the identity −1 (T − z)−1 = (G − z)−1 I + S(G − z)−1 implies that T has compact resolvent too.
for z ∈ (G) ∩ (T )
2
The statement about the asymptotic shape of the spectrum of T can be refined as follows: Remark 4.6. To obtain a condition for z ∈ (T ), we consider without loss of generality the case σ (G) ∩ Ω(2ϕ) ⊂ R0 , 0 < ϕ π/2, and z = x + iy ∈ Ω(ϕ). Then dist(z, σ (G)) |y| and, in
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221
view of (7), b(1 + |z|/|y|)p |y|p−1 < 1 is sufficient to get z ∈ (T ). For p > 0 this leads to the condition x<
|y| b
1/p
1 − 2b1/p |y|1−1/p ,
which is asymptotically better than x < (|y|/α)1/p from the theorem since 1 − 2b1/p |y|1−1/p → 1 as |y| → ∞. For p = 0 we obtain the optimal condition b < |y|. For p > 0, the estimates of Markus [17, Lemma 5.2] lead to asymptotics which are even slightly better. Also note that simply taking the limit α → b in Theorem 4.5 is not possible since then also r0 → ∞. 5. Estimates for Riesz projections In this section G is always a normal operator with compact resolvent on a Hilbert space H such that σ (G) ∩ Ω(2ϕ) ⊂ R0
with 0 < ϕ
π 2
and T = G + S with S p-subordinate to G and p < 1. The first two lemmas can be found in the book of Markus [17], for the special case α = 4b. Since his proofs literally apply to the general situation, we omit them here; see also [27, §3.3]. Lemma 5.1. (See [17, Lemma 6.6].) Let G be normal with compact resolvent and σ (G) ∩ Ω(2ϕ) ⊂ R0 with 0 < ϕ π/2. Then for 0 p < 1, α > 0 there exists r0 > 0 such that the contours
Γ± = x + iy ∈ C x r0 , y = ±αx p
(13)
satisfy Γ± ⊂ (G) ∩ Ω(ϕ) and we have
p
|z|
−1
(G − z)
2 u |dz| C1 u2 ,
Γ±
2 |z|p−2 G(G − z)−1 u |dz| C2 u2
Γ±
for all u ∈ H with some constants C1 , C2 0. Lemma 5.2. (See [17, Lemma 6.7].) Let G be normal with compact resolvent and σ (G) ∩ Ω(2ϕ) ⊂ R0 with 0 < ϕ π/2. Let (xk )k1 be a sequence of positive numbers, 0 p < 1, p−1 and α, c1 , c2 > 0 such that αx1 tan ϕ and 1−p
xn
1−p
− xk
c1 (n − k)
for n > k,
p dist xk , σ (G) c2 xk
for k 1.
Then the lines p
γk = xk + iy ∈ C |y| αxk
(14)
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satisfy γk ⊂ (G) ∩ Ω(ϕ) and we have ∞ k=1
p
xk
(G − z)−1 u2 |dz| C1 u2 ,
∞
p−2
k=1
γk
xk
G(G − z)−1 u2 |dz| C2 u2
γk
for all u ∈ H with some constants C1 , C2 0. With the previous resolvent estimates at hand, we derive an estimate for a sequence of Riesz projections associated with the parabola Γ± and the lines γk : Lemma 5.3. Let G be normal with compact resolvent, σ (G) ∩ Ω(2ϕ) ⊂ R0 with 0 < ϕ π/2, S p-subordinate to G with bound b, 0 p < 1, and T = G + S. Let α > b, let (xk )k1 , γk be as in Lemma 5.2, and suppose that there is a constant M 0 such that γk ⊂ (T )
and S(T − z)−1 M
for all z ∈ γk , k 1.
Then there exist r0 > 0, k0 1 such that xk0 r0 and the following holds: If Γ± is as in (13) and Γk with k k0 is the positively oriented boundary contour of the region enclosed by γk , Γ− , γk+1 , Γ+ , then Γk ⊂ (T ). If Pk is the Riesz projection of T associated with Γk , then ∞ (Pk u|v) Cuv for all u, v ∈ H
(15)
k=k0
with some constant C 0. Proof. We want to apply Lemmas 4.4, 5.1 and 5.2, and choose ε ∈ ]b/α, 1[ and r0 accordingly. The assumptions on the sequence (xk )k imply that it tends monotonically to infinity and we choose k0 such that xk0 r0 . By Lemma 4.4, S(T − z)−1 is uniformly bounded on Γ± . We thus have Γk ⊂ (G) ∩ (T )
and S(T − z)−1 M0
for all z ∈ Γk , k k0 ,
with some M0 0. Consider now the Riesz projections Qk of G associated with Γk , which are orthogonal since G is normal. It is easy to see that, to prove (15), it suffices to prove ∞ (Pk − Qk )uv Cuv. k=k0
Now i Pk − Qk = 2π
Γk
and hence
−i (T − z)−1 − (G − z)−1 dz = 2π
Γk
(T − z)−1 S(G − z)−1 dz
C. Wyss / Journal of Functional Analysis 258 (2010) 208–240
(Pk − Qk )uv 1 2π
223
S(G − z)−1 u(T − z)−∗ v |dz|.
Γk
Then, with the help of (T − z)−1 = (G − z)−1 I − S(T − z)−1 ⇒ (T − z)−∗ v 1 + S(T − z)−1 (G − z)−∗ v M0
and (G − z)−∗ v = (G − z)−1 v (since G is normal), we find ∞ ∞ (Pk − Qk )uv 1 + M0 S(G − z)−1 u(G − z)−1 v |dz| 2π k=k0 k=k0 Γ k
∞ 1 + M0 S(G − z)−1 u(G − z)−1 v |dz|. + +2 2π Γ+
Γ−
k=k0 γk
Using p-subordination, Lemma 5.1, and (for p = 0) Hölder’s inequality, we estimate
S(G − z)−1 u(G − z)−1 v |dz|
Γ±
1/2 1/2 2 2 |z|−p S(G − z)−1 u |dz| |z|p (G − z)−1 v |dz| ,
Γ±
Γ±
C1 v2
2 |z|−p S(G − z)−1 u |dz|
Γ±
b
2
p−2
|z|
−1
G(G − z)
p 1−p 2 p −1 2 u |dz| |z| (G − z) u |dz|
Γ± 2 p 1−p b C2 C1 u2 .
Γ±
In the same way, with Lemma 5.2, we see that S(G − z)−1 u(G − z)−1 v |dz| k γk
k γk
1/2 1/2 2 2 −p p xk S(G − z)−1 u |dz| xk (G − z)−1 v |dz|
k γk
C1 v2
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and
2 −p xk S(G − z)−1 u |dz|
k γk
b
2
b
2
p
2 p−2 xk G(G − z)−1 u |dz|
k γk p 1−p C2 C1 u2 .
1−p
2 p xk (G − z)−1 u |dz|
k γk
2
To proceed, we need the concept of the determinant for operators, see [17, §2.5], [9, Chapter VII] and [10, §IV.1]. For an operator A of finite rank m, the determinant of I + A is defined by det(I + A) = det (I + A)|R(A)
(16)
and it satisfies (i) |det(I + A)| (1 + A)m ; (ii) I + A is invertible if and only if det(I + A) = 0, and in this case m (I + A)−1 (1 + A) ; |det(I + A)|
(iii) if the operator-valued function B : Ω → L(H ) is analytic on a domain Ω ⊂ C, then z → det(I + AB(z)) is analytic on Ω too. We also use the following auxiliary result from complex analysis, cf. [17, Lemma 1.6], [16, Theorem I.11]: Lemma 5.4. Let U ⊂ C be a bounded, simply connected domain, F ⊂ U compact, z0 an interior point of F , and η > 0. Then there exists a constant C > 0 such that the following holds: If a, b ∈ C and f : aU + b → C with f (az0 + b) = 0 is holomorphic and bounded, then there is a set E ⊂ C being the union of finitely many discs with radii summing up to at most |a|η such that 1+C f (z) |f (az0 + b)| f C aU +b,∞
for all z ∈ (aF + b) \ E.
The next proposition permits us to estimate the resolvent of the perturbed operator even close to its eigenvalues by artificially creating a gap in the spectrum of G. The method is taken from Lemma 5.6 in [17], which may be obtained from our proposition as a corollary. We denote by N+ (r1 , r2 , G) the sum of the multiplicities of all the eigenvalues of G in the open interval ]r1 , r2 [, N+ (r1 , r2 , G) =
dim L(λ).
(17)
λ∈σp (G)∩]r1 ,r2 [
Proposition 5.5. Let G be normal with compact resolvent, σ (G) ∩ Ω(2ϕ) ⊂ R0 with 0 < ϕ π/2, S p-subordinate to G with bound b, 0 p < 1, and T = G + S.
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Let l > b, 0 l0 < l − b and η > 0. Then there are constants C0 , C1 , r0 > 0 such that for every r r0 there is a set Er ⊂ C with the following properties: (i) Er is the union of finitely many discs with radii summing up to at most ηr p . (ii) For every z ∈ Ω(ϕ) \ Er with | Re z − r| l0 r p we have z ∈ (T )
C0 C1m , and (T − z)−1 rp
S(T − z)−1 C0 C m 1
where m = N+ (r − lr p , r + lr p , G). Proof. We choose l1 ∈ ]l0 , l − b[ and α, b˜ such that b < b˜ < α < l − l1 . Let r r0 . We may assume that r − lr p > 0 by choosing r0 large enough. Let λ1 , . . . , λn be the eigenvalues of G in r = ]r − lr p , r + lr p [ , P1 , . . . , Pn the orthogonal projections onto the corresponding eigenspaces, and Kr =
n (λj − λ˜ j )Pj
with λ˜ j =
j =1
r − lr p
if λj < r,
r + lr p
if λj r.
Then Gr = G − Kr is a normal operator with σ (Gr ) ∩ Ω(2ϕ) ⊂ R0 and r ⊂ (Gr ). Kr has rank m and satisfies Kr lr p . Noting that λj /λ˜ j r/(r − lr p ) for all j , it is straightforward to show that Gu
r Gr u. r − lr p
˜ we conclude Since 1 − lr p−1 → 1 as r → ∞ and b < b,
Su bGu u p
1−p
1 b 1 − lr p−1
p
˜ r up u1−p , Gr up u1−p bG
˜ provided r0 is sufficiently large. Thus S is p-subordinate to Gr with bound less or equal than b. Next, we want to prove that |x − r| l1 r p
⇒
]x − αx p , x + αx p [ ⊂ (Gr )
(18)
for r0 sufficiently large. Let |x − r| l1 r p . Since the function x → x − αx p is monotonically increasing for large x, we have p x − αx p r − l1 r p − α r − l1 r p r − l1 r p − αr p > r − lr p for r0 large enough. Furthermore α(1 + l1 r p−1 )p l − l1 holds for large r and we obtain
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p x + αx p r + l1 r p + α r + l1 r p r + lr p . In view of r ⊂ (Gr ), (18) is proved. Now we aim to apply Lemma 5.4 to the function d(z) = det I + Kr (Gr + S − z)−1 ,
z ∈ (Gr + S),
and the sets
Ur = x + iy |x − r| < l1 r p , |y| < 4br p ,
Fr = x + iy |x − r| l0 r p , |y| 3br p . For r0 sufficiently large we have Ur ⊂ Ω(ϕ). Using (18), we can apply Lemma 4.4 to Gr + S ˜ with some ε ∈ ]b/α, 1[; we obtain Ur ⊂ (Gr + S) and, for z ∈ Ur , dist z, σ (Gr ) lr p − l1 r p > αr p and −1 (Gr + S − z)−1 (1 − ε) , p αr
S(Gr + S − z)−1
ε . 1−ε
Then −1 m d(z) 1 + Kr (Gr + S − z)−1 m 1 + l(1 − ε) = c0m α on Ur with c0 > 0. For z ∈ (T ) ∩ Ur we have I = I + Kr (Gr + S − z)−1 I − Kr (T − z)−1 . Applying Lemma 4.4 (now with ε = 2/3) to the operator T and zr = r + i · 2br p ∈ Fr , we obtain zr ∈ (T ) and (T − zr )−1
3 2br p
and thus 1 d(z
m = det I − Kr (T − zr )−1 1 + 3l = c1m ) 2b r
with c1 > 0. Lemma 5.4 then yields a constant C > 0 depending only on b, l0 , l1 and η such that for every r r0 there exists a union Er of discs with radii summing up to at most ηr p and d(z) c−mC c−m(1+C) 0
1
for all z ∈ Fr \ Er .
For z ∈ Fr \ Er , we thus obtain that I + Kr (Gr + S − z)−1 is invertible with cm I + Kr (Gr + S − z)−1 −1 0 (c0 c1 )(1+C)m . |d(z)|
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Consequently z ∈ (T ) with −1 (T − z)−1 = (Gr + S − z)−1 I + Kr (Gr + S − z)−1 and m −1 (T − z)−1 (1 − ε) (c0 c1 )(1+C)m C0 C1 , αr p rp S(T − z)−1 ε (c0 c1 )(1+C)m C0 C m 1 1−ε
with appropriate constants C0 , C1 depending on b, l, l0 , l1 , η, α, ε only. Finally, we consider z = x + iy ∈ Ω(ϕ) with |x − r| l0 r p and |y| 3br p . Then p 2bx p 2b r + l0 r p 3br p |y| holds for r0 sufficiently large. Applying Lemma 4.4 (again with ε = 2/3), we obtain z ∈ (T ) and m (T − z)−1 3 1 C0 C1 , |y| br p rp
for C0 max{2, b−1 } and C1 1.
S(T − z)−1 2 C0 C m 1
2
Corollary 5.6. Let G be normal with compact resolvent, σ (G) ∩ Ω(2ϕ) ⊂ R0 with 0 < ϕ π/2, S p-subordinate to G with bound b, 0 p < 1, and T = G + S. Then for l0 , q > 0 there are constants C0 , C1 , r0 > 0 such that for every r r0 the following holds: For every z = x + iy with |x − r| l0 r p , |y| 2bx p there exists q1 ∈ ]0, q[ such that |w − z| = q1 r p
⇒
w ∈ (T ),
m (T − w)−1 C0 C1 , rp
where m = N+ (r − lr p , r + lr p , G) with l = b + 2(l0 + q). Proof. We use Proposition 5.5 with l = b + 2(l0 + q), l0 + q replacing l0 , and η = q/3. For z as in the claim and |w − z| qr p we have |arg w| ϕ (for r0 large enough) and | Re w − r| (l0 + q)r p . Now the sum of the diameters of the discs in Er is at most 2ηr p < qr p . Hence there / Er for |w − z| = q1 r p and the claim is proved. 2 exists q1 ∈ ]0, q[ such that w ∈ Under certain assumptions on the distribution of the eigenvalues of G on the positive real axis, we now obtain a sequence of closed integration contours in (T ) of the form in Lemma 5.3 and estimates for the associated Riesz projections. Proposition 5.7. Let G be normal with compact resolvent, σ (G) ∩ Ω(2ϕ) ⊂ R0 with 0 < ϕ π/2, S p-subordinate to G with bound b, 0 p < 1, and T = G + S. Assume that there is a sequence (rk )k1 of positive numbers tending monotonically to infinity and some l > b, m ∈ N1 such that p p (19) N+ rk − lrk , rk + lrk , G m for all k 1.
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Then there are constants C, r0 > 0, α > b, and a sequence (xk )k1 in R0 tending monotonically to infinity such that the following holds: (i) z ∈ Ω(ϕ) with Re z = xk implies z ∈ (T ), (T − z)−1 C. (ii) The contours Γ± , γk from (13) and (14) satisfy Γ± , γk ⊂ (T ). (iii) If Pk is the Riesz projection of T associated with the region enclosed by γk , Γ− , γk+1 , Γ+ , then ∞ (Pk u|v) Cuv for all u, v ∈ H. k=1
Proof. We apply Proposition 5.5 with l0 = (l − b)/2 and η = l0 /2 to r = rk , k k0 , k0 app propriate. Since the sum of the diameters of the discs in Er is at most l0 rk and the interval p p [rk − l0 rk , rk + l0 rk ] contains at most m eigenvalues of G, we can find an xk such that p
|xk − rk | l0 rk ,
l0 p r , dist xk , σ (G) 3m k
and that z ∈ Ω(ϕ) with Re z = xk implies z ∈ (T ),
m (T − z)−1 C0 C1 , p rk
S(T − z)−1 C0 C m . 1
Then xk /rk → 1 as k → ∞ and we obtain p dist xk , σ (G) c2 xk
for k k0
with c2 > 0 and k0 appropriately chosen. Since xk → ∞, for every k1 there exists k2 > k1 such 1−p 1−p that xk2 − xk1 1. Passing to an appropriate subsequence, we can thus assume that 1−p
1−p
1 for all k ∈ N,
1−p
n−k
xk+1 − xk which yields 1−p
xn
− xk
for n > k.
Now an application of Lemma 5.3 with α = 2b and the sequence (xk )kk0 , k0 large enough, completes the proof. 2 If the spectrum of G has sufficiently large gaps on R0 , then the spectrum of T has corresponding gaps (cf. Fig. 3): Proposition 5.8. Let G be normal with compact resolvent, σ (G) ∩ Ω(2ϕ) ⊂ R0 with 0 < ϕ π/2, S p-subordinate to G with bound b, 0 p < 1, and T = G + S.
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229
Fig. 3. A large gap in σ (G) yields a gap in σ (T ).
Assume that there is a sequence (rk )k1 of non-negative numbers tending monotonically to infinity and constants β 0, α > b, l > β + α such that σ (G) ∩ R0 ⊂
p p rk − βrk , rk + βrk
(20)
k1
and p
p
rk + lrk rk+1 − lrk+1 for almost all k. Then there are constants C, r0 > 0, k0 1 such that the following holds: (i) The contours Γ± from (13) and
p with k k0 γk± = x + iy x = rk ± lrk , |y| αx p − as well as the regions enclosed by γk+ , γk+1 , Γ+ , Γ− belong to (T ). p (ii) z ∈ Ω(ϕ) with Re z = rk + lrk , k k0 , implies (T − z)−1 C. (iii) If Pk and Qk are the Riesz projections of T and G, respectively, associated with the region enclosed by γk− , γk+ , Γ+ , Γ− , then ∞ (Pk u|v) Cuv for all u, v ∈ H k=k0
and dim R(Pk ) = dim R(Qk )
for k k0 .
− − rk+1 . Consider s ∈ [sk+ , sk+1 ] with k k0 . Proof. We set sk± = rk ± lrk so that rk sk+ sk+1 Then p
− + αrk+1 = rk+1 − (l − α)rk+1 rk+1 − βrk+1 . s + αs p sk+1 p
p
p
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Furthermore we have p s − αs p sk+ − α sk+ for k0 large enough, since the left-hand side is monotonically increasing in s for large s. In addition, the equivalent inequalities p p sk+ − α sk+ rk + βrk
⇔
p−1
hold for k0 sufficiently large since 1 + lrk k k0 , − s ∈ sk+ , sk+1
⇒
p p p p lrk − α rk + lrk βrk
→ 1. Using (20), we have thus proved that, for
s − αs p , s + αs p ⊂ (G).
− , With r0 and k0 appropriately chosen, Lemma 4.4 implies that the region enclosed by γk+ , γk+1 + − −1 Γ , and Γ as well as the contours itself belong to (T ) for k k0 . Moreover, (T − z) and S(T − z)−1 are uniformly bounded for z ∈ Ω(ϕ) with Re z = sk+ , k k0 . We also have dist(sk+ , G) α(sk+ )p and
p p p + sk+1 − sk+ = rk+1 − rk + l rk+1 − rk 2lrk+1 . The mean value theorem then yields p
+ 1−p + 1−p 2l(1 − p)rk+1 + −p + sk+1 sk+1 − sk+ − sk (1 − p) sk+1 , p (rk+1 + lrk+1 )p + 1−p i.e., (sk+1 ) − (sk+ )1−p l(1 − p) for k k0 , k0 sufficiently large. We can thus apply Lemma 5.3 with xk = sk+ to get the estimate for the sum over the Riesz projections. The final claim is a consequence of Lemmas 4.1 and 4.4. 2
6. Existence of Riesz bases of invariant subspaces Let G be an operator with compact resolvent. Recall that we denote by N+ (r1 , r2 , G) the sum of the multiplicities of the eigenvalues of G in the interval ]r1 , r2 [, see (17). Similarly, we write
N (r, G) =
dim L(λ)
(21)
λ∈σp (G)∩Br (0)
for the sum of the multiplicities of all the eigenvalues λ with |λ| r and N (K, G) =
dim L(λ)
for every set K ⊂ C.
(22)
λ∈σp (G)∩K
Our first existence result for Riesz bases of invariant subspaces improves a theorem due to Markus and Matsaev ([18], [17, Theorem 6.12]); there, condition (23) was formulated with lim sup instead of lim inf.
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231
Theorem 6.1. Let G be a normal operator with compact resolvent whose spectrum lies on a finite number of rays from the origin. Let S be p-subordinate to G with 0 p < 1. If lim inf r→∞
N (r, G) < ∞, r 1−p
(23)
then T = G + S admits a Riesz basis of finite-dimensional T -invariant subspaces. Proof. Let eiθj R0 with 0 θ1 < · · · < θn < 2π be the rays containing the eigenvalues of G and let S be p-subordinate to G with bound b. From Theorem 4.5 we know that T has a compact resolvent and that almost all of its eigenvalues lie inside sectors of the form
Ωj = z ∈ C |arg z − θj | < ψj
with 0 < ψj
π , 4
where ψj can be chosen such that these sectors are disjoint. Lemma 4.4 shows that (T − z)−1 is uniformly bounded for z ∈ / Ω1 ∪ · · · ∪ Ωn , |z| r0 . From the assumption (23) it can be shown that for each sector Ωj there is a sequence (rj k )k1 of positive numbers tending monotonically to infinity such that p p sup N+ rj k − 2brj k , rj k + 2brj k , e−iθj G < ∞, k
see [17, Lemma 6.11]. By Proposition 5.7 we thus obtain a corresponding sequence (xj k )k1 such that (T − z)−1 is uniformly bounded for z ∈ Ωj , Re(e−iθj z) = xj k . Corollary 2.4 implies that the system of root subspaces of T is complete. Furthermore, if (Pj k )k1 are the Riesz projections from Proposition 5.7 corresponding to the eigenvalues λ ∈ Ωj of T with Re(e−iθj λ) > xj 1 and P0 is the Riesz projection for the (finitely many) remaining eigenvalues, then n ∞ (Pj k u|v) Cuv (P0 u|v) + j =1 k=1
with some constant C 0. Now Proposition 3.7 shows that the ranges of the projections P0 , (Pj k )j,k form a Riesz basis. 2 Replacing condition (23) by an assumption on the localization of the spectrum of G on the rays, we obtain our second perturbation theorem. Theorem 6.2. Let G be a normal operator with compact resolvent on a Hilbert space H and S p-subordinate to G with bound b and 0 p < 1. Suppose that the spectrum of G lies on sequences of line segments on rays from the origin, σ (G) ⊂
n j =1 k1
Lj k ,
p
Lj k = eiθj x x 0, |x − rj k | βj rj k ,
(24)
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Fig. 4. The situation of Theorem 6.2.
where βj 0, 0 θ1 < · · · < θn < 2π , and (rj k )k1 are monotonically increasing sequences of non-negative numbers such that p
p
rj k + lj rj k rj,k+1 − lj rj,k+1
(25)
for almost all k with some constants lj > βj + b. Then T = G + S has compact resolvent; for b < α < min{l1 − β1 , . . . , ln − βn } almost all eigenvalues of T lie inside the regions
p (26) Kj k = eiθj (x + iy) x 0, |x − rj k | (βj + α)rj k , |y| αx p , j = 1, . . . , n, k 1 (cf. Fig. 4); the spectral subspaces of T corresponding to the regions Kj k together with the subspace corresponding to σ (T ) \ j,k Kj k form a Riesz basis of H ; and we have N (Lj k , G) = N (Kj k , T )
for almost all pairs (j, k).
(27)
Moreover, if there are constants m, q > 0 such that for almost all pairs (j, k) the assertions (i) N(Lj k , G) m and p (ii) λ1 , λ2 ∈ σ (T ) ∩ Kj k , λ1 = λ2 ⇒ |λ1 − λ2 | > qrj k hold, then the root subspaces of T form a Riesz basis of H . Proof. We apply Theorem 4.5 and, for each ray, Proposition 5.8 with α and l replaced by α˜ = (α + b)/2 and l˜j = βj + α, respectively. This shows that T has a compact resolvent and that almost all of its eigenvalues lie inside regions
iθ p ˜ p ⊂ Kj k . e j (x + iy) x 0, |x − rj k | < l˜j rj k , |y| < αx
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233
By Lemma 4.4, (T − z)−1 is uniformly bounded outside certain disjoint sectors Ωj around the rays for |z| large enough. For each ray, Proposition 5.8 yields a sequence (xj k )k∈N tending monotonically to infinity such that (T − z)−1 is bounded for z ∈ Ωj , Re(e−iθj z) = xj k . With Corollary 2.4 we conclude that the system of root subspaces of T is complete. Moreover, we have n ∞ (Pj k u|v) Cuv (P0 u|v) + j =1 k=1
for some C 0 where Pj k is the Riesz projection associated with Kj k and P0 the one associated with σ (T ) \ j k Kj k ; Proposition 3.7 then yields the Riesz basis property. Finally, if Qj k is the spectral projection of G associated with Lj k , Proposition 5.8 implies dim R(Qj k ) = dim R(Pj k ) for almost all (j, k) and thus (27). Now suppose that with m, q > 0 the additional assumptions (i) and (ii) hold for almost all pairs (j, k). We aim to show that the root subspaces corresponding to the eigenvalues of T in each Kj k form a Riesz basis of R(Pj k ) with constant c independent of (j, k). Without loss of generality we may assume θj = 0 and βj + α + q lj . We want to apply Corollary 5.6 with l0 = βj + α and set l accordingly. From the relation p rj,k+1 − rj k 2lj rj k it is easy to verify that the number of elements rj k in the interval [r − lr p , r + lr p ] is at most 2l/ lj for r sufficiently large. Hence there is a constant m0 such that N+ r − lr p , r + lr p , G m0
for r sufficiently large.
Let λ be an eigenvalue of T in Kj k . By Corollary 5.6 there exists q1 ∈ ]0, q[ such that the points p m −p w on the circle around λ with radius q1 rj k satisfy (T − w)−1 C0 C1 0 rj k . In addition, this p circle lies inside the strip | Re z − rj k | lj rj k and assumption (ii) thus implies that λ is the only possible eigenvalue of T inside that circle. Therefore, the Riesz projection Pλ for λ satisfies m
p
Pλ 2πq1 rj k
C0 C1 0 p rj k
m
2πqC0 C1 0 .
If λ1 , . . . , λm1 are the eigenvalues of T in Kj k , we have m1 N (Kj k , T ) m and conclude m1 (Pλ u|v) 2πmqC0 C m0 uv. s 1 s=1
According to Proposition 3.7, the subspaces R(Pλs ), s = 1, . . . , m1 , form a Riesz basis of R(Pj k ) with constant c independent of k. This is true for almost all pairs (j, k), and hence an application of Lemma 3.3 shows that the root subspaces of T form a Riesz basis of H . 2 Remark 6.3. If almost all eigenvalues of G are simple and almost all line segments Lj k contain one eigenvalue only, then Theorem 6.2 yields a Riesz basis of eigenvectors and finitely many
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Jordan chains for T . Indeed almost all spectral subspaces corresponding to the Kj k are onedimensional in this case, and the Riesz basis of subspaces is thus equivalent to the existence of a Riesz basis of eigenvectors and finitely many Jordan chains. Remark 6.4. It can be shown [27, Lemma 3.4.9] that, if G satisfies the spectral condition (24) p p with some βj > 0 such that rj k + βj rj k rj,k+1 − βj rj,k+1 and N (Lj k , G) is bounded in (j, k), then supr1 N (r, G)r p−1 < ∞; in particular the spectral condition (23) of Theorem 6.1 holds. However, the first part of Theorem 6.2 is applicable even if N (Lj k , G) is unbounded and (23) does not hold. The condition (25) can be reformulated for sequences with a certain asymptotic behavior: Lemma 6.5. Consider the sequence of non-negative numbers given by rk = ck q + dk k q−1 with c > 0, q 1 and a converging sequence (dk )k∈N . Then for l, p 0 the relation p
p
rk + lrk rk+1 − lrk+1 holds for almost all k ∈ N if (i) p < 1 − 1/q, or (ii) p = 1 − 1/q and l < qc1/q /2. Proof. This can be shown in a straightforward way by a Taylor series expansion of rk+1 in k.
2
The next proposition reverses the assertions of Theorems 4.5, 6.1 and 6.2 to some extend. As a consequence, the assumptions in these theorems can be relaxed. Proposition 6.6. Let G be an operator on a Hilbert space H with compact resolvent and a Riesz basis of Jordan chains. Suppose that 0 p < 1, α 0, 0 θj < 2π , j = 1, . . . , n, such that either (i) there exists r0 > 0 with σ (G) ⊂ Br0 (0) ∪
n
eiθj (x + iy) x > 0, |y| αx p ,
or
j =1
(ii) almost all eigenvalues of G lie inside regions
Kj k = eiθj (x + iy) rj−k x rj+k , |y| αx p ,
j = 1, . . . , n, k 1,
− where (rj±k )k1 are sequences of positive numbers satisfying rj−k rj+k < rj,k+1 .
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235
Then there is an isomorphism J : H → H , a normal operator G0 on H with compact resolvent, and an operator S0 p-subordinate to G0 such that J GJ −1 = G0 + S0 .
J D(G) = D(G0 ),
In case (i), all eigenvalues of G0 lie on the rays eiθj R0 and we have N (r, G0 ) = N (r, G)
for r 1.
In case (ii), all eigenvalues of G0 lie on the line segments
Lj k = eiθj x rj−k x rj+k , and N(Lj k , G0 ) = N (Kj k , G) holds for almost all pairs (j, k). Moreover, if S is p-subordinate to G, then J SJ −1 is p-subordinate to G0 . Proof. The idea is to use the isomorphism J to transform the Riesz basis of Jordan chains of G to an orthonormal basis of eigenvectors of G0 . Then one associates with each eigenvalue λ = eiθj (x + iy) of G an eigenvalue μ = eiθj w of G0 with w > 0 such that the assertions on the spectrum hold. A complete proof can be found in [27, §3.4]. 2 Remark 6.7. Theorems 4.5 and 6.1 also hold if G is as in the previous proposition and satisfies condition 6.6(i). Indeed we have J (G + S)J −1 = G0 + S0 + J SJ −1 in this case, S0 + J SJ −1 is p-subordinate to G0 , and the theorems can be applied to the righthand side. Analogously, Theorem 6.2 also holds if G satisfies 6.6(ii). In both cases, b is now the p-subordination bound of S0 + J SJ −1 to G0 . 7. Examples and applications To illustrate our theory, we first consider ordinary differential operators. Example 7.1. Let g0 , . . . , gn−2 ∈ L2 ([a1 , a2 ]), gn−1 ∈ L∞ ([a1 , a2 ]), and consider the differential operator T on L2 ([a1 , a2 ]) given by T = G + S,
Gu = i u
n (n)
,
Su =
n−1
gl u(l) ,
l=0
D(T ) = D(G) = D(S) = u ∈ W n,2 [a1 , a2 ] V1 (u) = · · · = Vn (u) = 0 , where W n,2 ([a1 , a2 ]) denotes the Sobolev space of n times weakly differentiable, square integrable functions, and the boundary conditions Vj (u) =
n−1 l=0
αj l u(l) (a1 ) + βj l u(l) (a2 ) = 0
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C. Wyss / Journal of Functional Analysis 258 (2010) 208–240
are regular in the sense of Naimark [20, §4.8] and such that G is selfadjoint. Then the resolvent of G is compact, and its spectrum consists of at most two sequences of eigenvalues of the form λj k = cj k n + dj k k n−1 ,
k kj 0 , j = 1, 2,
with cj = 0 and converging sequences (dj k )kkj 0 , see [20, §4.9]. As the multiplicity of every eigenvalue is at most n, Lemma 6.5 together with Remark 6.4 then implies that N (r, G) n−1 . < ∞ with p = 1−p n r r1 sup
For 0 ∈ (G) we can now show that S is (n − 1)/n-subordinate to G: Using Sobolev and interpolation inequalities, see [2], we can find constants b1 , b2 , b3 0 such that n−2 gl L2 u(l) ∞ b1 uW n−1,2 SuL2 gn−1 ∞ u(n−1) L2 + l=0
1/n (n−1)/n b2 uL2 uW n,2
(n−1)/n 1/n b3 uL2 uL2 + u(n) L2
(n−1)/n 1/n (n−1)/n b3 G−1 + 1 uL2 GuL2
for u ∈ D(G); for more details see [27, §3.2]. Consequently, Theorems 4.5 and 6.1 can be applied to T = G + S. In particular, T has a Riesz basis of finite-dimensional T -invariant subspaces. In the case that 0 ∈ σ (G), we choose any τ ∈ (G) ∩ R. Then S + τ is (n − 1)/n-subordinate to G − τ and Theorems 4.5 and 6.1 can be applied to the decomposition T = G − τ + S + τ . Finally, if the boundary conditions are such that almost all eigenvalues of G are simple and the norms of the coefficient functions gl are sufficiently small, then Theorem 6.2 is applicable and yields a Riesz basis of eigenvectors and finitely many Jordan chains for T , see also [27, §3.5]. Now we apply Theorems 4.5, 6.1 and 6.2 to two classes of diagonally dominant block operator matrices. For many results about the spectral theory of block operator matrices see [24,25]. Theorem 7.2. Let A(H1 → H1 ) and D(H2 → H2 ) be normal operators with compact resolvents on Hilbert spaces such that the spectra of A and D lie on finitely many rays from the origin and lim inf r→∞
N (r, A) < ∞, r 1−p
lim inf r→∞
N (r, D) <∞ r 1−p
with 0 p < 1. Suppose that the operators C(H1 → H2 ) and B(H2 → H1 ) are p-subordinate4 to A and D, respectively, Cu bu1−p Aup
for u ∈ D(A) ⊂ D(C),
Bv bv1−p Dvp
for v ∈ D(D) ⊂ D(B).
4 This notion of p-subordination is more general than the one from Section 4, since the operators B and C map from one Hilbert space into a (possibly) different one.
C. Wyss / Journal of Functional Analysis 258 (2010) 208–240
237
Then the block operator matrix T=
A C
B D
acting on H1 × H2 has a compact resolvent, admits a Riesz basis of finite-dimensional T invariant subspaces, and for every α > b there is a constant r0 0 such that σ (T ) ⊂ Br0 (0) ∪
n iθ
e j (x + iy) x 0, |y| αx p . j =1
Here θ1 , . . . , θn with 0 θj < 2π are the angles of the rays on which the spectra of A and D lie. Proof. We apply Theorems 4.5 and 6.1 to the decomposition T =G+S
with G =
A 0 0 , S= 0 D C
B 0
.
Indeed G is normal with compact resolvent and σ (G) = σ (A) ∪ σ (D),
N(r, G) = N (r, A) + N (r, D).
Moreover, using Hölder’s inequality, we find 2 S u = Bv2 + Cu2 b2 v2(1−p) Dv2p + b2 u2(1−p) Au2p v 1−p p Au2 + Dv2 b2 u2 + v2 for u ∈ D(A), v ∈ D(D), i.e. Sw bw1−p Gwp S is p-subordinate to G.
for w ∈ D(G) = D(A) × D(D);
2
In the next theorem, a symmetry of the operator matrix with respect to an indefinite inner product yields a gap in the spectrum around the imaginary axis. This makes it possible to apply the second part of Theorem 6.2. Theorem 7.3. Let A be a skew-adjoint operator with compact resolvent on a Hilbert space H . Let B, C : H → H be bounded, selfadjoint and uniformly positive, B, C γ > 0. Write (irk )k∈Λ for the sequence of eigenvalues of A where Λ ∈ {Z+ , Z− , Z} and (rk )k is increasing. Suppose that almost all eigenvalues irk are simple and that for some l > b = max{B, C} we have rk+1 − rk 2l
for almost all k ∈ Λ.
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Then the block operator matrix T=
A C
B A
has a compact resolvent, its spectrum is symmetric with respect to the imaginary axis and satisfies
σ (T ) ⊂ z ∈ C |z − irk | b for some k, | Re z| γ . Moreover almost all eigenvalues are simple and T admits a Riesz basis of eigenvectors and finitely many Jordan chains. Proof. We consider the decomposition T = G + S,
G=
A 0
0 0 , S= A C
B 0
.
G is skew-adjoint with compact resolvent, σ (G) = {irk | k ∈ Λ}, and almost all of its eigenvalues have multiplicity 2. S is bounded with S = b. By Theorem 4.5 T has a compact resolvent. If z is a point outside the discs Dk with radius b around the irk , then dist(z, σ (G)) > b and (G − z)−1 < b−1 ; thus z ∈ (T ) by Lemma 4.1. Now we use the indefinite inner products (Jj · |·) on H × H given by the fundamental symmetries 0 −iI 0 I J1 = , J2 = , iI 0 I 0 where (·|·) is the standard scalar product on H × H . We refer to [4] for a treatment of indefinite inner product spaces and operators therein. It is easy to verify that T is J1 -skew-adjoint (i.e., J1 T is skew-adjoint), which implies that σ (T ) is symmetric with respect to iR. On the other hand, for an eigenvalue λ of T with eigenvector w an easy calculation yields γ w2 Re(J2 T w|w) | Re λ|(J2 w|w) | Re λ|w2 ; hence | Re λ| γ , which shows the asserted shape of the spectrum. Finally we apply Theorem 6.2 with p = β1 = β2 = 0, θ1 = π/2, θ2 = 3π/2. It shows that N(Dk , T ) = 2 for almost all discs Dk . Consequently, almost all Dk contain only one skewconjugate pair of simple eigenvalues λ, −λ with | Re λ| γ . The second part of the theorem thus implies that the root subspaces of T form a Riesz basis. Since almost all root subspaces have dimension one, this is equivalent to the existence of a Riesz basis of eigenvectors and finitely many Jordan chains. 2 The previous results can be used to prove the existence of infinitely many solutions of an operator Riccati equation (see [27] for more details): Let A, B, C be operators on a Hilbert space, A closed and densely defined, B and C symmetric and non-negative. Consider the Hamiltonian operator matrix A B T= C −A∗
C. Wyss / Journal of Functional Analysis 258 (2010) 208–240
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and suppose that it has a Riesz basis of finite-dimensional T -invariant subspaces, e.g. obtained through Theorem 7.2. Using this Riesz basis of subspaces, one can show that for any subset of the point spectrum σ ⊂ σp (T ) the closed subspace U generated by all the corresponding root subspaces of T , i.e.
U = span x ∈ L(λ) λ ∈ σ , is both T - and (T − z)−1 -invariant for all z ∈ (T ). Moreover, T is J1 -skew-symmetric, which implies that σp (T ) is symmetric with respect to the imaginary axis. If now σp (T ) ∩ iR = ∅ and σ contains one eigenvalue from each skew-conjugate pair in σp (T ), then U is J1 -neutral, i.e., (J1 x|x) = 0 for all x ∈ U . Under additional assumptions on A and B it follows that U is the graph of a selfadjoint operator X, which in turn implies that X is a solution of the Riccati equation A∗ X + XA + XBX − C = 0. If in addition the assumptions of Theorem 7.3 are fulfilled, then X is even bounded and boundedly invertible. Acknowledgment The author wishes to thank Alexander Markus for some comments on the conditions in Theorem 6.1 and Christiane Tretter for many valuable suggestions. The author is also grateful for the support of Deutsche Forschungsgemeinschaft DFG, grant No. TR368/6-1, and of Schweizerischer Nationalfonds SNF, grant No. 15-486. References [1] V. Adamjan, V. Pivovarchik, C. Tretter, On a class of non-self-adjoint quadratic matrix operator pencils arising in elasticity theory, J. Operator Theory 47 (2) (2002) 325–341. [2] R.A. Adams, Sobolev Spaces, Academic Press, New York–London, 1975. [3] T.Y. Azizov, A. Dijksma, L.I. Sukhocheva, On basis properties of selfadjoint operator functions, J. Funct. Anal. 178 (2) (2000) 306–342. [4] T.Y. Azizov, I.S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, John Wiley & Sons, Chichester, 1989. [5] C. Clark, On relatively bounded perturbations of ordinary differential operators, Pacific J. Math. 25 (1968) 59–70. [6] R.F. Curtain, H.J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Springer, New York, 1995. [7] E.B. Davies, Pseudo-spectra, the harmonic oscillator and complex resonances, Proc. R. Soc. Lond. A 455 (1999) 585–599. [8] N. Dunford, J.T. Schwartz, Linear Operators, Part III, Wiley–Interscience, New York, 1971. [9] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, vol. I, Birkhäuser, Basel, 1990. [10] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Amer. Math. Soc., Providence, 1969. [11] B. Jacob, C. Trunk, M. Winklmeier, Analyticity and Riesz basis property of semigroups associated to damped vibrations, J. Evol. Equ. 8 (2) (2008) 263–281. [12] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1980. [13] M.V. Keldysh, On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators, Uspekhi Mat. Nauk 26 (4) (1971) 15–41, English transl. in Russian Math. Surveys 26 (4) (1971) 15–44. [14] S.G. Krein, Linear Differential Equations in Banach Space, Amer. Math. Soc., Providence, 1971.
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[15] C.R. Kuiper, H.J. Zwart, Connections between the algebraic Riccati equation and the Hamiltonian for Riesz-spectral systems, J. Math. Systems Estim. Control 6 (4) (1996) 1–48. [16] B.J. Levin, Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence, 1980. [17] A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer. Math. Soc., Providence, 1988. [18] A.S. Markus, V.I. Matsaev, On the convergence of eigenvector expansions for an operator which is close to being selfadjoint, Mat. Issled. 61 (1981) 104–129 (in Russian). [19] M. Marletta, A. Shkalikov, C. Tretter, Pencils of differential operators containing the eigenvalue parameter in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 133 (4) (2003) 893–917. [20] M.A. Naimark, Linear Differential Operators, Part 1, Frederick Ungar Publishing Co., New York, 1967. [21] A.A. Shkalikov, On the basis problem of the eigenfunctions of an ordinary differential operator, Uspekhi Mat. Nauk 34 (5) (1979) 235–236, English transl. in Russian Math. Surveys 34 (5) (1979) 249–250. [22] A.A. Shkalikov, The basis problem of the eigenfunctions of ordinary differential operators with integral boundary conditions, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 37 (6) (1982) 12–21, English transl. in Moscow Univ. Math. Bull. 37 (6) (1982) 10–20. [23] I. Singer, Bases in Banach Spaces II, Springer, Berlin, 1981. [24] C. Tretter, Spectral issues for block operator matrices, in: Differential Equations and Mathematical Physics, Birmingham, AL, 1999, in: AMS/IP Stud. Adv. Math., vol. 16, Amer. Math. Soc., Providence, 2000, pp. 407–423. [25] C. Tretter, Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London, 2008. [26] V.N. Vizitei, A.S. Markus, On convergence of multiple expansions in the eigenvectors and associated vectors of an operator pencil, Mat. Sb. 66 (108) (1965) 287–320, English transl. in Amer. Math. Soc. Transl. (2) 87 (1970) 187–227. [27] C. Wyss, Perturbation theory for Hamiltonian operator matrices and Riccati equations, PhD thesis, University of Bern, 2008. [28] G.Q. Xu, S.P. Yung, The expansion of a semigroup and a Riesz basis criterion, J. Differential Equations 210 (1) (2005) 1–24. [29] H. Zwart, Riesz basis for strongly continuous groups, arXiv:0808.3447 [math.FA], 2008.
Journal of Functional Analysis 258 (2010) 241–254 www.elsevier.com/locate/jfa
Equivariant Yamabe problem and Hebey–Vaugon conjecture Farid Madani Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, France Received 17 March 2009; accepted 2 October 2009 Available online 12 October 2009 Communicated by Paul Malliavin
Abstract In their study of the Yamabe problem in the presence of isometry groups, E. Hebey and M. Vaugon announced a conjecture. This conjecture generalizes T. Aubin’s conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In this paper, we generalize Aubin’s theorem and we prove the Hebey–Vaugon conjecture in dimensions less or equal to 37. © 2009 Elsevier Inc. All rights reserved. Résumé Dans leur étude du probleme de Yamabe équivariant, E. Hebey et M. Vaugon annonçaient une conjecture. Cette conjecture généralise la conjecture de T. Aubin qui a été déjà démontrée et est suffisante pour résoudre le probleme de Yamabe. Dans cet article, nous généralisons un théoreme de T. Aubin et nous démontrons que cette conjecture de Hebey–Vaugon est vraie jusqu’à la dimesion 37. © 2009 Elsevier Inc. All rights reserved. Keywords: Conformal metric; Isometry group; Scalar curvature; Yamabe problem
1. Introduction Let (M, g) be a compact Riemannian manifold of dimension n 3. Denote by I (M, g), C(M, g) and Rg the isometry group, the conformal transformations group and the scalar curvature, respectively. Let G be a subgroup of the isometry group I (M, g). E. Hebey and M. Vaugon [6] considered the following problem: E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.002
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Hebey–Vaugon problem. Is there some G-invariant metric g0 which minimizes the functional Rg dv(g ) J (g ) = M n−2 ( M dv(g )) n where g belongs to the G-invariant conformal class of metrics g defined by: [g]G := g˜ = ef g/f ∈ C ∞ (M), σ ∗ g˜ = g˜ ∀σ ∈ G . The positive answer would have two consequences. The first is that there exists an I (M, g)invariant metric g0 conformal to g such that the scalar curvature Rg0 is constant. The second is that the A. Lichnerowicz’s conjecture [8], stated below, is true. By the works of J. LelongFerrand [7] and M. Obata [10], we know that if (M, g) is not conformal to (Sn , gcan ) (the unit sphere endowed with its standard metric gcan ), then C(M, g) is compact and there exists a conformal metric g to g such that I (M, g ) = C(M, g). This implies that the first consequence is equivalent to the A. Lichnerowicz conjecture. For every compact Riemannian manifold (M, g) which is not conformal to the unit sphere Sn endowed with its standard metric, there exists a metric g˜ conformal to g for which I (M, g) ˜ = C(M, g), and the scalar curvature Rg˜ is constant. To such metrics correspond functions which are necessarily solutions of the Yamabe equation. 4 In other words, if g˜ = ψ n−2 g, ψ is a G-invariant smooth positive function then ψ satisfies n+2 4(n − 1) g ψ + Rg ψ = Rg˜ ψ n−2 . n−2
The classical Yamabe problem, which consists in finding a conformal metric with constant scalar curvature on a compact Riemannian manifold, is the particular case of the problem above when G = {id}. Denote by OG (P ) the orbit of P ∈ M under G, Wg the Weyl tensor associated to the manifold (M, g) and ωn the volume of the unit sphere Sn . Following E. Hebey and M. Vaugon [5,6], we define the integer ω(P ) at the point P as ω(P ) = +∞ if ∀i ∈ N, ∇ i Wg (P ) = 0 . ω(P ) = inf i ∈ N/∇ i Wg (P ) = 0 Hebey–Vaugon conjecture. Let (M, g) be a compact Riemannian manifold of dimension n 3 and G be a subgroup of I (M, g). If (M, g) is not conformal to (Sn , gcan ) or if the action of G has no fixed point, then the following inequality holds 2/n 2/n inf card OG (Q) inf J g < n(n − 1)ωn .
g ∈[g]G
Q∈M
(1)
Remarks 1.1. 1. This conjecture is the generalization of the former T. Aubin’s conjecture [1] for the Yamabe problem corresponding to G = {id}, where the constant in the right side of the inequality is equal to infg ∈[gcan ] J (g ) for Sn . In this case, the conjecture is completely proved. 2. The inequality is obvious if infg ∈[g]G J (g ) is nonpositive, it is the case when there exists a Yamabe metric with nonpositive scalar curvature. 3. If for any Q ∈ M, card OG (Q) = +∞ then this conjecture is also obvious.
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243
The only results known about this conjecture are given in the following theorem: Theorem 1.1 (E. Hebey and M. Vaugon). Let (M, g) be a smooth compact Riemannian manifold of dimension n 3 and G be a subgroup of I (M, g). We always have: 2/n 2/n inf J g n(n − 1)ωn inf card OG (Q)
g ∈[g]G
Q∈M
and inequality (1) holds if one of the following items is satisfied. 1. The action of G on M is free. 2. 3 dim M 11. 3. There exists a point P with minimal orbit ( finite) under G such that ω(P ) > (n − 6)/2 or ω(P ) ∈ {0, 1, 2}. The case ω = 3 was studied by A. Rauzy (private communication). In this paper we prove the following results: Main theorem. The Hebey–Vaugon conjecture holds if there exists a point P ∈ M with minimal orbit ( finite) for which ω(P ) 15 or if the degree of the leading part of Rg is greater or equal to ω(P ) + 1, in the neighborhood of this point P . Corollary 1.1. Hebey–Vaugon conjecture holds for every smooth compact Riemannian manifold (M, g) of dimension n ∈ [3, 37]. To prove the main theorem, we need to construct a G-invariant test function φ such that 2/n
Ig (φ) < n(n − 1)ωn
2/n inf card OG (Q) .
Q∈M
Thus, all the difficulties are in the construction of a such function. For some cases, we can use the test functions constructed by T. Aubin [1] and R. Schoen [11] in the case of Yamabe problem. They have been already proven by E. Hebey and M. Vaugon [6]. The item 3, presented in Theorem 1.1, uses test functions different than T. Aubin and R. Schoen ones. We multiply T. Aubin’s test function uε,P by a function as follows: ϕε (Q) = 1 − r ω+2 f (ξ ) uε,P (Q),
n−2 n−2 ε ε 2 −( ) 2 if Q ∈ BP (δ), δ 2 +ε 2 uε,P (Q) = ( r 2 +ε2 ) 0 if Q ∈ M − BP (δ)
(2) (3)
for all Q ∈ M, where r = d(Q, P ) is the distance between P and Q. (r, ξ j ) is a geodesic coordinates system in the neighborhood of P and BP (δ) is the geodesic ball of center P with radius δ fixed sufficiently small. f is a function depending only on ξ , chosen such that j Sn−1 f dσ = 0. Without loss of generality, we suppose that in the coordinates system (r, ξ ) we m have det g = 1 + o(r ) for m 1. In fact, E. Hebey and M. Vaugon proved that there exists g˜ ∈ [g]G for which det g˜ = 1 + o(r m ) and infg ∈[g]G J (g ) does not depend on the conformal G-invariant metric.
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2. Computation of
M
Rg ϕε2 dv
Let be δ/ε Iab (ε) = 0
tb dt (1 + t 2 )a
and Iab = lim Iab (ε) ε→0
then Ia2a−1 (ε) = log ε −1 + O(1). If 2a − b > 1 then Iab (ε) = Iab + O(ε 2a−b−1 ) and by integration by parts, we establish the following relationships: Iab =
b−1 b − 1 b−2 2a − b − 3 b = I b−2 = I I , 2a − b − 1 a 2a − 2 a−1 2a − 2 a−1
4(n − 2)Inn+1 (Inn−2 )(n−2)/n
= n.
(4)
Using the inequality (a − b)β a β − βa β−1 b for 0 < b < a, we have for β 2, 0 α < (n − 2)(β − 1) − n
β α+n−1 α+n−β(n−2)/2 r α uε,P dv = ωn−1 I(n−2)β/2 ε + O ε n−2 .
(5)
M
This integral appears frequently in the following computations, and it allows us to neglect the constant term in the expression of uε , when we choose δ sufficiently small and ε smaller than δ. Denote by Ig the Yamabe functional defined for all ψ ∈ H 1 (M) by Ig (ψ) =
|∇g ψ|2 dv + M
(n − 2) 4(n − 1)
Rg ψ 2 dv ψ −2 N
(6)
M
where N = 2n/(n − 2) and ∇g is the gradient of the metric g. The second integral of the functional Ig with the scalar curvature term needs a special consideration. Let μ(P ) be an integer defined as follows: |∇β Rg (P )| = 0 for all |β| < μ(P ) and there exists γ ∈ Nμ(P ) such that |∇γ Rg (P )| = 0 then Rg (Q) = R¯ + O r μ(P )+1 , where R¯ = r μ(P ) |β|=μ ∇β Rg (P )ξ β is a homogeneous polynomial of degree μ(P ), the β are multi-indices. For simplicity, we drop the letter P in ω(P ) and μ(P ). By E. Hebey and M. Vaugon [6] results: Lemma 2.1. μ ω, gij = δij + O(r ω+2 ) and ¯ S(r) Rg = O(r 2ω+2 ) which implies that ¯ S(r) R dσ = 0 when μ < 2ω + 2.
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245
¯ denotes the average. Then
Rg ϕε2 dv M
=
Rg u2ε,P
dv − 2
M
= ε 2ω+4 ωn−1
¯
f u2ε,P Rg r ω+2 dv
+
M
f 2 u2ε,P Rg r 2ω+4 dv M
n+2ω+1 r −2ω−2 Rg dσ In−2 (ε)
S(r) ω+μ+n+1
− 2ε ω+μ+4 In−2
(ε)ωn−1
¯
r −μ f (ξ )R¯ dσ (ξ ) + O ε n−2 .
(7)
S(r)
Moreover T. Aubin [2] proved that: Theorem 2.1. If μ ω + 1 then there exists C(n, ω) > 0 such that ¯
R dσ = C(n, ω)(−g )ω+1 R(P )r 2ω+2 + o r 2ω+2 ,
Sn−1 (r)
(−g )ω+1 R(P ) is negative. Then Ig (uε,P ) <
n(n−2) 2/n 4 ωn−1 .
From now until the end of this section, we make the assumption that μ = ω. Now, we recall some results obtained by T. Aubin in his papers [3,4]: R¯ is homogeneous polynomial of degree ω then E R¯ is homogeneous of degree ω − 2 and E R¯ = r −2 s R¯ − ω(n + ω − 2)R¯ ¯ where E is the Euclidean Laplacian and s is the Laplacian on the sphere Sn−1 . k−1 E R is homogeneous of degree ω − 2k + 2 and −2k ¯ kE R¯ = r −2 (s − νk id)k−1 E R=r
k
(S − νp id)R¯
p=1
with νk = (ω − 2k + 2)(n + ω − 2k).
(8)
The sequence of integers (νk ){1k[ω/2]} is decreasing. It will play the role of the eigenvalues of the Laplacian on the sphere Sn−1 . It is known that the eigenvalues of the geometric Laplacian are non-negative and increasing. Our νk are in the opposite order. [ω/2] We know by T. Aubin’s paper [2] that E R¯ = 0 and S(r) R¯ dσ = 0, then q = min k ∈ N/kE R¯ = 0 q is well defined and r −ω R¯ ∈ k=1 Ek , with Ek the eigenspace associated to the positive eigenvalues νk of the Laplacian s on the sphere Sn−1 . If j = k, then Ek is orthogonal to Ej , for the
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standard scalar product in H12 (Sn−1 ). Moreover, since functions of s ) such that R¯ = r ω s
q
ϕk = r
ω
k=1
R¯ dσ = 0 there exist ϕk ∈ Ek (eigen-
q
νk ϕk .
(9)
k=1
According to Lemma 2.1, we can split the metric g in the following way: g=E +h
(10)
where E is the Euclidean metric and h is a symmetric 2-tensor defined in our geodesic coordinates system by hij = r ω+2 g¯ ij + r 2(ω+2) gˆ ij + h˜ ij
and hir = hrr = 0
(11)
where g, ¯ gˆ and h˜ are symmetric 2-tensors defined on the sphere Sn−1 . We denote by s the standard metric on the sphere, ∇, are the associated gradient and Laplacian on Sn−1 . By straightforward computations, Aubin [3] proved that: Lemma 2.2. R¯ = ∇ ij g¯ ij r ω
¯
and
R dσ = B/2 − C/4 − (1 + ω/2)2 Q r 2(ω+1) + o r 2(ω+1)
Sn−1 (r)
where B = ¯ Sn−1 ∇ i g¯ j k ∇j g¯ ik dσ , C = ¯ Sn−1 ∇ i g¯ j k ∇i g¯ j k dσ and Q = ¯ Sn−1 g¯ ij g¯ ij dσ . For further details refer to [9]. The integrals Q, B and C are given in terms of the tensor g. ¯ Our goal is to compute them using the eigenfunctions ϕk above. Let us define bij =
q k=1
1 (n − 1)∇ij ϕk + νk ϕk sij (n − 2)(νk + 1 − n)
and aij such that g¯ ij = aij + bij then, according to (9), we check that R¯ = R¯ b = ∇ ij bij r ω
and R¯ a = ∇ ij aij r ω = 0.
If g¯ ij = aij then R¯ = R¯ a = 0 and μ ω + 1. By Theorem 2.1 ¯ Sn−1 (r)
R dσ =
¯ Sn−1 (r)
Ra dσ < 0.
(12)
F. Madani / Journal of Functional Analysis 258 (2010) 241–254
247
If g¯ ij = bij then ¯
R dσ =
Sn−1 (r)
¯
Rb dσ = Bb /2 − Cb /4 − (1 + ω/2)2 Qb r 2(ω+1) + o r 2(ω+1)
Sn−1 (r)
where Bb , Cb and Qb are the same integrals defined in Lemma 2.2 when the considered tensor g¯ ij = bij . We compute them in terms of ϕk
Qb =
¯ Sn−1
¯ q νk n−1 ϕk2 dσ, b¯ij b¯ ij dσ = n−2 νk − n + 1 k=1
Bb = −(n − 1)Qb +
q k=1
Cb = −(n − 1)Qb +
Sn−1
¯
νk
ϕk2 dσ,
Sn−1
¯ q n−1 νk ϕk2 dσ. n−2 k=1
Sn−1
q To find these expressions, we used several times the identity ∇ i bij = − k=1 ∇j ϕk and Stokes formula (more details are given in [3,4] and [9]). In the general case, we deduce that Lemma 2.3. If μ = ω and g¯ ij = aij + bij , where bij is defined above, ¯
¯
R dσ =
Sn−1 (r)
Ra + Rb dσ
Sn−1 (r)
Bb /2 − Cb /4 − (1 + ω/2)2 Qb r 2(ω+1) + o r 2(ω+1)
(13)
and Bb /2 − Cb /4 − (1 + ω/2)2 Qb =
q k=1
uk
¯
ϕk2 dσ
(14)
Sn−1
with uk =
(n − 1)2 + (n − 1)(ω + 2)2 n−3 − νk . 4(n − 2) 4(n − 2)(νk − n + 1)
uk is obtained using the expressions of Qb , Bb and Cb above.
(15)
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F. Madani / Journal of Functional Analysis 258 (2010) 241–254
3. Generalization of T. Aubin’s theorem Theorem 3.1. If there exists P ∈ M such that ω(P ) (n − 6)/2 then there exists f ∈ C ∞ (Sn−1 ) with vanishing mean integral such that Ig (ϕε ) <
n(n − 2) 2/n ωn−1 . 4
The case ω = 0 of the this theorem has already been proven by T. Aubin [1]. He also proved the theorem when μ ω + 1 (see Theorem 2.1). From now until the end of this paper, we drop the letter P in ω(P ) and μ(P ). Proof. If μ ω + 1 then the inequality holds by Theorem 2.1. So we suppose that μ = ω until the end of the proof. We start by computing the first integral of the Yamabe functional (6) with ψ = ϕε . Using formula |∇g ϕε |2 = (∂r ϕε )2 + r −2 |∇s ϕε |2 , we obtain:
δ
|∇g ϕε | dv =
|∇g uε,P | dv +
2
M
2
M
(ω+2) 2 ∂r r uε,P r n−1 dr
δ
|∇f |2 dσ.
u2ε,P r n+2ω+1 dr 0
f 2 dσ
Sn−1
0
+
Sn−1
The substitution t = r/ε gives 2ω+n+1 |∇g ϕε |2 dv = (n − 2)2 ωn−1 Inn+1 (ε) + ε 2ω+4 |∇f |2 dσ In−2 (ε) M
+
Sn−1
f 2 dσ (ω − n + 4)2 In2ω+n+5 (ε)
Sn−1
.
+ 2(ω + 2)(ω − n + 4)In2ω+n+3 (ε) + (ω + 2)2 In2ω+n+1 (ε) For ϕε −2 N , we need to compute the Taylor expansion of: 2ω+4 N N (N − 1) 2ω+4 2 N ω+2 ϕε (Q) = 1 − N r r uε,P . f (ξ ) + f (ξ ) + o r 2 Using the fact that Sn−1 f dσ (ξ ) = 0 and formula (5), we conclude that ϕε N N
δ N (N − 1) 2(ω+2) 2 1+ r = f (ξ ) + o r 2ω+4 r n−1 uN ε,P dr dσ (ξ ) 2 0 Sn−1
= ωn−1 Inn−1
N (N − 1) 2(ω+2) ε + 2
Sn−1
f 2 dσ In2ω+n+3 + o ε 2ω+4
(16)
F. Madani / Journal of Functional Analysis 258 (2010) 241–254
249
then n−1 −2/N ϕε −2 N = ωn−1 In × 1 − (N − 1)ε 2(ω+2) f 2 dσ In2ω+n+3 / ωn−1 Inn−1 + o ε 2ω+4 .
(17)
Sn−1
By Eqs. (16), (17), (7) and the relationship (4), if n > 2ω + 6 then: Ig (ϕε ) =
−2/N n+2ω+1 2ω+4 n(n − 2) 2/n ωn−1 + ωn−1 Inn−1 In−2 ε 4 (n − 2)ωn−1 ¯ −2ω−2 n−2 × r Rg dσ − f (ξ )R¯ dσ + |∇f |2 dσ 4(n − 1) 2(n − 1) S(r)
−
n(n − 2)2
− (ω + 2)2 (n2
+ n + 2)
(n − 1)(n − 2)
Sn−1
Sn−1
f 2 dσ + o ε 2ω+4 .
Sn−1
If n = 2ω + 6 then Ig (ϕε ) =
−2/N 2ω+4 n(n − 2) 2/n ωn−1 + ωn−1 Inn−1 ε log ε −1 4 (n − 2)ωn−1 ¯ −2ω−2 n−2 × r Rg dσ − f (ξ )R¯ dσ 4(n − 1) 2(n − 1) S(r)
+
|∇f |2 dσ + (ω + 2)2
Sn−1
Sn−1
f 2 dσ + O ε 2ω+4 .
Sn−1
For further details refer to [9]. Let IS be the functional defined for a function f on the sphere Sn−1 , with zero mean integral, by IS (f ) =
¯
4(n − 1)(n − 2)|∇f |2 − 4n(n − 2)2 − 4(ω + 2)2 n2 + n + 2 f 2
Sn−1
− 2(n − 2)2 f R¯ dσ. This implies that if n > 2ω + 6 2/n
Ig (ϕε ) =
n+2ω+1 2ω+4 ε ωn−1 In−2 n(n − 2) 2/n ωn−1 + 4 4(n − 1)(n − 2)(Inn−1 )2/N ¯ 2 −2ω−2 × (n − 2) r Rg dσ + IS (f ) + o ε 2ω+4 S(r)
(18)
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and if n = 2ω + 6 n+2ω+1 2ω+4 ε log ε −1 ωn−1 In−2 n(n − 2) 2/n Ig (ϕε ) = ωn−1 + 4 4(n − 1)(n − 2)(Inn−1 )2/N ¯ 2 −2ω−2 × (n − 2) r Rg dσ + IS (f ) + O ε 2ω+4 . 2/n
(19)
S(r)
Notice that if k = j then IS (ϕk + ϕj ) = IS (ϕk ) + IS (ϕj ). Indeed, ϕk and ϕj are orthogonal for the standard scalar product in H12 (Sn−1 ). 2 ¯ 2 2 2 ϕk dσ IS (ck νk ϕk ) = dk ck − 2(n − 2) ck νk
=−
(n − 2)4 dk
νk2
¯
Sn−1
ϕk2 dσ
Sn−1
where dk = 4 (n − 1)(n − 2)νk − n(n − 2)2 + (ω + 2)2 n2 + n + 2
and ck =
(n − 2)2 . dk
Using(8), we can check easily that dk is positive for any 1 k [ω/2]. Now, let us consider q f = 1 ck νk ϕk . Then IS (f ) = −
q (n − 2)4
dk
1
¯
νk2
ϕk2 dσ
Sn−1
and by Lemma 2.3 (n − 2)
2
¯
r
−2ω−2
Rg dσ + IS (f )
q 1
S(r)
The following lemma implies that Ig (ϕε ) <
(n − 2)4 2 uk (n − 2) − νk dk 2
¯ Sn−1
n(n−2) 2/n 4 ωn−1 .
2
Lemma 3.1. For any k q [ω/2] the following inequality holds uk −
(n − 2)2 2 νk < 0. dk
Proof. Recall the expression of νk given in (8). The sequence (Uk ) defined by uk (n − 2)3 νk Uk := (νk − n + 1)dk (n − 2) − νk dk
ϕk2 dσ + o(1).
F. Madani / Journal of Functional Analysis 258 (2010) 241–254
251
is polynomial decreasing in νk when νk 0. In fact, Uk = P (νk ) with P the decreasing polynomial in R+ , defined by P (x) = (n − 1)(n − 2)x − n(n − 2)2 + (ω + 2)2 n2 + n + 2 × (n − 3)(x − n + 1) − (n − 1)2 − (n − 1)(ω + 2)2 − (n − 2)3 x 2 − (n − 1)x . The derivative of P is P (x) = −2(n − 2)x − 2n(n − 2)3 + 2 n2 − 3n − 2 (ω + 2)2 . By assumption ω + 2 (n − 2)/2 then P is decreasing in R+ . Hence Uk = P (νk ) P (νω/2 ) = Uω/2 for all k ω/2. It easy to check that uω/2 is negative so Uk Uω/2 < 0.
2
4. Proof of the main theorem By Remarks 1.1, we consider only the positive case (i.e., infg ∈[g]G J (g ) > 0) and the case when there exists P ∈ M such that OG (P ) = {Pi }1im , ω
m = card OG (P ) = inf card OG (Q), Q∈M
n−6 2
and P1 = P .
Let ϕ˜ ε,i be a function defined as follows: ϕ˜ ε,i (Q) = 1 − riω+2 fi (ξ ) uε,Pi (Q)
(20)
where ri = d(Q, Pi ), the function uε,Pi is defined as in (3) and fi is defined by: −1 fi (Q) = cri−ω ∇gω R(Pi ) exp−1 Pi Q, . . . , expPi Q ,
(21)
expPi is the exponential map. In a geodesic coordinates system {r, ξ j } with origin P , induced by the exponential map f1 = cr
−ω
R¯ = c
q
νk ϕk
k=1
¯ ϕk and νk are defined in Section 2. Thus the functions fi are defined on the sphere Sn−1 . where R, The choice of the constant c is important.
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F. Madani / Journal of Functional Analysis 258 (2010) 241–254
Lemma 4.1. Suppose that ω (n − 6)/2. If ω ∈ [3, 15] or if deg R¯ ω + 1 then there exists c ∈ R such that the corresponding functions ϕ˜ε,i satisfy: 1 2/n Ig (ϕ˜ε,i ) < n(n − 2)ωn . 4
(22)
Remarks 4.1. 1. We proved inequality of this lemma for any ω (n−6)/2, using test function ϕε (see Theorem 3.1). We notice that the difference between ϕε and ϕ˜ε,i is on the construction of the corresponding functions f and fi , respectively. From ϕ˜ε,i we define a G-invariant function (see proof of the main theorem below), this property is not possible with the function ϕε . 2. For ω = 16 and n sufficiently big, we can check that for any c ∈ R, inequality (22) is false. Proof. 1. If deg R¯ ω + 1, then by Theorem 2.1 Ig (uε,Pi ) <
n(n − 2) 2/n ωn . 4
It is sufficient to take c = 0, hence ϕ˜ε,i = uε,Pi . 2. If deg R¯ = ω. Using estimates given in the proof of Theorem 3.1 (see (18), (19)), it is sufficient to show that there exists c ∈ R such that ¯ IS (f1 ) + (n − 2)2 r −2ω−2 Rg dσr < 0. (23) S(r)
We keep the notations used in the proof of Theorem 3.1. Thus IS (f1 ) =
q
¯ IS (cνk ϕk ) = dk c2 − 2(n − 2)2 c νk2 ϕk2 dσ
k=1
and
Sn−1
¯ S(r)
r −2ω−2 Rg dσr =
q k=1
uk
¯
ϕk2 dσ.
Sn−1
To prove inequality (23), it is sufficient to prove that ∀k q
uk dk c2 − (n − 2)c + (n − 2) 2 < 0. 2(n − 2) 2νk
(24)
The left side of the inequality above is a second degree polynomial with variable c, his discriminant is: k = (n − 2)2 −
dk uk . νk2
(25)
Using Lemma 3.1, we deduce that for any k q, k > 0. Hence, the polynomial above admits two different roots denoted xk < yk and given by √ √ (n − 2)2 − (n − 2) k (n − 2)2 + (n − 2) k , yk = . xk = dk dk
F. Madani / Journal of Functional Analysis 258 (2010) 241–254
253
Inequality (24) holds if and only if q
(xk , yk ) = ∅.
(26)
k=1
The sequence (dk )k[ω/2] decreases. It is easy to check that ω xk < yj . ∀k < j 2
(27)
Hence intersection (26) is not empty if ∀k < j
ω xj < yk . 2
(28)
We also check that if ω is even, uω/2 < 0, which implies xω/2 < 0. i. If ω = 3 then q = 1, intersection above is not empty. It is sufficient to take c = (x1 + y2 )/2. ii. If ω = 4 then k ∈ {1, 2}, x2 < 0 (because u2 < 0) and 0 < x1 < y2 . Hence intersection ]x1 , y1 [ ∩ ]x2 , y2 [ is not empty. iii. If 5 ω 15, it is sufficient to prove (28) which is equivalent to prove that ω (n − 2)(dj − dk ) + dk j + dj k > 0. (29) ∀k < j 2 Notice that k given by (25) is a rational fraction in n. By straightforward computations, we check that there exists reel numbers ak , bk , ek , hk and sk which depend on k and ω such that sk hk + , n − 2 νk + 1 − n
√ bk . k > ak n + 2ak
k = a k n 2 + b k n + e k +
(30) (31)
Inequality (29) holds if we use (31). The expressions of the reel numbers above are known explicitly (we used the software Maple to compute them, see [9]). For simplicity, we omit to give these expressions. 2 Proof of the main theorem. The orbit of P under the action of G is supposed to be minimal (i.e. card OG (P ) = infQ∈M card OG (Q)). Without loss of generality, we suppose that 3 ω (n − 6)/2, because if ω > (n − 6)/2 or ω 2, we conclude using Theorem 1.1. From functions ϕ˜ε,i defined by (20), we define the function φε as follows: φε =
m
ϕ˜ε,i ,
k=1
φε is G-invariant. In fact, for any σ ∈ G, such that σ (Pi ) = Pj uε,Pi = uε,Pj ◦ σ
and fi = fj ◦ σ,
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fi are defined by (21), we deduce that ϕ˜ε,i = ϕ˜ ε,j ◦ σ. The support of ϕ˜ε,i is included in the ball BPi (δ). We choose δ sufficiently small such that for all integers i = j in [1, m], intersection BPj (δ) ∩ BPi (δ) = ∅. Thus 2/n Ig (ϕε ). Ig (φε ) = card OG (P ) By Lemma 4.1, we conclude that Ig (φε ) <
2/n n(n − 2) 2/n ωn−1 card OG (P ) . 4
4/(n−2)
It remains to notice that if g˜ = φε J (g) ˜ =4
g then
2/n n−1 2/n Ig (φε ) < n(n − 1)ωn−1 card OG (P ) n−2
where ε is sufficiently smaller than δ.
2
Proof of Corollary 1.1. Suppose that the orbit of P under the action of G is minimal (otherwise the conjecture is obvious). If ω = ω(P ) > [(n − 6)/2], we conclude using Theorem 1.1. If ω [(n − 6)/2] 15, we conclude using main theorem. 2 References [1] T. Aubin, Équations différentielles non linéaires et problème de Yamabe, J. Math. Pures Appl. 55 (1976) 269–296. [2] T. Aubin, Sur quelques problèmes de courbure scalaire, J. Funct. Anal. 240 (2006) 269–289. [3] T. Aubin, Solution complète de la C 0 compacité de l’ensemble des solutions de l’équation de Yamabe, J. Funct. Anal. 244 (2007) 579–589. [4] T. Aubin, On the C 0 compactness of the set of the solutions of the Yamabe equation, Bull. Sci. Math. (2008). [5] E. Hebey, M. Vaugon, Courbure scalaire prescrite pour des variétés non conformément difféomorphes à la sphere, C. R. Acad. Sci. Paris 316 (3) (1993) 281–282. [6] E. Hebey, M. Vaugon, Le probleme de Yamabe équivariant, Bull. Sci. Math. 117 (1993) 241–286. [7] J. Lelong-Ferrand, Mém. Acad. Royale Belgique, Classe des Sciences 39 (1971). [8] A. Lichnerowicz, Sur les transformations conformes d’une variété riemannienne compacte, C. R. Acad. Sci. Paris 259 (1964). [9] F. Madani, Le probléme de Yamabe avec singularités et la conjecture de Hebey–Vaugon, PhD thesis, Université Pierre et Marie Curie, 2009. [10] M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971) 247–258. [11] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984) 479–495.
Journal of Functional Analysis 258 (2010) 255–259 www.elsevier.com/locate/jfa
The Dixmier problem, lamplighters and Burnside groups Nicolas Monod a,∗,1 , Narutaka Ozawa b,2 a EPFL, Switzerland b The University of Tokyo, Japan
Received 22 March 2009; accepted 20 June 2009 Available online 17 July 2009 Communicated by S. Vaes
Abstract J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. This gives a new characterisation of amenability. Furthermore, we deduce that certain Burnside groups are non-unitarisable, answering a question raised by G. Pisier. © 2009 Elsevier Inc. All rights reserved. Keywords: Unitarisable representation; Amenable group; Burnside group
1. Introduction A group G is said to be unitarisable if every uniformly bounded representation π of G on a Hilbert space H is unitarisable, i.e. there is an invertible operator S on H such that Sπ(·)S −1 is a unitary representation. Dixmier [3] proved that all amenable groups are unitarisable and asked whether unitarisability characterises amenability. Since unitarisability passes to subgroups and non-commutative free groups are not unitarisable, every group containing a noncommutative free group is non-unitarisable. For these facts and more background, we refer to Pisier [10,11]. * Corresponding author.
E-mail address:
[email protected] (N. Monod). 1 Supported in part by the Swiss National Science Foundation. 2 Supported in part by the Japan Society for the Promotion of Science.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.029
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N. Monod, N. Ozawa / Journal of Functional Analysis 258 (2010) 255–259
Recently, a criterion was discovered [4] that lead to examples without free subgroups (see [4,9]). We shall improve a strategy proposed in [7] in order to apply ergodic methods to the problem. Now are our browes bound with Victorious Wreathes3 Let G and A be groups. Recall that the associated (restricted) wreath product, or lamplighter group, is the group AG=
G
A G,
wherein G A is the restricted product indexed by G upon which G acts by permutation. We shall be interested in the case where A and hence also G A is abelian. Theorem 1. For any group G, the following assertions are equivalent. (i) The group G is amenable. (ii) The wreath product A G is unitarisable for all abelian groups A. (iii) The wreath product A G is unitarisable for some infinite abelian group A. The above theorem leads to a partial answer to a question of G. Pisier, namely whether free Burnside groups are unitarisable (see e.g. [11]). Theorem 2. Let m, n, p be integers with m, n 2, p 665 and n, p odd. Then the free Burnside group B(m, np) of exponent np with m generators is non-unitarisable. 2. Proofs Let G be a group and (π, H ) be a unitary representation of G. We write L (H ) for the algebra of bounded operators of H . A map D : G → L (H ) is called a derivation if it satisfies the Leibniz rule D(gh) = D(g)π(h) + π(g)D(h), or equivalently if the map πD defined by πD (g) =
π(g) 0
D(g) π(g)
∈ L (H ⊕ H )
is a group homomorphism. In that case, πD is a uniformly bounded representation if and only if D is a bounded derivation. Moreover, πD is unitarisable if and only if D is inner, i.e. there is T ∈ L (H ) such that D(g) = π(g)T − T π(g). (See Lemma 4.5 in [10] for a proof of this fact.) To set up a cohomological framework for studying this problem, we will view L (H ) as a coefficient G-module whose G-action is given by the conjugation g · T = π(g)T π(g)∗ . Then, the space of bounded derivations modulo inner derivations is canonically isomorphic to the first bounded cohomology group H1b (G, L (H )). Hence, to prove non-unitarisability of G, it suffices to produce a unitary G-representation (π, H ) for which H1b (G, L (H )) = 0. 3 Shakespeare, Richard III, 1:1 (we quote from the 1623 First Folio).
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We now undertake the proof of Theorem 1. It suffices to show that if A is infinite abelian and G is non-amenable, then the wreath product H = A G is non-unitarisable. We can and shall assume that A and G are countable. Indeed, since amenability is preserved under direct limits, G contains some countable non-amenable group G0 . Further, if A0 is a countable subgroup of A, then A0 G0 is a subgroup of A G. Thus our claim follows since unitarisability passes to subgroups. Let F be a countable non-commutative free group. The proof relies on the following two facts. (1) H1b (F, L (2 F)) = 0, see the proof of Theorem 2.7∗ in [10], or [2]. (2) Every nonamenable countable group admits a free type II1 action whose orbits contain the orbits of a free F-action [5], as described below. The strategy of the proof is to induce H1b (F, L (2 F)) through this “randembedding” in the sense of [7]. We henceforth consider a non-amenable countable group G and the corresponding Bernoulli shift action on the compact metrisable product space X = [0, 1]G endowed with the product of the Lebesgue measures. Gaboriau and Lyons prove in [5] that the resulting equivalence relation R ⊆ X × X contains the equivalence relation of some free measure-preserving F-action upon X. In particular, we have commuting G- and F-actions on R given by the action on the first, respectively the second coordinate. These actions preserve the σ -finite measure on R provided by integrating over X the counting measure on orbits. Each of these actions admits a fundamental domain; let Y ⊆ R be a fundamental domain for F. We may now forget the orbit equivalence relation and view R just as a standard measure space with a measure-preserving G × F-action such that G admits a fundamental domain X of finite measure and F admits a fundamental domain Y . We identify R with Y × F in such a way that t −1 y ∈ R corresponds to (y, t) ∈ Y × F. Then, s ∈ F acts on Y × F by s(y, t) = (y, ts −1 ) and g ∈ G acts by g(y, t) = (g · y, α(g, y)t), where g · y ∈ Y is the (essentially) unique element in Fgy ∩ Y ⊂ R and α(g, y) ∈ F is the (essentially) unique element such that α(g, y)gy = g · y. It follows that α satisfies the cocycle relation α(gh, y) = α(g, h · y)α(h, y). We now consider any countable infinite abelian group A. We claim that A has a representation into the unitaries of the von Neumann algebra L∞ (Y ) whose image generates L∞ (Y ) as a von Neumann algebra. By construction, Y is a standard Borel space with a σ -finite non-atomic measure. Furthermore, as far as the present claim is concerned, we may temporarily assume this measure finite since only its measure class is of relevance. Since A is countably infinite, its Pon (for A endowed with the discrete topology) is a non-discrete compact metrisable tryagin dual A endowed group. In other words, we have reduced to the case where we may assume that Y is A ∞ and the with a Haar measure. Fourier transform establishes an isomorphism between L (A) group von Neumann algebra L(A) ⊆ L (2 A), which is by definition generated by the unitary regular representation of A; this proves the claim. Returning to the main argument, we view A in the unitary group of L∞ (Y ) ∼ = L∞ (Y )⊗C1F ⊂ ∞ −1 ∞ L (R). Since A and gAg ⊂ L (Y ) commute, this gives rise to a unitary representation of H = A G on L2 (R). We will prove that H1b (H, L (L2 (R))) = 0. We write N = G A. Since N is amenable and L (L2 (R)) is a dual module, a weak-∗ averaging argument shows that there is a canonical isomorphism N H∗b H, L L2 (R) ∼ = H∗b G, L L2 (R) (see Corollary 7.5.10 in [6]). With the identification R = Y × F, one has N ¯ L (2 F) ∼ L L2 (R) = N ∩ L L2 (R) = L∞ (Y ) ⊗ = L∞ Y, L (2 F)
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(see Theorem IV.5.9 in [13]). Keeping track of the G-representation, one sees that g ∈ G acts on L∞ (Y, L (2 F)) by (g · f )(y) = τα(g,g −1 ·y) (f (g −1 · y)), where τ denotes the F-action on L (2 F). For ease of notation, we denote the coefficient F-module L (2 F) by V . Then, one further has a G-isomorphism L∞ (Y, V ) ∼ = L∞ (R, V )F , where f ∈ L∞ (Y, V ) corresponds to f˜ ∈ L∞ (R, V )F defined by f˜(y, t) = τt−1 (f (y)). Now, F acts on L∞ (R, V ) by (s · F )(z) = τs (F (s −1 z)) and G acts by (g · F )(z) = F (g −1 z). Since both the F-action and the G-action on R admit a fundamental domain, Proposition 4.6 in [8] implies that H∗b G, L∞ (R, V )F ∼ = H∗b F, L∞ (R, V )G ∼ = H∗b F, L∞ (X, V ) . (See also Proposition 5.8 in [7].) Since X = R/G has a finite F-invariant measure, the inclusion V → L∞ (X, V ) has a G-equivariant left inverse. It follows that the corresponding morphism H∗b (F, V ) −→ H∗b F, L∞ (X, V ) is an injection. Therefore, putting all identifications together, we conclude that there are injections H∗b F, L (2 F) −→ H∗b H, L L2 (R) in all degrees. Since H1b (F, L (2 F)) = 0, this completes the proof. Analysing the proof at the level of derivations, one observes that the above injection maps D : F → L (2 F) to D˜ : H → L (L2 (Y, 2 F)) defined by ˜ D(ag)ξ (y) = a(y)D α g, g −1 · y ξ g −1 · y , where a ∈ N is viewed as an element of L∞ (Y ), g ∈ G and ξ ∈ L2 (Y, 2 F). Proof of Theorem 2. By a theorem of Adyan [1], the free Burnside group G = B(2, p) is nonamenable. Therefore, Theorem 1 implies that ( N Z/nZ) G is non-unitarisable. Notice that this wreath product is a countably generated group of exponent np. Therefore, by the universal property of free Burnside groups, it is a quotient of B(ℵ0 , np). In particular, the latter is non-unitarisable. It was shown by Širvanjan [12] that B(ℵ0 , np) embeds into B(2, np) which is therefore also non-unitarisable. Finally, each B(m, np) surjects onto B(2, np) as long as m 2, concluding the proof. 2 Acknowledgments The essential part of this work was done during the authors’ stay at the Institute of Mathematical Sciences in Chennai. The authors would like to thank Professor V.S. Sunder and IMSc for their very kind hospitality.
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References [1] S.I. Adyan, Random walks on free periodic groups, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982) 1139–1149, 1343. [2] M. Bo˙zejko, G. Fendler, Herz–Schur multipliers and uniformly bounded representations of discrete groups, Arch. Math. (Basel) 57 (1991) 290–298. [3] J. Dixmier, Les moyennes invariantes dans les semi-groups et leurs applications, Acta Sci. Math. (Szeged) 12 (1950) 213–227. [4] I. Epstein, N. Monod, Non-unitarisable representations and random forests, preprint arXiv:0811.3422v1, Int. Math. Res. Not. (2009), in press. [5] D. Gaboriau, R. Lyons, A measurable-group-theoretic solution to von Neumann’s problem, preprint arXiv: 0711.1643v1, Invent Math. (2009), in press. [6] N. Monod, Continuous Bounded Cohomology of Locally Compact Groups, Lecture Notes in Math., vol. 1758, Springer-Verlag, Berlin, 2001. [7] N. Monod, An invitation to bounded cohomology, in: International Congress of Mathematicians, vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1183–1211. [8] N. Monod, Y. Shalom, Orbit equivalence rigidity and bounded cohomology, Ann. of Math. (2) 164 (2006) 825–878. [9] D. Osin, L2 -Betti numbers and non-unitarizable groups without free subgroups, preprint arXiv:0812.2093, Int. Math. Res. Not. (2009), in press. [10] G. Pisier, Similarity Problems and Completely Bounded Maps, second, expanded edition, includes the solution to “The Halmos problem”, Lecture Notes in Math., vol. 1618, Springer-Verlag, Berlin, 2001. [11] G. Pisier, Are unitarizable groups amenable? in: Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, in: Progr. Math., vol. 248, Birkhäuser, Basel, 2005, pp. 323–362. [12] V.L. Širvanjan, Embedding the group B(∞, n) in the group B(2, n), Math. USSR Izv. 10 (1976) 181–199. [13] M. Takesaki, Theory of Operator Algebras. I, reprint of the first 1979 ed., Encyclopaedia Math. Sci., vol. 124, Springer-Verlag, Berlin, 2002, Operator Algebras and Non-commutative Geometry 5.
Journal of Functional Analysis 258 (2010) 260–278 www.elsevier.com/locate/jfa
Stable isomorphism of dual operator spaces ✩ G.K. Eleftherakis a , V.I. Paulsen b,∗ , I.G. Todorov c a Department of Mathematics, University of Athens, Penepistimioupolis 157 84, Athens, Greece b Department of Mathematics, University of Houston, Houston, TX 77204-3476, USA c Department of Pure Mathematics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom
Received 23 March 2009; accepted 29 June 2009 Available online 16 July 2009 Communicated by D. Voiculescu
Abstract We prove that two dual operator spaces X and Y are stably isomorphic if and only if there exist completely isometric normal representations φ and ψ of X and Y , respectively, and ternary rings of operators M1 , M2 ∗ ∗ such that φ(X) = [M2∗ ψ(Y )M1 ]−w and ψ(Y ) = [M2 φ(X)M1∗ ]−w . We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. Consequently, we obtain that certain complex domains are biholomorphically equivalent if and only if their algebras of bounded analytic functions are Morita equivalent in our sense. Finally, we provide examples motivated by the theory of CSL algebras. © 2009 Elsevier Inc. All rights reserved. Keywords: Operator space; Stable isomorphism; Biholomorphic equivalence; Morita equivalence
1. Introduction K. Morita [15] developed an equivalence for rings based on their categories of modules and proved three central theorems explaining this equivalence relation. A parallel Morita theory for C ∗ - and W ∗ -algebras was introduced by Rieffel in [18]. Later Brown, Green and Rieffel [7] ✩ The research of the second named author was partially supported by NSF grant DMS-0600191. The first and the third named authors were supported by EPSRC grant D050677/1. * Corresponding author. E-mail addresses:
[email protected] (G.K. Eleftherakis),
[email protected] (V.I. Paulsen),
[email protected] (I.G. Todorov).
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.034
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introduced the idea of stable isomorphism and proved that two C ∗ -algebras with strictly positive elements are strongly Morita equivalent if and only if they are stably isomorphic in the sense that the two C ∗ -algebras obtained by tensoring with the C ∗ -algebra of all compact operators on a separable Hilbert space are *-isomorphic. This type of stable isomorphism theorem is often referred to as the fourth Morita theorem, and can often be used as an efficient way to prove some of the first three Morita theorems. After the advent of the theory of operator spaces and operator algebras, a parallel Morita theory for non-selfadjoint operator algebras was developed by Blecher, Muhly and the second named author in [4]. Many of the technical results needed to extend this theory to the setting of dual operator algebras appear in the book of Blecher and Le Merdy [3]. In [11] the first named author developed a version of Morita theory for dual operator algebras using a relation called -equivalence, together with a certain category of modules over the algebras, and analogues of the first three Morita theorems were proved. In [13] the first and second named authors developed the fourth part of the Morita theory, stable isomorphism, for -equivalence. A different Morita theory for dual operator algebras has been formulated and studied by Blecher and Kashyap [2,14]. They have shown that their equivalence relation is a coarser equivalence relation than -equivalence, and have successfully proved the first three Morita theorems in their theory. In [12] the first author proved that the equivalence relation of Blecher and Kashyap is strictly coarser and that consequently, the usual stable isomorphism theorem cannot hold in their setting. We conjecture that their equivalence relation is equivalent to (1 + )-stable isomorphism. In this paper we extend the results of [11] and [13] to dual operator spaces. We define -equivalence for dual operator spaces and show that two dual operator spaces are stably isomorphic if and only if they are -equivalent. Thus, we are able to develop parts of the Morita theory in a setting where the basic objects of study are not even rings. This result and several of its corollaries are included in Section 2. We end this section by applying our results for spaces to obtain some new results about algebras. In Section 3 we provide examples arising from the theory of CSL algebras. Our notation is standard. If H and K are Hilbert spaces we denote by H ⊗ K their Hilbert space tensor product. For a subset S ⊆ B(H, K) we denote by S the commutant of S, by [S] w∗ the linear span of S and by [S] the w ∗ -closed hull of [S]. If H ⊆ H is a closed subspace we let PH be the orthogonal projection from H onto H . By Ball(X) we denote the unit ball of a Banach space X. For an operator algebra A we denote by pr(A) the set of all projections in A. Throughout the paper, we use extensively the basics of Operator Space Theory and we refer the reader to the monographs [3,9,16,17] for further details. 2. Stably isomorphic dual operator spaces Let X be a dual operator space. A normal representation of X is a completely contractive w ∗ -continuous map φ : X → B(K, H ) where K and H are Hilbert spaces. A normal representation φ : X → B(K, H ) is called non-degenerate if φ(X)K = H and φ(X)∗ H = K and degenerate, otherwise. Note that if φ is a degenerate normal representation and if we set H = φ(X)K, K = φ(X)∗ H and define φ : X → B(K , H ) by φ (x) = PH φ(x)|K , then φ is a non-degenerate normal representation, which we shall refer to as the non-degenerate representation obtained from φ. If φ is completely isometric then φ is completely isometric as well. If A is a unital dual operator algebra, a normal representation of A is a unital completely contractive w ∗ -continuous homomorphism α : A → B(H ) for some Hilbert space H .
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If A and B are unital operator algebras and X is an operator space, X is called an operator A − B-module if there exist completely contractive bilinear maps (in the sense of Christensen– Sinclair) A × X → X and X × B → X. Recall that a bilinear map is completely contractive in the sense of Christensen–Sinclair if and only if the induced linear map is completely contractive when the domain is endowed with the Haagerup tensor norm. In this case there exist Hilbert spaces H , K, completely contractive unital homomorphisms π : A → B(H ), σ : B → B(K) and a complete isometry φ : X → B(K, H ) such that φ(axb) = π(a)φ(x)σ (b) for all a ∈ A, x ∈ X, b ∈ B [16, Corollary 16.10]. The triple (π, φ, σ ) is called a CES representation of the operator A − B-module X. Moreover, replacing the original π and σ by their direct sums with completely isometric representations, if necessary, one may assume that π and σ are completely isometric. In this case the triple (π, φ, σ ) is called a faithful CES representation. We recall that one surprising consequence of the existence of the CES representation is that an operator A − Bmodule, is automatically an A − B-bimodule, that is, the associativity condition, (ax)b = a(xb), follows from the other assumptions. If X and Y are dual operator spaces, we call a mapping φ : X → Y a dual operator space isomorphism if it is a surjective complete isometry which is also a w ∗ -homeomorphism. If there exists such a mapping, we say that X and Y are isomorphic dual operator spaces. Similarly, if A and B are dual operator algebras, we call a mapping φ : A → B a dual operator algebra isomorphism if it is a surjective complete isometry which is also a homomorphism and a w ∗ -homeomorphism. If there exists such a mapping, we say that A and B are isomorphic dual operator algebras. In the case that A and B are unital dual operator algebras and X is a dual operator space, X is called a dual operator A − B-module if it is an operator A − B-module and the module actions are separately w ∗ -continuous. In this case the triple (π, φ, σ ) can be chosen with the property that π , φ and σ be w ∗ -continuous completely isometric maps [3, Theorem 3.8.3]. We call such a triple a faithful normal CES representation. Note that since X is an A − B-bimodule the set C = A0 BX is naturally endowed with a product making it into an algebra and every CES representation (π, φ, σ ) as above yields a representation a x φ(x) ρ : C → B(H ⊕ K) defined by ρ 0 b = π(a) . When (π, φ, σ ) is a faithful CES repre0 σ (b) sentation, then the representation ρ endows C with the structure of an operator algebra. In the case A and B are unital C ∗ -algebras, X is an operator A − B-module and (π, φ, σ ) is a faithful CES representation, this induced operator algebra structure on C is unique; that is, any two faithful CES representations give rise to the same matrix norm structures. This fact was first pointed out in [5, p. 11] and follows from the uniqueness of the operator system structure on C + C ∗ as can be seen from [20] (see also [3, 3.6.1]). In case A and B are W ∗ -algebras the image of the faithful normal CES representation is ∗ w -closed and C can be equipped with a dual operator algebra structure. We isolate the following useful consequence of the above remarks. Proposition 2.1. Let A1 , A2 , B1 , B2 be W ∗ -algebras and X1 (resp. X2 ) be a dual A1 − B1 - (resp. A2 − B2 -) module. Let π : A1 → A2 , σ : B1 → B2 be normal *-isomorphisms and φ : X1 → X2 be a dual operator space isomorphism which is a bimodule map in the sense that φ(lxr) = π(l)φ(x)σ (r), Then the map
l ∈ A1 , x ∈ X1 , r ∈ B1 .
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Φ:
A1 0
X1 B1
→
A2 0
X2 B2
263
l x π(l) φ(x) : → 0 r 0 σ (r)
is a dual operator algebra isomorphism. We recall some definitions from [11] and [13]. Let I be a set and 2I be the Hilbert space of all square summable families indexed by I . Recall that if H is a Hilbert space we may identify B( 2I ⊗ H ) with the space MI (B(H )) of all matrices of size |I | × |I | with entries from B(H ) which define bounded operators on 2I ⊗ H . If X ⊆ B(H ) is an operator space we let MI (X) ⊆ MI (B(H )) denote the space of those operators whose matrices have entries from X. This space is denoted MIw (X) in [6]. We define similarly MI,J (X) where I and J are (perhaps different) index sets. In particular, the column (resp. row) operator space CI (X) (resp. RI (X)) over X is defined as MI,1 (X) (resp. M1,I (X)). If X ⊆ B(H ) is a w ∗ -closed subspace, then it is easy to see that MI (X) is a w ∗ -closed subspace of MI (B(H )). Moreover, if X is a w ∗ -closed subalgebra of B(H ), then MI (X) is a w ∗ -closed subalgebra of MI (B(H )). Definition 2.1. (i) [11] Let H and K be Hilbert spaces. Two w ∗ -closed subalgebras A ⊆ B(H ) and B ⊆ B(K) are called TRO-equivalent if there exists a ternary ring of operators (TRO) w∗ M ⊆ B(H, K), i.e., a subspace satisfying MM ∗ M ⊆ M, such that A = [M ∗ BM] and B = ∗ w [MAM ∗ ] . (ii) [11] Two dual operator algebras A and B are called -equivalent if they possess completely isometric normal representations whose images are TRO-equivalent. (iii) [13] Two dual operator algebras A and B are called stably isomorphic (as algebras), if there exists a cardinal I such that the algebras MI (A) and MI (B) are isomorphic as dual operator algebras. It is clear that stable isomorphism is an equivalence relation and it is easy to see that the same holds for TRO-equivalence. While it is obvious that the relation of -equivalence is reflexive and symmetric, it is not apparent that it is transitive. Nonetheless, the results of [11] show that it is equivalent to a certain category equivalence and hence it is also an equivalence relation. The results of [11] and [13] show that the relations of -equivalence and stable isomorphism coincide. In this paper we generalize this result to the case of dual operator spaces. We begin with the relevant definitions. Definition 2.2. (i) Let X ⊆ B(K1 , K2 ) and Y ⊆ B(H1 , H2 ) be w ∗ -closed operator spaces. We say that X is TRO-equivalent to Y if there exist TRO’s M1 ⊆ B(H1 , K1 ) and M2 ⊆ B(H2 , K2 ) w∗
w∗
such that X = [M2 Y M1∗ ] and Y = [M2∗ XM1 ] . (ii) Let X and Y be dual operator spaces. We say that X is -equivalent to Y if there exist completely isometric normal representations φ and ψ of X and Y , respectively, such that φ(X) is TRO-equivalent to ψ(Y ). (iii) Let X and Y be dual operator spaces. We say that X and Y are stably isomorphic if there exists a cardinal J and a w ∗ -continuous, completely isometric map from MJ (X) onto MJ (Y ), i.e., if they are isomorphic as dual operator spaces.
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Blecher and Zarikian [6, Section 6.2] define two dual operator spaces X and Y to be weakly Morita equivalent if MI1 ,J1 (X) and MI2 ,J2 (Y ) are completely isometrically isomorphic as dual operator spaces. Note that if MI1 ,J1 (X) is completely isometrically isomorphic to MI2 ,J2 (Y ) for some cardinals I1 , I2 , J1 , J2 , then for a large enough cardinal J the spaces MJ (X) and MJ (Y ) are completely isometrically isomorphic. Thus, their definition of weakly Morita equivalent is the same as our stable isomorphism. Since one goal of our research is to prove that stable isomorphism is equivalent to a type of Morita equivalence, we believe that our terminology is clearer in our context. It is obvious that the relation of TRO-equivalence of w ∗ -closed operator subspaces is reflexive and symmetric. We shall now prove that it is in fact an equivalence relation. First we note that the spaces involved can always be assumed to act non-degenerately. Proposition 2.2. Let X and Y be dual operator spaces, φ : X → B(K1 , K2 ), and ψ : Y → B(H1 , H2 ) be completely isometric normal representations with TRO-equivalent images. If φ : X → B(K1 , K2 ), and ψ : Y → B(H1 , H2 ) are the non-degenerate completely isometric normal representations obtained from φ and ψ , respectively, then the images of φ and ψ are TRO-equivalent. Proof. Recall that to make φ and ψ non-degenerate, we restrict to subspaces, K2 = [φ(X)K1 ], K1 = [φ(X)∗ K2 ], H2 = [ψ(Y )H1 ], H1 = [ψ(Y )∗ H2 ], where [E] denotes the closed linear subspace spanned by E . Note that K1 = [φ(X)∗ K2 ] = [M1 ψ(Y )∗ M2∗ K2 ] ⊆ [M1 ψ(Y )∗ H2 ] ⊆ [M1 H1 ]. We also have that [M1 H1 ] = [M1 ψ(Y )∗ H2 ] = [M1 M1∗ φ(X)∗ M2 H2 ]. Since M1 M1∗ φ(X)∗ = M1 M1∗ M1 ψ(Y )∗ M2∗ ⊆ M1 ψ(Y )∗ M2∗ ⊆ φ(X)∗ , we have that [M1 H1 ] ⊆ [φ(X)∗ K2 ] = K1 . Thus, it follows that K1 = [M1 H1 ] and it can be easily checked that M1 = PK1 M1 |H , is a TRO. ! Similarly, one shows that M2 = PK2 M2 |H2 is a TRO. It is now easy to verify that M1 and M2 implement a TRO-equivalence of φ (X) and ψ (Y ). 2 Proposition 2.3. TRO-equivalence of w ∗ -closed operator spaces is an equivalence relation. Proof. We need to prove that TRO-equivalence is a transitive relation. Assume that X ⊆ B(K1 , K2 ), Y ⊆ B(H1 , H2 ) and Z ⊆ B(R1 , R2 ) are w ∗ -closed subspaces such that X is TROequivalent to Y and Y is TRO-equivalent to Z. By Proposition 2.2, we may assume that (the identity representations of) X, Y and Z are non-degenerate. We fix TRO’s M1 ⊆ B(H1 , K1 ),
M2 ⊆ B(H2 , K2 ),
N1 ⊆ B(H1 , R1 )
and N2 ⊆ B(H2 , R2 )
w∗ Y = M2∗ XM1 ,
w∗ Y = N2∗ ZN1
w∗ and Z = N2 Y N1∗ .
such that w∗ X = M2 Y M1∗ ,
By [10, Theorem 3.2], there exist *-isomorphisms and χ : N2∗ N2 → N2 N2∗ φ : M2∗ M2 → M2 M2∗ such that M2 = T ∈ B(H2 , K2 ): T P = φ(P )T , for each P ∈ pr M2∗ M2
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265
and N2 = T ∈ B(H2 , R2 ): T P = χ(P )T , for each P ∈ pr N2∗ N2 . Let S = pr((M2∗ M2 ) ∩ (N2∗ N2 ) ),
2 = T : T P = φ(P )T , for each P ∈ S M and
2 = T : T P = χ(P )T , for each P ∈ S . N
2 and N
2 are TRO’s containing M2 and N2 , respectively. From [10, Lemma 2.2] Observe that M it follows that ∗ w∗ ∗ w∗
2 M
2 N
2
2 . M = S = N w∗
2 M
2 ∗ ] We let L2 = [N
⊆ B(K2 , R2 ). The space L2 is a TRO since
2 M
2 S S M
2 M
2 ∗ M
2 N
2 ∗ N
2 M
2 ∗ ⊆ N
2 ∗ ⊆ N
2 ∗ ⊆ L2 . N
1 ⊇ M1 , N
1 ⊇ N1 such Similarly, if T = pr((M1∗ M1 ) ∩ (N1∗ N1 ) ) then there exist TRO’s M w∗
w∗
1 ∗ M
1 ∗ N
1 ] = T = [N
1 ] that [M S Y T ⊆ Y we have
2 ∗ M
1 ∗ M
2 Y M
1 ⊆ Y M
w∗
Since IK2 ∈ [M2 M2∗ ] w∗
w∗
1 M
1 ∗ ] . As above, the space L1 = [N
is a TRO. Since
∗
⇒
1 M1 ⊆ Y
2 Y M M2∗ M
⇒
∗
1 ∗ M1 M1∗ ⊆ M2 Y M1∗ ⊆ X.
2 Y M M2 M2 M
and IK1 ∈ [M1 M1∗ ]
w∗
2 Y M
1 ∗ ⊆ X and hence X = we have M
2 Y M
1 ∗ ] . [M Similarly, we can show that ∗ w∗
2 X M
1 , Y= M w∗
Now, writing ABC for [ABC]
w∗
2 Y N
1 ∗ Z= N w∗
and AB for [AB] ∗
∗
w∗
2 ∗ Z N
1 ] . and Y = [N
we have ∗
2 M
1 N
2 Y N
1 = Z
2 X M
1 = N L2 XL∗1 = N and
2 N
1 M
2 Y M
1 ∗ = X.
2 ∗ Z N
1 ∗ = M L∗2 ZL1 = M
2
We will show later that -equivalence of dual operator spaces is an equivalence relation. Note that if A and B are dual operator algebras, then they could be stably isomorphic as algebras
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(which requires that the map implementing the stable isomorphism be an algebra homomorphism) or simply stably isomorphic as dual operator spaces. However, by the operator algebra generalization of the Banach–Stone theorem [3, Theorem 4.5.13] these two conditions are equivalent. We recall the following main result from [13]: Theorem 2.4. Two dual operator algebras are -equivalent if and only if they are stably isomorphic as algebras. In this section we shall generalize this result to the case of dual operator space. Namely, we will prove the following: Theorem 2.5. Two dual operator spaces are -equivalent if and only if they are stably isomorphic. We now present the proof of one of the directions of Theorem 2.5 showing that -equivalence of dual operator spaces implies stable isomorphism. Assume, without loss of generality, that X ⊆ B(H1 , H2 ) and Y ⊆ B(K1 , K2 ) are concrete w ∗ -closed operator spaces which are TRO-equivalent and non-degenerate. Let M1 ⊆ B(H1 , K1 ) w∗
w∗
and M2 ⊆ B(H2 , K2 ) be w ∗ -closed TRO’s such that [M2 XM1∗ ] = Y and [M2∗ Y M1 ] = X. Let w∗ w∗ [M2∗ M2 ] X [M2 M2∗ ] Y A= and B = w∗ w∗ . 0 0 [M1∗ M1 ] [M1 M1∗ ] Since ∗ ∗ M2 M2 X M1 M1 ⊆ M2∗ Y M1 ⊆ X, the space X is an [M2∗ M2 ]
w∗
w∗
− [M1∗ M1 ]
-module and hence A is a subalgebra of B(H2 ⊕ H1 ). w∗
w∗
Since Y (resp. X) is non-degenerate, the relation [M2 XM1∗ ] = Y (resp. [M2∗ Y M1 ] = X) implies that M2 H2 = K2 (resp. M2∗ K2 = H2 ). Thus, M2 is non-degenerate. Taking adjoints we w∗
obtain the relations [M1 X ∗ M2∗ ]
w∗
= Y ∗ and [M1∗ Y ∗ M2 ]
= X ∗ which imply that M1 is nonw∗
w∗
degenerate. It follows that the (selfadjoint) algebras [M2∗ M2 ] and [M1∗ M1 ] are unital, and so A is unital. One sees similarly that B is a unital w ∗ -closed subalgebra of B(K2 ⊕ K1 ). Let M2 0 ⊆ B(H2 ⊕ H1 , K2 ⊕ K1 ). M= 0 M1 Then M is a w ∗ -closed TRO and it is easily verified that w∗
[MAM ∗ ]
=B
and [M ∗ BM]
w∗
= A.
By Theorem 2.4, A and B are stably isomorphic. Thus, there exists a cardinal I and a dual operator algebra isomorphism Φ : MI (A) → MI (B). We have that
G.K. Eleftherakis et al. / Journal of Functional Analysis 258 (2010) 260–278
MI (A)
w∗
267
MI ([M2∗ M2 ] )
MI (X)
0
MI ([M1∗ M1 ] )
w∗
and MI (B)
w∗
MI ([M2 M2∗ ] )
MI (Y )
0
MI ([M1 M1∗ ] )
w∗
.
onto thediagonal of MI (B) (see It is well known that Φ must carry I (A) diagonal I 0 of M I 0the 0 0 e.g. [3, 2.1.2]). We claim that Φ 0 0 = 0 0 and Φ 0 I = 00 I0 . To show this, note that Φ I0 00 is a projection in the diagonal of MI (B) and hence there exist projections Q and P 0 0 acting on K2 and K1 , respectively, such that Φ I0 00 = Q . Then Φ 00 I0 = I −Q . 0 I −P 0 P Let x ∈ MI (X). Then Φ
0 x 0 0
=Φ = ⊆
Q 0
0 0 x 0 0 0 0 0 0 I 0 0 x I −Q Φ P 0 0 0
I 0
QMI (B(K2 ))(I − Q) 0
0 I −P
QMI (Y )(I − P ) . P MI (B(K1 ))(I − P )
Since Φ is surjective and Y is non-degenerate, it follows that Q= I and P = 0. The claim is proved. Since Φ is a homomorphism, we have that Φ 00 MI0(X) ⊆ 00 MI0(Y ) and since Φ is onto, the last inclusion is actually an equality. It follows that there exists a normal complete isometry between MI (X) and MI (Y ). In order to prove the converse direction of Theorem 2.5 we need the notion of multipliers of an operator space [3,16]. Let X be an operator space and Ml (X) be the space of all completely bounded linear maps u on X for which there exist Hilbert spaces H and K, a complete isometry ι : X → B(H, K) and an operator T ∈ B(K) such that T ι(X) ⊆ ι(X) and u(x) = ι−1 (T ι(x)), x ∈ X. Then Ml (X) can be endowed with an operator algebra structure in a canonical way and is called the left multiplier algebra of X. Similarly one defines the right multiplier algebra Mr (X) of X. The operator space X is an operator Ml (X) − Mr (X)-module; for l ∈ Ml (X), r ∈ Mr (X) and x ∈ X we write lx = l(x) and xr = r(x). If X is a dual operator space then Ml (X) and Mr (X) are dual operator algebras [3, Theorem 4.7.4]. Their diagonals Al (X) = Ml (X) ∩ Ml (X)∗ and Ar (X) = Mr (X) ∩ Mr (X)∗ are thus W ∗ -algebras. Since the maps Al (X) × X → X : (l, x) → lx,
X × Ar (X) → X : (x, r) → xr
are completely contractive and separately w ∗ -continuous bilinear maps [3, Lemma 4.7.5], the space Ω(X) =
X Al (X) 0 Ar (X)
(2.1)
can be canonically endowed with the structure of a dual operator algebra (see Proposition 2.1).
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Proposition 2.6. Let X and Y be isomorphic dual operator spaces. Then the algebras Ω(X) and Ω(Y ) are isomorphic dual operator algebras. Proof. Assume that φ : X → Y is a dual operator space isomorphism. We let σ : Ml (X) → Ml (Y ) be given by σ (u) = φ ◦ u ◦ φ −1 . Then σ is a completely isometric homomorphism [3, Proposition 4.5.12] and we can easily check that it is w ∗ -continuous. Also, σ (Al (X)) = Al (Y ) and φ(ux) = φ u(x) = φ ◦ u ◦ φ −1 φ(x) = σ (u) φ(x) = σ (u)φ(x) for all u ∈ Al (X), x ∈ X. Similarly, the completely isometric surjection τ : Mr (X) → Mr (Y ) given by τ (w) = φ ◦ w ◦ φ −1 satisfies the identity φ(xw) = φ(x)τ (w). The conclusion now follows from Proposition 2.1. 2 In order to complete the proof of Theorem 2.5 we will also need the following lemma. Lemma 2.7. Let CX = B0X AXX and CY = B0Y AYY be concrete operator algebras acting on the Hilbert spaces H2 ⊕ H1 and K2 ⊕ K1 , respectively. Suppose that BX , AX , BY and AY are von Neumann algebras. (i) If CX and CY are TRO-equivalent, then there exist TRO’s M1 ⊆ B(H1 , K1 ) and w∗
M2 ⊆ B(H2 , K2 ) such that Y = [M2 XM1∗ ] and X = [M2∗ Y M1 ] M1∗ AY M1 ⊆ AX , M2 BX M2∗ ⊆ BY , M2∗ BY M2 ⊆ BX . (ii) If CX and CY are -equivalent, then X and Y are -equivalent.
w∗
, M1 AX M1∗ ⊆ AY ,
Proof. Suppose that (i) holds and assume that CX and CY are -equivalent. Then there exˆ ist normal completely isometric algebra homomorphisms, α : CX → B(Hˆ ) and β : CY → B(K) such that α(CX ) and β(CY ) are TRO-equivalent. Note that α(CX ) (resp. β(CY )) has the form BT T BZ Z ˆ ˆ ˆ ˆ ˆ ˆ 0 AZ (resp. 0 AT ) for a suitable decomposition H = H2 ⊕ H1 , K = K2 ⊕ K1 , von Neu∗ mann algebras BZ , AZ , BT , AT and w -closed subspaces Z, T that are isomorphic to X and Y , respectively, as dual operator spaces. Thus, (ii) follows from (i). We now prove (i). Let PX (resp. PY ) denote the projection from H2 ⊕ H1 onto H1 (resp. ˜ where DX = CX ∩ C ∗ and X˜ = (I − PX )CX PX from K2 ⊕ K1 onto K1 ). Write CX = DX + X, X is isomorphic to X as a dual operator space. Similarly, we decompose CY = DY + Y˜ . Let w∗ M ⊆ B(H2 ⊕ H1 , K2 ⊕ K1 ) be a non-degenerate TRO such that [MCX M ∗ ] = CY and w∗ [M ∗ CY M] = CX . By [10, Proposition 2.8], we may also choose M to be a DY − DX -bimodule. Set M1 = PY MPX ⊆ B(H1 , K1 ) and M2 = (I − PY )M(I − PX ) ⊆ B(H2 , K2 ). Since M is a DY − DX -module, we have that M1 M1∗ M1 = PY MPX M ∗ PY MPX ⊆ PY (MM ∗ M)PX ⊆ M1 , and hence M1 is a TRO. Similarly, we see that M2 is a TRO. Note that since [MDX M ∗ ]∗ = [MDX M ∗ ], we have that MDX M ∗ ⊆ CY ∩ CY∗ = DY , and hence M1 AX M1∗ ⊆ AY and M2 BX M2∗ ⊆ BY . Similarly, M1∗ AY M1 ⊆ AX and M2∗ BY M2 ⊆ BX . Finally, PY [MDX M ∗ ](I − PY ) = 0, and it follows that
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w∗
˜ ∗ ] PY Y˜ = (I − PY )CY PY = (I − PY )[MDX M ∗ + M XM w∗ w∗ ˜ ∗ ] PY = M2 XM ˜ ∗ . = (I − PY )[M XM 1 w∗
Similarly, X˜ = [M2∗ Y˜ M1 ]
, and hence X and Y are TRO-equivalent.
2
Now we are ready to complete the proof of Theorem 2.5. Suppose that X and Y are dual operator spaces and that there exists a cardinal J such that MJ (X) ∼ = MJ (Y ) as dual operator spaces. We recall the unital dual operator algebras Ω(X) and Ω(Y ) defined as in (2.1) and note that MJ (X) MJ (Al (X)) ∼ . MJ Ω(X) = 0 MJ (Ar (X)) By [6, Theorem 5.46(ii)], the algebras MJ (Al (X)) and Al (MJ (X)) are isomorphic as dual operator algebras, and it can be deduced from its proof that the isomorphism is given by sending
a matrix u = (ui,j ) ∈ MJ (Al (X)) to the multiplier Tu of MJ (X) given by Tu ((xi,j )) = ( k ui,k (xk,j ))i,j . It follows from Proposition 2.1 that
MJ (Al (X)) 0
MJ (X) MJ (Ar (X))
∼ =
Al (MJ (X)) 0
MJ (X) . Ar (MJ (X))
By Proposition 2.6, the algebra on the right-hand side is isomorphic as a dual operator algebra to
Al (MJ (Y )) 0
MJ (Y ) . Ar (MJ (Y ))
By the same arguments, this algebra is isomorphic to MJ (Ω(Y )). It follows from Theorem 2.4 that the algebras Ω(X), Ω(Y ) are -equivalent as algebras. By Lemma 2.7(ii), X and Y are -equivalent. The proof of Theorem 2.5 is now complete. We note several immediate corollaries. Corollary 2.8. If A and B are unital dual operator algebras then the following are equivalent: (i) (ii) (iii) (iv)
A and B A and B A and B A and B
are -equivalent as dual operator algebras; are stably isomorphic as dual operator algebras; are -equivalent as dual operator spaces; are stably isomorphic as dual operator spaces.
Proof. By Theorem 2.4, (i) is equivalent to (ii), while by Theorem 2.5, (iii) is equivalent to (iv). The equivalence of (ii) and (iv) follows from the generalized Banach–Stone theorem [3, Theorem 4.5.13]. 2 Since stable isomorphism is an equivalence relation we conclude: Corollary 2.9. -equivalence of dual operator spaces is an equivalence relation.
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Let A ⊆ B(H ) and B ⊆ B(K) be w ∗ -closed unital operator algebras and M ⊆ B(H, K) be a TRO such that A = [M ∗ BM]
w∗
w∗
and B = [MAM ∗ ] . w∗
def
w∗
def
We define the B − A-bimodule X = [MA] = [BM] and the A − B-bimodule Y = w∗ w∗ [AM ∗ ] = [M ∗ B] . These bimodules are important in the theory of -equivalence. In [11] they “generate” the functor of equivalence between the categories of normal representations of A and B. Also, it is proved in [13] that B X ⊗σAh Y and A Y ⊗σBh X, where the tensor products are quotients of the corresponding normal Haagerup tensor products. Corollary 2.10. The spaces A, B, X, Y defined above are stably isomorphic. Proof. Observe that M ∗ MAC ⊆ A; hence M ∗ XC ⊆ A
and MAC ⊆ X.
It follows that X and A are TRO-equivalent. Similarly, we obtain that Y and B are TROequivalent. The claim now follows from Theorem 2.5. 2 In the special case of selfadjoint algebras we recapture the following known result: Corollary 2.11. Let A be a W ∗ -algebra and M be a w ∗ -closed TRO such that A = [M ∗ M] Then A and M are stably isomorphic.
w∗
.
Proof. Observe that w∗
MAC ⊆ [MM ∗ M]
⊆M
and M ∗ MC ⊆ A.
It follows that A and M are TRO-equivalent. By Theorem 2.5, A and M are stably isomorphic. 2 In the next result we link the -equivalence of two dual operator spaces X and Y to that of the corresponding algebras Ω(X) and Ω(Y ). Theorem 2.12. The dual operator spaces X and Y are -equivalent if and only if the algebras Ω(X) and Ω(Y ) are -equivalent. Proof. If X and Y are -equivalent then there exists a cardinal I such that MI (X) and MI (Y ) are isomorphic as dual operator spaces. Hence, Ω(MI (X)) and Ω(MI (Y )) are isomorphic as dual operator algebras. As in the proof of Theorem 2.5, using Proposition 2.1 and [6, Theorem 5.46(ii)], we conclude that Ω MI (X) =
Al (MI (X)) 0
MI (X) Ar (MI (X))
∼ =
MI (Al (X)) 0
MI (X) MI (Ar (X))
∼ = MI Ω(X)
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and, similarly, Ω(MI (Y )) ∼ = MI (Ω(Y )). Thus, Ω(X) and Ω(Y ) are stably isomorphic as algebras. By Theorem 2.4, Ω(X) and Ω(Y ) are -equivalent. Conversely, if Ω(X) and Ω(Y ) are -equivalent then, by Lemma 2.7(ii), X and Y are -equivalent. 2 Theorem 2.13. Let X and Y be -equivalent dual operator spaces. If (π, φ, σ ) is a normal CES representation of the dual operator Al (X) − Ar (X)-module X and φ is a complete isometry, then there exists a normal completely isometric representation ψ of Y such that φ(X) is TROequivalent to ψ(Y ). Proof. The CES triple (π, φ, σ ) defines a normal representation Φ of the algebra Ω(X). If l ∈ Al (X) with π(l) = 0 then φ(lx) = 0 and hence lx = 0 for all x ∈ X. This implies that l = 0, and so π is one-to-one. Similarly σ is one-to-one. Thus, (π, φ, σ ) is a faithful CES representation and induces the unique operator algebra structure on Ω(X). Thus, Φ is a normal completely isometric representation of the dual operator algebra Ω(X). By Theorem 2.12, Ω(X) and Ω(Y ) are -equivalent; by [12, Theorem 2.7], there exists a normal completely isometric representation Ψ of Ω(Y ) such that Φ(Ω(X)) is TRO-equivalent to Ψ (Ω(Y )). Let ψ be the restriction of Ψ to Y . By Lemma 2.7(i), the spaces φ(X) and ψ(Y ) are TROequivalent. 2 By [11], -equivalence for dual operator algebras can be equivalently defined in terms of a special type of isomorphism between certain categories of representations of the algebras. These types of category isomorphisms are in the spirit of Morita equivalence. Thus, one would like to claim that the representations of Ω(X) and of Ω(Y ) define certain special families of representations of X and Y such that X and Y are stably isomorphic if and only if these classes of representations are isomorphic. Unfortunately, the correspondence between representations of Ω(X) and representations of X is not one-to-one. We finish this section with some applications of the above theorems. Definition 2.3. An operator space X is called rigid if Ml (X) = Mr (X) = C and *-rigid if Al (X) = Ar (X) = C. Note that if X is rigid, then it is *-rigid. There are many examples of rigid and *-rigid operator spaces. For example, the spaces MAX( n1 ) by a result of Zhang [21] (see also [16, Exercise 14.3]) can be identified with the subspace of the full group C ∗ -algebra of the free group on n − 1 generators, C ∗ (Fn−1 ), spanned by the identity and the n − 1 generators. Moreover, Zhang argues that I (MAX( n1 )) = I (C ∗ (Fn−1 )) and since C ∗ (Fn−1 ) is a C ∗ -subalgebra of its injective envelope it follows from [5, Theorem 1.9] that any left multiplier of MAX( n1 ) necessarily belongs to I (C ∗ (Fn−1 )) and multiplies the subspace MAX( n1 ) back into itself in the usual product. Since the identity belongs to the subspace, this forces the multiplier to be an element of the subspace and then it is easily seen that in fact it must be a multiple of the identity. A similar argument applies for right multipliers. Thus, MAX( n1 ) is rigid. The argument given in the previous paragraph applies equally well to any subspace X of a unital C ∗ -algebra A which contains the identity and for which I (X) = I (A). In this case, the left (and right) multipliers are simply the elements of the subspace X that leave the subspace
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invariant under the algebra multiplication, and so it is often quite easy to determine whether X is rigid or *-rigid. Theorem 2.14. Let X and Y be *-rigid dual operator spaces. Then X and Y are stably isomorphic if and only if they are isomorphic as dual operator spaces. Proof. If X and Y are stably isomorphic, then they are -equivalent. Hence, by Theorem 2.12, Ω(X) and Ω(Y ) have completely isometric representations whose images are TRO-equivalent. The images of these representations are two concrete operator algebras CX and CY of the type considered in Lemma 2.7, with AX , BX , AY and BY all scalar multiplies of the identity and X and Y replaced by images of normal completely isometric representations, say φ(X) and ψ(Y ). Hence, the TRO’s M1 and M2 appearing in the conclusions of Lemma 2.7(i), satisfy M1∗ M1 = M1 M1∗ = M2 M2∗ = M2 M2∗ = C. Now it readily follows that the spaces M1 and M2 are each the span of a single unitary. Let Mi = CUi , i = 1, 2, for some unitaries U1 and U2 . Applying Lemma 2.7 again, we see that ψ(Y ) = U2∗ φ(X)U1 and the claim follows. 2 Corollary 2.15. Let A and B be dual operator algebras for which A ∩ A∗ = B ∩ B ∗ = C. Then A and B are stably isomorphic as operator spaces if and only if they are isomorphic as dual operator algebras. Proof. Since B is a unital algebra, we have that Ml (B) = Mr (B) = B and hence, Al (B) = Ar (B) = B ∩ B ∗ = C. Hence, B, and similarly A, is a *-rigid operator space. Thus, by Theorem 2.14, A and B are stably isomorphic if and only if they are isomorphic as dual operator spaces. By the generalized Banach–Stone theorem [3, Theorem 3.8.3], A and B are isomorphic as dual operator algebras. 2 It is interesting to note that the hypotheses and conclusions of the above corollary are really special to non-selfadjoint operator algebras. In fact, we now turn our attention to a special family of non-selfadjoint operator algebras to which our theory applies. Definition 2.4. Let G ⊆ Cn be a bounded, connected, open set, i.e., a complex domain, and let H ∞ (G) ⊆ L∞ (G) denote the dual operator algebra of bounded analytic functions on G. We shall call G holomorphically complete if every weak*-continuous multiplicative linear functional on H ∞ (G) is given by evaluation at some point in G. Recall that two complex domains Gi ⊆ Cni , i = 1, 2, are called biholomorphically equivalent if there exists a holomorphic homeomorphism, ϕ : G1 → G2 whose inverse is also holomorphic. Corollary 2.16. Let Gi , i = 1, 2, be complex domains that are holomorphically complete. Then the following are equivalent: (i) (ii) (iii) (iv)
G1 and G2 are biholomorphically equivalent, H ∞ (G1 ) and H ∞ (G2 ) are isometrically weak*-isomorphic algebras, H ∞ (G1 ) and H ∞ (G2 ) are isometrically weak*-isomorphic dual Banach spaces, H ∞ (G1 ) and H ∞ (G2 ) are stably isomorphic dual operator spaces,
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(v) H ∞ (G1 ) and H ∞ (G2 ) are -equivalent dual operator algebras, (vi) H ∞ (G1 ) and H ∞ (G2 ) are -equivalent dual operator spaces. Proof. Since G1 and G2 are connected sets, we have that H ∞ (Gi ) ∩ H ∞ (Gi )∗ = C, i = 1, 2. Also, since these algebras are subalgebras of commutative C ∗ -algebras, every contractive map between them is automatically completely contractive. Thus, the equivalence of (ii)–(vi) follows from the previous results. Given a biholomorphic map ϕ : G1 → G2 , composition with φ defines the weak*-continuous isometric isomorphism between the algebras. Thus, (i) implies (ii). Conversely, given a weak*-continuous isometric algebra isomorphism, π : H ∞ (G1 ) → ∞ H (G2 ), let w ∈ G2 , and let Ew : H ∞ (G2 ) → C denote the weak*-continuous, multiplicative linear functional given by evaluation at w. Then Ew ◦ π : H ∞ (G1 ) → C is a weak*-continuous, multiplicative linear functional and hence is equal to Ez for some z ∈ G1 . If we assume that G1 ⊆ Cn , let z1 , . . . , zn denote the coordinate functions on G1 and set ϕi = π(zi ) ∈ H ∞ (G2 ), then it readily follows that ϕ = (ϕ1 , . . . , ϕn ) : G2 → Cn satisfies, ϕ(w) = z. Hence, ϕ : G2 → G1 . Since each of the mappings ϕi is holomorphic, a similar argument with the inverse of π shows that ϕ is a biholomorphic equivalence. Thus, (ii) implies (i). 2 Recalling that -equivalence is originally defined in terms of a Morita-type equivalence of categories, we see that the equivalence of (i) and (v) shows that two domains have “equivalent” categories of representations in this sense if and only if they are biholomorphically equivalent. One does not need the full force of the rigidity result, Corollary 2.15, to prove the above result, since H ∞ (Gi ) is the center of MI (H ∞ (Gi )) and any stable isomorphism must carry centers to centers. In fact, using the “isomorphism of centers” result of [14], one can replace (v) by the coarser Blecher–Kashyap equivalence. 3. Applications and examples In this section we prove that whenever two dual operator algebras A and B are -equivalent, there exists a dual operator space X such that A is completely isometrically isomorphic to Ml (X) and B is completely isometrically isomorphic to Mr (X). We then give an example of a dual operator space Y for which Ml (Y ) and Mr (Y ) are not stably isomorphic and hence not -equivalent. We also give some examples which emphasize the difference between dual operator spaces arising from non-synthetic CSL algebras and those arising from synthetic ones. Let A ⊆ B(H ) be a unital w ∗ -closed algebra, (A) = A ∩ A∗ be its diagonal and M ⊆ w∗ B(K, H ) be a non-degenerate TRO such that MM ∗ ⊆ A. We call the space X = [AM] the M-generated A-module. In this section we fix A and M as above and we investigate some w∗ properties of X. Since MM ∗ ⊆ A the space B = [M ∗ AM] ⊆ B(K) is a unital algebra and w∗ XB ⊆ X. Note that if we set Y = [M ∗ A] , then A, X, Y , B form the four corners of what could potentially be a “linking” algebra of a Morita context. For this reason, we shall call (A, M, X, B) a generating tuple. Theorem 3.1. Let (A, M, X, B) be a generating tuple. (i) Ml (X) is isomorphic as a dual operator algebra to A and Mr (X) is isomorphic as a dual operator algebra to B.
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(ii) The algebra Ω(X) is isomorphic as a dual operator algebra to the algebra D(X) = (A) X 0 (B) . Proof. Since X ⊆ B(K, H ), A ⊆ B(H ) and AX ⊆ X, by the definition of left multipliers, the map λ(a) : X → X given by λ(a)(x) = ax, is a left multiplier. It follows that the map λ : A → Ml (X) : a → λ(a) is contractive. It is also w ∗ -continuous by [3, Theorem 4.7.4]. We now prove that λ is an isometric surjection. Using analogous arguments, we can show that the map ρ : B → Mr (X),
ρ(b)(x) = xb
is w ∗ -continuous and contractive. Let u be in Ml (X). By [3, Lemma 8.5.23] there
exists a family (mi )i∈I ⊆ M of partial isometries such that mi m∗i ⊥ mj m∗j for i = j and IH = i∈I mi m∗i , the series converging in the strong operator topology. Let x ∈ X, ξ ∈ K and F ⊆ I be finite. Since the operators on X from Ml (X) commute with those from Mr (X) and since M ⊆ X and M ∗ X ⊆ B, we have
u(mi )m∗i x(ξ ) =
i∈F
ρ m∗i x u(mi ) (ξ ) = u ρ m∗i x mi (ξ ) i∈F
i∈F
= u mi m∗i x (ξ ) = u mi m∗i x (ξ ). i∈F
i∈F
Since u is w ∗ -continuous [3, Theorem 4.7.1] we have that lim F
u(mi )m∗i x(ξ ) = u(x)(ξ ),
ξ ∈ K.
(3.1)
i∈F
Observe that if F = {i1 , . . . , in } ⊆ I then t ∗ u(mi )mi = u (mi1 , . . . , min ) mi1∗ , . . . , m∗in i∈F
t uMl (X) (mi1 , . . . , min ) mi1∗ , . . . , m∗in uMl (X) .
Hence, the net ( i∈F u(mi )m∗i )F is bounded. Since X is non-degenerate the limit of the net
( i∈F u(mi )m∗i (ξ ))F exists for all ξ ∈ H . We let a = i∈I u(mi )m∗i , the series converging in the strong operator topology. Since XM ∗ ⊆ A, we have that a ∈ A. Observe that a uMl (X) .
(3.2)
By (3.1), ax = u(x) for all x ∈ X and so u = λ(a). We proved that λ is onto. By standard arguments, Eq. (3.2) implies that λ is isometric.
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Let n ∈ N and N = M ⊕ · · · ⊕ M. Then the N -generated Mn (A)-module is equal to Mn (X) = [Mn (A)N ]
w∗
n
. By the arguments above, the map σ : Mn (A) → Ml Mn (X) : σ (a)(x) = ax
is a surjective isometry. It follows from [6, Theorem 5.46(iii)] and its proof (see also [6, Eq. (2.4)]) that the mapping L : Mn Ml (X) → Ml Mn (X) : L (uij )i,j (xij )i,j = uik (xkj ) k
i,j
is a complete isometry. Since λ(n) = L−1 ◦ σ : Mn (A) → Mn (Ml (X)), we have that λ is n-isometric. We have thus shown that λ is a completely isometry. Similarly, we can prove that ρ is completely isometric and surjective. By Proposition 2.1, the map a x λ(a) x Φ : D(X) → Ω(X) : → 0 b 0 ρ(b) is a dual operator algebra isomorphism.
2
Corollary 3.2. If C and D are -equivalent unital dual operator algebras then there exists a dual operator space X such that C ∼ = Mr (X) as dual operator algebras. = Ml (X) and D ∼ Proof. The algebras C and D have completely isometric normal representations which are TROequivalent. Letting A be the image of C, letting M be the TRO that induces the equivalence and applying Theorem 3.1 to the corresponding generating tuple completes the proof. 2 Remark 3.3. The converse of Corollary 3.2 does not hold. Example 3.9 shows that there exists a dual operator space Y such that Ml (Y ) and Mr (Y ) are not stably isomorphic. Proposition 3.4. Let (A, M, X, B) be a generating tuple. If Y is a dual operator space which is -equivalent to the dual operator space X, then there exists a normal completely isometric representation ψ of Y such that X is TRO-equivalent to ψ(Y ). Proof. By Theorem 3.1, (A) is isomorphic to Al (X), and (B) is isomorphic to Ar (X). Thus, there is a normal CES representation of the form (π, idX , σ ) of the dual operator Al (X) − Ar (X)module X. Now apply Theorem 2.13. 2 We recall some definitions and concepts that we will need in the rest of the paper, see [8]. A commutative subspace lattice (CSL) is a strongly closed projection lattice L whose elements mutually commute. A CSL algebra is the algebra Alg L of operators leaving invariant all projections belonging to a CSL L. In the special case where L is totally ordered we call L a nest and the algebra Alg L a nest algebra. There exists a smallest w ∗ -closed algebra contained in A which contains the diagonal (A) of A and whose reflexive hull is A [1,19] (for the definition of the reflexive hull of an operator algebra see [8]). We denote this algebra by Amin . If A = Amin the CSL algebra is called A synthetic.
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Proposition 3.5. Let A and D be CSL algebras and M and N be TRO’s such that MM ∗ ⊆ A w∗ w∗ and NN ∗ ⊆ D. Set X = [AM] and Y = [DN] . Then X and Y are -equivalent if and only if they are TRO-equivalent. Proof. Suppose that X and Y are -equivalent. Since Ω(X) and Ω(Y ) are -equivalent, the algebras D(X) and D(Y ) defined as in Theorem 3.1(ii) are -equivalent. Assuming that D(X) and D(Y ) are CSL algebras, it follows from [12, Theorem 3.2] that D(X) and D(Y ) are TROequivalent. By Lemma 2.7(i), X and Y are TRO-equivalent. We now prove that D(X) is a CSL algebra, the proof for D(Y ) is similar. Denote by pr(M) ∗ the set of all projections belonging to a von Neumann algebra M. Let B = [M ∗ AM]−w . ∗ ∗ By [10, Proposition 2.8], we may assume that (B) = [M ∗ M]−w , (A) = [MM ∗ ]−w . Since (A) contains a maximal abelian selfadjoint algebra (MASA), we can easily check that (B) also contains a MASA and so the algebra D(X) contains a MASA. It now suffices to prove that D(X) is a reflexive space. Suppose that A ⊆ B(H ), B ⊆ B(K). Since the invariant subspace lattice Lat(D(X)) of D(X) is contained in (A) ⊕ (B) , we can verify that the reflexive hull Ref(D(X)) of D(X) is the space of w ∈ B(H ⊕ K) satisfying ∀ei ∈ pr (A) , fi ∈ pr (B) , (e1 ⊕ f1 )D(X)(e2 ⊕ f2 ) = {0} If w =
u a 0v
⇒
(e1 ⊕ f1 )w(e2 ⊕ f2 ) = 0.
∈ Ref(D(X)) and e1 , e2 ∈ pr((A) ) such that e2 (A)e1 = 0, then
(e2 ⊕ 0)D(X)(e1 ⊕ 0) = {0}
⇒
(e2 ⊕ 0)w(e1 ⊕ 0) = 0
⇒
e2 ue1 = 0.
Hence, u ∈ (A). Similarly, v ∈ (B). If e ∈ pr((A) ), f ∈ pr((B) ) satisfy eXf = {0}, then (e ⊕ 0)D(X)(0 ⊕ f ) = {0}
⇒
eaf = 0.
Thus, it follows that a ∈ Ref(X) = X. We have shown that w ∈ D(X), and hence Ref(D(X)) = D(X). Thus, D(X) is reflexive and contains a MASA, and hence, D(X) is a CSL algebra. 2 Example 3.6. We now give an example of spaces which are not -equivalent. Let A be a CSL algebra, B be a non-synthetic, separably acting CSL algebra and M and N be TRO’s w∗ w∗ such that MM ∗ ⊆ A and N N ∗ ⊆ B. Then the spaces X = [AM] and Y = [Bmin N ] are not -equivalent. Indeed, if they were, they would be stably isomorphic. On the other hand, Corollary 2.10 implies that X is stably isomorphic to A and Y is stably isomorphic to Bmin . Thus, the algebras A and Bmin would be stably isomorphic, hence -equivalent. This contradicts [12, Theorem 3.4]. Let N1 and N2 be nests acting on separable Hilbert spaces H1 and H2 , respectively. Recall [8] that N1 and N2 are called similar if there exists an invertible operator y : H1 → H2 such that N2 = {yn(H1 ): n ∈ N1 }. In this case there exists an order isomorphism θ : N1 → N2 which preserves the dimension of the atoms of N1 and N2 , namely, θ (n) can be taken to be equal to the projection onto yn(H1 ), for all n ∈ N1 . We say that the invertible operator y ∈ B(H1 , H2 ) implements θ . Let Y = y ∈ B(H1 , H2 ): I − θ (n) yn = 0, ∀n ∈ N1
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and Z = x ∈ B(H2 , H1 ): (I − n)xθ (n) = 0, ∀n ∈ N1 . If C = Alg N1 and D = Alg N2 one can easily verify that w∗
C = [ZY ] ,
w∗
D = [Y Z] ,
CZD ⊆ Z
and DY C ⊆ Y.
We will need the Similarity Theorem [8, Theorem 13.20]: Theorem 3.7. For every δ > 0 there exists an invertible operator y ∈ Y which implements θ such that y < 1 + δ and y −1 < 1 + δ. Theorem 3.8. ∼ C, Mr (Z) ∼ (i) Ml (Z) = = D as dual operator algebras. Z (ii) The algebra Ω(Z) is isomorphic as a dual operator algebra to the algebra (C) 0 (D) . Proof. We can easily check that the map τ : C2 (Z) → C2 (Z) : (x1 , x2 )t → (ax1 , x2 )t is completely contractive for all a ∈ C. So by [3, Theorem 4.5.2] the linear map λ : C → Ml (Z) given by λ(a)(x) = ax is contractive. Moreover, λ is one-to-one. To see this, suppose that λ(a) = 0 for some a ∈ C. Then ax = 0 for all x ∈ Z, and hence axy = 0 for all x ∈ Z and w∗ all y ∈ Y . Since C = [ZY ] , this implies that a = 0. Similarly, the map ρ : D → Mr (Z) given by ρ(b)(x) = xb is contractive. The maps λ and ρ are w ∗ -continuous by [3, Theorem 4.7.4]. Let u be in Ml (Z). By Theorem 3.7 for every δ > 0 there exists yδ ∈ Y such that yδ−1 ∈ Z, yδ < 1 + δ and yδ−1 < 1 + δ. Since the operators of Ml (Z) and Mr (Z) commute, for all x ∈ X we have u yδ−1 yδ x = ρ(yδ x) u yδ−1 = u ρ(yδ x)yδ−1 = u yδ−1 yδ x = u(x). If aδ = u(yδ−1 )yδ ∈ C then for all δ > 0 we have that λ(aδ ) = u. It follows that λ is surjective. Since λ is one-to-one, aδ = a, for all δ and a = u yδ−1 yδ uMl (Z) (1 + δ)2 ,
for all δ > 0,
we have that a uMl (Z) . Thus, λ is isometric. If n ∈ N the algebras Mn (C), Mn (D) are similar nest algebras. Repeating the above arguments we can show that λ is n-isometric. Hence λ, and similarly ρ, are completely isometric. 2 Example 3.9. The above result shows that there exists a dual operator space Z such that Ml (Z) and Mr (Z) not stably isomorphic. Indeed, from [12, Example 3.7] there exist similar nest algebras C and D which are not stably isomorphic. The claim now follows from Theorem 3.8.
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References [1] W.B. Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974) 433–532. [2] D.P. Blecher, U. Kashyap, Morita equivalence of dual operator algebras, J. Pure Appl. Algebra, in press. [3] D.P. Blecher, C. Le Merdy, Operator Algebras and Their Modules—An Operator Space Approach, Oxford University Press, 2004. [4] D.P. Blecher, P.S. Muhly, V.I. Paulsen, Categories of operator modules (Morita equivalence and projective modules), Mem. Amer. Math. Soc. 143 (2000), no. 681. [5] D.P. Blecher, V.I. Paulsen, Multipliers of operator spaces, and the injective envelope, Pacific J. Math. 200 (2001) 1–17. [6] D.P. Blecher, V. Zarikian, The calculus of one-sided M-ideals and multipliers in operator spaces, Mem. Amer. Math. Soc. 179 (2006), no. 842. [7] L. Brown, P. Green, M. Rieffel, Stable isomorphism and strong Morita equivalence of C ∗ -algebras, Pacific J. Math. 71 (1977) 349–363. [8] K.R. Davidson, Nest Algebras, Longman Scientific & Technical, Harlow, 1988. [9] E. Effros, Z.-J. Ruan, Operator Spaces, Clarendon Press, Oxford, 2000. [10] G.K. Eleftherakis, TRO equivalent algebras, preprint, arXiv:math.OA/0607488. [11] G.K. Eleftherakis, A Morita type equivalence for dual operator algebras, J. Pure Appl. Algebra 212 (5) (2008) 1060–1071. [12] G.K. Eleftherakis, Morita type equivalences and reflexive algebras, J. Operator Theory, in press. [13] G.K. Eleftherakis, V.I. Paulsen, Stably isomorphic dual operator algebras, Math. Ann. 341 (1) (2008) 99–112. [14] U. Kashyap, A Morita theorem for dual operator algebras, J. Funct. Anal. 256 (11) (2009) 3545–3567. [15] K. Morita, Duality of modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 6 (1958) 85–142. [16] V.I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. [17] G. Pisier, Introduction to Operator Space Theory, Cambridge University Press, 2003. [18] M. Rieffel, Morita equivalence for C ∗ -algebras and W ∗ -algebras, J. Pure Appl. Algebra 5 (1974) 51–96. [19] V.S. Shulman, L. Turowska, Operator synthesis. I. Synthetic sets, bilattices, tensor algebras, J. Funct. Anal. 209 (2004) 293–331. [20] C.-Y. Suen, Completely bounded maps on C ∗ -algebras, Proc. Amer. Math. Soc. 93 (1985) 81–87. [21] C. Zhang, Representations of operator spaces, J. Operator Theory 33 (1995) 327–351.
Journal of Functional Analysis 258 (2010) 279–306 www.elsevier.com/locate/jfa
Brownian and fractional Brownian stochastic currents via Malliavin calculus Franco Flandoli a,∗ , Ciprian A. Tudor b a Dipartimento di Matematica Applicata, Universita di Pisa, Via Bonnano 25B, I-56126, Pisa, Italy b SAMOS/MATISSE, Centre d’Economie de La Sorbonne, Université de Panthéon-Sorbonne Paris 1,
90, rue de Tolbiac, 75634 Paris Cedex 13, France Received 25 March 2009; accepted 4 May 2009 Available online 19 May 2009 Communicated by Paul Malliavin
Abstract By using Malliavin calculus and multiple Wiener–Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in the Watanabe spaces. © 2009 Elsevier Inc. All rights reserved. Keywords: Currents; Multiple stochastic integrals; Brownian motion; Fractional Brownian motion; Malliavin calculus
1. Introduction The concept of current is proper of geometric measure theory. The simplest example is the functional T ϕ →
ϕ γ (t) , γ (t) Rd dt
0
* Corresponding author.
E-mail addresses:
[email protected] (F. Flandoli),
[email protected] (C.A. Tudor). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.001
280
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
defined over the set of all smooth compact support vector fields ϕ : Rd → Rd , with γ : [0, T ] → Rd being a rectifiable curve. This functional defines a vector valued distribution. Let us denote it by T
δ x − γ (t) γ (t) dt.
(1)
0
We address for instance to the books [4,24,20,8], for definitions, results and applications. The stochastic analog of 1-currents is a natural concept, where the deterministic curve (γ (t))t∈[0,T ] is replaced by a stochastic process (Xt )t∈[0,T ] in Rd and the stochastic integral must be properly interpreted. Several works deal with random currents, see for instance [17,10, 23,11,19,18,15,6,16,7]. Random currents have potential links with the area of stochastic analysis and differential geometry, see for instance [1] and some of the above mentioned works. The difference between classical integration theory and random currents is the attempt to understand the latter as random distribution in the strong sense: random variables taking values 1 in the space of distributions. The question is not simply how to define 0 ϕ(Xt ), dXt Rd for every given test functions, but when this operation defines, for almost every realization of the process X, a continuous functional on some space of test functions. This is related to (a particular aspect of) T. Lyons theory of rough paths: a random current of the previous form is a concept of pathwise stochastic integration. The degree of regularity of the random analog of expression (1) is a fundamental issue. In this work we study the mapping defined as ξ(x) =
δ(x − Bs ) dBs ,
x ∈ Rd , T > 0.
(2)
[0,T ]N
Here the integrator process is a d-dimensional Wiener process with multidimensional time parameter or a d-dimensional fractional Brownian motion. A first direction of investigation is the regularity with respect to the variable x ∈ Rd of the mapping given by (2) in the (deterministic) Sobolev spaces H −r (Rd ; Rd ). This is the dual space to H r (Rd , Rd ), or equivalently the space of all vector valued distributions ϕ such that Rd (1 + 2 dx < ∞, ϕˆ being the Fourier transform of ϕ. Some results on this have been |x|2 )−r |ϕ(x)| ˆ recently obtained by [6], [5] or [7] using different techniques of the stochastic calculus. We propose here a new approach to this problem. The main tool is constituted by the Malliavin calculus based on the Wiener–Itô chaos expansion. Actually the delta Dirac function δ(x − Bs ) can be understood as a distribution in the Watanabe sense (see [25]) and it has been the object of study by several authors, as [22], [12] or [13] in the Brownian motion case, or [2], [3] or [9] in the fractional Brownian motion case. In particular, it is possible to obtain the decomposition of the delta Dirac function into an orthogonal sum of multiple Wiener–Itô integrals and as a consequence it is easy to get the chaos expansion of the integral (2) since the divergence integral acts as a creation operator on Fock spaces (see Section 2). As it will be seen, the chaos expansion obtained will be useful to obtain the regularity properties of the mapping ξ . Another advantage of this method is the fact that it can be relatively easily extended to multidimensional settings. Besides the estimation of the Sobolev regularity with respect to x, we are also interested to study the regularity with respect to ω, that is, as a functional in the Watanabe sense, of the
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integral (2). Our interest comes from the following observation. Consider N = d = 1. It is well known (see Nualart and Vives [22] ) that for fixed x ∈ R we have δ(x − Bs ) ∈ D−α,2
for any α >
1 2
where D−α,2 are the Watanabe (or Sobolev–Watanabe) spaces introduced in Section 2. On the other side, more or less surprisedly the same order of regularity holds with respect to x; for fixed ω, the mapping g(x) = δ(x − Bs (ω)) belongs to the negative Sobolev space H −r (R; R) ˆ = e−ixBs we have for every r > 12 . Indeed, since the Fourier transform of g is g(x) |g|2H −r (R;R)
=
2 −r g(x) ˆ 1 + x2 dx
R
and this is finite if and only if r > 12 . One can ask the question if this similarity of the order of regularity in the deterministic and stochastic Sobolev space still holds for the functional ξ defined by 2; and actually in the case of dimension d = 1 the above property still holds: we have the same regularity of ξ both with respect to x and with respect to ω. One can moreover ask if it holds for other Gaussian processes. The answer to this question is negative, since we show that (actually, this has also been proved in [7] but using another integral with respect to fBm) in the fractional Brownian motion case the mapping (2) belongs to 1 1 − 12 (as a function in x) and to D−α,2 with α > 32 − 2H (as a function H −r (R; R) for any r > 2H on ω). We organized our paper as follows. Section 2 contains some preliminaries on Malliavin calculus and multiple Wiener–Itô integrals. Section 3 contains a discussion about random distribution where we unify the definition of the quantity δ(x − Xs (ω)) (X is a Gaussian process on Rd ) which in principle can be understood as a distribution with respect to x and also as a distribution in the Watanabe sense when it is regarded as o function of ω. In Section 4 we study the existence and the regularity of the stochastic currents driven by a N parameter Brownian motion in Rd while in Section 5 concerning the same problem when the driving process is the fractional Brownian motion. In Section 6 we give the regularity of the integral (2) with respect to ω and we compare it with its regularity in x. 2. Preliminaries Here we describe the elements from stochastic analysis that we will need in the paper. Consider H a real separable Hilbert space and (B(ϕ), ϕ ∈ H) an isonormal Gaussian process on a probability space (Ω, A, P ), that is a centered Gaussian family of random variables such that E(B(ϕ)B(ψ)) = ϕ, ψH . Denote by In the multiple stochastic integral with respect to B (see [21]). This In is actually an isometry between the Hilbert space H n (symmetric tensor product) equipped with the scaled norm √1 · H⊗n and the Wiener chaos of order n which is n! defined as the closed linear span of the random variables Hn (B(ϕ)) where ϕ ∈ H, ϕ H = 1 and Hn is the Hermite polynomial of degree n 1 Hn (x) =
2 n 2
x d (−1)n x exp exp − , n n! 2 dx 2
x ∈ R.
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The isometry of multiple integrals can be written as: for m, n positive integers, E In (f )Im (g) = n!f, gH⊗n if m = n, E In (f )Im (g) = 0 if m = n.
(3)
It also holds that In (f ) = In (f˜) 1 where f˜ denotes the symmetrization of f defined by f˜(x1 , . . . , xx ) = n! σ ∈Sn f (xσ (1) , . . . , xσ (n) ). We recall that any square integrable random variable which is measurable with respect to the σ -algebra generated by B can be expanded into an orthogonal sum of multiple stochastic integrals
F=
In (fn )
(4)
n0
where fn ∈ H n are (uniquely determined) symmetric functions and I0 (f0 ) = E[F ]. Let L be the Ornstein–Uhlenbeck operator LF = −
nIn (fn )
n0
if F is given by (4). For p > 1 and α ∈ R we introduce the Sobolev–Watanabe space Dα,p as the closure of the set of polynomial random variables with respect to the norm
F α,p = (I − L)α/2 Lp (Ω) where I represents the identity. In this way, a random variable F as in (4) belongs Dα,2 if and only if
2
(1 + n)α In (fn ) L2 (Ω) = (1 + n)α n! fn 2H⊗n < ∞.
n0
n0
We denote by D the Malliavin derivative operator that acts on smooth functions of the form F = g(B(ϕ1 ), . . . , B(ϕn )) (g is a smooth function with compact support and ϕi ∈ H) DF =
n ∂g B(ϕ1 ), . . . , B(ϕn ) ϕi . ∂xi i=1
The operator D is continuous from Dα,p into Dα−1,p (H). The adjoint of D is denoted by δ and is called the divergence (or Skorohod) integral. It is a continuous operator from Dα,p (H) into Dα−1,p . For adapted integrands, the divergence integral coincides to the classical Itô integral. We will use the notation
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283
T δ(u) =
us dBs . 0
Let u be a stochastic process having the chaotic decomposition us = n0 In (fn (·, s)) where ⊗n for every s. One can prove that u ∈ Dom δ if and only if f˜ ∈ H⊗(n+1) for every fn (·, s) ∈ H n 2 (Ω). In this case, ˜ I ( f ) converges in L n 0, and ∞ n+1 n n=0 δ(u) =
∞
In+1 (f˜n )
∞ 2 and Eδ(u) = (n + 1)! f˜n 2H⊗(n+1) .
n=0
n=0
In the present work we will consider divergence integral with respect to a Brownian motion in Rd as well as with respect to a fractional Brownian motion in Rd . Throughout this paper we will denote by ps (x) the Gaussian kernel of variance s > 0 given x2 by ps (x) = √ 1 e− 2s , x ∈ R and for x = (x1 , . . . , xd ) ∈ Rd by psd (x) = di=1 ps (xi ). 2πs
3. Random distributions We study now the regularity of the stochastic integral given by (2). Our method is based on the Wiener–Itô chaos decomposition. Let X be isonormal Gaussian process with variance R(s, t). We will use the following decomposition of the delta Dirac function (see Nualart and Vives [22], Imkeller et al. [12], Kuo [14], Eddahbi et al. [3]) into orthogonal multiple Wiener–Itô integrals δ(x − Xs ) =
n0
− n2
R(s)
pR(s) (x)Hn
x In 1⊗n [0,s] 1/2 R(s)
(5)
where R(s) := R(s, s), pR(s) is the Gaussian kernel of variance R(s), Hn is the Hermite polynomial of degree n and In represents the multiple Wiener–Itô integral of degree n with respect to the Gaussian process X as defined in the previous section. A key element of the entire work is the quantity δ(x − Bs ). This can be understood as a generalized random variable in some Sobolev–Watanbe space as well as a generalized function with respect to the variable x. The purpose of this section is to give an unitary definition of the delta Dirac as a random distribution. Let D(Rd ) be the space of smooth compact support functions on Rd and let D (Rd ) be its dual, the space of distributions, endowed with the usual topologies. We denote by S, ϕ the S ∈ D (Rd ) is of dual pairing between S ∈ D (Rd ) and ϕ ∈ D(Rd ). We say that a distribution p class Lloc (Rd ), p 1, if there is f ∈ L1loc (Rd ) such that S, ϕ = Rd f (x)ϕ(x) dx for every ϕ ∈ D(Rd ). In this case we also say that the distribution is given by a function. Let (Ω, A, P ) be a probability space, with expectation denoted by E. Definition 1. We call random distribution (on Rd , based on (Ω, A, P )) a measurable mapping ω → S(ω) from (Ω, A) to D (Rd ) with the Borel σ -algebra. Given a random distribution S, for every ϕ ∈ D(Rd ) the real valued function ω → S(ω), ϕ is measurable. The converse is also true.
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Similarly to the deterministic case, we could say that a random distribution S is of class L0 (Ω, L1loc (Rd )) if there exists a measurable function f : Ω × Rd → R, f (ω, ·) ∈ L1loc (Rd ) for P -a.e. ω ∈ Ω, such that S(ω), ϕ = Rd f (ω, x)ϕ(x) dx for every ϕ ∈ D(Rd ). In this case we also say that the distribution is given by a random field. However, this concept is restrictive if we have to deal with true distributions. There is an intermediate concept, made possible by the simultaneous presence of the two variables ω ∈ Ω and x ∈ Rd . One could have that the random distribution is given by a function with respect to the x variable, but at the price that it is distribution-valued (in the Sobolev– Watanabe sense) in the ω variable. To this purpose, assume that a real separable Hilbert space H is given, an isonormal Gaussian process (W (h), h ∈ H) on (Ω, A, P ) is given. For p > 1 and α ∈ R, denote by Dα,p the Sobolev–Watanabe space of generalized random variables on (Ω, A, P ), defined in terms of Wiener chaos expansion. Definition 2. Given p 1 and α 0, a random distribution S is of class L1loc (Rd ; Dα,p ) if E[|S, ϕ|p ] < ∞ for all ϕ ∈ D(Rd ) and there exists a function f ∈ L1loc (Rd ; Dα,p ) such that S, ϕ =
ϕ(x)f (x) dx Rd
for every ϕ ∈ D(Rd ). Let us explain the definition. The integral Rd ϕ(x)f (x)dx is of Bochner type (the integral of the Dα,p -valued function x → ϕ(x)f (x)). The integral Rd ϕ(x)f (x) dx is thus, a priori, an element of Dα,p . The random variable ω → S(ω), ϕ, due to the assumption E[|S, ϕ|p ] < ∞, is an element of Dα ,p for all α 0. The definition requires that they coincide, as elements α,p of D , a priori. A fortiori, since ω → S(ω), ϕ is an element of D0,p , the same must be true for Rd ϕ(x)f (x) dx. Thus, among the consequences of the definition there is the fact that 0,p although f (x) lives only in Dα,p . Rd ϕ(x)f (x) dx ∈ D The following theorem treats our main example. Given any d-dimensional random variable X on (Ω, A, P ), let S be the random distribution defined as
S(ω), ϕ := ϕ X(ω) ,
ϕ ∈ D Rd .
We denote this random distribution by δ x − X(ω) and we call it the delta Dirac at X(ω). Denote Hermite polynomials by Hn (x), multiple Wiener integrals by In , Gaussian kernel of variance σ 2 by pσ 2 (x). If H1 , . . . , Hd are real and separable Hilbert spaces, a d-dimensional isonormal process is defined on the product space (Ω, A, P ), Ω = Ω1 ×· · ·×Ωd , A = A1 ⊗· · ·⊗Ad , P = P1 ⊗· · ·⊗ Pd as a vector with independent components ((W 1 (h1 ), . . . , W d (ϕd )), h1 ∈ H1 , . . . , hd ∈ Hd ) where for every i = 1, . . . , d, (W i (hi ), hi ∈ Hi ) is an one-dimensional isonormal process on (Ωi , Ai , Pi ). We denote by Ei the expectation on (Ωi , Ai , Pi ) and by E the expectation on (Ω, A, P ).
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285
Theorem 1. Let Hi , i = 1, . . . , d, be real separable Hilbert spaces. Consider ((W 1 (h1 ), . . . , W d (ϕd )), h1 ∈ H1 , . . . , hd ∈ Hd ) a d-dimensional isonormal Gaussian process on (Ω, A, P ) as above. Then, for every hi ∈ Hi , i = 1, . . . , d, the random distribution δ(· − W (h)) is of class L1loc (Rd ; Dα,2 ) for some α < 0 and the associated element f of L1loc (Rd ; Dα,2 ) is d
f (x) =
−ni
|hi |
p|hi |2 (xi )Hni
n1 ,...,nd 0 i=1
xi I i h⊗ni |hi | ni i
where x = (x1 , . . . , xd ) ∈ Rd and Ini i , i = 1, . . . , d, denotes the multiple Wiener–Itô integral with respect to the Wiener process W (i) . We denote this Dα,2 -valued function f (x) by δ(x − W (h)). Proof. Let us consider first the case d = 1. In this case f (x) =
−n
|h|
p|h|2 (x)Hn
n0
x In h⊗n . |h|
We have to prove that ϕ W (h) =
ϕ(x)f (x) dx R
for every test function ϕ. We have ϕ(x)f (x) dx =
−n
|h|
In h⊗n
n0
R
=
ϕ(x)p1 R
−n+1
|h|
In h⊗n
n0
x x Hn dx |h| |h|
ϕ |h|x Hn (x)p1 (x) dx.
R
On the other hand, by Stroock’s formula we can write 1 (n) 1 ⊗n (n) In D ϕ W (h) = In h E ϕ W (h) ϕ W (h) = n! n! n0
n0
and E ϕ (n) W (h) =
ϕ
(n)
R
= (−1) R
(x)p|h|2 (x) dx = |h|
ϕ (n) x|h| p1 (x) dx
R
ϕ (n−1) x|h| p1 (x) dx
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F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
= ···
= (−1) |h| n
n−1
(n) ϕ x|h| p1 (x) dx.
R
By using the recurrence formula for the Gaussian kernel p1 (n) p1 (x) = (−1)n n!p1 (x)Hn (x) we get ϕ W (h) =
ϕ x|h| Hn (x)p1 (x) dx =
n0 R
ϕ(x)f (x) dx. R
Concerning the case d 2, note that ϕ(x)f (x) dx Rd
=
d
|hi |−ni Ini i
⊗ni hi
n1 ,...,nd 0 i=1
=
d
dx ϕ(x)p|hi |2 (xi )Hni
Rd
|hi |−ni +1 Ini i
n1 ,...,nd 0 i=1
⊗ni hi
xi |hi |
ϕ |h1 |x1 , . . . , |hd |xd p1 (xi )Hni (xi ) dx .
Rd
Let us apply Stroock’s formula in several steps. First we apply Stroock’s formula with respect to the component W i 1 1 (n),1 1 In1 E1 D ϕ W (h) = ϕ W 1 (h1 ), . . . , W d (hd ) = ϕ W (h1 ), . . . , W d (hd ) n1 ! n1 0
n1
1 ⊗n1 ∂ ϕ 1 1 d = I h E1 W (h1 ), . . . , W (hd ) n1 ! n1 1 ∂x1n1 n1 0
and then with respect to W 2 n1 +n2
1 2 ⊗n2 ∂ ∂ n1 ϕ 1 ϕ 1 d W (h1 ), . . . , W d (hd ) . In2 h2 E2 n1 W (h1 ), . . . , W (hd ) = n1 n2 n2 ! ∂x1 ∂x1 x2 n 0 2
We will obtain
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ϕ W (h) =
287
n1 +···+nd
d 1 i ⊗ni ϕ 1 ∂ d Ini hi E W (h ), . . . , W (h ) 1 d n ni ! ∂x1n1 . . . xd d
n1 ,...,nd 0 i=1
=
n1 ,...,nd
d 1 i ⊗ni ∂ n1 +···+nd ϕ (x)p|hi |2 (xi ) dx Ini hi ni ! ∂x1n1 . . . xdnd 0 i=1 Rd
and now it suffices to follow the case d = 1 by integrating by parts and using the recurrence formula for the Gaussian kernel. 2 4. Regularity of Brownian currents with respect to x Let (Bt )t∈[0,T ]N be a Gaussian isonormal process. For every n 1, x ∈ R and s = (s1 , . . . , sN ) ∈ [0, T ]N we will use the notation n
anx (s) = R(s)− 2 pR(s) (x)Hn
x R(s)1/2
(6)
with R(s) = EBs2 = s1 . . . sN . We will start by the following very useful calculation which plays an important role in the sequel. Lemma 1. Let anx be given by (6) and denote by anxˆ (s) the Fourier transform of the function x → anx (s). Then it holds that anxˆ (s) = e−
x 2 R(s) 2
(−i)n x n . n!
(7)
Proof. Using formula (5) one can prove that the Fourier transform of g(x) = δ(x − Bs ) admits the chaos expansion g(x) ˆ =
anxˆ (s)In 1⊗n [0,s]
(8)
n0
where anxˆ (s) denotes the Fourier transform of the function x → anx (s). ˆ − Bs ) = e−ixBs and using Stroock’s formula to deBut on the other hand, we know that δ(x compose in chaos a square integrable random variable F infinitely differentiable in the Malliavin sense F=
1 In E D (n) F n!
n0
where D (n) denotes the nth Malliavin derivative and by the trivial relation Du e−ixBs = −ixe−ixBs 1[0,s] (u) which implies
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Du(n) e−ixBs = (−ix)n e−ixBs 1⊗n [0,s] (u1 , . . . , un ) 1 ,...,un one obtains n n (−i)n x n ⊗n x 2 R(s) (−i) x In 1[0,s] = e− 2 In 1⊗n g(x) ˆ = E e−ixBs . [0,s] n! n! n0
(9)
n0
Now, by putting together (8) and (9) we get (7).
2
4.1. The one-dimensional Brownian currents Let B be in this part a the Wiener process. Then by (6) δ(x − Bs ) =
s
− n2
ps (x)Hn
n0
x s 1/2
x In 1⊗n an (s)In 1⊗n [0,s] = [0,s] ,
(10)
n0
here In representing the multiple Wiener–Itô integral of degree n with respect to the Wiener process. In this case we have n anx (s) = s − 2 ps (x)Hn
x s 1/2
.
We prove first the following result, which is already known (see [6,7]) but the method is different and it can also used to the fractional Brownian motion case and to the multidimensional context. Proposition 1. Let ξ be given by (2). For every x ∈ R, it holds that 2 Eξˆ (x) = T . As a consequence ξ ∈ H −r (R, R) if and only if r > 12 . Proof. We can write, by using the expression of the Skorohod integral via chaos expansion ξ(x) =
∼ In+1 anx (s)1⊗n [0,s] (·)
(11)
n0 ∼ where (anx (s)1⊗n [0,s] (·)) denotes the symmetrization in n + 1 variables of the function (s, t1 , . . . , tn ) → anx (s)1⊗n [0,s] (t1 , . . . , tn ). It can be shown that
ξˆ (x) =
n0
By using (7),
∼ . In+1 anxˆ (s)1⊗n [0,s] (·)
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
ξˆ (x) =
In+1
n0
289
∼ x 2 s ∼ (−i)n x n − x 2 s ⊗n (−i)n x n = e 2 1[0,s] (·) In+1 e− 2 1⊗n (·) [0,s] n! n! n0
and therefore, by the isometry of multiple integrals (3), 2 x 2n
x 2 s ∼ 2
2 (n + 1)! e− 2 1⊗n . Eξˆ (x) = [0,s] (·) L ([0,T ]n+1 ) 2 (n!)
(12)
n0
Since ∼ − x 2 s ⊗n e 2 1[0,s] (·) (t1 , . . . , tn+1 ) =
1 − x 2 ti ⊗n e 2 1[0,ti ] (t1 , . . . , tˆi , . . . , tn+1 ) n+1 n+1 i=1
where tˆi means that the variable ti is missing, we obtain 2 Eξˆ (x) = n0
=
n!
ds
n!
x2 s 2 dt1 . . . dtn e− 2 1⊗n [0,s] (t1 , . . . , tn )
[0,T ]n
0
T x 2n n0
L ([0,T ]n+1 )
i=0
T x 2n n0
=
n+1
2
2 x 2n
− x 2ti ⊗n
e 1[0,ti ] (t1 , . . . , tˆi , . . . , tn+1 )
2 (n + 1)!
ds e
−x 2 s n
T
s ds =
0
e−x
2s
0
(x s s)n n0
n!
ds = T .
Finally, E|ξ |2H −r (R;R) =
2 −r 1 + x 2 Eξˆ (x) dx =
R
and this is finite if and only if r > 12 .
−r 1 + x 2 T dx
R
2
Remark 1. The above result gives actually a rigorous meaning of the formal calculation T T E( 0 δ(x − Bs ) dBs )2 = E 0 |δ(x − Bs )|2 ds = T . 4.2. The multidimensional multiparameter Brownian currents Let us consider now the multidimensional situation. In this part we will actually treat the regularity of the function δ(x − Bs ) dBs [0,T ]N
(13)
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F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
where x ∈ Rd and B = (B 1 , . . . , B d ) is a d-dimensional Wiener sheet with parameter t ∈ [0, T ]N . Here the term (13) represents the vector given by
[0,T ]N
δ(x − Bs ) dBs1 , . . . , [0,T ]N
:= ξ1 (x), . . . , ξd (x)
δ(x − Bs ) dBs =
ξ(x) =
δ(x − Bs ) dBsd
[0,T ]N
for every x ∈ Rd . The method considered above based on the chaos expansion of the delta Dirac function has advantage that, in contrast with the approaches in [7] or [5], it can be immediately extended to time parameter in RN . Moreover, we are able to compute explicitly the Sobolev norm of the vector (13) and to obtain an “if and only if” result. Proposition 2. Let (Bt )t∈[0,T ]N be a d-dimensional Wiener process with multidimensional pa rameter t ∈ [0, T ]N . Then for every w ∈ Ω the mapping x → [0,T ]N δ(x − Bs ) dBs belongs to the negative Sobolev space H −r (Rd , Rd ) if and only if r > d2 . Proof. We need to estimate
2 −r 1 + |x|2 Eξˆ (x) dx
Rd
with |ξˆ (x)|2 = |ξˆ1 (x)|2 + · · · + |ξˆd (x)|2 where |ξˆi (x)| denotes the complex modulus of ξˆi (x) ∈ C. We can formally write (but it can also written in a rigorous manner by approximating the delta Dirac function by Gaussian kernels with variance ε → 0) δ(x − Bs ) =
d j δ xj − Bs = δk (x − Bs )δ xk − Bsk j =1
where we denoted δk (x − Bs ) =
d
j δ xj − Bs ,
k = 1, . . . , d.
(14)
j =1,j =k
Let us compute the kth component of the vector ξ(x). We will use the chaotic expansion for the delta Dirac function with multidimensional parameter s ∈ [0, T ]N (see (5)) xj j j ⊗n δ xj − Bs = anj (s)Inj 1[0,s]j (·)
(15)
nj 0
j
where In denotes the Wiener–Itô integral of order n with respect to the component B j and for s = (s1 , . . . , sN )
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
nj xj x (s1 . . . sN )− 2 . anjj (s) = ps1 ...sN (xj )Hnj √ s1 . . . sN
291
(16)
It holds, for every k = 1, . . . , d, by using formula (10)
ξk (x) =
δ(x
− Bs ) dBsk
=
[0,T ]N
δk (x − Bs )δ xk − Bsk dBsk
[0,T ]N
=
δk (x − Bs )
nk 0
[0,T ]N
k k anxkk (s)Inkk 1⊗n [0,s] (·) dBs .
Since the components of the Brownian motion B are independent, the term δk (x − Bs ) is viewed as a deterministic function when we integrate with respect to B k . We obtain, ξk (x) =
δ(x − Bs ) dBsk =
nk 0
[0,T ]N
∼ k . Ink +1 δk (x − Bs )anxkk (s)1⊗n [0,s] (·)
⊗n
Here (δk (x − Bs )ankk (s)1[0,s]k (·))∼ denotes the symmetrization in nk + 1 variables of the function x
⊗n
(s, t1 , . . . , tnk ) = δk (x − Bs )anxkk (s)1[0,s]k (t1 , . . . , tnk ). Let us denote by ξˆk (x) the Fourier transform of ξk (x) = [0,T ]N δ(x − Bs ) dBsk . As in the previous section one can show that (that is, the Fourier transform with respect to x “goes inside” the stochastic integral) ξˆk (x) =
nk 0
∼ k Inkk +1 δk (xˆ − Bs )anxˆkk (s)1⊗n [0,s] (·)
where δk (xˆ − Bs )anxˆkk (s) is the Fourier transform of R x = (x1 , . . . , xd ) → δk (x d
− Bs )anxkk (s) =
d
j δ xj − Bs anxkk (s).
j =1,j =k
Now clearly δk (xˆ − Bs ) = e−i
d
l l=1,l=k xl Bs
and by (7) anxˆkk (s) =
(−i)nk xknk −ixk B k (−i)nk xknk xk2 |s| s = E e e 2 nk ! nk !
(17)
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F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
with |s| = s1 . . . sN if s = (s1 , . . . , sN ). Thus relation (17) becomes ξˆk (x) =
nk 0
Inkk +1
e−i
d
l l=1,l=k xl Bs
∼
(−i)nk xknk −ixk B k ⊗nk s 1 . E e (·) [0,s] nk !
(18)
Taking the modulus in C of the above expression, we get ∼ 2 d nk x nk xk2 |s| 2 k (−i) ⊗n k − 2 ξˆk (x) = e Ink +1 cos xl Bsl 1[0,s]k (·) nk ! l=1,l=k
nk 0
∼ 2 d nk nk x 2 |s| ⊗nk k l (−i) xk − k2 e + Ink +1 sin xl Bs 1[0,s] (·) nk ! l=1,l=k
nk 0
and using the isometry of multiple stochastic integrals (3) and the independence of components we get as in the proof of Proposition 1 (we use the notation |s| = s1 . . . sN )
d
2
xk2 |s| 2 xk2nk
⊗n l − k xl Bs e 2 1[0,s] (·) Eξˆk (x) =
cos
2 nk !
L ([0,T ]N(nk +1) )
l=1,l=k
nk 0
d
2
x 2 |s| x 2nk
⊗nk k l − k2 + xl Bs e 1[0,s] (·)
sin
2 nk !
=
x 2nk k nk !
nk 0
= [0,T ]N
L ([0,T ]N(nk +1) )
l=1,l=k
nk 0
e−xk |s| |s|nk ds 2
[0,T ]N
e−xk |s| 2
2 nk
(xk |s|) ds = nk ! nk 0
ds = T N .
[0,T ]N
This implies
2 −r 1 + |x|2 Eξˆ (x) dx =
Rd
2 2 −r 1 + |x|2 E ξˆ1 (x) + · · · + ξˆd (x) dx
Rd
= dT
N
−r 1 + |x|2 dx
Rd
which is finite if and only if 2r > d.
2
Remark 2. It is interesting to observe that the dimension N of the time parameter does not affect the regularity of currents. This is somehow unexpected because it is known that this dimension N influences the regularity of the local time of the process B which can be formally written as [0,T ]N δ(x − Bs ) ds (see [12]).
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
293
5. Regularity of fractional currents with respect to x 5.1. The one-dimensional fractional Brownian currents We will consider in this paragraph a fractional Brownian motion (BtH )t∈[0,T ] with Hurst parameter H ∈ ( 12 , 1). That is, B H is a centered Gaussian process starting from zero with covariance function R H (t, s) :=
1 2H t + s 2H − |t − s|2H , 2
s, t ∈ [0, T ].
Let us denote by HH the canonical Hilbert space of the fractional Brownian motion which is defined as the closure of the linear space generated by the indicator functions {1[0,t] , t ∈ [0, T ]} with respect to the scalar product 1[0,t] , 1[0,s] HH = R H (t, s),
s, t ∈ [0, T ].
One can construct multiple integrals with respect to B H (the underlying space L2 [0, T ] is replaced by HH ) and these integrals are those who appear in the formula (10). We prove the following result. Proposition 3. Let B H be a fractional Brownian motion with H ∈ ( 12 , 1) and let ξ be given by (2). Then for every w ∈ Ω, we have ξ ∈ H −r (R; R) for every r >
1 2H
− 12 .
Proof. Using the above computations, we will get that the Fourier transform of g(x) = δ(x − BsH ) is equal, on one hand, to g(x) ˆ =
H anxˆ s 2H InB 1⊗n [0,t] (·)
n0 H
(here InB denotes the multiple stochastic integral with respect to the fBm B H and the function anx is defined by (6)), and on the other hand, n H (−ix) InB 1⊗n (·) g(x) ˆ = E e−ixBs [0,t] n! n0
and so n x 2 s 2H (−ix) . anxˆ s 2H = e− 2 n!
Moreover
(19)
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F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
(−i)n x n
ξˆ (x) =
n!
n0
− x 2 s 2H ⊗n ∼ BH 2 In+1 e 1[0,s] (·) .
We then obtain E ξ 2H −r (R;R)
=E R
1 ξˆ (x)2 dx (1 + x 2 )r
x 2n
− x 2 s 2H ⊗n ∼ 2 1
e 2 (n + 1)! 1[0,s] (·) H⊗n+1 dx 2 r 2 (1 + x ) (n!)
=
n0
R
n+1 x 2n x 2 u2H 1 i dx e− 2 1⊗n ˆ i , . . . , un+1 ), [0,ui ] (u1 , . . . , u (n + 1)! (1 + x 2 )r
=
i=1
n0
R
n+1
e
−
x 2 vj2H 2
j =1
1⊗n [0,vj ] (v1 , . . . , vˆ j , . . . , vn+1 )
. HH
Using the fact that for regular enough functions f and g in HH their product scalar is given by T T (see e.g. [21], Chapter 5) f, gHH = H (2H − 1) 0 0 f (x)g(y)|x − yβ|2H −2 dx dy and thus for regular enough function f, g ∈ H⊗n n f, gH⊗n = H (2H − 1) f (u1 , . . . , un )g(v1 , . . . , vn ) H
[0,T ]n [0,T ]n n
×
|ui − vi |2H −2 du1 . . . dun dv1 . . . dvn
i=1
we find (by dui duj we mean below du1 . . . dun+1 dv1 . . . dvn+1 ) E ξ 2H −r (R;R) n+1 x 2n 1 dx H (2H − 1) = (n + 1)! (1 + x 2 )r n0
R
×
i,j =1
× e− = R
×
n+1
n+1
[0,T ]n+1 [0,T ]n+1
x 2 u2H i 2
e−
x 2 vj2H 2
|ul − vl |2H −2 dui dvj
l=1
1⊗n ˆ i , . . . , un+1 )1⊗n [0,ui ] (u1 , . . . , u [0,vj ] (v1 , . . . , vˆ j , . . . , vn+1 )
n+1 x 2n 1 dx H (2H − 1) 2 r (n + 1)! (1 + x ) n0
n+1 i=1
[0,T ]n+1 [0,T ]n+1
e−
x 2 u2H i 2
e−
x 2 vi2H 2
n+1 l=1
|ul − vl |2H −2 dui dvj
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
295
n+1 x 2n 1 2H (2H − 1) dx 2 r (n + 1)! (1 + x )
+
n1
R
n+1
×
i,j =1; i=j
× e−
x 2 u2H i 2
n+1
[0,T ]n+1 [0,T ]n+1
e−
x 2 vj2H
|ul − vl |2H −2 dui dvj
l=1
1⊗n ˆ i , . . . , un+1 )1⊗n [0,ui ] (u1 , . . . , u [0,vj ] (v1 , . . . , vˆ j , . . . , vn+1 )
2
:= A + B. Let us compute first the term A. Using the symmetry of the integrand and the fact that t s 2H (2H − 1)
|u − v|2H −2 du dv = R(t, s),
s, t ∈ [0, T ]
(20)
0 0
we can write
n+1 x 2n 1 dx 2H (2H − 1) n! (1 + x 2 )r
A=
n0
R
×
n+1 i=1
× e−
n+1
[0,T ]n+1 [0,T ]n+1
x 2 u2H 1 2
e−
x 2 v12H 2
|ul − vl |2H −2 dui dvj
l=1
⊗n 1⊗n [0,u1 ] (u2 , . . . , un+1 )1[0,v1 ] (v2 , . . . , vn+1 )
and by integrating with respect to du2 . . . dun+1 A=
dx R
1 x 2n dx 2H (2H − 1) n! (1 + x 2 )r n0
T T ×
e−
x 2 u2H 2
e−
x 2 v 2H 2
|u − v|2H −2 R(u, v)n du dv
0 0
= H (2H − 1) R
= H (2H − 1) R
= H (2H − 1) R
1 dx (1 + x 2 )r 1 dx (1 + x 2 )r 1 (1 + x 2 )r
T T e
−x
2 u2H 2
e
−x
2 v 2H 2
2H −2
|u − v|
n0
0 0
T T
e−
x 2 u2H 2
e−
x 2 v 2H 2
|u − v|2H −2 ex
0 0
T T 0 0
e−
x 2 |u−v|2H 2
|u − v|2H −2 du dv.
x 2n R(u, v)n du dv n!
2 R(u,v)
du dv
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F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
Now, by classical Fubini, T T
2H −2
|u − v|
A = H (2H − 1)
du dv R
0 0
2 2H 1 − x |u−v| 2 e dx (1 + x 2 )r
and by using the change of variable x|u − v|H = y in the integral with respect to, T T A = H (2H − 1)
|u − v|2H −2+2H r−H
e−
y2 2
−r |u − v|2H + y 2 dy
R
0 0
and this is finite when 2H − 2 + 2H r − H > −1 which gives r > The term denoted by B can be treated as follows B= R
x 2n n+1 1 n(n + 1) 2H (2H − 1) 2 r (n + 1)! (1 + x ) n1
× e−
x 2 u2H 1 2
e−
x 2 v22H 2
1 2H
− 12 . dui dvj
[0,T ]n+1 [0,T ]n+1
1[0,u1 ] (u2 , . . . , un+1 )⊗n 1[0,v2 ] (v1 , v3 , . . . , vn+1 )⊗n
n+1
|ul − vl |2H −2
l=1
2 = 2H (2H − 1)
R
×e
x 2 u2H − 21
e
x 2 v 2H − 22
T T T T x 2n 1 du1 du2 dv1 dv2 (n − 1)! (1 + x 2 )r n1
0 0 0 0
1[0,u1 ] (u2 )1[0,v2 ] (v1 )R(u1 , v2 )n−1 |u1 − v1 |2H −2 |u2 − v2 |2H −2 .
Since n1
x 2n R(u1 , v2 )n x 2n 2 R(u1 , v2 )n−1 = x 2 = x 2 ex R(u1 ,v2 ) (n − 1)! n! n0
we obtain 2 B = 2H (2H − 1)
R
×e
x 2 |u1 −v2 |2H − 2
1 x2 (1 + x 2 )r
T T T T du1 du2 dv1 dv2 0 0 0 0
|u1 − v1 |2H −2 |u2 − v2 |2H −2 1[0,u1 ] (u2 )1[0,v2 ] (v1 ).
We calculate first the integral du2 and dv1 and then in dx we use the change of variables x|u1 − v2 |H = y and we get
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
B =H
2 R
1 x2 (1 + x 2 )r
T T
du1 dv2 e−
297
x 2 |u1 −v2 |2H 2
0 0
× |u1 − v2 |2H −1 − u12H −1 |u1 − v2 |2H −1 − v22H −1 T = 0
du1 dv2 |u1 − v2 |2H −1 − u12H −1 |u1 − v2 |2H −1 − v22H −1 |u1 − v2 |2H r−2H −H
×
y 2 e−
y2 2
|u1 − v2 |2H + y 2 dy
R
which is finite if 4H − 2 + 2H r − 3H > −1 and this implies again r >
1 2H
− 12 .
2
5.2. The multidimensional fractional currents We will also consider the multidimensional case of the fractional Brownian motion B H in Rd . It is defined as a random vector B H = (B H1 , . . . , B Hd ) where B Hi are independent onedimensional fractional Brownian sheets. We will assume that Hi ∈ ( 12 , 1) for every i = 1, . . . , d. In this case ξ(x) =
δ x − BsH dB H1 , . . . ,
[0,T ]N
δ x − BsH dB Hd
[0,T ]N
where the above integral is a divergence (Skorohod) integral with respect to the fractional Brownian motion. We mention that the canonical Hilbert space HHk of the fractional Brownian sheet B Hk is now the closure of the linear space of the indicator functions with respect to the inner product N 1[0,t] , 1[0,s] HHk = E BtHk BsHk := R Hk (t, s) = R Hk (ti , si ) i=1
if t = (t1 , . . . , tN ) and s = (s1 , . . . , sN ). In this case we have Proposition 4. Let B H be as above. Then for every ω the fractional Brownian current ξ belong to the Sobolev space H −r (Rd ; Rd ) for every r > maxk=1,...,d ( d2 − 1 + 2H1 k ). Proof. We write ξk (x) = [0,T ]N
δ x − BsH dBsHk =
δk x − BsH δ xk − BsHk dBsHk
[0,T ]N
= [0,T ]N
x H k k δk x − BsH ankk (s)InHkk 1⊗n [0,s] (·) dBs nk 0
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F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
and ξˆk (x) =
nk 0
InHkk+1
∼
nk nk H x |s|2Hk ⊗nk −i dl=1,l=k xl Bs l (−i) xk − 2 2 1[0,s] (·) e . e nk !
Here again the integral dB Hk denotes the Skorohod integral with respect to the fractional Brownian sheet B Hk and I Hk denotes the multiple integral with respect to B Hk . Then ∼ 2 d n k 2 2Hk Hk ⊗nk Hl xk − x |s|2 ξˆk (x)2 = e Ink +1 cos xl Bs 1[0,s] (·) nk ! l=1, l=k
nk 0
∼ 2 d n k 2 2Hk Hk ⊗nk Hl xk − x |s|2 e + Ink +1 sin xl Bs 1[0,s] (·) nk ! l=1, l=k
nk 0
and
d ∼ 2 2nk
2 x 2 |s|2Hk ⊗n x
Eξˆk (x) = (nk + 1)! k 2 cos xl BsHl e− 2 1[0,s]k (·)
(nk !)
+
(HHk )⊗(nk +1)
l=1, l=k
nk 0
d ∼ 2
2 2Hk xk2nk
⊗nk Hl − x |s|2 sin (nk + 1)! x B 1 (·) e
l s [0,s]
(nk !)2
.
(HHk )⊗(nk +1)
l=1, l=k
nk 0
Note that, if f, g are two regular functions of N variables N f, gHHk = Hk (2Hk − 1)
f (u)g(u)
N := Hk (2Hk − 1)
|ui − vi |2Hk −2 du dv
i=1
[0,T ]N [0,T ]N
N
f (u)g(v) u − v 2Hk −2 du dv
[0,T ]N [0,T ]N
if u = (u1 , . . . , uN ) and v = (v1 , . . . , vN ) and if F, G are regular functions of N (nk + 1) variables, F, G(HH
k
=
n k +1
)⊗(nk +1)
N Hk (2Hk − 1)
k=1
×
F u1 , . . . , unk +1 G v 1 , . . . , v nk +1
[0,T ]N(nk +1) [0,T ]N(nk +1)
n k +1
i
u − v i 2Hk −2 du1 . . . dunk +1 dv 1 . . . dv nk +1
i=1
and thus, by symmetrizing the above function and taking the scalar product in (HHk )⊗(nk +1)
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
299
2 Eξˆk (x) =
nk 0
×
2n N (nk +1) xk k Hk (2HK − 1) (nk + 1)!
n k +1
i,j =1
[0,T ]N(nk +1)
du
d
× cos
l=1, l=k
n k +1
dv
l=1
[0,T ]N(nk +1)
xl BuHi l
d
cos
l
u − v l 2Hk −2
l=1, l=k
e−
xl BvHjl
xk2 |ui |2Hk 2
e−
xk2 |v j |2Hk 2
⊗n ⊗n × 1[0,uki ] u1 , . . . , uˆ i , . . . , unk +1 1[0,vkj ] v 1 , . . . , vˆ j , . . . , v nk +1 +
nk 0
×
N (nk +1) xk2nk Hk (2HK − 1) (nk + 1)!
n k +1
i,j =1
[0,T ]N(nk +1)
du
d
× sin
l=1, l=k
n k +1
dv
l=1
[0,T ]N(nk +1)
xl BuHi l
sin
l
u − v l 2Hk −2
d l=1, l=k
xl BvHjl
e−
xk2 |ui |2Hk 2
e−
xk2 |v j |2Hk 2
⊗n ⊗n × 1[0,uki ] u1 , . . . , uˆ i , . . . , unk +1 1[0,vkj ] v 1 , . . . , vˆ j , . . . , v nk +1 . Here |ui |2Hk = (ui1 . . . uiN )2Hk and 1[0,ui ] = 1[0,ui ] . . . 1[0,ui ] if ui = (ui1 , . . . , uiN ). The next step N
1
is to majorize |(cos(u) cos(v)| by 12 (cos2 (u) + cos2 (v)) and similarly for the sinus. 2 Eξˆk (x) cst.
nk 0
×
N (nk +1) xk2nk Hk (2HK − 1) (nk + 1)!
i=1
× e−
du
[0,T ]N(nk +1)
x 2 |ui |2Hk 2
+ cst.
nk 0
×
n k +1
e−
dv [0,T ]N(nk +1)
x 2 |v i |2Hk 2
n k +1
l
u − v l 2Hk −2
l=1
1 1 k k u , . . . , uˆ i , . . . , unk +1 1⊗n v , . . . , vˆ j , . . . , v nk +1 1⊗n [0,ui ] [0,v j ]
N (nk +1) xk2nk Hk (2HK − 1) (nk + 1)!
n k +1
i,j =1; i=j
[0,T ]N(nk +1)
du
dv [0,T ]N(nk +1)
n k +1 l=1
l
u − v l 2Hk −2
300
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306 xk2 |ui |2Hk 2
× e−
e−
xk2 |v j |2Hk 2
⊗n ⊗n 1[0,uki ] u1 , . . . , uˆ i , . . . , unk +1 1[0,vkj ] v 1 , . . . , vˆ j , . . . , v nk +1
:= Ck + Dk and the two terms above can be treated as the terms A and B from the one-dimensional case. For example the term denotes by Ck (we illustrate only this term because Dk is similar to the term B in the one-dimensional case) gives
Ck =
nk 0
n k +1
l
u − v l 2Hk −2
[0,T ]N(nk +1) [0,T ]N(nk +1)
× e− =
x 2nk N (nk +1) k Hk (2HK − 1) nk ! xk2 |u1 |2Hk 2
e−
xk2 |v 1 |2Hk 2
l=1
2 2 k k u , . . . , . . . , unk +1 1⊗n v , . . . , . . . , v nk +1 1⊗n [0,u1 ] [0,v 1 ]
x 2nk N k Hk (2Hk − 1) nk !
nk 0
e−
×
xk2 |u1 |2Hk 2
[0,T ]N [0,T ]N
N = Hk (2Hk − 1)
xk2 |v 1 |2Hk 2
e−
2
exk R(u
1
u − v 1 2Hk −2 R Hk u1 , v 1 nk d 1 dv 1
1 ,v 1 )
e−
xk2 |u1 |2Hk 2
e−
xk2 |v 1 |2Hk 2
1
u − v 1 2Hk −2 d 1 dv 1
[0,T ]N [0,T ]N H
H
1 ). But R Hk (u, v)−u2Hk −v 2Hk = E(B k B k )− with R Hk (u1 , v 1 ) = R Hk (u11 , v11 ) . . . R Hk (u1N , vN v u Hk 2 Hk 2 Hk Hk 2 n E(Bu ) − E(Bv ) = −E(Bu − Bv ) for any u, v ∈ R and this implies
−r dx 1 + |x|2 Ck = cst.
Rd
−r dx 1 + |x|2
Rd
e−
H H xk2 E(Bu k −Bv k )2 2
u − v 2Hk −2 du dv.
[0,T ]N [0,T ]N
The case N = 1 can be easily handled. Indeed, in this case
−r dx 1 + |x|2 Ck =
|u − v|2Hk −2
[0,T ] [0,T ]
Rd
=
Rd
1 2Hk
+
d 2
−r xk2 |u−v|2Hk 2 1 + |x| e− dx
Rd
|u − v|2Hk −2+2Hk r−dHk
[0,T ] [0,T ]
which is finite if r >
e−
yk2 2
−r |u − v|2Hk + y 2 dy
Rd
− 1. When N 2 we will have
−r dx 1 + |x|2 Ck =
[0,T ]N [0,T ]N
2 d/2+r u − v 2Hk −2 E BuHk − BvHk
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
301
2 −r 2 e−yk /2 E BuHk − BvHk + y 2 dx
× Rd
and this is finite if r >
1 2Hk
+
d 2
− 1.
2
Remark 3. In [7] the authors obtained the same regularity in the case of the pathwise integral with respect to the fBm. They considered only the case Hk = H for every k = 1, . . . , d and N = 1. On the other hand [7], the Hurst parameter is allowed to be lesser than one half, it is assumed to be in ( 14 , 1). 6. Regularity of stochastic currents with respect to ω 6.1. The Brownian case We study now, for fixed x ∈ R, the regularity of the functional ξ in the sense of Watanabe. We would like to see if, as for the delta Dirac function, the stochastic integral ξ(x) keeps the same regularity in the Watanabe spaces (as a function of ω) and in the Sobolev spaces as a function of x. We consider B a one-dimensional Wiener process and denote T δ(x − Bs ) dBs
ξ(x) = a
with a > 0. Recall that by (10) ξ(x) =
∼ In+1 anx (s)1⊗n [0,s] (·)1[a,T ] (s)
n0
(we consider the integral from a > 0 instead of zero to avoid a singularity) where x ∼ an (s)1⊗n [0,s] (·)1[a,T ] (s) (t1 , . . . , tn+1 ) =
1 x ˆ an (ti )1⊗n [0,ti ] (t1 , . . . , ti , . . . , tn+1 )1[a,T ] (ti ) n+1 n+1 i=1
where as above tˆi means that the variable ti is missing. We recall that F is a random variable having the chaotic decomposition F = n Ini (fn ) then its Sobolev–Watanabe norm is given by F 22,α =
2
(n + 1)α In (fn ) L2 (Ω) . n
We will get
∼ 2
ξ(x) 2 =
2 (n + 2)α (n + 1)! anx (s)1⊗n [0,s] (·)1[a,T ] (s) 2,α L [0,T ]n+1 n0
=
T (n + 2)α n!
ni 0
a
2 x ps (x)Hn √ ds. s
(21)
302
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
We use the identity 2
Hn (y)e
− y2
[n/2]
= (−1)
2 2 n!π n 2
∞
√ 2 un e−u g uy 2 du
(22)
0
where g(r) = cos(r) if n is even and g(r) = sin(r) if n is odd. Since |g(r)| 1, we have the bound
Hn (y)e−y 2 /2 2 n2 2 n + 1 := cn . n!π 2
(23)
Then
ξ(x) 2 cst. (n + 2)α n!cn2 2,α
T
n
1 ds s
(24)
a
2 √1 and this is finite for α < −1 2 since by Stirling’s formula n!cn behaves as cst. n . We summarize the above discussion.
Proposition 5. For any x ∈ R the functional ξ(x) = Watanabe space D−α,2 for any α > 12 .
T 0
δ(x − Bs ) dBs belongs to the Sobolev–
Remark 4. As for the delta Dirac function, the regularity ξ is the same with respect to x and with respect to ω. 6.2. The fractional case In this paragraph the driving process is a fractional Brownian motion (BtH )t∈[0,T ] with Hurst parameter H ∈ ( 12 , 1). We are interested to study the regularity as a functional in Sobolev– Watanabe spaces of T ξ(x) =
δ x − BsH dBsH
(25)
0
when x ∈ R is fixed. We will that now the order of regularity in the Watanabe spaces changes and it differs from the order of regularity of the same functional with respect to the variable x. We prove the following result. Proposition 6. Let (BtH )t∈[0,T ] be a fractional Brownian motion with Hurst parameter H ∈ ( 12 , 1). For any x ∈ R the functional ξ(x) (2) is an element of the Sobolev–Watanabe space 1 D−α,2 with α > 32 − 2H .
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
303
Proof. We will have in this case
∼ 2
ξ(x) 2 =
⊗n+1 (n + 2)α (n + 1)! anx (s)1⊗n [0,s] (·) 2,α H H
n0
where we denoted by HH the canonical Hilbert space of the fractional Brownian motion. As in the proof of Proposition 2 we obtain
x
a (s)1⊗n (·) ∼ 2 ⊗n+1 n [0,s] H H
=
n+1 − 1))n+1
(H (2H (n + 1)2
i,j =1
du1 . . . dun+1 dv1 . . . dvn+1
n+1
|ul − vl |2H −2
l=1
[0,T ]n+1 [0,T ]n+1
× anx (ui )anx (vj )1⊗n ˆ i , . . . , un+1 )1⊗n [0,ui ] (u1 , . . . , u [0,vj ] (v1 , . . . , vˆ j , . . . , vn+1 ) n+1 (H (2H − 1))n+1 = (n + 1)2 i=1
du1 . . . dun+1 dv1 . . . dvn+1
n+1
|ul − vl |2H −2
l=1
[0,T ]n+1 [0,T ]n+1
× anx (ui )anx (vi )1⊗n ˆ i , . . . , un+1 )1⊗n [0,ui ] (u1 , . . . , u [0,vi ] (v1 , . . . , vˆ i , . . . , vn+1 ) +
n+1
(H (2H − 1))n+1 (n + 1)2
du1 . . . dun+1 dv1 . . . dvn+1
i=j ; i,j =1
n+1
|ul − vl |2H −2
l=1
[0,T ]n+1 [0,T ]n+1
× anx (ui )anx (vj )1⊗n ˆ i , . . . , un+1 )1⊗n [0,ui ] (u1 , . . . , u [0,vj ] (v1 , . . . , vˆ j , . . . , vn+1 ) := A(n) + B(n). The first term A(n) equals, by symmetry, (H (2H − 1))n+1 A(n) = n+1
T T
du dv anx (u)anx (v)|u − v|2H −2
0 0
u v
n
2H −2
|u − v |
du dv
0 0
and using equality (20) we get (H (2H − 1)) A(n) = n+1
T T du dv 0 0
2H −2
× |u − v| Using the identity (22)
n −H n −H n
R(u, v) u
v
pu2H (x)pv 2H (x)Hn
x x Hn H . uH v
304
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
(n + 2)α (n + 1)!A(n)
n0
= c(H )
T T (n + 2) n! α
n0
0 0 n −H n −H n
× R(u, v) u
= c(H )
du dv |u − v|2H −2
v
pu2H (x)pv 2H (x)Hn
T T (n + 2)
α
du dv |u − v|2H −2 R(u, v)n u−H n v −H n u−H v −H .
n!cn2
n0
x x Hn H uH v
0 0
By the selfsimilarity of the fBm we have R(u, v) = u−2H R(1, uv ) and by the change of variables v = zu we obtain
(n + 2)α (n + 1)!A(n)
n0
c(H )
1
T (n + 2)α n!cn2
u1−2H du
n0
0
R(1, z)n (1 − z)2H −2 dz. zH n zH
(26)
0
Using Lemma 2 (actually a slightly modification of it) in [3] 1 0
R(1, z)n (1 − z)2H −2 1 dz c(H ) 1 H n H z z n 2H
(27)
and the right-hand side of (26) is bounded, modulo a constant, by
1
(n + 2)α n!cn2 n− 2H
n0 1 1 and since n!cn2 behaves as √1n , the last sum is convergent if 2H − α + 12 > 1, or −α > − 2H + 12 . Let us regard now the sum involving the term B(n). It will actually decide the regularity of the functional ξ(x). Following the computations contained in the proof of Proposition 3
(n + 2)α (n + 1)!B(n)
n0
= c(H )
n0
n(n + 1) (n + 2) (n + 1)! (n + 1)2
T T T T
α
× 1[0,u1 ] (u2 )1[0,v2 ] (v1 )R(u1 , v2 )
du1 du2 dv1 dv2 0 0 0 0
n−1
|u1 − v1 |2H −2 |u2 − v2 |2H −2 anx (u1 )anx (v2 )
F. Flandoli, C.A. Tudor / Journal of Functional Analysis 258 (2010) 279–306
c(H )
305
(n + 2)α nn!cn2
n0
T T ×
n −H n du1 dv2 R(u1 , v2 )n−1 u−H v2 1
0 0
c(H )
u1
2H −2
|u2 − v2 |
v2
|u1 − v1 |
du2
0
T T (n + 2)α nn!cn2
n0
2H −2
0
n −H n du1 dv2 R(u1 , v2 )n−1 u−H v2 . 1
0 0
Using again Lemma 2 in [3], we get that the integral T T
n −H n du1 dv2 R(u1 , v2 )n−1 u−H v2 c(H )n− 2H 1 1
0 0
and since the sequence n!cn2 behaves when n → ∞ as (n + 1)!B(n) converges if −α − Remark 5. Only when H = and as a function of ω.
1 2
1 2
+
1 2H
√1 n
we obtain that the sum
> 1 and this gives −α >
3 2
−
1 2H
.
α n0 (n+2)
×
2
we retrieve the same order of regularity of (25) as a function of x
References [1] H. Airault, P. Malliavin, Intégration géométrique sur l’espace de Wiener, Bull. Sci. Math. (2) 112 (1) (1988) 3–52. [2] L. Coutin, D. Nualart, C.A. Tudor, The Tanaka formula for the fractional Brownian motion, Stochastic Process. Appl. 94 (2) (2001) 301–315. [3] M. Eddahbi, R. Lacayo, J.L. Sole, C.A. Tudor, J. Vives, Regularity of the local time for the d-dimensional fractional Brownian motion with N -parameters, Stoch. Anal. Appl. 23 (2) (2001) 383–400. [4] H. Federer, Geometric Measure Theory, Springer, Berlin, 1969. [5] F. Flandoli, M. Gubinelli, Random currents and probabilistic models of vortex filaments, in: Proceedings Ascona 2002, Birkhäuser, 2002. [6] F. Flandoli, M. Gubinelli, M. Giaquinta, V. Tortorelli, Stochastic currents, Stochastic Process. Appl. 115 (2005) 1583–1601. [7] F. Flandoli, M. Gubinelli, F. Russo, On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model, Ann. Inst. H. Poincaré Probab. Statist. 45 (2) (2009) 545–576. [8] M. Giaquinta, G. Modica, J. Soucek, Cartesian Currents in the Calculus of Variation I, Springer, Berlin, 1988. [9] Y.Z. Hu, B. Oksendhal, Chaos expansion of local time of fractional Brownian motions, Stoch. Anal. Appl. 20 (4) (2003) 815–837. [10] N. Ikeda, Limit theorems for a class of random currents, in: Probabilistic Methods in Mathematical Physics, Katata/Kyoto, 1985, Academic Press, Boston, MA, 1987, pp. 181–193. [11] N. Ikeda, Y. Ochi, Central limit theorems and random currents, in: Lecture Notes in Control and Inform. Sci., vol. 78, 1986, pp. 195–205. [12] P. Imkeller, P. Weisz, The asymptotic behavior of local times and occupation integrals of the N -parameter Wiener process in Rd , Probab. Theory Related Fields 98 (1) (1994) 47–75. [13] P. Imkeller, V. Perez-Abreu, J. Vives, Chaos expansion of double intersection local time of Brownian motion in Rd and renormalization, Stochastic Process. Appl. 56 (1) (1995) 1–34. [14] H.H. Kuo, White Noise Distribution Theory, CRC Press, Boca Raton, 1996. [15] K. Kuwada, Sample path large deviations for a class of random currents, Stochastic Process. Appl. 108 (2003) 203–228.
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[16] K. Kuwada, On large deviations for random currents induced from stochastic line integrals, Forum Math. 18 (2006) 639–676. [17] S. Manabe, Stochastic intersection number and homological behavior of diffusion processes on Riemannian manifolds, Osaka J. Math. 19 (1982) 429–450. [18] S. Manabe, Large deviation for a class of current-valued processes, Osaka J. Math. 29 (1) (1992) 89–102. [19] S. Manabe, Y. Ochi, The central limit theorem for current-valued processes induced by geodesic flows, Osaka J. Math. 26 (1) (1989) 191–205. [20] F. Morgan, Geometric Measure Theory – A Beginners Guide, Academic Press, Boston, 1988. [21] D. Nualart, Malliavin Calculus and Related Topics, Springer, 1995. [22] D. Nualart, J. Vives, Smoothness of Brownian local times and related functionals, Potential Anal. 1 (3) (1992) 257–263. [23] Y. Ochi, Limit theorems for a class of diffusion processes, Stochastics 15 (1985) 251–269. [24] L. Simon, Lectures on Geometric Measure Theory, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 3, Austral. Nat. Univ., Canberra, 1983. [25] S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Springer-Verlag, 1994.
Journal of Functional Analysis 258 (2010) 307–327 www.elsevier.com/locate/jfa
The Yang–Mills functional and Laplace’s equation on quantum Heisenberg manifolds Sooran Kang Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO 80309, United States Received 26 March 2009; accepted 30 September 2009
Communicated by S. Vaes
Abstract In this paper, we discuss the Yang–Mills functional and a certain family of its critical points on quantum Heisenberg manifolds using noncommutative geometrical methods developed by A. Connes and M. Rieffel. In our main result, we construct a certain family of connections on a projective module over a quantum Heisenberg manifold that gives rise to critical points of the Yang–Mills functional. Moreover, we show that there is a relationship between this particular family of critical points of the Yang–Mills functional and Laplace’s equation on multiplication-type, skew-symmetric elements of quantum Heisenberg manifolds; recall that Laplacian is the leading term for the coupled set of equations making up the Yang–Mills equation. © 2009 Elsevier Inc. All rights reserved. Keywords: The Yang–Mills functional; Quantum Heisenberg manifolds; Laplace’s equation
Since Alain Connes initiated noncommutative differential geometry in his ground-breaking paper [4], the theory has flourished in different areas and, has motivated new ideas in various fields. Connes’ and Marc Rieffel’s Yang–Mills theory for the noncommutative torus [6] is one of these examples, using the framework of noncommutative geometry to extend Yang–Mills theory to finitely generated projective modules over noncommutative C ∗ -algebras. This generalization seems to be natural, but Connes’ and Rieffel’s Yang–Mills theory for the noncommutative torus seems to be the only specific example of such an application so far. In this paper, using the same framework developed in [6], we attempt to develop Yang–Mills theory on quantum Heisenberg E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.024
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S. Kang / Journal of Functional Analysis 258 (2010) 307–327
c, } ∗ manifolds, {Dμν ∈R , which are a different type of noncommutative C -algebra first constructed by Marc Rieffel [12]. The main difference in our case from that of the theory of the noncommutative torus is the c, is constructed by following. First of all, for fixed μ, ν and c, the projective module over Dμν c, realizing Dμν as a generalized fixed point algebra of a certain crossed product C ∗ -algebra, and thereby developing a bimodule structure. Also we use a particular Grassmannian connection to produce a compatible connection on the projective module. The method of finding such a nontrivial connection is related to the technique of finding Rieffel projections in noncommutative tori, a method not employed by Connes and Rieffel in [6] and [13]. In our case, the last step of finding actual solutions of the Yang–Mills equation is related to solving an elliptic partial differential equation, which is very different from the approach of [6] and [13]. This paper is organized as follows. In Section 1, we begin with the definition of quantum c, -D c, projective Heisenberg manifolds, and we give a specific formula for a particular Eμν μν c, is isomorphic in a fashbimodule Ξ described in [2] and [1]. In Section 2, we show that Eμν
ion preserving the bimodule structure to D c, 1
ν 4μ , 2μ
, using the technique of crossed products by
Hilbert C ∗ -bimodules described in [7,3]. In Section 3, we describe the noncommutative geometrical framework for Yang–Mills theory, and we produce a special function R that gives a nontrivial Grassmannian connection and curvature. In Sections 4 and 5, we introduce the notion of “multiplication-type” element, and we describe a certain set of critical points of the Yang– Mills functional on quantum Heisenberg manifolds. In the last section, we show that the set of critical points that we found can be described as a set of solutions to Laplace’s equation on multiplication-type, skew-symmetric elements of quantum Heisenberg manifolds. 1. Projective modules over quantum Heisenberg manifolds Let G be the Heisenberg group, parametrized by ⎛
1 y ⎝ (x, y, z) = 0 1 0 0
⎞ z x⎠ 1
so that when we identify G with R3 the product is given by (x, y, z) x , y , z = x + x , y + y , z + z + yx . For any positive integer c, let Dc denote the subgroup of G consisting of those (x, y, z) such that x, y, and cz are integers. Then the Heisenberg manifold, Mc , is the quotient G/Dc , on which G acts on the left. c, } In [12], Rieffel constructed strict deformation quantizations {Dμν ∈R of Mc in the direction of the Poisson bracket Λμν , determined by two real parameters μ and ν, where μ2 + ν 2 = 0. He recognized that these noncommutative C ∗ -algebras could be described as generalized fixedpoint algebras of certain crossed product C ∗ -algebras under proper actions. Then Abadie showed in [2] that it is possible to construct a (finitely generated) projective bimodule over two generalized fixed-point algebras, under appropriate conditions. As an example in [2] and [1], she stated
S. Kang / Journal of Functional Analysis 258 (2010) 307–327
309
explicit formulas for a finitely generated projective module over two generalized fixed point alc, , the quantum Heisenberg manifold. We give here more details of gebras, one of which is Dμν the specific construction, which will be used for further discussion in later sections. First, we introduce the reparametrization of Heisenberg group described in [12]. For a given positive integer c, we reparametrize the Heisenberg group G as ⎛
1 ⎝ (x, y, z) = 0 0
y 1 0
⎞ z/c x ⎠. 1
(1)
Then the product on R3 becomes (x, y, z) x , y , z = x + x , y + y , z + z + cyx , and Dc becomes the subgroup with integer entries. Let Ec = {(0, m, n) ∈ Dc } be the normal subgroup of Dc . Then we can check that for f ∈ C ∞ (G), the operator corresponding to right translation of f by (k, m, n) ∈ Dc is given by f (x, y, z) → f (x + k, y + m, z + n + cky). To obtain the Heisenberg manifold, we consider the quotient, Nc , of G by the right action of Ec . Then this quotient looks like R × T2 . If we define an action ρ of Z on Nc by (ρk f )(x, y, z) = f (x + k, y, z + cky),
(2)
for (x, y, z) ∈ R × T2 and a smooth function f on Nc , then the Heisenberg manifold Mc is the quotient of Nc by ρ. Also the action ρ of Z can be viewed as (k, 0, 0) ∈ Dc acting on the right on Nc , which means the following. f (x, y, z) · (k, 0, 0) = f (x + k, y, z + cky) = (ρk f )(x, y, z). Thus we can consider functions on Mc as functions on Nc which are invariant under the action ρ. Now we describe the action of G on the left on Nc . For g = (r, s, t) ∈ G and (x, y, z) ∈ Nc = R × T2 , define the left action of G on C ∞ (Nc ) by (g · f )(x, y, z) = f (r, s, t)−1 · (x, y, z) = f x − r, y − s, z − t − sc(x − r) ,
(3)
where f ∈ C ∞ (Nc ) and (r, s, t)−1 is the inverse of (r, s, t) in G. Then a straightforward calculation shows that this action of G on the left on Nc commutes with the action ρ. To obtain a strict deformation quantization of the Heisenberg manifold, Rieffel first formed a deformation quantization on Nc ∼ = R × T in [12], denoted by A , by using the Fourier transform, and showed that the action ρ on this quantization is proper. Then he recognized that A can be identified with a certain crossed product C ∗ -algebra under the map J given in [12, p. 547], and a strict deformation quantization of C ∞ (Mc ), denoted by D , via the above isomorphism, it is possible to view as the generalized fixed-point algebra of this crossed product C ∗ -algebra under the action ρ. See more about proper actions and generalized fixed point algebras in [14]. The
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corresponding action ρ and the action of the Heisenberg group on A are given as follows. Take the Fourier transform in the third variable in Eqs. (2) and (3); then we have, for φ ∈ S(R × T × Z), (ρk φ)(x, y, p) = e(ckpy)φ(x + k, y, p),
(4)
and the formula for the action of the Heisenberg group on the same deformed algebra is given by L(r,s,t) φ(x, y, p) = e p t + cs(x − r) φ(x − r, y − s, p),
(5)
where S(R × T × Z) is Schwartz space, the set of functions on R × T × Z which go to zero at infinity faster than any polynomial grows. Since these two actions commute on S(R × T × Z), a dense subalgebra of A , the same formula L gives the action of the Heisenberg group on the generalized fixed point algebra for ρ, D . Now we state the specific formula for a particular projective module over the quantum Heisenc, , shown in [2] and [1] as follows. berg manifolds, Dμν Let M = R × T and let λ and σ be the commuting actions of Z on M defined by λp (x, y) = (x + 2pμ, y + 2pν)
and σp (x, y) = (x − p, y),
where is Planck’s constant, μ, ν ∈ R, and p ∈ Z. Then construct the crossed product C ∗ -algebras Cb (R × T) λ Z and Cb (R × T) σ Z with usual star-product and involution. Here Cb (R × T) is a set of bounded functions on R × T, and ρ and γ denote the actions of Z on Cb (R × T) λ Z and Cb (R × T) σ Z given by, for Φ, Ψ ∈ Cc (R × T × Z), (ρk Φ)(x, y, p) = e ckp(y − pν) Φ(x + k, y, p), (γk Ψ )(x, y, p) = e cpk(y − kν) Ψ (x − 2kμ, y − 2kν), where k, p ∈ Z, and e(x) = exp(2πix) for any real number x. Then these actions ρ, γ are proper. c, , is the The generalized fixed point algebra of Cb (R × T) λ Z by the action ρ, denoted by Dμν closure of ∗-subalgebra D0 in the multiplier algebra of Cb (R × T) λ Z consisting of functions Φ ∈ Cc (R × T × Z), which have compact support on Z and satisfy ρk (Φ) = Φ for all k ∈ Z. Remark 1. The above formula of ρ on Cb (R × T) λ Z can be obtained from Eq. (4) under the c, as the corresponding generalized fixed point map J given in [12, p. 547], and we consider Dμν algebra, D , under the same map J . c, from Eq. (5) via the map J , given We can obtain the action of the Heisenberg group on Dμν
by (L(r,s,t) Φ)(x, y, p) = e p t + cs(x − r − pμ) Φ(x − r, y − s, p),
(6)
for Φ ∈ D0 . Similarly, the generalized fixed point algebra of Cb (R × T) σ Z by the action γ , denoted c, , is the closure of ∗-subalgebra E in the multiplier algebra of C (R × T) Z consisting by Eμν 0 b σ
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of functions Ψ ∈ Cc (R × T × Z), with compact support on Z and satisfying γk (Ψ ) = Ψ for all k ∈ Z. c, and E c, According to the main theorem in [2], these generalized fixed point algebras Dμν μν c, c, are strongly Morita equivalent. Let Ξ be the left-Eμν and right-Dμν bimodule constructed as follows. Ξ is the completion of Cc (R × T) with respect to either one of the norms induced by c, and E c, -valued inner products, ·,· and ·,· respectively, given by one of the Dμν D E μν f, g D (x, y, p) =
e ckp(y − pν) f (x + k, y)g(x − 2pμ + k, y − 2pν), k∈Z
f, g E (x, y, p) = e cpk(y − kν) f (x − 2kμ, y − 2kν)g(x − 2kμ + p, y − 2kν), k∈Z c, and D c, on Ξ are where f, g ∈ Cc (R × T) and k, p ∈ Z. Also the left and right actions of Eμν μν given by
(Ψ · f )(x, y) = (g · Φ)(x, y) =
Ψ (x, y, q)f (x + q, y),
q∈Z
g(x + 2qμ, y + 2qν)Φ(x + 2qμ, y + 2qν, q),
q∈Z
for Ψ ∈ E0 , Φ ∈ D0 and f, g ∈ Ξ . 2. Morita equivalence of quantum Heisenberg manifolds It has been shown that the generalized fixed-point algebra of a certain crossed C ∗ -algebra constructed by Rieffel in [14] can be generated by the fixed-point algebra and the first spectral subspace for the action of T on the crossed product C ∗ -algebra. Also, it is known that the first spectral subspace has a natural bimodule structure over the fixed-point algebra. Thus, we can classify generalized fixed-point algebras by examining each fixed-point algebra, its first spectral subspace, and the bimodule structure. We follow the same technique that Abadie introduced in c, can be identified with a quantum Heisenberg the papers [3] and [7] in order to prove that Eμν 1 ν , 2μ ). Note that Abadie did not indicate that the generalized fixedmanifold with parameters ( 4μ c, can be identified with D c, in the papers [3,7]. point algebra Eμν 1 ν 4μ 2μ
c, is the generalized fixed-point As we described earlier, the quantum Heisenberg manifold Dμν algebra of Cb (R × T) λ Z under the action ρ. In particular,
c, Dμν = span Φ ∈ Cc (R × T × Z) e ckp(y − pν) Φ(x + k, y, p) = Φ(x, y, p) for all k ∈ Z
= span φδp e ckp(y − pν) φ(x + k, y) = φ(x, y) for all k ∈ Z . c, by Now choosing k = 1, then we can write Dμν
c, = span φδp e cp(y − pν) φ(x + 1, y) = φ(x, y) , Dμν
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c, carries a natural dual action of T. Let ς be the action for φ ∈ Cb (R × T). Also notice that Dμν c, of T on Dμν given by
(ςz Φ)(x, y, p) = zp Φ(x, y, p) = e(pz)Φ(x, y, p)
for z ∈ T.
c, is given as follows. Thus, the nth spectral subspace of Dμν
c, Dμν n = f δn f ∈ Cb (R × T), e cn(y − nν) f (x + 1, y) = f (x, y) .
(7)
c, ) be the fixed point algebra of D c, under the action ς , and let (D c, ) be the first Let (Dμν 0 μν μν 1 c, spectral subspace of Dμν for ς , i.e.
c, c,
φ(x, y) = φ(x + 1, y) , Dμν 0 = φδ0 ∈ Dμν
c, c,
Dμν 1 = gδ1 ∈ Dμν g(x, y) = e c(y − ν) g(x + 1, y) . c, , mentioned in the previous section, is the generalized fixed-point algebra of Similarly, Eμν Cb (R × T) σ Z under γ . Thus, it carries a natural action of T given by
(z Ψ )(x, y, p) = zp Ψ (x, y, p)
for z ∈ T.
c, ) be the fixed point algebra of E c, under the action , and let (E c, ) be the first Let (Eμν 0 μν μν 1 c, for , i.e. spectral subspace of Eμν
c, c,
ψ(x, y) = ψ(x − 2μ, y − 2ν) , Eμν 0 = ψδ0 ∈ Eμν
c, c,
f (x, y) = e c(y − ν) f (x − 2μ, y − 2ν) . Eμν 1 = f δ1 ∈ Eμν According to Proposition 1.2 in [3] by Abadie, the C ∗ -algebras Cb (M/α) Xβα,u and β,u∗
Cb (M/β) Xα are Morita equivalent, where α, β are free and proper commuting actions β,u∗ of Z on a locally compact Hausdorff space M, u is a unitary in Cb (M), and Xβα,u and Xα β,u∗
are C ∗ -bimodules. Thus, there is a left-Cb (M/α) Xβα,u and right-Cb (M/β) Xα bimodule, c, is and we denote it by Ξ . By choosing appropriate actions α and β, Abadie showed that Dμν strongly Morita equivalent to D c, 1
ν 4μ , 2μ
in [3]. We give the specific formulas that we use later in
this section as follows. 1 Consider the actions α and β on M = R × T given by α(x, y) = (x + 2μ , y) and β(x, y) = ∗ (x + 1, y + 2ν). Let u(x, y) = e(c(y − ν)), so u (x, y) = e(c(y − ν)). Then Cb (M/α) ∼ = C(T2 ) and Cb (M/β) ∼ = C(T2 ). Note that the unitary u here is different from the unitary given in Proposition 2.2 in [3]. With the unitary u and the commuting actions α and β as above, we can β,u∗ write the corresponding C ∗ -bimodules Xβα,u and Xα as follows.
1
Xβα,u = f ∈ Cb (R × T) f x − , y = e c(y − ν) f (x, y) . 2μ
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Then Xβα,u is a bimodule over Cb (R × T/α) ∼ = C(T2 ) with following formulas. For ψ ∈ α,u Cb (R × T/α) and f, g ∈ Xβ , (ψ · f )(x, y) = ψ(x, y)f (x, y), f, g L (x, y) = f (x, y)g(x, y),
(f · ψ)(x, y) = f (x, y)ψ(x − 1, y − 2ν), f, g R (x, y) = f (x + 1, y + 2ν)g(x + 1, y + 2ν).
Here the subscripts R and L stand for the right and left inner products, respectively. Also we can β,u∗ write Xα in the following way.
∗ Xαβ,u = g ∈ Cb (R × T/β) g(x − 1, y − 2ν) = e c(y − ν) g(x, y) . ∗
β,u Then Xα is a bimodule over Cb (R × T/β) ∼ = C(T2 ) with the following formulas. For φ ∈ β,u∗ Cb (R × T/β) and f, g ∈ Xα ,
(φ · f )(x, y) = φ(x, y)f (x, y), f, g L (x, y) = f (x, y)g(x, y),
1 (f · φ)(x, y) = f (x, y)φ x − ,y , 2μ
1 f, g R (x, y) = f x + , y g(x, y). 2μ
c, -E c, bimodule Ξ given in the previous section, and the C (M/α) X α,u For the Dμν b μν β β,u∗
Cb (M/β) Xα
equivalent to D c, 1
c, is strongly Morita bimodule Ξ above, it is not hard to verify that Dμν
ν 4μ , 2μ
c, is isomorphic to C (R × T/β) X α,u and D c, by showing that Dμν b 1 β
ν 4μ , 2μ
β,u∗
is
isomorphic to Cb (R × T/β) Xα in a fashion preserving the bimodule structures as shown c, is isomorphic to D c, . in [3]. Thus we only need the following lemma to show that Eμν 1 ν 4μ , 2μ
c, is isomorphic in a fashion preserving the bimodule structure to C (R × Lemma 1. Eμν b β,u∗
T/β) Xα
.
Proof. In this proof, we absorb the Planck constant into the parameters μ and ν for simplicity. Define maps S and H by c, ∗ S : Xαβ,u → Eμν , H : Cb (R × T/β) → C T2 , 1
2
cy x x S(f )(x, y) = e f − , −y , H (φ)(x, y) = φ − , −y , 2ν 2μ 2μ β,u∗
for f ∈ Xα and φ ∈ Cb (R × T/β). The straightforward calculations show that S and H are c, ) . Also, it is not hard to show that S and H satisfy the following bijections and S(f ) ∈ (Eμν 1 conditions, S(φ · f ) = H (φ) · S(f ), S(f ), S(g) L = H f, g L ,
S(f · φ) = S(f ) · H (φ), S(f ), S(g) R = H f, g R .
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c, is isomorphic to C (R × T/β) X Therefore Eμν b α structure. 2
in a fashion preserving the bimodule
c, -D c, bimodule given in the previous section. Then E c, is isomorProposition 2. Let Ξ be Eμν μν μν
phic in a fashion preserving the bimodule structure to D c, 1
ν 4μ , 2μ
Proof. Since D c, 1 isomorphic to
ν 4μ , 2μ c, D1 ν . 4μ , 2μ
.
β,u∗
is isomorphic to Cb (R × T/β) Xα
c, is , Lemma 1 shows that Eμν
2
3. Grassmannian connections and curvature on quantum Heisenberg manifolds As mentioned in [5], projective modules are the proper generalizations of vector bundles when taking the view point of noncommutative geometry. By the Gelfand–Naimark theorem we can identify a commutative unital C ∗ -algebra A with C(X) for a compact space X, and the Serre– Swan theorem says that there is a contravariant equivalence of categories between category of vector bundles and bundle maps over a compact space X and the category of finitely generated projective modules and module morphisms over the commutative algebra C(X). So, many times, a geometric property on X can be understood in terms of a corresponding algebraic property on the algebra C(X). J.L. Koszul established an algebraic version of differential geometry, in particular, algebraic concepts for connections and curvature on a projective module over a commutative associative algebra in [9]. Before we state Connes’ definitions of connections and curvature on a projective module over a noncommutative C ∗ -algebra, we mention the general algebraic definition of the connection on a module described in [9]. Let k be a commutative ring with unit and let A be a commutative, associative algebra over k having unit element. Then a derivation X is an element of Homk (A, A) with the condition, X(ab) = (Xa)b + a(Xb). Denote the set of derivations by D; then D is obviously an A-module with usual operations and has a natural Lie algebra structure. A connection on A-module (commutative case) is described as a derivation law in [9]. Definition 1. (See [9].) A derivation law ∇ is an element of HomA (D, Homk (M, M)), where M is a unitary A-module, satisfying 1. ∇X+Y = ∇X + ∇Y , ∇aX = a∇X , 2. ∇X (au) = (Xa)u + a∇X u, for a ∈ A, u ∈ M and X, Y ∈ D. As can be seen, ∇X is defined on an A-module M. So ∇X can be viewed as a differentiation of differentiable sections of a bundle in a certain direction, since the set of differentiable sections of a vector bundle has an obvious module structure. Thus, this derivation law corresponds to a connection on a projective module in modern geometry, and Connes’ definition of a connection can be viewed as a noncommutative extension of Koszul’s derivation law. Now we state the setting developed by Connes in [4]. Let G be a Lie group with Lie algebra g, and let α be an action of G as automorphisms of a C ∗ -algebra A. We let A∞ = {a ∈ A |
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g → αg (a) is smooth in norm}. Then the infinitesimal form of α gives an action, δ, of the Lie algebra g of G, as derivations of A∞ . By Lemma 1 in [4], given a finitely generated projective A-module Ξ , there is a dense A∞ submodule Ξ ∞ ⊂ Ξ , such that Ξ ∞ is finitely generated and projective over A∞ and Ξ is isomorphic to Ξ ∞ ⊗A∞ A. Furthermore, Ξ ∞ is unique up to isomorphism as an A∞ -module. Note that we say “projective” when we mean “finitely generated projective”. Also, we will denote Ξ ∞ and A∞ by Ξ and A for notational simplicity for the general definitions. Also, we can always equip Ξ with an A-valued positive definite inner product ·,· A , called a Hermitian metric, such that ξ, η ∗A = η, ξ A , ξ, ηa A = ξ, η A a, for ξ, η ∈ Ξ and a ∈ A. Definition 2. (See [4].) Let Ξ , A and g be as above. A connection ∇ is a linear map from Ξ to Ξ ⊗ g∗ such that ∇X (ξ a) = ∇X (ξ ) a + ξ δX (a) , for all X ∈ g, ξ ∈ Ξ and a ∈ A. We say that the connections are compatible with the Hermitian metric if δX ξ, η A = ∇X ξ, η A + ξ, ∇X η A . We denote the set of compatible connections by CC(Ξ ). According to Connes’ theory, we can always define a compatible connection on a projective module over A as follows. For a given unital C ∗ -algebra A and a projection Q ∈ A, QA is a projective right A-module in an obvious way. As described in [4], we define a connection on QA, called “Grassmannian connection”, by 0 ∇X (ξ ) = QδX (ξ ) ∈ QA
for all ξ ∈ QA and X ∈ g.
Obviously, this is a compatible connection with the canonical Hermitian metric on QA, such that ξ, η = ξ ∗ η for ξ, η ∈ QA. For given right A-module Ξ , let E = EndA (Ξ ). Then the following facts are known in [6]. If is an element of E, for each X ∈ g. If ∇ and ∇ ∇ and ∇ are any two connections, then ∇X − ∇X is a skew-symmetric element are both compatible with the Hermitian metric, then ∇X − ∇X of E for each X ∈ g. Thus, once we have a compatible connection ∇, every other compatible connection ∇ is of the form ∇ + μ, where μ is a linear map from g into E s , the set of skewsymmetric elements of E, such that μX ∗ = −μX for X ∈ g. The curvature of a connection ∇ is defined to be the alternating bilinear form Θ∇ on g, given by Θ∇ (X, Y ) = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] , for X, Y ∈ g. It is not hard to check that the values of Θ are in E for a connection ∇, and the values of Θ are in E s if a connection ∇ is compatible with the Hermitian metric. For given A-valued inner product ·,· A , we can define an E-valued inner product ·,· E by ξ, η E ζ = ξ η, ζ A , for ξ, η, ζ ∈ Ξ . So there is a natural bimodule structure (left E-right A) on Ξ .
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If A has a faithful α-invariant trace, τ , then τ determines a faithful trace, τE , on E, defined by τE ξ, η E = τ η, ξ A . To define the Yang–Mills functional on CC(Ξ ), we need a bilinear form on the space of alternating 2-forms with values in E. Let {Z1 , . . . , Zn } be a basis for g. We define a bilinear form {·,·}E by {Φ, Ψ }E =
Φ(Zi ∧ Zj )Ψ (Zi ∧ Zj ),
i<j
for alternating E-valued 2-forms Φ, Ψ . Clearly, its values are in E. Then the Yang–Mills functional, YM, is defined on CC(Ξ ) by YM(∇) = −τE {Θ∇ , Θ∇ }E .
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c, and right-D c, bimodule described in Section 1. Let G be the Now let Ξ be the left-Eμν μν reparametrized Heisenberg Lie group in (1) with the Lie algebra g; then for a positive integer c, the basis of g is given by {X, Y, Z} with [X, Y ] = cZ, where X = (0, 1, 0), Y = (1, 0, 0), Z = c,h¯ given by the formula (6), we calculate (0, 0, 1). Using the Heisenberg group action G on Dμν c,h¯ in the following way. For Φ ∈ (D c,h¯ )∞ , the infinitesimal form, δ, of the action g on Dμν μν
d L(exp(k(r ,s ,t ))) Φ(x, y, p)
dk k=0 ∂Φ ∂Φ (x, y, p) − s (x, y, p), = 2πip t + cs(x − h¯ pμ) Φ(x, y, p) − r ∂x ∂y
(δ(r,s,t) Φ)(x, y, p) =
where (r , s , t ) ∈ G and (r, s, t) ∈ g. Then the associated derivations δX , δY , δZ are given by (δX Φ)(x, y, p) = 2πicp(x − h¯ pμ)Φ(x, y, p) −
∂Φ (x, y, p), ∂y
∂Φ (x, y, p), ∂x (δZ Φ)(x, y, p) = 2πipΦ(x, y, p). (δY Φ)(x, y, p) = −
To find a compatible connection on the projective module Ξ ∞ , we use the canonical Grassc, )∞ for a projection Q ∈ (D c, )∞ . As we know, QD c, and Ξ mannian connection ∇ 0 on Q(Dμν μν μν c, on the right. So we can construct a modare projective modules over the same C ∗ -algebra Dμν c, and Ξ , which preserves the dense subalgebras Q(D c, )∞ and Ξ ∞ as ule map between QDμν μν c, -D c, bimodule Ξ , there exists a follows. As shown in Proposition 2.1 in [11], for given Eμν μν c, since E c, function R ∈ Ξ such that R, R E = IdE and Q = R, R D is a projection of Dμν μν c, both have identity elements. We will show that we can choose R smooth later in this and Dμν c, is given by φ(f ) = R, f , and the corresection. The module isomorphism φ : Ξ → QDμν D ∞ sponding Grassmannian connection on Ξ is given by
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0 ∇X (f ) = R · δX R, f D ,
317
(9)
for X ∈ g, f ∈ Ξ ∞ , and R ∈ Ξ ∞ such that Q = R, R D . It is not hard to check that ∇ 0 in (9) is compatible with ·,· D . Remark 2. Notice that, since R, R E = IdE , the trace of the projection Q = R, R D is 2μ. c, )∞ and (E c, )∞ by Ξ , D c, and E c, in the For notational simplicity, we denote Ξ ∞ , (Dμν μν μν μν rest of this section. We now compute the corresponding curvature Θ∇0 (X, Y ) of the Grassmannian connection ∇ 0 c, -valued, using formula (9) we can calculate Θ 0 (X, Y ) · f for X, Y ∈ g. Since Θ∇0 (X, Y ) is Eμν ∇ as follows.
0 0 0 0 0 0 0 0 0 ∇Y − ∇Y0 ∇X − ∇[X,Y Θ∇0 (X, Y ) · f = ∇X ] · f = ∇X ∇Y (f ) − ∇Y ∇X (f ) − ∇[X,Y ] (f ) = R · δX R, R D δY R, f D − δY R, R D δX R, f D , for f ∈ Ξ , since R · R, R D = R and δ is a Lie algebra homomorphism, i.e. [δX , δY ] = δ[X,Y ] . At first glance, it seems that the curvature Θ∇0 (X, Y ) would depend on the value f at which it c, )s , the set of skew-symmetric elements is evaluated. Since the values of the curvature lie in (Eμν c, of Eμν , we have the following property. Lemma 3. For the element R constructed as above, R · δX R, R D δY R, g D − δY R, R D δX R, g D = R · δX R, R D δY R, R D − δY R, R D δX R, R D · R, g D , where g ∈ Ξ and X, Y ∈ g. c, )s , Θ 0 (X, Y ) satisfies Proof. Since the values of Θ∇0 are in (Eμν ∇
Θ∇0 (X, Y ) · f, g
D
+ f, Θ∇0 (X, Y ) · g D = 0,
for f, g ∈ Ξ and X, Y ∈ g. Then a straightforward calculation shows that f, R D δX R, R D δY R, g D − f, R D δY R, R D δX R, g D = δX f, R D δY R, R D R, g D − δY f, R D δX R, R D R, g D . Now let f = R; then we have R, R D δX R, R D δY R, g D − R, R D δY R, R D δX R, g D = δX R, R D δY R, R D R, g D − δY R, R D δX R, R D R, g D . By applying R on both sides, we obtain the desired equation.
2
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c, )s -valued Θ 0 in closed form that does not Therefore, we can develop a formula for the (Eμν ∇ depend on f ∈ Ξ as follows.
Proposition 4. For given basis elements of the Heisenberg Lie algebra g, {X, Y, Z} with [X, Y ] = cZ, where c ∈ Z+ , we have Θ∇0 (X, Y ) = R · δX R, R D δY R, R D − δY R, R D δX R, R D , R E , Θ∇0 (X, Z) = R · δX R, R D δZ R, R D − δZ R, R D δX R, R D , R E , Θ∇0 (Y, Z) = R · δY R, R D δZ R, R D − δZ R, R D δY R, R D , R E . Proof. For given basis elements {X, Y, Z} with [X, Y ] = cZ of the Heisenberg Lie algebra g, the previous lemma implies that 0 0 0 0 ∇Y (f ) − ∇Y0 ∇X (f ) − ∇[X,Y Θ∇0 (X, Y ) · f = ∇X ] (f ) = R · δX R, R D δY R, f D − δY R, R D δX R, f D = R · δX R, R D δY R, R D − δY R, R D δX R, R D · R, f D = R · δX R, R D δY R, R D − δY R, R D δX R, R D , R E · f. Since this equation holds for every f ∈ Ξ , we have Θ∇0 (X, Y ) = R · δX R, R D δY R, R D − δY R, R D δX R, R D , R E . Similarly, we can establish the second and the third equalities.
2
The above proposition suggests that the curvature of the Grassmannian connection on Ξ can be computed in terms of a smooth function R ∈ Ξ . In what follows, we will give a specific example of nontrivial smooth function R ∈ Ξ that produces a Grassmannian curvature Θ∇0 that is nonzero. In fact, R can be given by an explicit formula. Since R is chosen to give a projection Q c, such that Q = R, R and R, R = Id , using the technique for construction of of Dμν D E E Rieffel projections we let R, R D (x, y, p) = g(x, y)δ1 (p) + h(x, y)δ0 (p) + g(x + 2μ, y + 2ν)δ−1 (p), where g, h ∈ Ξ , and h is real-valued. We also use the formula for ·,· D , and obtain h(x, y) =
(a-1)
R(x + k, y) 2 , k
(a-2)
g(x, y) = e ck(y − hν) R(x + k, y)R(x − 2μ + k, y − 2ν). k
Using the fact that R, R D is idempotent, we obtain (b-1)
g(x, y)g(x − 2μ, y − 2ν) = 0,
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g(x, y) 1 − h(x, y) − h(x − 2μ, y − 2ν) = 0,
g(x, y) 2 + g(x + 2μ, y + 2ν) 2 = h(x, y) − h2 (x, y).
(b-2) (b-3)
We also want Q to be in D0 , so we require h(x, y) = h(x + k, y),
(c-1)
g(x, y) = e ck(y − ν) g(x + k, y).
Since R, R E = IdE , i.e. e cpk(y − kν) R(x − 2kμ, y − 2kν)R(x − 2kμ + p, y − 2kν) = IdΞ (x, y)δ0 (p), k
we have the following.
R(x − 2kμ, y − 2kν) 2 = 1 if p = 0,
(d-1)
k
(d-2) e cpk(y − kν) R(x − 2kμ, y − 2kν)R(x − 2kμ + p, y − 2kν) = 0 if p = 0. k
The technique that we will use to find a specific function R is somewhat related to the way to find a scaling function, which is continuous and compactly supported, in wavelet theory, in particular, a Meyer-type wavelet. (See more details in [10].) So we assume that R is a compactly supported real-valued function of one variable x for simplicity. As suggested in the paper [10], we define R as follows: let R(x) be 0 on (−2μ, −μ], smooth on (−μ, − 12 μ) and 1 on [− 12 μ, 0], where |2μ| < 12 , and define R on [0, 2μ) by R(x) = 1 − |R(x − 2μ)|2 . Then the function R is smooth. Then this definition implies that h(x, y) and g(x, y) in (a-1) and (a-2) should have only one term that is not equal to zero, although which term is nonzero depends on x. So we have, for x ∈ (− 12 , 12 ), (A-1)
h(x) = R 2 (x),
g(x, y) = R(x)R(x − 2μ).
Also, the condition in (c-1) suggests that we should extend h and g by (A-2)
h(x) = R 2 (x + k),
g(x, y) = e k(y − ν) R(x + k)R(x + k − 2μ),
for x ∈ (− 12 − k, 12 − k). With these functions, h(x) and g(x, y), we can rewrite Eqs. (b-1)–(b-3) as follows. (B-1)
R 2 (x)R(x − 2μ)R(x + 2μ) = 0,
(B-2)
R 2 (x) + R 2 (x − 2μ) = 1,
(B-3)
R 2 (x − 2μ) + R 2 (x + 2μ) + R 2 (x) − 1 = 0.
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The above conditions (A-1)–(B-3) imply the following equation. R(x)R(x − 2lμ) = 0 if |l| 2.
(C-1)
Also, Eqs. (d-1) and (d-2) give the following.
R(x − 2kμ) 2 = 1.
(C-2)
k
R(x)R(x + j ) = 0
(C-3)
if j = 0,
where j ∈ Z.
With this special function R, we calculate the corresponding Grassmannian curvature as follows. Lemma 5. For the function R described above, Θ∇0 (X, Y )(x, y, p) = f1 (x)δ0 (p), Θ∇0 (X, Z)(x, y, p) = 0, Θ∇0 (Y, Z)(x, y, p) = f2 (x)δ0 (p), where f1 and f2 are smooth skew-symmetric periodic functions in the sense that f1 (x) = −f1 (x) and f2 (x) = −f2 (x), and f1 (x − 2kμ) = f1 (x) and f2 (x − 2kμ) = f2 (x) for any integer k. Proof. By the formula in Proposition 4, we have, for X, Y ∈ g, Θ∇0 (X, Y ) = R · δX R, R D δY R, R D − δY R, R D δX R, R D , R E . The main reason that Grassmannian curvature is only supported at p = 0 comes from the formula of ·,· E and Eq. (C-3). By the related formulas and properties, the Grassmannian curvature with the special function R is given as follows. Θ∇0 (X, Y )(x, y, p) =− 2πiq x − 2kμ + 2qμ − q μ R 2 (x − 2kμ)R 2 (x − 2kμ + 2qμ) k
q
q
× R x − 2kμ + 2qμ − 2q μ R x − 2kμ + 2qμ − 2q μ + R(x − 2kμ)R (x − 2kμ)R 2 (x − 2kμ + 2qμ)R x − 2kμ + 2qμ − 2q μ × R x − 2kμ + 2qμ − 2q μ e cq k(y − kν) 2πi q − q x − 2kμ + q − q μ + k
q
q
× R 2 (x − 2kμ)R(x − 2kμ + 2qμ)R (x − 2kμ + 2qμ) × R 2 x − 2kμ + 2qμ − 2q μ + R 2 (x − 2kμ)R 2 (x − 2kμ + 2qμ) × R x − 2kμ + 2qμ − 2q μ R x − 2kμ + 2qμ − 2q μ = 0.
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321
According to Eq. (C-1), we know that for fixed integer k, R 2 (x − 2kμ)R 2 (x − 2kμ + 2qμ) is not necessarily zero unless |q| 2. So choose k = 0 and q = 0 for simplicity. Then the first triple term in the above expression for Θ∇0 (X, Y ) becomes zero and the curvature Θ∇0 (X, Y )(x, y, p) becomes the following expression.
4πiq(x + qμ)R 2 (x)R 3 (x + 2qμ)R (x + 2qμ).
q
Since R(x)R(x + 2qμ) = 0 for |q| 2, the previous expression becomes the following.
4πiq(x + qμ)R 2 (x)R 3 (x + 2qμ)R (x + 2qμ)
q
= −4πi(x − μ)R 2 (x)R 3 (x − 2μ)R (x − 2μ) + 4πi(x + μ)R 2 (x)R 3 (x + 2μ)R (x + 2μ) = 0. For each k, we will have a similar expression as above that is not necessarily zero. Thus the Grassmannian curvature Θ∇0 (X, Y )(x, y, p) is not trivial. As seen above, Θ∇0 (X, Y )(x, y, p) is only supported at p = 0. Also it is not hard to see that Θ∇0 (X, Y )(x, y, p) is a one-variable, periodic, complex valued function. So we denote Θ∇0 (X, Y )(x, y, p) by f1 (x)δ0 (p) such that f1 (x − 2lμ) = f1 (x) for an integer l. It is clear that f1 (x) is a skew-symmetric function, i.e. f 1 (x) = −f1 (x). Thus we conclude that Θ∇0 (X, Y )(x, y, p) = f1 (x)δ0 (p) for a skew-symmetric periodic function f1 . Similarly, we can show that Θ∇0 (Y, Z)(x, y, p) = f2 (x)δ0 (p) for a periodic function f2 . Also another similar direct calculation shows that Θ∇0 (X, Z)(x, y, p) = 0. 2 c,
4. Multiplication-type elements of Eμν
c, and right-D c, projective bimodule given in Section 1. Let Ξ be the left-Eμν μν c, by Definition 3. For an element G ∈ C ∞ (T2 ), define a multiplication-type element G of Eμν
G(x, y, p) = G(x, y)δ0 (p). c, means γ (G) = G for all k ∈ Z. This implies G(x − 2kμ, y − 2kν) = Remark 3. G ∈ Eμν k G(x, y), for (x, y) ∈ R × T. (Here we are identifying T2 with R2 /(2πμZ × 2πνZ).) Thus c, . Also any multiplication-type eleG has to be defined on T2 to produce an element G of Eμν c, ment G is a smooth element of Eμν since the corresponding function G is smooth. c, . Then G is skew-symmetric, i.e. Lemma 6. Let G be a multiplication-type element of Eμν ∗ G = −G if and only if the corresponding function G ∈ C ∞ (T2 ) is also skew-symmetric, i.e. G(x, y) = −G(x, y).
Proof. The proof is left to the reader.
2
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c, with a corresponding function Proposition 7. Let G be a multiplication-type element of Eμν ∞ 2 G ∈ C (T ), i.e. G(x, y, p) = G(x, y)δ0 (p). Then G is skew-symmetric if and only if G acts on Ξ ∞ as a (skew-symmetric) multiplication operator, i.e.
(G · f )(x, y) = −G(x, y)f (x, y)
for f ∈ Ξ ∞ .
c, , i.e. G(x, y, p) = G(x, y)δ (p), for Proof. Let G be a multiplication-type element of Eμν 0 G ∈ C ∞ (T2 ). If G is skew-symmetric, then the corresponding function G is skew symmetric by the previous lemma. Thus G(x, y) = −G(x, y). So (G · f )(x, y) = q G(x, y, q) f (x +q, y) = q G(x, y)δ0 (q)f (x +q, y) = G(x, y)f (x, y) = −G(x, y)f (x, y). Now assume c, acts on Ξ ∞ by (G · f )(x, y) = −G(x, y)f (x, y) that a multiplication-type element G of Eμν ∞ for f ∈ Ξ . Then q G(x, y, q)f (x + q, y) = −G(x, y)f (x, y). This implies that q G(x, y) δ0 (q)f (x + q, y) = G(x, y)f (x, y) = −G(x, y)f (x, y). Thus G(x, y) = −G(x, y). Therefore, G∗ = −G by Lemma 6. 2
With this notation and Lemma 5, we can view the Grassmannian curvature Θ∇0 as a c, with the corresponding skew-symmetric functions f , 0 multiplication-type element of Eμν 1 0 and f2 . So we write Θ∇ as follows. Θ∇0 (X, Y ) = f1 ,
Θ∇0 (X, Z) = 0,
Θ∇0 (Y, Z) = f2 ,
(10)
where f1 (x, y, p) = f1 (x)δ0 (p), f2 (x, y, p) = f2 (x)δ0 (p) and f 1 (x) = −f1 (x), f 2 (x) = −f2 (x), f1 , f2 ∈ C ∞ (T). Proposition 8. Let X, Y , Z be the basis of the Heisenberg Lie algebra g with [X, Y ] = cZ. Let ∇ 0 be the Grassmannian connection on Ξ ∞ given in (9) and let G be a multiplication-type, c, as defined above, corresponding to the smooth skew-symmetric skew-symmetric element of Eμν ∞ 2 function G ∈ C (T ). Then for f ∈ Ξ ∞ , ∂ G (x, y)f (x, y), ∂y
0 ∂ ∇Y , G · f (x, y) = G (x, y)f (x, y), ∂x 0 ∇Z , G · f (x, y) = 0.
0 ∇X , G · f (x, y) =
Proof. Let G(x, y, p) = G(x, y)δ0 (p) for a skew-symmetric function G ∈ C ∞ (T2 ). Then for f ∈ Ξ ∞, 0 0 0 ∇X , G (f )(x, y) = ∇X ◦ G (f )(x, y) − G ◦ ∇X (f )(x, y) 0 0 = ∇X (G · f )(x, y) − G · ∇X (f ) (x, y) 0 = R · δX R, G · f D (x, y) − G(x, y) ∇X (f ) (x, y)
S. Kang / Journal of Functional Analysis 258 (2010) 307–327
=
323
R(x + 2qμ)δX R, G · f D (x + 2qμ, y + 2h¯ qν, q)
q
+ G(x, y) R · δX R, f D (x, y). The important equation that we use in this proof is given in (C-3), R(x)R(x + j ) = 0 if j = 0. By related formulas and equations, we obtain the following.
0 ∂ ∇X , G (f )(x, y) = G(x, y) f (x, y). R 2 (x + 2qμ) ∂y q 0 , G] · f )(x, y) = ( ∂ G)(x, y)f (x, y). Similarly, we can obtain By Eq. (C-2), it follows that ([∇X ∂y the second equation and the third equation. We leave the details to the reader. 2
The previous two propositions imply the following. Corollary 9. Let X, Y, Z be the basis of the Heisenberg Lie algebra g with [X, Y ] = cZ. Let ∇ 0 be the Grassmannian connection given in (9) and let G be a multiplication-type, skew-symmetric c, . Then element of Eμν 0 0 ∂ ∂ ∇X , G (x, y, p) = − G(x, y, p), ∇Y , G (x, y, p) = − G(x, y, p), ∂y ∂x 0 ∇Z , G = 0. Proof. It is obvious by Propositions 8 and 7.
2
For notational simplicity we write the equations in Corollary 9 as follows. 0 ∂ ∇Y , G = − G, ∂x
0 ∂ ∇X , G = − G, ∂y
0 ∇Z , G = 0.
Proposition 10. The Grassmannian connection ∇ 0 given in (9) generates an infinite dimensional 0 , ∇ 0 , ∇ 0 is infinite Lie algebra, in the sense that the Lie algebra of operators generated by ∇X Y Z dimensional. Proof. Using the formula for the curvature and the notation for the Grassmannian curvature in (10), we have the following. 0 0 ∇X , ∇Y = Θ∇0 (X, Y ) + ∇Z0 = f1 + ∇Z0 , 0 0 ∇X , ∇Z = Θ∇0 (X, Z) = 0 and ∇Y0 , ∇Z0 = Θ∇0 (Y, Z) = f2 . Also, by the previous corollary we have 0 ∇X , f1 = 0, 0 ∇X , f2 = 0,
0 ∂ ∇Y , f1 = − f1 , ∂x 0 ∂ ∇Y , f2 = − f2 , ∂x
0 ∇Z , f1 = 0,
∇Z0 , f2 = 0.
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∂ ∂ 0 , ∇0 , ∇0 The proof of Lemma 5 shows that − ∂x f1 = 0 and − ∂x f2 = 0, in general. Thus ∇X Y Z generate an infinite dimensional Lie algebra. 2
5. Critical points of the Yang–Mills functional on quantum Heisenberg manifolds The Yang–Mills problem is mainly about determining the nature of the set of the critical points for the Yang–Mills functional YM, and the critical points where YM attains its minimum. According to differential calculus, ∇ is a critical point of YM if D(YM(∇)) = 0, i.e. the derivative of YM at ∇ is zero. Also we have
d
YM(∇ + tμ) = D YM(∇) · μ,
dt t=0 where D is the derivative of YM. So ∇ is a critical point of YM if we have, for all linear maps μ : g → Es ,
d
YM(∇ + tμ) = 0. dt t=0 Thus, as given in [13], for given Lie algebra g, ∇ is a critical point of YM if for all Zi ∈ g, i ∇Zi , Θ∇ (Zi ∧ Zj ) − cj k Θ∇ (Zj ∧ Zk ) = 0, j
(11)
j
where cji k are structure constants of g. Recall that our Lie algebra g is the Heisenberg Lie algebra with three basis elements {X, Y, Z} satisfying [X, Y ] = cZ for a positive integer c. This together with (11) gives the following. The connection ∇ will be a critical point if ∇Y , Θ∇ (X, Y ) + ∇Z , Θ∇ (X, Z) = 0, ∇X , Θ∇ (Y, X) + ∇Z , Θ∇ (Y, Z) = 0, ∇X , Θ∇ (Z, X) + ∇Y , Θ∇ (Z, Y ) − c · Θ∇ (X, Y ) = 0.
(12) (13) (14)
It is now easy to see that the Grassmannian connection ∇ 0 given in (9) is not a critical point of YM. But we know that any other compatible connection can be obtained from the Grassmanc, )s , the set of skew-symmetric nian connection ∇ 0 by adding a linear map μ from g into (Eμν c, element of Eμν . It is not hard to find a concrete example of a smooth skew-symmetric element c, , but it is difficult to compute the left-hand sides of (12), (13) and (14) in general. So we of Eμν will try the simplest form of such a linear map whose range is skew-symmetric. In particular, we c, )s whose range lies in the set of multiplication-type element will use a linear map μ : g → (Eμν c, of Eμν introduced in the previous section. For given Grassmannian connection ∇ 0 with the curvature Θ∇0 as before, let ∇ = ∇ 0 + μ, c, )∞ for X ∈ g. Then the corresponding curvature Θ of ∇ is the where μ∗X = −μX ∈ (Eμν ∇ following. For X, Y, Z ∈ g with [X, Y ] = cZ,
S. Kang / Journal of Functional Analysis 258 (2010) 307–327
0 Θ∇ (X, Y ) = Θ∇0 (X, Y ) + ∇X , μY − ∇Y0 , μX + [μX , μY ] − μ[X,Y ] , 0 , μZ − ∇Z0 , μX + [μX , μZ ], Θ∇ (X, Z) = Θ∇0 (X, Z) + ∇X Θ∇ (Y, Z) = Θ∇0 (Y, Z) + ∇Y0 , μZ − ∇Z0 , μY + [μY , μZ ].
325
(15) (16) (17)
Now consider the case where μX is a multiplication-type, skew-symmetric element. Let c, for X ∈ g with the corresponding G ∈ μX = GX , μY = GY and μZ = GZ for GX ∈ Eμν i C ∞ (T2 ) such that Gi (x, y) = −Gi (x, y) for i = 1, 2, 3. Using the formulas in the proof of Proposition 10, we can write the curvature Θ∇ in terms of the Grassmannian curvature Θ∇0 and the multiplication-type elements GX , GY and GZ . Then the proposed conditions for obtaining critical points, (12), (13) and (14) give the following two equations. c · G3 (x, y) = −
∂ ∂ G2 (x, y) + G1 (x, y) + f1 (x) + c1 · i ∂y ∂x
for some real number c1 ,
∂2 ∂2 ∂ f2 (x) + c · c1 · i. G (x, y) + G3 (x, y) = 3 ∂x ∂y 2 ∂x 2
(18) (19)
∂ Let w(x) = ∂x f2 (x) + c · c1 · i. Then it is obvious that w(x) = −w(x) since f2 is skewsymmetric. To solve the elliptic equation (19), take the Fourier transform on both sides in (19). Then
3 (n, m) = w −m2 − n2 G (n),
3 (n, m) = − so G
w (n) , (m2 + n2 )
for (m, n) = (0, 0). Therefore G3 (x, y) =
3 (n, m)e(nx)e(my), G
n,m∈Z, (n,m)=(0,0)
= where e(x) = exp(2πiμx) and e(y) = exp(2πiνy). Notice that G3 is periodic and w(n) ∞ 2 w (−n), so G3 is a skew-symmetric element of C (T ), i.e. G3 (x, y) = −G3 (x, y). With the solution G3 as above, we can show that Eq. (18) allows to choose nontrivial G1 and G2 . We finally state the main theorem. Theorem 11. Let g be the Heisenberg Lie algebra with the basis X, Y, Z satisfying [X, Y ] = cZ c, -D c, bimodule given as before. Let ∇ 0 be the for a positive integer c. Let Ξ be the Eμν μν ∞ Grassmannian connection on Ξ produced by a special function R with the Grassmannian curvature Θ∇0 such that Θ∇0 (X, Y ) = f1 , Θ∇0 (X, Z) = 0 and Θ∇0 (Y, Z) = f2 , where f1 (x, y, p) = f1 (x)δ0 (p), f2 (x, y, p) = f2 (x)δ0 (p) for smooth periodic functions f1 and f2 . Let G be a linear map on g whose range lies in a set of the multiplication-type, skew-symmetric elements of c, . Then ∇ = ∇ 0 + G is a critical point of the Yang–Mills functional if and EndD c, (Ξ ) = Eμν μν only if the corresponding skew-symmetric periodic functions G1 , G2 and G3 satisfy the following equations.
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∂ ∂ G1 (x, y) − G2 (x, y) = c · G3 (x, y) − f1 (x), ∂x ∂y
(20)
∂2 ∂2 ∂ G (x, y) + G3 (x, y) = f2 (x) + c · a0 , 3 ∂x ∂y 2 ∂x 2
(21)
where GX (x, y, p) = G1 (x, y)δ0 (p), GY (x, y, p) = G2 (x, y)δ0 (p), G3 (x, y)δ0 (p), and f1 (x) = f1 (x) − a0 and a0 = T f1 (x) dx. Proof. The proof follows from the previous argument.
GZ (x, y, p) =
2
Remark 4. We note that these critical points are not minima, but inflection points. 6. Laplace’s equation on quantum Heisenberg manifolds In [15], J. Rosenberg described the Euler–Lagrangian equation for critical points of the energy functional on self adjoint elements a of the noncommutative torus as Laplace’s equation a = 0. Inspired by the works in [15] and [8], we show that those two equations (20) and (21) in the main theorem of this paper can be expressed in terms of this Laplacian and derivations on quantum Heisenberg manifolds, by using a result of Morita equivalence for quantum Heisenberg manifolds. First we give the explicit formula of the Laplacian first defined by N. Weaver in [16] c, )∞ , as follows. For Φ ∈ (Dμν (Φ)(x, y, p) = (δX )2 (Φ)(x, y, p) + (δY )2 (Φ)(x, y, p) = −4πip 2 (x − pμ)2 Φ(x, y, p) − 2πip(x − pμ) − 2πip(x − pμ)
∂Φ (x, y, p) ∂y
∂ 2Φ ∂Φ ∂ 2Φ (x, y, p) + (x, y, p) + (x, y, p). ∂y ∂y 2 ∂x 2
When p = 0, then the above Laplacian becomes the following: (Φ)(x, y, 0) =
∂ 2Φ ∂ 2Φ (x, y, 0) + (x, y, 0), ∂y 2 ∂x 2
c, )∞ . Thus, if we restrict the Laplacian on a smooth element H of D c, that is only for Φ ∈ (Dμν μν supported at p = 0, i.e. H(x, y, p) = H (x, y)δ0 (p), H ∈ C ∞ (T2 ), then we have
(H)(x, y, p) =
∂ 2H ∂ 2H (x, y, p) + (x, y, p). ∂y 2 ∂x 2
c, is isomorphic to a quanAs shown in Section 2, the generalized fixed point C ∗ -algebra Eμν
tum Heisenberg manifold with different parameters, in particular D c, 1 ν , so we can treat an 4μ 2μ
c, -valued multiplication operator G as a D c, -valued multiplication operator in the main Eμν 1 ν 4μ 2μ
c, )∞ -valued 2-form now can be theorem. Also, the Grassmannian curvature Θ∇0 that is an (Eμν
∞ considered as a (D c, 1 ν ) -valued 2-form. Thus we obtain the following corollary. 4μ 2μ
S. Kang / Journal of Functional Analysis 258 (2010) 307–327
327
Corollary 12. Let g and Ξ be as before. Let δ be the infinitesimal form of the Heisenberg group 2 + δ 2 for X, Y ∈ g. Let ∇ 0 be the Grassmannian connecaction given as before and = δX Y tion on Ξ ∞ produced by the special function R given in (9). Let Θ∇0 be the corresponding Grassmannian curvature such that Θ∇0 (X, Y ) and Θ∇0 (Y, Z) are nontrivial multiplication-type c, , and Θ 0 (X, Z) = 0. Let G be a linear map on g whose range lies in a set elements of Eμν ∇ c, . Then, ∇ = ∇ 0 + G of the multiplication-type skew-symmetric elements of EndD c, (Ξ ) = Eμν μν is a critical point of Yang–Mills functional if and only if GX , GY and GZ satisfy the following equations. δX (GY ) − δY (GX ) = c · GZ − Θ∇0 (X, Y ) + a0 , (GZ ) = −δY Θ∇0 (Y, Z) + c · a0 , where a0 = tion f1 (x).
T f1 (x) dx,
and Θ∇0 (X, Y )(x, y, p) = f1 (x)δ0 (p) for a smooth periodic func-
Acknowledgments I would like to take this opportunity to thank my thesis advisor, Judith Packer, for her constant patience and encouragement, as well as a number of helpful suggestions and comments. References [1] B. Abadie, “Vector Bundles” over quantum Heisenberg manifolds, in: Algebraic Methods in Operator Theory, Birkhäuser, Boston, 1994, pp. 307–315. [2] B. Abadie, Generalized fixed-point algebras of certain actions on crossed products, Pacific J. Math. 171 (1) (1995) 1–21. [3] B. Abadie, Morita equivalence for quantum Heisenberg manifolds, Proc. Amer. Math. Soc. 133 (12) (2005) 3515– 3523. [4] A. Connes, C ∗ -algebres et geometrie differentielle, C. R. Acad. Sci. Paris Ser. A–B 290 (13) (1980) A559–A604. [5] A. Connes, Noncommutative Geometry, Academic Press, 1994. [6] A. Connes, M. Rieffel, Yang–Mills for non-commutative two-tori, in: Contemp. Math., vol. 62, 1987, pp. 237–266. [7] R. Exel, B. Abadie, S. Eilers, Morita equivalence for crossed products by Hilbert C ∗ -bimodules, Trans. Amer. Math. Soc. 350 (8) (1998) 3043–3054. [8] A. Konechny, A. Schwarz, Introduction to M(atrix) theory and noncommutative geometry, Phys. Rep. 360 (5–6) (2002) 353–465. [9] J.L. Koszul, Lectures on Fibre Bundles and Differential Geometry, Notes by S. Ramanan, Tata Inst. Fund. Res. Lect. Math., vol. 20, 1965. [10] J. Packer, M. Rieffel, Projective multi-resolution analyses for L2 (R2 ), J. Fourier Anal. Appl. 10 (5) (2004) 439–464. [11] M. Rieffel, C ∗ -algebras associated with irrational rotations, Pacific J. Math. 93 (2) (1981) 415–429. [12] M. Rieffel, Deformation quantization of Heisenberg manifolds, Comm. Math. Phys. 122 (4) (1989) 521–562. [13] M. Rieffel, Critical points of Yang–Mills for non-commutative two-tori, J. Differential Geom. 31 (2) (1990) 535– 546. [14] M. Rieffel, Proper actions of groups on C ∗ -algebras, in: Mappings of Operator Algebras, in: Progr. Math., vol. 84, 1990, pp. 141–182. [15] J. Rosenberg, Noncommutative variations on Laplace’s equation, Anal. PDE 1 (1) (2008) 95–114. [16] N. Weaver, Sub-Riemannian metrics for quantum Heisenberg manifolds, J. Operator Theory 43 (2) (2000) 223–242.
Journal of Functional Analysis 258 (2010) 328–356 www.elsevier.com/locate/jfa
Positivity of Riesz functionals and solutions of quadratic and quartic moment problems Lawrence Fialkow a,1 , Jiawang Nie b,∗,2 a Department of Computer Science, State University of New York, New Paltz, NY 12561, United States b Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093,
United States Received 31 March 2009; accepted 16 September 2009 Available online 8 October 2009 Communicated by D. Voiculescu
Abstract We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence y, we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set K ⊆ Rn if and only if the associated Riesz functional Ly is K-positive. For a determining set K, we prove that if Ly is strictly K-positive, then y admits a representing measure supported in K. As a consequence, we are able to solve the truncated K-moment problem of degree k in the cases: (i) (n, k) = (2, 4) and K = R2 ; (ii) n 1, k = 2, and K is defined by one quadratic equality or inequality. In particular, these results solve the truncated moment problem in the remaining open cases of Hilbert’s theorem on sums of squares. © 2009 Elsevier Inc. All rights reserved. Keywords: Truncated moment sequence; Riesz functional; (Strict) K-positivity; Determining set; Moment matrix; Representing measure
* Corresponding author.
E-mail addresses:
[email protected] (L. Fialkow),
[email protected] (J. Nie). 1 Partially supported by NSF grant DMS-0758378. 2 Partially supported by NSF grants DMS-0757212, DMS-0844775 and Hellman Foundation Fellowship.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.015
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329
1. Introduction Denote by Z+ the set of nonnegative integers and let |α| = α1 + · · · + αn for α ≡ (α1 , . . . , αn ) ∈ Zn+ . Let y = (yα )α∈Zn+ , |α|k be a real multisequence of degree k in n variables (also referred to as a truncated moment sequence), and let K ⊆ Rn be a closed set. The truncated K-moment problem of degree k concerns conditions on y such that it has a K-representing measure, i.e., a positive Borel measure μ on Rn , supported in K, such that yα =
x α dμ(x),
∀α ∈ Zn+ : |α| k.
(1.1)
Rn
(Here, x α = x1α1 · · · xnαn for x ≡ (x1 , . . . , xn ) ∈ Rn .) For K = Rn , we refer to (1.1) simply as the truncated moment problem and to μ as a representing measure. Let Pk ⊂ R[x1 , . . . , xn ] denote the polynomials of degree at most k. Corresponding to the sequence y of degree k is the Riesz functional Ly : Pk → R defined by
Ly (p) =
α∈Zn+ :
pα y α ,
|α|k
∀p ≡
α∈Zn+ :
pα x α ∈ P k .
|α|k
Ly is said to be K-positive if Ly (p) 0,
∀p ∈ Pk , p|K 0.
Further, Ly is strictly K-positive if Ly is K-positive and Ly (p) > 0,
∀p ∈ Pk , p|K 0, p|K ≡ 0.
For K = Rn we say simply that Ly is positive or strictly positive. K-positivity is a necessary condition for K-representing measures, for if μ is a K-representing measure and p ∈ Pk with p|K 0, then Ly (p) = K p dμ 0. The proof of Tchakaloff’s theorem [21] shows that if K is compact, then K-positivity is actually sufficient for K-representing measures, but this is not so in general (see below). Nevertheless, in [10] R.E. Curto and the first-named author obtained the following solution to the truncated K-moment problem expressed in terms of K-positivity. Theorem 1.1. (See [10, Theorem 1.2].) A multisequence y of degree 2d or 2d + 1 admits a Krepresenting measure if and only if y can be extended to a sequence y˜ of degree 2d + 2 such that Ly˜ is K-positive. A significant issue associated with Theorem 1.1 is that in general it is quite difficult to establish that Ly or Ly˜ is K-positive. We show in Section 2 (Theorem 2.2) that Ly is K-positive if and only if limm→∞ y − y (m) = 0 for a sequence {y (m) } in which each truncated moment sequence y (m) has a K-representing measure μ(m) . In this case, for each α, we have yα = limm→∞ K x α dμ(m) (x), and we say that {μ(m) } is a sequence of approximate representing measures for y. This leads us to identify some cases of interest, including certain multivariate quadratic and quartic moment problems, in which we can utilize such approximating sequences to establish K-representing measures for y or K-positivity for Ly . To explain our results further,
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consider K = Rn . For k = 2d, the moment sequence y is associated with the dth order moment matrix Md (y) defined by Md (y) = (yα+β )(α,β)∈Zn+ ×Zn+ : |α|,|β|d . (We sometimes refer to a representing measure for y as a representing measure for Md (y).) A basic necessary condition for positivity of Ly (and hence for the existence of a representing measure) is that My be positive semidefinite (Md (y) 0). To see this, observe that Md (y) is uniquely determined by the relation
Md (y)p, ˆ qˆ = Ly (pq),
p, q ∈ Pd ,
(1.2)
where rˆ denotes the coefficient vector of r ∈ Pd relative to the basis for Pd consisting of the monomials in degree-lexicographic order. Thus, if Ly is positive, then Md (y)p, ˆ p ˆ = Ly (p 2 ) 0. It is known that if Ly is positive and Md (y) is singular, then y need not have a representing measure; the simplest such example occurs with n = 1, d = 2 and M2 (y) of the form
a M2 (y) = a a
a a a
a a , b
with b > a > 0 (cf. [10, Example 2.1]). Nevertheless, the following question, essentially asked in [10, Question 2.9], remains unsolved. Question 1.2. Let k = 2d. If Ly is K-positive and Md (y) is positive definite, does y have a K-representing measure; equivalently, does Ly admit a K-positive extension Ly˜ : P2d+2 → R? In the sequel we say that K is a determining set (of degree k) if whenever p ∈ Pk and p|K ≡ 0, then p ≡ 0 (i.e., p(x) = 0, ∀x ∈ Rn ); sets K with nonempty interior are clearly determining. It follows readily from (1.2) that if K is a determining set and Ly is strictly K-positive, then Md (y) 0. Our main tool in establishing K-representing measures is the following result, which complements Theorem 1.1 and partially answers Question 1.2. Theorem 1.3. Suppose K is a determining set of degree k and let y be a truncated moment sequence of degree k in n variables. If Ly is strictly K-positive, then y admits a K-representing measure. To discuss concrete applications of Theorem 1.3, we consider the following property: (Hn,d ) Each p ∈ P2d admits a sum-of-squares decomposition, p = als pi ∈ Pd (which depend on p).
pi2 , for certain polynomi-
If (Hn,d ) holds and we set k = 2d, then positivity for Ly is equivalent to positivity of Md2(y); n , then L (p) = indeed, in this case, if M (y) 0 and p ∈ P is nonnegative on R Ly (pi ) = d 2d y Md (y)pˆi , pˆi 0. A well-known theorem of Hilbert (cf. [17,18]) shows that (Hn,d ) holds if and only if n = 1, n = d = 2, or n > 1 and d = 1. In these cases, whether or not y has
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a representing measure, Theorem 2.2 (cf. Section 2) implies that if Md (y) 0, then y has a sequence of approximate representing measures. For n = 1, the truncated moment problem has been solved (cf. [4]): a multisequence y of degree 2d has a representing measure if and only if Md (y) is positive semidefinite and recursively generated (see below for terminology concerning moment matrices). In the sequel we address the truncated moment problem in the other cases covered by Hilbert’s theorem. Consider first the bivariate quartic moment problem (n = d = 2). For the case when M2 (y) is singular, concrete necessary and sufficient conditions for representing measures are known (cf. [7,9]): y has a representing measure if and only if M2 (y) 0,
M2 (y) is recursively generated, and rank M2 (y) card V M2 (y) ,
(1.3)
where V(M2 (y)) is the algebraic variety associated to M2 (y) (see definition (1.5)) and card denotes the cardinality of a set. When 2 is replaced by d, the conditions of (1.3) apply more generally to any bivariate sequence y of degree 2d for which M2 (y) is singular, i.e., the first 6 columns of M2 (y) are dependent (cf. [9, Theorem 1.2]). Subsequent to [7], the case M2 (y) 0 has been open (cf. [15]). In this case, it is easy to find a moment matrix extension M3 (y) ˜ 0, but an example of [5] shows that for such y, ˜ Ly˜ need not be positive, so Theorem 1.1 cannot be applied to yield a representing measure for y. Instead, in Section 3 we will use Theorem 1.3, together with Hilbert’s theorem, to establish that such y does indeed have a representing measure. This provides a positive answer to Question 1.2 for n = d = 2, with K = R2 . Consider next the case of the multivariate quadratic moment problem, where n 1 and d = 1. For n = 1, 2, it was shown in [4] that if M1 (y) 0, then y has a rank M1 (y)-atomic representing measure, and in Section 4, Theorem 4.5, we prove the same result for n 1. In the sequel, let Rn,k (K) denote the convex set of n-variable moment sequences of degree k which admit Krepresenting measures, and let Rn,k (K) denote the closure of Rn,k (K) in Rη , where η = dim Pk . Now let q be a quadratic polynomial, and define the quadratic variety E(q) = {x ∈ Rn : q(x) = 0} and the quadratic semialgebraic set S(q) := {x ∈ Rn : q(x) 0}. We are interested in determining whether y has a representing measure supported in E(q) or in S(q). It is obvious that if y has a representing measure supported in E(q) (resp., S(q)), then M1 (y) 0,
Ly (q) = 0
resp., Ly (q) 0 .
(1.4)
For the case when S(q) is compact, we will show in Theorem 4.7 that if y satisfies (1.4), then y ∈ Rn,2 (E(q)) (resp., y ∈ Rn,2 (S(q))). For the general case, we show in Theorem 4.8 that if (1.4) holds, then y ∈ Rn,2 (E(q)) (resp., y ∈ Rn,2 (S(q))). In Theorem 4.10, we further show that if M1 (y) 0 and Ly (q) = 0 (resp., Ly (q) > 0), then y ∈ Rn,2 (E(q)) (resp., y ∈ Rn,2 (S(q))); this result implies an affirmative answer to Question 1.2 for d = 1 and K = E(q) (resp., K = S(q)). The preceding concrete results all concern the positive cases of Hilbert’s theorem. In some cases where sums-of-squares are not available, it is still possible to use a sequence of approximate representing measures to establish positivity of a functional Ly : P2d → R. In Example 2.5, for n = 2, d = 3, k = 6, we will use this approach to illustrate a multisequence y of degree 6 such that Ly is positive (whence M3 (y) 0), but y has no representing measure. We believe this is the first such example in a case where the positivity of Ly cannot be established by sums-of-squares, via positivity of Md (y).
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We recall some additional terminology and results from [4,8] concerning moment matrices and representing measures. Let [x]k denote the column vector of all n-variable monomials up to degree k in degree-lexicographic order, that is, [x]Tk = 1 x1
...
x12
xn
x1 x2
...
xnk .
Throughout this paper, the superscript T denotes the transpose of a matrix or vector. Note that if k = 2d and μ is a representing measure for y, then Md (y) =
[x]d [x]Td dμ(x), R2
which shows again that Md (y) 0 is a necessary condition for representing measures. Moreover, in this case, card supp μ rank Md (y) [4] (where supp μ denotes the closed support of μ). We denote the successive columns of Md (y) by 1, X1 , . . . , Xn , X12 , X1 X2 , . . . , Xn2 , . . . , Xnd , . . . , Xnd . For p =
α∈Zn+ : |α|d
pα x α ∈ Pd , we define an element p(X) of the column space of Md (y) by
p(X) =
α∈Zn+ :
pα X α .
|α|d
Md (y) is recursively generated if, whenever p ∈ Pd and p(X) = 0, then (pq)(X) = 0 for q ∈ Pd with deg pq d; recursiveness is a necessary condition for representing measures [4]. The algebraic variety associated to Md (y) is defined by
V Md (y) :=
x ∈ Rn : p(x) = 0 ;
(1.5)
p∈Pd , p(X)=0
if y has a representing measure μ, then supp μ ⊆ V(Md (y)) [4], whence
rank Md (y) card V Md (y) .
(1.6)
p Recall that a measure ν is p-atomic if it is of the form ν = i=1 λi δui , where λi > 0 and δui is the unit-mass measure supported at ui ∈ Rn . For k = 2d, a fundamental result of [4,8] shows that y admits a rank Md (y)-atomic representing measure if and only Md (y) is positive semidefinite ˜ in this case y˜ and Md (y) admits a flat (i.e., rank-preserving) moment matrix extension Md+1 (y); has a unique (and computable) representing measure, which is rank Md (y)-atomic, with support ˜ More generally, y admits a finitely atomic representing measure if and precisely V(Md+1 (y)). ˜ (for some m 0), which in turn admits a only if Md (y) admits a positive extension Md+m (y) flat extension Md+m+1 [8]. A remarkable result of Bayer and Teichmann [1] implies that a multisequence y of degree k admits a K-representing measure if and only if y admits a finitely atomic K-representing measure μ (with card supp μ dim Pk ), so the preceding moment matrix criterion provides a complete characterization of the existence of representing measures when k = 2d.
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This characterization is more concrete than the criterion of Theorem 1.1, because it provides algebraic coordinates for constructing representing measures, although precise conditions for flat extensions are presently known only in special cases. For the case when K is a closed semialgebraic set, analogues of the preceding results appear in [8]. The papers [7,9,13] describe various concrete existence theorems for representing measures based on flat extensions. These results usually assume that Md (y) is positive semidefinite and singular, so that any representing measure is necessarily supported in the nontrivial algebraic variety V(Md (y)). By contrast, for the case when Md (y) is positive definite, very few results are known concerning the existence of representing measures. Our solutions to the positive definite cases of the bivariate quartic moment problem and the multivariate quadratic moment problem provide two such results. A notable feature of the proofs of these results is that they do not rely on flat extension techniques. For this reason, the results which depend on Theorem 1.3 (or Lemma 2.1) are purely existential and do not provide a procedure for explicitly computing representing measures (cf. Question 3.5 below). This paper is organized as follows. Section 2 contains an analysis of positivity of Riesz functionals, leading to a proof of Theorem 1.3. Section 3 shows that every bivariate quartic moment sequence y with M2 (y) 0 admits a representing measure supported in R2 . Section 4 gives a complete solution of quadratic K-moment problems when K = Rn , or when K ≡ S(q) or K ≡ E(q) is defined by a quadratic multivariate polynomial q(x). 2. Positivity, approximation, and representing measures In this section we will prove Theorem 1.3. Let Mn,k = y ≡ (yα )α∈Zn+ : |α|k , the set of n-variable multisequences of degree k, and let α Rn,k (K) = y ∈ Mn,k : yα = x dμ(x), μ 0, supp(μ) ⊆ K , K
the multisequences with K-representing measures. When K = Rn , we simply write Rn,k (Rn ) = Rn,k . Note that Rn,k (K) is a convex cone in Mn,k (K) and that Mn,k can be identified with the
η affine space Rη , where η ≡ dim Pk = n+k k . R is equipped with the usual Euclidean norm · , although we sometimes employ · 1 as well. Note also that for x ∈ K, the truncated moment sequence y ≡ [x]k is an element of Rn,k (K), since δx is a K-representing measure. The truncated moment sequence y is said to be in the interior of Rn,k if there exists > 0 such that for any truncated moment sequence y ∗ having the same degree as y, y ∗ ∈ Rn,k whenever y ∗ − y < . Equivalently, the interior of Rn,k is defined in the standard way for a subset of the space Rη . Let us begin with a well-known fact about the interior and closure of convex sets. Lemma 2.1. If C ⊂ RN is a convex set, then int(C) = int(C). The above lemma is a consequence of Theorem 25.20(iii) of Berberian [2], which actually applies to convex sets in general topological vector spaces.
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In the sequel, let Fn,k (K) denote the moment sequences y ∈ Mn,k having finitely atomic K-representing measures. Fn,k (K) is clearly a convex subset of Rn,k (K), and the Bayer– Teichmann theorem [1, Theorem 2], [14, Theorem 5.8] shows that Fn,k (K) = Rn,k (K). The following result, which is implicit in the proof of [10, Theorem 2.4], is the basis for our approximation approach to K-positivity for Riesz functionals. Theorem 2.2. For y ∈ Mn,k , the following are equivalent: (i) Ly is K-positive. (ii) y ∈ Fn,k (K). (iii) y ∈ Rn,k (K). measure μ, then Ly is Proof. We begin with (iii) ⇒ (i). If y ∈ Rn,k (K), with K-representing K-positive; indeed, if p ∈ Pk and p|K 0, then Ly (p) = K p dμ 0. Since the K-positive linear functionals form a closed positive cone in the dual space Pk∗ (equipped with the usual norm topology), it follows that if y ∈ Rn,k (K), then Ly is K-positive. Since (ii) ⇒ (iii) is clear, it suffices to show (i) ⇒ (ii), which we prove by contradiction. / Fn,k (K). Since Fn,k (K) is a closed convex cone in Rη , it Suppose Ly is K-positive, but y ∈ follows from the Minkowski separation theorem [2, (34.2)] that there exists a nonzero vector p ∈ Rη such that p T y < 0,
and p T w 0,
∀w ∈ Fn,k (K).
Now define the nonzero polynomial p˜ in Pk by p(x) ˜ = p T [x]k . Since, for each x ∈ K, the monomial vector [x]k is an element of Fn,k (with K-representing ˜ is nonnegative on K. However, we have measure δx ), then p(x) Ly (p) ˜ = p T y < 0, which contradicts the K-positivity of Ly . Therefore, we must have y ∈ Fn,k (K).
2
Lemma 2.3. Let K be a determining set of degree k and let y ∈ Mn,k . If the Riesz functional Ly is strictly K-positive, then there exists > 0 such that Ly˜ is also strictly K-positive whenever y˜ − y 1 < . Proof. We equip Pk with the norm p = max |pα | α
p≡
pα x α ∈ P k .
α∈Zn+ : |α|k
A sequence {p (i) } in Pk that is norm-convergent to p ∈ Pk is also pointwise convergent, so if p (i) |K 0 for each i, then p|K 0. It follows that the set
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T := p ∈ Pk : p|K 0, p = 1 is compact. Note that since K is a determining set, if p ∈ T , then p|K ≡ 0. Thus, Ly˜ is strictly K-positive if and only if Ly˜ (p) > 0 for every p ∈ T . Since T is compact and Ly : Pk → R is a norm-continuous functional on T , there exists > 0 such that Ly (p) 2,
∀p ∈ T .
For any p ∈ T , we have Ly (p) − Ly˜ (p) y − y ˜ 1. So, if y − y ˜ 1 < , then Ly˜ (p) Ly (p) − y − y ˜ 1 > 0, whence Ly˜ is strictly positive. Thus, the lemma is proved.
∀p ∈ T ,
2
We now prove Theorem 1.3, which we can restate as follows for convenience. Theorem 2.4. Let K be a determining set of degree k. If y ∈ Mn,k and Ly is strictly K-positive, then y ∈ Rn,k (K). Proof. By Theorem 2.2, we have y ∈ Rn,k (K). Lemma 2.3 implies that y lies in the interior of Rn,k (K). Lemma 2.1 tells us that Rn,k (K) and Rn,k (K) have the same interior. Therefore we must have y ∈ int(Rn,k (K)) ⊂ Rn,k (K). 2 Although we believe that the hypothesis that K is a determining set cannot be omitted from Theorem 2.4, at present we do not have an example illustrating this. We next present an example which shows how a sequence of approximate representing measures can be used to establish positivity of a functional Ly : P2d → R in a case where y has no representing measure and the positivity of Ly cannot be derived from the positivity of Md (y) via sums-of-squares arguments. Let n = 2 and consider the bivariate moment matrix Md (y). Denote the rows and columns by 1, X1 , X2 , X12 , X1 X2 , X22 , . . . , X1d , X1d−1 X2 , . . . , X1 X2d−1 , X2d ; j
then yij is precisely the entry in row X1i , column X2 , the moment corresponding to the monomial j x1i x2 . Example 2.5. Let n = 2 and d = 3. We consider the general form of a moment matrix M3 (y) with a column relation X2 = X13 (normalized with y00 = 1):
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⎡
1 ⎢a ⎢ ⎢b ⎢ ⎢c ⎢ ⎢e M ≡ M3 (y) = ⎢ ⎢d ⎢ ⎢b ⎢ ⎢f ⎣ g x
a c e b f g e d h j
b e d f g x d h j k
c b f e d h f g x u
e f g d h j g x u v
d g x h j k x u v w
b e d f g x d h j k
f d h g x u h j k r
g h j x u v j k r s
⎤ x j⎥ ⎥ k⎥ ⎥ u⎥ ⎥ v⎥ ⎥. w⎥ ⎥ k⎥ ⎥ r⎥ ⎦ s t
(2.1)
For suitable values of the moment data, M satisfies the following properties: M 0,
X2 = X13 ,
rank M = 9;
(2.2)
this is the case, for example, with a = b = f = g = u = v = w = 0, c = 1, e = 2, d = 5, h = 14, j = 42, k = 132, r = 429, s = 1442, t = 4798, x = 0.
(2.3)
In [13] we solved the truncated K-moment problem for K = {(x1 , x2 ) ∈ R2 : x2 = x13 }. In particular, [13] provides a numerical test, that we next describe, for the existence of K-representing measures whenever M as in (2.1) satisfies (2.2). From [1] we know that if M admits a representing measure, then M admits a finitely atomic measure, and thus M admits a positive, recursively generated extension M4 (y). ˜ In any such extension, the moments must be consistent with the relation x2 = x13 , so in particular, we must have y44 = y15 (≡ s). To insure positivity of M4 (y), ˜ we require a lower bound for the diagonal element y44 , which we may derive as in [13]. Let J denote the compression of M obtained by deleting row X13 and column X13 ; thus, J 0. Let us write
N J= UT
U ,
where N is the compression of J to its first 8 rows and columns, U is a column vector, and ≡ y06 (≡ t) > 0. Consider the corresponding block decomposition of J −1 , which is of the form J
−1
P = VT
V
,
where P 0 and > 0. In extension M4 (y), ˜ we have X14 = X1 X2 and X13 X2 = X22 , so by moment matrix structure, after deleting the element in row X13 , the first 8 remaining elements of column X12 X22 must be W ≡ ( h, x, u, j, k, r, v, w )T . Let ω = P W, W and define ψ(y) :=
ω − V , W 2 .
(2.4)
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In [13] we showed that in M4 (y) ˜ we must have y44 ψ(y), and [13, Theorem 2.4] implies that M has a representing measure if and only if y15 ≡ s > ψ(y). A calculation shows that for M as in (2.1) and satisfying (2.2), with appropriate values of the moment data we can also have ψ(y) independent of s and t. This is the case, for exam1 ple, if we modify (2.3) so that x = 10 , r = 600, s is arbitrary and t is chosen sufficiently large so as to preserve positivity and the property rank M3 (y) = 9. More generally, this is the case if we modify (2.3) so that x, k, u, v, w, r, s, t are chosen, successively, to maintain positivity and the rank M = 9 property. (We conjecture that whenever M3 (y) satisfies (2.2), then ψ(y) is independent of s and t.) For any such M, with ψ(y) independent of s and t, we now specify s ≡ y1,5 = ψ(y) and we adjust t (if necessary) so that M continues to be posi1 tive with rank M = 9. (For a specific example, we may modify (2.3) so that x = 10 , r = 600, 526 337 068 574 699 ≈ 709 722, and t 11 319 100 143 (cf. [13, Example 3.2]).) s ≡ ψ(y) = 741 609 900 We claim that Ly is K-positive for K = {(x1 , x2 ) ∈ R2 : x2 = x13 }, and thus positive. Since y1,5 = ψ(y), positivity for Ly cannot be derived from the existence of a representing measure, since [13, Theorem 2.4] shows that y has no representing measure. Moreover, positivity for Ly cannot be established from the positivity of M via sums-of-squares arguments because, by Hilbert’s theorem, there exist degree 6 bivariate polynomials that are everywhere nonnegative but are not sums of squares. To prove that Ly is K-positive, we employ a sequence of approximate representing measures. Since J 0, t ≡ > U T N −1 U . Thus, there exists δ > 0 such that if we replace s(= ψ(y)) by s + m1 (with m1 < δ), then the resulting moment matrix, M3 (y (m) ), remains positive, with rank M3 (y (m) ) = 9 and X2 = X13 . Since ψ(y (m) ) is independent of y15 [y (m) ] and y06 [y (m) ], we have ψ(y (m) ) = ψ(y) = s < s + m1 = y15 [y (m) ]. It now follows from [13, Theorem 2.4] that y (m) has a K-representing measure, whence Ly (m) is K-positive. Since y (m) − y = m1 → 0, we conclude that Ly is K-positive, and thus positive. Remark 2.6. We have previously noted an example of [5, Section 4] (based on a construction of Schmüdgen [19]) which illustrates a case where, with n = 2, M3 (y) 0 but Ly is not positive. Example 2.5 shows that if M3 (y) 0 and Ly is positive, y need not have a representing measure. Whether this can happen with M3 (y) 0 is the content of Question 1.2. Now we introduce a variety associated to Ly that provides a finer tool than V(Md (y)) for studying issues related to Question 1.2. For a moment sequence y of degree 2d, we define the variety of Ly by V (Ly ) :=
Z(p).
p∈P2d , p|V (Md (y))0, Ly (p)=0
Proposition 2.7. If y has a representing measure μ, then supp μ ⊆ V (Ly ). Proof. Suppose there exists u ∈ supp μ such that u ∈ / V (Ly ). Then there exists some p ∈ P2d , such that p|V(Md (y)) 0 and Ly (p) = 0, but p(u) = 0. Since supp μ ⊆ V(Md (y)), we have p|supp μ 0, and hence p(u) > 0. Thus, it follows that Ly (p) = supp μ p(t) dμ(t) > 0, which contradicts Ly (p) = 0. 2 Proposition 2.8. For each truncated moment sequence y, V (Ly ) ⊆ V(Md (y)).
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Proof. Let p be an arbitrary polynomial such that p ∈ Pd and p(X) = 0 in the column space ˆ p ˆ = 0. Since p 2 |V(Md (y)) 0, it follows that V (Ly ) ⊆ of Md (y). Then Ly (p 2 ) = Md (y)p, 2 Z(p ) = Z(p). By definition of V(Md (y)) in (1.5), the result is proved. 2 In view of Proposition 2.8, the following result refines the necessary condition rank Md (y) card V(Md (y)). Corollary 2.9. If y has a representing measure, then rank Md (y) card V (Ly ). Proof. Let μ be a representing measure for y. Then rank Md (y) card supp μ (see relation (1.6) in Section 1), and the result follows from Proposition 2.7. 2 We conclude this section with an example which shows that V (Ly ) may be a proper subset of V(Md (y)) (in a case where y has a representing measure). Example 2.10. For n = 2, d = 3, consider the moment matrix ⎡
8 ⎢0 ⎢ ⎢0 ⎢ ⎢6 ⎢ ⎢0 M3 (y) := ⎢ ⎢6 ⎢ ⎢0 ⎢ ⎢0 ⎣ 0 0
0 6 0 0 0 0 6 0 4 0
0 0 6 0 0 0 0 4 0 6
6 0 0 6 0 4 0 0 0 0
0 0 0 0 4 0 0 0 0 0
6 0 0 4 0 6 0 0 0 0
0 6 0 0 0 0 6 0 4 0
0 0 4 0 0 0 0 4 0 4
0 4 0 0 0 0 4 0 4 0
⎤ 0 0⎥ ⎥ 6⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥. 0⎥ ⎥ 0⎥ ⎥ 4⎥ ⎦ 0 6
A calculation shows that M3 (y) 0, with rank M3 (y) = 8. V(M3 (y)) is determined by the column relations X1 = X13 and X2 = X23 , and thus consists of the 9 points u1 = (0, 0), u2 = (0, 1), u3 = (0, −1), u4 = (−1, 0), u5 = (−1, 1), u6 = (−1, −1), u7 = (1, 0), u8 = (1, 1), u9 = (1, −1). Observe that y has the 8-atomic representing measure μ := 9i=2 δui , and we will show that V (Ly ) = supp μ, so that V (Ly ) is a proper subset of V(M3 (y)). To see this, we consider the dehomogenized Robinson polynomial, r(x1 , x2 ) = x16 + x26 − x14 x22 − x12 x24 − x14 − x24 − x12 − x22 + 3x12 x22 + 1. It is known that r(x1 , x2 ) is nonnegative on R2 and has exactly 8 zeros in the affine plane, namely the points in supp μ (cf. [18]). A calculation shows that Ly (r) = 0, so V (Ly ) ⊆ Z(r) = supp μ ⊆ V (Ly ) (by Proposition 2.7), so V (Ly ) = supp μ and thus V (Ly ) is a proper subset of V(M3 (y)). a sum of squares (cf. [18]); to see this using variety methods, It is known that r(x1 , x2 ) is not suppose to the contrary that r = i ri2 , with each ri ∈ P3 . Then supp μ = Z(r) = i Z(ri ), whence supp μ ⊆ Z(ri ) for each i. It now follows from [4] that for each i, ri (X1 , X2 ) = 0 in the column space of M3 (y). Thus, we have V(M3 (y)) ⊆ i Z(ri ) = supp μ, a contradiction. This example also illustrates a moment sequence y with a rank Md (y)-atomic representing measure and rank Md (y) < card V(M3 (y)) < +∞; the first such example appears in [12].
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3. Solution of the bivariate quartic moment problem Throughout this section, we consider bivariate quartic moment problems, that is, n = 2 and the degree 2d = 4. Let y ∈ M2,4 be a truncated moment sequence of degree 4, which is associated with the second order moment matrix ⎡
y00 ⎢ y10 ⎢ ⎢y M2 (y) := ⎢ 01 ⎢ y20 ⎣ y11 y02
y10 y20 y11 y30 y21 y12
y01 y11 y02 y21 y12 y03
y20 y30 y21 y40 y31 y22
y11 y21 y12 y31 y22 y13
⎤ y02 y12 ⎥ ⎥ y03 ⎥ ⎥. y22 ⎥ ⎦ y13 y04
As noted in Introduction (cf. (1.3)), if M2 (y) is singular, then y has a representing measure if and only if M2 (y) is positive semidefinite, recursively generated, and rank M2 (y) card V(M2 (y)). Example 3.1. Consider ⎡
8 ⎢0 ⎢ ⎢0 M2 (y) = ⎢ ⎢4 ⎣ 0 4
⎤ 0 0 4 0 4 4 0 2 0 −2 ⎥ ⎥ 0 4 0 −2 0 ⎥ ⎥. 2 0 11 0 a ⎥ ⎦ 0 −2 0 a 0 −2 0 a 0 b
With a = 1 and b = 3, M2 (y) is positive and recursively generated, with column relations X1 = 1 − 2X22 and X2 = −2X1 X2 , and rank M2 (y) = 4. A calculation shows that x1 = 1 − 2x22 and x2 = −2x1 x2 have only 3 common zeros, so 3 = card V(M2 (y)) < rank M2 (y) = 4, whence (1.3) implies that y has no representing measure. We will show below how to approximate y with truncated moment sequences having representing measures. For the case when M2 (y) 0, it has been an open question as to whether y admits a representing measure. The aim of this section is to give an affirmative answer to this question. We begin, however, by showing that when M2 (y) is merely positive semidefinite, then y admits approximate representing measures. Theorem 3.2. If y ∈ M2,4 and M2 (y) 0, then y ∈ R2,4 . Proof. Let y ∈ M2,4 be such that M2 (y) 0. To show y ∈ R2,4 , by Theorem 2.2, it suffices to show that the Riesz functional Ly is positive. If a polynomial p(x) ∈ P4 is nonnegative on the plane R2 , then by Hilbert’s theorem it must be a sum of squares, so there exist bivariate quadratic polynomials q1 (x), . . . , qm (x), deg qi 2 (1 i m), such that p(x) = q1 (x)2 + · · · + qm (x)2 . Hence, since M2 (y) 0, we have
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2 = M2 (y)qˆ1 , qˆ1 + · · · + M2 (y)qˆm , qˆm 0, Ly (p) = Ly q12 + · · · + Ly qm so Ly is positive. It now follows from Theorem 2.2 that y ∈ R2,4 .
2
Note that if M2 (y) is positive and singular, and y does not have a representing measure, then Ly is positive, but not strictly positive. Indeed, positivity follows from Theorem 3.2. Since M2 (y) is singular, there exists p ∈ P2 , p ≡ 0, such that M2 (y)pˆ = 0; then p 2 0 and Ly (p 2 ) = M2 (y)p, ˆ p ˆ = 0, so Ly is not strictly positive. We now turn to the positive definite case. The following result provides an affirmative answer to Question 1.2 for the case n = d = 2, K = R2 . Theorem 3.3. If M2 (y) 0, then y has a representing measure. Proof. Clearly R2 is a determining set. By Theorem 2.4, it suffices to show that Ly is strictly positive. Proceeding as in the previous proof, if p ∈ P4 is nonnegative on R2 and not identically zero, then p is of the form p(x) = q1 (x)2 + · · · + qm (x)2 , with deg qi 2 (1 i m) and every 2 ) = M (y)qˆ , qˆ + · · · + qi ≡ 0. Since M2 (y) 0, we have Ly (p) = Ly (q12 ) + · · · + Ly (qm 2 1 1 M2 (y)qˆm , qˆm > 0, and the result follows. 2 Remark 3.4. Theorem 3.3 shows that if n = 2 and M2 (y) 0, then y has a representing measure, whence [1] implies that y has a representing measure μ with card supp μ dim P4 = 15. We do not have a better upper bound for the size of the support, and it remains an open problem as to whether, in this case, M2 (y) actually has a flat extension M3 (y), ˜ with a corresponding 6-atomic representing measure for y. In the case when n = 2 and M2 (y) is positive semidefinite and singular, y has a representing measure if and only if the conditions of (1.3) hold, and in this case, the results of [9] show that either M2 (d) has a flat extension M3 (y), ˜ or M2 (y) admits a positive extension M3 (y) ˜ satisfying rank M3 (y) ˜ = 1 + rank M2 (y), and M3 (y) ˜ has a flat extension. This leads to our next question (cf. [7,15]). Question 3.5. If y ∈ M2,4 and M2 (y) 0, does M2 (y) have a flat extension? Does y have an extension y˜ ∈ M2,6 such that M3 (y) ˜ is positive and has a flat extension? We next present two examples which illustrate Theorem 3.2 in cases where y has no representing measure. Example 3.6. Consider the moment sequence y ∈ M2,4 such that ⎡
1 ⎢1 ⎢ ⎢1 M2 (y) = ⎢ ⎢1 ⎣ 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 2 2 2
1 1 1 2 2 2
⎤ 1 1⎥ ⎥ 1⎥ ⎥. 2⎥ ⎦ 2 2
Clearly, M2 (y) 0. Since X1 = 1 but X12 = X1 , M2 (y) is not recursively generated, so y has no representing measure. However, by Theorem 3.2, y lies in the closure of moment sequences
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having representing measures. To see this explicitly, define the moment sequence y() via the moment matrix ⎡
1
⎢ 1 + 3/4
⎢ 1 + 3/4 M2 y() := ⎢ ⎢ 1 + 1/2 ⎣ 1 + 1/2 1 + 1/2
− − − − −
1 + 3/4 − 1 + 1/2 − 1 + 1/2 − 1 + 1/4 − 1 + 1/4 − 1 + 1/4 −
1 + 3/4 − 1 + 1/2 − 1 + 1/2 − 1 + 1/4 − 1 + 1/4 − 1 + 1/4 −
1 + 1/2 − 1 + 1/4 − 1 + 1/4 − 2− 2− 2−
1 + 1/2 − 1 + 1/4 − 1 + 1/4 − 2− 2− 2−
⎤ 1 + 1/2 − 1/4 1+ −⎥ 1 + 1/4 − ⎥ ⎥. ⎥ 2− ⎦ 2− 2−
A calculation shows that y() has the 2-atomic representing measure (1 − )δ(1,1) + δ( −1/4 , −1/4 ) , and obviously y() → y as → 0. Example 3.7. Let us return to Example 3.1. With a = 1 and b = 3, M2 (y) is positive semidefinite, so although y has no representing measure, Theorem 3.2 implies that y can be approximated by moment sequences having measures. One way to do this is to replace b = 3 by b = 3 + m1 . The resulting moment sequence y (m) satisfies M2 (y (m) ) 0 and M2 (y (m) ) is recursively generated. Further, V(M2 (y (m) )) = {(x1 , x2 ): x2 = −2x1 x2 }, and since the variety is infinite, (1.3) implies that y (m) has a representing measure. Following [9, Proposition 3.6], a calculation shows (m) ) (so y (m) has no 5-atomic representthat although M2 (y (m) ) admits no flat extension M3 (y (m) (m) ), with rank M (y (m) ) = 6, ing measure), M2 (y ) does admit a positive extension M3 (y 3 (m) ) has a flat extension M (y (m) ). Thus, y (m) has a 6-atomic representing measuch that M (y 3
4
sure. 1 Another approach is to replace a = 1 by a = 1 + m1 and b = 3 by b = 3 + 4m 2 . Then the (m) (m) (m) has M2 (y ) 0, so y has a representing measure by Theresulting moment sequence y (m) (m) (m) orem 3.3. Indeed, a Mathematica calculation shows that with y 4,1 = y 2,3 = y 1,4 = (m) (m) (m) y 0,5 = 0, M2 (y ) admits two distinct flat extensions M3 (y ) (and corresponding 6-atomic representing measures for y (m) ). We conclude this section with an application of Theorem 3.3 to a solution to the bivariate cubic moment problem, with y of the form y = {y00 , y10 , y01 , y20 , y11 , y02 , y30 , y21 , y12 , y03 }, with y00 > 0. To such a sequence we may associate M1 (y) and the block
y20 B(2) := y30 y21
y11 y21 y12
y02 y12 . y03
Theorem 3.8. Suppose y ∈ M2,3 . If y has a representing measure, then M1 (y) 0. Conversely, suppose M1 (y) 0. (i) If M1 (y) 0, then y has a representing measure.
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(ii) If rank M1 (y) = 2, then y has a representing measure if and only if Ran B(2) ⊆ Ran M1 (y) and [ M1 (y) B(2) ] is recursively generated. (iii) If rank M1 (y) = 1, then y has a representing measure if and only if Ran B(2) ⊆ Ran M1 (y). Proof. Since a representing measure for y is, in particular, a representing measure for M1 (y), the necessity of the condition M1 (y) 0 is clear. Conversely, suppose M1 (y) 0. For (i), if M1 (y) 0, then it is not difficult to see that M1 (y) admits a positive definite moment matrix extension M2 , of the form
M1 (y) M2 ≡ B(2)T
B(2) , C(2)
where
y40 C(2) = y31 y22
y31 y22 y13
y22 y13 . y04
Indeed, by choosing y40 , y22 , and y04 successively, and sufficiently large, we can insure that C(2) B(2)T M1 (y)−1 B(2). By Theorem 3.3, M2 then has a representing measure, which is obviously a representing measure for y. Suppose next that y has a representing measure. It follows from [1] that y has a finitely atomic representing measure μ, and thus M2 [μ] is a positive semidefinite and recursively generated extension of M1 (y). In particular, we must have Ran B(2) ⊆ Ran M1 (y) and [ M1 (y) B(2) ] must be recursively generated. Now suppose that these conditions hold and that rank M1 (y) = 2. Since y00 > 0, we may assume without loss of generality that there exist scalars α and β so that in the column space of M1 (y) we have a column dependence relation X2 = α1 + βX1 . Since [ M1 (y)
(3.1)
B(2) ] is recursively generated, we then have the column relations X1 X2 = αX1 + βX12 ,
(3.2)
X22 = αX2 + βX1 X2 .
(3.3)
Since Ran B(2) ⊆ Ran M1 (y), there is a matrix W such that B(2) = M1 (y)W , and we may thus define a positive, rank-preserving extension M of M1 (y) by M :=
M1 (y) B(2)T
B(2) , C
where C := B(2)T W (= W T M1 (y)W ). It is straightforward to check that the columns of M satisfy (3.1)–(3.3), from which it also follows that M has the form of a moment matrix M2 . Thus M is a flat, positive moment matrix extension of M1 (y), whence [8] implies the existence of a representing measure for M, and thus for y. The proof of (iii) is similar to the proof of (ii), but simpler. It is straightforward to check that if rank M1 (y) = 1 and Ran B(2) ⊆ Ran M1 (y), then the dependence relations in the columns of
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M1 (y) propagate recursively so as to define a rank one (flat, positive) moment matrix extension M2 (y) of M1 (y). The result follows as above. 2 4. Quadratic moment problems Let y = (yα )α∈Zn+ : |α|2 be a quadratic moment sequence such that M1 (y) 0. Does y have a representing measure? For this question, we may assume without loss of generality that y0 = 1 and we may write M1 (y) as M1 (y) =
v1T U
1 v1
,
where v1 ∈ Rn . Since M1 (y) 0, then U − v1 v1T 0, so the Spectral Theorem implies that there exist vectors v2 , . . . , vr in Rn such that U = v1 v1T + v2 v2T + · · · + vr vrT . A calculation now shows that we have 1 M1 (y) = r −1 r
i=2
For i = 2, . . . , r, let u+ i = v1 +
1 v1
1 v1
T
0 + (r − 1) vi
0 vi
T ! .
√ √ r − 1vi , u− i = v1 − r − 1vi . Then we have the representation
1 M1 (y) = 2(r − 1) r
i=2
1 u+ i
1 u+ i
T
1 + − ui
1 u− i
T ! ,
and hence we know y has a (2r − 2)-atomic representing measure μ=
r i=2
1 (δ + + δu− ). i 2(r − 1) ui
In the sequel, we will show that y actually has a rank M1 (y)-atomic representing measure (equivalently, M1 (y) admits a flat extension M2 (y)). ˜ Now we turn to the quadratic truncated moment problem on an algebraic set E(q) := {x ∈ Rn : q(x) = 0} or a semialgebraic set S(q) := {x ∈ Rn : q(x) 0}, where q(x) is a quadratic polynomial in x. If y ∈ Mn,2 has a representing measure supported in E(q), it is necessary that M1 (y) 0,
Ly (q) = 0.
Is the above also sufficient for y to have a representing measure supported in E(q)? If y ∈ Mn,2 has a representing measure supported in S(q), it is necessary that M1 (y) 0,
Ly (q) 0.
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Is the above also sufficient for y to have a representing measure supported in S(q)? These questions will be answered affirmatively under certain suitable conditions. Throughout this section, we will employ a well-known connection between nonnegative polynomials and positive semidefinite real symmetric matrices (cf. [14]), which we apply in the case of quadratic polynomials. Let p(x) = α∈Zn+ : |α|2 pα x α . Let yp denote the degree 2 moment sequence whose moment corresponding to a monomial of degree 1, or to a monomial of the form x α = xi xj (i = j ), is pα /2, and whose moment corresponding to a monomial of degree 0, or of the form x α = xi2 , is pα . A calculation shows that p(x) = [x]T1 M1 (yp )[x]1 .
(4.1)
From this it follows immediately that p(x) is nonnegative on Rn if and only if there exists a matrix P such that P = P T , P 0, and p(x) = [x]T1 P [x]1
x ∈ Rn .
(4.2)
In the case when p(x) is a homogeneous quadratic, by compressing M1 (yp ) to the rows and columns indexed by the variables xi , and similarly for [x]1 , we see that p(x) admits a representation of the form p(x) = x T P x
x ∈ Rn ,
(4.3)
where P = P T ; further, p(x) is nonnegative on Rn if and only if P 0. In the sequel, for m × m real matrices R ≡ (rij ) and S ≡ (sij ), we denote by R • S the Frobe nius inner product, defined by R • S = Trace(RS T ) = 1i,j m rij sij . A calculation shows that if p has a representation as in (4.2) and y is a quadratic moment sequence, then Ly (p) = P • M1 (y).
(4.4)
If R = R T 0 and S = S T 0, then R = LLT and S = MM T , and thus
R • S = Trace LLT MM T = Trace M T LLT M
T T T
= M L • M L 0. = Trace M T L M T L It now follows that if R = R T 0 and S = S T 0,
then R • S 0.
(4.5)
4.1. Quadratic polynomials nonnegative on quadratic sets A useful tool in quadratic moment theory, which we will employ repeatedly, is the following matrix decomposition developed by Sturm and Zhang [20]. In the sequel, let Mm denotes the space of real m × m matrices, endowed with the norm induced by the Frobenius inner product.
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Proposition 4.1. (See [20, Corollary 4].) Let Q ∈ Mm be a symmetric matrix. If X ∈ Mm is symmetric positive semidefinite and has rank r, then there exist nonzero vectors u1 , . . . , ur ∈ Rm such that X = u1 uT1 + · · · + ur uTr ,
uT1 Qu1 = · · · = uTr Qur =
Q•X . r
We will also utilize the following representation of quadratic polynomials that are nonnegative on S(q). Proposition 4.2 (S-Lemma). (See Yakubovich (1971) [22].) Let f (x), q(x) be two quadratic polynomials in x. Suppose there exists ξ ∈ Rn such that q(ξ ) > 0. Then f (x) 0 for all x ∈ S(q) if and only if there exists t 0 such that f (x) − tq(x) 0,
∀x ∈ Rn .
When f (x) and g(x) are homogeneous and quadratic, if f (x) is nonnegative on the algebraic set E(q) = {x ∈ Rn : q(x) = 0}, then a certificate like that provided by S-Lemma holds, but without requiring t 0, as pointed out by Luo, Sturm and Zhang [16]. However, we are not able to find a complete proof from [16] and the references therein. Moreover, this result can also be generalized to the case when f (x) and g(x) are non-homogeneous. So here we summarize these results and include a proof for completeness. Proposition 4.3. Let f (x), q(x) be two quadratic polynomials in x, and assume E(q) = ∅. Suppose f (x) 0 for all x ∈ E(q), and suppose there exist ξ, ζ ∈ Rn such that q(ξ ) > 0 > q(ζ ). Then there exists t ∈ R such that f (x) − tq(x) 0,
∀x ∈ Rn .
Proof. Step 1. Consider first the case when both f and g are homogeneous quadratics. From (4.3), we may write f (x) = x T F x and q(x) = x T Qx for symmetric matrices F, Q ∈ Mn . In the sequel we view Mn as a locally convex normed real vector space, with norm induced by the Frobenius inner product. By finite-dimensionality, each linear functional on Mn is of the form F → F • X for some X ∈ Mn . Let E = {S + tQ: S T = S 0, t ∈ R}. Obviously E is a convex set, and we claim that E is also closed. To see this, let {Ak ≡ Sk + tk Q} ⊂ E be sequence such that Ak → A. Note that every Sk 0 and thus ξ T Ak ξ = ξ T Sk ξ + tk ξ T Qξ tk ξ T Qξ,
ζ T Ak ζ = ζ T Sk ζ + tk ζ T Qζ tk ζ T Qζ.
From this, and the hypothesis ξ T Qξ = q(ξ ) > 0 > q(ζ ) = ζ T Qζ , it follows that ζ T Ak ζ ξ T Ak ξ tk . q(ζ ) q(ξ ) Since {Ak } is bounded, {ζ T Ak ζ } and {ξ T Ak ξ } are also bounded, whence the sequence {tk } is bounded too. Thus {Sk } is also bounded, so we may assume Sk → S∗ 0 and tk → t∗ , whence Ak → A = S∗ + t∗ Q ∈ E .
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Now we show that F belongs to the closed convex set E . Suppose to the contrary that F ∈ / E. It follows from a version of the Hahn–Banach theorem [3, Proposition 14.15] that there exist a nonzero symmetric matrix X and a scalar η such that F • X < η,
(S + tQ) • X η,
∀S T = S 0, t ∈ R.
By choosing S = 0, we see that Q • X = 0. Thus S • X η, ∀S T = S 0, whence X 0. The preceding implies that Q • X = 0,
X 0,
η 0.
Then, by Proposition 4.1, there exist vectors u1 , . . . , ur such that X = u1 uT1 + · · · + ur uTr , From
r
T i=1 ui F ui
uTi Qui = 0 (1 i r).
= F • X < η 0, we see that at least one ui satisfies uTi Qui = 0.
uTi F ui < 0,
Thus, q(ui ) = 0, but f (ui ) < 0, which is a contradiction. So F must belong to E. With F = S + tQ, for some S T = S 0 and t ∈ R, we have f (x) = s(x) + tq(x) for some nonnegative quadratic s(x) corresponding to S via (4.3), so the result follows in this case. Step 2. We next consider the case when at least one of q and f is non-homogeneous, and without loss of generality in the following argument we may assume both are non-homogeneous. Since E(q) = ∅, we may further assume that q(0) = 0 (for if q(a) = 0, we may replace q and f by q(x + a) and f (x + a)). Let q(x ˜ 0 , x) = x02 q(x/x0 ) (resp. f˜(x0 , x) = x02 f (x/x0 )) be the homogenization of q(x) (resp. f (x)). Denote x˜ T := [ x0 x T ]T , and note that f˜(x) ˜ = x02 f0 + x0 f1 (x) + f2 (x),
q( ˜ x) ˜ = x0 q1 (x) + q2 (x),
where every fi and qi are homogeneous of degree i. Now we claim that f˜(x) ˜ 0,
∀x: ˜ q( ˜ x) ˜ = 0.
(4.6)
From the hypothesis that f (x) 0 whenever q(x) = 0, (4.3) follows easily from the homogenization formulas when x0 = 0. For the case when x0 = 0, we need to prove f2 (x) 0,
∀x: q2 (x) = 0.
Let u be an arbitrary point such that q2 (u) = 0. Consider the equation q(, ˜ x) = q1 (x) + q2 (x) = 0. If q1 (u) = 0, then q(αu) = 0 for all real α. Thus αu ∈ E(q) and f (αu) 0 for all α, which implies f2 (u) 0. If q1 (u) = 0, then the rational function
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(x) = −
347
q2 (x) q1 (x)
is continuous in a neighborhood Ou of u. Choose a sequence {u(i) } ⊂ Ou such that q2 (u(i) ) = 0 (i) ), u(i) ) = (u(i) )q (u(i) ) + and u(i) → u. Then (u(i) ) = 0 and (u(i) ) → 0. Since q((u ˜ 1 u(i) u(i) (i) q2 (u ) = 0, it follows that q( (u(i) ) ) = 0. The hypothesis now implies that f ( (u (i) ) ) 0, (i) (i) whence f˜((u ), u ) 0. Letting i → ∞, we get f2 (u) = f˜(0, u) 0. Therefore, claim (4.6) is proved. The existence of ξ, ζ ∈ Rn such that q(ξ ) > 0 > q(ζ ) implies that q(1, ˜ ξ) > 0 > q(1, ˜ ζ ). Now the homogeneous case can be applied to yield t ∈ R such that f˜(x0 , x) − t q(x ˜ 0 , x) 0, ∀(x0 , x) ∈ Rn+1 , and the result follows by setting x0 = 1. 2 In Proposition 4.3, if there do not exist ξ, ζ ∈ Rn such that q(ξ ) > 0 > q(ζ ), then the conclusion might fail. For instance, for polynomials f (x) = x1 x2 and q(x) = −x12 , the summation f (x) − tq(x) is never globally nonnegative for any scalar t. However, Proposition 4.3 can be weakened as follows. Proposition 4.4. Let f (x), q(x) be two quadratic polynomials. (a) If S(q) = ∅ and f (x) 0 for all x ∈ S(q), then for any > 0 there exists t 0 such that
f (x) + 1 + x 22 − tq(x) 0,
∀x ∈ Rn .
(b) If E(q) = ∅ and f (x) 0 for all x ∈ E(q), then for any > 0 there exists t ∈ R such that
f (x) + 1 + x 22 − tq(x) 0,
∀x ∈ Rn .
Proof. As in (4.1), write f (x) and q(x) as f (x) =
T f0 f1T 1 1 , x f 1 F2 x " #$ %
q(x) =
T 1 q0 q1T 1 . x x q1 Q2 " #$ %
F
Q
(a) If there exists ξ ∈ Rn such that q(ξ ) > 0, then we are done by Proposition 4.2. So we need only consider the case when q(x) 0 for every x ∈ Rn . Since S(q) = ∅, without loss of generality we may assume that the origin belongs to S(q), which implies that q0 = 0. Let E = S + tQ: S T = S 0, t 0 . Note that E is a convex set (but not necessarily closed). We claim that for each > 0,
f + F () := F + In+1 = 0 f1
f1T ∈ E. F2 + In
Suppose to the contrary that F () ∈ / E for some > 0. Then as in the proof of Proposition 4.3, there exist a nonzero symmetric matrix X and a scalar η such that
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F () • X η,
(S + tQ) • X η,
∀S 0, ∀t 0.
The above implies that Q • X 0,
X 0,
η 0.
Then, by Proposition 4.1, there exist nonzero vectors u1 , . . . , ur such that X = u1 uT1 + · · · + ur uTr ,
uTi Qui =
Q•X 0 (1 i n). r
Write every ui as τi , ui = vi
τi ∈ R, vi ∈ Rn .
Order ui such that τi = 0 (1 i k), and τk+1 = · · · = τr = 0 (the nonzero terms may be absent, or the zero terms may be absent). For every i = 1, . . . , k if k > 0, we have τi2 q(vi /τi ) = uTi Qui 0. Thus every vi /τi ∈ S(q) and hence f (vi /τi ) 0. For every i = k + 1, . . . , r, we have viT Q2 vi = uTi Qui 0. Then we must have viT Q2 vi = 0, because otherwise q(αvi ) > 0 for α > 0 big enough contradicts the assumption that q(x) 0 for all x ∈ Rn at the beginning. So q(αvi ) = 2αq1T vi . Replacing ui by −ui if necessary, we may assume that q1T vi 0. So q(αvi ) 0 and αvi ∈ S(q) for all α > 0. Then f (αvi ) 0 for all α > 0, and hence viT F2 vi 0. So we have F () • X =
r
ui F ()ui
i=1
=
k
r
T
vi F2 vi + vi 22 τi2 f (vi /τi ) + 1 + vi /τi 22 +
i=1
k i=1
i=k+1
τi2 +
r
vi 22 .
i=k+1
Since every ui is nonzero, we have either τi > 0 or vi = 0. Thus we must have F () • X > 0,
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349
which contradicts that F () • X η 0. So F () must belong to E, and the result follows. (b) If there exist ξ, ζ such that q(ξ ) > 0 > q(ζ ), then we are done by applying Proposition 4.3. Replacing q by −q if necessary, we may thus assume that q(x) 0 for all x ∈ Rn . Let us recall the decomposition q(x) = q0 + 2q1T x + x T Q2 x given just before the proof of (a). Since E(q) = ∅, we may assume that the origin belongs to E(q), i.e., q0 = q(0) = 0. Since we assumed q(x) 0 for all x ∈ Rn , the origin is a minimizer of q(x), whence ∇q(0) = 0. Thus it follows that 1 q1 = ∇q(0) = 0. 2 We now proceed to derive a contradiction similar to that used in (a), but now we define E as E = S + tQ: S T = S 0, t ∈ R . As in part (a), if F () ∈ / E, then there exist a nonzero symmetric matrix X and a scalar η such that F () • X η,
(S + tQ) • X η,
∀S T = S 0, ∀t ∈ R,
which implies Q • X = 0,
X 0,
η 0.
Again, applying Proposition 4.1, we get nonzero vectors u1 , . . . , ur such that X = u1 uT1 + · · · + ur uTr ,
uT1 Qu1 = · · · = uTr Qur =
Q•X = 0. r
As before, write ui as τi ui = , vi
and reorder the ui so that τi = 0 (1 i k), and τk+1 = · · · = τr = 0. For i = 1, . . . , k, we have τi2 q(vi /τi ) = uTi Qui = 0, so vi /τi ∈ E(q), and hence f (vi /τi ) 0. For every i = k + 1, . . . , r, we have that for all α ∈ R, 0 = α 2 uTi Qui = α 2 viT Q2 vi = q(αvi ). Thus we get 0 f (αvi ) = f0 + 2αf1T vi + α 2 viT F2 vi , whence viT F2 vi 0 for i = k + 1, . . . , r. As in part (a), we have
∀α ∈ R,
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F () • X =
k
r
T
vi F2 vi + vi 22 τi2 f (vi /τi ) + 1 + vi /τi 22 +
i=1
k i=1
i=k+1 r
τi2 +
vi 22 > 0,
i=k+1
which contradicts F () • X 0. So F () must belong to E, and the result follows.
2
4.2. Quadratic moment problems We now apply the preceding results to quadratic moment problems. Recall from [4] that for n = 1, 2, if M1 (y) 0, then M1 (y) has a flat extension, and thus y has admits a rank M1 (y)atomic representing measure. We begin by generalizing the latter result to n 1. Theorem 4.5. If y ∈ Mn,2 and M1 (y) 0, then y has a rank M1 (y)-atomic representing measure. Proof. Without loss of generality, we may normalize y so that y0 = 1. Write the moment matrix M1 (y) as follows: M1 (y) =
1 z
zT , W
where z ∈ Rn . Since y0 = 1, we can choose a number α > 0 small enough such that the matrix Q=
1 0
0 −αIn
satisfies Q • M1 (y) 0. Then, by Proposition 4.1, there exist nonzero (column) vectors u1 , . . . , ur ∈ Rn+1 (r = rank M1 (y)) such that M1 (y) = u1 uT1 + · · · + ur uTr ,
uT1 Qu1 = · · · = uTr Qur =
Q • M1 (y) 0. r
Write the vectors ui as τi , ui = wi
τi ∈ R, wi ∈ Rn .
Then uTi Qui 0 implies that τi2 α wi 22 . So, if τi = 0, then wi = 0. Note that u 2i = τi2 + wi 22 . Since all ui are nonzero, every τi = 0, and hence we can write ui as ui = τi Thus, we have
1 , vi
vi ∈ Rn .
L. Fialkow, J. Nie / Journal of Functional Analysis 258 (2010) 328–356
M1 (y) = τ12
1 v1
1 v1
T
+ · · · + τr2
The above gives an r-atomic representing measure for y.
1 vr
1 vr
351
T .
(4.7)
2
We pause to give an application of Theorem 4.5 to the multivariable degree one moment problem. Corollary 4.6. A degree one multisequence y has a representing measure if and only if y0 > 0. Proof. Note that if v denotes the vector of moments in y, in degree-lexicographic order, then v T v has the form of a positive moment matrix M1 , so the existence of a representing measure follows from Theorem 4.5. 2 We next turn to the quadratic K-moment problem where q is a quadratic polynomial and K = E(q) or K = S(q). For the case when n = 2 and q(x) = 1 − x 22 , it is known that the conditions M1 (y) 0 and Ly (q) = 0 (resp., Ly 0) imply the existence of representing measures supported in E(q) [6, Theorem 3.1] (resp., S(q) [6, Theorem 1.8]). This can be generalized to n 1 and S(q) compact. Theorem 4.7. Suppose q(x) is quadratic and S(q) is compact and nonempty. (a) y ∈ Mn,2 has a representing measure supported in E(q) if and only if M1 (y) 0,
Ly (q) = 0.
(b) y ∈ Mn,2 has a representing measure supported in S(q) if and only if M1 (y) 0,
Ly (q) 0.
Proof. We write q(x) as q(x) = q0 + 2q1T x
T q0 q1T 1 1 . + x Q2 x = x q1 Q2 x " #$ % T
(4.8)
Q
Since S(q) is nonempty, we can assume 0 ∈ S(q), i.e., q0 0, without loss of generality. From the compactness of S(q), we know q(x) must be strictly concave, that is, Q2 must be negative definite (Q2 ≺ 0). To see this, suppose otherwise, i.e., that Q2 is not negative definite. Then there exists a nonzero u ∈ Rn such that uT Q2 u 0. We can also further choose u so that q1T u 0 (otherwise replace u by −u). Thus, for any t > 0, we have q(tu) 0, which implies S(q) is unbounded. However, this contradicts the compactness of S(q). Therefore, Q2 must be negative definite. (a) We need only prove the sufficiency direction. Suppose y ∈ Mn,2 and let X = M1 (y). Then we have X 0,
Q • X = Ly (q) = 0.
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By Proposition 4.1, there exist nonzero vectors u1 , . . . , ur ∈ Rn+1 such that X=
r
uT1 Qu1 = · · · = uTr Qur =
ui uTi ,
i=1
Write ui = [ τi that
Q•X = 0. r
wiT ]T for some scalar τi and some vector wi ∈ Rn . Then uTi Qui = 0 implies q0 τi2 + 2τi q1T wi + wiT Q2 wi = 0.
(4.9)
If τi = 0 for some i, then wiT Q2 wi = 0, and hence wi = 0 because of negative definiteness of Q2 . Since ui is nonzero, it follows that every τi = 0, and we can write ui = τi [1viT ]T . (4.8) and (4.9) now imply that q(vi ) = 0, so vi ∈ E(q). Therefore, we have M1 (y) = τ12
1 v1
1 v1
T
+ · · · + τr2
1 vr
1 vr
T ,
and it follows that μ ≡ ri=1 τi2 δvi is a representing measure for y supported in E(q). (b) The proof is very similar to part (a). Suppose y ∈ Mn,2 and let X = M1 (y). Then X 0,
Q • X = Ly (q) 0.
By Proposition 4.1, there exist nonzero vectors u1 , . . . , ur ∈ Rn+1 such that X=
r
uT1 Qu1 = · · · = uTr Qur =
ui uTi ,
i=1
Write ui = [ τi
Q•X 0. r
wiT ]T for some wi ∈ Rn . Then uTi Qui 0 implies that q0 τi2 + 2τi q1T wi + wiT Q2 wi 0.
(4.10)
If τi = 0 for some i, then wiT Q2 wi 0 and hence wi = 0 because of negative definiteness of Q2 . But this is also impossible, since otherwise ui = [ τi wiT ]T is a zero vector. Thus, every τi = 0. So we can further write ui = τi [ 1 viT ]T . Then (4.8) and (4.10) imply that q(vi ) 0, and so vi ∈ S(q). Hence we get M1 (y) = τ12
1 v1
1 v1
T
+ · · · + τr2
1 vr
1 vr
T ,
and it follows as above that y has a representing measure supported in S(q).
2
When E(q) or S(q) is not compact, the conclusions of Theorem 4.7 might fail. However, we can get a sightly weakened version. Theorem 4.8. Let y ∈ Mn,2 and let q(x) be a quadratic polynomial.
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353
(i) Suppose E(q) = ∅. Then M1 (y) 0 and Ly (q) = 0 if and only if y ∈ Rn,2 (E(q)). (ii) Suppose S(q) = ∅. Then M1 (y) 0 and Ly (q) 0 if and only if y ∈ Rn,2 (S(q)). Proof. (i) The sufficiency direction is obvious, so we only need prove necessity. Suppose to / Rn,2 (E(q)). Since Rn,2 (E(q)) is a closed the contrary that M1 (y) 0 and Ly (q) = 0, but y ∈ convex cone, Minkowski’s separation theorem implies that there exists a nonzero polynomial p ∈ P2 such that Ly (p) ≡ pˆ T y < 0,
and pˆ T w 0,
∀w ∈ Rn,2 E(q) .
For 1 i n, let y2ei denote the element of y corresponding to the monomial xi2 . Choose > 0 small enough so that pˆ T y + 1 +
n
! y2ei Ly (1) < 0,
(4.11)
i=1
and define the nonzero polynomial
p(x) ˜ = pˆ T [x]2 + 1 + x 22 .
(4.12)
Since, for each x ∈ E(q), the monomial vector [x]2 belongs to Rn,2 (E(q)) (with E(q)representing measure δx ), the polynomial pˆ T [x]2 is nonnegative on E(q). By Proposition 4.4(b), there exists t ∈ R such that
pˆ T [x]2 + 1 + x 22 − tq(x) 0,
∀x ∈ Rn .
It follows from (4.2) that there exists a matrix P , with P = P T 0, such that
pˆ T [x]2 + 1 + x 22 − tq(x) = [x]T1 P [x]1 ,
∀x ∈ Rn ,
whence p(x) ˜ = [x]T1 P [x]1 + tq(x). Since M1 (y) 0 and Ly (q) = 0, applying Ly on both sides of the above (see Eq. (4.4)) implies that Ly (p) ˜ = P • M1 (y) + tLy (q) = P • M1 (y) 0. However, from (4.11)–(4.12) we have Ly (p) ˜ = pˆ y + 1 + T
n
! y2ei Ly (1) < 0,
i=1
which is a contradiction. So we must have y ∈ Rn,2 (E(q)).
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(ii) Sufficiency is again obvious, so we focus on necessity. The proof is very similar to the argument of (i), but we replace E(q) by S(q). Thus, the polynomial pˆ T [x]2 is now nonnegative on S(q). Using Proposition 4.4(a), it follows as above that there exists t 0 and a matrix P with ˜ = [x]T1 P [x]1 + tq(x). Since t 0 and Ly (q) 0, it follows as P = P T 0, such that p(x) ˜ 0, which leads to the same contradiction as in (i). 2 before that Ly (p) Theorem 4.8 implies that if q is a quadratic polynomial and if M1 (y) 0 and Ly (q) = 0 (resp. Ly (q) 0), then y is in the closure of the quadratic moment sequences which admit representing measures supported in E(q) (resp. S(q)). But this does not necessarily imply that y admits a representing measure supported in E(q) or S(q), as the following example shows. Example 4.9. Let n = 2 and let y ∈ M2,2 be the quadratic moment sequence such that
1 1 1 M1 (y) = 1 1 1 . 1 1 2 Let 1, X1 , X2 denote the columns of M1 (y). Obviously, M1 (y) is positive semidefinite with rank M1 (y) = 2, so y admits 2-atomic representing measures by Theorem 4.5. Since 1 = X1 , Proposition 3.1 of [4] implies that any representing measure μ must be supported in the variety {(x1 , x2 ) ∈ R2 : x1 = 1}. Let q(x) = x2 − x12 . Then S(q) is convex but noncompact, and E(q) is nonconvex and noncompact. Note that Ly (q) = y01 − y20 = 0, so of course Ly (q) 0. But y does not have a representing measure μ supported in either E(q) or S(q). Indeed, suppose a representing measure μ with supp μ ⊆ S(q) exists. For any x = (x1 , x2 ) ∈ supp μ ⊆ S(q), we must have x1 = 1 and x2 1. Then the relation x2 dμ(x) = y01 = 1, R2
together with y00 = 1, implies that x2 = 1 on the support of μ. So μ is supported at the single point (1, 1), which is obviously false. Therefore, y does not have a representing measure μ supported in S(q) or E(q). In keeping with Theorem 4.7, we next show that an arbitrarily small perturbation can be applied to make the perturbed y have a representing measure supported in E(q)(⊂ S(q)). For 1 > > 0, let the moment sequence y() ¯ be defined by T T 1 1 1 1
−1/4 −1/4 ¯ = (1 − ) 1 + M1 y() 1 1 1 −1/2 −1/2 1 1 − + 3/4 1 + 1/2 − 3/4 1/2 1/4 = 1−+ 1+ − 1+ − . 2− 1 + 1/2 − 1 + 1/4 − We see that y() ¯ → y as → 0, and y() ¯ has the 2-atomic E(q)-representing measure (1 − )δ(1,1) + δ
(
− 41
,
− 21
)
.
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355
Despite the preceding example, if, in Theorem 4.8, the quadratic moment sequence y is such that M1 (y) 0 and Ly (q) = 0 (resp. Ly (q) > 0), then y does have a representing measure supported in E(q) (resp. S(q)). The following result thus provides some affirmative evidence for Question 1.2. Theorem 4.10. Let y ∈ Mn,2 and let q(x) be a quadratic polynomial. (i) If E(q) = ∅, M1 (y) 0 and Ly (q) = 0, then y ∈ Rn,2 (E(q)). (ii) If S(q) = ∅, M1 (y) 0 and Ly (q) > 0, then y ∈ Rn,2 (S(q)). Proof. (i) Define the affine subspace N (q) and set FE as follows: N (q) = y ∈ Mn,2 : Ly (q) = 0 ,
FE = y ∈ N (q): M1 (y) 0 .
Note that Rn,2 (E(q)) and FE are both convex sets contained in the space N (q). Theorem 4.8 says that FE = Rn,2 (E(q)). If M1 (y) 0, then y lies in the interior of FE . By Lemma 2.1, we know y ∈ Rn,2 (E(q)). (ii) Let FS be the following convex set FS = y ∈ Mn,2 : M1 (y) 0, Ly (q) 0 . Theorem 4.8 says that FS = Rn,2 (S(q)). If M1 (y) 0 and Ly (q) > 0, then y lies in the interior of FS . Hence Lemma 2.1 implies y ∈ Rn,2 (S(q)). 2 Using Theorem 4.10, we can now show that Question 1.2 has an affirmative answer when d = 1 and K = E(q) or K = S(q) for a quadratic polynomial q(x). Corollary 4.11. Let y ∈ Mn,2 and let q(x) be a quadratic polynomial. (i) Suppose E(q) = ∅. If M1 (y) 0 and Ly is E(q)-positive, then y has an E(q)-representing measure. (ii) Suppose S(q) = ∅. If M1 (y) 0 and Ly is S(q)-positive, then y has an S(q)-representing measure. Proof. (i) From Theorem 4.10(i), it suffices to show that Ly (q) = 0. Since Ly is E(q)-positive, we have Ly (q) 0 and Ly (−q) 0, so Ly (q) = 0. (ii) Suppose first that E(q) = ∅. Since Ly is S(q)-positive, Ly (q) 0. If Ly (q) = 0, Theorem 4.10(i) implies that y has a representing measure supported in E(q) ⊆ S(q). If Ly (q) > 0, then Theorem 4.10(ii) shows that y has a representing measure supported in S(q). Suppose next that E(q) = ∅. Since S(q) = ∅, then S(q) = Rn , so in this case the result follows from Theorem 4.5. 2 Remark 4.12. In a note [11] completed after the acceptance of this paper, C. Easwaran and the first-named author proved the conjecture mentioned in the body of Example 2.5.
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Journal of Functional Analysis 258 (2010) 357–372 www.elsevier.com/locate/jfa
Fixed point property for Banach algebras associated to locally compact groups Anthony To-Ming Lau a,1 , Peter F. Mah b,∗ a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1 b Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON, Canada, P7B 5E1
Received 2 April 2009; accepted 20 July 2009 Available online 7 August 2009 Communicated by K. Ball Dedicated to our teacher, Professor Edmond E. Granirer, on the occasion of his 75th birthday with admiration and respect
Abstract In this paper we investigate when various Banach algebras associated to a locally compact group G have the weak or weak∗ fixed point property for left reversible semigroups. We proved, for example, that if G is a separable locally compact group with a compact neighborhood of the identity invariant under inner automorphisms, then the Fourier–Stieltjes algebra of G has the weak∗ fixed point property for left reversible semigroups if and only if G is compact. This generalizes a classical result of T.C. Lim for the case when G is the circle group T . © 2009 Elsevier Inc. All rights reserved. Keywords: Group C ∗ -algebra; Group von Neumann algebra; Fourier algebra; Fourier–Stieltjes algebra; Weak∗ uniform Kadec–Klee property; Weak∗ normal structure; Weak∗ fixed point property; Nonexpansive mapping; Left reversible semigroup; Commutative semigroup
1. Introduction Let E be a Banach space and K be a nonempty bounded closed convex subset of E. We say that K has the fixed point property if every nonexpansive mapping T : K → K (i.e. T x − T y x − y for all x, y ∈ K) has a fixed point. We say that E has the weak fixed point property if * Corresponding author.
E-mail addresses:
[email protected] (A.T.-M. Lau),
[email protected] (P.F. Mah). 1 This research is supported by NSERC Grant A-7679.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.07.011
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every weakly compact convex subset of E has the fixed point property. A dual Banach space E is said to have the weak∗ fixed point property if each weak∗ compact convex subset of E has the fixed point property. As we have a need to refer to results in [26,27] occasionally, we should point out that weak fixed point property and weak∗ fixed point property in there are denoted by FPP (or fpp) and FPP∗ (or fpp∗ ), respectively. Let S be a semitopological semigroup, i.e., S is a semigroup with a Hausdorff topology such that for each a ∈ S, the mappings s → as and s → sa from S into S are continuous. S is called left reversible if aS ∩ bS = ∅ for any a, b ∈ S, where, in general, K denotes the closure of the set K. Clearly abelian semigroups and groups are left reversible. Let CB(S) be the C ∗ algebra of bounded continuous complex-valued functions on S and for a ∈ S, let la be the left translation operator on CB(S) be defined by (la f )(t) = f (at) for all f ∈ CB(S) and for all t ∈ S. Then S is left amenable if there is an m ∈ CB(S)∗ such that m = m(1) = 1 and m(la f ) = m(f ) for all f ∈ CB(S) and a ∈ S. If the topology on S is normal and S is left amenable, then S is left reversible. In particular, if S is left amenable as a discrete semigroup, then S is left reversible. Left reversible semigroups have played an important role in the study of common fixed point theorems and ergodic type theorems for semigroups of nonexpansive mappings (see [18,22,23,30–32,34–36]). Let S be a semitopological semigroup, and K be a topological space. An action of S on K is a map ψ from S × K to K, denoted by ψ(s, k) = sk, s ∈ S, k ∈ K, such that s1 s2 (k) = s1 (s2 k), for all s1 , s2 ∈ S, and k ∈ K. The action is separately continuous if ψ is continuous in each of the variables when the other is kept fixed. Lau showed in [22] that if E is a Banach space and S = {Ts : s ∈ S} is a continuous representation of a left reversible semitopological semigroup S as nonexpansive self-maps on a compact convex subset K of E, then K contains a common fixed point for S. We say a Banach space E has the weak fixed point property for left reversible semigroups if whenever S is a left reversible semitopological semigroup and K is a nonempty weakly compact convex subset of E for which the action of S on K (with the norm topology) is separately continuous and nonexpansive, then K has a common fixed point for S. Similarly a dual Banach space E has the weak∗ fixed point property for left reversible semigroups if whenever S is a left reversible semitopological semigroup and K is a nonempty weak∗ compact convex subset of E for which the action of S on K is separately continuous and nonexpansive, then K has a common fixed point for S. In general, a weakly compact convex set of a Banach space need not have the fixed point property for left reversible semigroups, not even commutative semigroups. Indeed, Alspach [1] (see also [3, Theorem 4.2], [4,8]) showed there is a weakly compact convex subset K in L1 [0, 1] and an isometry T : K → K without a fixed point. Hence if S = (N, +) and S = {T n : n ∈ N}, then K does not have a common fixed point for S. However, Bruck showed in [5] that a Banach space E having the weak fixed point property has the weak fixed point property for commutative semigroups, and Lim showed in [34] that a Banach space with weak normal structure has the weak fixed point property for left reversible semigroups. For dual Banach spaces, it is known (see [34,35]) that 1 and any uniformly convex Banach space have the weak∗ fixed point property for left reversible semigroups. This paper is organized as follows. In Section 3, we shall establish a technical lemma that we shall need for our result on weak∗ fixed point property for left reversible semigroups. In Section 4, we prove our main results concerning the weak∗ fixed point property for left reversible semigroups on the Fourier–Stieltjes algebra of a locally compact group and its relations with other geometric properties. In Section 5, we shall study the weak fixed point property for left reversible semigroups or commuting semigroups on various Banach algebras associated to a locally compact group. In Section 6, we shall discuss some open problems arising from this work.
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2. Some preliminaries Let K be a bounded closed convex subset of a Banach space E. A point x in K is called a diametral point if sup x − y: y ∈ K = diam(K), where diam(K) denotes the diameter of K. The set K is said to have normal structure if every nontrivial (i.e., contains at least two points) convex subset H of K contains a non-diametral point of H (see [15,20]). A Banach space E has weak normal structure if every nontrivial weakly compact convex subset of E has normal structure. A dual Banach space E has weak∗ normal structure if every nontrivial weak∗ compact convex subset of E has normal structure. A Banach space E is said to have property UKK (uniformly Kadec–Klee property) if for any ε > 0 there is a 0 < δ < 1 such that whenever (xn ) is a sequence in the unit ball of E converging weakly to x and satisfying sep((xn )) ≡ inf{xn − xm : n = m} > ε, then x δ. A dual Banach space E is said to have property UKK∗ (weak∗ uniformly Kadec–Klee property) if for any ε > 0 there is a 0 < δ < 1 such that whenever A is a subset of the closed unit ball of E containing a sequence (xn ) with sep((xn )) > ε, then there is an x in the weak∗ closure of A such that x δ. The property UKK∗ was introduced by van Dulst and Sims [11]. They proved that if E has property UKK∗ , then E has weak∗ normal structure and hence has the weak∗ fixed point property. Let G be a locally compact group with a fixed left Haar measure λ. Let L1 (G) be the group algebra of G with convolution product. We define C ∗ (G), the group C ∗ -algebra of G, to be the completion of L1 (G) with respect to the norm f ∗ = supπf , where the supremum is taken over all nondegenerate ∗-representations π of L1 (G) as a ∗-algebra of bounded operators on a Hilbert space. Let B(L2 (G)) be the set of all bounded operators on the Hilbert space L2 (G) and ρ be the left regular representation of G, i.e., for each f ∈ L1 (G), ρ(f ) is the bounded operator in B(L2 (G)) defined by ρ(f )(h) = f ∗ h, the convolution of f and h in L2 (G). Denote by Cρ∗ (G) the completion of L1 (G) with the norm ρ(f ), f ∈ L1 (G), and denote by VN(G) the closure of {ρ(f ): f ∈ L1 (G)} in the weak operator topology in B(L2 (G)). In the case when G is left amenable, which is the case when G is compact, then C ∗ (G) is isometric isomorphic to Cρ∗ (G). Denote the set of continuous positive definite functions on G by P (G), and the set of continuous functions on G with compact support by C00 (G). We define the Fourier–Stieltjes algebra of G, denoted by B(G), to be the linear span of P (G). Then B(G) is a Banach algebra with the norm of each φ ∈ B(G) defined by φ =
sup f ∈L1 (G), f ∗ 1
f (t)φ(t) dλ(t).
The Fourier algebra of G, denoted by A(G), is defined to be the closed linear span of P (G) ∩ C00 (G). Clearly, A(G) = B(G) when G is compact. It is known that C ∗ (G)∗ = B(G), where the duality is given by f, φ = f (t)φ(t) dλ(t), f ∈ L1 (G), φ ∈ B(G), and A(G)∗ = VN(G) (see [13] for details).
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3. Technical lemma In preparation for our results on the weak∗ fixed point property for left reversible semigroups for B(G), we first establish the following lemma. Lemma 3.1. Let G be a compact group, and let {Dα : α ∈ Λ} be a decreasing net of bounded subsets of B(G), and {φm : m ∈ M}, be a weak∗ convergent sequence with weak∗ limit φ. Then lim sup lim sup φm − ψ: ψ ∈ Dα = lim sup φ − ψ: ψ ∈ Dα m
α
α
+ lim supφm − φ.
(3.1.1)
m
Proof. Since G is compact, it follows from Lemma 4.1 and Theorem 4.2 in [29] that C ∗ (G) is a c0 -sum of finite-dimensional C ∗ -algebras. But each finite-dimensional C ∗ -algebra is a finite direct sum of full matrix algebras [37, Theorem 11.2]. Thus we may write C ∗ (G) = c0 −
K(Hi )
and B(G) = l1 −
i∈I
T (Hi ),
(3.1.2)
i∈I
where K(Hi ) and T (Hi ) are the compact operators and the trace class operators on the finitedimensional Hilbert space Hi , respectively. Let ψ ∈ B(G) and write ψ = (ψ(i)), ψ(i) ∈ T (Hi ). Clearly for each ψ ∈ Dα , we have φ − ψ + φm − φ φm − ψ. Hence if α0 is fixed, then for each m ∈ M, sup φ − ψ: ψ ∈ Dα + φm − φ sup φm − ψ: ψ ∈ Dα , αα0
αα0
and so lim sup φ − ψ: ψ ∈ Dα + φm − φ lim sup φm − ψ: ψ ∈ Dα . α
α
Thus the right side is greater or equal to the left side in (3.1.1). To prove the reverse inequality, we first show that we can assume {Dα : α ∈ Λ} is a decreasing sequence {Dn : n 1} of bounded sets. For each α ∈ Λ, let ρα := sup φ − ψ: ψ ∈ Dα ∈ [0, ∞), and let ρ := lim ρα = inf ρα ∈ [0, ∞). α∈Λ
α∈Λ
For each k ∈ N, we can choose αk ∈ Λ such that
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1 ρ ραk < ρ + . k Moreover, we may choose the sequence (αk ) so that αk+1 αk for all k ∈ N. Note that for α ∈ Λ with α αk , 1 ρ ρα ραk < ρ + . k Thus {Dαk : k ∈ N} is a decreasing sequence and ρ := lim ρα = lim ραk . α∈Λ
k∈N
Next, for each m ∈ M and each α ∈ Λ, let ναm := sup φm − ψ: ψ ∈ Dα ∈ [0, ∞) and ν m := lim ναm = inf ναm . α∈Λ
α∈Λ
Then, as above, we can choose an increasing sequence (αkm )k∈N in Λ so that 1 ν m ναmk < ν m + , k ν m := lim ναm = lim ναmm , α∈Λ
k∈N
k
and for α ∈ Λ with α αkm , 1 ν m ναm ναmm < ν m + . k k Since (3.1.1) is obvious when M is finite, we may assume M is infinite. Since the set Λ is directed, we can choose t1 ∈ Λ so that t1 α1 , α11 . For k 2, choose tk tk−1 , αk , αki , i = 1, . . . , k. Then (tk )k∈N is an increasing sequence and for each m ∈ M, ρ = lim ρtk kN
and ν m = lim νtmk . k∈N
Let ε > 0 and let m ∈ M be fixed. Choose k0 so large that k0 > m and k > k0 , tk > tk0 αk0 , and so ρ ρtk ρtk0 ραk0 < ρ + and since tk > tk0 αkm0 ,
1 < ρ + ε; k0
1 < ε. Then for all ko
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ν m νtmk νtmk ναmm < ν m + 0
k0
1 < ν m + ε. k0
Thus we may assume we have a decreasing sequence {Dn : n 1} of bounded sets. Choose ψn ∈ Dn such that lim supφ − ψn = lim sup φ − ψ: ψ ∈ Dn . n
n
It follows that it suffices to prove the following inequality: lim supφ − ψn + lim supφm − φ lim sup lim supφm − ψn . n
m
m
n
We may assume, without loss of generality, that φ = 0 and that limψn , q := limφm , and r := limm lim supn φm − ψn exist. Suppose, on the contrary, that for some p > 0 we have limψn = r − q + p.
(3.1.3)
We will show that for each ε > 0 we can find two sequences N1 < N2 < · · · and finite subsets σ1 ⊂ σ2 ⊂ · · · of I such that for n Nk ,
ψn (i) > (p − ε)/2,
with σ0 = ∅.
i∈σk \σk−1
This would contradict the boundedness of (ψn ) because for n Nk , ψn >
ψn (i) k(p − ε)/2. i∈σk
To show how the two sequences can be constructed, let ε > 0 be arbitrary. Using (3.1.3), there exist an m1 ∈ M and an N1 such that for all n N1 , φm1 > q − ε/4; φm1 − ψn < r + ε/4; ψn > r − q + p − ε/4. For this m1 choose a finite set σ1 ⊂ I such that φm (i) < ε/8. 1 i ∈σ / 1
Then for all n N1 ,
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r + ε/4 > φm1 − ψn φm (i) − ψn (i) + φm (i) − ψn (i) = 1 1 i ∈σ / 1
i∈σ1
φm (i) − ψn (i) + ψn (i) − φm (i) 1 1 i ∈σ / 1
i∈σ1
φm (i) + ψn − 2 ψn (i) = φm1 − 2 1 i ∈σ / 1
i∈σ1
> q − ε/4 − /4 + r − q + p − ε/4 − 2 ψn (i) . = r + p − 3ε/4 − 2
ψn (i)
i∈σ1
i∈σ1
Thus ψn (i) > (p − ε)/2. i∈σ1
Next, since (φm ) converges to 0 in the weak∗ topology, it follows from (3.1.2) that for each i ∈ I , (φm (i)) is weak∗ convergent to 0 in T (Hi ), and as Hi is finite-dimensional, it is norm-convergent to 0. Using this, we can find an m2 ∈ M and an N2 > N1 such that for all n N2 , φm (i) < ε/10; 2 i∈σ1
φm2 > q − ε/5; ψn − φm2 < r + ε/5; ψn > r − q + p − ε/5. For this m2 , we can find a finite subset σ2 ⊃ σ1 such that φm (i) < ε/10. 2 i ∈σ / 2
Then for all n N2 , r + ε/5 > φm2 − ψn φm (i) − ψn (i) + φm (i) − ψn (i) + φm (i) − ψn (i) = 2 2 2 i∈σ1
i∈σ2 \σ1
i ∈σ / 2
ψn (i) − φm (i) + φm (i) − ψn (i) 2 2 i∈σ1
i∈σ2 \σ1
ψn (i) − φm (i) + 2 i ∈σ / 2
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= φm2 − 2
φm (i) − 2 φm (i) + ψn − 2 ψn (i) 2 2 i ∈σ / 2
i∈σ1
i∈σ2 \σ1
ψn (i) > q − ε/5 − ε/5 − ε/5 + r − q + p − ε/5 − 2 i∈σ2 \σ1
ψn (i) . = r + p − 4ε/5 − 2 i∈σ2 \σ1
Thus ψn (i) > (p − ε)/2. i∈σ2 \σ1
We repeat the above steps to find an m3 ∈ M, an N3 > N2 and a finite subset σ3 ⊃ σ2 such that for all n N3 , φm (i) < ε/10; 3 i∈σ2
φm (i) < /10; 3
i ∈σ / 3
φm3 > q − ε/5; ψn − φm3 < r + ε/5; ψn > r − q + p − ε/5. From these inequalities we obtain, as before, ψn (i) > (p − ε)/2. i∈σ3 \σ2
We continue this process to obtain the desired sequences.
2
Corollary 3.2. Suppose G is a compact group. If (ψn ) is a bounded net in B(G), and (φm ) is a sequence that converges to φ in the weak∗ topology, then lim supψn − φ + lim supφm − φ = lim sup lim supψn − φm . n
m
m
n
A dual Banach space E is said to have the lim–sup property if whenever (φn ) is a sequence in E that converges to 0 in the weak∗ topology and limn φn exists, then limn φn − ψ = limn φn + φ − ψ for any ψ ∈ E. In [35], Lim showed that 1 has this property and as a consequence, 1 has weak∗ normal structure. Corollary 3.3. Let G be a compact group. If (φm ) is a sequence in B(G) that converges to φ in the weak∗ topology, then for all ψ ∈ B(G),
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lim supφm − ψ = ψ − φ + lim supφm − φ. m
m
In addition, if limm φm − φ exists, then limφm − ψ = limφm − φ + φ − ψ. m
m
In particular, B(G) has the lim–sup property. 4. Weak∗ fixed point property for left reversible semigroups We are now ready to state and prove our results for B(G) of a separable compact group G. Let C be a nonempty subset of a Banach space X and {Dα : α ∈ Λ} be a decreasing net of bounded nonempty subsets of X. For each x ∈ C, and α ∈ Λ, let rα (x) = sup x − y: y ∈ Dα , r(x) = lim rα (x) = inf rα (x), α α r = inf r(x): x ∈ C . The set (possibly empty)
AC {Dα : α ∈ Λ} = x ∈ C: r(x) = r is called the asymptotic center of {Dα : α ∈ Λ} with respect to C and r is called the asymptotic radius of {Dα : α ∈ Λ} with respect to C. Theorem 4.1. Let G be a separable compact group. Let C be a nonempty weak∗ closed convex subset of B(G) and {Dα : α ∈ Λ} be a decreasing net of nonempty bounded subsets of C. Let r(x) be as defined above. Then for each s 0, {x ∈ C: r(x) s} is weak∗ compact and convex, and the asymptotic center of {Dα : α ∈ Λ} with respect to C is a nonempty norm compact convex subset of C. Proof. First, we observe that since G is separable, the group C ∗ -algebra C ∗ (G) is separable, and so the weak∗ topology on bounded subsets of B(G) is metrizable. Next, we show the convex function r(φ) is weak∗ lower semi-continuous. To this end, it suffices to prove the level set Ks := {x ∈ C: r(x) s} is weak∗ closed for each s. We may assume that s 0. Let (ψm ) be a sequence in Ks which converges to ψ in the weak∗ topology. By Lemma 3.1 r(ψ) = lim sup r(ψm ) − lim supψm − ψ s. m
m
Hence ψ ∈ Ks , and Ks is weak∗ closed. Now denote the asymptotic centre by K and the asymptotic radius by r. For s > r, let r0 := inf{r(x): x ∈ C ∩ Ks } and K0 = {x ∈ C ∩ Ks : r(x) = r0 }. We have r = r0 and K = K0 . Now, the set Ks is norm bounded and weak∗ closed, so it is weak∗ compact. Thus the weak∗ lower semi-continuous convex functional must attain its minimum on the set C ∩ Ks . Hence K = ∅.
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Next, we prove K compact. Suppose (φn ) is a sequence in K that converges to φ in the weak∗ topology. Then r(φn ) = r(φ) = r and, from Lemma 3.1, we must have lim supn φn − φ = 0. Thus the sequence is norm convergent; hence K is compact. 2 Theorem 4.2. Let G be a separable compact group. Then B(G) has the weak∗ fixed point property for left reversible semigroups. Proof. Let S be a left reversible semitopological semigroup, and C be a weak∗ compact convex nonempty subset of B(G) for which the action of S on (C, · ) is separately continuous and nonexpansive. Let S be directed by a b iff aS ⊆ bS. For a fixed u ∈ C, let Ws = sS(u) for all s ∈ S. Then {Ws : s ∈ S} is a decreasing net of subsets of C. Let K be the asymptotic center of {Ws : s ∈ S} with respect to C. By Theorem 4.1, K is a nonempty compact convex subset of C. Moreover, it is S-invariant. For, let x ∈ K, s ∈ S, and > 0 be arbitrary. Since x ∈ K, there exists t ∈ S such that tS(u) ⊂ Wt ⊂ B[x, r + ], where r is the asymptotic radius and B[x, r] denotes the closed ball of radius r centered at x. Since s is nonexpansive, we have stS(u) ⊂ B[s(x), r + ], so that Wst ⊂ B[s(x), r + ]. Thus, s(x) ∈ K. It now follows from Corollary 1 in [18] that K, and hence C, contains a common fixed point for S. 2 Remark 4.3. For a locally compact group G, denote the set of equivalence classes of irre When G = T , the circle group, then G is the dual ducible unitary representation of G by G. group isomorphic to the integers Z. In this case, as is well known, C ∗ (G) is isometric isomorphic to c0 (Z) via the Fourier transform, and B(G) is isometric isomorphic to 1 (Z) via Bochner’s Theorem. See Examples 1.9 and 2.5 in [13]. Thus our Theorem 4.2 can be seen to be a generalization of Lim’s result that 1 has the weak∗ fixed point property for left reversible semigroups. Remark 4.4. We were not able to remove separability from the hypothesis of Theorem 4.2. As far as we know, it is even unknown whether 1 (Γ ) has the weak∗ fixed point property for left reversible semigroups when Γ is uncountable. Remark 4.5. Let G be a separable locally compact group. Then the measure algebra M(G) has the weak∗ fixed point property for left reversible semigroups if and only if G is discrete. If M(G) has the weak∗ fixed point property for left reversible semigroups then it has the weak∗ fixed point property, and so G must be discrete by [26, Theorem 1]. The other direction follows from Lim’s result [35, Theorem 4]. A locally compact group G is called an [IN]-group if there is a compact neighbourhood of the identity e in G which is invariant under inner automorphisms. The class of [IN]-group contains all discrete groups, abelian groups and compact groups. Every [IN]-group is unimodular. Theorem 4.6. Let G be a separable [IN]-group. Then the following are equivalent: (a) (b) (c) (d)
G is compact. B(G) has property UKK∗ . B(G) has weak∗ normal structure. B(G) has weak∗ fixed point property for left reversible semigroups.
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(e) (f) (g) (h)
367
B(G) has the weak∗ fixed point property. B(G) has the lim–sup property. B(G) is separable. is countable. G
Proof. (a) ⇒ (b) was proved in [26]. (b) ⇒ (c) was proved in [11]. (c) ⇒ (e) was in proved in [35]. (e) ⇒ (a) Since every weakly compact convex subset of A(G) is weakly compact convex in B(G), it follows that it is a weak∗ compact convex subset of B(G). Thus by assumption (e), A(G) has the weak fixed point property. Since G is an [IN]-group, it follows from [24, Corollary 4.2] that G is compact. (a) ⇒ (f) follows from Corollary 3.3. (f) ⇒ (c) If G is separable, then, as we observe in the proof of Lemma 3.1, C ∗ (G) is separable. If in Theorem 2 in [35] Lim defines the function δ by δ(r, s) = r + s, then it is easy to see that δ satisfies conditions (i) and (ii), and by (f), δ satisfies (iii). And so it follows from that theorem that B(G) has weak∗ normal structure. (a) ⇒ (d) follows from Theorem 4.2, and (d) ⇒ (e) is trivial. Finally, the equivalence of (g) and (h) to the compactness of G follows from Theorem 6.1 and Lemma 6.2 in [17]. 2 Remark 4.7. (a) If G is separable then so is A(G), and conversely. See [17, Corollary 6.9]. (b) There is a non-compact locally compact group G, the so-called Fell’s group, for which is countable. See [2]. The Fell’s group G is the semi-direct product of the additive p-adic G number field Qp and the multiplicative compact group of p-adic units for a fixed prime p. So G is solvable and hence amenable. The unit ball of B(G) is weak∗ sequentially compact. So by [26, Theorem 5], B(G) cannot have property UKK∗ . By Proposition 5.1 in Section 5, B(G) has the weak fixed point property for left reversible semigroups. However, it is unknown whether B(G) even has the weak∗ fixed point property. (c) It is known that if G is an [IN]-group, then G is compact iff B(G) has the Radon–Nikodym property iff B(G) has the Krein–Milman property. See [12,16,24]. 5. Weak fixed point property for a semigroup We now investigate the weak fixed point property for a semigroup. A group G is said to be an [AU]-group if the von Neumann algebra generated by every continuous unitary representation of G is atomic. It is an [AR]-group if the von Neumann algebra VN(G) is atomic. Since VN(G) is the von Neumann algebra generated by the regular representation, it is clear that every [AU]-group is an [AR]-group. It was shown in [27, Lemma 3.1] that if the predual M∗ of a von Neumann algebra M has the Radon–Nikodym property, then M∗ has the weak fixed point property. In fact, since the property UKK is hereditary, the proof there actually showed M∗ has property UKK, and hence has weak normal structure. For the two preduals A(G) and B(G), we know from [38, Theorem 4.1 and Theorem 4.2] that the class of groups for which A(G) and B(G) have the Radon–Nikodym property are precisely the [AR]-groups and [AU]-groups, respectively. Thus by Lim’s result [34, Theorem 3] we have
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Proposition 5.1. Let G be a locally compact group. (a) If G is an [AR]-group, then A(G) has the weak fixed point property for left reversible semigroups. (b) If G is an [AU]-group, then B(G) has the weak fixed point property for left reversible semigroups. If [compact] denote the class of compact groups, etc., then we have the inclusions [compact] ⊂ [AU] ⊂ [AR], so that A(G) and B(G) have the weak fixed point property for left reversible semigroups when G is compact. Moreover, the inclusions are proper. For example, if G is the Fell’s group, then G is a non-compact group for which B(G) (and hence A(G)) has the weak fixed point property for left reversible semigroups. See [38, Remark 4.6]. In view of [24, Corollary 4.2], we have the following for [IN]-groups. Proposition 5.2. Let G be an [IN]-group. Then the following are equivalent: (a) (b) (c) (d) (e) (f) (g)
G is compact. A(G) has property UKK. A(G) has weak normal structure. A(G) has the weak fixed point property for left reversible semigroups. A(G) has the weak fixed point property. A(G) has the Radon–Nikodym property. A(G) has the Krein–Milman property.
Proposition 5.3. Let G be a locally compact group. Then the group algebra L1 (G) has the weak fixed point property for left reversible semigroups if and only if G is discrete. Proof. If L1 (G) has the weak fixed point property for left reversible semigroup, then it has the weak fixed point property. If G is not discrete, then L1 (G) contains an isometric copy of L1 [0, 1] (see [21, p. 136]), which contradicts Alspach’s result in [1]. Conversely, if G is discrete, then L1 (G) has weak∗ normal structure [26, Theorem 1]. It follows from [34, Theorem 3] that L1 (G) has the weak fixed point property for left reversible semigroups. 2 It is well known that the weak fixed point property is separably determined, i.e., a Banach space X has the weak fixed point property if and only if all its separable closed subspaces do. See [15, p. 35]. In [14], García-Falset defined the coefficient R(X) of a Banach space X by
R(X) := sup lim infxn + x , n→∞
where the supremum is taken over all weakly null sequences (xn ) of the unit ball and all points x of the unit ball, and then showed that any Banach space X with R(X) < 2 has the weak fixed point property. In a private communication he has informed us that he has shown that if X ∗ has the UKK∗ property and the unit ball is weak∗ sequentially compact, then R(X) < 2. The authors would like to express our gratitude to him for sending us the proof of this fact. Now for K(H0 ), the compact operators on an arbitrary Hilbert space H0 , Lennard proved in [33] its
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dual has the UKK∗ property. It follows from the results of García-Falset mentioned above that if, in addition, H0 is separable, then R(K(H0 )) < 2, and so K(H0 ) has the weak fixed point property. Alternatively, this fact also follows from [10, Theorem 3]. See also the note on page 775 in [10]. Now if H is any Hilbert space, and Y is any separable closed subspace of K(H), then as the proof of Theorem 5 in [9] shows, every separable subspace Y of K(H) can be embedded as a subspace of K(H0 ), where H0 is a separable Hilbert subspace of H. Thus Y and hence K(H) has the weak fixed point property. Now let G be a compact group. Then the group C ∗ -algebra C ∗ (G) is a C ∗ -subalgebra of K(L2 (G)). So C ∗ (G) has the weak fixed point property when G is compact. This answers Questions 3 in [27]. See [7] for results on fixed point property for C ∗ -algebras and in particular for C ∗ (G). Combining this result with Bruck’s result [5], we have the following Proposition 5.4. If G is a compact group, then C ∗ (G) has the weak fixed point property for commutative semigroups. We note that Lim’s result cannot be applied here to conclude that C ∗ (G) has the weak fixed point property for left reversible semigroups since C ∗ (G) does not have weak normal structure unless G is finite. However, we do not know whether it is possible for C ∗ (G) to have the weak fixed point property for left reversible semigroups without having weak normal structure. Nor do we know if the converse of Proposition 5.4 is true. However, we do know that G must be an [AU]-group (see [27, Corollary 4.2] and [6, Theorem 3]). Proposition 5.5. VN(G) has the weak (weak∗ ) fixed point property for left reversible semigroups if and only if G is finite. It was shown in [27, Corollary 4.4] that VN(G) has the weak fixed point property if and only if G is finite. It follows from this that if VN(G) has the weak fixed point property for left reversible semigroups then G must be finite. Conversely, if G is finite, then VN(G) is finite-dimensional, and so it has the weak fixed point property for left reversible semigroups by [18, Corollary 1]. 6. Remarks and open problems Remark 6.1. Lemma 3.1 does not hold for general dual Banach spaces. Suppose, on the contrary, that the conclusion of Lemma 3.1 is true in a dual Banach space E, that is, whenever {Dα : α ∈ Λ} is a decreasing net of bounded subsets of E, and (φm ) is a weak∗ convergent sequence with weak∗ limit φ, then (3.1.1) holds. If for all α ∈ Λ we take Dα = {ψ} and assume that limm φm − φ = s, then we would have ψ − φ + s = lim supm φm − ψ, that is, E satisfies the lim–sup property. The proof of Theorem 5 in [25] showed that T (H), the trace-class operators on a Hilbert space H, does not have the lim–sup property if H is infinite-dimensional. Thus we cannot hope to extend Lemma 3.1 to T (H) for infinite-dimensional H. Open Problem 6.2. Bruck proved in [5] a Banach space E has the weak fixed point property for commuting semigroups if it has the weak fixed point property. If a dual Banach space E has the weak∗ fixed point property, does E have the weak∗ fixed point property for commuting semigroups, or left reversible semigroups? Open Problem 6.3. Lim proved in [34] a Banach space has the weak fixed point property for left reversible semigroups if it has weak normal structure. Let E be a dual Banach space with weak∗
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normal structure. Does E have the weak∗ fixed point property for left reversible semigroups? In particular, note that as a consequence of the result in [33], T (H) has weak∗ normal structure and hence the weak∗ fixed point property. Does T (H) have the weak∗ fixed point property for left reversible semigroups? We have the following result. Proposition 6.4. Let E be a dual Banach space which has weak∗ normal structure. If G is a group of isometric self-maps of a weak∗ compact convex subset K, then K has a common fixed point for G. Proof. An application of Zorn’s Lemma gives the existence of a minimal G-invariant nonempty weak∗ compact convex subset X ⊆ K. A second application of Zorn’s Lemma yields a minimal G-invariant nonempty weak∗ -compact subset M ⊆ X. If M is a singleton then we have a common fixed point. Suppose M contains more than one point. First, we show g(M) = M for each g ∈ G. It suffices to show that g is onto. If m ∈ M is arbitrary, then m = gg −1 (m) ∈ gM, provided we can show e(m) = m for the group identity e. For each g ∈ G, g is an isometry and so e(m) − m = g(em) − g(m) = g(m) − g(m) = 0. Thus e(m) = m, and so g(M) = M. Next, since K has weak∗ normal structure, M contains a point u ∈ w∗ -clco(M) := M1 such that ρ := sup u − y: y ∈ M < diam(M1 ) = diam(M). For each y ∈ M, let Yy := {x ∈ X: y − x ρ} andY = y∈M Yy . Then (i) Y = ∅ since u ∈ Y ; (ii) Y is weak∗ -compact and convex since Y = y∈M (X ∩ B[y, ρ]), where B[y, ρ] is the closed ball centred at y with radius ρ; (iii) Y is a proper subset of X since if X ⊆ Y then M ⊆ X ⊆ B[y, ρ], contradicting that diam(M) > ρ. Thus the set Y contradicts the minimality of X. Consequently, M must be a singleton. 2 Open Problem 6.5. Let E be a Banach space with the weak fixed point property. Does E have the weak fixed point property for left reversible semigroups? Open Problem 6.6. If B(G) has any of the properties UKK∗ , weak∗ normal structure, the weak∗ fixed point property, the weak∗ fixed point property for left reversible semigroups, or the lim–sup property, does it follow that G is compact? Open Problem 6.7. Is there a fixed point property for groups of isometries on weak∗ compact convex sets in a dual Banach space which characterizes G-amenability of von Neumann algebras of a locally compact group G as defined in [28]? See also [19]. References [1] D. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981) 423–424. [2] L. Baggett, A separable group having a discrete dual space is compact, J. Funct. Anal. 10 (1972) 131–148. [3] T.D. Benavides, M.A. Japon Pineda, S. Prus, Weak compactness and fixed point property for affine maps, J. Funct. Anal. 209 (2004) 1–15. [4] T.D. Benavides, M.A. Japon Pineda, Fixed points of nonexpansive mappings in spaces of continuous functions, Proc. Amer. Math. Soc. 133 (2005) 3037–3046.
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[5] R.E. Bruck, A common fixed point theorem for a commutative family of nonexpansive mappings, Pacific J. Math. 53 (1974) 59–71. [6] C.H. Chu, A note on scattered C ∗ -algebras and the Radon–Nikodym property, J. London Math. Soc. 24 (1981) 533–536. [7] S. Dhompongsa, W. Fupinwong, W. Lawton, Fixed point property for C ∗ -algebra, preprint. [8] P.N. Dowling, C.J. Lennard, B. Turett, The fixed point property for subsets of some classical Banach spaces, Nonlinear Anal. 49 (2002) 141–145. [9] P.N. Dowling, N. Randrianantoanina, Spaces of compact operators on a Hilbert space with the fixed point property, J. Funct. Anal. 168 (1999) 111–120. [10] P.N. Dowling, B. Randrianantoanina, B. Turett, The fixed point property via dual space properties, J. Funct. Anal. 255 (2008) 768–775. [11] D. van Dulst, B. Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), in: Banach Space Theory and Its Applications, Proceedings Bucharest, 1981, in: Lecture Notes in Math., vol. 991, Springer-Verlag, 1983. [12] J. Diestel, J.J. Uhl Jr., Vector Measures, Math. Survey, vol. 15, Amer. Math. Soc., 1977. [13] P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964) 181– 236. [14] J. García-Falset, The fixed point property in Banach spaces with the NUS-property, J. Math. Anal. Appl. 215 (2) (1997) 532–542. [15] K. Goebel, W.A. Kirk, Topics in Metric, Fixed Point Theory, University Press, Cambridge, 1990. [16] E.E. Granirer, M. Leinert, On some topologies which coincide on the unit sphere of the Fourier–Stieltjes algebras of B(G) and of the measure algebra M(G), Rocky Mountain J. Math. 11 (3) (1981) 459–472. [17] C.C. Graham, A.T.-M. Lau, M. Leinert, Separable translation-invariant subspaces of M(G) and other dual spaces on locally compact groups, Colloq. Math. 55 (1) (1988) 131–145. [18] R.D. Holmes, A.T.-M. Lau, Nonexpansive actions of topological semigroups and fixed points, J. London Math. Soc. 5 (2) (1972) 330–336. [19] P. Jolissaint, Invariant states and a conditional fixed point property for affine actions, Math. Ann. 304 (3) (1996) 561–579. [20] W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965) 1004–1006. [21] E.H. Lacey, The Isometric Theory of Classical Banach Spaces, Springer-Verlag, New York, 1974. [22] A.T.-M. Lau, Invariant means on almost periodic functions and fixed point properties, Rocky Mountain J. Math. 3 (1973) 69–76. [23] A.T.-M. Lau, Semigroup of nonexpansive mappings on a Hilbert space, J. Math. Anal. Appl. 105 (1985) 514–522. [24] A.T.-M. Lau, M. Leinert, Fixed point property and Fourier algebra of a locally compact group, Trans. Amer. Math. Soc. 360 (12) (2008) 6389–6402. [25] A.T.-M. Lau, P.F. Mah, Quasi-normal structures for certain spaces of operators on a Hilbert space, Pacific J. Math. 121 (1986) 109–118. [26] A.T.-M. Lau, P.F. Mah, Normal structure in dual Banach spaces associated with a locally compact group, Trans. Amer. Math. Soc. 310 (1988) 341–353. [27] A.T.-M. Lau, P.F. Mah, A. Ülger, Fixed point property and normal structure for Banach spaces associated to locally compact groups, Proc. Amer. Math. Soc. 125 (1997) 2021–2027. [28] A.T.-M. Lau, A.L.T. Paterson, Group amenability properties for von Neumann algebras, Indiana Univ. Math. J. 55 (4) (2006) 1363–1388. [29] A.T.-M. Lau, A. Ülger, Some geometric properties on the Fourier and Fourier–Stieltjes algebras of locally compact groups, Arens regularity and related problems, Trans. Amer. Math. Soc. 337 (1993) 321–359. [30] A.T.-M. Lau, W. Takahashi, Weak convergence and nonlinear ergodic theorems for reversible semigroups of nonexpansive mappings, Pacific J. Math. 126 (1987) 277–294. [31] A.T.-M. Lau, W. Takahashi, Invariant submeans and semigroups of non-expansive mappings on Banach spaces with normal structure, J. Funct. Anal. 25 (1996) 79–88. [32] A.T.-M. Lau, Yong Zhang, Fixed point properties of semigroups of non-expansive mappings, J. Funct. Anal. 254 (2008) 2534–2554. [33] C. Lennard, C1 is uniformly Kadec–Klee, Proc. Amer. Math. Soc. 109 (1990) 71–77. [34] T.C. Lim, Characterization of normal structures, Proc. Amer. Math. Soc. 43 (1974) 313–319. [35] T.C. Lim, Asymptotic centres and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math. 90 (1980) 135–143.
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[36] T. Mitchell, Fixed points of left reversible semigroups of non-expansive mappings, Kodai Math. Sem. Rep. 22 (1970) 322–323. [37] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York, 1979. [38] K. Taylor, Geometry of the Fourier algebras and locally compact groups with atomic representations, Math. Ann. 262 (1983) 183–190.
Journal of Functional Analysis 258 (2010) 373–396 www.elsevier.com/locate/jfa
The classification problem for nonatomic weak Lp spaces ✩ Denny H. Leung a,∗,1 , Rudy Sabarudin b a Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543 b Mathematics Support Group, Temasek Engineering School, Temasek Polytechnic, 21 Tampines Avenue 1,
Singapore 529757 Received 3 April 2009; accepted 4 August 2009 Available online 15 August 2009 Communicated by K. Ball
Abstract The aim of the paper is to study the isomorphic structure of the weak Lp space Lp,∞ (Ω, Σ, μ) when (Ω, Σ, μ) is a purely nonatomic measure space. Using Maharam’s classification of measure algebras, it is shown that every such Lp,∞ (Ω, Σ, μ) is isomorphic to a weak Lp space defined on a weighted direct sum of product measure spaces of the type 2κ . Several isomorphic invariants are then obtained. In particular, it is found that there is a notable difference between the case 1 < p < 2 and the case where 2 p < ∞. Applying the methods developed, we obtain an isomorphic classification of the purely nonatomic weak Lp spaces in a special case. © 2009 Elsevier Inc. All rights reserved. Keywords: Weak Lp spaces; Isomorphism of Banach spaces
✩ The contents of the article constitute part of the second author’s PhD thesis prepared under the supervision of the first author and submitted to the National University of Singapore. * Corresponding author. E-mail addresses:
[email protected] (D.H. Leung),
[email protected] (R. Sabarudin). 1 Research of the first author was partially supported by AcRF project No. R-146-000-086-112.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.08.001
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1. Introduction Let 1 < p < ∞ and let (Ω, Σ, μ) be a measure space. The weak Lp space Lp,∞ (Ω, Σ, μ) is the space of all measurable functions f on (Ω, Σ, μ) so that 1/p < ∞. f = sup c μ ω: f (ω) > c c>0
A good source of information regarding the weak Lp spaces, and more generally, the Lorentz spaces Lp,q , is [1]. While · is only a quasinorm on Lp,∞ (Ω, Σ, μ), it is equivalent to a norm; in fact, if we set 1− 1 |||f ||| = sup μ(σ ) p σ
|f |, σ
where the sup is taken over all sets σ ∈ Σ with 0 < μ(σ ) < ∞, then ||| · ||| is a norm on Lp,∞ (Ω, Σ, μ) so that f |||f |||
1 1−
1 p
f .
(1)
(To see the inequality, use the fact that f 1 if and only if the decreasing rearrangement f ∗ −1
of |f | satisfies f ∗ (t) t p for t ∈ (0, ∞). See [1, Chapter 4, §4], in particular, Lemma 4.5.) As we will be concerned exclusively with the isomorphic structure of weak Lp spaces, we will primarily utilize the quasinorm · . The weak Lp spaces arise naturally in interpolation theory, and find applications in harmonic analysis, probability theory and functional analysis. As a class, they share many of the properties of the classical Lebesgue Lp spaces and yet are different in many respects. Thus it is a natural and interesting problem to try to understand the isomorphic structure of the class of weak Lp spaces. In [7], the first author gave a complete isomorphic classification of the atomic weak Lp spaces. In [8], however, it was shown that, in general, Lp,∞ (Ω, Σ, μ) behaves differently for atomic and nonatomic measure spaces. In the present paper, we will attempt to classify isomorphically all purely nonatomic weak Lp spaces. While the attempt is only wholly successful for a special subclass, many interesting results have been thrown up along the way. In particular, the bifurcation in behavior between the cases where 1 < p < 2 and where 2 p < ∞ is quite unexpected and does not occur for atomic weak Lp spaces. The classification of the Lebesgue spaces Lp (Ω, Σ, μ) is classical (an exposition may be found in [5]) and is based on Maharam’s classification of measure algebras [9]. In Section 2, we make use of Maharam’s result to show that if (Ω, Σ, μ) is a purely nonatomic measure space, then Lp,∞ (Ω, Σ, μ) is isomorphic to Lp,∞ ( α<τ aα · 2κα ), where α<τ aα · 2κα denotes a weighted direct sum of the product measure spaces 2κα . The representation is further refined in Theorem 1. In Section 3, several isomorphic invariants are obtained. By an isomorphic invariant, we mean a parameter, defined only in terms of sequences (aα )α<τ and (κα )α<τ mentioned in the representation above, that depend solely on the isomorphism class of the space Lp,∞ ( α<τ aα · 2κα ). Essentially, the invariants obtained measure either the complexity (sup κα ) or the “width” (|τ |) of
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the measure space α<τ aα · 2κα , or a combination of both. However, the surprising fact emerges that the width is only an invariant when 2 p < ∞. In Section 4, making use of some results of Carothers and Dilworth[2–4] and arguments of a probabilistic flavor, it is shown that the weak Lp spaces defined on α<κ 2ℵ0 ⊕ 2κ and 2κ respectively are isomorphic if 1 < p < 2. In the final section, we make use of the methods developed in the preceding sections to give an isomorphic classification of nonatomic weak Lp spaces in a special case. The paper ends with a list of several open problems. 2. Reduction to standard form The main objective of this section is to show that every weak Lp space is isomorphic to a weak space defined on a measure space of a special form. This runs in parallel to the situation in the Lebesgue spaces Lp . The argument relies on Maharam’s classification of measure algebras. Let us establish some notation regarding measure spaces that will be used throughout the rest of the paper. By 2 we denote the two point measure space {−1, 1}, where each of the one-point sets {−1} and {1} is assigned a measure of 1/2. If κ is a cardinal, let 2κ be the product measure space of κ copies of 2. If (Ω, Σ, μ) is a measure space and a is a positive real number, denote by a· (Ω, Σ, μ) the measure space (Ω, Σ, aμ). Given a family of measure spaces (Ωα , Σα , μα ), let α (Ωα , Σα , μα ) be the measure space (Ω, Σ, μ), where Ω = α Ωα (we assume here that the sets Ωα are pairwise disjoint; otherwise, replace them with pairwise disjoint copies) and Σ is the smallest σ -algebra generated by α Σα . Note that σ ⊆ Ω belongs to Σ if and only if σ ∩ Ωα ∈ Σα for all α and either σ ∩ Ωα = ∅ for all but countably
many α or σ ∩ Ωα = Ωα for all but countably many α. For σ ∈ Σ, μ(σ ) is defined to be α μα (σ ). We can now state the main result of this section. Lp
Theorem 1. Let (Ω, Σ, μ) be a purely nonatomic measure space and let 1 < p < ∞. Then Lp,∞ (Ω, Σ, μ) is isomorphic to a direct sum of at most three spaces E ⊕ F ⊕ G, where, if nontrivial, ∞ ∞
p,∞ κα p,∞ κn p,∞ −n κn E=L 2 2 ; 2 ·2 G=L . ; F =L α<ω1 ·τ
n=1
n=1
Here κα , κn and κn are infinite cardinals so that, when present, κα1 κα2 < κn 1 κn 2 < κn 1 κn 2 if α1 < α2 and n1 < n2 . Let (Ω, Σ, μ) be a measure space. Define an equivalence relation on Σ by σ1 ∼ σ2 if μ(σ1 σ2 ) = 0, where denotes the symmetric difference. Write the equivalence class containing σ as σˆ and let the set of equivalence classes be denoted by Σˆ . Clearly, Σˆ is a Boolean algebra under the operations σˆ 1 ∨ σˆ 2 = (σ1 ∪ σ2 )ˆ,
σˆ 1 ∧ σˆ 2 = (σ1 ∩ σ2 )ˆ,
¬σˆ = (Ω \ σ )ˆ.
We may also transfer the measure μ over to Σˆ by defining μ( ˆ σˆ ) = μ(σ ) for all σ ∈ Σ . The subset of Σ consisting of all σ with μ(σ ) < ∞ is denoted by Σ0 . Let Σˆ 0 = {σˆ : σ ∈ Σ0 }.
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Now suppose that (Ω , Σ , μ ) is another measure space, with the corresponding objects Σˆ , μˆ and Σˆ . A finite measure isomorphism is a bijection Φ : Σˆ 0 → Σˆ such that 0
0
Φ(σˆ 1 ∨ σˆ 2 ) = Φ(σˆ 1 ) ∨ Φ(σˆ 2 ),
Φ(σˆ 1 ∧ σˆ 2 ) = Φ(σˆ 1 ) ∧ Φ(σˆ 2 ),
μˆ Φ(σˆ ) = μ( ˆ σˆ ).
If such a finite measure isomorphism exists, we say that the measure spaces (Ω, Σ, μ) and (Ω , Σ , μ ) are finitely measure isomorphic. The words “finite” and “finitely” are suppressed if the measure spaces under consideration are finite measure spaces. The first proposition is well known. Proposition 2. Suppose that 1 < p < ∞ and that the measure spaces (Ω, Σ, μ) and (Ω , Σ , μ ) are finitely measure isomorphic. Then the spaces Lp,∞ (Ω, Σ, μ) and Lp,∞ (Ω , Σ , μ ) are isometrically lattice isomorphic. Sketch of that f is a nonnegative function in Lp,∞ (Ω, Σ, μ). For any C > 1,
Proof. Suppose k1 C let g = ∞ {C k f
Then g f Cg. Therefore, the space X ak 1σk , where ak ∈ R and (σk ) is a pairwise of all functions in Lp,∞ (Ω, Σ, μ) of the form p,∞ of L (Ω, Σ, μ). Let Φ : Σˆ 0 → Σˆ 0 be a finite disjoint sequence in Σ0 , is a dense sublattice
measure isomorphism. The map T : ak 1σk → ak 1τk , where τˆk = Φ(σˆ k ), is an isometric lattice isomorphism from X onto a dense sublattice of Lp,∞ (Ω , Σ , μ ). 2 Theorem 3 (Maharam). (See [9].) Let (Ω, Σ, μ) be a purely nonatomic finite measure space. Then there are a sequence of positive real numbers (an )κand a sequence of infinite cardinals (κn ) such that (Ω, Σ, μ) is measure isomorphic to an · 2 n . Proposition 4. Let (Ω, Σ, μ) be a purely nonatomic measure space. There exist positive real numbers aα and infinite cardinals κα so that Lp,∞ (Ω, Σ, μ) is isometrically lattice isomorphic p,∞ ( aα · 2κα ). to L Proof. By Zorn’s Lemma, there exists a family of sets (Ωα ) in Σ0 so that (i) μ(Ωα ∩ Ωβ ) = 0 if α = β and (ii) if σ ∈ Σ0 and μ(σ ∩ Ωα ) = 0 for all α, then μ(σ ) = 0. Let Σα = {σ ∩ Ωα : σ ∈ Σ} and μα = μ|Σα . Denote by (Ω , Σ , μ ) the measure space α (Ωα , Σα , μα ). It is straightfor ward to check that the map Φ : Σˆ 0 → Σˆ 0 , σˆ → ( α∈A (σ ∩ Ωα ))ˆ, A = {α: μ(σ ∩ Ωα ) > 0}, is a finite measure By Maharam’s Theorem, each (Ωα , Σα , μα ) is measure isomor isomorphism. κα,n . It follows easily that (Ω, Σ, μ) is finitely measure isomorphic to a · 2 phic to some n α,n κα,n . The desired conclusion follows from Proposition 2. 2 α n aα,n · 2 Suppose ((Ωα , Σα , μα ))α and ((Ωα , Σα , μ α ))α are families of finite measure spaces. Assume that for each α, there is a measure space (Ωα , Σα , μ α ) measure isomorphic to (Ωα , Σα , μα ) so that (1) Ωα is a subset of Ωα and Ωα ∈ Σα ; (2) Σα is a sub-σ -algebra of Σα ∩ Ωα = {σ ∩ Ωα : σ ∈ Σα }; (3) μ α = μ α|Σ . α
Then we write α (Ωα , Σα , μα ) → α (Ωα , Σα , μ α ). For each α, the operator Eα that maps each f ∈ L1 (Ωα , Σα ∩ Ωα , μ α |Σ ∩Ω ) to its Radon–Nikodym derivative with respect to the α
α
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measure μ α (defined on the sub-σ -algebra Σα of Σα ∩ Ωα ) is a norm 1 projection from L1 (Ωα , Σα ∩ Ωα , μ α |Σ ∩Ω ) onto L1 (Ωα , Σα , μ α ). Keeping in mind the form of the norm ||| · ||| defined α α in Section 1, we see that Eα is also a norm 1 projection from Lp,∞ (Ωα , Σα ∩ Ωα , μ α |Σ ∩Ω ) α α onto Lp,∞ (Ωα , Σα , μ α ), provided both spaces are equipped with the norm ||| · |||. When both of these spaces are equipped with the quasinorm · , it follows from inequality (1) in Section 1 that Eα (1− p1 )−1 . Denote by Pα the operator from Lp,∞ (Ωα , Σα , μ α ) onto Lp,∞ (Ωα , Σα , μ α ) given by Pα f = Eα (f|Ωα ). Then Pα is a projection of norm (1 − p1 )−1 . It is easy to ver ify that the map fα → Pα fα is a bounded projection from Lp,∞ ( α (Ωα , Σα , μ α )) onto Proposition 2, Lp,∞ ( α (Ωα , Σα , μα )) is isomorphic to a comLp,∞ ( α (Ωα , Σα , μ α )). By p,∞ ( α (Ωα , Σα , μ α )). plemented subspace of L
Proposition 5. Consider the measure spaces α aα · 2κα and β bβ · 2κβ , where (aα ), (bβ ) are positive real numbers and (κα ), (κβ ) are infinite cardinals. Suppose that for each α, there is a set
J (α) of indices β so that the sets (J (α))α are pairwise disjoint, β∈J (α) bβ aα for all α and κβ κα for all β ∈ J (α). Then Lp,∞ ( α aα · 2κα ) is isomorphic to a complemented subspace of Lp,∞ ( β bβ · 2κβ ).
aα = β∈J (α) cβ Proof. For each α, choose nonnegative real numbers (cβ )β∈J (α) so that κα and and each β. Now aα · 2κα is measure isomorphic to β∈J (α) cβ · 2 that cβ κbβ for κ κ α α α is finitely measure isomorphic to (α) cβ · 2 → β∈J (α) bβ · 2 . Hence α aα · 2 β∈J κα ) and ( c · 2 β α β∈J (α) α
cβ · 2
κα
→
α
β∈J (α)
bβ · 2
κα
→
b β · 2 κβ ,
α β∈J (α)
β∈J (α)
where the final “ →” follows from the fact that κα κβ for all β ∈ J (α). By Proposition 2 and the discussion above, Lp,∞ ( α aα · 2κα ) is isomorphic to a complemented subspace of Lp,∞ ( α β∈J (α) bβ · 2κβ ), which in turn is clearly isomorphic to a complemented subspace of Lp,∞ ( β bβ · 2κβ ). 2 By Proposition 4, given a purely nonatomic measure space, there exist an ordinal τ , positive real numbers (aα )α<τ and infinite cardinals (κα )α<τ so that Lp,∞ (Ω, Σ, μ) is isometrically p,∞ ( α<τ aα · 2κα ). We may also assume that κα κβ
if α β < τ . We lattice isomorphic to L now further refine this representation. Let τ1 be the smallest ordinal such that τ1 α<τ aα < ∞ if such an ordinal exists. Otherwise, let τ1 = τ . Write τ1 = ω1 · τ2 + γ , where γ is a countable ordinal. Now Lp,∞ (Ω, Σ, μ) is isomorphic to the direct sum L
p,∞
α<ω1 ·τ2
aα · 2
κα
⊕L
p,∞
ω1 ·τ2 α<τ1
aα · 2
κα
⊕L
p,∞
. aα · 2 κα
(2)
τ1 α<τ
Any one (but not all) of the three terms may be trivial. Let (κα ) be a family of infinite For each α, the measure space 2κα is measure iso cardinals. κ κ κ κα α α α ⊕ 2 . Hence a · 2 is finitely measure isomorphic to ( morphic to 2 α α α aα · 2κ ) ⊕ κ p,∞ κ p,∞ α α ( α aα · 2 ) is isomorphic to L ( α aα · 2 α ) ⊕ ( α aα · 2 ). By Proposition 2, L
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Lp,∞ ( α aα · 2κα ). Combined with the representation (2), we see that Lp,∞ (Ω, Σ, μ) is isomorphic to its square Lp,∞ (Ω, Σ, μ) ⊕ Lp,∞ (Ω, Σ, μ) for any purely nonatomic measure space (Ω, Σ, μ). Lemma 6. Suppose that η is a nonzero ordinal and (γα )α<ω1 ·η is an increasing sequence of ordinals so that (i) γ0 = 0 and for all α < ω1 · η, γα+1 = γα + λα for some countable ordinal λα , and (ii) γα = supξ <α γξ for all limit ordinals α < ω1 · η. Then γα < ω1 · η for all α < ω1 · η. Proof. Since the supremum of countably many countable ordinals is countable, γα < ω1 for all α < ω1 . It follows that the lemma holds for η = 1. Suppose that the lemma holds for all η < η0 . If α < ω1 · η0 , we can write α = ω1 · η + β + m, where η < η0 , β is either a countable limit ordinal or 0, and m < ω. By the inductive hypothesis, γω1 ·η ω1 · η. If ω1 · η α < ω1 · η + β, let γα = γω1 ·η + ξα . Then ξα is countable and hence supω1 ·ηα <ω1 ·η+β ξα is countable. Therefore, γω1 ·η+β < ω1 · (η + 1). It follows that γα < ω1 · (η + 1) ω1 · η0 . 2 Recall that a well-known variant of Pełczy´nski’s Decomposition Method states that if E and F are Banach spaces so that E is isomorphic to E ⊕ E, F is isomorphic to F ⊕ F , E is isomorphic to a complemented subspace of F and F is isomorphic to a complemented subspace of E, then E and F are isomorphic. Proposition 7. If τ2 > 0, then Lp,∞ ( α<ω1 ·τ2 aα · 2κα ) is isomorphic to Lp,∞ ( α<ω1 ·τ2 2κα ). Proof. Let (aα )α<ω1 ·τ2 and (bα )α<ω1 ·τ2 be any two transfinite sequences of positive real numbers. Set γ0 = 0. If γα < ω1 · τ2 has
been chosen for some α < ω1 · τ2 , there exists γα+1 = γα + λα for some countable λα such that ξ ∈[γα ,γα+1 ) bξ aα . In particular, γα+1 < ω1 · τ2 . If α < ω1 · τ2 is a limit ordinal and γξ < ω1 ·τ2 has been defined for all ξ < α, let γα = supξ <α γξ . By Lemma 6, γα < ω1 · τ2 . Thus γα is defined for all α < ω1 · τ2 . Take J (α) to be the set [γα , γα+1 ) for each α. Since γ Proposition 5 that α α for all α, κβ κγ0 κγα for all β ∈ J (α). It follows from Lp,∞ ( α<ω1 ·τ2 aα ·2κα ) is isomorphic to a complemented subspace of Lp,∞ ( α<ω1 ·τ2 bα ·2κα ). The conclusion of the proposition follows by symmetry and by Pełczy´nski’s Decomposition Method. 2 Proposition 8. Suppose that ω1 · τ2 < τ1 . (1) If (κα )ω1 ·τ2 α<τ1 has a maximum κ, then Lp,∞ ( ω1 ·τ2 α<τ1 aα · 2κα ) is isomorphic to the κ space Lp,∞ ( ∞ n=1 2 ). (2) If (κα )ω1 ·τ2 α<τ1 does not have a maximum, then for any sequence of cardinals κ1 < κ2 < · · · p,∞ ( ω1 ·τ2 α<τ1 aα · 2κα ) is isomorphic such that supn κn = supω1 ·τ2 α<τ1 κα , the space L ∞ κ p,∞ ( n=1 2 n ). to L Proof. (1) Since [ω1 · τ2 , τ1 ) is countable and each aα is finite, we can choose pairwise disjoint finite subsets of N, J (α), ω1 · τ2 α < τ1 , so that |J (α)| aα for all α. By Proposition 5, κ ). 2 Lp,∞ ( ω1 ·τ2 α<τ1 aα · 2κα ) is isomorphic to a complemented subspace of Lp,∞ ( ∞ n=1 ∈ [ω · τ , τ ) be such that κ = κ. By the definition of τ , On the other hand, let τ 1 2 1 1 τ
, τ ) a = ∞. Thus, there is a sequence (J (n)) of pairwise disjoint subsets of [τ α 1 n∈N τ α<τ
1 κ so that α∈J (n) aα 1 for all n. By Proposition 5, Lp,∞ ( ∞ n=1 2 ) is isomorphic to a comple-
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mented subspace of Lp,∞ ( ω1 ·τ2 α<τ1 aα · 2κα ). The conclusion (1) follows by Pełczy´nski’s Decomposition Method. (2) Let j be an injection from [ω1 ·τ2 , τ1 ) into N and let (Ni ) be a sequence of pairwise disjoint infinite subsets of N. For each α ∈ [ω1 · τ2 , τ1 ), there exists a finite subset J (α) of Nj (α) such that p,∞ κn κα for all n ∈ J (α) and that |J (α)| aα . By Proposition 5, L ( ω1 ·τ2 α<τ1 aα · 2κα ) ∞ κ p,∞ ( n=1 2 n ). Conversely, using the fact that is isomorphic to a complemented subspace of L
∈ [ω · τ , τ ), one can find pairwise disjoint intervals (J (n)) a = ∞ for all τ 1 2 1 n∈N in τ α<τ1 α
[ω1 · τ2 , τ1 ) such that κα κn for all α ∈ J (n) and that α∈J (n) aα 1. Again, by Proposition 5, κn p,∞ ( κα Lp,∞ ( ∞ ω1 ·τ2 α<τ1 aα · 2 ). n=1 2 ) is isomorphic to a complemented subspace of L Apply Pełczy´nski’s Decomposition Method to complete the proof. 2 A similar idea applied to the last term in (2) results in a simplification of that term as well. Proposition 9. Suppose that τ1 < τ . (1) If (κα )τ1 α<τ has a maximum κ, then Lp,∞ ( τ1 α<τ aα · 2κα ) is isomorphic to the space Lp,∞ (2κ ). (2) If (κα )τ1 α<τ does not have a maximum, then there exists a sequence of infinite cardinals κ1 κ2 · · · < sup κn = sup κα such that Lp,∞ ( τ1 α<τ aα · 2κα ) is isomorphic to −n · 2κn ). Lp,∞ ( ∞ n=1 2 Proof. (1) Suppose that κ = κα0 and let b = Clearly aα0 · 2κ →
τ1 α<τ
aα . Then b is finite by the choice of τ1 .
aα · 2κα → b · 2κ .
τ1 α<τ
Since Lp,∞ (a · 2κ ) is isomorphic to Lp,∞ (2κ ) for any positive real number a, the desired conclusion follows
once again by Pełczy´nski’s Decomposition Method. (2) Let a = τ1 α<τ aα . By definition of τ1 , a is a positive real number. If λ is an infinite cardinal, then (b + c) · 2λ is measure isomorphic to b · 2λ ⊕ c · 2λ for any positive reals b and c. Splitting the appropriate terms aα · 2κα in this manner if necessary,
we may assume thatnthere is an increasing sequence of ordinals (βn )∞ so that β = τ and 0 1 βn−1 α<βn aα = a/2 for all n=0 n ∈ N. The last condition ensures that sup βn = τ . Define κn = κβn for all n ∈ N. Then sup κn = for all m. We have the following chain of relationships: sup κα > κm
a α · 2 κα =
τ1 α<τ
aα · 2κα →
n βn−1 α<βn
= →
n βn α<βn+1
aα · 2κβn
n βn−1 α<βn
a · 2 κn = n 2 n n
2aα · 2κn
βn α<βn+1
2aα · 2κα →
τ1 α<τ
2aα · 2κα .
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Since Lp,∞ ( τ1 α<τ aα · 2κα ) and Lp,∞ ( τ1 α<τ 2aα · 2κα ) are isomorphic, Pełczy´nski’s De composition Method yields that Lp,∞ ( τ1 α<τ aα · 2κα ) is isomorphic to Lp,∞ ( n 2an · 2κn ), which in turn is isomorphic to Lp,∞ ( n 21n · 2κn ). 2 The representation given by (2) before Lemma 6, together with Propositions 7, 8 and 9, yield Theorem 1. 3. Invariants In this section, we only consider weak Lp spaces represented in the form Lp,∞ ( α<τ aα · 2κα ), where (aα ) is a transfinite sequence of positive real numbers and (κα )α<τ is a nondecreasing sequence of infinite cardinals. Let π = π(τ, (κα ), (aα )) be a parameter that depends on the constants arising from the representation. We call π an isomorphic invariant if π(τ, (κα ), (aα )) = π(τ , (κα ), (aα )) whenever Lp,∞ ( α<τ aα · 2κα ) and Lp,∞ ( α<τ aα · 2κα ) are isomorphic as Banach spaces. In this section, we will show that the following parameters are isomorphic invariants. The symbol |τ | denotes the cardinality of the ordinal τ . (1) max{|τ |, sup κα }. (2) sup κα . (3) For 2 p < ∞, max{|τ |, ℵ0 }. Let us reiterate that the third parameter is only an isomorphic invariant for p in the range [2, ∞). We will show in the next section that it fails to be an isomorphic invariant if 1 < p < 2. Of course, if 2 p < ∞, the fact that the first parameter is an invariant is a consequence of the second and the third. However, the assertion is that the first parameter is an invariant for the entire range p ∈ (1, ∞). It will also be shown that, subject to some constraints, whether the set of cardinals (κ α )α<τ has a maximum element is also determined by the isomorphic class of the space Lp,∞ ( α<τ aα · 2κα ). In the rest of the section, we assume that 1 < p < ∞ unless expressly stated otherwise. The exception will only occur when we discuss the third parameter. By inequality (1) in Section 1, we have
1− p1
|f | dμ |||f |||μ(σ )
1 1− p
−1
1− p1
f μ(σ )
σ
for all f ∈ Lp,∞ (Ω, Σ, μ) and all sets σ ∈ Σ of finite measure. Theorem 10. If Lp,∞ ( α<τ aα · 2κα ) is isomorphic to Lp,∞ ( β<τ aβ · 2κβ ), then max{|τ |, sup κα } = max{|τ |, sup κβ }.
κβ Proof. Recall that Σ0 denotes the set of all measurable subsets of with fiβ<τ aβ · 2 κ β by ν. For each σ ∈ Σ , x (f ) = nite measure. Denote the measure on σ β<τ aβ · 2 0 κβ −1+1/p p,∞ (ν(σ )) ( β<τ aβ · 2 ). Moreover, σ f dν defines a bounded linear functional on L
1 −1 1 f sup xσ (f ) 1 − f 2 p σ ∈Σ 0
(3)
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for all f ∈ Lp,∞ ( β<τ aβ · 2κβ ). There is a subset S of Σ0 of cardinality max{|τ |, sup κβ } so that for all σ ∈ Σ0 , inf{ν(σ θ ):θ ∈ S } = 0. It follows that (3) holds with S in place of Σ0 . Transferring over to X = Lp,∞ ( α<τ aα · 2κα ) via the assumed isomorphism and normalizing, we obtain a normalized subset S of X , the dual space of X, of cardinality max{|τ |, sup κβ } and c > 0 so that supx ∈S |x (f )| > cf for all f ∈ X. Suppose that A is a subset of normalized elements of X of cardinality greater than max{|τ |, sup κβ }. For each f ∈ A, there exists xf ∈ S so that |xf (f )| > c. Now there is an x ∈ S so that xf = x for infinitely many f ∈ A. It follows that for any n ∈ N,
there is a subset F of A having exactly n elements and a choice of signs (ηf )f ∈F so that f ∈F ηf f > nc.
−1/p
Suppose that τ > max{|τ |, sup κβ }. For each α < τ , let fα = aα 1α , where 1α denotes the characteristic function of the component aα · 2κα in γ <τ aγ · 2κγ . The elements (fα )α<τ are
pairwise disjoint and normalized. Hence α∈F ηα fα n1/p for any subset F of [0, τ ) with n elements and any choice of signs (ηα ). This contradicts the previous paragraph. Suppose that supα κα > max{|τ |, sup κβ }. Choose α0 < τ such that κα0 > max{|τ |, sup κβ }. κα0 , let εγ be the projection of We identify κα0 with the set of ordinals less than κα0 . For each γ ∈ 2κα0 onto the γ -th component. We may regard εγ as a function on α<τ aα · 2κα by defining it to −1/p be 0 on all components aα · 2κα , α = α0 . The set (aα0 εγ )γ ∈κα0 is a normalized set of functions in X. By Khintchine’s inequality, it is equivalent to the unit vector basis of 2 (κα0 ). Once again, this contradicts the conclusion of the paragraph before last. We have shown that max{|τ |, sup κα } max{|τ |, sup κβ }. The desired conclusion follows by symmetry. 2
For each α0 < τ , let Pα0 f denote the restriction of f ∈ Lp,∞ ( α<τ aα · 2κα ) to the compoκα 0 nent aα0 · 2 . Lemma 11. Let κ be an uncountable cardinal and let (κα )α<τ be a set of infinite cardinals so for all countable subsets A of [0, τ ). Suppose that (gγ )γ ∈κ is a transfinite that supα∈A κα < κ sequence in Lp,∞ ( α<τ aα · 2κα ) that is dominated by the unit vector basis of 2 (κ). For any countable subset A of [0, τ ), there exists γ ∈ κ such that Pα gγ = 0 for α ∈ A. Consequently, there exists (γη )η<ω1 such that (gγη )η<ω1 is a pairwise disjoint sequence of functions. Proof. Assume that the first conclusion of thelemma fails for a countable subset A of [0, τ ). κα For each γ , let hγ be the restriction of gγ to α∈A aα · 2 . Then (hγ )γ ∈κ is a set of nonzero κ > κ0 ≡ functions in Lp,∞ ( α∈A aα · 2κα ) that is dominated by the 2 (κ) basis. Note that supα∈A κα · |A|. There exists a subset S of cardinality κ0 in the dual X of X = Lp,∞ ( α∈A aα · 2κα ) such that x (h) = 0 for all x ∈ S implies h = 0. In particular, for each γ ∈ κ, there exists (x , n) ∈ S × N so that |x (hγ )| > 1/n. Since κ > |S × N|, there exist an infinite subset Γ of κ and an element (x0 , n0 ) ∈ S × N so that |x0 (hγ )| > 1/n0 for all γ ∈ Γ . For every finite subset F of Γ ,
|F | 1/2 < x0 sgn x0 (hγ )hγ x0 sgn x0 (hγ )hγ C|F | n0 γ ∈F
γ ∈F
for some fixed constant C since (hγ )γ ∈κ is dominated by the unit vector basis of 2 (κ). This is clearly impossible.
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The second statement of the lemma follows from the first by induction. Indeed, choose γ0 arbitrarily. Assume that γη has been chosen for all η < ρ for some ρ < ω1 . For each γ , gγ has σ finite support and hence {α: Pα gγ = 0} is countable. Thus A = {α: Pα gγη = 0 for some η < ρ} is countable. By the first part of the lemma, there exists γρ such that Pα gγρ = 0 for all α ∈ A. This completes the inductive choice of the sequence (γη )η<ω1 . It is clear from the inductive definition that the sequence consists of pairwise disjoint functions. 2 Lemma 12. Let (Ω, Σ, μ) be a measure space. Suppose that p = 2. Then no transfinite pairwise disjoint sequence (gγ )γ <ω1 of length ω1 can be equivalent in Lp,∞ (Ω, Σ, μ) to the unit vector basis of 2 (ω1 ). Proof. Suppose that (gγ )γ <ω1 is a pairwise disjoint sequence in Lp,∞ (Ω, Σ, μ) that is equivalent to the unit vector basis of 2 (ω1 ). Every normalized pairwise disjoint sequence in Lp,∞ (Ω, Σ, μ) is dominated by the unit vector basis of p of the appropriate dimension. On the other hand, (gγ )γ <ω1 is equivalent to the unit vector basis of 2 (ω1 ). Thus we must have p 2. On the other hand, (gγ )γ <ω1 is bounded away from 0. There exists δ > 0 so that for each γ , there exists a rational number cγ such that cγ (μ{|gγ | > cγ })1/p > δ. Since ω1 is uncountable, there exist c > 0 and an infinite subset Γ of [0, ω1 ) so that c(μ{|gγ | > c})1/p > δ for all γ ∈ Γ . Recall that the sets {|gγ | > c}, γ < ω1 , are pairwise disjoint. For any finite subset F of Γ , 1/p
1/p c μ gγ > c =c μ |gγ | > c > δ|F |1/p . γ ∈F
γ ∈F
Hence γ ∈F gγ δ|F |1/p . But (gγ )γ ∈κ is equivalent to the unit vector basis of 2 (κ). Thus, p 2. Since we are assuming that p = 2, we have a contradiction. 2 Denote by M p,∞ (2κ ) the closure of L∞ (2κ ) in the space Lp,∞ (2κ ). Lemma 13. Let κ be an uncountable cardinal and let (κα )α<τ be a set of infinite cardinals so that supα∈A κα < κ for all countable subsets A of [0, τ ). Then M p,∞ (2κ ) does not embed p,∞ ( α<τ aα · 2κα ). isomorphically into L Proof. We first consider the case where p = 2. The sequence of Rademacher functions (εγ )γ ∈κin M p,∞ (2κ ) is equivalent to the 2 (κ) basis. If M p,∞ (2κ ) embeds isomorphically in Lp,∞ ( α<τ aα · 2κα ), then the latter space contains a sequence (gγ )γ ∈κ equivalent to the 2 (κ) basis. By Lemma 11, there is a sequence (γη )η<ω1 such that (gγη )η<ω1 is a pairwise disjoint sequence of functions. However, this contradicts Lemma 12. Now suppose that T : M 2,∞ (2κ ) → L2,∞ ( α<τ aα · 2κα ) is an isomorphic embedding. Let (εγ )γ ∈κ be the sequence of Rademacher functions in M 2,∞ (2κ ). Let h0 be the constant function 1 / A0 . Since (T εγ )γ ∈κ is on 2κ . There is a countable subset A0 of [0, τ ) such that Pα T h0 = 0 if α ∈ equivalent to the unit vector basis of 2 (κ), by Lemma 11, there exists γ1 so that Pα T εγ1 = 0 for all α ∈ A0 . Set h1 = εγ1 . Next, there exists a countable subset A1 of [0, τ ) such that Pα T hi = 0 if i = 0, 1 and α ∈ / A1 . Then one can find γ2 = γ1 so that Pα T (1{h1 =−1} · εγ2 ) = 0 for all α ∈ A1 . Set h2 = 1{h1 =−1} · εγ2 . Choose a countable subset A2 of [0, τ ) such that Pα T hi = 0 if i = 0, 1, 2 / {γ1 , γ2 } so that Pα T (1{h1 =1} · εγ3 ) = 0 for all α ∈ A2 . Set and α ∈ / A2 . Then one can find γ3 ∈
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h3 = 1{h1 =1} · εγ3 . Continuing in this manner, we obtain a sequence (hi ) in M 2,∞ (2κ ) equivalent to the sequence of Haar functions in M 2,∞ [0, 1]. At the same time, (hi ) is equivalent to the pairwise disjoint sequence (T hi ) in L2,∞ ( α<τ aα · 2κα ). This is impossible according to [8, Proposition 8]. 2 Theorem 14. If Lp,∞ ( α<τ aα · 2κα ) is isomorphic to Lp,∞ ( β<τ aβ · 2κβ ), then sup κα = sup κβ .
Proof. By symmetry, it suffices to show that sup κβ sup κα . Assume on the contrary that there κ
exists β0 with κβ 0 > sup κα . Note that, in particular, κβ 0 > ℵ0 . By Lemma 13, Lp,∞ (2 β0 ) does not embed isomorphically into Lp,∞ ( α<τ aα · 2κα ). Consequently, the spaces Lp,∞ ( α<τ aα · 2κα ) and Lp,∞ ( β<τ aβ · 2κβ ) cannot be isomorphic. 2 We say that an infinite cardinal κ has uncountable cofinality if it is not equal to the supremum of a countable set of smaller cardinals. Theorem 15. Suppose that (κα )α<τ and (κβ )β<τ are sequences of cardinals with uncountable cofinality. If the spaces Lp,∞ ( α<τ aα · 2κα ) and Lp,∞ ( β<τ aβ · 2κβ ) are isomorphic and the sequence (κβ )β<τ has a maximum, then so does the sequence (κα )α<τ . Proof. Let κ = maxβ<τ κβ . Assume that the spaces Lp,∞ ( α<τ aα ·2κα ) and Lp,∞ ( β<τ aβ ·
2κβ ) are isomorphic. Then κ = supα<τ κα by Theorem 14. Suppose that the sequence (κα )α<τ does not have a maximum. Then κ > κα for all α < τ . Since κ is assumed to have uncountable p,∞ (2κ ) does cofinality, κ > supα∈A κα for all countable subsets A of [0, τ ). By Lemma 13, L not embed isomorphically into Lp,∞ ( α<τ aα · 2κα ). This contradicts the isomorphism of the spaces Lp,∞ ( α<τ aα · 2κα ) and Lp,∞ ( β<τ aβ · 2κβ ). 2 Let κ be an infinite cardinal. We have already encountered the projections εγ : 2κ → {−1, 1}, γ ∈κ (Rademacher functions). For any finite subset F of κ, we define the Walsh function WF to be γ ∈F εγ , where the empty product is taken to be the constant function 1. Denote the set of all Walsh functions WF , |F | n, by Wn . Lemma 16. Let 1 q < ∞. From any infinite set of Walsh functions in Wn , one can extract an infinite sequence that is equivalent in the norm of Lq (2κ ) to the 2 basis. Proof. We prove the lemma by induction on n. The case n = 1 is Khintchine’s inequality. Assume that the lemma holds for some n. Let (Fk ) be an infinite sequence of pairwise distinct finite subsets of κ, with |Fk | n + 1 for all k. By considering a suitable subsequence, we may assume that either (Fk ) is pairwise disjoint, or that there exists γ ∈ Fk . In the former case, (Fk ) has the some joint distribution as the Rademacher functions and the result follows from Khintchine’s inequality. In the latter case, for any finitely supported sequence (ak ), ak WFk = εγ ak WFk \{γ } ak WFk \{γ } = q
q
and the desired result follows by the inductive assumption.
q
2
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Lemma 17. If f ∈ Lp,∞ (2κ ), then
f WF = 0 for all but countably many Walsh functions WF .
Proof. Suppose that the lemma fails. There exist f ∈ Lp,∞ (2κ ), a rational δ > 0, n ∈ N and an infinite set (WFk ) in Wn so that | f WFk | > δ for all k. Choose 1 < p < p and let q = pp −1 . By
Lemma 16, considered as a sequence in Lq (2κ ), (WFk ) has a subsequence equivalent to the 2 basis. We may assume that (WFk ) ⊆ Lq (2κ ) is equivalent to the 2 basis. Let ηk = sgn f WFk . There is a constant C < ∞ so that for any m ∈ N, m m √ mδ < f ηk WFk f p ηk WFk C mf p . k=1
k=1
This is impossible since Lp,∞ (2κ ) ⊆ Lp (2κ ).
q
2
Corresponding to each Walsh function WF , we define a bounded linear functional xF on via xF (f ) = f WF . Note that xF 1 − p1 . A well-known fact, easily verified, is that for every finite subset F of κ, each function on 2κ that is measurable with respect to the set of coordinates F lies in the span of {WG : G ⊆ F }. As a result, the only function f ∈ Lp,∞ (2κ ) that satisfies xF (f ) = 0 for all finite subsets F of κ is the 0 function. For any set Ξ , p,∞ (Ξ ) is the weak Lp space defined on the measure space consisting of the set Ξ endowed with the counting measure. Lp,∞ (2κ )
Lemma 18. Suppose that 2 p < ∞ and that (fξ )ξ ∈Ξ is a set of functions in Lp,∞ (2κ ) dominated by the set of coordinate unit vectors in p,∞ (Ξ ). The set fξ WF = 0 for some finite F ⊆ κ Ξ = ξ: is countable. Proof. Suppose that the set Ξ is uncountable. We will select inductively a transfinite sequence (ξγ )γ <ω1 from Ξ so that the sets fξγ WF = 0 Fγ = F : F is a finite subset of κ with are pairwise disjoint. Choose ξ0 ∈ Ξ arbitrarily. Assume that ζ0 < ω 1 and that ξζ has been chosen for all ζ < ζ0 . Since ζ0 is countable, by Lemma 17, the set ζ <ζ0 Fγζ is countable. Suppose that for each ξ ∈ Ξ \ {ξζ : ζ < ζ0 }, there exists F ∈ ζ <ζ0 Fγζ such that fξ WF = 0. Then there are a particular F ∈ ζ <ζ0 Fγζ and a rational δ > 0 so that | fξ WF | > δ for all ξ in an infinite subset Ξ of Ξ \ {ξζ : ζ < ζ0 }. If A is a finite subset of Ξ and ηξ = sgn fξ WF , then δ|A| <
ξ ∈A
ηξ
fξ WF = xF
ξ ∈A
ηξ f ξ
1 ηξ f ξ 1− . p ξ ∈A
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This contradicts the fact that (fξ )ξ ∈Ξ is dominated by the set ofcoordinate unit vectors in p,∞ (Ξ ). Thus we can choose ξζ0 ∈ Ξ \ {ξζ : ζ < ζ0 } so that fξζ0 WF = 0 for all F ∈ ζ <ζ0 Fγζ . The inductive selection is complete. For each γ < ω1 , choose Fγ ∈ Fγ so that fξγ WFγ = 0. There exist a rational r > 0 and an infinite subset Γ of ω1 so that | fξγ WFγ | > r for all γ ∈ Γ . Since the sets Fγ are pairwise disjoint, the sets Fγ are pairwise distinct. Let 1 < p < p and q = is a sequence (γk ) in Γ so that, in the norm of
Lq (2κ ),
p
p p −1 .
By Lemma 16, there
(WFγk ) is equivalent to the 2 basis.
Since the formal inclusion L (2κ ) ⊆ Lp,∞ (2κ ) is bounded, it follows that the sequence (xF γ ) k
is dominated by (WFγk ) ⊆ Lq (2κ ), and hence dominated by the 2 basis. Let (bk ) and (ck ) be finitely supported real sequences. For each k, let sk = sgn(bk ck fξγk WFγk ). Note that if k = j , / Fγj and hence fξγj WFk = 0. Now then Fk ∈ sk bk ck fξγk WFγk r bk ck < =
sk bk xF γ k
cj fξγj
j
sk bk xF γ
k
c f j ξγj j
C (bk ) 2 cj fξγj j
for some fixed constant C. This implies that j cj fξγj Cr (cj )2 . We have a contradiction since (fξγj ) is dominated by the unit vectors in p,∞ and 2 p < ∞. 2 Theorem 19. Suppose that 2 p < ∞. If Lp,∞ ( α<τ aα · 2κα ) is isomorphic to Lp,∞ ( β<τ aβ · 2κβ ), then max{|τ |, ℵ0 } = max{|τ |, ℵ0 }. Proof. By symmetry, it suffices to show that max{|τ |, ℵ0 } max{|τ |, ℵ0 }. Assume on the contrary that |τ | > max{|τ |, ℵ0 }. There exists n ∈ N such that |Ξ | > max{|τ |, ℵ0 }, where Ξ= {β < τ : 1/n aβ n}. Note that p,∞ (Ξ ) is isomorphic to a subspace of Lp,∞ ( β<τ aβ · 2κβ ) and hence to a subspace of Lp,∞ ( α<τ aα · 2κα ). Let (fξ )ξ ∈Ξ be a set of functions in Lp,∞ ( α<τ aα · 2κα ) equivalent to the set of coordinate unit vectors in p,∞ (Ξ ). For each p,∞ α0 < τ , let Pα0 f be the restriction of f ∈ L ( α<τ aα · 2κα ) to the component aα0 · 2κα0 . By Lemma 18, for each α < τ , there is a countable subset Ξα of Ξ so that Pα fξ WF = 0 for all ξ ∈ Ξ \ Ξα and all Walsh functions WF . Now | α<τ Ξα | |τ | · ℵ0 < |Ξ |. Thus there exists ξ ∈ Ξ \ α<τ Ξα . For this ξ , Pα fξ = 0 for all α < τ and hence fξ = 0, which is absurd. 2 Remark. It is known [6] that Lp,∞ ( α<ω 2ℵ0 ) is isomorphic to Lp,∞ (2ℵ0 ). Thus it is not possible in Theorem 19 to conclude that |τ | = |τ |.
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4. The big squeeze: 1 < p < 2 In this section, we will show that the third invariant from the last section does not apply in the range 1 < p < 2; equivalently, Theorem 19 does not hold for 1 < p < 2. For a Banach space E, denote by ∞ (E) the space of all bounded sequences (xn ) in E with the norm (xn ) = supn xn . We will first show that for 1 < p < 2 and any infinite cardinal κ, ∞ (p,∞ (κ)) is isomorphic to a complemented of Lp,∞ (2κ ). Then, subspace viewing p,∞ ℵ p,∞ n L ( α<κ 2 0 ) as a “limit” of the spaces L ( α<κ 2 ), it is shown that Lp,∞ ( α<κ 2ℵ0 ) is isomorphic to a complemented subspace of ∞ (p,∞ (κ)). It is then easy to deduce that the spaces Lp,∞ ( α<κ 2ℵ0 ) ⊕ Lp,∞ (2κ ) and Lp,∞ (2κ ) are isomorphic. The main idea is to use probabilistic independence to replicate disjointness (in the lattice sense). The strategy has been used, for example, to embed q isometrically isomorphically into Lp , 1 p < q < 2. In our case, simply using iid random variables will not do the job. (This assertion can be formulated in a precise way.) Instead, we use random variables that are independent in sections. The argument relies vitally on certain norm estimates of square functions in Lorentz spaces due to Carothers and Dilworth [2–4]. A set of random variables (fi ) on [0, 1] is said to be symmetric if for each finite subset F , the joint distribution of (fi )i∈F is the same as that of (±fi )i∈F . Theorem 20. (See [2, Lemma 2.2].) Let X be a rearrangement invariant space on [0, 1]. There is a finite positive constant D so that for every symmetric sequence (fi ) in X, and all scalars (ai )ni=1 , n n 1/2 n 1/2 |ai fi |2 ai fi D |ai fi |2 D −1 . i=1
i=1
i=1
For the remainder of this section, let 1 < p < 2 and let κ be an infinite cardinal. Denote by κ μ the usual product set of functions f1 , . . . , fn
in Lp,∞ (2κ ), the
n measure on 2 . Given a finite p,∞ disjoint sum i=1 fi is any function f on L [0, ∞) so that λ{|f | > t} = ni=1 μ{|fi | > t} for all t > 0, where λ denotes Lebesgue measure. Theorem 21. (See [3, Corollary 2.7].) For 1 < p < 2, there is a finite positive constant C such that n 1/2 2 |fi | i=1
p,∞
n fi C i=1
p,∞
for any f1 , . . . , fn ∈ Lp,∞ [0, 1]. For any probability space (Ω, Σ, ν), Theorems 20 and 21 apply to functions in Lp,∞ (Ω, Σ, ν), since any finite set of measurable functions on (Ω, Σ, ν) has a copy on [0, 1] with the same joint distribution. Let η0 > 0 be an absolute constant so that 1 − e−x x/2 for all x ∈ [0, η0 ].
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Proposition 22. Suppose that N ∈ N, 0 < b < 1/2, N b η0 and that (fα )α<κ is a set of iid random variables in Lp,∞ (2κ ) so that each fα has the same distribution as b−1/p (1[0,b/2) − 1[b/2,b) ). Then for all scalars (aα ) and all subsets J of κ of cardinality at most N ,
1/2 2 . a f |a f | 2−1/p (aα )α∈J p,∞ = 2−1/p α α α α α∈J
α∈J
Proof. Let I be a finite subset of κ with cardinality at most N . For any 0 < c < b−1/p , μ
|fα | c = μ |fα | c = (1 − b)|I | e−b|I | .
α∈I
α∈I
Hence μ
|fα | > c
1 − e−b|I |
α∈I
b|I | 2
since b|I | < η0 . Then
1/2 2 |fα | α∈I
sup 0
b−1/p
1/2 1/p 2 c μ |fα | >c
b|I | 2
1/p
α∈I
= 2−1/p |I |1/p .
It is clear that α∈J aα fα = (aα )α∈J p,∞ . Thus, if α∈J aα fα > 1, then there exists a subset I of J so that |aα | |I |−1/p for all α ∈ I . Therefore,
1/2
1/2 2 −1/p 2 2−1/p . |aα fα | |fα | |I | α∈J
2
α∈I
Choose positive sequences (wn ) and (bn ) so that
(1) wn = 1, ∞ (2) n=m+1 wn wm bm for all m, (3) nbn η0 for all n. For each n ∈ N, let (F˜α (n))α∈κ be iid random variables on 2κ (with respect to the product measure μ) so that each F˜α (n) has the same distribution as (wn bn )−1/p (1[0,bn /2) − 1[bn /2,bn ) ). Let Fα (n) be formally the same function as F˜α (n), but regarded as a function on the space wn · 2κ . κ We view 2 as the direct sum n wn · 2κ . With respect to the direct sum, a function f on 2κ will be written as fn , where each fn is a function on the component wn · 2κ . In the proof of the next proposition, we will use the fact that since 1 < p < 2, any (aα ) in the ball of p,∞ (κ)
−2/p 2 1/2 2 1/2 satisfies ( |aα | ) ( k )1/2 ( 2−p ) .
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Proposition 23. Assume that for each n ∈ N, a real sequence (aα (n)) α∈κ is given so that supn (aα (n))α∈κ p,∞ < ∞. Let I be a finite subset of κ and set f = n α∈I aα (n)Fα (n). Then 2−1/p D −1 sup aα (n) α∈I p,∞ f C D sup aα (n) α∈I p,∞ , n|I |
n∈N
2 p/2 1/p where C = [C p + ( 2−p ) ] and D, C are the constants from Theorems 20 and 21 respectively.
Proof. By homogeneity, it suffices to prove the proposition assuming that sup aα (n) α∈I p,∞ = 1. n
For each α ∈ κ, let Gα = n aα (n)Fα (n). First we estimate ( α∈I |Gα |2 )1/2 . Given c > 0, let
n0 be the smallest natural number so that μ{( α∈I |aα (n0 )Fα (n0 )|2 )1/2 > c} = 0. In particular, c<
1/2 aα (n0 )2 (wn0 bn0 )−1/p
2 2−p
1/2
(wn0 bn0 )−1/p .
By Theorem 21,
1/2 C aα (n0 )Fα (n0 )2 aα (n0 )Fα (n0 ) α∈I
α∈I
C aα (n0 )
p,∞
C
and hence
p 2 1/2 C aα (n0 )Fα (n0 ) μ >c . c α∈I
Now μ
1/2 |Gα |2
α∈I
∞ 1/2 aα Fα (n0 )2 >c μ >c + wn n=n0 +1
α∈I
p p
p/2 C C 2 1 + wn0 bn0 + . c c 2−p cp Thus
1/2 1/p |Gα |2 >c Cp + c μ α∈I
2 2−p
p/2 1/p .
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Since c > 0 is arbitrary, we have ( α∈I |Gα |2 )1/2 C . In particular, this shows that Gα ∈ Lp,∞ (2κ ) for all α and hence f ∈ Lp,∞ (2κ ). Since (Gα )α∈κ is a set of symmetric random variables in Lp,∞ (2κ ),
1/2
1/2 2 2 C D = f D |G | G |G | D −1 α α α α∈I
α∈I
α∈I
by Theorem 20. Suppose that n |I |. With respect to the measure μ,
1/2 1/p 2−1/p aα (n) aα (n)F˜α (n)2 wn α∈I α∈I
by Proposition 22. Therefore,
1/2 −1 2 f = Gα D |Gα | α∈I
α∈I
2 1/2 −1 D aα (n)Fα (n) α∈I
Lp,∞ (wn ·μ)
1/2 1/p aα (n)F˜α (n)2 = D −1 wn p,∞ L (μ) −1/p −1 aα (n) α∈I . 2 2 D For 1 < p < ∞, q = p/(p − 1) and an infinite cardinal κ, we refer to [1, Definition 4.1] for the definition of the Lorentz space Lq,1 (2κ ). By [1, Corollary 4.8], Lp,∞ (2κ ) is naturally isomorphic to the dual of Lq,1 (2κ ). The span of the characteristic functions of measurable subsets of 2κ is ) → 0, then 1σn → 1σ in Lq,1 (2κ ). Let (fα )α∈κ be a family of dense in Lq,1 (2κ ). If μ(σn σ
p,∞ κ ∗ fα we mean the weak∗ limit of the finite sums α∈I fα , I ⊆ κ, (2 ). By w functions in L |I | < ∞, along the Fréchet filter on κ. Proposition 24. Let ((aα (n))α∈κ )∞ ∞ (p,∞ (κ)). For all α ∈ κ, let Gα = n aα (n)Fα (n). n=1 ∈
Then the sum w ∗ Gα exists and w ∗ Gα C D supn (aα (n))α∈κ p,∞ . Moreover, ∗ Gα 2−1/p D −1 aα (n) α∈I w for all n ∈ N and all subsets I of κ with |I | n.
Proof. For all finite subsets I of κ, α∈I Gα C D supn (aα (n))p,∞ by Proposition 23. the functions Suppose that I is a finite subset of κ and g∈ Lq,1 (2κ ) is measurable with respectto
) . Since (G ) is symmetric, G g = 0 for all β ∈ κ \ I . Hence (G α α∈I α α∈κ β α∈J Gα g =
G g for all finite subsets J of κ containing I . Thus, letting Y be the space of all functions α∈I α
Gα g g ∈ Lq,1 (2κ ) measurable with respect to (Gα )α∈I for some finite set I ⊆ κ, we see that
Gα g exists for all g ∈ Y . exists for all g ∈ Y . Since { α∈I Gα : I ⊆ κ, |I | < ∞} is bounded, ˜ If σ ∈ Σ˜ , then there exists a sequence (σn ) Denote the σ -algebra generated by (Gα )α∈κ as Σ.
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such that 1σn ∈ Y for all n and μ(σ σn ) → 0. Thus 1σ ∈ Y . Hence all Σ˜ -measurable functions κ in Lq,1 (2κ ) belong to Y . Let h ∈ Lq,1 (2 ) and let g be the conditional
expectation of h with ˜ Then g ∈ Y and Gα h = Gα g for all α ∈ κ. Hence respect to Σ. Gα h exists. This proves
that w ∗ Gα exists. Obviously, w ∗ Gα C D supn (aα (n))p,∞ . q,1 κ For any finite subset I of κ and any δ > 0, there exists
a normalized g ∈ L (2 ), measurable with respect to (Gα )α∈I , such that G g G − δ. Then α∈I α α∈I α
Gα g =
α∈J
Gα g G α − δ
α∈I
α∈I
for all J ⊇ I , |J | < ∞. Hence ∗ −1/p −1 Gα Gα D sup aα (n) α∈I p,∞ w 2 n|I |
α∈I
by Proposition 23.
2
∞ ∞ Partition N into a sequence ofinfinite subsets (Mm )m=1 . Suppose that a = ((aα (n))α∈κ )n=1 ∈ Define Hα (a) = m n∈Mm aα (m)Fα (n) for all α ∈ κ.
∞ (p,∞ (κ)).
Theorem 25. Suppose that 1 < p < 2 and that κ is an infinite cardinal. Then Lp,∞ (2κ ) contains a complemented subspace isomorphic to ∞ (p,∞ (κ)).
Proof. By Proposition 24, for all a ∈ ∞ (p,∞ (κ)), w ∗ Hα (a) exists and w ∗ Hα (a) C D supn (aα (n))α∈κ p,∞ . The map T : ∞ (p,∞ (κ)) → Lp,∞ (2κ ), T a = w ∗ Hα (a) is a bounded linear operator. Let I be a finite subset of κ. For any m ∈ N, there exists n ∈ Mm with n |I |. By Proposition 24, w ∗ Hα (a) 2−1/p D −1 (aα (m))α∈I . It follows that T a 2−1/p D −1 supm (aα (m))α∈I p,∞ . Thus T is an (into) isomorphism. To complete the proof, we require a bounded linear map Q : Lp,∞ (2κ ) → ∞ (p,∞ (κ)) such ∞ p,∞ (κ)). For each m, fix a free ultrafilter U on M . Assume that QT identity m m is the p,∞ on ( that fn ∈ L ( wn · 2κ ) has norm at most 1. For each n and each α ∈ κ, observe that μ{Fα (n) = 0} = wn bn . Hence
−1 fn Fα (n) (wn bn )−1/p 1 − 1 fn μ Fα (n) = 0 p 1
q(wn bn ) q
− p1
.
1 1 − ( fm ) = (wn bn ) p q fn Fα (n) defines a bounded linear functional on Lp,∞ ( wm · Thus xα,n . We 2κ ) of norm at most q. For each (α, m) ∈ κ × N, let yα,m be the weak∗ limit limn→Um xα,n ∞ claim that the map we desire. map Qf = ((yα,m (f ))α∈κ )m=1 is the Let f = fn be a normalized element in Lp,∞ ( n wn · 2κ ). Suppose that m ∈ N, c > 0 and (f )| > c for all α ∈ I . There is a sufficiently I = {α1 , . . . , αk } is a finite subset of κ so that |yα,m 1/q large n 2 in Mm so that q(2bn k) c/2 and that |xα j ,n (f )| > c for all j k. For each j k and each n ∈ N, let σj,n , respectively, σ˜ j,n , be the support of Fαj (n), respectively, F˜αj (n). (The
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391
two sets are the same; but they are associated with different measures.) If 1 j k, then using the independence of (F˜α (n))α∈κ , μ σ˜ j,n ∩
j −1
σ˜ i,n
j −1
1 − μ(σ˜ i,n )
= μ(σ˜ j,n ) 1 −
i=1
i=1
= bn 1 − (1 − bn )j −1 bn 1 − e−2bn (j −1) since 2bn η0 2bn2 (j − 1) 2bn2 k.
Thus
|f | q μ σj,n ∩ σj,n ∩(
j −1 i=1
j −1
1/q σi,n
i=1
σi,n )
1/q c q wn 2bn2 k (wn bn )1/q . 2 Now c < x
−1/q αj ,n (f ) (wn bn )
|fn |. σj,n
Hence j −1
σj,n \
i=1
c |f | > (wn bn )1/q . 2 σi,n
Therefore, 1/q
(kwn bn )
=
k
1/q μ(σj,n )
μ
j =1
>
1 q
k
1/q σj,n
j =1
1 q k
|f | = k j =1 σj,n
|f |
j =1 j −1 σj,n \ i=1 σi,n
c k(wn bn )1/q . 2q
So we have c|I |1/p = ck 1/p < 2q. This proves that Qf 2q.
1
Observe that xα,n can be identified with the function (wn bn ) p p,∞ κ L (2 ). Hence, if n ∈ Mm , then
− q1
Fα (n) in the predual of
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D.H. Leung, R. Sabarudin / Journal of Functional Analysis 258 (2010) 373–396 xα,n (T a) =
Hβ (a) xα,n
β∈κ
=
1 1 − (wn bn ) p q aβ (m) Fβ (n)Fα (n) = aα (m). β∈κ
(T a) = a (m) and QT is the identity map. Thus yα,m α
2
N Identify ℵ0 with N. For each n ∈ N, denote by εn the projection from the n-th com2 onto p,∞ ℵ0 ) is written as ponent. Suppose that κ is an infinite cardinal. A function f ∈ L ( 2 α<κ fα is a function on the α-th copy of 2ℵ0 . For each β < κ and each n ∈ N, α<κ fα , where each let εβ,n be the function α<κ )nk=1 ∈ 2n n fα , where fβ = εn and fα = 0 if αn= β. Ifnn ∈ N, ϕ = (ϕkm and α < κ, denote the set k=1 {εα,k = ϕk } by σα,ϕ . If ϕ = (ϕk )k=1 ∈ 2 and ψ = (ψk )k=1 ∈ 2m , let (ϕ, ψ) be the element (ϕ1 , . . . , ϕn , ψ1 , . . . , ψm ) in 2n+m .
Theorem 26. Suppose that 1 < p < 2 and that κ is an infinite cardinal. Then ∞ (p,∞ (κ)) p,∞ ℵ0 p,∞ (2κ ) contains a contains a complemented subspace isomorphic to L ℵ ( α<κ 2 ) and L p,∞ complemented subspace isomorphic to L ( α<κ 2 0 ). Proof.The second statement follows from the first ∞ (p,∞ (κ)) because of Theorem 25. Identify p,∞ ( ℵ0 ). For each n ∈ N with ( n p,∞ (κ × 2n ))∞ . Suppose that f = α<κ f ∈ L 2 α<κ α and each (α, ϕ) ∈ κ × 2n , let aα,ϕ = aα,ϕ (f ) = 2n/q σα,ϕ fα . If I is a finite subset of κ × 2n for some n ∈ N so that |aα,ϕ | > c for all (α, ϕ) ∈ I , then c|I | <
2
(α,ϕ)∈I
2n/q qμ
fα = 2n/q
n/q
σα,ϕ
|f |
(α,ϕ)∈I
σα,ϕ
1/q σα,ϕ
f = q|I |1/q f .
(α,ϕ)∈I
Thus (aα,ϕ )(α,ϕ)∈κ×2n p,∞ qf . Hence the map T : Lp,∞ ( α<κ 2ℵ0 ) → ( n p,∞ (κ × ∞ n 2 ))∞ , Tf = ((aα,ϕ (f ))(α,ϕ)∈κ×2n )n=1 is bounded. p,∞ Now suppose that b = ((bα,ϕ )(α,ϕ)∈κ×2n )∞ (κ × 2n ))∞ . Let n ∈ N and consider n=1 ∈ ( n the function gn =
α∈κ
Then gn Lp,∞ (
2n/p
bα,ϕ 1σα,ϕ .
ϕ∈2n
ℵ0 = (bα,ϕ )(α,ϕ)∈κ×2n p,∞ . Let U be a free ultrafilter on N and define α<κ 2 ) ∗ ∗ ℵ0 g = w limn→ weak∗ topology on Lp,∞ ( α<κ as the U gn . Here w refers to the 2 ), identified q,1 ℵ p,∞ n p,∞ 0 (κ × 2 ))∞ → L ( α<κ 2ℵ0 ), Qb = g, is dual to L ( α<κ 2 ). The map Q : ( n ℵ0 a bounded linear operator. Suppose that b = Tf for some f ∈ Lp,∞ ( α<κ that 2 ). Assume m (β, ψ) ∈ κ × 2 for some m. If n m, we have by direct computation that σβ,ψ gn = σβ,ψ fβ = m m (κ × 2 ), span a dense σβ,ψ f . Since (gn ) is bounded and the functions 1σβ,ψ , (β, ψ) ∈
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393
subspace of Lq,1 ( α<κ 2ℵ0 ), we deduce that w ∗ lim gn = f . Therefore, QT is the identity map on Lp,∞ ( α<κ 2ℵ0 ). 2 By Theorem 26 and Pełczy´nski’s Decomposition Method, Lp,∞ (2κ ) and Lp,∞ ( α<κ 2ℵ0 ) ⊕ Lp,∞ (2κ ) are isomorphic if 1 < p < 2. In particular, Theorem 19 does not extend to the range 1 < p < 2. 5. Isomorphic classification: A special case We have seen in Theorem 1 that if 1 < p < ∞ and (Ω, Σ, μ) is a purely nonatomic measure p,∞ p space, then L (Ω, Σ,κμ) is isomorphic to the weak L space defined on a measure space of the form α<τ aα · 2 α . In this section, we give, for a subclass of measure spaces in such “standard form” and 2 p < ∞, a complete isomorphic classification of the corresponding weak Lp spaces. Precisely, we prove the following theorem. Theorem 27. Suppose that (κα )α<τ and (κβ )β<τ are sequences of cardinals. Assume that each κα has uncountable cofinality and that if the sequence (κβ )β<τ has a maximum, then the maximum is attained an infinite number of times. Consider the following statements: κα for all α < τ . (1) There is an injection i : [0, τ ) → [0, τ ) such that κi(α) p,∞ κ α (2) The space L ( α<τ 2 ) is isomorphic to a complemented subspace of the space Lp,∞ ( β<τ 2κβ ). (3) The space Lp,∞ ( α<τ 2κα ) is isomorphic to a subspace of the space Lp,∞ ( β<τ 2κβ ).
If 1 < p < ∞, then (1) ⇒ (2) ⇒ (3). If 2 q < ∞, then all three statements are equivalent. Corollary 28. Suppose that (κα )α<τ and (κβ )β<τ are sequences of cardinals with uncountable cofinality. Assume that if either sequence has a maximum, then the maximum is attained an infinite number of times. If 2 p < ∞, then the spaces Lp,∞ ( α<τ 2κα ) and Lp,∞ ( β<τ 2κβ ) are isomorphic if and only if there are injections i : [0, τ ) → [0, τ ) and j : [0, τ ) → [0, τ ) such κα for all α < τ and κj (β) κβ for all β < τ . that κi(α) Suppose that condition (1) holds. Then it follows from Proposition 5 that the space Lp,∞ ( α<τ 2κα ) is isomorphic to a complemented subspace of Lp,∞ ( β<τ 2κβ ). The implication (2) ⇒ (3) is obvious. Thus, to complete the proof of Theorem 27, it suffices to show that (3) ⇒ (1) when 2 p < ∞. Also, Corollary 28 is an immediate consequence of Theorem 27. Recall that M p,∞ (2κ ) is the closure of L∞ (2κ ) in Lp,∞ (2κ ). Lemma 29. Suppose that (κα )α<τ and (κβ )β<τ are sequences of cardinals, where each κα has uncountable cofinality. If 2 p < ∞ and Lp,∞ ( α<τ 2κα ) isomorphically embeds into the κα for space Lp,∞ ( β<τ bβ · 2κβ ), then there exists a map k : [0, τ ) → [0, τ ) such that κk(α) −1 all α < τ and |k {β}| ℵ0 for all β < τ . Proof. Let T : Lp,∞ ( α<τ 2κα ) → Lp,∞ ( β<τ bβ · 2κβ ) be an isomorphic embedding. For each g ∈ Lp,∞ ( β<τ bβ · 2κβ ) and each β < τ , let Pβ g denote the restriction of g to the com-
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ponent bβ · 2κβ . Let α < τ and consider the set Bα consisting of all β < τ such that κβ < κα . By Lemma 13, M p,∞ (2κα ) does not embed isomorphically into Lp,∞ ( β∈Bα bβ · 2κβ ). Thus, for each α < τ , there exist fα ∈ M p,∞ (2κα ), regarded as a subspace of Lp,∞ ( α<τ 2κα ), and βα ∈ / Bα , so that Pβα Tfα = 0. By a small perturbation, we may as well assume that fα ∈ L∞ (2κα ). Consider the correspondence k : α → βα . By definition of Bα , κk(α) κα for −1 all α < τ . Suppose there exists β0 < τ such that |k {β0 }| > ℵ0 . Then there exist M < ∞ and an uncountable subset A of k −1 {β0 } such that fα ∞ M for all α ∈ A. Since the functions are also pairwise disjoint, (fα )α∈A (as a sequence of functions in Lp,∞ ( α<τ 2κα )) is dominated by κβ
the unit vectors in p,∞ (A). Denote by (WF ) the set of Walsh functions on 2 the set
0
. By Lemma 18,
A = α ∈ A: Pβ0 Tfα · WF = 0 for some finite F ⊆ κ
is countable. Let α0 ∈ A \ A . Then Pβ0 Tfα = 0, contrary to the fact that k(α) = β0 .
2
Lemma 30. Let A be a set, τ be a limit ordinal and let k : A → [0, τ ) be a function so that |k −1 {β}| ℵ0 for all β < τ . Then there is an injection i : A → [0, τ ) such that i(α) k(α) for all α ∈ A. Proof. Suppose that γ ∈ [0, τ ) is either 0 or a limit ordinal. The set Aγ = n<ω k −1 {γ + n} is countable. Hence there is an injection iγ : Aγ → [γ , γ + ω) such that iγ (α) k(α) for all α ∈ Aγ . Consider the map i = iγ : A = Aγ → [0, τ ), where the unions are taken over all γ that is either 0 or a limit ordinal in [0, τ ). Clearly i(α) k(α) for all α ∈ A. Suppose that α, α ∈ A and that i(α) = i(α ). Express α uniquely as γα + nα , where γα is either 0 or a limit ordinal, and nα < ω. Similarly, let α = γα + nα . Then i(α) = i(α ) ∈ [γα , γα + ω) ∩ [γα , γα + ω) and hence γα = γα . But then i(α) = iγα (α) and i(α ) = iγα (α ). Since i(γα ) is injective, α = α . 2 Completion of proof of Theorem 27. As discussed above, it suffices to prove the implication (3) ⇒ (1). Suppose that condition (3) holds. First of all, we may assume that the sequence of cardinals (κβ )β<τ is arranged in nondecreasing order. Taking note of the condition on the maximum (if any) of the sequence (κβ )β<τ , we may further assume that τ is a limit ordinal. By Lemma 29, there exists a map k : [0, τ ) → [0, τ ) such that κk(α) κα for all α < τ and |k −1 {β}| ℵ0 for all β < τ . By Lemma 30, there is an injection i : [0, τ ) → [0, τ ) such that i(α) k(α) for all α ∈ [0, τ ). In particular, κi(α) κk(α) κα for all α < τ . 2 In Theorem 1, it was shown that if (Ω, Σ, μ) is a purely nonatomic measure space, then Lp,∞ (Ω, Σ, μ) has a representation E ⊕ H , where E has the form Lp,∞ ( α<ω1 ·τ 2κα ) for a nondecreasing sequence of cardinals (κα )α<ω1 ·τ and some ordinal τ (τ = ρ0 is allowed here, in n which case E = {0}) and H is either {0} or has the form Lp,∞ ( ∞ n=1 an · 2 ), with ρn κα for all n and all α. Making use of the method of proof of Theorem 27, we show that the factor E in the representation is uniquely determined up to isomorphism if 2 p < ∞ and the ordinals κα have uncountable cofinality.
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Theorem 31. Suppose that the spaces E1 ⊕ H1 and E2 ⊕ H2 are isomorphic, where E1 = Lp,∞ ( α<ω1 ·τ 2κα ), E2 = Lp,∞ ( β<ω1 ·τ 2κβ ), with nondecreasing sequences of infinite car dinals (κα )α<ω1 ·τ and (κβ )β<ω1 ·τ , and H1 , respectively H2 , are either {0} or Lp,∞ ( ∞ n=1 an · ∞ ρ p,∞ ρ n 2 ) (ρn κα ) and L ( n=1 bn · 2 n ) (ρn κβ ) respectively. If 2 p < ∞ and the cardinals κα and κβ have uncountable cofinality, then E1 is isomorphic to E2 . Moreover, if H1 = {0}, then so is H2 . Proof. First, suppose that E1 = {0}, so that H1 is isomorphic to E2 ⊕ H2 . By Theorem 19, ℵ0 |ω1 · τ |. Thus τ = 0, i.e., E2 = {0}. Now suppose that E1 = {0} and thus τ > 0. By the foregoing argument, we must have τ > 0 as well. Since E1 isomorphically embeds into E2 ⊕ H2 , by Lemma 29, there exists a map k : [0, ω1 · τ ) → {κβ : β < ω1 · τ } ∪ {ρn : n < ω} such that k(α) κα for all α < ω1 · τ and that |k −1 {κβ }|, |k −1 {ρn }| ℵ0 for each β and each ρn . Suppose that there exists α0 < ω1 · τ so that κα > κβ for all β < ω1 · τ . Then k(α) ∈ {ρn : n < ω} for all α ∈ [α0 , ω1 · τ ). Hence [α0 , ω1 · τ ) ⊆ n k −1 {ρn }. Since the latter set is countable, we have a contradiction. Therefore, for each α < ω1 · τ , there exists βα < ω1 · τ such that κβ α κα . Define j : [0, ω1 · τ ) → {κβ : β < ω1 · τ } by j (α) =
k(α) κβα
if k(α) ∈ {κβ : β < ω1 · τ }, if α ∈ n k −1 {ρn }.
Since j differs from k at only countably many α, |j −1 {κβ }| ℵ0 for all β. By Lemma 30, there exists an injection i : [0, ω1 · τ ) → {κβ : β < ω1 · τ } such that i(α) j (α) κα for all α < ω1 · τ . It follows from Proposition 5 (or Theorem 27) that E1 is isomorphic to a complemented subspace of E2 . By symmetry and Pełczy´nski’s Decomposition Method, E1 and E2 are isomorphic. Now suppose that H1 = {0} = H2 . It was shown in the previous paragraph that for each α < ω1 · τ , there exists βα < ω1 · τ such that κβ α κα . In particular, sup κα sup κβ < sup({κβ } ∪ {ρn }). However, this contradicts Theorem 14. 2 We conclude the paper with several of the main open problems that need to be resolved on the way to a complete isomorphic classification of nonatomic weak Lp spaces. Open problems. κ p,∞ (2κ ) isomor(1) Let κ be an uncountable cardinal. Are the spaces Lp,∞ ( ∞ n=1 2 ) and L phic? Note that the answer is yes if κ = ℵ0 [6]. (2) Let κ be an uncountable ordinal of countable cofinality. Can Lp,∞ (2κ ) be isomorphic to a p,∞ ( α<τ aα · 2κα ), where κα < κ for all α? space of the form L (3) If 1 < p < 2 and κ is an uncountable cardinal, are the space Lp,∞ (2κ ) and Lp,∞ ( α<κ 2κ ) isomorphic? generally, does Lp,∞ (2κ ) contain a complemented subspace of the form More p,∞ ρ ( α<κ 2 ) for some uncountable cardinal ρ? L References [1] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, 1988. [2] N.L. Carothers, S.J. Dilworth, Geometry of Lorentz spaces via interpolation, in: Longhorn Notes, University of Texas Functional Analysis Seminar, 1985–1986, pp. 107–133.
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[3] [4] [5] [6] [7]
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N.L. Carothers, S.J. Dilworth, Subspaces of Lp,q , Proc. Amer. Math. Soc. 104 (1988) 537–545. N.L. Carothers, S.J. Dilworth, Equidistributed random variables in Lp,q , J. Funct. Anal. 84 (1989) 146–159. H.E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer-Verlag, 1974. D.H. Leung, Isomorphism of certain weak Lp spaces, Studia Math. 104 (1993) 151–160. D.H. Leung, Isomorphic classification of atomic weak Lp spaces, in: N.J. Kalton, E. Saab, S.J. Montgomery-Smith (Eds.), Interaction Between Functional Analysis, Harmonic Analysis and Probability, Marcel Dekker, 1996, pp. 315– 330. [8] D.H. Leung, Purely non-atomic weak Lp spaces, Studia Math. 121 (1997) 55–66. [9] D. Maharam, On homogeneous measure algebras, Proc. Natl. Acad. Sci. USA 28 (1942) 108–111.
Journal of Functional Analysis 258 (2010) 397–420 www.elsevier.com/locate/jfa
The road to equal-norm Parseval frames Bernhard G. Bodmann a,∗,1 , Peter G. Casazza b,2 a Department of Mathematics, University of Houston, Houston, TX 77204, United States b Department of Mathematics, University of Missouri, Columbia, MO 65211, United States
Received 6 April 2009; accepted 27 August 2009 Available online 5 September 2009 Communicated by K. Ball
Abstract The construction of equal-norm Parseval frames is fundamental for many applications of frame theory. We present a construction method based on a system of ordinary differential equations, which generates a flow on the set of Parseval frames that converges to equal-norm Parseval frames. We developed this method to address a question posed by Vern Paulsen: How close is a nearly equal-norm, nearly Parseval frame to an equal-norm Parseval frame? The distance estimate derived here can be used to substantiate numerically found, approximate constructions of equal-norm Parseval frames. The estimate is valid for a fairly general class of frames — requiring that the dimension of the Hilbert space and the number of frame vectors is relatively prime. In addition, we re-phrase our distance estimate to show that certain projection matrices which are nearly constant on the diagonal are close in Hilbert–Schmidt norm to ones which have a constant diagonal. © 2009 Elsevier Inc. All rights reserved. Keywords: Frames; Nearly equal-norm; Nearly Parseval; Switching equivalence; Bures distance; ODE system; Frame energy
1. Introduction A family of vectors {fj }j ∈J is a frame for a Hilbert space H if it provides a stable embedding of H in 2 (J ) when each vector in H is mapped to the sequence of its inner prod* Corresponding author.
E-mail addresses:
[email protected] (B.G. Bodmann),
[email protected] (P.G. Casazza). 1 The author was supported in part by NSF grant DMS-0807399. 2 The author was supported in part by NSF grant DMS-0704216.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.08.015
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ucts with the frame vectors. Frames were defined by Duffin and Schaeffer [16] to address some deep questions in non-harmonic Fourier series. Traditionally, frames were most popular in signal processing [20], but today, frame theory has an abundance of applications in pure mathematics, applied mathematics, engineering, medicine and even quantum communication [8,15,20,26,1,5]. Many of these applications give rise to design problems in frame theory, the construction of frames with certain desired properties. Digital transmissions of analog signals, for example, often rely on frames because of their built-in resilience to data loss [19,18], and it has been shown that encoding with equal-norm Parseval frames has certain optimality properties for this purpose [11] (see also [23,6]). Moreover, the use of frames for compensating quantization errors has relied on equal-norm Parseval frames as well [4,7]. Despite their popularity, we know only a few ways to construct such frames analytically [22,3,9,13], mostly with the help of group actions. Success has been claimed for generating a special type of equal-norm Parseval frames with numerical methods [30], however, the analytic verification of convergence remains wanting. The use of frame potentials [3,10] shows the existence of large numbers of equal-norm Parseval frames, but offers little control over additional properties (see [23,13]). Finally, there is an algorithm due to Holmes and Paulsen [23] for turning a Parseval frame into an equal-norm Parseval frame in finitely many moves. Unfortunately, to the best of the authors’ abilities, it cannot be combined with the numerical results to provide the existence of an equal-norm Parseval frame in the close vicinity of a nearly equal-norm and nearly Parseval frame, because it does not include a distance estimate. Here, the metric on the set of frames is induced by the norm on the Hilbert space when frames are viewed as vector-valued, square summable functions (see Section 2 for precise definitions). The closest Parseval frame to a frame {fj }j ∈J is known [2,9,12,24]. Also, the closest equalnorm frame to a given frame can be found easily [9]. However, despite a significant amount of effort, so far we knew very little about the closest equal-norm Parseval frame to a given frame. This question is known in the field as the Paulsen problem. The main problem here is that finding a close equal-norm frame to a given frame involves a geometric condition while finding a close Parseval frame involves an algebraic or spectral condition. We will present the first method for finding an equal-norm Parseval frame in the vicinity of a given frame which gives quantitative estimates for the distance. The new technique we introduce is a system of vector-valued ODEs which induces a flow on the set of Parseval frames that converges to equal-norm Parseval frames. We then bound the arc length traversed by a frame by an integral of the so-called frame energy. With an exponential bound on the frame energy, we derive a quantitative estimate for the distance between our initial, -nearly equal-norm and -nearly Parseval frame F = {f1 , f2 , . . . , fn } for a d-dimensional real or complex Hilbert space and the equal-norm Parseval frame G = {g1 , g2 , . . . , gn } obtained as the limit of the flow governed by the ODE system,
n fj − gj 2 j =1
1/2
29 2 d n(n − 1)8 . 8
We also show that the order of in this estimate cannot be improved. For our method to work, we must assume that the dimension d of the Hilbert space and the number n of frame vectors are relatively prime. We will use a tensor product technique to show that if our main goal is to produce equal-norm Parseval frames, this is not a serious restriction.
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Finally, we show that the Paulsen problem is equivalent to a fundamental problem in matrix theory, and so we find an answer for the corresponding case of this problem. We believe that the techniques introduced in this paper will have application to other “nearness” questions in frame theory, in particular, to the famous equiangular tight frame problem [23,29]. Finding and classifying such frames, or even the easier problem of finding equiangular lines through the origin in Rn or Cn , started in 1948 by Haantjes [21,14], still leaves a lot to be done. This type of equal-norm Parseval frames is particularly important because of their applications to signal processing [31,29,6,32,25] and to quantum information theory [33,28,17,5]. 2. Preliminaries In this section, we introduce the notation and terminology used throughout the paper. Definition 2.1. A family of vectors F = {fj }j ∈J is a frame for a Hilbert space H if there are constants 0 < A B < ∞ so that Ax2
x, fj 2 Bx2 ,
for all x ∈ H.
j ∈J
We call the largest A and smallest B the lower and upper frame bounds respectively. If we can choose A = B then F is a tight frame and if A = B = 1 it is a Parseval frame. If all the frame vectors have the same norm, it is an equal-norm frame. The analysis operator of the frame is the map V : H → 2 (J ) given by (V x)j = x, fj . Its adjoint is the synthesis operator which maps a ∈ 2 (J ) to V ∗ (a) = j ∈J aj fj . The frame operator is the positive, self-adjoint invertible operator S = V ∗ V on H and the Grammian is the matrix G with entries Gj,k = fj , fk so that Gj,k = (V V ∗ )k,j , k, j ∈ {1, 2, . . . , n}. Definition 2.2. (1) A frame {fj }nj=1 for a d-dimensional real or complex Hilbert space H is -nearly equal-norm with constant c if (1 − )c fj (1 + )c,
for all j ∈ {1, 2, . . . , n}.
(2) The frame is -nearly Parseval if the frame constants can be chosen as A = 1 − and B = 1 + , so for all x ∈ H, (1 − )x2
x, fj 2 (1 + )x2 . j ∈J
If a frame satisfies either of these properties (1) or (2) with = 0 then we say that it is an equal-norm frame or a Parseval frame, respectively. Remark 2.3. The operators V and V ∗ allow us to state the above properties in an alternative fashion. If a frame {fj }nj=1 is -nearly equal-norm, then the diagonal entries of the Grammian, Gj,j = (V V ∗ )j,j = fj 2 , are bounded above and below by (1 + )2 c2 and (1 − )2 c2 , respectively. If a frame is -nearly Parseval, then the operator inequalities (1 − )I V ∗ V (1 + )I hold between V ∗ V and the identity I on H.
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When frames are interpreted as a vector-valued functions, they carry a natural metric. Definition 2.4. The 2 -distance between two frames F = {fj }nj=1 and G = {gj }nj=1 for a Hilbert space H is defined by F − G =
n
1/2 fj − gj
2
.
j =1
Two frames F and G are -close if F − G . We can now state the main problem we address in this paper. Problem 2.5 (V. Paulsen). Let H be a real or complex Hilbert space of dimension d. Given > 0 and an integer n d, find the largest number δ > 0 so that whenever {fj }nj=1 is a δ-nearly equal-norm, δ-nearly Parseval frame for a Hilbert space H, there is an equal-norm Parseval frame {gj }nj=1 whose 2 -distance to {fj }nj=1 is less than . The existence of such a δ is assured by an argument of Don Hadwin. Proposition 2.6 (D. Hadwin). Given a real or complex Hilbert space H of dimension d and an integer n d, then for every > 0 there is a δ > 0 so that whenever a frame {fj }nj=1 for H is δ-nearly equal-norm and δ-nearly Parseval, then {fj }nj=1 is -close to an equal-norm Parseval frame. Proof. We proceed by way of contradiction. If the assertion is false, then there exists some > 0 (m) and a sequence {δm }∞ m=1 converging to zero and a sequence of frames {fj : 1 j n, m ∈ (m)
{1, 2, . . .}} so that each {fj }nj=1 is δm -nearly equal-norm and δm -nearly Parseval but for any equal-norm Parseval frame {gj }nj=1 we have n (m) 2 f − gj 2 . j
j =1
By compactness and switching to a subsequence we may assume that the sequence of frame (m) vectors {fj }∞ m=1 has a limit for each fixed j ∈ {1, 2, . . . , n}, (m) lim f m→∞ j
= fj .
By continuity of the spectrum of V ∗ V in the frame vectors and of the entries in V V ∗ , it follows (m) that {fj }nj=1 is an equal-norm Parseval frame and that its distance to {fj }nj=1 goes to zero as (m) n }j =1
m → ∞ which is in contradiction with the assumption that the distance between each {fj and any equal-norm Parseval frame was bounded below by > 0. 2
The diagonal entries of V V ∗ and the operator inequalities for V ∗ V are not affected when the frame vectors are multiplied by unimodular constants, because then V V ∗ is simply conjugated
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by a diagonal unitary, and V ∗ V is invariant. Therefore, we can form equivalence classes of frames which share the same nearly equal-norm and nearly Parseval properties. A similar, coarser equivalence relation has already proven useful in the study of frames for erasure coding [18,23]. Definition 2.7. We define two frames F = {fj }nj=1 and G = {gj }nj=1 for a real or complex Hilbert space to be switching equivalent if the frame vectors fj and gj are collinear and fj = gj for each j ∈ {1, 2, . . . , n}. Accordingly, we speak of switching a frame F to a frame G, also denoted F (c) , if we multiply each frame vector by an unimodular constant, gj = cj fj with |cj | = 1 for j ∈ {1, 2, . . . , n}. Note that unlike the (nearly) equal-norm or Parseval properties, the 2 -distance between two frames is not preserved when one of them is switched. We now define another distance for frames which does not depend on which particular representative of an equivalence class is chosen. Definition 2.8. The Bures distance between two frames F = {fj }nj=1 and G = {gj }nj=1 for a real or complex Hilbert space H is defined by dB (F , G) =
1/2 n fj 2 + gj 2 − 2fj , gj . j =1
Two frames F and G are -close in the Bures distance if dB (F , G) . The Bures distance is only a pseudo-metric on the set of frames, because dB (F , G) = 0 only implies fj = cj gj with |cj | = 1 for all j ∈ {1, 2, . . . , n}. We have extended its usual definition for a pair of normalized √ vectors f and g in a real or complex Hilbert space, which assigns their Bures distance to be 2 − 2|f, g|, to the setting of vector-valued functions. This way of extending the Bures distance is natural when it is viewed as the solution of a minimization problem. Lemma 2.9. Let H be a Hilbert space over the field of real or complex numbers, hereafter denoted by F. The value dB (F , G) is the solution of the minimization problem
n fj − cj gj 2 dB (F , G) = minn c∈T
1/2 ,
j =1
where Tn = {c ∈ Fn : |cj | = 1 for all 1 j n}. Proof. The equivalence between these two definitions of dB is seen from the inequality fj − cj gj 2 = fj 2 + gj 2 − 2 cj fj , gj fj 2 + gj 2 − 2fj , gj , which is saturated (i.e. gives equality) when each cj is chosen so that cj fj , gj = |fj , gj |. Here, cj denotes the complex conjugate of cj . 2 The Bures distance is therefore the quotient metric obtained from the 2 -metric when passing from frames to their equivalence classes. From the fact that equal-norm and Parseval properties are switching-invariant, we get an immediate consequence for the closeness of frames.
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Corollary 2.10. A frame F = {fj }nj=1 is -close to an equal-norm Parseval frame G = {gj }nj=1 in Bures distance if and only if it is -close to an equal-norm Parseval frame G = {gj }nj=1 in 2 -distance. Proof. The “only if” part follows from choosing the 2 -distance minimizing equal-norm Parseval frame G in the equivalence class of G. For this frame, F − G = dB (F , G ) = dB (F , G) . The “if” part is clear from the inequality dB (F , G ) F − G .
2
As a final remark before the main part of the paper, we will see in Section 3.4 that the Paulsen problem is equivalent to a problem in matrix theory. Problem 2.11. Let the field F be either the real or complex numbers, and assume Fn is equipped with the canonical inner product. Given > 0, find the largest number γ > 0 so that whenever P is an orthogonal rank-d projection matrix on Fn with nearly constant diagonal, meaning there is c > 0 such that (1 − γ )c Pj,j (1 + γ )c,
for all j ∈ {1, 2, . . . , n},
then there exists an orthogonal projection Q satisfying 1. Qj,j = dn for all j ∈ {1, 2, . . . , n}, and 2. ( nj,k=1 |Pj,k − Qj,k |2 )1/2 < . 3. Construction of equal-norm Parseval frames 3.1. First steps towards an equal-norm Parseval frame We begin by first finding the closest Parseval frame to a given nearly equal-norm and nearly Parseval frame. Proposition 3.1. Let {fj }nj=1 be an -nearly Parseval frame for a d-dimensional Hilbert space H, with frame operator S = V ∗ V , then {S −1/2 fj }nj=1 is the closest Parseval frame to {fj }nj=1 and n √ −1/2 2 S fj − fj d(2 − − 2 1 − ) d 2 /4. j =1
Proof. It is known that {S −1/2 fj }nj=1 is the closest Parseval frame to {fj }nj=1 [2,9,12,24]. We summarize the derivation of this fact. The squared 2 -distance between {fj }nj=1 and {gj }nj=1 can be expressed in terms of their analysis operators V and W as
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F − G2 = tr (V − W )(V − W )∗ = tr[V V ∗ ] + tr[W W ∗ ] − 2 tr[V W ∗ ]. Choosing a Parseval frame {gj }nj=1 is equivalent to choosing the isometry W . To minimize the distance over all choices of W , consider the polar decomposition V = U P , where P is positive and U is an isometry. In fact, S = V ∗ V implies P = S 1/2 , which means its eigenvalues are bounded away from zero. Since P is positive and bounded away from zero, the term tr[V W ∗ ] = tr[U P W ∗ ] = tr[W ∗ U P ] is an inner product between W and U . Its magnitude is bounded by the Cauchy Schwarz inequality, and thus it has a maximal real part if W = U which implies W ∗ U = I . In this case, V = W P = W S 1/2 , or equivalently W ∗ = S −1/2 V ∗ , and we conclude gj = S −1/2 fj for all j ∈ {1, 2, . . . , n}. After choosing W = V S −1/2 , the 2 -distance is expressed in terms of the eigenvalues {λk }dk=1 of S = V ∗ V by
F − G2 = tr[S] + tr[I ] − 2 tr S 1/2 =
d
λk + d − 2
k=1
d
λk .
k=1
√ If 1 − λ 1 + for all j ∈ {1, 2, . . . , n}, calculus shows that the maximum of λ − 2 λ is achieved when λ = 1 − . Consequently, √ F − G2 2d − d − 2d 1 − . √ Estimating 1 − by the first three terms in its decreasing power series gives the inequality F − G2 d 2 /4. 2 Remark 3.2. Examining the proof shows that the first inequality in Proposition 3.1 saturates (i.e., √ equality holds) if {gj }nj=1 is a Parseval frame and fj = 1 − gj . This means, the inequality cannot be improved further. For the Paulsen problem, this implies that we cannot expect to find an estimate for the distance between an -nearly equal-norm, -nearly Parseval frame F = {f1 , f2 , . . . , fn } in a d-dimensional Hilbert space H and the closest equal-norm Parseval frame G in the form F − G C √ where C is smaller than d/2. In the following, we show that we can derive an estimate of the above form, where C depends on d and n. We have an upper bound for the distance between a frame and the closest Parseval frame, and for sufficiently small , we have control over how much of the “nearly equal-norm” property we lose.
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Proposition 3.3. Fix 0 1/2 and let {fj }nj=1 be an -nearly equal-norm frame with constant c which is also an -nearly Parseval frame with frame operator S = V ∗ V , then {S −1/2 fj }nj=1 is a Parseval frame and for all j ∈ {1, 2, . . . , n} we have (1 − 3)c2
2 (1 + )2 2 (1 − )2 2 c S −1/2 fj c (1 + 7)c2 . 1+ 1−
Proof. Since the frame operator S = V ∗ V is by assumption bounded by (1 − )I S (1 + )I we have via the spectral theorem 1 1 I S −1/2 √ I. √ 1+ 1− √ √ This means that the image of any unit vector has norm between 1/ 1 + and 1/ 1 − , and for the frame vectors with norm bounds (1 − )c fj (1 + )c, we get 2 (1 + )2 2 (1 − )2 2 c S −1/2 fj c . 1+ 1− Further, convexity and elementary estimates give together with the assumption 1/2 the bounds 2 (1 − 3)c2 S −1/2 fj (1 + 7)c2 .
2
Corollary 3.4. Fix 0 1/2 and let {fj }nj=1 be an -nearly equal-norm frame with constant c which is also an -nearly Parseval frame with frame operator S = V ∗ V , then the norm of each vector S −1/2 fj , j ∈ {1, 2, . . . , n}, is bounded by 3 (1 − )3 d S −1/2 fj 2 (1 + ) d . (1 + )3 n (1 − )3 n
Proof. By summing the square-norms of the frame vectors, and using the fact that the Grammian and the frame operator have the same eigenvalues, except possibly for zero, we obtain n (1 − )d fj 2 (1 + )d. j =1
The nearly equal-norm condition gives (1 − )d (1 + )2 c2 n and (1 + )d (1 − )2 c2 n.
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This bounds the value of c by (1 − )d (1 + )d c2 . (1 + )2 n (1 − )2 n Now we combine this with the preceding proposition to obtain 3 (1 − )3 d S −1/2 fj 2 (1 + ) d . (1 + )3 n (1 − )3 n
2
In the next section, we turn the resulting nearly equal-norm Parseval frame {S −1/2 fj }nj=1 into an equal-norm Parseval frame while measuring the distance between them. 3.2. On the road to an equal-norm Parseval frame We begin with a dilation argument. We observe that if {fj }nj=1 is a Parseval frame for a real or complex Hilbert space, then the Grammian G = (fj , fk )nj,k=1 is an orthogonal projection matrix and we have the expression Gj,k = Gej , Gek = V ∗ ej , V ∗ ek with the canonical orthonormal basis {ej }nj=1 on 2 ({1, 2, . . . , n}) and V ∗ , the adjoint of the analysis operator of {fj }nj=1 . Proposition 3.5. Let G be the Grammian of a Parseval frame for a real or complex Hilbert space H, then the system of ODEs n d Gej (t)2 − Gek (t)2 ek (t), ej (t) = dt
j ∈ {1, 2, . . . , n},
(3.1)
k=1
for the vector-valued functions {ej : R+ → 2 ({1, 2, . . . , n})} with the canonical basis vectors as initial values {ej (0)}nj=1 has a unique, global solution on R+ . Moreover, there exists t 0 such that ej (t) = 0 for all j ∈ {1, 2, . . . , n} if and only if there is a c > 0 such that Gej (t) = c for all j ∈ {1, 2, . . . , n}. Proof. To simplify terminology in the proof, we write Fn instead of the Hilbert space 2 ({1, 2, . . . , n}), where F stands for R or C, depending on whether the Hilbert space H is real or complex. Moreover, we identify a family of vectors {ej (t)}nj=1 in Fn with a vector
2 (e1 (t), e2 (t), . . . , en (t)) ∈ nj=1 Fn ≡ Fn . With this identification, the system of ODEs for
{ej (t)}nj=1 combines to an ODE for a single vector-valued function E : R+ → Fn . Since the 2
2
velocity vector field of the combined ODE is smooth on any bounded set in Fn , we have local existence and uniqueness of the solution in a sufficiently small neighborhood of t = 0. We first prove that these local solutions preserve orthonormality of {ej (t)}nj=1 , and then conclude the existence of global solutions. Since nj=1 ej (0) ⊗ ej∗ (0) = I we only have to prove that n d ej (t) ⊗ ej∗ (t) = 0. dt j =1
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Denoting dej (t)/dt = ej (t) and dropping the argument of the vector-valued functions, we compute n n ∗ d ej ⊗ ej∗ + ej ⊗ ej ej ⊗ ej∗ = dt j =1
j =1
=
n Gej 2 − Gek 2 ek ⊗ ej∗ + Gej 2 − Gek 2 ej ⊗ ek∗ j,k=1
= 0. The last step follows from swapping the summation indices in the second term. Now we invoke that these local solutions are uniformly bounded, because {ej (t)}nj=1 is orthonormal for each t 0. This implies that the local solution stays inside the compact set 2 S n = {(e1 , e2 , . . . , en ): ej = 1 for all j } ⊂ Fn . The existence of a unique global solution now follows from the boundedness of the velocity vector field on the compact set S n , because otherwise the maximal domain [0, a) for a solution would yield a limiting value at a inside S, which we could again use as initial value to find a local solution in the neighborhood of a, and then by the uniqueness of local solutions extend the domain [0, a) to include a neighborhood of a, contradicting the maximality assumption. For more details on this argument, see [27, Section 2.4]. Finally, we observe that ej (t) = 0 for all j ∈ {1, 2, . . . , n} implies by orthonormality that Gej (t)2 − Gek (t)2 = 0 for all j and k and thus the family {Gej }nj=1 is equal-norm. Conversely, it follows directly from the definition of the ODE system that all orthonormal bases which G projects to an equal-norm family are fixed points. 2 By mapping the evolving orthonormal basis with the synthesis operator of a Parseval frame, we obtain a family of Parseval frames which solves a corresponding ODE system. Proposition 3.6. Let G be the Grammian of a Parseval frame for a real or complex Hilbert space H, let V : H → 2 ({1, 2, . . . , n}) be the analysis operator of the frame, and consider the solution {ej : R+ → 2 ({1, 2, . . . , n})}nj=1 of the initial value problem given in the preceding proposition, then fj (t) = V ∗ ej (t) defines a family of Parseval frames {fj : R+ → H}nj=1 which satisfies the ODE system n d fj (t)2 − fk (t)2 fk (t), fj (t) = dt
j ∈ {1, 2, . . . , n},
(3.2)
k=1
and V is the analysis operator of {fj (0)}nj=1 . Conversely, each solution of this ODE system, with a Parseval frame {fj (0)}nj=1 having analysis operator V as initial value, is globally defined and unique, and to each such solution corresponds a unique solution for the ODE (3.1) starting at the canonical basis of 2 ({1, 2, . . . , n}) such that V ∗ ej (t) = fj (t) for all t 0. Proof. We use the two facts that (1) the projection of any orthonormal basis {ej }nj=1 with the Grammian G is a Parseval frame for the range of G and that (2) the analysis operator V of a Parseval frame is an isometry, which implies by orthonormality of {ej (t)}nj=1 that for any t 0, F (t) = {V ∗ ej (t)}nj=1 is a Parseval frame for H. Moreover, from the identity Gej (t) =
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V ∗ ej (t) = fj (t) for all j ∈ {1, 2, . . . , n} and from applying V ∗ to both sides of the ODE system (3.1), we deduce that F : R+ → nj=1 H defines a family of Parseval frames which solves the ODE system (3.2). The initial value problem for (3.2) has a unique solution, which is seen by repeating the ar gument of the preceding proposition with the vector-valued function F : R+ → nj=1 H instead
of E and with the sphere S = {(f1 , f2 , . . . , fn ): nj=1 fj 2 = d} ⊂ nj=1 H instead of S n . The set S is preserved under the flow because each {fj (t)}nj=1 is a Parseval frame, so the trace of its Grammian is equal to its rank, nj=1 fj 2 = d, independent of the choice of t 0. Since the solution of the initial value problem (3.2) is unique, and F (t) = {V ∗ ej (t)}nj=1 provides a solution when the orthonormal basis evolves under (3.1), each solution of (3.2) can be lifted to a unique solution of (3.1) which has as its initial value {ej (0)}nj=1 , the canonical orthonormal basis of 2 ({1, 2, . . . , n}). 2 The reason for introducing the dilation argument with the ODE system for the basis vectors is that the fixed points of (3.1) are as desired, whereas the set of fixed points of (3.2) contains more than all equal-norm Parseval frames, see the example below. Proposition 3.7. Given a family of n vector-valued functions {fj : R+ → H}nj=1 , satisfying (3.2), with {fj (0)}nj=1 a Parseval frame, then fj (0) = 0 for all j ∈ {1, 2, . . . , n} if and only if the frame is equal-norm or the following zero-summing conditions hold: n
fj (0) =
j =1
n fj (0)2 fj (0) = 0. j =1
Proof. In the proof we again omit the explicit time dependence of the frame vectors. From the ODEs system for the frame vectors, we see that if d fj = fj 2 − fk 2 fk = 0, dt n
k=1
then fj 2
n k=1
fk =
n
fk 2 fk .
k=1
Hence, if d d fj = fm = 0, dt dt for j = m ∈ {1, 2, . . . , n}, then fj 2
n k=1
fk = fm 2
n k=1
fk .
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That is, fj = fm or
n
fk = 0.
k=1
Consequently, if
d dt fj
= 0 for all j ∈ {1, 2, . . . , n} then the frame is equal-norm or n
fk =
k=1
fk 2 fk = 0.
k
d Conversely, if the zero-summing conditions hold, then dt fj = 0 follows for all j ∈ {1, 2, . . . , n} directly from the definition of the ODE system (3.2). 2
Example 3.8. Given a real or complex Hilbert space H of dimension d and an orthonormal basis {e1 , e2 , . . . , ed } for H, we can construct a Parseval frame {fj }2d+1 j =1 by ⎧ 1 ⎪ ⎨ √2 e j , fj = − √1 ej −d , ⎪ 2 ⎩ 0,
1 j d, d + 1 j 2d, j = 2d + 1.
It is straightforward to check that this frame satisfies the zero-summing conditions in the preceding proposition, and is thus a fixed point for the ODE (3.2), but it is not an equal-norm Parseval frame. It has been observed numerically that using an example of this type as initial value and dilating the Parseval frame to an orthonormal basis leads to an oscillating behavior of the basis vectors evolving under the ODE system (3.1). Therefore, one cannot hope to use these ODEs alone to achieve convergence to equal-norm Parseval frames. Definition 3.9. We define the frame energy of a frame F = {fj }nj=1 by U (F ) =
n 2 fj 2 − fk 2 . j,k=1
We will show below that with an appropriate use of intermittent switching, the energy of Parseval frames obtained from piecewise solutions of the ODE (3.2) decreases rapidly (in fact, exponentially) in time. Together with the following arc length estimate, this amounts to showing a rate of convergence to an equal-norm Parseval frame. Definition 3.10. Given a family of differentiable vector-valued functions F = {fj : R+ → H}nj=1 and 0 t1 t2 , the arc length traversed by the family between time t1 and t2 is defined by 1/2 t2 n 2 f (t) s= dt. j
t1
j =1
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The arc length traversed by the vector-valued function F evolving under (3.2) is bounded by an energy integral. Theorem 3.11. The arc length traversed by the solution F : R+ → tem (3.2) between time t1 and t2 is bounded by the energy integral t2 s
n
j =1 H
of the ODE sys-
1/2 U F (t) dt.
t1
Proof. We pass from the solution of (3.2) to the orthonormal basis E = {ej : R+ → Fn }nj=1 evolving under (3.1), giving V ∗ ej (t) = fj (t), where V ∗ is the synthesis operator of {fj (0)}nj=1 . Denoting by G the Grammian, we have by orthonormality, 2
n d 2 2 ej = Gej 2 − Gek 2 dt k=1
where we have suppressed the explicit time dependence of the orthonormal basis vectors. Summing over all j gives n n d 2 2 ej = Gej 2 − Gek 2 = U F (t) . dt j =1
j,k=1
∗ x for each x ∈ 2 ({1, 2, . . . , n}), yields Finally, again using the Parseval property, Gx = V d ∗ d d fj = dt V ej = dt Gej dt ej , and we have nj=1 fj (t)2 U (F (t)). Now the definition of arc length provides the desired estimate. 2
Proposition 3.12. An alternative expression for the frame energy of a Parseval frame F = {fj }nj=1 is U (F ) = 2n
n fj 4 − 2d 2 , j =1
where d is the dimension of H. Proof. We use the antisymmetry of fj 2 − fk 2 in j and k to write U (F ) = 2
n fj 2 − fk 2 fj 2 . j,k=1
Now we can sum over k. Since {fk }nk=1 is Parseval, the square-norms sum to d = dim(H). The result is U (F ) = 2
n nfj 4 − dfj 2 . j =1
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Again, we can split the two terms into separate sums and carry out the sum over j for the second term to get d again. 2 Next, we give a closed expression for the time derivative of the frame energy while the frame F evolves under (3.2). Lemma 3.13. If F = {fj : R+ → H} is a solution of (3.2) with a Parseval frame {fj (0)}nj=1 as initial value, then n 2 2 2 d U F (t) = 4n fj (t), fk (t) fj (t) − fk (t) . dt j,k=1
Proof. Defining Gj,k (t) = Gej (t), ek (t) = fj (t), fk (t) and proceeding with the lifted ODE ej (t) =
n Gj,j (t) − Gk,k (t) ek (t), k=1
we have n 2 d Gj,j (t) = 2Gj,j (t) Gj,k (t) Gj,j (t) − Gk,k (t) . dt k=1
Summing over j and antisymmetrizing Gj,j (t) with Gk,k (t) gives n 2 d U F (t) = 4n Gj,k (t) Gj,j (t) − Gk,k (t) . dt j,k=1
In terms of the frame vectors, this is precisely the claimed expression.
2
Definition 3.14. We define σn to be the uniform probability measure on the n-torus Tn = {c ∈ Fn : |cj | = 1 for all j }, where F is R or C. In the complex case, these are all n-tuples of unimodular complex numbers and in the real case n-tuples of ±1’s. We also denote diagonal unitaries {D(c)}, parametrized by the diagonal entries (D(c))j,j = cj , |cj | = 1 for all j ∈ {1, 2, . . . , n}. For later notational convenience, we define W (F ) = 4n
n
2 fj , fk fj 2 − fk 2 .
(3.3)
j,k=1
We recall the definition of two frames being switching equivalent, meaning the two families consist of vectors that are pairwise collinear and of the same norm. Remark 3.15. We observe that if F = {fj }nj=1 and G = {gj }nj=1 are switching equivalent, then U (F ) = U (G). Switching a frame amounts to conjugating the Grammian with a diagonal unitary matrix. (If the Hilbert space is over the reals, then this is just a diagonal matrix with eigenvalues ±1.)
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So while U (F ) is switching-invariant, W (F ) generally depends on which representative of a switching-equivalence class is chosen. We now use the switching dependence of W to our advantage. Proposition 3.16. Given a Parseval frame F = {fj }nj=1 , then there is a choice c ∈ Tn such that W F (c) 0. Proof. Let G denote the Grammian of F . For the switched frame F (c) , we have n cj ck∗ Gj,k (Gj,j − Gk,k )2 . W F (c) = 4n j,k=1
Integrating over the torus Tn with respect to the switching-invariant measure σn gives
cj∗ ck dσn (c) = δj,k .
Tn
Thus we note, since terms with j = k have a vanishing contribution in W (F (c) ),
W F (c) dσ (c) = 0.
Tn
Since the average is equal to zero, there must be a choice of c which gives W (F (c) ) 0.
2
Next, we compute a lower bound for the variance of W (F (c) ). Proposition 3.17. For a fixed Parseval frame F , the variance of W (F (c) ) with respect to the probability measure σ on the torus {c ∈ Tn } is
(c) 2 W F dσ (c) = 16n2 |Gj,k |2 (Gj,j − Gk,k )4 . j,k
Tn
Proof. Similar to the preceding proposition, with the help of
∗ cj ck∗ cl cm dσ (c) = δj,k δl,m + δj,m δk,l .
Tn
Let n, d ∈ N be relatively prime, and define d d1 η = min − n1
2
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then we have a lower bound η
1 . n(n − 1)
This follows immediately from the fact that since d, n are relatively prime, dn1 − d1 n is a nonzero integer. Since n1 < n and d1 < d we have d 1 − d1 = dn1 − nd1 1 . n n nn nn n(n − 1) 1 1 1 Lemma 3.18. Let n 2, η as defined above, and let F = {fj }nj=1 be a Parseval frame for a d-dimensional Hilbert space, then the variance of the random variable W : c → W (F (c) ) on the torus Tn equipped with the uniform probability measure σn is bounded below by 2 (c) 2 16η U (F ) W F dσn . (n − 1)7 Tn
Proof. Without loss of generality we can number the frame vectors so that their norms decrease, f1 f2 · · · fn . If U (F ) does not vanish then f1 > fn and there is at least one j ∈ {1, 2, . . . , n − 1} such that fj 2 − fj +1 2 f1 2 − fn 2 /(n − 1). This means, if j j and j j + 1, then also fj 2 − fj 2 f1 2 − fn 2 /(n − 1). Thus we have partitioned the frame vectors into two sets, and the difference of square-norms between any pair of vectors containing one from each of these sets is bounded below by (f1 2 − fn 2 )/(n − 1). Therefore, the matrix A containing entries Aj,k = (fj 2 − fk 2 )4 is entry-wise bounded below by a matrix (block notation) A =
0 J ∗
J 0
where J is a block containing all 1’s and = (f1 2 − fn 2 )4 /(n − 1)4 . If we form the corresponding blocks in the Grammian G=
G11 G21
G12 G22
then we know 0 G11 I , meaning the eigenvalues of G11 are contained in the closed interval [0, 1]. Since G is an orthogonal projection, G11 = G211 + G12 G21 which means
tr[G12 G21 ] = tr G11 − G211 .
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But that is exactly the squared Frobenius norm of the block G12 . Hence,
|Gj,k |2 Aj,k 2 tr G11 − G211 .
j,k
The smallest number of non-zero entries in A is achieved when J contains only one row. If n and d are relatively prime and the vectors are sufficiently near equal-norm, then the diagonal entries of G11 are close to d/n and summing them does not give an integer. Therefore, not all eigenvalues are 0 or 1. In fact, a lower bound for the Hilbert–Schmidt square-norm of G12 is tr[G11 − G211 ] η/(2n − 2). This is because at least one of the eigenvalues has distance η/(n − 1) from {0, 1} and the function x → x(1 − x) is bounded below by x → x/2 on [0, 1/2] and by x → 1/2 − x/2 on [1/2, 1]. Consequently, (f1 2 − fn 2 )4 η . |Gj,k |2 Aj,k (n − 1)4 (n − 1) j,k
Using the equivalence of norms again, n 2 2 fj 2 − fk 2 n(n − 1) f1 2 − fn 2 j,k=1
and then applying Proposition 3.17 n 2 16η d (c) 2 2 2 U F U (F ) 16n |G | A = dσ (c). j,k j,k (n − 1)7 dt j,k=1
2
Tn
Next we will bound |W (F (c) )| by the frame energy. Lemma 3.19. For a fixed Parseval frame F , the random variable W : c → W (F (c) ) on the torus Tn is bounded, (c) W F dU (F ). Proof. Let B denote the matrix with entries Bj,k = (fj 2 − fk 2 )2 , and G(c) = D(c)GD ∗ (c), then W (F (c) ) = tr[G(c) B]. Estimating the inner product between G(c) and B gives (c) (c)
W F = tr G B max tr P |B| P
where the maximum is over all rank-d orthogonal projections P , and the spectral theorem de√ ∗ B. According to Perron–Frobenius, the largest eigenvalue of |B| is bounded by fines |B| = B maxj nk=1 Bi,j . Hence,
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Finally, we observe that U (F ) = U (F (c) ) =
k=1
j,k=1
n
j,k=1 Bj,k .
2
To finish the quantitative bound on the distance from our initial Parseval frame to our equalnorm Parseval frame, we will find an exponential upper bound on the frame energy. Theorem 3.11 will then give the needed quantitative upper bound on the arc length. Lemma 3.20. Let W : Ω → [−a, a], a > 0 be a bounded random variable on a probability space, which induces a a normalized Borel measurema on [−a, a]. If the expectation and variance of W are E[W ] = −a x dm(x) = 0 and E[W 2 ] = −a x 2 dm(x) = σ 2 > 0, then the support of m contains a point in the set {x ∈ [−a, a]: x −σ 2 /a}. Proof. We consider the polynomial given by p(x) = (x − a)(x + b), then
E p(W ) =
a
2 x + (b − a)x − ab dm(x) = σ 2 − ab.
−a
Choosing b = σ 2 /a gives E[p(W )] = 0, and so supp m ∩ x ∈ [−a, a]: p(x) 0 = ∅. The subset of [−a, a] where p is non-negative is [−a, −b].
2
Now we are able to bound W (F (c) ) from above by a strictly negative quantity. Theorem 3.21. Let n 2, η as defined above, and let F = {fj }nj=1 be a Parseval frame for a d-dimensional Hilbert space, then there exists c ∈ Tn such that W F (c) −
16η U (F ). (n − 1)7 d
Proof. We have that a = dU (F ) bounds the magnitude of W (F (c) ) and its variance σ 2 is 16η 2 bounded below by σ 2 (n−1) 7 (U (F )) . The preceding lemma then establishes that there is a choice for {cj }nj=1 such that W F (c) −
16η σ2 − U (F ). dU (F ) (n − 1)7 d
2
Theorem 3.22. Let H be a real or complex Hilbert space of dimension d, and let F = {f1 , f2 , . . . , fn } be an -nearly equal-norm Parseval frame, with n 2 and n and d relatively prime, then there exists an equal-norm Parseval frame G at 2 -distance F − G U (F )1/2
n(n − 1)8 d . 8
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Proof. We let the frame F serve as the initial value F (0) for the ODE system (3.2). Assuming that for each t, we pick c(t) which yields the desired estimate for W , then naively integrating the differential inequality 16η d (c(t)) U F U F (t) (t) = W F (c(t)) (t) − dt (n − 1)7 d obtained in the preceding theorem gives 7 U F (c(t)) (t) U F (0) e−16ηt/(n−1) d . However, we note that there is no guarantee that c is a measurable function. To achieve this, we relax the constant governing the exponential decay. Choose 0 < α < 1. We know that for any Parseval frame there is at least one choice of c which gives 16αη d (c) 16η U F U (F ) < − U (F ). − dt (n − 1)7 d (n − 1)7 d
(3.4)
By the continuity of U and dU/dt in F , we can cover the space of Parseval frames with open sets for which the strict inequality holds with the choice of a corresponding c. To finish the argument we need to patch together the local flows in each open set. We define a global flow by the appropriate choice of c in each subset. Upon exiting a set at time t, we choose one of the open sets of which the frame G(t) is an element and continue with the respective flow given by the corresponding choice of c in this subset. Since the cover is open, c is piecewise constant and right continuous. In the complex case, we choose a countable number of c’s which are dense in the torus. By continuity of U and W , for any frame there is a choice in this countable set of c’s such that again the strict differential inequality (3.4) is satisfied. Moreover, the countable family of open sets corresponding to all c’s cover the space of all Parseval frames. By the Heine Borel property of the compact set of Parseval frames, there is a finite sub-cover and we can repeat the argument as in the real case. We recall that switching affects the 2 -distance. Piecewise integrating the differential inequality, including switching when necessary, gives that the frame energy of {F (c(t)) (t)}t∈R+ decays exponentially in time. Then using the inequality between arc length and frame energy in Theorem 3.11, we obtain that the sequence {F (c(m)) (m)}∞ m=0 is Cauchy in the Bures metric, ∞ (c(m)) (c(m+1)) (m), F (m + 1)) is dominated by a geometric series, because the series m=0 dB (F and hence summable. Passing to a subsequence converging to an accumulation point G then yields that the equalnorm Parseval frame G is within Bures distance dB F (0), G
∞
1/2 U F (c(t)) (t) dt
0
∞ 0
1/2 −8ηαt/(n−1)7 d U F (0) e dt
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1/2 (n − 1)7 d = U F (0) . 8ηα However, we recall that we can always choose G in the equivalence class of G which minimizes the 2 -distance to F , and obtain the same result for the 2 -distance. To finish the proof, we recall η 1/n(n − 1) and use the fact that the set of equal-norm Parseval frames is closed in the compact set of all Parseval frames. Therefore, choosing a sequence of values for α converging to one, we obtain a sequence of frames with an accumulation point within the desired 2 -distance. 2 Now, putting together the distances we computed above, and taking into account that in the first step we moved from our nearly equal-norm, nearly Parseval frame to the closest Parseval frame, we can give the distance estimate for the Paulsen problem. Theorem 3.23. Let n, d ∈ N be relatively prime, n 2, let 0 < < 12 , and assume F = {fj }nj=1 is an -nearly equal-norm and -nearly Parseval frame for a real or complex Hilbert space of dimension d, then there is an equal-norm Parseval frame G = {gj }nj=1 such that F − G
29 2 d n(n − 1)8 . 8
Proof. After passing to the closest Parseval frame to the given frame, denoted by G(0) = {S −1/2 fj }, we have by the lower and upper bound for the norms of {S −1/2 fj } in Corollary 3.4 a bound for the frame energy d 2 (n − 1) (1 + )3 (1 − )3 2 U G(0) − . n (1 − )3 (1 + )3 Using convexity and elementary estimates, we infer for < 1/2 that U G(0) < 272 d 2 2 . Now using the preceding theorem, we obtain that there is an equal-norm Parseval frame G at distance G(0) − G 27 d 2 n(n − 1)8 . 8 To complete the proof, we use the triangle inequality, d(F , G) d F , G(0) + d G(0), G
√ 27 d + d 2 n(n − 1)8 , 2 8
and then combine the two contributions after estimating
√ d/2 d 2 /2 d 2 n(n − 1)8 /4.
2
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3.3. Construction of equal-norm Parseval frames without the relative prime condition We conclude with an observation which allows us to reduce the construction of equal-norm Parseval frames to the special case discussed in the previous section. Lemma 3.24. Given two Hilbert spaces H1 and H2 over the real or complex numbers and equal-norm Parseval frames F = {f1 , . . . , fn1 } and G = {g1 , . . . , gn2 }, then the family of vectors F ⊗ G = {fi ⊗ gj : 1 i n1 , 1 j n2 } is an equal-norm Parseval frame for H1 ⊗ H2 . Proof. The Parseval property of F ⊗ G is equivalent to the identity x= x, fi ⊗ gj fi ⊗ gj i,j
for all x ∈ H1 ⊗ H2 . From the Parseval property of both frames it is clear that this identity holds for any x = a ⊗ b with a ∈ H1 and b ∈ H2 . Linearity then establishes the result for general x ∈ H1 ⊗ H2 . The equal-norm property follows from f ⊗ g = f g for any pair (f, g) ∈ F × G and from the equal-norm property of the individual frames.
2
Corollary 3.25. The construction of an equal-norm Parseval frame of n vectors in a d-dimensional real or complex Hilbert space H can be reduced to the case of d and n being relatively prime. Proof. If their greatest common divisor is not one, say gcd(n, d) = m, then we can proceed as follows. Consider the Hilbert space H = H1 ⊗ H2 , where dim(H1 ) = d/m and dim(H2 ) = m. Now choose an orthonormal basis {e1 , e2 , . . . , em } for H2 and construct an equal-norm Parseval frame of n/m vectors {f1 , f2 , . . . , fn/m } for H1 . The resulting family of tensor products {fi ⊗ ej : 1 i n/m, 1 j m} is an equal-norm Parseval frame for H. 2 3.4. The Paulsen problem in matrix theory In this section we will show that the estimate for the special case of the Paulsen problem provides a partial answer for Problem 2.11 in matrix theory. Proposition 3.26. If {fj }j ∈I , {gj }j ∈I are frames for H with analysis operators V1 , V2 respectively, then V1 fj − V2 gj 2 < 2 V1 2 + V2 2 fj − gj 2 . j ∈I
j ∈I
Proof. Note that for all j ∈ I , V1 fj =
fj , fi ei , i∈I
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and V2 gj =
gj , gi ei .
i∈I
Hence, V1 fj − V2 gj 2 =
fj , fi − gj , gi 2 i∈I
fj , fi − gi + fj − gj , gi 2 = i∈I
2
fj , fi − gi 2 + 2 fj − gj , gi 2 . i∈I
i∈I
Summing over j gives j ∈I
V1 fj − V2 gj 2 2
fj , fi − gi 2 + 2 fj − gj , gi 2 j ∈I i∈I
j ∈I i∈I
fj , fi − gi 2 + 2V2 2 =2 fj − gj 2 i∈I j ∈I
j ∈I
= 2V1 2 fi − gi 2 + 2V2 2 fj − gj 2 j ∈I
i∈I
= 2 V1 2 + V2 2 fj − gj 2 .
2
j ∈I
Corollary 3.27. Let {fj }j ∈I , {gj }j ∈I be Parseval frames for H with analysis operators V1 , V2 respectively. If fj − gj 2 < 2 , j ∈I
then
V1 fj − V2 gj 2 < 4 2 .
j ∈I
Proof. The analysis operators V1 and V2 are isometries, so the preceding proposition simplifies to the desired estimate. 2 Corollary 3.28. Let H be a Hilbert space having two Parseval frames F = {fj }nj=1 and G = {gj }nj=1 at 2 -distance F − G , then their Grammians G and Q satisfy G − QH S ≡
n j,k=1
1/2 |Gj,k − Qj,k |
2
< 2.
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419
Corollary 3.29. Let n, d ∈ N be relatively prime, n 2, and let 0 < < 1/2. If G is a rank-d orthogonal n × n projection matrix over R or C and there is c > 0 such that the diagonal entries satisfy (1 − )2 c2 Gj,j (1 + )2 c2 for all j ∈ {1, 2, . . . , n}, then there is an orthogonal rank-d projection Q with diagonal Qj,j = and G − QH S
d n
29 2 d n(n − 1)8 . 4
Proof. The matrix G is the Grammian of a nearly equal-norm Parseval frame. Using the distance estimate in Theorem 3.23 and the preceding corollary, we obtain the desired estimate for the Hilbert–Schmidt distance. 2 Acknowledgment The authors would like to thank the referee for detailed suggestions which lead to substantial improvements in transparency and readability. References [1] D.M. Appleby, SIC-POVMs and the extended Clifford group, J. Math. Phys. 46 (2005), 052107/1–29. [2] R. Balan, Equivalence relations and distances between Hilbert frames, Proc. Amer. Math. Soc. 127 (8) (1999) 2353–2366. [3] J.J. Benedetto, M. Fickus, Finite normalized tight frames, Adv. Comput. Math. 18 (2003) 357–385. [4] J.J. Benedetto, A.M. Powell, O. Yilmaz, Sigma-Delta quantization and finite frames, IEEE Trans. Inform. Theory 52 (2006) 1990–2005. [5] B.G. Bodmann, D.W. Kribs, V.I. Paulsen, Decoherence-insensitive quantum communication by optimal C ∗ encoding, IEEE Trans. Inform. Theory 53 (2007) 4738–4749. [6] B.G. Bodmann, V.I. Paulsen, Frames, graphs and erasures, Linear Algebra Appl. 404 (2005) 118–146. [7] B.G. Bodmann, V.I. Paulsen, Frame paths and error bounds for sigma-delta quantization, Appl. Comput. Harmon. Anal. 22 (2007) 176–197. [8] P.G. Casazza, Modern tools for Weyl–Heisenberg (Gabor) frame theory, Adv. Imaging Electron Phys. 115 (2001) 1–127. [9] P.G. Casazza, Custom building finite frames, in: Wavelets, Frames and Operator Theory, College Park, MD, 2003, in: Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 61–86. [10] P.G. Casazza, M. Fickus, J. Kovaˇcevi´c, M. Leon, J. Tremain, A physical interpretation of tight frames, in: C. Heil (Ed.), Harmonic Analysis and Applications, Birkhäuser, Boston, 2006, pp. 51–76. [11] P. Casazza, J. Kovaˇcevi´c, Equal-norm tight frames with erasures. Frames, Adv. Comput. Math. 18 (2003) 387–430. [12] P. Casazza, G. Kutyniok, A generalization of Gram Schmidt orthogonalization generating all Parseval frames, Adv. Comput. Math. 18 (2007) 65–78. [13] P. Casazza, N. Leonhard, Classes of finite equal norm Parseval frames, Contemp. Math. 451 (2008) 11–31. [14] P.G. Casazza, D. Redmond, J. Tremain, Real equiangular frames, in: Information Sciences and Systems (42nd Annual Conference on), Princeton, NJ, 2008, pp. 715–720. [15] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003. [16] R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366. [17] S.T. Flammia, On SIC-POVMs in prime dimensions, J. Phys. A 39 (2006) 13483–13493. [18] V.K. Goyal, J. Kovaˇcevi´c, J.A. Kelner, Quantized frame expansions with erasures, Appl. Comput. Harmon. Anal. 10 (2001) 203–233. [19] V.K. Goyal, M. Vetterli, N.T. Thao, Quantized overcomplete expansions in RN : Analysis, synthesis and algorithms, IEEE Trans. Inform. Theory 44 (1998) 16–31.
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[20] K. Gröchenig, Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2001. [21] J. Haantjes, Equilateral point-sets in elliptic two- and three-dimensional spaces, Nieuw Arch. Wiskd. 22 (1948) 355–362. [22] D. Han, D. Larson, Bases, Frames and Group Representations, vol. 697, Memoirs, Amer. Math. Soc., Providence, RI, 2000. [23] R.B. Holmes, V.I. Paulsen, Optimal frames for erasures, Linear Algebra Appl. 377 (2004) 31–51. [24] A.J.E.M. Janssen, Zak transforms with few zeroes and the tie, in: H.G. Feichtinger, T. Strohmer (Eds.), Advances in Gabor Analysis, Birkhäuser, Boston, 2002, pp. 31–70. [25] D. Kalra, Complex equiangular cyclic frames and erasures, Linear Algebra Appl. 419 (2006) 373–399. [26] J. Kovaˇcevi´c, A. Chebira, An introduction to frames, in: Foundations and Trends in Signal Processing, NOW publishers, 2008. [27] L. Perko, Differential Equations and Dynamical Systems, second ed., Springer, New York, 1996. [28] J.M. Renes, R. Blume-Kohout, A.J. Scott, C.M. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys. 45 (2004) 2171–2180. [29] Th. Strohmer, R.W. Heath Jr., Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal. 14 (2003) 257–275. [30] J.A. Tropp, I.S. Dhillon, R. Heath Jr., T. Strohmer, Structured tight frames via an alternating projection method, IEEE Trans. Inform. Theory 51 (2005) 188–209. [31] L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory IT-20 (1974) 397–399. [32] P. Xia, Sh. Zhou, G.B. Giannakis, Achieving the Welch bound with difference sets, IEEE Trans. Inform. Theory 51 (2005) 1900–1907. [33] G. Zauner, Quantendesigns – Grundzüge einer nichtkommutativen Designtheorie, Doctorial thesis, University of Vienna, 1999.
Journal of Functional Analysis 258 (2010) 421–457 www.elsevier.com/locate/jfa
New solutions for Trudinger–Moser critical equations in R2 Manuel del Pino a , Monica Musso b , Bernhard Ruf c,∗ a Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile b Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avda. Vicuña Mackenna 4860, Macul, Chile c Dipartimento di Matematica, Universitá di Milano, Via Saldini 50, 20133 Milan, Italy
Received 7 April 2009; accepted 13 June 2009 Available online 10 July 2009 Communicated by J. Coron
Abstract Let Ω be a bounded, smooth domain in R2 . We consider critical points of the Trudinger–Moser type 2 functional Jλ (u) = 12 Ω |∇u|2 − λ2 Ω eu in H01 (Ω), namely solutions of the boundary value problem 2
u + λueu = 0 with homogeneous Dirichlet boundary conditions, where λ > 0 is a small parameter. Given k 1 we find conditions under which there exists a solution uλ which blows up at exactly k points in Ω as λ → 0 and Jλ (uλ ) → 2kπ . We find that at least one such solution always exists if k = 2 and Ω is not simply connected. If Ω has d 1 holes, in addition d + 1 bubbling solutions with k = 1 exist. These results are existence counterparts of one by Druet in [O. Druet, Multibump analysis in dimension 2: Quantification of blow-up levels, Duke Math. J. 132 (2) (2006) 217–269] which classifies asymptotic bounded energy levels of blow-up solutions for a class of nonlinearities of critical exponential growth, including this one as a prototype case. © 2009 Elsevier Inc. All rights reserved. Keywords: Trudinger–Moser inequality; Blowing-up solutions; Singular perturbations
* Corresponding author.
E-mail addresses:
[email protected] (M. del Pino),
[email protected] (M. Musso),
[email protected] (B. Ruf). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.018
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1. Introduction and statement of main results Let Ω be a bounded domain in R2 with smooth boundary and λ > 0. This paper is concerned with the analysis of solutions to the boundary value problem
2
u + λueu = 0, u=0
u > 0 in Ω, on ∂Ω,
(1.1)
where λ > 0 is a small parameter. This problem is the Euler–Lagrange equation for the functional Jλ (u) =
1 2
|∇u|2 −
λ 2
Ω
2
eu ,
u ∈ H01 (Ω),
(1.2)
Ω
which corresponds to the free energy associated to the critical Trudinger embedding (in the sense of Orlicz spaces) [18,22,23] 2
H01 (Ω) u −→ eu ∈ Lp (Ω)
∀p 1,
which is connected to the critical Trudinger–Moser inequality C(Ω) = sup
e
4πu2
/u ∈ H01 (Ω),
Ω
|∇u| = 1 < +∞, 2
Ω
[17]. Observe that, in general, critical points of the above constrained variational problem satisfy, after a simple scaling, an equation of the form (1.1). The Trudinger–Moser embedding is critical, involving loss of compactness analogous to that of the Sobolev embeddings in dimension N 3, 2N
H01 (Ω) u −→ u ∈ L N−2 (Ω), for which the problem analogous to (1.1) is
N+2
u + λu + u N−2 = 0, u=0
u > 0 in Ω, on ∂Ω,
(1.3)
for λ 0, whose associated energy is Iλ (u) =
1 2
|∇u|2 − Ω
λ 2
u2 − Ω
N −2 2N
2N
|u| N−2 ,
u ∈ H01 (Ω).
Ω
Loss of compactness in H01 (Ω) for the functionals Jλ or Iλ translates into the presence of nonconvergent Palais–Smale (PS) sequences. Let us consider for instance a sequence λn → λ0 0, and a sequence un with ∇Iλn (un ) → 0, Iλn (un ) → c. Then, by the result in [20], un decomposes asymptotically into a finite sum of blowing-up standard bubbles and a critical point u0 of Iλ0 yielding in particular that
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Iλn (un ) = Iλ0 (u0 ) + kSN + o(1) for some k 1, where SN is a positive constant. Existence of solutions, namely critical points of Iλ with the above property and u0 = 0 is known under suitable assumptions, see [7,9,16,19]. For the Trudinger– Moser functional (1.2), a clean classification of all PS sequences for Jλ does not seem possible after the results in [3]. Actually PS holds as long as c < 2π , see [1,11]. On the other hand, for solutions more is known. In [2,14] a class of nonlinearities is considered for which the one in (1.1) may be regarded as the prototype. From the result in [14], we have the following fact: Assume that un solves problem (1.1) for λ = λn , with Jλn (un ) bounded and λn → 0. Then, passing to a subsequence, there is an integer k 0 such that Jλn (un ) = 2kπ + o(1).
(1.4)
When k = 1 a more precise answer is obtained in [2]: the solution un has for large n only one isolated maximum, which blows up around a point x0 ∈ Ω which is characterized as follows: Let G(x, y) be Green’s function of the problem −x G = 4πδy (x), G(x, y) = 0,
x ∈ Ω,
x ∈ ∂Ω,
and H its regular part defined as H (x, y) = 4 log
1 − G(x, y). |x − y|
(1.5)
Then from [2], it follows that x0 is a critical point of Robin’s function x → H (x, x). It is natural to ask whether or not solutions satisfying (1.4) exist. In fact the existence and multiplicity question seems much more difficult than its critical Sobolev exponent counterpart. Some results are known: From the result in [3], it follows that there is a λ0 > 0 such that a solution to (1.1) exists whenever 0 < λ < λ0 (this is in fact true for a larger class of nonlinearities with critical exponential growth). By construction this solution falls, as λ → 0, into the bubbling category (1.4) with k = 1. No solution other than this one is known. Struwe in [21] built in the case of a domain with a sufficiently small hole (in the sense of Bahri and Coron [6,10]) a solution taking advantage of the presence of topology. This solution exists for a class of nonlinearities, 2 perturbation of the Trudinger–Moser one, that also include λueu −u for which no solution exists for small λ, in a disk, see [4,12]. It is reasonable to believe that the construction of Struwe in reality produces a second solution of Eq. (1.1), but this is so far not known. In this paper we will address the issue of existence and multiplicity of solutions of problem (1.1) when Ω is not contractible to a point. More precisely, we provide conditions for the existence of solutions of problem (1.1) for small λ which satisfy the bubbling condition (1.4), at the same time giving a precise characterization of its bubbling location. In particular our main result implies the following: if Ω has a hole “of any size”, namely Ω is not simply connected, then a solution blowing up at exactly two points and satisfying property (1.4) with k = 2 indeed exists. We expect this result to be true for any k 1, provided that the domain is not contractible to a point.
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Theorem 1. Assume that Ω is not simply connected. Then there exists a family of solutions uλ to problem (1.1) such that 1 2
λ |∇uλ | − 2
Ω
2
euλ = 4π + o(1)
2
Ω
where o(1) → 0 as λ → 0. The location of the bubbling points (which are exactly two) for the solutions in this result can be thoroughly described. To this end, let us introduce the following functional of k distinct points ξ1 , ξ2 , . . . , ξk ∈ Ω and k positive numbers m1 , m2 , . . . , mk ,
ϕk (ξ, m) = b
k
m2j + 2
j =1
+
k
k
m2j log m2j
j =1
H (ξj , ξj )m2j −
G(ξi , ξj )mi mj .
(1.6)
i =j
j =1
Here G and H are the Green function for the Laplacian on Ω with Dirichlet boundary condition and its regular part, as defined above and b is an absolute constant which we will specify later. As λ approaches 0, the solution in Theorem 1 satisfies, up to subsequences,
uλ (x) ∼
2 √ λ mj G(x, ξj ) j =1
where (m1 , m2 , ξ1 , ξ2 ) is a critical point of ϕ2 . Let us consider an open set D compactly contained in the domain of the functional ϕk , namely D¯ ⊂ (ξ, m) ∈ Ω k × Rk+ /ξi = ξj ∀i = j . We say that ϕk has a stable critical point situation if there exists a δ > 0 such that for any ¯ with g 1 ¯ < δ, the perturbed functional ϕk + g has a critical point in D. g ∈ C 1 (D) C (D ) Theorem 2. Let k 1 and assume that there is an open set D where ϕk has a stable critical point situation. Then, for all small λ > 0 there exists a solution uλ of problem (1.1) such that 1 2
|∇uλ |2 − Ω
λ 2
2
euλ = 2kπ + o(1) Ω
where o(1) → 0 as λ → 0. Moreover, passing to a subsequence, there exists (ξ, m) ∈ D such that ∇ϕk (ξ, m) = 0 and
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k √ uλ (x) = λ mj G(x, ξj ) + o(1) j =1
where o(1) → 0 on each compact subset of Ω¯ \ {ξ1 , . . . , ξk }. As a direct consequence of the above result, we find in addition that the presence of topology of the domain induces multiplicity of solutions with a single blow-up point: when k = 1 a bubbling solution around a critical point of the function H (x, x) exists. It is standard that H (x, x) → +∞ as x → ∂Ω, thus we always have such a solution bubbling near a global minimizer of this function. In addition, if Ω is not simply connected, Ljusternik–Schnirelmann theory yields the presence of at least cat(Ω) = d + 1 such solutions, where d is the number of holes of Ω. Theorem 1 follows by showing via a topological construction that ϕ2 has a stable critical point situation. We strongly believe that such a situation is in reality present for ϕk for any k 3, but the construction appears to be much harder. It is reasonable to conjecture that any family of solutions satisfying (1.4) must have the concentration behavior described in the above theorem, in further precision of the result in [14]. It is interesting to mention the link of the above discovered concentration phenomena with the related Liouville equation u + λeu = 0 under Dirichlet boundary condition in a bounded domain Ω in R2 , see [8,13,15] and references therein. The fine blow-up structure very close to the bubbling points is similar to that in the present problem however scalings and intermediate regimes are much more subtle here. Our choice of first approximations to bubbling solutions is inspired by the discovery of the blow-up shapes first in [5] then in [2,14]. However more accurate information is needed, in particular the discovery of the role of the distinct weights mj , which marks a strong difference with the blow-up structure in Liouville’s equation. As in other elliptic problems involving point concentration phenomena, our strategy of proof involves linearization about a first approximation, to later reduce the problem to a finite dimensional variational one of adjusting the bubbling centers and corresponding weights. The critical character of this nonlinearity is very much reflected in the delicate error terms left by the first approximation, which makes the linear elliptic theory needed fairly subtle because of the multiple-regime in the error 2 size. While we restrict our investigation here to the nonlinearity λueu , we expect that similar analysis can be carried out for a broader class of critical exponential growth nonlinearities. For simplicity in the exposition we shall only consider the prototype case. 2. A first approximation and outline of the argument It is convenient for our purposes to rewrite problem (1.1) by replacing u = problem becomes
u˜ + λue ˜ λu˜ = 0, u˜ = 0 2
u > 0 in Ω, on ∂Ω.
√ λu, ˜ so that the
(2.1)
Let us consider k distinct points ξ1 , ξ2 , . . . , ξk in Ω and k positive numbers m1 , m2 , . . . , mk . We choose a sufficiently small but fixed number δ > 0 and assume that for j = 1, . . . , k,
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dist(ξj , ∂Ω) δ,
|ξl − ξj | δ
for l = j, δ < mj <
1 δ
(2.2)
for some given δ > 0. We shall build an approximation U˜ (x) which away from the points ξj satisfies, in agreement with the statement of Theorem 1, U˜ (x) =
k
mj G(x, ξj ) + o(1) as λ → 0.
(2.3)
j =1
Near each point ξj , we consider positive numbers μj , εj , to be chosen in dependence from the values of λ, ξ and m, and the function U˜ (x) =
k
mj log
j =1
1 − Hj (x) (μ2j εj2 + |x − ξj |2 )2
(2.4)
where Hj is a harmonic function so that the boundary condition zero is satisfied, that is
Hj = 0, Hj (x) = log
in Ω, 1 , (μ2j εj2 +|x−ξj |2 )2
for x ∈ ∂Ω.
Let us observe that from elliptic estimates Hj (x) = H (x, ξj ) + O εj2 μ2j , uniformly in Ω, where H is the regular part of Green’s function with zero Dirichlet boundary condition in Ω, as defined in (1.5). Hence log
(μ2j εj2
1 − Hj (x) = G(x, ξj ) + O εj2 μ2j , 2 2 + |x − ξj | )
(2.5)
and the desired outer expansion (2.3) indeed holds for this U˜ . Let us examine U˜ in a small neighborhood of a given ξj . We write, for |x − ξj | < δ, with sufficiently small but fixed δ, U˜ (x) = mj wj (x) + log εj−4 + βj + θ (x) where mj βj := −mj log 8μ2j − mj H (ξj , ξj ) +
mi G(ξi , ξj ),
i =j k mj θ (x) = O |x − ξj | + O εi2 i=1
and
(2.6)
M. del Pino et al. / Journal of Functional Analysis 258 (2010) 421–457
wj (x) := wμj
x − ξj εj
427
with wμ (y) := log
8μ2 . (μ2 + |y|2 )2
(2.7)
The functions wμ , μ > 0, are the radially symmetric solutions of the Liouville equation w + ew = 0 in R2 . The idea is to choose the numbers μj , εj in such a way that the error of approximation for U˜ is small around each point ξj . This error is by definition R(x) = U˜ + f (U˜ ).
(2.8)
Here and in what follows f denotes the nonlinearity f (u) ˜ = λue ˜ λu˜ . 2
(2.9)
Let us observe that for |x − ξj | < δ we have −U˜ (x) = mj εj−2 ewj +
k
O εi2 .
i=1
On the other hand 1 ˜ f (U ) = λmj log 4 + λmj wj + O(1) εj ×e
2m2j λ(βj +θ) log
1 εj4
e
2m2j λ(log
1 +βj )wj εj4
2
2
eλwj mj e
λm2j log2
1 εj4
eλmj (βj +θ) 2
2 +2λm2 θw j j
.
Let us make the following choice of εj , 1 2m2j λ log 4 + βj = 1, εj
(2.10)
so that λU˜ =
1 1 + 2m2j λ wj + O(1) 2mj
(2.11)
and 2 2 ˜2 eλU = eβj /2 εj−2 ewj eλmj wj 1 + O(θ ) 1 + O(λ)wj .
(2.12)
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Thus, in order to match f (U˜ ) and −U˜ at main order near ξj we must fix μj so that the number βj satisfies eβj /2 = 2m2j ,
(2.13)
namely we require that μj satisfies log 8μ2j = −2 log 2m2j − H (ξj , ξj ) +
mi m−1 j G(ξi , ξj ).
(2.14)
i =j
Then we get 2 2 f (U˜ ) = mj 1 + 2λm2j wj + O(λ) ewj εj−2 eλmj wj 1 + O(θ ) 1 + O(λ)wj . Now, it is easily checked that there is a C > 0 such that for all |x − ξj | < δ we have θ (x) = O |x − ξj | + εi2 i
C
log εj−4
|x − ξj | log 1 + +1 εj
and hence 1 + O(θ ) 1 + O(λwj ) 1 + Cλ|wj | . Hence, the error of approximation is given near ξj by 2 2 2 O εi . R(x) = mj εj−2 ewj 1 − 1 + 2m2j λwj + O(λ) emj λwj 1 + O(λwj ) + i
Observe that for |x − ξj | = O(ε) we have that R(x) ∼ λεj−2 ewj . On the other hand, for |x − ξj | > δ for all j we clearly have that |R(x)| Cλ. Hence the error of approximation satisfies the global bound R(x) Cλρ(x), where ρ(x) :=
k
ρi χBδ (ξj ) (x) + 1;
j =1
here χ denotes the characteristic function and ρj :=
2 2 1 1 + 2m2j λ(wj + 1) 1 + λ 1 + |wj | emj λwj − 1 εj−2 ewj . 2 2mj λ
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For later reference let us notice that ρj (x) = cγj
w2 2γj 1 1 1 + (wj + 1) 1 + 1 + |wj | e j − 1 εj−2 ewj , γj γj
(2.15)
where γj = log εj−4 . This motivates us to introduce the following L∞ -weighted norm for bounded functions defined in Ω. Let us set ρ(x) :=
k
ρi χBδ (ξj ) (x) + 1
j =1
and define h∗ = sup ρ(x)−1 h(x),
(2.16)
R∗ Cλ.
(2.17)
x∈Ω
so that
In the rest of this paper we will look for a solution u˜ of problem (2.1) of the form u˜ = U˜ + φ, where U˜ is defined as above in (2.4), and we aim at finding a solution for which φ is small, provided that the points ξj and scalars mj are suitably chosen. For small φ it is natural to rewrite problem (2.1) as a nonlinear perturbation of its linearization, namely,
φ + f (U˜ )φ = −R − [f (U˜ + φ) − f (U˜ ) − f (U˜ )φ] φ=0
in Ω, on ∂Ω.
(2.18)
Let us observe that ˜2 f (U˜ ) = λ 2λU˜ 2 + 1 eλU = O(λ) away from the points ξj , so the linearized operator is a small perturbation of the Laplacian away from the concentration points. Using relations (2.11), (2.12), similarly to the computation for f (U˜ ) we find that very close to the points ξj , at least for distances O(εj ) from ξj , we have f (U˜ ) ≈ εj−2 ewj . Moreover, repeating the corresponding computation for R, we readily get that k ˜ εj−2 ewj Cλ. f (U ) − j =1
∗
(2.19)
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For this reason, it is more convenient to rewrite problem (2.18) in the form
L(φ) := φ + [ φ = 0,
k
−2 wj (x) ]φ j =1 εj e
= −[R + N (φ)],
in Ω, on ∂Ω,
(2.20)
where k −2 ωj (x) ˜ ˜ ˜ ˜ N(φ) = f (U + φ) − f (U ) − f (U )φ + f (U ) − εj e φ.
(2.21)
j =1
What we hope for is to find a small solution φ to problem (2.21) which respects the size just defined for the error, namely so that φ∞ Cλ. Let us observe that the function εj−2 ewj is actually very concentrated near ξj : It has size O(εj2 ) away from the point, while it globally integrates to 8π in R2 . L is therefore a nontrivial perturbation of the Laplacian near the points while it is essentially this operator in most of the domain. Unlike the Laplacian, the operator L has an approximate kernel which in principle prevents any form of bounded invertibility. In fact L can be approximately regarded as a superposition of the linear operators Lj (φ) = φ + εj−2 ewj φ. The problem Lj (φ) = 0 has bounded solutions originating in the natural invariances of the equation w + ew = 0. Let us consider the family of solutions wμ (y) given by (2.7). Then the functions z0j (y) = ∂μ wμj (y), satisfy the equation Z + e
wμj
zlj (y) = ∂yl wμj (y),
l = 1, 2,
Z = 0. Hence the functions
Zij (x) := zij
x − ξj , εj
i = 0, 1, 2,
(2.22)
are bounded solutions of Lj (Z) = 0 in all of R2 . It is known that these actually span the space of all bounded solutions of this equation, see [8] for a proof. We want to solve, in a uniformly bounded way, problems of the form L(φ) = h. This is possible, but only for a restricted class of right-hand sides. We identify them by considering the problem projected to a suitable orthogonal of the “almost-kernel” for L. To formulate this problem, let us consider now a large but fixed number R0 > 0 and a nonnegative function ζ (ρ) with ζ (ρ) = 1 if ρ < R0 and χ(ρ) = 0 if ρ > R0 + 1. We denote x − ξj . ζj (x) = εj−2 ζ ε
(2.23)
j
Given h ∈ L∞ (Ω), we consider the linear problem of finding a function φ such that for certain scalars cij , i = 0, 1, 2, j = 1, . . . , k, it satisfies
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L(φ) = h +
2 k
cij Zij ζj ,
431
in Ω,
(2.24)
on ∂Ω,
(2.25)
for all i, j.
(2.26)
i=0 j =1
φ = 0,
Zij ζj φ = 0, Ω
Consider the norm φ∞ = sup φ(x). x∈Ω
Proposition 2.1. Let δ > 0 be fixed. There exist positive numbers λ0 and C, such that for any points ξj , j = 1, . . . , m in Ω, parameters mj , j = 1, . . . , k, satisfying (2.2), μj given by (2.14), and h ∈ L∞ (Ω), there is a unique solution φ := Tλ (h) to problem (2.24)–(2.26) for all λ < λ0 . Moreover φ∞ Ch∗ .
(2.27)
We will prove this result in the next section. It is worth mentioning that the criticality of the Moser Trudinger situation is fairly delicate compared with other problems of concentration phenomena. Not only is the form of the seeked solutions quite nonobvious, but also, even with the right ansatz, the error is not small, and the invertibility theory in which the weight ρ enters is indeed very tight. Let us consider now the projected version of problem (2.20), L(φ) = −R − N (φ) +
2 k
cij Zij ζj ,
in Ω,
(2.28)
on ∂Ω,
(2.29)
for all i, j.
(2.30)
i=0 j =1
φ = 0, Zij ζj φ = 0,
Ω
To solve this problem in L∞ (Ω), we recast it in fixed point form φ = Tλ −R − N (φ) := A(φ)
(2.31)
where Tλ is the operator in Proposition 2.1. Using estimate (2.19) and the easily checked fact that f (U˜ )∗ C we find that N (φ) Cφ2 + Cλφ∞ . ∞ ∗ This estimate, Proposition 2.1 and estimate (2.17) imply that A(B) ⊂ B where B = {φ/φ∞ Mλ} for a sufficiently large and fixed M and all small λ. Besides, it is directly checked that the operator A has a small Lipschitz constant in B for all small λ. Thus, the contraction mapping principle leads us to the following fact.
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Proposition 2.2. Under the assumptions of Proposition 2.1 there exist positive numbers C and λ0 , such that for all 0 < λ < λ0 problem (2.28)–(2.30) has a solution φ = φ(ξ, m) which defines a continuous map into L∞ (Ω) and satisfies φ∞ C λ. The constant C is uniform on all (ξ, m) satisfying the constraints (2.2). As a function of points and parameters, this φ is actually smooth. We will check this in Section 4. Evaluating at this φ in problem (2.28)–(2.30), the constants cij define functions cij = cij (ξ, m). Thus we need to find a solution (ξ, m) of the 3k × 3k system cij (ξ, m) = 0 for all i = 0, 1, 2, j = 1, . . . , k.
(2.32)
To solve this problem, we formulate it in variational form. Let Jλ be the energy functional of problem (1.1) defined in (1.2). As we will see in Lemma 5.1, if (ξ, m) is a critical point of the functional (ξ, m) −→ Jλ
√ λ U˜ (ξ, m) + φ(ξ, m) =: Iλ (ξ, m),
(2.33)
then it automatically satisfies system (2.32). In Lemma 6.1 we will show that at main order we have Iλ (ξ, m) = 2kπ + aλ + λ4πϕk (ξ, m) + o(λ) where a is a fixed constant and ϕk is the functional introduced in (1.6) and o(1) → 0 in the C 1 -sense as λ → 0, uniformly on (ξ, m) satisfying (2.2). From here, using the definition of nontrivial critical point situation, the result of Theorem 1 immediately follows. Theorem 2 will be established as a special case of this result in Section 7. In the remainder of this paper we will carry out the above outlined construction. A main step in solving problem (2.20) for small φ under a suitable choice of the points ξj and the parameters mj is that of a solvability theory for the linear operator L. This is the content of next section. 3. Analysis of the linearized operator We will prove here Proposition 2.1. At the very core of the proof is the following estimate for the Laplacian. Let us consider fixed positive numbers R and M and for ε > 0 the annular region M Aε = x/R < |y| < ε and the function 2 w w+1 |w| ρε |y| = γ ew e 2γ 1 + 1+ −1 γ γ
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where w |y| = log
(μ2
8μ2 , + |y|2 )2
γ = log ε −4 .
Let us consider the problem
−Ψ = ρε Ψ =0
in Aε , on ∂Aε .
(3.1)
Lemma 3.1. There exist constants C, ε0 depending only on uniform upper and away from zero lower bounds for R, M, such that for all ε < ε0 the solution Ψε to problem (3.1) satisfies Ψε ∞ C. Proof. The solution Ψ is radial, say Ψ = Ψ (r), r = |y|. Let us consider the change of variables ψ(t) = Ψ (et ). Then it is straightforward to check that ψ satisfies the two-point boundary value problem −ψ (t) = e2t ρε et ,
t ∈ [log R, log M + γ /4],
ψ(log R) = ψ(log M + γ /4) = 0.
Since w(r) = −4 log r + O(1) in all the considered range, we find that we can estimate e2t ρε et Ce−2t
1+
t M + γ γ
1+
2 2t t e γ − 1 =: g(t). γ
Thus, in order to prove the desired result it suffices to show that the solution of the problem −ψ˜ (t) = g(t),
t ∈ [log R, log M + γ /4],
˜ ˜ ψ(log R) = ψ(log M + γ /4) = 0
is uniformly bounded. Here and in the rest of the proof, C denotes a generic constant independent of large γ . ˜ Since ψ˜ is concave and positive, it suffices to show that the quantity ψ(a) is uniformly bounded at some point a distant of order γ from the boundary, let us say a = γ8 . Let G(t, s) be the Green’s function of −ψ with Dirichlet boundary conditions in the interval. Then γ γ C min t − log R, log M + − t . G t, 8 4 Hence, since γ ˜ ψ = 8
log M+ γ4
log R
we get
γ g(t) dt G t, 8
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M. del Pino et al. / Journal of Functional Analysis 258 (2010) 421–457 γ
8 ψ˜ γ C (t − log R)g(t) dt 8 log R log M+ γ4
+C γ 8
γ log M + − t g(t) dt. 4
(3.2)
The first integral in (3.2) can be estimated as follows γ
8
γ
(t − log R)g(t) dt C
log R
8
e−2t e
2t 2 γ
1 + t 4 dt
log R γ
8
e−t 1 + t 4 dt C,
C log R
since in this region |g(t)| ∼ γ e−2t e
2t 2 γ
2t 2 γ
(1 − γt ), so we get
log M+ γ4
γ 8
t. Concerning the second integral in (3.2), we observe that in that region
γ log M + − t g(t) dt C 4
γ 4 +O(1)
e
2
−2t+ 2tγ
2 γ − t + O(1)
γ 8 γ 4 +O(1)
C
2 e−s s + O(1) ds C,
O(1)
where we have used the change of variable s = γ − t. We have thus established the existence of a positive constant C independent of ε such that Ψ (y) C, and the proof of the lemma is concluded.
for R < |y| <
M , ε
2
An immediate consequence of the above estimate is the following. Corollary 1. Let us consider now the weight ρj (x) defined in (2.15), and the solution Ψj to the problem −Ψj (x) = ρj , Ψj = 0 on ∂Aε .
in Rεj < |x − ξj | < M, (3.3)
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Then there exists a C > 0, independent of all small εj such that Ψj (x) C,
for all Rεj < |x − ξj | < M.
The proof of Proposition 2.1 makes use of an a priori estimate. Lemma 3.2. Under the assumptions of Proposition 2.1, there exist positive numbers λ0 and C, such that for any points ξj , j = 1, . . . , m in Ω, and positive numbers mj , μj , for j = 1, . . . , k, which satisfy relations (2.2) and (2.14), and any solution φ to (2.24)–(2.26), one has φ∞ Ch∗ ,
(3.4)
for all λ < λ0 . Proof. We will carry out the proof of the a priori estimate (3.4) by contradiction. We assume then the existence of sequences λn → 0, points ξjn ∈ Ω and numbers mnj , μnj which satisfy relations (2.2) and (2.14), functions hn with hn ∗ → 0, φn with φn ∞ = 1, and constants cij n such that L(φn ) = hn +
2 k
cij n Zij ζj ,
in Ω,
(3.5)
on ∂Ω,
(3.6)
for all i, j.
(3.7)
i=0 j =1
φn = 0,
Zij ζj φn = 0, Ω
We will prove that in reality under the above assumption we must have that φn → 0 uniformly in Ω, which is a contradiction that concludes the result of the lemma. Passing to a subsequence we may assume that the points ξjn approach limiting, distinct points ∗ ξj in Ω. We claim that φn → 0 in C 1 local sense on compacts of Ω¯ \ {ξ1∗ , . . . , ξk∗ }. Indeed, let us observe that hn → 0 locally uniformly, away from the points ξj . Away from the ξj ’s we have then φn → 0 uniformly on compact subsets on Ω¯ \ {ξ1∗ , . . . , ξk∗ }. Since φn is bounded it follows also that passing to a further subsequence, φn approaches in C 1 local sense on compacts of Ω¯ \ {ξ1∗ , . . . , ξk∗ } a limit φ ∗ which is bounded and harmonic in Ω \ {ξ1 , . . . , ξk } and φ ∗ = 0 on ∂Ω. Hence φ ∗ extends smoothly to a harmonic function in Ω, so that φ ∗ = 0, and the claim follows. For notational convenience, we shall omit the explicit dependence on n in the rest of the proof. Next we claim that cij → 0 for all i = 0, 1, 2 and j = 1, . . . , k. Fix j and multiply Eq. (3.5) x−ξ against Zij (x) = zij ( εj j ), with zij given by (2.22). We integrate the resulting relation in B(ξj , δ) and obtain hZij + B(ξj ,δ)
2 l=0
clj
Zij Zlj ζj =
L(φ)Zij
B(ξj ,δ)
= B(ξj ,δ)
φL(Zij ) + ∂B(ξj ,δ)
[Zij ∇φ · ν − φ∇Zij · ν].
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Let us observe that L(Zij ) = l =j εl2 = o(1) and that the C 1 convergence to zero of φ on ∂B(ξj , δ) imply that the boundary integral in the above relation also approaches zero. On the other hand, we have
hZij Ch∗ ,
Zij Zlj ζj = κi δlj
and
B(ξj ,δ)
where κi is a positive universal constant. Then we get cij → 0, as desired. Observe then that n h˜ n := hn + cij Zij ζj satisfies h˜ n ∗ → 0. Our next claim is that φn approaches zero uniformly, close to the concentration points. More precisely, we have that sup
|x−ξj |
φ(x) → 0 for all R > 0.
Let us assume the opposite, so that there exists an index j and R > 0 such that, for all n, sup |x−ξjn |
φn (x) κ > 0.
Let us set φˆ n (z) = φn (ξjn + εj z). Elliptic estimates imply that φˆ n converges uniformly over compacts to a bounded solution φˆ = 0 of the problem in R2 φ +
8μ2j (μ2j + |z|2 )2
φ = 0.
According to the nondegeneracy result in [8], φˆ is therefore a linear combination of the functions zij , i = 0, 1, 2. However, our assumed orthogonality conditions on φn pass to the limit and yield ζ (|z|)zij φˆ = 0, for i = 0, 1, 2, thus a contradiction from which the claim follows. Let us fix such a number R > 0 which we may take larger whenever it is needed and consider the quantity φi =
k
sup
|φ|.
j =1 B(ξj ,Rεj )
The desired result, φn ∞ → 0 is a consequence of the following fact: there is a uniform constant C > 0 such that φ∞ C φi + h∗ .
(3.8)
We will establish this with the use of barriers. Indeed, a crucial fact is that the operator L satisfies the maximum principle in Ω outside balls centered at the points ξj of radius Rεj , for some fixed R > 0. Let us check this. Given a > 0, consider the function
M. del Pino et al. / Journal of Functional Analysis 258 (2010) 421–457
|x − ξj | , z0 a εj
x ∈ Ω,
is the radial solution in R2 of z0 +
8 z (1+r 2 )2 0
Z(x) =
k j =1
where z0 (r) =
r 2 −1 1+r 2
−Z =
k
εj−2
j =1
(3.9)
= 0. Then we have
8a 2 (a 2 εj−2 |x − ξj |2 − 1) (1 + a 2 εj−2 |x − ξj |2 )3
437
.
So that for |x − ξj |2 > 100a −2 εj2 for all j ,
−Z 2
k
a 2 εj−2
j =1
(1 + a 2 εj−2 |x − ξj |2 )2
m
j =1
εj2 a 2 |x − ξj |4
.
On the other hand, in the same region,
k
εj−2 eω˜ j Z C
j =1
m j =1
εj2 |x − ξj |4
.
Hence if a is taken small and fixed, and R> 0 is chosen sufficiently large depending on this a, ˜ then we have that L(Z) < 0 in Ω˜ := Ω \ m j =1 B(ξj , Rεj ) and Z > 0 in Ω. Thus in this region ˜ L satisfies the maximum principle, namely if L(ψ) 0 in Ω˜ and ψ 0 on ∂ Ω˜ then ψ 0 in Ω. Let us consider the function Ψj in Corollary 1. Since Ψj is uniformly bounded, it is easy to see that if R is chosen sufficiently large then the uniformly bounded function ψj (x) := Ψj (4x) satisfies −L(ψj ) > ρj for Rεj < |x − ξj | < M, where we fix M > 0 such that Ω ⊂ B(ξj , M). Let us set ˜ φ(y) = 2φi Z(x) + h∗
k
ψj (x),
j =1
where Z is the function defined above in (3.9). Then, it iseasily checked that, choosing R larger ˜ h, φ˜ φ on ∂ Ω, ˜ where Ω˜ = Ω \ m ˜ ˜ if necessary, −L(φ) j =1 B(ξj , Rεj ). Hence |φ| φ on Ω and estimate (3.8) follows. The proof of the lemma is concluded. 2 We have now the main ingredient to prove Proposition 2.1. Proof of Proposition 2.1. It only remains to prove the solvability assertion. To this purpose we consider the space 1 H = φ ∈ H0 (Ω): ζj Zij φ = 0 for i = 0, 1, 2, j = 1, . . . , k , Ω
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endowed with the usual inner product [φ, ψ] = Ω ∇φ∇ψ. Problem (2.24)–(2.26) expressed in weak form is equivalent to that of finding a φ ∈ H , such that [W φ − h]ψ dx,
[φ, ψ] =
for all ψ ∈ H,
Ω
where W = kj =1 εj −2 ewj . With the aid of Riesz’s representation theorem, this equation gets ˜ for certain h˜ ∈ H , where K is a compact rewritten in H in the operator form φ = K(φ) + h, operator in H . Fredholm’s alternative guarantees unique solvability of this problem for any h provided that the homogeneous equation φ = K(φ) has only the zero solution in H . This last equation is equivalent to (2.24)–(2.26) with h ≡ 0. Thus existence of a unique solution follows from the a priori estimate (2.27). This finishes the proof. 2 Proposition 2.1 says in particular that the unique solution φ = T (h) of (2.24)–(2.26) defines a continuous linear map from the Banach space C∗ of all functions h in L∞ for which h∗ < ∞, into L∞ , with norm bounded uniformly in λ. 4. Differentiability with respect to parameters Let φ be the solution to the nonlinear projected problem (2.28)–(2.30), whose existence is guaranteed by Proposition 2.2. This section is devoted to study the dependence of φ on (ξ, m) = (ξ1 , . . . , ξk , m1 , . . . , mk ), where the points ξj and the parameters mj satisfy the constraints (2.2). A direct consequence of the fixed point characterization of φ given by Proposition 2.2 together with the fact that the error term R in the right-hand side of Eq. (2.28) depends continuously (in ¯ is continuous (in the the ∗-norm) on (ξ, m), is that the map (ξ, m) → φ into the space C(Ω) ∞-norm). We analyze next the differentiability of this map, say with respect to ξ11 . We start with the following fact: Fix h ∈ C∗ and let φ = Tλ (h) be the solution to the linear projected problem (2.24)–(2.26) whose existence is guaranteed by Proposition 2.1. Then, under the same assumptions of Proposition 2.1, there exist positive constants λ0 and C such that, for all λ < λ0 , ∂ξ Tλ (h) Ch∗ . 11 ∞
(4.1)
Indeed, we have in general ∂ξ Tλ (h) Ch∗ , hl ∞
∂m Tλ (h) Ch∗ . j ∞
(4.2)
Differentiating Eq. (2.24) and the orthogonality condition (2.26), we get that Z := ∂ξ11 φ satisfies
L(Z) = −∂ξ11
k j =1
εj−2 eω˜ j (x) φ +
i,j
cij ∂ξ11 (Zij ζj ) +
i,j
dij Zij ζj ,
M. del Pino et al. / Journal of Functional Analysis 258 (2010) 421–457
439
with dij = ∂ξ11 (cij ), and
Zij ζj Z = −
∂ξ11 (Zij ζj )φ.
Ω
Define αab =
φ∂ξ11 (Zab ζb ) , (Zab ζb )2
for a = 0, 1, 2 and b = 1, . . . , k. We have Z˜ = Tλ (f ),
f = −∂ξ11
where k
εj−2 eω˜ j (x)
Z˜ = Z +
αab Zab ζb
a,b
˜ ij ζj = 0 for all i, j . Furthermore ZZ
φ+
j =1
cij ∂ξ11 (Zij ζj ) +
i,j
αab L(Zab ζb ).
ab
˜ ∞ Ch∗ and hence the validity of (4.1). Using the result of Proposition 2.1, we get Z Let φ be now the solution to (2.28)–(2.30). Since φ = Tλ (−(N (φ) + R)), we have formally that ∂ξ11 φ = (∂ξ11 Tλ ) − N (φ) + R + Tλ − ∂ξ11 N (φ) + ∂ξ11 R . From (4.1) we get (∂ξ Tλ ) − N (φ) + R C N (φ) + R Cλ. 11 ∞ ∗ On the other hand, a direct computation gives k −2 ω˜ j (x) εj e ∂ξ11 N (φ) = f (U + φ) − f (U ) − f (U )φ ∂ξ11 U + ∂ξ11 f (U ) − φ j =1
k −2 ω˜ j (x) εj e ∂ξ11 φ. + f (U + φ) − f (U ) ∂ξ11 φ + f (U ) − j =1
Thus, using (2.19) ∂ξ N (φ) C λ−1 φ2 + λφ∞ + λ∂ξ φ∞ + λ∂ξ φ∞ . ∞ 11 11 11 ∗ Since ∂ξ11 R∗ Cλ, we can conclude that ∂ξ11 φ∞ Cλ. Analogous computation holds true if we differentiate with respect to mj .
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The above computation can be made rigorous by using the implicit function theorem and the fixed point representation (2.31) which guarantees C 1 regularity in (ξ, m). Thus we have the validity of the following: ¯ where φ is the unique solution Lemma 4.1. Consider the map (ξ, m) → φ into the space C(Ω), to the nonlinear projected problem (2.28)–(2.30), whose existence is guaranteed by Proposition 2.2. Under the assumptions of Proposition 2.1 the derivative Dξ φ (or Dm φ) exists and defines a continuous function of (ξ, m). Besides, there is a constant C > 0, such that Dξ φ∗ Cλ,
Dm φ∗ Cλ.
After problem (2.28)–(2.30) has been solved, we will find solutions to the full problem (2.20) (or equivalently (1.1)) if we manage to adjust (ξ, m) in such a way that cij (ξ, m) = 0 for all i, j . A nice feature of this system of equations is that it turns out to be equivalent to finding critical points of a functional of (ξ, m) which is close, in an appropriate sense, to the energy of the first approximation U . We make this precise in the next sections. 5. Variational reduction As we have said, after problem (2.28)–(2.30) has been solved, we find a solution to problem (2.20) and hence to the original problem if ξ and m is such that cij (ξ, m) = 0
for all i, j.
(5.1)
This problem is indeed variational: it is equivalent to finding critical points of a function of ξ and m. To see this let us recall the energy functional Jλ associated to problem (1.1), namely 1 Jλ (u) = 2
λ |∇u| dx − 2
2
Ω
2
eu dx.
(5.2)
Ω
We define, as in (2.33), Iλ (ξ, m) ≡ Jλ
√ λ U˜ (ξ, m) + φ(ξ, m) ,
(5.3)
where φ is the solution of problem (2.28)–(2.30) given by Proposition 2.1. Critical points of Iλ correspond to solutions of (5.1) for small λ, as the following result states. Lemma 5.1. Under the assumptions of Proposition 2.1, the functional Iλ (ξ, m) is of class C 1 . Moreover, for all λ > 0 sufficiently small, Dξ,m Iλ (ξ, m) = 0
⇒
cij (ξ, m) = 0
for all i, j.
Proof. A direct consequence of the results obtained in Section 4 and of the definition of the function U˜ is the fact that the map (ξ, m) → Iλ (ξ, m) is of class C 1 .
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Furthermore, thanks to Lemma 4.1, we have that √ √ λ U (ξ, m) + φ(ξ, m) λDξ,m U (ξ, m) √ √ λDξ,m φ(ξ, m) + DJλ λ U (ξ, m) + φ(ξ, m) √ √ λDξ,m U (ξ, m) 1 + o(1) . = DJλ λ U (ξ, m) + φ(ξ, m)
Dξ,m Iλ (ξ, m) = DJλ
(5.4)
Let u(x) ˜ = U (ξ, m)(x) + φ(ξ, m)(x). For any l define Il (v) =
m2l 2
|Dv|2 − Ωl
where Ωl =
Ω−ξl εl .
2 2
ev eλml v ,
(5.5)
Ωl
l If we perform the change of variables u(x) ˜ = ml vl ( x−ξ εl ) +
1 2λml ,
we get
√ ˜ := Jλ ( λu) = Il (v) J˜λ (u) and, as a direct consequence of (2.6), (2.10) and (2.13), vl (y) = ωμl (y) +
O |εl y + ξl − ξj | + O εj2
for |y|
j
δ . εl
(5.6)
Furthermore, we have u˜ + λue ˜ λu˜ = 2
cij ζj Zij (x),
x ∈ Ω,
ij
and that v˜l (y) solves in Ωl ml εl−2 v˜l
λm2 v˜ 2 εl y + ξl − ξj εl y + ξl − ξj −2 2 l l εj Zij . = + e 1 + 2λml v˜l e cij ζ εj εj v
ij
Thus we start with the computation of Dm1 Iλ (ξ, m). From (5.4), we get Dm1 Iλ (ξ, m) = Dm1 Il (v˜l ) = DIl (v˜l )[Dm1 v˜l ] εl y + ξl − ξj εl y + ξl − ξj −2 = εj Zij Dm1 vl dy cij . ζ εj εj ij
Ωl
Fix now i and j . To compute the coefficient in front of cij in the above expression, we choose l = j and obtain Ωl
∂μj εl y + ξl − ξj εl y + ξl − ξj −2 2 εj Zij Dm1 vl dy = ζ z0j (y) dy 1 + o(1) . εj εj ∂m1
R2
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Thus we conclude that, for any h = 1, . . . , k, Dmh Iλ (ξ, m) =
∂μj 2 Z0j (y) dy c0j 1 + o(1) . ∂mh j
R2
A direct argument shows on the other hand that, for a = 1, 2, b = 1, . . . , k, Dξab Iλ (ξ, m) =
∂μj j
∂ξab
2 Z0j (y) dy c0j
+
R2
2 Z1j (y) dy
cab 1 + o(1) .
R2
We can conclude that Dξ,m Iλ (ξ, m) = 0 implies the validity of a system of equations of the form j
∂μj c0j 1 + o(1) = 0, ∂mh
∂μj A c0j + cab 1 + o(1) = 0, ∂ξab
h = 1, . . . , k,
(5.7)
a = 1, 2, b = 1, . . . , k,
(5.8)
j
for some fixed constant A, with o(1) small in the sense of the L∞ norm as λ → 0. The conclusion ∂μ of the lemma follows if we show that the matrix ( ∂mjh ) of dimension k × k is invertible in the range of the points ξj and parameters mj we are considering. Indeed, this fact implies unique solvability of (5.7). Inserting this in (5.8) we get unique solvability of (5.8). Consider the definition of the μj ’s, in terms of mj ’s and points ξj given in (2.14). These relations correspond to the gradient Dm F (m, ξ ) of the function F defined as follows F (m, ξ ) =
k 1 2 mj −2 log 2m2j − log 8μ2j − 1 − H (ξj , ξj ) + G(ξi , ξj )mi mj . 2 j =1
i =j
It is natural to perform the change of variable sj = m2j . With abuse of notation, the above function now reads as follows F (s, ξ ) =
k 1 √ sj −2 log 2sj − log 8μ2j − 1 − H (ξj , ξj ) + G(ξi , ξj ) si sj . 2 j =1
i =j
This is a strictly convex function of the parameters sj , for parameters sj uniformly bounded and uniformly bounded away from 0 and for points ξj in Ω uniformly far away from each other 2F ) is invertible in the range of parameters and from the boundary. For this reason, the matrix ( ∂s∂i ∂s j and points we are considering. Thus, by the implicit function theorem, relation (2.14) defines a ∂μ diffeomorphism between μj and mj . This fact gives the invertibility of ( ∂mjl ) we were aiming at. This concludes the proof of the lemma. 2
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In order to solve √for critical points of the functional Iλ , a key step is its expected closeness to the functional Jλ ( λU˜ ), which we will analyze in the next section. Lemma 5.2. The following expansion holds √ Iλ (ξ, m) = Jλ ( λU˜ ) + θλ (ξ, m), where |θλ | + |∇θλ | = O λ3 , uniformly on points and parameters satisfying the constraints in Proposition 2.1. √ Proof. Taking into account DJλ ( λ(U˜ + φ))[φ] = 0, a Taylor expansion gives Jλ
√ √ λ(U˜ + φ) − Jλ ( λU˜ ) 1 =λ
D 2 Jλ
√
λ(U˜ + tφ) [φ]2 (1 − t) dt
(5.9)
0
1 =λ 0
N (φ) + R φ +
Ω
f (U˜ ) − f (U˜ + tφ) φ 2 (1 − t) dt.
(5.10)
Ω
Since φ∞ Cλ, we get Jλ
√
√ λ(U˜ + φ) − Jλ ( λU˜ ) = θ˜λ = O λ3 .
Let us differentiate with respect to ξ . We use the representation (5.9) and differentiate directly under the integral sign, thus obtaining, for each j = 1, 2, l = 1, . . . , k, √ √ ∂ξj l Jλ λ(U˜ + φ) − Jλ ( λU˜ ) 1 =λ 0
∂ξj l N (φ) + R φ +
Ω
∂ξj l
2 ˜ ˜ f (U ) − f (U + tφ) φ (1 − t) dt.
Ω
Using the fact that ∂ξ φ∞ Cλ and the computations in the proof of Lemma 4.1 we get √ √ ∂ξj l Jλ λ(U˜ + φ) − Jλ ( λU˜ ) = ∂ξj l θ˜λ = O λ3 . In a very analogous way one gets √ √ ∂mj Jλ λ(U˜ + φ) − Jλ ( λU˜ ) = O λ3 . The continuity in ξ and m of all these expressions is inherited from that of φ and its derivatives in ξ and in m in the L∞ norm. The proof is complete. 2
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6. Asymptotics of energy of approximate solution The purpose of this section is to give an asymptotic estimate of Jλ (U ) where U (x) = The function U is defined as U (x) =
k √ λ mj log j =1
1 − H (x) j (μ2j εj2 + |x − ξj |2 )2
√ λU˜ .
(6.1)
(see (2.4)) and Jλ is the energy functional associated to (1.1), whose definition we recall below 1 Jλ (u) = 2
λ |∇u| dx − 2
2
Ω
2
eu dx. Ω
We have the following result. Lemma 6.1. Let δ > 0 be a fixed small number and U be the function defined in (6.1). With the choice (2.14) for the parameters μj , the following expansion holds Jλ (U ) = 2kπ + aλ + 4πλϕk (ξ, m) + λ2 Θλ (ξ, m)
(6.2)
where the function ϕk (ξ, m) = ϕk (ξ1 , . . . , ξk , m1 , . . . , mk ) is defined by ϕk (ξ, m) = b
k
m2j + 2
j =1
+
k
k
m2j log m2j
j =1
H (ξj , ξj )m2j −
G(ξi , ξj )mi mj .
(6.3)
i =j
j =1
Here G and H are the Green function for the Laplacian on Ω with Dirichlet boundary condition and its regular part, as defined in Section 1, and a, b are absolute constants. In (6.2), Θλ is a smooth function of (ξ, m) = (ξ1 , . . . , ξk , m1 , . . . , mk ), bounded together with its derivatives, as λ → 0, uniformly on points ξ1 , . . . , ξm ∈ Ω and parameters (m1 , . . . , mk ) ∈ (R+ )k satisfying (2.2). Remark 6.1. In the sequel, by θλ , Θλ we will denote generic functions of ξ and m that are bounded, together with its derivatives, in the region dist(ξi , ∂Ω) > δ and |ξi − ξj | > δ, and δ < mj < 1δ . Proof. We write U (x) =
k j =1
Uj (x),
with Uj (x) =
√ λmj uj (x) − Hj (x)
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and uj (x) = log
(μ2j εj2
1 . + |x − ξj |2 )2
We start with 1 2
m 1 |∇U |2 dx = |∇Uj |2 dx + ∇Uj ∇Ui dx . 2
Ω
(6.4)
j =i Ω
j =1 Ω
Fix j . We have 1 2
|∇Uj | dx 2
Ω
= λm2j
1 1 2 2 |∇uj | dx − ∇uj ∇Hj dx + |∇Hj | dx 2 2 Ω
Ω
Ω
Ω
∂Ω
∂Hj ∂Hj 1 2 1 2 dσ + dσ = λmj |∇uj | dx − uj Hj 2 ∂ν 2 ∂ν
∂Hj 1 1 = λm2j dσ |∇uj |2 dx − Hj 2 2 ∂ν Ω
∂Ω
(6.5)
∂Ω
where ν denotes the unitary outer normal of ∂Ω. In the above equation we used the facts that Hj is harmonic in Ω and Uj is zero on the boundary ∂Ω. We will now evaluate Ω |∇uj |2 dx. Let δ˜ > 0 be small and fixed. We split the previous integral into two pieces, namely
|∇uj |2 dx =
|∇uj |2 dx +
˜ B(ξj ,δ)
Ω
|∇uj |2 dx.
˜ Ω\B(ξj ,δ)
Direct computations show, using (2.10), (2.13) and (2.14) |∇uj |2 dx ˜ B(ξj ,δ)
|x − ξj |2
= 16 ˜ B(ξj ,δ)
(μ2j εj2 + |x − ξj |2 )2
= 16 ˜ j j
|y|2 dy (1 + |y|2 )2
dx
x − ξj y= εj μj
B(0, μ δε )
= 16π −2 log εj μj − 1 + log (εj μj )2 + δ˜2 +
(εj μj )2 (εj μj )2 + δ˜2
(6.6)
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= 16π −2 log εj − log 8μ2j + log 8 − 1 + log (εj μj )2 + δ˜2 + = 16π
(εj μj )2 + δ˜2
βj 1 2 − + H (ξ , ξ ) − mi m−1 + 2 log 2m j j j j G(ξi , ξj ) + log 8 − 1 2 2 4mj λ i =j
+ log (εj μj )2 + δ˜2 + = 16π
(εj μj )2
(εj μj )2 (εj μj )2 + δ˜2
1 + log 2m2j + H (ξj , ξj ) − mi m−1 j G(ξi , ξj ) + log 8 − 1 2 4mj λ i =j
+ log (εj μj )2 + δ˜2 +
(εj μj )2 (εj μj )2 + δ˜2
.
(6.7)
On the other hand, taking into account that for the fundamental solution Γ we have 4 ∇Γ (x, ξ ) = |x−ξ | and that H = Γ on ∂Ω, we have
|x − ξj |2
|∇uj |2 dx = 16 ˜ Ω\B(ξj ,δ)
(μ2j εj2 + |x − ξj |2 )2
˜ Ω\B(ξj ,δ)
=
dx
∇Γ (x, ξj )2 dx + (εj μj )2 Θ ˜ (ξj ), δ
˜ Ω\B(ξj ,δ)
=
H (x, ξj )
∂Ω
∂Γ 1 (x, ξj ) dσ − 32π log + (εj μj )2 Θδ˜ ∂ν δ˜
=
H (x, ξj )Γ (x, ξj ) dx +
Ω
Γ (x, ξj )
∂H (x, ξj ) dσ ∂ν
∂Ω
1 + (εj μj )2 Θδ˜ δ˜ ∂H (x, ξj ) dσ = −8πH (ξj , ξj ) + H (x, ξj ) ∂ν − 32π log
∂Ω
1 + (εj μj )2 Θδ˜ . δ˜
− 32π log
(6.8)
In the above formula Θδ˜ (ξj ) is a function dependent on δ˜ which is uniformly bounded, together with its derivatives, in the region dist(ξj , ∂Ω) > δ. Noticing that the integral on the left-hand side in (6.6) is independent from δ˜ and that H (x, ξj ) ∂Ω
∂H (x, ξj ) dσ − ∂ν
∂Ω
Hj (x)
∂Hj (x) dσ = O (εj μj )2 , ∂ν
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from (6.5)–(6.8) we thus conclude that, for j = 1, . . . , k,
|∇Uj |2 dx = λm2j 16π Ω
−
1 1 + log 2m2j + H (ξi , ξi ) 2 4m2j λ
2 mi m−1 j G(ξi , ξj ) + log 8 − 1 + εj log
i =j
1 O(1) . εj
(6.9)
We next deal with the mixed term in (6.4). Fix i = j . Notice that √ Ui (x) = − λmi
8μ2i εi2 . ((εi μi )2 + |x − ξi |2 )2
Moreover Ui = 0 on ∂Ω. Hence we can write
√ ∇Ui ∇Uj dx = λmi
Ω
Ω
8μ2i εi2 Uj (x) dx ((εi μi )2 + |x − ξi |2 )2
8 1 log dy 2 2 2 (1 + |y| ) (εj μj ) + |εi μi y + ξi − ξj |2
= λmi mj 1 εi μi
(Ω−ξi )
8 Hj (εi μi y + ξi + ε) dy (1 + |y|2 )2
− λmi mj 1 εi μi
(Ω−ξi )
= 8πλmi mj G(ξi , ξj ) + O
εj2 log
1 1 + εi2 log εj εi
+ O εi2 + εj2 , (6.10)
where the O(·) terms have uniform bounds in ξ in the region considered. Summing up all the previous information contained in (6.9)–(6.10) and using the definition (2.14) for μj , we finally get the estimate for (6.4), namely 1 2
|∇U | dx = 2πk + 4πλ (2 log 8 − 2) 2
k
m2j + 2
j =1
Ω
+
k
m2j H (ξj , ξj ) −
k
m2j log 2m2j
j =1
mi mj G(ξi , ξj ) +
i =j
j =1
k j =1
εj2 log
1 O(1) . (6.11) εj
Let us now evaluate the second term in the energy. We have
λ
e Ω
U2
dx = λ
k
j =1 B(ξ ,δ √ε ) j j
e
U2
dx + Aλ .
(6.12)
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Taking into account (2.3), we first observe that Aλ = λ |Ω| + λΘλ (ξ, m)
(6.13)
with Θλ a function, uniformly bounded together with its derivatives, as λ → 0. Now, we write
2
√ B(ξj ,δ εj )
2
eU dx =
eU dx +
2
eU dx
√ B(ξj ,δ εj )\B(ξj ,δεj | log εj |)
B(ξj ,δεj | log εj |)
= I1 + I2 . We will show next that I1 = 16πm2j + λΘλ (ξ, m),
I2 = λΘλ (ξ, m)
(6.14)
for some function Θλ , uniformly bounded together with its derivatives, as λ → 0. Indeed, perx−ξ forming the change of variables y = εj j and using the notations Vj (y) = 2γj U (εj y +ξj )−2γj2
and γj = log εj−4 , we have
I1 = εj2 e
γj2
Vj (y)+
e
Vj2 (y) 4γj2
dy
B(0,δ| log εj |)
= 2m2j
R2
8 dy + λΘλ (ξ, m) = 16πm2j 1 + λΘλ (ξ, m) . 2 2 (1 + |y| )
On the other hand − 21
δεj
|I2 | C δ| log εj |
1 e r4
log2 r γj2
r dr
(t = log r) R2 +
γj2 4
=C R1 +log γj2
e
2 −2t+ 4t2 γj
R2 +
γj2 4
dt C
e−t dt = O(λ).
R1 +log γj2
We can thus conclude that estimate (6.2) holds true in C 0 -sense. The C 1 -closeness is a direct consequence of the fact that Θλ (ξ, m) is bounded together with its derivatives in the considered region. 2
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7. Proofs of the theorems In this section we carry out the proofs of our main results. Proof of Theorem 2. Let D be the open set such that D¯ ⊂ (ξ, m) ∈ Ω k × Rk+ : ξi = ξj , ∀i = j , where ϕk has a stable critical point situation. Then any C 1 -perturbation of ϕk has a critical point in D. Thanks to the results contained in Lemma 5.2 and Lemma 6.1, we thus conclude that the function Iλ (ξ, m), which is C 1 -close to ϕk (ξ, m) when λ is small enough, has a critical point (ξ¯λ , m ¯ λ ) in D, for all such λ. From Lemma 5.1 we have then that cij (ξ¯λ , m ¯ λ) = 0
for all i, j,
and therefore uλ (x) =
√ λ U˜ (ξ¯λ , m ¯ λ ) (x) + φ(ξ¯λ , m ¯ λ )(x)
is a solution to our problem (1.1). The qualitative properties of this solution predicted by Theorem 2 are a direct consequence of our construction. This concludes the proof. 2 Proof of Theorem 1. We shall apply the result of Theorem 2 for the case k = 2. Thus we want to prove that the function ϕ2 has a stable critical point situation in some open set D, compactly contained in Ω 2 × R2+ . We make the change of variables sj = m2j . With slight abuse of notation we write ϕ2 (ξ, s) =
√ bsj + 2sj log sj + H (ξj , ξj )sj − 2G(ξ1 , ξ2 ) s1 s2 . j =1,2
To establish Theorem 1 we need to show the existence of a stable critical point situation for ϕ2 (ξ, s). To do so we shall show the existence of a critical point for ϕ2 obtained through a min– max characterization, which is in fact preserved for small C 1 perturbations of the functional. The rest of the section is devoted to carry out this construction. Let us fix a small number δ > 0 to be chosen later. We define D to be D = R2+ × Ωδ2 ,
where Ωδ2 = y ∈ Ω 2 / dist y, ∂Ω 2 > δ .
Denote by Ω1 a bounded nonempty component of R2 \ Ω¯ and assume that 0 ∈ Ω1 . Consider a closed, smooth Jordan curve γ contained in Ω which encloses Ω1 . We let S be the image of γ and B = [δ, δ −1 ]2 × S × S. Thus B is a closed and connected subset of D. Let Γ be the class of all maps Φ ∈ C(B, D) with the property that there exists a function Ψ ∈ C([0, 1] × B, D) such that Ψ (0, ·) = IdB ,
Ψ (1, ·) = Φ.
(7.1)
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Then we define C = inf sup ϕ2 Φ(z) .
(7.2)
C −K
(7.3)
Φ∈Γ z∈B
We will show that
for some fixed constant K independent of δ, and also that if δ > 0 is chosen sufficiently small, then for all (ξ, s) ∈ ∂D such that ϕ2 (ξ, s) = C there exists a vector τ tangent to ∂D at (ξ, s) such that ∇ϕ2 (ξ, s) · τ = 0.
(7.4)
Under the conditions (7.3) and (7.4), a critical point (ξ¯ , s¯ ) for ϕ2 with ϕ2 (ξ¯ , s¯ ) = C exists, as a standard deformation argument involving the negative gradient flow of ϕ2 shows. This structure ¯ is clearly preserved for small C 1 (D)-perturbations of ϕ2 and hence a stable critical point situation for this functional is established. We begin with proving inequality (7.3). Lemma 7.1. There exists K > 0, independent of the small number δ used to define D such that C −K. Proof. We need to prove the existence of K > 0 independent of small δ such that if Φ ∈ Γ , then there exists a point z¯ ∈ B for which ϕ2 Φ(¯z) −K.
(7.5)
We write z = (z1 , z2 , z3 , z4 ),
Φ(z) = Φ1 (z), Φ2 (z), Φ3 (z), Φ4 (z) ,
with (z1 , z2 ), Φ1 (z), Φ2 (z) ∈ R2+ ,
(z3 , z4 ), Φ3 (z), Φ4 (z) ∈ Ω 2 .
We claim that for any (z1 , z2 ) ∈ R2+ there exists a zˆ ∈ S × S such that Φ3 (z1 , z2 , zˆ ) and Φ4 (z1 , z2 , zˆ ) have antipodal directions, more precisely Φ3 (z1 , z2 , zˆ ) Φ4 (z1 , z2 , zˆ ) = Rπ , |Φ3 (z1 , z2 , zˆ )| |Φ4 (z1 , z2 , zˆ )|
(7.6)
where Rπ denotes a rotation in the plane of an angle π . This fact clearly implies that the existence of a number M > 0 depending only on Ω such that G(Φ3 (z1 , z2 , zˆ ), Φ4 (z1 , z2 , zˆ )) M. Thus
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ϕ2 Φ(z1 , z2 , zˆ ) 2 Φj (z1 , z2 , zˆ ) log Φj (z1 , z2 , zˆ ) j =1,2
− 2M Φ1 (z1 , z2 , zˆ )Φ2 (z1 , z2 , zˆ ) √ min 2(r log r + s log s) − 2M rs −K, r>0,s>0
for some explicit number K, which depends on M, but it is independent of δ. This gives the validity of estimate (7.5). We will prove (7.6) by means of a degree argument. Fix (z1 , z2 ). Let us consider an orientation-preserving homeomorphism h : S 1 → S and the map f : S 1 × S 1 → S 1 × S 1 defined as f (ζ ) = (f1 (ζ ), f2 (ζ )) with f1 (ζ1 , ζ2 ) =
Φ3 (z1 , z2 , h(ζ1 ), h(ζ2 )) , |Φ3 (z1 , z2 , h(ζ1 ), h(ζ2 ))|
Φ4 (z1 , z2 , h(ζ1 ), h(ζ2 )) . |Φ4 (z1 , z2 , h(ζ1 ), h(ζ2 ))|
f2 (ζ1 , ζ2 ) =
If we show that f is onto, we get in particular the validity of (7.6). By definition, there exists a Ψ ∈ Γ such that Ψ (1, ·) = Φ. If we denote by Ψi (t, ·) the components of the map Ψ and Ψ˜ i (t, ξ1 , ξ2 ) = Ψi (t, z1 , z2 , ξ2 , ξ2 ), it follows that, for i = 3, 4, Ψ˜ i ∈ C([0, 1] × S 1 × S 1 , Ωδ2 ), Ψ˜ i (0, ·) = IdS 1 ×S 1 and Ψ˜ i (1, ·) = Φi . Define a homotopy F : [0, 1] × S 1 × S 1 → S 1 × S 1 by F1 (t, ζ ) =
Ψ˜ 3 (t, h(ζ1 ), h(ζ2 )) |Ψ˜ 3 (t, h(ζ1 ), h(ζ2 ))|
and F2 (t, ζ ) =
Ψ˜ 4 (t, h(ζ1 ), h(ζ2 )) . |Ψ˜ 4 (t, h(ζ1 ), h(ζ2 ))|
Let us notice that F (1, ζ ) = f (ζ ) and F (0, ζ ) =
h(ζ1 ) h(ζ2 ) , . |h(ζ1 )| |h(ζ2 )|
This function defines a homeomorphism of S 1 × S 1 , which we regard as embedded in R3 , parametrized as follows: ζ : (θ1 , θ2 ) ∈ [0, 2π)2 → (ρ1 cos θ1 , ρ1 sin θ1 , 0) + (0, ρ2 cos θ2 , ρ2 sin θ2 ), for 0 < ρ2 < ρ1 . The map f defined above can be read in the introduced variables as f (ζ ) = ρ1 f1 (ζ ), 0 + 0, ρ2 f2 (ζ ) . The function f can be extended to a continuous map f˜ : T → T , where T is the solid torus described by (θ1 , θ2 , ρ) ∈ [0, 2π)2 × [0, ρ2 ] → (ρ1 cos θ1 , ρ1 sin θ1 , 0) + (0, ρ cos θ2 , ρ sin θ2 ) and f˜(ζ, ρ) = ρ1 f1 (ζ ), 0 + 0, ρf2 (ζ ) .
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The function f˜ is homotopic to a homeomorphism of T , along a deformation which maps ∂T = S 1 × S 1 into itself. Thus deg(f˜, T , P ) = 0 for all P in the interior of T . We see next how this fact implies that f is onto. Take (θ1∗ , θ2∗ ) ∈ [0, 2π)2 and ρ ∗ ∈ (0, ρ2 ) then there exist ζ ∗∗ ∈ S 1 × S 1 and ρ ∗∗ ∈ (0, ρ2 ) such that ρ1 f1 ζ ∗∗ , 0 + 0, ρ ∗∗ f2 ζ ∗∗ = ρ1 cos θ1∗ , ρ1 sin θ1∗ , 0 + 0, ρ ∗ cos θ2∗ , ρ ∗ sin θ2∗ . Thus we get f1 (ζ ∗∗ ) = (cos θ1∗ , sin θ1∗ ), f2 (ζ ∗∗ ) = (cos θ2∗ , sin θ2∗ ) and ρ ∗ = ρ ∗∗ . It then follows that f is onto. This concludes the proof. 2 We now prove (7.4). Lemma 7.2. There exists a sufficiently small δ > 0 with the following properties: If (ξ¯ , s¯ ) ∈ ∂Dδ is such that ϕ2 (ξ¯ , s¯ ) = C, then there exists a vector τ , tangent to ∂Dδ at the point (ξ¯ , s¯ ), so that ∇ϕ2 (ξ¯ , s¯ ) · τ = 0. Proof. Let us assume there exist a sequence δ = δn → 0 and points (ξn , sn ) ∈ Dδ such that (omitting the subscript n), (ξ, s) → (ξ¯ , s¯ ) ∈ Ω¯ 2 × R2+ and ϕ2 (ξ, s) → C < 0. We shall show that there exists a tangent vector τ , tangent to Ω¯ 2 × R2+ , such that ∇ϕ2 (ξ¯ , s¯ ) · τ = 0. Assume first that ξ¯ ∈ Ω 2 . If |s| → ∞, then ϕ2 (ξ, s) → ∞. Thus we may assume that |s| is bounded. Let us observe now the following fact: for any points ξ = (ξ1 , ξ2 ) fixed and far from each other and from the boundary, the function ϕ2 (ξ, s) =
√ bsj + 2sj log sj + H (ξ, ξj )sj − 2G(ξ1 , ξ2 ) s1 s2 j =1,2
is strictly convex as a function of s, and it is bounded below. Hence it has a unique minimum point, which we denote by (¯s1 , s¯2 ). Then each component s¯i of s¯ is a function of ξ1 and ξ2 , namely s¯i = s¯i (ξ1 , ξ2 ) for i = 1, 2. Furthermore, a direct computation shows that ϕ2 (ξ, s¯ ) = −2(¯s1 + s¯2 )
(7.7)
and H (ξi ,ξi ) b+2 ϕ2 (ξ, s¯ ) min min ϕ2 (ξ, s) −2e− 2 min e− 2 . i=1,2 si =0
i=1,2
(7.8)
Assuming again that ξ¯ ∈ Ω 2 , if s = (s1 , s2 ) → (0, 0), then we would get that C = 0, which is impossible. On the other hand, if say s1 is far away from 0 and s2 → 0, then ∂s2 ϕ2 (ξ, s1 , s2 ) → −∞, and then we can take τ = ∂s2 ϕ2 . Let us consider now the case in which dist(ξ2 , ∂Ω) = δ. As δ → 0, this fact implies that H (ξ2 , ξ2 ) → ∞, then we must also have that |ξ1 − ξ2 | → 0 to keep the value of ϕ2 bounded. By construction we have dist(ξ1 , ∂Ω) δ. Two cases arise: if ∇s ϕ2 (ξ, s) = 0, then we can chose τ parallel to ∇s ϕ2 (ξ, s). Otherwise, we are in the case in which ∇s ϕ2 (ξ, s) = 0. It remains to analyze this case.
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Formula (7.7) leads us to change variables as r = s1 + s2 ,
rt = s1
with 0 < r < ∞, 0 < t < 1.
In these new variables the function ϕ2 gets rewritten as ϕ2 (ξ, r, t) = r b + 2t log t + 2(1 − t) log(1 − t) + 2r log r + r H (ξ1 , ξ1 )t + H (ξ2 , ξ2 )(1 − t) − 2G(ξ1 , ξ2 ) t (1 − t) . The relation ∂ϕ2∂r(r,t) = 0 gives r = e− for 0 < t < 1 is given by
C+h(ξ,t) 2
, where C is an explicit positive number and h(ξ, t),
h(ξ, t) = H (ξ1 , ξ1 )t + H (ξ2 , ξ2 )(1 − t) − 2G(ξ1 , ξ2 ) t (1 − t) + 2t log t + 2(1 − t) log(1 − t). (7.9) To get the minimum value of ϕ2 in the variable s is thus equivalent to get the minimum of the function h as a function of t in the interval (0, 1). Differentiating h(ξ, t) with respect to t we get t − 12 t 1 H (ξ2 , ξ2 ) − H (ξ1 , ξ1 ) =√ log . − 2G(ξ1 , ξ2 ) 1−t t (1 − t) G(ξ1 , ξ2 )
(7.10)
This relation defines uniquely the value of t. Thus the relation ∇s ϕ2 (ξ, s) = 0 implies that ϕ2 (ξ, s) = −2r = −2e−
C+h(ξ,t) 2
(7.11)
with t uniquely defined by (7.10) and h given by (7.9). Next we want to analyze the dependence of this t on the points ξ1 and ξ2 . Our first claim is that t is away both from 0 and 1. This fact is a direct consequence of the following statement: there exists a positive number C such that H (ξ2 , ξ2 ) − H (ξ1 , ξ1 ) C. (7.12) 2G(ξ1 , ξ2 ) We show the validity of (7.12). We assume by contradiction that H (ξ2 , ξ2 ) − H (ξ1 , ξ1 ) → +∞. 2G(ξ1 , ξ2 )
(7.13)
We have δ = dist(ξ2 , ∂Ω). Let us denote d1 = dist(ξ1 , ∂Ω), and d = |ξ1 − ξ2 |. Condition (7.13) implies that d1 and d → 0, with δ d1 and δ = o(d). Let us consider the expanded domain Ω˜ = δ −1 Ω and observe that for this domain its associated Green’s function and regular part are given by H˜ (x1 , x2 ) = 4 log δ + H (δx1 , δx2 ),
˜ 1 , x2 ) = G(δx1 , δx2 ). G(x
(7.14)
˜ = 1. After a rotation and translation, we Furthermore, dist(ξ2 , ∂Ω) = δ implies dist( ξδ2 , ∂ Ω) ξ2 assume that δ = (0, 1) and as δ → 0 the domain Ω˜ becomes the half-plane x2 > 0. We denote
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respectively by G0 and H0 Green’s function and its regular part, associated to the half plane x2 > 0. The expressions for G0 and H0 are explicit: H0 (x, y) = 4 log
1 , |x − y| ¯
G0 (x, y) = 4 log
1 1 − 4 log . |x − y| |x − y| ¯
y¯ = (y1 , −y2 )
where y = (y1 , y2 ), and
We thus compute the expression in (7.13) ξ ξ ξ ξ H (ξ2 , ξ2 ) − H (ξ1 , ξ1 ) H˜ ( δ2 , δ2 ) − H˜ ( δ1 , δ1 ) = ˜ ξ1 , ξ2 ) 2G(ξ1 , ξ2 ) 2G( δ
=
δ
H0 ((0, 1), (0, 1)) − 4 log |ξ
δ
1 −ξ¯1 |
+ o(1)
δ 4 log |ξ1 −δ(0,1)|
= O(1),
but this is in contradiction with (7.13). The next step is to study the dependence of t on the points ξi . Let us call μ=
H (ξ2 , ξ2 ) − H (ξ1 , ξ1 ) , 2G(ξ1 , ξ2 )
λ=
1 . G(ξ1 , ξ2 )
Let us analyze how t depends on μ and λ. Let us set z = t − 12 . Then Eq. (7.10) defines uniquely z = z(μ, λ) μ=
z 1 4
− z2
+ λf (z).
(7.15)
We observe that z(μ, 0) =
H (ξ2 , ξ2 ) − H (ξ1 , ξ1 ) 1 . 2 G(ξ1 , ξ2 )2 + (H (ξ2 , ξ2 ) − H (ξ1 , ξ1 ))2
Differentiating expression (7.15) with respect to μ and to λ we get that |zμ | + |zλ | is bounded if μ and λ are bounded. Now we replace the values of t = t (μ, λ), defined by the relation t = z + 12 , in (7.9). We thus get a function h = h(H (ξ1 , ξ1 ), H (ξ2 , ξ2 ), G(ξ1 , ξ2 ), t (μ, λ)). Our next claim is that the derivatives of h with respect to H (ξ1 , ξ1 ), to H (ξ2 , ξ2 ), and to G(ξ1 , ξ2 ) are bounded above and below away from 0. We show this fact for ∂G(ξ∂1 ,ξ2 ) h. We have ∂ h H (ξ1 , ξ1 ), H (ξ2 , ξ2 ), G(ξ1 , ξ2 ), t (μ, λ) ∂G(ξ1 , ξ2 )
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∂h ∂t ∂μ ∂t ∂λ ∂ h+ + ∂G(ξ1 , ξ2 ) ∂t ∂μ ∂G(ξ1 , ξ2 ) ∂λ ∂G(ξ1 , ξ2 ) 1 1 = O(1) + O . = − t (1 − t) + O G(ξ1 , ξ2 ) G(ξ1 , ξ2 )
=
The above conclusion holds true since we have that |tμ | + |tλ | is bounded. Furthermore, 1 ∂λ =− = o(1), ∂G(ξ1 , ξ2 ) G(ξ1 , ξ2 )2
H (ξ1 , ξ1 ) − H (ξ2 , ξ2 ) ∂μ =− = o(1). ∂G(ξ1 , ξ2 ) G(ξ1 , ξ2 )2
Finally, we Taylor expand ∂h H (ξ1 , ξ1 ), H (ξ2 , ξ2 ), G(ξ1 , ξ2 ), t (μ, λ) ∂t ∂h H (ξ1 , ξ1 ), H (ξ2 , ξ2 ), G(ξ1 , ξ2 ), t (μ, 0) = ∂t ∂ 2h + 2 H (ξ1 , ξ1 ), H (ξ2 , ξ2 ), G(ξ1 , ξ2 ), t˜(μ, λ) t (μ, λ) − t (μ, 0) ∂ t = O(1), since ∂ 2h H (ξ1 , ξ1 ), H (ξ2 , ξ2 ), G(ξ1 , ξ2 ), t˜(μ, λ) = O(1)G(ξ1 , ξ2 ) 2 ∂ t and t (μ, λ) − t (μ, 0) ∼ λ ∼
1 . G(ξ1 , ξ2 )
In a similar fashion we get that the quantities ∂h , ∂H (ξ1 , ξ1 )
∂h ∂H (ξ2 , ξ2 )
are bounded above and below away from 0 in the considered region. We have now the tools to conclude the proof of our lemma. We recall that the case we are discussing is the following: dist(ξ2 , ∂Ω) = δ, ξ1 → ξ2 , with dist(ξ1 , ∂Ω) δ, ∇s ϕ2 (ξ, s) = 0 which implies the validity of (7.11), namely ϕ2 (ξ, s) = −2e−
C+h(ξ,t) 2
with t uniquely defined by (7.10) and h given by (7.9). We argue by contradiction, assuming that (7.4) does not hold. Then we have in particular that ∇ξ2 ϕ2 (ξ, s) · τ = 0
(7.16)
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for any vector τ tangent to ∂Ωδ at ξ2 , where Ωδ = {x ∈ Ω: dist(x, ∂Ω) > δ}. Note that ∇ξ2 ϕ2 (ξ, s) = e− We recall that
∂h ∂G(ξ1 ,ξ2 )
C+h(ξ,t) 2
and
∂h ∂h ∇ξ2 G(ξ1 , ξ2 ) + ∇ξ2 H (ξ2 , ξ2 ) . ∂G(ξ1 , ξ2 ) ∂H (ξ2 , ξ2 )
∂h ∂H (ξ2 ,ξ2 )
are bounded above and below away from 0. We denote
ρ = |ξ1 − ξ2 | → 0. Only two cases may occur, namely ρδ → ∞ or We shall show that in both cases relation (7.16) is impossible. Let us assume first that ρδ → ∞ and define xj =
ξj − ξ1 ρ
δ ρ
c0 , for some constant c0 .
for j = 1, 2,
and x˜j = limδ→0 xj . Let us define ϕ(x ˜ 1 , x2 ) = ϕ2 (ξ1 + ρx1 , ξ1 + ρx2 , s). After rotation we may assume that in (7.16) we have τ = (0, 1), and hence (writing ξ2 = (ξ21 , ξ22 )) lim ∂ 2 ϕ(x ˜ 1 , x2 ) = lim ρ ∂ξ 2 ϕ2 (ξ1 2 δ→0 x2 δ→0
+ ρx1 , ξ1 + ρx2 , s) = 0.
On the other hand, since away from the boundary the function H (x, x) is bounded, we get lim ∂ 2 ϕ(x ˜ 1 , x2 ) = −C∂x 2 2 δ→0 x2
log
1
= 0, |x˜1 − x˜2 |
a contradiction. Thus, we necessarily have that ρδ is bounded. The interesting case is when ξ1 ∈ ∂Ωδ . If not, we can reproduce the argument above to reach a contradiction. Let us assume first that δ = o(ρ). In this case we find that (7.13) holds true, which leads us to a contradiction. Let us assume then that ρδ → c. We consider the scaled domain Ω˜ = δ −1 Ω, whose associated ˜ and regular part H˜ are given by (7.14). Furthermore, in this scaled domain Green’s function G the number t defined by relation (7.10) remains away from 0 and 1, since the quantity H˜ (ξ2 , ξ2 ) − H˜ (ξ1 , ξ1 ) ˜ 1 , ξ2 ) 2G(ξ remains bounded. Furthermore, after a rotation and translation, we may assume that ξ˜2 := ξδ2 → (0, 1), ξ˜1 := ξδ1 → (a, 1), for some a > 0, as δ → 0 and the domain Ω˜ becomes the half-plane x2 > 0. Under this condition, we see that the derivative of ϕ2 in the direction e = (0, 1) is not 0, reaching again a contradiction with (7.16), and the proof is concluded. 2 Acknowledgments This research has been partly supported by Fondecyt Grants 1070389, 1080099 and Fondecyt Grant-International Cooperation 7070150, Chile.
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References [1] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the nLaplacian, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 17 (1990) 393–413. [2] Adimurthi, O. Druet, Blow-up analysis in dimension 2 and a sharp form of Trudinger–Moser inequality, Comm. Partial Differential Equations 29 (2004) 295–322. [3] Adimurthi, S. Prashanth, Failure of Palais–Smale condition and blow-up analysis for the critical exponent problem in R2 , Proc. Indian Acad. Sci. Math. Sci. 107 (1997) 283–317. [4] Adimurthi, P.N. Srikanth, S.L. Yadava, Phenomena of critical exponent in R2 , Proc. Roy. Soc. Edinburgh Sect. A 199 (1991) 19–25. [5] Adimurthi, M. Struwe, Global compactness properties of semilinear elliptic equations with critical exponential growth, J. Funct. Anal. 175 (2000) 125–167. [6] A. Bahri, J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988) 255–294. [7] A. Bahri, Y.-Y. Li, O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations 3 (1995) 67–93. [8] S. Baraket, F. Pacard, Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations 6 (1) (1998) 1–38. [9] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983) 437–477. [10] J.M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Math. Acad. Sci. Paris Ser. I 299 (1984) 209–212. [11] D.G. de Figueiredo, O. Miyagaki, B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995) 139–153. [12] D.G. de Figueiredo, B. Ruf, Existence and non-existence of radial solutions for elliptic equations with critical exponent in R2 , Comm. Pure Appl. Math. 48 (1995) 1–17. [13] M. del Pino, M. Kowalczyk, M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations 24 (2005) 47–81. [14] O. Druet, Multibump analysis in dimension 2: Quantification of blow-up levels, Duke Math. J. 132 (2) (2006) 217–269. [15] P. Esposito, M. Grossi, A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2) (2005) 227–257. [16] Z.-C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (2) (1991) 159–174. [17] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/1971) 1077–1092. [18] S.I. Pohozhaev, The Sobolev embedding in the case pl = n, in: Proc. Tech. Sci. Conf. on Adv. Sci. Research 1964– 1965, Mathematics Section, Moskov. Ènerget. Inst., Moscow, 1965, pp. 158–170. [19] O. Rey, The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1) (1990) 1–52. [20] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984) 511–517. [21] M. Struwe, Positive solutions of critical semilinear elliptic equations on non-contractible planar domains, J. Eur. Math. Soc. 2 (2000) 329–388. [22] N.S. Trudinger, On embedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473–483. [23] V.I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSRR 138 (1961) 805–808 (in Russian); English transl.: Soviet Math. Dokl. 2 (1961) 746–749.
Journal of Functional Analysis 258 (2010) 458–503 www.elsevier.com/locate/jfa
Multiple-end solutions to the Allen–Cahn equation in R2 Manuel del Pino a,∗ , Michał Kowalczyk a , Frank Pacard b , Juncheng Wei c a Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS),
Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile b Université Paris Est and Institut Universitaire de France, France c Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
Received 7 April 2009; accepted 20 April 2009 Available online 18 August 2009 Communicated by H. Brezis
Abstract We construct a new class of entire solutions for the Allen–Cahn equation u + (1 − u2 )u = 0, in 2 R (∼ C). Given k 1, we find a family of solutions whose zero level sets are, away from a compact set, asymptotic to 2k straight lines (which we call the ends). These solutions have the property that there exist θ0 < θ1 < · · · < θ2k = θ0 + 2π such that limr→+∞ u(reiθ ) = (−1)j uniformly in θ on compact subsets of (θj , θj +1 ), for j = 0, . . . , 2k − 1. © 2009 Published by Elsevier Inc. Keywords: Allen–Cahn equation; Toda system; Multiple-end solutions; Infinite-dimensional Lyapunov–Schmidt reduction; Moduli spaces
1. Introduction and statement of main results 1.1. Introduction In this paper, we are interested in the construction of a new class of solutions, in the entire space RN , for the semilinear elliptic equation * Corresponding author.
E-mail addresses:
[email protected] (M. del Pino),
[email protected] (M. Kowalczyk),
[email protected] (F. Pacard),
[email protected] (J. Wei). 0022-1236/$ – see front matter © 2009 Published by Elsevier Inc. doi:10.1016/j.jfa.2009.04.020
M. del Pino et al. / Journal of Functional Analysis 258 (2010) 458–503
u + 1 − u2 u = 0,
459
(1.1)
known as the Allen–Cahn equation. This problem has its origin in the gradient theory of phase transitions [2], a model in which two distinct phases (represented by the values u = ±1) try to coexist in a domain Ω while minimizing their interaction which is proportional to the (N − 1)dimensional volume of the interface. Idealizing the phase as a regular function which takes values close to ±1 in most of the domain, except in a narrow transition layer of width ε, one defines the Allen–Cahn energy, ε Jε (u) := 2
1 |∇u| dx + 4ε
2
Ω
2 1 − u2 dx,
Ω
whose critical points satisfy the Euler–Lagrange equation ε 2 u + 1 − u2 u = 0
in Ω.
(1.2)
u + 1 − u2 u = 0 in ε −1 Ω.
(1.3)
Replacing u by u(·/ε) we obtain the equation
Therefore, (1.1) appears as the limit problem in the blow up analysis of (1.2) as ε tends to 0. The relation between interfaces of least volume and critical points of Jε was first established by Modica in [23] (see also [18]). Let us briefly recall the main results in this direction: If uε is a family of local minimizers of Jε for which sup Jε (uε ) < +∞,
(1.4)
ε>0
then, up to a subsequence, uε converges in L1 to 1Λ − 1Λc , where ∂Λ has minimal volume. Here 1Λ (resp. 1Λc ) is the characteristic function of the set Λ (resp. Λc = Ω − Λ). Moreover, Jε (uε ) → √1 HN −1 (∂Λ). 2 For critical points of Jε which satisfy (1.4), a related assertion is proven in [17]. In this case, the convergence of the interface holds with certain integer multiplicity to take into account the possibility of multiple transition layers converging to the same minimal hypersurface. These results provide a link between solutions of (1.1) and the theory of minimal hypersurfaces which has been widely explored in the literature. For example, solutions concentrating along non-degenerate, minimal hypersurfaces of a compact manifold were found in [25] (see also [20]). As far as multiple transition layers are concerned, given a minimal hypersurface Γ (subject to some additional property on the sign of the potential of the Jacobi operator about Γ , which holds on manifolds with positive Ricci curvature) and given an integer k 1, solutions of (1.2) with multiple transitions near Γ were built in [11] (see [11] for the 2-dimensional case, and [9] for the euclidean case), in such a way that Jε (uε ) → √k HN −1 (Γ ). 2 Recall that, in dimension 1, finite energy solutions of (1.1) are given by translations of the function H which is the unique solution of the problem H + 1 − H 2 H = 0,
with H (±∞) = ±1 and H (0) = 0.
(1.5)
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In fact, the function H is explicitly given by y H (y) = tanh √ . 2 Then, in any dimension and for all a ∈ RN with |a| = 1 and for all b ∈ R, the function u(x) = H (a · x + b) solves (1.1). A celebrated conjecture due to De Giorgi states that, in dimension N 8, these solutions are the only ones which are bounded and monotone in one direction. Let us recall that the monotonicity property is related to the fact that solutions u are local minimizers [12,13]. In dimensions N = 2, 3, De Giorgi’s conjecture has been proven in [3,14] and (under some extra assumption) in the remaining dimensions in [26] (see also [12,13]). When N = 2, the monotonicity assumption can even be replaced by a weaker stability assumption [16]. Finally, counterexamples in dimension N 9 have recently been built in [10], using the existence of nontrivial minimal graphs in higher dimensions. In light of these results, it is natural to study the set of entire solutions of (1.1). The functions u(x) = H (a · x + b) are obvious solutions. In dimension N = 2, nontrivial examples (whose nodal set is the union of two perpendicular lines) were built in [5] using the following strategy: A positive solution to (1.1) in the quadrant {(x, y): x > |y|} with zero boundary conditions is built by constructing appropriate super and subsolutions. This solution is then extended by odd reflections through the lines x = y and x = −y to yield u2 , a solution of (1.1) in all R2 . The function u2 is a solution of (1.1), whose 0-level set is the union of the two axis. It can easily be generalized to obtain solutions with dihedral symmetry by considering, for k 3, the π } and extending it by corresponding solution within the sector {(r cos θ, r sin θ ): r > 0, |θ | < 2k 2k − 1 consecutive reflections to yield a solution uk (we refer to [15] for the details, see also [4] where higher dimensional versions of this construction is given). The zero level set of uk is constituted outside any ball by 2k infinite half lines with dihedral symmetry. To our knowledge, no other nontrivial examples of solutions are known in dimension N = 2 (up to the action of rigid motions). 1.2. Statement of the result We assume from now on that the dimension is equal to N = 2. Definition 1. We say that u, solution of (1.1), has 2k-ends if, away from a compact set, its nodal set is given by 2k connected curves which are asymptotic to 2k oriented half lines aj · x + bj = 0, j = 1, . . . , 2k (for some choice of aj ∈ R2 , |aj | = 1 and bj ∈ R) and if, along these curves, the solution is asymptotic to either H (aj · x + bj ) or −H (aj · x + bj ). Given any k 1, we prove in this paper the existence of a wealth of 2k-ended solutions of (1.1). In a forthcoming paper [6], we will complete this analysis and show that the solutions we construct in the present paper belong to some smooth 2k-parameter family of 2k-ended solutions of (1.1). To state our result in precise way, we assume that we are given a solution q := (q1 , . . . , qk ) of the Toda system c0 qj = e
√ 2(qj −1 −qj )
−e
√
2(qj −qj +1 )
,
(1.6)
M. del Pino et al. / Journal of Functional Analysis 258 (2010) 458–503
for j = 1, . . . , k, where c0 =
461
√
2 24
and we agree that
q0 ≡ −∞ and qk+1 ≡ +∞. The Toda system (1.6) is a classical example of integrable system which has been extensively studied. It models the dynamics of finitely many mass points on the line under the influence of an exponential potential. We recall in the next section some of the results which are concerned with the solvability of (1.6) and which will be needed for our purposes. We refer to [19,24] for the complete description of the theory. Of importance for us is the fact that solutions of (1.6) can be described (almost explicitly) in terms of 2k parameters. Moreover, if q is a solution of (1.6), then the long term behavior (i.e. long term scattering) of the qj at ±∞ is well understood and it is known that, for all j = 1, . . . , k, there exist aj+ , bj+ ∈ R and aj− , bj− ∈ R, all depending on the solution q, such that qj (t) = aj± |t| + bj± + OC ∞ (R) e−τ0 |t| ,
(1.7)
as t tends to ±∞, for some τ0 > 0. Moreover, aj±+1 > aj± for all j = 1, . . . , k − 1. Given ε > 0, we define the vector valued function qε , whose components are given by qj,ε (x) := qj (εx) −
√
k+1 log ε. 2 j− 2
(1.8)
It is easy to check that the qj,ε are again solutions of (1.6). Observe that, according to the description of the asymptotics of the functions qj , the graphs of the functions qj,ε are asymptotic to oriented half lines at infinity. In addition,√for ε > 0 small enough, these graphs are disjoint and in fact their mutual distance is given by − 2 log ε + O(1) as ε tends to 0. It will be convenient to agree that χ + (resp. χ − ) is a smooth cutoff function defined on R which is identically equal to 1 for x > 1 (resp. for x < −1) and identically equal to 0 for x < −1 (resp. for x > 1) and additionally χ − + χ + ≡ 1. With these cutoff functions at hand, we define the 4-dimensional space D := Span x → χ ± (x), x → xχ ± (x) ,
(1.9)
2,μ
and, for all μ ∈ (0, 1) and all τ ∈ R, we define the space Cτ (R) of C 2,μ functions h which satisfy h C 2,μ (R) := (cosh x)τ h C 2,μ (R) < ∞. τ
Keeping in mind the above notations, we have: Theorem 1.1. For all ε > 0 sufficiently small, there exists an entire solution uε of the Allen– Cahn equation (1.1) whose nodal set is the union of k disjoint curves Γ1,ε , . . . , Γk,ε which are the graphs of the functions x → qj,ε (x) + hj,ε (εx),
462
M. del Pino et al. / Journal of Functional Analysis 258 (2010) 458–503 2,μ
for some functions hj,ε ∈ Cτ (R) ⊕ D satisfying hj,ε C 2,μ (R)⊕D Cε α , τ
for some constants C, α, τ, μ > 0 independent of ε > 0. In other words, given a solution of the Toda system, we can find a one parameter family of 2k-ended solutions of (1.1) which depend on a small parameter ε > 0. As ε tends to 0, the nodal sets of the solutions we construct become close to the graphs of the functions qj,ε . Going through the proof, one can be more precise about the description of the solution uε . If Γ ⊂ R2 is a curve in R2 which is the graph over the x-axis of some function, we denote by dist(·, Γ ) the signed distance to Γ which is positive in the upper half of R2 \ Γ and is negative in the lower half of R2 \ Γ . Then, we have: Proposition 1.1. The solution of (1.1) provided by Theorem 1.1 satisfies εα|x| e ˆ uε − u∗ ε
L∞ (R2 )
Cε α¯ ,
for some constants C, α, ¯ αˆ > 0 independent of ε, where u∗ε :=
k
1 (−1)j +1 H dist(·, Γj,ε ) − (−1)k + 1 . 2
(1.10)
j =1
It is interesting to observe that, when k 3, there are solutions of (1.6) whose graphs have no symmetry and our result yields the existence of entire solutions of (1.1) without any symmetry provided the number of ends is larger than or equal to 6. 1.3. Comments and open problems Our result raises some interesting questions: (i) The classification of entire solutions of (1.1) remains an important and rather unexplored problem. In particular, the classification of entire solutions with finite Morse index is certainly an interesting problem (the Morse index of an entire solution u being defined as the supremum of the dimension of the space of smooth functions with compact support over which the quadratic form φ →
|∇φ|2 − 1 − 3u2 φ 2 dx
RN
is negative definite). In dimension N = 2, we believe that these solutions are precisely the solutions with finitely many ends. In addition, there is strong evidence that the solutions with 2k ends we construct have Morse index equal to the Morse index of the Toda system.
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(ii) Still in dimension N = 2, the understanding of the moduli space of all 2k-ended solutions is far from being complete: the result in Theorem 1.1 (see also [5]) implies that this space is nonempty and contains smooth families of solutions. Moreover, the result of [6] shows that this moduli space has formal dimension equal to 2k (the formal dimension is the dimension of the moduli space close to any non-degenerate solution). The main result of the present paper asserts that, there is a one to one correspondence between an open set of solutions of (1.6) and solutions of (1.1). In particular, this result provides a 2k-dimensional family of solutions (even if it is not clear from our construction that this family is smooth) and this dimension count is in agreement with the result of [6]. Let us also mention that some balancing conditions on the directions of the ends is available (see [15]), it states that the sum of the unit vectors of the ends (oriented toward the ends) has to be 0. (iii) It is tempting to conjecture that the solution uk (whose nodal set has dihedral symmetry and whose construction is described in [15] and outlined before the statement of Definition 1) and the solutions given in Theorem 1.1 belong to the same connected component of the moduli space of 2k-ended solutions. (iv) When k = 2, it turns out that solutions of (1.6) are symmetric with respect to the reflections through two perpendicular lines. Equivalently, one can prove that, when k = 2, the solutions of (1.1) which are provided by Theorem 1.1 also share this symmetry. In fact, we believe that any solution of (1.1) with 4 ends is symmetric with respect to reflections through two perpendicular lines. These questions hint towards the classification of finite Morse index entire solutions of (1.1), a program on generalizing De Giorgi’s conjecture. 1.4. Description of the proof Let us briefly describe the proof of Theorem 1.1. The method is based on an infinitedimensional version of the standard Lyapunov Schmidt reduction argument, as introduced in [25] or in [9] (see also [8,11,20–22]). Given a solution q of (1.6), we first build some infinite-dimensional family of approximate solutions uε,h , which depend on a small parameter ε > 0 and a some (small) vector valued func2,μ tion h = (h1 , . . . , hk ) whose components belong to Ca (R) ⊕ D, for some a > 0, where D has been defined in (1.9). In essence, these approximate solutions are defined as in (1.10), the curves Γj,ε,h being the graphs of the functions qj,ε + hj (ε·). For all ε small enough, we explain how these approximate solutions can be perturbed into genuine solutions of (1.1). To do so, we look for a solution of (1.1) of the form u := uε,h + φ, where the function φ is small in a sense to be made precise. Substituting this expression of u in (1.1), we reduce the problem to the solvability of the following nonlinear equation + 1 − 3u2ε,h φ + S(uε,h ) − N (uε,h , φ) = 0, where we have defined S(u) := u + 1 − u2 u,
(1.11)
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and N (u, φ) := φ 3 + 3uφ 2 . One of the important task will be to analyze, as ε tends to 0, the mapping properties of the linear operator + 1 − 3u2ε,h which appears on the left-hand side of (1.11). It turns out that this analysis is quite delicate and involves some carefully designed weighted spaces. It also requires some Lyapunov–Schmidt type reduction argument. To set up the analysis of the linearized operator + 1 − 3u2ε,h , we let ρj be a cutoff function such that ρj ≡ 1 in a tubular neighborhood of Γj,ε,h and identically equal to 0 outside some larger tubular neighborhood of Γj,ε,h . We will show that for all f in a suitable weighted function space, there exists a function φ : R2 → R and, for j = 1, . . . , k, a function κj : R2 → R which is defined in a tubular neighborhood of Γj,ε,h and only depend on the projection onto Γj,ε,h , solutions of k
2 + 1 − 3uε,h φ + κj ρj H dist(·, Γj,ε,h ) = f,
(1.12)
j =1
and whose norms are uniformly controlled as ε tends to 0. Observe that we have introduced new unknown functions κj . These will be needed to overcome the fact that the solution of ( + 1 − 3u2ε,h )φ = f blows up as ε tends to 0 unless some orthogonality conditions are imposed on the function f . In view of this result, instead of solving (1.11), we will look for φ and functions κj , for j = 1, . . . , k, solutions of the following nonlinear problem k
κj ρj H dist(·, Γj,ε,h ) + S(uˆ ε,h ) − N (uε,h , φ) = 0. + 1 − 3u2ε,h φ +
(1.13)
j =1
Now, a solution of (1.13) is a solution of (1.11) provided all functions κj are identically equal to 0. At this stage, it is worth remembering that our approximate solution uε,h depends on the vector valued functions h and we will see that it is possible to choose h appropriately so that the solution of (1.13) satisfies κj = 0, for j = 1, . . . , k. This will complete the proof of the result. 2. The Toda system and its linearization In this section, we gather some information about the theory which is necessary for solving (1.6) since this system is at the heart of our construction. 2.1. The Toda system We are interested in the understanding of the solutions of the Hamiltonian system c0 qj = e where c0 =
√
2 24
√ 2(qj −1 −qj )
−e
√
2(qj −qj +1 )
and we agree that q0 ≡ −∞ and qk+1 ≡ +∞.
,
(2.14)
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We introduce the functions rj :=
√ c0 2(qj +1 − qj ) + log √ , 2
(2.15)
for j = 1, . . . , k − 1. It is easy to check that, if q is a solution of (2.14), then r := (r1 , . . . , rk−1 ) is a solution of the following nonlinear system r − Me−r = 0,
(2.16)
where the (k − 1) × (k − 1) matrix M is given by ⎛ 2 −1 ⎜ −1 . . . ⎜ ⎜ .. M := ⎜ 0 . ⎜ . .. ⎝ . . . 0 ...
0 .. . .. . .. . 0
... 0 ⎞ .. ⎟ .. . . ⎟ ⎟ .. , . 0 ⎟ ⎟ .. ⎠ . −1 −1 2
and where e−r is the vector whose entries are given by e−r := e−r1 , . . . , e−rk−1 . Conversely, given a solution r of (2.16) and p, ¯ q¯ ∈ R, the functions 1 qj = k
j −1
iri −
i=0
c0 1 k−1 − j log √ , irk−i + pt ¯ + q¯ + √ 2 2 2 i=0
k−j
(2.17)
for j = 1, . . . , k (we agree that r0 = rk ≡ 0), are solutions of (2.14). The system (2.16) is an integrable system which has been extensively studied for example by J. Moser [24] and B. Kostant [19]. Some explicit formula of all solutions of (2.16) is available as well as a precise description of the asymptotic behavior of the solutions as t tends to ±∞. We briefly recall the main features of this theory. The expression of the solutions of (2.16) can be found in Section 7.7 of [19]. To describe it, we need to be given w := (w1 , . . . , wk ) ∈ Rk such that k
wj = 0,
j = 1, . . . , k − 1,
and wj +1 > wj ,
(2.18)
j =1
and g := (g1 , . . . , gk ) ∈ Rk such that k
gj = 1,
and gj > 0,
for j = 1, . . . , k.
j =1
Finally, for j = 2, . . . , k − 1, we define the function
(2.19)
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Φj (g, w; t) :=
Ri1 ...ij (w)gi1 . . . gij e
−t (wi1 +···+wij )
1ii <···
(see formula 7.7.10 in [19]) where Ri1 ...ij are rational functions of the entries of the vector w whose precise form can be found in Section 7.5 of [19]. We also agree that Φ0 = Φk ≡ 1. It is proven in [19] that all solutions of (2.14) are of the form rj (t) = − log Φj −1 (g, w; t) + 2 log Φj (g, w; t) − log Φj +1 (g, w; t)
(2.20)
for some choice of g and w. Observe that we have a 2k family of solutions of (2.16) since g and w provide 2(k − 1) independent parameters to which we have to add the parameters p¯ and q. ¯ The next result is also borrowed from [19,24]. It describes the asymptotics of the solutions of (2.16) (see Theorem 7.7.2 of [19]): Lemma 2.1. Let τ0 > 0 be defined by τ0 :=
min
j =1,...,k−1
(wj +1 − wj ).
(2.21)
Then, for j = 1, . . . , k − 1, the following expansion holds rj (t) = cj t − dj + ej+ (c) + OC ∞ (cosh t)−τ0 , as t tends to +∞ and rj (t) = −ck−j t + dk−j + ej− (c) + OC ∞ (cosh t)−τ0 , as t tends to −∞, where, for j = 1, . . . , k − 1, cj := wj +1 − wj ,
dj := log gj +1 − log gj ,
(2.22)
and where ej± are smooth functions of c := (c1 , . . . , ck−1 ). Proof. Thanks to (2.20), we can write as t tends to +∞ Φj (g, w; t) = R1...j (w)g1 . . . gj e−(w1 +···+wj )t 1 + OC ∞ (cosh t)−τ0 , while we can write, as t tends to −∞ Φj (g, w; t) = Rk−j ...k−1 (w)gk . . . gk−j +1 e−(wk +···+wk−j +1 )t 1 + OC ∞ (cosh t)−τ0 . The expansions follow at once from elementary computations together with the definition of rj . We leave the details to the reader. 2
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2.2. The linearized Toda system We assume that q = (q1 , . . . , qk ) is a solution of (2.14) described in the previous section. The linearized system associated to linearization of (2.14) about the solution q, reads as c0 v + Nv = z,
(2.23)
where the k × k matrix N has coefficients which are exponentially decaying at ±∞ (this follows from Lemma 2.1 which implies that the functions rj tend to +∞ as t tends to ±∞). We ana,μ lyze the solvability of the above linear problem in the space Cτ (R; Rk ) of C ,μ vector valued functions v which satisfy v C ,μ (R;Rk ) := (cosh x)τ v C ,μ (R) < ∞.
(2.24)
τ
We take advantage of the fact that the solution q, as described in (2.17), depends smoothly ¯ Differenon the parameters c1 , . . . , ck−1 and d1 , . . . , dk−1 as well as the parameters q¯ and p. tiating with respect to any of these parameters yields 2k linearly independent solutions of the homogeneous problem c0 v + Nv = 0. We will write
vj := ∂cj q
and vj := ∂dj q,
for j = 1, . . . , k − 1, and
vk := ∂p¯ q and vk := ∂q¯ q.
It follows from the result of Lemma 2.1 that the vector valued functions vj are linearly grow
ing at ±∞ while the vector valued functions vj are bounded. More precisely, it follows from Lemma 2.1 that
Lemma 2.2. As t tends to ±∞, the vector valued functions vj and vj can be decomposed as vj = aj,± t + bj,± + OC ∞ (cosh t)−τ0 , and vj = bj,± + OC ∞ (cosh t)−τ0 ,
where aj,± and bj,± , bj,± are fixed vectors in Rk . Moreover, {aj,ι : j = 1, . . . , k} and
{bj,ι : j = 1, . . . , k} are basis of Rk , for ι = ±. We now define the deficiency space D := Span χ ± vj , χ ± vj : j = 1, . . . , k ,
(2.25)
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where we recall that χ + (resp. χ − ) is a cutoff function identically equal to 1 for t > 1 (resp. for t < −1) and identically equal to 0 for t < −1 (resp. for t > 1) and χ + + χ − ≡ 1. Observe that D is 4k-dimensional and contains K := Span vj , vj : j = 1, . . . , k , which is the 2k-dimensional space of homogeneous solutions of c∗ v + Nv = 0. Therefore, we can certainly decompose D = K ⊕ E,
(2.26)
where E is a complement of K in D. With this decomposition at hand, we have the following result which follows from standard arguments in ordinary differential equations. Lemma 2.3. Assume that τ > 0. Then the mapping 2,μ
T : Cτ
0,μ R; Rk ⊕ E → Cτ R; Rk , v → c0 v + Nv
is an isomorphism. Proof. Standard arguments in ordinary differential equations imply that there exists a unique solution of (2.23) which satisfies v(0) = v (0) = 0. We will denote v = S0 (z). 2,μ We now prove that v ∈ Cτ (R; Rk )⊕D. To do so, we observe that one can also find a (unique) solution v¯ of (2.23) which satisfies v¯ (t) Ceτ t z
Cτ0,μ (R;Rk )
,
in (−∞, 0]. Indeed, using the variation of parameters formula it is easy to show the existence of a unique solution decaying to 0 at −∞ at some exponential rate. Integrating the equation 2,μ twice over (−∞, t] shows that in fact v¯ ∈ Cτ ((−∞, 0]; Rk ). Then v − v¯ is a linear combination of the functions vj and vj . This proves that, in (−∞, 0], the vector valued function v can be decomposed into the sum of a linear combination of elements in D and a vector valued function which is bounded by a constant times eτ t . A similar decomposition can be derived on [0, +∞). Once this decomposition is proven, the estimates for the Hölder norm of v follow at once. 0,μ 2,μ In other words, S0 : Cτ (R; Rk ) → Cτ (R; Rk ) ⊕ D is a right inverse for T . The decomposition D = K ⊕ E induces the decomposition S0 (z) = S¯0 (z) + e(z) + k(z), where S¯0 (z) ∈ 2,μ Cτ (R; Rk ), e(z) ∈ E and k(z) ∈ K. The operator S := S0 − k is also a right inverse of T and 2,μ maps onto Cτ (R; Rk ) ⊕ E as desired. This completes the proof of the lemma. 2 3. Linearized operator for a single interface In this section we develop the relevant analysis which will allow us to find a right inverse for the operator which will appear in the linearization of (1.1) about an approximate solution.
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3.1. Injectivity result We start by considering the linearized operator about H , namely L0 := ∂y2 + 1 − 3H 2 . First, we recall that L0 has a one-dimensional kernel spanned by H since L0 H = 0 as can be checked by taking the derivative of H + (1 − H 2 )H = 0. Since H > 0 this implies that 0 is the bottom of the spectrum of −L0 . In fact more is known and we recall the following result from [1]: Lemma 3.1. (See [1].) The spectrum of the operator −L0 is the union of the point spectrum, √ given by 0 (associated to the eigenfunction H ) and 32 (associated to the eigenfunction H H ) and the continuous spectrum given by [2, +∞). In particular, for all ξ = 0, given f ∈ L2 (R), the problem L0 − ξ 2 φ = f,
(3.27)
is uniquely solvable in H 1 (R). Let us consider operator L := ∂x2 + L0 , acting on functions defined in the plane. Obviously, we still have LH = 0. Our first result shows any bounded solution of Lφ = 0 is colinear to H . The proof of this fact follows the method first introduced in [25]. Lemma 3.2. Let φ be a bounded solution of Lφ = 0,
(3.28)
in R2 . Then φ is colinear to H . ˆ y) the Fourier Proof. Let assume that φ is a bounded solution of Lφ = 0. We denote by φ(ξ, transform of φ(x, y) in the x variable. This distribution is defined by ˆ f = φ(·, y), fˆ = φ,
φ(x, y)fˆ(x) dx,
R
where f is any smooth rapidly decreasing function and where fˆ is its Fourier transform. Let us now consider a smooth rapidly decreasing function of the two variables ψ(ξ, y). It follows from Lφ = 0 that ˆ y), L0 ψ − ξ 2 ψ dy = 0. φ(·, (3.29) R
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Let ϕ(y) and μ(ξ ) be smooth and compactly supported functions such that 0 does not belong to the support of f . Then we can solve the family of equations (parameterized by ξ ∈ R) L0 − ξ 2 ψ(ξ, y) = f (ξ )ϕ(y), and obtain a smooth, rapidly decreasing function ψ(ξ, y) such that ψ(ξ, y) = 0 whenever ξ is not the support of the function f . The fact that y → ψ(ξ, y) decays exponentially is standard and left to the reader. Using ψ in (3.29), we conclude that
ˆ y), f ϕ(y) dy = 0. φ(·,
R
ˆ y), f = 0 for all f whose support does not meet 0. Since ϕ is arbitrary, we have proven that φ(·, ˆ This implies that the support of φ(·, y) is included in {0}. ˆ y) is a linear combination (with coefficients depending on y) of derivaIt follows that φ(·, tives up to a finite order of Dirac masses at 0. Taking the inverse Fourier transform, we get that φ(x, y) = Py (x), where for each y ∈ R, Py is a polynomial in x. Since φ is assumed to be bounded, we conclude that Py (x) is a constant polynomial and hence φ(x, y) = φ(y) is a bounded function which satisfies L0 φ = 0. Therefore, φ is colinear to H . 2 3.2. A priori estimates Making use of the previous lemma, we now obtain a priori estimates for solutions of the problem Lφ = f,
(3.30)
in R2 . The results of Lemma 3.2 shows that such an a priori estimate will not be possible without imposing any extra conditions on the solution φ. The classification of the bounded solutions of Lφ = 0 suggests to impose the following orthogonality condition on the function φ
φ(x, ·)H dy = 0,
(3.31)
R
for all x ∈ R. With these restrictions imposed we have the following a priori estimates for this problem. Lemma 3.3. There exists a constant C > 0 such that φ L∞ (R2 ) C Lφ L∞ (R2 ) , provided φ ∈ L∞ (R2 ) satisfies (3.31). Proof. The proof of the lemma is by contradiction (it is actually similar to the proof of Lemma 2.2 in [7]). If the result were not true, there would exist sequences of bounded functions φn and fn satisfying
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Lφn = fn , φn H dy = 0,
in R2 , for all x ∈ R,
471
(3.32) (3.33)
R
with limn→∞ fn L∞ = 0 while φn L∞ = 1. For each n ∈ N we pick a point (xn , yn ) ∈ R2 such that φn (xn , yn ) 1/2.
(3.34)
We now consider the sequence of functions φ˜ n (x, y) = φn (x + xn , y + yn ). Using elliptic estimates together with Ascoli’s theorem, we can assume (up to a subsequence) that the sequence φ˜ n converges, uniformly on compact sets, to a function φ˜ which is defined in R2 and which is either a solution of ( − 2)φ˜ = 0, if the sequence (yn )n tends to ±∞ or a solution of ˜ · − y∞ ) = 0, + 1 − 3H 2 φ(x, if (yn )n converges to y∞ . Moreover, φ˜ is bounded and φ˜ is not identically equal to 0 since (3.34) ˜ guaranties that φ(0) 1/2. Finally, in the latter case, we can pass to the limit in (3.33) to get
˜ · − y∞ )H dy = 0, φ(x,
R
for all x ∈ R. The maximum principle implies that the former case does not occur and the result of Lemma 3.2 implies that the latter case does not occur either. Having found a contradiction in all cases, this completes the proof of the result. 2 Using the maximum principle, we also get a priori estimates in weighted space. √ Lemma 3.4. Assume that σ ∈ [0, 2) is fixed. There exists C > 0 such that (cosh y)σ φ
C 2,μ (R2 )
C φ L∞ (R2 ) + (cosh y)σ Lφ C 0,μ (R2 ) .
(3.35)
Proof. Since we have assumed that σ 2 < 2, we can choose ν > 0 so that σ 2 + 4ν 2 2. We consider the auxiliary function Wν (x, y) := e−σy + νeσy cosh(νx). We have
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( − 2)Wν = − 2 − σ 2 − ν 2 Wν . The potential in L is given by 1 − 3H 2 , hence, for |y| large enough, say |y| yσ , we can write 2 − σ2 2 − ν Wν . LWν − 2
Therefore, we get 2 − σ 2 −σ |y| e LWν − , 4
in this range. We can now use the barrier Wν and the maximum principle, to conclude that sup Wν−1 φ C φ L∞ (R2 ) + (cosh y)σ f L∞ (R2 ) .
|y|yσ
Letting ν tend to 0 yields the desired estimate.
2
For the time being, we have only considered the decay behavior of the solution in the y variable. The next result shows that some a priori weighted estimate with both decay in the x and y variables is also available. The key observation is that, according to Lemma 3.1, the least nonzero eigenvalue of −L0 is 32 and its continuous spectrum starts at 2, hence, if φ ∈ H 1 (R) satisfies φH dy = 0, R
we have the inequality R
3 |∂y φ|2 − 1 − 3H 2 φ 2 dy 2
φ 2 dy.
(3.36)
R
Using this, we can prove the: √ Lemma 3.5. Assume that σ ∈ (0, 2) is fixed. For all a ∈ [0, √1 ) such that 2
σ 2 + a 2 < 2, there exists a constant Ca > 0, which depends on a but remains bounded as a tends to 0, such that (cosh x)a (cosh y)σ φ
L∞ (R2 )
Ca φ L∞ (R2 ) + (cosh x)a (cosh y)σ Lφ L∞ (R2 ) ,
provided φ ∈ L∞ (R2 ) satisfies (3.31).
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Before we proceed with the proof of the result, let us emphasize that the key property is that the constant Ca remains bounded as a tends to 0, we shall further comment on this at the end of this section. Also, the range in which the parameter a can be chosen is not optimal and it follows
from the analysis of [6] that the optimal range is [0, present paper.
3 2)
but we will not need this result in the
Proof. We already have proven the appropriate decay in the y direction. We will now prove that, under the assumptions of the lemma, the function φ has the appropriate decay in the x variable provided y remains in some compact set. Then, the result will follow from the use of suitable barrier functions as in the proof of the previous lemma. We consider the function ψ(x) := φ 2 (x, y) dy, R
which, thanks to the result of Lemma 3.4, is well defined (notice that here we implicitly use the fact that σ > 0). We can compute ψ (x) = 2
|∂x φ|2 dy + 2
R
φ∂x2 φ dy, R
where denote the derivative with respect to x. Using the fact that Lφ = f , we also have, using some integration by parts,
φ∂x2 φ dy = R
|∂y φ|2 + 1 − 3H 3 φ 2 + φf dy.
(3.37)
R
Collecting this together with (3.36), which holds since we have assumed that the orthogonality condition (3.31) was true for all x ∈ R, we conclude easily that ψ (x) 2
|∂x φ|2 dy + 3
R
φ 2 dy + 2
R
φf dy. R
Using Cauchy–Schwarz inequality to estimate the last term on the right-hand side, we find that ψ satisfies the following differential inequality ψ (x) 2ψ(x) −
f 2 (x, y) dy. R
Therefore, we conclude that 2 −ψ (x) + 2ψ(x) Ce−2a|x| (cosh x)a (cosh y)σ f L∞ (R2 ) , for some constant C > 0. Observe that, thanks to the results of Lemma 3.2 and Lemma 3.4, we know that ψ is bounded and we have
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ψ(x) C (cosh x)a (cosh y)σ f 2 ∞
L (R2 )
.
Now, we can use the auxiliary function 2 ψ¯ ν (x) := M (cosh x)a (cosh y)σ f L∞ (R2 ) e−2ax + νe2ax , where the constant M > 0 is chosen sufficiently large and ν > 0 is arbitrary small. If a ∈ [0, √1 ), 2 this function can be used as a barrier and the maximum principle implies that 0 ψ ψ¯ ν for all y 0 and letting ν tend to 0 we conclude that 2 ψ(x) C (cosh x)a (cosh y)σ f L∞ (R2 ) e−2ax , for all x 0. A similar argument yields the corresponding estimate for x 0. Hence we have obtained the bound 2a
(cosh x)
2 φ 2 (x, y) dy C (cosh x)a (cosh y)σ f L∞ (R2 ) .
R
Local elliptic estimates then imply that, for all y0 > 0, there exists a constant C > 0 (depending on the choice of y0 ) such that φ(x, y) C (cosh x)a (cosh y)σ f 2 ∞
L (R2 )
(cosh x)−a ,
uniformly in x ∈ R and |y| y0 . Having established such a decay in the x variable, the relevant estimate in the complementary region can be found using appropriately designed barriers. For instance, enlarging y0 if this is necessary, in the quadrant {(x, y): x > 0, y > y0 } we may consider a barrier of the form 2 x y φ˜ ν (x, y) := Me−(ax+σy) (cosh x)a (cosh y)σ f L∞ (R2 ) + νe 2 + 2 , with ν > 0 arbitrarily small. Fixing M large enough (depending on y0 ) and letting ν tend to 0 yields the desired estimate in the right upper quadrant of the plane. Similar argument also provide the relevant estimate in the other three quadrants, we leave the details to the reader. 2 3.3. Surjectivity result As far as the existence of solutions of (3.30)–(3.31) is concerned, provided we assume that
f (x, ·)H dy = 0,
R
for all x ∈ R, we have the following result whose proof relies on the previous analysis:
(3.38)
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√ Proposition 3.1. Assume that σ ∈ (0, 2) is fixed. For all a ∈ [0, √1 ) such that 2
σ 2 + a 2 < 2, there exists a constant Ca > 0, which depends on a but remains bounded as a tends to 0, such that, for all f satisfying the orthogonality condition (3.38) and (cosh x)a (cosh y)σ f
C 0,μ (R2 )
< +∞,
there exists a unique function φ, solution of (3.30)–(3.31), which satisfies (cosh x)a (cosh y)σ φ
C 2,μ (R2 )
Ca (cosh x)a (cosh y)σ f C 0,μ (R2 ) .
Proof. We first consider the equation on functions which are ζ -periodic in the x variable for some fixed ζ > 0. Observe that 0 is in the spectrum of the operator −L and the corresponding kernel is spanned by the function H . The remaining part of the spectrum of −L is positive and (according to Lemma 3.1) is larger than or equal to 32 , hence
3 |∇φ| + H 1 − 3H 3 φ 2 dx 2
2
R2ζ
φ 2 dx, R2ζ
for any function φ satisfying
φH dx = 0,
(3.39)
R2ζ
where R2ζ := (R/ζ Z) × R. As a consequence, given f ∈ L2 (R2ζ ) satisfying
f H dx = 0,
R2ζ
there exists a unique solution φ ∈ H 1 (R2ζ ), also satisfying (3.39), of Lφ = f and φ H 1 (R2 ) ζ C f L2 (R2 ) . Elliptic regularity theory then implies that ζ
φ L∞ (R2 ) C f L∞ (R2 ) + f L2 (R2 ) . ζ
ζ
ζ
Now, let us assume that, in addition the function f satisfies (3.38). Multiplying the equation Lφ = f by functions of the form ψ(x)H (y) and integrating by parts, one checks that ζ 0
R
φH dy ∂x2 ψ dx = 0,
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for any ζ -periodic function ψ . This implies that the function x →
φH dy
R
does not depend on x and, since its integral over [0, ζ ] is 0, we conclude that φ satisfies (3.31). We can now apply the result of Lemmas 3.3 and 3.4 to get the estimate (cosh y)σ φ
L∞ (R2ζ )
< C (cosh y)σ f L∞ (R2 ) , ζ
where the constant C > 0 does not depend on ζ . Now, given a function f satisfying the assumptions of the proposition, we define fζ to be the restriction of f to [0, ζ ] × R which is extended by periodicity in the x variable. Let φζ be the corresponding solution of Lφζ = fζ obtained above. Elliptic estimates together with a simple compactness argument allows one to pass to the limit as ζ tends to ∞ to get the existence of φ, a bounded solution of (3.30)–(3.31). The estimate of φ follows from Lemma 3.5 together with classical elliptic estimates and the uniqueness of φ follows from Lemma 3.2. 2 We end up this section with some comment on the orthogonality condition we impose on the function f . Given any (bounded) function f , with the appropriate decay as in the statement of Proposition 3.1, we want to solve the equation Lφ = f . We can certainly find a function x → c(x) such that f − cH satisfies (3.38). And then, we can apply the result of Proposition 3.1 to solve Lφ = f − cH . Therefore, it just remains to solve the equation Lψ = cH , but this is rather easy since it is enough to look for ψ of the form ψ(x, y) = d(x)H (y) in which case the equation reduces to the solvability of the equation d = c. Observe that, it is not possible to find a solution to this ordinary differential equation which decays exponentially at ±∞ unless the function c satisfies
c(x) dx =
R
xc(x) dx = 0. R
In fact this solution is explicitly given by x d(x) = x −∞
x c(z) dz −
zc(z) dz.
−∞
Now if c is bounded by a constant times (cosh x)−a , and satisfies the two conditions above, it is easy to check that d is also bounded by a constant (independent of a ∈ (0, 1)) times a −2 (cosh x)−a . In particular, this solution blows up as a tends to 0. In the next section we will need to invert L on functions spaces corresponding to a tending to 0 and, in order to get a right inverse whose norm does not blow up, is will be necessary to impose the restriction (3.38) on the functions f .
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4. The approximate solutions and the general set up 4.1. Description of the nodal curves of the approximate solutions We keep the notations introduced in the introduction and in Section 2 to describe an infinitedimensional family of approximate solutions to our problem. We first choose the data which allow us to describe the curves which will be very close to the nodal sets of our solutions. Remark 4.1. In order to simplify notations, if ζ → Ξ (π1 , . . . , πm ; ζ ) is a function or operator acting on ζ , which depends on parameters π1 , . . . , πm (which might be integers, real numbers, functions, . . . .), we agree that we simply write Ξ instead of Ξ (π1 , . . . , πm ; ·) when no confusion is possible. Let us assume that we are given a solution q := (q1 , . . . , qk ) of the Toda system (1.6), we define qε to be the vector valued function whose components are given by qj,ε (x) := qj (εx) −
√
k+1 log ε. 2 j− 2
We also assume that we are given v := (v1 , . . . , vk ) ∈ E (see Section 2 for a precise definition of E) such that v E δ1 ε α1 ,
(4.40)
where the constants α1 ∈ (0, 2) and δ1 > 0 will be fixed later, independent of ε ∈ (0, 1/2]. Remark 4.2. In the following we have to estimate various quantities Ξ (ε, v, h; ·) which depend on ε, v or h. In general, we will prove statements of the following form: there exist constants C0 , β0 > 0 which do not depend on the choice of the parameters δ1 and α1 such that Ξ (ε, v, h; ·) C0 β0 , provided ε is chosen small enough, say ε ∈ (0, ε0 ). And in general, ε0 does depend on δ1 and α1 . The idea behind this type of estimates is that there exist constants C0 , β0 > 0 such that Ξ (ε, 0, 0; ·) C20 β0 , while Ξ (ε, v, h; ·) C20 β0 + C1 ε β1 provided (4.40) is satisfied. Here C1 and β1 do depend on δ1 and α1 but β1 > β0 and hence, for ε small enough, the term C1 ε β1 is certainly controled by C20 β0 and this explains the general claim. With these data at hand, we define the planar curve Γ¯j (ε, v) to be the image of γj (x) := x, qj,ε (x) + vj (εx) . Even though the definition of Γ¯j also depends on the choice of q, the solution of the Toda system, we shall not make this dependence explicit in the notation since we will assume from now on that q is fixed. Roughly speaking, the curves Γ¯j will describe the nodal sets of our solution, or at least they will be close to them. For each j = 1, . . . , k, we introduce the Fermi coordinates (xj , yj ) which are associated to the curve Γ¯j . More precisely, we consider the parameterization of a tubular neighborhood of Γ¯j by Xj = Xj (ε, v; ·)
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Xj (xj , yj ) := γj (xj ) + yj nj (xj ),
(4.41)
where nj is the normal vector about Γ¯j (the curves are assumed to be positively oriented). Observe that the coordinate yj is nothing but the signed distance to Γ¯j . In the sequel, we will make use of the convenient notation Xj∗ f (xj , yj ) = (f ◦ Xj )(xj , yj ), where f is a function defined in a neighborhood of Γ¯j . 4.2. An infinite-dimensional family of approximate solutions Now that we have described the possible candidates for the nodal sets of our approximate solution, the basic idea is to consider the approximate solution which is close to the function ±H (dist(·, Γ¯j )) (with alternative signs according to wether j is odd or even). A possible choice could be the function k
1 (−1)j +1 H dist(·, Γ¯j ) − (−1)k+1 + 1 . 2
(4.42)
j =1
We need to take care of two technical problems. The first one concerns the regularity of the distance function to the curves Γ¯j . This distance function is smooth in the neighborhood of Γ¯j but is not smooth in the whole plane. More precisely, it is a simple exercise to check that, there exists Cq > 0 (only depending on q) such that the distance function to Γ¯j , is smooth in the set V := (x, y) ∈ R2 : |y| Cq ε −1 1 + |x|2 .
(4.43)
This follows at once from the structure of q at infinity which implies that the curve Γ¯j is exponentially close to half lines at infinity. Observe that the constant Cq > 0 can be chosen independently of ε ∈ (0, 1/2) and also observe that Γ¯j ⊂ V ,
(4.44)
for ε small enough. To overcome the regularity issue, we take advantage of the fact that the function H is almost constant (equal to either +1 or −1) away from 0 and we make use of an appropriate cutoff function to connect the approximate solution (4.42) to the constant functions ±1 away from the curves Γ¯j . The second problem we have to face is more delicate to explain. As we will see shortly, it takes its origin in the orthogonality condition (3.38) we have to impose to produce a right inverse of L whose norm does not blow up as the weight parameter a tends to 0. This problem translates into the fact that, even though the nodal sets of the solutions we will construct are close to the curves Γ¯j (say in Hausdorff topology), this topology is not refined enough to perform the construction. Hence, in some sense we need to improve the definition of the nodal sets of the approximate solutions by allowing more flexibility in the definition of the curves Γ¯j . This is the reason why we have already introduced the vector valued function v in the definition of Γ¯j .
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Unfortunately this is not quite enough and we need to introduce another vector valued function h := (h1 , . . . , hk ) ∈ C 2,μ (R; Rk ) satisfying h C 2,μ (R;Rk ) := (cosh x)τ h C 2,μ (R;Rk ) δ1 ε α1 , τ
(4.45)
where τ > 0 and the constants α1 ∈ (0, 2) and δ1 > 0 will be fixed later on (independently of ε). It will be convenient to define the functions Hj = Hj (ε, v, h; ·) by the identity Xj∗ Hj (xj , yj ) := H yj − hj (εxj ) .
(4.46)
With these data and notations, we are now in a position to define a multiple-end approximate solution of (1.1). We start with the definition of u¯ 0 = u¯ 0 (ε, v, h; ·) given by u¯ 0 :=
k
1 (−1)j +1 Hj − (−1)k+1 + 1 . 2 j =1
We let t → η(t) be a smooth cutoff function such that η(t) ≡ 1 for |t| 1/2 and η(t) ≡ 0 for |t| 1 and we define for all ε > 0 small enough the function ηε (x, y) := η
εy
Cq 1 + |x|2
,
where the constant Cq is the one introduced in the definition of V . The cutoff function ηε is now used to smooth this function and define the approximate solution u¯ = u(ε, ¯ v, h; ·) in the following way u¯ := ηε u¯ 0 + (1 − ηε )
u¯ 0 . |u¯ 0 |
Let us emphasize that the approximate solution u¯ depends on the choice of ε, v ∈ E and 2,μ h ∈ Cτ (R; Rk ). 4.3. The set up of the nonlinear problem We now define an appropriate weighted norm for functions defined in R2 . For all σ, a > 0, we need to build a weight function Wσ,a = Wσ,a (ε, v; ·) which is defined to be equal to Wσ,a :=
k
Wσ,a,j ,
j =1
where Xj∗ Wσ,a,j (xj , yj ) = (cosh xj )−a (cosh yj )−σ , in V . In the lower part of R2 \ V , the weight function Wσ,a is designed in such a way that
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ce−(a|x1 |+σ |y1 |) Wσ,a (x, y) Ce−(a|x1 |+σ |y1 |) , for all (x1 , y1 ) ∈ R2 such that x1 is coordinate in Γ¯1 of the point which realizes the (signed) distance y1 from the point (x, y) to Γ¯1 . Here c < 1 < C are fixed constants. This being understood, we have: ,μ
Definition 2. Given σ, a > 0, we define Cσ,a (R2 ) to be the space of C ,μ functions for which the following norm is finite −1 φ C ,μ (R2 ) := sup Wσ,a (x) φ C ,μ (B1 (x)) . σ,a
(4.47)
x∈R2
In other words, σ is related to the rate of decay of the functions in the direction transverse to the curves Γ¯j and a is related to the rate of decay of the functions along the curves Γ¯j . Observe that these definitions depend on ε even though this is not clear in the notations. Granted the above notations and definitions, the equation we want to solve reads (u¯ + φ) + u¯ + φ − (u¯ + φ)3 = 0, 2,μ
(4.48) 2,μ
where u¯ = u(ε, ¯ v, h; ·) for some φ ∈ Cσ,ετ (R), some vector valued function h ∈ Cτ (R; Rk ) and some v ∈ E. We can then formally rewrite (4.48) as Lφ = Q(φ), where the linear operator L = L(ε, v, h; ·) is defined by L := + 1 − 3u¯ 2 , and where the nonlinear operator Q = Q(ε, v, h; ·) is defined by ¯ 2. Q(φ) := − u¯ + 1 − u¯ 2 u¯ + φ 3 + 3uφ
(4.49)
We now study the mapping properties of the linear operator L and the nonlinear operator Q when defined between appropriate weighted function spaces. 5. The linear theory for multiple interfaces 5.1. Laplacian in Fermi coordinates It will be useful to have the expression of the Laplacian in the above defined Fermi coordinates. Observe that in the coordinates (xj , yj ) the Euclidean metric reads Xj∗ dx 2 + dy 2 = Aj dxj2 + dyj2 , where the function Aj is explicitly given by
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Aj := 1 + ε 2 Bj2 − 2yj
ε 2 Cj (1 + ε 2 Bj2 )1/2
+ yj2
ε 4 Cj2 (1 + ε 2 Bj2 )2
481
,
where Bj (xj , yj ) := (qj + vj ) (εxj ), and Cj (xj , yj ) := (qj + vj ) (εxj ). In these coordinates, the expression of the Laplacian is given by = ∂x2j + ∂y2j +
1 1 ∂yj Aj 1 ∂xj Aj − 1 ∂x2j + ∂yj − ∂xj . Aj 2 Aj 2 A2j
Observe that, there exists a constant c > 0 such that ε 2 |yj |(cosh xj )−τ0 C, in V , uniformly as ε tends to 0. Using this, it is an easy exercise to check that the following estimates hold in V Aj = 1 + OC ∞ (V ) ε 2 + OC ∞ (V ) ε 2 |yj |e−τ0 ε|xj | and hence
1 1− Aj
∂yj Aj Aj
1/2 = OC ∞ (V ) ε 2 + OC ∞ (V ) ε 2 1 + yj2 (cosh xj )−τ0 , = OC ∞ (V ) ε 2 (cosh xj )−τ0
and ∂xj Aj A2j
1/2 = OC ∞ (V ) ε 3 1 + yj2 (cosh xj )−τ0 .
We will also need the elementary fact which follows from the definition of the curves Γ¯j and the Fermi coordinates together with elementary geometry. In V we have √ yi = (i − j ) 2 log ε + OC ∞ (V ) (1) + 1 + OC ∞ (V ) ε 2 yj + ε aj± − ai± + OC ∞ (V ) δ1 ε α1 + OC ∞ (V ) ε 2 xj ,
(5.50)
as ε tends to 0 (the superscript ± is equal to + (resp. −)) when xj 0 (resp. xj 0). Recall that the parameters aj± have been defined in (1.7). In other words, we evaluate the sign distance to Γ¯i in therm of the Fermi coordinates associated to Γ¯j .
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Observe that the term OC ∞ (V ) (δ1 ε α1 ) depends on δ1 and α1 and since we assume that α1 ∈ (0, 2), we can absorb the term OC ∞ (V ) (ε 2 ) into it, keeping in mind that the estimate does depend on δ1 and α1 . 5.2. Linear theory for multiple interfaces We now want to study the mapping properties of the operator L := + 1 − 3u¯ 2 , where the potential is built using the approximate solution u¯ = u(ε, ¯ v, h; ·). The idea is to glue together parametrices which have been obtained in the previous section for the model operator L = + 1 − 3H 2 , using a perturbation argument. We make use of the weighted function spaces 2,μ 2,μ Ca,σ (R2 ), Ca (R; Rk ) which have already been defined in (4.47) and (2.24), respectively. Following (4.46), we introduce the functions Hj = Hj (ε, v, h; ·) by the identity Xk∗ Hj (xj , yj ) := H yj − hj ( xj ) . We also define the cutoff functions ρj = ρj (ε, v, h; ·) by Xj∗ ρj (xj , yj ) := ρε yj − hj ( xj ) , where ρε (t) :=
√
4t 2 log 1ε
,
(5.51)
and where ρ is a cutoff function identically equal to 1 on |t| < 12 and identically equal to 0 for |t| > 1. (Remember that the distance between two consecutive curves Γ¯j and Γ¯j +1 can be √ estimated by − 2 log ε + O(1), so the supports of the cutoff functions ρj are disjoint for ε small.) We will consider the solvability of the linear problem Lφ +
k
κj ρj Hj = f,
(5.52)
j =1
in R2 , where the unknowns are the function φ and the functions κj which are defined in V in such a way that Xj∗ κj only depends on xj . To keep notations short, we set L(φ, κ) := Lφ +
k
κj ρj Hj ,
j =1
where we have set κ := (κ1 , . . . , κk ). Here, one has to keep in mind that L, ρj and Hj all depend on ε, v and h and hence so does L. We will always assume that
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h + v C 2,μ (R;Rk )⊕E δ1 ε α1 ,
483
(5.53)
τ
for some constants α1 ∈ (0, 2) and δ1 > 0 which will be fixed later on. Building on the analysis of the previous section, we prove: √ Proposition 5.1. Assume that σ ∈ (0, 2) and τ > 0 are fixed and assume that (5.53) is satisfied for some fixed α1 and δ1 . Then, there exists ε0 > 0 (depending on α1 and δ1 ) such that for all ε ∈ (0, ε0 ), there exists a linear operator G = G(ε, v, h; ·) 2 2 0,μ 2,μ 0,μ R → Cσ,ετ R × Cετ R; Rk , G : Cσ,ετ whose norm is bounded by a constant (independent of ε, δ1 and α1 ), such that, (φ; κ) := G(f ) is the unique solution of (5.52) which satisfies Xj∗ (ρj Hj φ) dyj = 0, (5.54) R
for all xj ∈ R. The main idea in the proof of this proposition is to first handle the case where h = 0. In this case we glue together parametrices of L which were obtained in Proposition 3.1 to get an approximate right inverse of L which is then perturbed into a genuine right inverse of L. The general case, when h = 0, can then be handled using a simple perturbation argument. We decompose the proof of this proposition in a sequence of intermediate results. We start by considering the case where h = 0 and v ∈ E is fixed and prove the existence of G(ε, v, 0; ·) in this case. This is the content of the following: Lemma 5.1. Assume that h = 0. Then, for all ε > 0 small enough, the existence of G(ε, v, 0) satisfying the statement of Proposition 5.1 holds. Proof. We decompose the proof in three steps. Step 1. We make use of Proposition 3.1 to get the existence of φj solution of 2 ∂xj + ∂y2j + 1 − 3H 2 Xj∗ φj = ρε Xj∗ f − κj0 H , where H , H and ρε are functions of yj and κj0 are functions of xj . The functions κj0 are chosen so that the right-hand side of this equation satisfies the orthogonality condition (3.38), hence κj0 (xj ) ρε (H )2 dyj = ρε H Xj∗ f dyj . R
R
Observe that Xj∗ κj0 only depends on xj . It is easy to check that k
+ ρ φ 0,μ j j k Cετ (R;R )
0 κ
j =1
2,μ Cσ,ετ (R2 )
C f C 0,μ (R2 ) , σ,ετ
(5.55)
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for some constant C > 0 independent of ε, δ1 and α1 . The estimate for κj0 follows at once from the definition while the estimate for kj =1 ρj φj follows directly from the result of Proposition 3.1. Observe that, by construction, we have
H Xj∗ φj dyj = 0.
(5.56)
R
We define f0 := f − L
k
j =1
ρj φj
−
k
κj0 ρj Hj .
j =1
Observe that there are two main reasons why f0 is not identically equal to 0. The first being the effect of the cutoff function which implies that, away from the support of the functions ρj , we have f0 = f . The second being that, close to the curves Γ¯j , even though ρj = 1, there is a small discrepancy between the Laplacian and the operator ∂x2j + ∂y2j . We now give a more quantitative statement of these two facts. First we compute f0 = 1 −
k
ρj f −
j =1
k k
(φj ρj + 2∇ρj ∇φj ) + ρj ∂x2j + ∂y2j − φj . j =1
j =1
It is easy to check that we have f0 C 0,μ (R2 ) C f C 0,μ (R2 ) . σ,ετ
σ,ετ
Moreover, in the region where ρj ≡ 1 we simply have f0 = (∂x2j + ∂y2j − )φj and still using the expression of the Laplacian in Fermi coordinates, once can check that the operator −(∂x2j +∂y2j ) is a second order differential operator in ∂xj and ∂yj whose coefficients are bounded by a constant times ε 2 log 1ε in this region. Hence, we get 1 χj f0 C 0,μ (R2 ) Cε 2 log f C 0,μ (R2 ) , σ,ετ σ,ετ ε
(5.57)
where the cutoff function χ1 , . . . , χk are defined by Xj∗ χj (xj , yj ) := ρε (2yj ). Step 2. We now solve ( − 2)ψ = f0 .
(5.58)
The existence of ψ , bounded solution of this equation, is straightforward. We claim that ψ C 2,μ (R2 ) C f C 0,μ (R2 ) , σ,ετ
σ,ετ
(5.59)
for some constant C > 0 independent of ε, α1 and δ1 . Indeed, the maximum principle immediately implies that
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ψ L∞ (R2 ) C f C 0,μ (R2 ) . σ,ετ
Next, arguing as in the proof of Lemma 3.4, we define the auxiliary function W¯ σ,ετ,ν by Xj∗ W¯ σ,ετ,ν = e−σyj (cosh xj )−ετ + ν(cosh xj )ετ , and, using once more the expression of the Laplacian in Fermi coordinates, we check that 1 2 2 ¯ ( − 2)Wσ,ετ,ν = − 2 − σ + O ε log W¯ σ,ετ,ν , ε
(5.60)
in the region V¯j where yj −ετ |xj | and yj +1 ετ |xj +1 | (i.e. in a region which slightly encompasses the region between the curves Γ¯j and Γ¯j +1 ). The maximum principle can then be used in V¯j to prove that ψ is bounded by a constant (independent on ν) times W¯ σ,ετ,ν times the norm of f in V¯j . Letting ν tend to 0 we obtain the estimate (5.59). A similar analysis can be carried out in the region of the plane which is above Γ¯k or below Γ¯1 . We define the cutoff functions χˆ 1 , . . . , χˆ k by Xj∗ χˆ j (xj , yj ) := ρε (4yj ). Observe that we also have the following estimate
1 χˆ j ψ C 2,μ (R2 ) C ε log + ε 0,ετ ε
√
2
2σ 16
f C 0,μ (R2 ) ,
(5.61)
σ,ετ
which again follows from the maximum principle, using the barrier function, W¯ 0,ετ,ν together with (5.57) to evaluate the right-hand side in (5.58) and (5.59) to evaluate ψ the boundary of the √ set {Xj (xj , yj ): |yj | 162 log 1ε }. Step 3. We set φ¯ := ψ +
k
ρj φj −
j =1
k
λj ρj Hj ,
j =0
and κ¯ j := κj0 + λj , where the functions λ1 , . . . , λk are defined by the identity Xj∗ λj (xj , yj )
ρε2 (H )2 dyj
R
=
ρε H
Xj∗
R
ψ+
k
j =1
Observe that Xj∗ λj only depends on xj . We consider the operator ¯ ) := (φ, ¯ κ). G(f ¯ It follows from (5.55), (5.59) that
ρj φj dyj .
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2 2 0,μ 2,μ 0,μ ¯ : Cσ,ετ R → Cσ,ετ R × Cετ R; Rk , G is well defined and has norm bounded by a constant independent of ε, δ1 and α1 . We compute Lφ¯ +
k
k
κ¯ j ρj Hj = f + 3 1 − u¯ 2 ψ − 2 ∇(λj ρj )∇Hj
j =1
j =1
−
k
λj ρj LHj
k
− (λj ρj + 2∇λj ∇ρj )Hj .
j =1
j =1
Using (5.56), we can estimate λj C 2,μ (R) Cε α f C 0,μ (R2 ) , ετ
σ,ετ
and using (5.61) together with the fact that σ < 1 − u¯ 2 ψ
0,μ Cσ,ετ (R2 )
√ 2, we check that Cε α f C 0,μ (R2 ) , σ,ετ
for some α > 0 (independent of ε and f ). Then, it is easy to check that L ◦ G(f ¯ )−f
0,μ Cσ,ετ (R2 )
Cε α f C 0,μ (R2 ) , σ,ετ
for all ε small enough. When h = 0, the existence of G(ε, v, 0; ·) follows at once from a standard perturbation argument. This completes the proof of the lemma. 2 We now assume that h = 0 and, using the previous lemma together with a perturbation argument, we prove: Lemma 5.2. For all ε > 0 small enough, the existence of G(ε, v, h; ·) satisfying the statement of Proposition 5.1 holds. Proof. Again, the proof of this result relies on some perturbation argument. To distinguish the operators when h = 0 and h = 0, we adorn them with the subscript h writing for example Lh , Gh , Hj,h , . . . instead of L(ε, v, h; ·), G(ε, v, h; ·), Hj (ε, v, h; ·), . . . . ¯ h by G ¯ h (f ) := (φ, ¯ κ) ¯ where We set (φ, κ) := G0 (f ) and define the operator G φ¯ := φ −
k
λj ρj,h Hj,h
and κ¯ j := κj + λj ,
j =0
and where the functions λ1 , . . . , λk are defined by the identity Xj∗ λj (xj , yj ) ρε2 (H )2 dyj = Xj∗ ρj,h Hj,h φ dyj . R
R
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Observe that Xj∗ λj only depends on xj and, by construction, we have
Xj∗ ρj,0 Hj,0 φ dyj = 0,
R
hence we can also write Xj∗ λj (xj , yj )
ρε2 (H )2 dyj =
R
φ dyj . Xj∗ ρj,h Hj,h − ρj,0 Hj,0
R
Since we already know that φ C 2,μ (R2 ) C f C 0,μ (R2 ) , we get σ,ετ
σ,ετ
λj C 2,μ (R;Rk ) C h C 0,μ (R;Rk ) f C 0,μ (R2 ) . ετ
ετ
σ,ετ
(5.62)
In particular, this implies that 2 2 0,μ 2,μ 0,μ ¯ h : Cσ,ετ R → Cσ,ετ R × Cετ R; Rk G is well defined and has norm bounded by a constant independent of ε, α1 and δ1 . We claim that L h ◦ G ¯ h (f ) − f
0,μ Cσ,ετ (R2 )
C h C 0,μ (R;Rk ) f C 0,μ (R2 ) . ετ
σ,ετ
Assuming we have already proved the claim, the existence of G(ε, v, h; ·) follows again from a standard perturbation argument. Therefore, it remains to prove the claim. To this aim, we compute k
¯ κ) Lh (φ, ¯ − f = 3 u¯ 20 − u¯ 2h φ + κj ρj,h Hj,h − ρj,0 Hj,0 j =1
−2
k
∇(λj ρj,h )∇Hj,h −
j =1
−
k
k
j =1
λj ρj,h + 1 − 3u¯ 2h¯ Hj,h
(λj ρj,h + 2∇λj ∇ρj,h )Hj,h .
j =1
Using the result of the previous proposition to evaluate the norm of f and κ in terms of the norm of f and using (5.62), it is straightforward to check that Lh (φ, ¯ κ) ¯ −f
0,μ Cσ,ετ (R2 )
C h C 0,μ (R2 ) f C 0,μ (R2 ) , σ,ετ
σ,ετ
for some constant C > 0 which does not depend on ε. This completes the proof of the claim.
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Finally, it remains to prove the uniqueness of G. This is the content of: Lemma 5.3. For all ε > 0 small enough, the operator G described in the statement of Proposition 5.1 is unique. Proof. The proof is decomposed into two steps. Step 1. We first prove an a priori estimate for the solutions of the homogeneous problem L(φ, κ) = 0 satisfying (5.54). More precisely, we claim that there exists a constant C > 0 and α > 0 (independent of ε, φ and κ) such that κ C 2,μ (R;Rk ) Cε α φ C 2,μ (R2 ) , ετ
σ,ετ
for any such solution. To simplify notations, we identify Xj∗ φ with φ and Xj∗ u¯ with u. ¯ We start by multiplying L(φ, κ) = 0 by ρj Hj and integrate over yj to get with little work −Xj∗ κj
ρε2 (H )2 dyj =
R
ρj Hj ∂x2j φ dyj +
R
ρj Hj ∂y2j + 1 − 3Hj2 φ dyj
R
ρj Hj Hj2 − u¯ 2 φ dyj +
+3 R
ρj Hj − ∂x2j − ∂y2j φ dyj .
R
We evaluate each consecutive term. Observe that thanks to (5.54) we can write
ρj Hj ∂x2j φ dyj = −
R
φ∂x2j ρj Hj dyj − 2
R
∂xj φ∂xj ρj Hj dyj .
R
Since ∂xj ρj Hj = −εhj ρj Hj + ρj Hj , and 2 ∂x2j ρj Hj = ε 2 hj ρj Hj + 2ρj Hj + ρj Hj − ε 2 hj ρj Hj + ρj Hj , it is easy to check that ρj H ∂ 2 φ dyj j xj R
2,μ Cετ (R)
Cε α φ C 2,μ (R2 ) , σ,ετ
for some α > 0 which does not depend on ε, φ and κ. Using an integration by parts and the fact that (∂y2j + 1 − 3Hj3 )Hj = 0, we see that the second term can also be written as
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ρj Hj
2 ∂yj + 1 − 3Hj3 φ dyj =
R
489
ρj Hj + 2ρj Hj φ dyj
R
from which it follows at once that (reducing α if this is necessary) ρj H ∂ 2 + 1 − 3H 3 φ dyj j yj j
2,μ Cετ (R)
R
Cε α φ C 2,μ (R2 ) . σ,ετ
Using the fact that the approximate solution u¯ is close to Hj near Γ¯j , we check that (reducing α if this is necessary) ρj H H 2 − u¯ 2 φ dyj j j
2,μ Cετ (R)
R
Cε α φ C 2,μ (R2 ) . σ,ετ
Finally, using the expansion of the Laplacian in Fermi coordinates, we check that (reducing α if this is necessary) ρj H − ∂ 2 − ∂ 2 φ dyj xj yj j
2,μ Cετ (R;Rk )
R
Cε α φ C 2,μ (R2 ) . σ,ετ
Collecting these estimates completes the proof of the claim. 2,μ
2,μ
Step 2. We now assume that φ ∈ Cσ,ετ (R2 ) and κ ∈ Cετ (R; Rk ) satisfy L(φ, κ) = 0. We prove that φ = 0 and κ = 0 provided ε is close to 0. The proof is by contradiction and close to the proof of Lemma 3.3. Assume that for a sequence εn tending to 0 there exist φn = 0 and κn solution of L(φn ; κn ) = 0. We normalize φn so that −1 W σ,ετ φn L∞ (R2 ) = 1. −1 (x , y )φ (x , y ) 1 . We define the sequence We pick up a point (xn , yn ) ∈ R2 such that Wσ,ετ n n n n n 2 φ˜ n by −1 φ˜ n (x, y) := Wσ,ετ (xn , yn )φn (x − xn , y − yn ).
Using elliptic estimates together with Ascoli–Arzela’s theorem, we can assume that (up to a subsequence) the sequence φ˜ n converges uniformly, as n tends to +∞, to some function φ˜ ˜ 0) 1 and hence is not on compacts of R2 . The choice of the point (xn , yn ) implies that φ(0, 2 ˜ we distinguish two cases according identically equal to 0. To identify the equation satisfied by φ, to the behavior of the sequence (xn , yn ). If, for some subsequence, (xn , yn ) stays at finite distance from any curve Γ¯j , then φ˜ satisfies + 1 − 3H 2 (· − y0 ) φ˜ = 0, for some y0 ∈ R. Moreover
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˜ (· − y0 ) dy = 0. φH
R
˜ C(cosh y)−σ in R2 . However, the result of Lemma 3.2 shows that φ˜ = 0, which is Finally, |φ| a contradiction. If, for no subsequence (xn , yn ) stays at finite distance from the curves Γ¯j , then φ˜ satisfies ( − 2)φ˜ = 0. ˜ C(cosh y)σ (or |φ| ˜ Ceσy or |φ| ˜ Ce−σy ) in R2 . We then consider the Finally, either |φ| function W˜ a,b (x, satisfies ( − 2)W˜ a,b = −(2 − a 2 − b2 )W˜ a,b . √y) := cosh(ax) cosh(by) which 2 2 Taking a ∈ (σ, 2) and b > 0 such that a + b < 2, we can use W˜ a,b as a barrier to prove ˜ ν W˜ a,b for all ν > 0. Letting ν tend to 0 we conclude that φ ≡ 0 which is again a that |φ| contradiction. Having reached a contradiction in all cases, the proof of the claim is complete. 2 ˜ ·) from Observe that, thanks to the uniqueness result, one can also obtain G(ε, v, h; G(ε, v, h; ·) using a perturbation argument as in the proof of Lemma 5.3. Hence we obtain: Corollary 5.1. There exists a constant C > 0 (independent of ε, α1 and δ1 ) such that, G(ε, v, h; ˜ f ) − G(ε, v, h; f )
2,μ Cσ,ετ (R2 )
C h˜ − h C 2,μ (R,Rk ) f C 0,μ (R2 ) , σ,ετ
σ,ετ
provided ε > 0 is small enough. 5.3. Estimates We now measure how far the function u¯ = u(ε, ¯ v, h; ·) is from a genuine solution of (1.1). To do so, we analyze the nonlinear operator Q(ε, v, h; 0) which has been defined in (4.49). Recall that Q(ε, v, h; 0) = − u¯ + u¯ − u¯ 3 , where u¯ = u(ε, ¯ v, h; ·). The following result is close to the corresponding analysis performed in [9]. √ Proposition 5.2. Assume that σ ∈ (0, 2] and τ > 0 are fixed so that τ0 τ<√ . 2 Further assume that δ1 and α1 (defined in (5.53)) are fixed. Then, there exists a constant C > 0 independent of ε, α1 and δ1 and there exists ε0 > 0 such that, for all ε ∈ (0, ε0 ), we have: Q(ε, v, 0; 0)
0,μ Cσ,ετ (R2 )
Cε
2− √σ
2
,
(5.63)
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and Q(ε, v, h; ˜ 0) − Q(ε, v, h; 0)
0,μ Cσ,ετ (R2 )
Cε
2− √σ
2
h˜ − h C 2,μ (R;Rk ) . ετ
(5.64)
Proof. The proof is fairly technical and, in order to enlighten the key points and the ideas involved as clearly as possible, we will assume that k = 2. The estimates in the general case follow from similar considerations but notations are more involved. We first derive the estimates where the cutoff function ηε = 1. In this case, we simply have u¯ = H1 − H2 − 1, and, we can reorganize Q(ε, v, h; 0) as follows u¯ + 1 − u¯ 2 u¯ = H1 + H1 − H13 − H2 + H2 − H23 − (H1 − H2 − 1)3 + H13 − H23 − 1. We now restrict our attention to the subregion V− in V where y1 + y2 0 (similar estimates are available in the region where y1 + y2 0). In V− , we write √ (H1 − H2 − 1)3 − H13 + H23 + 1 = 3(H2 + 1)2 (H1 − 1) + 3 2H1 (H2 + 1) √ since 1 − H12 = 2H1 . Taking advantage of the fact that H + H − H 3 = 0, and using the expansion of the Laplacian in Fermi coordinates, we realize that u¯ + 1 − u¯ 2 u¯ =
√ 1 ∂y1 A1 1 ∂x1 A1 2 h1 −ε −3 2(H2 + 1) + h H1 2 A1 A1 2 A21 1 1 ∂y2 A2 1 ∂x2 A2 2 h2 − −ε + h H2 2 A2 A2 2 A22 2 1 2 1 2 2 2 h H1 − h H2 , − 3(H2 + 1) (H1 − 1) + ε A1 1 A2 2
where we have defined ∗ Xj,ε Hj (xj , yj ) := H (yj )
∗ and Xj,ε Hj (xj , yj ) := H (yj ).
To evaluate these terms, we will use the following facts √ √ and H2 + 1 = OC ∞ (−∞,0) e 2y2 H2 = OC ∞ (−∞,0) e 2y2 while √ H1 − 1 = OC ∞ (0,∞) e− 2y1
and H1 − 1 = OC ∞ (−∞,0) (1).
And we also make use of (5.50) which gives y2 in terms of y1 and x1
(5.65)
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y2 = 1 + OC ∞ (V ) ε 2 y1 + ε a1± − a2± + OC ∞ (V ) δ1 ε α1 |x1 | √ + 2 log ε + OC ∞ (V ) (1)
(5.66)
with ± according to wether x1 0 or x1 0 (remember that α1 ∈ (0, 2)). We find with some work −1 u¯ ε + 1 − u¯ 2ε u¯ ε C 0,μ (B sup Wσ,ετ
1
x∈V
Cε (x))
2− √σ
2
.
(5.67)
Let us now explain where the estimate comes from. It turns √ out that the parameters σ and τ which define the weights have to be chosen so that σ ∈ (0, 2) and also τ ∈ (0, √τ0 ). This is 2 needed to ensure that the function we evaluate has the appropriate decay in both the x and y directions so that its weighted norm is finite. With this choice, a quick inspection of the structure of u¯ ε + (1 − u¯ 2ε )u¯ ε shows that, to estimate the norm of this function, there are two region of interest (namely regions where the norm is actually achieved): the region close to the curve defined by y1 = 0 (namely the curve Γ¯1 ) and the region close to the curve defined by y1 + y2 = 0. It turns out that the estimate comes from the evaluation of the term (H2 + 1)2 H1 along the curve y1 + y2 = 0. Indeed, we have −1 (H2 + 1)2 H1 ∼ e Wσ,ετ
√
2y2
√
(cosh y1 )σ −
2
(cosh x1 )ετ ,
when y1 + y2 0. Therefore, we find that √ 2)y1
−1 (H2 + 1)2 H1 ∼ e(σ −2 Wσ,ετ
(cosh x1 )ετ ,
when y1 + y2 = 0. Now, along this curve, we have from (5.66) y1 =
ε ± 1 a2 − a1± + O δ1 ε α1 |x1 | − √ log ε + O(1), 2 2
again with ± according to wether x1 0 or x1 0. Therefore, we conclude that −1 sup Wσ,ετ (H2 + 1)2 H1 Cε
2− √σ
y1 +y2 =0
2
.
Observe that we have implicitly used the fact that τ<
√
σ ± a2 − a1± , 2− 2
so that the above supremum is finite. Since, by definition of τ0 we have a2± − a1± τ0 and since √ we assume that σ ∈ (0, 2), then one can check that this inequality holds provided τ < √τ0 . Using similar arguments, we find that the terms
2
Hj
contribute to the estimate by at
− 1) contributes to the estimate by at most most a constant times and the term a constant times ε 2 . All other quantities involving the functions hj give a contribution of size ε2
(H2 + 1)2 (H1
∂yj Aj Aj
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a constant (depending on δ1 ) times ε 2+α1 to the estimate, and hence this contribution can be 2− √σ 2 provided ε is chosen small enough. absorbed into Cε We finally have to take into account the effect of the cutoff function ηε . We denote by V¯ ⊂ V the set where ηε is not equal to either 0 or 1. It is easy to check that −1 u¯ ε + u¯ ε 1 − u¯ 2ε C 0,μ (B sup Wσ,ετ
1 (x))
x∈V¯
Cε 2 .
(5.68)
2
The estimate then follows from (5.67) and (5.68).
We are now interested in the estimates of the functions u¯ + u¯ − u¯ 3 ρj Hj dyj , Fj (ε, v, h; ·) := R
as functions of x (or xj ). As we will see in the proof of the next result, there exists β > 0 such that √ √ Fj (ε, 0, 0; x) = −ε 2 c∗ qj + c∗ e 2(qj −qj +1 ) − e 2(qj −1 −qj ) (εx) + O ε 2+β ,
on any compact of R. Here the constants c∗ and c∗ are given by √ √2t 2 c := 6 2 e H (t) dt = 12 e2t (cosh t)−4 dt = 32, ∗
R
R
and c∗ :=
√ 2 H (t) dt = 2
R
(cosh t)−4 dt =
R
4√ 2. 3
The estimate we have obtained in the previous proposition is quite general and does not use the fact that the functions qj are required to be solutions to the Toda system (1.6). In contrast, this expansion shows that the estimates of Fj strongly relies on this assumption and indeed, Fj (ε, 0, 0; ·) = O(ε 2+β ) if q is a solution of (1.6). It will be convenient to define √ √ Fj0 (ε, v, h; x) := −ε 2 c∗ (vj + hj ) + c∗ 2 e 2(qj −qj +1 ) (vj + hj − vj +1 − hj +1 ) √ − e 2(qj −1 −qj ) (vj −1 + hj −1 − vj − hj ) (εx), and we finally define F˚ := (F˚1 , . . . , F˚k ) where F˚j := Fj − Fj0 . We have:
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√ Proposition 5.3. Assume that σ ∈ (0, 2) and τ ∈ (0, τ0 ) are fixed. Further assume that α1 and δ1 are fixed. Then, there exists β1 ∈ (0, 1) and C > 0 (which neither depend on ε, α1 and δ1 , nor on σ and τ ) such that the following estimates hold F(ε, ˚ v, h; ·)
0,μ Cετ (R;Rk )
Cε 2+β1 ,
and F(ε, ˚ v, h; ˚ v, h; ·) ˜ ·) − F(ε,
0,μ Cετ (R;Rk )
Cε 2+β1 h˜ − h C 2,μ (R;Rk ) , ετ
for all ε small enough and provided v, h and h˜ satisfy (5.53). Proof. Again, we only consider the case where k = 1 since this simplifies the notations. The starting point if the formula (5.65) which was obtained in the proof of the previous proposition. The result then follows at once from the integration of this formula against ρ1 H1 . Let us mention the most important aspects of this computation. For brevity we will denote q˜ = q + v. We can write R
ε 2 q˜j 1 ∂yj Aj 2 1 2 Hj ρj dyj = − H ρj dyj 2 2 1/2 2 Aj Aj j (1 + ε (q˜j ) ) R
+
ε 4 (q˜j )2 (1 + ε 2 (q˜j )2 )2
R
1 2 yj Hj ρj dyj . Aj
Since Aj is close to 1, we can estimate
1 ∂yj Aj 2 Hj ρj dyj = −ε 2 q˜j 2 Aj
R
Now
R
2 Hj ρj dyj + O ε 4 (cosh x1 )−2ετ0 .
R
2 Hj ρj dyj =
(H )2 dy + O ε β ,
R
where β > 0 is fixed (and in fact depends on the definition of the cutoff function ρε see (5.51)). Hence, reducing β if this is necessary, we conclude that 1 ∂y1 Aj 2 Hj ρj dyj = −ε 2 q˜j (H )2 dy + O ε 2+β (cosh xj )−ετ . (5.69) 2 Aj R
R
Similarly, we have 1 2 Hj ρj dyj = −ε 2 hj (H )2 dy + O ε 2+β (cosh x1 )−ετ , −ε 2 hj Aj R
for some β > 0.
R
(5.70)
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Considering in (5.65) the remaining terms which carry the factor H1 , we have
2 (H2 + 1) H1 ρ1 dy1 =
R
2 H y2 − h2 (εx2 ) + 1 H y1 − h1 (εx1 ) ρ1 dy1 .
R
Elementary geometry and the fact that q˜j ∼ εxj at ±∞ yields the following estimates (please compare with (5.50)) y2 = q˜1 (εx1 ) − q˜2 (εx1 ) +
√
2 log ε + y1 1 + O ε 2 + O ε 3 |x1 | ,
and 1 , x2 = 1 + O ε 2 x1 + O(εy1 ) + O ε log ε in the region where y1 + y2 0. Using this together with the estimate H2 + 1 ∼ 2e holds in a tubular neighborhood of Γ¯1 , we conclude that
√ 2y2 ,
which
2 H y2 − h2 (εx1 ) + 1 H y1 − h1 (εx1 ) ρ1 dy1
R
= −2ε
2
√
e
2y
(H )2 dy e
√ 2(q1 −q2 )(εx1 )
R
√ √ 2 √2y 2 − 2 2ε e (H ) dy e 2(q1 −q2 )(εx1 ) (v1 + h1 − v2 − h2 )(εx1 ) R
+ O ε 2+β (cosh x1 )−ετ ,
(5.71)
for some constant β > 0. As already mentioned, the fact that qj is a solution of the Toda system implies that the leading parts in (5.69), (5.70) and (5.71) cancel. The other terms resulting from multiplication of (5.65) by H1 ρ1 can easily be estimated by O(ε 2+β (cosh x1 )−ετ ) and similar estimates can be obtained for the Hölder derivatives, completing the proof of the first estimate. The other estimate follows using similar arguments. 2 5.4. Solvability of the nonlinear problem We are now in a position to apply a first fixed point theorem, to find, close to the approximate solution u¯ a solution of (1.1) which has the desired features. First, we assume that we are given 2,μ v ∈ E and h ∈ Cτ (R; Rk ) satisfying (5.53) and we look for a function φ = φ(ε, v, h; ·) solution of L(ε, v, h; φ, κ) = Q(ε, v, h; φ).
(5.72)
Thanks to the result of Proposition 5.1, this equation can be rewritten as a fixed point problem
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(φ, κ) = G ε, v, h; Q(ε, v, h; φ) .
(5.73)
√ We choose σ ∈ (0, 2) and τ ∈ (0, √τ0 ) so that the results of the previous sections apply for 2 ε small enough. Collecting the results of the previous sections, we prove: √ Proposition 5.4. Assume that σ ∈ (0, 2) and τ ∈ (0, √τ0 ) are fixed. Further assume that α1 2 and δ1 are fixed. Then, there exists C0 > 0 (independent of the choice of α1 and δ1 ) and there 2,μ 2,μ exists ε0 > 0 such that, for all ε ∈ (0, ε0 ), there exists a unique (φ, κ) ∈ Cσ,ετ (R2 ) × Cετ (R; Rk ) solution of (5.72) which satisfies φ C 2,μ (R2 ) + κ C 2,μ (R;Rk ) C0 ε σ,ετ
2− √σ
2
ετ
.
Proof. The result of Proposition 5.2 and Proposition 5.1 show that G ε, v, h; Q(ε, v, h; 0)
2,μ 2,μ Cσ,ετ (R2 )×Cετ (R;Rk )
¯ Cε
2− √σ
2
¯ Next, observe for some constant C¯ > 0 which does not depend on ε. We now choose C0 = 2C. ¯ 2 and it is easy to check that the nonlinearity with respect to φ in Q is simply given by φ 3 + 3uφ that Q(ε, v, h; φ) ˜ − Q(ε, v, h; φ)
0,μ Cσ,ετ (R2 )
2− √σ
Cε
2− √σ
2
φ˜ − φ C 0,μ (R2 ) , σ,ετ
2 ˜ φ are both in the ball of radius C0 ε 2 in C provided φ, σ,ετ (R ). It is now standard to prove that, provided ε is chosen small enough, (5.73) has a solution which can be obtained as a fixed point for contraction mapping in this ball. 2 2,μ
The solution we have obtained in the previous proposition will be denoted by (φ(ε, v, h; ·), κ(ε, v, h; ·)). It is standard to check that, reducing ε0 if this is necessary, φ depends smoothly on the parameter h and, in some sense to be made precise, also depends continuously on v. However, more will be needed and, with little work, we can estimate the Lipschitz dependence of this solution with respect to h. This is the content of the following: Lemma 5.4. Under the assumptions of the previous proposition, there exists C > 0 such that the following estimate holds √ φ(ε, v, h; ˜ ·) − φ(ε, v, h; ·) 2,μ 2 Cε 2− 2σ h˜ − h 2,μ . C (R ) C (R;Rk ) σ,ετ
ετ
Proof. To distinguish the operators depending on different values of h we will adorn the operators and functions with a subscript h, writing Lh , u¯ h , . . . instead of L, u, ¯ . . . . We also write Qh = Q(ε, v, h; φh ). Taking the difference between the equation satisfied by φh and the equation satisfied by φh˜ , we find ˜ Q ˜ ). (φh˜ − φh , κh˜ − κh ) = G(ε, v, h; Qh − Qh˜ ) + G(ε, v, h; Qh˜ ) − G(ε, v, h; h
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We have Qh − Qh˜ = u¯ h + u¯ h − u¯ 3h − u¯ h˜ − u¯ h˜ + u¯ 3˜ h + φ 3˜ − φh3 + 3u¯ h φh2˜ − φh2 + 3 u¯ 2h˜ − u¯ 2h φh2˜ . h
We evaluate each term on the right-hand side. Making use of the bound of the solutions of (5.72) which have been obtained in Proposition 5.4, we can write 2 3 u¯ − u¯ 2 φ 2 h˜
h
Cε
0,μ h˜ Cσ,ετ (R2 )
2− √σ
2
h˜ − h C 2,μ (R;Rk ) . ετ
Similarly, we get σ 3 φ − φ 3 + 3u¯ h φ 2 − φ 2 0,μ 2 Cε 2− √2 φh − φ ˜ 2,μ h C h ˜ h C (R;Rk ) . (R ) h˜
h
σ,ετ
ετ
Finally, Proposition 5.2 yields σ u¯ h + u¯ h − u¯ 3 − u¯ ˜ − u¯ ˜ + u¯ 3 0,μ 2 Cε 2− √2 h˜ − h 2,μ . h ˜ C h h (R ) C (R;Rk )
h
σ,ετ
ετ
Therefore, we conclude that Qh − Qh˜ C 0,μ (R2 ) Cε
2− √σ
2
σ,ετ
φh − φh˜ C 2,μ (R2 ) + h˜ − h C 2,μ (R;Rk ) . σ,ετ
ετ
On the other hand, using Corollary 5.1 and Proposition 5.2, we get 2− √σ ˜ G(ε, v, h; Q ˜ ) − G(ε, v, h; ˜ Q ˜ ) 2,μ 2 2,μ 2 h − h 2,μ Cε . h h C (R )×C (R;Rk ) C (R,Rk ) σ,ετ
ετ
σ,ετ
Summarizing the above we have: φh − φh˜ C 2,μ (R2 ) Cε σ,ετ
2− √σ
2
φh − φh˜ C 2,μ (R;Rk ) + h˜ − h C 2,μ (R;Rk ) . ετ
The desired estimate follows by taking ε small enough.
ετ
2
We now explain in which sense the solution φ(ε, v, h; ·) depends continuously on v. To this aim, let us denote by Xj,v instead of Xj the parameterization defined in (4.41) so that its de pendence with respect to v becomes apparent. Similarly, we will write ρj,v , instead of ρj , Hj,v instead of Hj , . . . . We define a family of diffeomorphism Yv smoothly depending on v ∈ E (satisfying (4.45)) and designed in such a way that Y0 ≡ Id and that ∇(Yv − Id)
C ∞ (R2 )
Cε v E ,
and, for all j = 1, . . . , k, Yv Xj,v (xj , yj ) = Xj,0 (xj , yj ),
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for all (xj , yj ) such that |yj |
√ 4 2 3
log 1ε . Observe that, with this choice ρj,v = ρj,0 ◦ Yv ,
and Hj,v = Hj,0 ◦ Yv ,
on the support of ρj . Lemma 5.5. The mapping 2 2,μ v ∈ E → φv ◦ Yv−1 ∈ Cσ,ετ R , is continuous (beware that the weighted space of the right-hand side is the one corresponding to v = 0). Proof. We denote by φv the solution obtained in Proposition 5.72. We can write + 1 − 3u¯ 2v φv = − u¯ v + uv − u3v + φv3 + 3u¯ v φv2 . We can write φv = φ˜ v ◦ Yv and, composing with Yv−1 , we can write the equation satisfied by φ˜ v as + 1 − 3u¯ 20 φ˜ v = − u¯ v + uv − u3v ◦ Yv−1 + φ˜ v3 + 3u¯ v ◦ Yv−1 φ˜ v2 + 3 u¯ 2v ◦ Yv−1 − u¯ 20 φ˜ v + (φ˜ v ◦ Yv ) ◦ Yv−1 − φ˜ v .
(5.74)
By definition of Yv , we see that φ˜ v satisfies the orthogonality condition (5.54) with v = 0. It is easy to check that φ˜ v is also the unique solution of (5.74) whose norm is bounded by a constant 2− √σ 2 and which can be obtained as a fixed point for contraction mapping (this implicitly times ε uses the uniqueness result of Lemma 5.3). Observe that we are now working in a fixed function 2,μ space Cσ,ετ (R2 ) whose definition corresponds to v = 0. Using this formulation we can check that the mapping v ∈ E → φv ◦ Yv−1 is continuous. 2 We now explain how to choose v and h so that κ(ε, v, h; ·) = 0.
(5.75)
Observe that, multiplying (5.72) by ρj Hj we see that the equation κ(ε, v, h; ·) = 0 can be written as 3 u¯ + u¯ − u¯ 3 ρj Hj dyj + φ + φ − 3u¯ 2 φ ρj Hj dyj = φ + 3uφ ¯ 2 ρj Hj dyj , R
R
R
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for j = 1, . . . , k. It is worth mentioning that both u¯ and φ depend on ε, v and h. We study each of the three terms which compose this equation. First we observe that in Proposition 5.3 we have already derived an estimate for F = (F1 , . . . , Fk ), where Fj (ε, v, h; ·) :=
u¯ + u¯ − u¯ 3 ρj Hj dyj .
R
Next, let us define E := (E1 , . . . , Ek ) where
Ej (ε, v, h; ·) :=
φ + φ − 3u¯ 2 φ ρj Hj dyj .
R
We have the following: √ Lemma 5.6. Assume that σ ∈ (0, 2) and τ ∈ (0, √τ0 ) are fixed. Further assume that α1 and δ1 2 are fixed. Then, there exist a constant β2 > 0 (which does not depend on ε, σ , α1 and δ1 ) and a constant C > 0 such that E(ε, v, h; ·)
Cε
0,μ Cετ (R;Rk )
2− √σ +β2 2
,
and E(ε, v, h; ˜ ·) − E(ε, v, h; ·)
0,μ Cετ (R;Rk )
Cε
2− √σ +β2 2
h˜ − h C 2,μ (R;Rk ) , ετ
for all ε small enough, provided v, h and h˜ satisfy (5.53). Proof. The proof is very close to the analysis we have already performed in the proof of Lemma 5.3. Indeed, following a similar computation we can rewrite Ej as Ej = 2εhj
ρj Hj + ρj Hj ∂xj φ dyj + ε 2 hj
R
2 − ε 2 hj + R
+
ρj Hj + ρj Hj φ dyj
R
ρj Hj + 2ρj Hj + ρj Hj φ dyj
R
ρj Hj + 2ρj Hj φ dyj + 3
ρj Hj Hj2 − u¯ 2 φ dyj
R
ρj Hj − ∂x2j − ∂y2j φ dyj .
R
Instead of going through a technical proof, we simply explain where the estimate comes from. We observe that, thanks to the result of Proposition 5.4, all the terms which carry a factor of hj , (hj )2 or hj in front can be estimated by a constant times ε
3− √σ +α1 2
.
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Using the definition of the cutoff function ρj and the exponential decay of functions 1 ± H , H and H , we can estimate 2 2 ρj Hj + 2ρj Hj φ dyj + 3 ρj Hj Hj − u¯ φ dyj R
0,μ Cετ (R;Rk )
R
Cε
2− √σ +β 2
,
where β > 0 only depends on the definition of ρε given in (5.51). The estimate for the last term in the expression for Ej is straightforward using the expression of the Laplacian in Fermi coordinates. The second estimate follows from similar consideration together with the result of Lemma 5.4. We leave the details to the reader. 2 ¯ := (E¯ 1 , . . . , E¯ k ) where Let us define E E¯ j (ε, v, h; ·) :=
3 φ + 3uφ ¯ 2 ρj Hj dyj .
R
We have: Lemma 5.7. Under the assumptions of the previous lemma, there exists C > 0 such that the following estimates hold E(ε, ¯ v, h; ·)
√
0,μ Cετ (R;Rk )
Cε 4−
2σ
,
and E(ε, ¯ v, h; ˜ ·) − E(ε, ¯ v, h; ·)
0,μ Cετ (R;Rk )
√
Cε 4−
2σ
h˜ − h C 0,μ (R;Rk ) , ετ
for all ε small enough, provided v, h and h˜ satisfy (5.53). Proof. The proof follows at once from Lemma 5.4 and the estimates for the solutions of (5.72) provided by Proposition 5.4. 2 We are now in a position √ to explain how the constant α1 , which√was used in (5.53), is fixed. We fist assume that σ ∈ (0, 2) and μ ∈ (0, 1) are chosen so that 2 − 2 − μ > 0, − √σ + β2 − μ > 0 2 and β1 − μ > 0, where β1 and β2 are the constants which appear in the last lemmas. Observe that it is crucial that β2 > 0 could be chosen not to depend on σ . Then we define √ σ α1 = min 2 − 2 − μ, − √ + β2 − μ, β1 − μ . 2 With this choice, it follows from Proposition 5.3, Lemma 5.6 and Lemma 5.7 that the condition (5.75) is equivalent to ε 2 c0 (v + h) + N(v + h) j = Eˆ j (ε, v, h; ·)
(5.76)
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where N is the matrix of the linearized Toda system associated to linearization of (2.14) about ˆ := (Eˆ 1 , . . . , Eˆ k ) satisfies the solution q (see (2.23)), and where E Lemma 5.8. Assume that σ , μ and τ are fixed as above. Then, there exists a constant C1 > 0 (independent of ε and δ1 ) such that the following estimates hold E(ε, ˆ v, h; ·)
0,μ Cετ (R;Rk )
C1 ε 2+α1 +μ ,
and E(ε, ˆ v, h; ˜ ·) − E(ε, ˆ v, h; ·)
0,μ Cετ (R;Rk )
C1 ε 2+α1 +μ h˜ − h C 2,μ (R;Rk ) ετ
for all ε small enough, provided v, h and h˜ satisfy (5.53). In view of the result of Lemma 2.3 and the previous lemma, it is natural to solve (5.76) in the 2,μ space Cτ (R; Rk ) ⊕ E. At this point, it is worth mentioning that for a function g: R → Rk we have the obvious estimate g(ε·)
,μ Cετ (R;Rk )
C g C ,μ (R;Rk ) , τ
while on the other hand we have g C ,μ (R;Rk ) Cε −−μ g(ε·) C ,μ (R;Rk ) . τ
ετ
Collecting the previous analysis, it is easy to check that: Lemma 5.9. There exists δ1 > 0 such that, for all v ∈ E satisfying v E δ1 ε α1 and for all ε > 0 2,μ small enough, there exists a unique h ∈ Cετ (R; Rk ) and v¯ ∈ E satisfying ˆ v, h; ·), ε 2 c0 (¯v + h) + N(¯v + h) = E(ε,
(5.77)
and ¯v + h C 2,μ (R;Rk )⊕E τ
δ1 α1 ε . 2
Moreover v¯ depends continuously on v. Proof. The proof of this lemma follows immediately from the theory developed in Section 2 and more specifically Lemma 2.3, the result of Lemma 5.8 and the use of a fixed point theorem for contraction mapping. Using the result of Lemma 2.3, we can rewrite the equation we want to solve as ˆ v, h; ·) . v¯ + h = ε −2 T −1 E(ε, Thanks to the result of Lemma 5.8 and the above remark, we can estimate for all ε > 0 small enough
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−2 −1 ε T ˆ v, 0; ·) E(ε,
Cτ2,μ (R;Rk )
C¯ 1 ε α1 ,
for some constant C1 > 0 which does not depend on the choice of δ1 . In particular, we can choose, we can choose δ1 = 4C¯ 1 and the previous estimate will be valid provided we take ε > 0 small enough. Let us denote by Π the projection Π : Cτ2,μ R; Rk ⊕ E → Cτ2,μ R; Rk . Using Lemma 5.8 together with a fixed point theorem for contraction mapping, we get the existence of a (unique) fixed point h, for the mapping ˆ v, h; ˜ ·) , h˜ → ε −2 ΠT −1 E(ε, in the ball of radius the identify
δ1 α1 2ε
2,μ
in Cτ (R; Rk ). This fixed point h then induces a (unique) v¯ ∈ E by ˆ v, h; ·) − h. v¯ := ε −2 T −1 E(ε,
We clearly v¯ + h is a solution of (5.77) and we have the estimate ¯v + h C 2,μ (R;Rk )⊕E τ
δ1 α1 ε . 2
This completes the proof of the result. Continuity with respect to v follows from Lemma 5.5.
2
We will write v¯ = v¯ (ε, v) for the element of E which is given by the previous lemma. Therefore, in order to complete the proof of the result it remains to find v such that v = v¯ (ε, v). This can be easily achieved by using Browder’s fixed point theorem in the ball of radius δ1 ε α1 in E. Observe that we do not apply a fixed point theorem for contraction mapping to determine v since this would require to prove Lipshitz dependence of all solutions with respect to v. Even though this Lipshitz dependence holds, it would require some extra work and will complicate the notations. Therefore, we have chosen to solve this last equation using some topological fixed point result instead of a fixed point theorem for contraction mapping. Acknowledgments This work has been partly supported by chilean research grants Fondecyt 1070389, 1050311, 1090103, FONDAP, an ECOS-CONICYT contract C05E05 and an Earmarked Grant from RGC of Hong Kong and Focused Research Scheme of CUHK of Hong Kong. The third author was partially supported by the ANR-08-BLAN-0335-01. References [1] N. Alikakos, G. Fusco, V. Stefanopoulos, Critical spectrum and stability of interfaces for a class of reaction– diffusion equations, J. Differential Equations 126 (1) (1996) 106–167. [2] S. Allen, J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27 (1979) 1084–1095.
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[3] L. Ambrosio, X. Cabré, Entire solutions of semilinear elliptic equations in R3 and a conjecture of de Giorgi, J. Amer. Math. Soc. 13 (2000) 725–739. [4] X. Cabré, J. Terra, Saddle shaped solutions of bistable diffusion equations in all R2m , preprint, 2008, arXiv:0801.3379. [5] H. Dang, P.C. Fife, L.A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys. 43 (1992). [6] M. del Pino, M. Kowalczyk, F. Pacard, Moduli space theory for some class of solutions to the Allen–Cahn equation in the plane, preprint, 2008. [7] M. del Pino, M. Kowalczyk, F. Pacard, J. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math. (2009), in press. [8] M. del Pino, M. Kowalczyk, J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math. 70 (2007) 113–146. [9] M. del Pino, M. Kowalczyk, J. Wei, The Toda system and clustering interfaces in the Allen–Cahn equation, Arch. Ration. Mech. Anal. 190 (2008) 141–187. [10] M. del Pino, M. Kowalczyk, J. Wei, A conjecture by de Giorgi in large dimensions, preprint, 2008. [11] M. del Pino, M. Kowalczyk, J. Wei, J. Yang, Interface foliation on Riemannian manifolds with positive Ricci curvature, preprint, 2008. [12] A. Farina, Symmetry for solutions of semilinear elliptic equations in RN and related conjectures, Ric. Mat. 48 (1999) 129–154. [13] A. Farina, E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems, in: H. Ishii, W.-Y. Lin, Y. Du (Eds.), Recent Progress on Reaction–Diffusion Systems and Viscosity Solutions, World Scientific, 2008. [14] N. Ghoussoub, C. Gui, On a conjecture of de Giorgi and some related problems, Math. Ann. 311 (1998) 481–491. [15] C. Gui, Hamiltonian identities for elliptic partial differential equations, J. Funct. Anal. 254 (4) (2008) 904–933. [16] C. Gui, Allen Cahn equation and its generalizations, preprint, 2008. [17] J.E. Hutchinson, Y. Tonegawa, Convergence of phase interfaces in the van der Waals–Cahn–Hilliard theory, Calc. Var. 10 (2000) 49–84. [18] R.V. Kohn, P. Sternberg, Local minimizers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 69–84. [19] B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. Math. 34 (1979) 195–338. [20] M. Kowalczyk, On the existence and Morse index of solutions to the Allen–Cahn equation in two dimensions, Ann. Mat. Pura Appl. (4) 184 (1) (2005) 17–52. [21] A. Malchiodi, M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math. 55 (12) (2002) 1507–1568. [22] A. Malchiodi, M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J. 124 (1) (2004) 105–143. [23] L. Modica, Convergence to minimal surfaces problem and global solutions of u = 2(u3 − u), in: Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, Pitagora, Bologna, 1979, pp. 223–244. [24] J. Moser, Finitely many mass points on the line under the influence of an exponential potential — an integrable system, in: Dynamical Systems, Theory and Applications, Rencontres, Battelle Res. Inst., Seattle, Wash., 1974, in: Lecture Notes in Phys., vol. 38, Springer, Berlin, 1975, pp. 467–497. [25] F. Pacard, M. Ritoré, From the constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differential Geom. 64 (3) (2003) 356–423. [26] O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2) 169 (1) (2009) 47–78.
Journal of Functional Analysis 258 (2010) 504–533 www.elsevier.com/locate/jfa
Gradient estimates for the subelliptic heat kernel on H-type groups Nathaniel Eldredge Department of Mathematics, Cornell University, 593 Malott Hall, Ithaca, NY 14853, USA Received 8 April 2009; accepted 28 August 2009 Available online 4 September 2009 Communicated by L. Gross
Abstract We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups G of H-type: |∇Pt f | KPt |∇f | , where Pt is the heat semigroup corresponding to the sublaplacian on G, ∇ is the subelliptic gradient, and K is a constant. This extends a result of Li (2006) [10] for the Heisenberg group. The proof is based on pointwise heat kernel estimates, and follows an approach used by Bakry et al. (2008) [3]. © 2009 Elsevier Inc. All rights reserved. Keywords: Heat kernel; Subelliptic; Hypoelliptic; Heisenberg group; Gradient
1. Introduction In [10], H.-Q. Li proved the following gradient inequality for the heat kernel on the classical Heisenberg group H1 of real dimension 3: |∇Pt f | KPt |∇f | , E-mail address:
[email protected]. URL: http://www.math.cornell.edu/~neldredge/. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.08.012
(1.1)
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where Pt is the heat semigroup corresponding to the usual sublaplacian on H1 , ∇ is the corresponding subgradient, K is a constant, and f is any appropriate smooth function on H1 . This was the first extension of (1.1) to a subelliptic setting; the elliptic case was shown by Bakry [1,2], and in the case of a Riemannian manifold corresponds to a lower bound on the Ricci curvature. The proof in [10] relies on pointwise upper and lower estimates for the heat kernel, and a pointwise upper estimate for its gradient, both of which were obtained in [11] in the context of Heisenberg groups of any dimension. Ref. [3] contains two alternate proofs of (1.1) for the classical Heisenberg group H1 , also depending on the pointwise heat kernel estimates from [11]. Earlier, Driver and Melcher in [5] had shown a partial result: that for any p > 1 there exists a constant Kp such that |∇Pt f |p Kp Pt |∇f |p .
(1.2)
Their argument proceeded probabilistically via methods of Malliavin calculus and did not depend on heat kernel estimates, but they also showed that it could not produce (1.1), which is the corresponding estimate with p = 1. Ref. [13] extended the “Lp -type” inequality (1.2) to the case of a general nilpotent Lie group, at the cost of replacing the constant Kp with a function Kp (t). In [6], we were able to show that pointwise heat kernel estimates analogous to those of [11] (see (2.8)–(2.10)) hold for Lie groups of H type, a class which generalizes the Heisenberg groups while retaining some rather strong algebraic properties. (H-type groups were introduced by Kaplan in [9]; a useful reference and primer is Chapter 18 of [4].) The purpose of the present article is to show that given these heat kernel estimates, the first proof from [3] can be adapted to establish the inequality (1.1) in the setting of H-type groups. Our proof approximately follows the structure of the first proof from [3] but may be read independently of it, and is more explicitly detailed. 2. Definitions and notation In order to fix notation, we give a definition of H-type groups and accompanying concepts. Our notation, where applicable, matches that of [6]. A finite-dimensional Lie algebra g (with nonzero center z), together with an inner product ·,·, is said to be of H type or Heisenberg type if the following conditions hold: 1. [z⊥ , z⊥ ] = z; and 2. for each z ∈ z, the map Jz : z⊥ → z⊥ defined by Jz x,y = z,[x, y]
for x, y ∈ z⊥ ,
(2.1)
is an orthogonal map when z, z = 1. A connected, simply connected Lie group G is said to be of H type if its Lie algebra g is equipped with an inner product satisfying the above conditions. It is easy to see that an H-type Lie algebra (respectively, Lie group) is a step 2 stratified nilpotent Lie algebra (Lie group). The special case m = 1 produces the isotropic Heisenberg or Heisenberg–Weyl groups, and the case n = m = 1 gives the classical Heisenberg group H1 of dimension 3 discussed in [3].
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As usual, G can be identified as a set with g, taking the exponential map to be the identity. By fixing an orthonormal basis for g = z⊥ ⊕ z, we can identify G and g with Euclidean space equipped with an appropriate bracket, as the following proposition states. (The proof is uncomplicated.) Proposition 2.1. If G is an H-type Lie group identified with its Lie algebra g, then there exist integers n, m > 0, a bracket operation [·,·] on R2n+m = R2n × Rm , and a map T : G → R2n+m such that T : g → (R2n+m , [·,·]) is a Lie algebra isomorphism, T z = 0 × Rm , and T is an isometry with respect to the inner product ·,· on g and the usual Euclidean inner product on R2n+m . If we define a group operation on R2n+m as usual via v w = v + w + 12 [v, w], then T : G → (R2n+m , ) is a Lie group isomorphism, which maps the center of G to 0 × Rm . The identity of G is 0 and the group inverse is given by g −1 = −g. Henceforth we make this identification, and assume that our Lie group G is just R2n+m with an appropriate bracket [·,·] and corresponding group operation . We let {e1 , . . . , e2n } denote the , um } the standard orthonormal basis standard orthonormal basis for R2n × 0 ⊂ G, and {u1 , . . . for 0 × Rm ⊂ G, and write elements of G as g = (x, z) = i x i ei + j zj uj . The maps Jz can then be identified with skew-symmetric 2n × 2n matrices, which are orthogonal when |z| = 1. We remark a few obvious consequences of (2.1): Proposition 2.2. 1. 2. 3. 4.
Jz depends linearly on z; |Jz x| = |z||x|, and by polarization Jz x,Jw x = z,w|x|2 and Jz x,Jz y = |z|2 x,y; Jz x,x = 0, so Jz∗ = −Jz ; Jz2 = −|z|2 I .
We note that Lebesgue measure m on R2n+m = G is bi-invariant under the group operation, and thus m can be taken as the Haar measure on the locally compact group G. For i = 1, . . . , 2n, let Xi be the unique left-invariant vector field on G, and Xˆ i the unique right-invariant vector field, such that Xi (0) = Xˆ i (0) = ∂x∂ i . We can write Xi f (g) =
d f g (sei , 0) , ds s=0
d Xˆ i f (g) = f (sei , 0) g . ds s=0
(2.2)
A straightforward calculation shows ∂ 1 ∂ + Juj x,ei j , i ∂x 2 ∂z m
Xi =
j =1
∂ 1 ∂ Xˆ i = i − Juj x,ei j . ∂x 2 ∂z m
(2.3)
j =1
We note that [Xi , Xˆ j ] = 0 for all i, j . As a consequence of the H-type property, the collection {Xi (g), [Xi , Xj ](g): i, j = 1, . . . , 2n} ⊂ Tg G spans Tg G for each g ∈ G. Such a collection is said to be bracket-generating.
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The left-invariant subgradient ∇ on G is given by ∇f = (X1 f, . . . , X2n f ), with the rightinvariant ∇ˆ defined analogously. We shall also use the notation ∇x f := ( ∂x∂ 1 f, . . . , ∂x∂2n f ) and ∇z f := ( ∂z∂ 1 f, . . . , ∂z∂m f ) to denote the usual Euclidean gradients in the x and z variables, respectively. Note that ∇z is both left- and right-invariant. From (2.3) it is easy to verify that 1 ∇f (x, z) = ∇x f (x, z) + J∇z f (x,z) x, 2 1 ˆ (x, z) = ∇x f (x, z) − J∇z f (x,z) x. ∇f 2
(2.4)
In particular, since Jz depends linearly on z and is orthogonal for |z| = 1, we have (∇ − ∇)f ˆ (x, z) = |x|∇z f (x, z).
(2.5)
We shall make use of this fact later. The left-invariant sublaplacian L is the second-order differential operator defined by L = 2 ; L is subelliptic but not elliptic. By a renowned theorem due to Hörmander [7], X12 + · · · + X2n the bracket-generating condition implies that L is hypoelliptic, so that if Lf ∈ C ∞ then f ∈ C ∞ ; the same holds for the heat operator L − ∂t∂ . L is an essentially self-adjoint operator on L2 (m), and we let Pt := etL be the heat semigroup corresponding to L. Pt has a convolution kernel pt , so that Pt f (g) = f (g k)pt (k) dm(k). (2.6) G
By hypoellipticity, pt is a smooth function on G. An explicit formula for pt is known: −m
pt (x, z) = (2π)
−n
(4π)
Rm
e
iλ,z− 14 |λ| coth(t|λ|)|x|2
|λ| sinh(t|λ|)
n dλ.
(2.7)
See, among others, [15] for a derivation of (2.7). We note in particular that pt is a radial function; i.e. pt (x, z) is a function of |x|, |z|. This is unsurprising in light of the fact, easily verified, that L maps radial functions to radial functions. For α > 0, define the dilation ϕα : G → G by ϕα (x, z) = (αx, α 2 z); then ϕα is a group automorphism of G. A straightforward computation shows that Xi (f ◦ ϕα ) = α(Xi f ) ◦ ϕα , and Pt (f ◦ ϕα ) = (Pα 2 t f ) ◦ ϕα . We now make some definitions concerning the geometry of G. An absolutely continuous path γ : [0, 1] → G is said to be horizontal if there exist absolutely continuous ai : [0, 1] → R such that γ˙ (t) = 2n i=1 ai (t)Xi (γ (t)). In such a case the speed of γ is given by γ˙ (t) := 2 )1/2 . (This corresponds to taking a subriemannian metric on G such that {X } ( 2n a (t) i i i=1 are an orthonormal frame for the horizontal bundle; see [14] for an exposition of these ideas 1 from subriemannian geometry.) The length of γ is defined as [γ ] := 0 γ˙ (t) dt. The Carnot– Carathéodory distance between two points g, h ∈ G is
d(g, h) := inf [γ ]: γ horizontal, γ (0) = g, γ (1) = h .
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By the left-invariance of the vector fields Xi , it follows that d(g, h) = d(kg, kh). By Chow’s theorem, the bracket-generating condition implies that d(g, h) < ∞ for all g, h ∈ G. An explicit formula for d and for length-minimizing paths (geodesic) can be found in [6]. For the moment we note that d(0, (x, z)) |x| + |z|1/2 , where the symbol is defined as follows. Notation 2.3. If X is a set, and a, b : X → R are real-valued functions on X, we write a b to mean that there exist positive finite constants C1 , C2 such that C1 b(x) a(x) C2 b(x) for X
all x ∈ X. We will also write a b if the domain where the estimates hold is not obvious from context. We will make extensive use of the following precise pointwise estimates on the heat kernel pt , which were obtained in [6] by using the explicit formula (2.7): p1 (x, z)
1 + (d(0, (x, z)))2n−m−1 n− 12
1
e− 4 d(0,(x,z)) ,
1 + (|x|d(0, (x, z))) ∇p1 (x, z) C 1 + d 0, (x, z) p1 (x, z), ∇z p1 (x, z) Cp1 (x, z).
2
(2.8) (2.9) (2.10)
We can combine (2.9) and (2.10) using (2.5) to obtain ∇p ˆ 1 (x, z) C 1 + d 0, (x, z) p1 (x, z).
(2.11)
Let C be the class of f ∈ C 1 (G) for which there exist constants M 0, a 0, and ∈ (0, 1) such that f (g) + ∇f (g) + ∇f ˆ (g) Mead(0,g)2−
for all g ∈ G. By the heat kernel bounds (2.8), the convolution formula (2.6) makes sense for all f ∈ C, and thus we shall treat (2.6) as the definition of Pt f for f ∈ C. It is easy to see, by the translation invariance of the Haar measure m, that Pt remains left-invariant under this definition. The main theorem of this article is the following: Theorem 2.4. There exists a finite constant K such that for all f ∈ C, |∇Pt f | KPt |∇f | .
(2.12)
Following an argument found in [5], by left-invariance of Pt and ∇, we see that in order to establish (2.12) it suffices to show that it holds at the identity, i.e. to show (∇Pt f )(0) KPt |∇f | (0).
(2.13)
It also suffices to assume t = 1. This can be seen by taking t = 1 in (2.13) and replacing f by f ◦ ϕs 1/2 .
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Therefore, in order to prove Theorem 2.4, it will suffice to show |(∇P1 f )(0)| KP1 (|∇f |)(0). We may replace ∇ by ∇ˆ on the left side, since ∇ = ∇ˆ at 0. Since [Xi , Xˆ j ] = 0, we expect that ∇ˆ should commute with Pt , which we now verify. ˆ t f (0) = (Pt ∇f ˆ )(0). Proposition 2.5. For f ∈ C, ∇P Proof. By (2.2) and (2.6) we have d ˆ Xi Pt f (0) = Pt f (sei , 0) ds s=0 d = f (sei , 0) k pt (k) dm(k). ds s=0
G
We now differentiate under the integral sign, which can be justified because d f (sei , 0) k = d f (s + σ )ei , 0 k dσ ds σ =0 d = f (σ ei , 0) (sei , 0) k dσ σ =0 = Xˆ i f (sei , 0) k Mead(0,(sei ,0)k)
2−
.
But d 0, (sei , 0) k = d (sei , 0)−1 , k = d (−sei , 0), k d 0, (−sei , 0) + d(0, k) = |s| + d(0, k). Thus for all s ∈ [−1, 1] we have d f (sei , 0) k Mea(1+d(0,k))2− M ea d(0,k)2−
ds for some M , a , and therefore by the heat kernel bounds (2.8) we have d sup f (sei , 0) k pt (k) dm(k) < ∞ s∈[−1,1] ds G
which justifies differentiating under the integral sign. Thus d f (sei , 0) k pt (k) dm(k) Xˆ i Pt f (0) = ds s=0 G
= G
Xˆ i f (k)pt (k) dm(k)
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= Pt Xˆ i f (0). This completes the proof.
2
Thus Theorem 2.4 reduces to showing (P1 ∇f ˆ )(0) KP1 |∇f | (0)
(2.14)
(∇f ˆ )p1 dm K |∇f |p1 dm
(2.15)
or in other words
G
G
for which it suffices to show K |∇f |p1 dm. ˆ (∇ − ∇)f p dm 1 G
(2.16)
G
A similar argument can be used to verify the following integration by parts formula. Proposition 2.6. If f ∈ C, then
(∇f )p1 dm = −
G
(∇p1 )f dm, G
ˆ )p1 dm = − (∇f
G
ˆ 1 )f dm. (∇p
(2.17)
G
Proof. Tentatively, we have
(Xi f )p1 + f Xi p1 dm =
G
Xi (fp1 ) dm G
= G ?
=
d (fp1 ) g (sei , 0) dm(g) ds s=0
d (fp1 ) g (sei , 0) dm(g) ds s=0
d = ds
G
(fp1 )(g) dm(g) = 0 s=0
G
by right-invariance of Haar measure m. It remains to justify the differentiation under the integral sign in the third line. We note that
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
G
d sup (fp1 ) g (sei , 0) dm(g) = ds
s∈[−1,1]
G
sup Xi (fp1 ) g (sei , 0) dm(g) s∈[−1,1]
sup (Xi f )p1 g (sei , 0) dm(g)
G
511
s∈[−1,1]
+ G
sup (f Xi p1 ) g (sei , 0) dm(g). s∈[−1,1]
The first integral is easily seen to be finite by the definition of C and the heat kernel estimate (2.8), by similar logic to that in the proof of Proposition 2.5. The second integral is similar; we may bound |∇p1 | using the estimates (2.9) and (2.8). ˆ the same argument applies, using instead the leftTo show the second identity, involving ∇, ˆ 1 | using (2.11) and (2.8). 2 invariance of Haar measure. We can bound |∇p We now introduce an alternate coordinate system on G, similar but not exactly analogous to the so-called “polar coordinate” system used in [3]. As shown in [6], there is a unique (up to reparametrization) shortest horizontal path or geodesic from the identity 0 to each point (x, z) ∈ G with x, z nonzero; it has as its projection onto R2n × 0 an arc of a circle lying in the plane spanned by x and Jz x, with the origin as one endpoint, and x as the other. The region in this plane bounded by the arc and the straight line from 0 to x has area equal to |z|. The projection of the geodesic onto 0 × Rm is a straight line from 0 to z. Our new coordinate system will identify a point (x, z) with the point u ∈ R2n which is the center of the arc, and a vector η ∈ Rm which is parallel to z and whose magnitude equals the angle subtended by the arc. The change of coordinates (u, η) → (x, z) will be denoted by
Φ : (u, η) ∈ R2n+m : 0 < |η| < 2π → (x, z) ∈ G: x = 0, z = 0 ,
(2.18)
where
|u|2 sin|η| 1− η I − eJη u, 2 |η|
sin|η| sin|η| |u|2 = 1 − cos|η| u + Jη u, 1− η |η| 2 |η|
Φ(u, η) :=
(2.19) (2.20)
by Proposition 2.2, items 3 and 4. To visualize this, let us consider the special case of the Heisenberg group H1 , with n = m = 1. It is convenient to identify the subspace R2n × 0 with C; in this case, Jη = iη. (Note η ∈ R.) We then have
|u|2 iη (η − sin η) . Φ(u, η) = 1 − e u, 2
(2.21)
See Fig. 1 for an illustration of the relationship between (u, η) and Φ(u, η) in H1 . Note that the (u, η) coordinate system omits the set {z = 0} = R2n × 0 ⊂ G, for which the arc degenerates into a straight line and has “infinite radius,” as well as the set {x = 0} = 0 × Rm ,
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Fig. 1. Illustration of the change of coordinates Φ in the classical Heisenberg group H1 . The bold line is a geodesic, whose projection into the x-plane is an arc of a circle with center u and subtending an angle η. The z coordinate of Φ(u, η) is equal to the area of the shaded circular segment.
for which the arc becomes a circle whose center u is no longer uniquely determined. These sets are of Haar measure zero and hence will be neglected in the argument without further comment. Estimates which are shown to hold off these sets will also hold on them, by continuity. Φ has the property that for each (u, η), the path s → Φ(u, sη) traces the shortest horizontal path between any two of its points, and has constant speed |u||η|. In particular, d 0, Φ(u, η) = |u||η|.
(2.22)
d f Φ(u, sη) |u||η|∇f Φ(u, sη) . ds
(2.23)
Also, for any f ∈ C 1 (G),
Note that if (x, z) = Φ(u, η), we have |x|2 = |u|2 2 − 2 cos|η| ,
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
|z| =
513
|u|2 |η| − sin|η| . 2
To compare this with the “polar coordinates” (u, s) used in [3], take u = u and s = |u|η. Let B := {g ∈ G: d(0, g) < 1} denote the unit ball of the Carnot–Carathéodory distance. To express B in (u, η) coordinates, we note by (2.22) that Φ(u, η) ∈ B iff |u||η| 1; combining this with the constraints on u and η given in (2.18), we have 1 B = Φ(u, η): u ∈ R2n , |η| < 2π ∧ |u|
(2.24)
1 1 B C = Φ(u, η): |u| , |η| < 2π 2π |u|
(2.25)
and conversely
modulo the null sets {x = 0} and {z = 0}, as usual. In (u, η) coordinates, the heat kernel estimate (2.8) reads p1 Φ(u, η)
1 1 + (|u||η|)2n−m−1 2 e− 4 (|u||η|) √ 1 1 + (|u|2 |η| 2 − 2 cos|η|)n− 2
1 + (|u||η|)2n−m−1 1 + (|u|2 |η|2 (2π
1
n− 12
− |η|))
e− 4 (|u||η|)
2
(2.26)
(2.27)
since 1 − cos θ θ 2 (2π − θ )2 for θ ∈ [0, 2π]. We will often abuse notation and write p1 (u, η) for p1 (Φ(u, η)), when no confusion will result. 3. Proof of the gradient estimate We now begin the proof of Theorem 2.4, which occupies the rest of this article. We begin by computing the Jacobian determinant of the change of coordinates Φ, so that we can use (u, η) coordinates in explicit computations. Lemma 3.1. Let A(u, η) denote the Jacobian determinant of Φ, so that dm = A(u, η) du dη. Then
A(u, η) = |u|
2m
1 sin|η| − 2 2|η|
m−1
n−1 2 − 2 cos|η| 2 − 2 cos|η| − |η| sin|η| .
(3.1)
Note that A(u, η) depends on u, η only through their absolute values |u|, |η|. By an abuse of notation we may occasionally use A with u or η replaced by scalars, so that A(r, ρ) means A(r u, ˆ ρ η) ˆ for arbitrary unit vectors u, ˆ η. ˆ For the Heisenberg group H1 with n = m = 1, this reduces to A(u, η) = |u|2 2 − 2 cos|η| − |η| sin|η| .
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The analogous expression appearing in [3] is slightly incorrect. However, it does have the same asymptotics as the correct expression (see Corollary 3.2), which is sufficient for the rest of the argument in [3], so that its overall correctness is not affected. ∼ R2n+m as follows. Let uˆ be a Proof. Fix u, η. Form an orthonormal basis for T(u,η) Φ −1 (G) = unit vector in the direction of (u, 0), vˆ a unit vector in the direction of (Jη u, 0). For i = 1, . . . , ˆ v, ˆ wˆ j , yˆj , 1 j < i, n − 1 let wˆ i , yˆi ∈ R2n × 0 be unit vectors such that wˆ i is orthogonal to u, ˆ v, ˆ wˆ j , yˆj , 1 j < i as well and let yˆi be in the direction of Jη wˆ i so that yˆi is orthogonal to u, as to wˆ i . (To see this, note that if x,y = 0 and x,Jz y = 0, then Jz x,y = 0 and Jz x,Jz y = −|z|2 x,y = 0.) Let ηˆ be a unit vector in the direction of (0, η), and let ζˆk , k = 1, . . . , m − 1 ˆ Then {u, ˆ v, ˆ wˆ i , yˆi , η, ˆ ζˆk } form an be orthonormal vectors in 0 × Rm which are orthogonal to η. ˆ Jη vˆ = −|η|u, ˆ Jη wˆ i = |η|yˆi , Jη yˆi = −|η|wˆ i . orthonormal basis for R2n+m . Note Jη uˆ = |η|v, Then ˆ ∂uˆ Φ(u, η) = 1 − cos|η| uˆ + sin|η|vˆ + |u| |η| − sin|η| η, ∂vˆ Φ(u, η) = 1 − cos|η| vˆ − sin|η|u, ˆ ∂wˆ i Φ(u, η) = 1 − cos|η| wˆ i + sin|η|yˆi , ∂yˆi Φ(u, η) = 1 − cos|η| yˆi − sin|η|wˆ i , |u|2 1 − cos|η| η, ˆ ∂ηˆ Φ(u, η) = |u| sin|η| uˆ + |u| cos|η| vˆ + 2
sin|η| sin |η| |u|2 ∂ζˆk Φ(u, η) = Jζˆk u + 1− ζˆk . |η| 2 |η| In this basis, the Jacobian matrix has the form ⎛ 1 − cos|η| − sin|η| 0 sin|η| 1 − cos|η| 0 ⎜ ⎜ J =⎜ 0 0 B ⎝ |u|(|η| − sin|η|) 0 0 0 0 0 where
|u| sin|η| |u| cos|η| 0 |u|2 (1 − cos|η|) 2 0
⎞ 0 0⎟ ⎟ , ∗⎟ ⎠ 0 D (2n+m)×(2n+m) ⎞
⎛
1 − cos|η| − sin|η| 1 − cos|η| ⎜ sin|η| ⎜ B := ⎜ ⎜ ⎝
(3.2)
..
. 1 − cos|η| sin|η|
is a block-diagonal matrix of 2 × 2 blocks, and ⎛ |u|2 sin|η| 2 (1 − |η| ) ⎜ .. D := ⎝ .
− sin|η| 1 − cos|η|
⎟ ⎟ ⎟ ⎟ ⎠
(3.3)
2(n−1)×2(n−1)
⎞ ⎟ ⎠ |u|2 2
(1 −
sin|η| |η| )
is diagonal. Note |B| = (2 − 2 cos|η|)n−1 and |D| = ( |u|2 (1 − 2
(m−1)×(m−1) sin|η| m−1 . |η| ))
(3.4)
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515
So factoring out |D| and expanding about the ηˆ row, we have 0 |u| sin|η| − sin|η| |J | = |D| |u| |η| − sin|η| 1 − cos|η| 0 |u| cos|η| 0 B 0 0 1 − cos|η| − sin|η| |u|2 + 1 − cos|η| sin|η| 1 − cos|η| 0 2 0 0 B
2
n−1 |u| sin|η| m−1 = |u| |η| − sin|η| −|u| sin|η| 2 − 2 cos|η| 1− 2 |η| 2 n |u| 1 − cos|η| 2 − 2 cos|η| + 2
2
n−1 sin|η| m−1 2 |u| 1− |η| − sin|η| − sin|η| = |u| 2 − 2 cos|η| 2 |η| 2 + 1 − cos|η|
n−1 1 sin|η| m−1 − = |u|2m 2 − 2 cos|η| 2 − 2 cos|η| − |η| sin|η| . 2 2 2|η|
Corollary 3.2. 2n−1 A(u, η) |u|2m |η|2(m+n) 2π − |η| .
(3.5)
Proof. The asymptotic equivalence near |η| = 0 and |η| = 2π follows from a routine Taylor series computation. It then suffices to show that A(u, η) > 0 for all 0 < |η| < 2π . We have 12 − sin|η| 2|η| > 0 for all |η| > 0, since x > sin x for all x > 0. We also have 2 − 2 cos|η| > 0 for all 0 < |η| < 2π . Finally, to show f (|η|) := 2 − 2 cos|η| − |η| sin|η| > 0, let θ = 12 |η|. Using double-angle identities, we have f (2θ ) = 4 sin θ (sin θ − θ cos θ ). For 0 < θ < π we have sin θ > 0 so it suffices to show g(θ ) := sin θ − θ cos θ > 0. But we have g(0) = 0 and g (θ ) = θ sin θ > 0 for 0 < θ < π. 2 The heat kernel estimates will be used to prove a technical lemma regarding integrating the heat kernel along a geodesic. The proof requires the following simple fact from calculus, of which a close relative appears in [6]. Lemma 3.3. For any q ∈ R, a0 > 0 there exists a constant C = Cq,a0 such that for any a a0 we have t=∞
t q e−(at) dt C 2
t=1
Proof. Make the change of variables s = t 2 to get
1 −a 2 e . a2
(3.6)
516
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533 t=∞
q −(at)2
t e
s=∞
1 dt = 2
t=1
where q =
q−1 2 .
s q e−a s ds, 2
s=1
For q 0 (i.e. q 1), we have s q 1 and thus s=∞
q −a 2 s
s e
s=∞
e−a s ds =
ds
s=1
2
1 −a 2 e . a2
s=1
For q > 0, notice that integration by parts gives s=∞
q −a 2 s
s e
1 q 2 ds = 2 e−a + 2 a a
s=1
s=∞
2
2
s q −1 e−a s ds
s=1
1 q 2 2 e−a + 2 a a0 whereupon the result follows by induction.
s=∞
s q −1 e−a s ds,
s=1
2
Lemma 3.4. For each q ∈ R there exists a constant Cq such that for all u, η with Φ(u, η) ∈ B C , i.e. |u||η| 1, we have t= 2π |η|
p1 (u, tη)A(u, tη)t q dt
Cq p1 (u, η)A(u, η) (|u||η|)2
(3.7)
t=1
Cq p1 (u, η)A(u, η).
(3.8)
Note that (3.8) follows immediately from the stronger statement (3.7), since by assumption |u||η| 1. In fact, we shall only use (3.8) in the sequel. Proof. Assume throughout that |u||η| 1 and 0 < |η| < 2π . (See (2.25).) The proof involves the fact that a geodesic passes through (up to) three regions of G in which the estimates for p1 and A simplify in different ways. We define these regions, which partition B C , as follows. See Fig. 2. 1. Region R1 is the set of Φ(u, η) such that 0 < |η| π . (This corresponds to having |x|2 |z|.) In this region we have |u| π1 and π 2π − |η| < 2π . Therefore (2.27) becomes −m − 1 (|u||η|)2 R1 e 4 p1 (u, η) |u||η| and Corollary 3.2 yields
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
517
Fig. 2. The regions R1 , R2 , R3 , seen in the |u|–|η| plane. The dark lines indicate examples of the geodesic paths of integration used in (3.7). R1
A(u, η) |u|2m |η|2(n+m) so that R1
1
p1 (u, η)A(u, η) |u|m |η|2n+m e− 4 (|u||η|) =: F1 (u, η). 2. Region R2 is the set of Φ(u, η) such that π < |η| 2π − |x|2
|z| and
|x|2 |z| 1.)
In this region, we have
2
1 . |u|2
|u|2 |η|2 (2π
(This corresponds to having
− |η|) π 2 , so that
−n+ 1 − 1 (|u||η|)2 R2 2e 4 p1 (u, η) |u|−m 2π − |η| , 2n−1 R2 , A(u, η) |u|2m 2π − |η| n− 1 − 1 (|u||η|)2 R2 2e 4 =: F2 (u, η) p1 (u, η)A(u, η) |u|m 2π − |η| 1 R2 n− 2 − 1 (|u||η|)2 |u|m |η|2n+m 2π − |η| e 4 =: F˜2 (u, η). R2 R2 We shall use the estimates F2 , F˜2 at different times. Although F2 F˜2 (since |η| 1), they are not equivalent on R1 . 3. Region R3 is the set of Φ(u, η) such that |η| > max(π, 2π − |u|1 2 ). (This corresponds to
having |x|2 |z| and |x|2 |z| 1.) In this region, we have |u|2 |η|2 (2π − |η|) < (2π)2 , so that R3
1
p1 (u, η) |u|2n−m−1 e− 4 (|u||η|) , 2n−1 R3 A(u, η) |u|2m 2π − |η| , 2n−1 − 1 (|u||η|)2 R3 p1 (u, η)A(u, η) |u|2n+m−1 2π − |η| e 4 =: F3 (u, η). 2
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We observe that a geodesic starting from the origin (given by t → Φ(u, tη) for some fixed (u, η)) passes through these regions in order, except that it skips Region 2 if |u| < π −1/2 . We now estimate the desired integral along a portion of a geodesic lying in a single region. Claim 3.5. Let q ∈ R. Suppose that F : G → R is given by γ 1 2 F (u, η) = |u|α |η|β 2π − |η| e− 4 (|u||η|) R
for some nonnegative powers α, β, γ , and that there is some region R ⊂ G such that F p1 A. Then there is a constant C depending on q, F , R such that for all u, η, τ0 , τ1 , τ2 satisfying • |u||η| 1; • 1 τ0 τ1 τ2 2π |η| ; and • Φ(u, tη) ∈ R for all t ∈ [τ1 , τ2 ] we have t=τ 2
q−1
τ0 F (u, τ0 η). (|u||η|)2
p1 (u, tη)A(u, tη)t q dt C
(3.9)
t=τ1
Proof. We have t=τ 2
t=τ 2
p1 (u, tη)A(u, tη)t dt C q
t=τ1
F (u, tη)t q dt
t=τ1 t=τ 2
C
F (u, tη)t q dt
t=τ0 t=τ 2
γ 1 2 t q+β 2π − t|η| e− 4 (t|u||η|) dt
= C|u| |η| α
β t=τ0
γ C|u| |η| 2π − τ0 |η| α
t=τ 2
1
t q+β e− 4 (t|u||η|) dt
β
t=τ0
since t τ0 . We now make the change of variables t = t τ0 : =∞ t
γ q+β+1 C|u|α |η|β 2π − τ0 |η| τ0
1
t q+β e− 4 (t τ0 |u||η|) dt 2
t =1
γ q+β+1 C |u|α |η|β 2π − τ0 |η| τ0
1 1 2 e− 4 (τ0 |u||η|) 2 (τ0 |u||η|)
2
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
= C
519
q−1
τ0 F (u, τ0 η), (|u||η|)2
where in the second-to-last line we applied Lemma 3.3 with a = 12 τ0 |u||η|, a0 = 12 . (Note that |u||η| 1 and τ0 1 by assumption, so a a0 .) Claim 3.5 is proved. 2 Now for fixed u, η, let
π , t2 := max 1, |η|
1 1 t3 := max t2 , 2π − 2 |η| |u| so that Φ(u, tη) ∈ R1
for 1 < t t2 ,
Φ(u, tη) ∈ R2
for t2 < t < t3 , 2π . for t3 t < |η|
Φ(u, tη) ∈ R3
We divide the remainder of the proof into cases, depending on the region where Φ(u, η) resides. Case 1. Suppose that Φ(u, η) ∈ R1 . We have t= 2π |η|
t= 2π
t=t3 t=t2 |η| q p1 (u, tη)A(u, tη)t dt = + + .
t=1
t=1
t=t2
t=t3
For the first integral, where Φ(u, tη) ∈ R1 , we have by Claim 3.5 (taking τ0 = τ1 = 1, τ2 = t2 , R = R1 , F = F1 ) that t=t2 p1 (u, tη)A(u, tη)t q dt
C C F (u, η) p1 (u, η)A(u, η) 1 (|u||η|)2 (|u||η|)2
t=1 R1
since F1 p1 A. For the second integral, where Φ(u, tη) ∈ R2 , we take τ0 = 1, τ1 = t2 , τ2 = t3 , R = R2 , F = F˜2 in Claim 3.5 to obtain t=t3 p1 (u, tη)A(u, tη)t q dt t=t2
However, for Φ(u, η) ∈ R1 we have
C F˜2 (u, η). (|u||η|)2
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N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
n− 1 1 F˜2 (u, η) 2 (2π)n− 2 . = 2π − |η| F1 (u, η) Thus t=t3 p1 (u, tη)A(u, tη)t q dt
C F1 (u, η) (|u||η|)2
t=t2
C p1 (u, η)A(u, η). (|u||η|)2
The third integral is more subtle. We apply Claim 3.5 with τ0 = τ1 = t3 , τ3 = F = F3 : t= 2π |η|
2π |η| ,
R = R3 ,
q−1
p1 (u, tη)A(u, tη)t q dt C
t3 F3 (u, t3 η). (|u||η|)2
t=t3
Then q−1
t3
2n−1 − 1 (|u||η|)2 (t 2 −1) F3 (u, t3 η) q−1 3 = t3 |u|2n−1 |η|−2n−m 2π − t3 |η| e 4 . F1 (u, η)
(3.10)
We must show that this ratio is bounded. Fix some > 0. If |u| (π − )−1/2 > π −1/2 , we have 1 2π − |u|1 2 > π + and thus t3 = |η| (2π − |u|1 2 ). Then
1 2 |η|2 t32 − 1 = 2π − 2 − |η|2 |u| (π + )2 − π 2 = 2π + 2 . So in this case (3.10) becomes q−1
t3
q−1
1 2n−1 − 1 (|u||η|)2 (t 2 −1) 1 1 2n−1 −2n−m 3 2π − 2 |u| |η| e 4 |η| |u| |u|2
1 q−1 −2n+1 −2n−m−q+1 − 1 (|u||η|)2 (t 2 −1) 3 = 2π − 2 |u| |η| e 4 |u|
F3 (u, t3 η) = F1 (u, η)
1
(2π)q−1 |u|m+q e− 4 (2π +
2 )|u|2
1 since |η| |u| . This is certainly bounded by some constant. On the other hand, if |u| −1/2 1/2 , so that the right side of (3.10) is (π − ) , then |η| (π − )1/2 and 1 t3 ( π+
π− ) clearly bounded.
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
521
Thus we have t= 2π |η|
p1 (u, tη)A(u, tη)t q dt
C F1 (u, η) (|u||η|)2
t=t3
C p1 (u, η)A(u, η). (|u||η|)2
This completes the proof of this case. Case 2. Suppose that Φ(u, η) ∈ R2 . We have t= 2π |η|
t= 2π
t=t3 |η| q p1 (u, tη)A(u, tη)t dt = + .
t=1
t=1
t=t3
Note that in this region we have 1 t3 2. Again by Claim 3.5, with τ0 = τ1 = 1 and τ2 = t3 , we have t=t3 p1 (u, tη)A(u, tη)t q dt
C C F (u, η) p1 (u, η)A(u, η). 2 (|u||η|)2 (|u||η|)2
t=1
For the second integral, we apply Claim 3.5 with τ0 = 1, τ1 = t3 , τ2 = t= 2π |η|
p1 (u, tη)A(u, tη)t q dt
2π |η|
to get
C F3 (u, η). (|u||η|)2
t=t3
But |η| 2π −
1 |u|2
on R3 , so we have n− 1 F3 (u, η) 2 = |u|2n−1 2π − |η| F2 (u, η)
1 1 n− 2 |u|2n−1 = 1. |u|2
Thus t= 2π |η|
p1 (u, tη)A(u, tη)t q dt
C F2 (u, η) (|u||η|)2
t=t3
C p1 (u, η)A(u, η). (|u||η|)2
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Case 3. Suppose Φ(u, η) ∈ R3 ; we apply Claim 3.5 with τ0 = τ1 = 1, τ2 = t= 2π |η|
p1 (u, tη)A(u, tη)t q dt C
2π |η|
to get
1 F3 (u, η) C p1 (u, η)A(u, η). (|u||η|)2
t=1
2
The three cases together complete the proof of Lemma 3.4. Notation 3.6. For f ∈ C 1 (G), let mf :=
f dm B , B dm
where B is the Carnot–Carathéodory unit ball.
To continue to follow the line of [3], we need the following Poincaré inequality. This theorem can be found in [8], and is a special case of a more general theorem appearing in [12]. Theorem 3.7. There exists a constant C such that for any f ∈ C ∞ (G),
|f − mf | dm C
B
|∇f | dm.
(3.11)
B
Corollary 3.8. There exists a constant C such that for any f ∈ C ∞ (G),
|f − mf |p1 dm C
B
|∇f |p1 dm.
(3.12)
B
Proof. p1 is bounded and bounded below away from 0 on B.
2
Lemma 3.9. (See Akin to Lemma 5.2 of [3].) There exists a constant C such that for all f ∈ C,
|f − mf |p1 dm C
BC
|∇f |p1 dm.
(3.13)
G
Proof. Changing to (u, η) coordinates, we wish to show
f Φ(u, η) − mf p1 Φ(u, η) A(u, η) dη du C
|∇f |p1 dm. G
1 1 |u| 2π |u| |η|<2π
(3.14) (The limits of integration are as described in (2.25).) By an abuse of notation we shall write f (u, η) for f (Φ(u, η)), p1 (u, η) for p1 (Φ(u, η)), ∇f (u, η) for (∇f )(Φ(u, η)), et cetera. 1 η Let g(u, η) := f (u, min(|η|, |u| ) |η| ). Then g = f on B (in particular mg = mf ), g is bounded,
the function s → g(u, sη) is absolutely continuous, and
d ds g(u, sη) = 0
for s >
1 |u||η| .
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
Now |f − mf | |f − g| + |g − mf |. We first observe that for |u||η| 1 we have s=1
d d f (u, η) − g(u, η) = f (u, sη) − g(u, sη) ds ds ds 1 s= |u||η|
s=1
∇f (u, sη)|u||η| ds
1 s= |u||η|
by (2.23). Thus
f (u, η) − g(u, η)p1 (u, η)A(u, η) dη du,
|f − g|p1 dm = BC
1 1 |u| 2π |η| |u|
where the limits of integration come from the conditions |u||η| 1, |η| < 2π ;
s=1
∇f (u, sη)|u||η|p1 (u, η)A(u, η) ds dη du
1 1 1 |u| 2π |η| |u| s= |u||η|
s=1
= 1 s=0 |u| 2π
∇f (u, sη)|u||η|p1 (u, η)A(u, η) dη ds du
1 s|u| |η|2π
by Tonelli’s theorem. We now make the change of variables η = sη to obtain s=1
∇f (u, η )|u| 1 |η |p1 u, 1 η A u, 1 η 1 dη ds du s s s sm
= 1 s=0 |u| 2π
1 |u| |η |2πs
= 1 |u| 2π
∇f (u, η )|u||η |
1 |u| |η |2π
s=1 ×
1 1 1 p1 u, η A u, η ds dη du. s s s m+1
| s= |η 2π
Make the further change of variables t =
1 s
to get
523
524
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533 2π
t= |η | m−1 ∇f (u, η )|u||η | p1 (u, tη )A(u, tη )t dt dη du.
1 |u| 2π
1 |u| |η |2π
=
t=1
Applying Lemma 3.4 to the bracketed term gives
1 |u| 2π
1 |u| |η |2π
C
C
1 ∇f (u, η )p1 (u, η )A(u, η ) dη du |u||η |
|∇f |p1 dm
BC converting back from geodesic coordinates and using the fact that |u||η | 1. To complete the proof, we must show that B C |g − mf |p1 dm G |∇f |p1 dm. Note that for 1 Φ(u, η) ∈ B C , i.e. |u||η| 1, we have g(u, η) = f (u, |u||η| η), so
1 |u| 2π
1 |u| |η|2π
|g − mf |p1 dm = BC
f u, 1 η − mf p1 (u, η)A(u, η) dη du. |u||η| (3.15)
Change the η integral to polar coordinates by writing η = ρ η, ˆ where ρ 0 and |η| ˆ = 1. Note that p1 (u, η), A(u, η) depend on η only through ρ and not η. ˆ
1 |u| 2π
η∈S ˆ m−1
=C
ρ=2π f u, 1 ηˆ − mf p1 (u, ρ)A(u, ρ)ρ m−1 dρ d ηˆ du. |u|
(3.16)
1 ρ= |u|
Now, for any s ∈ [0, 1] we have
f u, 1 ηˆ − mf f u, 1 ηˆ − f u, s ηˆ + f u, s ηˆ − mf . |u| |u| |u| |u|
(3.17)
Let s=1 D(u) := s=0
By multiplying both sides of (3.17) by
s m−1 s ds. A u, |u|m |u|
1 s m−1 s D(u) |u|m A(u, |u| )
and integrating we obtain
(3.18)
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
525
s=1
f u, 1 ηˆ − f u, s ηˆ + f u, s ηˆ − mf f u, 1 ηˆ − mf 1 |u| D(u) |u| |u| |u| s=0
s × A u, ds. |u|m |u| s m−1
(3.19)
Let 1 R(u) := D(u)
ρ=2π
p1 (u, ρ)A(u, ρ)ρ m−1 dρ.
(3.20)
1 ρ= |u|
Then substituting (3.19) into (3.16) and using (3.20) we have |g − mf |p1 dm I1 + I2 ,
(3.21)
BC
where
m−1
s=1
s f u, 1 ηˆ − f u, s ηˆ s ds R(u) d ηˆ du, A u, |u| |u| |u|m |u|
I1 :=
1 η∈S ˆ m−1 s=0 |u| 2π
(3.22)
1 |u| 2π
η∈S ˆ m−1
m−1
s=1
s f u, s ηˆ − mf s |u|m A u, |u| ds R(u) d ηˆ du. |u|
I2 :=
(3.23)
s=0
We now show that I1 , I2 can each be bounded by a constant times following claim. Claim 3.10. There exists a constant C such that for all |u|
1 2π
G |∇f |p1 dm,
we have
1 2n−1 R(u) C 2π − (2π)2n−1 C. |u| Proof. First, by Corollary 3.2 we have s=1 D(u) := s=0
s m−1 s ds A u, |u|m |u|
s=1 C s=0
using the
s 2n−1 s m−1 2m s 2(m+n) 2π − |u| ds |u|m |u| |u|
(3.24)
526
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
−2n−m
= C|u|
s=1 s 2n−1 3m+2n−1 s ds 2π − |u|
s=0 −2n−m
C|u|
s=1 2n−1 s 3m+2n−1 2π(1 − s) ds
since u
1 2π
s=0
−2n−m
= C |u|
since the s integral is a positive constant independent of u. t shows On the other hand, making the change of variables ρ = |u| ρ=2π
p1 (u, ρ)A(u, ρ)ρ 1 ρ= |u|
m−1
−m
t=2π|u|
dρ = |u|
p1 t=1
t t u, A u, t m−1 dt |u| |u|
1 1 A u, C|u|−m p1 u, |u| |u|
1 1 in Lemma 3.4. Now p1 (u, |u| ) is the heat kernel evaluated at a point on the by taking |η| = |u| unit sphere of G, so this is bounded by a constant independent of u. Thus by Corollary 3.2 we have ρ=2π
p1 (u, ρ)A(u, ρ)ρ m−1 dρ C|u|−m |u|2m
1 ρ= |u|
1 |u|
2(m+n)
1 2n−1 2π − |u|
1 2n−1 −2n−m C 2π − |u| . |u|
Combining this with the estimate on D(u) proves the claim.
2
To estimate I1 (see (3.22)), we observe that
t=1
1 s t d f u, ηˆ − f u, ηˆ = f u, ηˆ dt |u| |u| dt |u| t=s
t=1 t=s
d f u, t ηˆ dt dt |u|
t=1
∇f u, t ηˆ dt |u| t=s
by (2.23). Thus
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
m−1
s=1 t=1
s ∇f u, t ηˆ s A u, dt ds R(u) d ηˆ du |u| |u|m |u|
I1
527
(3.25)
1 η∈S ˆ m−1 s=0 t=s |u| 2π
t=1
s=t 1 t s m−1 ∇f u, ds dt d ηˆ du. s A u, R(u) ηˆ |u| |u|m |u|
=
1 η∈S ˆ m−1 t=0 |u| 2π
s=0
(3.26) Now by Claim 3.10 and Corollary 3.2, we have for all t ∈ [0, 1]: s=t R(u)
s
m−1
s=0
s ds A u, |u|
2(m+n)
s=t s s 2n−1 1 2n−1 m−1 2m 2π − C 2π − s |u| ds |u| |u| |u| s=0
s=t t 2n−1 2n−1 −2n (2π) |u| s 3m+2n−1 ds C 2π − |u|
t |u|
= C 2π −
s=0
|u|−2n t 3m+2n
t 2n−1 2m t 2(m+n) m = C 2π − |u| t |u| |u|
t tm C A u, |u|
t t m−1 . C A u, |u|
2n−1
Thus
m−1 t=1
∇f u, t ηˆ A u, t t dt d ηˆ du. |u| |u| |u|m
1 |u| 2π
η∈S ˆ m−1
t=0
Make the change of variables r =
t |u| :
I1 C
1 r= |u|
=C 1 η∈S ˆ m−1 r=0 |u| 2π
∇f (u, r η) ˆ A(u, r)r m−1 dr d ηˆ du
(3.27)
(3.28)
528
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
1 r= |u|
C
∇f (u, r η) ˆ A(u, r)r m−1 dr d ηˆ du
(3.29)
u∈R2n η∈S ˆ m−1 r=0
=C
|∇f | dm B
C infB p1
C
(3.30)
|∇f |p1 dm
(3.31)
B
|∇f |p1 dm,
(3.32)
G
where we have used the fact that p1 is bounded away from 0 on B. For I2 (see (3.23)), we have by Claim 3.10 that
1 |u| 2π
η∈S ˆ m−1
I2 C
m−1
s=1
s f u, s ηˆ − mf s |u|m A u, |u| ds d ηˆ du. |u|
s=0
Make the change of variables r =
(3.33)
s |u| : 1 r= |u|
=C
f (u, r η) ˆ − mf r m−1 A(u, r) dr d ηˆ du
(3.34)
1 η∈S ˆ m−1 r=0 |u| 2π
1 r= |u|
C
f (u, r η) ˆ − mf r m−1 A(u, r) dr d ηˆ du
(3.35)
u∈R2n η∈S ˆ m−1 r=0
=C
|f − mf | dm
(3.36)
|∇f | dm
(3.37)
B
C B
by Theorem 3.7. The inequalities (3.30)–(3.32) now show that I2 C sired. 2
G |∇f |p1 dm,
as de-
Corollary 3.11. There exists a constant C such that for all f ∈ C,
|f − mf |p1 dm C
G
|∇f |p1 dm. G
(3.38)
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
529
2
Proof. Add (3.12) and (3.13).
We can now prove some cases of the desired gradient inequality (2.16). Notation 3.12. Let D(R) = {(x, z): |x| R} denote the “cylinder about the z axis” of radius R. Lemma 3.13. For fixed R > 0, (2.16) holds, with a constant C = C(R) depending on R, for all f ∈ C which are supported on D(R) and satisfy mf = 0. Proof. ˆ ˆ (∇ − ∇)f p1 dm = f (∇ − ∇)p1 dm by integration by parts (2.17) G
G
ˆ 1 dm |f |(∇ − ∇)p
G
=
|f ||x||∇z p1 | dm by (2.5) G
|f |p1 dm by (2.10); note |x| R on the support of f
CR G
|∇f |p1 dm by Corollary 3.11.
C R
2
G
Notation 3.14. If T : G → M2n×2n is a matrix-valued function on G, with k th entry ak , let ∇ · T : G → R2n be defined as ∇ · T (g) :=
2n
X ak (g)ek .
(3.39)
k, =1
Note that for f : G → R we have the product formula ∇ · (f T ) = T ∇f + f ∇ · T .
(3.40)
Lemma 3.15. For fixed R > 1, (2.16) holds, with a constant C = C(R) depending on R, for all f ∈ C which are supported on the complement of D(R). Proof. Applying (2.4) we have 1 ∇p1 (x, z) = ∇x p1 (x, z) + J∇z p1 (x,z) x. 2 Now p1 is a “radial” function (that is, p1 (x, z) depends only on |x| and |z|). Thus we have that ∇x p1 (x, z) is a scalar multiple of x, and also that ∇z p1 (x, z) is a scalar multiple of z, so that J∇z p1 (x,z) x is a scalar multiple of Jz x.
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N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
For nonzero x ∈ R2n , let T (x) ∈ M2n×2n be orthogonal projection onto the m-dimensional subspace of R2n spanned by the orthogonal vectors Ju1 x, . . . , Jum x. (Recall Jui x,Juj x = −ui ,uj x 2 = −δij x 2 .) Thus for any z ∈ Rm , T (x)Jz x = Jz x, and T (x)x = 0; in particular, 1 1 ˆ 1 (x, z). T (x)∇p1 (x, z) = J∇z p1 (x,z) x = (∇ − ∇)p 2 2
(3.41)
Explicitly, we have T (x) =
m 1 Juj x(Juj x)T . |x|2 j =1
Note that |T (x)| = 1 (in operator norm) for all x = 0, and a routine computation verifies that C |∇ · T (x)| = |∇x · T (x)| |x| . Indeed, the k th entry of T (x) is ak (x) =
m 1 Juj x,ek Juj x,e |x|2 j =1
2
3m(2n) so that |Xk ak (x)| = | ∂x∂ k ak (x)| 3m |x| ; thus |∇ · T (x)| |x| . Since p1 decays rapidly at infinity, we have the integration by parts formula 0 = ∇ · (fp1 T ) dm = (fp1 ∇ · T + f T ∇p1 + p1 T ∇f ) dm. G
(3.42)
G
Thus ˆ ˆ (∇ − ∇)f p1 dm = f (∇ − ∇)p1 dm G
G
= 2 f T ∇p1 dm G
= 2 fp1 (∇ · T + T ∇f ) dm G
2
|f ||∇ · T |p1 dm + 2
G
2C R
|f |p1 dm + 2 G
|T ||∇f |p1 dm G
|∇f |p1 dm G
C C since on the support of f , we have |∇ · T | |x| R , and |T | = 1. The second integral is the desired right side of (2.16). The first integral is bounded by the same by Corollary 3.11, where we note that mf = 0 because f vanishes on D(R) ⊃ B. 2
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
531
We can now complete the proof of Theorem 2.4. Proof of Theorem 2.4. We prove (2.16) for general f ∈ C. By replacing f by f − mf ∈ C, we can assume mf = 0. Let ψ ∈ C ∞ (G) be a smooth function such that ψ ≡ 1 on D(1) and ψ is supported in D(2). Then f = ψf + (1 − ψ)f . ψf is supported on D(2), so Lemma 3.13 applies to ψf . (Note that mψf = 0 since ψ ≡ 1 on D(1) ⊃ B.) We have (∇ − ∇)(ψf C ∇(ψf )p1 dm ˆ )p dm 1 G
G
C
|∇ψ||f |p1 dm +
G
G
G
C sup|∇ψ|
|ψ||∇f |p1 dm
|f |p1 dm + C sup|ψ| G
G
|∇f |p1 dm. G
The second integral is the right side of (2.16), and the first is bounded by the same by Corollary 3.11. Precisely the same argument applies to (1 − ψ)f , which is supported on the complement of D(1), by using Lemma 3.15 instead of Lemma 3.13. 2 4. The optimal constant K We observed previously that the constant K in (2.12) can be taken to be independent of t. We now show that the optimal constant is also independent of t > 0, and is discontinuous at t = 0. This distinguishes the current situation from the elliptic case, in which the constant is continuous at t = 0; see, for instance [2, Proposition 2.3]. This fact was initially noted for the Heisenberg group H1 in [5], and the proof here is similar to the one found there. Proposition 4.1. For t 0, let |(∇Pt f )(g)| Kopt (t) := sup : f ∈ C, g ∈ G, Pt |∇f | (g) = 0 . Pt (|∇f |)(g)
(4.1)
Then Kopt (0) = 1, and for all t > 0, Kopt (t) ≡ Kopt > 1 is independent of t, so that Kopt (t) is discontinuous at t = 0. In particular, Kopt
3n+5 3n+1 .
Proof. It is obvious that Kopt (0) = 1. As before, by the left-invariance of Pt and ∇, it suffices to take g = 0 on the right side of (4.1). ˜ To show independence of t > 0, fix t, s > 0. If f ∈ C, then f˜ := f ◦ ϕs−1 1/2 ∈ C and f = f ◦ ϕs 1/2 . Then
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N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
|(∇Pt f )(0)| |(∇Pt (f˜ ◦ ϕs 1/2 ))(0)| = Pt (|∇f |)(0) Pt (|∇(f˜ ◦ ϕs 1/2 )|)(0) =
|(∇(Pst f˜) ◦ ϕs 1/2 )(0)| Pt (s 1/2 |∇ f˜| ◦ ϕs 1/2 )(0)
=
s 1/2 |(∇Pst f˜)(ϕs 1/2 (0))| Kopt (st). s 1/2 Pst (|∇ f˜|)(ϕs 1/2 (0))
Taking the supremum over f shows that Kopt (t) Kopt (st). s was arbitrary, so Kopt (t) is constant for t > 0. In order to bound the constant, we explicitly compute a related ratio for a particular choice of function f . The function used is an obvious generalization of the example used in [5] for the Heisenberg group H1 . Fix a unit vector u1 in the center of G, i.e. u1 ∈ 0 × Rm ⊂ R2n+m . We note that the operator L and the norm of the gradient |∇f |2 = 12 (L(f 2 ) − 2f Lf ) are independent of the orthonormal basis {ei } chosen to define the vector fields {Xi }, so without loss of generality we suppose that Ju1 e1 = e2 . Then take f (x, z) := x,e1 + z,u1 x,e2 = x 1 + z1 x 2 , k(t) :=
|(∇Pt f )(0)| . Pt (|∇f |)(0)
Note that k(t) Kopt for all t. By the Cauchy–Schwarz inequality, k(t)2 k2 (t) :=
|(∇Pt f )(0)|2 . Pt (|∇f |2 )(0) 2
Since f is a polynomial, we can compute Pt f by the formula Pt f = f + 1!t Lf + t2! L2 f + · · · since the sum terminates after a finite number of terms (specifically, two). The same is true of |∇f |2 , which is also a polynomial (three terms are needed). The formulas (2.3) are helpful in carrying out this tedious but straightforward computation. We find k2 (t) =
(1 + t)2 1 − 2t + (3n + 2)t 2
2 which, by differentiation, is maximized at tmax = 3n+3 , with k2 (tmax ) = √ 3n+5 k(tmax ) k2 (tmax ) = 3n+1 , this is the desired bound. 2
3n+5 3n+1 .
Since Kopt
5. Consequences and possible extensions Section 6 of [3] gives several important consequences of the gradient inequality (1.1). The proofs given there are generic (see their Remark 6.6); with Theorem 2.4 in hand, they go through without change in the case of H-type groups. These consequences include: • Local Gross–Poincaré inequalities, or ϕ-Sobolev inequalities;
N. Eldredge / Journal of Functional Analysis 258 (2010) 504–533
533
• Cheeger type inequalities; and • Bobkov type isoperimetric inequalities. We refer the reader to [3] for the statements and proofs of these theorems, and many references as well. It would be very useful to extend the gradient inequality (1.1) to a more general class of groups, such as the nilpotent Lie groups. However, this is likely to require a proof which is divorced from the heat kernel estimates (2.8)–(2.10). Such precise estimates are currently not known to hold in more general settings, and could be difficult to obtain. A key difficulty is the lack of a convenient explicit heat kernel formula like (2.7). Acknowledgments The author would like to express his sincere thanks to his advisor, Bruce Driver, for a great many helpful discussions during the preparation of this article. The author would also like to thank the anonymous referee for several helpful corrections and comments, notably the suggestion to include Fig. 1. This research was supported in part by NSF Grants DMS-0504608 and DMS-0804472, as well as an NSF Graduate Research Fellowship. References [1] D. Bakry, Transformations de Riesz pour les semi-groupes symétriques, II, Étude sous la condition Γ2 0, in: Séminaire de probabilités, XIX, 1983/1984, in: Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 145– 174. [2] D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, in: New Trends in Stochastic Analysis, Charingworth, 1994, World Sci. Publ., River Edge, NJ, 1997, pp. 43–75. [3] Dominique Bakry, Fabrice Baudoin, Michel Bonnefont, Djalil Chafaï, On gradient bounds for the heat kernel on the Heisenberg group, J. Funct. Anal. 255 (8) (2008) 1905–1938. [4] A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their sub-Laplacians, Springer Monogr. Math., Springer, Berlin, 2007. [5] Bruce K. Driver, Tai Melcher, Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal. 221 (2) (2005) 340–365. [6] Nathaniel Eldredge, Precise estimates for the subelliptic heat kernel on H-type groups, J. Math. Pures Appl. 92 (2009) 52–85. [7] Lars. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967) 147–171. [8] David Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53 (2) (1986) 503–523. [9] Aroldo Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1) (1980) 147–153. [10] Hong-Quan Li, Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg, J. Funct. Anal. 236 (2) (2006) 369–394. [11] Hong-Quan Li, Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg, C. R. Math. Acad. Sci. Paris 344 (8) (2007) 497–502. [12] P. Maheux, L. Saloff-Coste, Analyse sur les boules d’un opérateur sous-elliptique, Math. Ann. 303 (4) (1995) 713– 740. [13] Tai A. Melcher, Hypoelliptic heat kernel inequalities on Lie groups, PhD thesis, University of California, San Diego, 2004. [14] Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math. Surveys Monogr., vol. 91, American Mathematical Society, Providence, RI, 2002. [15] Jennifer Randall, The heat kernel for generalized Heisenberg groups, J. Geom. Anal. 6 (2) (1996) 287–316.
Journal of Functional Analysis 258 (2010) 534–558 www.elsevier.com/locate/jfa
Quasi-potentials of the entropy functionals for scalar conservation laws Giovanni Bellettini a,b,∗ , Federica Caselli c , Mauro Mariani d a Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientifica, 00133 Roma, Italy b Laboratori Nazionali di Frascati, INFN, Via E. Fermi 40, 00044 Frascati (Roma), Italy c Dipartimento di Ingegneria Civile, Università di Roma ‘Tor Vergata’, Via del Politecnico 1, 00100 Roma, Italy d CEREMADE, UMR-CNRS 7534, Université de Paris-Dauphine, Place du Marechal de Lattre de Tassigny,
F-75775 Paris Cedex 16, France Received 13 April 2009; accepted 6 July 2009 Available online 22 July 2009 Communicated by C. Villani
Abstract We investigate the quasi-potential problem for the entropy cost functionals of non-entropic solutions to scalar conservation laws with smooth fluxes. We prove that the quasi-potentials coincide with the integral of a suitable Einstein entropy. © 2009 Elsevier Inc. All rights reserved. Keywords: Quasi-potential; Conservation laws; Entropy
* Corresponding author at: Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientifica, 00133 Roma, Italy. E-mail addresses:
[email protected] (G. Bellettini),
[email protected] (F. Caselli),
[email protected] (M. Mariani).
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.07.003
G. Bellettini et al. / Journal of Functional Analysis 258 (2010) 534–558
535
1. Introduction For a real function f , consider the scalar conservation law in the unknown u ≡ u(t, x) ut + f (u)x = 0
(1.1)
where t ∈ [0, T ] for some T > 0, x ∈ T (the one-dimensional torus), and subscripts denote partial derivatives. Eq. (1.1) does not admit in general classical solutions for the associated Cauchy problem, even if the initial datum is smooth. On the other hand, if f is non-linear, there exist in general infinitely many weak solutions. An admissibility condition, the so-called entropic condition, is then required to recover uniqueness for the Cauchy problem in the weak sense [6]. The unique solution satisfying such a condition is called the Kruzkhov solution. A classical result [6, Chapter 6.3] states that the Kruzkhov solution can be obtained as limit for ε ↓ 0 of the solution uε to the Cauchy problem associated with the equation ut + f (u)x =
ε D(u)ux x 2
(1.2)
provided that the initial data also converge. Here the diffusion coefficient D is a uniformly positive smooth function, and we remark that convergence takes place in the strong Lp ([0, T ] × T) topology. The Kruzkhov solution to (1.1) has also been proved to be the hydrodynamical limit of the empirical density of stochastic particles systems under hyperbolic scaling, when the number of particles diverges to infinity [11, Chapter 8]. These results legitimize the Kruzkhov solution as the physically relevant solution to (1.1), and the entropic condition as the appropriate selection rule between the infinitely many weak solutions to (1.1). Provided the flux f and the diffusion coefficient D are chosen appropriately (depending on the particles system considered), one may say that (1.2) is a continuous version for the evolution of the empirical density of particles system, in which the small stochastic effects are neglected (or averaged) and ε is the inverse number of particles. The convergence of both (1.2) and the empirical measure of the density of particles to the same solution of (1.1) confirms somehow that this approximation is reliable. In [10,15], the long standing problem of providing a large deviations principle for the empirical measure of the density of stochastic particles systems under hyperbolic scaling is addressed. In particular, the totally asymmetric simple exclusion process is investigated (which in particular corresponds to f (u) = u(1 − u) in the hydrodynamical limit equation (1.1)), and the large deviations result partially established. Roughly speaking, when the number of particles N diverges to infinity, the asymptotic probability of finding the density of particles in a small neighborhood of JV a path u : [0, T ] × T → R is e−N H (u) , where H JV is a suitable large deviations rate functional (see Section 2). A continuous mesoscopic mean field counterpart of this large deviations result is provided in [2,14]. In [14] a large deviations principle for a stochastic perturbation to (1.2) (driven by a fluctuation coefficient σ ) is investigated in the limit of jointly vanishing stochastic noise and (deterministic) diffusion. In [2] a purely variational problem is addressed, namely the investigation of the Γ -limit of a family of functionals Hε associated with (1.2) (see Section 2). The candidate large deviations functional H introduced in [14] and the candidate Γ -limit introduced in [2] coincide, and in the case f (u) = u(1 − u) they are expected to coincide with the functional H JV introduced in [10,15] (the equality can be proved on functions of bounded variations, but
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it is missing in the general case). The functional H thus provides a generalization of the functional H JV , for arbitrary fluxes f (in particular, not necessarily convex or concave), diffusion coefficients D and fluctuation coefficients σ . The functionals Hε , H and H JV are nonnegative; Hε vanishes only on solutions to (1.2), so that Hε can be interpreted as the cost of violating the flow (1.2). On the other hand, H and H JV are +∞ off the set of weak solutions to (1.1), they vanish only on Kruzkhov solution to (1.1), and thus they can be interpreted as the cost of violating the entropic condition for the flux (1.1). Section 2 of the paper is devoted to the precise definition of the functionals Hε , H and H JV . Redirecting the reader to Section 3 for a more detailed discussion, here we briefly recall a general definition of the quasi-potential associated with a family of functionals. Suppose we are given a topological space U , and for each T > 0 a set XT ⊂ C([0, T ]; U ) and a functional IT : XT → [0, +∞]. For the sake of simplicity, let us also fix a point m ∈ U . The quasi-potential V : U → [0, +∞] associated with {IT } is then defined as V (u) := infT >0 infw IT (w), where the infimum is carried over all the w ∈ XT such that w(0) ≡ m and w(T ) = u. A natural choice for the reference point m should be an attractive point for the minima of the functionals IT (see e.g. Theorems 4.4 and 5.5 for the case of Hε , H and H JV ). Indeed, in such a case, the investigation of the quasi-potential is a classical subject both in dynamical optimal control theory and in large deviations theory, as it quantifies “the cheapest cost” to move from the stable point m to a general one u. Moreover, from the optimal control theory point of view, the quasi-potential describes the long time limit of the functionals IT , see e.g. [5]. Furthermore, there is a broad family of stochastic processes for which the quasi-potential is expected to be the large deviations rate functional of their invariant measures, provided IT is the large deviations rate functional of the laws of such processes up to time T > 0 (see e.g. [9, Theorem 4.4.1] for the classical finitedimensional case, and [4] for a more general discussion and applications to particles systems). Moreover, see [9, Chapter 4], the quasi-potential of the large deviations rate functionals provides a valuable tool to investigate long time behavior of the processes (e.g. average time to be waited for the process to leave an attractive point, and the path to follow when the process performs such a deviation). Finally, in the context of non-equilibrium statistical mechanics in which the functionals Hε , H and H JV have been introduced, the quasi-potential has been proposed as a dynamical definition of the free energy functional for systems out of equilibrium [3]. Since Hε is a functional associated with a control problem (see [2]) and it can be also retrieved as large deviations rate functional of some particles systems (e.g. weakly asymmetric particle systems, see [11]) and stochastic PDEs (see [13]), the quasi-potential problem is relevant for such a functional. Similarly, H and H JV are the (candidate) large deviations rate functionals for both particles systems processes and stochastic PDEs, see [10,15,14]. Given a bounded measurable map ui : T → R, it is well known that the (entropic) solutions to the Cauchy problems for (1.1) and (1.2) with initial datum ui will converge to the constant m = T dx ui (x), namely constant profiles are attractive points for the zeros of the functionals Hε , H and H JV . Given m ∈ R and two positive smooth maps on T, interpreted as the diffusion coefficient D and fluctuation coefficient σ , the Einstein entropy is defined as the unique nonnegative function hm on R such that hm (m) = 0 and σ h m = D. In this paper, we establish an explicit formula for the quasi-potential problem associated with the functionals Hε , H and H JV (which of course will depend on a time parameter T ) with reference point the constant maps on the torus, proving that these three quasi-potential functionals coincide and are equal to the integral of the Einstein entropy. More precisely, given uf ∈ L∞(T), the quasi-potential V (m, uf ) of Hε , H and H JV with reference constant m ∈ R is equal to T dx hm (uf (x)) if T dx uf (x) = m and it is +∞ otherwise (see Theorem 3.1).
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As remarked above, both the large deviations results in [10,15,14] and the Γ -limit results in [2] are incomplete, due to little knowledge of structure theorems and regularity results for weak solutions to conservation laws (1.1). These results on the quasi-potential give therefore an additional heuristic argument supporting the actual identification of H as the Γ -limit of Hε . Similarly, since it is easily seen that the large deviations rate functional (in the hydrodynamical limit) of the invariant measures of the totally asymmetric simple exclusion process is also given by the integral of the Einstein entropy, these results also support the conjecture that H and H JV may coincide at least in the case f (u) = u(1 − u) and that they are in fact the large deviations rate functional of the totally asymmetric simple exclusion process. Finally, we remark that the integral of the Einstein entropy is expected to rule the long time behavior of the well-behaving physical systems, and the result provided in this paper thus also supports the universality of the Jensen and Varadhan functional H JV (or in general of H ) as a relevant universal entropy functional for asymmetric conservative, closed physical systems. In Section 2 we recall some preliminary notions. In Section 3 the main definitions and results are stated, while Sections 4 and 5 are devoted to the proofs. The techniques used to prove the results vary from calculus of variations (Remark 3.2 and Corollary 5.4), large deviations theory (Lemmas 4.2, 4.3 and 5.1) and conservation laws (Lemma 5.8). 2. Preliminaries Our analysis will be restricted to equibounded “densities” u, and for the sake of simplicity we let u take values in [−1, 1]. Let U denote the compact Polish space of measurable functions u : T → [−1, 1], equipped with the H −1 (T) metric dU := sup u, ϕ , ϕ ∈ Cc∞ (T), ϕx , ϕx + ϕ, ϕ = 1 where ·,· denotes the scalar product in L2 (T). Given T > 0, let XT be the Polish space C([0, T ]; U ) endowed with the metric dXT (u, v) := sup dU u(t), v(t) + u − v L1 ([0,T ]×T) . t∈[0,T ]
Hereafter we assume f a Lipschitz function on [−1, 1]. Moreover we let D, σ ∈ C([−1, 1]) with D uniformly positive, and σ strictly positive in (−1, 1). 2.1. The functional Hε For ε > 0, T > 0, we define Hε;T : XT → [0, +∞] as (hereafter we may drop the explicit dependence on integration variables inside integrals when no misunderstanding is possible) ⎧ T ε ⎪ ⎨ supϕ∈Cc∞ ((0,T )×T) ε −1 [− 0 dt u, ϕt + f (u), ϕx − 2 D(u)ux , ϕx T Hε;T (u) := (2.1) − 12 0 dt σ (u)ϕx , ϕx ] if ux ∈ L2 ([0, T ] × T), ⎪ ⎩ +∞ otherwise. Note that Hε;T (u) = 0 iff u ∈ XT is a weak solution to (1.2). Hε;T is a lower-semicontinuous and coercive functional on XT (see [2, Proposition 3.3, Theorem 2.5]). Moreover if Hε;T (u) < +∞ then u ∈ C([0, T ]; L1 (T)) (see [2, Lemma 3.2]).
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2.2. Entropy-measure solutions We say that u ∈ XT is a weak solution to (1.1) iff for each ϕ ∈ Cc∞ ((0, T ) × R) T
dt u, ϕt + f (u), ϕx = 0.
0
A function η ∈ C 2 ([−1, 1]) is called an entropy, and its conjugated entropy flux q ∈ C([−1, 1]) w is defined up to an additive constant by q(w) := dv η (v)f (v). For u ∈ XT a weak solution to (1.1), for (η, q) an entropy – entropy flux pair, the η-entropy production is the distribution ℘η,u acting on Cc∞ ((0, T ) × R) as T ℘η,u (ϕ) := −
dt η(u), ϕt + q(u), ϕx .
0
Let Cc2,∞ ([−1, 1] × (0, T ) × T) be the set of compactly supported maps ϑ : [−1, 1] × (0, T ) × R (v, t, x) → ϑ(v, t, x) ∈ R, that are C 2 in the v variable, with derivatives continuous up to the boundary of [−1, 1] × (0, T ) × T, and C ∞ in the (t, x) variables. For ϑ ∈ Cc2,∞ ([−1, 1] × (0, T ) × T) let ϑ and ϑ denote its partial derivatives with respect to the v variable. We say that a function ϑ ∈ Cc2,∞ ([−1, 1] × (0, T ) × T) is an entropy sampler, and its conjugated entropy flux sampler : [−1, 1] × (0, T ) × T is defined up to an additive function of (t, x) by w Q Q(w, t, x) := ϑ (v, t, x)f (v) dv. Finally, given a weak solution u to (1.1), the ϑ -sampled entropy production Pϑ,u is the real number Pϑ,u := − dt dx (∂t ϑ) u(t, x), t, x + (∂x Q) u(t, x), t, x . (2.2) (0,T )×T
If ϑ(v, t, x) = η(v)ϕ(t, x) for some entropy η and some ϕ ∈ Cc∞ ((0, T ) × T), then Pϑ,u = ℘η,u (ϕ). The next proposition introduces a suitable class of solutions to (1.1) which will be needed in the sequel. We denote by MT the set of finite measures on (0, T ) × T that we consider equipped with the weak∗ topology. In the following, for ∈ MT we denote by ± the positive and negative part of . Proposition 2.1. (See [2, Proposition 2.3], [7].) Let u ∈ XT be a weak solution to (1.1). The following statements are equivalent: (i) for each entropy η, the η-entropy production ℘η,u can be extended to a Radon measure on (0, T ) × T, namely ℘η,u TV := sup{℘η,u (ϕ), ϕ ∈ Cc∞ ((0, T ) × T), |ϕ| 1} < +∞; (ii) there exists a bounded measurable map u : [−1, 1] v → u (v; dt, dx) ∈ MT such that for any entropy sampler ϑ dv u (v; dt, dx) ϑ (v, t, x). (2.3) Pϑ,u = [−1,1]×(0,T )×T
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We say that a weak solution u ∈ XT is an entropy-measure solution to (1.1) iff it satisfies the equivalent conditions of Proposition 2.1. The set of entropy-measure solutions to (1.1) is denoted by ET ⊂ XT . In general ET BV([0, T ] × T), the main regularity result for ET being ET ⊂ C([0, T ]; L1 (T)), provided f ∈ C 2 ([−1, 1]) is such that there is no interval in which f is affine (see [2, Lemma 5.1]). A Kruzkhov solution to (1.1) is a weak solution u ∈ C([0, T ]; L1 (T)) such that ℘η,u 0 in distributional sense, for each convex entropy η. Since a weak solution u such that ℘η,u 0 can be shown to be an entropy-measure solution, the entropic condition is equivalent to u (v; dt, dx) 0 for a.e. v ∈ [−1, 1]. 2.3. Γ -entropy cost of non-entropic solution For T > 0, we introduce the functional HT : XT → [0, +∞] as HT (u) :=
+ dv D(v) σ (v) u (v; dt, dx) if u ∈ ET , +∞ otherwise.
(2.4)
In [2, Proposition 2.6] it is proved that HT is coercive and lower semicontinuous, and that it vanishes only on Kruzkhov solutions to (1.1). As noted in [2, Remark 2.7], if u ∈ XT ∩ BV([0, T ] × T) is a weak solution to (1.1), then u ∈ ET . Let Ju be the jump set of u ∈ ET ∩ BV([0, T ] × T), H1 Ju the one-dimensional Hausdorff measure restricted to Ju and, at a point (s, y) ∈ Ju , let n = (nt , nx ) ≡ n(s, y) be the normal to Ju and u− ≡ u− (s, y) (respectively u+ ≡ u+ (s, y)) be the left (respectively right) trace of u (these are well defined H1 Ju a.e., since nx can be chosen uniformly positive, see [2, Remark 2.7]). Then HT (u) = Ju
dH1 nx
dv
D(v) ρ + (v, u+ , u− ) σ (v) |u+ − u− |
(2.5)
where ρ v, u+ , u− := f u− u+ − v + f u+ v − u− − f (v) u+ − u− 1[u− ∧u+ , u− ∨u+ ] (v)
(2.6)
and ρ + denotes the positive part of ρ. In [2] a suitable set ST ⊂ ET of entropy-splittable solutions to (1.1) is also introduced, and the next result is proved. Theorem 2.2. (See [2, Theorem 2.5].) For each T > 0, the following statements hold. (i) The sequence of functionals {Hε;T } satisfies the Γ -liminf inequality Γ -limε Hε;T HT on XT . (ii) Assume that there is no interval where f is affine. Then the sequence of functionals {Hε;T } is equicoercive on XT .
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(iii) Assume furthermore that f ∈ C 2 ([−1, 1]), and D, σ ∈ C α ([−1, 1]) for some α > 1/2. Define H T (u) := inf lim HT (un ), {un } ⊂ ST : un → u in XT . n
Then the sequence of functionals {Hε;T } satisfies the Γ -limsup inequality Γ -limε Hε;T H T on XT . Note that Γ -limsup inequality is not complete, as it is not known that H T = HT . 2.4. The Jensen and Varadhan functional Suppose that σ is such that there exists h ∈ C 2 ([−1, 1]) such that σ h = D, and let g be such that g = h f . For T > 0, we also introduce the Jensen and Varadhan functional HTJV : XT → [0, +∞] as ⎧ ⎨ supϕ∈C ∞ ([0,T ]×T;[0,1]) { T dx [h(u(T , x))ϕ(T , x) − h(u(0, x))ϕ(0, x)] T (2.7) HTJV (u) := ⎩ − 0 dt h(u), ϕt + g(u), ϕx } if u is a weak solution to (1.1), +∞ otherwise. Note that the definition of HTJV does not depend on the choice of h, provided it satisfies σ h = D. This functional has been introduced in [10] (in the case D ≡ 1 and f (u) = σ (u) = u(1 − u)). In [2] it is proved that HTJV HT , that HTJV (u) = HT (u) if f is convex or concave and u has bounded variation, and that HTJV < HT if f is neither convex or concave. 3. Quasi-potentials We want to study the quasi-potentials Vε , V , V JV : [−1, 1] × U → [0, +∞] associated respectively with Hε;T , HT and HTJV , and defined as Vε (m, uf ) := inf Hε;T (u), T > 0, u ∈ XT : u(0) ≡ m, u(T ) = uf , V (m, uf ) := inf HT (u), T > 0, u ∈ XT : u(0) ≡ m, u(T ) = uf , V JV (m, uf ) := inf HTJV (u), T > 0, u ∈ XT : u(0) ≡ m, u(T ) = uf .
(3.1) (3.2) (3.3)
JV Note that, if uf ≡ m, then Vε (m, uf ) = V (m, uf ) = V (m, uf ) = 0. On the other hand, whenever m = +1 or m = −1, if uf ≡ m, then dx uf (x) = m and thus Vε (m, uf ) = V (m, uf ) = JV since Hε;T (u) = HT (u) = HTJV (u) = +∞ whenever u ∈ XT is such that V (m, uf ) = +∞, T dx u(s, x) = T dx u(t, x) for some s, t ∈ [0, T ]. Therefore, in the following we focus on the case m ∈ (−1, 1). Our main result is the following. For m ∈ (−1, 1) define the Einstein entropy hm ∈ C([−1, 1]; [0, +∞]) ∩ C 2 ((−1, 1)) as the unique function such that σ (v)h m (v) = D(v) for v ∈ (−1, 1), hm (m) = 0, h m (m) = 0, and let
Wm (uf ) := T
dx hm uf (x) ∈ [0, +∞].
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Note that, if formula
541
T dx uf (x) = m, Wm (uf ) can also be written by the more explicit but less evocative
u f (x)
D(w) . dw uf (x) − w σ (w)
dx T
m
Theorem 3.1. (i) Assume lim α
2
α↓0
1 1 + = 0. σ (−1 + α) σ (1 − α)
(3.4)
Then
Vε (m, uf ) = Wm (uf ) +∞
if T dx uf = m, otherwise
for any ε > 0, for any m ∈ (−1, 1) and uf ∈ U . (ii) Assume f ∈ C 2 ([−1, 1]) is such that there is no interval in which f is affine. Assume also that for some δ0 > 0 the set {v ∈ [−1, 1]: f (v) = 0} ∩ [m − δ0 , m + δ0 ] is finite. Then V (m, uf ) = Wm (uf ) +∞
if T dx uf = m, otherwise
for any m ∈ (−1, 1) and uf ∈ U . (iii) Assume the same hypotheses of (ii) and furthermore that there exists h ∈ C 2 ([−1, 1]) such that σ h = D. Then V JV (m, uf ) = Wm (uf ) if T dx uf = m, +∞ otherwise for any m ∈ (−1, 1) and uf ∈ U . Note that (3.4) is verified if σ does not vanish, or vanishes slower than quadratically in −1 and +1. Observe that Hε;T has a quadratic structure (see (4.1)), so that the proof of Theorem 3.1(i) is an infinite-dimensional version of Freidlin–Wentzell theorem [9, Theorem 4.3.1]. However this is not the case for HT . In particular, since HT (u) = +∞ if u is not an (entropy-measure) solution to (1.1), the main technical difficulty in the proof of Theorem 3.1(ii) is to show that one can find a solution u to (1.1) such that u connects in finite time a profile v ∈ U close in L∞ (T) to the constant profile m, to m itself. We remark that the quasi-potential problem for HT is at this time being addressed in [1] in the case of Dirichlet boundary conditions. While this setting is quite similar to ours, the difficulties are completely different. In the boundary driven case, the entropic evolution connects a non-constant profile to a constant in finite time, so for T large it is not difficult to solve the minimization problem (3.2) far from the boundaries. On the other hand, new challenging difficulties appear in (3.2) when dealing with weak solutions to (1.1) featuring
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discontinuities at the boundary (boundary layers). Of course, this problem does not appear at all on a torus. Remark 3.2. Let T1 , T2 > 0, and let u1 ∈ XT1 , u2 ∈ XT2 . Define the measurable function u : [0, T1 + T2 ] × T → [−1, 1] by u(t, x) = u1 (t, x) if t ∈ [0, T1 ], and u(t, x) = u2 (t − T1 , x) if t ∈ (T1 , T1 + T2 ]. Then u ∈ XT1 +T2 iff u1 (T1 ) = u2 (0) and in such a case Hε;T1 +T2 (u) = Hε;T1 (u1 ) + Hε;T2 (u2 ). Furthermore if the hypotheses of Theorem 3.1(ii) hold, then HT1 +T2 (u) = HT1 (u1 ) + HT2 (u2 ). Proof. A change of variables in the definition (2.1) shows that Hε;T1 (u1 ) + Hε;T2 (u2 ) can be still written in the form (2.1) with T = T1 + T2 , where now the supremum is carried over all the test functions ϕ ∈ Cc∞ ((0, T1 ) ∪ (T1 , T1 + T2 ) × T). However, if u1 (T1 ) = u2 (0), the supremum carried over such test functions coincides with the supremum carried over Cc∞ ((0, T1 + T2 ) × T). Namely, Hε;T1 (u1 ) + Hε;T2 (u2 ) equals the definition of Hε;T1 +T2 (u). By (2.4) it follows that HT1 +T2 (u) = +∞ whenever HT1 (u1 ) = +∞ or HT2 (u2 ) = +∞. Assume instead HT1 (u1 ), HT2 (u2 ) < ∞. Under the assumptions of Theorem 3.1(ii), the boundedness of HT implies strong continuity in L1 (T) as remarked below Proposition 2.1. Therefore if u1 (T1 ) = u2 (0) then u ∈ C([0, T1 + T2 ]; L1 (T)) and u ∈ ET1 +T2 . By (2.2), (2.3) and the L1 (T) continuity of u1 , u2 and u, it follows that u (v; {T1 } × T) = u1 (v; {T1 } × T) =
u2 (v; {0} × T) = 0 for a.e. v ∈ [−1, 1]. Thus u (v; dt, dx) = u1 (v; dt, dx) in [0, T1 ] × T and u (v; dt, dx) = u2 (v; d(t − T1 ), dx) in [T1 , T1 + T2 ] × T, and the equality follows from (2.4). 2 Since Hε;T (m) = HT (m) = 0, by Remark 3.2, the infima in (3.1), (3.2) are attained in the limit T → +∞. 4. Proof of Theorem 3.1 for Vε 1 Given a bounded measurable function a 0 on [0, T ] × T let Da;T be the Hilbert space ∞ obtained by identifying and completing the functions ϕ ∈ C ([0, T ] × T) with respect to the T −1 seminorm [ 0 dt ϕx , aϕx ]1/2 . Let Da;T be its dual space. The corresponding norms are denoted respectively by · D1 and · D−1 . a;T
a;T
Remark 4.1. Let a 0 be a bounded measurable function on [0, T ] × T. Let F, G ∈ −1 L2 ([0, T ] × T) be such that Fx , (aG)x ∈ Da;T . Assume that T dx G(t, x) = 0 for a.e. t ∈ [0, T ]. Then Fx , (aG)x D−1 =
T dt F, G
a;T
0 −1 where (·,·)D−1 denotes the scalar product in Da;T . a;T
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By a standard application of the Riesz representation theorem (see [2, Lemma 3.1]), we have that if Hε;T (u) < +∞ then Hε;T (u) =
2 ε −1 ut + f (u)x − ε D(u)ux x −1 2 2 D
(4.1)
.
σ (u);T
If T dx uf (x) = m, then Theorem 3.1(i) follows from the conservation of the total mass of functions u ∈ XT with Hε;T (u) < +∞. On the other hand, if T dx uf (x) = m, the proof of the theorem is a consequence of the following lemmas. In fact from Lemma 4.2 we get Vε (m, uf ) Wm (uf ), and from Lemmas 4.2 and 4.3 we have Vε (m, uf ) Wm (uf ) + γ for each γ > 0. Lemma 4.2. Assume (3.4), let T > 0 and u ∈ XT be such that Hε;T (u) < +∞, u(0, x) ≡ m, u(T , x) = uf (x). Then T dx hm (uf (x)) < +∞, ut + f (u)x , (D(u)ux )x ∈ Dσ−1(u);T and Hε;T (u) = T
2 ε −1 ε ut + f (u)x + D(u)ux x dx hm uf (x) + −1 2 2 D
.
σ (u);T
Lemma 4.3. For each γ > 0, there exist T > 0 and u ∈ XT such that Hε;T (u) < +∞, u(0) ≡ m, u(T ) = uf and 2 ε −1 ut + f (u)x + ε D(u)ux x −1 2 2 D
γ.
σ (u);T
Proof of Lemma 4.2. We first assume that there exists δ > 0 such that for a.e. (t, x) ∈ [0, T ] × T, −1 + δ u(t, x) 1 − δ, so that σ (u) is uniformly positive. It follows that (D(u)ux )x , f (u)x ∈ Dσ−1(u);T so that, since Hε;T (u) < +∞, by (4.1) we also have ut ∈ Dσ−1(u);T . In particular there exists θ ∈ L2 ([0, T ] × T) such that ut = θx weakly. Therefore ε −1 ut + f (u)x − Hε;T (u) = 2 ε −1 ut + f (u)x + = 2 ε −1 ut + f (u)x + = 2 T −
2 ε D(u)ux x −1 2 Dσ (u);T 2 ε D(u)ux x − θx + f (u)x , D(u)ux x D−1 σ (u);T 2 Dσ−1(u);T 2 ε D(u)ux x −1 2 D σ (u);T
D(u) D(u) ux + f (u), ux dt θ, σ (u) σ (u)
0
where in the last line we used Remark 4.1, as for each t ∈ [0, T ] D(u(t, x)) ux (t, x) = dx h m u(t, x) x = 0. dx σ (u(t, x)) T
T
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Similarly we have f (u(t)), D(u(t)) σ (u(t)) ux (t) = 0 and integrating by parts: T −
T T
D(u) ux = dt θx , hm (u) = dt ut , h m (u) dt θ, σ (u)
0
0
=
0
dx hm u(T , x) − hm u(0, x) .
T
Lemma 4.2 is therefore established for each u ∈ XT bounded away from −1 and +1. For a general u ∈ XT such that u(0, ·) ≡ m ∈ (−1, 1), and δ > 0, let us define uδ (t, x) = (1 − δ)u(t, x) + δm. Provided (3.4) holds, the sequence {uδ } ⊂ XT converges to u as δ → 0, and is such that: for δ > 0, uδ is bounded away from −1 and +1; uδ (0, ·) ≡ m, T dx h(uδ (T , x)) → T dx h(u(T , x)); Hε;T (uδ ) → Hε;T (u); δ 2 u + f uδ + ε D uδ uδ x x t x 2 D −1
σ (uδ );T
2 ε u D(u)u → + f (u) + x x x t 2 D −1
.
σ (u);T
Therefore, since Lemma 4.2 holds for uδ for each δ > 0, it also holds for u.
2
The following result is well known [8]. Theorem 4.4. Let uf ∈ U and let v : [0, ∞) × T → R be the solution to (1.2) with initial datum uf . Then limt→∞ v(t) − m L∞ ([0,T ]×T) = 0 where m = T dx uf (x). Proof of Lemma 4.3. Let v : [0, ∞) × T → R be the solution to (1.2) with initial datum v(0, x) = uf (−x), and for T1 , T2 > 0 let u ∈ XT1 +T2 be defined as u(t, x) =
(1 − Tt1 )m + Tt1 v(T2 , −x) for t ∈ [0, T1 ], for t ∈ [T1 , T1 + T2 ]. v(T1 + T2 − t, −x)
Since u satisfies ut + f (u)x + 2ε (D(u)ux )x = 0 for t ∈ [T1 , T1 + T2 ], we have by Remark 3.2 2 ε −1 ut + f (u)x + ε D(u)ux x −1 2 2 D
σ (u);T1 +T2
2 ε −1 ut + f (u)x + ε D(u)ux = x −1 2 2 D
3ε −1 2
ut
σ (u);T1
2
Dσ−1(u);T
1
2 + f (u)x
Dσ−1(u);T
1
ε 2 + D(u)ux x −1 2 D
σ (u);T1
.
(4.2)
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Let now δ > 0 (to be chosen below) be small enough to have −1 < m − δ < m + δ < 1, and define 1 < +∞, v∈[m−δ,m+δ] σ (v) f (u(t, x)) cf (t) := dx σ u(t, x) , dx σ (u(t, x)) Cσ,δ :=
max
T
Cf,δ :=
T
max
v∈[m−δ,m+δ]
f (v) −
min
v∈[m−δ,m+δ]
f (v),
D(v)2 . v∈[−1,1] 2
CD := max
Let also θ ∈ L2 ([0, T1 ] × T) be defined by θx (t, x) = T
v(T2 , −x) − m , T1
θ (t, x) = 0. σ (u(t, x))
By Theorem 4.4, there exists τδ > 0 such that v(t) − m L∞ (T) δ for each t τδ . By Remark 4.1 and (4.2), since ut = θx weakly, we have for each T2 τδ 2 ε −1 ut + f (u)x + ε D(u)ux x −1 2 2 D
σ (u);T1 +T2
3ε −1 2
T1
f (u) − cf θ D(u)ux + f (u) − cf , + D(u)ux , dt θ, σ (u) σ (u) σ (u)
0
3ε −1 Cσ,δ 2
T1 2 dt θ, θ + T1 Cf,δ
+ T1 CD v(T2 )x , v(T2 )x .
(4.3)
0
By standard parabolic estimates we have +∞ dt vx , vx < +∞. 0
In particular there exists T2,δ > τδ such that v(T2,δ )x , v(T2,δ )x < δ. Note that, as δ → 0, Cσ,δ stays bounded, while Cf,δ , θ, θ and v(T2,δ )x , v(T2,δ )x vanish. Therefore the right-hand side of (4.3) can be made arbitrarily small provided δ is small enough. 2
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5. Proof of Theorem 3.1 for V and V JV Define the parity operator P : U → U by P u(x) = u(−x) and for T > 0 the time–space parity operator P T : XT → XT by P T u(t, x) = u(T − t, −x). Define the time reversed quasi-potential V : U × [−1, 1] → [0, ∞] as V (ui , m) := inf HT (u), T > 0, u ∈ XT : u(0) = ui , u(T ) ≡ m . Lemma 5.1. Assume f ∈ C 2 ([−1, 1]) is such that there is no interval in which f is affine. Let T > 0, uf ∈ U and m = T dx uf (x). Then V (m, uf ) = V (P uf , m) + Wm (uf ). Proof. By the assumptions on f , as remarked below Proposition 2.1, ET ⊂ C([0, T ]; L1 (T)). In particular Eqs. (2.2)–(2.3) extend to any ϑ ∈ C 2,∞ ([−1, 1] × [0, T ] × T) (now ϑ(0) and ϑ(T ) need not to vanish) as
dx ϑ u(T , x), T , x − ϑ u(0, x), 0, x
T
dt dx (∂t ϑ) u(t, x), t, x + (∂x Q) u(t, x), t, x
− [0,T ]×T
dv u (v; dt, dx) ϑ (v, t, x).
=
(5.1)
[−1,1]×[0,T ]×T
Note that for u ∈ ET and v ∈ [−1, 1]
P T u (v; dt, dx) = − u v; d(T − t), d(−x) . Therefore assuming also u(0) ≡ m, u(T ) = uf , we have for each η ∈ C 2 ([−1, 1]) with η(m) = 0
dv η (v) u+ (v; dt, dx) − =
dv η
= =
dv η (v) P+T u (v; dt, dx)
dv η (v) u+ (v; dt, dx) −
=
(v) u+ (v; dt, dx) −
dv η (v) u (v; dt, dx) = dx η uf (x)
dv η (v) u− v; d(T − t), −dx dv η (v) u− (v; dt, dx) dx η u(T , x) − η u(0, x) (5.2)
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where we used (5.1) with ϑ(v, t, x) = η(v). If σ is bounded away from 0, then (5.2) evaluated for η = hm immediately yields HT (u) = HT P T u + Wm (uf ). (5.3) If σ vanishes at −1 or +1, then (5.3) is obtained by monotone convergence, when considering in (5.2) a sequence {ηn } ⊂ C 2 ([−1, 1]) such that: ηn (m) = 0; 0 (ηn ) h m ; and for all v ∈ [−1, 1], ηn (v) ↑ hm (v) and (ηn ) (v) ↑ h m (v). Optimizing in (5.3) over T and u we get V (m, uf ) V (P uf , m) + Wm (uf ). Replacing uf by P uf and thus P uf by P (P uf ) = uf , we get the reverse inequality. 2 Definition 5.2. We say that ui ∈ U is piecewise constant iff there is a finite partition of T in intervals such that ui is constant on each interval. For T > 0, we say that u ∈ XT is piecewise constant iff u ∈ C([0, T ]; L1 (T)) and there exists a finite partition of [0, T ] × T in connected sets with Lipschitz boundary such that u is constant on each set of these. The following lemma is the main technical difficulty of this paper, and its proof is postponed at the end of this section. Lemma 5.3. Assume the same hypotheses of Theorem 3.1(ii). For each γ > 0, there exist Tγ , δ γ > 0 such that the following holds. For each piecewise constant ui ∈ U satisfying T dx ui (x) = m and ui − m L∞ (T) δ γ , there exists uγ ∈ XT γ such that uγ (0) = ui , uγ (T γ ) ≡ m and HT γ (uγ ) γ . The next corollary relaxes the condition in Lemma 5.3 requiring ui to be piecewise constant. Corollary 5.4. Assume the same hypotheses of Theorem 3.1(ii). For each γ > 0, there exist T γ , δ γ > 0 such that the following holds. For each ui ∈ U satisfying T dx ui (x) = m and ui − m L∞ (T) δ γ , there exists uγ ∈ XT γ such that uγ (0) = ui , uγ (T γ ) ≡ m and HT γ (uγ ) γ . Proof. For a fixed γ > 0, let T γ and δ γ > 0 be as in Lemma 5.3. For ui ∈ U such that n ui − m L∞ (T) δ γ , where m = T dx ui (x), let {u be a sequence of piecewise coni} ⊂ U stant functions converging to ui in U and satisfying T dx uni (x) = m and ui − m L∞ (T) δ γ . For each n, by Lemma 5.3 there exist un,γ such that un,γ (0) = uni , un,γ (T γ ) ≡ m and HT γ (un,γ ) γ . Therefore, since HT γ has compact sublevel sets (see [2, Proposition 2.6]), there is a (not relabeled) subsequence {un,γ } converging to a uγ in XT γ , and HT γ (uγ ) γ . By the definition of convergence in XT γ , un,γ (0) and un,γ (T γ ) converge in U to uγ (0) and uγ (T γ ) respectively, and thus uγ (0) = ui and uγ (T γ ) ≡ m. 2 We recall a result in [6, Chapter 11.5], [12]. Theorem 5.5. Assume f ∈ C 2 ([−1, 1]), and that there is no interval in which f is affine. Let ui ∈ U and let u : [0, ∞) × T → R be the Kruzkhov solution to (1.1) with initial datum ui ∈ U . Then lim u(t) − mL
t→∞
where m =
T dx ui (x).
∞ ([0,T ]×T)
=0
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Proof of Theorem 3.1(ii). Fix ui ∈ U and γ > 0. Let T γ and δ γ be as in Corollary 5.4, let u : [0, +∞) → U be the Kruzkhov solution to (1.1) with initial datum ui , and let m = γ γ γ T dx ui (x). By Theorem 5.5, there exists τ such that |u(τ ) − m|L∞ (T) δ . By Corollary 5.4 γ γ ˜ = u(τ ), u(T ˜ ) ≡ m and HT γ (u) ˜ γ . Define u ∈ Xτ γ +T γ there exists u˜ ∈ XT γ such that u(0) by u(t, x) :=
u(t, x) u(t ˜ − τ γ , x)
if t τ γ , if τ γ t τ γ + T γ .
Then, by Remark 3.2, Hτ γ +T γ (u) = Hτ γ (u) + HT γ (u) ˜ γ . Therefore V (ui , m) = 0 and the proof is thus complete since Lemma 5.1 holds. 2 The remaining of this section is devoted to the proof of Lemma 5.3. Remark 5.6. Let m ∈ (−1, 1), assume the same hypotheses of Theorem 3.1(ii), and let δ0 be defined accordingly. Then, taking perhaps a smaller δ0 , one can assume [m − δ0 , m + δ0 ] ⊂ (−1, 1) and that one (and only one) of the following holds: (A) in the interval [m − δ0 , m + δ0 ], f is either strictly convex or strictly concave. (B) f is either strictly convex in [m − δ0 , m] and strictly concave in [m, m + δ0 ], or strictly concave in [m − δ0 , m] and strictly convex in [m, m + δ0 ]. With no loss of generality, we will assume f convex in [m − δ0 , m + δ0 ] if case (A) holds, and f concave in [m − δ0 , m] and convex in [m, m + δ0 ] if (B) holds. Remark 5.7. Let T > 0 and assume u ∈ ET to be piecewise constant according to Definition 5.2. Then the jump set of u consists of a finite number of segments in [0, T ] × T. In particular there exist a finite sequence 0 = T 0 < T 1 < · · · < T n = T , and, for k = 1, . . . , n, finite sequences {wjk }j =1,...,Nk ⊂ [−1, 1] such that for t ∈ (T k−1 , T k ), u(t) is piecewise constant with jump set consisting of a finite set of points {xjk (t)}k=1,...,Nk ∈ T, and the traces of u(t) at xjk (t) are wjk k ). (from the right) and wjk−1 (from the left, where we understand w0k ≡ wN k In particular, by (2.5) we have that HT (u) =
k−1 Nk k n + k D(v) ρ (v, wj , wj ) T − T k−1 . dv σ (v) |wjk − wjk−1 | k=1
(5.4)
j =1
If u ∈ ET is piecewise constant, and u− , u+ are the left and right traces of u at a given point in the jump set of u, we say that the shock between u− and u+ is entropic iff ρ(v, u− , u+ ) 0 for almost every v, while it is anti-entropic iff ρ(v, u− , u+ ) 0 for almost every v. If f is convex or concave, each shock is either entropic or anti-entropic, but in the general case the sign of ρ(v, u− , u+ ) may depend on v. Lemma 5.8. Let m ∈ (−1, 1), and δ0 ≡ δ0 (m) > 0 be as in Remark 5.6. Let ui ∈ U be piecewise constant and such that T dx ui (x) = m and ui − m L∞ (T) δ0 . Then for each T , γ > 0 there exists w ∈ XT piecewise constant such that w − m L∞ ([0,T ]×T) ui − m L∞ (T) , w(0) = ui , and HT (w) γ .
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Fig. 1. In the figure, we have f (u) = u3 − u, m = 0 and M = 2. Consider a discontinuity between the values u− 1 = −δ and u+ 1 = δ. Then U1 is chosen as the abscissa of the point at which a line passing in (−δ, f (−δ)) is tangent to the graph k (u+ − U ) for k = 1, 2 are the abscissas of the intersections of the dashed lines with the graph of f . The values U1 + M 1 1 of f .
The proof of Lemma 5.8 will be divided in three steps. The main idea is to construct a piecewise smooth weak solution w, by splitting each shock appearing in the initial datum in an entropic part and an anti-entropic part, the anti-entropic part being split itself in M small anti-entropic shocks, with M a large integer, see Fig. 1. For such a solution to exist, the points at which the shocks are split have to be carefully chosen. We are then able to define w up to the first time at which two (or more) shocks collide. Defining then w recursively, we prove that there can be only a finite number of times at which the shocks collide, and thus w is well defined globally in time. Finally, we show that HT (w) can be made arbitrarily small by choosing M large. Proof of Lemma 5.8. As noted in Remark 5.6, we assume f to be strictly convex in [m, m + δ0 ]. Hereafter we let δ := ui − m L∞ (T) δ0 . Step 1 (Evolution of shocks). Let x1 , . . . , xN ∈ T be the points at which the discontinuities of ui are located. With a little abuse of notation, we also denote by ui : R → [−1, 1] and xj ∈ [0, 1] ⊂ R the lift of ui and xj to R, and we assume xj < xj +1 for j = 1, . . . , N − 1. For + j = 1, . . . , N and n ∈ Z, let xj +nN = xj + n ∈ R, and for j ∈ Z let u− j and uj be respectively the left and right traces of ui at xj . Define Uj :=
+ − − + max{w ∈ [u− j , uj ]: ρ(v, w, uj ) 0, ∀v ∈ [−1, 1]} if uj < uj , − − min{w ∈ [u+ j , uj ]: ρ(v, w, uj ) 0, ∀v ∈ [−1, 1]}
− if u+ j < uj .
+ + + − + Since f is convex in [m, m + δ0 ], if u− j < uj and Uj v v uj , or if uj < uj and uj v v Uj then
f (u− j ) − f (Uj ) u− j − Uj
f (Uj ) − f (v) f (v) − f (v ) Uj − v v − v
(w) where we understand f (w)−f = f (w) for w ∈ [−1, 1]. w−w u Therefore, fixed an integer M 2, it is possible to define the map wj i : [0, +∞) × R → [m − δ, m + δ] as (see Fig. 2)
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Fig. 2. In the figure, we have f (u) = u3 − u, m = 0, and the initial datum ui having two jumps between the values −δ and δ. The figure shows w at different times t ∈ [0, T (ui )].
⎧ ⎪ ⎪ u− ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Uj ⎪ ⎪ ⎪ ⎪ ⎨ ui wj (t, x) := Uj + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u+ ⎩
if x − xj
f (u− j )−f (Uj )
if x − xj ∈ [ + k M (uj
− Uj )
u− j −Uj
f (u− j )−f (Uj )
if x − xj ∈ [M M
t,
u− j −Uj
f (Uj )−f (Uj + t, M Uj −u+ j
u+ j −Uj M
)
t],
+ + k f (Uj + k−1 M (uj −Uj ))−f (Uj + M (uj −Uj ))
Uj −u+ j
k k+1 + f (Uj + M (u+ j −Uj ))−f (Uj + M (uj −Uj ))
Uj −u+ j
t, (5.5)
t]
for k ∈ {1, . . . , M − 1}, if x − xj M
j
+ + f (Uj + M−1 M (uj −Uj ))−f (uj )
Uj −u+ j
t.
+ − Note that this definition makes sense in the case Uj = u− j or Uj = uj . We also let Xj (t) :=
xj +
f (u− j )−f (Uj ) u− j −Uj
t, Xj+ (t) := xj +
+ + f (Uj + M−1 M (uj −Uj ))−f (uj )
Uj −u+ j
t and
T (ui ) := inf t 0: min Xj− (t) − Xj+−1 (t) = 0 . j
We next define w ui : [0, T (ui )] × R → [m − δ, m + δ] as w ui (t, x) = wjui (t, x)
if x ∈ Xj+ (t), Xj−+1 (t) .
w ui is a weak solution to (1.1) in [0, T (ui )] × R, since it is piecewise constant and satisfies the Rankine–Hugoniot condition along the shocks. Since it is also 1-periodic on R, it defines a map w ui : [0, T (ui )] × T → [m − δ, m + δ] such that w ui (0) = ui and w ui ∈ ET (ui ) .
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Step 2 (There is a finite number of shocks merging). We next define recursively, for k 1, T k ∈ [0, T ] and w k : [T k−1 , T k ] × T → [m − δ, m + δ] (where T 0 = 0) by T 1 := T (ui ) ∧ T , w 1 := w ui , T k := T k−1 + T w k−1 T k−1 ∧ T for k 2, k−1 k−1 w k (t, x) := w w (T ) t − T k−1 , x for k 2. We want to show that there exists a K ∈ N such that T K = T . By definition, for each k 1 and t ∈ (T k−1 , T k ), the discontinuities of w k (t) are either entropic or non-entropic. On the other hand, at the times T k at which two or more shocks collide, one and only one of the following may happen. – At a point y ∈ T, two or more entropic shocks of w k merge at time T k . Then w k+1 has one entropic shock in [T k , T k+1 ] starting at y. – At a point y ∈ T one or more entropic shocks of w k merge with one or more anti-entropic shocks. Then either w k+1 has one entropic shock starting at y, or w k+1 has a anti-entropic shocks starting at y, for some integer a, 0 a M. Note that, at a time T k , one or more of the merging here described may happen at different points y ∈ T, but at no point there can be a shock merging involving only anti-entropic shocks. The last statement can be proved by exhaustion, and – roughly speaking – it is a consequence of the well-known instability of anti-entropic shocks. Let us detail the case of convex f (corresponding to (A) in Remark 5.6). Anti-entropic shocks are then increasing, and according to the Rankine–Hugoniot condition, the higher the shock the higher its speed. Therefore, two or more anti-entropic shocks that are close enough, will separate as time increases rather than colliding. The case (B) of Remark 5.6 is treated similarly. Summarizing the previous remark, at a given shocks merging: either the number of entropic shocks stays constant and the number of anti-entropic shocks decreases by at least one; or the number of entropic shocks decreases by at least one, and the number of anti-entropic shocks may either decrease, or increase (by at most M). It follows that there can be at most a finite number of shocks merging, and in particular a finite number of times at which shocks merge. Recalling that N was the total number of discontinuity points of ui , and that by definition w 1 has at most N entropic shocks and N M anti-entropic shocks, it follows that for each k, w k has at most (2 N − 1)M anti-entropic shocks, the remaining shocks being entropic. Therefore the sequence {T k } has no accumulation points before T , and it will hit T for k large enough. Step 3 (Computing HT ). We can thus define w : [0, T ] × T → [m − δ, m + δ] by requiring w(t, x) = w k (t, x) for t ∈ [T k−1 , T k ]. w is piecewise constant and it satisfies the Rankine– Hugoniot condition along the shocks, therefore, since w k−1 (T k ) = w k (T k ), w ∈ ET . As noted above, in each time interval [T k−1 , T k ], w has at most (2 N − 1)M shocks. Moreover, by the definition (5.5), the left and the right traces of w at an anti-entropic shock differ at most by 2δ0 /M. Therefore we can bound the sum in (5.4) as
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HT (w) T (2 N − 1) M
sup
dv
u− ,u+ ∈[m−δ0 ,m+δ0 ] 2δ |u+ −u− | M0
D(v) ρ + (v, u+ , u− ) σ (v) |u+ − u− |
D(v) T (2 N − 1) M max v∈[m−δ0 ,m+δ0 ] σ (v) u+ − |u+ − u− | + f u +f u × sup − dv f (v) 2 2δ0 + −
|u −u |
M
= T (2 N − 1) M
max
v∈[m−δ0 ,m+δ0 ]
×
sup |u+ −u− |
2δ0 M
D(v) σ (v)
u−
|u+ − u− | + f u − f u− − f u− u+ − u− 2
u+ − − − − dv f (v) − f u − f u v − u u−
CT (2 N − 1)M −2 where in the last inequality we used the standard Taylor remainder estimate and C is a constant depending only on f , D, σ . Namely, HT (w) is arbitrarily small provided M is chosen sufficiently large. 2 In the following, whenever m + δ, m + δ ∈ [−1, 1], we introduce the short hand notation R(δ, δ ) for the Rankine–Hugoniot velocity of a shock between the values m + δ and m + δ , namely R(δ, δ ) :=
f (m + δ) − f (m + δ ) δ − δ
and we understand R(δ, δ) = f (m + δ). We also introduce
C(δ, δ ) :=
dv
ρ(v, m + δ, m + δ ) |δ − δ |
|δ − δ | f (m + δ) + f (m + δ ) − = 2
m+δ
dv f (v).
m+δ
The following lemma introduces an explicit solution to (1.1), with initial datum having only two discontinuities and final datum being constant. Lemma 5.9. Assume the same hypotheses of Theorem 3.1(ii) and let γ > 0. Let m ∈ (−1, 1) and let δ0 = δ0 (m) be defined as in Remark 5.6. Then for each δ1 ∈ (0, δ0 ) there exists δ 2 ≡ δ 2 (δ1 ) ∈ (0, δ0 ) such that for each δ2 ∈ (0, δ 2 ) the following holds. For a fixed arbitrary x0 ∈ T
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let ud ≡ uδd1 ,δ2 ∈ U be defined as
ud (x) := m + δ1 m − δ2
if |x − x0 | otherwise,
δ2 2(δ1 +δ2 ) ,
and let τ ≡ τ δ1 ,δ2 :=
1 . |R(δ1 , 0) − R(0, −δ2 )|
Then τ < ∞, and there exists u ∈ Xτ such that u(0) = ud , u(τ ) ≡ m and D(v) C(δ1 , 0)+ + C(0, −δ2 )+ . Hτ (u) max v∈[m−δ0 ,m+δ0 ] σ (v) |R(δ1 , 0) − R(0, −δ2 )|
(5.6)
Proof. Fix δ1 ∈ (0, δ0 ). By the definition of δ0 , R(δ1 , 0) = f (m) and assuming f strictly convex in [m, m + δ0 ] (see Remark 5.6), we have R(δ1 , 0) > f (m). Recalling the definition of ρ in (2.6), still by the convexity of f in [m, m + δ0 ], we have ρ(v, m, m + δ1 ) < 0 for v ∈ (m, m + δ1 ) and C(δ1 , 0) > 0. In particular there exists δ 2 small enough such that for each δ2 ∈ (0, δ 2 ) and each v ∈ (m − δ2 , m + δ1 ) R(δ1 , 0) − R(0, −δ2 ) > 0,
(5.7)
ρ(v, m − δ2 , m + δ1 ) < 0.
(5.8)
Let us now fix δ2 ∈ (0, δ 2 ). By (5.7) τ δ1 ,δ2 is finite. With no loss of generality we may assume 2 x0 = 2(δ1δ+δ , as the general case is obtained by a space translation of the solution u given below 2) by the quantity x0 −
δ2 2(δ1 +δ2 ) .
⎧ m ⎪ ⎪ ⎪ ⎨m + δ 1 u(t, x) := ⎪ ⎪ ⎪ ⎩ m − δ2
Define
if |x − [R(δ1 , 0) + R(0, −δ2 )] 2t | [R(δ1 , 0) − R(0, −δ2 )] 2t , δ2 t 2(δ1 +δ2 ) − [R(δ1 , 0) + R(δ1 , −δ2 )] 2 | δ2 t 2(δ1 +δ2 ) − [R(δ1 , 0) − R(δ1 , −δ2 )] 2 ,
if |x −
otherwise.
It follows that u(0) = ud and u(τ ) ≡ m. Moreover u is a piecewise constant weak solution to (1.1). For a fixed t ∈ (0, T ), u(t) has three discontinuity points, where its value jumps from m to m + δ1 , from m + δ1 to m − δ2 and from m − δ2 to m. In particular Hτ (u) can be calculated by (5.4). The shock between the values m + δ1 and m − δ2 is entropic by (5.8), and thus it gives no contributions to the sum (5.4). By the convexity assumption on f in [m, m + δ0 ], the shock between m and m + δ1 is anti-entropic, namely ρ(v, m + δ1 , m) 0. Moreover the shock between m − δ2 and m is either entropic (if case (A) in Remark 5.6 holds) or anti-entropic (if case (B) in Remark 5.6 holds). Therefore (5.4) yields
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D(v) ρ + (v, m + δ1 , m) D(v) ρ + (v, m, m − δ2 ) + dv σ (v) δ2 σ (v) δ1 D(v) ρ(v,m,m−δ2 ) + D(v) ρ(v,m+δ1 ,m) [ dv σ (v) ] + dv σ (v) δ2 δ1
Hτ (u) = τ =
dv
R(δ1 , 0) − R(0, −δ2 ) ρ(v,m,m−δ2 ) + 1 ,m) ] + dv ρ(v,m+δ D(v) [ dv δ2 δ1 max v∈[m−δ0 ,m+δ0 ] σ (v) R(δ1 , 0) − R(0, −δ2 )
namely (5.6).
2
Remark 5.10. Let s1 , . . . , sn : [0, T ] → T be a finite collection of Lipschitz maps, and let F : [0, T ] × T → R be a bounded function, such that F is Lipschitz in each connected component of [0, T ] × T \ ni=1 Graph(si ) (F may feature discontinuities on the graphs of the curves si , i = 1, . . . , n). Assume either inft,x F (t, x) ess supi,t s˙i (t) or supt,x F (t, x) ess infi,t s˙i (t). Then, for each fixed s0 ∈ T, there exists a unique Lipschitz map s : [0, T ] → T such that s(0) = s0 and s˙ (t) = F (t, s) for a.e. t ∈ [0, T ]. Proof of Lemma 5.3. Fix γ > 0. Recall the definition of δ0 ≡ δ0 (m) in Remark 5.6; as noted in Remark 5.6 we may assume f to be strictly convex in [m, m + δ0 ]. We thus have R(δ1 , 0) > f (m), ρ(v, m + δ1 , m) 0 for each δ1 ∈ (0, δ0 ). Then by explicit computation |C(δ1 , δ )| + |C(δ , −δ2 )| + |C(δ1 , −δ2 )| = 0. δ1 ↓0 δ2 ↓0 δ↓0 δ ,δ ∈[−δ,δ] R(δ1 , 0) − f (m) lim lim lim
sup
γ
γ
In particular, defining δ 2 (·) as in Lemma 5.9, there exist δ1 ≡ δ1 ∈ (0, δ0 ), δ2 ≡ δ2 ∈ (0, δ 2 (δ1 )) and δ ≡ δ γ ∈ (0, δ1 ∧ δ2 ) such that
D(v) max v∈[m−δ0 ,m+δ0 ] σ (v)
C(δ1 , 0) + |C(δ , −δ2 )| + C(δ1 , −δ2 ) γ , R(δ1 , 0) − f (m) 8 δ ∈[−δ,δ] sup
R(δ1 , 0) − f (m) R(δ1 , 0) − R(0, −δ2 ) 4 2 inf R(δ1 , δ ) − R(δ , −δ2 ), δ ,δ ∈[−δ,δ]
inf
(5.10)
R(δ , −δ2 ) − R(δ , δ ) > 0,
(5.11)
R(δ1 , δ ) − R(δ , δ ) > 0,
(5.12)
δ ,δ ∈[−δ,δ]
inf
(5.9)
δ ,δ ∈[−δ,δ]
ρ(v, m − δ, m + δ1 ) > 0 for v ∈ (m − δ, m + δ1 ), ρ(v, m + δ, m − δ2 ) > 0 for v ∈ (m − δ2 , m + δ).
(5.13) (5.14)
Let now ui ∈ U be an arbitrary piecewise constant profile such that ui − m L∞ (T) δ. Fix T :=
4 . R(δ1 , 0) − f (m)
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By Lemma 5.8 there exists a piecewise constant map w ≡ w T ,γ /4 : [0, T ] × T → [m − δ, m + δ] such that w(0) = ui and HT (w) γ /4. Let the Lipschitz map s1 , s2 : [0, +∞) → T be defined as the solutions to the Cauchy problems ⎧ f (m + δ1 ) − f (w(t, s1 (t))) ⎨ s˙1 (t) = ≡ R δ1 , w t, s1 (t) − m , m + δ1 − w(t, s1 (t)) ⎩ s1 (0) = 0, ⎧ f (m − δ2 ) − f (w(t, s2 (t))) ⎨ ≡ R w t, s2 (t) − m, −δ2 , s˙2 (t) = m − δ2 − w(t, s2 (t)) ⎩ s2 (0) = 0. Despite the right-hand sides are discontinuous, these equations are well posed since w is piecewise constant and conditions (5.11)–(5.12) hold, so that Remark 5.10 applies. With a little abuse of notation, we also denote by s1 and s2 the lift of s1 and s2 on R. Note that, by (5.10), s1 (t) − s2 (t) is increasing in t and letting T > 0 be the first time t at which s1 (t) − s2 (t) = 1, we have still by (5.10) T T.
(5.15)
We also set x0 := s1 (T ) ≡ s2 (T ) ∈ T, and let ud ≡ uδd1 ,δ2 , τ ≡ τ δ1 ,δ2 be defined as in Lemma 5.9 (with δ1 , δ2 and x0 defined as above in this proof), and let v ∈ Xτ be the solution to (1.1) whose existence is proved in Lemma 5.9. We finally let (see Figs. 3 and 4) ⎧ m + δ1 ⎪ ⎨ m − δ2 uγ (t, x) := ⎪ ⎩ w(t, x) v(t − T , x)
if t ∈ [0, T ] and x ∈ A1 (t), if t ∈ [0, T ] and x ∈ A2 (t), if t ∈ [0, T ] and x ∈ / A1 (t) ∪ A2 (t), if t ∈ [T , T + τ ]
where for t 0 x − 1 s1 (t) + R(δ1 , −δ2 )t 2 1 s1 (t) − R(δ1 , −δ2 )t , 2 1 A2 (t) := x ∈ T: x − R(δ1 , −δ2 )t + s2 (t) 2 1 R(δ1 , −δ2 )t − s2 (t) . 2 A1 (t) := x ∈ T:
γ
Note that u|[0,T ] ∈ XT is piecewise constant, and it is the gluing of solutions to (1.1) satisfying the Rankine–Hugoniot condition at the borders of {(t, x) ∈ [0, T ] × T: x ∈ Ai (t)} (for i = 1, 2). γ We thus have u|[0,T ] ∈ ET and uγ ∈ ET +τ . γ In order to calculate HT (u|[0,T ] ), we will use Remark 5.7. Note that for each t ∈ [0, T ] the set of discontinuity points of uγ (t) consists of the discontinuity points of w(t), and the discontinuities at s1 (t), at s2 (t) and at R(δ1 , −δ2 )t. Because of assumptions (5.11)–(5.12), there is at most
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Fig. 3. In the figure, we have f (u) = u3 − u, m = 0, and the initial datum ui having two jumps between the values −δ and δ, so that w is the same as in Fig. 2. Here the figure shows uγ at a small time 0 < t < T and at time T .
Fig. 4. In the figure, we have f (u) = u3 − u, m = 0, and the initial datum ui having two jumps between the values −δ and δ. The figure shows uγ at different times t ∈ (T , T + τ ].
a finite number of times t ∈ [0, T ] at which s1 (t) and s2 (t) may overlap with a discontinuity of w(t, ·). Note that assumption (5.13) implies ρ(v, w, m + δ1 ) 0, for each v ∈ [−1, 1] and w ∈ [m − δ, m + δ], so that the shock of uγ at s1 is entropic and it does not appear in the sum (5.4). Conversely ρ(v, m − δ2 , m + δ1 ) 0, so that the shock along the curve t → R(δ1 , −δ2 )t appears in the sum (5.4). Finally, by (5.14), ρ(v, w(t, s2 (t)), m − δ2 ) is either negative or positive for each t ∈ [0, T ] and v ∈ [−1, 1], depending on whether case (A) or (B) of Remark 5.6 holds for f . By Remark 3.2 and recalling that v satisfies (5.6)
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γ HT +τ uγ = Hτ (v) + HT u|[0,T ] Hτ (v) + HT (w)
T +
dt
dv
0
D(v) ρ + (v, m − δ2 , w(t, s2 (t))) σ (v) δ2
D(v) ρ + (v, m + δ1 , m − δ2 ) + dv σ (v) δ1 + δ 2 D(v) C(δ1 , 0)+ + C(0, −δ2 )+ γ max + v∈[m−δ0 ,m+δ0 ] σ (v) 4 |R(δ1 , 0) − R(0, −δ2 )|
T +
dt 0
D(v) ρ(v, m − δ2 , w(t, s2 (t))) dv σ (v) δ2
+
D(v) ρ(v, m + δ1 , m − δ2 ) σ (v) δ1 + δ 2 D(v) γ + max v∈[m−δ0 ,m+δ0 ] σ (v) 4 C(δ1 , 0)+ + C(0, −δ2 )+ + T C(δ1 , −δ2 ) + T × |R(δ1 , 0) − R(0, −δ2 )| +
dv
+
sup C(δ , −δ2 )
δ ∈[−δ,δ]
.
By (5.15) and (5.10) we thus obtain γ HT +τ uγ + 4
D(v) max v∈[m−δ0 ,m+δ0 ] σ (v)
2C(δ1 , 0)+ + 2C(0, −δ2 )+ + 4C(δ1 , −δ2 ) + 4 supδ ∈[−δ,δ] C(δ , −δ2 )+ R(δ1 , 0) − f (m) γ D(v) +6 max v∈[m−δ0 ,m+δ0 ] σ (v) 4 ×
×
C(δ1 , 0) + C(δ1 , −δ2 ) + supδ ∈[−δ,δ] |C(δ , −δ2 )| . R(δ1 , 0) − f (m)
Therefore HT +τ (uγ ) γ by (5.9).
2
Proof of Theorem 3.1(iii). We assume T dx uf (x) = m, the proof being trivial otherwise. Since HTJV HT we have V JV Wm (uf ) by Theorem 3.1(ii). The converse inequality is obtained by taking ϕ ≡ 1 in the very definition of H JV . 2 Acknowledgments We are indebted to Lorenzo Bertini and Matteo Novaga for enlighting discussions.
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References [1] C. Bahadoran, Non-local entropies for conservation laws with open boundaries, in preparation. [2] G. Bellettini, L. Bertini, M. Mariani, N. Novaga, Γ -entropy cost functional for scalar conservation laws, Arch. Ration. Mech. Anal., in press. [3] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim, Large deviation approach to non-equilibrium processes in stochastic lattice gases, Bull. Braz. Math. Soc. (N.S.) 37 (2006) 611–643. [4] L. Bertini, A. Faggionato, D. Gabriele, in preparation. [5] A. Bressan, B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. [6] C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, second ed., Springer-Verlag, Berlin, 2005. [7] C. De Lellis, F. Otto, M. Westdickenberg, Structure of entropy solutions for multi-dimensional scalar conservation laws, Arch. Ration. Mech. Anal. 170 (2) (2003) 137–184. [8] L.C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, American Mathematical Society, 1997. [9] M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems, second ed., Springer-Verlag, New York, 1984. [10] L.H. Jensen, Large deviations of the asymmetric simple exclusion process in one dimension, PhD thesis, Courant Institute NYU, 2000. [11] C. Kipnis, C. Landim, Scaling Limits of Interacting Particle Systems, Springer-Verlag, Berlin, 1999. [12] S.N. Kruzhkov, N.S. Petrosyan, Asymptotic behaviour of the solutions of the Cauchy problem for non-linear first order equations, Russian Math. Surveys 42 (5) (1987) 1–47. [13] M. Mariani, Large deviations for stochastic conservations laws and their variational counterpats, PhD thesis, Sapienza Università di Roma, 2008. [14] M. Mariani, Large deviations principle for perturbed conservations laws, Probab. Theory Related Fields, in press. [15] S.R.S. Varadhan, Large deviations for the simple asymmetric exclusion process, in: Stochastic Analysis on Large Scale Interacting Systems, in: Adv. Stud. Pure Math., vol. 39, 2004, pp. 1–27.
Journal of Functional Analysis 258 (2010) 559–603 www.elsevier.com/locate/jfa
On the singularity probability of discrete random matrices Jean Bourgain a , Van H. Vu b,∗,1 , Philip Matchett Wood b a Institute for Advanced Study, 1 Einstein dr., Princeton, NJ 08540, USA b Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
Received 16 April 2009; accepted 23 April 2009 Available online 13 May 2009 Communicated by Paul Malliavin
Abstract Let n be a large integer and Mn be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this paper is to prove a general upper bound for the probability that Mn is singular. For a constant 0 < p < 1 and a constant positive integer r, we will define a property p-bounded of exponent r. Our main result shows that if the entries of Mn satisfy this property, then the probability that Mn is singular is at most (p 1/r + o(1))n . All of the results in this paper hold for any characteristic zero integral domain replacing the complex numbers. In the special case where the entries of Mn are “fair coin flips” (taking the values +1, −1 each with probability 1/2), our general bound implies that the probability that Mn is singular is at most ( √1 + o(1))n , improving on 2
the previous best upper bound of ( 34 + o(1))n , proved by Tao and Vu [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628]. In the special case where the entries of Mn are “lazy coin flips” (taking values +1, −1 each with probability 1/4 and value 0 with probability 1/2), our general bound implies that the probability that Mn is singular is at most ( 12 + o(1))n , which is asymptotically sharp. Our method is a refinement of those from [Jeff Kahn, János Komlós, Endre Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1) (1995) 223–240; Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628]. In particular, we make a critical use of the structure theorem from [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628], which was obtained using tools from additive combinatorics. © 2009 Elsevier Inc. All rights reserved. * Corresponding author.
E-mail addresses:
[email protected] (J. Bourgain),
[email protected] (V.H. Vu),
[email protected] (P.M. Wood). 1 Supported by NSF Career Grant 0635606 and by an AFORS grant. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.04.016
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Keywords: Discrete random matrix; Singularity
1. Introduction Let n be a large integer and Mn be an n by n random matrix whose entries are independent (but not necessarily identically distributed) discrete random variables taking values in the complex numbers. The problem of estimating the probability that Mn is singular is a basic problem in the theory of random matrices and combinatorics. The goal of this paper is to give a bound that applies to a large variety of distributions. The general statement (Theorem 2.2) is a bit technical, so we will first discuss a few corollaries concerning special cases. The most famous special case is when the entries of Mn are independent identically distributed (i.i.d.) Bernoulli random variables (taking values ±1 with probability 1/2). The following conjecture has been open for quite some time: Conjecture 1.1. For M±1,n an n by n matrix with each entry an i.i.d. Bernoulli random variable taking the values +1 and −1 each with probability 1/2, n 1 + o(1) . Pr(M±1,n is singular) = 2 It is easy to verify that the singularity probability is at least (1/2)n by considering the probability that there are two equal rows (or columns). Even in the case of i.i.d. Bernoulli random variables, proving that the singularity probability is o(1) is not trivial. It was first done by Komlós in 1967 [5] (see also [6]; [8] generalizes Komlós’s bound to other integer distributions). The first exponential bound was proven by Kahn, Komlós, and Szemerédi [4], who showed that Pr(M±1,n is singular) 0.999n . This upper bound was improved upon by Tao and Vu in [9] to 0.958n . A more significant improvement was obtained by the same authors in [10]: n 3 + o(1) . (1) Pr(M±1,n is singular) 4 This improvement was made possible through the discovery of a new theorem [10, Theorem 5.2] (which was called the structure theorem in [10]), which gives a complete characterization of a set with certain additive properties. The structure theorem (to be more precise, a variant of it) will play a critical role in the current paper as well. Our general result has the following corollary in the Bernoulli case: n 1 (2) Pr(M±1,n is singular) √ + o(1) , 2 √ which gives a slight improvement over Inequality (1) (since 1/ 2 ≈ 0.7071 < 0.75). Let us now discuss a more general class of random matrices. Consider the random variable γ (μ) defined by ⎧ ⎨ +1 with probability μ/2, with probability 1 − μ, (3) γ (μ) := 0 ⎩ −1 with probability μ/2,
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561
(μ)
and let M±1,n be an n by n matrix with each entry an independent copy of γ (μ) . The random (μ)
variable γ (μ) plays an important role in [4,9,10], and the matrices M±1,n are of interest in their own right. In fact, giving zero a large weight is a natural thing to do when one would like to (randomly) sparsify a matrix, a common operation used in randomized algorithms (the values of ±1, as the reader will see, are not so critical). Our general result implies the following upper bounds: (μ) n 1 Pr M±1,n is singular 1 − μ + o(1) for 0 μ , 2 n (μ) 2μ + 1 1 Pr M±1,n is singular + o(1) for μ 1, 4 2 n (μ) 3 Pr M±1,n is singular 1 − 2μ + μ2 + o(1) for 0 μ 1. 2
(4) (5) (6)
Note that Inequality (5) implies Inequality (1) and that Inequality (6) implies Inequality (2) (in both cases setting μ = 1). Fig. 1 summarizes the upper bounds from Inequalities (4), (5), and (6) and also includes the following lower bounds: n (μ) 1 − μ + o(1) Pr M±1,n is singular for 0 μ 1, n (μ) 3 1 − 2μ + μ2 + o(1) Pr M±1,n is singular for 0 μ 1. 2
(7) (8)
These lower bounds can be derived by computing the probability that one row is all zeros (Inequality (7)) or that there is a dependency between two rows (Inequality (8)). Note that in the case where μ 1/2, the upper bound in Inequality (4) asymptotically equals the lower bound in Inequality (7), and thus our result is the best possible in this case. We also used a Maple program to derive the formulas for lower bounds resulting from a dependency between three, four, or five rows; however, these lower bounds were inferior to those in Inequality (7) and Inequality (8). We will now present another corollary of the main theorem that has a somewhat different flavor. In this corollary, we treat partially random matrices, which may have many deterministic rows. Our method allows us to obtain exponential bounds so long as there are still at most c ln n random rows, where c > 0 is a particular constant. Corollary 1.2. Let p be a real constant between 0 and 1, let c be any positive constant less than 1/ ln(1/p), and let S ⊂ C be a set of complex numbers having cardinality |S| O(1). Let Nf,n be an n by n complex matrix in which f c ln n rows contain fixed, non-random elements of S and where the other rows contain entries that are independent random variables taking values in S. If the fixed rows are linearly independent and if for every random entry α, we have maxx Pr(α = x) p, then Pr(Nf,n is singular)
√ n p + o(1) .
Notice that the case f = 0 and p = 1/2 also implies Inequality (2).
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Asymptotic upper and lower bounds for Pr(M±1,n is singular)1/n for 0 μ 1
(μ)
(μ)
Fig. 1. Let P (μ) := limn→∞ Pr(M±1,n is singular)1/n , where M±1,n is the n by n matrix with independent random entries taking the value 0 with probability 1 − μ and the values +1 and −1 each with probability μ/2. The solid lines denote the upper bounds on P (μ) given by Inequalities (4), (5), and (6), and the dashed lines denote the lower bounds given by Inequalities (7) and (8). The upper and lower bounds coincide for 0 μ 12 , and the shaded area shows the
difference between the best known upper and lower bounds for 12 μ 1. The straight line segments from the point (0, 1) to (1/2, 1/2) and from the point (1/2, 1/2) to (1, 3/4) represent the best upper bounds we have derived using the ideas in [10], and the curve 1 − 2μ + 32 μ2 for 0 μ 1 represents a sometimes-better upper bound we have derived by adding a new idea. Note that the upper bounds given here also apply to the singularity probability of a random matrix with independent entries having arbitrary symmetric distributions in a set S of complex numbers, so long as each entry is 0 with probability 1 − μ and the cardinality of S is |S| O(1) (see Corollary 3.1).
Remark 1.3 (Other exponential bounds). The focus of this paper is optimizing the base of the exponent in bounds on the singularity probability for discrete random matrices. One main tool in this optimization is the use of a structure theorem similar to [10, Theorem 5.2] (see Theorem 6.1 below); however, using such a theorem requires additional assumptions to be placed on the values that can appear as entries, and in particular, this is why we assume in Corollary 1.2 that the set S has cardinality |S| O(1) and that f c ln n. If one is interested in an exponential bound where there are no conditions on f or on the set S (at the expense of having an unspecified constant for the base of the exponential), one can follow the analysis in [9], which does not make use of a structure theorem, along with ideas in this paper to get a result of the following form:
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Theorem 1.4. For every > 0 there exists δ > 0 such that the following holds. Let Nf,n be an n by n complex matrix in which f rows contain fixed, non-random entries and where the other rows contain entries that are independent discrete random variables. If the fixed rows have co-rank k and if for every random entry α, we have maxx Pr(α = x) 1 − , then for all sufficiently large n Pr(Nf,n has co-rank > k) (1 − δ)n−f . Note that Theorem 1.4 holds for any f and k, and so in particular, an exponential bound on the singularity probability is achieved whenever k = 0 and f cn, where c < 1 is a constant. Also note that the theorem allows the random entries to have discrete distributions taking infinitely many values. Corollary 3.6 proves a version of Theorem 1.4 with a much better exponential bound, given some additional conditions. The structure of the rest of the paper is as follows. In Section 2 we define p-bounded of exponent r and state the main theorem of this paper. In Section 3, we discuss some corollaries of Theorem 2.2. In particular, we will: (A) prove Inequalities (4), (5), and (6); (B) prove general bounds on the singularity probability for discrete random matrices with entries that have symmetric distributions and with entries that have asymmetric distributions; (C) prove a version of Corollary 1.2 (namely, Corollary 3.5) that holds for up to o(n) fixed rows, assuming that the entries in the fixed rows take integer values between −C and C for any positive constant C; and (D) prove that the probability that random matrices with integer entries have a rational eigenvalue is exponentially small. In Section 4, we discuss Lemma 4.1, a result that is proved in [12] using standard tools from algebraic number theory and algebraic geometry. Lemma 4.1 reduces the question of bounding the singularity probability of a random matrix with entries in C to a question of bounding the singularity probability of a random matrix with entries in Z/QZ for some large prime Q (in fact, it is possible to replace C with any characteristic zero integral domain). The proof of Theorem 2.2 is outlined in Section 5, where we also prove some of the easier lemmas needed for the theorem. In Section 6, we state a structure theorem (Theorem 6.1) that completes the proof of our Theorem 2.2 and that is very similar to [10, Theorem 5.2] (which is the structure theorem in [10]). We discuss the proof of Theorem 6.1, which uses discrete Fourier analysis and tools from additive combinatorics, in Sections 7 and 8. Finally, in Section 9 we show that the entire argument proving Theorem 2.2 can be generalized to random complex matrices with f rows of the matrix containing fixed, non-random entries, so long as f c ln n for a particular constant c > 0 (this leads to Corollary 1.2). 2. The general theorem To prove the results in Inequalities (1) and (2) (and also the results in [4] and [9]), one basic idea is to replace entries of a random matrix with independent copies of the random variable γ (μ) or 2γ (μ) (see Eq. (3)). One key idea in proving the more general results of the current paper is replacing the entries of a random matrix with more complicated symmetric discrete random variables.
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A generalized arithmetic progression of rank r is a set of the form {v0 + m1 v1 + · · · + mr vr : |mi | Mi /2}, where the vi are elements of a Z-module and the mi and Mi > 0 are integers. Note that whenever the term “symmetric” is used in this paper, it will apply to the distribution of a random variable or to a generalized arithmetic progression; in particular, the term will never apply to matrices. Also, throughout this paper we will use the notation e(x) := exp(2πix). The following definition lies at the heart of our analysis. Definition 2.1 (p-bounded of exponent r). Let p be a positive constant such that 0 < p < 1 and let r be a positive integer constant. A random variable α taking values in the integers (or, respectively, the integers modulo some large prime Q) is p-bounded of exponent r if (i) maxx Pr(α = x) p, and if there exists a constant q where 0 < q p and a Z-valued (or, respectively, a Z/QZ-valued) symmetric random variable β (μ) taking the value 0 with probability 1 − μ = p such that the following two conditions hold: (ii) q minx Pr(β (μ) = x) and maxx Pr(β (μ) = x) p, and (iii) the following inequality holds for every t ∈ R:
E e(αt) r E e β (μ) t . Here, if the values of α and β (μ) are in Z/QZ, we view those values as integers in the range (−Q/2, Q/2) (note that each element in Z/QZ has a unique such integer representation). We will define p-bounded of exponent r for collections of random variables below, but first we note that the conditions above are easy to verify in practice. In particular, if we have a symmetric random variable ⎧ b ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎨ b1 β (μ) = 0 ⎪ ⎪ ⎪ −b1 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎩. −b
with probability μp /2, .. . with probability μp1 /2, with probability 1 − μ, with probability μp1 /2, .. .
(9)
with probability μp /2,
where bs ∈ Z for all s (or, respectively, bs ∈ Z/QZ for all s), then condition (iii) becomes
E e(αt) r E e β (μ) t = 1 − μ + μ ps cos 2πbs t, s=1
where the equality on the right-hand side is a simple expected value computation.
(10)
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We say that a collection of random variables {αj k }nj,k=1 is p-bounded of exponent r if each αj k is p-bounded of exponent r with the same constants p, q, and r; and, importantly, the same value of μ = 1 − p. We also make the critical assumption that the set of all values that (μ) can be taken by the βj k has cardinality O(1) (a relaxation of this assumption is discussed in (μ)
Remark 8.5). However, the definition of βj k is otherwise allowed to vary with j and k. Also, we will use S to denote the set of all possible values taken by the random variables αj k , and we will assume that the cardinality of S is at most |S| no(n) . If α takes non-integer values in C, we need to map those values to a finite field of prime order so that we may use Definition 2.1, and for this task we will apply Lemma 4.1, which was proved in [12]. We say that α is p-bounded of exponent r if and only if for each prime Q in an infinite sequence of primes produced by Lemma 4.1, we have φQ (α) is p-bounded of exponent r, where φQ is the ring homomorphism described in Lemma 4.1 that maps S, the finite set of all possible values taken by the αj k , into Z/QZ in such a way that for any matrix Nn := (sj k ) with entries in S, the determinant of Nn is zero if and only if the determinant of φQ (Nn ) := (φQ (sj k )) is zero. Theorem 2.2. Let p be a positive constant such that 0 < p < 1, let r be a positive integer constant, and let S be a generalized arithmetic progression in the complex numbers with rank O(1) (independent of n) and with cardinality at most |S| no(n) . Let Nn be an n by n matrix with entries αj k , each of which is an independent random variable taking values in S. If the collection of random variables {αj k }1j,kn is p-bounded of exponent r, then n Pr(Nn is singular) p 1/r + o(1) . In the motivating examples of Section 1 (excluding Corollary 1.2), we discussed the case where the entries of the matrix are i.i.d.; however, in general the distributions of the entries are allowed to differ (and even depend on n), so long as the entries all take values in the same structured set S described above. The condition that S has additive structure seems to be an artifact of the proof (in particular, at certain points in the proof of Theorem 6.1, we need the set
{ nj=1 xj : xj ∈ S for all j } to have cardinality at most no(n) ). The easiest way to guarantee that S has the required structure is to assume that the set of values taken by all the αj k has cardinality at most O(1), and this is the approach we take for the corollaries in Section 3, since it also makes it easy to demonstrate that the collection of entries is p-bounded of exponent r. Remark 2.3 (Strict positivity in Inequality (10)). Note that the constants μ, ps , bs must be such that the right-hand side of Eq. (10) is non-negative. It turns out for the proof of Theorem 2.2 that we will need slightly more. At one point in the proof, we will apply Lemma 7.3, for which (μ) we must assume that there exists a very small constant −1 > 0 such that E(e(βj k t)) > −1 (μ)
for all t and for all βj k used in the definition of p-bounded of exponent r. Of course, if the expectations are not strictly larger than −1 , we can simply reduce μ by −1 > 0. Then, since we are assuming 1 − μ = p, we clearly have that all the αj k are (p + −1 )-bounded of exponent r (by (μ− ) (μ− ) (μ) using βj k −1 instead of βj k ) and we have that E(e(βj k −1 t)) > −1 > 0. Since Theorem 2.2 would thus yield a bound of ((p + −1 )1/r + o(1))n for every −1 > 0, we can conclude a bound of (p 1/r + o(1))n by letting −1 tend to 0. Thus, without loss of generality, we will assume that (μ) (μ) E(e(βj k t)) > −1 for all t and for all βj k used in the definition of p-bounded of exponent r.
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3. Some corollaries of Theorem 2.2 In this section, we will state a number of corollaries of Theorem 2.2, starting with short proofs of Inequalities (4), (5), and (6). The two most interesting results in this section will be more general: first (in Section 3.2), we will show an exponential bound on the singularity probability for a matrix with independent entries each a symmetric random variable taking values in S ⊂ C, where |S| O(1) and assuming that each entry takes the value 0 with probability 1 − μ; and second (in Section 3.3), we will describe a similar (and sometimes better) bound when the condition that the random variables have symmetric distributions is replaced with the assumption that no entry takes a value with probability greater than p. In the first case, the bound will depend only the value of μ, and in the second case, the bound will depend only on the value of p. In Section 3.4, we will show an exponential bound on the singularity probability for an n by n matrix with f = o(n) fixed rows containing small integer values and with the remaining rows containing independent random variables taking values in S ⊂ C, where |S| O(1) (this is similar to Corollary 1.2, which is proved in Section 9). Finally, in Section 3.5, we will prove an exponential upper bound on the probability that a random integer matrix has a rational eigenvalue. In each corollary, we will use the definition of p-bounded of exponent 1 and of exponent 2. The definition of p-bounded of exponent 2 is particularly useful, since then the absolute value on the left-hand side of Inequality (10) is automatically dealt with; however, when μ is small (for example whenever μ 1/2), one can get better bounds by using p-bounded of exponent 1. We have not yet found an example where the best possible bound from Theorem 2.2 is found by using p-bounded of an exponent higher than 2. 3.1. Proving Inequalities (4), (5), and (6) To prove Inequality (4), we note for 0 μ
1 2
that (using the definition in Eq. (3) of γ (μ) )
(μ)
E e γ t = 1 − μ + μ cos(2πt), and thus γ (μ) is (1 − μ)-bounded of exponent 1 (i.e., take β (μ) := γ (μ) ), and so Inequality (4) follows from Theorem 2.2. To prove Inequality (5), we note for 12 μ 1 that
(μ)
E e γ t = 1 − μ + μ cos(2πt)
2μ + 1 2μ − 1 + (1 − μ) cos(2πt) + cos(4πt) 4 4
(the inequality above may be checked by squaring both sides and expanding as polynomials in cos(2πt)). Thus, we can take
β (μ) :=
⎧ ⎪ +2 with probability ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −2 with probability +1 with probability ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 with probability ⎪ ⎩ 0 with probability
2μ−1 8 , 2μ−1 8 , 1−μ 2 , 1−μ 2 , 2μ+1 4
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to see that γ (μ) is ( 2μ+1 4 )-bounded of exponent 1, and so Inequality (5) follows from Theorem 2.2. To prove Inequality (6), we note for 0 μ 1 that
(μ) 2
E e γ t = 1 − μ + μ cos(2πt) 2 = 1 − 2μ + 3 μ2 + 2(1 − μ)μ cos(2πt) 2 2 μ cos(4πt). + 2 Thus, we can take ⎧ ⎪ +2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −2 β (μ) := +1 ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎪ ⎩ 0
with probability
μ2 4 , μ2 4 ,
with probability with probability (1 − μ)μ, with probability (1 − μ)μ, with probability 1 − 2μ + 32 μ2 ,
3 to see that γ (μ) is (1 − 2μ + μ2 )-bounded of exponent 2, and so Inequality (6) follows from 2 Theorem 2.2. 3.2. Matrices with entries having symmetric distributions (μ)
In this subsection, we will prove a singularity bound for an n by n matrix Nn for which each entry is a symmetric discrete random variable taking the value 0 with probability 1 − μ. (μ)
Corollary 3.1. Let S be a set of complex numbers with cardinality |S| O(1). If Nn is an n by n matrix in which each entry is an independent symmetric complex random variable taking values in S and taking the value 0 with probability 1 − μ, then ⎧ n for 0 μ 12 , ⎪ ⎪ (1 − μ + o(1)) n (μ) ⎨ 2μ+1 for 12 μ 1, Pr Nn is singular 4 + o(1) ⎪ ⎪ ⎩ 1 − 2μ + 3 μ2 + o(1) n for 0 μ 1. 2
In particular, the same upper bounds as in Inequalities (4), (5), and (6) (which are shown in (μ) Fig. 1) apply to the singularity probability for Nn . (μ)
Proof. Let αij be an entry of Nn . Since αij is symmetric and takes the value 0 with probability (μ) (μ) 1 − μ, we may write αij = γij ηij , where γij is an independent copy of γ (μ) as defined in Eq. (3) and ηij is a random variable that shares no values with −ηij . This description of αij was inspired by [1], and it allows us to condition on ηij and then use the remaining randomness in (μ) γij to get a bound on the singularity probability. In particular,
(μ) Pr Nn(μ) is singular = Pr Nn is singular {ηij = cij } Pr {ηij = cij } , (cij )
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2 where the sum runs
over all (n )-tuples (cij )1i,j n of possible values taken by random variables ηij . Since (cij ) Pr({ηij = cij }) = 1, we can complete the proof by proving an exponential (μ)
bound on Pr(Nn is singular | {ηij = cij }), and we will use Theorem 2.2 for this task. (μ) (μ) Consider the random matrix Nn |{ηij =cij } , where the i, j entry is the random variable cij γij (μ)
for some constant cij . Note that the entries of Nn |{ηij =cij } take values in S, a set with cardinality (μ)
O(1), and let φQ be the map from Lemma 4.1, which lets us pass to the case where Nn |{ηij =cij } has entries in Z/QZ. Defining θij := 2πφQ (cij ), we compute
Ee φQ cij γ (μ) t
ij
= 1 − μ + μ cos(θij t)
⎧ ⎪ 1 − μ + μ cos(θij t) ⎪ ⎨ 2μ+1 cos(2θij t) + (1 − μ) cos(θij t) + 2μ−1 4 4 ⎪ ⎪ ⎩ 1 − 2μ + 3 μ2 + 2(1 − μ)μ cos(θ t) + μ2 cos(2θ t)1/2 ij ij 2 2
for 0 μ 12 , for
1 2
μ 1,
and
for 0 μ 1.
(μ)
We have thus shown that the entries of Nn |{ηij =cij } are 1 (1 − μ)-bounded of exponent 1 for 0 μ , 2 1 2μ + 1 -bounded of exponent 1 for μ 1, and 4 2 3 1 − 2μ + μ2 -bounded of exponent 2 for 0 μ 1. 2 Applying Theorem 2.2 completes the proof.
2
Corollary 3.1 is tight for 0 μ 12 , since the probability of a row of all zeroes occurring is (1 − μ + o(1))n ; however, for any specific case, Theorem 2.2 can usually prove better upper bounds than those given by Corollary 3.1. (μ) For example, consider the case of a matrix M{±2,±1},n with each entry an independent copy of the symmetric random variable ⎧ +2 with probability μ4 , ⎪ ⎪ ⎪ μ ⎪ ⎪ ⎨ −2 with probability 4 , α (μ) := +1 with probability μ4 , ⎪ ⎪ ⎪ −1 with probability μ4 , ⎪ ⎪ ⎩ 0 with probability 1 − μ. (μ)
Corollary 3.2. For M{±2,±1},n as defined above, we have ⎧ n (μ) ⎨ (1 − μ + o(1)) Pr M{±2,±1},n is singular ⎩ 1 − 2μ + 5 μ2 + o(1) n 4
for 0 μ
16 25 ,
for 0 μ 1.
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Proof. By the definition of α (μ) we have
(μ)
Ee α t = 1 − μ + μ cos(2πt) + μ cos(4πt), 2 2
for 0 μ
16 25
(i.e., the right-hand side of the equation above is non-negative for such μ), which proves the first bound. Also, we have
(μ) 2
Ee α t = 1 − 2μ + 5 μ2 + μ − 3 μ2 cos(2πt) + μ − 7 μ2 cos(4πt) 4 4 8 +
μ2 μ2 cos(6πt) + cos(8πt) 4 8
for 0 μ 1, which proves the second bound.
2 (μ)
We also have the following lower bounds for the singularity probability of M{±2,±1},n : n 1 − μ + o(1)
(from one row of all zeroes), n (from a two-row dependency). 1 − 2μ + 5μ2 /4 + o(1)
(11) (12)
The results of Corollary 3.2 and the corresponding lower bounds are shown in Fig. 2, and one should note that the upper bounds are substantially better than those guaranteed by Corollary 3.1. 3.3. Random matrices with entries having arbitrary distributions A useful feature of the definition of p-bounded of exponent 2 is that it lets one bound the singularity probability of matrices with independent discrete random variables that are asymmetric. Corollary 3.3. Let p be a constant such that 0 < p 1 and let S ⊂ C be a set with cardinality |S| O(1). If Nn is an n by n matrix with independent random entries taking values in S such that for any entry α, we have maxx Pr(α = x) p, then √ n Pr(Nn is singular) p + o(1) . We will need the following slightly more general corollary in Section 3.4. For a set A and an m := { m a : a ∈ A}. integer m, we will use the notation mA := { m a : a ∈ A} and A j j j j =1 j =1 j Corollary 3.4. Let p be a constant such that 0 < p 1, let S ⊂ C be a set with cardinality |S| O(1), and let Xn be an n by n matrix with fixed, non-random entries in no(n) (S ∪ {−1, 0, 1})O(1) . If Nn is an n by n matrix with independent random entries taking values in S such that for any entry α, we have maxx Pr(α = x) p, then n √ Pr(Xn + Nn is singular) p + o(1) . Note that Corollary 3.4 implies Corollary 3.3 by taking Xn to be the matrix of all zeroes.
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Asymptotic Upper and Lower Bounds for Pr(M{±2,±1},n is singular)1/n for 0 μ 1
(μ)
(μ)
Fig. 2. Let P (μ) := limn→∞ Pr(M{±2,±1},n is singular)1/n , where M{±2,±1},n is the n by n matrix with independent random entries taking the value 0 with probability 1 − μ and the values +2, −2, +1, −1 each with probability μ/4. This figure summarizes the upper bounds on P (μ) from Corollary 3.2 and the lower bounds from Displays (11) and (12). The best upper bounds (shown in thick solid lines) match the best lower bounds (thick dashed lines) for 0 μ 16 25 ; and it is not hard to improve the upper bound a small amount by finding a bound (of exponent 1) to bridge the discontinuity. One should note that even as stated above, the upper bounds are substantially better than those given by Corollary 3.1 (which are shown in Fig. 1). The shaded area represents the gap between the upper and lower bounds.
Proof of Corollary 3.4. Let αij be an entry in Nn . Our goal is to describe αij in a two-step random process, condition on one of the steps, and then use the randomness in the other step to bound the singularity probability. The conditioning approach is the same as that used in the symmetric case (Corollary 3.1) and was inspired by [1]. The conditioning argument is useful since some entries of the random matrix may take some values with very small probability (i.e. probability less than any constant); recall that while the entries of the random matrix always take values in a fixed set S of cardinality O(1), the distributions of those random variables within S are allowed to vary with n. (Note that making use of Remark 8.5 would provide an alternate way of dealing with entries that take some values with very small probability.) Say that αij takes the values v1 , . . . , vt with probabilities 1 , . . . , t , respectively, where 1 2 · · · t . Define new random variables ηij k such that for some i0 and i1 , the values taken by
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ηij k are vi0 , vi0 +1 , . . . , vi0 +i1 with corresponding probabilities i0 /pk , i0 +1 /pk , . . . , i0 +i1 /pk ,
1 i0 +i . Thus, we can write where pk := ii=1
αij =
⎧ ηij 1 ⎪ ⎪ ⎪ ⎨η ij 2
.. ⎪ . ⎪ ⎪ ⎩ ηij
with probability p1 , with probability p2 , .. . with probability p .
(13)
Furthermore, the ηij k can be constructed so that pk p for every k, so that p/2 pk for 1 k − 1, and so that no two ηij k with different k’s ever take the same value. There are two cases to consider for the technical reason that p is not necessarily bounded below by a constant. Let > 0 be a very small constant, so for example p/2 > . Case 1 is when p , and in this case each pk is bounded below by and above by p. We will consider case 1 first and then discuss the small changes needed to deal with case 2. As in the proof of Corollary 3.1, we will condition on the values taken by the ηij k in order to prove a bound on the singularity probability. We have that Pr(Xn + Nn is singular) =
Pr Xn + Nn is singular {ηij k = cij k } Pr {ηij k = cij k } ,
(cij k )
where the sum runs over all possible values (cij k ) that the ηij k can take. Thus, it is sufficient to prove a bound on the singularity probability for the random matrix Xn + Nn |{ηij k =cij k } which has random entries ⎧ xij + cij 1 with probability p1 , ⎪ ⎪ ⎪ ⎨ xij + cij 2 with probability p2 , xij + α˜ ij = .. .. ⎪ . . ⎪ ⎪ ⎩ xij + cij with probability p , where xij and the cij k are constants. Note the entries of Xn + Nn |{ηij k =cij k } take values in no(n) (S ∪ {−1, 0, 1})O(1) , a generalized arithmetic progression with rank O(1) and cardinality at most no(n) , and let φQ be the map from Lemma 4.1, which lets us pass to the case where Xn + Nn |{ηij k =cij k } has entries in Z/QZ. Defining θij k := 2πφQ (cij k ) and letting α˜ ij be an i.i.d. copy of α˜ ij , we compute
Ee φQ (xij + α˜ ij )t 2 = Ee φQ xij + α˜ ij − xij − α˜ t = Ee φQ α˜ ij − α˜ t ij ij =
k=1
pk2 + 2
pk1 pk2 cos (θij k1 − θij k2 )t .
1k1
Thus, xij + α˜ ij is ( k=1 pk2 )-bounded of exponent 2 (using the constant q = 2 in Definition 2.1, so q does not depend on n). Given that 0 < pk p for every k, it is not hard to show that k=1 pk2 p < p + , and so from Definition 2.1, we see that the collection {xij + α˜ ij : α˜ ij has corresponding probability p } is (p + )-bounded of exponent 2. We are thus finished with case 1.
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Case 2 is when the decomposition of αij given in Eq. (13) has p < . In this case we need
to modify Eq. (13) slightly, deleting ηij and replacing ηij (−1) with a new variable ηij (−1) that takes all the values previously taken by ηij and by ηij (−1) with the appropriate probabilities. Thus, in case 2, we have that p/2 pk < p + for all 1 k − 1, showing that each pk is bounded below by a constant and is bounded above by p + (here we are using p−1 to denote
the probability that αij draws a value from the random variable ηij (−1) ). For case 2, we use exactly the same reasoning as in case 1 above to show that such entries
−1 2 of Xn + Nn |{ηij k =cij k } are ( k=1 pk )-bounded of exponent 2 (using the constant q = 2 < p 2 /4
−1 2 pk < p + and using Defin Definition 2.1, so q does not depend on n). Noting that k=1 inition 2.1, we see that the collection {xij + α˜ ij : α˜ ij has corresponding probability p < } is (p + )-bounded of exponent 2. Combining case 1 and case 2, we have that the collection {xij + α˜ ij } is (p + )-bounded of exponent 2, and so by and by Theorem 2.2 we have that Pr(Xn + Nn |{ηij k =cij k } is singular) √ ( p + + o(1))n . The constant > 0 was chosen arbitrarily, and so letting tend to zero, we get that
√ n Pr Xn + Nn is singular {ηij k = cij k } p + o(1) .
2
3.4. Partially random matrices In this subsection, we prove a bound on the singularity probability for partly random matrices where many rows are deterministic. Corollary 3.5. Let p be a real constant between 0 and 1, let K be a large positive constant, and let S ⊂ C be a set of complex numbers having cardinality |S| K. Let Nf,n be an n by n matrix in which f rows contain fixed, non-random integers between −K and K and where the other rows contain entries that are independent random variables taking values in S. If f o(n), if the f fixed rows are linearly independent, and if for every random entry α, we have maxx Pr(α = x) p, then Pr(Nf,n is singular)
√ n−f p + o(1) .
Corollary 3.5 applies to partly random matrices with f = o(n) fixed, non-random rows containing integers bounded by a constant and with random entries taking at most O(1) values in the complex numbers. Corollary 1.2, on the other hand, holds with the fixed entries also allowed to take values in the complex numbers and gives a slightly better bound, but additionally requires f O(ln n) (which is far smaller in general than o(n)). Proving Corollary 1.2 requires mirroring the entire argument used to prove the main theorem (Theorem 2.2) in the case where f rows contain fixed, non-random entires, and we discuss this argument in Section 9. Proving Corollary 3.5, however, can be done directly from Theorem 2.2, as we will show below. First, we will state a generalization of Corollary 3.5. Corollary 3.6. Let p be a real constant between 0 and 1, let K be a large positive constant, and let S ⊂ C be a set of complex numbers having cardinality |S| K. Let Nf,n be an n by n matrix in which f rows contain fixed, non-random integers between −K and K and where the other
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rows contain entries that are independent random variables taking values in S. If f o(n), if the fixed rows have co-rank k, and if for every random entry α, we have maxx Pr(α = x) p, then Pr(Nf,n has co-rank > k)
√ n−f p + o(1) .
To obtain Corollary 3.6 from Corollary 3.5, find a collection C of f − k linearly independent rows among the deterministic rows. Replace the rest of the deterministic rows with a collection C of rows containing integer values between −K and K such that C is linearly independent from C. Finally, apply Corollary 3.5 to the new partially random matrix whose deterministic rows are from C ∪ C , thus proving Corollary 3.6. Proof of Corollary 3.5. By reordering the rows and columns, we may write Nf,n =
A B C D
,
where A is an f by f non-random invertible matrix, B is an f by n − f non-random matrix, C is an n − f by f random matrix, and D is an n − f by n − f random matrix. Note that Nf,n is singular if and only if there exists a vector v such that Nf,n v = 0. Let v1 be the first f coordinates of v and let v2 be the remaining n − f coordinates. Then Nf,n v = 0 if and only if
Av1 + Bv2 = 0,
and
Cv1 + Dv2 = 0. Since A is invertible, these two equations are satisfied if and only if (−CA−1 B + D)v2 = 0, that is, if and only if the random matrix −CA−1 B + D is singular. We want to show that every entry that can appear in −CA−1 B is an element of no(n) × (S ∪ {−1, 0, 1})O(1) . By the cofactor formula for A−1 , we know that the i, j entry of A−1 is (−1)i+j (det Aij )/ det A, where Aij is the f − 1 by f − 1 matrix formed by deleting the ith row where the i, j entry of A is (−1)i+j det Aij . By the and j th column of A. Thus, A−1 = det1 A A, volume formula for the determinant, we know that | det A| is at most the product of the lengths of the row vectors of A; and thus | det A| no(n) (here we need that A has integer entries between −K and K, where K is a constant, and that f o(n)). Similarly, we have | det Aij | no(n) . is thus in no(n) {−1, 0, 1}, every entry of C is in S, and every entry of B is in Every entry of A is an element of no(n) (S ∪ {−1, 0, 1}). O(1){−1, 0, 1}; thus, every entry of −C AB Conditioning on the values taken by all the entries in C, we have Pr(Nf,n is singular) = Pr −CA−1 B + D is singular
Pr −CA−1 B + D is singular C = (cij ) Pr C = (cij ) , (14) = (cij )
where the sum runs over all possible matrices (cij ) that C can produce. Considering the entries in C = (cij ) to be fixed (note that A and B are fixed by assumption), we now need to bound + (det A)D is singular . Pr −(cij )A−1 B + D is singular = Pr −(cij )AB
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is an element of no(n) (S ∪ {−1, 0, 1})O(1) and that the random Note that every entry of −(cij )AB matrix (det A)D has entries that take values in the fixed set {(det A)s: s ∈ S} having cardinality O(1). Thus, by Corollary 3.4, we have that + (det A)D is singular √p + o(1) n−f . Pr −(cij )AB 2
Plugging this bound back into Eq. (14) completes the proof. 3.5. Integer matrices and rational eigenvalues
Let ηk be the random variable taking the values −k, −k + 1, . . . , k − 1, k each with equal probability, and let Mn be the n by n matrix where each entry is an independent copy of ηk . In [7], Martin and Wong show that for any > 0, Pr(Mn has a rational eigenvalue)
c(n, ) , k 1−
where c(n, ) is a constant depending on n and . (One goal in [7] is to study this bound as k goes to ∞ while n is fixed, which is why c(n, ) is allowed to depend on n.) Below, we prove a similar result for random integer matrices with entries between −k and k (with k fixed), where we allow each entry to have a different (independent) distribution and we also allow the distributions to be very general. Corollary 3.7. Fix a positive integer k, and let Mk,n be a random integer matrix with independent entries, each of which takes values in the set {−k, −k + 1, . . . , k − 1, k}. Let c be a constant such that for every entry α, we have max−kxk Pr(α = x) c/k. Then Pr(Mk,n has a rational eigenvalue)
n/2 c + o(1) , k
where the o(1) term goes to zero as n goes to ∞. For example, in the case where each independent entry has the uniform distribution on {−k, −k + 1, . . . , k − 1, k} (as in [7]), one can set c = 1/2 in the corollary above. Proof. The proof given below follows the same outline as the main theorem of [7], with Corollary 1.2 replacing an appeal to [7, Lemma 1]. The characteristic polynomial for Mk,n is monic with integer coefficients, and thus the only possible rational eigenvalues are integers (by the rational roots theorem). Every eigenvalue of Mk,n has absolute value at most nk (see [7, Lemma 4]); thus, the only possible integer eigenvalues are between −nk and nk. The matrix Mk,n has λ as an eigenvalue if and only if Mk,n − λI is singular (where I is the n by n identity matrix). By Corollary 1.2 (with f = 0), we have n c + o(1) . Pr(Mk,n − λI is singular) k Using the union bound, we have
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Pr(Mk,n has a rational eigenvalue) = Pr Mk,n − λI is singular, for some λ ∈ {−nk, . . . , nk}
nk
Pr(Mk,n − λI is singular)
λ=−nk
n c (2nk + 1) + o(1) k n/2 c + o(1) . 2 k 4. Random matrices with complex entries: A reduction technique The original work on discrete random matrices in [4,5,9,10] is concerned with matrices having integer entries, which can also be viewed as matrices with entries in Z/QZ where Q is a very large prime. In this section we show that one can pass from a (random) matrix with entries in C to one with entries in Z/QZ where Q is an arbitrarily large prime number, all without affecting the probability that the determinant is zero, thanks to the following lemma. Lemma 4.1. (See [12].) Let S be a finite subset of C. There exist infinitely many primes Q such that there is a ring homomorphism φQ : Z[S] → Z/QZ satisfying the following two properties: (i) the map φQ is injective on S, and (ii) for any n by n matrix (sij )1i,j n with entries sij ∈ S, we have det (sij )1i,j n = 0 if and only if
det φQ (sij ) 1i,j n = 0.
In order to apply this lemma, let us point out that the proof of Theorem 2.2, which is discussed in Sections 5 through 8, works exclusively in Z/QZ; though at various points, it is necessary to assume Q is extremely large with respect to n and various constants. For this paper, S will be the set of all possible values taken by the random variables αj k . Recall that by assumption, |S| no(n) , so in particular, S is finite. Remark 4.2 (On the size of Q). When we apply Lemma 4.1, we will take Q > exp(exp(Cn)) for some constant C in order for Freiman-type theorems such as [10, Theorem 6.3] (which is restated in Theorem 8.1 below) to apply, and we will also choose Q large enough so that the integral approximation in Inequality (A.1) holds and so that Q is large with respect to various constants. One should note that while Q can be taken arbitrarily large with respect to n, we cannot choose Q so that it is arbitrarily large with respect to φQ (s) for all s ∈√S, where S is the set of all values that could appear in the given random matrix. For example, √ √ the smallest positive √ if 2 ∈ S, then integer representative for φQ ( 2) must be larger than Q (since (φQ ( 2))2 = 2 in Z/QZ). Finally, if we were in a situation where S ⊂ Q, then we could avoid using Lemma 4.1 altogether by clearing denominators to pass to Z and then take Q ≈ exp(exp(Cn)), as is done in [10]. Lemma 4.1 is a corollary of the main theorem of [12] and its proof is given in detail in [12, Section 6]. The paper [12] also contains further applications of the method used to prove Lemma 4.1,
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for example proving a sum-product result for the complex numbers and proving a Szemerédi– Trotter-type result for the complex numbers, where the applications follow from the analogous results for Z/Q where Q is a prime (see [3]). The results in [12], including Lemma 4.1, all go through with the complex numbers being replaced by any characteristic zero integral domain. Thus, the results stated in Sections 1, 2, and 3 above for the complex numbers C also all go through with C replaced by any characteristic zero integral domain. For example, Corollary 3.3 becomes Corollary 4.3. Let p be a constant such that 0 < p 1 and let D be a characteristic zero integral domain. Let S ⊂ D have cardinality |S| O(1). If Nn is an n by n matrix with independent random entries, each taking values in S, such that for every entry α, we have maxx Pr(α = x) p, then √ n Pr(Nn is singular) p + o(1) . 5. Proof of the main theorem (Theorem 2.2) The proof of Theorem 2.2 very closely follows the proof of [10, Theorem 1.2]. Our goal is to highlight the changes that need to be made to generalize the proof in [10] so that it proves Theorem 2.2. A reader interested in the details of the proof of Theorem 2.2 should read this paper alongside of [10]. Throughout the proof, we will assume that n is sufficiently large, and we will allow constants hidden in the o(·) and O(·) notation to depend on the constants −1 , 0 , 1 , 2 , p, q, r, cMedDim , cLgDim , cLO , and cm . The constants −1 , 0 , 1 , 2 should be considered very small, and, in fact, we will let them tend to zero to prove the full strength of Theorem 2.2. The constants p, q, r, cMedDim , cLgDim , cLO , and cm can be thought of as absolute, except possibly for depending on each other. 5.1. Definitions and preliminaries Given an n by n matrix Nn with entries αij , we assume that the collection of independent random variables {αij }1i,j n is p-bounded of exponent r for some fixed constants p, q, and r (here, q is the constant from Definition 2.1 which is independent of n). We also assume that each αij takes at most no(n) distinct values. Using Lemma 4.1, we may assume without loss of generality that each αij takes values in Z/QZ for some very large prime Q. The entirety of the proof will take place over the field Z/QZ, and so terminology such as “linearly independent”, “span”, “dimension”, “rank” and so forth will always be with respect to the field Z/QZ. Let Xi := (αi,1 , . . . , αi,n ) denote the ith row of Nn . We note that Nn has determinant zero if and only if there is a linear dependency among the rows Xi . It has been shown (see [9, Lemma 5.1] and also [4]) that the dominant contribution to the singularity probability comes from the Xi spanning a hyperplane (of dimension n − 1). In particular, Pr(Nn is singular) = p −o(n)
Pr(AV ),
(15)
V a non−trivial hyperplane in (Z/QZ)n
where AV denotes the event that X1 , . . . , Xn span V , and non-trivial means that V contains the origin, V is spanned by vectors in S n (where S is the set of all possible values that can occur in Nn ), and Pr(Xi ∈ V ) > 0 for all i.
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As in [10], we will divide the non-trivial hyperplanes into n2 classes, since it is then sufficient to show that the sum of Pr(AV ) over all V in a particular class is at most (p 1/r + o(1))n . Definition 5.1 (Combinatorial dimension). Let D := { an : 0 a n2 , a ∈ Z}. For any d± ∈ D such that d± n1 , we define the combinatorial Grassmannian Gr(d± ) to be the set of all nontrivial hyperplanes V in (Z/QZ)n such that p n−d± +1/n < max Pr(Xi ∈ V ) p n−d± . 1in
(16)
For d± = 0, we define Gr(0) to be the set of all non-trivial hyperplanes such that max Pr(Xi ∈ V ) p n .
1in
We will refer to d± as the combinatorial dimension of V . Note that Gr(d± ) = ∅ for d± n − 1 + 1/n (by Lemma B.1). We will consider hyperplanes V with combinatorial dimension in three main regions: d± small, d± medium-sized, and d± large. The two lemmas and the proposition below suffice to prove Theorem 2.2. Lemma 5.2 (Small combinatorial dimension [4,9,10]). For any δ > 0 we have Pr(AV ) nδ n . d± ∈D s.t. p n−d± δ n V ∈Gr(d± )
In proving Theorem 2.2, we will take δ = (p + cMedDim 0 )1/r to take care of all small d± not covered by Proposition 5.4 below. Proof. The reasoning here is the same as in [10, Lemma 2.3], making use of fact that Pr(Xi ∈ V ) max1in Pr(Xi ∈ V ) p n−d± δ n . In particular, Pr(AV )
n
Pr {Xj }1j n \ {Xi } spans V Pr(Xi ∈ V ),
i=1
which completes the proof since the summing the right-hand side over all V is at most n maxi Pr(Xi ∈ V ) (note that an instance of the vectors {Xj }1j n \ {Xi } can span at most one hyperplane). 2 Lemma 5.3 (Large combinatorial dimension, [4,9,10]). We have n Pr(AV ) p + o(1) d± ∈D s.t.
cLgDim √ p n−d± n
V ∈Gr(d± )
Here we choose the constant cLgDim so that cLgDim cLO p −1/n 2r q , where cLO is the constant from the Littlewood–Offord inequality (see Lemma A.1 in Appendix A) and q is the constant from Definition 2.1.
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Proof. Our proof is essentially the same as [10, Lemma 2.4]. Fix V ∈ Gr(d± ), where p n−d± .
cLgDim √ n
Let imax be an index such that Pr(Ximax ∈ V ) = max1in Pr(Xi ∈ V ). By assumption, Pr(Ximax
c 2r LgDim . ∈ V ) p n−d± +1/n √ p 1/n cLO qn n
Noting that Ximax ∈ V if and only if Ximax is orthogonal to the normal vector for V , we have by Lemma A.1 that r , Pr(Ximax ∈ V ) cLO qk where k is the number of nonzero coordinates in the normal vector to V . Combining the two inequalities above shows that k n/2. Thus, we have ⎫⎞ ⎛⎧ ⎨ there exists a vector v with at ⎬ Pr(AV ) Pr ⎝ most n/2 nonzero coordinates ⎠ ⎩ ⎭ cLgDim such that Nn · v = 0 d± ∈D s.t. √ p n−d± V ∈Gr(d± ) n
n p + o(1)
(by Lemma A.2).
(Lemma A.2 is a natural generalization of [4, Section 3.1]; see also [6], [9, Lemma 5.1], and [2, Lemma 14.10].) 2 Proposition 5.4 (Medium combinatorial dimension estimate). Let 0 < 0 be a constant much cLgDim smaller than 1, and let d± ∈ D be such that (p + cMedDim 0 )n/r < p n−d± < √ . Then n Pr(AV ) o(1)n . V ∈Gr(d± ) 1 Here we choose the constant cMedDim so that cMedDim > ( 100 +cm ), where cm is some absolute 1 constant such that 0 < cm < 1 (the 100 here comes from μ as defined in Section 5.2 below; 1 ). in [10], it happens that the constant cm is also taken to be 100 To prove Theorem 2.2, we can simply combine Lemma 5.2 with δ = (p + cMedDim 0 )1/r , Lemma 5.3, and Proposition 5.4. Thus, proving Proposition 5.4 will complete the proof of Theorem 2.2. To prove Proposition 5.4, as in [10, Proposition 2.5], we will separate hyperplanes V of medium combinatorial dimension into two classes, which we will call exceptional and unexceptional (see Definition 5.5). See [10, Section 3] for motivation. The unexceptional case will be proved in the remainder of this section, and the exceptional case will be proved in Sections 6, 7, and 8. The results in [9] and [4] were derived using the ideas that we will use for the unexceptional medium combinatorial dimension case. The idea of considering the exceptional case separately in [10] (and using tools from additive combinatorics in the exceptional case) is what lead to the improvement of Inequality (1), which gives a bound of asymptotically ( 43n ), over the 0.999n bound in [4].
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5.2. Proof of the medium combinatorial dimension Before defining exceptional and unexceptional hyperplanes, we will need some new notation. By assumption, the collection of random variables {αij }1i,j n is p-bounded of exponent r (μ) with a constant μ = 1 − p, with random variables βij corresponding to each αij , and with a constant 0 < q p (see Definition 2.1). We also define a constant slightly smaller than μ, 0 (μ) (μ) . We will let Yi := (yi,1 , . . . , yi,n ) := (βi,1 , . . . , βi,n ) denote another row namely μ := μ − 100 (μ) vector that corresponds to the row vector Xi (βi,j comes from the definition of p-bounded of exponent r). Also, we will let kstart −1 zeroes
n−kend zeroes
∗ := (0, . . . , 0, yi,kstart , . . . , yi,kend , 0, . . . , 0), Zi,k
(17)
∗ can be thought of as the kth where kstart := (k − 1) nr + 1 and kend := k nr . The vector Zi,k ∗ are both defined using segment of Yi (out of r roughly equal segments). Note that Yi and Zi,k 0 μ := μ − 100 , not μ. Finally, let 1 be a positive constant that is small with respect to 0 , cm , and r.
Definition 5.5 (Exceptional and unexceptional). Consider a hyperplane V of medium combinatorial dimension (that is, d± satisfies the condition in Proposition 5.4). We say V is unexceptional if there exists an i0 where 1 i0 n and there exists a k0 where 1 k0 r such that max Pr(Xj ∈ V ) < 1 Pr Zi∗0 ,k0 ∈ V . 1j n
We say V is exceptional if for every i where 1 i n and for every k where 1 k r we have ∗ ∈ V max Pr(Xj ∈ V ) . (18) 1 Pr Zi,k 1j n
In particular, there exists imax such that Pr(Ximax ∈ V ) = max1j n {Pr(Xj ∈ V )}; and so if V is exceptional, then (19) 1 Pr Zi∗max ,k ∈ V Pr(Ximax ∈ V ) for every k. We will refer to Ximax as the exceptional row. Inequality (10) following Definition 2.1 can be used to give another relationship between Pr(Zi∗max ,k ∈ V ) and Pr(Ximax ∈ V ) that, together with Inequality (19), will be of critical importance in Section 7. Proposition 5.4 follows from the two lemmas below, so long as 1 is chosen suitably small with respect to 0 , cm , and r. Lemma 5.6 (Unexceptional space estimate). We have c n/r Pr(AV ) p −o(n) 2n 1m 0 . V ∈Gr(d± ): V is unexceptional
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Lemma 5.7 (Exceptional space estimate). We have
Pr(AV ) n− 2 +o(n) . n
V ∈Gr(d± ): V is exceptional
We will prove Lemma 5.6 in Section 5.3, and we will prove Lemma 5.7 in Section 6. 5.3. The unexceptional medium combinatorial dimension case The general idea for the case of an unexceptional hyperplane V is to replace some of the rows Xi in the matrix Nn with rows that concentrate more sharply on the subspace V . In the case (μ)
(μ)
where the exponent r = 1, replacing a row Xi with Yi := (βi,1 , . . . , βi,n ) is successful; however, in the exponent r = 2 case, for example, replacing the entire row results in a bound that is off by an exponential factor. We solve this problem by replacing Xi with only half of Yi (with the other half of the entries being zero). This idea easily extends to any integer r 2 and is the motivation ∗ to have all zeros except for roughly n/r coordinates, as is done in for defining the vectors Zi,k Eq. (17). The basic utility of Zi∗0 ,k0 (from Definition 5.5) is that it concentrates more sharply on the unexceptional subspace V than the vector Xi for any i. Let Zi∗0 ,k0 be the vector from the definition of unexceptional (Definition 5.5) such that Pr(Xi ∈ V ) < 1 Pr(Zi∗0 ,k0 ∈ V ) for every i, and set Z := Zi∗0 ,j0 . Let m be the closest integer to cmr0 n , where cm is a small positive absolute constant (for example, in [10], cm is taken to 1 be 100 ). Finally, let Z1 , . . . , Zm be copies of Z, independent of each other and of X1 , . . . , Xn . Lemma 5.8. (See Lemma 4.4 in [10].) Let BV ,m be the event that Z1 , . . . , Zm are linearly independent and lie in V . Then, Pr(BV ,m ) p o(n)
max1in Pr(Xi ∈ V ) 1
m .
Proof. The argument follows the same reasoning as [10, Lemma 4.4], however, the quantity 2d± −n in [10] should be replaced by max1in Pr(Xi ∈ V ). Details are provided in Appendix B. 2 To conclude the proof of Lemma 5.6, we follow the “row-swapping” argument at the end of [10, Section 4], with the small change of bounding Pr(Xi ∈ V ) by max1in Pr(Xi ∈ V ), which we use in place of the quantity 2d± −n . Details are provided in Appendix B. 6. Analyzing the exceptional medium combinatorial dimension case The approach for exceptional V in [10] is very different from that used in the unexceptional case or in the large or small combinatorial dimension cases. Using some powerful tools from additive combinatorics, the general idea is to put exceptional hyperplanes V in correspondence with a particular additive structure called a generalized arithmetic progression, and then to show that the number of the particular generalized arithmetic progression s that arise in this way is exceedingly small. The key to this approach is a structure theorem—namely, [10, Theorem 5.3]. In this section, we state a slightly modified structure theorem (Theorem 6.1), and then we show
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how to use Theorem 6.1 to prove Lemma 5.7. In the beginning of Section 7, we outline the changes needed to prove the structure theorem for our current context, and in Sections 7 and 8 we provide details. Before stating the structure theorem, we need some definitions and notation. A generalized arithmetic progression of rank r is a set of the form P = v0 + m1 v1 + · · · + mr : |mi | Mi /2 , where the basis vectors v0 , v1 , . . . , vr are elements of a Z-module (here, Z/QZ) and where the dimensions M1 , . . . , Mr are positive integers. We say that vi has corresponding dimension Mi . For a given element a = v0 + m1 v1 + · · · + mr in P , we refer to m1 , . . . , mr as coefficients for a. A generalized arithmetic progression P is symmetric if v0 = 0, and P is proper if for each a ∈ P , the there is a unique r-tuple (m1 , . . . , mr ) with |mi | < Mi /2 that gives the coefficients for a. If P is proper and symmetric, we define the P -norm aP of an element a ∈ P to be aP :=
!1/2 r mi 2 i=1
Mi
.
We will use the notation mP , where m is a positive integer, to denote the set { m i=1 xi : xi ∈ P } x : x and the notation P m , where m is a positive integer, to denote the set { m i ∈ P }. If P is i=1 i a generalized arithmetic progression of rank r, then so is mP , while P m , on the other hand, is a generalized arithmetic progression of rank at most rm . Also note that |mP | mr |P | and that |P m | |P |m . Let V be an exceptional hyperplane of medium combinatorial dimension in Gr(d± ) and let Ximax = (α1 , . . . , αn ) be the exceptional row (here we are using αj as shorthand for αimax ,j ). Let (μ) (μ) (β1 , . . . , βn ) be the row of random variables corresponding to Ximax from the definition of p(μ) bounded of exponent r, and let bj,s with 1 j n and 1 s j be the values taken by βj (μ)
(see Eq. (9) for the definition of βj ). Given an exceptional hyperplane V , there exists a representation of the form V = (x1 , x2 , . . . , xn ) ∈ (Z/QZ)n : x1 a1 + x2 a2 + · · · + xn an = 0 for some elements a1 , a2 , . . . , an ∈ Z/QZ. We will call a1 , a2 , . . . , an the defining coordinates of V . Finally, let a˜ j := bj,1 aj . We will refer to (a˜ 1 , . . . , a˜ n ) as the scaled defining coordinates of V . Note that once imax is fixed, so are the elements bj,1 . We should also note that the choice (μ) of bj,1 among bj,s for 1 s j is arbitrary—since βj takes the values bj,s each with probability at least q, any value of s will do; and so we have taken s = 1 for convenience. Let H denote the highly rational numbers, that is, those numbers in Z/QZ of the form a/b (mod Q) where a, b are integers such that |a|, |b| no(n) and b = 0. The highly rational numbers were defined in [10, Section 8], and we will need a small extension for the current paper, due to the fact that we are using the scaled defining coordinates of V instead of simply the defining coordinates of V . If we were to assume that bj,1 was an O(1) integer for all j and that every possible value taken by αij was an O(1) integer for all i, j , then we could still use the same definition of highly rational as in [10]. However, if there is a bj,1 or an entry αij in the matrix Nn that ever takes an irrational value, then when we pass to Z/QZ using Lemma 4.1 we have to account for values possibly on the order of Q (see Remark 4.2), and the highly rational numbers are
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not sufficient for this task. We can overcome this difficulty by extending to the highly T -rational numbers, which contain the highly rational numbers along with all the values in a structured set T (described below). We will now give a rigorous definition the highly T -rational numbers. Let T be a generalized arithmetic progression in Z/QZ with rank O(1) and having cardinality at most no(n) . As in the definition of p-bounded of exponent r (Definition 2.1), we will take S to be the generalized arithmetic progression containing all possible values in Z/QZ taken by the random variables αij that are the entries of Nn ; thus, by assumption |S| no(n) . By the definition (μ) of p-bounded of exponent r, we know that all of the random variables βij take values in a set with cardinality O(1). Thus, there is a symmetric generalized arithmetic progression T with rank O(1) and cardinality |T | no(n) such that T contains S, such that T contains the set {−1, 0, 1}, (μ) and such that T contains all the values taken by the βij . To construct T from S, one can, for example, add each distinct value taken by a βij as a new basis vector v with corresponding dimension M := 3 (say). A highly T -rational number h is any element of Z/QZ of the form a/b, where a, b ∈ no(n) T O(1) . Note that therefore, the cardinality of the highly T -rational numbers is at most (ndo(n) |T |)O(1) = no(n) , where d = O(1) is the rank of T (here we used the fact that |T | no(n) ). (μ)
Theorem 6.1 (Structure theorem). There is a constant C = C(−1 , 0 , 1 , 2 , q, r, μ) such that the following holds. Let V be an exceptional hyperplane and let a˜ 1 , . . . , a˜ n be its scaled defining coordinates (as described above). Then there exist integers 1rC and M1 , . . . , Mr 1 with the volume bound M1 · · · Mr C Pr(Ximax ∈ V )−1 and nonzero elements v1 , . . . , vr ∈ Z/QZ such that the following holds • (i) (Scaled defining coordinates lie in a progression) The symmetric generalized arithmetic progression P := {m1 v1 + · · · + mr vr : −Mi /2 < mi < Mi /2} is proper and contains all of the a˜ j . • (ii) (Bounded norm) The a˜ j have small P -norm: n
a˜ j 2P C.
j =1
• (iii) (Rational T -commensurability) The set {v1 , . . . , vr } ∪ {a˜ 1 , . . . , a˜ n } is contained in the set {hv1 : h is highly T -rational}.
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Note that unlike [10], part (iii) above does not necessarily place {v1 , . . . , vr } ∪ {a˜ 1 , . . . , a˜ n } in a simple arithmetic progression. We will discuss the proof of the structure theorem in Sections 7 and 8. In the remainder of this section, we will discuss how to use the structure theorem to prove Lemma 5.7. Fix d± of medium combinatorial dimension (see Proposition 5.4). Using independence of the rows, we have
Pr(AV )
V ∈Gr(d± ): V is exceptional
n "
Pr(Xi ∈ V )
V ∈Gr(d± ): i=1 V is exceptional
$n
# V ∈ Gr(d± ): V is exceptional · max Pr(Xi ∈ V ) . 1in
(20)
In [10, Section 5], it is shown using Theorem 6.1(i) and (ii) and Gaussian-type methods (and the fact that r is bounded by a constant) that o(n)
V ∈ Gr(d± ): V is exceptional n Q−1
n 1 + n−1/2 M1 · · · Mr ,
r,{M1 ,...,Mr } {v1 ,...,vr }
where the sum runs over all possible values for r, for the Mi , and for v1 , . . . , vr . By Theorem 6.1, we know that r C = O(1) and that Mi M1 M2 · · · Mr C Pr(Ximax ∈ V )−1 O(1/p n );thus, there are at most no(n) choices for r and the Mi . Furthermore, there are at most Q − 1 choices for v1 (since v1 = 0), and once the value for v1 has been fixed, (iii) tells us that there are at most no(n) choices for {v2 , . . . , vr } (since |no(n) T O(1) | no(n) ). Thus, the sum runs over at most no(n) terms. (This is the point in the proof where it is essential that no(n) T O(1) has cardinality no(n) .) Plugging the volume bound on M1 · · · Mr into the previous displayed inequality, we have
V ∈ Gr(d± ): V is exceptional no(n) 1 + n− 12 C Pr(Xi ∈ V )−1 n max = n− 2 +o(n) Pr(Ximax ∈ V )−n , n
(21)
c
√ , which is a consequence of d± being of medium comusing the fact that Pr(Ximax ∈ V ) LgDim n binatorial dimension. Plugging in Inequality (21) into Inequality (20) and summing over all d± of medium combinatorial dimension completes the proof of Lemma 5.7 (recall that by assumption max1in Pr(Xi ∈ V ) = Pr(Ximax ∈ V )).
7. Halász-type arguments The proof of the structure theorem has two main ingredients: tools from additive combinatorics, and Halász-type arguments using discrete Fourier analysis. Our proof of Theorem 6.1 will follow the proof of [10, Theorem 5.2] very closely. We will use results about additive combinatorics from [10, Section 6] directly, and we will discuss below the extent to which the Halász-type arguments of [10, Section 7] need to be modified to work for our current context. The proof of Theorem 6.1 will be given in Section 8 using results from the current section, [10, Section 6],
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[10, Section 7], and [10, Section 8]. Our Section 8 follows [10, Section 8] closely, with a few modifications to prove rational T -commensurability instead of only rational commensurability. In this section we discuss modifications to the lemmas in [10, Section 7] that are needed in order to prove Theorem 6.1. We will use eQ (·) to denote the primitive character eQ (x) := exp(2πix/Q). Let imax be the index of the exceptional row, so for every 1 k r we have 1 Pr(Zi∗max ,k ∈ V ) Pr(Ximax ∈ V ),
(22)
and recall that by Definition 5.5 we have Pr(Ximax ∈ V ) = maxi Pr(Xi ∈ V ). Let (α1 , . . . , αn ) := (μ) (μ) Ximax with the corresponding random variables (β1 , . . . , βn ) from the definition of p-bounded of exponent r (see Definition 2.1 and Eq. (9)), and let (a1 , . . . , an ) be the defining coordinates of V . Then, using the Fourier expansion, we can compute
Pr(Ximax
1 ∈ V ) = E(1{Ximax ∈V } ) = E Q
1 Q 1 Q
1 = Q 1 Q
n "
eQ
n
!! αj aj ξ
j =1
ξ ∈Z/QZ
E eQ (αj aj ξ )
ξ ∈Z/QZ j =1
n " (μ) 1/r E eQ βj aj ξ
ξ ∈Z/QZ j =1
n "
1−μ+μ
j
ξ ∈Z/QZ j =1
s=1
n "
j
1−μ+μ
ξ ∈Z/QZ j =1
r "
!1/r pj,s cos(2πbj,s aj ξ/Q)
(23)
!1/r pj,s cos(2πbj,s aj ξ/Q)
s=1
1/r Pr Zi∗max ,k ∈ V ,
(24)
k=1
where the last line is an application of Hölder’s inequality. Define f (ξ ) :=
n "
1−μ+μ
j =1
fj (ξ ) := 1 − μ + μ
j
!1/r pj,s cos(2πbj,s aj ξ/Q)
,
(25)
and
(26)
s=1 j s=1
!1/r pj,s cos(2πbj,s aj ξ/Q)
,
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gk (ξ ) :=
"
1−μ+μ
(k−1) nr <j k nr
j
585
!1/r pj,s cos(2πbj,s aj ξ/Q)
,
(27)
s=1
0 where μ := μ − 100 , as defined in Section 5.2. Note that f (ξ ) = We will need the following analog of [10, Lemma 7.1]:
n
j =1 fj (ξ ).
Lemma 7.1. For all ξ ∈ Z/QZ, we have n "
fj (ξ )rμ/μ
j =1
r "
gk (ξ ).
k=1
Proof. This inequality may be proven pointwise (for each j after expanding out the definition of gk ) using the convexity of the log function, just as in the proof of [10, Lemma 7.1] (see also [9, Lemma 7.1]. 2 Let 2 be sufficiently small compared to 1 (we will specify how small in Inequality (33) while proving Lemma 7.2). Following [10], we define the spectrum Λ ⊂ Z/QZ of {b1,1 a1 , . . . , bn,1 an } = {a˜ 1 , . . . , a˜ n } (the scaled defining coordinates of V ) to be Λ := ξ ∈ Z/QZ: f (ξ ) 2 . (28) Let xR/Z denote the distance from x ∈ R to the nearest integer. Using the elementary inequality 1 x2R/Z , we have cos(2πx) 1 − 100 n j μ pj,s bj,s aj ξ/Q2R/Z f (ξ ) exp − 100r j =1 s=1 ! n q 2 bj,1 aj ξ/QR/Z exp − 50r
!
(29)
j =1
(μ)
(μpj,1 2q since minx Pr(βj = x) q by Definition 2.1). Thus, there is a constant C(2 , q, r) such that !1/2 !1/2 n n 2 2 a˜ j ξ/QR/Z = bj,1 aj ξ/QR/Z C(2 , q, r), j =1
(30)
j =1
for every ξ ∈ Λ. (E.g., the constant C(2 , q, r) :=
50r q
1/2 ln( 12 ) suffices.)
Lemma 7.2. There exists a constant C depending on −1 , 0 , 1 , 2 , q, r, and μ such that C −1 Q Pr(Ximax ∈ V ) |Λ| CQ Pr(Ximax ∈ V ). Furthermore, for every integer k 4 we have C+k−3 |kΛ| CQ Pr(Ximax ∈ V ). k−2
(31)
(32)
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Proof. Our goal is to bound ξ ∈Λ f (ξ ) from above and below, and then pass to bounds on |Λ| using the fact that 2 f (ξ ) 1 for all ξ ∈ Λ. Note that 1 Q
by Eq. (25) and Eq. (23) .
f (ξ ) Pr(Ximax ∈ V )
ξ ∈Z/QZ
Also, 1 1 " 1−μ/μ 1 " f (ξ ) = fj (ξ ) 2 fj (ξ )μ/μ Q Q Q n
ξ ∈Λ /
n
ξ ∈Λ / j =1
1 Q
1−μ/μ
2
1−μ/μ
gk (ξ )1/r
ξ ∈Z/QZ k=1
1−μ/μ 1 2 Q
2
ξ ∈Λ / j =1
r "
r "
(Lemma 7.1)
!1/r (Hölder’s inequality)
gk (ξ )
k=1 ξ ∈Z/QZ
1 Pr(Ximax ∈ V ) 1
by Inequality (22) .
For the lower bound, we have
f (ξ ) =
ξ ∈Λ
f (ξ ) −
f (ξ )
ξ ∈Λ /
ξ ∈Z/QZ
1−μ/μ
Q Pr(Ximax ∈ V ) − = Q Pr(Ximax
2
1
Q Pr(Ximax ∈ V )
1−μ/μ 2 . ∈V) 1− 1
We can choose 2 sufficiently small with respect to 1 and 1 − μ/μ so that, for example, 1−μ/μ
1−
2
1
1 . 2
(33)
For the upper bound, we have ξ ∈Λ
f (ξ )
f (ξ )
ξ ∈Z/QZ
Q
r "
1/r Pr Zi∗max ,k ∈ V
Inequality (24)
k=1
Q
1 Pr(Ximax ∈ V ) 1
Inequality (22) .
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Thus, we have shown that ξ ∈Λ f (ξ ) = Θ(Q Pr(Ximax ∈ V )). Since 2 f (ξ ) 1 for all ξ ∈ Λ, we have proven Inequality (31). Making use of [10, Lemma 6.4], we can prove Inequality (32) by showing |4Λ| C|Λ| for some constant C. Using Lemma 7.3 below (for which we need to assume strict positivity (μ) of E(e(βj t))—see Remark 2.3), we have that there exists a constant c := c(−1 , 2 ) such that f (ξ ) c(−1 , 2 ), for every ξ ∈ 4Λ. Thus, |4Λ|
1 c(−1 , 2 )
f (ξ )
ξ ∈Z/QZ
1
Q Pr(Ximax ∈ V ) = C|Λ|, c(−1 , 2 ) 1
for some constant C. This completes the proof of Lemma 7.2.
2
We now state and prove a lemma showing that f (ξ ) is at least a constant for all ξ ∈ 4Λ. In [10], the lemma below is unnecessary because an inequality following from [10, Inequality (30)] (which corresponds to Inequality (30)) and the triangle inequality suffices. Lemma 7.3. Let Λ and f be defined as in Eq. (28) and Eq. (25), respectively. If ξ ∈ 4Λ, then ln(1/ ) 320000 f (ξ ) 2 −1 2 =: c(−1 , 2 ). Note that c(−1 , 2 ) is a constant. Proof. Note that Inequality (29) implies that for any ξ ∈ Λ we have j n
!1/2 pj,s bj,s aj ξ
j =1 s=1
/Q2R/Z
1/2 1 100r ln . μ 2
Thus, by the triangle inequality, we have for any ξ ∈ 4Λ that j n j =1 s=1
!1/2 pj,s bj,s aj ξ/Q2R/Z
4
1/2 1 100r ln . μ 2
(34)
Fix ξ ∈ 4Λ. Let k0 be the number of indices j such that 100μ
j s=1
1 pj,s bj,s aj ξ/Q2R/Z > , 2
and without loss of generality, say that these indices are j = 1, 2, . . . , k0 . Squaring Inequalk0 1 1600r ity (34), we see that 200μ μ ln( 2 ), and so we have k0 320000r ln
1 , 2
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which is a constant. Thus, for the vast majority of the indices j , namely j = k0 + 1, k0 + 2, . . . , n, we have 100μ
j s=1
1 pj,s bj,s aj ξ/Q2R/Z . 2
(35)
We may now compute that f (ξ ) :=
n "
1−μ+μ
j =1 k0 /r −1
j
!1/r pj,s cos(2πbj,s aj ξ/Q)
s=1 n "
1−μ+μ
j =k0 +1
j
!1/r pj,s cos(2πbj,s aj ξ/Q)
s=1
since f (ξ ) −1 for any ξ by the assumption of strict positivity—see Remark 2.3 !1/r j n " k0 /r −1 pj,s bj,s aj ξ/Q2R/Z 1 − 100μ j =k0 +1
s=1
since cos(2πx) 1 − 100x2R/Z and the factors are all positive by Inequality (35) ! j n 200μ k0 /r 2 pj,s bj,s aj ξ/QR/Z −1 exp − 1 − x e−2x for 0 x 0.79 r j =k0 +1 s=1
320000 ln( 1 ) 2
−1
1 by Inequality (34) exp −320000 ln 2
ln(1/ ) 320000 = 2 −1 2 . This completes the proof.
2
We have shown that the spectrum Λ has small doubling, and the next step is to use this fact to show that a set containing most of the scaled defining coordinates a˜ j also has small doubling. Towards that end, we will use the Λ-norm from [10], which is defined as follows: for x ∈ Z/QZ, let xΛ be defined by 1/2 %2 1 %
% % x(ξ − ξ )/Q R/Z xΛ := . |Λ|2
ξ,ξ ∈Λ
Note that 0 xΛ 1 for all x and that the triangle inequality holds: x +yΛ xΛ +yΛ . We also have that 1/2 1/2 1 1 2
2 xξ/Q + xξ /Q xΛ R/Z R/Z |Λ|2
|Λ|2
=2
ξ,ξ ∈Λ
1 xξ/Q2R/Z |Λ| ξ ∈Λ
ξ,ξ ∈Λ
1/2 .
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Thus, squaring Inequality (30) and summing over all ξ ∈ Λ, we have n
a˜ j 2Λ 4C(2 , q, r) =: C .
(36)
j =1
We will now show that the set of all x with small Λ-norm, which by Inequality (36) includes most of the a˜ j , has small doubling. Lemma 7.4. (See [10, Lemma 7.4].) There is a constant C such that the following holds. Let A ⊆ Z/QZ denote the “Bohr set”: & 1 . A := x ∈ Z/QZ: xΛ < 100 Then we have C −1 Pr(Ximax ∈ V )−1 |A| |A + A| C Pr(Ximax ∈ V )−1 . The proof of Lemma 7.4 is the same as in [10], with the small modification that aj should be replaced with a˜ j := bj,1 aj and the quantity 2d± −n should be replaced with Pr(Ximax ∈ V ) (and, of course, the field F in [10] should be replaced with Z/QZ). Also, one should note that [10, Inequality (30)], [10, Inequality (31)], and [10, Inequality (32)] correspond to, respectively, Inequalities (30), (31), and (32). In the next section, we will complete the proof of the structure theorem using the lemma above. 8. Proof of the structure theorem (Theorem 6.1) The key to proving the structure theorem is an application of Freiman’s Theorem for finite fields. Theorem 8.1. (See Lemma 6.3 in [10].) For any constant C there are constants r and δ such that the following holds. Let A be a non-empty subset of Z/QZ, a finite field of prime order Q, such that |A + A| C|A|. Then, if Q is sufficiently large depending on |A|, there is a symmetric generalized arithmetic progression P of rank r such that A ⊂ P and |A|/|P | δ. Note that by Lemma 4.1 we can assume that Q is sufficiently large with respect to |A| C Pr(Ximax ∈ V )−1 C(1/p)n (this follows from V being of medium combinatorial dimension). The set A from Lemma 7.4 satisfies |A + A| C 2 |A|, where C O(1), and also contains / A implies that a˜ j Λ 1/100 and all but O(1) of the scaled defining coordinates a˜ j , since a˜ j ∈ Inequality (36) shows that there can be at most 100C = O(1) such a˜ j . By Theorem 8.1, there exists a symmetric generalized arithmetic progression P = {m1 v1 + · · · + mr vr : |mi | < Mi /2} containing A and satisfying the bounds: rank(P ) = r O(1) and |P | M1 M2 · · · Mr O Pr(Ximax ∈ V )−1 .
(37) (38)
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The symmetric generalized arithmetic progression P is close to what is needed for Theorem 6.1, since it satisfies the required volume and rank bounds. We will show below that P can be altered in ways that preserve Inequalities (37) and (38) (except possibly for changing the implicit constants) so that P satisfies conditions (i), (ii), and (iii) of Theorem 6.1. To show Theorem 6.1(i), we will first add the remaining scaled defining coordinates {a˜ 1 , . . . , a˜ n } \ P (i.e., those a˜ j such that a˜ j Λ 1/100) as new basis vectors vk with corresponding dimensions Mk equal to (say) 3. The resulting generalized arithmetic progression, which we will continue to call P by abuse of notation, satisfies both Inequalities (37) and (38), since there are only O(1) of the a˜ j with a˜ j Λ 1/100 (by Inequality (36)). Second, we need to ensure that P is proper, for which we will use the following lemma: Lemma 8.2. (Cf. Lemma 9.3 in [10].) There is an absolute constant C0 1 such that the following holds. Let P be a symmetric progression of rank r in a abelian group G, such that every 3 nonzero element of G has order at least rC0 r |P |. Then there exists a proper symmetric generalized arithmetic progression P of rank at most r containing P such that |P | rC0 r |P |. 3
Furthermore, if P is not proper and r 2, then P can be chosen to have rank an most r − 1. One can conclude Lemma 8.2 from the proof of [10, Lemma 9.3] (the only difference is noting that the rank can be reduced by at least 1 if P is not proper to begin with). Note that we 3 can always choose Q larger than rC0 r |P | O( p1 )n . Applying Lemma 8.2 gives us a proper symmetric generalized arithmetic progression, which again we call P by abuse of notation, that contains all the a˜ j and satisfies both Inequalities (37) and (38). The next task is to show that P can be further altered so to meet the condition (ii) in Theorem 6.1. Note that there are only O(1) scaled defining coordinates a˜ j such that a˜ j Λ 1/100, and so these a˜ j contribute only a constant to the sum nj=1 a˜ j 2P . On the other hand, for any a˜ j with a˜ j Λ < 1/100, we have that k a˜ j ∈ A ⊂ P for every positive integer k < 1001a˜ j Λ . We will exploit this fact, and to do so will need the following notation. Let ΦP : P → Zr be the map sending a point m1 v1 + · · · + mr vr in the proper generalized arithmetic progression P to the unique r-tuple of coefficients (m1 , . . . , mr ). If the representation for a˜ j in P is a˜ j = m1 v1 + · · · + mr vr and k a˜ j is in P , we would like to be able to say that the representation for k a˜ j is km1 v1 + · · · + kmr vr ; i.e., we hope that ΦP (k a˜ j ) is equal to kΦP (a˜ j ). If this were true, then we would have |kmi | Mi for 1 i r, which, if k is large, would show that a˜ j P is small. However, at this point we may well have ΦP (k a˜ j ) = kΦP (a˜ j ). A priori, changing this to equality would require replacing P with kP and then applying Lemma 8.2 to get a proper symmetric generalized arithmetic progression, but since k may be large, this would increase the volume of P too much, violating Inequality (38). Luckily, the lemma below provides a way around this difficulty. We will say that P is (kj , xj )-proper if ΦP (kj xj ) = kj ΦP (xj ). Lemma 8.3. There exists an absolute constant C1 such that the following holds. Let P be a symmetric proper generalized arithmetic progression with rank r containing elements x1 , . . . , xm , and let k1 , . . . , km be positive integers such that j xj ∈ P for every 1 j kj and for every j .
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Then, there exists a proper symmetric generalized arithmetic progression P of rank at most r such that P contains P , |P | rC1 r |P |, 4
and P is (kj , xj )-proper for every j .
Furthermore, if r 2 and if there is some j for which P is not (kj , xj )-proper, then P can be chosen to have rank at most r − 1. The proof of this lemma relies on an application of Lemma 8.2 to 2P (which contains P ) along with the fact that if a˜ j Λ < 1/100 then k a˜ j ∈ P for every 1 k < 1001a˜ j Λ . Proof. We proceed by induction on the rank r. For the base case, let r = 1 and consider xj ∈ P such that kj xj ∈ P . Since P has rank 1 in this case, we have that xj = ΦP (xj )v1 and kj xj = ΦP (kj xi )v1 . Combining these two equations we have kj ΦP (xj )v1 = ΦP (kj xj )v1 , and dividing by v1 (note that we may assume that v1 = 0), we see that kj ΦP (xj ) = ΦP (kj xj ). Thus P is (kj , xj )-proper for every j . For r 2, we may assume that there is some j0 such that kj0 ΦP (xj0 ) = ΦP (kj0 xj0 ) (i.e., we assume that P is not (kj0 , xj0 )-proper). We may assume that P has the form {m1 v1 + · · · + mr vr : |mi | < Mi /2}. Let M := (M1 , . . . , Mr ), and let (−M/2, M/2) denote the box {(m1 , . . . , mr ): |mi | < Mi /2}. Let k be the largest integer such that ΦP (kxj0 ) = kΦP (xj0 ), so 1 k < kj0 and ΦP ((k + 1)xj0 ) = (k + 1)ΦP (xj0 ). Since kxj0 ∈ P and xj0 ∈ P , we know that ΦP (xj0 ) ∈ (−M/2, M/2) and ΦP (kxj0 ) = kΦP (xj0 ) ∈ (−M/2, M/2); and thus, (k + 1)ΦP (xj0 ) ∈ (−M, M). This shows that 2P , which has dimensions 2M = (2M1 , . . . , 2Mr ), is not proper, since it has two distinct representations for (k + 1)xj0 . We can now apply Lemma 8.2 to 2P , thus finding a proper symmetric generalized arithmetic progression P of rank at most r − 1 containing 2P (which contains P ) such that |P | rC0 r |2P | r2C0 r |P |. 3
3
Since P has rank at most r − 1, we have by induction that there exists P
a proper symmetric generalized arithmetic progression of rank at most r − 1 containing P and such that |P
| (r − 1)C1 (r−1) |P | rC1 (r−1) r2C0 r |P |, 4
4
3
and such that P
is (kj , xj )-proper for every j . Choosing C1 2C0 (for example) guarantees 4 3 4 that rC1 (r−1) r2C0 r rC1 r , which completes the induction. 2 Applying Lemma 8.3, we can generate a new proper symmetric generalized arithmetic progression, which again we will call P by abuse of notation, such that P contains the a˜ j , satisfies Inequalities (37) and (38), and is (kj , a˜ j )-proper for every a˜ j such that a˜ j Λ < 1/100, where kj := 2001a˜ j Λ 1. We will now show that such P satisfies part (ii) of Theorem 6.1. For a˜ j such that P is (kj , a˜ j )-proper, we have that |kj mi | Mi for each 1 i r, and so a˜ j P =
r mi 2 i=1
Mi
r r 2 1 2 200a˜ j Λ = 40000ra˜ j 2Λ . kj i=1
i=1
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Thus, part (ii) of Theorem 6.1 follows from Inequality (36), since P is (kj , a˜ j )-proper for all but O(1) of the a˜ j . The next step is to make further alterations to P so that we can prove part (iii) of Theorem 6.1. The key property that we will use for (iii) is to have the set of vectors {ΦP (a˜ j ): 1 j n} span all of Rr , and we will use a rank reduction argument on P to produce a new proper symmetric generalized arithmetic progression satisfying this full rank property. Lemma 8.4. (See [10].) Let P be a proper symmetric generalized arithmetic progression of rank r containing a set B such that the set of vectors ΦP (B) does not span Rr . Then there exists a symmetric generalized arithmetic progression P containing P such that rank(P ) r − 1
and |P | |P |.
Note that the resulting P is not necessarily proper or (kj , a˜ j )-proper, even if P had these properties. Proof. We use the same proof here as appears in [10, Section 8]. If {ΦP (a˜ j ): 1 j n} does not have rank r, then it is contained is a subspace of Rr of dimension r − 1. Thus, there exists an integer vector (α1 , . . . , αr ) with all the αi coprime such that (α1 , . . . , αr ) is orthogonal to every vector in {ΦP (a˜ j ): 1 j n}. Thus, for every w ∈ Z/QZ and any a˜ j = m1 v1 + · · · + mr vr , we have that a˜ j = m1 v1 + · · · + mr vr = m1 (v1 − wα1 ) + · · · + mr (vr − wαr ). Since not all the αi are zero, we may assume that αr = 0. Setting w = vr /αr so that vr −wαr = 0, we see that P is contained in the symmetric generalized arithmetic progression
P := m 1 v1 + · · · + m r−1 vr−1 : |m i | < Mi /2 with rank r − 1, dimensions M1 , . . . , Mr−1 (which are the same as the corresponding dimensions for P ), and basis vectors vi := vi − αi vr /αr . By construction |P | |P |. 2 We can now run the following algorithm to create a generalized arithmetic progression with all the desired properties. As the input, we take the generalized arithmetic progression P that we arrived at after applying Lemma 8.3, thus the input P contains all the a˜ j , satisfies Inequalities (37) and (38), and is (kj , a˜ j )-proper for every a˜ j such that a˜ j Λ < 1/100; however, we do not yet know whether ΦP ({a˜ j : 1 j n}) spans Rr . 1. If ΦP ({a˜ j : 1 j n}) spans Rr , then do nothing; otherwise apply Lemma 8.4. 2. If P is proper, then do nothing; otherwise apply Lemma 8.2. 3. If for every a˜ j with a˜ j Λ < 1/100 we have that P is (kj , a˜ j )-proper, then do nothing; otherwise apply Lemma 8.3. 4. If P satisfies the three properties given in steps 1, 2, and 3, halt; otherwise, return to step 1. Each application of a lemma in the algorithm may disrupt some property that other two lemmas preserve; however, we also know that each step in the algorithm either does not change P or reduces the rank of P by at least 1. Since the original input P has rank O(1), the algorithm
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must terminate in O(1) steps, giving us a generalized arithmetic progression of rank r that satisfies Inequalities (37) and (38), satisfies conditions (i) and (ii) of Theorem 6.1, and satisfies the condition that ΦP ({a˜ j : 1 j n} spans all of Rr . Thus, all that is left to prove is part (iii), the claim of rational T -commensurability. Though we will not need it in the current section, one should recall that Theorem 6.1 is only useful when |no(n) T O(1) | = no(n) , where T is the symmetric generalized arithmetic progression containing (μ) {−1, 0, 1} and all possible values taken by the βij and the αij (see Section 6). We say that a set W economically T -spans a set U if each u ∈ U can be represented as a highly T -rational linear combination of elements in W , where each coefficient may be expressed as a/b where a, b ∈ no(n) T O(1) and where the implicit constants in the o(·) and O(·) notation are uniform over U . Comparing our definitions with those from [10, Section 8], we note that “highly rational” means the same thing as “highly {−1, 0, 1}-rational”, and “economically spans” means the same thing as “economically {−1, 0, 1}-spans”. Thus, it is clear that any highly rational number is also highly T -rational for any T containing {−1, 0, 1}, and also the statement “W economically spans U ” implies “W economically T -spans U ” for any set T containing {−1, 0, 1}. The remainder of this section paraphrases (with some notational changes) the latter portion of [10, Section 8]. We know that ΦP ({a˜ j : 1 j n} spans Rr . Thus, there exists a subset U ⊂ {a˜ 1 , . . . , a˜ n } of cardinality r such that ΦP (U ) spans Rr . Renumbering if necessary, we can write U = {a˜ 1 , . . . , a˜ r }. It will be important later on that U has cardinality O(1). The set {v1 , . . . , vr } of basis vectors for P economically {−1, 0, 1}-spans {a˜ 1 , . . . , a˜ n } by the definition of P (note that Mi O(Pr(Ximax ∈ V )−1 ) O(p −n ) = no(n) ), and so by Cramer’s rule, the vectors ΦP (U ) economically {−1, 0, 1}-span the standard basis vectors {e1 , . . . , er } for Rr . Applying ΦP−1 (recall that ΦP is a bijection since P is proper) shows that U economically {−1, 0, 1}-spans {v1 , . . . , vr }. Following this paragraph, we will show that there exists a single vector vi0 where 1 i0 r such that vi0 economically T -spans U , which will show by transitivity that vi0 economically T -spans {a˜ 1 , . . . , a˜ n } (since U economically T -spans {v1 , . . . , vr } which economically T -spans {a˜ 1 , . . . , a˜ n }; the relation “economically T -spans” is transitive here since the sets U and {v1 , . . . , vr } have cardinality O(1)). Let s be the smallest integer such that there exists a subset of cardinality s of {v1 , . . . , vr } (by renumbering, say the set is {v1 , . . . , vs }) so that for some nonzero d ∈ no(n) T O(1) and some cij ∈ no(n) T O(1) we have d a˜ i =
s
cij vj
for every 1 i n.
(39)
j =1
Note that d does not depend on i, and so this statement is slightly stronger than having {v1 , . . . , vs } economically T -span {a˜ 1 , . . . , a˜ n }. Also, note that Eq. (39) holds (for example) with s = r by the definition of P and since T contains {−1, 0, 1}. We now consider two cases: • The n × s matrix C = (cij ) has rank 1 in Z/QZ. In this case, a˜ i1 /a˜ i2 is highly T -rational for all i1 , i2 (since all the cij are highly T -rational). We know that U economically T -spans {v1 , . . . , vr }, and so the numbers vi1 /vi2 are also highly T -rational (note that it is critical
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here that U has cardinality O(1)). This means that v1 (for example) economically T -spans {v1 , . . . , vr }, and so by transitivity v1 economically T -spans U . • The matrix C has rank at least 2. Recall that (a1 , . . . , an ) is the normal vector for V and that V is spanned by (n − 1) linearly independent vectors with entries in S (recall that S contains all possible values taken by the αij ). We can scale the j th coordinate of each of these vectors −1 by bj,1 to get a set of n − 1 linearly independent vectors each of which is orthogonal to a˜ := (a˜ 1 , . . . , a˜ n ). Among these (n − 1) linearly independent vectors that are orthogonal to −1 −1 (a˜ 1 , . . . , a˜ n ), we can find at least one, say w = (b1,1 w1 , . . . , bn,1 wn ) that is not orthogonal to every column of C (since C has column rank at least 2). Let B := {bj,1 : 1 j n}, and let w˜ := w b∈B b = (w˜ 1 , . . . , w˜ n ). Thus w˜ is orthogonal to a˜ and every coordinate w˜ i of w˜ is an element of T O(1) (since T contains S and B and |B| = O(1) by the definition of p-bounded of exponent r). Remark 8.5. Note that the line above is the only place in the proof where we use the assumption (μ) from the definition of p-bounded of exponent r that the βij take values in a set with cardinality O(1). As is evidenced here, the following weaker assumption suffices instead: say that for (μ) (μ) (μ) each 1 i n there exists a set Bi such that |Bi | = O(1) and such that βi1 , βi2 , . . . , βin each take a nonzero value in Bi with probability at least q. In fact, this weaker assumption also (μ) replaces the assumption in the definition of p-bounded of exponent r that q minx Pr(βij = x) (μ)
for every i, j : It suffices for each βij to take one value in Bi with probability at least q, instead of taking every value with probability at least q. We may now compute:
0 = d a˜ · w˜ =
n
d a˜ i w˜ i =
i=1
n s
cij vj w˜ i =
i=1 j =1
s n j =1
! cij w˜ i vj .
i=1
Since w˜ is not orthogonal to every column of C = (cij ), we can assume (reordering if necessary), that the coefficient for vs above is nonzero, and thus we have ! n s−1 −1 vs = n cj w˜ vj . ˜ =1 cs w j =1
=1
Plugging this last equation into Eq. (39), we arrive at
d
n =1
! cs w˜ a˜ i =
s−1 j =1
cij
n =1
cs w˜ − cis
n
! cj w˜ vj .
=1
Since the coefficient for a˜ i on the left is an element of no(n) T O(1) and the coefficient for each vj on the right is an element of no(n) T O(1) , we have contradicted the minimality of s. Thus, we have completed the proof of the structure theorem (Theorem 6.1).
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9. A generalization: f rows have fixed, non-random values In this section, we will give a generalization of Theorem 2.2 to the case where the random matrix Nn has f O(ln n) rows that are assumed to be linearly independent and contain fixed, non-random entries. The proof of the generalized result is very similar to the proof of Theorem 2.2, and we will sketch the main differences in the two proofs below. Definition 9.1 (A random matrix Nf,n with entries in S). Let f be an integer between 1 and n, let S be a subset of a ring, and let Nf,n be an n by n matrix defined as follows. For 1 i f and 1 j n, let the entries sij of Nf,n be fixed (non-random) elements of S such that the rows (si,1 , . . . , si,n ) for 1 i f are linearly independent. For f + 1 i n and 1 j n, let the entries αij of Nf,n be discrete finite random variables taking values in S. Thus, ⎛ s 1,1 ⎜ .. ⎜ . ⎜ ⎜ sf,1 ⎜ ⎜α Nf,n := ⎜ f+1,1 ⎜ αf+2,1 ⎜ ⎜ αf+3,1 ⎜ . ⎝ . . αn,1
s1,2
···
··· ···
··· ··· ··· ··· ··· .. . ···
αf+1,2 αf+2,2 αf+3,2 .. . αn,2
⎫ s1,n ⎞ ⎪ .. ⎟ ⎬ Fixed rows; assumed to be linearly . ⎟ ⎟ ⎪ independent sf,n ⎟ ⎭ ⎟ αf+1,n ⎟ ⎫ ⎟ ⎪ ⎪ αf+2,n ⎟ ⎪ ⎬ ⎟ ⎪ αf+3,n ⎟ Random rows .. ⎟ ⎠ ⎪ ⎪ ⎪ . ⎪ ⎭ αn,n
Theorem 9.2. Let p be a positive constant such that 0 < p < 1, let r be a positive integer constant, and let S be a generalized arithmetic progression in the complex numbers with rank O(1) (independent of n) and with cardinality at most |S| no(n) . Consider the matrix Nf,n r with entries in S (see Definition 9.1 above), where f ( 2 ln(1/p) − o(1)) ln n. If the collection of random variables {αj k }f+1j n,1kn is p-bounded of exponent r, then n n−f . Pr(Nf,n is singular) max p 1/r + o(1) , p + o(1) Note that the bound on the singularity probability of Nf,n for r 2 is the same as in Theorem 2.2 (since for r 2, we have n/r n − c ln n = n − f). This is a reflection of the fact that only the large dimension case uses the randomness in all the rows simultaneously, and in that case the exponential bound does not depend on r. Generally speaking, the best known lower bounds on the singularity probability of a discrete random matrix come from a dependency among at most two random rows, and since Nf,n certainly has more than two random rows, the upper bounds given in Theorem 9.2 seem reasonable. Theorem 9.2 leads to Corollary 1.2 by following a conditioning argument very similar to that given in Section 3.3. 9.1. Outline of the proof of Theorem 9.2 The proof of Theorem 9.2 follows the same lines of reasoning as that of Theorem 2.2. In this subsection, we will state the main lemmas with the necessary modifications, and we will mention a few important considerations when making the modifications.
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Note that Eq. (15), which reduces the question of singularity to one of the rows spanning non-trivial hyperplane of dimension n − 1 holds in the current context, using the same definition of AV and “non-trivial hyperplane” (both are defined after Eq. (15) in Section 5.1). Definition 9.3 (Combinatorial dimension with f fixed rows). Let D := { an : 0 a n2 , a ∈ Z}. For any d± ∈ D, we define the combinatorial Grassmannian Grf (d± ) to be the set of all nontrivial hyperplanes V in (Z/QZ)n such that p n−d± +1/n <
max Pr(Xi ∈ V ) p n−d± .
f+1in
For d± = 0, we define Grf (0) to be the set of all non-trivial hyperplanes such that max Pr(Xi ∈ V ) p n .
f+1in
We will refer to d± as the combinatorial dimension of V . Lemma 9.4 (Small combinatorial dimension, with f fixed rows). For any δ > 0 we have
Pr(AV ) (n − f)δ n .
d± ∈D s.t. T d± q n δ n V ∈Grf (d± )
Proof. The proof is the same as that for Lemma 5.2; also see [4,9,10].
2
Lemma 9.5 (Large combinatorial dimension, with f fixed rows). We have
d± ∈D s.t.
cLgDim T d± q n n1/2
n−f Pr(AV ) p + o(1) .
V ∈Grf (d± )
Here, cLgDim is the same as in Lemma 5.3. Proof. The proof is the same as that for Lemma 5.3, except now we appeal to Lemma A.2 with f > 0. Note that we must assume f n/2 in order to apply Lemma A.2. See also [4,9,10]. 2 Proposition 9.6 (Medium combinatorial dimension estimate, with f fixed rows). Let 0 < 0 be a constant much smaller than 1, and let d± ∈ D be such that (p + cMedDim f 0 )n/r < T d± q n < cLgDim r √ . If f ( 2 ln(1/p) − o(1)) ln n, then n
n/r Pr(AV ) p + o(1) .
V ∈Grf (d± ) 1 Here we choose the constant cMedDim f so that cMedDim f > (cm + cf + 100 ), where cm and cf c f 0 n are positive absolute constants (in particular, we need cf such that f r , which is true for any positive constant cf since f O(ln n)). As before, we will prove this proposition by separating V with medium combinatorial dimension into two cases: exceptional and unexceptional, which ∗ from Eq. (17) (this definition is the same as in are defined below using the definition of Zi,k
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Definition 5.5 with the small change that i and j are required to be between f + 1 and n instead of between 1 and n). Definition 9.7. Consider a hyperplane V of medium combinatorial dimension (that is, d± satisfies the condition in Proposition 9.6). We say V is unexceptional if there exists an i0 where f + 1 i0 n and there exists a k0 where 1 k0 r such that max
Pr(Xj ∈ V ) < 1 Pr Zi∗0 ,k0 ∈ V .
f+1j n
We say V is exceptional if for every i where f + 1 i n and for every k where 1 k r we have ∗ 1 Pr Zi,k ∈V
max
Pr(Xj ∈ V ) .
(40)
f+1j n
In particular, there exists imax such that Pr(Ximax ∈ V ) = maxf+1j n {Pr(Xj ∈ V )}; and so if V is exceptional, then 1 Pr Zi∗max ,k ∈ V Pr(Ximax ∈ V )
for every k.
(41)
We will refer to Ximax as the exceptional row. Lemma 9.8 (Unexceptional space estimate, with f fixed rows). If f constant cf , then we have
Pr(AV ) p −o(n) 2n 1m
c f 0 n r
c 0 n/r
for some positive
.
V ∈Grf (d± ): V is unexceptional
Notice that the bound is the same as in Lemma 5.6, except that we replaced cMedDim with cMedDim f when defining “unexceptional”. Proof. The proof follows in the same way as that for Lemma 5.6; however, when replacing rows Xi of Nf,n with rows Zi that concentrate more sharply on V , we must take care to only replace random rows of Nf,n (i.e., rows X1 , . . . , Xf must not be replaced by Zi ). See Appendix B for details. 2 In the exceptional case, The same structure theorem (Theorem 6.1) holds, leading to the following lemma. r − o(1)) ln n, then Lemma 9.9 (Exceptional space estimate, with f fixed rows). If f ( 2 ln(1/p)
Pr(AV ) p n/r .
(42)
V ∈Gr(d± ): V is exceptional
Note that this upper bound is dramatically worse than the analogous upper bound in n Lemma 5.7 of n− 2 +o(n) .
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Proof. As in Lemma 5.7, the main step in the proof is applying the structure theorem (Theorem 6.1). In the current context, Inequality (20) holds with n − f as the exponent instead of n (since there are only n − f random rows). If we combine this modified version of Inequality (20) with Inequality (21), then we have the bound n Pr(AV ) n− 2 +o(n) Pr(Ximax ∈ V )−n Pr(Ximax ∈ V )n−f V ∈Grf (d± ): V is exceptional n
= n− 2 +o(n) Pr(Ximax ∈ V )−f , where by assumption Ximax is the random row such that Pr(Ximax ∈ V ) = maxf+1in Pr(Xi ∈ V ). In order for this upper bound to achieve the desired bound in Inequality (42), it is sufficient to have n
n− 2 +o(n) Pr(Ximax ∈ V )−f p n/r .
(43)
Using the assumption that Pr(Ximax ∈ V ) (p + cMedDim f 0 )n/r > p n/r (since V is of medium combinatorial dimension), we see that Inequality (43) holds whenever r f − o(1) ln n, 2 ln(1/p) which completes the proof.
2
Acknowledgments We would like to thank Kevin Costello for helpful conversations on the conditioning argument in Sections 3.2 and 3.3. Also the third author would like to thank the National Defense Science and Engineering Fellowship and the National Science Foundation Graduate Research Fellowship for helping fund this work. Appendix A. Two background results A.1. A version of the Littlewood–Offord result in Z/QZ If S ⊂ Q, then we can clear denominators and prove (as in [10, Lemma 2.4]) the large combinatorial dimension estimate in R instead of working in Z/QZ, in which case we can also use the Littlewood–Offord result over R (see [11, Corollary 7.13]), instead of the version over Z/QZ given here in Lemma A.1. When working in R, the integral approximation of Inequality (A.1) can be replaced by a limit going to infinity, and we do not need any extra assumptions on Q. In particular, we may take Q ≈ exp(exp(Cn)) (see Remark 4.2). For Q sufficiently large with respect to q, r, and n, it is clear that we have ) k/r k/r 1 1 − 2q + 2q cos(2πξ/Q) 1 − 2q + 2q cos(2πt) dt + , (A.1) n 1
1 Q
ξ ∈Z/QZ
for all 1 k n.
0
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Lemma A.1. Let Q be sufficiently large to satisfy Inequality (A.1), and let v1 , . . . , vn ∈ Z/QZ be such that v1 , . . . , vk are nonzero. Let {αj }nj=1 be a collection of random variables that are p-bounded of exponent r, and let Xv := α1 v1 + · · · + αn vn . Then, for every x ∈ Z/QZ we have √ 1 cLO r =O √ , Pr(Xv = x) √ qk k where cLO is an absolute constant. (μ)
Proof. Our proof is closely modeled on the proof of [11, Corollary 7.13]. Let βj be the symmetric random variables from the definition of p-bounded of exponent r corresponding to αj (see Eq. (9)). Then, we can compute 1 Pr(Xv = x) Q
k "
E eQ (αj aj ξ ) (note that aj = 0 for j > k) ξ ∈Z/QZ j =1
k " 1 j =1
1 Q
1 Q =
1 Q 1 Q
Q
1/k
E eQ (αj aj ξ ) k
(Hölder’s inequality)
ξ ∈Z/QZ
E eQ (αj aj ξ ) k 0 0 ξ ∈Z/QZ
(where j0 corresponds to the largest factor in the previous line)
!k/r
1−μ+μ
ξ ∈Z/QZ
j0
pj0 ,s cos(2πbj0 ,s vj0 ξ/Q)
(since αj0 is p-bounded of exponent r)
s=1
k/r 1 − 2q + 2q cos(2πbj0 ,1 vj0 ξ/Q)
(since μpj0 ,1 2q)
ξ ∈Z/QZ
k/r 1 − 2q + 2q cos(2πξ/Q)
(by reordering the sum).
ξ ∈Z/QZ
Combining the above inequalities with Inequality (A.1) and following the proof of [11, Corollary 7.13] to bound the integral, we have )1 Pr(Xv = x) 0
√ k/r 1 1 cLO r =O √ , 1 − 2q + 2q cos(2πt) dt + = √ n qk k
where cLO is an absolute constant.
2
A.2. A generalization of a lemma due to Komlós [6] This lemma is a generalization of the result in [6] (see also [2, Lemma 14.10], [4, Section 3.1], and [9, Lemma 5.3]).
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Lemma A.2. Fix n, and let p be a positive constants such that 0 < p < 1 and let r be a positive integer constant. Consider the matrix Nf,n taking values in Z/QZ, where f n/2 and Q is large enough to satisfy Inequality (A.1). If the collection of random entries in Nf,n is p-bounded of exponent r, then n−f Pr(there exists v ∈ Ω1 such that Nf,n · v = 0) p + o(1) , where & c + 1 of the vi are nonzero \ {0}, Ω1 := (v1 , . . . , vn ) ∈ Z/QZ: at most (n − f) 1 − ln n where the constant c can be taken to be c 2 ln(100/p), and where 0 denotes the zero vector. Proof. Let Ek = {there exists v ∈ Ω1 with at most k nonzero coordinates such that Nf,n ·v = 0}. Clearly,
Pr(there exists v ∈ Ω1 such that Nf,n · v = 0)
Pr(Ek \ Ek−1 ).
1k(n−f)(1− lncn )+1
Let S be the set of all possible values that could appear as entries in Nf,n , and let Nf,n |j1 ,...,jk be the n by k matrix consisting of columns j1 , . . . , jk of Nf,n . Following [6, Lemma 2] (see also [2, Lemma 14.10] and [9, Lemma 5.3]) we can write Pr(Ek \ Ek−1 )
1j1 <··· ···<jk n
1i1 <··· ···
Pr(RwSpni1 ,...,ik−1 ,H ) Pr(RwIni1 ,...,ik−1 ,H ),
H a (k−1)dimensional hyperplane spanned by S k
where RwSpni1 ,...,ik−1 ,H := {rows i1 , . . . , ik−1 of Nf,n |j1 ,...,jk span H },
and
RwIni1 ,...,ik−1 ,H := {all rows of Nf,n |j1 ,...,jk except i1 , . . . , ik−1 are in H }. Let U (k, p, q) be a uniform upper bound for Pr(row i is in H ), where f + 1 i n and q is the constant from Definition 2.1 (here, we mean uniform with respect to the index sets {j1 , . . . , jk } and {i1 , . . . , ik }). Then we have Pr(Ek \ Ek−1 ) U (k, p, q)
n−k−f+1
n n , k k−1
since k − 1 fixed rows of Nf,n |j1 ,...,jk can span at most 1 hyperplane H of dimension k − 1. 28 c2 r
For k p2LO (a constant), we can set U (k, p, q) = p by the Weighted Odlyzko Lemma (see q Lemma B.1), giving us a bound of n−f Pr(Ek \ Ek−1 ) p + o(1) .
(A.2)
J. Bourgain et al. / Journal of Functional Analysis 258 (2010) 559–603 28 c2 r
For p2LO < k (n − f)(1 − q n n 22n k k−1 n we thus have
c ln n ) + 1,
601
we use Lemma A.1 to set U (k, p, q) =
√ cLO √ r. qk
Since
2 n−k−f+1 2 r 1 2n cLO Pr(Ek \ Ek−1 ) 2 . n qk As a function of k, this upper bound has strictly positive second derivative; thus, the largest upper bound will occur at one of the extremal values of k = a bit of computation shows that
2 r 28 cLO 2 p q
or k = (n − f)(1 −
c ln n ) +
1 Pr(Ek \ Ek−1 ) O p n−f . n
1, and
(A.3)
Summing the bounds in Inequalities (A.2) and (A.3) completes the proof.
2
Appendix B. The unexceptional case with f fixed rows This section is adapted from the proof of [10, Lemma 4.1], and proves Lemma 5.6 by setting c n f = 0. Assume that f f r0 , and let m be the closest integer to cmrn . Let Z1 , . . . , Zm be i.i.d. copies of the unexceptional row vector Zi∗0 ,k0 from Definition 9.7, so 1 Pr(Zi ∈ V ) > Pr(Xi ∈ V ) for all f + 1 i n. We will need the following version of the Weighted Odlyzko Lemma: Lemma B.1. (Cf. [10, Lemma 4.3] or [4, Section 3.2].) For 1 i, let Wi−1 be an (f + i − 1)dimensional subspace containing X1 , . . . , Xf (which are fixed, linearly independent row vectors). Then
0 Pr(Zi ∈ Wi−1 ) p + 100
n −f−i+1 r
.
Proof. Since Wi−1 has dimension f + i − 1, there exists a set of f + i − 1 “determining” coordinates such that if a vector V ∈ Wi−1 , then the f + i − 1 “determining” coordinates determine the values of the remaining n − f − i + 1 coordinates. Since the maximum probability that any of the 0 n/r random coordinates in Zi takes a given value is at most 1 − μ = p + 100 , and since there n are at least r − f − i + 1 of the random coordinates in Zi that are not among the “determining” coordinates, we have the desired upper bound. 2 Let V0 := Span{X1 , . . . , Xf }, the space spanned by the f fixed rows, and for 1 i m let BV ,i be the event that Z1 , . . . , Zm are linearly independent in V \ V0 . We have the following analog of Lemma 5.8 (and also [10, Lemma 4.4]): Lemma B.2. (See Lemma 4.4 in [10].) Let m, f, and BV ,m be as defined above. Then, Pr(BV ,m ) p
o(n)
maxf+1in Pr(Xi ∈ V ) 1
m .
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Proof. Using Bayes’ Identity, we have Pr(BV ,m ) =
m "
Pr(BV ,i | BV ,i−1 ),
(B.1)
i=1
where BV ,0 denotes the full space of the Zi . Conditioning on a particular instance of Z1 , . . . , Zi−1 in BV ,i−1 , we have that Pr(BV ,i |BV ,i−1 ) = Pr(Zi ∈ V ) − Pr(Zi ∈ Wi−1 ), where Wi−1 denotes the (f + i − 1)-dimensional space spanned by X1 , . . . , Xf and Z1 , . . . , Zi−1 . We will now establish a uniform bound that does not depend on which particular instance of Z1 , . . . , Zi−1 in BV ,i−1 that we fixed by conditioning. By the definition of unexceptional, we have Pr(Zi ∈ V ) >
1 1
max Pr(Xi ∈ V ),
f+1in
and by the Weighted Odlyzko Lemma (see Lemma B.1), we have n n 0 r −f−i+1 0 r (1−(cm +cf )0 ) p+ . Pr(Zi ∈ Wi−1 ) p + 100 100 Using Taylor’s Theorem with remainder (for example), one can show that n 0 r (1−(cm +cf )0 ) 1 1 p+ (p + cMedDim 0 )n/r max Pr(Xi ∈ V ), 100 2n n f+1in 1 1 so long as cMedDim > 100 + cm + cf > 100 + (cm + cf )p ln( p1 ) and n is sufficiently large (the second inequality in the display above is the definition of medium combinatorial dimension). Thus $ 1# 1 , max Pr(Xi ∈ V ) 1 − Pr(BV ,i | BV ,i−1 ) 1 f+1in n
and plugging this estimate back into Inequality (B.1) we get m o(n) maxf+1in Pr(Xi ∈ V ) Pr(BV ,m ) p . 1
2
To conclude Lemma 9.8 (which implies Lemma 5.6 by setting f = 0), we will proceed as in the proof for [10, Lemma 4.1]. Let Z1 , . . . , Zm be i.i.d. copies of Zi∗0 ,k0 that are independent of the random rows Xf+1 , . . . , Xn . Using independence and Bayes’ Identity we have Pr(AV ) = Pr(AV | BV ,m ) = Pr(AV ∧ BV ,m )p
Pr(AV ∧ BV ,m ) Pr(BV ,m )
−o(n)
1 maxf+1in Pr(Xi ∈ V )
m .
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Because the Zi are linearly independent in V \ V0 , we know that there is a subset I ⊂ {f + 1, / I } spans V . Let CV ,I be f + 2, . . . , n} of cardinality |I | = m, such that {Z1 , . . . , Zm } ∪ {Xi : i ∈ / I } spans V . Then we have the event that {Z1 , . . . , Zm } ∪ {Xi : i ∈ Pr(AV ∧ BV ,m ) Pr CV ,I ∧ {Xi ∈ V : i ∈ I } I ⊂{f+1,...,n} |I |=m
#
$m max Pr(Xi ∈ V )
f+1in
Summing the above inequality over all unexceptional V (note that bining with the bound for Pr(AV ) above gives us unexceptional V
Pr(CV ,I ).
I ⊂{f+1,...,n} |I |=m
V
Pr(CV ,I ) 1) and com-
m $m n − f 1 −o(n) p Pr(AV ) max Pr(Xi ∈ V ) m maxf+1in Pr(Xi ∈ V ) f+1in #
p −o(n) 2n 1m . This completes the proof of the estimate for unexceptional V . References [1] Michael Aizenman, Francois Germinet, Abel Klein, Simone Warzel, On Bernoulli decompositions for random variables, concentration bounds, and spectral localization, Probab. Theory Related Fields 143 (2009) 219–238. [2] Béla Bollobás, Random Graphs, second ed., Cambridge Stud. Adv. Math., vol. 73, Cambridge Univ. Press, Cambridge, 2001. [3] Jean Bourgain, Nets Katz, Terence Tao, A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14 (1) (2004) 27–57. [4] Jeff Kahn, János Komlós, Endre Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1) (1995) 223–240. [5] János Komlós, On the determinant of (0, 1) matrices, Studia Sci. Math. Hungar. 2 (1967) 7–21. [6] János Komlós, Circulated manuscript, edited version available online at http://www.math.rutgers.edu/~komlos/ 01short.pdf, 1977. [7] Greg Martin, Erick B. Wong, Almost all integer matrices have no integer eigenvalues, Amer. Math. Monthly 116 (7) (2009) 588–597. [8] Arkadii Slinko, A generalization of Komlós’s theorem on random matrices, New Zealand J. Math. 30 (1) (2001) 81–86. [9] Terence Tao, Van Vu, On random ±1 matrices: Singularity and determinant, Random Structures Algorithms 28 (1) (2006) 1–23. [10] Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628. [11] Terence Tao, Van Vu, Additive Combinatorics, Cambridge Stud. Adv. Math., vol. 105, Cambridge Univ. Press, Cambridge, 2006. [12] Van Vu, Melanie Matchett Wood, Philip Matchett Wood, Mapping incidences, arXiv:0711.4407v1 [math.CO], November 28, 2007.
Journal of Functional Analysis 258 (2010) 604–615 www.elsevier.com/locate/jfa
On the unconditional subsequence property ✩ Edward Odell, Bentuo Zheng ∗ Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, United States Received 18 April 2009; accepted 25 July 2009 Available online 7 August 2009 Communicated by N. Kalton
Abstract We show that a construction of Johnson, Maurey and Schechtman leads to the existence of a weakly null sequence (fi ) in ( Lpi )2 , where pi ↓ 1, so that for all ε > 0 and 1 < q 2, every subsequence of (fi ) admits a block basis (1 + ε)-equivalent to the Haar basis for Lq . We give an example of a reflexive Banach space having the unconditional subsequence property but not uniformly so. Published by Elsevier Inc. Keywords: Unconditional subsequence property; Unconditionally saturated; Unconditional tree property
1. Introduction If every normalized weakly null sequence in a Banach space X has an unconditional subsequence, X is said to have the unconditional subsequence property (USP). If, for some K < ∞, every such sequence admits a K-unconditional subsequence, X has the (K-USP). The first example of a space without the (USP) was constructed in 1977 by Maurey and Rosenthal [16]. This construction later played a role in the work of Gowers and Maurey [4] where they gave an example of a reflexive space not containing an unconditional basic sequence. Subsequent work by S. Argyros and others has shown that such spaces are plentiful. Thus having the (USP) is by no means automatic. In [16] it was asked if L1 [0, 1] has the (USP). In 2007 Johnson, Maurey and Schechtman [8] showed that L1 fails the (USP). Moreover, for 1 p < 2, they constructed ✩
Edward Odell’s research is supported in part by NSF grant DMS-0700126 and Bentuo Zheng’s research is supported in part by NSF grant DMS-0800061. * Corresponding author. E-mail addresses:
[email protected] (E. Odell),
[email protected] (B. Zheng). 0022-1236/$ – see front matter Published by Elsevier Inc. doi:10.1016/j.jfa.2009.07.015
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a weakly null sequence in Lp [0, 1] so that for all ε > 0, every subsequence admits a block basis (1+ε)-equivalent to the Haar basis for Lp . Since the Haar basis for Lp is unconditional for p > 1 but the unconditional constant blows up as p → 1, it follows that Lp has the (Cp -USP) for all 1 < p < ∞, but limp→1+ Cp = ∞. The situation is different for p > 2. Lp has the (2 + ε-USP) for all p 2 and ε > 0 (see [9] for p ∈ 2N, [5] for the general case). In this note we present two examples. In Section 2 we show how the [8] construction easily yields the following. Let 2 > p1 > p2 > · · · with limi pi = 1. Then X = ( ∞ i=1 Lpi )2 contains a weakly null sequence such that for all ε > 0 and 1 < q 2, every subsequence admits a block basis that is 1 + ε-equivalent to the Haar basis for Lq [0, 1]. X is reflexive and for every infinite dimensional subspace of X and ε > 0, some further subspace Z satisfies d(Z, p ) < 1 + ε for some 1 < p 2. In Section 3 we construct a reflexive space X with the (USP) which fails the (K-USP) for all K. This solves problem 3 in [17]. X = ( Xn )2 where each Xn is isomorphic to 2 but √ n fails the ( 3 -USP). The Xn ’s are a modification of an example in [16]. We will show that every normalized weakly null sequence in X admits an 2 subsequence. This example contrasts with the result that if every normalized weakly null sequence in a Banach space admits a subsequence equivalent to the unit vector basis of c0 , then this is uniformly so [12] (see [3] for a more general uniformity theorem). It is worth mentioning that the (USP) is a weak version of the (UTP): X has the unconditional tree property if every normalized weakly null tree in X admits an unconditional branch (see [10]). In this case it is automatic that X has the (K-UTP) for some K [18]. The (UTP), rather than the (USP) is the property that ensures a space embeds into one with an unconditional basis if X is reflexive. A reflexive space with the (UTP) embeds into a reflexive space with an unconditional basis [10]. If X ∗ is separable and X has the (ω∗ -UTP), i.e., every normalized weak* null tree in X ∗ admits an unconditional branch, then X embeds into a space with a shrinking unconditional basis [11]. The almost isometric version of this result is given in [2]. Results on the (USP) date back to the 1970’s. In [6] and [17] it was proved that a quotient X of a space with a shrinking unconditional basis has the (USP). From [11] we have more, namely X embeds into a space with a shrinking unconditional basis. We use standard Banach space notation [7]. X, Y , and Z will denote separable real infinite dimensional Banach spaces. SX and BX denote the unit sphere and unit ball, respectively, of X. [N]<ω denotes the finite subsets of X, [N]2 = {(i, j ): i < j, i, j ∈ N}. For E, F ∈ [N]<ω , E < F means max E < min F and |E| is the cardinality of E. c00 is the linear space of finitely supported sequences of reals. For x, y ∈ c00 x < y denotes supp(x) < supp(y) and we use the same notation ∞ for x, y ∈ span(ei )∞ i=1 where (ei )i=1 is a basic sequence. Lp = Lp [0, 1] and m(E) denotes the Lebesgue measure of E. We thank the referee for pointing out an embarrassing error in our original proof of Lemma 3.4. 2. A weakly null sequence in ( ∞ i=1 Lpi )2 Example 2.1. Let2 > p1 > p2 > · · · with limi→∞ pi = 1. There exists a weakly null sequence ∞ ˜ ∞ (f˜i )∞ i=1 in X = ( i=1 Lpi )2 with the following property. If 1 < p 2, ε > 0 and (fni )i=1 is ∞ ∞ ˜ ˜ any subsequence of (fi )i=1 , then some block basis of (fni )i=1 is (1 + ε)-equivalent to the Haar basis of Lp .
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For p < q 2, Lq embeds isometrically into Lp . Moreover the Haar basis for Lq is precisely reproducible [14]. This means that if Lq ⊆ Z, a space with a basis (zi )∞ i=1 , and ε > 0 then some ∞ block basis of (zi )i=1 is (1+ε)-equivalent to the Haar basis for Lq . Thus it will suffice to produce (f˜i )∞ i=1 satisfying the desired property for each Lpi . Let 1 < p1 < 2. We begin by recalling some specific aspects of the [8] construction in Lp1 . Let A be the algebra generated by the dyadic subintervals of [0, 1]. Let (Ei )∞ i=1 be a listing of all elements in A so that for each E ∈ A, M(E) ≡ {j ∈ N : Ej = E} is infinite. In [8] a cer∞ tain sequence (kn )∞ n=1 , of powers of 2, along with a sequence (an )n=1 ⊆ (0, ∞) and for n ∈ N, ∞ a sequence (hi,n )i=1 of functions on [0, 1] are constructed to satisfy the following: i) ii) iii) iv) v) vi)
|hi,n | = 1Ai,n , Ai,n ⊆ En ; hi,n = 0; m(Ai,n ) = m(En )/kn ; hi,n is A-measurable; 1 are independent random variables in the probability space (En , m(E m); (hi,n )∞ n) ∞ i=1 n=1 an hi,n p1 < ∞.
The desired sequence in Lp1 [0, 1] is defined by fi = ∞ n=1 an hi,n . is weakly null in L Note that each sequence (hi,n )∞ p 1 . Thus by passing to subsequences, i=1 using a diagonal argument, we may assume that (hi,n )i,n∈N is, in some order, a perturbation of a block basis of the Haar basis for Lp1 and hence is unconditional. In particular there exists C = C(p1 ) < ∞ so that if (gi )∞ i=1 is a disjointly supported sequence in span{hi,n : i, n ∈ N} w.r.t. the coordinates (hi,n ) then (see e.g. [1])
∞ i=1
1/2
gi 2p1
∞ C gi . i=1
(2.1)
p1
The arguments of [8] yield the following. Let R be an infinite subsequence of N so that for all E ∈ A, M(E) ∩ R is infinite. Let E ∈ A and ε > 0. We shall say h is a Haar function on E if |h| = 1E , h = 0 and h is A-measurable. Then for all infinite M ⊆ N there exists f ∈ span(fi : i ∈ M), say f = bi fi , and a Haar function h on E so that f − h p1 < ε. Moreover, writing ∞ ⎧
⎪ ⎪ ⎪ f = b a h a b h = ⎪ i n i,n n i i,n ⎪ ⎪ ⎪ i n∈R i n=1 ⎨
⎪ ⎪ ≡ f (R) + f (N \ R) + a b h n i i,n ⎪ ⎪ ⎪ ⎪ n ∈R / i ⎪ ⎩ we have f (R) − h p1 < ε and f (N \ R) p1 < ε.
(2.2)
It follows from this that (fi )∞ i=1 has the property that every subsequence admits a block basis 1 + ε-equivalent to the Haar basis in Lp1 .
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We next define (f˜i )∞ i=1 ⊆ X. First partition M(E), for E ∈ A, into infinitely many infinite j ∞ subsets (ME,j )j =1 . For j ∈ N let Rj = E∈A ME,j . Let hi,n be hi,n regarded as an element of Lpj [0, 1]. For i ∈ N, define f˜i =
∞
j
an hi,n .
j =1 n∈Rj
Note that (f˜i )∞ i=1 is seminormalized in X since
f˜i =
2 1/2 2 1/2 ∞ ∞ j 1 an hi,n an hi,n C fj p1 pj
j =1 n∈Rj
j =1 n∈Rj
p1
by (2.1), since · pj · p1 for all j . (f˜i ) is weakly null by vi) and the fact that each (hi,n )∞ i=1 is weakly null. Let E ∈ A, ε > 0 and let M be an infinite subsequence of N. Let j ∈ N and let f = bi fi ∈ span(fi : i ∈ M) and let h be a Haar function on E so that (2.2) holds for R = Rj . Let hj be h regarded as an element of Lpj [0, 1]. Set f˜ = bi f˜i . Then f˜ =
an
n∈Rj
j bi hi,n + bi an hki,n ≡ f˜(Rj ) + f˜(N \ Rj ). k =j n∈Rk
i
i
Now f˜(Rj ) = f (Rj ). By (2.1) and (2.2), f˜(N \ Rj ) = X
2 1/2 k bi an hi,n C f (N \ Rj )p < Cε. 1 k =j
n∈Rk
i
pk
Thus f˜ − hj 2 = f˜(Rj ) − hj 2 + f˜(N \ Rj )2 f (Rj ) − h2 + C 2 ε 2 < 1 + C 2 ε 2 . p X p j
1
As in [8], the fact that every subsequence of (f˜i ) admits a block basis (1 + ε)-equivalent to the Haar basis of Lp follows readily. Each Lpj is stable and from this it is easy to check that X is stable. Thus for every infinite dimensional subspace of X and ε > 0 some further subspace Z satisfies d(Z, q ) < 1 + ε, for some 1 < q 2 [13]. 3. The (USP) does not imply the (K-USP) Example 3.1. There exists a reflexive space X = ( ∞ n=1 Xn )2 satisfying the following: i) For all i ∈ N, Xn is isomorphic to 2 but fails the (
√
n 3 -USP).
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ii) Every normalized weakly null sequence in X admits a subsequence equivalent to the unit vector basis of 2 . iii) For all ε > 0 and subspace Y of X some subspace Z of Y satisfies d(Z, 2 ) < 1 + ε. defined in [16] (and attributed to W.B. Johnson). It is The Xn ’s are slight variants of spaces √ n shown there that those spaces fail the ( 3 -USP) and the same argument holds for our variants. Our work will be in establishing ii). We begin with the definition. Definition of Xn . Let εi ↓ 0 such that
εmax(i,j ) < 1.
(3.1)
{(i,j )∈[N]2 : i =j }
Let M = (mi )∞ i=1 be a subsequence of N so that for all i = j,
min(mi , mj ) < εmax(i,j ) √ √ mi mj
(3.2)
and if Ej ∈ [N]<ω
∞ 1 Ej |Ej | j =1
with |Ej | = mj
for j ∈ N, then
is 2-equivalent to the unit vector basis of 2 in 2 .
(3.3)
Let G = {(E1 , . . . , Ej ): j ∈ N, φ = Ei ∈ [N]<ω for i j and E1 < · · · < Ej }. Choose an injection Φ : G → M so that Φ(E1 , . . . , Ej ) > Φ(E1 , . . . , Ej −1 ) for all (E1 , . . . , Ej ) ∈ G, j 2. For n ∈ N, let n 1 1 Ei Fn = √ : (E1 , . . . , En ) ∈ G, |E1 | = 1, |Ej +1 | = Φ(E1 , . . . , Ej ) for j < n . √ n |Ei | i=1
We also let 1E F0 = √ : E ∈ [N]<ω . |E| For x ∈ c00 let
x Fn = sup f, x: f ∈ Fn . Xn is the completion of c00 under 1
x n = √ x 2 ∨ x Fn ∨ x F0 . n
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Note that since Fn ⊆ S2 , for x ∈ Xn 1 √ x 2 x n x 2 . n to 2 and the unit vector basis (ei ) of c00 is a normalized monotone basis Thus Xn is isomorphic for Xn . X = ( Xn )2 is then reflexive and has a monotone basis, namely the bases for each Xn , properly ordered. The spaces defined in [16] were given by 1
x = √ x 2 ∨ x Fn ∨ x ∞ . n Our example requires the · F0 term. We also have added the lacunary condition (3.3). Lemma 3.2. (See [16].) For n ∈ N, Xn fails the (
√
n 3 -USP).
The proof is the same as that given in [16]. The idea is to consider any subsequence (ei )i∈M of (ei ) and form vectors 1 1 Ei x=√ ∈ Fn √ n |Ei | n
and
i=1
1 1E y=√ (−1)i+1 √ i n |Ei | n
with Ei ⊆ M
for i n.
i=1
√ One then shows that x
√ ∼ 1 and y ∼ 1/ n using (3.1) and (3.2). Also x F0 and y F0 are both of the order 1/ n so the proof remains valid in our modified space. We need (3.3) and
· F0 to prove that X has the (USP) and, in fact, satisfies ii). Lemma 3.3. Let n ∈ N and ε > 0. There exists K(n, ε) < ∞ so that if x 2 |{mj : there exists E ∈ [N]<ω , |E| = mj and | √1mE j , x| ε}| K(n, ε). 1E
√ n, then
1E
Proof. Let |Ei | = mji for i K with ji = j if i = K and | √mij , x| = δi √mij , x ε i i K √ 1Ei Kε √ √ for i K. Let g = √1 δ ∈ 2B , by (3.3). Then g(x) and
x
n so i 2 2 i=1 m ji K K √ Kε 4n √ g(x) 2 n which yields K 2 . 2 ε K
Lemma 3.4. Let n ∈ N and let (xi )∞ i=1 be a normalized block basis of Xn . There exists a subse∞ satisfying, for all (a ) ∈ c , quence (yi )∞ of (x ) i i=1 i 00 i=1 ∞ ai yi i=1
Fn
4
∞ i=1
1/2 ai2
.
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1E As above we will use the notation f = √1n n=1 fi ∈ Fn to signify that fi = √|Ei | , i E1 < · · · < En , |E1 | = 1 and |Ej +1 | = Φ(E1 , . . . , Ej ) for j < n. We will write f (x) for f, x. Proof of Lemma 3.4. Given x ∈ c00 , let x ∗ denote the decreasing rearrangement of (|x(i)|)∞ i=1 . Passing to a subsequence, and relabeling, we may assume √ that (xi∗ )∞ i=1 converges pointwise to some x0 . x0 could be identically 0 but, in any event, x0 ∈ nB2 . By passing to a further subsequence, and perturbing, we may assume, in the case x0 = 0, that xi = xi (1) + xi (2) where xi (1)∗ = x0 |[1,mi ] , for some mi ↑ ∞, xi (2) ∞ → 0 as i → ∞ and |xi (1)| ∧ |xi (2)| = 0. Fix ε > 0 and define A(ε) = {(i, j ) ∈ [N]2 : there exists f = √1n n=1 fi ∈ Fn with |f1 (xi )| ε and |f2 (xj )| ε for some 1 < 2 n}. Passing again to a subsequence, using Ramsey’s Theorem, we may assume A(ε) = ∅ or A(ε) = [N]2 . We will show that the latter is impossible. Assume A(ε) = [N]2 . Passing to a further subsequence, using Ramsey’s Theorem, we may assume the integers 1 < 2 in the definition of A(ε) are fixed, independently of (i, j ). Moreover, we may assume we have one of three cases. (i,j ) (i,j ) Case (1): For all i < j there exists f (i,j ) = √1n n=1 f ∈ Fn with |f1 (xi (1))| ε/2 and (i,j )
|f2
(xj )| ε.
Case (2, s): for s ∈ {1, 2}: For all i < j there exists f (i,j ) = (i,j ) |f1 (xi (2))| ε/2
and
√1 n
n
(i,j ) =1 f
∈ Fn with
i,j |f2 (xj (s))| ε/2.
Assume Case (1) holds. Now |{j : there exists E ∈ [N]<ω , |E| = mj and | √1mE j , x0 | ε/2} < ∞ by Lemma 3.3. Thus, passing to a subsequence and using Ramsey’s Theorem, we 1
(i,j )
may assume that for some fixed j0 , f1
=
(i,j ) E 1 m j0
(i,j )
, where |E1 | = mj0 , for i < j . We claim
√
(i,j )
that if j is sufficiently large then at least K(n, ε) + 1 of the functions {f1 tinct. Hence at least K(n, ε) + 1 of the functions (i,j ) E 1
(i,j ) {f2 :
: i < j } are dis-
i < j } are distinct and this contradicts
could intersect the support of at most mj0 xi ’s. Thus the claim is Lemma 3.3. Each set evident and Case (1) is impossible. Next assume Case (2, 1) holds. As we argued in Case (1), we may similarly assume that for some fixed j1 , (i,j )
f 2
1E (i,j ) =√ 2 , mj1
(i,j ) E = mj , 1 2
for i < j.
1 (i,j )
Since xi (2) ∞ → 0 it follows that for f1
=
(i,j ) E 1
1
(i,j ) E 1
(i,j )
, |E1 | → ∞ as i → ∞. This forces
(i,j )
|E2 | → ∞ as i → ∞ which is a contradiction. Finally we assume Case (2, 2) holds. By passing to a subsequence we may assume for all f = √1n n=1 f ∈ Fn , for all < n, for all j ∈ N, if max(supp f ) max supp(xj (2)) then ∞ f+1 xi (2) < ε 2
i=j +1
and
∞ f1 xi (2) < ε . 2 i=1
(3.4)
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Indeed | supp(f1 )| = 1, | supp(f+1 )| is determined by (f1 , . . . , f ) and xi (2) ∞ → 0. Thus the last statement is achieved by taking the subsequence so that xi (2) < ε/2 for all i. Now fixing x1 (2) we can consider all f ’s and < n so that max supp(f ) max sup(x1 (2)) and pass to a subsequence to achieve (3.4) for j = 1. Let x2 (2) be the first term of this new subsequence and repeat, and so on. (i ,j ) (i ,j ) (i ,j ) Fix j0 and let i1 < i2 < j0 . By (3.4) f1 1 0 = f1 2 0 since max supp f1 1−10 max supp(xi1 ). (i ,j )
(i ,j )
Hence f2 1 0 = f2 2 0 as well for i1 < i2 < j0 . Thus if j0 > K(n, ε) + 1 we again contradict Lemma 3.3. Let δ > 0 and let εi ↓ 0 with ∞ √ n εi < δ.
(3.5)
i=1 ∞ We use the above to inductively pass to a subsequence (yi )∞ i=1 of (xi )i=1 , so that for all m ∈ N, 2 A(m) ≡ A(εm ) ∩ {(i, j ) ∈ [N] : i m} = ∅ where A(εm ) is defined as above with respect to the sequence (yi ). Let f = √1n n=1 f ∈ Fn and (ai ) ∈ c00 . We shall estimate |f ( aj yj )| by breaking it into 3 sums. First note that
1E if g = √ ∈ F0 , |E|
∞ then g(yj ) j =1 ∈ B2 .
(3.6)
Indeed, let |E| = m and |E ∩ supp(yj )| = kj for j ∈ N. Thus ∞ j =1 kj m. Moreover, since is coordinatewise dominated by
yj F0 1 for all j , | 1E , yj | kj . Hence (|g(yj )|)∞ j =1 ∞ 1 √ ( m kj )j =1 . But ∞ 1/2 √
∞ 1 m 1 =√ √ kj kj √ = 1. m m m j =1 2 j =1
Let K = {i: f (yi ) = 0 for at most one n} and let L = N \ K. We split L into two sets. If L = {i1 , i2 , . . .} in increasing order, let L0 = {i1 , i3 , . . .} and Le = {i2 , i4 , . . .}. Thus ∞
a i y i f a i y i + f a i y i + f ai yi . f i=1
i∈L0
i∈Le
i∈K
From the definition of K and (3.6), the vectors (fi (yj ))j ∈K form a block basis in B|K| , over the 2 index i, and thus (f (yj ))j ∈K ∈ B|K| . We then have 2
1/2 f (aj yj ) aj2 . j ∈K
j ∈K
(3.7)
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Let L0 = {j1 , j2 , . . .} in increasing order. Note that if for some n, f (yji ) = 0 then for i = i , f (yji ) = 0. Let m1 = inf m: f (yjm ) εjm for some n . ∞ √ If no such m exists, then for all i |f (yji )| √1n nεji and so ∞ i=1 |f (yji )| i=1 nεji < δ, √ by (3.5). Otherwise, |f (yji )| nεji for i < m1 and |f (yjm1 )| 1. Furthermore, since √ A(jm1 ) = ∅, |f (yjm )| n εjm1 for m > m1 . Let m2 = inf{m > m1 : |f (yjm )| εjm for some n}. If m2 does not exist, we argue as in the case above where m1 does not exist. We have for m1 < m < m2 that √ f (yj ) n εj m m
√ and f (yjm2 ) n εjm1 .
Continuing in this fashion we obtain ∞ ∞ √ f (yj ) = f (yj ) 1 + n εji < 1 + δ. i j ∈L0
A similar estimate holds for
i=1
(3.8)
i=1
|f (yj )|. Thus from (3.7) and (3.8) we have
j ∈Le
1/2 aj yj aj2 + 2(1 + δ)(aj )∞ . f j ∈K
Taking δ <
1 2
yields the lemma.
2
Corollary 3.5. Let n ∈ N, δ > 0 and let (xi )∞ i=1 be a normalized block sequence in Xn . There ∞ satisfying exists a subsequence (yi )∞ of (x ) i i=1 i=1 ∞ ∞ 1/2 ai yi 4 ai2 i=1
i=1
for all (ai ) ∈ c00 . Proof. Note that ∞ 1/2 ∞ 1/2 ∞ 1/2 ∞ 1 1 2 2 2 2 ai xi = √ ai xi 2 ai xi n = ai2 . √ n n i=1
2
i=1
i=1
Thus the corollary follows from Lemma 3.4, (3.6) and the definition of · n .
i=1
2
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Remark 3.6. Corollary 3.5 can be improved in the case xi ∞ → 0. Given δ > 0 we can choose (yi ) to satisfy ∞ ∞ 1/2 ai yi (1 + δ) ai2 i=1
i=1
for all scalars (ai ). To do this we choose a suitable δi ↓ 0 and then choose (yi ) so that if |f (yi )| δi for some n, then j >i |f (yi )| < δi . Thus the argument reduces to that used in the estimation of |f ( j ∈K aj yj )|. Proposition 3.7. Let (xi )∞ i=1 be a normalized weakly null sequence in X and let δ > 0. ∞ a) If xi ∞ → 0, there exists a subsequence (yi )∞ i=1 of (xi )i=1 satisfying ∞ ∞ 1/2 ai yi (1 + δ) ai2 i=1
i=1
for all (ai ) ∈ c00 . b) There exists a subsequence (yi ) of (xi ) satisfying ∞ ∞ 1/2 ai yi 5 ai2 i=1
i=1
for all scalars (ai ). Proof. We prove a) only, using Remark 3.6. The proof of b) is similar, using Corollary 3.5. Let Pn : X → Xn be the natural restriction projection. If I is an interval in N, let PI = n∈I Pn . Passing to a subsequence and perturbing we may assume that (xi )∞ i=1 is a block basis of the ba 2 sis for X, and, for n ∈ N,limi→∞ Pn xi ≡ λn exists. We have that ∞ n=1 λn 1 since xi = 1. ∞ 2 Choose λ0 0 so that n=0 λn = 1. Let δ > 0. Passing to a subsequence, and perturbing, we may assume that we have ε¯ , 0 < δ¯ < δ, and integers 0 = n0 < n1 < n2 < · · · , so that ¯ < λ0 (1 + δ), if λ0 > 0, ε¯ + λ0 + ε¯ (1 + δ) ¯ 2 + ε¯ + ε¯ (1 + δ) ¯ 2 < (1 + δ)2 , (1 + δ) ∞
(3.9) (3.10)
λ2n < ε¯ 2 ,
(3.11)
n=n1 +1
Pn (xj ) = λn
for all j ∈ N, j i and n ni , (3.12) (3.13) xi = P[1,ni+1 ] (xi ) and P(ni ,ni+1 ] (xi ) − λ0 < ε¯ for all i, ∞ ∞ 1/2 ¯ n for j ∈ N, nj −1 < n nj and (ai ) ∈ c00 ai Pn (xi ) (1 + δ)λ ai2 . (3.14) i=j
Eq. (3.14) comes from applying Corollary 3.5 and (3.12).
i=j
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Let (ai ) ∈ S2 . Then, using (3.13), 2 ∞ ∞ 2 n1 ∞ ai xi = ai Pn (xi ) + i=1
n=1
i=1
∞ n1 n=1
+
j =2 n=nj −1 +1
i=j
¯ 2 ai2 λ2n (1 + δ)
i=1
∞
2 ∞ ai Pn (xi ) aj −1 Pn (xj −1 ) +
nj
nj
j =2 n=nj −1 +1
∞ 2 ai xi , |aj −1 | Pn (xj −1 ) + Pn
by (3.14).
i=j
Now the right most term above is, by (3.13) and the triangle inequality in 2 ,
∞
2 j ,nj +1 ] (xj )
a 2 P(n j
1/2 +
j =1
∞
1/2 aj2 (λ0
+ ε¯ )
+
2
j =1
∞
nj
j =2 n=nj −1 +1 ∞
nj
j =2 n=nj −1 +1
∞
∞ 2 1/2 2 ai xi Pn
ai2
i=j
1/2 2
¯ 2 λ2n (1 + δ)
i=j
by (3.13) and (3.14),
1/2 2
∞
¯ (λ0 + ε¯ ) + (1 + δ)
λ2n
n=n1 +1
¯ ε 2, (λ0 + ε¯ ) + (1 + δ)¯
by (3.11).
We thus have, if λ0 > 0, ∞ 2 n1 ¯ 2 ai xi (1 + δ) λ2n + λ20 (1 + δ)2 , i=1
by (3.9),
n=1
< (1 + δ)2 . If λ0 = 0 then 2 ∞ n1 2 ¯ ¯ 2 ai xi (1 + δ) λ2n + ε¯ + ε¯ (1 + δ) i=1
n=1
< (1 + δ)
2
by (3.10).
2
Corollary 3.8. Let (xi )∞ i=1 be a normalized weakly null sequence in X. Then some subsequence is equivalent to the unit vector basis of 2 .
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Proof. By Proposition 3.7 we may assume that (xi ) satisfies an upper 2 estimate. If for some n, limi→∞ Pn (xi ) > 0 then we easily obtain a lower 2 estimate for some subsequence. If this never happens then some subsequence (yi ) of (xi ) satisfies lim P(ni ,ni+1 ] (yi ) − yi = 0 for some n1 < n2 < · · · , i→∞
and so we also easily obtain a lower 2 estimate.
2
Corollary 3.9. Let Y be a subspace of X and ε > 0. There exists a subspace Z of Y with d(Z, 2 ) < 1 + ε. Proof. By Corollary 3.8 we may assume Y contains a basic sequence (yi ) which is equivalent to the unit vector basis of 2 . Replacing (yi ) by a suitable block sequence of long averages we may assume that (yi ) is a block basis of X with yi ∞ → 0. By James’ argument ∞that 1 is ) of (y ) satisfies
not distortable (see e.g. [15]) some normalized block basis (z i i i=1 ai zi 2 1/2 . Passing to a subsequence of (z )∞ , using Proposition 3.7a), we obtain (1 + ε)−1 ( ∞ i i=1 i=1 ai ) the corollary. 2 References [1] D. Alspach, E. Odell, Lp spaces, in: Handbook of the Geometry of Banach Spaces, vol. I, North-Holland, Amsterdam, 2001, pp. 123–159. [2] S.R. Cowell, N.J. Kalton, Asymptotic unconditionality, Q. J. Math. (2009), in press, doi: 10.1093/qmath/han036. [3] D. Freeman, Weakly null sequences with upper estimates, Studia Math. 184 (1) (2008) 79–102. [4] W.T. Gowers, B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (4) (1993) 851–874. [5] S. Guerre, Types and symmetric sequences in Lp , 1 p < +∞, p = 2, Israel J. Math. 53 (2) (1986) 191–208. [6] W.B. Johnson, On quotients of Lp which are quotients of lp , Compos. Math. 34 (1) (1977) 69–89. [7] W.B. Johnson, J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, in: Handbook of the Geometry of Banach Spaces, vol. I, North-Holland, Amsterdam, 2001, pp. 1–84. [8] W.B. Johnson, B. Maurey, G. Schechtman, Weakly null sequences in L1 , J. Amer. Math. Soc. 20 (1) (2007) 25–36. [9] W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (217) (1979) v+298. [10] W.B. Johnson, B. Zheng, A characterization of subspaces and quotients of reflexive Banach spaces with unconditional bases, Duke Math. J. 141 (3) (2008) 505–518. [11] W.B. Johnson, B. Zheng, Subspaces and quotients of Banach spaces with shrinking unconditional bases, preprint. [12] H. Knaust, E. Odell, On c0 sequences in Banach spaces, Israel J. Math. 67 (2) (1989) 153–169. [13] J.-L. Krivine, B. Maurey, Espaces de Banach stables, Israel J. Math. 39 (4) (1981) 273–295. [14] J. Lindenstrauss, A. Pełczy´nski, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971) 225–249. [15] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Ergeb. Math. Grenzgeb., vol. 92, Springer-Verlag, New York, 1977. [16] B. Maurey, H.P. Rosenthal, Normalized weakly null sequence with no unconditional subsequence, Studia Math. 61 (1977) 77–98. [17] E. Odell, On quotients of Banach spaces having shrinking unconditional bases, Illinois J. Math. 36 (1992) 681–695. [18] E. Odell, Th. Schlumprecht, A. Zsák, On the structure of asymptotic p spaces, Quart. J. Math. 58 (2008) 85–122.
Journal of Functional Analysis 258 (2010) 616–649 www.elsevier.com/locate/jfa
Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise Carlo Marinelli a,∗ , Claudia Prévôt b , Michael Röckner b,c a Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D-53115 Bonn, Germany b Fakultät für Mathematik, Universität Bielefeld, Postfach 100 131, D-33501 Bielefeld, Germany c Departments of Mathematics and Statistics, Purdue University, 150 N. University St., West Lafayette, IN 47907-2067,
USA Received 19 April 2009; accepted 23 April 2009 Available online 13 May 2009 Communicated by Paul Malliavin
Abstract We prove existence, uniqueness and Lipschitz dependence on the initial datum for mild solutions of stochastic partial differential equations with Lipschitz coefficients driven by Wiener and Poisson noise. Under additional assumptions, we prove Gâteaux and Fréchet differentiability of solutions with respect to the initial datum. As an application, we obtain gradient estimates for the resolvent associated to the mild solution. Finally, we prove the strong Feller property of the associated semigroup. © 2009 Elsevier Inc. All rights reserved. Keywords: Stochastic PDE with jumps; Maximal inequalities; Strong Feller property
1. Introduction We shall consider the mild formulation of a stochastic PDE of the form du(t) = Au(t) + f t, u(t) dt + B t, u(t) dW (t) +
G t, u(t), z μ(dt, ¯ dz),
Z
u(0) = x, * Corresponding author.
E-mail addresses:
[email protected] (C. Marinelli),
[email protected] (M. Röckner). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.04.015
(1.1)
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where W and μ¯ are a Wiener process and a compensated Poisson measure, respectively, on a Hilbert space, thus including a large class of equations driven by Hilbert space-valued Lévy noise, thanks to the Lévy–Itô decomposition theorem. Precise assumptions on the data of (1.1) are given in the next section. The first main contribution of this paper is global well-posedness (i.e. existence, uniqueness, and continuous dependence on the initial datum of a mild solution on any time interval [0, T ], T < ∞) for (1.1) in spaces of càdlàg predictable processes whose supremum (in time) has finite pth moments. While the L2 result was fully settled by Kotelenez [18] about twenty-five years ago, the lack of an Lp theory has been pointed out more recently in [2]. The new tool allowing the development of such a theory is an infinite-dimensional Bichteler–Jacod inequality, which also holds for stochastic convolutions. The Lp existence result allows us to prove the second main result of this work, that is first and second order Fréchet differentiability of the solution with respect to the initial datum (for first order Gâteaux differentiability the L2 theory is enough, see also [1]). Moreover, these differentiability results are a key tool to prove that, as long as the noise has a Brownian component, the semigroup associated to the SPDE is regularizing, in particular, that it has the strong Feller property. An essential ingredient to obtain this result is a formula of Bismut–Elworthy type, which only holds under non-degeneracy assumptions on B. Finally, we also obtain gradient estimates on the resolvent associated to the SPDE. The issues considered in this paper are by now classical for stochastic PDE with Wiener noise (see e.g. [5–7,11] and references therein), but comparable results do not seem to be available in the more general jump case considered here. In fact, it is fair to say that the theory of stochastic PDEs driven by jump noise is not yet fully developed, even though recent years have witnessed a growing interest in the area: let us just mention, without any claim of completeness, the recent monograph [24], where the semigroup approach is discussed, [20] for an analytic approach based on generalized Mehler semigroups, as well as the earlier important contributions [12,22] for the variational approach. Let us also mention that differentiability properties of the solution to stochastic PDE play an essential role in the probabilistic approach to infinite-dimensional Kolmogorov equations, including cases with quite general monotone nonlinearities. This direction of research, while thoroughly pursued in the case of Wiener noise (see e.g. [5,6,29]), is still in its infancy for equations with jumps, and our results provide a basis for further developments. The paper is organized as follows: in Section 2 precise assumptions on the SPDE (1.1) are given, and the main results on well-posedness and regular dependence on the initial datum are stated. In Section 3 we prove a Bichteler–Jacod inequality for infinite-dimensional stochastic integrals with respect to Poisson random measures, and we extend it to corresponding stochastic convolutions. This result is essential in order to obtain Lp well-posedness, and it could be interesting in its own right. We also recall some results on the differentiability of implicit functions in Banach spaces, on which the proofs of regular dependence heavily rely. Section 4 contains the proofs of the well-posedness and differentiability results. In Section 5 we obtain an analytic consequence of these results, that is gradient estimates for the resolvent associated to the (solution of the) SPDE. Finally, in Section 6 we show that the semigroup associated to the SPDE is strong Feller, if B is not degenerate. Finally, we would like to mention that a part of the results of this paper has been announced in [25] (based on [26]). There was, however, an error both in the formulation and proof in what was called there Burkholder–Davis–Gundy inequality for Poisson integrals, on which all subsequent results depended. One point of this paper is to correct this error. The corresponding
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inequality is contained in Proposition 3.3 below. Then, as described above, among other things we prove that all results announced in [25] hold. Let us conclude this introduction with some words about notation. By a b we mean that there exists a constant N such that a N b. To emphasize that the constant N depends on a parameter p we shall write N (p) and a p b. Generic constants, which may change from line to line, are denoted by N . Given two separable Banach spaces E, F we shall denote the space of linear bounded operators from E to F by L(E, F ). Similarly, if H and K are Hilbert spaces, we shall denote the space of trace-class and Hilbert–Schmidt operators from K to H by L1 (K, H ) and L2 (K, H ), respectively. L+ 1 stands for the subset of L1 consisting of all positive operators. We shall write L1 (H ) in place of L1 (H, H ), and similarly for the other spaces. Q Given a self-adjoint operator Q ∈ L+ 1 (K), we denote by L2 (K, H ) the set of all (possibly unbounded) operators B : Q1/2 K → H such that BQ1/2 ∈ L2 (K, H ). The norms in L2 (K, H ) and Q L2 (K, H ) will be denoted by | · |2 and | · |Q , without explicitly indicating the dependence on the spaces K and H . Lebesgue measure is denoted by Leb, without mentioning the underlying space if no misunderstanding can arise. Given a function φ : E → F , we set [φ]1 := sup x,y∈E x=y
|φ(x) − φ(y)| , |x − y|
and we denote by ∂φ : E × E → F the map (x, y) → ∂y φ(x), where the directional derivative ∂y φ(x) is defined by φ(x + hy) − φ(x) . h→0 h
∂y φ(x) := lim Qhy φ(x) := lim h→0
We shall also use the symbol ∂φ(x) to denote the Gâteaux derivative, so ∂φ(x) ∈ L(E, F ), defined by y → ∂y φ(x). Analogously, given a function φ : E1 × E2 → F , where E1 , E2 are further Banach spaces, we define the following directional derivatives φ(x1 + hy, x2 ) − φ(x1 , x2 ) , h→0 h→0 h φ(x1 , x2 + hz) − φ(x1 , x2 ) , ∂2,z φ(x1 , x2 ) = lim Qh2,z φ(x1 , x2 ) := lim h→0 h→0 h
∂1,y φ(x1 , x2 ) = lim Qh1,y φ(x1 , x2 ) := lim
and the corresponding maps ∂1 φ : E1 × E2 × E1 (x1 , x2 , y) → ∂1,y φ(x1 , x2 ) ∈ F, ∂2 φ : E1 × E2 × E2 (x1 , x2 , z) → ∂2,z φ(x1 , x2 ) ∈ F. Partial Gâteaux derivatives are denoted by the same symbols. Fréchet differentials are denoted by D, with subscripts if necessary. Moreover, in view of the canonical isomorphism between L(E, L(E, F )) and L⊗2 (E, F ), the space of bilinear maps from E to F , we can and will consider D 2 φ as a map from E to L⊗2 (E, F ). The space of k times continuously differentiable maps from E to F will be denoted by C k (E, F ), and simply by C k (E) if F = R. We shall occasionally use the following standard notation for stochastic integrals with respect to semimartingales and random measures:
C. Marinelli et al. / Journal of Functional Analysis 258 (2010) 616–649
t φ · M(t) :=
619
t φ(s) dM(s),
φ μ(t) :=
0
φ(s, z) μ(ds, dz). 0
2. Main results Let us begin stating our precise assumptions on Eq. (1.1). We are given two real separable Hilbert spaces H , K and a filtered probability space (Ω, F , F, P), F = (Ft )t∈[0,T ] , on which a Wiener process with covariance operator Q ∈ L+ 1 (K) is defined. Moreover, we are given a measure space (Z, Z, m) and a Poisson measure μ on [0, T ] × Z, independent of W , defined on the same stochastic basis. The compensator (dual predictable projection) of μ is Leb ⊗ m, and the compensated measure μ¯ is μ¯ := μ − Leb ⊗ m. Denoting the predictable σ -field by P, we shall assume throughout the paper that the following assumptions are satisfied: (i) A is the generator of a strongly continuous semigroup on H ; Q (ii) f : Ω × [0, T ] × H → H and B : Ω × [0, T ] × H → L2 (K, H ) are P × B(H )-measurable functions; (iii) G : Ω × [0, T ] × H × Z → Z is a P × B(H ) × Z-measurable function; (iv) x is an H -valued F0 -measurable random variable. Further assumptions on the data of the problem will be specified when needed. For simplicity, we shall suppress explicit dependence on ω ∈ Ω of all random elements, if no confusion can arise. Let us also recall that, by (i), there exist M, σ 0, such that |etA | Meσ t . We set, for future reference, MT := Meσ T . The concept of solution we shall work with and the spaces where solutions are sought are defined next. Definition 2.1. A predictable process u : [0, T ] → H is a mild solution of (1.1) if it satisfies t u(t) = e x + tA
e 0
+
(t−s)A
f s, u(s) ds +
t
e(t−s)A B s, u(s) dW (s)
0
e(t−s)A G s, u(s), z μ(ds, ¯ dz)
(0,t] Z
P-a.s. for all t ∈ [0, T ], where at the same time we assume that all integrals on the right-hand side exist. We shall also write u(x) to emphasize the dependence on the initial datum, and u(t, x) will stand for the value of u(x) at time t ∈ [0, T ]. t In the following, for simplicity of notation, we shall often write 0 in place of (0,t] . Definition 2.2. Let p 2. We shall denote by Hp (T ) and Hp (T ) the spaces of all predictable processes u : [0, T ] → H such that
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1/p [u] := sup Eu(t)p < ∞, p t∈[0,T ]
and p 1/p
u p := E sup u(t) < ∞, t∈[0,T ]
respectively. For reasons that will become apparent later, we shall also need to consider the same spaces endowed with the equivalent norms [u]
:= p,λ
p 1/p sup Ee−λt u(t) ,
p 1/p
u p,λ := E sup e−λt u(t) ,
t∈[0,T ]
t∈[0,T ]
with λ > 0. We shall also use the notation Lp to denote the set Lp (Ω, F , P; H ). The following well-posedness result in H2 (T ) is quite simple to prove and it essentially relies only on the isometric formula for stochastic integrals with respect to compensated Poisson measures (see also [1,24] for similar results). Q
Theorem 2.3. Assume that x ∈ L2 , esA B(t, x) ∈ L2 (K, H ), esA G(t, x, ·) ∈ L2 (Z, m) for all (s, t, x) ∈ [0, T ]2 × H , and there exist h ∈ L1 ([0, T ]) and a ∈ H such that sA e f (t, x) − f (t, y) 2 + esA B(t, x) − B(t, y) 2 Q sA 2 + e G(t, x, z) − G(t, y, z) m(dz) Z
h(s)|x − y|2 , sA e f (t, a)2 + esA B(t, a)2 + Q
(2.1)
sA e G(t, a, z)2 m(dz) h(s)
(2.2)
Z
P-a.s. for all x, y ∈ H and s, t ∈ [0, T ]. Then Eq. (1.1) admits a unique mild solution in H2 (T ). Moreover, the solution map x → u(x) is Lipschitz from L2 to H2 (T ). As briefly mentioned above, this theorem is stated and proved for its simplicity, even though a more refined result holds true. In fact, one can look for mild solutions of (1.1) in the smaller (and more regular) spaces Hp (T ), obtaining also that solutions have càdlàg paths. The price to pay is that one has to find suitable estimates to replace the isometry of the stochastic integral. In order to obtain such estimates, we need to assume that A is η–m-dissipative, i.e. that A − ηI is m-dissipative for some η 0 (this is equivalent to assuming that |etA | eηt for all t 0, i.e. that the semigroup generated by A is of quasi-contraction type). On the other hand, Theorem 2.3 above holds without the quasi-dissipativity condition on A. Our first main result is the following theorem, where the solution map is defined from Lp to Hp (T ).
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Theorem 2.4. Let p 2. Assume that A is η–m-dissipative, x ∈ Lp , and there exist h ∈ Lp ([0, T ]) and a ∈ H such that sA e f (t, x) − f (t, y) + B(s, x) − B(s, y) Q + max G(s, x, ·) − G(s, y, ·) L (Z,m) , G(s, x, ·) − G(s, y, ·)L 2
p (Z,m)
h(s)|x − y|, sA e f (t, a)2 + B(s, a)2 + Q
(2.3)
G(s, a, z)2 + G(s, a, z)p m(dz) h(s)
(2.4)
Z
P-a.s. for all x, y ∈ H and s, t ∈ [0, T ]. Then Eq. (1.1) admits a unique càdlàg mild solution in Hp (T ). Moreover, the solution map x → u(x) is Lipschitz from Lp to Hp (T ). Remark 2.5. Before we proceed to state other results, we would like to make the following remarks: (i) Much more general existence and uniqueness results in H2 (T ) were proved by Kotelenez [18], where noise terms driven by general locally square integrable martingales are allowed, as well as locally Lipschitz coefficients with linear growth. (ii) One could also consider equations driven by martingales with independent increments, “embedding” equations driven by compensated Poisson random measures, using the equivalence result of Gyöngy and Krylov [13]. This approach, however, even though very powerful, would be less transparent, and for this reason we prefer to work directly with equations driven by a Wiener process and a compensated Poisson measure. Let us also recall that, if one only wants to obtain results in H2 (T ), then general stochastic martingale measures are also allowed, appealing to the results in [18] and to the above mentioned procedure developed in [13]. (iii) If the coefficients of (1.1) are independent of ω ∈ Ω, then one can obtain the Markov property of solutions in a standard way, e.g. following the method of [19, Sect. 2.9] – see also [12,24]. (iv) It is not difficult to prove that mild solutions are weak solutions (in the sense of PDEs), as it follows, roughly speaking, by a suitable stochastic version of Fubini’s theorem. More details can be found e.g. in [24, Sect. 9.3]. Under the additional assumption that the coefficients f , B and G are Gâteaux differentiable, we obtain that the solution map enjoys the same property. For this to hold, the simpler H2 (T ) well-posedness suffices. In particular, no quasi-m-dissipativity assumption on A is needed. From here until the end of this section we assume, for simplicity only, that f , B and G are deterministic maps that do not depend on time. Given a Banach space E, we shall denote the p space of functions φ : Z → E such that Z |φ|E dm < ∞ by Lp (Z, m; E). Theorem 2.6. Under the hypotheses of Theorem 2.3, assume that (i) f is Gâteaux differentiable with ∂f ∈ C(H × H, H ); Q (ii) B is Gâteaux differentiable and ∂B ∈ C(H × H, L2 (K, H ));
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(iii) the map x → G(x, z) is Gâteaux differentiable for all s ∈ ]0, T ] and z ∈ Z, and x → esA ∂1,y G(x, z) ∈ C(H, H ) for all s ∈ ]0, T ], y ∈ H , and z ∈ Z; (iv) one has x → esA ∂1,y G(x, ·) ∈ C H, L2 (Z, m; H ) for all s ∈ ]0, T ] and y ∈ H . Then the solution map x → u(x) : L2 → H2 (T ) is Gâteaux differentiable and ∂u : (x, y) → ∂y u(x) ∈ C(L2 × L2 , H2 (T )) is the mild solution of dv(t) = Av(t) dt + ∂f u(t, x) v(t) dt + ∂B u(t, x) v(t) dW (t) ¯ dz), v(0) = y. + ∂1 G u(t, x), z v(t) μ(dt,
(2.5)
Z
Moreover, one has ∂y u(x) N |y|L 2 2 for all x, y ∈ L2 , where the constant N , which does not depend on x and y, is the Lipschitz constant of the solution map L2 x → u(x) ∈ H2 (T ). On the other hand, in order to obtain Fréchet differentiability of the solution map, the full Hp (T ) well-posedness result is needed. At this point we would like to stress that the following two theorems cannot be proved, to the best of our knowledge, on the basis of the already known H2 (T ) well-posedness, even if one is interested only in the Fréchet differentiability of the solution map from H to H2 (T ). Theorem 2.7. Let q > p 2, and assume that the hypotheses of Theorem 2.4 are satisfied with p replaced by q. Moreover, assume that (i) f ∈ C 1 (H, H ) and B ∈ C 1 (H, L(K, H )); Q (ii) x → etA DB(x) ∈ C(H, L(H, L2 (K, H ))) for all t ∈ ]0, T ]; (iii) x → G(x, z) ∈ C 1 (H, H ) for all z ∈ Z. Then the solution map Lp x → u(x) ∈ Hp (T ) is Gâteaux differentiable and (x, y) → ∂y u(x) ∈ C(Lp × Lp , Hp (T )) is the mild solution of (2.5) in Hp (T ). Moreover, one has ∂y u(x) N |y|L p p for all x, y ∈ Lp , where N denotes the Lipschitz constant of Lp x → u(x) ∈ Hp (T ). Finally, if x ∈ Lq , then x → u(x) ∈ C 1 (Lq , Hp (T )). In particular, x → u(x) ∈ C 1 (H, Hp (T )).
C. Marinelli et al. / Journal of Functional Analysis 258 (2010) 616–649
623
Theorem 2.8. Let q > 2p 4. Under the hypotheses of Theorem 2.4 with p replaced by q, assume that (i) f ∈ C 2 (H, H ), B ∈ C 2 (H, L(K, H )), and there exists C1 > 0 such that 2 D f (x) + D 2 B(x) C1
∀x ∈ H ;
(ii) the map x → G(x, z) : H → H is twice Fréchet differentiable for all z ∈ Z and x → D12 G(x, z) ∈ C H, L H, L(H ) for all z ∈ Z; (iii) there exists h1 ∈ Lp (Z, m) ∩ L2 (Z, m) such that 2 D G(x, z)(y1 , y2 ) h1 (z)|y1 ||y2 | 1
for all x, y1 , y2 ∈ H and z ∈ Z; (iv) there exists k ∈ L2 ([0, T ]) such that tA 2 e D B(x)(y, z) k(t)|y||z|. Q Then the Fréchet derivative Du : Lq → L(Lq , Hp (T )) is Gâteaux differentiable. Let x, y1 , y2 ∈ Lq , and w := [∂Du(x)](y1 , y2 ) ≡ [∂ 2 u(x)](y1 , y2 ), v1 = Du(x)y1 , v2 = Du(x)y2 . Then w is the mild solution of dw(t) = Aw(t) + Df u(t) w(t) + D 2 f u(t) v1 (t), v2 (t) dt D1 G u(t), z w(t) + D12 G u(t), z v1 (t), v2 (t) μ(dt, ¯ dz), + Z
w(0) = 0. Moreover, there exists a constant N = N (T , p, q) > 0 such that ∂Du(x)(y1 , y2 ) N |y1 |L |y2 |L q q p for all y1 , y2 ∈ Lq . Finally, if q > 4p 8, then x → u(x) ∈ C 2 Lq , Hp (T ) . In particular, the solution map belongs to C 2 (H, Hp (T )).
(2.6)
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3. Auxiliary results 3.1. Lp estimates for stochastic convolutions In order to prove Theorem 2.4 we need to establish a maximal inequality for stochastic convolutions with respect to a compensated Poisson measure, which may be of independent interest. For related estimates (which hold only for stochastic integrals) in the finite-dimensional case see [4] and references therein, and for the special case of stochastic integrals with respect to Lévy processes [16,28]. Maximal inequalities for stochastic convolutions can be found e.g. in [15,17,18]. None of the latter results, however, seems to be useful to obtain the estimates we need to establish well-posedness in Hp (T ). Let us begin with a Bichteler–Jacod inequality for Poisson integrals. Lemma 3.1. Let p 2. Assume that g : [0, T ] × Z → H is a predictable process such that the expectation on the right-hand side of (3.1) below is finite. Then one has p E sup g(s, z) μ(ds, ¯ dz) tT
(0,t] Z
T NE Z
0
g(s, z)p m(dz) +
p/2 g(s, z)2 m(dz) ds,
(3.1)
Z
where N = N(p, T ), and (p, T ) → N is continuous. Proof. Setting φ : H → R, φ(x) = |x|p , we have that φ is twice Fréchet differentiable with derivatives φ (x) : η → p|x|p−2 x, η and φ (x) : (η, ζ ) → p(p − 2)|x|p−4 x, ηx, ζ + p|x|p−2 η, ζ ,
x = 0,
φ (0) = 0. Let us set X = g μ. ¯ Then Itô’s formula (see e.g. [21]) yields X(t)p = p
t
X(s−)p−2 X(s−), dX(s)
0
+
X(s)p − X(s−)p − p X(s−)p−2 X(s−), X(s)
(3.2)
st
P-a.s. for all t T , where, as usual, X(s) := X(s) − X(s−). Applying Taylor’s formula to the function φ we obtain
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625
X(s)p − X(s−)p − p X(s−)p−2 X(s−), X(s) p−4 2 1 = p(p − 2)X(s−) + ξ X(s) X(s−) + ξ X(s), X(s) 2 p−2 1 X(s)2 + p X(s−) + ξ X(s) 2 p−2 1 X(s)2 , p(p − 1)X(s−) + ξ X(s) 2 where ξ ≡ ξ(s) ∈ ]0, 1[ (see e.g. [10, Thm. 4.18.1]). Since |X(s−) + ξ X(s)| |X(s−)| + | X(s)|, we also have X(s−) + ξ X(s)p−2 p X(s−)p−2 + X(s)p−2 X ∗ (s−)p−2 + X(s)p−2 , where X ∗ (s) := suprs |X(r)|. Let us now assume, for the time being, that X is bounded P-a.s. Then the first term on the right-hand side of (3.2) is a martingale with expectation zero, and we obtain p p 2 EX(t) N (p)E X ∗ (s−)p−2 X(s) + X(s) . st
Therefore, recalling that the compensator of μ is m ⊗ Leb and using Young’s inequality p
ab
a p−2 p p−2
+
bp/2 , p/2
we get p EX(t) p E
t
2 ∗ X (s−)p−2 g(s, ·)L
2 (Z,m)
p + g(s, ·)L
p (Z,m)
ds
0
t p E
p ∗ p X (s) + g(s, ·)L
p + g(s, ·)L
p ∗ p X (s) + g(s, ·)L
p + g(s, ·)L
2 (Z,m)
p (Z,m)
ds.
0
Doob’s inequality then yields ∗
t
EX (t) p E p
2 (Z,m)
p (Z,m)
ds,
0
hence, thanks to Gronwall’s inequality, we obtain (3.1). In order to remove the assumption that X is bounded almost surely, we shall proceed in two steps. Assume first that |g(s, z)| N a.s. for all (s, z) ∈ [0, T ] × Z, and define the stopping times τn = inf t 0: X(t) > n ∧ T .
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We clearly have τn → T a.s. as n → ∞ because X is bounded on compact intervals. We also have X(t)
sup
(s,z)∈[0,T ]×Z
g(s, z) N
a.s.
hence, setting Yn = (1]0,τn ] g) μ¯ ≡ X(t ∧ τn ), one easily sees that Yn (t) Yn (t−) + sup X(t) n + N, tT
and, by Fatou’s lemma and passing to the limit, EX ∗ (T )p lim inf EYn∗ (T )p n→∞
T p lim inf E n→∞
Z
0
T E 0
1]0,τ ] g(s, z)p m(dz) + n
p/2 1]0,τ ] g(s, z)2 m(dz) ds n
Z
g(s, z)p m(dz) +
Z
p/2 g(s, z)2 m(dz) ds,
Z
which proves the claim if g is a.s. bounded. The general case can be proved by setting gn (s, z) :=
g(s, z), g(s,z) n |g(s,z)| ,
if |g(s, z)| n, if |g(s, z)| > n,
and Xn := gn μ, ¯ from which it is easy to prove that, by (3.1), {Xn }n∈N is a Cauchy sequence ¯ Using again Fatou’s lemma and recalling in Hp (T ) and Xn → X in Hp (T ), with X = g μ. that (3.1) holds for bounded integrands, we have EX ∗ (T )p lim inf EXn∗ (T )p n→∞
T p lim inf E n→∞
0
T E 0
which concludes the proof.
gn (s, z)p m(dz) +
Z
g(s, z)p m(dz) +
Z
p/2 gn (s, z)2 m(dz) ds
Z
p/2 g(s, z)2 m(dz) ds,
Z
2
Remark 3.2. The “usual” Bichteler–Jacod inequality for Lévy integrals (see e.g. [28]) follows immediately by (3.1) and the Lévy–Itô decomposition. Moreover, the proof we gave is different from the ones in the literature and, apart from holding also in infinite dimensions, has the peculiarity of avoiding completely the use of the Burkholder–Davis–Gundy’s inequality.
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Inequality (3.1) can be extended also to stochastic convolutions, even though in general they are not martingales. Proposition 3.3. Let A be m-dissipative on H and g satisfies the hypotheses of Lemma 3.1. Then for all p ∈ [2, ∞) there exists a constant N such that t p e(t−s)A g(s, z) μ(ds, ¯ dz) E sup tT 0 Z
T NE 0
g(s, z)p m(dz) +
Z
p/2 g(s, z)2 m(dz) ds,
(3.3)
Z
where N depends continuously on p and T only. Proof. We shall follow the approach of [14]. In particular, by Sz.-Nagy’s theorem on unitary dilations, there exists a Hilbert space H¯ , with H isometrically embedded into H¯ , and a unitary strongly continuous group T (t) on H¯ such that πT (t)x = etA x for all x ∈ H , t ∈ R, where π denotes the orthogonal projection from H¯ to H . Then we have, recalling that the operator norms of π and T (t) are less than or equal to one, t p E sup e(t−s)A g(s, z) μ(ds, ¯ dz) tT
H
0 Z
p t = E sup πT (t) T (−s)g(s, z) μ(ds, ¯ dz) ¯ tT
H
0 Z
t p p |π|p sup T (t) E sup T (−s)g(s, z) μ(ds, ¯ dz) ¯ tT tT
H
0 Z
t p E sup T (−s)g(s, z) μ(ds, ¯ dz) . ¯ tT H
0 Z
Since the integral in the last expression is a martingale, inequality (3.1) implies that there exists a constant N = N(p, T ) such that t p (t−s)A E sup e g(s, z) μ(ds, ¯ dz) tT 0 Z
T NE 0
Z
T (−s)g(s, z)p m(dz) +
Z
p/2 T (−s)g(s, z)2 m(dz) ds
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T NE 0
g(s, z)p m(dz) +
Z
p/2 g(s, z)2 m(dz) ds,
Z
where we have used again that T (t) is a unitary group and that the norms of in H¯ and H are equal. 2 Corollary 3.4. Let A be η–m-dissipative. Then inequality (3.3) holds, with N a continuous function of p, T , and η. Proof. Follows by exactly the same arguments used above applied to the m-dissipative operator A − ηI . 2 3.2. Differentiability of implicit functions In order to prove regular dependence of solutions with respect to the initial datum, we shall need the following versions of the implicit function theorem. Similar results can be found in the literature (see e.g. [5,8,9]), but we have included the complete statements here for the reader’s convenience. A proof of these specific versions can be found in [11]. Let E, Λ be two Banach spaces, and Φ : Λ × E → E a function such that Φ(λ, x) − Φ(λ, y) α|x − y| for all λ ∈ Λ and all x, y ∈ E, with α ∈ [0, 1[. Banach’s fixed point theorem implies the existence and uniqueness of a function φ : Λ → E such that Φ(λ, φ(λ)) = φ(λ) for all λ ∈ Λ. Theorem 3.5. Assume that λ → Φ(λ, x) is continuous for all x ∈ E. Then φ ∈ C(Λ, E). Moreover, if Φ is Lipschitz with respect to λ uniformly over x ∈ E, then φ is Lipschitz. Theorem 3.6. Assume that Φ(·, x) : Λ → E is continuous for all x ∈ E, and that the maps ∂1 Φ : Λ × E × Λ → E, ∂2 Φ : Λ × E × E → E are continuous. Then φ is Gâteaux differentiable and (λ, μ) → ∂μ φ(λ) is continuous from Λ × Λ to E. Moreover, one has −1 ∂1,μ Φ λ, φ(λ) . ∂μ φ(λ) = I − ∂2 Φ λ, φ(λ) In the formulation of the following theorems we shall denote by Λ0 and E0 two Banach spaces continuously embedded in Λ and E, respectively. Moreover, Λ1 will denote a further Banach space continuously embedded in Λ0 . Theorem 3.7. Assume that Φ satisfies the hypotheses of Theorem 3.6, also with Λ0 and E0 replacing Λ and E, respectively. Moreover, assume that ∂1 Φ ∈ C(Λ0 × E0 , L(Λ0 , E)) and ∂2 Φ ∈ C(Λ0 × E0 , L(E0 , E)). Then ∂φ ∈ C(Λ0 , L(Λ0 , E)), hence φ ∈ C 1 (Λ0 , E). Theorem 3.8. Assume that both Φ : Λ × E → E and Φ : Λ0 × E0 → E0 satisfy the hypotheses of Theorem 3.6. If Φ : Λ0 × E0 → E admits second-order directional derivatives, then φ : Λ0 → E is twice Gâteaux differentiable with ∂ 2 φ ∈ C(Λ30 , E) and
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−1 2 ∂1 Φ λ0 , φ(λ0 ) (μ0 , ν0 ) ∂ 2 φ(λ0 ) : (μ0 , ν0 ) → I − ∂2 Φ λ0 , φ(λ0 ) + ∂1 ∂2 Φ λ0 , φ(λ0 ) ∂μ0 φ(λ0 ), ν0 + ∂2 ∂1 Φ λ0 , φ(λ0 ) μ0 , ∂ν0 φ(λ0 ) + ∂22 Φ λ0 , φ(λ0 ) ∂μ0 φ(λ0 ), ∂ν0 φ(λ0 ) . Theorem 3.9. Assume that both Φ : Λ × E → E and Φ : Λ0 × E0 → E0 satisfy the hypotheses of Theorem 3.6. Moreover, assume that Φ ∈ C 2 (Λ0 × E0 , E) and that φ ∈ C 1 (Λ1 , E0 ). Then the Fréchet derivative Dφ : Λ1 → L(Λ1 , E) is Gâteaux differentiable. Furthermore, if ∂Dφ can be realized as a map Λ1 → L(Λ1 , L(Λ1 , E0 )), then φ ∈ C 2 (Λ1 , E). Corollary 3.10. Let Φ be as in the previous theorem and φ ∈ C 1 (Λ1 , E0 ). Moreover, assume that Dφ and Di Dj Φ, i, j ∈ {1, 2}, are bounded. Then ∂Dφ : Λ1 → L(Λ1 , L(Λ1 , E)). 3.3. Some regularization results We record for future reference some simple regularization and approximation results which are used in the proofs of the main results. Proposition 3.11. Let u be the mild solution of (1.1) in H2 (T ), and uλ the strong solution of the equation du(t) = Aλ u(t) + f u(t) dt + B u(t) dW (t) +
G u(t−), z μ(dt, ¯ dz),
Z
u(0) = x,
(3.4)
where Aλ stands for the Yosida approximation of A. Then uλ → u in H2 (T ) as λ → 0. Proof. We sketch the proof only, as it resembles the corresponding proof for equations driven by Wiener noise only. In fact, the strong solution uλ of (3.4) is an adapted càdlàg process, and the predictable process t → uλ (t−) is a mild solution of (3.4). Recalling that, for a fixed t ∈ [0, T ], one has uλ (t) − uλ (t−) = 0 almost surely (no jumps at a fixed time can occur), we can proceed along the lines of e.g. [6, Thm. 3.5]. 2 In the following proposition we take f , fε , B, Bε , G, Gε independent of t ∈ [0, T ] and ω ∈ Ω. Proposition 3.12. Let u and uε be, respectively, the mild solutions in H2 (T ) of (1.1) and of the equation obtained replacing f , B, and G with fε , Bε , and Gε in (1.1), where fε (x) → f (x) in Q H , etA Bε (x) → etA B(x) in L2 (H ), and
tA e Gε (x, z) − G(x, z) 2 m(dz) → 0
Z
for all x ∈ H as ε → 0. Moreover, assume that there exists K > 0 such that
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[fε ]21
2 + etA Bε (x) − Bε (y) Q +
tA e Gε (x, z) − Gε (y, z) 2 m(dz) K|x − y|2
Z
for all t ∈ [0, T ]. Then uε → u in H2 (T ). Proof. We have 2 Euε (t) − u(t) E
t
(t−s)A 2 e fε uε (s) − f u(s) ds
0
t +E
(t−s)A 2 e Bε uε (s) − B u(s) Q ds
0
t +E
(t−s)A 2 e Gε uε (s), z − G u(s), z m(dz) ds
0 Z
=: I1 (ε) + I2 (ε) + I3 (ε), and t
(t−s)A 2 e Bε uε (s) − Bε u(s) Q ds
I2 (ε) E 0
t +E
(t−s)A 2 e Bε u(s) − B u(s) Q ds
0
t KE
uε (s) − u(s)2 ds + δ2 (ε),
0
where δ2 (ε) → 0 as ε → 0, in view of the assumptions on Bε and by dominated convergence. Completely similar estimates can be obtained for I1 (ε) and I3 (ε). We thus get 2 Euε (t) − u(t) N
t
2 Euε (t) − u(t) + δ(ε),
0
with δ(ε) → 0 as ε → 0, and the conclusion follows by Gronwall’s lemma.
2
4. Proofs Proof of Theorem 2.3. We sketch the proof only, as we follow the well-known approach based on Banach’s fixed point theorem. We have to prove that the mapping F : H2 (T ) → H2 (T ) defined by
C. Marinelli et al. / Journal of Functional Analysis 258 (2010) 616–649
t Fu(t) = e x + tA
e
(t−s)A
f s, u(s) ds +
0
t
+
t
631
e(t−s)A B s, u(s) dW (s)
0
e(t−s)A G s, u(s), z μ(ds, ¯ dz)
(4.1)
0 Z
is well defined and is a contraction, after which the result follows easily. Let us show that, for any u ∈ H2 (T ), Fu admits a predictable modification such that |[Fu]|2 < ∞. Predictability of Fu follows by the mean-square continuity of the stochastic convolution term with respect to μ¯ t (t−s)A in (4.1): in fact, setting MA (t) = 0 Z e G(s, u(s), z) μ(ds, ¯ dz), a simple calculation shows that, for 0 s t T , 2 EMA (t) − MA (s) E
s
(t−r)A 2 2 e − e(s−r)A G r, u(r), z m(dz) dr
0 Z
t +E
(t−r)A 2 e G r, u(r), z 2 m(dz) dr,
s Z
which converges to zero as s → t. Moreover, we have t 2 2 tA 2 [Fu] sup Ee x + sup E e(t−s)A f s, u(s) ds 2 tT tT 0
t 2 (t−s)A + sup E e B s, u(s) dW (s) tT 0
t 2 + sup E e(t−s)A G s, u(s), z μ(ds, ¯ dz) . tT 0 Z
Using the isometry for stochastic integrals, and noting that hypotheses (2.2) and (2.1) imply the estimate Z
we have
sA e G(t, x, z)2 m(dz) N h(s) 1 + |x| 2 ,
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t 2 sup E e(t−s)A G s, u(s), z μ(ds, ¯ dz) tT 0 Z
t
(t−s)A 2 e G s, u(s), z m(dz) ds sup E
= sup E tT
tT
0 Z
2|h|L1
2
1 + sup Eu(t) < ∞.
t
2 h(t − s) 1 + u(s) ds
0
tT
Analogous estimates for the remaining terms in (4.1) are classical (see e.g. [8]), hence |[Fu]|22 < ∞. We shall now prove that there exists λ such that |[Fu − Fv]|2,λ N |[u − v]|2,λ , with N < 1. In fact we have 2 t 2 −2λt (t−s)A [Fu − Fv] sup e f s, u(s) − f s, v(s) ds E e 2,λ tT 0
+ sup e
−2λt
tT
2 t (t−s)A B s, u(s) − B s, v(s) dW (s) E e 0
2 t −2λt (t−s)A G s, u(s), z − G s, v(s), z μ(ds, ¯ dz) , + sup e E e tT 0 Z
and t 2 E ¯ dz) e(t−s)A G s, u(s), z − G s, v(s), z μ(ds, 0 Z
t =E
(t−s)A 2 e G s, u(s), z − G s, v(s), z m(dz) ds
0 Z
t E
e
2λs
h(t − s)e
−2λs
2 u(s) − v(s) ds |u − v|22,λ
0
t e2λs h(t − s) ds 0
t e2λt |u − v|22,λ
e−2λs h(s) ds,
0
which implies that the third summand on the right-hand side of the previous estimate of T |[Fu − Fv]|22,λ is bounded by |u − v|22,λ 0 e−2λs h(s) ds, which converges to zero as λ → ∞. Completely analogous calculations for the other summands show that there exists N = N(T , h, λ) such that |[Fu − Fv]|22,λ N |[u − v]|22,λ , and that one can find λ0 > 0 so that
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N(T , h, λ0 ) < 1, thus obtaining, by Banach’s fixed point theorem, existence and uniqueness of a mild solution to (1.1). Finally, Lipschitz continuity of the solution map follows by Theorem 3.5. 2 Remark 4.1. One can also prove by a direct calculation that x → u(x) is Lipschitz. This method has the advantage of yielding explicit estimates on the Lipschitz constant, and will be useful to establish the strong Feller property. In fact, one has t u(t, x) − u(t, y) = e (x − y) + tA
e(t−s)A f s, u(s, x) − f s, u(s, y) ds
0
t +
e(t−s)A B s, u(s, x) − B s, u(s, y) dW (s)
0
t
¯ dz), e(t−s)A G s, u(s, x), z − G s, u(s, y), z μ(ds,
+ 0 Z
hence, squaring both sides and taking expectations, 2 Eu(t, x) − u(t, y) 2M 2 e2ηt E|x − y|2 + (2t + 1)
t
2 h(t − s)Eu(s, x) − u(s, y) ds,
0
which yields, via Gronwall’s inequality, √ u(x) − u(y) 2Me(η+|h|1 )T +|h|1 /2 |x − y|L . 2 2 Proof of Theorem 2.4. We shall use a fixed point argument in the space Hp (T ). In particular, we want to prove that the mapping F defined as in (4.1) is a well-defined contraction on Hp (T ). Here we limit ourselves to prove that there exists N < 1 such that Fu − Fv p,λ N u − v p,λ for all u, v ∈ Hp (T ), with a suitably chosen λ 0. In fact, this implies
Fu p u − a p + Fa p < ∞ for all u ∈ Hp (T ), thanks to (2.4). Moreover, predictability of Fu, u ∈ Hp (T ), follows as in the proof of Theorem 2.3. We have p
Fu − Fv p,λ
p t −λt (t−s)A p E sup e e f s, u(s) − f s, v(s) ds tT 0
p t −λt (t−s)A B s, u(s) − B s, v(s) dW (s) + E sup e e tT 0
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p t −λt ¯ dz) + E sup e e(t−s)A G s, u(s), z − G s, v(s), z μ(ds, tT 0 Z
=: A1 + A2 + A3 . The term A1 on the right-hand side is bounded from above, thanks to (2.3) and Hölder’s inequality, by
T
p−1
E sup e
−pλt
tT
T T
p−1
p−1
t
p h(t − s)p epλs e−λs u(s) − v(s) ds
0
p
u − v p,λ
t sup tT
h(t − s)p e−pλ(t−s) ds
0
p p |hλ |Lp ([0,T ]) u − v p,λ ,
where hλ (s) := e−λs h(s). Moreover, since t p A3 = E sup ¯ dz) e(t−s)(A−λI ) e−λs G s, u(s), z − G s, v(s), z μ(ds, tT 0 Z
and, by a slight modification of the proof of (3.3), t p E sup e(t−s)(A−λI ) φ(s, z) μ(ds, ¯ dz) tT 0 Z
p,T ,η e
−λpT
T E
p eλps φ(s, ·)L
p (Z,m)
p + φ(s, ·)L
2 (Z,m)
ds,
0
we obtain, thanks to (2.3),
A3 p,T ,η e
−λpT
T E
p eλps e−λps G s, u(s), · − G s, v(s), · L
2 (Z,m)
ds
0
T +
e
p G s, u(s), · − G s, v(s), · L
λps −λps
e
p (Z,m)
ds
0
p
u − v p,λ e−λpT
T 0
eλps h(s)p ds |h˜ λ |Lp ([0,T ]) u − v p,λ , p
p
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635
where h˜ λ (s) := e−λs h(T − s). Classical maximal inequalities for stochastic convolutions with respect to Wiener processes yield a completely analogous estimate for A2 . Observing that the norms of hλ and h˜ λ appearing in the above estimates tend to zero as λ → ∞, we conclude that there exists a constant N = N (λ, T , p, η) such that Fu−Fv p,λ N u−v p,λ , with N < 1 for some λ > 0 sufficiently large. The existence and uniqueness of a solution, as well as its Lipschitz continuity with respect to the initial datum, follows then by Banach’s fixed point theorem, as for Theorem 2.3, and by the equivalence of the norms · p,λ for λ 0. 2 Remark 4.2. The Lipschitz continuity of the solution map, in analogy to the previous remark, could also be obtained by a direct calculation. However, in this case the norm of Hp (T ) is somewhat more difficult to work with. In Section 5 below we shall obtain some estimates for the Lipschitz constant of the solution map under additional assumptions on the coefficient of the SPDE. Proof of Theorem 2.6. It is enough to prove the statements in the case B ≡ 0. We are going to apply Theorem 3.6, with Λ = L2 , E = H2 (T ). The latter space needs to be endowed with a norm |[ · ]|2,λ , where λ > 0 is chosen in such a way that F : L2 ×H2 (T ) → H2 (T ) is a contraction in its second argument. However, in view of the equivalence of the norms |[ · ]|2,λ , we shall perform the calculations assuming λ = 0, without loss of generality. It is immediate that the directional derivative ∂1,y F(x, u) coincides with the map t → etA y, which clearly belongs to H2 (T ). Moreover, we have · · h (·−s)A (·−s)A ∂v(s) f u(s) ds − e ∂1,v(s) G u(s), z μ(ds, ¯ dz) Q2,v F(x, u) − e 0 Z
0
· h (·−s)A Qv(s) f u(s) − ∂v(s) f u(s) ds e 0
2
2
· h (·−s)A Q1,v(s) G u(s), z − ∂1,v(s) G u(s), z μ(ds, + e ¯ dz) . 2
0 Z
The first term on the right-hand side of this inequality tends to zero as h → 0 by obvious estimates and the dominated convergence theorem. Using the isometric property of the stochastic integral, the square of the second term is equal to T E
(T −s)A h e Q
2 u(s), z − e(T −s)A ∂1,v(s) G u(s), z m(dz) ds.
1,v(s) G
0 Z
In view of assumptions (ii)–(iv), a simple computation shows that etA G(x + hy, ·) − etA G(x, ·) → etA ∂1,y G(x, ·) h
∀x ∈ H
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in L2 (Z, m; H ) as h → 0, whence we obtain convergence to zero of the second term in the above estimate again by dominated convergence. Therefore we have ∂2,v F(x, u) (t) =
t e
(t−s)A
∂v(s) f u(s) ds +
t
e(t−s)A ∂1,v(s) G u(s), z μ(ds, ¯ dz).
0 Z
0
The continuity of ∂1 F and ∂2 F, considered as maps L2 × H2 (T ) × L2 → H2 (T ) and L2 × H2 (T ) × H2 (T ) → H2 (T ), respectively, can be proved in a completely similar way, and we omit the details. Let us now prove the second assertion of the theorem: let x, y in L2 . Then Theorem 3.6 yields −1 ∂1,y F x, u(x) , ∂y u(x) = I − ∂2 F x, u(x) thus also ∂y u(x) = ∂1,y F x, u(x) + ∂2 F x, u(x) ∂y u(x), and the result follows substituting in the previous formula the expressions for the directional derivatives of F found above. The last assertion of the theorem is a direct consequence of the definition of directional derivative and the fact that the solution map x → u(x) is Lipschitz. 2 Proof of Theorem 2.7. One can prove that the solution map is Gâteaux differentiable as in the proof of Theorem 2.6, except for the fact that one cannot use the isometric property of the stochastic integral, but has to rely on the estimate (3.3). In particular, one has · p h (t−s)A Q1,v(s) G u(s), z − ∂1,v(s) G u(s), z μ(ds, e ¯ dz) p
0 Z
T E
h Q
p u(s), · − ∂1,v(s) G u(s), · L
1,v(s) G
p (Z,m)
ds
0
T +E
h Q
p u(s), · − ∂1,v(s) G u(s), · L
1,v(s) G
2 (Z,m)
ds,
0
which converges to zero by dominated convergence, thanks to the assumptions on G. In order to prove that x → u(x) is also Fréchet differentiable, let us set Λ0 = Lq , Λ = Lp , E0 = Hq (T ), and E = Hp (T ), and apply Theorem 3.7. We are going to prove that the partial Gâteaux derivatives ∂1 F : Lq × Hq (T ) → L Lq , Hp (T ) , ∂2 F : Lq × Hq (T ) → L Hq (T ), Hp (T )
C. Marinelli et al. / Journal of Functional Analysis 258 (2010) 616–649
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are continuous, which implies that F is Fréchet differentiable (see e.g. [3]). Since ∂1 F : (x, u) → etA x is clearly continuous, it suffices to show that ∂2 F is continuous. To this purpose, let {xn } ⊂ Lq , x ∈ Lq , {un } ⊂ Hq (T ), u, w ∈ Hq (T ) such that xn → x in Lq , un → u in Hp (T ), and w q 1. We shall prove that ∂2 F(xn , un )w − ∂2 F(x, u)w → 0 p as n → ∞. In fact, the above expression is no greater than · e(·−s)A Df un (s) w(s) − Df u(s) w(s) ds 0
p
· (·−s)A D1 G un (s), z w(s) − D1 G u(s), z w(s) μ(ds, + e ¯ dz) 0 Z
p
=: I1 (n) + I2 (n), and, using Hölder’s inequality with conjugate exponents r = q/p and r , T I1 (n) E p
Df un (s) − Df u(s) w(s)p ds
0
1/r T 1/r T pr pr Df un (s) − Df u(s) w(s) ds ds E E 0
0
q−p T q pq p w q E Df un (s) − Df u(s) q−p ds . 0
Since |Df (un (s)) − Df (u(s))| 2[f ]1 , the dominated convergence theorem and the continuity of Df imply that I1 (n) → 0 as n → ∞. Similarly, using the maximal inequality (3.3), we obtain p t I2 (n)p E sup e(t−s)A D1 G un (s), z w(s) − D1 G u(s), z w(s) μ(ds, ¯ dz) tT 0 Z
T E
D1 G un (s), · − D1 G u(s), · p
Lp (Z,m)
0
p + D1 G un (s), · − D1 G u(s), · L
2 (Z,m)
w(s)p ds,
which converges to zero as n → ∞ by arguments completely analogous to the above ones.
2
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Proof of Theorem 2.8. We are going to apply Theorem 3.9, with Λ = Lp , Λ0 = Lq , Λ1 = Lq , and E = Hp (T ), E0 = Hq (T ). Here q ∈ ]2p, q[. In analogy to what we have done before, we shall endow E and E0 with norms · q ,λ(q ) and · p,λ(p) , respectively, where the λ are chosen in a such a way that F : Lr × Hr (T ) → Hr (T ) are contractions in the second argument, but we shall perform the calculations assuming λ ≡ 0, without loss of generality. It is clear that, in view of Theorem 2.7, it is enough to prove that F ∈ C 2 (Lq × Hq (T ), Hp (T )). Since ∂1 D1 F ≡ ∂1 D2 F ≡ ∂2 D1 F ≡ 0, we only have to consider ∂2 D2 F. Let us first prove that ∂2,v D2 F(x, u)w (t) =
t
e(t−s)A D 2 f u(s) v(s), w(s) ds
0
t +
e(t−s)A D12 G u(s), z v(s), w(s) μ(ds, ¯ dz),
t ∈ [0, T ],
0 Z
(4.2) for all v, w ∈ Hq (T ). In fact, the pth power of the Hp (T ) norm of the difference between Qh2,v D2 F(x, u)w and the right-hand side of (4.2) is not greater than a constant times t p h (t−s)A 2 Qv(s) Df u(s) w(s) − D f u(s) v(s), w(s) ds E sup e tT 0
t p h (t−s)A 2 Q1,v(s) D1 G u(s), z w(s) − D1 G u(s), z v(s), w(s) μ(ds, + E sup e ¯ dz) tT 0 Z
=: I1 (h) + I2 (h). We have, by Hölder’s inequality, T I1 (h) E
h Q Df u(s) − D 2 f u(s) v(s), · w(s)p ds v(s)
0 p w q
1/2 T h 2p , E Qv(s) Df u(s) − D 2 f u(s) v(s), · L(H ) ds 0
which converges to zero as h → 0 by dominated convergence, thanks to the boundedness of D 2 f . Similarly, using inequality (3.3), we get T I2 (h) E
h Q
p u(s), · w(s) − D12 G u(s), · v(s), w(s) L
1,v(s) D1 G
0
p (Z,m)
ds
C. Marinelli et al. / Journal of Functional Analysis 258 (2010) 616–649
T +E
h Q
639
p u(s), · w(s) − D12 G u(s), · v(s), w(s) L
1,v(s) D1 G
2 (Z,m)
ds
0
= I21 (h) + I22 (h), and, by Hölder’s inequality, T I21 (h) E
h Q
p u(s), · − D12 G u(s), · v(s), · L
1,v(s) D1 G
p (Z,m;L(H ))
p ds w(s) ds
0 p w q
T E
h Q
2p u(s), · − D12 G u(s), · v(s), · L (Z,m;L(H )) ds.
1,v(s) D1 G
p
0
By hypothesis (ii) we have that h Q
2p u(s), z − D12 G u(s), z v(s), · L(H ) → 0
1,v(s) D1 G
as h → 0 for all (s, z) ∈ [0, T ] × Z, and, by (iii), T E
h Q
1,v(s) D1 G
2p u(s), · − D12 G u(s), · v(s), · L (Z,m;L(H )) ds p
0 2p |h1 |Lp (Z,m) E
T
v(s)2p ds |h1 |2p
p Lp (Z,m) v q
< ∞,
0
hence, by dominated convergence, I21 (h) → 0 as h → 0. In a completely similar way one can prove that I22 (h) → 0 as h → 0 as well. We have thus proved that I2 (h) → 0, hence that (4.2) holds. Let us now show that v → ∂2,v D2 F(x, u) ∈ L Hq (T ), L Hq (T ), Hp (T ) for all x ∈ Lq and u ∈ Hq (T ). In fact, for w ∈ Hq (T ), we have ∂2,v D2 F(x, u)w p E
T
p
2 D f u(s) v(s), w(s) p ds
0
T +E
2 D G u(s), z v(s), w(s) p m(dz) ds 1
0 Z
(4.3)
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T +E 0
T E
p/2 2 D G u(s), z v(s), w(s) 2 m(dz) ds 1
Z
v(s)p w(s)p ds
0
p p + |h1 |Lp (Z,m) + |h1 |L2 (Z,m) E
T
v(s)p w(s)p ds
0
p p p p 1 + |h1 |Lp (Z,m) + |h1 |L2 (Z,m) v q w q , which establishes the continuity of (v, w) → ∂2,v D2 F(x, u)w, and hence ensures that (4.3) holds true. Our next goal is to prove that u → ∂2 D2 F(x, u) ∈ C Hq (T ), L⊗2 Hq (T ), Hp (T ) ,
(4.4)
for all x ∈ Lq , which implies the twice continuous differentiability of F by a well-known criterion. Let un → u in Hq (T ). Then we have ∂2 D2 F(x, un )(v, w) − ∂2 D2 F(x, u)(v, w) p p
t p E sup e(t−s)A D 2 f un (s) v(s), w(s) − D 2 f u(s) v(s), w(s) ds tT 0
t + E sup e(t−s)A D12 G un (s), z v(s), w(s) tT 0 Z
p u(s), z v(s), w(s) μ(ds, ¯ dz)
− D12 G
=: I1 (n) + I2 (n), where, using Hölder’s inequality with conjugate exponents q /(2p) and q /(q − 2p), T I1 (n) E
2 D f un (s) v(s), w(s) − D 2 f u(s) v(s), w(s) p ds
0
T 2p q −2p T pq q q q /2 2 −2p q 2 v(s) w(s) E D f un (s) − D f u(s) L⊗2 ds E 0
0
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641
T q −2p pq q 2 −2p p p E D f un (s) − D 2 f u(s) Lq ⊗2
v q w q , 0
which converges to zero as n → ∞ by dominated convergence, thanks to the assumption of boundedness of D 2 f . Applying again inequality (3.3) yields T I2 (n) E
2 D G un (s), z v(s), w(s) − D 2 G u(s), z v(s), w(s) p m(dz) ds 1
1
0 Z
T +E
p/2 2 D G un (s), z v(s), w(s) − D 2 G u(s), z v(s), w(s) 2 m(dz) ds 1
0
1
Z
=: I21 (n) + I22 (n), where, again by Hölder’s inequality, T I21 (n) E
v(s)p w(s)p D 2 G un (s), · − D 2 G u(s), · p 1
1
Lp (Z,m;L⊗2 )
ds
0
q −2p T pq q p p ds , v q w q E D12 G un (s), · − D12 G u(s), · Lq −2p (Z,m;L⊗2 ) p
0
which converges to zero as n → ∞ by continuity of D12 G in its first argument and dominated convergence, thanks to hypothesis (iii). An analogous argument shows that I22 (n) → 0 as n → ∞. We have thus proved (4.4). This concludes the proof that F ∈ C 2 , hence that the Fréchet derivative Du : Lq → L(Lq , Hp (T )) is Gâteaux differentiable. By Theorem 3.8 we have −1 2 ∂Du(x)(y1 , y2 ) = I − D2 F x, u(x) D2 F x, u(x) Du(x)y1 , Du(x)y2 , hence ∂Du(x)(y1 , y2 ) = D2 F x, u(x) ∂Du(x)(y1 , y2 ) + D22 F x, u(x) Du(x)y1 , Du(x)y2 , and (2.6) now follows substituting in the previous identity the expressions for D2 F and D22 F obtained above and in the proof of Theorem 2.7. The bound for the bilinear form ∂Du(x) can be established as an application of Corollary 3.10. Since Du(x) is bounded by Theorem 2.7, it is enough to show that D22 F : Lq × Hq (T ) → L⊗2 (Hq (T ), Hp (T )) is bounded. In fact, by a computation completely analogous to the above ones, we have
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2 D F x, u(x)(v, w) p p,T ,η 2
T
p
2 D f u(s) v(s), w(s) p ds
0
T +
v(s)p w(s)p D 2 B u(s), · p 1 L
p (Z,m;L
0 p p p T C1 v q w q
⊗2 )
p + D12 B u(s), · L
2 (Z,m;L
⊗2 )
ds
p p p p + |h1 |Lp (Z,m) + |h1 |L2 (Z,m) v q w q .
Let us now assume that q > 4p 8, and take q ∈ ]2p, q/2[. We shall deduce the twice Fréchet differentiability of the solution map applying Theorem 3.9, with all spaces defined as before. Since q > 2q , the first part of the proof guarantees that Du : Lq → L(Lq , Hq (T )) is Gâteaux differentiable with derivative ∂Du : Lq → L(Lq , L(Lq , Hq (T ))). Therefore, by the last statement of Theorem 3.9, we infer that x → u(x) ∈ C 2 (Lq , Hp (T )). 2 5. Application: Gradient estimates for the resolvent In this section we assume that the coefficients f , B, and G do not depend on t and ω. This assumption allows us to define the semigroup and resolvent associated to the mild solution: ∞
Pt ϕ(x) = Eϕ u(t, x) ,
Rα ϕ(x) =
e−αt Pt ϕ(x) dt,
0
where ϕ ∈ Cb (H ) and α > 0. In order to prove gradient estimate for the resolvent Rα we need the following lemma, which gives an explicit bound on the Lipschitz constant of the solution map. Lemma 5.1. Let A be η–m-dissipative, and set [B]1,Q = sup x,y∈H x=y
|B(x) − B(y)|Q , |x − y|
[G]1,m = sup x,y∈H x=y
|G(x, z) − G(y, z)|L2 (Z,m) . |x − y|
Then we have Eu(t, x) − u(t, y) eω1 t |x − y|L2 ,
(5.1)
where 1 1 ω1 := η + [f ]1 + [B]21,Q + [G]21,m . 2 2 Proof. Let Aλ be the Yosida approximation of A, Aλ → A as λ → 0. Let uλ be the solution of duλ (t) = Aλ uλ (t) + f uλ (t) dt + B uλ (t) dW (t) +
Z
G uλ (t), z μ(dt, ¯ dz).
C. Marinelli et al. / Journal of Functional Analysis 258 (2010) 616–649
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Then Itô’s formula yields uλ (t)2 = uλ (0)2 + 2
t
Aλ uλ (s), uλ (s) ds + 2
0
t +2
t
f uλ (s) , uλ (s) ds
0
uλ (s), B uλ (s) dM(s) + B(uλ ) · W (t) + G(uλ ) μ¯ (t)
0
hence 2 Euλ (t, x) − uλ (t, y) = E|x − y|2 + 2E
t
Aλ uλ (s, x) − Aλ uλ (s, y), uλ (s, x) − uλ (s, y) ds
0
t + 2E
f uλ (s, x) − f uλ (s, y) , uλ (s, x) − uλ (s, y) ds
0
t +E
B uλ (s, x) − B uλ (s, y) 2 ds Q
0
t +E
G uλ (s, x), · − G uλ (s, y), · 2
L2 (Z,m)
ds
0
E|x − y|2 + 2η + 2[f ]1 + [B]21,Q + [G]21,m t ×
2 Euλ (s, x) − uλ (s, y) ds
0
and Gronwall’s inequality implies that 2 Euλ (t, x) − uλ (t, y) e2ω1 t E|x − y|2 , therefore, in view of Proposition 3.11, we can pass to the limit as λ → 0, and applying Cauchy– Schwartz’ inequality, we obtain (5.1). 2 Remark 5.2. Note that (5.1) also implies that |u(x) − u(y)|H2 (T ) eω1 T |x − y|H2 and Eu(t, x) − u(t, y) eω1 t |x − y| if x, y ∈ H are nonrandom. Moreover, if the noise is additive, i.e. if B is constant, then one can prove, solving differential inequalities ω-by-ω, that |u(t, x) − u(t, y)| eω1 t |x − y| almost surely.
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Obtaining an explicit estimate for the Lipschitz constant of the solution map in the Hp (T ) setting seems considerably more difficult in the general case of multiplicative noise. It reduces instead to a simple computation in the case of additive noise, as we show next. Lemma 5.3. Let A be η–m-dissipative and assume that B and G do not depend on x. If (1.1) is well-posed in Hp (T ), then for any x, y ∈ Lp we have p Eu(t, x) − u(t, y) epω1 t E|x − y|p , where ω1 = η + [f ]1 . Proof. One has d u(t, x) − u(t, y) = Au(t, x) − Au(t, y) + f u(t, x) − f u(t, y) dt P-a.s., hence, multiplying both sides by |u(t, x) − u(t, y)|p−2 (u(t, x) − u(t, y)), we obtain p 1 d u(t, x) − u(t, y) p dt p−2 u(t, x) − u(t, y) = Au(t, x) − Au(t, y), u(t, x) − u(t, y) p−2 + u(t, x) − u(t, y) f u(t, x) − f u(t, y) , u(t, x) − u(t, y) p−2 p−2 Au(t, x) − u(t, y) 2 u(t, x) − u(t, y) , u(t, x) − u(t, y) 2 u(t, x) − u(t, y) p + [f ]1 u(t, x) − u(t, y) p η + [f ]1 u(t, x) − u(t, y) . Writing in integral form and taking expectations, we obtain p Eu(t, x) − u(t, y) E|x − y|p + p η + [f ]1
t
p Eu(s, x) − u(s, y) ds,
0
from which the result follows by Gronwall’s inequality.
2
The following gradient estimate for the resolvent associated to the mild solution of the stochastic PDE is a consequence of (5.1). Theorem 5.4. Assume that ϕ ∈ Cb (H ) is Gâteaux differentiable and Lipschitz, and that f , B, G satisfy the assumptions of Theorem 2.6. Let α > ω1 . Then x → Rα ϕ(x) is Gâteaux differentiable with ∞ ∂y Rα ϕ(x) = 0
e−αt E ∂ϕ u(t, x) ∂2,y u(t, x) dt,
(5.2)
C. Marinelli et al. / Journal of Functional Analysis 258 (2010) 616–649
645
and it satisfies the estimate ∂Rα ϕ(x) H
1 [ϕ]1 . α − ω1
Proof. We have, setting v(t) = ∂2,y u(t, x), R ϕ(x + hy) − R (x) ∞ α α −αt − e E ∂ϕ u(t, x) ∂2,y u(t, x) dt h 0
∞
e−αt h−1 E ϕ u(t, x + hy) − ϕ u(t, x) − h∂ϕ u(t, x) v(t) dt
0
∞
e−αt Eh−1 ϕ u(t, x + hy) − ϕ u(t, x) + hv(t) dt
0
∞ +
e−αt Eh−1 ϕ u(t, x) + hv(t) − ϕ u(t, x) − h∂ϕ u(t, x) v(t) dt
0
=: I1 (h) + I2 (h), and ∞ I1 (h) [ϕ]1
e−αt Eh−1 u(t, x + hy) − u(t, x) − ∂2,y u(t, x) dt.
0
Since, by Cauchy–Schwartz’ inequality and the differentiability of the solution map from H to H2 (T ), we have Eh−1 u(t, x + hy) − u(t, x) − ∂2,y u(t, x) 2 1/2 Eh−1 u(t, x + hy) − u(t, x) − ∂2,y u(t, x) → 0, we conclude that I1 (h) → 0 as h → 0 by dominated convergence. On the other hand, I2 (h) converges to zero as h → 0 by definition of directional derivative and dominated convergence. This establishes (5.2). Note that x → Pt ϕ(x) is Lipschitz: in fact, the previous lemma yields Pt ϕ(x) − Pt ϕ(y) Eϕ u(t, x) − ϕ u(t, y) [ϕ]1 Eu(t, x) − u(t, y) [ϕ]1 eω1 t |x − y|. Therefore
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Rα ϕ(x + hy) − Rα ϕ(x) = h
∞
e−αt h−1 Pt ϕ(x + hy) − Pt ϕ(x) dt
0
∞ [ϕ]1 |y|
eω1 t e−αt dt < ∞,
0
hence, in view of (5.2), ∂y Rα ϕ(x) thus also |∂Rα ϕ(x)| (α − ω1 )−1 [ϕ]1 .
1 [ϕ]1 |y|, α − ω1
2
6. Strong Feller property The purpose of this section is to establish a Bismut–Elworthy formula for the semigroup associated to (1.1), and to deduce from it the strong Feller property, adapting an argument of [27] to the infinite-dimensional case. We would like to emphasize that the proof depends essentially on the second order differentiability of the solution with respect to the initial datum established in Theorem 2.8 above. In the following we shall denote the set of bounded Borel functions from H to R by Bb (H ). Theorem 6.1. Assume that Q ∈ L+ 1 (K), B(x) : K → U is a linear bounded invertible operator with |B(x)−1 | C for all x ∈ H , for some C > 0, and the hypotheses of Theorem 2.6 are satisfied. Then the semigroup Pt is strong Feller, i.e. ϕ ∈ Bb (H ) implies Pt ϕ ∈ Cb (H ). Proof. We first assume that the coefficients f , B and G satisfy the hypotheses of Theorem 2.8, so that x → u(x) ∈ C 2 (H, H2 (T )). This assumption will be removed in the last part of the proof. A formal application of Itô’s formula shows that the generator L of the semigroup Pt associated to the mild solution of (1.1) takes the form, for ϕ ∈ Cb2 (H ), 1 Lϕ(x) = Ax + f (x), Dϕ(x) + Tr QB(x)B ∗ (x)D 2 ϕ(x) 2 ϕ x + G(x, z) − ϕ(x) − Dϕ(x), G(x, z) m(dz). + Z
Let uλ be the solution of (1.1) with A replaced by its Yosida approximation Aλ , Ptλ the associated semigroup, and Lλ the generator of Ptλ . Then the action of Lλ on ϕ ∈ Cb2 (H ) is exactly λ ϕ(x) ≡ as for L, with A replaced by Aλ . Let ϕ ∈ Cb2 (H ), s ∈ [0, t], and set v(s, x) = Pt−s 1,2 Eϕ(uλ (t − s, x)). Then v ∈ C ([0, T ] × H ) and Itô’s formula implies
C. Marinelli et al. / Journal of Functional Analysis 258 (2010) 616–649
v s, uλ (s) = v(0, x) +
s
(∂r + Lλ )v r, uλ (r) dr +
0
s +
s
647
Dv r, uλ (r) , B uλ (r) dW (r)
0
v r−, uλ (r−) + G uλ (r−), z − v r−, uλ (r−) μ(dr, ¯ dz).
0 Z
Since (∂t + Lλ )v = 0, the previous identity evaluated at s = t implies ϕ uλ (t) = Ptλ ϕ(x) + M1λ (t) + M2λ (t), where t M1λ (t) =
λ DPt−r ϕ uλ (r) , B uλ (r) dW (r) ,
0
t M2λ (t) =
λ λ Pt−s ϕ uλ (s−) + G uλ (s−) z − Pt−s ϕ uλ (s−) μ(ds, ¯ dz).
0 Z
Letting λ → 0 and recalling Proposition 3.11 we obtain ϕ u(t) = Pt ϕ(x) + M1 (t) + M2 (t),
(6.1)
with M1 and M2 defined in the obvious way. Moreover, setting w(t) = ∂2,y u(t, x) and t M3 (t) =
B −1 u(s) w(s), dW (s) ,
0
multiplying both sides of (6.1) by M3 (t) and taking expectations yields Eϕ u(t) M3 (t) = EM1 (t)M3 (t) = E
t
DPt−s ϕ u(s) , w(s) ds
0
t =E
D Pt−s ϕ u(s) y ds =
0
t DPt ϕ(x)y ds = tDPt ϕ(x)y. 0
Here EM2 (t)M3 (t) = 0 because W and μ¯ are independent, and we have used the Markov property of solutions in the second to last step. In particular, we have proved the Bismut–Elworthytype formula t −1 1 B u(s, x) ∂2,y u(s, x), dW (s) . DPt ϕ(x)y = E ϕ u(t, x) t 0
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We shall now remove the assumptions on f , B and G. Let us assume for a moment that we can find sequences fε , Bε , Gε satisfying the hypotheses of Theorem 2.8 and Proposition 3.12, and denote the mild solution of du(t) = Au(t) + fε u(t) dt + Bε u(t) dW (t) + Gε u(t), z μ(dt, ¯ dz), Z
u(0) = x, by uε , so that x → uε (x) ∈ C 2 (H, H2 (T )) and uε → u in H2 (T ). In particular we also have Ptε ϕ(x) → Pt ϕ(x) for all x ∈ H and t T , where Ptε ϕ(x) := Eϕ(uε (t, x)), ϕ ∈ Cb (H ). Then Cauchy–Schwartz’ inequality yields DP ε ϕ(x)y 2 1 |ϕ|2 C 2 E t t2 ∞
t
∂2,y uε (s, x)2 ds,
0
where |ϕ|∞ := supx∈H |ϕ(x)|. In view of Remark 4.1 it is not difficult to see that there exists a constant N , which does not depend on x, y, and ε, such that |[uε (x1 ) − uε (x2 )]|2 N |x1 − x2 |H , hence, by Theorem 2.6, |∂2,y uε (s, x)| N|y|. We obtain DP ε ϕ(x) N C |ϕ|∞ , t t 1/2 thus also |Ptε ϕ(x1 ) − Ptε ϕ(x2 )| t −1/2 N C|ϕ|∞ |x1 − x2 |, and letting ε → 0, Pt ϕ(x1 ) − Pt ϕ(x2 ) t −1/2 N C|ϕ|∞ |x1 − x2 |. The same Lipschitz property continues to hold also for ϕ ∈ Bb (H ) by a simple regularization argument (see e.g. [23, Lemma 2.2]). In order to complete the proof, we have to show that we can find sequences fε , Bε , Gε satisfying the hypotheses of Theorem 2.8 and Proposition 3.12. The existence of such fε and Bε is well known (see e.g. [6, Sect. 3.3.1], [23]), and the construction of Gε can be carried out in a completely similar way, hence we omit it. 2 Acknowledgments The first author was partially supported by the SFB 611, Bonn, by the ESF through grant AMaMeF 969, by the Centre de Recerca Matemàtica, Barcelona, through an EPDI fellowship, and by the EU through grant MOIF-CT-2006-040743. The two last named authors were supported by SFB 701, NSF Grant 0606615 and the BiBoS Research Centre in Bielefeld. References [1] S. Albeverio, V. Mandrekar, B. Rüdiger, Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise, Stochastic Process. Appl. 119 (3) (2009) 835–863. [2] S. Albeverio, J.-L. Wu, T.-S. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastic Process. Appl. 74 (1) (1998) 21–36, MR MR1624076 (99c:60124).
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Journal of Functional Analysis 258 (2010) 650–664 www.elsevier.com/locate/jfa
Universal coefficient theorems for the stable Ext-groups Changguo Wei School of Mathematical Sciences, Ocean University of China, Qingdao, 266071, PR China Received 28 April 2009; accepted 19 October 2009
Communicated by D. Voiculescu
Abstract Let A be a unital separable nuclear C ∗ -algebra and let B be a stable C ∗ -algebra. Using K-theory and KK-theory we establish universal coefficient theorems for the stable Ext-groups of unital extensions of A by B when A and B have certain properties, which generalize a result of L. Brown and M. Dadarlat for the strong Ext-groups. The class of extensions being studied are also enlarged. © 2009 Elsevier Inc. All rights reserved. Keywords: Extension; Ext-group; UCT
1. Introduction Since Brown, Douglas and Fillmore gave the famous BDF theory [4] in 1970’s, classifications of extensions of C ∗ -algebras have been studied deeply ([9–12,15,16,21], etc.). The theory of C ∗ algebra extensions becomes an important tool for classifications of C ∗ -algebras together with K-theory and KK-theory. The original BDF theory deals with the strong Ext-group of unital extensions of commutative C ∗ -algebras by the compact operators K. In order to compute the strong Ext-groups, Brown [2] proved a universal coefficient theorem (UCT) for A = C(X), 0 → Ext K0 (A), Z → Extus (A) → Hom K1 (A), Z → 0. Based on the work of Brown, Rosenberg and Schochet [21] gave a more general UCT 0 → Ext K∗ (A), K∗ (B) → Ext(A, B) → Hom K∗ (A), K∗+1 (B) → 0. E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.009
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Since then, by KK-theory and the UCT, one can compute extension groups of many C ∗ -algebras. But unlike the BDF theory, Ext(A, B) is the group of extensions under stably unitary equivalence, so it cannot provide enough information for classifications of unital extensions. On the other hand, the classification of amenable C ∗ -algebras has developed rapidly since 1990’s, and a number of classification results appeared ([5,13,14,17], etc.). Extension algebras form an important class of C ∗ -algebras and there are many classification results of such C ∗ -algebras [3,11,19]. The group Ext(A, B) in KK-theory provides little information for the classification of extension algebras. So one has to compute strong Ext-groups. Given C ∗ -algebras A with modest hypotheses and B = K, Brown and Dadarlat [3, Theorem 2] have proved a UCT for the strong Ext-groups of unital extensions of the form 0 → Ext K0 (A), [1]0 , Z → Extus (A, B) → Hom K1 (A), Z → 0. Using this result one can compute almost all strong Ext-groups of extensions by K. But when the ideal is not K, this formula does not hold. Motivated by [3], in this paper we use the method of group action, combining the UCT of Rosenberg and Schochet, to prove universal coefficient theorems for the stable Ext-groups of unital extensions of A by B when A and B have certain properties (Theorem 3.12 and Theorem 3.15). Specifically, we establish two universal coefficient theorems of the following forms which determine the stable Ext-groups in terms of K-theory: 0 → Ext K∗ (A), K∗ (B) → Extuw (A, B) → Γ ⊕ Hom K1 (A), K0 (B) → 0 and 0 → Σ → Extus (A, B) → Γ ⊕ Hom K1 (A), K0 (B) → 0, where Γ = f ∈ Hom K0 (A), K0 Q(B) : f [1A ]0 = 0 and Σ = Ext K0 (A), [1A ]0 , K0 (B) ⊕ Ext K1 (A), K1 (B) . 2. Preliminaries In this section we give some notations and results used in this paper. One can see [1] for more details of C ∗ -algebra extensions. Let A and B be C ∗ -algebras. Recall that an extension of A by B is a short exact sequence α
β
0 → B → E → A → 0. Denote this extension by e or (E, α, β) and the set of all such extensions by E(A, B). The extension (E, α, β) is called trivial, if the above sequence splits, i.e. if there is a homomorphism γ : A → E such that β ◦ γ = idA .
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We call (E, α, β) essential, if α(A) is an essential ideal in E. α
β
Let 0 → B → E → A → 0 be an extension of A by B. Then there is a unique homomorphism σ : E → M(B) such that σ ◦ α = ι, where M(B) is the multiplier algebra of B, and ι is the inclusion map from B into M(B). It is known that σ is injective if and only if the extension is essential. The Busby invariant of (E, α, β) is a homomorphism τ from A into the corona algebra Q(B) = M(B)/B defined by τ (a) = π(σ (b)) for a ∈ A, where π : M(B) → Q(B) is the quotient map, and b ∈ E such that β(b) = a. If A is unital and the Busby invariant is unital, then (E, α, β) is called unital. Denote by Ext eu (A, B) the set of unital essential extensions of A by B if A is unital. There are several equivalence relations of extensions of A by B. Let ei : 0 → B → Ei → A → 0 be two extensions with Busby invariants τi for i = 1, 2. Definition 2.1. Two extensions e1 and e2 are called strongly unitarily equivalent, denoted by s e1 ∼ e2 , if there exists a unitary u ∈ M(B) such that τ2 (a) = π(u)τ1 (a)π(u)∗ for all a ∈ A. Denote by Ext(A, B) or Exts (A, B) the set of strongly unitary equivalence classes of extensions of A by B. It should be noted that e1 is strongly unitarily equivalent to e2 if and only if there is a unitary u in M(B) and homomorphism φ : E1 → E2 making the following diagram commute: 0 −−−−→ B −−−−→ ⏐ ⏐ Ad u
E1 −−−−→ ⏐ ⏐φ
A −−−−→ 0
0 −−−−→ B −−−−→ E2 −−−−→ A −−−−→ 0. Definition 2.2. Two extensions e1 and e2 are called weakly unitarily equivalent, denoted by w e1 ∼ e2 , if there exists a unitary v ∈ Q(B) such that τ2 (a) = vτ1 (a)v ∗ for all a ∈ A. Denote by Extw (A, B) the set of equivalence classes of extensions of A by B under weakly unitary equivalence. Let H be a separable infinite dimensional Hilbert space and K the ideal of compact operators in B(H ). If B is a stable C ∗ -algebra (i.e. B ⊗ K ∼ = B), then the sum of two extensions τ1 and τ2 is defined to be the homomorphism τ1 ⊕ τ2 , where τ1 ⊕ τ2 : A → Q(B) ⊕ Q(B) ⊆ M2 (Q(B)) ∼ = Q(B). One can see [7, 1.3.8] for details of the definition of the inner isomorphism θ between M2 (M(B)) and M(B). Denote by θ˜ the inner isomorphism between M2 (Q(B)) and Q(B) induced by θ . It is easy to see that Exts (A, B) and Extw (A, B) are commutative semigroups with respect to the above addition, and the sets of equivalence classes of trivial extensions are subsemigroups, respectively. eu The notations Extus (A, B), Extuw (A, B), Exteu s (A, B) and Extw (A, B) are defined analogously. The superscript e denotes essential extensions and u denotes unital extensions if A is unital. One can see [1, 15.6.3] for details. When A is unital, a trivial extension τ of A by B is called strongly unital if τ lifts to a unital homomorphism from A to M(B).
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Definition 2.3. Denote by Ext(A, B) or Exts (A, B) [resp. Extw (A, B)] the quotient of Exts (A, B) [resp. Extw (A, B)] by the subsemigroup of trivial extensions. If A is unital, Extus (A, B) [resp. Extuw (A, B)] is the quotient of the subsemigroup of unital extensions by the subsemigroup of strongly unital trivial extensions. The equivalence class of an extension τ in Exteu s (A, B) is denoted by [τ ]. The equivalence class of τ in Ext(A, B) or Extus (A, B) is denoted by [τ ]ss . If [τ1 ]ss = [τ2 ]ss , then we write ss τ1 ∼ τ2 . This is equivalent to that τ1 and τ2 are stably strongly unitarily equivalent. sw Similarly, for stably weakly unitary equivalence, there are notations [τ ]sw and τ1 ∼ τ2 in Extw (A, B) and Extuw (A, B). Let D be a C ∗ -algebra. Denote by P (D) and U(D) all projections and all unitaries in D, respectively. The notations of equivalences of projections and unitaries are referred to [20]. For example, for any unitary u and any projection p, [u]1 and [p]0 are the images of u and p in K1 (D) and K0 (D) respectively, and [p] is the Murray–von Neumann equivalence class of p. Denote by p ∼ q when two projections p, q are Murray–von Neumann equivalent. A non-zero projection p in a C ∗ -algebra A is said to be properly infinite if there are mutually orthogonal subprojections e, f of p such that p ∼ e ∼ f . A unital C ∗ -algebra A is said to be properly infinite if 1A is a properly infinite projection. When B is a stable C ∗ -algebra, it is clear that M(B) and Q(B) are properly infinite. Let A be a C ∗ -algebra. An element a in A is said to be full if it is not contained in any proper ideal in A. Definition 2.4. Let A and B be C ∗ -algebras [with A unital]. An [unital] extension τ of A by B is called absorbing [unital-absorbing] if τ is strongly unitarily equivalent to τ ⊕ σ , for any trivial extension [strongly unital trivial extension] σ . Definition 2.5. Recall that [16] a C ∗ -algebra B has the property (P) if for every full element b ∈ Q(B) there exist x, y ∈ Q(B) such that xby = 1. It follows that every full projection in M(B) is equivalent to 1M(B) when B has the property (P). By [16] the algebra B0 ⊗ K has the property (P) when B0 is one of the C ∗ -algebras in the following: (1) unital AF-algebra; (2) unital purely infinite simple C ∗ -algebra; (3) unital simple C ∗ -algebra with real rank zero, stable rank one and weakly unperforated K0 group; (4) C(X), where X is a compact Hausdorff space with covering dimension d. Definition 2.6. [21] Let N be the smallest class of separable nuclear C ∗ -algebras with the following properties: (1) N contains all separable commutative C ∗ -algebras. (2) N is closed under inductive limits. (3) If 0 → A → D → B → 0 is an exact sequence, and two of the terms are in N , then so is the third.
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Note that N is closed under tensor products and N is also closed under KK-equivalence (see [1, 22.3.5]). In particular, N contains all type I C ∗ -algebras. Theorem 2.7 (Universal Coefficient Theorem (UCT)). (See [21].) Let A and B be separable C ∗ -algebras, with A ∈ N . Then there is a short exact sequence δ γ 0 → Ext K∗ (A), K∗ (B) → KK ∗ (A, B) → Hom K∗ (A), K∗ (B) → 0. The map γ has degree 0 and δ has degree 1. 3. Main results Throughout this section, we assume that B0 is a separable unital C ∗ -algebra and B is the stabilization B0 ⊗ K of B0 . By [1, 12.2] one has K1 Q(B) = U1 Q(B) /U1 Q(B) 0 . h
Hence, when u1 , u2 are unitaries in Q(B), it follows that [u1 ]1 = [u2 ]1 is equivalent to u1 ∼ u2 , that is, u1 and u2 are homotopic. Suppose A is a unital C ∗ -algebra. For any x ∈ K1 (Q(B)), choose a unitary u ∈ Q(B) such that x = [u]1 . Define a map eu εx : Exteu s (A, B) → Exts (A, B),
such that εx ([τ ]) = [Ad u ◦ τ ]. One can check that εx is a bijection on the set Exteu s (A, B) for every x ∈ K1 (Q(B)). This defines a map ε from K1 (Q(B)) to the set of bijections on Exteu s (A, B). Then one can check the following proposition. Proposition 3.1. The map ε is a group action of the additive group K1 (Q(B)) on Exteu s (A, B). Definition 3.2. Suppose that S is an abelian semigroup and ρ is a bijection on S. If there exist c, d ∈ S such that ρ(a) + c = a + d for any a ∈ S, then we call ρ a translation transformation on S. When S is a group, it is equivalent to ρ(a) = a + (d − c), so ρ is a group translation in the usual sense. If ε is a group action on S such that every map is a translation transformation on S, then we call ε a translation action on S. Lemma 3.3. (See [20, p. 148].) Let A be a unital C ∗ -algebra, u a unitary and S an isometry in A. Then SuS ∗ + 1 − SS ∗ is a unitary such that [u]1 = [SuS ∗ + 1 − SS ∗ ]1 .
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Lemma 3.4. Let A be a unital C ∗ -algebra. Suppose that for any n there are isometries S1 , S2 , . . . , Sn in A such that the range projections are mutually orthogonal and ni=1 Si Si∗ = 1. Let θ : Mn (A) → A be the inner isomorphism induced by the isometries, that is n
Si aij Sj∗ . θ [aij ] = i,j =1
Then K1 (θ ) = id. Proof. Note that A is properly infinite. It follows from [20, p. 149] that K1 (A) = {[u]1 : u ∈ U(A)}, where U(A) is the set of unitaries in A. For any x ∈ K1 (A), choose a unitary u in A such that x = [u]1 = [u ⊕ In−1 ]1 . Then we have θ (u ⊕ In−1 ) = S1 uS1∗ +
n
Si Si∗ .
i=2
It follows from Lemma 3.3 that [θ (u ⊕ In−1 )]1 = [S1 uS1∗ + 1 − S1 S1∗ ]1 = [u]1 . Therefore, we have K1 (θ ) = id. 2 Theorem 3.5. The map ε is a translation action of K1 (Q(B)) on the semigroup Exteu s (A, B). Proof. (1) We first show that εx1 ([τ1 ]) + εx2 ([τ2 ]) = εx1 +x2 ([τ1 ] + [τ2 ]). Let xi = [ui ]1 for i = 1, 2. Choose a unitary u3 in Q(B) such that [u3 ]1 = x1 + x2 . Note that εx1 [τ1 ] + εx2 [τ2 ] = θ˜ ◦ Ad(u1 ⊕ u2 ) ◦
τ1 τ2
and εx1 +x2 [τ1 ] + [τ2 ] = Ad(u3 ) ◦ θ˜ ◦
τ1 τ2
.
So it is sufficient to show that ϕ and ψ are unitarily equivalent by a liftable unitary in Q(B), where ϕ = θ˜ ◦ Ad(u1 ⊕ u2 ) and ψ = Ad u3 ◦ θ˜ . Note that ϕ and ψ are inner isomorphisms from M2 (Q(B)) to Q(B) induced by isometries {π(S1 )u1 , π(S2 )u2 } and {u3 π(S1 ), u3 π(S2 )} respectively, where S1 , S2 are isometries which induce the inner isomorphism θ : M2 (M(B)) → M(B). Set u = π(S1 )u1 π(S1∗ )u∗3 + π(S2 )u2 π(S2∗ )u∗3 . Then ϕ = Ad u ◦ ψ . Since u = θ˜ (u1 ⊕ u2 )u∗3 , we have
θ˜ (u1 ⊕ u2 ) 1 = θ˜ (u1 ⊕ 1) 1 + θ˜ (1 ⊕ u2 ) 1 = [u1 ]1 + [u2 ]1 by Lemma 3.4. Hence [u]1 = [u1 ]1 + [u2 ]1 − [u3 ]1 = 0. Therefore, u is liftable. (2) Let x2 = 0 and x1 = x for any x ∈ K1 (Q(B)). By (1) and ε0 = id, we have εx ([τ1 ])+[τ2 ] = εx ([τ1 ] + [τ2 ]). Exchange [τ1 ] and [τ2 ], we obtain [τ1 ] + εx ([τ2 ]) = εx ([τ1 ] + [τ2 ]). Hence εx [τ1 ] + [τ2 ] = [τ1 ] + εx [τ2 ] .
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Let c = [τ2 ] and d = εx ([τ2 ]) for a fixed [τ2 ]. Then for any a ∈ Exteu s (A, B), we have εx (a) + c = a + d. Therefore, ε is a translation action of K1 (Q(B)) on Exteu s (A, B). 2 Proposition 3.6. The map ε induces a group action of K1 (Q(B)) on the set Extus (A, B). ss
Proof. Let τi ∈ Ext eu (A, B) such that τ1 ∼ τ2 . Then there exist strongly unital trivial extensions τ1 , τ2 and a unitary w ∈ M(B) such that τ2 ⊕ τ2 = Ad π(w) ◦ (τ1 ⊕ τ1 ). For any u ∈ U(Q(B)), ss
we have Ad u ◦ τ1 ∼ Ad u ◦ τ2 . In fact, since (Ad u ◦ τ2 ) ⊕ τ2
= θ˜ ◦ Ad(u ⊕ 1) ◦
τ2
τ2
s
∼ Ad u ◦ θ˜ ◦
τ2
τ2
and θ˜ ◦
τ2
τ2
= Ad π(w) ◦ θ˜ ◦
τ1
τ1
,
it follows that s
(Ad u ◦ τ2 ) ⊕ τ2 ∼ Ad u ◦ Ad π(w) ◦ θ˜ ◦
τ1
= Ad uπ(w)u∗ ◦ Ad u ◦ θ˜ ◦ s
∼ θ˜ ◦ Ad(u ⊕ 1) ◦
τ1
τ1
τ1
τ1
τ1
= (Ad u ◦ τ1 ) ⊕ τ1 . h
Similarly, we have [Ad u1 ◦ τ ]ss = [Ad u2 ◦ τ ]ss in Extus (A, B) when u1 ∼ u2 . As in the proof of Proposition 3.1, one can check that εx is bijection on the set Extus (A, B) and εx1 +x2 = εx1 ◦ εx2 . So ε is a group action of K1 (Q(B)) on Extus (A, B). 2 By Theorem 3.5 and Proposition 3.6, we have the following theorem. Theorem 3.7. When Extus (A, B) is a group, the group action ε in Proposition 3.6 is a translation action of K1 (Q(B)) on the group Extus (A, B). When Extus (A, B) is a group, the equivalence class of unital trivial extensions is the identity. Define a map by: α : K1 Q(B) → Extus (A, B);
x → εx [σ ]ss ,
where σ is an essential unital trivial extension. It follows from Theorem 3.7 that α is a group homomorphism and Ker α = {x ∈ K1 (Q(B)): εx = id}. Let O(ε) be the orbit space of the action ε. Then we have
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Extus (A, B)/Im α ∼ = O(ε) ∼ = Extuw (A, B), so the sequence of groups K1 Q(B) → Extus (A, B) → Extuw (A, B) → 0 is exact. In order to obtain a short exact sequence of groups, we need to characterize the kernel of α. We need two groups defined below. Definition 3.8. Suppose A, B are C ∗ -algebras with A unital and B stable. Define two groups as follows: Γ = f ∈ Hom K0 (A), K0 Q(B) : f [1A ]0 = 0 and G = ρ[1A ]0 : ρ ∈ Hom K0 (A), K0 (B) . Note that K0 (B) ∼ = K1 (Q(B)). Then one can reckon G as a subgroup of K1 (Q(B)). Remark 3.9. It should be noted that the group G has appeared in [18]. A main result in [18] is that there is a bijection between K0 (B)/G onto the set of strongly unitary equivalence classes of unital full essential extensions σ such that [σ ] = [τ ] in KK 1 (A, B) when B has the property (P). Theorem 3.10. Let A be a unital separable nuclear C ∗ -algebra with A ∈ N . If K1 (B) = 0, or B has the property (P), or A is an infinite C ∗ -algebra, there is a short exact sequence of groups 0 → K1 Q(B) /G → Extus (A, B) → Extuw (A, B) → 0. Proof. By Kasparov’s theorems [8,9], there exists an essential unital trivial extension σ of A by B, which is unital-absorbing. For any x = [u]1 ∈ Ker α, there are strongly unital trivial extens sions σ0 , σ0 such that (Ad u ◦ σ ) ⊕ σ0 ∼ σ ⊕ σ0 . Note that s σ ˜ = Ad u ◦ (σ ⊕ σ0 ) (Ad u ◦ σ ) ⊕ σ0 ∼ Ad u ◦ θ ◦ σ0 s
s
and σ ⊕ σ0 ∼ σ ∼ σ ⊕ σ0 . Hence s
Ad u ◦ (σ ⊕ σ0 ) ∼ σ ⊕ σ0 . Set τ = σ ⊕ σ0 . Then τ is a unital-absorbing and trivial extension, so there is a unitary w in M(B) such that Ad(π(w)u) ◦ τ = τ . We note that [π(w)u]1 = [u]1 = x, so we obtain Ad u ◦ τ = τ by replacing π(w)u by u.
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(1) Similar to the proof in [3], we can show that Ker α ⊂ G. Define a homomorphism ϕ : C(T) → Q(B) such that ϕ(z) = u, where z is the identity map of T. Since every element of Im(τ ) commutes with every element of Im(ϕ), there is a homomorphism δ = τ ⊗ ϕ : A ⊗ C(T) → Q(B) such that δ(a ⊗ b) = τ (a)ϕ(b) for any a ∈ A and any b ∈ C(T). Let β : K0 (A) → K1 (SA) → K1 (A ⊗ C(T)) be the Bott map. Then β([1A ]0 ) = [1 ⊗ z]1 . Set ρ = K1 (δ) ◦ β : K0 (A) → K1 (Q(B)) ∼ = K0 (B). Then we have ρ [1A ]0 = K1 (δ) [1 ⊗ z]1 = [u]1 = x. Therefore, it follows that Ker α ⊂ G. (2) Next, we show that G ⊂ Ker α. We first need to show that the homomorphism γ1 : Extuw A ⊗ C(T), B → Hom K1 A ⊗ C(T) , K1 Q(B) is surjective, where γ1 ([τ ]) = K1 (τ ). Case (1). B has a trivial K1 -group. By [1, 15.14.2], we have Ext(A ⊗ C(T), B) ∼ = Extuw (A ⊗ C(T), B). Hence γ1 is surjective by the UCT for Ext(A ⊗ C(T), B). Case (2). B has the property (P) or A is an infinite C ∗ -algebra. For every η ∈ Hom(K1 (A⊗C(T)), K1 (Q(B))), choose ξ ∈ Hom(K0 (A⊗C(T)), K0 (Q(B))) such that ξ([1A⊗C(T) ]0 ) = 0. Hence (ξ, η) ∈ Hom K0 A ⊗ C(T) , K0 Q(B) ⊕ Hom K1 A ⊗ C(T) , K1 Q(B) . By [16, 8.6] and the UCT for Ext(A ⊗ C(T), B), there exists a full essential extension τ1 : A ⊗ C(T) → Q(B) such that K0 (τ1 ) = ξ and K1 (τ1 ) = η. Set p = τ1 (1A⊗C(T) ). Then [p]0 = ξ([1A⊗C(T) ]0 ) = 0 = [1]0 , where 1 is the unit of Q(B). When B has the property (P), there exists z ∈ Q(B) such that zpz∗ = 1. Let q = pz∗ zp. Then q ∼ 1 and q p. Hence 1 p q ∼ 1. Since 1 is properly infinite, p is properly infinite. Since q ∼ 1 and 1 is full, q is a full projection. Note that p q. This implies that p is a full projection. Therefore, by [20, p. 75] and [p]0 = [1]0 we have p ∼ 1. When A is an infinite C ∗ -algebra, the projection p is infinite in Q(B). By [1, 12.2] we have K0 (Q(B)) = {[p]: p ∈ Q(B) is infinite}. Hence p ∼ 1. Therefore, there is a coisometry V such that V ∗ V = p when B has the property (P) or A is an infinite C ∗ -algebra. Set τ2 = Ad V ◦ τ1 . Then τ2 is a unital essential extension of A ⊗ C(T) by B. Since pQ(B)p ∼ = Q(B), we have K1 (pQ(B)p) = {[u]1 : u ∈ U(pQ(B)p)}. For any u ∈ pQ(B)p, we have [u]1 = [u + 1 − p]1 under the inclusion map. It follows from Lemma 3.3 that
Ad V (u) 1 = V ∗ Ad V (u)V + 1 − V ∗ V 1 = [u + 1 − p]1 . Hence, K1 (τ2 ) = K1 (τ1 ) = η. Therefore, γ1 : Extuw (A ⊗ C(T), B) → Hom(K1 (A ⊗ C(T)), K1 (Q(B))) is surjective in the two cases.
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For every x ∈ G, let x = η([1A ]0 ) for some η ∈ Hom(K0 (A), K0 (B)). One can identify K0 (B) with K1 (Q(B)) and reckon K0 (A) as a direct summand of K1 (A ⊗ C(T)). Since η ∈ Hom(K0 (A), K0 (B)) ⊂ Hom(K1 (A ⊗ C(T)), K1 (Q(B))), there is a unital essential extension τ : A ⊗ C(T) → Q(B) such that γ1 (τ ) = η. Set τ = τ |A and u = τ (1 ⊗ z). Then for every a ∈ A we have τ (a)u = uτ (a) and [u]1 = K1 τ [1 ⊗ z]1 = η [1A ]0 = x. Furthermore, εx ([τ ]ss ) = [τ ]ss . Since εx is a translation action on Extus (A, B), we have εx = id. Therefore, G ⊂ Ker α. 2 Lemma 3.11. Let A be a separable, nuclear C ∗ -algebra with unit. If K1 (B) = 0, or B has the property (P), or A is an infinite C ∗ -algebra, then Extuw (A, B) → Ext(A, B) is an injective homomorphism. ss
Proof. Let τ1 , τ2 be unital essential extensions such that τ1 ∼ τ2 . Let σ0 be an absorbss ing extension. Then τ1 ⊕ σ0 ∼ τ2 ⊕ σ0 . Hence there are trivial extension σ1 , σ2 such that s s s s τ1 ⊕ σ0 ⊕ σ1 ∼ τ2 ⊕ σ0 ⊕ σ2 . Since σ0 ⊕ σ1 ∼ σ0 and σ0 ⊕ σ2 ∼ σ0 , τ1 ⊕ σ0 ∼ τ2 ⊕ σ0 . Hence there is a liftable unitary u1 ∈ Q(B) such that τ1 ⊕ σ0 = Ad u1 (τ2 ⊕ σ0 ). Set p = σ0 (1). Since σ0 is trivial, K0 (σ0 ) = 0. Therefore, [p]0 = [1Q(B) ]0 = 0. As in the V
proof of Theorem 3.10, by the assumption there is a coisometry V ∈ Q(B) such that p ∼ 1. Since s τ u τ1 τ2 ∼ θ˜ ◦ Ad 1 θ˜ ◦ = Ad u1 ◦ θ˜ 2 , σ0 σ0 1 σ0 s s τ2 u1 τ2 τ1 . ∼ Ad ∼ σ0 1 σ0 σ0 Hence, there is a liftable unitary u ∈ M2 (Q(B)) such that
τ1 σ0
Let w =
τ = Ad u 2
and
σ0
1 p
1 = Ad u
.
p
1 1 u V ∗ . Since V ∗ V = p, we have V 1 Ad
τ1 σ0
V
1 = Ad w ◦ Ad
τ2 σ0
V
.
Hence
τ1
Ad V ◦ σ0
= Ad w
τ2
Ad V ◦ σ0
.
Furthermore, θ˜ ◦
τ1
Ad V ◦ σ0
= θ˜ ◦ Ad w ◦ θ˜ −1 ◦ θ˜
τ2
Ad V ◦ σ0
.
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Since θ˜ = Ad U and θ˜ −1 = Ad U ∗ , where U = (π(S1 ), π(S2 )), we have θ˜ ◦ Ad w ◦ θ˜ −1 = Ad U wU ∗ = Ad θ˜ (w) . Note that Ad V (p) = 1 and θ˜ (w) is a unitary in Q(B). So Ad V ◦ σ0 is a unital extension. w Finally, by τ1 ⊕ Ad V ◦ σ0 ∼ τ2 ⊕ Ad V ◦ σ0 , we have [τ1 ]sw + [Ad V ◦ σ0 ]sw = [τ2 ]sw + [Ad V ◦ σ0 ]sw in Extuw (A, B). Therefore, [τ1 ]sw = [τ2 ]sw holds in Extuw (A, B).
2
Theorem 3.12. Let A be a unital separable nuclear C ∗ -algebra with A ∈ N . If K1 (B) = 0, or B has the property (P), or A is an infinite C ∗ -algebra, then there is a short exact sequence 0 → Ext K∗ (A), K∗ (B) → Extuw (A, B) → Γ ⊕ Hom K1 (A), K0 (B) → 0, where Γ = {f ∈ Hom(K0 (A), K0 (Q(B))): f ([1A ]0 ) = 0}. Proof. Let γ0 be the quotient homomorphism in the UCT for Ext(A, B). Define a homomorphism γ : Extuw (A, B) → Γ ⊕ Hom(K1 (A), K1 (Q(B))) such that γ ([τ ]) = (K0 (τ ), K1 (τ )). From the proof of Theorem 3.10, it follows that γ is a surjective homomorphism. Then there is a commutative diagram: γ
0 −−−−→ Ker γ −−−−→ Extuw (A, B) −−−−→ Γ ⊕ Hom(K1 (A), K1 (Q(B))) −−−−→ 0 ⏐ ⏐ ⏐ ⏐φ ⏐ι ⏐η γ0
0 −−−−→ Ker γ0 −−−−→ Ext(A, B) −−−−→
Hom(K∗ (A), K∗ (Q(B)))
−−−−→ 0,
where φ is the homomorphism in Lemma 3.11, ι is the inclusion map, and η : Ker γ → Ker γ0 is the restriction map of φ. It follows from Lemma 3.11 that η is injective. In the following we need to show that η is an isomorphism. In the case of K1 (B) = 0, η is an isomorphism since Extuw (A, B) ∼ = Ext(A, B). When B has the property (P) or A is an infinite C ∗ -algebra, for every x ∈ Ker γ0 , there is an essential extension τ : A → Q(B) such that x = [τ ]sw . Since x ∈ Ker γ0 , K∗ (τ ) = 0. If τ is not unital, let p = τ (1A ). Then [p]0 = [1]0 and 1 − p = 0. As in the proof of Theorem 3.10, there is a coisometry v0 such that v0∗ v0 = p. Let τ = Ad v0 ◦ τ : A → Q(B). Then τ is unital. Set V = v0 ⊕ 0. Then V ∗ V = p ⊕ 0 and V V ∗ = 1 ⊕ 0. Therefore, we have
τ
0
= Ad V
τ 0
.
Similarly, since [1 − p]0 = 0 = [1]0 , we have 1 − p ∼ 1. Hence (1 − p) ⊕ 1 ∼ 1 ⊕ 1 ∼ 1 ∼ 0 ⊕ 1.
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It follows that there is a partial isometry W ∈ M2 (Q(B)) such that W ∗ W = (1 − p) ⊕ 1 and W W ∗ = 0 ⊕ 1. Let U = V + W . Then U is a unitary in M2 (Q(B)) and Ad U
τ 0
= Ad V
τ 0
=
τ
0
.
w
Hence, τ ⊕ 0 ∼ τ ⊕ 0. Therefore, sw
w
sw
τ ∼ τ ⊕ 0 ∼ τ ⊕ 0 ∼ τ ⊕ 0. We conclude that there is a unital extension τ such that [τ ]sw = [τ ]sw in Ext(A, B). Since K∗ (τ ) = K∗ (τ ) = 0, we have [τ ]sw ∈ Ker γ . Therefore, η is a surjective homomorphism. By the UCT for Ext(A, B), we have Ker γ0 ∼ = = Ext(K∗ (A), K∗ (B)). Hence, Ker γ ∼ Ext(K∗ (A), K∗ (B)). Therefore, there is a short exact sequence 0 → Ext K∗ (A), K∗ (B) → Extuw (A, B) → Γ ⊕ Hom K1 (A), K0 (B) → 0.
2
Definition 3.13. Let H be an abelian group, h0 ∈ H , and K an abelian group. We denote by Ext((H, h0 ), K) the set of equivalence classes of all extensions of abelian groups with base point of the form φ
0 → K → (G, g0 ) → (H, h0 ) → 0. Recall that the natural map Ext((H, h0 ), K) → Ext(H, K) has kernel isomorphic to K/ f (h0 ) f ∈ Hom(H, K) . See [3] for more details. Lemma 3.14. Let A be a unital separable nuclear C ∗ -algebra with A ∈ N . If K1 (B) = 0, or B has the property (P), or A is an infinite C ∗ -algebra, then there is an isomorphism θ : Ker γ → Σ such that the following diagram is commutative 0 −−−−→ K1 (Q(B))/G −−−−→ Ker γ −−−−→ ⏐ ⏐ ⏐ ⏐ θ id 0 −−−−→ K1 (Q(B))/G −−−−→
Σ
Ker γ ⏐ ⏐η
−−−−→ 0
−−−−→ Ext(K∗ (A), K∗ (B)) −−−−→ 0,
where γ : Extus (A, B) → Γ ⊕ Hom(K1 (A), K0 (B)) such that γ ([τ ]ss ) = (K0 (τ ), K1 (τ )), Σ = Ext((K0 (A), [1A ]0 ), K0 (B)) ⊕ Ext(K1 (A), K1 (B)), and γ is the homomorphism defined in Theorem 3.12. Proof. Let γ be the map Extus (A, B) → Γ ⊕ Hom(K1 (A), K0 (B)) such that γ ([τ ]ss ) = (K0 (τ ), K1 (τ )). Then, from the proof of Theorem 3.10 it follows that γ is a surjective homomorphism.
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Set Σ = Ext((K0 (A), [1A ]0 ), K0 (B)) ⊕ Ext(K1 (A), K1 (B)). Let e : 0 → B → E → A → 0 be an essential unital extension with Busby invariant τ . If [τ ] ∈ Ker γ , then the associated six term exact sequence in K-theory breaks into two short exact sequences εj of the forms 0 → Kj (B) → Kj (E) → Kj (A) → 0. Define a map θ : Ker γ → Σ by θ ([τ ]) = θ0 ([τ ]) ⊕ θ1 ([τ ]) where θ0 ([τ ]) and θ1 ([τ ]) are the equivalence classes of ε0 and ε1 in Ext((K0 (A), [1A ]0 ), K0 (B)) and Ext(K1 (A), K1 (B)), respectively. ψi
Let ei : 0 → B → Ei → A → 0 be essential unital extensions with Busby invariant τi for i = 1, 2. If [τ1 ]ss = [τ2 ]ss in Ker γ , then there are two strongly unital trivial extension σ1 , σ2 s such that τ1 ⊕ σ1 ∼ τ2 ⊕ σ2 . By [19] we have θ1 ([τ1 ]) = θ1 ([τ2 ]).
Let τ1 = τ1 ⊕ σ1 and σ1 = π ◦ ϕ for some unital homomorphism ϕ : A → M(B). Choose two isometries s1 , s2 ∈ M(B) such that s1 s1∗ + s2 s2∗ = 1. Then τ1 (a) = π(s1 )τ1 (a)π s1∗ + π(s2 )σ1 (a)π s2∗ for any a ∈ A. Let e : 0 → B → E → A → 0 be an essential unital extensions with Busby invariant τ . As in the proof of [19, 2.1], let λ : E1 → M(B) and β : B → B be given by λ(x) = s1 xs1∗ + s2 ϕ ◦ ψ1 (x) s2∗ ,
β(b) = s1 bs1∗
for any x ∈ E1 and any b ∈ B. Then λ(E1 ) ⊂ E2 and there is a commutative diagram e1 : 0 −−−−→ B −−−−→ ⏐ ⏐β
E1 −−−−→ ⏐ ⏐ λ
A −−−−→ 0
e1 : 0 −−−−→ B −−−−→ E1 −−−−→ A −−−−→ 0. Since τ1 , τ1 are essential and unital, we have 1E1 = 1E2 = 1M(B) . Since ϕ, ψ1 are unital, λ(1E1 ) = s1 s1∗ + s2 s2∗ = 1E2 . Hence, λ is unital. Note that K∗ (β) = id, so we have θ0 ([τ1 ]) = s
θ0 ([τ1 ]). Similarly, θ0 ([τ2 ]) = θ0 ([τ2 ⊕ σ2 ]). By τ1 ⊕ σ1 ∼ τ2 ⊕ σ2 we have θ0 [τ1 ] = θ0 [τ1 ⊕ σ1 ] = θ0 [τ2 ⊕ σ2 ] = θ0 [τ2 ] .
Therefore, the map θ is well defined and one can check that θ is a homomorphism. Hence, by Theorem 3.10, there is a short exact sequence 0 → K1 Q(B) /G → Ker γ → Ker γ → 0, where G = {ρ[1A ]0 : ρ ∈ Hom(K0 (A), K0 (B))} and γ : Extuw (A, B) → Γ ⊕ Hom K1 (A), K1 (Q(B)) defined in Theorem 3.12. Therefore, the following diagram is commutative
C. Wei / Journal of Functional Analysis 258 (2010) 650–664
0 −−−−→ K1 (Q(B))/G −−−−→ Ker γ −−−−→ ⏐ ⏐ ⏐ ⏐ θ id 0 −−−−→ K1 (Q(B))/G −−−−→
Ker γ ⏐ ⏐η
663
−−−−→ 0
−−−−→ Ext(K∗ (A), K∗ (B)) −−−−→ 0.
Σ
By Theorem 3.12 we see that η is an isomorphism. It follows that θ is an isomorphism from the five lemma. 2 Combining 3.10, 3.12 and 3.14, we have a commutative diagram as follows 0 ⏐ ⏐
0 ⏐ ⏐
0 ⏐ ⏐
S ⏐ ⏐
←−−−−−
Σ ⏐ ⏐
←−−−−− Ker γ −−−−−→ Extus (A, B) −−−−−→ Γ ⊕ Hom(K1 (A), K0 (B)) −−−−−→ 0 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐
S ⏐ ⏐
−−−−−→
S ⏐ ⏐ γ
γ
Ext(K∗ (A), K∗ (B)) ←−−−−− Ker γ −−−−−→ Extw s (A, B) −−−−−→ Γ ⊕ Hom(K1 (A), K0 (B)) −−−−−→ 0 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 0
0
0
where S = K1 (Q(B))/G. By the above diagram, we have the following theorem immediately. Theorem 3.15. Let A be a unital separable nuclear C ∗ -algebra with A ∈ N . If K1 (B) = 0, or B has the property (P), or A is an infinite C ∗ -algebra, then there is a short exact sequence of groups 0 → Σ → Extus (A, B) → Γ ⊕ Hom K1 (A), K0 (B) → 0, where Σ = Ext K0 (A), [1A ]0 , K0 (B) ⊕ Ext K1 (A), K1 (B) . When B = K, we have Γ = 0. Furthermore, we have Extus (A, B) ∼ = Extus (A, B) by Voiculescu’s theorem [22]. Hence, by Theorem 3.15 we obtain the UCT of L. Brown and M. Dadarlat for the strong Ext-group. Corollary 3.16. (See [3].) Let A be a unital separable nuclear C ∗ -algebra with A ∈ N . If B = K, then there is a short exact sequence of groups 0 → Ext K0 (A), [1]0 , Z → Extus (A, B) → Hom K1 (A), Z → 0. By the proof of [16, 8.6], for every x ∈ Extus (A, B), there is a unital full extension τ such that x = [τ ]. When B has the property (P), by [6] every unital full extension is unital-absorbing. So
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u ∼ it follows that Exteu s (A, B) = Exts (A, B) if A is simple or Q(B) is simple. Then we have the following corollary by Theorem 3.15.
Corollary 3.17. Suppose that A is a unital separable nuclear C ∗ -algebra with A ∈ N and B has the property (P). If A is simple or Q(B) is simple, then there is a short exact sequence 0 → Σ → Exteu s (A, B) → Γ ⊕ Hom K1 (A), K0 (B) → 0, where Σ = Ext K0 (A), [1A ]0 , K0 (B) ⊕ Ext K1 (A), K1 (B) . References [1] B. Blackadar, K-Theory for Operator Algebras, second ed., Math. Sci. Res. Inst. Publ., vol. 5, Cambridge University Press, Cambridge, 1998. [2] L.G. Brown, The universal coefficient theorem for Ext and quasidiagonality, in: Operator Algebras and Group Representations, vol. I, Neptun, 1980, in: Monogr. Stud. Math., vol. 17, Pitman, Boston, MA, 1984, pp. 60–64. [3] L.G. Brown, M. Dadarlat, Extensions of C ∗ -algebras and quasidiagonality, J. London Math. Soc. (2) 53 (1996) 582–600. [4] L.G. Brown, R.G. Douglas, P.A. Fillmore, Extensions of C ∗ -algebras and K-homology, Ann. of Math. (2) 105 (2) (1977) 265–324. [5] G.A. Elliott, G. Gong, On the classification of C ∗ -algebras of real rank zero. II, Ann. of Math. (2) 144 (3) (1996) 497–610. [6] G.A. Elliott, D. Kucerovsky, An abstract Brown–Douglas–Fillmore absorption theorem, Pacific J. Math. 198 (2001) 385–409. [7] K.K. Jensen, K. Thomsen, Elements of KK-Theory, Birkhäuser, Boston, Basel, Berlin, 1991. [8] G.G. Kasparov, Topological invariants of elliptic operators. I. K-homology, Izv. Akad. Nauk SSSR Ser. Mat. 39 (4) (1975) 796–838; transl. in: Math. USSR Izv. 9 (1976) 751–792. [9] G.G. Kasparov, The operator K-functor and extensions of C ∗ -algebras, Math. USSR Izv. 16 (1981) 513–572. [10] H. Lin, Extensions by C ∗ -algebras of real rank zero. II, Proc. London Math. Soc. (3) 71 (3) (1995) 641–674. [11] H. Lin, On the classification of C ∗ -algebras of real rank zero with zero K1 , J. Operator Theory 35 (1) (1996) 147–178. [12] H. Lin, Extensions by C ∗ -algebras of real rank zero. III, Proc. London Math. Soc. (3) 76 (3) (1998) 634–666. [13] H. Lin, Classification of simple C ∗ -algebras and higher dimensional noncommutative tori, Ann. of Math. (2) 157 (2003) 521–544. [14] H. Lin, Classification of simple C ∗ -algebras of tracial topological rank zero, Duke Math. J. 125 (1) (2004) 91–119. [15] H. Lin, An approximate universal coefficient theorem, Trans. Amer. Math. Soc. 357 (2005) 3375–3405. [16] H. Lin, Full extensions and approximate unitary equivalences, Pacific J. Math. 229 (2007) 389–428. [17] H. Lin, Simple nuclear C ∗ -algebras of tracial topological rank one, J. Funct. Anal. 251 (2007) 601–679. [18] H. Lin, Unitary equivalences for essential extensions of C ∗ -algebras, Proc. Amer. Math. Soc. 137 (10) (2009) 3407–3420. [19] M. Rørdam, Classification of extensions of certain C ∗ -algebras by their six term exact sequences in K-theory, Math. Ann. 308 (1) (1997) 93–117. [20] M. Rørdam, F. Larsen, N. Laustsen, An Introduction to K-Theory for C ∗ -Algebras, London Math. Soc. Stud. Texts, vol. 49, Cambridge University Press, Cambridge, 2000. [21] J. Rosenberg, C. Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke Math. J. 55 (2) (1987) 431–474. [22] D. Voiculescu, A non-commutative Weyl–von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976) 97–113.
Journal of Functional Analysis 258 (2010) 665–712 www.elsevier.com/locate/jfa
Keller–Osserman conditions for diffusion-type operators on Riemannian manifolds Luciano Mari a , Marco Rigoli a , Alberto G. Setti b,∗ a Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133 Milano, Italy b Dipartimento di Fisica e Matematica, Università dell’Insubria – Como, via Valleggio 11, I-22100 Como, Italy
Received 1 May 2009; accepted 12 October 2009
Communicated by L. Gross
Abstract In this paper we obtain essentially sharp generalized Keller–Osserman conditions for wide classes of differential inequalities of the form Lu b(x)f (u)(|∇u|) and Lu b(x)f (u)(|∇u|) − g(u)h(|∇u|) on weighted Riemannian manifolds, where L is a non-linear diffusion-type operator. Prototypical examples of these operators are the p-Laplacian and the mean curvature operator. The geometry of the underlying manifold is reflected, via bounds for the modified Bakry–Emery Ricci curvature, by growth conditions for the functions b and . A weak maximum principle which extends and improves previous results valid for the ϕ-Laplacian is also obtained. Geometric comparison results, valid even in the case of integral bounds for the modified Bakry–Emery Ricci tensor, are presented. © 2009 Elsevier Inc. All rights reserved. Keywords: Keller–Osserman condition; Diffusion-type operators; Weak maximum principles; Weighted Riemannian manifolds; Quasi-linear elliptic inequalities
1. Introduction Consider the Poisson-type inequality on Euclidean space Rm u f (u),
(1.1)
* Corresponding author.
E-mail addresses:
[email protected] (L. Mari),
[email protected] (M. Rigoli),
[email protected] (A.G. Setti). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.008
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L. Mari et al. / Journal of Functional Analysis 258 (2010) 665–712
where f ∈ C 0 ([0, +∞)), f (0) = 0 and f (t) > 0 if t > 0. By an entire solution of (1.1) we mean a C 1 function u satisfying (1.1) on Rm in the sense of distributions. Let t F (t) =
f (s) ds.
(1.2)
0
It is well know that if f satisfies the Keller–Osserman condition 1 ∈ L1 (+∞), √ F (t)
(1.3)
then (1.1) has no non-negative entire solutions except u ≡ 0. Note that in the case where f (t) = t q the integrability condition expressed by (1.3) is equivalent to q > 1. But (1.3) is sharper than the condition on powers it is implied by. For instance (1.3) holds if f (t) = t logβ (1 + t) with β > 2. As a matter of fact, if the Keller–Osserman condition fails, that is, if 1 ∈ / L1 (+∞), √ F (t)
(1.4)
then inequality (1.1) admits positive solutions. Indeed, consider the ODE problem
m−1 α = f (α), r α(0) = αo > 0, α (0) = 0.
α +
(1.5)
General theory yields the existence of a solution in a maximal interval [0, R) and a first integration of (1.5) gives α > 0 on (0, R). Suppose by contradiction that R < +∞. Using the maximality condition and the monotonicity of α we obtain lim α(r) = +∞.
r→R −
(1.6)
On the other hand, it follows from (1.5) that α α f (α)α , whence integrating over [0, r], 0 < r R, changing variables in the resulting integral, and taking square roots we obtain √
√ α 2. F (α)
A further integration over [0, r] with 0 < a < r < R yields α(r)
α(a)
√ dt 2(r − a) √ F (t)
L. Mari et al. / Journal of Functional Analysis 258 (2010) 665–712
667
and letting r → R − and using (1.6) we contradict (1.4). This shows that the function α is defined on [0, +∞). Setting u(x) = α(r(x)) (r(x) = |x|) gives rise to a radial positive entire solution of (1.1). Note however that any non-negative solution of (1.1) must diverge at infinity sufficiently fast. Indeed, it follows from [17], Corollary 16, that if u 0 is an entire solution of (1.1) satisfying u(x) = o r(x)σ
as r(x) → +∞,
with 0 σ < 2, and f is non-decreasing, then u ≡ 0. Note that this latter conclusion can be hardly deduced from (1.4). We also observe that differential inequalities of the type (1.1) often appear in connection with geometrical problems on complete manifolds and, in fact, R. Osserman introduced condition (1.3) in [13] in his investigation on the type of a Riemann surface. For a number of further examples we refer, for instance, to [16]. Motivated by the above considerations, from now on we will denote with (M, , ) a complete, non-compact, connected Riemannian manifold of dimension m 2. We fix an origin o in M and we let r(x) = dist(x, o) be the Riemannian distance from the chosen reference point, and we denote by Br the geodesic ball of radius r centered at o and with ∂Br its boundary. 1 + Given a positive function D(x) ∈ C 2 (M) and a non-negative function ϕ ∈ C 0 (R+ 0 ) ∩ C (R ), where, as usual R+ = (0, +∞) and R+ 0 = [0, +∞), we consider the diffusion-type operator defined on M by the formula LD,ϕ u =
1 div D|∇u|−1 ϕ |∇u| ∇u . D
For instance, if D ≡ 1 and ϕ(t) = t p−1 , p > 1, or ϕ(t) = √ t
1+t 2
we recover the usual p-
Laplacian and the mean curvature operator, respectively. If b(x) ∈ C 0 (M) and ∈ C 0 (R+ 0 ), we will be interested in solutions of the differential inequality LD,ϕ u b(x)f (u) |∇u| .
(1.7)
By an entire classical weak solution of (1.7) we mean a C 1 function u on M which satisfies the inequality in the sense of distributions, namely, − |∇u|−1 ϕ |∇u| ∇u, ∇ϕ D dV b(x)f (u) |∇u| ψD dV (1.8) for every non-negative function ψ ∈ Cc∞ (M), where we have denoted with dV the Riemannian volume element. Since we are dealing with a diffusion-type operator, the interplay between analysis and geometry will be taken into account by means of the modified Bakry–Emery Ricci tensor that we now introduce. Following Z. Qian [20], for n > m let n−m−1 1 1 Hess D + dD ⊗ dD D n − m D2 1 1 dD ⊗ dD = Ricc(LD ) − n − m D2
Riccm,n (LD ) = RiccM −
(1.9)
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be the modified Bakry–Emery Ricci tensor, where Ricc(LD ) is the usual Bakry–Emery Ricci tensor, RiccM is the Ricci tensor of (M, , ) (see D. Bakry and P. Emery [2]), and where, to simplify notation, we have denoted with LD the operator LD,ϕ for ϕ(t) = t. We introduce some more terminology. Definition 1.1. Let g be a real valued function defined on R+ . We say that g is C-increasing on R+ if there exists a constant C 1 such that sup g(s) Cg(t) ∀t ∈ R+ .
(1.10)
s∈(0,t]
It is easily verified that the above condition is equivalent to inf
s∈[t,+∞)
g(s)
1 g(t) C
∀t ∈ R+ ,
and both formulations will be used in the sequel. Clearly, (1.10) is satisfied with C = 1 if g is non-decreasing on R+ . In general, the validity of (1.10) allows a controlled oscillatory behavior such as, for instance, that of g(t) = t 2 (2 + sin t). In order to state our next result, we introduce the following set of assumptions. (Φ0 ) (F1 ) (L1 ) (L2 ) (ϕ) (θ )
ϕ > 0 on R+ . f ∈ C(R), f (0) = 0, f (t) > 0 if t > 0 and f is C-increasing on R+ . + ∈ C 0 (R+ 0 ), (t) > 0 on R . + is C-increasing on R . tϕ (t) 1 + 1 lim inft→0+ ϕ(t) (t) = 0, (t) ∈ L (0 ) \ L (+∞). There exists θ ∈ R such that the functions t→
ϕ (t) θ t (t)
and t →
ϕ(t) θ−1 t (t)
are C-increasing on R+ . Clearly the last two conditions relate the operator LD,ϕ to the gradient term , and, in general, they are not independent. As we shall see below, in favorable circumstances (θ ) implies (ϕ). This is the case, for instance, in the next Theorem A when θ < 1. For a better understanding of these two assumptions, we examine the special but important case where (t) = t q , q 0. First we consider the case of the p-Laplacian, so that ϕ(t) = t p−1 , p > 1. Then, given θ ∈ R, (ϕ) and (θ ) are simultaneously satisfied provided p > q + 1 and θ q − p + 2. 2
If we consider ϕ(t) = tet (which, when D ≡ 1, gives rise to the operator associated to the exponentially harmonic functions, see [5] and [6]), then (ϕ) and (θ ) are both satisfied provided q < 1 and q θ.
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If ϕ = √ t
1+t 2
669
, which, for D ≡ 1, corresponds to the “mean curvature operator”, then (ϕ) does
not hold for any q 0. However, a variant of our arguments will allow us to analyze this situation, see Section 4 below. + Because of (L1 ) and (ϕ) we may define a C 1 -diffeomorphism K : R+ 0 → R0 by the formula t K(t) =
sϕ (s) ds. (s)
(1.11)
0 −1 Since K is increasing on R+ 0 so is its inverse K . Moreover, when ≡ 1 then
(t), K (t) = H where (t) = tϕ(t) − H
t ϕ(s) ds 0
t
is the pre-Legendre transform of t → 0 ϕ(s) ds. Having defined F as in (1.2) we are ready to introduce our first generalized Keller–Osserman condition: 1 K −1 (F (t))
∈ L1 (+∞).
(KO)
It is clear that, in the case of the Laplace–Beltrami operator (or more generally, of the p-Laplacian) and for ≡ 1, (KO) is equivalent to the classical Keller–Osserman condition (1.3). After this preparation we are ready to state Theorem A. Let (M, , ) be a complete manifold satisfying β/2 , Riccn,m (LD ) H 2 1 + r 2
(1.12)
for some n > m, H > 0 and β −2. Let also b(x) ∈ C 0 (M) be a non-negative function such that b(x)
C r(x)μ
if r(x) 1,
(1.13)
for some C > 0 and μ 0. Assume that (Φ0 ), (F1 ), (L1 ), (L2 ), (ϕ), (θ ) and (KO) hold, and suppose that θ < 1 − β/2 − μ or θ = 1 − β/2 − μ < 1 if μ > 0, (θβμ) θ < 1 − β/2 if μ = 0. Then any entire classical weak solution u of the differential inequality (1.7) is either non-positive or constant. Furthermore, if u 0 and (0) > 0, then u ≡ 0.
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We remark that letting β < −2 in (1.12) yields the same estimates valid for β = −2, which roughly correspond to the Euclidean behavior. Correspondingly, the conclusion of Theorem A is not improved by such a strengthening of the assumption on the modified Bakry–Emery Ricci curvature. To better appreciate the result and the role played by geometry, we state the following consequence for the p-Laplace operator p . Corollary A1. Let (M, , ) and b(x) be as in the statement of Theorem A and satisfying (1.12) with D ≡ 1 (so that Riccn,m = Ricc) and (1.13). Let f satisfy (F1 ) and let (t) = t q , for some q 0. Assume that p and μ satisfy p > q + 1,
0 μ p − q,
β 2(p − q − μ − 1).
If 1 ∈ L1 (+∞), F (t)1/(p−q)
(KO)
then any entire classical weak solution u of the differential inequality p u b(x)f (u)|∇u|q is either non-positive or constant. Note that if p = 2 and q = μ = 0, then the maximum amount of negative curvature allowed is obtained by choosing β = 2. In particular, the result covers the cases of Euclidean and hyperbolic space. We observe in passing that the choice β = 2 is borderline for the stochastic completeness of the underlying manifold. To include in our analysis the case of the mean curvature operator we state the following consequence of Theorem 4.4. Corollary A2. Let (M, , ) and b(x) be as in the statement of Theorem A and satisfying (1.12) with D ≡ 1 and (1.13). Let f satisfy (F1 ) and let (t) = t q , for some q 0. Assume μ 0 and that 0q <−
β − μ. 2
If 1 ∈ L1 (+∞), F (t)1/(1−q) then any non-negative, entire classical weak solution u of the differential inequality
∇u
div 1 + |∇u|2 is constant.
b(x)f (u)|∇u|q
(KO)
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Note that, contrary to Corollary A1, the case of hyperbolic space, which corresponds to β = 0, is not covered by Corollary A2. On the other hand, if β = −2, which, as already mentioned, roughly corresponds to a Euclidean behavior, the conditions on the parameters become μ 0,
0 q < 1 − μ,
and they are clearly compatible. This is one of the instances where the interaction between geometry and differential operators comes into play. As briefly remarked at the beginning of this introduction, the failure of the Keller–Osserman condition may yield existence of non-constant non-negative entire solutions. The next result shows that such solutions, if they exist, have to go to infinity sufficiently fast depending on the geometry of M and, of course, of the relevant parameters in the differential inequality satisfied. To state our result we introduce the following set of assumptions. (Φ1 ) (F0 ) (L3 ) (b1 )
(i) ϕ(0) = 0; (ii) ϕ(t) At δ on R+ , for some A, δ > 0. f ∈ C 0 (R+ 0 ). + 0 ∈ C (R0 ), (t) Ct χ on R+ , for some C > 0, χ 0. C b ∈ C 0 (M), b(x) > 0 on M, b(x) r(x) μ if r(x) 1, for some C > 0, μ ∈ R.
Theorem B. Let (M, , ) be a complete Riemannian manifold, and assume that conditions (Φ1 ), (F0 ), (L3 ) and (b1 ) hold. Given σ 0, let η = μ − (1 + δ − χ)(1 − σ ) and suppose that σ η,
0 χ < δ.
Let u be a non-constant entire classical weak solution of LD,ϕ u b(x)f (u) |∇u| ,
(1.7)
σ > 0, lim inf f (t) > 0 and t→+∞
as r(x) → +∞, u+ (x) = max u(x), 0 = o r(x)σ
(1.14)
and suppose that either
or σ =0
and u∗ = sup u < +∞.
(1.15)
M
Assume further that either lim inf
log
Br
D(x) dV (x)
Br
D(x) dV (x)
r→+∞
r σ −η
< +∞ if σ − η > 0
(1.16)
< +∞ if σ − η = 0.
(1.17)
or lim inf r→+∞
log
log r
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Then u∗ < +∞ and f (u∗ ) 0. In particular, if we also assume that f (t) > 0 for t > 0, and that u(xo ) > 0 for some x ∈ M, then u is constant on M, and if in addition f (0) = 0 and (0) > 0, then u ≡ 0 on M. Observe that the growth condition (1.14) is sharp. Indeed, we consider the case of the p-Laplace operator on Euclidean space, for which D ≡ 1 and δ = p − 1, and suppose that χ = μ = 0 and σ = η. Since η = p(σ − 1), the latter condition amounts to σ = p , the Hölder conjugate exponent of p. Since condition (1.17), which now reads lim inf r→+∞
log vol Br < +∞, log r
is clearly satisfied, all assumptions of Theorem B hold. On the other hand, a simple computation shows that the function u(x) = p1 r(x)p is a classical entire weak solution of p u = m, for which (1.14) barely fails to be met. We also stress that while in Theorem A the main geometric assumption is the radial lower bound on the modified Bakry–Emery Ricci curvature expressed by (1.12), in Theorem B we consider either (1.16) or (1.17), which we interpret as follows. Let dVD = D dV be the measure with density D(x), so that, for every measurable set Ω, volD (Ω) =
D(x) dV , Ω
and consider the weighted Riemannian manifold (M, , , dVD ). With this notation, we may rewrite, for instance (1.16), in the form log volD Br <∞ r→+∞ r σ −η
lim inf
if σ > η,
(1.18)
and interpret it as a control from above on the growth of the weighted volume of geodesic balls with respect to Riemannian distance function. This is a mild requirement, which is implied, via a version of the Bishop–Gromov volume comparison theorem for weighted manifolds, by a lower bound on the modified Bakry–Emery Ricci curvature in the radial direction. Indeed, as we shall see in Section 2 below, the latter yields an upper estimate on LD r which in turn gives the volume comparison estimate. In fact, we shall prove there that an Lp -condition on the modified Bakry– Emery Ricci curvature implies a control from above on the weighted volume of geodesic balls. On the contrary, as in the classical case of Riemannian geometry, volume growth restrictions do not provide in general a control on LD r. This in turn prevents the possibility of constructing radial super-solutions of (the equation corresponding to) (1.7), that could be used, as in the proof of Theorem A, as suitable barriers to study the existence problem via comparison techniques. This technical difficulty forces us to devise a new approach in the proof of Theorem B, based on a generalization of the weak maximum principle introduced by the authors in [22,16] (see Section 5). In Section 6 we implement our techniques to analyze differential inequalities of the type LD,ϕ u b(x)f (u) |∇u| − g(u)h |∇u| ,
(1.19)
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where g and h are continuous functions. Our first task is to find an appropriate form of the Keller–Osserman condition. To this end, we let ρ ∈ C 0 R+ 0 ,
ρ(t) 0 on R+ 0,
(ρ)
(t) = F ρ,ω depending on the real parameter ω by the formula and define the function F ρ,ω (t) = F
t f (s)e(2−ω)
s 0
ρ(z) dz
ds.
(1.20)
0
is well defined because of our assumptions. We assume that tϕ / ∈ L1 (0+ ) \ Note that F + 1 L (+∞), define K as in (1.11) and let K −1 : R+ 0 → R0 be its inverse. The new version of the Keller–Osserman condition that we shall consider is t
e 0 ρ(z) dz ∈ L1 (+∞). (t)) K −1 (F
(ρKO)
Of course, when ρ ≡ 0 we recover condition (KO) introduced above. As we shall see in Section 5, the two conditions are in fact equivalent if ρ ∈ L1 under some mild additional conditions. We prove Theorem C. Let (M, , ) be a complete manifold satisfying β/2 Riccn,m (LD ) H 2 1 + r 2 ,
(1.12)
for some n > m, H > 0 and β −2. Assume that (Φ0 ), (F1 ), (L1 ), (L2 ), (ϕ), (θ ) and (b1 ) hold with μ 0, θ 1 and θ < 1 − β/2 − μ if θ 1, μ > 0, θ = 1 − β/2 − μ if θ < 1, μ > 0, θ < 1 − β/2 if θ 1, μ = 0.
(θβμ )
Suppose also that + 2 (h) h ∈ C 0 (R+ 0 ), 0 h(t) Ct ϕ (t) on R0 , for some C > 0, + + (g) g ∈ C 0 (R0 ), g(t) Cρ(t) on R0 , for some C > 0,
, then any entire classical and ρ satisfying (ρ). If (ρKO) holds with ω = θ in the definition of F weak solution u of the differential inequality (1.19) either non-positive or constant. Moreover, if u 0 and (0) > 0 then u ≡ 0. As already observed, (ϕ) is not satisfied by the mean curvature operator; however, a version of Theorem C can be given to handle this case, see Section 6 below. As mentioned earlier, in some circumstances (ρKO) is equivalent to (KO). This is the case, for instance, in the next
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Corollary C1. Let (M, , ) be as in Theorem C. Assume that (g), (F1 ), (L1 ), (L2 ), (ϕ), (θ ) and (b1 ) hold with ϕ(t) = t p−1 . Suppose also that
g+ (t) = max 0, g(t) ∈ L1 (+∞). If (KO) holds, then any entire classical weak solution u of p u b(x)f (u) |∇u| − g(u)|∇u|p is either non-positive or constant. Moreover, if u 0 and (0) > 0 then u ≡ 0. We conclude this introduction by observing that in the literature have recently appeared other methods to obtain Liouville-type results for differential inequalities such as (1.7) or (1.19). Among them we mention the important technique developed by E. Mitidieri and S.I. Pohozaev, see, e.g., [11], which proves to be very effective when the ambient space is Rm . Their method, which involves the use of cut-off functions in a non-local way, may be adapted to a curved ambient space, but is not suitable to deal with situations where the volume of balls grows superpolynomially. The paper is organized as follows: 1 2 3 4 5 6
Introduction. Comparison results. Proof of Theorem A and related results. A further version of Theorem A. The weak maximum principle and non-existence of solutions with controlled growth. Proof of Theorem C. In the sequel C will always denote a positive constant which may vary from line to line.
2. Comparison results In this section we consider the diffusion operator LD u =
1 div(D∇u), D
D ∈ C 2 (M), D > 0,
(2.1)
and denote by r(x) the distance from a fixed origin o in an m-dimensional complete Riemannian manifold (M, , ). The Riemannian metric and the weight D give rise to a metric measure space, with measure D dV , dV denoting the usual Riemannian volume element. For ease of notation in the sequel we will drop the index D and write LD = L. The purpose of this section is to collect the estimates for Lr and for the weighted volume of Riemannian balls, that will be used in the sequel. The estimates are derived assuming an upper bound for a family of modified Ricci tensors, which account for the mutual interactions of the geometry and the weight function. Although most of the material is available in the literature (see, e.g., D. Bakry and P. Emery [2], Bakry [1], A.G. Setti [24], Z. Qian [20], Bakry and Qian [3], J. Lott [10], X.-D. Li [8]), we
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675
are going to present a quick derivation of the estimates for completeness and the convenience of the reader. We note that our method is somewhat different from that of most of the above authors. In addition we will be able to derive weighted volume estimates under integral type conditions on the modified Bakry–Emery Ricci curvature, which extend to this setting results of S. Gallot [7], P. Petersen and G. Wei [14], and S. Pigola, M. Rigoli and A.G. Setti [18]. For n > m we let Ricc(L) and Riccn,m (L) denote the Bakry–Emery and the modified Bakry– Emery Ricci tensors defined in (1.9). The starting point of our considerations is the following version of the Bochner–Weitzenböck formula for the diffusion operator L. Lemma 2.1. Let u ∈ C 3 (M), then 1 L |∇u|2 = |Hess u|2 + ∇Lu, ∇u + Ricc(L)(∇u, ∇u). 2
(2.2)
Proof. It follows from the definition of L and the usual Bochner–Weitzenböck formula that L |∇u|2 = |∇u|2 + D −1 ∇D, ∇|∇u|2
= 2|Hess u|2 + 2 ∇u, ∇u + 2 Ricc(∇u, ∇u) + D −1 ∇D, ∇|∇u|2 .
Now computations show that D −1 ∇D, ∇|∇u|2 = 2D −1 Hess u(∇u, ∇D) and
∇u, ∇u = ∇ Lu − D −1 ∇D, ∇u , ∇u = ∇(Lu), ∇u + D −2 ∇u, ∇D 2 − D −1 Hess u(∇u, ∇D) − D −1 Hess D(∇u, ∇u), so that substituting yields the required conclusion.
2
Lemma 2.2. Let (M, , ) be a complete Riemannian manifold of dimension m. Let r(x) be the Riemannian distance function from a fixed reference point o, and denote with cut(o) the cut locus of o. Then for every n > m and x ∈ / {o} ∪ cut(o) 1 (Lr)2 + ∇Lr, ∇r + Riccn,m (∇r, ∇r) 0. n−1
(2.3)
Proof. We use u = r(x) in the generalized Bochner–Weitzenböck formula (2.2). Since Hess r(∇r, X) = 0 for every vector field X, by taking an orthonormal frame in the orthogonal complement of ∇r, and using the Cauchy–Schwarz inequality we see that |Hess r|2
1 (r)2 . m−1
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Using the elementary inequality (a − b)2
1 2 1 2 a − b , 1+
a, b ∈ R, > 0,
we estimate 2 (u)2 = Lu − D −1 ∇D, ∇u
1 1 (Lu)2 − D −2 ∇D, ∇u 2 . 1+
Now, the required conclusion follows substituting into (2.2), using |∇r| = 1, choosing in such a way that (1 + )(m − 1) = n − 1, and recalling the definition of Riccn,m . 2 We are now ready to prove the weighted Laplacian comparison theorem. Versions of this results have been obtained by Setti [24], for the case where n = m + 1 and later by Qian [20] in the general case where n > m (see also [3] which deals with the case where the drift term is not even assumed to be a gradient). We present a proof modeled on the proof of the Laplacian comparison theorem described in [16]. Proposition 2.3. Let (M, , ) be a complete Riemannian manifold of dimension m. Let r(x) be the Riemannian distance function from a fixed reference point o, and denote with cut(o) the cut locus of o. Assume that Riccn,m (∇r, ∇r) −(n − 1)G(r)
(2.4)
for some G ∈ C 0 ([0, +∞)), let h ∈ C 2 ([0, +∞)) be a solution of the problem
h − Gh 0, h(0) = 0, h (0) = 1,
(2.5)
and let (0, R), R +∞, be the maximal interval where h(r) > 0. Then for every x ∈ M we have r(x) R, and the inequality Lr(x) (n − 1)
h (r(x)) h(r(x))
(2.6)
holds pointwise in M \ (cut(o) ∪ {o}) and weakly on M. Proof. Next let x ∈ M \ (cut(o) ∪ {o}), let γ : [0, r(x)] → M be the unique minimizing geodesic parametrized by arc length joining o to x, and set ψ(s) = (Lr) ◦ γ (s). It follows from (2.3) and γ˙ = ∇r that d (Lr ◦ γ )(s) = ∇Lr, ∇r ◦ γ ds 1 − (Lr ◦ γ )(s)2 + (n − 1)G(s) n−1 on (0, r(x)). Moreover,
(2.7)
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(Lr ◦ γ )(s) =
m−1 + O(1) s
as s → 0+ ,
677
(2.8)
which follows from the fact that (Lr ◦ γ )(s) = r + D −1 ∇D, ∇r ◦ γ (s) and the second summand is bounded as s → 0+ , while, by standard estimates, r(x) =
m−1 + o(1). r(x)
Because of (2.8), we may set g(s) = s
m−1 n−1
s exp 0
(Lr ◦ γ )(t) m − 1 1 − dt , n−1 n−1 t
(2.9)
so that g is defined in [0, r(x)], g(s) > 0 in (0, r(x)), and it satisfies (n − 1)
g = Lr ◦ γ , g
g(0) = 0,
m−1 g(s) = s n−1 1 + o(1) as s → 0+ .
(2.10)
It follows from this and (2.7) that g satisfies the problem
g Gg, g(0) = 0,
m−n g (s) = s n−1 1 + o(1) as s → 0+ .
(2.11)
Recalling that, by assumption h satisfies (2.5), we now proceed as in the standard Sturm comparison theorems, and consider the function z(s) = h (s)g(s) − h(s)g (s). Then
g h − 0 z (s) = gh h g in the interval (0, τ ), τ = min{r(x), R}, where g is defined and h is positive. Also, it follows from the asymptotic behavior of g and h that m−1
h (s)g(s) s n−1 ,
h(s)g (s)
m − 1 m−1 s n−1 , n−1
so that z(s) → 0+ We conclude that z(s) 0 and therefore
as s → 0+ .
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g (s) h (s) g(s) h(s) in the interval (0, τ ). Integrating between and s, 0 < < s < τ , yields g(s)
g() h(s), h()
showing that h must be positive in (0, τ ), and therefore r(x) R. Since this holds for every x ∈ M we deduce that if R < +∞ then M is compact and diam(M) 2R. Moreover, in (0, r(x)) we have h g (Lr ) γ r(x) = (n − 1) r(x) (n − 1) r(x) . g h This shows that the inequality (2.6) holds pointwise in M \ (cut(o) ∪ {o}). The weak inequality now follows from standard arguments (see, e.g., [16], Lemma 2.2, [18], Lemma 2.5). 2 As in the standard Riemannian case, the estimate for Lr allows to obtain weighted volume comparison estimates (see [24,20,3,8]). Theorem 2.4. Let (M, , ) be as in the previous proposition, and assume that the modified Bakry–Emery Ricci tensor Riccn,m satisfies (2.4) for some G ∈ C 0 ([0, +∞)). Let h ∈ C 2 ([0, +∞)) be a solution of the problem (2.5), and let (0, R) be the maximal interval where h is positive. Then, the functions volD ∂Br (o) h(r)n−1
(2.12)
volD Br (o) r → r n−1 dt 0 h(t)
(2.13)
r → and
are non-increasing a.e., respectively non-increasing in (0, R). In particular, for every 0 < ro < R, there exists a constant C depending on D and on the geometry of M in Bro (o) such that m if 0 r ro , r (2.14) volD Br (o) C r n−1 dt if r r. h(t) o 0 Proof. By Proposition 2.3, inequality (2.6) holds weakly on M, so for every 0 ϕ ∈ Lipc (M), we have h (r(x)) D(x) dV . (2.15) − ∇r, ∇ϕ D(x) dV (n − 1) ϕ h(r(x)) For any ε > 0, consider the radial cut-off function
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679
−n+1 ϕε (x) = ρε r(x) h r(x) ,
(2.16)
where ρε is the piecewise linear function ⎧0 ⎪ ⎪ t−r ⎪ ⎪ ⎨ ε ρε (t) = 1 ⎪ ⎪ R−t ⎪ ⎪ ⎩ ε 0
if t ∈ [0, r), if t ∈ [r, r + ε), if t ∈ [r + ε, R − ε), if t ∈ [R − ε, R), if t ∈ [R, ∞).
(2.17)
Note that −n+1 h (r(x)) χR−ε,R χr,r+ε ∇ϕε = − + − (n − 1) ρε h r(x) ∇r, ε ε h(r(x)) for a.e. x ∈ M, where χs,t is the characteristic function of the annulus Bt (o) \ Bs (o). Therefore, using ϕε into (2.15) and simplifying, we get
1 ε
−n+1 1 h r(x) ε
BR (o)\BR−ε (o)
−n+1 h r(x) .
Br+ε (o)\Br (o)
Using the co-area formula we deduce that 1 ε
R
−n+1
vol ∂Bt (o)h(t) R−ε
1 ε
r+ε vol ∂Bt (o)h(t)−n+1 r
and, letting ε 0, volD ∂BR (o) volD ∂Br (o) h(R)m−1 h(r)m−1
(2.18)
for a.e. 0 < r < R. The second statement follows from the first and the co-area formula, since, as noted by M. Gromov (see [4]), for general real valued functions f (t) 0, g(t) > 0, t f f (t) if t → is decreasing, then t → 0t is decreasing. g(t) 0g
2
We next consider the situation where the modified Bakry–Emery Ricci curvature satisfies some Lp -integrability conditions and extends results obtained in [18] for the Riemannian volume which in turn slightly generalize previous results by P. Petersen and G. Wei [14] (see also [7] and [9]). Since we will be interested in the case the underlying manifold is non-compact, we assume that G is a non-negative, continuous function on [0, +∞) and that h(t) ∈ C 2 ([0, +∞)) is the solution of the problem
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h (t) − G(t)h(t) = 0, h(0) = 0, h (0) = 1.
The assumption that G 0 implies that h 1 on [0, +∞) and therefore h > 0 on (0, +∞). For ease of notation, in the course of the arguments that follow we set r AG,n (r) = h(r)
n−1
and VG,n (r) =
h(t)n−1 dt
(2.19)
0
so that AG,n (r) and VG,n (r) are multiples of the measures of the sphere and of the ball of radius r centered at the pole in the n-dimensional model manifold MG with radial Ricci curvature equal to −(n − 1)G. Using an exhaustion of Eo = M \ cut(o) by means of starlike domains one shows (see, e.g., [18], p. 35) that for every non-negative test function ϕ ∈ Lipc (M),
∇r, ∇ϕ D dV
− M
(2.20)
ϕLrD dV . Eo
We outline the argument for the convenience of the reader. Let Ωn be such an exhaustion of Eo , so that, if νn denotes the outward unit normal to ∂Ωn , then νn , ∇r 0. Integrating by parts shows that − ∇r, ∇ϕ D dV = − lim ∇r, ∇ϕ D dV n
M
Ωn
1 = lim ϕ r + ∇D, ∇r D dV − ϕ ∇r, νn D dσ n D Ωn
lim n
Ωn
∂Ωn
ϕLD rD dV =
ϕLD rD dV , Eo
where the inequality follows from ∇r, νn 0, and the limit on the last line exists because, by Proposition 2.3, Lr is bounded above by some positive integrable function g on the relatively compact set Eo ∩ supp ϕ (namely, if Riccm,n −(n − 1)H 2 on Eo ∩ supp ϕ for some H > 0, we can choose g = H coth(H r)). Applying the above inequality to the test function −n+1 , ϕ (x) = ρ r(x) h r(x) already considered in (2.16), arguing as in the proof of Theorem 2.4, and using the fact that AG,n (r) = h(r)n−1 is non-decreasing, we deduce that for a.e. 0 < r < R 1 volD ∂BR volD ∂Br − AG,n (R) AG,n (r) AG,n (r)
ψD dV , BR \Br
(2.21)
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where we have set h (r(x)) ψ(x) = max{0, Lr(x) − (n − 1) h(r(x)) } if x ∈ Eo , 0 if x ∈ cut(o).
(2.22)
Note by virtue of the asymptotic behavior of Lr and h / h as r(x) → 0, ψ vanishes in a neighborhood of o. Moreover, if Riccn,m (∇r, ∇r) −(n − 1)G(r(x)), then, by the weighted Laplacian comparison theorem, ψ(x) ≡ 0, and we recover the fact that the function r→
volD ∂Br AG,n (r)
(2.23)
is non-increasing for a.e. r. Using the co-area formula, inserting (2.21), and applying Hölder inequality with exponents 2p and 2p/(2p − 1) to the right-hand side of the resulting inequality we conclude that
VG,n (R) volD ∂BR − AG,n (R) volD BR d volD BR (o) = dR VG,n (R) VG,n (R)2 −2
R
= VG (R)
AG,n (r) vol ∂BR − AG,n (R) volD ∂Br dr
0
1/2p volD BR 1−1/2p RAG,n (R) 2p ψ D dV . VG,n (R)1+1/2p VG,n (R)
(2.24)
BR
Now we define
ρ(x) = − min 0, Riccn,m (∇r, ∇r) + (n − 1)G r(x) = Riccn,m (∇r, ∇r) + (n − 1)G r(x) − .
(2.25)
We will need to estimate the integral on the right-hand side of (2.24) in terms of ρ. This is achieved in the following lemma, which is a minor modification of [14], Lemma 2.2, and [18], Lemma 2.19. Lemma 2.5. For every p > n/2 there exists a constant C = C(n, p) such that for every R 2p ψ D dV C ρ p D dV , BR
BR
with ρ(x) defined in (2.25). Proof. Integrating in polar geodesic coordinates we have
f D dV =
BR
S m−1
min{R,c(θ)}
dθ
f (tθ )(Dω)(tθ ) dt, 0
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where ω is the volume density with respect to Lebesgue measure dt dθ , and c(θ ) is the distance from o to the cut locus along the ray t → tθ . It follows that it suffices to prove that for every θ ∈ S m−1 min{R,c(θ)}
ψ
min{R,c(θ)} 2p
(tθ )(Dω)(tθ ) dt C
0
ρ p (tθ )(Dω)(tθ ) dt.
(2.26)
0
An easy computation which uses (2.7) yields 2 ∂ h (Lr)2 h h . Lr − (n − 1) − − Riccn,m (∇r, ∇r) − (n − 1) − ∂t h n−1 h h Thus, recalling the definitions of ψ and ρ, we deduce that the locally Lipschitz function ψ satisfies the differential inequality h ψ2 + 2 ψ ρ, n−1 h
ψ +
on the set where ρ > 0 and a.e. on (0, +∞). Multiplying through by ψ 2p−2 Dω, and integrating we obtain r
ψψ
2p−2
0
r 1 h 2p−1 2p ψ +2 ψ + Dω ρψ 2p−2 Dω. n−1 h 0
On the other hand, integrating by parts, and recalling that (Dω)−1 ∂(Dω)/∂t = Lr ψ + (n − 1)
h h
and that ψ(tθ ) = 0 if t c(θ ), yield r
ψψ 0
2p−2
1 1 ψ(r)2p−1 (Dω)(rθ ) − ω= 2p − 1 2p − 1 −
1 2p − 1
r ψ 2p−1
r ψ 2p−1 LrDω 0
h ψ + (n − 1) Dω. h
0
Substituting this into (2.27), and using Hölder inequality we obtain
1 1 − n − 1 2p − 1
r ψ 0
2p
r h n−1 Dω + 2 − ψ 2p−1 Dω 2p − 1 h 0
(2.27)
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683
r
ρψ 2p−2 Dω 0
r
1/p r p
ρ Dω
(p−1)/p ψ
0
2p
Dω
,
0
and, since the coefficient of the first integral on the left-hand side is positive, by the assumption on p, while the second summand is non-negative, rearranging and simplifying we conclude that (2.26) holds with
C(n, p) =
1 1 − n − 1 2p − 1
−p
2
.
We are now ready to state the announced weighted volume comparison theorem under assumptions on the Lp -norm of the modified Bakry–Emery Ricci curvature. Theorem 2.6. Keeping the notation introduced above, let p > n/2 and let 1/2p
f (t) =
Cn,p tAG,n (t) VG,n (t)1+1/2p
1/2p
ρ p D dV
(2.28)
,
Bt
where Cn,p is the constant in Lemma 2.5. Then for every 0 < r < R,
volD BR (o) VG,n (R)
1/2p
−
volD Br (o) VG,n (r)
1/2p
1 2p
R (2.29)
f (t) dt. r
Moreover for every ro > 0 there exists a constant Cro such that, for every R ro 2p R volD BR (o) 1 Cr o + f (t) dt , VG,n (R) 2p
(2.30)
ro
and 2p R volD ∂BR (o) 1 Cr o + f (t) dt AG,n (R) 2p ro
R + VG,n (R)1/2p
1/2p
ρ BR
p
1 Cr o + 2p
2p−1
R f (t) dt ro
.
(2.31)
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Proof. Set volD Br (o) . VG,n (r)
y(r) =
According to (2.24), Lemma 2.5 and (2.28) we have
y (t) f (t)y(t)1−1/2p , y(t) ∼ cm t m−n as t → 0+ ,
y(t) > 0 if t > 0,
whence, integrating between r and R we obtain
1/2p
y(R)
− y(r)
1/2p
1 2p
R f (t) dt, r
that is, (2.29), and (2.30) follows at one with Cro = ( to (2.24) and Lemma 2.5, volD ∂BR volD BR R + AG,n (R) VG,n (R) VG,n (R)1/2p
volD Bro (o) 1/2p . On the other hand, according VG,n (ro ) )
1/2p
ρ p D dV
volD BR VG,n (R)
1−1/2p
Bt
2
and the conclusion follows inserting (2.30).
Keeping the notation introduced above, assume, for instance, that G = B 2 0, so that n−1 AG,n (t) = t −1 (B sinh Bt)n−1
if B = 0, if B > 0
and suppose that ρ = Riccn,m + (n − 1)B 2 − ∈ Lp (M, D dV ), for some p > n/2. Then, arguing as in the proof of [18], Corollary 2.21, we deduce that for every ro sufficiently small there exist constants C1 and C2 , depending on ro , B, m, p and on the Lp (M, D dV )-norm of ρ, such that, for every R ro ,
2p volD BR C1 R(n−1)BR e
if B = 0, if B > 0
and
2p−1 volD ∂BR C2 R(n−1)BR e
if B = 0, if B > 0.
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3. Proof of Theorem A and further results The aim of this section is to give a proof of a somewhat stronger form of Theorem A (see Theorem 3.5 below), together with a version of the result valid when (KO) fails. The idea of proof of Theorem A is to construct a function v(x) defined on an annular region BR¯ \ Bro , with 0 < ro < R¯ sufficiently large, with the following properties: for fixed ro < r1 < R¯ and 0 < < η
v(x) = v(x) η v(x) → +∞
on ∂Bro , on Br1 \ Bro , as r(x) → +∞,
(3.1)
and v is a weak super-solution on BR¯ \ Bro of LD,ϕ w = b(x)f (w) |∇w| .
(3.2)
This is achieved by taking v of the form v(x) = α r(x) ,
(3.3)
where α is a suitable super-solution of the radialized inequality (3.2), whose construction depends in a crucial way on the validity of the Keller–Osserman condition (KO). The conclusion is then reached comparing v with the solution of (1.7). To this end, we will extend a comparison technique first introduced in [15]. Finally, in Theorem 3.6 below we will consider the case where the Keller–Osserman condition fails, that is, 1 ∈ / L1 (+∞). K −1 (F (t))
(3.4)
Its proof is based on a modification of the previous arguments and uses (3.4) in a way which is, in some sense, dual to the use of (KO) in the proof of Theorem A. We begin with the following simple Lemma 3.1. Assume that f , and ϕ satisfy the assumptions (F1 ), (L1 ) and (ϕ)2 , and let σ > 0. Then (KO) holds if and only if 1 ∈ L1 (+∞). K −1 (σ F (s))
(KOσ )
Proof. We consider first the case 0 < σ 1. Since K −1 is non-decreasing, +∞
ds K −1 (F (s))
+∞
1 . K −1 (σ F (s))
On the other hand, if C 1 is such that supst f (s) Cf (t), then, for every 0 < σ 1, f (Cσ −1 t) C −1 f (t) and
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Ct F σ
Ct
σ =
C f (z) dz = σ
0
t
Cξ f σ
1 dξ σ
0
t f (ξ ) dξ =
1 F (t), σ
0
so, using the monotonicity of K −1 , we obtain +∞
C ds = −1 K (σ F (s)) σ
+∞
C Ct −1 K (σ F ( σ )) σ dt
+∞
dt K −1 (F (t))
,
showing that (KO) and (KOσ ) are equivalent in the case σ 1. Consider now the case σ > 1, and set fσ = σf , Fσ = σ F . Since (KOσ ) is precisely (KO) for Fσ , and since σ −1 1, by what we have just proved it is equivalent to 1 1 = ∈ L1 (+∞), K −1 (σ −1 Fσ (s)) K −1 (F (s)) as required.
2
We note for future use that the conclusion of the lemma depends only on the monotonicity of K −1 and the C-monotonicity of f . Before proceeding toward our main result we would like to explore the mutual connections between (θ ) and (ϕ). To simplify the writing, with the statement “(θ )1 holds” we will mean that the first half of condition (θ ) is valid. Proposition 3.2. Assume that conditions (Φ0 ) and (L1 ) hold. Then (θ )1 with θ < 2 implies (ϕ)2 , and (θ )2 with θ < 1 implies (ϕ)1 . As a consequence, (θ ) with θ < 1 implies (ϕ). Proof. Assume (θ )1 , that is, the function t → exists C 1 such that 0 < sθ
ϕ (t) θ (t) t
ϕ (t) ϕ (st) C (st) (t)
is C-increasing on R+ . By definition there
∀t ∈ R+ , s ∈ (0, 1],
or, equivalently, sθ
ϕ (st) ϕ (t) C −1 (st) (t)
∀t ∈ R+ , s ∈ [1, +∞).
(s) Letting t = 1, we deduce that if θ < 2 then sϕ(s) ∈ L1 (0+ ) \ L1 (+∞), which is (ϕ)2 . In an entirely similar way, if (θ )2 holds, that is,
ϕ(t) θ−1 ϕ(st) (st)θ−1 C (t) (st) (t) ∞ and θ < 1, then s θ−1 ϕ(s) (s) ∈ L ((0, 1)), and
∀t ∈ R+ , s ∈ (0, 1],
(3.5)
L. Mari et al. / Journal of Functional Analysis 258 (2010) 665–712
lim
s→0+
which implies (ϕ)1 .
687
ϕ(s) = 0, (s)
2
Remark 3.1. Note that the above argument also shows that if (θ )2 holds with θ < 2 then L1 (0+ ) \ L1 (+∞).
ϕ(t) (t)
∈
Proposition 3.3. Assume that conditions (Φ0 ) and (L1 ) hold, and let F be a positive function defined on R+ 0 . If (θ )1 holds with θ < 2, then there exists a constant B 1 such that, for every σ 1 we have B σ 1/(2−θ) K −1 (σ F (t)) K −1 (F (t))
on R+ .
(3.6)
Proof. Observe first of all that according to Proposition 3.2, (θ )1 with θ < 2 implies (ϕ)2 , so that K −1 is well defined on R+ 0. Changing variables in the definition of K, and using (3.5) above, for every λ 1 and t ∈ R+ , we have λt K(λt) =
ϕ (s) ds = λ2 s (s)
0
C −1 λ2−θ
t s
ϕ (λs) ds (λs)
0
t s
ϕ (s) ds = C −1 λ2−θ K(t), (s)
0
where C 1 is the constant in (θ )1 . Applying K −1 to both sides of the above inequality, and setting t = K −1 (σ F (s)) we deduce that λK −1 σ F (s) K −1 λ2−θ σ C −1 F (s) , whence, setting λ = (C/σ )1/(2−θ) 1, the required conclusion follows with B = C 1/(2−θ) .
2
Remark 3.2. We note for future use that the estimate holds for any positive function F on R+ , without any monotonicity property, and it depends only on the fact that the integrand ψ(s) = sϕ (s)/(s) in the definition of K satisfies the C-monotonicity property ψ(λs) C −1 λ1−θ ψ(s)
∀s ∈ R+ , ∀λ 1.
In order to state the next proposition we introduce the following assumption ˜ ∈ C 1 (R+ ), b(t) ˜ > 0, b˜ (t) 0 for t 1, and b˜ λ ∈ (b) b(t) / L1 (+∞) for some λ > 0. 0 Proposition 3.4. Assume that conditions (Φ0 ), (F1 ), (L1 ), (L2 ) (ϕ)1 , (θ ), (KO) hold, and let b˜ be a function satisfying assumption (b), A > 0, and β ∈ [−2, +∞). If λ and θ are the constants specified in (b) and (θ ), assume also that
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λ(2 − θ ) 1 and either (i) t
β/2 ˜
t λ(1−θ)−1
b(t)
˜ λ ds C b(s)
for t t0 ,
or
1
(ii) t
β/2 ˜
λ(1−θ)−1
b(t)
C
for t t0 and θ < 1.
(3.7)
Then there exists T > 0 sufficiently large such that, for every T t0 < t1 and 0 < < η, there exist T¯ > t1 and a C 2 function α : [t0 , T¯ ) → [, +∞) which is a solution of the problem
˜ (α)(α) on [t0 , T¯ ), ϕ α α + At β/2 ϕ α b(t)f α > 0 on [t0 , T¯ ), α(t0 ) = , α(t) → +∞ as t → T¯ −
(3.8)
and satisfies on [t0 , t1 ].
αη
(3.9)
Proof. Note first of all, that the first condition in (3.7) forces θ < 2, and (ϕ)2 follows from (θ )1 . ˜ > 0 and b˜ (t) 0 on [T , +∞). Since (b) and We choose T > 0 large enough that, by (b), b(t) ˜ we may assume without loss of generality that b˜ 1 on (3.7) are invariant under scaling of b, [T , ∞). Let t0 , t1 , , η be as in the statement of the proposition, and, for a given σ ∈ (0, 1], set +∞ Cσ =
ds K −1 (σ F (s))
(3.10)
,
˜ ∈ which is well defined in view of (KO) and Lemma 3.1. Since b(t) / L1 (+∞), there exists Tσ > t0 such that Tσ Cσ =
˜ λ ds. b(s)
t0
We note that, by monotone convergence, Cσ → +∞ as σ → 0+ , and we may therefore choose σ > 0 small enough that Tσ > t1 . We let α : [t0 , Tσ ) → [, +∞) be implicitly defined by the equation Tσ
˜ λ ds = b(s)
t
∞
ds K −1 (σ F (s))
,
α(t)
so that, by definition, α(t0 ) = ,
α(t) → +∞
as t → Tσ− .
(3.11)
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689
Differentiating (3.11) yields ˜ λ K −1 σ F α(t) , α (t) = b(t)
(3.12)
so that α > 0 on [t0 , Tσ ), and σ F (α) = K α /b˜ λ . Differentiating once more, using the definition of K and (3.12), we obtain
α ϕ (α /b˜ λ ) α . σf (α)α = K α /b˜ λ α /b˜ λ = b˜ λ (α /b˜ λ ) b˜ λ
(3.13)
Since f (t) > 0 on (0, ∞), α > 0 and b˜ 0, we have (α /b˜ λ ) 0 and α /b˜ λ is non-decreasing. Moreover,
α b˜ λ
= α /b˜ λ − λ α b˜ /b˜ λ+1 α /b˜ λ .
Inserting this into (3.13), using the fact that b˜ −λ 1 and (θ )1 (in the form of (3.5)), and rearranging we obtain
˜ (α) α on [t0 , Tσ ). ϕ α α Cσ b˜ λ(2−θ) bf
(3.14)
In order to estimate the term At β/2 ϕ(α ) we rewrite (3.13) in the form ϕ α /b˜ λ α /b˜ λ = σ b˜ λ f (α) α /b˜ λ
on [t0 , Tσ ),
integrate between t0 and t ∈ (t0 , Tσ ), use the fact that α and α/b˜ λ are increasing, and f and are C-increasing to deduce that ϕ α /b˜ λ ϕ α /b˜ λ (t0 ) + Cσf (α) α /b˜ λ
t
˜ λ ds, b(s)
t0
for some constant C 1. On the other hand, since t θ−1 ϕ(t)/(t) is C-increasing and b˜ 1, we have ˜λ ϕ(α ) ˜ λ(1−θ) ϕ(α /b ) C b (α ) (α /b˜ λ ) t /b˜ λ )(t ) ϕ(α 0 λ(1−θ) λ ˜ + σf (α) b(s) C b˜ (α /b˜ λ ) t0
˜ λ(1−θ)−1
Cb
ϕ(α /b˜ λ )(t0 ) +σ f ()(α /b˜ λ )(t0 )
t
˜ b(s)
λ
t0
˜ (α), bf
(3.15)
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where the second inequality follows from the fact that α and α /b˜ λ are increasing, and f and are C-increasing. Using (3.14) and (3.15), and recalling that, by (3.12), (α /b˜ λ )(t0 ) = K −1 (σ F ()), we obtain ˜ (α) α , ϕ α α + At β/2 ϕ α Nσ (t)bf
(3.16)
where Nσ (t) = Cσ b˜ λ(2−θ)−1 + ACt β/2 b˜ λ(1−θ)−1 + ACσ t β/2 b˜ λ(1−θ)−1
t
ϕ(K −1 (σ F ())) (K −1 (σ F ()))f ()
˜ λ = (I )(t) + (II)(t) + (III)(t). b(s)
(3.17)
t0
Since b˜ 1, and λ(2 − θ ) − 1 0 by (3.7), we see that (I )(t) → 0 uniformly on [t0 , +∞) as σ → 0. As for (II), according to (3.7) t β/2 b˜ λ(1−θ)−1 C
on [t0 , +∞),
so that, using (φ)1 , we deduce that lim inf σ →0+
−1 (σ F ())) ϕ(K = 0. −1 (σ F ())) f ()(K
Thus (II)(t) → 0 uniformly on [t0 , +∞) along a sequence σk → 0. It remains to analyze (III). Clearly, if (3.7)(i) holds, then (III)(t) → 0 uniformly on [t0 , +∞) as σ → 0. Assume therefore that (3.7)(ii) holds, so that t (III)(t) ACσ
˜ λ ds. b(s)
t0
By the definition of α(t), Proposition 3.3, and (KO) t
˜ ds = b(s)
α(t)
λ
t0
ds K −1 (σ F (s))
Bσ −1/(2−θ)
+∞
ds Cσ −1/(2−θ) K −1 (F (s))
(3.18)
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691
in [t0 , Tσ ). Since θ < 1 we conclude that (III)(t) Cσ 1−1/(2−θ) → 0
uniformly in [t0 , Tσ ) as σ → 0.
Putting together the above estimates, we conclude that we can choose σ small enough that Nσ (t) 1, showing that α(t) satisfies the differential inequality in (3.8). In order to complete the proof we only need to prove that α(t) η for t0 t t1 . Again from the definition of α we have t1
˜ ds = b(s)
α(t 1)
λ
t0
ds K −1 (σ F (s))
,
so if we choose σ ∈ (0, 1] small enough to have t1
˜ ds b(s)
η
λ
t0
ds K −1 (σ F (s))
,
then clearly α(t1 ) η, and, since α is increasing, this finishes the proof.
2
We are now ready to prove Theorem 3.5. Let (M, , ) be a complete Riemannian manifold satisfying β/2 Riccn,m (LD ) H 2 1 + r 2 ,
(1.12)
for some n > m, H > 0 and β −2 and assume that (Φ0 ), (F1 ), (L1 ), (L2 ), (ϕ)1 , and (θ ) hold. Let b(x) ∈ C 0 (M), b(x) 0 on M and suppose that b(x) b˜ r(x)
for r(x) 1,
(3.19)
where b˜ satisfies assumption (b) and (3.7). If the Keller–Osserman condition 1 K −1 (F (t))
∈ L1 (+∞)
(KO)
holds then any entire classical weak solution u of the differential inequality LD,ϕ u b(x)f (u) |∇u|
(1.7)
is either non-positive or constant. Furthermore, if u 0, and (0) > 0, then u vanishes identically.
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Proof. If u 0 then there is nothing to prove. We argue by contradiction and assume that u is non-constant and positive somewhere. We choose T > 0 sufficiently large that (3.19) holds in M \ BT and for every ro T we have 0 < u∗o = sup u u∗ = sup u. Bro
M
We consider first the case where u∗ < +∞. We claim that u∗o < u∗ . Otherwise there would exist xo ∈ B ro such that u(xo ) = u∗ , and by (1.7) and assumptions (F1 ) and (1 ), LD,ϕ u 0 in the connected component Ωo of {u 0} containing xo . By the strong maximum principle [19], u would then be constant and positive on Ωo . Since u = 0 on ∂Ωo this would imply that Ωo = M and u is a positive constant on M, contradicting our assumption. / B ro satisfying u(x) ˜ > u∗ − η. Next, we choose η > 0 small enough that u∗o + 2η < u∗ and x˜ ∈ ˜ Because of (1.12), Proposition 2.3 and [18], Proposition 2.11, there We let t0 = ro and t1 = r(x). exists A = A(T ) > 0 such that LD r Ar β/2
on M \ BT .
According to Proposition 3.4 there exist T¯ > t1 and a C 2 function α : [t0 , T¯ ) → [, +∞) which satisfies
˜ ϕ α α + At β/2 ϕ α (2C)−1 b(t)f (α)(α) on [t0 , T¯ ), α > 0 on [t0 , T¯ ), α(t0 ) = , α(t) → +∞ as t → T¯ −
and αη
on [t0 , t1 ],
where C is the constant in the definition of C-monotonicity of f. It follows that the radial function defined on BR¯ \ Bro by v(x) = α(r(x)) satisfies the differential inequality LD,ϕ v (2C)−1 b(x) f (α) α r(x)
(3.20)
pointwise in (BR¯ \ B ro ) \ cut(o) and weakly in BR¯ \ B ro . Furthermore v satisfies (3.1), and u(x) ˜ − v(x) ˜ > u∗ − 2η. Since u(x) − v(x) u∗o − < u∗ − 2η − and
on ∂Bro
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u(x) − v(x) → −∞
693
as x → ∂BR¯ ,
we deduce that the function u − v attains a positive maximum μ in BR¯ \ B ro . We denote by Γμ a connected component of the set
x ∈ BR¯ \ B ro : u(x) − v(x) = μ and note that Γμ is compact. We claim that for every y ∈ Γμ we have u(y) > v(y),
∇u(y) = α r(y) .
(3.21)
Indeed, this is obvious if y is not in the cut locus cut(o) of o, for then ∇u(y) = ∇v(y) = α (r(y))∇r(y). On the other hand, if y ∈ cut(o), let γ be a unit speed minimizing geodesic joining o to y, let o = γ () and let r (x) = d(x, o ). By the triangle inequality, r(x) r (x) +
∀x ∈ M,
with equality if and only if x lies on the portion of the geodesic γ between o and y (recall that γ ceases to be minimizing past y). Define v (x) = α( + r (x)), then, since α is strictly increasing, v (x) v(x) with equality if and only if x lies on the portion of γ between o and y. We conclude that ∀x ∈ BR \ B ro , (u − v )(y) = (u − v)(y) (u − v)(x) (u − v )(x), and u − v attains a maximum at y. Since y is not on the cut locus of o , v is smooth there, and ∇u(y) = ∇v (y) = α + r (y) ∇r (y) = α r(y) , as claimed. Since f is C-increasing, 1 b(y)f u(ξ ) |∇u|(y) b(y)f v(y) α r(y) C and by continuity the inequality 1 b(x)f (u) |∇u| b(x)f v(x) α r(x) 2C holds in a neighborhood of y. It follows from this and the differential inequalities satisfied by u and v that LD,ϕ u LD,ϕ v
(3.22)
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weakly in a sufficiently small neighborhood U of Γμ . Now fix y ∈ Γμ and for ζ ∈ (0, μ) let Ωy,ζ be the connected component containing y of the set
x ∈ BR¯ \ B ro : u(x) > v(x) + ζ . By choosing ζ sufficiently close to μ we may arrange that Ω y,ζ ⊂ U, and, since u = v + ζ on ∂Ωy,ζ , (3.22) and the weak comparison principle (see, e.g., [16], Proposition 6.1) implies that u v + ζ on Ωy,ζ , contradicting the fact that y ∈ Ωy,ζ . The case where u∗ = +∞ is easier, and left to the reader. 2 ˜ Remark 3.3. Theorem A is a special case of Theorem 3.5 with the choice b(r) = C/r μ for r 1. Assume first that μ > 0. Choosing λ = 1/μ, it follows that
t
β/2 ˜
λ(1−θ)−1
b(t)
= O t θ−1+β/2+μ
t and
˜ λ ds = O(log t). b(s)
1
Then (θβμ) (and β −2) implies first that λ(2 − θ ) − 1 μ−1 (1 + β/2) 0, and then that either (i) or (ii) in (3.7) holds. Thus Theorem 3.5 applies. On the other hand, if μ = 0 and θ < 1 − β/2, then θ < 1 − β/2 − μo for sufficiently small μo > 0, and the conclusion follows from the previous case. The next example shows that the validity of the generalized Keller–Osserman condition (KO) is indeed necessary for Theorem 3.5 to hold. Since (KO) is independent of geometry, we consider the most convenient setting where (M, , ) is Rm with its canonical flat metric. We further simplify our analysis by considering the differential inequality p u f (u) |∇u| ,
(3.23)
for the p-Laplacian p , where f is increasing and satisfies f (0) = 0, f (t) > 0 for t > 0, is + non-decreasing and satisfies (L1 ), and (ϕ) and (θ ) hold. We let K : R+ 0 → R0 be defined as in (1.11), and assume that 1 K −1 (F (t))
∈ / L1 (+∞).
(¬KO)
Define implicitly the function w on R+ 0 by setting w(t)
t=
ds . K −1 (F (s))
(3.24)
1
Note that w is well defined, w(0) = 1, and (¬KO) implies that w(t) → +∞ as t → ∞. Differentiating (3.24) yields w = K −1 F w(t) > 0,
(3.25)
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and a further differentiation gives
w
p−2
w =
1 f (w) |∇w| . p−1
(3.26)
We fix t¯ > 0 to be specified later, and let u1 (x) be the radial function defined on Rm \ Bt¯ by the formula u1 (x) = w |x| . Using (3.25) and (3.26) we conclude that u1 satisfies p−2 m − 1 p−1 w p u1 = (p − 1) w w + f (u1 ) |∇u1 | |x|
(3.27)
on Rm \ B t¯. Next we fix constants βo , Λ > 0, and, denoting with p the conjugate exponent of p, we let β(t) =
Λ p t + βo . p
Noting that β (0) = 0, we deduce that the function u2 (x) = β |x| is C 1 on Rm , and an easy calculation shows that p u2 = Λp−1 div |x|x = mΛp−1 .
(3.28)
Since β 0, and f and are monotonic, it follows that, if mΛp−1 f β(t¯) β (t¯) ,
(3.29)
then p u2 f (u2 ) |∇u2 |
on Bt¯.
(3.30)
The point now is to join u1 and u2 in such a way that the resulting function u is a classical C 1 weak sub-solution of p u = f (u) |∇u| . This is achieved provided we may choose the parameters t¯, Λ, βo , in such a way that (3.29) and
β(t¯) = w(t¯), β (t¯) = w (t¯)
(3.31)
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are satisfied. Towards this end, we define t¯ =
λ
ds > 0, K −1 (F (s))
(3.32)
1
where 1 < λ 2. Note that, by definition, w(t¯) = λ, and, by the monotonicity of K −1 and F λ−1 K −1 (F (2))
t¯
λ−1 K −1 (F (1))
(3.33)
,
so that, in particular, t¯ → 0 as λ → 1+ . Putting together (3.29) and (3.31) and recalling the relevant definitions we need to show the following system of inequalities ⎧ −1 F (λ) t¯/p + β = λ, ⎪ o ⎨ (i) K −1 p −1 (ii) Λt¯ F (λ) , =K ⎪ ⎩ (iii) mΛp−1 f (λ) K −1 F (λ) .
(3.34)
Since, by (3.33), t¯ 1 K −1 (F (2)) (λ − 1) K −1 F (λ) −1 p p K (F (1)) for λ sufficiently close to 1 the first summand on the left-hand side of (i) is strictly less that 1, and therefore we may choose βo > 0 in such a way that (i) holds. Next we let Λ be defined by (ii), and note, that Λ = K −1 F (λ) t¯1−p K −1 F (1) → +∞
as λ → 1+ .
Therefore, since f (λ) K −1 F (Λ) f (2) K −1 F (2) , if λ is close enough to 1 then (iii) is also satisfied. Summing up, if λ is sufficiently close to 1, the function u(x) =
u1 (x) u2 (x)
on Rm \ Bt¯, on Bt¯
(3.35)
is a classical weak solution of (3.23). We remark that we may easily arrange that assumptions (ϕ) and (θ ) are also satisfied. Indeed, if we choose, for instance, (t) = t q with q 0, then, as already noted in the Introduction, () holds for every p > 1 + q and (θ ) is verified for every θ ∈ R such that p 2 + q − θ. We also stress that the solution u of (3.23) just constructed is positive and diverges at infinity. Indeed the method used in the proof of Theorem 3.5 may be adapted to yield non-existence of non-constant, non-negative bounded solutions even when (¬KO) holds. This is the content of the next
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Theorem 3.6. Maintain notation and assumptions of Theorem 3.5, except for (KO) which is replaced by (¬KO). Then any non-negative, bounded, entire classical weak solution u of the differential inequality (1.7) is constant. Furthermore, if (0) > 0, then u is identically zero. The proof of the theorem follows the lines of that of Theorem 3.5 once we prove the following Proposition 3.7. In the assumptions of Proposition 3.4, with (KO) replaced by (¬KO), there exists T > 0 large enough that for every T t0 < t1 , and 0 < < η, there exists a C 2 function α : [t0 , +∞) → [, +∞) which solves the problem
˜ (α)(α) on [t0 , T¯ ), ϕ α α + At β/2 ϕ α b(t)f α > 0 on [t0 , T¯ ), α(t0 ) = , α(t) → +∞ as t → +∞
(3.36)
and satisfies αη
on [t0 , t1 ].
(3.37)
Proof. The argument is similar to that of Proposition 3.4. The main difference is in the definition of α which now proceeds as follows. We fix T > 0 large enough that (b) holds on [T¯ , +∞). For t0 , t1 , , η as in the statement, and σ ∈ (0, 1] we implicitly define α : [t0 , +∞) → [, +∞) by setting t
˜ ds = b(s)
α(t)
λ
t0
ds , K −1 (σ F (s))
so that α(t0 ) = , and, by (b) and (¬KO), α(t) → +∞ as t → +∞. The rest of the proof proceeds as in Proposition 3.4. 2 Summarizing, the differential inequality (1.7) may admit non-constant, non-negative entire classical weak solutions only if (¬KO) holds, and possible solutions are necessarily unbounded. We shall address this case in Section 5. 4. A further version of Theorem A As mentioned in the Introduction, condition (ϕ) fails, for instance, when ϕ is of the form t ϕ(t) = √ 1 + t2 which, when D(x) ≡ 1, corresponds to the mean curvature operator. Because of the importance of this operator, in Geometry as well as in Analysis, it is desirable to have a version of Theorem A valid when (ϕ)2 fails. To deal with this situation we consider an alternative form of the Keller– Osserman condition, and correspondingly, modify our set of assumptions. We therefore replace assumption (ϕ)2 with
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(Φ2 ) There exists C > 0 such that ϕ(t) Ctϕ (t) on R+ . 1 + 1 (ϕ)3 ϕ(t) (t) ∈ L (0 ) \ L (+∞). As noted in Remark 3.1, (ϕ)3 is implied by (θ )2 with θ < 2. It is easy to verify that in the case of the mean curvature operator, tϕ (t) =
t ϕ(t) (1 + t 2 )3/2
and ϕ(t) ∼
t as t → 0+ , 1 as t → +∞,
/ L1 (+∞). By conso that (Φ2 ) holds, and (ϕ)3 is satisfied provided t−1 ∈ L1 (0+ ) and −1 ∈ trast, the choice 2
ϕ(t) = tet , corresponding to the operator of exponentially harmonic functions, does not satisfy (Φ2 ). by According to (ϕ)3 , we may define a function K = K(t)
t
ϕ(s) ds (s)
(4.1)
0 1 which is well defined on R+ 0 , tends to +∞ as t → +∞ and therefore gives rise to a C -diffeo+ morphism of R0 onto itself. The variant of the generalized Keller–Osserman condition mentioned above is then
1 ∈ L1 (+∞). −1 K (F (t))
( KO)
Analogues of Lemma 3.1, Propositions 3.3 and 3.4 are also valid in this setting. Lemma 4.1. Assume that f , and ϕ satisfy the assumptions (F1 ), (L1 ) and (ϕ)3 , and let σ > 0. Then ( KO) holds if and only if 1 −1 (σ F (s)) K
∈ L1 (+∞).
( KOσ )
Indeed, the proof of Lemma 3.1 depends only on the monotonicity of K and the C-monoton icity of f , and can be repeated without change replacing K with K. Similarly, using Remark 3.2, one establishes the following Proposition 4.2. Assume that conditions (Φ0 ) and (L1 ) hold, and let F be a positive function defined on R+ 0 . If (θ )2 holds with θ < 2, then there exists a constant B > 1 such that, for every σ 1 we have σ 1/(2−θ) B −1 −1 (σ F (t)) K (F (t)) K
on R+ .
(4.2)
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699
Finally, we have KO) hold, let b˜ Proposition 4.3. Assume that (Φ0 ), (Φ2 ), (F1 ), (L1 ), (L2 ), (ϕ)1 , (θ )2 and ( be a function satisfying assumption (b), and let A > 0, and β ∈ [−2, +∞). If λ and θ are the constants specified in (b) and (θ ), assume also that λ(2 − θ ) 1 (i) t
β/2 ˜
and either t
λ(1−θ)−1
b(t)
˜ λ ds C b(s)
for t t0 ,
or
1
(ii) t
β/2 ˜
λ(1−θ)−1
b(t)
C
for t t0 and θ < 1.
(3.7)
Then there exists T > 0 sufficiently large such that, for every T t0 < t1 and 0 < < η, there exist T¯ > t1 and a C 2 function α : [t0 , T¯ ) → [, +∞) which is a solution of the problem
˜ (α)(α) on [t0 , T¯ ), ϕ α α + At β/2 ϕ α b(t)f ¯ α > 0 on [t0 , T ), α(t0 ) = , α(t) → +∞ as t → T¯ −
(3.8)
and satisfies αη
on [t0 , t1 ].
(3.9)
instead of K in the Proof. The proof is a small variation of that of Proposition 3.4, using K definition of α. Note first of all that (3.7) forces θ < 2, so that (ϕ)3 is automatically satisfied. Arguing as in Proposition 3.4, one deduces that α > 0 and α satisfies σf (α)α =
ϕ(α /b˜ λ ) ˜ λ α /b , (α /b˜ λ )
(4.3)
so, again, α /b˜ λ is increasing on [t0 , Tσ ). From this, using the fact that t θ−1 ϕ(t)/(t) is ˜ −λ > 1, we obtain C-increasing (assumption (θ )2 ), ϕ(t) Ctϕ (t) (assumption (Φ2 )), and b(t) ϕ α α Cσ b˜ λ(2−θ)−1 bf (α) α
(4.4)
on [t0 , Tσ ), for some constant C > 0. On the other hand, applying (Φ2 ) to (4.3), rearranging, integrating over [t0 , t], and using (F1 ), (L2 ) and the fact that α and α /b˜ λ are increasing, we deduce that ϕ α /b˜ λ ϕ α /b˜ λ (t0 ) + Cσf (α) α /b˜ λ
t
˜ λ ds. b(s)
t0
Finally, using (F1 ), (L2 ), the fact that α and α /b˜ λ are non-decreasing, α(t0 ) = and (θ )2 we obtain
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t /b˜ λ )(t ) ϕ(α ϕ(α ) 0 ˜ λ bf ˜ (α). + σ b(s) C b˜ λ(1−θ)−1 (α ) f ()(α /b˜ λ )(t0 )
(4.5)
t0
Combining (4.4) and (4.5) we conclude that ˜ (α) α ϕ α α + At β/2 ϕ α Nσ bf with Nσ (t) defined as in (3.17). The proof now proceeds exactly as in the case of Proposition 3.4.
(4.6) 2
We then have the following version of Theorem 3.5: Theorem 4.4. Let (M, , ) be a complete Riemannian manifold satisfying β/2 Riccn,m (LD ) H 2 1 + r 2 ,
(1.12)
for some n > m, H > 0 and β −2 and assume that (Φ0 ), (Φ2 ), (F1 ), (L1 ), (L2 ), (ϕ)1 , (ϕ)2 and (θ )2 hold. Let b(x) ∈ C 0 (M), b(x) 0 on M and suppose that b(x) b˜ r(x) for r(x) 1, (3.19) where b˜ satisfies assumption (b) and (3.7). If the modified Keller–Osserman condition 1 ∈ L1 (+∞) −1 K (F (t)) holds then any entire classical weak solution u of the differential inequality LD,ϕ u b(x)f (u) |∇u|
( KO)
(1.7)
is either non-positive or constant. Furthermore, if u 0, and (0) > 0, then u vanishes identically. ˜ = C/t μ for t 1 where According to Remark 3.3, Theorem 4.4 holds if we assume that b(t) μ 0 and θ < 1 − β/2 − μ or θ = 1 − β/2 − μ < 1 if μ > 0, (θβμ) θ < 1 − β/2 if μ = 0. We note that in the model case of the mean curvature operator with (t) = t q ,
q 0,
then assumptions (Φ0 ), (Φ2 ), (ϕ)1 and (θ )2 hold provided (0 )q < 1,
θ 1+q
and the above restrictions are compatible with (θβμ).
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701
5. The weak maximum principle and non-existence of solutions with controlled growth As shown in Section 3 above, the failure of the Keller–Osserman condition allows to deduce existence of solutions of the differential inequality (1.7). The solutions thus constructed diverge at infinity. This is no accident. Indeed, Theorem B shows that under rather mild conditions on the coefficients and on the geometry of the manifold, if solutions exist, they must be unbounded, and in fact, must go to infinity sufficiently fast. The proof of Theorem B depends on the following weak maximum principle for the diffusion operator LD,ϕ which improves on the weak maximum principle for the ϕ-Laplacian already considered in [21,23,22,16]. It is worth pointing out that, besides allowing the presence of a term depending on the gradient of u, we are able to deal with C 1 functions, removing the requirement that u ∈ C 2 (M) and that the vector field |∇u|−1 ϕ(|∇u|)∇u be C 1 . In order to formulate our version of the weak maximum principle, we note that if X is a C 1 vector field, and v a positive continuous function on an open set Ω, then the following two statements hold: (i) infΩ v −1 div X Co , (ii) if div X Cv on Ω for some constant C, then C Co . Since (ii) is meaningful in distributional sense, we may take it as the weak definition of (i), and apply it to the case where X is only C 0 (L∞ loc would suffice), and v is only assumed to be nonnegative and continuous. Indeed, it is precisely the implication stated in (ii) that will allow us to prove Theorem B. In view of applications to the case of the diffusion operator LD,ϕ , it may also be useful to 1,1 is enough if X is observe that, if the weight function D(x) is assumed to be C 1 (indeed, Wloc ∞ assumed to be merely in Lloc ), then the weak inequality D(x)−1 div X Cv is in fact equivalent to the inequality div X CD(x)v. Theorem 5.1. Let (M, , ) be a complete Riemannian manifold, let D(x) ∈ C 0 (M) be a positive weight on M, and let ϕ satisfy (Φ1 ). Given σ , μ, χ ∈ R, let η = μ + (σ − 1)(1 + δ − χ), and assume that σ 0,
σ − η 0,
and 0 χ < δ.
Let u ∈ C 1 (M) be a non-constant function such that u(x) < +∞, σ r(x) r(x)→+∞
uˆ = lim sup
(5.1)
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and suppose that either log volD Br = d0 < +∞ if σ − η > 0 r σ −η
(5.2)
log volD Br = d0 < +∞ log r
(5.3)
lim inf r→+∞
or lim inf r→+∞
if σ − η = 0.
Suppose that γ ∈ R is such that the superset Ωγ = {x ∈ M: u(x) > γ } is not empty, and that the weak inequality −μ |∇u|χ D(x) div D(x)|∇u|−1 ϕ |∇u| ∇u K 1 + r(x)
(5.4)
holds on Ωγ . Then the constant K satisfies ˆ 0}δ−χ , K C(σ, δ, η, χ, d0 ) max{u,
(5.5)
where C = C(σ, δ, η, χ, δ0 ) is given by 0 C=
Ad0 (σ − η)1+δ−χ Ad0 σ δ−χ (σ − η)
if σ = 0, if σ > 0, η < 0, if σ > 0, η 0,
(5.6)
if σ − η > 0 and by C=
0 if σ = 0 or σ > 0, δ(σ − 1) + d0 − 1 0, Aσ δ−χ [δ(σ − 1) + d0 − 1] if σ > 0, δ(σ − 1) + d0 − 1 > 0
(5.7)
if σ − η = 0. Remark 5.1. According to what observed before the statement, if u in C 2 , the vector field |∇u|−1 ϕ(|∇u|)∇u is C 1 and χ = 0, then the conclusion of the theorem is that μ ˆ 0}δ , inf 1 + r(x) LD,ϕ u C(σ, δ, η, χ, δ0 ) max{u, Ωγ
and we recover an improved version of Theorem 4.1 in [16]. Proof of Theorem 5.1. The proof is an adaptation of that of Theorem 4.1 in [16]. Clearly we may assume that K > 0, for otherwise there is nothing to prove. Note also that since u is assumed to be non-constant, then it cannot be constant on any connected component Eo of Ωγ . Indeed, if u were constant in Eo , then ∅ = ∂Eo ⊆ ∂Ωγ . Since, by continuity, u = γ on ∂Ωγ , we would conclude that u ≡ γ on Eo ⊂ Ωγ , contradicting the fact that u > γ on Ωγ . Next, because both the assumptions and the conclusions of the theorem are left unchanged by adding a constant to u, arguing as in the proof of Theorem 4.1 in [16] shows that given b > max{u, ˆ 0}, we may assume that
L. Mari et al. / Journal of Functional Analysis 258 (2010) 665–712
(i)
u
703
and (ii) u(xo ) > 0 for some xo ∈ Ωγ .
(5.8)
Further, we observe that if (5.5) follows from (5.4) for some γ then the conclusion holds for any γ γ . Thus, by increasing γ if necessary, we may also suppose that γ > 0. We fix θ ∈ (1/2, 1) and choose R0 > 0 large enough that |∇u| ≡ 0 on the nonempty set BR0 ∩ Ωγ . Given R > R0 , let ψ ∈ C ∞ (M) be a cut-off function such that 0 ψ 1,
ψ ≡1
on BθR ,
ψ ≡ 0 on M \ BR ,
|∇ψ|
C , R(1 − θ )
(5.9)
for some absolute constant C > 0. Let also λ ∈ C 1 (R) and F (v, r) ∈ C 1 (R2 ) be such that 0 λ 1,
λ > 0, λ 0 on (γ , +∞),
λ = 0 on (−∞, γ ],
(5.10)
and F (v, r) > 0,
∂F (v, r) < 0 ∂v
(5.11)
on [0, +∞) × [0, +∞), where v is given by v = α(1 + r)σ − u,
(5.12)
and α is a constant greater than b, so that v > 0 on Ωγ . Indeed, according to (5.8), and the assumption that γ 0, so that u > 0 on Ωγ , we have (α − b)(1 + r)σ v α(1 + r)σ
on Ωγ .
(5.13)
By definition of the weak inequality (5.4), for every non-negative test function 0 ρ ∈ H01 (Ωγ ), −
∇ρ, |∇u|−1 ϕ |∇u| ∇u D(x) dx K
Ωγ
ρ(1 + r)−μ |∇u|χ D(x) dx.
Ωγ
We use as test function the function ρ = ψ 1+δ λ(u)F (v, r) which is non-negative, Lipschitz, compactly supported in M and vanishes on M \ (Ωγ ∩ BR (o)). Inserting the expression for ∇ρ in the above integral inequality, using the conditions λ > 0, F (v, r) > 0, ∂F /∂v < 0, and |∇u| A−1/δ ϕ(|∇u|)1/δ , which in turn follows from the structural condition ϕ(t) At δ , after some computations we obtain (1 + δ) where
ψ δ λ(u)F (v, r)ϕ |∇u| |∇ψ|D(x) dx
∂F B(u, r)D(x) dx, (5.14) ψ 1+δ λ(u) ∂v
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1+1/δ B(u, r) = A−1/δ ϕ |∇u| χ/δ F (v, r) (1 + r)−μ ϕ |∇u| |∂F /∂v|
∂F /∂r σ −1 |∇u|−1 ϕ |∇u| ∇r, ∇u . + − ασ (1 + r) |∂F /∂v| + KA−χ/δ
(5.15)
Now one needs to consider several cases separately. We treat in detail only the case where M satisfies the volume growth condition (5.2), σ > 0, and η < 0. In this case we let F (v, r) = exp −qv(1 + r)−η , where q > 0 is a constant that will be specified later. An elementary computation which uses the estimate for v given in (5.13) shows that 0
∂F ∂r (v, r) ∂F | ∂v (v, r)|
− ασ (1 + r)σ −1 −α(σ − η)(1 + r)σ −1
(5.16)
1 (1 + r)η . q
(5.17)
and F (v, r) | ∂F ∂v (v, r)|
=
Inserting (5.16) and (5.17) into (5.15), and using the Cauchy–Schwarz inequality we deduce that χ/δ B(u, r) ϕ |∇u|
δ+1−χ 1 K ϕ |∇u| δ + (1 + r)(1+δ−χ)(σ −1) 1/δ qAχ/δ A δ−χ − α(σ − η)(1 + r)σ −1 ϕ |∇u| δ .
(5.18)
In order to estimate the right-hand side of (5.18) we use the following calculus result (see [16], Lemma 4.2): let ν, ρ, β, ω be positive constants, and let f be the function defined on [0, +∞) by f (s) = ωs 1+ν + ρ − βs ν . Then the inequality f (s) Λs 1+ν holds on [0, +∞) provided Λω−
νβ 1+1/ν . (1 + ν)1+1/ν ρ 1/ν
(5.19)
Applying this result with ν = δ − χ and s = ϕ(|∇u|)1/δ , and recalling the definition of η we deduce that the estimate 1+1/δ B(u, r) Λϕ |∇u|
(5.20)
holds provided Λ
νq 1/ν Aχ/δν [α(σ − η)]1+1/ν 1 − . 1/δ A (1 + ν)1+1/ν K 1/ν
(5.21)
L. Mari et al. / Journal of Functional Analysis 258 (2010) 665–712
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In particular, given τ ∈ (0, 1) if we let Λ=
1−τ A1/δ
and q =
τ ν (1 + ν)1+ν K, ν ν A[α(σ − η)]1+ν
(5.22)
then Λ is positive, and satisfies (5.21) with equality. Inserting (5.20) and the expression for ∂F /∂v into (5.14), we deduce that qΛ 1+δ
1+1/δ ψ 1+δ λ(u)F (v, r)(1 + r)−η ϕ |∇u| D(x) dx
Ωγ ∩BR
ψ δ λ(u)F (v, r)|∇ψ|ϕ |∇u| D(x) dx.
Ωγ ∩BR
Now the proof proceeds as in [16]: applying Hölder inequality with conjugate exponents 1 + δ and 1 + 1/δ to the integral on the right-hand side, and simplifying we obtain
qΛ 1+δ
1+δ
1+1/δ ψ 1+δ λ(u)F (v, r)(1 + r)−η ϕ |∇u| D(x)
Ωγ ∩BR
λ(u)F (v, r)(1 + r)ηδ |∇ψ|1+δ D(x).
(5.23)
Ωγ ∩BR
By the volume growth assumption (5.2), for every d > d0 , there exists a diverging sequence Rk ↑ +∞ with R1 > 2R0 such that σ −η
log vol BRk dRk
(5.24)
.
Since θ Rk > Rk /2 > R0 , we may let R = Rk in (5.23), and use the support properties of ψ , the estimate for |∇ψ|, and the fact that λ 1, η < 0 to show that
E=
C
qΛ 1+δ 1+δ
1+δ
1+1/δ λ(u)F (v, r)ϕ |∇u| D(x)
Ωγ ∩BR0
−(1+δ) (1 + θ Rk ) (1 − θ )Rk
ηδ
F (v, r)D(x).
(5.25)
Ωγ ∩(BRk \BθRk )
Now, since |∇u| ≡ 0 on Ωγ ∩ BR0 , then E > 0. On the other hand, using the bound (5.13) for v, and the expression of F we get F (v, r) exp −q(α − b)(1 + θ Rk )σ −η on Ωγ ∩ (BRk \ BθRk ), so inserting this into the right-hand side of (5.25) we conclude that
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L. Mari et al. / Journal of Functional Analysis 258 (2010) 665–712 δη−(1+δ)
0 < E CRk
σ −η exp dRk − q(α − b)(1 + θ Rk )σ −η ,
(5.26)
where C is a constant independent of k. In order for this inequality to hold for every k, we must have d (α − b)qθ σ −η , whence, letting θ tend to 1, d (α − b)q. We set α = tb, insert the definition (5.22) of q in the above inequality, solve with respect to K, and then let τ tend to 1 to obtain K Adbν (σ − η)1+ν
νν t 1+ν . (1 + ν)1+ν t − 1
The conclusion is then obtained minimizing with respect to t > 1, letting d → d0 and b → max{u, ˆ 0} and recalling that ν = δ − χ . The other cases are treated adapting the arguments carried out in the proof of [16], Theorem 4.1, cases II and III, and of Theorem 4.3 for the case of polynomial volume growth. 2 Proof of Theorem B. We begin by showing that if under the assumptions of the theorem, u is necessarily bounded above. Indeed, assume by contradiction that u∗ = +∞, so that, by (1.14), σ > 0, and there exist γo and C > 0 such that f (t) > C for t γ . Keeping into account the assumptions on b and , we deduce that u satisfies the differential inequality −μ div D(x)|∇u|−1 ϕ |∇u| ∇u K 1 + r(x) |∇u|χ D(x) weakly on Ωγo , with a constant K > 0. On the other hand, because of growth assumption on u, the constant uˆ in the statement of Theorem 5.1 is equal to zero, and this shows that K = 0, and the contradiction shows that u∗ < +∞ is bounded above. Assume now that f (u∗ ) > 0. Since f (t) > 0 for t > 0, by continuity there exists γo such that f (u) C > 0 on Ωγo , and a contradiction is reached as above. The final statement follows immediately from this and from the assumptions. 2 6. Proof of Theorem C The aim of this section is to prove Theorem C in the Introduction together with a version covering the case of the mean curvature operator. Before proceeding, we analyze the Keller– Osserman condition t
e 0 ρ(z) dz ∈ L1 (+∞), (t)) K −1 (F
(ρKO)
L. Mari et al. / Journal of Functional Analysis 258 (2010) 665–712
707
+ where ρ ∈ C 0 (R+ 0 ) is non-negative on R0 and F (t) = Fρ,ω is defined in (1.20), namely,
t Fρ,ω (t) =
f (s)e(2−ω)
s 0
ρ(z) dz
(1.20)
ds.
0
Lemma 6.1. Assume that (F1 ), (L1 ) and the first part of (θ )1 with θ < 2 hold, and let ω = θ and σ ∈ R+ . Then (ρKO) is equivalent to t
e 0 ρ(z) dz ∈ L1 (+∞). (t)) K −1 (σ F
(ρKOσ )
Proof. Assume first that σ 1. Since K −1 is non-decreasing, 1 (t)) K −1 (F
1 (t)) K −1 (σ F
and (ρKOσ ) implies (ρKO). On the other hand, according to Proposition 3.3 and Remark 3.2 there exists a constant B 1 such that σ 1/(2−θ) B −1 −1 (t)) K (F (t)) K (σ F
on R+ ,
and (ρKO) implies (ρKOσ ). Thus the stated equivalence holds when σ 1. Then the case σ 1 follows as in Lemma 3.1. 2 We observe that in favorable circumstances (KO) and (ρKO) are indeed equivalent. For instance we have Proposition 6.2. Assume that (F1 ), (L1 ), (ϕ)2 and (ρ) hold. If ρ ∈ L1 (+∞) and ω 2 then (ρKO) is equivalent to (KO). Proof. Observe first of all that since 0 ρ ∈ L1 ((0, +∞)) (ρKO) is equivalent to 1 ∈ L1 (+∞). (t)) K −1 (F
(6.1)
Since ω 2 we also have 1 e(2−ω)
s 0
ρ(z) dz
Λ,
and therefore t F (t) = 0
(t) = f (s) ds F
t f (s)e(2−ω) 0
s 0
ρ(z) dz
ΛF (t).
(6.2)
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L. Mari et al. / Journal of Functional Analysis 258 (2010) 665–712
Recalling that K −1 is increasing, the left-hand side inequality in (6.2) shows that +∞
dt (t)) K −1 (F
+∞
dt K −1 (F (t))
and, by (6.1), (KO) implies (ρKO). On the other hand, since, by (F1 ), f is C-increasing with constant C 1, so is also the , and therefore the right-hand side inequality in (6.2) and the integrand in the definition of F , show that argument in the proof of Lemma 3.1, with σ = Λ−1 and F replaced by F +∞
ds −1 K (F (s))
+∞
ds CΛ −1 (s)) K (Λ−1 F
and, again by (6.1), (ρKO) implies (KO).
+∞
dt (t)) K −1 (F
,
(6.3)
2
Remark 6.1. The above proposition generalizes Proposition 6.1 in [12]. Proposition 6.3. Assume that (Φ0 ), (F1 ), (L1 ), (L2 ), (ϕ)1 , (θ ), (b), (ρ) and (ρKO) with ω = θ hold. Let A > 0, β −2, and, if λ > 0 and θ are the constants in (b) and (θ ), suppose that θ 1 and ⎧ ⎪ ⎪ ⎨ λ 1, ⎪ ⎪ ⎩
t
β/2 ˜
−1
t
b(t)
λ(2 − θ ) 1,
t
˜ λ ds C b(s)
∀t 1
1 β/2 ˜
b(t)λ(1−θ)−1 C
if θ = 1,
(6.4)
∀t 1 if θ < 1,
for come constant C > 0. The there exists T > 0 sufficiently large such that, for every T t0 < t1 and 0 < < η, there exist T¯ > t1 and a C 2 function α : [t0 , T¯ ) → [, +∞) which is a solution of the problem
2 ˜ (α)(α) − ρ(α)ϕ α α on [t0 , T¯ ), ϕ α α + At β/2 ϕ α b(t)f − α > 0 on [t0 , T¯ ), α(t0 ) = , α(t) → +∞ as t → T¯
(6.5)
and satisfies αη
on [t0 , t1 ].
(6.6)
Proof. The proof is a modification of that of Proposition 3.4 so we only sketch it. Note that since (θ )1 holds with θ 1, so does (ϕ)2 . Thus K defines a C 1 -diffeomorphism of R+ 0 and condition (ρKO) is meaningful. As in the proof of Proposition 3.4, we may assume that b˜ 1 for t large. Choose T > 0 large ˜ 1 in [T , +∞), let t0 , t1 , , η be as in the statement, use enough that b˜ (t) 0 and 0 < b(t) Lemma 6.1, (b) and condition (ρKO), to define Tσ by means of the formula
L. Mari et al. / Journal of Functional Analysis 258 (2010) 665–712
Tσ
+∞
˜ λ ds = b(s)
t0
709
s
e 0ρ , (s)) K −1 (σ F
and choose σ ∈ (0, 1] small enough to guarantee that Tσ > t1 . Next let α : [t0 , Tσ ) → [, +∞) be defined by the formula Tσ
+∞
˜ ds = b(s) λ
t
s
e 0ρ , (s)) K −1 (σ F
α(t)
so that and α Tσ− = +∞.
α(t0 ) = Differentiating we obtain
)e− α = b˜ λ K −1 (σ F
α 0
ρ
,
so that α > 0, and rearranging, differentiating once again, and simplifying we obtain
σf (α)e
(2−θ)
α 0
ρ
=
e
α 0
b˜ λ
α
ρ ϕ( α e 0 b˜ λ
(
ρ
α α e 0 ρ b˜ λ
) αe
α 0
ρ
b˜ λ
)
(6.7)
,
α
so that, in particular, (α e 0 ρ /b˜ λ ) > 0. α α We use the fact that e 0 ρ /b˜ 1 to apply (θ )1 , we expand the derivative of (α e 0 ρ /b˜ λ ), use b˜ 0, and rearrange to obtain 2 ϕ α α Cσf (α) α b˜ λ(2−θ) − ρ(α)ϕ α .
(6.8)
On the other hand, we rewrite (6.7) in the form ϕ
α
αe 0 b˜ λ
ρ αe
α 0
ρ
b˜ λ
αρ α αe0 e(1−θ) 0 ρ , = σ b f (α) b˜ λ ˜λ
integrate between t0 and t, and use the C-monotonicity of f and and (θ )2 to obtain
α
αe 0 ϕ b˜ λ
ρ
α
αe 0 −ϕ b˜ λ
ρ
(t0 ) Cσf (α)e
(1−θ)
α 0
ρ
α ρ t αe0 b˜ λ , b˜ λ 0
whence, rearranging and using the C-monotonicity of t θ−1 ϕ(t)/(t), f and , and the θ 1 shows that (see the argument that led to (3.15) in the proof of Proposition 3.4)
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L. Mari et al. / Journal of Functional Analysis 258 (2010) 665–712 α
α ρ θ−1 ϕ( α e 0 ρ ) e0 ϕ(α ) b˜λ C αρ λ (α ) b˜ ( α e 0 ) b˜ λ
˜ (α) σ b˜ λ(1−θ)−1 C bf
t
b˜ λ +
0
α 0 ρ
ϕ( α e˜ λ )(t0 ) b
α 0 ρ
f ()( α e˜ λ )
(t0 ) .
(6.9)
b
Thus, combining (6.8) and (6.9) and arguing as in Proposition 3.4 we deduce that ˜ (α) α − ρ(α)ϕ α α 2 ϕ α α + At β /2ϕ α N (σ )bf with Nσ (t) = Cσ b˜ λ(2−θ)−1 + ACt β/2 b˜ λ(1−θ)−1 + ACσ t
β/2 ˜ λ(1−θ)−1
t
b
ϕ(K −1 (σ F ())) (K −1 (σ F ()))f ()
˜ λ. b(s)
t0
The proof now proceeds exactly as in Proposition 3.4.
2
The next result is the analogue of Theorem 3.5 and Theorem C in the Introduction follows from it using Remark 3.3. Theorem 6.4. Let (M, , ) be a complete manifold satisfying β/2 Riccn,m (LD ) H 2 1 + r 2 ,
(6.10)
for some n > m, H > 0 and β −2, and assume that (h), (g), (ρ), (Φ0 ), (F1 ), (L1 ) (L2 ), (ϕ)1 and (θ ) hold. Let also b(x) ∈ C 0 (M) be strictly positive on M and such that b(x) b˜ r(x)
for r(x) 1,
(6.11)
with b˜ satisfying (b), and (6.4). Finally, suppose that (ρKO) holds with ω = θ in the definition . Then any entire classical weak solution of the differential inequality of F LD,ϕ u b(x)f (u) |∇u| − g(u)h |∇u|
(1.19)
is either non-positive or constant. Moreover, if u 0 and (0) > 0, then u ≡ 0. Proof. The proof is modeled on that of Theorem 3.5. However, in the case where u is bounded above, in order to prove that, if u takes on positive values and is non-constant then u∗o = sup u < sup u = u∗ , Bro
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711
we argue as follows. Assume that u attains its supremum u∗ > 0 and let Γ = {x: u(x) = u∗ }. Clearly Γ is closed and nonempty. We are going to show that it is also open so, by connectedness, Γ = M and u is constant. To this end, let xo ∈ Γ . We have b(x)f (u) 12 b(xo )f (u∗ ) > 0 and g(u) 2Cρ(u∗ ) in a suitable neighborhood U of xo . Moreover, by (θ )1 and (h), we may estimate h(s) Cs 2 ϕ (s) C
ϕ (1) 2−θ s (s) = Cs 2−θ (s) (1)
∀s 1,
so that, in U ,
b(xo ) ∗ ∗ 2−θ b(x)f (u) |∇u| − g(u)h |∇u| |∇u| . f u − Cρ u |∇u| 2 Since ∇u(xo ) = 0 it is now clear that there exists a neighborhood U ⊂ U of xo where the righthand side the above inequality is non-negative. Thus, LD,ϕ u 0 in U and u = u∗ in U by the strong maximum principle. We note in passing that if (0) > 0 the required conclusion may be obtained without having to appeal to condition (θ )1 . The rest of the proof proceeds as in Theorem 3.5 using Proposition 6.3 instead of Proposition 3.4. 2 As we did for Theorem 3.5 in Section 3, even in this case we can provide a version of the above result valid for a class of operators which include the mean curvature operator. In order to do this we need to introduce the appropriate Keller–Osserman condition. Given ω ∈ R, let ρ be defined in (4.1). be defined in (1.20). We assume (ϕ)3 holds and let K satisfy (ρ) and let F The version of Keller–Osserman condition we consider is then e
t 0
ρ
(t)) −1 (F K
∈ L1 (+∞).
(ρ KO)
Modifications of the arguments of Section 4 allow to obtain the following Theorem 6.5. Let (M, , ) be a complete manifold satisfying (6.10) for some n > m, H > 0 and β −2, and assume that (h), (g), (ρ), (Φ0 ), (F1 ), (L1 ) (L2 ), (ϕ)1 and (θ ) hold. Let also b(x) ∈ C 0 (M) be strictly positive on M and satisfying (6.11) with b˜ satisfying (b), and (6.4). . Then any entire classical Finally, suppose that (ρ KO) holds with ω = θ in the definition of F weak solution of the differential inequality (1.19) LD,ϕ u b(x)f (u) |∇u| − g(u)h |∇u| is either non-positive or constant. Moreover, if u 0 and (0) > 0, then u ≡ 0. We leave the details to the interested reader, and merely point out that, according to what remarked in the proof of Theorem 6.4, if (0) > 0 then it suffices to assume (θ )2 in the statement of Theorem 6.5.
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References [1] D. Bakry, L’hypercontractivité et son utilization en théorie des semigroups, Lecture Notes in Math. 1581 (1994) 1–114. [2] D. Bakry, P. Emery, Diffusions hypercontactives, Lecture Notes in Math. 1123 (1985) 177–206. [3] D. Bakry, Z. Qian, Volume comparison theorems without Jacobi fields, in: Current Trends in Potential Theory, in: Theta Ser. Adv. Math., vol. 4, Theta, Bucharest, 2005, pp. 115–122. [4] J. Cheeger, M. Gromov, M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1) (1982) 15–53. [5] D. Duc, J. Eells, Regularity of exponentially harmonic functions, Internat. J. Math. 2 (4) (1991) 395–408. [6] J. Eells, L. Lemaire, Some properties of exponentially harmonic maps, in: Partial Differential Equations, Part 1, Warsaw, 1990, in: Banach Center Publ., vol. 27, Polish Acad. Sci., Warsaw, 1992, pp. 129–136. [7] S. Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Asterisque 157–158 (1988) 191– 216. [8] X.-D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl. 84 (2005) 1295–1361. [9] P. Li, S.T. Yau, Curvature and holomorphic mappings of complete Kähler manifolds, Compos. Math. 73 (1990) 125–144. [10] J. Lott, Some geometric properties of the Bakry–Emery Ricci tensor, Comment. Math. Helv. 78 (2003) 865–883. [11] E. Mitidieri, S.I. Pohozaev, A priori estimates and blow up of solutions to non-linear partial differential equations and inequalities, Proc. Steklov Inst. Math. 234 (2001) 1–362. [12] M. Magliaro, L. Mari, P. Mastrolia, M. Rigoli, Keller–Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group, preprint. [13] R. Osserman, On the inequality u f (u), Pacific J. Math. 7 (1957) 1641–1647. [14] P. Petersen, G. Wei, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal. 7 (1997) 1031–1045. [15] S. Pigola, M. Rigoli, A.G. Setti, Maximum principle and singular elliptic inequalities, J. Funct. Anal. 113 (2002) 224–260. [16] S. Pigola, M. Rigoli, A.G. Setti, Maximum principle on Riemannian manifolds and applications, Mem. Amer. Math. Soc. 174 (822) (2005). [17] S. Pigola, M. Rigoli, A.G. Setti, Maximum principle at infinity on Riemannian manifolds: An overview, Mat. Contemp. 31 (2006) 81–128. [18] S. Pigola, M. Rigoli, A.G. Setti, Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the Bochner Technique, Progr. Math., vol. 266, Birkhäuser Verlag, Basel, 2008. [19] P. Pucci, J. Serrin, The Maximum Principle, Progr. Nonlinear Differential Equations Appl., vol. 73, Birkhäuser Verlag, Basel, 2007. [20] Z. Qian, Estimates for weighted volumes and application, Q. J. Math. 48 (1997) 235–242. [21] M. Rigoli, M. Salvatori, M. Vignati, A Liouville type theorem for a general class of operators on complete manifolds, Pacific J. Math. 194 (2000) 439–453. [22] M. Rigoli, M. Salvatori, M. Vignati, Some remarks on the weak maximum principle, Rev. Mat. Iberoamericana 21 (2005) 459–481. [23] M. Rigoli, A.G. Setti, Liouville type theorems for ϕ-subharmonic functions, Rev. Mat. Iberoamericana 17 (2001) 471–520. [24] A.G. Setti, Gaussian estimates for the heat kernel of the weighted Laplacian and fractal measures, Canad. J. Math. 44 (1992) 1061–1078.
Journal of Functional Analysis 258 (2010) 713–728 www.elsevier.com/locate/jfa
An existence result for superparabolic functions Juha Kinnunen a,∗ , Teemu Lukkari b , Mikko Parviainen a a Helsinki University of Technology, Institute of Mathematics, PO Box 1100, FI-02015 TKK, Finland b Norwegian University of Science and Technology, Department of Mathematical Sciences, N-7491 Trondheim, Norway
Received 6 May 2009; accepted 5 August 2009 Available online 2 September 2009 Communicated by H. Brezis
Abstract We study superparabolic functions related to nonlinear parabolic equations. They are defined by means of a parabolic comparison principle with respect to solutions. We show that every superparabolic function satisfies the equation with a positive Radon measure on the right-hand side, and conversely, for every finite positive Radon measure there exists a superparabolic function which is solution to the corresponding equation with measure data. © 2009 Elsevier Inc. All rights reserved. Keywords: Parabolic p-Laplace; Measure data; Superparabolic functions
1. Introduction This work provides an existence result for superparabolic functions related to nonlinear degenerate parabolic equations ∂u − div A(x, t, ∇u) = 0. ∂t
(1.1)
The principal prototype of such an equation is the p-parabolic equation ∂u − div |∇u|p−2 ∇u = 0 ∂t * Corresponding author.
E-mail addresses:
[email protected] (J. Kinnunen),
[email protected] (T. Lukkari),
[email protected] (M. Parviainen). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.08.009
(1.2)
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J. Kinnunen et al. / Journal of Functional Analysis 258 (2010) 713–728
with 2 p < ∞. Superparabolic functions are defined as lower semicontinuous functions that obey a parabolic comparison principle with respect to continuous solutions of (1.1). The superparabolic functions related to the p-parabolic equation are of particular interest because they coincide with the viscosity supersolutions of (1.2), see [5]. Thus there is an alternative definition in the theory of viscosity solutions and our results automatically hold for the viscosity supersolutions of (1.2) as well. By definition, a superparabolic function is not required to have any derivatives, and, consequently, it is not evident how to directly relate it to Eq. (1.1). However, by [8] a superparabolic function has spatial Sobolev derivatives with sharp local integrability bounds. See also [1,2,7]. Using this result we show that every superparabolic function u satisfies the equation with measure data ∂u − div A(x, t, ∇u) = μ, ∂t
(1.3)
where μ is the Riesz measure of u, see Theorem 3.9. A rather delicate point here is that the spatial gradient of a superparabolic function is not locally integrable to the natural exponent p. Consequently, the Riesz measure does not belong to the dual of the natural parabolic Sobolev space. For example, Dirac’s delta is the Riesz measure for the Barenblatt solution of the p-parabolic equation. We also consider the converse question. Indeed, for every finite nonnegative Radon measure μ, there is a superparabolic function which satisfies (1.3), see Theorem 5.8. This result is standard, provided that the measure belongs to the dual of the natural parabolic Sobolev space, but we show that the class of superparabolic functions is large enough to admit an existence result for general Radon measures. If the measure belongs to the dual of the natural parabolic Sobolev space, then uniqueness with fixed initial and boundary conditions is also standard. However, uniqueness questions related to (1.3) for general measures are rather delicate. For instance, the question whether the Barenblatt solution is the only solution of the p-parabolic equation with Dirac’s delta seems to be open. Hence, we will not discuss uniqueness of solutions here. 2. Preliminaries Let Ω be an open and bounded set in Rn with n 1. We denote ΩT = Ω × (0, T ), where 0 < T < ∞. For an open set U in Rn we write Ut1 ,t2 = U × (t1 , t2 ), where 0 < t1 < t2 < ∞. The parabolic boundary of Ut1 ,t2 is ∂p Ut1 ,t2 = ∂U × [t1 , t2 ] ∪ U × {t1 } . As usual, W 1,p (Ω) denotes the Sobolev space of functions in Lp (Ω), whose distributional gradient belongs to Lp (Ω). The space W 1,p (Ω) is equipped with the norm uW 1,p (Ω) = uLp (Ω) + ∇uLp (Ω) .
J. Kinnunen et al. / Journal of Functional Analysis 258 (2010) 713–728
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The Sobolev space with zero boundary values, denoted by W0 (Ω), is a completion of C0∞ (Ω) with respect to the norm of W 1,p (Ω). The parabolic Sobolev space Lp (0, T ; W 1,p (Ω)) consists of measurable functions u : ΩT → [−∞, ∞] such that for almost every t ∈ (0, T ), the function x → u(x, t) belongs to W 1,p (Ω) and 1,p
p |u| + |∇u|p dx dt < ∞.
(2.1)
ΩT 1,p
A function u ∈ Lp (0, T ; W 1,p (Ω)) belongs to the space Lp (0, T ; W0 (Ω)) if x → u(x, t) be1,p p 1,p longs to W0 (Ω) for almost every t ∈ (0, T ). The local space Lloc (0, T ; Wloc (Ω)) consist of functions that belong to the parabolic Sobolev space in every Ut1 ,t2 ΩT . We assume that the following structural conditions hold for the divergence part of our equation for some exponent p 2: (1) (2) (3) (4) (5)
(x, t) → A(x, t, ξ ) is measurable for all ξ ∈ Rn , ξ → A(x, t, ξ ) is continuous for almost all (x, t) ∈ Ω × R, A(x, t, ξ ) · ξ α|ξ |p for almost all (x, t) ∈ Ω × R and ξ ∈ Rn , |A(x, t, ξ )| β|ξ |p−1 for almost all (x, t) ∈ Ω × R and ξ ∈ Rn , (A(x, t, ξ ) − A(x, t, η)) · (ξ − η) > 0 for almost all (x, t) ∈ Ω × R and all ξ, η ∈ Rn , ξ = η.
Solutions are understood in the weak sense in the parabolic Sobolev space. 1,p
p
Definition 2.2. A function u ∈ Lloc (0, T ; Wloc (Ω)) is a weak solution of (1.1) in ΩT , if −
u
∂ϕ dx dt + ∂t
ΩT
A(x, t, ∇u) · ∇ϕ dx dt = 0
(2.3)
ΩT
for all test functions ϕ ∈ C0∞ (ΩT ). The function u is a weak supersolution if the integral in (2.3) is nonnegative for nonnegative test functions. In a general open set V of Rn+1 , the above notions are to be understood in a local sense, i.e. u is a solution if it is a solution in all sets Ut2 ,t2 V . By parabolic regularity theory, every weak solution has a locally Hölder continuous representative. The definition of a weak solution does not refer to the time derivative of u. We would, nevertheless, like to employ test functions depending on u, and thus the time derivative ∂u ∂t inevitably appears. The standard way to overcome this difficulty is to use a mollification procedure, for instance Steklov averages or convolution with the standard mollifier, in the time direction; see, e.g., [3]. Let uε denote the mollification of u. For each ϕ ∈ C0∞ (ΩT ), the regularized equation reads ΩT
∂uε ϕ dx dt + ∂t
ΩT
A(x, t, ∇u)ε · ∇ϕ dx dt = 0,
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for small enough ε > 0. The aim is to obtain estimates that are independent of the time derivatives of uε , and then pass to the limit ε → 0. 3. A-superparabolic functions We illustrate the notion of A-superparabolic functions by considering the Barenblatt solution Bp : Rn+1 → [0, ∞) first. It is given by the formula
−n/λ c − Bp (x, t) = t 0,
p−2 1/(1−p) |x| p/(p−1) (p−1)/(p−2) , p λ + t 1/λ
t > 0, t 0,
where λ = n(p − 2) + p, p > 2, and the constant c is usually chosen so that Bp (x, t) dx = 1 Rn
for every t > 0. The Barenblatt solution is a weak solution of the p-parabolic equation (1.2) in the open upper and lower half spaces, but it is not a supersolution in any open set that contains the origin. It is the a priori integrability of ∇Bp that fails, since 1
∇Bp (x, t)p dx dt = ∞,
−1 Q
where Q = [−1, 1]n ⊂ Rn . In contrast, the truncated functions min Bp (x, t), k , k = 1, 2, . . . , belong to the correct parabolic Sobolev space and are weak supersolutions in Rn+1 for every k. This shows that an increasing limit of weak supersolutions is not necessarily a weak supersolution. In order to include the Barenblatt solution in our exposition we define a class of superparabolic functions, as in [6]. Definition 3.1. A function u : ΩT → (−∞, ∞] is A-superparabolic in ΩT , if (1) u is lower semicontinuous, (2) u is finite in a dense subset, and (3) if h is a solution of (1.1) in Ut1 ,t2 ΩT , continuous in U t1 ,t2 , and h u on the parabolic boundary ∂p Ut1 ,t2 , then h u in Ut1 ,t2 . We say that u is A-hyperparabolic, if u satisfies properties (1) and (3) only. The class of A-superparabolic functions is strictly larger than that of weak supersolutions as the Barenblatt solution discussed above shows. If u and v are A-superparabolic functions, so are their pointwise minimum min(u, v), and the functions u + α for all α ∈ R. This is an immediate
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consequence of the definition. However, the functions u + v and αu are not superparabolic in general. This is well in accordance with the corresponding properties of weak supersolutions. In addition, the class of superparabolic functions is closed with respect to the increasing convergence, provided the limit function is finite in a dense subset. This is also a straightforward consequence of the definition. Theorem 3.2. Let (uj ) be an increasing sequence of A-superparabolic functions in ΩT . Then the limit function u = limj →∞ uj is always A-hyperparabolic, and A-superparabolic whenever it is finite in a dense subset. A much less straightforward property of A-superharmonic functions is the following theorem. Theorem 3.3. (See [9,10].) A locally bounded A-superparabolic function is a weak supersolution. These two results give a characterization of A-superparabolicity. Indeed, if we have an increasing sequence of continuous weak supersolutions and the limit function is finite in dense subset, then the limit function is A-superparabolic. Moreover, if the limit function is bounded, then it is a weak supersolution. On the other hand, the truncations min(v, k), k = 1, 2, . . . , of an A-superparabolic function v are weak supersolutions and hence every A-superparabolic function can be approximated by an increasing sequence of weak supersolutions. The reader should carefully distinguish between weak supersolutions and A-superparabolic functions. Notice that an A-superparabolic function is defined at every point in its domain, but weak supersolutions are defined only up to a set of measure zero. On the other hand, weak supersolutions satisfy the comparison principle and, roughly speaking, they are A-superparabolic, provided the issue about lower semicontinuity is properly handled. In fact, every weak supersolution has a lower semicontinuous representative as the following theorem shows. Hence every weak supersolution is A-superparabolic after a redefinition on a set of measure zero. Theorem 3.4. (See [11].) Let u be a weak supersolution in ΩT . Then there exists a lower semicontinuous weak supersolution that equals u almost everywhere in ΩT . Weak supersolutions have spatial Sobolev derivatives and they satisfy a differential inequality in a weak sense. By contrast, no differentiability is assumed in the definition of an Asuperparabolic function. The only tie to the differential equation is through the comparison principle. Nonetheless, [8] gives an integrability result with an exponent smaller than p. See also [1] and [2]. q
Theorem 3.5. Let u be A-superparabolic in ΩT . Then u belongs to the space Lloc (0, T ; 1,q Wloc (Ω)) with 0 < q < p − n/(n + 1). In particular, this shows that an A-superparabolic function u has a spatial weak gradient and that it satisfies ∂ϕ − u dx dt + A(x, t, ∇u) · ∇ϕ dx dt 0 ∂t ΩT
ΩT
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for all nonnegative test functions ϕ ∈ C0∞ (ΩT ). Note carefully that the integrability of the gradient is below the natural exponent p and hence u is not a weak supersolution. Although u satisfies the integral inequality, it seems to be very difficult to employ this property directly. A key ingredient in the proof of Theorem 3.5 is the following lemma, see [8, Lemma 3.14]. We will use it below. Lemma 3.6. Suppose that v is a positive function such that vk = min(v, k) belongs to 1,p Lp (0, T ; W0 (Ω)). If there is a constant M > 0, independent of k, such that
|∇vk |p dx dt + ess sup 0
ΩT
vk2 dx Mk,
k = 1, 2, . . . ,
Ω
then v and ∇v belong to Lq (ΩT ) for 0 < q < p − n/(n + 1) and their Lq norms have an estimate in terms of n, p, q, |ΩT |, and M. Next we study the connection between A-superparabolic functions and parabolic equations with measure data. First we define weak solutions to the measure data problem (1.3). Recall our assumption p 2. p−1
1,p−1
Definition 3.7. Let μ be a Radon measure on Rn+1 . A function u ∈ Lloc (0, T ; Wloc a weak solution of (1.3) in ΩT , if −
u
∂ϕ dx dt + ∂t
ΩT
(Ω)) is
A(x, t, ∇u) · ∇ϕ dx dt =
ΩT
ϕ dμ
(3.8)
ΩT
for all ϕ ∈ C0∞ (ΩT ). The Barenblatt solution satisfies ∂Bp − div |∇Bp |p−2 ∇Bp = δ ∂t in the weak sense of Definition 3.7, where the right-hand side is Dirac’s delta at the origin. In other words, Dirac’s delta is the Riesz mass of the Barenblatt solution. Theorem 3.5 implies the existence of the Riesz measure of any A-superparabolic function. Theorem 3.9. Let u be an A-superparabolic function. Then there exists a positive Radon measure μ such that u satisfies (1.3) in the weak sense. Proof. Theorem 3.5 implies that |u|p−1 , |∇u|p−1 ∈ L1loc (ΩT ). Let ϕ ∈ C0∞ (ΩT ) with ϕ 0 and denote uk = min(u, k). Then A(x, t, ∇uk ) · ∇ϕ → A(x, t, ∇u) · ∇ϕ
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pointwise almost everywhere as k → ∞ by continuity of ξ → A(x, t, ξ ), as ∇uk → ∇u almost everywhere. Using the structure of A, we have A(x, t, ∇uk ) · ∇ϕ C|∇uk |p−1 |∇ϕ| C|∇u|p−1 |∇ϕ|. The majorant established above allow us to use the dominated convergence theorem and the fact that the functions uk are weak supersolutions to conclude that −
u
∂ϕ dx dt + ∂t
ΩT
A(x, t, ∇u) · ∇ϕ dx dt
ΩT
∂ϕ dx dt + A(x, t, ∇uk ) · ∇ϕ dx dt 0. = lim − uk k→∞ ∂t ΩT
ΩT
The claim now follows from the Riesz representation theorem.
2
4. Compactness of A-superparabolic functions In this section we prove a compactness property of A-superparabolic functions. It will be essential in the proof of the fact that every finite Radon measure there exists a superparabolic function, which solves the corresponding equation with measure data. We use the following convergence result for weak supersolutions from [10]. Theorem 4.1. Let (uj ) be a bounded sequence of weak supersolutions in ΩT and assume that uj converges to a function u almost everywhere in ΩT . Then the limit function u is a weak supersolution, and ∇uj → ∇u almost everywhere. Note that a pointwise limit of weak supersolutions is not necessarily a weak supersolution if we drop the boundedness assumption, as illustrated by the Barenblatt solution discussed at the beginning of Section 3. We also use the following Caccioppoli estimate from [3]. The straightforward proof employs the test function −uϕ. Lemma 4.2. Let u 0 be a weak supersolution in ΩT , and ϕ ∈ C0∞ (ΩT ) with ϕ 0. Then
|∇u|p ϕ p dx dt C ΩT
|u|p |∇ϕ|p dx dt +
ΩT
∂ϕ |u|2 ϕ p−1 dx dt . ∂t
ΩT
Next we show that general superparabolic functions have a compactness property. Note that the limit function may very well be identically infinite. Theorem 4.3. Let (uj ) be a sequence of positive A-superparabolic functions in ΩT . Then there exist a subsequence (ujk ) and an A-hyperparabolic function u such that ujk → u almost everywhere in ΩT ,
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and
∇ujk → ∇u almost everywhere in (x, t) ∈ ΩT : u(x, t) < ∞ . Proof. Assume first that uj M < ∞. If we could extract a subsequence that converges pointwise almost everywhere to a function u, Theorem 4.1 would imply that u is a weak supersolution and that ∇uj → ∇u almost everywhere. By Theorem 3.4, we could then assume that u is lower semicontinuous and thus A-superparabolic. Once the result for bounded sequences is available, we can remove the boundedness assumption by a diagonalization argument. Indeed, we can find a subsequence (u1j ) and an Asuperparabolic function u1 such that min u1j , 1 → u1
and ∇ min u1j , 1 → ∇u1
almost everywhere in ΩT . We proceed inductively and pick a subsequence (ukj ) of (uk−1 j ) such that min ukj , k → uk
and ∇ min ukj , k → ∇uk
almost everywhere in ΩT . We observe that if l k and uk (x, t) < k, we have ul (x, t) = uk (x, t). Thus the sequence (uk ) is increasing, and we conclude that the limit u = lim uk k→∞
exists and defines the desired A-hyperparabolic function in ΩT . We note that by construction min(u, k) = uk , so that for the diagonal sequence (ukk ) it holds that ∇ukk → ∇u almost everywhere in the set
(x, t) ∈ ΩT : u(x, t) < ∞ .
To extract the pointwise convergent subsequence from a bounded sequence of weak supersolutions, we start by observing that the sequence (∇uj ) is bounded in Lp (τ1 , τ2 ; Lp (Ω )) for all subdomains Ωτ 1 ,τ2 = Ω × (τ1 , τ2 ) ΩT . This follows from Lemma 4.2 applied to uj − M and the boundedness of (uj ). Let μj denote the measure associated to uj by Theorem 3.9, and choose open polyhedra U U Ω and intervals (t1 , t2 ) (s1 , s2 ) (0, T ). If η ∈ C0∞ (Us 1 ,s2 ) with 0 η 1 and η = 1 in Ut1 ,t2 , we have μj (Ut1 ,t2 )
η dμj
Us
1 ,s2
=− Us
∂η uj dx dt + ∂t
1 ,s2
A(x, t, ∇uj ) · ∇η dx dt
Us
CM + C
1 ,s2
|∇uj |p dx dt Us
1 ,s2
1/p .
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Hence the sequence (μj (Ut1 ,t2 )) is bounded. For ϕ ∈ C0∞ (Ut1 ,t2 ), we have u , ϕ = − j
∂ϕ dx dt uj ∂t
Ut1 ,t2
= −
A(x, t, ∇uj ) · ∇ϕ dx dt +
Ut1 ,t2
Ut1 ,t2
C
ϕ dμj (x, t)
1/p |∇uj |p dx dt
+ μj (Ut1 ,t2 ) ϕL∞ (t
1,∞ (U )) 1 ,t2 ;W0
,
Ut1 ,t2
so that the sequence (u j ) is bounded in L1 (t1 , t2 ; W0−1,1 (U )). Recall that U is a polyhedron and hence W 1,p (U ) embeds compactly to L1 (U ) by the Rellich–Kondrachov compactness theorem. Moreover, L1 (U ) is contained in W0−1,1 (U ), so it follows from Theorem 5 in [14] that (uj ) is relatively compact in L1 (Ut1 ,t2 ). This allows us to pick a subsequence that converges pointwise almost everywhere in Ut1 ,t2 to a function u. To pass to the whole set Ω × (0, T ), we employ another diagonalization argument. Choose polyhedra U 1 U 2 · · · U j U j +1 · · · and intervals (t11 , t21 ) (t12 , t22 ) · · · so that ΩT =
∞
Utii ,t i . 1 2
i=1
The above reasoning allows us to pick a subsequence (u1j ) that converges pointwise almost everywhere in U 11
t1 ,t21
to a function u1 . We proceed inductively, and pick a subsequence (uk+1 j ) of
(ukj ) that converges almost everywhere in U k+1 k+1 uk = ul almost everywhere in U kk
t1
t1 ,t2k
to a function uk+1 . Since limits are unique,
,t2k+1
if l > k. Hence the diagonal sequence (ukk ) converges almost
everywhere in ΩT to a function u. As explained above, this completes the proof.
2
5. Existence of A-superparabolic solutions In this section we prove our main existence result, Theorem 5.8. Recall that a sequence of measures (μj ) converges weakly to a measure μ if
lim
j →∞ ΩT
ϕ dμj =
ϕ dμ
ΩT
for all ϕ ∈ C0∞ (ΩT ). The following well-known result asserts that for each finite positive Radon measure there exists an approximating sequence of functions in L∞ (ΩT ) in the sense of a weak convergence of measures. We repeat the proof given, for example, in [12] for the convenience of the reader.
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Lemma 5.1. Let μ be a finite positive Radon measure on ΩT . Then there is a sequence (fj ) of positive functions fj ∈ L∞ (ΩT ) such that fj dx dt μ(ΩT ) ΩT
and
lim
j →∞ ΩT
ϕfj dx dt =
ϕ dμ
ΩT
for every ϕ ∈ C0∞ (ΩT ). In other words, the sequence of measures (μj ) given by dμj (x, t) = fj dx dt converges weakly to μ. Proof. Let Qi,j , i = 1, . . . , Nj , be the dyadic cubes with side length 2−j such that Qi,j ΩT . We define fj (x, t) =
Nj μ(Qi,j ) i=1
|Qi,j |
χQi,j (x, t),
and show that the sequence (fj ) has the desired properties. Observe that fj dx dt =
Nj
μ(Qi,j ) μ(ΩT ),
i=1
ΩT
and thus the first property holds. Let then (xi,j , ti,j ) be the center of Qi,j . By the smoothness of ϕ, there is a constant C depending only on ϕ, such that ϕ(x, t) − ϕ(xi,j , ti,j ) C2−j for all (x, t) ∈ Qi,j . Hence, ϕ dμ − fj ϕ dx dt ΩT
ΩT
Nj μ(Qi,j ) ϕ dx dt ϕ dμ − ϕ(xi,j , ti,j ) dμ + ϕ(xi,j , ti,j ) dμ − |Qi,j | i=1 Q
i,j
C2−j
Nj
Qi,j
Qi,j
dμ C2−j μ(ΩT ).
i=1Q i,j
This proves the claim as j → ∞.
2
Qi,j
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In the proof of the next theorem we utilize the following standard existence result, see, e.g., Example 4.A in [13]. Suppose that f ∈ L∞ (ΩT ) has compact support in ΩT . Then there exists a unique function 1,p u ∈ Lp 0, T ; W0 (Ω) such that −
u
∂ϕ dx dt + ∂t
ΩT
A(x, t, ∇u) · ∇ϕ dx dt =
ΩT
ϕf dx dt
(5.2)
ΩT
for every ϕ ∈ C0∞ (ΩT ) and 1 lim t→0 t
t |u|2 dx dt = 0. 0 Ω
In particular, if f 0, then u is a weak supersolution. The following lemma provides us with a key estimate, cf. Lemma 3.6 above. Lemma 5.3. Let u be a solution of (5.2) with f 0. Then
∇ min(u, k)p dx dt + ess sup 0
ΩT
min(u, k)2 dx Ck Ω
f dx dt,
(5.4)
ΩT
for k = 1, 2, . . . . Proof. For each ϕ ∈ C0∞ (ΩT ), the mollification uε of u satisfies the regularized equation ΩT
∂uε ϕ dx dt + ∂t
A(x, t, ∇u)ε · ∇ϕ dx dt =
ΩT
f ε ϕ dx dt
(5.5)
ΩT
for small enough ε > 0. We prove the lemma by establishing a lower bound for the left-hand side, and an upper bound for the right-hand side. First, we choose a piecewise linear approximation χh , h > ε, of χ(0,T ) such that ⎧ ∂χh ⎪ ⎪ = 1/ h, ⎪ ⎪ ∂t ⎪ ⎨ χh = 1, ∂χh ⎪ ⎪ ⎪ = −1/ h, ⎪ ⎪ ⎩ ∂t χh = 0,
if h < t < 2h, if 2h < t < T − 2h, if T − 2h < t < T − h, otherwise,
and set uεk = min(uε , k). We use ϕ = uεk χh (here ϕ = 0, if t h or t T − h) as a test function, observing that χh gives enough room for the mollification because h > ε. We have
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∂uε ε ∂uεk ε ∂(uε − k)+ uk = uk + k . ∂t ∂t ∂t Thus the first term in the left-hand side of (5.5) becomes, after integration by parts,
1 ε 2 ∂χh u dx dt − 2 k ∂t
− ΩT
∂χh k uε − k + dx dt. ∂t
ΩT
Next, we would like to let ε → 0, but we only know that uεk converges to uk strongly for almost all real values of k. To deal with this, let us assume that an increasing sequence of numbers k such that the convergence holds has been chosen; then the conclusion of the lemma holds for these numbers, and this technicality plays no further role. We get the limit 1 − h
2h
1 2 1 uk dx dt + 2 h
h Ω
1 − h
2h
T−h T −2h Ω
1 2 u dx dt 2 k
1 k(u − k)+ dx dt + h
h Ω
T−h
k(u − k)+ dx dt T −2h Ω
as ε → 0. The negative terms in the above expression vanish as h → 0 by the initial condition while the positive terms can be ignored since we are proving a lower bound. The second term on the left-hand side reads
A(x, t, ∇u)ε · ∇ uεk χh dx dt.
ΩT
Here, we can simply let ε → 0, and then h → 0. This and the structure of A gives us the estimate α
|∇uk |p dx dt
ΩT
A(x, t, ∇uk ) · ∇uk dx dt.
ΩT
To deal with the right-hand side of (5.5), we note that
uk f χh dx dt
ΩT
uk f dx dt k
ΩT
f dx dt.
ΩT
Furthermore, the first term in the above estimate equals in the limit with the right-hand side of (5.5) as ε → 0. We have so far proved that
|∇uk |p dx dt Ck
ΩT
ΩT
f dx dt.
(5.6)
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To finish the proof, we repeat the above arguments with χ(0,T ) replaced by χ(0,τ ) , where 0 < τ < T is chosen so that 1 uk (x, τ ) dx ess sup uk (x, t) dx. 2 0
Ω
By the choice of τ , we obtain the inequality
|∇uk | dx dt + ess sup
uk (x, t) dx Ck 2
p
0
Ωτ
Ω
f dx dt.
(5.7)
ΩT
A combination of (5.6) and (5.7) now completes the proof.
2
Next we establish the existence of a solution to the measure data problem. Theorem 5.8. Let μ be a finite positive Radon measure in ΩT . Then there is an A-superparabolic function u in ΩT such that min(u, k) ∈ Lp (0, T ; W 1,p (Ω)) for all k > 0 and ∂u − div A(x, t, ∇u) = μ ∂t in the weak sense. Proof. Let (fj ) be the approximating sequence to μ obtained from Lemma 5.1 and denote by (uj ) the corresponding sequence of weak supersolutions satisfying (5.2). By Theorem 4.3, there is an A-hyperparabolic function u such that we can assume that uj → u
and ∇ min(uj , k) → ∇ min(u, k)
almost everywhere by passing to a subsequence. As the first step, we prove that u is finite almost everywhere, and thus u is A-superparabolic. To this end, according to Lemmas 5.3 and 5.1, we have
∇ min(uj , k)p dx dt Ck
ΩT
fj dx dt Cμ(ΩT )k.
(5.9)
ΩT 1,p
Since min(uj , k) ∈ Lp (0, T ; W0 (Ω)), the Sobolev–Poincaré inequality and (5.9) imply ΩT
min(uj , k)p dx dt C
∇ min(uj , k)p dx dt
ΩT
Cμ(ΩT )k,
(5.10)
where C is independent of k and j . Since uj → u almost everywhere, it follows from Fatou’s lemma and (5.10) that
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min(u, k)p dx dt Cμ(ΩT )k.
ΩT
This estimate implies that u is finite almost everywhere. Indeed, denoting
E = (x, t) ∈ ΩT : u(x, t) = ∞ , we have |E| =
1 kp
k p dx dt
1 kp
min(u, k)p dx dt Ck 1−p → 0
ΩT
E
as k → ∞. Thus, u is A-superparabolic and by Theorem 3.9, there exists a measure ν such that ∂u − div A(x, t, ∇u) = ν ∂t
(5.11)
in the weak sense. We will complete the proof by showing that μ = ν. The constants on the right-hand sides of (5.9) and (5.10) are independent of j . Thus Lemma 3.6 implies that the sequence (|∇uj |p−1 ) is bounded in Lq (ΩT ) for some q > 1. Next we use the structure of A, and obtain A(x, t, ∇uj )q dx dt C |∇uj |q(p−1) dx dt C. ΩT
ΩT
Thus the sequence (A(x, t, ∇uj )) is also bounded in Lq (ΩT ), and it follows from the pointwise convergence of ∇uj to ∇u, and the continuity of ξ → A(x, t, ξ ) that A(x, t, ∇uj ) → A(x, t, ∇u) pointwise almost everywhere, and thus weakly in Lq (ΩT ) at least for a subsequence, since the pointwise limit identifies the weak limit. Similarly, the sequence (uj ) is bounded in L(p−1)q (ΩT ) and thus a subsequence converges weakly in L(p−1)q (ΩT ). We use the weak convergences and (5.11) to conclude that ∂ϕ + A(x, t, ∇uj ) · ∇ϕ dx dt ϕ dμj = lim −uj lim j →∞ j →∞ ∂t ΩT
=
ΩT
−u
∂ϕ + A(x, t, ∇u) · ∇ϕ dx dt ∂t
ΩT
=
ϕ dν,
ΩT
which completes the proof.
2
Observe that we cannot directly deduce from the boundedness of gradients that (A(x, t, ∇uj )) converges weakly to A(x, t, ∇u) above. The additional information needed is the pointwise convergence of the gradients from Theorem 4.3 and the continuity of A with respect to the gradient variable.
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We close the paper by recording the following simple observation. Note that the current tools do not allow us to prove the claim for any solution of (5.13), since solutions to equations involving measures are not unique in general. Recall that in a general open set V of Rn+1 u is a solution if it is a solution in all sets Ut2 ,t2 V . Theorem 5.12. If u is a weak solution of ∂u − div A(x, t, ∇u) = μ ∂t
(5.13)
in ΩT given by Theorem 5.8, then u is a weak solution of ∂u − div A(x, t, ∇u) = 0 ∂t
(5.14)
in ΩT \ spt μ. Proof. The proof consists of verifying two facts. First, we must check that the limit has the right a priori integrability, and then show that it satisfies the weak formulation. Let (μj ) be the approximating sequence of μ from Lemma 5.1. From the proof of the lemma, we see that the support of μj is contained in the set
Ej = (x, t) ∈ ΩT : dist(z, spt μ) c 2−j , where the constant c is independent of j . Thus the corresponding supersolution uj is a nonnegative solution of (5.14) in ΩT \ E j . Pick any set Ut1 ,t2 ΩT \ spt μ. Then Ut1 ,t2 ΩT \ E j for all sufficiently large j . We take the subsequence from the proof of the previous theorem with uniform bounds in L(p−1)q (Ut1 ,t2 ), q > 1, converging to a limit u. We combine the bound for the sequence (uj ) in L(p−1)q (Ut1 ,t2 ) with a weak Harnack estimate (see [4] or [11]) to conclude that the sequence (uj ) is bounded in Ut1 ,t2 , and hence the limit function u is also bounded. The boundedness of u and Lemma 4.2 imply that u belongs to Lp (t1 , t2 ; W 1,p (U )). We are left with the task of checking the weak formulation. Recall from the proof of Theorem 5.8 that (uj ) and (A(x, t, ∇uj )) converge weakly in Lq (ΩT ) to u and A(x, t, ∇u), respectively. This implies that t2 t2 ∂ϕ 0 = lim − dx dt + uj A(x, t, ∇uj ) · ∇ϕ dx dt j →∞ ∂t t1 U
t2 =− t1 U
∂ϕ u dx dt + ∂t
t1 U
t2 A(x, t, ∇u) · ∇ϕ dx dt t1 U
for all ϕ ∈ C0∞ (Ut1 ,t2 ). Since Ut1 ,t2 ΩT \ spt μ was arbitrary, this implies that u is a weak solution in ΩT \ spt μ. 2
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References [1] L. Boccardo, A. Dall’Aglio, T. Gallouët, L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal. 147 (1) (1997) 237–258. [2] L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1) (1989) 149–169. [3] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. [4] E. DiBenedetto, U. Gianazza, V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math. 200 (2) (2008) 181–209. [5] P. Juutinen, P. Lindqvist, J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal. 33 (2001) 699–717. [6] T. Kilpeläinen, P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal. 27 (3) (1996) 661–683. [7] T. Kilpeläinen, J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 (4) (1992) 591–613. [8] J. Kinnunen, P. Lindqvist, Summability of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (1) (2005) 59–78. [9] J. Kinnunen, P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl. (4) 185 (3) (2006) 411–435. [10] R. Korte, T. Kuusi, M. Parviainen, A connection between a general class of superparabolic functions and supersolutions, J. Evol. Equ., in press. [11] T. Kuusi, Lower semicontinuity of weak supersolutions to nonlinear parabolic equations, submitted for publication. [12] P. Mikkonen, On the Wolff potential and quasilinear elliptic equations involving measures, Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996) 71. [13] R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surveys Monogr., vol. 49, American Mathematical Society, Providence, RI, 1997. [14] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl. (4) 146 (1987) 65–96.
Journal of Functional Analysis 258 (2010) 729–757 www.elsevier.com/locate/jfa
Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations C. Preda ∗,1 , P. Preda, A. Craciunescu West University of Timi¸soara, Bd. V. Parvan, no. 4, Timi¸soara 300223, Romania Received 7 May 2009; accepted 11 September 2009 Available online 24 September 2009 Communicated by J. Coron
Abstract We prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a “no past” exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow). Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift and therefore our approach addresses most of the possible pairs of admissible spaces. Complete characterizations for the exponential dichotomy of cocycles are also obtained. Moreover, we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the classical concept defined by Sacker and Sell (1994) in [23]. Our definition of exponential dichotomy follows partially the definition given by Chow and Leiva (1996) in [4] in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to O. Perron (1930) [14], J. Daleckij and M. Krein (1974) [7], J.L. Massera and J.J. Schäffer (1966) [11], N. van Minh, F. Räbiger and R. Schnaubelt (1998) [26]. © 2009 Elsevier Inc. All rights reserved. Keywords: (Nonlinear) (semi)flows; Exponentially bounded, strongly continuous cocycles (over (semi)flows); Exponential dichotomy
* Corresponding author.
E-mail addresses:
[email protected] (C. Preda),
[email protected] (P. Preda),
[email protected] (A. Craciunescu). 1 Part of this work was done while the corresponding author was an Adjunct Assistant Professor at the Department of Mathematics, University of California at Los Angeles (UCLA). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.002
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1. Introduction The study of linear systems seems to be more than classic by now but it still plays a central role in the qualitative theory of dynamical systems. This is mainly due to the fact that a comprehensive analysis of nonlinear systems using perturbation techniques requires a linear machinery, since, in most cases, the stability of solutions can be obtained from the linearization along the solution, the so-called variational equation. It is known for instance that the qualitative theory of (nonlinear) (semi)flows on (locally) compact spaces or (σ -)finite measure spaces relies heavily on notions like stability or exponential dichotomy for the associated linear skewproduct (semi)flow. It is worth to note that all truly infinite-dimensional situations, e.g. flows originating from partial differential equations and functional differential equations, only yield linear skew-product (semi)flows. For instance, it is known by now that well-known equations like Navier–Stokes, Bubnov–Galerkin, Taylor–Couette can be modeled asymptotically by associating a linear skew-product (semi)flow (for details we refer the reader to [16]). In this paper we investigate the existence of exponential dichotomies for linear skew-product semiflows (LSPS). An exponential dichotomy is one of the most basic concepts arising in the theory of dynamical systems. This topic, for example, plays a central role in the Hadamard– Perron theory of invariant manifolds for dynamical systems, and in many aspects of the theory of stability. Even in the context of bifurcation theory, the exponential dichotomy has a role. However in this context, the exponential dichotomy is represented by its younger sibling, the exponential trichotomy. In particular, topics such as the reduction principle and the center manifold theorem, the robustness of periodic solutions and invariant manifolds, as seen in the Poincaré–Melnikov scenario, are based on the theory of exponential trichotomies. The notion of exponential dichotomy of linear differential equations was introduced by Perron [14], which approaches the problem of conditional stability of a system x(t) ˙ = A(t)x and its connection with the existence of bounded solutions of the equation x(t) ˙ = A(t)x + f (x, t), where the state space is a Banach space X and the operator-valued function A(·) is bounded, continuous in the strong operator topology. An important contribution to these problems is the work done by Massera and Schäffer [11], Daleckij and Krein [7], Coppel [6], Sacker and Sell [22]. The need for a new approach came from the observation that for a time dependent linear differential equation with unbounded operator A(t), the solutions, generally speaking, either cannot be extended in the direction of the negative times, or can be extended, but not uniquely. All the problems above can be treated in the unified setting of a linear skew-product semiflow (LSPS). In [23] Sacker and Sell employ a notion of exponential dichotomy for skew-product semiflow with the restriction that the unstable subspace has finite dimension, and they point out a sufficient condition for the existence of exponential dichotomy for skew-product semiflow. In this work we use a concept of a no past exponential dichotomy for skew-product semiflow weaker than the concept used by Sacker and Sell. Our definition follows partially the definition (of exponential dichotomy) introduced by Chow and Leiva in [4] in the sense that we allow the unstable subspace to have infinite dimension. We go even more general and we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). We continue the approach initiated by Perron (the so-called “admissibility condition” or “test function method”) and we prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a (no past) exponential dichotomy. Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift (see Definition 2.1) and therefore our approach addresses most of the possible pairs of admissible spaces
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(see Theorem 5.4). Thus we generalize some known results due to O. Perron [14], J. Daleckij and M. Krein [7], J.L. Massera and J.J. Schäffer [11], N. van Minh, F. Räbiger and R. Schnaubelt [26]. 2. Preliminaries We now recall some preliminaries. We will use the symbol R+ to denote the set {t ∈ R: t 0}. Also, let X be real or complex Banach space and X ∗ its dual space. By M(R+ , X) we will denote the space of all strongly measurable functions from R+ to X. Furthermore B(X) denotes the Banach algebra of all bounded linear operators acting on the Banach space X. The norms on X, X ∗ , B(X) shall be denoted by the symbol · . As usual, we put Cb (R, X) = {f : R → X: f is continuous and bounded}; 1 f (t) dt < ∞, for each compact K from R+ ; Lloc (R+ , X) = f ∈ M(R+ , X): K
p Lp (R+ , X) = f ∈ M(R+ , X): f (t) dt < ∞ ,
where p ∈ [1, ∞);
R+
L∞ (R+ , X) = f ∈ M(R+ , X): ess sup f (t) < ∞ ; t∈R+
t+1 p M (R+ , X) = f ∈ M(R+ , X): sup f (s) ds < ∞ , p
t∈R+
where p ∈ [1, ∞);
t
T (R+ , X) is the space of all functions f ∈ L1loc (R+ , X) with the property that there exist (τn )n∈N and (an )n∈N two sequences of positive real numbers such that ∞
∞
an < ∞ and f (t) an χ[τn ,τn +1] (t)
n=0
a.e.
n=0
We recall that Cb (R, X), Lp (R+ , X), L∞ (R+ , X), M p (R+ , X), T (R+ , X) are Banach spaces endowed with the respectively norms: |||f ||| = sup f (t) ; t∈R
f p =
f (t)p dt
R+
f ∞ = ess sup f (t); t∈R+
1
p
;
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1 t+1 p p f (s) ds ; f M p (R+ ,X) = sup t∈R+
f T (R+ ,X) = inf
t ∞
an : where (an )n∈N satisfy the above inequality .
n=0
Definition 2.1. A Banach space (E(R+ , R), · E(R+ ,R) ) is said to be a scalar-valued Schäffer function space if the following conditions hold: (s1 ) E ⊂ L1loc (R+ , R) and for each compact K ⊂ R+ there is αK > 0 such that
f (t) dt αK f E(R
+ ,R)
,
for all f ∈ E(R+ , R).
K
(s2 ) φ[0,t] ∈ E(R+ , R), for each t 0, where φ[0,t] denotes the characteristic function (indicator) of the interval [0, t]. (s3 ) If f ∈ E(R+ , R) and h ∈ M(R+ , R) with |h| |f |, then h ∈ E(R+ , R) and hE(R+ ,R) f E(R+ ,R) . (s4 ) If f ∈ E(R+ , R), t 0, gt : R+ → R, gt (s) =
0, f (s − t),
s ∈ [0, t), s ∈ [t, ∞),
then gt ∈ E(R+ , R) and gt E(R+ ,R) = f E(R+ ,R) . Example 2.1. It is a routine to verify that M p (R+ , R), Lp (R+ , R), L∞ (R+ , R) and T (R+ , R), the spaces mentioned above are particular examples of scalar-valued Schäffer function spaces. One can easily remark that T (R+ , R) ⊂ E(R+ , R) ⊂ M 1 (R+ , R), for any scalar-valued Schäffer function space E(R+ , R). For details we refer the reader to [11, 23.G, p. 60]. Example 2.2. The class of scalar-valued Schäffer function spaces contains also the well-known class of scalar-valued Orlicz function spaces. For convenience we recall briefly to the reader the notion of a scalar-valued Orlicz function space. Let ϕ : R+ → R+ be a function which is non-decreasing, left-continuous, ϕ(t) > 0, for all t > 0. Define t Φ(t) =
ϕ(s) ds. 0
A function Φ of this form is called a Young function. For f : R+ → R a measurable function and Φ a Young function we define ∞ M (f ) = Φ
0
Φ f (s) ds.
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The set LΦ (R+ , R) of all f for which there exists a k > 0 that M Φ (kf ) < ∞ is clearly a vector space. Using the Luxemburg norm, Φ 1 f 1 , ρ (f ) = inf k > 0: M k Φ
we get that (LΦ (R+ , R), ρ Φ ) is a Banach space. It is trivial to see that (LΦ (R+ , R), ρ Φ ) verifies the conditions (s2 ), (s3 ), (s4 ). For checking (s1 ) we consider f ∈ LΦ (R+ , R), t > 0, k > 0 such that M Φ ( k1 f ) 1. It follows that
1 Φ kt
t
t f (s) ds 1 Φ 1 f (s) ds 1 , t k t
0
0
and so t
f (s) ds tΦ −1 1 k t
0
which implies that t
f (s) ds tΦ −1 1 ρ Φ (f ), t
0
for all f ∈ LΦ (R+ , R), t > 0, and hence the condition (s1 ) is also verified. Remark 2.1. Let now LΦ (R+ , R) be a scalar-valued Orlicz function space. We denote by LΦ (R+ , X) = f ∈ M(R+ , X) : t → f (t): R+ → R+ is in LΦ (R+ , R) . It is easy to check that LΦ (R+ , X) is a Banach space endowed with the norm f LΦ (R+ ,X) = f (·)ρ Φ . We will call LΦ (R+ , X) as a vector-valued Orlicz function space. Remark 2.2. LΦ (R+ , R) = Lp (R+ , R) if and only if Φ(t) = t p , for all t 0. For details we refer the reader to [20]. Remark 2.3. Take now E(R+ , R) to be a scalar-valued Schäffer function space. We define E(R+ , X) = f ∈ M(R+ , X) : t → f (t): R+ → R+ is in E(R+ , R) . Obviously that E(R+ , X) will be called as a vector-valued Schäffer function space.
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Remark 2.4. E(R+ , X) is a Banach space endowed with the norm f E(R+ ,X) = f (·)E(R
+ ,R)
.
For details we refer the reader to [19, Remark 2.1, p. 196]. Remark 2.5. If {fn }n∈N ⊂ E(R+ , X), f ∈ E(R+ , X), fn → f in E(R+ , X) when n → ∞, then there exists {fnk }k∈N a subsequence of {fn }n∈N such that fnk → f
a.e.
For the proof of this fact see [19, Remark 2.2, p. 197]. For a scalar-valued Schäffer function space E(R+ , R), we denote by α(·, E), β(·, E) : R+ → R+ the following functions:
t
α(t, E) = inf α > 0:
f (s) ds αf E(R
+ ,R)
, for all f ∈ E ,
0
β(t, E) = χ[0,t] E(R+ ,R) . Remark 2.6. It is known (see for instance [11, (23.1) from p. 61, and (23.K) from p. 62]) that α(·, E), β(·, E) are non-decreasing functions and t α(t, E)β(t, E) 2t,
for all t 0.
(1)
Remark 2.7. It is easy to compute the above numbers for Lp (R+ , R) and M p (R+ , R): α t, Lp = β t, Lp =
1− p1
t t,
,
p ∈ [1, ∞), p = ∞,
t 0,
1
t p , p ∈ [1, ∞), 1, p = ∞,
t 0.
1− 1
Also we can see that t α(t, M p ) [t] + {t} p , for each (p, t) ∈ [1, ∞) × R+ . Here [t] denotes the largest integer less than or equal t and {t} = t − [t]: 1 p β t, M p = t , 1,
t ∈ [0, 1), t 1.
Furthermore α(t, LΦ ) = tΦ −1 ( 1t ) and β(t, LΦ ) = (Φ −1 ( 1t ))−1 .
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3. Linear skew-product semiflows (LSPS) Consider now the trivial Banach bundle E = X × Θ, where X is a fixed Banach space (the state space) and Θ is a metric space (the base space). For details of Banach bundles we refer to [24, Chapter 4]. Definition 3.1. A (nonlinear) semiflow σ : Θ × R+ → Θ is defined by the properties: (i) σ (θ, 0) = θ , for all θ ∈ Θ; (ii) σ (θ, t + s) = σ (σ (θ, s), t), for all θ ∈ Θ and t, s ∈ R+ . If in addition (θ, t) → σ (θ, t) is continuous, then σ is called a continuous (nonlinear) semiflow on Θ. Also, if (ii) holds for any t, s ∈ R then σ is said to be a (nonlinear) flow on Θ. Definition 3.2. A family {T (t)}t0 of linear and bounded operators acting on X, is said to be a C0 -semigroup on X if the following conditions hold: (i) T (0) = I ; (ii) T (t + s) = T (t)T (s), for all t, s 0; (iii) there exists limt→0+ T (t)x = x, for all x ∈ X. If the second property holds for any t, s ∈ R then {T (t)}t∈R is called a C0 -group. For a general presentation of the theory of C0 -semigroups we refer the reader to [13]. Remark 3.1. It is known the connection between (nonlinear) (semi)flows, first order differential operators, and (linear) one-parameter (semi)groups. For instance, consider a continuously differentiable vector field F : Rn → Rn with supθ∈Rn DF(θ ) < ∞, for the derivative DF(θ ) of F and θ ∈ Rn . Take the first order differential operator on X := C0 Rn = f : Rn → Rn : f is continuous and vanishes at infinity corresponding to the vector field F , n
∂f Fi (θ ) (θ ), Af (θ ) = grad f (θ ), F (θ ) = ∂θi i=1
for f ∈ Cc1 (Rn ) = {f : Rn → Rn : f continuously differentiable, with compact support}, and θ ∈ Rn . For 0 = f ∈ X, the duality set J (f ) = {x ∗ ∈ X ∗ : x ∗ (f ) = f 2 = x ∗ 2 } contains all point measures supported by those points θ0 ∈ Rn where |f | reaches its maximum. More precisely,
f (θ0 )δθ0 : θ0 ∈ Rn , f (θ0 ) = f ⊂ J (f ).
Since ∂f∂θ(θi0 ) = 0 while |f (θ0 )| = f , it follows that A is dissipative (i.e. there exists j (f ) ∈ J (f ) such that Re j (f )(Af ) 0). Since F is globally Lipschitz it follows from standard argu-
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ments that there exists a continuous flow σ : Rn × R → Rn which solves the differential equation ∂ σ (θ, t) = F σ (θ, t) , ∂t
for all t ∈ R and θ ∈ Rn see [1, Thm. 10.3] .
To such a flow we associate a one-parameter group of linear operators on C0 (Rn ) given by T (t)f (θ ) := f σ (θ, t) ,
for θ ∈ Rn , t ∈ R,
the so-called group induced by the flow σ . It can be proved that the generator of the above group is the closure of the first order differential operator A. The domain of the generator will be D(A) = Cc1 (Rn ). For details we refer the reader to [8]. The general relation between (nonlinear) semiflows and linear semigroups is given in the example below. Example 3.1. Let Θ be a compact metric space and take X = C(Θ), where C(Θ) = {f : Θ → C: f continuous on Θ}. (i) The (nonlinear) semiflow σ is continuous if and only if it induces a strongly continuous semigroup {T (t)}t0 on X by the formula: T (t)f (θ ) := f σ (θ, t) ,
for θ ∈ Θ, t 0, f ∈ X.
(2)
(ii) The generator (A, D(A)) of {T (t)}t0 is a derivation. (iii) Every strongly continuous semigroup {T (t)}t0 on X that consists of algebra homomorphisms originates, via (2), from a continuous (nonlinear) semiflow on Θ (see [12, B-II, Thm. 3.4]). We will state in the next the basic definitions concerning cocycles, linear skew-product (semi)flows, and dichotomy. Before to state the formal definitions, let us recall the prototypical example of a linear skew-product flow; namely, the skew-product flow associated with the solutions of a nonautonomous differential equation u˙ = A(t)u on a Banach space X. For this case, consider the translation flow σ (θ, t) = θ + t on R and the trivial bundle X × R over R. A linear skew-product flow on X × R is defined by (u0 , θ, t) → (u(u0 , θ, t), θ + t) where t → u(u0 , θ, t) is the solution of the differential equation with the initial condition u(u0 , θ, θ ) = u0 . To avoid technical complications for the general case, we will define the notion of a cocycle and a linear skew-product (semi)flow in the setting of a trivial vector bundle. It is worth to mention that the theory is valid for general vector bundles, but the topology of nontrivial bundles plays no role in the analysis. In fact, the constructions of this section are local. They can always be carried out in a natural vector bundle chart. Definition 3.3. Let σ be a (nonlinear) continuous semiflow on Θ. A strongly continuous cocycle over the continuous semiflow σ is an operator-valued function Φ : Θ × R+ → B(X),
(θ, t) → Φ(θ, t),
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that satisfies the following properties: (i) Φ(θ, 0) = I (I – the identity operator in X), for all θ ∈ Θ, t ∈ R+ ; (ii) Φ(θ, ·)x is continuous for each θ ∈ Θ, x ∈ X; (iii) Φ(θ, t + s) = Φ(σ (θ, t), s)Φ(θ, t), for all t, s 0 and θ ∈ Θ (the cocycle identity). If, in addition, (iv) there exist constants M, ω such that Φ(θ, t) Meωt ,
for t 0, θ ∈ Θ,
then the strongly continuous cocycle is exponentially bounded. The linear skew-product semiflow (LSPS), associated with the above cocycle, is the dynamical system π = (Φ, σ ) on E = X × Θ defined by π : X × Θ × R+ → X × Θ,
(x, θ, t) → π(x, θ, t) = Φ(θ, t)x, σ (θ, t) .
Remark 3.2. Note that the operators in a strongly continuous cocycle are not assumed to be invertible. For this reason, the cocycle is parameterized by t 0, but not by t ∈ R. By the Uniform Boundedness Principle, if the base space Θ is compact, then a strongly continuous cocycle is exponentially bounded. Example 3.2. The classic example of a cocycle arises as the solution operator for a variational equation. Take σ to be a continuous flow on the locally compact metric space Θ, and {A(θ ): θ ∈ Θ} be a family of (possibly unbounded) densely defined closed operators on the Banach space X. A strongly continuous cocycle Φ(·, t)x is said to solve the variational equation u(t) ˙ = A σ (θ, t) u(t),
θ ∈ Θ, t ∈ R,
(3)
if, for every θ ∈ Θ, we can find a dense subset Zθ ⊂ D(A(θ )) such that, for every uθ ∈ Zθ ⊂ D(A(θ )), the function t → Φ(θ, t)uθ is differentiable (for t 0) and the values u(t) ∈ D(A(σ (θ, t))), and t → u(t) verifies the above differential equation. More restrictive definition can be given if we impose that Zθ = D(A(θ )) or even Zθ = D(A(θ )) = D; that is, Zθ is independent of θ . Characterizations of (exponential, discrete, pointwise) dichotomy for the solutions of the above variational systems were obtained through various techniques. For a complete presentation of these results we refer the reader to [2, Chapter 7]. Differential equations of type (3) arise from two reach (and essential) sources that we describe below. First, consider a nonlinear differential equation on X: x˙ = F (x),
(4)
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with F : X → X being Fréchet differentiable. Assume that (4) has a compact invariant set Θ ⊂ X (i.e. the solution t → x(θ, t), with x(θ, 0) = θ , has its values in Θ, for each t ∈ R, whenever the initial point θ ∈ Θ). Then the family of functions {σ (·, t) : θ → x(θ, t), t ∈ R} describes a flow on Θ. If θ ∈ Θ then for any other initial condition u0 ∈ X, the difference u(t) = x(u0 , t) − x(θ, t) such that u(t) ˙ = DF x(θ, t) u(t) + η(u, x),
η(u, x) = o(u),
|u| → 0.
Then the differential equation (3) with A(θ ) = DF(θ ), called the variational equation, determines the linearized flow of x˙ = F (x). It is worth to note that, in an infinite-dimensional context, the operators A(θ ) could be unbounded. Second, define the hull of a continuous function f : R → B(X) to be the set of operator-valued functions, given below: Hull(f ) = closure f (· + τ ): τ ∈ R . Under appropriate assumptions, the set Θ := Hull(f ) may be a compact set of operator-valued functions on R. For example, if f : R → B(Rn ) is almost-periodic and the closure is taken in the topology of uniform convergence on compact subsets of R, then, by Bochner’s Theorem, Θ is compact in the space of continuous matrix-valued functions. Consider now the flow (on Θ) given by σ (θ, t)(s) = θ (s + t), with t, s ∈ R. If we put also A(θ ) = θ (0) ∈ B(X), then we get in (3) all differential equations of the form u˙ = θ (t)u, where the function θ is in the hull of f . The next example shows how a cocycle arises from the linearization of a nonlinear partial differential equation. We will sketch extremely briefly the case of the linearized Navier–Stokes equation. Example 3.3. Consider the Navier–Stokes equations ∂v = νv − v, ∇ v − grad p + g, ∂t
div v = 0
on a bounded domain Ω ⊂ R2 with zero boundary conditions. As usual, v : Ω → R2 denotes the velocity of an incompressible fluid, ν measures the viscosity of the fluid, p : Ω → R represents the pressure, and the function g : R × Ω → R2 is a time-dependent forcing term. Take the orthogonal decomposition L2 (Ω; R2 ) = X ⊕ Hπ , with X being the closure in L2 (Ω; R2 ) of the C ∞ divergence-free (∇ · v = 0) vector fields with compact support in Ω, and Hπ being the closure in L2 (Ω; R2 ) of the gradients ∇p of all p ∈ C 1 (Ω; R), see for instance Constantin and Foias [5]. Let P : L2 (Ω; R2 ) → X be the corresponding orthogonal projection, and define A = P , B(v, u) = −P v, ∇ u, and f = P g. Thus, see for instance Temam [25], the Navier– Stokes equation can be rewritten as an abstract equation on the Hilbert space X: dv = νAv + B(v, v) + f, dt
v(0) = v0 .
(5)
The operator A with D(A) = X ∩ H 2 (Ω; R2 ) is a negative operator, and thus it generates an 1 analytic semigroup on X (see for instance [25]). Define now V = D((−A) 2 ). Suppose that the function F , defined by F (t) = f (t, ·), t ∈ R, is in Cb (R; X). Furthermore, suppose that the
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positive hull of F , H + (F ) = closureCb (R;X) Fτ = F (· + τ ): τ ∈ R+ is a compact subset of Cb (R; X). Therefore, we have that the omega-limit set ω(F ) = + τ 0 H (Fτ ) is nonempty and compact. Moreover, we can find a global compact attractor Θ ⊂ D(A) × ω(F ) for the semiflow generated by the strong solutions of the abstract equation (5). For details we refer the reader to Raugel and Sell [21, Sections 2.11–2.12] and the references therein. This attractor is invariant under the flow σ defined by (v, f ) → θτ = (vτ , fτ ),
τ ∈ R,
where fτ (t, ·) = f (t + τ, ·) and vτ (t, ·) = v(t + τ, ·) for the strong solution v(t, ·) of Eq. (5). If θ = (v0 , f ) ∈ Θ and v(t) = v(t; f, v0 ), t 0, is the corresponding strong solution of Eq. (5), then v(·; f, v0 ) ∈ C [0, ∞); V ∩ L∞ (0, ∞); V ∩ L∞ loc (0, ∞); D(A) . For details we refer the reader to [5]. The linearized Navier–Stokes equation along the solution v is given by dx = νAx + B v(t), x + B x, v(t) , dt
x(0) = x0 ∈ X.
(6)
Accordingly to [23], we have that if x0 ∈ V , then there is a unique strong solution x(t) = Φ(θ, t)x0 of the linearized equation (6) such that x(·) ∈ C [0, ∞); V ∩ L∞ loc (0, ∞); D(A) ,
xt (·) ∈ L2loc (0, ∞); X ,
where Φ(θ, t) is the solution operator of (6). Clearly Φ is a cocycle over the flow σ on Θ. Also, for the study of exponential dichotomies for the Navier–Stokes equations we refer the reader to [15]. Example 3.4. Let X be a Banach space and Θ a compact topological Hausdorff space. Consider the following linear dependent system x(t) ˙ = A σ (θ, t) x(t),
t > 0,
(7)
where A(σ (θ, t))x(t) = A + B(σ (θ, t)), A is the infinitesimal generator of the strongly continuous semigroup {T (t)}t0 and σ is a flow on Θ and B(θ ) ∈ B(X), θ ∈ Θ. To be precise in which sense the above equation generates a linear skew-product semiflow, we shall consider the following family of integral differential equations: t x(t) = T (t)x0 +
T (t − s)B σ (θ, s) x(s) ds,
t 0, θ ∈ Θ.
0
A solution x(t) = x(t, θ ) of Eq. (8) is called a mild solution of (7).
(8)
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Chow and Leiva establish in [3] that if A is the infinitesimal generator of the C0 -semigroup {T (t)}t0 on X and if B : Θ → B(X) is also strongly continuous, then for each θ ∈ Θ and x0 ∈ X the problem x(t) ˙ = A σ (θ, t) x(t) = A + B σ (θ, t) x(t);
x(0) = x0
has a unique mild solution given by t Φ(θ, t)x0 = T (t)x0 +
T (t − s)B σ (θ, s) Φ(θ, s)x0 ds.
0
Moreover if A0 is the infinitesimal generator of a strongly continuous semigroup {T0 (t)}t0 , and the mapping θ → A(θ ) − A0 : Θ → B(X) is strongly continuous and the equation x(t) ˙ = A(σ (θ, t))x(t) has an exponential dichotomy over Θ then there exists > 0 such that for any mapping θ → B(θ ) : Θ → B(X) strongly continuous and B(θ ) < , θ ∈ Θ, the equation x(t) ˙ = A σ (θ, t) + B σ (θ, t) x(t) has also exponential dichotomy. For details we refer the reader to [3]. Definition 3.4. A family of linear and bounded operators {U (t, t0 )}tt0 0 is said to be a twoparameter evolution family if the following conditions hold: (i) U (t, t) = I , for all t 0; (ii) U (t, s)U (s, t0 ) = U (t, t0 ), for all t s t0 0; (iii) U (·, t0 )x is continuous on [t0 , ∞), for all t0 0, x ∈ X; U (t, ·)x is continuous on [0, t], for all t 0, x ∈ X; (iv) there exist M, ω > 0 such that U (t, t0 ) Meω(t−t0 ) ,
for all t t0 0.
For a general presentation of the theory of two-parameter evolution families we refer the reader to [2] or [7]. Example 3.5. Let Θ = R+ , σ (θ, t) = θ + t and let {U (t, s)}ts be an evolution family on the Banach space X. We define ΦU (θ, t) = U (t + θ, θ ),
for all (θ, t) ∈ R+ × R+ .
Then {ΦU (θ, t)}θ∈Θ, t0 is an exponentially bounded, strongly continuous cocycle (over the above semiflow σ ). Therefore, we can say that the notion of a cocycle generalizes the classic notion of a two-parameter evolution family. An account of the results concerning the analysis of the exponential dichotomy for evolution families is given in [2, Chapter 4].
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4. (No past) exponential dichotomy and admissibility Let now (E(R+ , X), F (R+ , X)) be a pair of vector-valued Schäffer function spaces and take {Φ(θ, t)}θ∈Θ, t0 as an exponentially bounded, strongly continuous cocycle (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) and π = (Φ, σ ) as the associated linear skew-product semiflow on E. Since E = X × Θ is a trivial Banach bundle (see for instance [3, Remark 2.1], and [24, Chapter 4]), we define for any subset X ⊂ E the fibers X (θ ) = x ∈ X: (x, θ ) ∈ X ,
θ ∈ Θ.
In particular E(θ ) = X. Now we denote X1,F = (x, θ ) ∈ E: Φ(θ, ·)x ∈ F (R+ , X) . The corresponding fiber is X1,F (θ ) = {x ∈ X: (x, θ ) ∈ X1,F }, θ ∈ Θ. It can be seen that X1,F (θ ) is a vector subspace of X. In what follows X1,F (θ ) will be assumed complemented (i.e. X1,F (θ ) is closed and there exists X2,F (θ ) a closed subspace such that X = X1,F (θ ) ⊕ X2,F (θ )). Also we denote by PF (θ ) a projection onto X1,F (θ ) along X2,F (θ ) (that is PF (θ ) ∈ B(X), PF (θ )2 = PF (θ ) and Ker(PF (θ )) = X2,F (θ )) and by QF (θ ) = I − PF (θ ). Remark 4.1. If x ∈ X2,F (θ ), x = 0 then Φ(θ, t)x = 0, for all t 0, θ ∈ Θ. Proof. Assume for a contradiction that there exist t0 0 and θ0 ∈ Θ such that Φ(θ0 , t0 )x = 0. Then Φ(θ0 , t0 + s)x = Φ σ (θ0 , s), s Φ(θ0 , t0 )x = 0, for each s 0. It follows that x ∈ X1,F (θ0 ) and thus x = 0.
2
Definition 4.1. The pair (E(R+ , X), F (R+ , X)) is said to be admissible to an exponentially bounded, strongly continuous cocycle {Φ(θ, t)}θ∈Θ, t0 (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) if for each f ∈ E(R+ , X) and θ ∈ Θ, there exists x ∈ X such that t u(·; θ, x, f ) : R+ → X,
u(t; θ, x, f ) = Φ(θ, t)x +
Φ σ (θ, s), t − s f (s) ds
0
belongs to F (R+ , X). Definition 4.2. An exponentially bounded, strongly continuous cocycle {Φ(θ, t)}θ∈Θ, t0 (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) has an exponential dichotomy if there exists a family of projectors {P (θ )}θ∈Θ (i.e. P (θ ) ∈ B(X), and P 2 (θ ) = P (θ ), for each θ ∈ Θ) such that (i) Φ(θ, t)P (θ ) = P (σ (θ, t))Φ(θ, t), for all (θ, t) ∈ Θ × R+ (the invariance property). (ii) Φ(θ, t) : Ker P (θ ) → Ker P (σ (θ, t)) is an isomorphism, for each (θ, t) ∈ Θ × R+ .
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(iii) There exist constants N, ν > 0, such that • Φ(θ, t)P (θ )x N e−νt P (θ )x, (θ, t) ∈ Θ × R+ and x ∈ X; • Φ −1 (σ (θ, t))Q(θ )x N1 e−νt Q(θ )x, (θ, t) ∈ Θ × R+ and x ∈ X. The following definition of exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow) is weaker than Definition 4.2. Definition 4.3. An exponentially bounded, strongly continuous cocycle {Φ(θ, t)}θ∈Θ, t0 (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) has a no past exponential dichotomy if there exist a family of projectors {P (θ )}θ∈Θ and the constants N1 , N2 , ν1 , ν2 > 0 such that the following conditions hold: (i) Φ(θ, t)x N1 e−ν1 t x, for all t 0, θ ∈ Θ and x ∈ Im P (θ ); (ii) Φ(θ, t)x N2 eν2 t x, for all t 0, θ ∈ Θ and x ∈ Ker P (θ ). Remark 4.2. As it is known, exponential dichotomy means that X can be decomposed, at every θ ∈ Θ, as a direct sum between two subspaces such that solutions (of the variational equation (3)) starting in the first subspace (respectively, in the second one) decay exponentially in forward time (respectively, in backward time). Assuming the existence of an exponential dichotomy we practically force the solutions that starts in the second subspace to exist for negative time. However there are situations which require to drop off this requirement and to replace the exponential decay in negative time for the solutions starting in the second subspace with an exponential blow-up in positive time (we called ad hoc this behavior as a no past exponential dichotomy). Remark 4.3. It is obvious that the existence of an exponential dichotomy implies the existence of a no past exponential dichotomy. However, for infinite-dimensional subspaces Im Q(θ ), the inequality (ii) of Definition 4.3 does not imply the second inequality in condition (iii) of Definition 4.2. Assuming dim Im Q(θ ) < ∞ (and condition (i) in Definition 4.2) we get an equivalence between the two definitions. Lemma 4.1. Let h : R+ → R+ be a function with the property that there exist H > 0, δ > 0, η > 1 such that: (i) h(t) H h(t0 ), for all t ∈ [t0 , t0 + δ], t0 0; (ii) h(t0 + δ) ηh(t0 ), for all t0 0. Then there exist two constants N, ν > 0 such that h(t) N eν(t−t0 ) h(t0 ), Proof. See [11, 20C, p. 39].
for all t t0 0.
2
Lemma 4.2. If h1 , h2 : R+ → R+ satisfy the following conditions: (i) h1 (t) h1 (s)h2 (t − s) for all t s 0; (ii) supt∈[0,a] h2 (s) < ∞, for all a > 0; (iii) inft0 h2 (t) < 1,
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then there exist two constants N, ν > 0 such that h1 (t) N e−ν(t−s) h1 (s) Proof. See [19, Lemma 3.4, p. 202].
for all t s 0.
2
5. Results Proposition 5.1. If (E(R+ , X), F (R+ , X)) is admissible to an exponentially bounded, strongly continuous cocycle {Φ(θ, t)}θ∈Θ, t0 (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) then for every f ∈ E(R+ , X) and θ ∈ Θ there is a unique x2 ∈ X2,F (θ ) such that u(·; θ, x2 , f ) ∈ F (R+ , X). Proof. Let f ∈ E(R+ , X). Since (E(R+ , X), F (R+ , X)) is admissible to {Φ(θ, t)}θ∈Θ, t0 it follows that there exists x ∈ X such that t u(·; θ, x, f ) : R+ → X,
u(t; θ, x, f ) = Φ(θ, t)x +
Φ σ (θ, s), t − s f (s) ds
0
belongs to F (R+ , X), for each θ ∈ Θ. Denoting by v(t; θ, x) = Φ(θ, t)PF (θ )x and z(t; θ, x, f ) = u(t; θ, x, f ) − v(t; θ, x) we have that z(t; θ, x, f ) ∈ F (R+ , X) with t z(t; θ, x, f ) = Φ(θ, t)QF (θ )x +
Φ σ (θ, s), t − s f (s) ds.
0
The uniqueness follows easily using a simple proof by contradiction.
2
Given f ∈ E(R+ , X) we will denote, throughout of this paper, the unique vector x2 ∈ X2,F (θ ) by xf . Proposition 5.2. If (E(R+ , X), F (R+ , X)) is admissible to an exponentially bounded, strongly continuous cocycle {Φ(θ, t)}θ∈Θ, t0 (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) then for each θ ∈ Θ there exists K(θ ) > 0 such that u(·; θ, xf , f )
F (R+ ,X)
K(θ )f E(R+ ,X)
and xf K(θ )f E(R+ ,X) . Proof. Let θ ∈ Θ. We define Uθ : E(R+ , X) → X2,F (θ ) ⊕ F (R+ , X),
Uθ (f ) = xf , u(·; θ, xf , f ) .
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It is obvious that Uθ is a linear operator. We will show that Uθ is also closed. Let {fn }n∈N ⊂ E(R+ , X), f ∈ E(R+ , X), g ∈ F (R+ , X) such that E(R+ ,X)
fn −−−−−→ f,
X2,F (θ)⊕F (R+ ,X) Uθ (fn ) n −−−−−−−−−−−→ (y, g).
Then lim fn − f E(R+ ,X) = lim xfn − y = lim u(·; θ, xfn , fn ) − g F (R
n→∞
n→∞
n→∞
+ ,X)
= 0.
Taking into account that t t ωt fn (s) − f (s) ds Φ σ (θ, s), t − s fn (s) − f (s) ds Me 0
0
Me α(t, E)fn − f E(R+ ,X) , ωt
we have that t lim
n→∞
Φ σ (θ, s), t − s fn (s) ds =
0
t
Φ σ (θ, s), t − s f (s) ds.
0
By Remark 2.5 we have that there exists a subsequence (fnk ) ⊂ (fn ), fnk → f a.e. Since t u(·; θ, xfnk , fnk ) = Φ(θ, t)xfnk +
Φ σ (θ, s), t − s fnk (s) ds,
0
we have that t g(t) = Φ(θ, t)y +
Φ σ (θ, s), t − s f (s) ds.
0
This proves that u(·; θ, xf , f ) = g and xf = y. Thus Uθ is a closed linear operator and by the Closed-Graph Theorem it is also bounded. It follows that there exists K(θ ) > 0 such that u(·; θ, xf , f )
F (R+ ,X)
K(θ )f E(R+ ,X)
and xf K(θ )f E(R+ ,X) .
2
Definition 5.1. (E(R+ , X), F (R+ , X)) is said to be uniformly admissible to the exponentially bounded, strongly continuous cocycle {Φ(θ, t)}θ∈Θ, t0 (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) if supθ∈Θ K(θ ) = K < ∞.
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Theorem 5.1. If (E(R+ , X), F (R+ , X)) is uniformly admissible to an exponentially bounded, strongly continuous cocycle {Φ(θ, t)}θ∈Θ, t0 (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) and lim α(t, E)β(t, F ) = ∞,
t→∞
then {Φ(θ, t)}θ∈Θ, t0 has a no past exponential dichotomy. Proof. Let x ∈ X with QF (θ )x = 0, t0 0, and fθ (t) = ϕ[t0 ,t0 +1] (t)
Φ(θ, t)QF (θ )x . Φ(θ, t)QF (θ )x
By Remark 4.1 it follows that the above function is well defined. We can see that fθ (t) ϕ[t0 ,t0 +1] (t), for any t 0. This shows that fθ ∈ E(R+ , X) and fθ E(R+ ,X) β(1, E). But ∞ v(t; θ, x) = −
ϕ[t0 ,t0 +1] (s)
ds Φ(θ, t)QF (θ )x Φ(θ, s)QF (θ )x
ϕ[t0 ,t0 +1] (s)
ds Φ(θ, t)QF (θ )x Φ(θ, s)QF (θ )x
t
∞ =− 0
t +
Φ σ (θ, s), t − s fθ (s) ds
0
∞ = −Φ(θ, t) 0
t +
ds QF (θ )x ϕ[t0 ,t0 +1] (s) Φ(θ, s)QF (θ )x
Φ σ (θ, s), t − s fθ (s) ds
0
⎧ 0, ⎪ ⎨ t0 +1 = − t ⎪ ⎩ − t0 +1 t0
ds Φ(θ,s)QF (θ)x Φ(θ, t)QF (θ )x, ds Φ(θ,s)QF (θ)x Φ(θ, t)QF (θ )x,
t t0 + 1, t0 < t < t0 + 1, t t0 .
It follows that v(·; θ, x) ∈ F (R+ , X) and by using that v(0; θ, x) ∈ X2,F (θ ) we get that v(t; θ, x) = u(t; θ, xfθ , fθ ). Thus u(t; θ, xf , fθ ) θ F (R
+ ,X)
Kβ(1, E)
and xfθ Kβ(1, E).
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But t u(t; θ, xfθ , fθ ) = Φ(θ, t)xfθ +
Φ σ (θ, τ ), t − τ fθ (τ ) dτ
0
= Φ σ (θ, s), t − s Φ(θ, s)xfθ s
Φ σ σ (θ, τ ), s − τ , t − s Φ σ (θ, s), t − s fθ (τ ) dτ
+ 0
t
Φ σ (θ, τ ), t − τ fθ (τ ) dτ
+ s
= Φ σ (θ, s), t − s Φ(θ, s)xfθ s
Φ σ (θ, s), t − s Φ σ (θ, τ ), s − τ fθ (τ ) dτ
+ 0
t
Φ σ (θ, τ ), t − τ fθ (τ ) dτ
+ s
= Φ σ (θ, s), t − s u(s; θ, xfθ , fθ ) +
t
Φ σ (θ, τ ), t − τ fθ (τ ) dτ,
s
for each 0 s t. If t 1 and s ∈ [t − 1, t] we have that u(t; θ, xf , fθ ) Meω u(s; θ, xf , fθ ) + θ θ
t
Meω fθ (τ ) dτ.
s
Thus, for each t 1 we have that: u(t; θ, xf , fθ ) Meω θ
t
u(s; θ, xf , fθ ) ds + Meω α(1, E)fθ E(R ,X) θ +
t−1
Me α(1, F )u(·; θ, xfθ , fθ )F (R ω
+ ,X)
+ Meω α(1, E)fθ E(R+ ,X) .
Taking now t ∈ [0, 1] we have that u(t; θ, xf , fθ ) Meω θ
1
u(s; θ, xf , fθ ) ds + Meω α(1, E)fθ E(R ,X) θ +
0
Me α(1, F )u(·; θ, xfθ , fθ )F (R ω
+ ,X)
+ Meω α(1, E)fθ E(R+ ,X) .
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Thus, for any t 0 we have that: u(t; θ, xf , fθ ) Meω α(1, F )u(·; θ, xf , fθ ) + α(1, E)fθ E(R+ ,X) θ θ F (R+ ,X) L = Meω Kα(1, F ) + α(1, E) β(1, E). Thus t0 +1
t0
dτ Φ(θ, t)QF (θ )x L, Φ(θ, τ )QF (θ )x
for any t t0 . But Φ(θ, τ )QF (θ )x = Φ σ (θ, t0 ), τ − t0 Φ(θ, t0 )QF (θ )x Meω Φ(θ, t0 )QF (θ )x , for any t ∈ [t0 , t0 + 1]. It follows that 1 Meω Φ(θ, to )QF (θ )x
t0 +1
t0
dτ Φ(θ, τ )QF (θ )x
and from here we have that Φ(θ, t)QF (θ )x Meω LΦ(θ, t0 )QF (θ )x , for any 0 t t0 , or equivalent Φ(θ, t0 )QF (θ )x Meω LΦ(θ, t0 )QF (θ )x , for any 0 t0 t. Let now t0 0, δ > 0 and gδ (t) = ϕ[t0 ,t0 +δ] (t)
Φ(θ, t)QF (θ )x . Φ(θ, t0 + δ)QF (θ )x
We have that gδ (t) Meω Lϕ[t0 ,t0 +δ] (t), for any t 0, which shows that gδ ∈ E(R+ , X) and gδ E(R+ ,X) Meω Lβ(δ, E). Consider now ∞ z(t; θ, x) = −
ϕ[t0 ,t0 +δ] (s) t
∞
= −Φ(θ, t) 0
ds Φ(θ, t)QF (θ )x Φ(θ, t0 + δ)QF (θ )x
ds QF (θ )x ϕ[t0 ,t0 +δ] (s) Φ(θ, t0 + δ)QF (θ )x
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t +
Φ σ (θ, s), t − s gθ (s) ds
0
⎧ t t0 + δ, ⎪ ⎨ 0, Φ(θ,t)QF (θ)x = −(t0 + δ − t) Φ(θ,t0 +δ)QF (θ)x , t0 < t < t0 + δ, ⎪ ⎩ −δ Φ(θ,t)QF (θ)x , t t0 . Φ(θ,t0 +δ)QF (θ)x Then z(·; θ, x) ∈ F (R+ , X) and z(0; θ, x) ∈ X2,F (θ ). Thus it follows that z(·; θ, x)
F (R+ ,X)
KMeω Lβ(δ, E).
Integrating on [t0 , t0 + δ] and using (1), we have that δ 2 Φ(θ, t)QF (θ )x = 2 Φ(θ, t0 + δ)QF (θ )x
t0 +δ
(t0 + δ − s) ds t0
Φ(θ, t)QF (θ )x Φ(θ, t0 + δ)QF (θ )x
t0 +δ
MLe
z(s; θ, x) ds
ω t0
KLMeω α(δ, F )z(·; θ, x)F (R
+ ,X)
α(δ, F )K Me L β(δ, E) 2
2ω 2
4K 2 M 2 e2ω L2 δ 2 2δ 2δ K 2 Me2ω L2 = . β(δ, F ) α(δ, E) α(δ, E)β(δ, F )
Thus we have that Φ(θ, t0 + δ)QF (θ )x
1 8K 2 M 2 L2 e2ω
α(δ, E)β(δ, F )Φ(θ, t0 )QF (θ )x .
Therefore we can choose δ0 > 0 such that Φ(θ, t0 + δ0 )QF (θ )x 2Φ(θ, t0 )QF (θ )x , for each t0 0. Using Lemma 4.1 we have that there exist N2 , ν2 > 0 such that Φ(θ, t)QF (θ )x N2 eν2 t QF (θ )x , for all t 0 and x ∈ X. Let x ∈ X, t0 0 and θ ∈ Θ. We define the map fθ (t) = ϕ[t0 ,t0 +1] (t)Φ(θ, t)PF (θ )x.
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Thus, we have fθ (t) = ϕ[t
(t)Φ σ (θ, t0 ), t − t0 Φ(θ, t0 )PF (θ )x ϕ[t0 ,t0 +1] (t)Meω Φ(θ, t0 )PF (θ )x , for all t 0. 0 ,t0 +1]
Then fθ ∈ E(R+ , X) and fθ E(R+ ,X) Meω Φ(θ, t0 )PF (θ )xβ(1, E). Taking the function t v(t; θ ) =
Φ σ (θ, τ ), t − τ fθ (τ ) dτ
0
t =
ϕ[t0 ,t0 +1] (τ )Φ(θ, t)PF (θ )x dτ 0
=
0, (t − t0 )Φ(θ, t)PF (θ )x, Φ(θ, t)PF (θ )x,
t < t0 , t0 t t0 + 1, t > t0 + 1,
it follows that v(·; θ ) ∈ F (R+ , X). Taking into account that v(0; θ ) ∈ X2,F (θ ) we have that v(·; θ )
F (R+ ,X)
Kfθ E(R+ ,X) KMeω β(1, E)Φ(θ, t0 )PF (θ )x .
We can see that s v(t; θ ) =
Φ σ σ (θ, τ ), s − τ , t − s Φ σ (θ, τ ), s − τ fθ (τ ) dτ
0
t +
Φ σ (θ, τ ), t − τ fθ (τ ) dτ
s
= Φ σ (θ, s), t − s
s
Φ σ (θ, s), s − τ fθ (τ ) dτ
0
t +
Φ σ (θ, τ ), t − τ fθ (τ ) dτ,
s
for all t s 0. If we choose t 1 and s ∈ [t − 1, t], we have that v(t; θ ) Meω v(s; θ ) + Meω Me
t s
fθ (τ ) dτ
v(s; θ ) + Meω α(1, E)fθ E(R+ ,X) .
ω
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Integrating on [t − 1, t] we have that v(t; θ ) Meω
t
v(s; θ ) ds + Meω α(1, E)fθ E(R
+ ,X)
t−1
Me α(1, F )v(·; θ )F (R ,X) + Meω α(1, E)fθ E(R+ ,X) + ω Me α(1, F )K + α(1, E) fθ E(R+ ,X) . ω
If we choose now t t0 + 1 we obtain that Φ(θ, t)PF (θ )x Meω α(1, F )K + α(1, E) Meω Φ(θ, t0 )PF (θ )x . If we let t ∈ [t0 , t0 + 1) we have that Φ(θ, t)PF (θ )x = Φ σ (θ, t0 ), t − t0 Φ(θ, t0 )PF (θ )x Meω Φ(θ, t0 )PF (θ )x . Denoting by L = max{M 2 e2ω (α(1, F )K + α(1, E)), Meω } we have that Φ(θ, t)PF (θ )x L Φ(θ, t0 )PF (θ )x , for all t t0 0 and θ ∈ Θ. If δ > 0 and we set gθ (t) = ϕ[t0 ,t0 +δ] (t)Φ(θ, t)PF (θ )x, then we have gθ (t) = ϕ[t
0 ,t0 +δ]
(t)L Φ(θ, t0 )PF (θ )x .
Thus we have gθ ∈ E(R+ , X) and gθ E(R+ ,X) β(δ, E)L Φ(θ, t0 )PF (θ )x. We set t z(t; θ ) =
Φ σ (θ, τ ), t − τ gθ (τ ) dτ
0
⎧ ⎨ 0, = (t − t0 )Φ(θ, t)PF (θ )x, ⎩ δΦ(θ, t)PF (θ )x,
t < t0 , t0 t t0 + δ, t > t0 + δ.
It follows that z(·; θ ) ∈ F (R+ , X). Taking into account now that z(0; θ ) = 0 ∈ X2,F (θ ), we obtain that z(·; θ )
F (R+
Φ(θ, t0 )PF (θ )x . KLβ(δ, E) ,X)
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Using again (1) we have that δ2 Φ(θ, t0 + δ)PF (θ )x = 2
t0 +δ
(s − t0 )Φ(θ, t0 + δ)PF (θ )x ds
t0 t0 +δ
(s − t0 )L Φ(θ, s)PF (θ )x ds
t0
=L
t0 +δ
z(s; θ ) ds Lα(δ, F )z(·; θ )
F (R+ ,X)
t0
KLα(δ, F )β(δ, E)Φ(θ, t0 )PF (θ )x KL
4δ 2 Φ(θ, t0 )PF (θ )x . β(δ, F )α(δ, E)
Φ(θ, t0 + δ)PF (θ )x
8KL Φ(θ, t0 )PF (θ )x , α(δ, E)β(δ, F )
It follows that
for all t0 0, θ ∈ Θ and δ > 0. Thus there exists δ0 > 0 such that Φ(θ, t0 + δ0 )PF (θ )x 1 Φ(θ, t0 )PF (θ )x , 2 for each t0 0 and θ ∈ Θ. Using now Lemma 4.2 we have that there exist N1 , ν1 > 0 such that Φ(θ, t)PF (θ )x N1 e−ν1 t PF (θ )x , for each t 0, θ ∈ Θ and x ∈ X.
2
Remark 5.1. The reader will find an impressive list of papers by screening the literature regarding the connection between the admissibility of some function spaces, and the existence of an exponential dichotomy for dynamical systems. The milestone of this subject is the paper by O. Perron (see [14]) from 30’s, where he establishes for the first time an equivalence between the condition that the non-homogeneous equation has some bounded solution for every bounded “second member” on the one hand and a certain form of conditional stability of the solutions of the homogeneous equation on the other. This concept was called “admissibility” (or the “test function method” or “Perron’s method”) and it was extended in the more general framework of infinite-dimensional Banach spaces by J.L. Daleckij and M.G. Krein [7], J.L. Massera and J.J. Schäffer [11], and more recently by C. Chicone and Y. Latushkin [2], Nguyen van Minh, F. Räbiger and R. Schnaubelt [26], Nguyen Thieu Huy [9]. For more details, we also refer the reader to [17–20] and the references therein.
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Latest, it is known the equivalence between the admissibility of the pair (Lp (R+ , X), Lp (R+ , X)), 1 p q ∞ and (p, q) = (1, ∞) and the exponential dichotomy of a twoparameter evolution family {U (t, t0 )}tt0 0 when we assume a priori that there exists a family of projectors {P (t)}t∈R+ such that • U (t, t0 )P (t0 ) = P (t)U (t, t0 ), for all t t0 0; • U (t, t0 ) : Ker P (t0 ) → Ker P (t) is an isomorphism, for all t t0 0. The above equivalence has been proved by Nguyen van Minh, F. Räbiger and R. Schnaubelt in [26] by associating the evolution semigroup on (Lp (R+ , X). Also, a direct proof (i.e. by choosing the appropriate “test functions”) was obtained in [18, Theorem 3.9]. For related results we refer the reader to [19,20]. Theorem 5.1 extends the above results in few directions. First, we analyze the case of an exponentially bounded, strongly continuous cocycle (over a semiflow) which extends the classical notion of a two-parameter family (see Example 3.5). Also, most important is that we do not assume a priori that the family of projectors {P (θ )}θ∈Θ satisfy the restrictive requirements: • Φ(θ, t)P (θ ) = P (σ (θ, t))Φ(θ, t), for all (θ, t) ∈ Θ × R+ ; • Φ(θ, t) : Ker P (θ ) → Ker P (σ (θ, t)) is an isomorphism, for every (θ, t) ∈ Θ × R+ ; and still we succeed to prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a certain technical condition) implies the existence of a (no past) exponential dichotomy. Also it is worth to note that the class of vector-valued Schäffer function spaces is extremely large (see for instance Examples 2.1, 2.2) and this fact allows the reader to choose the “test functions” in various ways and in the same time it does not force the output (i.e. the solution of the inhomogeneous problem) to stay in Lp (R+ , X), as before. Moreover, this approach can provide “small” input spaces (i.e. the spaces consisting in “test functions”) which are obviously more convenient in the admissibility condition. Taking for instance Φ(t) = et − 1 in Example 2.2, we observe that the corresponding scalar-valued Orlicz function space LΦ (R+ , R) ⊂ Lp (R+ , R), for all p ∈ [1, ∞). Moreover, there is no p ∈ [1, ∞), such that LΦ (R+ , R) = Lp (R+ , R). We also prove that if there exists a pair of vector-valued Schäffer function spaces, (E(R+ , X), F (R+ , X)), which is uniformly admissible to {Φ(θ, t)}θ∈Θ, t0 and with the property that limt→∞ α(t, E)β(t, F ) = ∞ then the fiber X1,F (θ ) (which induces a (no past) exponential dichotomy) is always the same fiber X1,L∞ (θ ) (see Remark 5.2 below). More interesting is the result from Theorem 5.2 below, that is if we assume in addition the invariance property (i.e. condition (i) in Definition 4.2) then the above admissibility condition implies the invertibility of the operators {Φ(θ, t)} on the unstable fiber (i.e. the complement of X1,L∞ (θ )). Thus we can conclude with the following schema: • uniform admissibility ⇒ no past exponential dichotomy, • uniform admissibility + invariance property ⇒ exponential dichotomy. Equivalences are also established in Theorems 5.3 and 5.4. below. Let now (E(R+ , X), F (R+ , X)) be a pair of vector-valued Schäffer function spaces.
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Remark 5.2. If (E(R+ , X), F (R+ , X)) is uniformly admissible to an exponentially bounded, strongly continuous cocycle {Φ(θ, t)}θ∈Θ, t0 (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) and limt→∞ α(t, E)β(t, F ) = ∞, then X1,F (θ ) = X1,L∞ (θ ) = x ∈ X: Φ(θ, ·) ∈ L∞ (R+ , X) . Proof. Indeed if we choose x ∈ X1,F (θ ), we have (from the above theorem) that there exist N1 , ν1 > 0 such that Φ(θ, t)x N1 e−ν1 t x, for each t 0 and θ ∈ Θ. It follows that x ∈ X1,L∞ (θ ). Take now x ∈ X1,L∞ (θ ) and assume for a contradiction that QF (θ )x = 0. Then there exist N2 , ν2 > 0 such that Φ(θ, t)QF (θ )x N2 eν2 t QF (θ )x . It follows that Φ(θ, t)x Φ(θ, t)QF (θ )x − Φ(θ, t)PF (θ )x N2 eν2 t QF (θ )x − N1 e−ν1 t PF (θ )x . This shows that Φ(θ, ·) ∈ / L∞ (R+ , X) and thus we get the contradiction. Then QF (θ )x = 0 which implies that x ∈ X1,F (θ ). This ends the proof. 2 Let again (E(R+ , X), F (R+ , X)) be a pair of vector-valued Schäffer function spaces. Theorem 5.2. Assume that (E(R+ , X), F (R+ , X)) is uniformly admissible to an exponentially bounded, strongly continuous cocycle {Φ(θ, t)}θ∈Θ, t0 (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) with limt→∞ α(t, E)β(t, F ) = ∞. If Φ(θ, t)PL∞ (θ ) = PL∞ (σ (θ, t))Φ(θ, t) then Φ(θ, t) : Ker PL∞ (θ ) → Ker PL∞ σ (θ, t) , is invertible for all (θ, t) ∈ Θ × R+ . Proof. Let (θ, t0 ) ∈ Θ × R+ and x ∈ Ker PL∞ (θ ) such that Φ(θ, t0 )x = 0. From the above theorem it follows that there exist N2 , ν2 > 0 such that Φ(θ, t0 )x N2 eν2 t0 x. Thus we obtain that x = 0 and from here it follows that Φ(θ, t0 ) is one-to-one when the domain is restricted to Ker PL∞ (θ ). Take now y ∈ Ker PL∞ (σ (θ, t0 )), and set fθ (t) =
0, −Φ(σ (θ, t0 ), t − t0 )y, 0,
t ∈ [0, t0 ], t ∈ (t0 , t0 + 1], t > t0 + 1.
Since fθ (t) Meω yϕ[t0 ,t0 +1] (t), for each t 0, it follows that fθ ∈ E(R+ , X) and fθ E(R+ ,X) Meω yβ(1, E). Thus there exists a unique x ∈ Ker PL∞ (θ ) such that
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t u(t; θ, x, fθ ) = Φ(θ, t)x +
Φ σ (θ, τ ), t − τ fθ (τ ) dτ
0
0, = Φ(θ, t)x +
−(t − t0 )Φ(σ (θ, t0 ), t − t0 )y, −Φ(σ (θ, t0 ), t − t0 )y,
t ∈ [0, t0 ], t ∈ (t0 , t0 + 1], t > t0 + 1
belongs to F (R+ , X). Using a similar argument with the one from the proof of the above theorem we have that u(·; θ, x, fθ ) ∈ L∞ (R+ , X). If we choose t t0 we have that u(t; θ, x, fθ ) = Φ(θ, t)x − Φ σ (θ, t0 ), t − t0 y. Thus it follows that u(t; θ, x, fθ ) = Φ(θ, t)x − Φ σ (θ, t0 ), t − t0 y = Φ σ (θ, t0 ), t − t0 Φ(θ, t0 )x − Φ σ (θ, t0 ), t − t0 y = Φ σ (θ, t0 ), t − t0 Φ(θ, t0 )x − y N2 eν2 (t−t0 ) Φ(θ, t0 )x − y , for all t t0 + 1. Since u(·; θ, x, fθ ) is bounded, we have that Φ(θ, t0 )x = y. Thus Φ(θ, t) : Ker PL∞ (θ ) → Ker PL∞ (σ (θ, t)) is also onto. 2 Proposition 5.3. If the exponentially bounded, strongly continuous cocycle {Φ(θ, t)}θ∈Θ, t0 (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) has a no past exponential dichotomy then Im P (θ ) = X1,L∞ (θ ) (where {P (θ )}θ∈Θ is a family of projectors provided by Definition 4.2). Moreover supθ∈Θ P (θ ) < ∞. Proof. The first claim is obvious. For proving the second part we take x1 ∈ Im P (θ ) and x2 ∈ Ker P (θ ) with x1 = x2 = 1. Recall that the angular distance between Im P (θ ) and Ker P (θ ) is defined by x y : x ∈ Im P (θ ), y ∈ Ker P (θ ), x, y = 0 . γ Im P (θ ), Ker P (θ ) = inf − x y But
1 1 1 −νt νt x1 − x2 Φ(θ, t)x2 − Φ(θ, t)x1 Ne − e . Meωt Meωt N Choose t0 > 0 such that N eνt0 −
1 −νt0 Ne
= ψ0 > 0. Then
x1 − x2 ψ =
ψ0 , Meωt0
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and thus γ [Im P (θ ), Ker P (θ )] ψ , for all θ ∈ Θ. Taking into account that 2 1 γ Im P (θ ), Ker P (θ ) P (θ P (θ ) it follows that supθ∈Θ P (θ ) < ∞.
see [11, (11.D), p. 8]
2
Theorem 5.3. Let {Φ(θ, t)}θ∈Θ, t0 be an exponentially bounded, strongly continuous cocycle (over a semiflow {σ (θ, t)}θ∈Θ, t0 ). Assume that {Φ(θ, t)}θ∈Θ, t0 has an exponential dichotomy and that the corresponding family of projectors is strongly continuous (i.e. P (·)x is continuous for each x ∈ X). If E(R+ , X) is a vector-valued Schäffer function space then the pair (E(R+ , X), L∞ (R+ , X)) is uniformly admissible to {Φ(θ, t)}θ∈Θ, t0 . Proof. Let f ∈ E(R+ , X) and set t v(t; θ, f ) =
Φ σ (θ, τ ), t − τ P σ (θ, τ ) f (τ ) dτ
0
∞ −
Φ −1 σ (θ, t), τ − t Q σ (θ, τ ) f (τ ) dτ.
t
Denoting by x = v(0; θ, f ) = − t Φ(θ, t)x +
∞ 0
Φ −1 (θ, τ )Q(σ (θ, τ ))f (τ ) dτ we have that
Φ σ (θ, τ ), t − τ f (τ ) dτ
0
t =
Φ σ (θ, τ ), t − τ Q σ (θ, τ ) f (τ ) dτ
0
∞ −
Φ
−1
σ (θ, t), τ − t Q σ (θ, τ ) f (τ ) dτ +
t
t =
t
Φ σ (θ, τ ), t − τ f (τ ) dτ
0
Φ σ (θ, τ ), t − τ P σ (θ, τ ) f (τ ) dτ
0
∞ −
Φ −1 σ (θ, t), τ − t Q σ (θ, τ ) f (τ ) dτ
t
= u(t; θ, x, f ). It can bee seen that x ∈ Ker P (θ ) and by [11, (23.V), p. 69] (or alternatively [6, Lemma 1, p. 21]) it follows that u(·; θ, x, f ) belongs to L∞ (R+ , X). 2
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Theorem 5.4. Let {Φ(θ, t)}θ∈Θ, t0 be exponentially bounded, strongly continuous cocycle (over a semiflow {σ (θ, t)}θ∈Θ, t0 ) and assume that there exists {P (θ )}θ∈Θ a family of projectors with the following properties: • Φ(θ, t)P (θ ) = P (σ (θ, t))Φ(θ, t), for all (θ, t) ∈ Θ × R+ ; • P (·)x is continuous for each x ∈ X. Then {Φ(θ, t)}θ∈Θ, t0 has an exponential dichotomy if and only if one of the following statements is true: (i) There exists E(R+ , X) a vector-valued Schäffer function space such that (E(R+ , X), L∞ (R+ , X)) is uniformly admissible to {Φ(θ, t)}θ∈Θ, t0 ; (ii) There exist p, q, ∈ [1, ∞], (p, q) = (1, ∞) such that (Lp (R+ , X), Lq (R+ , X)) is uniformly admissible to {Φ(θ, t)}θ∈Θ, t0 ; (iii) (Lp (R+ , X), Lq (R+ , X)) is uniformly admissible to {Φ(θ, t)}θ∈Θ, t0 , for any p, q, ∈ [1, ∞], (p, q) = (1, ∞); (iv) there exists a vector-valued Orlicz function spaceLΦ(R+ , X) such that the pair (LΦ(R+ , X), LΦ (R+ , X)) is uniformly admissible to {Φ(θ, t)}θ∈Θ, t0 . Proof. It follows easily from Examples 2.1, 2.2, Remark 2.7 and above theorems.
2
Remark 5.3. It is worth to note that so far, it has been extensively analyzed the asymptotic behavior of exponentially bounded, strongly continuous cocycles over flows. The main results in this direction are focused on the characterization of exponential dichotomy of an exponentially bounded, strongly continuous cocycles over a flow in terms of Sacker–Sell spectral properties [23] or the hyperbolicity of the associated evolution semigroups and their generators [10]. In particular, a characterization of exponential dichotomy for cocycles over flows was given in [23] assuming the dimension of the unstable manifold to be finite. Meanwhile, in [10] a characterization is given through the hyperbolicity of the associated evolution semigroup and its generator. Another characterization in [3] uses a discrete cocycle over a discretized flow. In this paper we made an attempt to characterize the exponential dichotomy in a more general setting and we did consider an exponentially bounded, strongly continuous cocycles over a semiflow, i.e., there is only a semiflow on the base space. This setting is particularly appropriate in the infinitedimensional case since in this case the dynamical systems restricted to invariant manifolds are only semiflows in general. References [1] H. Amann, Ordinary Differential Equations. An Introduction to Nonlinear Analysis, de Gruyter Stud. Math., vol. 13, de Gruyter, 1990 (MR1071170)(91e:34001). [2] C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys Monogr., vol. 70, Amer. Math. Soc., 1999 (MR1707332)(2001e:47068). [3] S.N. Chow, H. Leiva, Existence and roughness of the exponential dichotomy for linear skew-product semiflow in Banach spaces, J. Differential Equations 120 (1995) 429–477 (MR1347351)(97a:34121). [4] S.N. Chow, H. Leiva, Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces, Proc. Amer. Math. Soc. 124 (1996) 1071–1081. [5] P. Constantin, C. Foias, Navier–Stokes Equations, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, 1988. [6] W.A. Coppel, Dychotomies in Stability Theory, Lecture Notes in Math., vol. 629, Springer-Verlag, Berlin, 1978.
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[7] J.L. Daleckij, M.G. Krein, Stability of Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI, 1974 (MR0352639)(50#5126). [8] K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, Berlin, 2000 (MR1721989)(2000i:47075). [9] Nguyen Thieu Huy, Existence and robustness of exponential dichotomy of linear skew-product semiflows over semiflows, J. Math. Anal. Appl. 333 (2007) 731–752 (MR2331690) (2008e:37072). [10] Y. Latushkin, R. Schnaubelt, Evolution semigroups, translation algebra and exponential dichotomy of cocycles, J. Differential Equations 159 (1999) 321–369 (MR1730724)(2000k:47054). [11] J.L. Massera, J.J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, New York, 1966. [12] R. Nagel, One-Parameter Semigroups of Positive Operators, Lecture Notes in Math., vol. 1184, Springer-Verlag, 1986 (MR0839450)(88i:47022). [13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol. 44, Springer-Verlag, 1983 (MR0710486)(85g:47061). [14] O. Perron, Die Stabilitätsfrage bei Differentialgeighungen, Math. Z. 32 (1930) 703–728. [15] V.A. Pliss, G.R. Sell, Robustness of exponential dichotomies in infinite-dimensional dynamical systems, J. Dynam. Differential Equations 11 (3) (1999) 471–513. [16] V.A. Pliss, G.R. Sell, Perturbations of normally hyperbolic manifolds with applications to the Navier–Stokes equation, J. Differential Equations 169 (2001) 396–492 (MR1808472)(2002a:37113). [17] P. Preda, A. Pogan, C. Preda, On the Perron problem for the exponential dichotomy of C0 -semigroups, Acta Math. Univ. Comenian. 72 (2) (2003) 207–212. [18] P. Preda, A. Pogan, C. Preda, (Lp , Lq )-admissibility and exponential dichotomy of evolutionary processes on the half-line, Integral Equations Operator Theory 49 (3) (2004) 405–418. [19] P. Preda, A. Pogan, C. Preda, Schäffer spaces and uniform exponential stability of linear skew-product semiflows, J. Differential Equations 212 (2005) 191–207. [20] P. Preda, A. Pogan, C. Preda, Schäffer spaces and exponential dichotomy for evolutionary processes, J. Differential Equations 230 (1) (2006) 378–391. [21] G. Raugel, G. Sell, Navier–Stokes equations on thin 3D domains, I: Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993) 503–568. [22] R.J. Sacker, G.R. Sell, Existence of dichotomies and invariant splitting for linear differential systems I, II, III, J. Differential Equations 15 (1974) 429–458, J. Differential Equations 22 (1976) 478–496, 497–525. [23] R.J. Sacker, G.R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations 113 (1994) 17–67 (MR1296160)(96k:34136). [24] G.R. Sell, Y. You, Dynamics of Evolutionary Equations, Appl. Math. Sci., vol. 143, Springer-Verlag, New York, 2002 (MR1873467)(2003f:37001b). [25] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. [26] Nguyen van Minh, F. Räbiger, R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equations Operator Theory 32 (1998) 332–353 (MR1652689)(99j:34083).
Journal of Functional Analysis 258 (2010) 758–784 www.elsevier.com/locate/jfa
Existence of densities of solutions of stochastic differential equations by Malliavin calculus Seiichiro Kusuoka 1 Department of Mathematics, Keio University, 3-14-1 Hiyoshi Kohoku-ku, Yokohama-shi, Kanagawa-ken 223-8522, Japan Received 8 May 2009; accepted 11 September 2009 Available online 2 October 2009 Communicated by Paul Malliavin
Abstract I considered if solutions of stochastic differential equations have their density or not when the coefficients are not Lipschitz continuous. However, when stochastic differential equations whose coefficients are not Lipschitz continuous, the solutions would not belong to Sobolev space in general. So, I prepared the class Vh which is larger than Sobolev space, and considered the relation between absolute continuity of random variables and the class Vh . The relation is associated to a theorem of N. Bouleau and F. Hirsch. Moreover, I got a sufficient condition for a solution of stochastic differential equation to belong to the class Vh , and showed that solutions of stochastic differential equations have their densities in a special case by using the class Vh . © 2009 Elsevier Inc. All rights reserved. Keywords: Stochastic differential equation; Malliavin calculus; Absolute continuity; Existence of densities; Existence of fundamental solutions
1. Introduction Malliavin calculus is well known as a method of researching the regularity of fundamental solutions of stochastic differential equations, and we can see that the fundamental solution has the regularity according to the smoothness of the coefficients of the stochastic differential equation. (See [3,7,10].) E-mail address:
[email protected]. 1 Research Fellow of the Japan Society for the Promotion of Science.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.009
S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
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Let T > 0, d and r be positive integers, (B(t)) be an r-dimensional Brownian motion, and σ = σji i=1,...,d, j =1,...,r ∈ Cb [0, T ] × R → Rd ⊗ Rr , b = bi i=1,...,d ∈ Cb [0, T ] × Rd → Rd . We consider the d-dimensional stochastic differential equation;
dX(t) = σ t, X(t) dB(t) + b t, X(t) dt, X(0) = x0 ∈ Rd .
We assume that this equation has some conditions about ellipticity, for example uniformly elliptic, Hörmander condition, and so on. Under the condition above, there is a result by S. Kusuoka and D. Stroock [6]. That is, if σ ∈ Cb0,n+2 [0, T ] × Rd → Rd ⊗ Rr ,
b ∈ Cb0,n+2 [0, T ] × Rd → Rd ,
then the distribution of the solution P ◦X(t)−1 has its density, and the density belongs to Cbn (Rd ). This theorem is proved by using Sobolev’s inequality associated to H -derivative. It’s one of the most famous result about Malliavin calculus. Moreover, in their paper, they also had some results in the case that the coefficients depend on the past. The equation of the case is often called Itô’s equation. Only about existence of the density, there is a result of N. Bouleau and F. Hirsch [1]. The result tells that if there is some conditions about ellipticity and there exists a constant K such that σ (t, x) − σ (t, y) + b(t, x) − b(t, y) K|x − y|,
for all x, y ∈ Rd and t ∈ [0, T ],
then the distribution of solution P ◦ X(t)−1 has its density. Besides, F. Hirsch showed a similar theorem about Itô’s equation in [2]. But, roughly speaking, if the coefficients satisfy some conditions about ellipticity, it seems that the solution has its density, even if the coefficients have no regularity. So I considered if the solution has its density or not when the coefficients are not Lipschitz continuous. However, when stochastic differential equations whose coefficients are not necessary Lipschitz continuous, the solutions would not belong to Sobolev space in general. Hence, I prepared the class Vh which is larger than Sobolev space, and considered the relation between absolute continuity of random variables and the class Vh . The relation is associated to a theorem of N. Bouleau and F. Hirsch. Moreover, I got a sufficient condition for solutions of stochastic differential equations to belong to the class Vh , and showed that solutions of stochastic differential equations have their densities in a special case by using the class Vh . We will see the analysis of the class Vh in Section 2, and the relation between the solution of stochastic differential equation and the class Vh in Section 3.
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2. Analysis of class Vh For the first, we define a class of random variables. When we consider stochastic differential equations whose coefficients are not necessary Lipschitz continuous, the solutions would not belong to Sobolev space in general. So we need a larger class than Sobolev space. Let (Ω, F , P ) be a probability space which is an orthogonal product measure space of an abstract Wiener space (B, H, μ) and another probability space (Ω , F , ν). Of course, this argument includes the case that Ω is trivial, for example, Ω = {0}. Through the paper we identify ω ∈ Ω as (x, ω ) ∈ B × Ω . Let F be a random variable on (Ω, F , P ). If the limit lim
ε→0
1 F (x + εh, ω ) − F (x, ω ) ε
exists for h ∈ H , then we denote the limit by Dh F (x, ω ). Dh is regarded as the derivative for the direction h. We prepare some notations. We fix h ∈ H and let {hk } be a complete orthonormal system of H ∗ such that h = h1 . Since B ∗ ⊂ H ∗ is a continuous embedding, B x → x, h1 , x, h2 , . . . ∈ R∞ is injection. Here we denote x, h in the sense of Wiener integral of 1-order. Hence let y = x, h1 ∈ R, x˜ = x, h2 , x, h3 , . . . ∈ R∞ , then we can identify x as (y, x). ˜ Next, we describe the measures of y and x. ˜ By the orthogonality of {hk } in H ∗ , if k = j , x, ˜ hk and x, ˜ hj are independent under μ. Since { x, hk } is a Gaussian system under μ, { x, hk } are independent. In particular, y = x, h1 and x˜ = ( x, h2 , x, h3 , . . .) are independent under μ. So, we regard the measure space (B, μ) as an orthogonal measure space for y and x. ˜ Moreover, we can decompose as following. B∼ = R × B, 2 y 1 μ∼ ˜ = √ e− 2 dy ⊗ μ. 2π Here, we used the fact that y = x, h1 has a normal distribution with mean 0 and variance 1 under μ. We denote partial derivative with respect to y by ∂y . We use these notations through this section. Definition 2.1. We define Vh (B × Ω ) by the total set of random variables F on (Ω, F , P ) such on (Ω, F , P ) satisfying that F = F a.s. and F (x + th, ω ) that there exists a random variable F is a function of bounded variation on any finite interval with respect to t for all x and ω . If Ω is trivial, for example, Ω = {0}, then we denote by Vh (B) simply. Now we give a criterion that a random variable belongs to the class Vh .
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Theorem 2.2. Let (Ω, F , P ) be a probability space which is an orthogonal product measure space of an abstract Wiener space (B, H, μ) and another probability space (Ω , F , ν). Let p > 1, h ∈ H , and F ∈ Lp (Ω, F , P ). We assume that there exists a sequence {Fn : n ∈ N} in Lp (Ω, F , P ) so that Fn converges to F almost surely, {Fn } are uniformly bounded in Lp (Ω, F , P ), Fn (x + th, ω ) is absolutely continuous in t with respect to the one-dimensional Lebesgue measure for all x and n, {Dh Fn } are uniformly bounded in L1 (Ω, F , P ). Then F ∈ Vh (B × Ω ). Proof. To simplify the notations, we assume that |h|H = 1. Since {Fn } are uniformly bounded in Lp (Ω, F , P ), {Fn } are uniformly integrable. Thus, since Fn converges to F almost surely, Fn also converges to F in L1∗ (Ω, F , P ). We give a positive number M, and define a function φ ∈ C ∞ (R) such that 0 φ 1, 0 φ 1,
φ(y) =
1, if |y| M, 0, if |y| M + 1.
Then, for t, s ∈ [−M, M] we have
F (y + t, x, ˜ ω )φ(y + t) − F (y + s, x, ˜ ω )φ(y + s)
R Ω B
2 1 y dy μ(d ˜ x) ˜ ν(dω ) × √ exp − 2 2π Fn (y + t, x, = lim ˜ ω )φ(y + t) − Fn (y + s, x, ˜ ω )φ(y + s) n→∞ R Ω B
2 1 y dy μ(d ˜ x) ˜ ν(dω ) × √ exp − 2 2π 1 Fn (y + t, x, ˜ ω )φ(y + t) lim inf √ n→∞ 2π R Ω B
˜ x) ˜ ν(dω ) − Fn (y + s, x, ˜ ω )φ(y + s) dy μ(d t
1 ˜ x) ˜ ν(dω ) = lim inf √ ˜ ω )φ(y + v) dv dy μ(d ∂y Fn (y + v, x, n→∞ 2π R Ω B
1 lim inf √ n→∞ 2π
t
s
∂y Fn (y + v, x, ˜ ω ) φ(y + v)
s Ω B R
˜ x) ˜ ν(dω ) dv + Fn (y + v, x, ˜ ω )φ (y + v) dy μ(d 1 = lim inf √ n→∞ 2π
t s Ω B R
∂y Fn (y, x, ˜ ω ) φ(y) + Fn (y, x, ˜ ω )φ (y)
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× dy μ(d ˜ x) ˜ ν(dω ) dv 1 √ |t − s| sup n 2π
(M+2)
∂y Fn (y, x, ˜ ω ) φ(y)
−(M+2) Ω B
+ Fn (y, x, ˜ x) ˜ ν(dω ) ˜ ω )φ (y) dy μ(d |t − s|e
(M+2)2 2
(M+2)
sup n
∂y Fn (y, x, ˜ ω ) + Fn (y, x, ˜ ω )
−(M+2) Ω B
2 1 y × √ exp − dy μ(d ˜ x) ˜ ν(dω ) 2 2π (M+2)2 e 2 |t − s| sup Fn L1 (Ω) + Dh Fn L1 (Ω) . n
Therefore, for s, t ∈ [−M, M],
F (y + t, x, ˜ ω )φ(y + t) − F (y + s, x, ˜ ω )φ(y + s)
R Ω B
2 1 y × √ exp − dy μ(d ˜ x) ˜ ν(dω ) 2 2π CM |t − s|,
(2.1)
where CM is a constant depending only on M and supn (Fn L1 (Ω) + Dh Fn L1 (Ω) ). We define a functions {Fφm } on B × Ω by 1
Fφm (y, x, ˜ ω ) := 2m
2m
F (y + v, x, ˜ ω )φ(y + v) dv,
0
Then,
∂y F m (y, x, ˜ ω ) dy φ
R 1 2m 2m ∂y F (y + u, x, ˜ ω )φ(y + u) du dy
R
0
1 y+ 2m m =2 F (u, x, ˜ ω )φ(u) du dy ∂y
R
y
m = 1, 2, . . . .
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1 1 ˜ ω )φ(y) dy =2 ˜ ω φ y + m − F (y, x, F y + 2m , x, 2 m
R
2m
1 1 1 F y + 1 , x, ˜ ω , x, ˜ ω φ y + − F y + φ y + 2m 2m 2m+1 2m+1
R
1 1 + F y + m+1 , x, ˜ ω φ y + m+1 − F (y, x, ˜ ω )φ(y) dy 2 2
1 1 m+1 =2 ˜ ω )φ(y) dy ˜ ω φ y + m+1 − F (y, x, F y + 2m+1 , x, 2 R 1 2 m+1 m+1 =2 F (y + u, x, ˜ ω )φ(y + u) du dy ∂y
=
0
R
∂y F m+1 (y, x, ˜ ω ) dy. φ
R
Hence, {
m ˜ ω )| dy} R |∂y Fφ (y, x,
are increasing in m. Thus, by (2.1), we have
m ˜ x) ˜ ν(dω ) ∂y Fφ (y, x, sup ˜ ω ) dy μ(d m
Ω B
R
= sup m
Ω B
R
= sup m
Ω B
∂y F m (y, x, ˜ x) ˜ ν(dω ) ˜ ω ) dy μ(d φ
1 1 ˜ ω )φ(y) dy 2m F y + m , x, ˜ ω φ y + m − F (y, x, 2 2
R
× μ(d ˜ x) ˜ ν(dω )
√ 1 1 (M + 1)2 m sup 2 2π exp ˜ ω φ y+ m F y + 2m , x, 2 2 m R Ω B
2 1 y dy μ(d ˜ x) ˜ ν(dω ) − F (y, x, ˜ ω )φ(y) √ exp − 2 2π
√ (M + 1)2 2π exp CM . 2 Therefore, for (μ˜ × ν)-almost all (x, ˜ ω ), sup m R
∂y F m (y, x, ˜ ω ) dy < ∞. φ
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On the other hand, by the definition of Fφm , for all (x, ˜ ω ) there exists a function F (·, x, ˜ ω ) so that lim Fφm (y, x, ˜ ω ) = F (y, x, ˜ ω )φ(y),
m→∞
F (·, x, ˜ ω ) = F (·, x, ˜ ω ),
dy-a.e.
Hence, by Corollary 5.3.4 of [13], we have F (·, x, ˜ ω )φ is a function of bounded variation on R ˜ ω), and for all M > 0, F (·, x, ˜ ω ) is for (μ˜ × ν)-almost all (x, ˜ ω ). So, for (μ˜ × ν)-almost all (x, a function of bounded variation on [−M, M]. Therefore, we conclude that F ∈ Vh (B × Ω ). 2 Example 2.3. Let (Ω, F , P ) be a probability space, L be a Lévy process on (Ω, F , P ), F L be a σ -field generated by L, and F be an F L -measurable random variable on the probability space. Then, by Lévy–Itô decomposition and Section 2 of Chapter A3 in [11], we can regard F as a random variable on a product space generated by a space (W, B(W ), μ) of the part of Brownian motion and a space (Ω , F , ν) of the Poisson part. We denote the Cameron–Martin space associated to (W, B(W ), μ) by H . Now let h ∈ H and p > 1, and we assume that there exists a sequence of random variables {Fn : n ∈ N} in Lp (W × Ω , μ ⊗ ν) so that {Fn } converges to F almost surely, and {Dh Fn } are uniformly bounded in L1 (W × Ω , μ ⊗ ν). Then F ∈ Vh (W × Ω ). Now we consider the merit of Vh . Next theorem tells the relation between the class Vh and absolute continuity. It is associated to that of N. Bouleau and F. Hirsch. Theorem 2.4. Let (Ω, F , P ) be a probability space which is an orthogonal product measure space of an abstract Wiener space (B, H, μ) and another probability space (Ω , F , ν). Let F is the modification of F appeared in the be a random variable such that F ∈ Vh (B × Ω ). If F definition of Vh (B × Ω ), then the measure |P ◦ F −1 |Dh F is absolutely continuous with respect to the one-dimensional Lebesgue measure. (·, x, Proof. Since F ˜ ω ) is a function of bounded variation on any finite interval, we can define ˜ ω ) by a function F (y, x, (y + ε, x, F (y, x, ˜ ω ) := lim F ˜ ω ). ε↓0
(·, x, The function F (·, x, ˜ ω ) is a right-continuous version of F ˜ ω ). Hence, we have F = F,
P -a.e.
× Ω Fix a constant M > 0. To slide the domain, we define a function F M (y, z) on [0, 2M] × B by F M (y, x, ˜ ω ) := F (y − M, x, ˜ ω ),
ω ∈ Ω . y ∈ [0, 2M], x˜ ∈ B,
S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
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Then, for every nonnegative continuous function f on R we have M
(y, x, (y, x, ˜ x) ˜ ν(dω ) f F ˜ ω ) ∂y F ˜ ω ) dy μ(d
−M Ω B
2M ˜ x) ˜ ν(dω ). = f F M (y, x, ˜ ω ) ∂y F M (y, x, ˜ ω ) dy μ(d Ω
(2.2)
0 B
Now we fill up the discontinuous points of F M (y, x, ˜ ω ) linearly with respect to y, and make a continuous function. For the first, we fix x˜ and ω , and let C ⊂ [0, 2M] be the set of contin˜ ω ) with respect to y, and {ξk } ⊂ [0, 2M] be discontinuous points of uous points of F M (y, x, F M (y, x, ˜ ω ) with respect to y. We define jx,ω ˜ (y) such that jx,ω ˜ (y) : [0, 2M] → R, jx,ω ˜ ω ) − F M (y−, x, ˜ ω ), ˜ (y) := F M (y, x, where F M (y−, x, ˜ ω ) := lim F M (y − ε, x, ˜ ω ). ε↓0
We define Jx,ω ˜ (y) by Jx,ω ˜ (y) : [0, 2M] → R, jx,ω Jx,ω ˜ (y) := ˜ (ξk ) . 0<ξk y
And we define τ by τ (y) ˜ :=
τ : 0, 2M + Jx,ω ˜ (2M) → [0, 2M], ˜ if y˜ ∈ [0, 2M + Jx,ω inf{u ∈ [0, 2M]; u + Jx,ω ˜ (u) > y}, ˜ (2M)), 2M, if y˜ = 2M + Jx,ω ˜ (2M).
Since τ is an inverse function of the increasing function · + Jx,ω ˜ (·), it is continuous and increasM (y, x, ˜ ω ) such that ing. The next, we define F
M (y, ˜ x, ˜ ω ) : 0, 2M + Jx,ω F ˜ (2M) × B → R, M (y, F ˜ x, ˜ ω + sgn jx,ω ˜ y˜ − Jx,ω ˜ + τ (y) ˜ , ˜ x, ˜ ω ) := F M τ (y), ˜ τ (y) ˜ τ (y) where ⎧ if u > 0, ⎨ 1, sgn(u) := 0, if u = 0, ⎩ −1, if u < 0.
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(y, Then, F ˜ x, ˜ ω˜ ) is continuous with respect to y. ˜ When τ (y) ˜ ∈ C, M (y, F ˜ x, ˜ ω . ˜ x, ˜ ω ) = F M τ (y), Therefore, by the fact that the total set of discontinuous points of a function of bounded variation is a null set with the Lebesgue measure, for every nonnegative continuous function f
2M f F M (y, x, ˜ ω ) ∂y F M (y, x, ˜ ω ) dy 0
2M = f F M (y, x, ˜ ω ) ∂y F M (y, x, ˜ ω )1C (y) dy 0
2M+Jx,ω ˜ (2M)
=
˜ x, ˜ ω (∂y F M ) τ (y), ˜ dτ (y) ˜ f F M τ (y), ˜ x, ˜ ω 1C τ (y)
0
2M+Jx,ω ˜ (2M)
=
M (y, ˜ ˜ (∂y F M ) τ (y), f F ˜ x, ˜ ω ) 1C τ (y) ˜ x, ˜ ω dτ (y)
0
2M+Jx,ω ˜ (2M)
M (y, ˜ dy˜ F M τ (y), ˜ x, ˜ ω f F ˜ x, ˜ ω ) 1C τ (y)
0
2M+Jx,ω ˜ (2M)
M (y, M (y, f F ˜ x, ˜ ω ) dy˜ F ˜ x, ˜ ω ).
(2.3)
0
By Theorem 6.4 of Chapter IX in [9], we have the next lemma. ψ
Lemma 2.5. Let ψ be a function of bounded variation on [a, b], and N[a,b] (c) be the number of crossing points on [a, b] between the graph ψ and the graph c for c ∈ R. Then,
b a
f ψ(x) dψ(x) =
∞
−∞
ψ
f (y)N[a,b] (y) dy,
f ∈ C(R).
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By this lemma, we have
∞
2M+Jx,ω ˜ (2M)
M (y, M (y, f F ˜ x, ˜ ω ) dy˜ F ˜ x, ˜ ω ) =
(·,x,ω F ˜ )
M f (u)N[0,2M+J
−∞
0
x,ω ˜ (2M)]
(u) du.
Hence, by (2.3) 2M ∞ M (·,x,ω F ˜ ) f F M (y, x, ˜ ω ) ∂y F M (y, x, ˜ ω ) dy f (u)N[0,2M+J (u) du. (2M)] −∞
0
x,ω ˜
This equation holds for (μ˜ × ν)-almost every (x, ˜ ω ). Therefore, by (2.2), we have M
(y, x, (y, x, ˜ x) ˜ ν(dω ) f F ˜ ω ) ∂y F ˜ ω ) dy μ(d
−M Ω B
∞
(·,x,ω F ˜ )
M f (u)N[0,2M+J
−∞ Ω B
x,ω ˜ (2M)]
(u) du μ(d ˜ x) ˜ ν(dω ).
(2.4)
This equation holds for every nonnegative continuous function f . Hence (2.4) also holds for every nonnegative Lebesgue measurable function f . So, let A be any null set on R with the Lebesgue measure, and replace f by 1A , then we have M
(y, x, (y, x, ˜ x) ˜ ν(dω ) = 0. ˜ ω ) ∂y F 1A F ˜ ω ) dy μ(d
−M Ω B
Since this equation holds for all M > 0,
(y, x, (y, x, ˜ x) ˜ ν(dω ) = 0. ˜ ω ) ∂y F 1A F ˜ ω ) dy μ(d
R Ω B
This means
)|Dh F | = 0. E 1A (F This completes the proof.
2
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3. Applications to stochastic differential equations Now we consider if solutions of stochastic differential equations whose coefficients are not Lipschitz continuous have their densities or not. For the first, we will show a lemma which makes the most important role in this paper. Lemma 3.1. Let r be a positive integer, (Ω, F , P ) be a probability space, (B(t)) be an rdimensional Brownian motion on (Ω, F , P ), (Ft ) be a reference family, σ = (σj )j =1,2,...,r be an Rr -valued measurable function on [0, T ] × Ω, and b be a measurable function on [0, T ] × Ω. We assume that a 1-dimensional (Ft )-adapted continuous process X = (X(t)) on (Ω, F , P ) satisfies the equation X(t) = x0 +
r
t
t
σj (s, ω)X(s) dB (s) + j
j =1 0
b(s, ω) ds, 0
where x0 is a constant. Moreover, we assume that max sup σj (t, ω) < ∞, j
t,ω
and there exist constants M, K and a finite measure η on [0, T ] satisfying that b(t, ω) M + K
t
X(s) dη(s) + X(t) ,
for all (t, ω) ∈ [0, T ] × Ω.
0
Then, there exists a constant C which depends on only T , x0 , M, K, and η([0, T ]) such that E X(t) C,
t ∈ [0, T ].
Proof. We choose 1 > a1 > a2 > · · · > 0 such that 1 a1
1 du = 1, u
a1
1 du = 2, u
a m−1
...,
a2
1 du = m, u
....
am
Then am → 0 as m → ∞. For {am }, we can define continuous functions ψm on [0, ∞) such that supp ψm ⊂ (am , am−1 ),
2 , 0 ψm (u) mu
a m−1
ψm (u) du = 1. am
Moreover, we define functions ϕm on R by |y| ϕm (y) =
z du
0
ψm (z) dz. 0
S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
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Then we have ϕ (y) 1, m
ϕm ∈ C 2 (R),
ϕm (y) y (m → ∞).
By Itô’s formula, r 1 E ϕm X(t) = ϕm (x0 ) + 2
t
2 2 X(s) σj (s, ω) X(s) ds E ϕm
j =1 0
t +
X(s) b(s, ω) ds. E ϕm
(3.1)
0
Then, from the definition of ϕm , we have r t 1 2 2 E ϕm X(s) σj (s, ω) X(s) ds 2 j =1 0
1 2 r
t
2 2 X(s) σj (s, ω) X(s) ds E ϕm
j =1 0
1 m r
t
2 E σj (s, ω) X(s) ds
j =1 0
→ 0 (m → ∞). And by the condition of b, we have t t
E ϕm X(s) b(s, ω) ds E b(s, ω) ds 0
0
t s Mt + K 0
E X(u) dη(u) + E X(s) ds
0
Mt + K η [0, T ] + 1
t 0
sup E X(u) ds. 0us
Therefore, let m → ∞ in (3.1), we have sup E X(s) |x0 | + Mt + K η [0, T ] + 1 0st
t 0
sup E X(u) ds. 0us
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By Gronwall’s inequality, sup E X(s) K |x0 | + Mt eK(η([0,T ])+1)t . 0st
Hence, there exists a constant C depending on T , x0 , M, K, and η([0, T ]) such that E X(t) C.
2
Now we give some notations. We denote Sobolev space with respect to H -derivative with index k and p by W k,p , and the total set of smooth functions on C([0, T ] → Rd ) by C ∞ (C([0, T ] → Rd )), where the smoothness is meant in the sense of Gâteau derivative. The precise definition of C ∞ (C([0, T ] → Rd )) can be seen in [6]. We define Cb∞ (C([0, T ] → Rd )) by the total set of the elements of C ∞ (C([0, T ] → Rd )) whose derivatives are bounded. We denote partial derivative with respect to spacial component by ∂x . For real numbers a and b, we define a ∨ b and a ∧ b by max{a, b} and min{a, b} respectably. Let r be a positive integer and T be a positive number. For fixed r and T , we set W := w ∈ C [0, T ] → Rr ; w(0) = 0 ,
T
H := h ∈ W ; h is absolute continuous and
˙hj (t)2 dt < ∞, j = 1, 2, . . . , r ,
0
and let μ be a Wiener measure on W . We call the triplet (W, H, μ) Wiener space. Clearly a Wiener space is an abstract Wiener space. The next lemma is a version of Lemma 3.1 about H -derivative of a stochastic differential equation. Lemma 3.2. We fix T > 0. Let d and r be positive integers, (W, H, P ) be an r-dimensional Wiener space, (B(t)) be an r-dimensional Brownian motion associated to (W, H, P ), B(W ) be a Borel σ -field of W , (Ft ) be a reference family, σ = σji i=1,...,d, j =1,...,r ∈ Cb [0, T ] × R → Rd ⊗ Rr , σji (t, ·) ∈ C ∞ (R), t ∈ [0, T ], b = bi i=1,...,d ∈ Cb [0, T ] × C [0, T ] → Rd → Rd , bi (t, ·) ∈ C ∞ C [0, T ] → Rd , t ∈ [0, T ]. We assume that a d-dimensional (Ft )-adapted continuous process X = (X(t)) on (W, B(W ),P ) satisfies the stochastic differential equation ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
dX i (t) =
r
σji t, X i (t) dB j (t) + bi (t, X) dt
j =1
X(0) = x0 ∈ Rd .
for all i = 1, 2, . . . , d,
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Moreover, we assume that there exist constants M, K and a finite measure η on [0, T ] satisfying that max σji (t, x) M, i,j t i i max b (t, w) − b (t, w ) K i
for all (t, x) ∈ [0, T ] × R, w(s) − w (s) dη(s) + w(t) − w (t) ,
0
for all t ∈ [0, T ], w, w ∈ C [0, T ] → Rd .
Then, for all t in [0, T ], k = 1, 2, . . . , and p 1, X(t) belongs to W k,p , and there exists a constant C which depends on only M, K, and η([0, T ]) such that E Dh X i (t) C|h|H ,
h ∈ H and i = 1, 2, . . . , d.
Proof. By the condition of the coefficients and [6], X can be expressed as a functional on (W, H, μ) which is the Wiener space generated by the Brownian motion (B(t)), and we have X(t) ∈ W k,p for any positive integer k and p 1. So, it’s sufficient to prove the existence of a constant C. We fix h ∈ H . Consider the H -differential of the stochastic differential equation for X, then we have Dh X (t) = i
r
t
∂x σji
r s, X i (s) Dh X i (s) dB j (s) +
j =1 0
t
h˙ j (s)σji s, X(s) ds
j =1 0
t +
Dh bi (s, X) ds,
i = 1, 2, . . . , d.
0
By the condition of b, it follows that for almost all w Dh bi t, X(·, w) 1 i b t, X(·, w + εh) − bi t, X(w) ε→0 ε t K X(s, w + εh) − X(s, w) dη(s) + X(t, w + εh) − X(t, w) lim ε→0 ε
= lim
0
t =K
Dh X(s, w) dη(s) + Dh X(t, w) .
0
Hence, we can show by similar discussion as the proof of Lemma 3.1.
2
Now we will show a sufficient condition for solutions of stochastic differential equations to belong to the class Vh . The advantage is that we assume only bounded on the diffusion coefficient σ .
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Theorem 3.3. Let d and r be positive integers, (B(t)) be an r-dimensional Brownian motion, σ = σji i=1,...,d, j =1,...,r ∈ Cb [0, T ] × R → Rd ⊗ Rr , b = bi i=1,...,d ∈ Cb [0, T ] × C [0, T ] → Rd → Rd , and we assume that there exist constants M, K and a Radon measure η on [0, T ] satisfying that max σji (t, x) M, i,j t i max b (t, w) − bi (t, w ) K i
for all (t, x) ∈ [0, T ] × R, w(s) − w (s) dη(s) + w(t) − w (t) ,
0
for all t ∈ [0, T ], w, w ∈ C [0, T ] → Rd .
We consider a d-dimensional stochastic differential equation; ⎧ r ⎪ ⎪ ⎨ dX i (t) = σ i t, X i (t) dB j (t) + bi (t, X) dt, j
⎪ ⎪ ⎩
i = 1, 2, . . . , d,
j =1
X(0) = x0 ∈ Rd ,
and we assume that the stochastic differential equation has pathwise uniqueness. Then, the solution (X(t)) can be defined on a Wiener space (W, H, μ), and X i (t) is in Vh (W ) for all t in [0, T ], i = 1, 2, . . . , d, and h ∈ H . i (t), then Moreover, if we denote the version of X i (t) appeared in Definition 2.1 by X Dh X i (t)μ ◦ X i (t)−1 is absolutely continuous to one-dimensional Lebesgue measure. Proof. Pathwise uniqueness of the equation implies that the solution X can be expressed as a functional on the Wiener space (W, H, μ) generated by the Brownian motion (B(t)). By Lemma 5.2 of [2], there exist the sequences {σn } and {bn } {σn } ⊂ Cb [0, T ] × R → Rd ⊗ Rr ,
{bn } ⊂ Cb [0, T ] × C [0, T ] → R → Rd ,
which satisfy that σn (t, ·) ⊂ Cb∞ (R), lim σn (t, ·) − σ (t, ·)C (R→Rd ×Rr ) = 0, b n→∞ i max σj (t, x) M, for all (t, x) ∈ [0, T ] × R, i,j
bn (t·, ·) ⊂ Cb∞ C [0, T ] → Rd , lim bn (t, w) − b(t, w) = 0, for all t ∈ [0, T ], w ∈ C [0, T ] → Rd ,
n→∞
S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
bn (t, w) − bn (t, w ) K
t
773
w(s) − w (s) dη(s) + w(t) − w (t) ,
0
for all t ∈ [0, T ], w, w ∈ C [0, T ] → Rd .
Let {Xn } be the strong solutions of the stochastic differential equations for the Brownian motion (B(t)) and coefficients σn and bn respectably. By [4], we have for all t ∈ [0, T ] Xn (t) → X(t)
a.s.
On the other hand, by a standard method of stochastic differential equations, we have for all t ∈ [0, T ] 2 sup E Xn (t) < ∞. n
Therefore, we can use Theorem 2.2, and we have X i (t) ∈ Vh (W ) for all t ∈ [0, T ] and i = 1, 2, . . . , d. The final assertion follows by using Theorem 2.4. 2 In the arguments above, we have no assumptions about ellipticity. In the case that the coefficients are Lipschitz continuous, it is known that some conditions about ellipticity of the stochastic differential equation tells the positivity of |det(DX i (t), DX j (t))H |. But when the coefficients are not necessary Lipschitz continuous, we cannot use these relations. However, in a special case we can show the positivity of |Dh X i (t)| for a special h as follows. Theorem 3.4. Let r be positive integer, and (B(t)) be an r-dimensional Brownian motion. We consider one-dimensional stochastic differential equation; ⎧ r ⎪ ⎨ dX(t) = σj t, X(t) dB j (t) + b t, X(t) dt, j =1 ⎪ ⎩ X(0) = x0 ∈ R, and we assume that the stochastic differential equation has pathwise uniqueness. On the coefficients we assume σ = (σj )j =1,...,r ∈ Cb [0, T ] × R → Rr , b ∈ Cb [0, T ] × R → R , there exist constants M and K satisfying that max σj (t, x) M, j
for all (t, x) ∈ [0, T ] × R,
b(t, x) − b(t, y) K|x − y|,
for all x, y ∈ R and t ∈ [0, T ].
0,2 There is a closed subset r S of [0, T ] × R satisfying that σj is in C t(([0, T ] × R) \ S) for all j = 1, 2, . . . , r, and j =1 σj is positive on ([0, T ] × R) \ S. We set S := {x; (t, x) ∈ S}.
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S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
Then, μ ◦ X(t)−1 R\S t is absolutely continuous to one-dimensional Lebesgue measure restricted on R \ S t for all t in [0, T ]. Proof. Since it is enough to prove the case that t = T , we set t = T . By Theorem 3.3, we have that the solution (X(t)) can be defined on a Wiener space (W, H, μ) and X(T ) ∈ Vh (W ). To simplify notations, we also denote X(T ) by the version appeared in the definition of Vh (W ). Now (n) we can choose σ (n) = (σj )j =1,2,...,r ∈ Cb ([0, T ] × R → Rr ) and b(n) ∈ Cb ([0, T ] × R → R) for n = 1, 2, . . . so that (n) σj (t, ·) ∈ Cb∞ R → Rr , j = 1, 2, . . . , r, and t ∈ [0, T ], (n) σ (t, x) − σj (t, x) = 0, j = 1, 2, . . . , r, lim sup j
n→∞ t∈[0,T ],x∈R
b(n) (t, ·) ∈ Cb∞ (R → R), (n) b (t, x) − b(t, x) = 0, sup
lim
n→∞ t∈[0,T ],x∈R
(n) b (t, x) − b(n) (t, y) K|x − y|,
for all x, y ∈ R and t ∈ [0, T ],
and for all I which is a closed interval included by R \ S lim
sup
∂x σ (n) (t, x) − ∂x σj (t, x) = 0,
for all j = 1, 2, . . . , r,
lim
sup
2 (n) ∂ σ (t, x) − ∂ 2 σj (t, x) = 0,
for all j = 1, 2, . . . , r,
n→∞ t∈[0,T ],x∈I n→∞ t∈[0,T ],x∈I
j
x j
lim
x
sup
n→∞ t∈[0,T ],x∈I
(n) ∂x b (t, x) − ∂x b(t, x) = 0.
Consider the stochastic differential equations: ⎧ r ⎪ (n) ⎨ dX (t) = σj t, Xn (t) dB j (t) + b(n) t, Xn (t) dt, n j =1 ⎪ ⎩ Xn (0) = x0 , and let Xn be the solution of each equation. Then, by [4] we have lim E
n→∞
2 sup Xn (t) − X(t) = 0.
t∈[0,T ]
Hence, we can choose a subsequence of {Xn } which converges to X in the topology of C([0, T ]) almost surely. For simplicity, we also denote the subsequence by {Xn } again. So we have
S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
lim
sup Xn (t) − X(t) = 0,
a.s.
n→∞ t∈[0,T ]
775
(3.2)
Now we consider the H -derivative of Xn (T ), then we have r
T
Dh Xn (T ) =
(n) ∂x σj s, Xn (s) Dh Xn (t) dB j (s) +
j =1 0
T +
r
T
(n) h˙ j (s)σj s, Xn (s) ds
j =1 0
∂x b(n) s, Xn (s) Dh Xn (s) ds.
0
If we take Xn as given, then we can regard the equation as a liner stochastic differential equation of DXn (·)[h]. Thus, by Problem 6.15 of Chapter 5 in [5] we have
Dh Xn (T ) =
T r
r T ˙hj (s)σj s, Xn (s) exp ∂x σj(n) u, Xn (u) dB j (u)
j =1
0
1 − 2
j =1 s
T T r 2 (n) (n) ∂x σj u, Xn (u) du + ∂x b u, Xn (u) du ds. s j =1
s
Through this proof, we set hj (t) := t,
for all t ∈ [0, T ] and j = 1, 2, . . . , r.
Then h ∈ H ,
Dh Xn (T ) =
T r 0
1 − 2
r T σj s, Xn (s) exp ∂x σj(n) u, Xn (u) dB j (u)
j =1
j =1 s
T T r 2 (n) (n) ∂x σj u, Xn (u) du + ∂x b u, Xn (u) du ds, s j =1
(3.3)
s
and Dh Xn (T ) 0,
n = 1, 2, . . . .
To get some information about the exponential part, we consider the time-reversal process of (Xn , B) by using the theory written in Section 4 of Chapter VII in [8]. Let Zn be an (r + 2)dimensional Markov process defined by
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S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
⎛
⎞ Xn (t) ⎜ t ⎟ ⎜ 1 ⎟ B (t) ⎟ Zn (t) := ⎜ ⎜ . ⎟, ⎝ .. ⎠
and Zn (t) := ,
if 0 < t < T ,
if t T .
B j (t) We denote the starting point of Zn by x˜0 . Clearly x˜0 = Zn (0) = (x0 , 0, . . . , 0). Let be the point of one point compactification of Rr+2 , W r+2 be C([0, ∞) → Rr+2 ∪ ), and B(W r+2 ) be a Borel σ -field of W r+2 . Let PZn be a probability measure on (W r+2 , B(W r+2 )) the law of Zn , and we define ζ by ζ (w) := inf t > 0; w 2 (t) > T ,
w ∈ W r+2 ,
where w 2 means the second component of w. It is clear that ζ is lifetime and ζ = T a.s. under PZn . Moreover, ζ becomes co-optional time, because of the definition of ζ . Since Zn is a Markov process, we can define a semi-group associated to PZn , and let {Tt } be the Feller semigroup on C∞ (Rr+2 ∪ ), where % $ C∞ Rr+2 ∪ := f ∈ C∞ Rr+2 ∪ ; lim f (x) = 0 . |x|→∞
We define a measure ν on Rr+2 ∪ by ∞
f (x) ν(dx) =
Ts f (x˜0 ) ds,
f ∈ C∞ Rr+2 ∪ .
0
Rr+2
By the definition of ζ , this integration is well-defined. Then, it is easy to see that {Tt } is a strong continuous, contractive semi-group on L2 (ν). Next, we define Tt by the dual operator of Tt on L2 (ν). Then we have the next lemma. Lemma 3.5. {Tt } is also a strong continuous, contractive semi-group on L2 (ν). Proof. It is clear that {Tt } is a semi-group on L2 (ν). Contractivity of {Tt } on L2 (ν) follows from that of {Tt }. Thus, it is enough to show the strong continuity on L2 (ν). Let f, g ∈ C∞ (Rr+2 ∪ ), then
f (Tt g) dν =
Rr+2 ∪
(Tt f )g dν,
Rr+2 ∪
→ Rr+2 ∪
Next, for all f ∈ C∞ (Rr+2 ∪ )
f g dν,
as t → 0.
(3.4)
S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
(Tt f )2 dν =
Rr+2 ∪
777
(Tt Tt f )f dν
Rr+2 ∪
& ∞
(Tt Tt f ) Zn (s) f Zn (s) ds
=E
'
0
& ∞ =E
E Tt f Zn (t + s) Fs f Zn (s) ds
'
0
& ∞
(Tt f ) Zn (t + s) f Zn (s) ds
=E
'
0
& ∞
(Tt f ) Zn (t + s) f Zn (t + s) ds
=E
'
0
& ∞
' (Tt f ) Zn (t + s) f Zn (t + s) − f Zn (s) ds .
−E 0
By the contractivity of {Tt } on L2 (ν) and (3.4), we have & ∞ ' & ∞ ' 2 (Tt f ) Zn (t + s) f Zn (t + s) ds − E f Zn (t + s) ds E 0
0
& ∞ ' = E (Tt − I )f Zn (t + s) f Zn (t + s) ds 0
& ∞ ' E (Tt − I )f Zn (s) f Zn (s) ds 0
& t ' + E (Tt − I )f Zn (s) f Zn (s) ds
0
(Tt − I )f f dν
Rr+2 ∪
& t +E 0
2 (Tt − I )f Zn (s) ds
' 1 & t '1 2 2 2 E f Zn (s) ds 0
(3.5)
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S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
√ f (Tt − I )f dν + (Tt − I )f L2 (ν) f ∞ t
Rr+2 ∪
→ 0,
as t → 0.
(3.6)
On the other hand, by the contractivity of {Tt } on L2 (ν), we have & ∞ ' (Tt f ) Zn (t + s) f Zn (t + s) − f Zn (s) ds E 0
& ∞ ' 1 & ∞ '1 2 2 2 2 E (Tt f ) Zn (t + s) ds E f Zn (t + s) − f Zn (s) ds 0
0
& ∞ ' & T '1 2 2 2 (Tt f ) Zn (s) ds E f Zn (t + s) − f Zn (s) ds E 1 2
0
0
= Tt f L2 (ν) E
& T
2 f Zn (t + s) − f Zn (s) ds
'1 2
0
→ 0 as t → 0.
(3.7)
Therefore, by (3.5), (3.6), and (3.7), we have lim t→0
(Tt f )2 dν − E
Rr+2 ∪
& ∞
' 2 f Zn (t + s) ds = 0.
0
But & ∞ ' & ∞ ' & t ' 2 2 2 f Zn (t + s) ds − E f Zn (s) ds = E f Zn (s) ds E 0
0
0
f 2∞ t Hence we have lim t→0
Rr+2 ∪
Therefore, by this equation and (3.4)
(Tt f )2 dν −
Rr+2 ∪
f dν = 0. 2
→ 0,
as t → 0.
S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
|Tt f − f |2 dν =
Rr+2 ∪
(Tt f )2 dν − 2
Rr+2 ∪
Rr+2 ∪
−2
f 2 dν
Rr+2 ∪
f 2 dν
Rr+2 ∪
(Tt f )f dν −
Rr+2 ∪
→ 0,
(Tt f )f dν +
Rr+2 ∪
(Tt f )2 dν −
=
779
f 2 dν
Rr+2 ∪
as t → 0.
Thus we finished to prove Lemma 3.5.
2
Now we continue to prove Theorem 3.4. By this lemma, we have a Markov process associated n taking values in Rr+2 ∪ by to {Tt }. We define a new process Z ⎧ ⎨ Zn (ζ −), Zn (t) := Zn (ζ − t), ⎩ ,
if t = 0, if 0 < t < ζ, if t ζ.
n be σ (Z n (s); s t). Then, by Theorem 4.5 of Chapter VII in [8] it follows that the Let F t n ) associated to transition semi-group {Tt }. process Zn is a Markov process with respect to (F t n (t); t ∈ [0, T ]) and (B(t); On the other hand, ζ = T . Therefore, if we define processes (X t ∈ [0, T ]) by n (t) := Xn (T − t), X := B(T − t), B(t)
t ∈ [0, T ], t ∈ [0, T ],
n )-adapted processes. Moreover, we n (t); t ∈ [0, T ]) and (B(t); then both (X t ∈ [0, T ]) are (F t define (B(t); t ∈ [0, T ]) by := B(t) − B(0). B(t) Since (B(t); t ∈ [0, T ]) is a Gaussian process, so is (B(t); t ∈ [0, T ]). By checking its mean n )-Brownian motion. By Exer and its covariance, it is easy to see that (B(t); t ∈ [0, T ]) is (F t cise (2.18) of Chapter IV in [8], we have T r
(n)
∂x σj
s, Xn (s) dB j (s)
j =1 T −t
=
r t j =1 0
(n)
∂x σj
j n (s) d B (s) T − s, X
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S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
+
r
t
(n)
n (s) σ (n) T − s, X n (s) ds T − s, X j
∂x2 σj
a.s.
j =1 0
=
r
t
(n)
j n (s) d B (s) T − s, X
∂x σj
j =1 0
+
r
t
(n)
∂x2 σj
n (s) σ (n) T − s, X n (s) ds T − s, X j
a.s.
(3.8)
j =1 0
Note that all of stochastic integrals here are in the sense of Itô integral. Let m be any positive integer and fix m. Let
τnm
(n) n (t) ∨ ∂x2 σ (n) T − t, X n (t) > m, := inf t > 0; max ∂x σj T − t, X j 1j r
r
σj(n) T
j =1
1 n (t) ∈ S ∧ T n (t) < , or T − t, X − t, X m
n )-stopping time for all m = 1, 2, . . . . Hence, for all m = 1, 2, . . . . Then τnm is an (F t τ nm ∧t 2 ' j (n) n (s) d B (s) m2 T , ∂x σj T − s, X E sup t∈[0,T ] &
j = 1, 2, . . . , r,
0
τ nm ∧t (n) n (s) σ (n) T − s, X n (s) ds MmT , ∂x2 σj T − s, X sup j t∈[0,T ]
j = 1, 2, . . . , r.
0
So we have τ nm ∧t 2 ' j (n) n (s) d B (s) m2 T , E lim inf sup ∂x σj T − s, X n→∞ t∈[0,T ] &
j = 1, 2, . . . , r,
0
τ nm ∧t (n) 2 (n) n (s) σ n (s) ds MmT , T − s, X ∂x σj T − s, X lim inf sup j n→∞ t∈[0,T ]
j = 1, 2, . . . , r.
0
Therefore, τ nm ∧t j (n) n (s) d B (s) < ∞ lim inf sup ∂x σj T − s, X n→∞ t∈[0,T ] 0
a.s., j = 1, 2, . . . , r,
S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
781
τ nm ∧t (n) (n) n (s) σ n (s) ds < ∞, T − s, X lim inf sup ∂x2 σj T − s, X j n→∞ t∈[0,T ] 0
j = 1, 2, . . . , r. But now, let
τ := inf t > 0; max ∂x σj T − t, X(T − t) ∨ ∂x2 σj T − t, X(T − t) > m, m
1j r
r j =1
1 σj T − t, X(T − t) < , or T − t, X(T − t) ∈ S ∧ T m
1 for m = 1, 2, . . . . Since rj =1 σj (T − t, X(T − t)) > 2m for all t in the neighbor of τ m , oscillam tion occurs in the neighbor of τ . Because of this fact, (3.2), and the definition of τnm , it follows that lim τ m n→∞ n
= τm
a.s.
Hence, by (3.2) again, there exists a subsequence {n(k)} of N such that m ∧t τn(k) j (n(k)) n(k) (s) d B (s) < ∞ a.s., lim sup T − s, X ∂x σj k→∞ t∈[0,T ]
0
j = 1, 2, . . . , r, m ∧t τn(k) (n(k)) (n(k)) n(k) (s) σ n(k) (s) ds < ∞ a.s., lim sup T − s, X T − s, X ∂x2 σj j k→∞ t∈[0,T ]
0
j = 1, 2, . . . , r. Therefore, if we set
Yn(k) (t, T ) :=
T r
(n(k))
∂x σj
s, Xn(k) (s) dB j (s),
j =1 T −t
then by (3.8) there is a random variable C such that for almost all w sup m (w),T ] t∈[τn(k)
Yn(k) (t, T )(w) < C(w),
for all k = 1, 2, . . . .
On the other hand, by the definition of τnm (w), for almost all w
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S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
sup m (w),T ] t∈[τn(k)
T r 2 (n(k)) ∂x σj u, Xn(k) (u, w) du rm2 T , t
for all k = 1, 2, . . . .
j =1
Thus, by (3.3) we have for almost all w T Dh Xn(k) (T , w)
r
(n(k))
s, Xn(k) (s, w)
σj
m (w) j =1 τn(k)
1 × exp Yn(k) (s, T ) − 2 T +
T r 2 (n(k)) ∂x σj u, Xn(k) (u, w) du s j =1
∂x b(n(k)) u, Xn(k) (u, w) du ds
s
T r 1 2 (n) exp −C(w) − rm T − KT σj s, Xn (s, w) ds 2 j =1 τ m (w) n
1 1 T − τnm (w) exp −C(w) − rm2 T − KT . 2m 2 Hence, for almost all w lim inf Dh Xn(k) (T , w) k→∞
1 1 exp −C(w) − rm2 T − KT lim inf T − τnm (w) . k→∞ 2m 2
m (w) < T , then Therefore, if w satisfies that X(T , w) ∈ S and lim infk→∞ τn(k)
lim inf Dh Xn(k) (T , w) > 0. k→∞
On the other hand, for almost all w with respect to μ 1 X(T , w + εh) − X(T , w) ε 1 = lim inf lim inf Xn(k) (T , w + εh) − Xn(k) (T , w) ε→0 k→∞ ε ε 1 Dh Xn(k) (T , w + uh) du = lim inf lim inf ε→0 k→∞ ε
Dh X(T , w) = lim inf ε→0
0
lim inf ε→0
1 ε
ε lim inf Dh Xn(k) (T , w + uh) du k→∞
0
S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
783
= lim inf Dh Xn(k) (T , w)h , k→∞
where lim infk→∞ Dh Xn(k) (T , w)h means a right-continuous version of lim inf Dh Xn(k) (T , w) k→∞
for direction h. Since Gaussian measure is absolute continuous to Lebesgue measure, we have Dh X(T ) lim inf Dh Xn(k) (T ) k→∞
a.s.
Therefore, there exists a null set N1 (m) such that if w satisfies that X(T , w) ∈ S, m lim inf τn(k) (w) < T , k→∞
and w ∈ / N1 (m), then Dh X(T , w) > 0. Now we define Sm by
T Sm
r 2 1 := x ∈ R; max ∂x σj (T , x) ∨ ∂x σj (T , x) > m, and σj (T , x) > . m 1j r j =1
m and (3.2), there exists a null set N (m) such that if X(T , w) ∈ Then, by the definition of τn(k) 2 m (w) < T . Thus, if w satisfies that X(T , w) ∈ S T and w ∈ T and w ∈ / N2 (m), lim infk→∞ τn(k) / Sm m N2 (m), then
Dh X(T , w) > 0. Hence, if we define N3 by
(
m∈N N2 (m),
then
−1 T S . Dh X(T , w) > 0 for w ∈ X(T ) So, by Theorem 3.3, we have the conclusion.
2
Example 3.6. Let (B(t)) be a one-dimensional Brownian motion, and consider a onedimensional stochastic differential equation;
) dX(t) = X(t) dB(t) + b t, X(t) dt, X(0) = x0 ∈ [0, ∞),
where b ∈ Cb [0, T ] × [0, ∞) → R ,
b(t, 0) > 0, t ∈ [0, T ],
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S. Kusuoka / Journal of Functional Analysis 258 (2010) 758–784
and there exists constant K satisfying that b(t, x) − b(t, y) K|x − y|,
for all x, y ∈ [0, ∞) and t ∈ [0, T ].
Then, the solution X(t) of the stochastic differential equation has its density function for all t in [0, T ]. In fact, the condition of coefficients implies that there exists a solution (X(t)) with state space [0, ∞). Moreover [12] tells that the stochastic differential equation has pathwise uniqueness. So we can use Theorem 3.4 with S = [0, T ] × {0}. Thus, we have μ ◦ X(t)−1 (0,∞) is absolutely continuous to one-dimensional Lebesgue measure restricted on (0, ∞) for all t in [0, T ]. However, because of the condition of coefficients, it can be seen that μ ◦ X(t)−1 ({0}) = 0. Therefore, we have the conclusion. Acknowledgment I would like to thank Associated Professor Yozo Tamura for helpful discussions and careful reading of the early version. References [1] N. Bouleau, F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, Walter de Gruyter, Berlin/New York, 1991. [2] F. Hirsch, Propriété d’absolue continuité pour les équations différentielles stochastiques dépendant du passé, J. Funct. Anal. 76 (1988) 193–216. [3] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, second ed., North-Holland/ Kodansha, 1989. [4] H. Kaneko, S. Nakao, A note on approximation for stochastic differential equations, in: Séminaire de Probabilités XXII, 1988, pp. 155–162. [5] I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin/Heidelberg, 1991. [6] S. Kusuoka, D. Stroock, Application of the Malliavin calculus, part I, in: Proceedings of the Taniguchi Intern. Symp. on Stochastic Analysis, Kyoto and Katata, 1982, pp. 271–306. [7] D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin/Heidelberg, 2006. [8] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, third ed., Springer-Verlag, Berlin/Heidelberg/ New York, 1999. [9] S. Saks, Theory of the Integral, second ed., Hafner, New York, Warszawa–Lwow, 1937. [10] I. Shigekawa, Stochastic Analysis, American Mathematical Society, 2004. [11] D. Williams, Probability with Martingales, Cambridge University, 1991. [12] T. Yamada, S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1) (1971) 155–167. [13] P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York/Tokyo, 1989.
Journal of Functional Analysis 258 (2010) 785–813 www.elsevier.com/locate/jfa
BV functions in abstract Wiener spaces Luigi Ambrosio a , Michele Miranda Jr. b , Stefania Maniglia c , Diego Pallara c,∗ a Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy b Dipartimento di Matematica, Università di Ferrara, via Machiavelli, 35, 44100 Ferrara, Italy c Dipartimento di Matematica “Ennio De Giorgi”, Università del Salento, C.P. 193, 73100 Lecce, Italy
Received 7 September 2009; accepted 11 September 2009 Available online 22 September 2009 Communicated by Paul Malliavin
Abstract Functions of bounded variation in an abstract Wiener space, i.e., an infinite-dimensional Banach space endowed with a Gaussian measure and a related differential structure, have been introduced by M. Fukushima and M. Hino using Dirichlet forms, and their properties have been studied with tools from analysis and stochastics. In this paper we reformulate, in an integral-geometric vein and with purely analytical tools, the definition and the main properties of BV functions, and investigate further properties. © 2009 Elsevier Inc. All rights reserved. Keywords: BV functions; Wiener space; Ornstein–Uhlenbeck semigroup
1. Introduction The spaces BV of functions of bounded variation in Euclidean spaces are by now a classical setting where several problems, mainly (but not exclusively) of variational nature, find their natural framework. Recently, generalizations in metric measure spaces have been studied (see e.g. [2]), but mostly under the hypothesis that the measure is doubling, which is not the case when dealing with probability measures in vector spaces, not even locally in infinite dimensions. However, there are several motivations for studying BV functions in this context, i.e., Banach * Corresponding author.
E-mail addresses:
[email protected] (L. Ambrosio),
[email protected] (M. Miranda),
[email protected] (S. Maniglia),
[email protected] (D. Pallara). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.008
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spaces, also of infinite dimensions, endowed with a probability measure. Among them, let us quote isoperimetric inequalities and mass concentration, see [21,22], infinite-dimensional analysis and semigroups (see e.g. [9,10]). Moreover, the importance of generalizing the classical notion of perimeter and variation has been pointed out in several occasions by E. De Giorgi: we refer to [13], where the infinite-dimensional context is explicitly mentioned. BV functions in an abstract Wiener space, i.e., a Banach space X endowed with a Gaussian measure γ and a related differential structure, have been defined by M. Fukushima and M. Hino in [17,18], relying upon Dirichlet forms theory, and studied also in [19,20]. The starting point of these papers has been a characterization of sets with finite perimeter in finite dimensions in terms of the behaviour of suitable stochastic processes (see [16]), and in fact some arguments in [17,18] come from stochastics. In this infinite-dimensional context, the main difficulty arises from the fact that the measures appearing in the integration by parts formula can’t be built by a direct approximation by smooth functions, since Riesz theorem is not available and the dual of Cb (X) strictly contains the class of signed measures; so, looking for instance at the approximation of u by the Ornstein–Uhlenbeck semigroup Tt u as t ↓ 0, one has to prove tightness of the family of measures ∇H Tt uγ as t ↓ 0. In the above mentioned papers this difficulty is overcome (after a reduction to nonnega tive functions u) by looking at the Dirichlet form Eu (v, v) = X u∇H v2H dγ and applying the capacity theory of Dirichlet forms. One of the aims of this paper is to study BV functions in abstract Wiener spaces with tools closer to those used in the Euclidean case. We analyse the connections between the distributional notion of (vector-valued) measure gradient and the approximation by smooth functions, as well as the relevant properties of the Ornstein–Uhlenbeck semigroup. Our methods rely basically on an integral-geometric approach, a viewpoint which has already proved to be very useful in the finite-dimensional theory (see also [15] for a recent contribution in the same direction in Wiener spaces). This approach allows to build directly the distributional derivative measure and to obtain the tightness property only as a consequence. Some of the main results of this paper have been presented, together with several open problems, in [3]. In the particular case when u = χE is the characteristic function of the set E, saying that u ∈ BV is equivalent to saying that E has finite perimeter. In this case, the polar decomposition of the measure DχE = ν|DχE | leads to defining a notion of measure-theoretic normal unit vector on the boundary of E. In the Euclidean case, it is known that |DχE | coincides with the 1-codimensional Hausdorff measure restricted to the reduced boundary ∂ ∗ E ⊂ ∂E and ν can be defined pointwise on ∂ ∗ E. Let us point out that coincidence of |DχE | with the 1-codimensional Gauss–Hausdorff (spherical) measure restricted to a suitable measure-theoretic boundary ∂∗ E has been recently proved in [20]. The paper is organized as follows: in Section 2 we describe the Wiener space setting and the tools useful to rephrase in this framework the characterizations of total variation that are known in the Euclidean case. A more detailed comparison with the Euclidean case is presented in [3]. In Section 3 we define BV functions in Wiener spaces and discuss their basic properties. In Section 4 we provide a complete characterization of BV functions in terms of integration by parts formulae and approximation by smooth functions. It is known that neither Sobolev nor BV spaces are compactly embedded in Lp spaces, but our integral-geometric viewpoint provides new natural compactness criteria, both in the Sobolev and BV classes. In Section 5, in the case when the Wiener space is Hilbert, we compare Sobolev and
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BV classes with those developed by Da Prato, Malliavin, Nualart (see [11]), proving compactness of these “stronger” Sobolev and BV classes. 2. Wiener space setting In this section we describe our setting: given an (infinite-dimensional) separable Banach space X, we denote by · X its norm and by BX (x, r) = {y ∈ X: y − xX < r} the open ball centred at x ∈ X and with radius r > 0 (also Br (x) if no confusion occurs). X ∗ denotes ∗ in X ∗ , we denote by the topological dual, with duality ·,· . Given the elements x1∗ , . . . , xm m ∗ ∗ Πx1 ,...,xm : X → R the map ∗ , Πx1∗ ,...,xm∗ x = x, x1∗ , . . . , x, xm
(1)
∗ . The also denoted by Πm : X → Rm if it is not necessary to specify the elements x1∗ , . . . , xm k symbol FCb (X) denotes the space of k times continuously differentiable cylindrical functions with bounded derivatives up to the order k, that is, u ∈ FCbk (X) if u(x) = v(Πm x) for some v ∈ Cbk (Rm ). We divide this section in some subsections; first of all we recall some notion of measure theory, with particular emphasis on the infinite-dimensional (i.e., nonlocally compact) setting, then we pass to the definition and description of abstract Wiener spaces. In the third subsection we discuss the integration by parts formula and recall the definitions of gradient and divergence. Finally, we introduce Sobolev classes and the Ornstein–Uhlenbeck semigroup together with some of their basic properties.
2.1. Infinite-dimensional measure theory We denote by B(X) the Borel σ -algebra; since X is separable, B(X) is generated by the cylindrical sets, that is by the sets of the form E = Πm−1 B with B ∈ B(Rm ), see [24, Theorem I.2.2]; this fact remains true even if we fix a sequence (xi∗ ) ⊂ X ∗ which separates the points in X and use only elements from that sequence to generate the maps Πm . We shall make later some special choices of (xi∗ ), induced by a Gaussian probability measure γ in X. We also denote by M (X, Y ) the set of countably additive measures on X with finite total variation with values in a Hilbert space Y , M (X) if Y = R. We denote by |μ| the total variation measure of μ, defined by ∞ ∞
μ(Bh ) : B = |μ|(B) := sup Bh , (2) Y
h=1
h=1
for every B ∈ B(X), where the supremum runs along all the countable disjoint unions. Notice that, using the polar decomposition, there is a unit |μ|-measurable vector field σ : X → Y such that μ = σ |μ|, and then the equality |μ|(X) = sup σ, φ d|μ|, φ ∈ Cb (X, Y ∗ ), φ(x) Y ∗ 1, ∀x ∈ X , X
holds, where ·,· denotes the duality between Y and Y ∗ . Note that, by the Stone–Weierstrass theorem, the algebra FCb1 (X) of C 1 cylindrical functions is dense in C(K) in sup norm, since it
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separates points, for all compact sets K ⊂ X. Since |μ| is tight, it follows that FCb1 (X) is dense in L1 (X, |μ|). Arguing componentwise, it follows that also the space FCb1 (X, Y ∗ ) of cylindrical functions with a finite-dimensional range is dense in L1 (X, |μ|, Y ∗ ). As a consequence σ can be approximated in L1 (X, |μ|, Y ∗ ) by a uniformly bounded sequence of functions in FCb1 (X, Y ∗ ), and we may restrict the supremum above to these functions only to get 1 ∗ |μ|(X) = sup σ, φ d|μ|, φ ∈ FCb (X, Y ), φ(x) Y ∗ 1, ∀x ∈ X . (3) X
We now recall a tightness criterion and we include its proof for the reader’s convenience. Lemma 2.1. Let (σn ) ⊂ M+ (X) be a bounded sequence, σ ∈ M+ (X) and assume that lim supn σn (X) σ (X), while lim inf σn (A) σ (A) n→∞
for all A ⊂ X open.
Then (σn ) is tight and σn → σ in the duality with Cb (X). Proof. Let (xi ) ⊂ X a dense sequence and ε > 0 be fixed. We claim that for all k 1 there exists N = N(k) such that sup σn X \ n
N
B 1/k (xi ) < ε2−k .
i=1
N (k) If this property holds, the totally bounded and closed set Kε := k 1 B 1/k (xi ) satisfies supn σn (X \ Kε ) < ε, proving the tightness of (σn ). We prove the claimed property by contradiction, assuming that for some k there exists a sequence n( ) such that σn( ) (X \ i=1 B 1/k (xi )) > ε2−k for all 1. Obviously n( ) → ∞ as
→ ∞ and for any 0 we have σ
0
i=1
B1/k (xi ) lim inf σn( )
→∞
0
B1/k (xi ) lim inf σn( )
→∞
i=1
B1/k (xi )
i=1
lim sup σn (X) − ε2−k σ (X) − ε2−k . n→∞
Letting 0 → ∞ gives a contradiction, since 10 B1/k (xi ) ↑ X. The last statement is a simple consequence of the Cavalieri formula, taking into account that σn (E) → σ (E) for all Borel sets E with σ (∂E) = 0, and that σ ({u = t}) = 0 with at most countably many exceptions for all u ∈ Cb (X). 2 Finally, let us define the sup of (the total variation of) an arbitrary family of measures {μα : α ∈ I } by setting ∞ |μα |(A) = sup |μαn |(An ) , α∈I
n=1
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where the supremum runs along all the countable pairwise disjoint partitions A = the choices of the sequence (αn ) ⊂ I .
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n An
and all
2.2. The abstract Wiener space Assume that a nondegenerate centred Gaussian measure γ is defined on X. This means that γ is a probability measure and for all x ∗ ∈ X ∗ the law x#∗ γ is a centred Gaussian measure on R, that is, the Fourier transform of γ is given by ∗
γˆ (x ) =
1 ∗ ∗ ∗ exp −ix, x dγ (x) = exp − Qx , x , 2
∀x ∗ ∈ X ∗ ,
X
where Q ∈ L(X ∗ , X) is the covariance operator. The nondegeneracy hypothesis means that γ is not concentrated on a proper subspace of X, in terms of Q this means that Qx ∗ , x ∗ > 0 for x ∗ = 0. The covariance operator is a symmetric and positive operator uniquely determined by the relation
(4) Qx ∗ , y ∗ = x, x ∗ x, y ∗ dγ (x), ∀x ∗ , y ∗ ∈ X ∗ ; X
we also write N (0, Q) for γ . The fact that the operator Q defined by (4) is bounded is a consequence of Fernique’s theorem (see e.g. [7, Theorem 2.8.5]), asserting that
exp αx2X dγ (x) < ∞
(5)
X
if and only if α −1 > σ := sup Qx ∗ , x ∗ 1/2 : x ∗ ∈ X ∗ , x ∗ X∗ 1 ; as another consequence of this we also get that any x ∗ ∈ X ∗ defines a function x → x, x ∗ that belongs to Lp (X, γ ) for all p 1. In particular, we can think of any x ∗ ∈ X ∗ as a continuous element of L2 (X, γ ). Let us denote by R ∗ : X ∗ → L2 (X, γ ) the embedding, R ∗ x ∗ (x) = x, x ∗ . The space H given by the closure of R ∗ X ∗ in L2 (X, γ ) is called the reproducing kernel of the Gaussian measure γ and obviously R ∗ X ∗ turns out to be dense in it. The above definition is motivated by the fact that if we consider the operator R : H → X whose adjoint is R ∗ , then R hˆ =
ˆ h(x)x dγ (x),
X
where the integral is understood in Bochner’s sense. As a consequence Q = RR ∗ : RR ∗ x ∗ , y ∗ = [R ∗ x ∗ , R ∗ y ∗ ]H =
X
x, x ∗ x, y ∗ dγ (x) = Qx ∗ , y ∗ .
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It can be easily proved that R is injective. In addition, the operator R is compact and even more, i.e., it belongs to the ideal γ (H , X) of γ -Radonifying, or Gaussian–Radonifying operators, see e.g. [23]. This remark shows that another presentation is possible: one can start with R ∈ γ (H , X) for some separable Hilbert space H and construct a Gaussian measure γ whose covariance operator is Q = RR ∗ . In any case, the measure γ built with this construction is concentrated on the separable subspace of X defined as the closure of RH in X. The space H = RH ⊂ X is called the Cameron–Martin space; it is a separable Hilbert space with inner product defined by [h1 , h2 ]H = [hˆ 1 , hˆ 2 ]H for all h1 , h2 ∈ H , where hi = R hˆ i , i = 1, 2. It is a dense subspace of X and the embedding of (H, · H ) in (X, · ) is compact since R is compact. In addition, γ (H ) = 0 if X is infinitedimensional [7, Theorem 2.4.7], while H = X if X is finite-dimensional. With this notation, the Fourier transform of the Gaussian measure γ reads 1 γˆ (x ∗ ) = exp − xˆ ∗ 2H , 2
∀x ∗ ∈ X ∗ ,
where xˆ ∗ = R ∗ x ∗ . Using the embedding R ∗ X ∗ ⊂ H , we shall say that a family {xj∗ } of elements of X ∗ is orthonormal if the corresponding family {R ∗ xj∗ } is orthonormal in H . Starting from a sequence (yj∗ ) in X ∗ whose image under R ∗ is dense in H , we may construct an orthonormal basis (R ∗ xj∗ ) in H (by the Gram–Schmidt procedure), hence hj = Qxj∗ = RR ∗ xj∗ provide an ⊥ = ker Π ∗ ∗ and Xm orthonormal basis of H . Set also Hm = span{h1 , . . . , hm }, and define Xm x1 ,...,xm the (m-dimensional) complementary space. Since the variables R ∗ xj∗ are Gaussian and uncorrelated, they are independent, hence the image γm of γ under Πx1∗ ,...,xm∗ is a standard Gaussian in Rm ; in addition it can be proved that we have a product decomposition γ = γm ⊗ γm⊥ of the measure γ , with γm⊥ Gaussian. Since R ∗ X ∗ is dense in H the following proposition is easily established by approximation: Proposition 2.2. Let hˆ 1 , . . . , hˆ m be in H . Then the law of the variable x → (hˆ 1 , . . . , hˆ m ) under γ is Gaussian. If hˆ i are orthonormal, the law is the standard Gaussian γm in Rm . One more property of Gaussian measures we shall use is rotation invariance, i.e., if : X × X → X × X is given by (x, y) = (cos ϑx + sin ϑy, − sin ϑx + cos ϑy) for some ϑ ∈ R, then # (γ ⊗ γ ) = γ ⊗ γ . We shall use, in particular, the following equality:
u(cos ϑx + sin ϑy) dγ (x) dγ (y) =
X X
u(x) dγ (x),
∀u ∈ L1 (X, γ ),
X
which is obtained by the above relation by integrating the function u ⊗ 1 on X × X.
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For every function u ∈ L1 (X, γ ) we define its canonical cylindrical approximations Em u ∗ }, as the conditional expectations relative to the σ -algebras generated by {x, x1∗ , . . . , x, xm denoted by Bm (X), i.e.,
u dγ = Em u dγ , ∀A ∈ Bm (X). (8) A
A
Then, Em u → u in L1 (X, γ ) and γ -a.e. (see e.g. [7, Corollary 3.5.2]). More explicitly, we have
Em u(x) =
u Πm x + (I − Πm )y dγ (y) =
u(Πm x + y ) dγm⊥ (y ),
⊥ Xm
X
where Πm is the projection onto Xm . Notice that Em u is invariant under translations along all the ⊥ , hence we may write E u(x) = v(Π x) for some function v, and, with an abuse vectors in Xm m m of notation, we may write Em u(xm ) instead of Em u(x). Finally, let us recall the Cameron–Martin theorem: the shifted measure γh (B) = γ (B − h),
B ∈ B(X),
also denoted by N (h, Q), is absolutely continuous with respect to γ if and only if h ∈ H and, ˆ we have, see e.g. [7, Corollary 2.4.3], in this case, with the usual notation h = R h, 1 2 ˆ (9) dγh (x) = exp h(x) − hH dγ (x). 2 It is also important to notice that if we define for any λ ∈ R the measure γλ (B) = γ (λB),
∀B ∈ B(X),
(10)
then γλ γσ if and only if |λ| = |σ | (see for instance [7, Example 2.7.4]). 2.3. Gradient, divergence and Sobolev spaces For a given function f : X → R and h ∈ H , we define f (x + th) − f (x) t→0 t
∂h f (x) := lim and
ˆ ∂h∗ f (x) = ∂h f (x) − f (x)h(x), ˆ We shall use the shorter notation ∂j = ∂hj , wherever this makes sense. Here, as usual, h = R h. ∗ ∗ ∂j = ∂hj . The gradient ∇H f : X → H of f is defined as ∇H f (x) :=
j ∈N
∂j f (x)hj .
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Notice that if f (x) = g(Πm x) with g ∈ C 1 (Rm ), then ∂h f (x) = ∇g(Πm x) · Πm h. The operator ∂h∗ is (up to a change of sign) the adjoint of ∂h with respect to L2 (X, γ ), namely
φ∂h f dγ = − X
f ∂h∗ φ dγ ,
∀φ, f ∈ FCb1 (X).
(11)
X
The divergence operator is defined for Φ : X → H as ∗ ∇H Φ(x) := ∂j∗ Φ(x), hj H . j ∈N
We define the space FCb1 (X, H ) of cylindrical H -valued functions as the vector space spanned by functions φh, where φ runs in FCb1 (X) and h in H . With this notation, the integration by parts formula (11) gives
∗ [∇H f, Φ]H dγ = − f ∇H Φ dγ (12) X
X
for every f ∈ FCb1 (X), Φ ∈ FCb1 (X, H ). Thanks to (12), the gradient ∇H is a closable operator in the topologies Lp (X, γ ), p L (X, γ , H ) for any p ∈ [1, ∞) and, as in [17], we denote by D1,p (X, γ ) the domain of its closure. Notice that the space denoted by D1,p (X, γ ) by Fukushima in [17] is denoted by W p,1 (X, γ ) in [7]. Anyway, these spaces coincide, see [7, Section 5.2] and (12) holds for every f ∈ D1,p (X, γ ), Φ ∈ FCb1 (X, H ). Let us recall the Gaussian isoperimetric inequality, see [21]. Let E ⊂ X, and set Br = {x ∈ H : xH < r}, Er = E + Br ; then Φ
−1
γ (Er ) Φ −1 γ (E) + r,
t with Φ(t) = −∞
e−s /2 ds. √ 2π 2
(13)
Then, setting U (t) = Φ ◦ Φ −1 (t) ≈ t 2 log(1/t),
as t → 0;
(14)
the inequality γ (Er ) γ (E) + rU γ (E) + o(r)
(15)
follows. The isoperimetric inequality implies the following Gauss–Sobolev inequality
∞ ∇H f L1 (X,γ ) 0
U γ |f | > s ds,
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which implies the continuous embedding of D1,1 (X, γ ) into the Orlicz space L log1/2 L(X, γ ) := u : X → R measurable: A1/2 λ|u| ∈ L1 (X, γ ) for some λ > 0 , endowed with the Luxembourg norm
uL log1/2 L := inf λ > 0: A1/2 |u|/λ dγ 1 , X
see [18, Proposition 3.2]. Here A1/2 is defined by
t A1/2 (t) :=
log1/2 (1 + s) ds. 0
Analogously, using the continuity of the map f → |f |p from D1,p (X, γ ) to D1,1 (X, γ ), one obtains that D1,p (X, γ ) embeds continuously into the Orlicz space Lp log1/2 L(X, γ ) := u : X → R measurable: A1/2 λ|u|p ∈ L1 (X, γ ) for some λ > 0 .
(17)
Finally we recall the Poincaré inequality, see [7, Theorem 5.5.11]: for any u ∈ D1,p (X, γ ), p 1, p
u − u dγ dγ Cp ∇H up dγ , (18) H X
X
X
where Cp depends only on p. 2.4. The Ornstein–Uhlenbeck semigroup Let us define the Ornstein–Uhlenbeck semigroup (Tt )t0 , by Mehler’s formula
Tt u(x) = u e−t x + 1 − e−2t y dγ (y)
(19)
X
for all u ∈ L1 (X, γ ), t 0. For our purposes, the following properties of the Ornstein–Uhlenbeck semigroup are relevant: Tt is a contraction semigroup in L1 (X, γ ) and Tt u ∈ D1,1 (X, γ ) for any u ∈ L log1/2 L(X, γ ), t > 0 (see [18, Proposition 3.6]). In addition, a direct consequence of (7) and of Jensen’s inequality is
Tt uY dγ uY dγ , for any u ∈ L1 (X, γ , Y ), (20) X
X
with Y Hilbert (here Tt is defined componentwise, namely Tt u, y = Tt u, y ).
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Stronger smoothing properties of Tt hold for bounded functions; in particular if f ∈ Lp (X, γ ) for any p > 1, then Tt f ∈ Dk,q (X, γ ) for any k ∈ N, q > 1 (see [7, Proposition 5.4.8]). Moreover, the following commutation relation holds (again componentwise) for any u ∈ D1,1 (X, γ ): ∇H Tt u = e−t Tt ∇H u,
t > 0.
(21)
Therefore, we get ∇H Tt+s u = ∇H Tt (Ts u) = e−t Tt ∇H Ts u,
t 0, s > 0,
for any u ∈ L log1/2 L(X, γ ), see [7, Proposition 5.4.8]. As a consequence, we obtain that the limit (possibly infinite)
(22) I(u) := lim ∇H Tt uH dγ t↓0
X
always exists for u ∈ L log1/2 L(X, γ ). Indeed, consider the map
s → ∇H Tt+s uH dγ = e−t Tt ∇H Ts uH dγ X
X
and observe that
∇H Tt uH dγ lim inf
∇H Tt+s uH dγ
s→0
X
X
= e−t lim inf
Tt ∇H Ts uH dγ
s→0
X
e−t lim inf
∇H Ts uH dγ
s→0
X
by (20), which obviously implies that the limit exists. It also follows from (21) and (12) that
∗ −t ∗ Tt f ∇H Φ dγ = e f ∇H (Tt Φ) dγ , X
(23)
X
for all f ∈ L log1/2 L(X, γ ), Φ ∈ FCb1 (X, H ). Indeed, we can assume by a density argument that f ∈ D1,1 (X, γ ) to get
∗ T t f ∇H Φ dγ = − [∇H Tt f, Φ]H dγ = −e−t [∇H f, Tt Φ]H dγ X
X
= e−t
X
X ∗ f ∇H Tt Φ dγ .
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Another important consequence of (21) is that if u ∈ D1,1 (X, γ ) then lim ∇H Tt u − ∇H uL1 (X,γ ) = 0.
(24)
t→0
Finally, notice that if Em u are the canonical cylindrical approximations of a function u ∈ L log1/2 L(X, γ ) defined in (8) then
∇H Tt Em uH dγ X
∇H Tt uH dγ ,
∀t > 0.
(25)
X
To prove (25), let us first notice that, by the rotational invariance of γm⊥ , Tt Em u = Em Tt u. Indeed,
Tt Em u(x) =
Em u e−t x + 1 − e−2t z dγ (z)
X
=
u e−t Πm x + 1 − e−2t Πm z + y dγm⊥ (y ) dγ (z)
⊥ X Xm
and by applying the rotation invariance of Gaussian measures (7) to γm⊥ we get
Em Tt u(x) =
Tt u(Πm x + w ) dγm⊥ (w )
⊥ Xm
=
u e−t (Πm x + w ) + 1 − e−2t z dγm⊥ (w ) dγ (z)
⊥ X Xm
=
u e−t Πm x + e−t w + 1 − e−2t zm
⊥ X⊥ Xm Xm m
+ 1 − e−2t z dγm⊥ (w ) dγm⊥ (z ) dγm (zm )
= u e−t Πm x + 1 − e−2t Πm z + y dγm⊥ (y ) dγ (z). ⊥ X Xm
From the above commutation relation it follows that the vector ∇H Tt Em u = ∇H Em Tt u coincides with its projection ∇m on Hm , since Em u depends only on xm ∈ Xm . Moreover, by Jensen’s inequality we have
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∇H Tt Em u(x) = Em (∇m Tt u)(x) = ∇m Tt u(Πm x + x ) dγ ⊥ (x ) m H H
⊥ Xm
H
∇m Tt u(Πm x + x ) dγ ⊥ (x ) = Em ∇m Tt u(x) . m H H
⊥ Xm
Since ∇m Tt u(x)H ∇H Tt u(x)H we can integrate both sides to get (25). 3. BV functions in infinite dimensions We have collected in the preceding section the tools we need in order to discuss BV functions in the Wiener space setting. The BV(X, γ ) class can be defined as follows. Definition 3.1 (BV space). Let u ∈ L log1/2 L(X, γ ). We say that u ∈ BV(X, γ ) if there exists a measure μ ∈ M (X, H ) such that for any φ ∈ FCb1 (X) we have
u(x)∂j∗ φ(x) dγ (x) = −
X
φ(x) dμj (x),
∀j ∈ N,
(26)
X
where μj = [hj , μ]H . In particular, if u = χE and u ∈ BV(X, γ ), then we say that E has finite perimeter. Equivalently, we may require the existence of measures μj as in (26) satisfying sup(μ1 , . . . , μm )(X) < ∞.
(27)
m
Indeed, if μj = [μ, hj ]H , then the total variation of the Rm valued measure (μ1 , . . . , μm ) in X is less than |μ|(X). Conversely, if (27) holds, then the measure μ := j μj hj is well defined and belongs to M (X, H ) (it suffices to consider m 2the densities fi of μi with respect to the measure σ := sup |(μ , . . . , μ )| to obtain 1 m m 1 fi 1 σ -a.e., hence (fi ) 2 1 σ -a.e. and μ = fj hj σ ). Remark 3.2. Notice that in the previous definition we have required that the measure μ is defined on the whole of B(X) and is σ -additive there. Since cylindrical functions generate the Borel σ -algebra the measure μ verifying (26) is unique, and will be denoted Dγ u. Using (3) the total variation of Dγ u is given by |Dγ u|(X) = sup Dγ u, Φ ; Φ ∈ Cb (X, H ), Φ(x) H 1, ∀x ∈ X ∗ u∇H Φ dγ ; Φ ∈ FCb1 (X, H ), Φ(x) H 1, ∀x ∈ X . = sup X
(28)
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797
Actually it will be convenient to define also an even smaller space FCc1 (X, H ) of vector fields Φ representable as follows: Φ(x) =
m
∗ hi φi x, x1∗ , . . . , x, xm
i=1 1 with hi = Qxi∗ and φi ∈ Cc1 (Rm ). The space ) has an additional technical advan FCc (X, H ∗ tage: the divergence ∇H Φ of Φ above is i ∂i φi − φi R ∗ xi∗ and is a bounded function because R ∗ xi∗ is bounded on the support of φi . By a further approximation, based on the fact that any Φ ∈ FCb1 (X, H ) is the pointwise limit of a uniformly bounded sequence (Φn ) ⊂ FCc1 (X, H ), we have also ∗ |Dγ u|(X) = sup u∇H Φ dγ ; Φ ∈ FCc1 (X, H ), Φ(x) H 1, ∀x ∈ X . (29) X
In the case u = χE , we write Pγ (E, ·) for the measure |Dγ χE | and we shall also write Pγ (E) for Pγ (E, X). Remark 3.3 (About the L log1/2 L(X, γ ) assumption). The L log1/2 L(X, γ ) membership hypothesis we made on u is necessary to give sense to the integral of the product u∂j∗ φ: indeed, this term is the sum of the function u∂j φ (which makes sense for u ∈ L1 (X, γ ) only) and uφ hˆ j , which makes sense by Orlicz duality if u ∈ L log1/2 L(X, γ ) and exp(c|hˆ j |2 ) ∈ L1 (X, γ ) for some c > 0. Since hˆ j = R ∗ xj∗ for some xj∗ ∈ X ∗ , by our construction of hj , this exponential integrability property follows by Fernique’s theorem (5). Nevertheless, we shall provide different equivalent definitions of BV where this extra integrability property is not needed (as in the finite-dimensional case) but rather derived as a consequence. These equivalent definitions will also show that Definition 3.1 is independent of the choice of the basis (hj ). Let us see an equivalent way of defining the BV class, with partial derivatives along all directions h ∈ H ; in this case, since hˆ is in L2 (X, γ ) and not better, we have to assume u ∈ L2 (X, γ ) to give a sense to the integration by parts formula, even when cylindrical test functions are involved. Proposition 3.4. Let u ∈ L2 (X, γ ). Then, u ∈ BV(X, γ ) if and only if for every h ∈ H there is a real measure μh such that
X
with
hH =1 |μh |
u(x)∂h∗ φ(x) dγ (x) = −
φ(x) dμh (x),
∀φ ∈ FCb1 (X),
(30)
X
finite. In this case, |Dγ u| =
hH =1 |μh |.
Proof. If u ∈ BV(X, γ ) then the existence of μh = [h, Dγ u]H for all h ∈ H follows from the linearity of the ∂h operator with respect to h, and the boundedness of |μh | from the finiteness of |Dγ u|. In particular, hH =1 |μh | |Dγ u|(X).
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Conversely, define μj = μhj and μ=
∞
μj hj ,
j =1
so that μh = [h, μ]H . The integration by parts (26) clearly holds; we have to prove that μ is a finite measure. If we fix a partition (Bn )n∈N of X, if μ(Bn ) = 0, we define αn =
μ(Bn ) μ(Bn )H
so that we obtain μ(Bn ) = μ(Bn ), αn αn = μα (Bn ) n H H H n∈N
n∈N
|μαn |(Bn )
n∈N
|μh |,
hH =1
n∈N
and then u ∈ BV(X, γ ) with Dγ u := μ and |μ|
hH =1 |μh |.
2
It is easy to verify that if u ∈ D1,1 (X, γ ), then u ∈ BV(X, γ ) with Dγ u = ∇H uγ . Now we relate BV functions in Rm with cylindrical functions in X. We denote as before by γm the standard Gaussian distribution on Rm and we point out that all directional derivatives ∂ν and their adjoints ∂ν∗ have the same meaning as in the infinite-dimensional case, but without restriction on directions, since H = X in this case; we shall try to use as much as possible a consistent notation, valid both for the finite-dimensional and the infinite-dimensional case. Proposition 3.5. Let u ∈ L log1/2 L(X, γ ) be a cylindrical function, u(x) = v(Πx1∗ ,...,xm∗ x), with R ∗ xi∗ orthonormal. Then v ∈ BV(Rm , γm ) if and only if u ∈ BV(X, γ ) and |Dγ u|(X) = |Dγm v| Rm . Proof. Recalling that the law of γ under Π is γm , we have |Dγ u|(X) = sup
Φ(x) 1, ∀x ∈ X H
∗
∈ FCb1 (X, H ),
∗
∈ FCb1 (X, Hm ),
u(x)∇ Φ(x) dγ (x): Φ
X
= sup
u(x)∇ Φ(x) dγ (x): Φ X
= sup Rm
Φ(x) 1, ∀x ∈ X H
m v(y)∇ ∗ Ψ (y) dγm (y): Ψ ∈ Cb1 Rm , Ψ ∞ 1 .
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As a particular example of Sobolev functions we can consider Lipschitz functions. Definition 3.6 (H -Lipschitz functions). A Borel function f : X → R is said to be H -Lipschitz if there exists a constant C such that for γ -a.e. x one has f (x + h) − f (x) ChH , ∀h ∈ H. (31) It can be proved that for an H -Lipschitz function f there exists a full-measure γ -measurable set X0 such that X0 + H = X0 and, for every x ∈ X0 , one has f (x + h) − f (x) ChH , ∀h ∈ H. In particular, f has a version such that the previous inequality is satisfied for every x ∈ X. By the arguments in [7, Section 5.11], it can be proved that H -Lipschitz functions belong to D1,p (X, γ ) for every p 1, and in particular to BV(X, γ ). An important result is the following coarea formula, which can be proved by following verbatim the proof of [14, Section 5.5]. Theorem 3.7. If u ∈ BV(X, γ ), then for every Borel set B ⊂ X the following equality holds:
(32) |Dγ u|(B) = Pγ {u > t}, B dt. R
As a corollary, we have that almost every ball has finite perimeter, since the distance function is H -Lipschitz. We do not know whether every ball has finite perimeter, because Pγ (Br (x)) is not trivially monotone with respect to r and no homothety argument can be used in view of (10). We now extend from finite dimensions to infinite dimensions some typical tools of the BV theory. The following definition is a very convenient tool in the theory of BV functions: Definition 3.8 (Total variation). We define total variation of a function v ∈ L1 (X, γ ) by ∗ 1 Vγ (v) := sup v∇H Φ dγ : Φ ∈ FCc (X, H ), Φ(x) H 1, ∀x ∈ X .
(33)
X
The name is justified by the following observation: if v ∈ BV(X, γ ), then (29) shows that Vγ (v) = |Dγ v|(X); on the other hand, if X is finite-dimensional and the supremum in (33) is finite, then a direct application of Riesz theorem provides us with an X-valued measure μ, with total variation less than Vγ (v), such that
∗ v∇H Φ dγ = −Φ, μ ,
∀Φ ∈ Cb1 (X, H ).
X
Hence μ = Dγ u and Vγ (u) = |Dγ v|(X). This equivalence is much less obvious in the infinitedimensional case, since Riesz theorem is not available, and it will be discussed in the next section. Notice that v → Vγ (v) is lower semicontinuous with respect to the L1 (X, γ ) convergence, since it is the supremum of a family of continuous functionals. Since v → Vγ (v) is easily seen to be
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nonincreasing under cylindrical approximation (with the same argument used in Proposition 3.5), we can combine this property with lower semicontinuity to get Vγ (v) = lim Vγ (Em v).
(34)
m→∞
Analogous definitions can be given for the total variation along a direction ν ∈ Hm for some m; in this case, in order to have a bounded adjoint derivative ∇ν∗ φ, we consider the space Fν Cc1 (X) of cylindrical functions φ with support contained in a strip {a < x, x ∗ < b}, where ν = Qx ∗ . Definition 3.9 (Directional total variation). Let ν ∈ variation of a function v ∈ L1 (X, γ ) along ν by Vγν (v) := sup
v∂ν∗ φ dγ :
φ∈F
ν
m Hm
Cc1 (X),
be a unit vector. We define total
φ(x) 1, ∀x ∈ X .
(35)
X
Again v → Vγν (v) is lower semicontinuous with respect to the L1 (X, γ ) convergence and, in finite-dimensional spaces, Riesz theorem shows that the quantity is finite if and only if the integration by parts formula
v∂ν∗ φ dγ = − φ dμ, ∀φ ∈ Cb1 (X), (36) X
X
holds for some real-valued measure μ with finite total variation, that we shall denote by Dγν v; if this happens, |μ|(X) coincides with Vγν (v). Finally, Vγν (v) = lim Vγν (Em v). m→∞
(37)
We can now discuss 1-dimensional sections of Gaussian BV functions in the same spirit as Section 3.11 of [1]. Notice that any u ∈ BV(Rm , γm ) is in the classical space BV loc (Rm ), so that we can use all the (local) properties known in Euclidean case, and Dγ u = Gm Du
(38)
where Gm (x) = (2π)−m/2 exp(−|x|2 /2) is the standard Gaussian kernel and Du stands for the classical derivative of u in the sense of distributions. Let us fix a unit direction ν = Qx ∗ ∈ H , let Π(x) = x, x ∗ be the induced projection and let us write x ∈ X as y + Π(x)ν. Then, denoting by K the kernel of Π , γ admits a product decomposition γ = γ ⊥ ⊗ γ1 with γ ⊥ Gaussian in K. For u : X → R and y ∈ K we define the function uy : R → R by uy (t) = u(y + tν). Theorem 3.10. Let u ∈ L log1/2 L(X, γ ) and let ν ∈ m Hm ; then
Vγν (u) = Vγ1 (uy ) dγ ⊥ (y). K
In particular Definition 3.9 is independent of the choice of the basis and makes sense for all h ∈ H.
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Proof. Let us fix φ ∈ Fν Cb1 (X) with φ∞ 1. Then
u(x)∂ν∗ φ(x) dγ (x) =
X
uy (t) φy (t) − tφy (t) dγ1 (t) dγ ⊥ (y) Vγ1 (uy ) dγ ⊥ (y),
R
K
K
whence
Vγν (u)
Vγ1 (uy ) dγ ⊥ (y).
K
For the reverse inequality we can assume that Vγν (u) is finite. First we prove the inequality in the finite-dimensional case X = Rm and γ = γm , and then we consider the general case. If X = Rm and γ = γm then the measure μ = Dγνm u in (36) is a real-valued measure with total variation in Rm less than Vγν (u). Let us show that we may find a sequence (un ) ⊂ C ∞ (Rm ) ∩ D1,1 (Rm , γm ) such that un → u in L1 (Rm , γm ) and
lim
n→∞ Rm
∂ν un (y) dγm (y) V ν (u). γm
For, set un = T1/n u and notice that for every φ ∈ Fν Cb1 (Rm ) with |φ| 1 φ∂ν un dγm = ∂ ∗ φun dγm → ∂ ∗ φu dγm V ν (u), ν ν γm Rm
Rm
Rm
then the sequence (∂ν un γm ) is bounded in M (Rm ), and (up to a subsequence which we do not relabel) weakly∗ converges to a measure μ. The above limit relation shows that μ = Dγνm and that the whole sequence is convergent. By Fubini theorem (possibly passing to a subsequence) for γ ⊥ -a.e. y ∈ K the sequence (un,y ) converges to uy in L1 (R, γ1 ) and then by semicontinuity we get
Vγν (u) lim inf
n→+∞ Rm
= lim inf
n→+∞
∂ν un (y) dγ (y) = lim inf
n→+∞
Vγ1 (un,y ) dγ ⊥ (y)
K
u (t) dγ1 (t) dγ ⊥ (y) n,y
K R
Vγ1 (uy ) dγ ⊥ (y).
K
In the infinite-dimensional case we consider the cylindrical approximations vm := Em u; since they converge in L1 (X, γ ) we can find a subsequence (mi ) such that vmi ,y → uy in L1 (R, γ1 ) for γ ⊥ -a.e. y ∈ K; then, lower semicontinuity of v → Vγ1 (v), Fatou’s lemma and monotonicity give
⊥
Vγ1 (uy ) dγ (y) lim inf i→∞
K
Vγ1 (umi y ) dγ ⊥ (y) sup Vγ (vm ) Vγ (u). m
K
2
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ˆ Corollary 3.11. Let h = R hˆ ∈ H and c ∈ R; then the sets E = {x ∈ X: h(x) c} have finite perimeter with 1 c2 . Pγ (E) = √ exp − 2h2H 2π
(39)
Proof. With no loss of generality we can assume that hH = 1. We prove the identity first in the case when hˆ = R ∗ x ∗ for some x ∗ ∈ X ∗ ; without loss of generality, we may assume that Qx ∗ H = 1. In this case the assertion simply follows by noticing that E is a cylindrical set of the form E = x ∈ X: x, x ∗ ∈ B , √ 2 with B = {s ∈ R: s c}. This implies that Pγ (E) = Pγ1 (B) = e−c /2 / 2π . In the general case, density of QX ∗ in H and lower semicontinuity of the perimeter provides the inequality in (39); to prove the inequality we fix φ ∈ Cc∞ (R) with φ(c) = 1, |φ| 1, k = Qk ∗ , kH = 1 and k − h2H < 2ε and the field Φε (x) = φ(x, k ∗ )k; then
Pγ (E) ∂k∗ φ x, k ∗ dγ . E
ˆ By Proposition2.2 and considering the map x → (h(x), x, k ∗ ), the right-hand side can be ∗ represented as {x1 c} ∂ φ(x2 ) dηε (x1 , x2 ), where ηε are Gaussian in R2 with γ1 as marginals and x1 x2 dηε (x1 , x2 ) > 1 − ε; as ε → 0 these Gaussians converge to the standard Gaussian on the diagonal of R2 , so that
φ(c) 2 Pγ (E) ∂ ∗ φ(z) dγ1 (z) = √ e−c /2 . 2 2π {zc}
In connection with the proof of the previous corollary, notice that it would be desirable to extend Theorem 3.10 even to the case when ν = R hˆ ∈ H ; however, this extension presents some difficulties, since hˆ is not really a linear map on X. 4. Main results We are now in a position to characterize BV functions in the same way as they are characterized in the classical case. Notice that in the classical case the original definition of BV given by E. De Giorgi in [12] was based on property (4) below, with the heat semigroup instead of the Ornstein–Uhlenbeck one. Theorem 4.1. Given u ∈ L1 (X, γ ), the following are equivalent: (1) u belongs to BV(X, γ ); (2) the quantity Vγ (u) in (33) is finite; L1 → u} < ∞; (3) Lγ (u) := inf{lim infn→∞ X ∇H un H dγ : un ∈ D1,1 (X, γ ), un − (4) u ∈ L log1/2 L(X, γ ) and the quantity I(u) in (22) is finite.
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Moreover, |Dγ u|(X) = Vγ (u) = Lγ (u) = I[u] and
∇H Tt uH dγ e−t |Dγ u|(X),
∀t > 0.
(40)
X
Proof. (1) ⇒ (2). Simply comparing the classes of competitors, we notice that Vγ (u) |Dγ u|(X). (2) ⇒ (3). Let tn ↓ 0 and un = Ttn u. Then, for all Φ ∈ FCb1 (X, H ) with Φ(x)H 1 for every x ∈ X, from (23) we deduce
[∇H un , Φ]H dγ = −e−tn
X
u∇ ∗ (Ttn Φ) dγ Vγ (u).
X
Therefore, ∇H un L1 (X,γ ) Vγ (u). In particular, we have proved that Lγ (u) Vγ (u). (3) ⇒ (4). Let a sequence (un )n∈N be fixed in such a way that un → u in L1 (X, γ ) and ∇H un L1 (X,γ ) → Lγ (u). Then,
∇H Tt uH dγ lim inf
∇H Tt un H dγ
n→∞
X
X
=e
−t
∇H un H dγ = Lγ (u).
lim inf n→∞
X
In addition, Fatou’s lemma and (16) yield
∞
U γ |f | > s ds Lγ (u) < ∞,
(41)
0
so that u ∈ L log1/2 L(X, γ ). Observe that in particular we have proved that I(u) Lγ (u). (4) ⇒ (1). We first prove that for all j the derivative μj along the direction hj exists, and then we prove (27) to conclude that u ∈ BV(X, γ ). Since Tt u ∈ D1,1 (X, γ ) for t > 0, we have
h h Vγ j (Tt u) = Dγ j Tt u = |∂j Tt u| dγ . X
In particular, setting ν = hj and adopting the same notation as in Theorem 3.10, we have
K
Vγ1 (Tt u)y dγ ⊥ (y) =
|∂j Tt u| dγ
X
∇H Tt uH dγ . X
Now we can find tn → 0 sufficiently fast in such a way that (Ttn u)y converge to uy in L1 (R, γ1 ) for γ ⊥ -a.e. y ∈ K and conclude, by the lower semicontinuity of v → Vγ1 (v), that
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Vγ1 (uy ) dγ ⊥ (y) I(u) < ∞.
K
It follows that for γ ⊥ -a.e. y ∈ K the function uy has bounded variation in R. By a Fubini argument, based on the factorization γ = γ ⊥ ⊗ γ1 , the 1-dimensional integration by parts formula yields that the measure μj = Dγ1 uy ⊗ γ ⊥ , i.e.
μj (A) = Dγ1 uy (Ay ) dγ ⊥ (y) K
(where Ay := {t: y + thj ∈ A} is the y-section of a Borel set A) provides the derivative of u along hj . Notice that μj is well defined, since we have just proved that K |Dγ1 uy |(R) dγ ⊥ is finite. Now, setting μi = Dγhi u, we check (27); by a density argument, it suffices to prove that m
φi dμi I(u)
i=1 X
for all φi ∈ FCb1 (X) with
i
φi2 1; by integration by parts, it suffices to show that lim sup t↓0
m
φi dDγhi Tt u I(u)
i=1 X
or equivalently lim sup t↓0
m
φi ∂i Tt u dγ I(u).
i=1 X
The latter inequality is trivial, since | m 1 φi ∂i Tt u| ∇H Tt uH . Passing to the limit as s ↓ 0 in the inequality
∇H Tt+s uH dγ e−t ∇H Ts uH dγ X
provides
X ∇H Tt uH
X
dγ e−t I(u) and concludes the proof.
2
Remark 4.2. Arguing as in the proof of the implication (2) ⇒ (3) we see that for all u ∈ BV(X, γ ) and all closed subspaces K ⊂ H the following inequality holds:
lim sup πK ∇H Tt H dγ DγK u(X). t↓0
X
Here πK : H → K is the orthogonal projection and DγK u = πK Dγ u.
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Remark 4.3. It is worth noticing that sets of finite perimeter E can also be characterized using the functional
U (Tt χE )2 + ∇H Tt χE 2H dγ , JE (t) = X
in the spirit of [5]. In fact, it can be proved that d JE (t) 0, dt
lim JE (t) = Pγ (E).
t→0
(42)
With minor modifications the proof of Theorem 4.1 allows also to show that, for all j , the family of measures ∂j Tt uγ is tight and the limit as t ↓ 0 is Dγ u, hj , in the duality with Cb (X); in addition, the limit of |Dγ u, hj γ | is |Dγ u, hj |. We give just a sketch of proof, since this result is a consequence of our characterization of BV functions, rather than a tool. We also notice that similar results could be stated and proved for the measures πK ∇H Tt uH γ and |πK Dγ u|, with K ⊂ H closed subspace. Theorem 4.4. Let u ∈ BV(X, γ ) and j 1. Then lim ∂j Tt uγ = Dγ u, hj , t↓0
lim |∂j Tt u|γ = Dγ u, hj t↓0
in the duality with Cb (X). Proof. Since, by the integration by parts formula, ∂j Tt uγ weakly converge to Dγ u, hj in the duality with FCb1 (X) as t ↓ 0, in order to show the convergence of ∂j Tt uγ it suffices to show that |∂j Tt u|γ is tight. Indeed, this ensures by Prokhorov theorem the compactness in the duality with Cb (X), and the weak limit must be the same as above, by the density of FCb1 (X) in Cb (X). By Remark 4.2 we have
lim sup t↓0
|∂j Tt u| dγ Dγ u, hj (X),
X
hence tightness is achieved, thanks to Lemma 2.1, by
lim inf t↓0
|∂j Tt u| dγ Dγ u, hj (A)
for all A ⊂ X open.
A
This inequality, in turn, can be derived as in the proof of the implication (3) ⇒ (4) in the proof of Theorem 4.1, using the fact that the sections Ay = {t ∈ R: y + thj ∈ A} are open, and the lower semicontinuity of v ∈ BV(R, γ1 ) → |Dγ1 v|(J ) with respect to the L1 (γ1 ) convergence, for all J ⊂ R open.
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From Theorem 4.1 the extension, by approximation, of many properties of Sobolev functions to BV follows: for instance BV(X, γ ) ∩ L∞ (X, γ ) is an algebra (and therefore sets of finite perimeter are stable under union and intersection), BV(X, γ ) is stable under left composition with Lipschitz maps f , and |Dγ f ◦ u|(X) Lip(f )|Dγ u|(X), etc. We need in particular the inequalities stated in the following proposition, a direct consequence of (16) and (18) for D1,1 (X, γ ) functions, and of the equality |Dγ u|(X) = Lγ (u). Proposition 4.5 (Sobolev and Poincaré inequalities). Let u ∈ BV(X, γ ). Then
+∞ U γ |u| > s ds |Dγ u|(X) 0
and
u − u dγ dγ C1 |Dγ u|(X), X
X
where C1 is the constant in the Poincaré inequality (18) for p = 1. The following approximation result for sets of finite perimeter is a consequence of the approximation in BV through smooth functions and the coarea formula. Proposition 4.6. Let E ⊂ X be a set with finite perimeter; then there exist cylindrical sets Ej = Πm−1j Bj , with Bj ∈ Rmj smooth sets, such that lim χEj − χE L1 (X,γ ) = 0 and
j →∞
lim Pγ (Ej ) = Pγ (E).
j →∞
Proof. According to Theorem 4.1, for every u ∈ BV(X, γ ) there is a sequence of D1,1 (X, γ ) functions such that the L1 norms of their gradients converge to the total variation of u. Moreover, smooth cylindrical functions are dense in D1,1 (X, γ ), hence there exists a sequence (uj ) of smooth cylindrical functions with
1 ∇H uj H dγ → Pγ (E). uj → χE in L (X, γ ) and X
Assuming with no loss of generality that 0 uj 1, the conclusion then follows from the coarea formula by taking smooth levels Bj of uj . 2 Due to the previous proposition, we say that E is a smooth set if E = Πm−1 B for some set B ∈ Rm with smooth boundary. Denoting by Hm−1 the Hausdorff (m − 1)-dimensional measure in Rm , since by (38) Dγm χB = Gm DχB and |DχB |(A) = Hm−1 (A ∩ ∂B) for all Borel sets A, for the sets Ej of the previous proposition we get
Pγ (E) = lim
j →+∞ ∂Bj
Gmj dHmj −1 .
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Remark 4.7. Using the results in [5], it is possible to repeat the same argument contained in [8] to show that the isoperimetric inequality and extremality of half-spaces hold in infinite dimension as well. That is, for any E ⊂ X with finite perimeter, Pγ (E) U γ (E) , with U defined in (14) with equality if and only if E is an arbitrary half-space. The proof is based on analysis of the quantity JE introduced in Remark 4.3; in particular, once one has (42), by using the fact that JE (t) is twice differentiable (see [7, Proposition 5.4.8]), one shows that if equality holds for a set E in the isoperimetric inequality, then χE is affine (see [8, Lemma 2.1]), that is E is a half-space. In the next corollary we consider a finite-dimensional subspace K ⊂ H . We shall denote by ΠK : X → K the canonical projection (induced by the choice of a basis of K) and set ΠK⊥ (x) = x − ΠK (x); since ΠK ◦ ΠK = ΠK and ΠK |K = Id we may write in a unique way x = x1 + x2 with x1 ∈ K and x2 ∈ Ker(ΠK ) and, accordingly, ux1 (x2 ) = u(x1 + x2 ). Setting X1 = K and X2 = Ker(ΠK ) (the closure of K ⊥ in X), we have also the factorization dγ (x1 , x2 ) = dγ1 (x1 ) ⊗ dγ2 (x2 ) with γ1 Gaussian in X1 and γ2 Gaussian in X2 ; furthermore, the Cameron–Martin spaces are respectively K and K ⊥ . The next proposition is the natural complement of Theorem 3.10: in that theorem the slices are 1-dimensional, and with minor changes the same result could be proved for finite-dimensional slices. Here we consider, instead, slices of finite codimension; for the sake of simplicity we state it assuming a priori that the map u is globally BV, hence without using variations, and we consider only one implication. In the sequel, for a given closed subspace L ⊂ H and u ∈ BV(X, γ ) we set, in accordance with Remark 4.2, DγL u := πL Dγ u ∈ M (X, L), where πL : H → L is the orthogonal projection. Proposition 4.8. Let u ∈ BV(X, γ ); then ux1 ∈ BV(X2 , γ2 ) for γ1 -a.e. x1 ∈ X1 and
K⊥ D u(X) = |Dγ ux |(X2 ) dγ1 (x1 ). γ 2 1
(43)
X1
Proof. Let un = Ttn u, with tn → 0, and assume with no loss of generality that (un )x1 converge to ux1 in L1 (X2 , γ2 ) for γ1 -a.e. x1 . We have
∇ ⊥ (un )x (x2 ) ⊥ dγ2 (x2 ) dγ1 (x1 ) = ∇ ⊥ un H dγ K K 1 K X1 X2
X
and, passing to the limit as n → ∞, Fatou’s lemma and Remark 4.2 give
⊥ Lγ2 (ux1 ) dγ1 (x1 ) DγK u(X), X1
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with Lγ2 as in (3) of Theorem 4.1. From Theorem 4.1 we deduce ux1 ∈ BV(X2 , γ2 ) for γ1 -a.e. x1 ∈ X1 and the inequality holds in (43). Arguing as in the first part of the proof of Theorem 3.10 we see that the factorization γ = γ1 ⊗ γ2 yields
h, Dγ u =
h, Dγ2 ux1 dγ1 (x1 ) X1
for all h ∈ K ⊥ (indeed, both measures satisfy the integration by parts formula in the direction h), hence
⊥ DγK u = Dγ2 ux1 dγ1 (x1 ). X1
This immediately gives
K⊥ D u(X)
|Dγ2 ux1 |(X2 ) dγ1 (x1 ).
γ
2
X1
Corollary 4.9. Let u ∈ BV(X, γ ) let K ⊂ H be finite-dimensional and EK u the conditional expectation relative to K. If K ⊥ is the complementary subspace of K, we have
⊥ |u − EK u| dγ C1 DγK u(X).
X
Proof. We write as in Proposition 4.8 x = x1 + x2 with x1 ∈ X1 = K and x2 ∈ X2 = Ker(ΠK ), ⊥ we get and denote by ux1 (x2 ) = u(x1 + x2 ); using the Poincaré inequality in XK
u(x) − EK u(x) dγ (x) =
u(x1 + x2 ) − EK u(x1 ) dγ1 (x1 ) dγ2 (x2 )
X1 ×X2
X
= ux1 (x2 ) − ux1 (z) dγ2 (z) dγ2 (x2 ) dγ1 (x1 ) X1 X2
C1
X2
K⊥ D ux (X2 ) dγ1 (x1 ) = C1 D K ⊥ u(X), γ2 γ 1
X1
where in the last line we have used Proposition 4.8.
2
In an analogous way one can prove that
|u − EK u|p dγ Cp
X
p
πK ⊥ ∇H uH dγ , X
∀u ∈ D1,p (X, γ ).
(44)
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The next theorem generalizes Theorem 5.12.5 of [7], for the part concerning Sobolev spaces. The part concerning BV functions is, to our knowledge, totally new. Theorem 4.10. The following statements hold. (i) For p > 1, let F be a bounded family of functions in D1,p (X, γ ); assume that for every ε > 0 there exists a finite-dimensional subspace K of H such that,
p
∇K ⊥ uH dγ ε,
∀u ∈ F;
(45)
X
then F is relatively compact in Lp (X, γ ). (ii) Let F be a bounded family of functions in BV(X, γ ); assume that for every ε > 0 there exists a finite-dimensional subspace K of H such that, |DK ⊥ u|(X) ε,
∀u ∈ F;
(46)
then F is relatively compact in L1 (X, γ ). Proof. We first discuss briefly the finite-dimensional case, X = Rm , γ = γm , for BV functions (for Sobolev functions, see [6, Theorem 9.3.19] or adapt the argument below, taking into account the continuous embedding of D1,p into the space Lp log1/2 L in (17)). Since the family F is bounded also in L log1/2 L(Rm , γm ), we obtain that F is equi-integrable in L1 (Rm , γm ), hence by a truncation argument we can assume with no loss of generality that F is uniformly bounded also in L∞ (Rm , γm ). Under this assumption relative compactness follows obviously from relative compactness in L1loc (Rm ); the latter is a consequence of the classical compact embedding of BV loc in L1loc and of the identity Dγm u = Gm Du, showing that supu∈F |Du|(K) is finite for any compact set K ⊂ Rm . We shall prove only (ii) by showing that F is totally bounded (the proof of (i) is analogous). By Corollary 4.9 we have
|u − EK u| dγ C1 ε,
∀u ∈ F.
X
Since the result holds in finite dimension, the family FK = {EK u: u ∈ F} is totally bounded in L1 (X, γ ) and the thesis follows. 2 Remark 4.11. Statement (i) is not true in D1,1 (X, γ ) under condition (45) with p = 1. In this case the family F is only pre-compact and the limit is in general only in BV(X, γ ). Moreover, bounded families in BV(X, γ ) are not in general pre-compact; as an example it suffices to consider the family F of characteristic functions of the sets x: x, x ∗ 0 ,
x ∗ ∈ X∗ ,
Qx ∗ H = 1.
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Condition (45) is satisfied if there exists a compact poperator K on H such that ∇H u ∈ K(H ) almost everywhere for all u ∈ F and X K−1 ∇H uH dγ is uniformly bounded on F. Indeed, a simple compactness argument proves that h − Πm hH → 0 uniformly on h ∈ K(H ): K−1 h H 1 , hence we may find ωm → 0 such that
p ∇H u − ∇Hm uH
dγ ωm
X
−1 K ∇H u p dγ . H
X
Analogously, in the BV case we have that (46) is fulfilled if there exists a compact operator K on H such that the Radon–Nikodym densities Dγ u/|Dγ u| belong to K(H ) |Dγ u|-a.e. for all u ∈ F and
−1 Dγ u d|Dγ u| K |Dγ u| H X
is uniformly bounded on F. Indeed, Hm⊥ Dγ u(X) = Dγ u − D Hm u(X) = γ
Dγ u Dγ u |D u| − πHm |D u| d|Dγ u| γ
γ
X
ωm
H
−1 Dγ u d|Dγ u|. K |D u| γ
H
H
5. The case when X is a Hilbert space The results presented above show that there are strict links between the notion of derivative ∂h , the semigroup Tt and the measure γ , which turns out to be invariant under Tt . Indeed, if X is a Hilbert space, different notions of derivative can be given, related to different semigroups still having γ as invariant measure (see Remark 5.2). In this section we briefly describe another point of view and confine ourselves to deriving the corresponding compact embedding theorem both for Sobolev and BV functions. In this section we assume that (X, ·,· X ) is a separable Hilbert space; let γ = N (0, Q) be as before. Identifying X and its dual X ∗ with the inner product, we fix an orthonormal basis (ek ) in X such that Qek = λk ek , with λk > 0 and that
k λk
∀k 1,
< ∞. If we set xk = x, ek , since R ∗ ek = xk and Rxk = λk ek it follows λk ek H = xk L2 (x,γ ) =
λk .
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1/2
As a consequence, an orthonormal basis in H is given by εk = λk ek , the Cameron–Martin norm is ( k xk2 /λk )1/2 and ∞ 2 x k H = Q1/2 X = x ∈ X: ∃y ∈ X with x = Q1/2 y = x ∈ X: <∞ . λk k=1
Notice that Q1/2 is still a compact operator on X. Setting for all k 1 Dk f (x) = ∂ek f (x) = lim
t→0
f (x + tek ) − f (x) , t
we define by linearity a gradient operator D : FCb1 (X) → FCb (X, X). The gradient turns out to be a closable operator with respect to the topologies Lp (X, γ ) and Lp (X, γ , X) for every p 1, and we denote by W 1,p (X, γ ) the domain of the closure in Lp (X, γ ), endowed with the norm u1,p =
u(x)p dγ +
X
p/2 1/p
∞ Dk u(x)2 dγ , X
k=1
where we keep the notation Dk also for the closure of the partial derivative operator. 1/2 As a consequence of the relation εk = λk ek we have also 1/2
∂εk = λk Dk ,
(47)
so that W 1,p (X, γ ) ⊂ D1,p (X, γ ), since ∇H f H = ( k λk |Dk f |2 )1/2 . Since eˆk = xk /λk , the integration by parts formula (11) becomes
1 g(x)Dk f (x) dγ (x) = − f (x)Dk g(x) dγ (x) + xk f (x)g(x) dγ (x) λk X
X
(48)
X
for f, g ∈ FCb1 (X). However, we point out that even though Dv f = i vi Di f makes sense for v ∈ X, the corresponding integration by parts along v does not, since at least convergence of |vk | 1 |vk | |xk | dγ = √ √ λk λk 2π k k is needed (this is ensured for v ∈ Q(X) = Q1/2 (H )). In this context, we may give a corresponding definition of BV functions. Definition 5.1. A function u ∈ L1 (X, γ ) belongs to BV X (X, γ ) if there exists ν u ∈ M (X, X) such that for any k ∈ N we have
1 u(x)Dk ϕ(x) dγ = − ϕ(x) dνku + xk u(x)ϕ(x) dγ , ϕ ∈ FCb1 (X), λk X
with νku = ν u , k X .
X
X
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L. Ambrosio et al. / Journal of Functional Analysis 258 (2010) 785–813
It is immediate to check that BV X (X, γ ) is contained in BV(X, γ ) and that 1/2
Dγ u, εi = λi νiu .
(49)
Remark 5.2. Even though we do not further pursue this point of view here, let us point out that the transition semigroup corresponding to the gradient D is given by
Rt f (x) =
−1/2 f e−tQ x + y dγt (y),
f ∈ Cb (X),
X
where γt = N (0, Qt ) and
t Qt = Q
e−sQ
−1
−1 ds = Q 1 − e−tQ .
0 −1
Notice that e−tQ e−ωt , t 0, where ω = inf λ1k . Therefore N (0, Qt ) → N (0, Q) = γ weakly as t → ∞, so that γ is invariant for Rt . Moreover Rt maps L1 (X, γ ) into W 1,1 (X, γ ) for every t > 0, see e.g. [10, Proposition 10.3.1], and this is coherent with the hypothesis u ∈ L1 (X, γ ) in Definition 5.1. In [4] we plan to investigate further relations between BV X (X, γ ) and the semigroup Rt . Let us show that both Sobolev and BV spaces in the present context are compactly embedded into the corresponding Lebesgue spaces, see [11] for the case p = 2. The following statement easily follows from Theorem 4.10 and the relation (47). Theorem 5.3. For every p 1, the embedding of W 1,p (X, γ ) into Lp (X, γ ) is compact. The embedding of BV X (X, γ ) into L1 (X, γ ) is compact. Proof. Let us prove the statement in the Sobolev case, that of BV is similar and uses (49). It suffices to show that every family F bounded in W 1,p (X, γ ) is totally bounded in Lp (X, γ ). If u1,p C for all u ∈ F, then in particular
∞ X
p/2 |Dk u|
2
dγ C p ,
∀u ∈ F,
k=1
whence by (47) 2 p/2
∞ 1 ∂ε u dγ C p , λ k k X
∀u ∈ F.
k=1
By applying Remark 4.11 with K = Q1/2 the thesis follows.
2
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References [1] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford Univ. Press, 2000. [2] L. Ambrosio, M. Miranda Jr., D. Pallara, Special functions of bounded variation in doubling metric measure spaces, in: D. Pallara (Ed.), Calculus of Variations: Topics from the Mathematical Heritage of Ennio De Giorgi, in: Quad. Mat., vol. 14, Dipartimento di Matematica della seconda Università di Napoli, 2004, pp. 1–45. [3] L. Ambrosio, S. Maniglia, M. Miranda Jr., D. Pallara, Towards a theory of BV functions in abstract Wiener spaces, Evolution Equations: A special issue of Physica D, forthcoming, http://cvgmt.sns.it/papers/ambmanmir08/ BVWiener.pdf. [4] L. Ambrosio, G. Da Prato, D. Pallara, in preparation. [5] D. Bakry, M. Ledoux, Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator, Invent. Math. 123 (1996) 259–281. [6] M. Bertoldi, L. Lorenzi, Analytical Methods for Markov Semigroups, Chapman and Hall, 2007. [7] V.I. Bogachev, Gaussian Measures, American Mathematical Society, 1998. [8] E.A. Carlen, C. Kerce, On the cases of equality in Bobkov’s inequality and Gaussian rearrangement, Calc. Var. Partial Differential Equations 13 (2001) 1–18. [9] G. Da Prato, An Introduction to Infinite-Dimensional Analysis, Springer, 2006. [10] G. Da Prato, J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Math. Soc. Lecture Note Ser., vol. 293, Cambridge Univ. Press, 2002. [11] G. Da Prato, P. Malliavin, D. Nualart, Compact families of Wiener functionals, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) 1287–1291. [12] E. De Giorgi, Su una teoria generale della misura (r − 1)-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954) 191–213; and also in: L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda, S. Spagnolo (Eds.), Ennio De Giorgi: Selected Papers, Springer, 2006, pp. 79–99; English translation in: L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda, S. Spagnolo (Eds.), Ennio De Giorgi: Selected Papers, Springer, 2006, pp. 58–78. [13] E. De Giorgi, Su alcune generalizzazioni della nozione di perimetro, in: G. Buttazzo, A. Marino, M.V.K. Murthy (Eds.), Equazioni differenziali e calcolo delle variazioni, Pisa, 1992, in: Quaderni U.M.I., vol. 39, Pitagora, 1995, pp. 237–250. [14] L.C. Evans, R.F. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions, CRC Press, 1992. [15] D. Feyel, A. de la Pradelle, Hausdorff measures on the Wiener space, Potential Anal. 1 (1992) 177–189. [16] M. Fukushima, On semimartingale characterization of functionals of symmetric Markov processes, Electron. J. Probab. 4 (1999) 1–32. [17] M. Fukushima, BV functions and distorted Ornstein–Uhlenbeck processes over the abstract Wiener space, J. Funct. Anal. 174 (2000) 227–249. [18] M. Fukushima, M. Hino, On the space of BV functions and a related stochastic calculus in infinite dimensions, J. Funct. Anal. 183 (2001) 245–268. [19] M. Hino, Integral representation of linear functionals on vector lattices and its application to BV functions on Wiener space, in: Stochastic Analysis and Related Topics in Kyoto in Honour of Kiyoshi Itô, in: Adv. Stud. Pure Math., vol. 41, 2004, pp. 121–140. [20] M. Hino, Sets of finite perimeter and the Hausdorff–Gauss measure on the Wiener space, J. Funct. Anal. (2009), doi:10.1016/j.jfa.2009.06.033, in press. [21] M. Ledoux, Isoperimetry and Gaussian analysis, in: Lectures on Probability Theory and Statistics, Saint Flour, 1994, in: Lecture Notes in Math., vol. 1648, Springer, 1996, pp. 165–294. [22] M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monogr., vol. 89, American Mathematical Society, 2001. [23] W. Linde, Probability on Banach Spaces, Teubner, 1983; Wiley, 1986. [24] N.N. Vakhania, V.I. Tarieladze, S.A. Chobanyan, Probability Distribution in Banach Spaces, Kluwer, 1987.
Journal of Functional Analysis 258 (2010) 814–851 www.elsevier.com/locate/jfa
Coercive inequalities on metric measure spaces ✩ W. Hebisch a , B. Zegarli´nski b,∗ a Institute of Mathematics, University of Wrocław, Wrocław, Poland b Department of Mathematics, Imperial College, London, UK
Received 11 May 2009; accepted 15 May 2009 Available online 30 May 2009 Communicated by L. Gross
Abstract In this paper we study coercive inequalities on finite dimensional metric spaces with probability measures which do not have the volume doubling property. Crown Copyright © 2009 Published by Elsevier Inc. All rights reserved. Keywords: Coercive inequalities; Logarithmic Sobolev; U -bounds; Heisenberg group
1. Introduction In this paper we study coercive inequalities on finite dimensional metric spaces with probability measures which do not have the volume doubling property. This class of inequalities includes the well-known Poincaré inequality Mμ|f − μf |q μ|∇f |q with some constants M ∈ (0, ∞), q ∈ (1, ∞) independent of the function f . The (metric) length of the gradient |∇f | is assumed to be well defined here. This class also includes a variety of stronger coercive inequalities with the variance on the left-hand side replaced by a functional ✩
Supported by EPSRC EP/D05379X/1 & Royal Society. The first author was partially supported by KBN grant 1 P03A 03029. * Corresponding author. E-mail address:
[email protected] (B. Zegarli´nski). 0022-1236/$ – see front matter Crown Copyright © 2009 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.05.016
W. Hebisch, B. Zegarli´nski / Journal of Functional Analysis 258 (2010) 814–851
815
with a stronger growth, as for example in case of the celebrated Log-Sobolev inequality, which is of the following form μf 2 log
f2 cμ|∇f |2 μf 2
with some constant c ∈ (0, ∞) independent of the function f . We are interested in probability measures on non-compact spaces, like for example finite products of real lines Rn , as well as certain non-compact groups, including the Heisenberg group. For probability measures on the real line the necessary and sufficient condition for Poincaré inequality characterising the density (of the absolutely continuous part with respect to the Lebesgue measure) were established a long time ago by Muckenhoupt [31] ([29]). More recently such criteria were established for other coercive inequalities (Log-Sobolev type: (LS2 ) [8], (LSq ) [11], for distributions with weaker tails [6], etc., and others). In the multidimensional case the situation is rather different and more intricate. First of all, since the inequalities of interest to us have a natural tensorisation property, there is a number of perturbative techniques which allow to obtain classes of interesting examples in higher and even in infinite dimensions (see e.g. [21,11,33,28], etc., [12,36,37,41,42], etc., and references given there). We would like to mention a work [34] in which the coercive inequalities for probability measures on Rn , n 3, with a variety of decay of the tails (slower as well as faster than the Gaussian) were systematically studied with the help of classical Sobolev inequalities providing in particular an effective sufficient criteria (in terms of certain non-linear differential inequalities for the log of the density function), for related coercive inequalities (see also reviews [35,18] and references therein). In the mid 80’ties Bakry and Emery [4] introduced a very effective criterion based on convexity (curvature) which allowed to enlarge the class of examples where a Log-Sobolev inequality holds, including the situation with measures on certain finite dimensional Riemannian manifolds (as well as some infinite dimensional cases, however with a compact configuration space [14]). Following a similar line of reasoning, in [3] the authors provided an effective criteria for (generalisation) of Brascamp–Lieb inequality [13] as well as Log-Sobolev inequality (with possibly more general entropy functional and weighted Dirichlet form dependent on the measure). More recently, in [9], certain convexity ideas (including Brunn–Minkowski inequality), were exploited to recover in the special case of the space Rn similar results as in [3] and obtained additionally inequalities (LSq ) which are naturally related to metrics different from the Euclidean metric (and in particular involving a different length of the gradient in the preceding inequalities). These results concerned principally the probability measures with tails decaying faster than the Gaussian. We point out that while such distributions were also discussed in [34], in [9] they involved in a natural way Lipschitz functions with respect to a non-Euclidean metric (while in Rosen’s work the emphasis of improvement was on different functionals on the left-hand side). The corresponding results for measures on Rn with slower distribution tails were obtained in [6] (see also references therein), which included in particular those of Rosen [34] for a similar class of measures. Part of the motivation for the current paper was provided by [28] in which the coercive inequalities involving Hörmander fields instead of the (non-degenerate full gradient) were studied. Such a situation is naturally related to a more general Carnot–Caratheodory metric associated to the family of fields and the interest here is to obtain coercive inequalities involving length of the corresponding metric gradient. While in [28] a rich family of examples on compact spaces
816
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was provided, non-compact situation is more difficult. In this paper we develop an efficient technology which not only recovers interesting results in Rn briefly reviewed in the above, but also allows us to extend to interesting metric spaces, such as certain non-compact Lie groups, including in particular the Heisenberg group. Part of our approach is directed at proving inequalities, which we call U -bounds, of the following form
|f |q U dμ C
|∇f |q dμ + D
|f |q dμ
with a suitable increasing unbounded function U of the metric and the length of the metric gradient |∇f |; see Section 2. We show later in Sections 3 and 4 that such an inequality implies corresponding Poincaré as well as other coercive inequalities; in fact as we illustrate in some of the cases the U -bounds are equivalent to the coercive inequalities. (This requires an extension of a result on a Gaussian exponential bound of [1] for other measures and functions with possibly unbounded gradient.) In Section 5 we explore also a family of weighted Poincaré and Log-Sobolev inequalities on Riemannian manifolds including measures with ultra slow tails. In such a context we can effectively employ a Laplacian comparison theorem (see e.g. [16]), which in particular allows us to extend recent results of [10] where convexity ideas in Euclidean spaces were used. As an application of our technique we also prove (see Sections 6–7) the Log-Sobolev inequality for the heat kernel measure on the Heisenberg group (a topic which attracted recently some extra attention [26,27,19]). 2. U -bounds By ∇ we denote a subgradient in RN , that is a finite collection of possibly non-commuting fields. It is assumed that the divergence of each of these fields with respect to the Lebesgue measure Λ on RN is zero. (While this provides some simplification in our expositions, it is possible to extend our arguments to a more general setting.) We begin with proving the following result. −βd p
Theorem 2.1. Let dμp = e Z dλ be a probability measure defined with β ∈ (0, ∞) and p ∈ (1, ∞) (Z being the normalisation constant). Suppose 0 < σ1 |∇d| 1, for some σ ∈ [1, ∞), and d K + βpεd p−1 outside the unit ball B ≡ {d(x) < 1} for some K ∈ [0, ∞) and ε ∈ [0, σ12 ). Then there exist constants C, D ∈ (0, ∞) such that the following bound is true:
|f |d p−1 dμp C
|∇f | dμp + D
|f | dμp .
(1)
Remark. In particular the assumptions of the theorem are satisfied for d being the Carnot– Caratheodory distance and ∇ the (horizontal) gradient of the Heisenberg group. Proof of Theorem 2.1. For a smooth function f 0 such that f = 0 on the unit ball, by the Leibniz rule we have p p p (∇f )e−βd = ∇ f e−βd + βpf d p−1 ∇d e−βd .
(2)
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Put α(·) ≡
(∇d)(·) dλ.
Acting with this functional on the expression (2) we get p p α (∇f )e−βd = α ∇ f e−βd + βp
f d p−1 |∇d|2 e−βd dλ. p
(3)
Using Hölder inequality, the left-hand side of (3) can be estimated from above as follows p α (∇f )e−βd =
(∇d) · (∇f )e−βd dλ p
|∇d||∇f |e−βd dλ p
|∇f |e−βd dλ p
(4)
where we have used the fact that |∇d| 1. The first term on the right-hand side of (3) can be treated with the help of integration by parts as follows p p (∇d) · ∇ f e−βd dλ = − (d)f e−βd dλ p −βd p −K f e dλ − βpε f d p−1 e−βd dλ
p = α ∇ f e−βd
(5)
where we have used the assumption that d K + βpεd p−1 . Combining (3), (4) and (5), we get βp
p f d p−1 |∇d|2 − ε e−βd dλ
from which the inequality (1) follows with C = [0,
1 ). σ2
|∇f |e−βd dλ + K p
1 (1/σ 2 −ε)βp
and D =
f e−βd dλ p
K , (1/σ 2 −ε)βp
provided ε ∈
Now, the estimate (1) is proven for smooth non-negative f which vanish on the unit ball. We can handle non-smooth functions approximating them by smooth ones (on compact sets via convolution and splitting f into compactly supported pieces using a smooth partition of unity — details are tedious but do not pose any essential difficulty). We can handle f of arbitrary sign replacing f by |f | and using equality ∇|f | = sgn(f )∇f . To handle f which are non-zero on the unit ball we write f = f0 + f1 where f0 = φf , f1 = (1 − φ)f and φ(x) = min(1, max(2 − d(x), 0)). Then
|f |d p−1 dμp =
|f |d p−1 dμp +
d(x)2
2
d(x)>2
|f | dμp +
p−1 d(x)2
2p−1
|f |d p−1 dμp
|f | dμp +
|f |1 d p−1 dμp
d(x)>2
|f |1 d p−1 dμp .
818
W. Hebisch, B. Zegarli´nski / Journal of Functional Analysis 258 (2010) 814–851
Next
|∇f1 | |∇f | + |f |, |f1 |d p−1 dμp C |∇f1 | dμp + D |f1 | dμp C |∇f | dμp + (D + C) |f | dμp .
Combining inequalities above we see that (1) is valid without restriction on the support of f if we replace D by D + 2p−1 + C. 2 Using our result and a perturbation technique we obtain the following generalisation. Theorem 2.2. Let dμ = tential W satisfying
e−W −V Z
dμθ be a probability measure defined with a differentiable po|∇W | δd p−1 + γδ
(6)
with some constants δ < 1/C and γδ ∈ (0, ∞), and suppose that V is a measurable function such that osc(V ) ≡ max V − min V < ∞. Then there exist constants C , D ∈ (0, ∞) such that the following bound is true: p−1 dμ C |∇f | dμ + D |f | dμ. (7) |f |d Remark. In particular the assumption (6) of the theorem is satisfied if W is a polynomial of lower order in d. Another example, in the spirit of [20] and [11], with deep wells is as follows W = ϑd p−1 cos(d) with a small constant ϑ > 0 (but ϑd p−1 cos(d 1+ε ) would not work for any ε > 0 no matter how small ϑ > 0 would be). Proof of Theorem 2.2. We consider first the case V = 0 and start from substituting f e−W in the inequality (1) for the measure μp . Using Leibniz rule |f |d
p−1 −W
e
dμp C
|∇f |e
−W
dμp + D
|f |e
−W
dμp + C
|f ||∇W |e−W dμp .
Now our assumption (6) about W implies |f ||∇W |e−W dμp δ |f |d p−1 e−W dμp + γδ |f |e−W dμp . Thus combining these bounds we arrive at |f |d p−1 e−W dμp C¯ |∇f |e−W dμp + D¯ |f |e−W dμp
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819
with C¯ ≡ C/(1 − Cδ)
and D¯ = (D + γδ )/(1 − Cδ).
Next we note that if V = 0 we have
e−W −V dμp Z e−W osc(V ) e dμp |f |d p−1 −W e dμp e−W e−W osc(V ) ¯ osc(V ) ¯ e dμp + e dμp C |∇f | −W D |f | −W e e dμp dμp e−W −V e−W −V 2osc(V ) ¯ 2osc(V ) ¯ e dμ + e dμp . 2 C |∇f | D |f | p Z Z |f |d p−1
Theorem 2.3. Let μ be a probability measure for which conclusion of Theorem 2.1 holds. Let p ∈ (1, ∞). Then for each q ∈ [1, ∞) there exist constants Cq , Dq ∈ (0, ∞) such that the following bound is true: |f |q d q(p−1) dμ Cq |∇f |q dμ + Dq |f |q dμ. (8) Proof. Let d1 (x) = max(1, d(x)). Enlarging constants D if necessary we may assume that p−1 |f |d1 dμ C |∇f | dμ + D |f | dμ. (p−1)(q−1)
Put h = |f |q d1
. We have
q(p−1) p−1 |f |q d1 dμ = hd1 dμ C |∇h| dμ + D h dμ.
|f |q d q(p−1) dμ
By Leibniz formula (q−1)(p−1)
|∇h| = q|∇f ||f |(q−1) d1
(q−1)(p−1)−1
+ (q − 1)(p − 1)|∇d1 ||f |q d1
and
(q−1)(p−1)
q|∇f ||f |(q−1) d1
1/q
q
|∇f | dμ q
αq
|∇f |q dμ +
dμ q−1 (q−1)(p−1) q/(q−1) dμ |f | d1
q −1 α q/(q−1)
q(p−1)
|f |q d1
dμ.
(q−1)/q
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Next
h dμ =
1/q
(q−1)(p−1) dμ |f |q d1
βq q
|f |q dμ +
q −1
|f | dμ q
q(p−1)
|f |q d1
β q/(q−1) q
q(p−1) dμ |f |q d1
(q−1)/q
dμ.
If (q − 1)(p − 1) 1, then (q
(q−1)(p−1)−1 − 1)(p − 1)|∇d1 ||f |q d1 dμ
|f |q dμ.
If (q − 1)(p − 1) > 1, then
(q−1)(p−1)−1
(q − 1)(p − 1)|∇d1 ||f |q d1 dμ (q−1)(p−1)−1 dμ (q − 1)(p − 1) |f |q d1 p/(q(p−1)) ((p−1)(q−1)−1)/(q(p−1)) q q q(p−1) dμ (q − 1)(p − 1) |f | d1 |f | dμ
(q − 1)p q(p−1)/p γ q
|f |q dμ +
(q − 1)2 (p − 1) − (q − 1) qγ q(p−1)/((q−1)(p−1)−1)
q(p−1)
|f |q d1
Combining inequalities above, if (q − 1)(p − 1) 1 we get 1−C
q −1
q −1
q(p−1)
− D q/(q−1) dμ |f |q d1 α q/(q−1) β q βq Cα q |∇f |q dμ + C + D |f |q dμ q
which gives the claim with Cq =
Cα q q−1 q−1 1 − C α q/(q−1) − D β q/(q−1) q
and q
Dq =
C + D βq
q−1 q−1 1 − C α q/(q−1) − D β q/(q−1) q
if α and β are big enough. Similarly, for (q − 1)(p − 1) > 1 we get the claim with Cq =
Cα q 2
q−1 (p−1)−(q−1) q−1 1 − C α q/(q−1) − C qγ(q−1) q(p−1)/((q−1)(p−1)−1) − D β q/(q−1) q
dμ.
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821
and q
Dq =
Cpγ q(p−1)/p + D βq 2
q−1 (p−1)−(q−1) q−1 1 − C α q/(q−1) − C qγ(q−1) q(p−1)/((q−1)(p−1)−1) − D β q/(q−1) q
if α, β and γ are big enough.
2
−βd p
Theorem 2.4. Let dμp = e Z dλ be a probability measure defined with β ∈ (0, ∞) and p ∈ [2, ∞) (Z being the normalisation constant). Suppose 0 < σ1 |∇d| 1, for some σ ∈ [1, ∞), and d K + βpεd p−1 outside the unit ball B ≡ {d(x) < 1} for some K ∈ [0, ∞) and ε ∈ [0, σ12 ). Suppose q1 + p1 = 1, then we have
|f |q d p dμ Cq
|∇f |q dμ + Dq
|f |q dμ.
(9)
Remark. In particular the assumptions of the theorem are satisfied for d being the Carnot– Caratheodory distance and ∇ the (horizontal) gradient of the Heisenberg group. Proof of Theorem 2.4. This is a special case of Theorem 2.3.
2
Extension to more general measures is as follows. Theorem 2.5. Let dμ = tential W satisfying
e−W −V Z
dμp be a probability measure defined with a differentiable po-
|∇W |q δd p + γδ
(10)
with some constants δ2q−1 q −q C < 1 and γδ ∈ (0, ∞), and suppose that V is a measurable function such that osc(V ) ≡ max V − min V < ∞. Then there exist constants C , D ∈ (0, ∞) such that the following bound is true:
with q such that
1 q
+
1 p
|f |q d p dμ C
|∇f |q dμ + D
|f |q dμ
(11)
= 1.
The proof is similar to that of Theorem 2.2. 2.1. U -bounds: Sub-quadratic case −βd θ
Theorem 2.6. Let dμθ = e Z dλ be a probability measure defined with β ∈ (0, ∞) and θ ∈ [1, 2] (Z being a normalisation constant). Suppose 0 < σ1 |∇d| 1, for some σ ∈ [1, ∞), and d K + βpεd p−1 outside the unit ball B ≡ {d(x) < 1} for some K ∈ [0, ∞) and ε ∈ [0, σ12 ).
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Then there exist constants Cθ , Dθ ∈ (0, ∞) such that the following bound is true
|f |2 d 2(θ−1) dμθ Cθ
|∇f |2 dμθ + Dθ
|f |2 dμθ .
(12)
Remark. In particular the assumptions of the theorem are satisfied for d being the Carnot– Caratheodory distance and ∇ the (horizontal) gradient of the Heisenberg group. Proof of Theorem 2.6. Again, this is a special case of Theorem 2.3.
2
Extension to more general measures is as follows. Theorem 2.7. Let dμ = tential W satisfying
e−W −V Z
dμθ be a probability measure defined with a differentiable po-
|∇W |2 δd 2(θ−1) + γδ
(13)
with some constants δC/2 < 1 and γδ ∈ (0, ∞), and suppose that V is a measurable function such that osc(V ) ≡ max V − min V < ∞. Then there exist constants C , D ∈ (0, ∞) such that the following bound is true.
|f |2 d θ dμ C
|∇f |2 dμ + D
|f |2 dμ.
(14)
Again, the proof is similar to that of Theorem 2.2. 3. Poincaré inequality Theorem 3.1. Suppose 1 q < ∞ and a measure λ satisfies the q-Poincaré inequality for every ball BR , that is there exists a constant cR ∈ (0, ∞) such that 1 |BR |
q f − 1 dλ cR 1 f |∇f |q dλ. |BR | |BR |
BR
BR
(15)
BR
Let μ be a probability measure on Rn which is absolutely continuous with respect to the measure λ and such that
f q η dμ C
|∇f |q dμ + D
f q dμ
(16)
with some non-negative function η and some constants C, D ∈ (0, ∞) independent of a function f . If for any L ∈ (0, ∞) there is a constant AL such that 1 dμ AL AL dλ
(17)
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823
on the set {η < L} and, for some R ∈ (0, ∞) (depending on L), we have {η < L} ⊂ BR , then μ satisfies the q-Poincaré inequality μ|f − μf |q cμ|∇f |q .
(18)
μ|f − μf |q 2q μ|f − a|q .
(19)
Proof. For any a we have
Next (20) μ|f − a|q μ|f − a|q χ(η < L) + μ|f − a|q χ(η L). Using our assumptions and putting a = |B1R | BR f , for the first term on the right-hand side of (20) we have q 1 q μ|f − a| χ(η < L) AL f − f dλ |BR | BR
BR
AL cR
|∇f |q dλ A2L cR μ|∇f |q .
(21)
BR
On the other hand for the second term on the right-hand side of (20) we get 1 μ|f − a|q η. L
(22)
C D μ|∇f |q + μ|f − a|q . L L
(23)
μ|f − a|q χ(η L) Hence, by (16), we obtain μ|f − a|q χ(η L) Combining (21) and (23), we get μ|f − a|
A2L cR
q
C D μ|∇f |2 + μ|f − a|q . + L L
Choosing L > D, simple rearrangement yields μ|f − a|q
A2L cR + 1−
D L
C L
μ|∇f |q .
This together with (19)–(21) yields μ|f − μf |q cμ|∇f |q with some constant c ∈ (0, ∞).
2
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Corollary 3.1. If we are on nilpotent Lie group the probability measure μq and μθ of Theorems 2.5 and 2.7, respectively, satisfies the Poincaré inequality. 4. From Sobolev inequalities to coercive inequalities with probability measure: The non-compact setting 4.1. Case p 2 e−U Z
Theorem 4.1. Let dμ =
dλ. Suppose the following Sobolev inequality is satisfied
|f |
q+ε
q q+ε
dλ
a
|∇f | dλ + b q
|f |q dλ
(24)
and the following bound is true ¯ ¯ μ |f |q |∇U |q + U Cμ|∇f |q + Dμ|f |q .
(25)
Then the following inequality is true fq q μ f log Cμ|∇f |q + Dμ|f |q . μf q
(26)
Moreover, if q ∈ (1, 2] and the following q-Poincaré inequality holds 1 μ|∇f |q , M
(27)
fq cμ|∇f |q μ f q log μf q
(28)
μ|f − μf |q then one has
with some constant c ∈ (0, ∞) independent of f . Proof. First we note that for f ≡ 0, we have
fq μ f log μf q
q
with g ≡ f · gets
−1U
e q Z 1/q
satisfying
q
g q log g q dλ + μ f q [U + log Z]
g q dλ = 1. Next, by arguments based on Jensen inequality, one
q g log g dλ = ε q
=μ fq
1 q+ε q(q + ε) q+ε log dλ g log g dλ g ε q
ε
whence, by the Sobolev inequality (24), one obtains
W. Hebisch, B. Zegarli´nski / Journal of Functional Analysis 258 (2010) 814–851
q +ε log g log g dλ ε q
with a ≡
q+ε ε a
q
and b ≡
q+ε ε b.
g
q+ε
q q+ε
dλ
a
|∇g|q dλ + b
825
g q dλ
Combining all the above we arrive at
q fq a μ f q log μ f μf q
− 1 U q q dλ + (b + log Z) f q dμ + μ f q U ∇ f e Z 1/q
and, by simple arguments, we obtain
fq μ f log μf q
q
2q−1 a
|∇f |q dμ + μ f q 2q−1 q −q a |∇U |q + U + b + log Z .
(29)
Now using our assumption (25) yields fq 2q−1 a + 2q−1 q −q a C¯ μ|∇f |q + (b + D¯ + log Z)μ|f |q . μ f q log q μf
(30)
Since for q ∈ (1, 2] one has [11]
fq μ f log μf q q
|f − μf |q q μ |f − μf | log + 2q+1 μ|f − μf |q μ|f − μf |q
(31)
using (30) we arrive at
fq μ f log μf q q
q−1 2q+1 (b + D¯ + log Z) q−1 −q ¯ 2 a +2 q a C + μ|∇f |q M
which ends the proof of the theorem.
2
Using Theorem 4.1 together with results of Section 3 (q-Poincaré inequality), we arrive at the following result. Corollary 4.1. The probability measures dμ = e−W −V dμp /Z , with p 2, described in Theorem 2.5 satisfies the following coercive inequality
|f |q μ |f | log μ|f |q q
with
1 q
+
1 p
cμ|∇f |q
= 1 and a constant c ∈ (0, ∞) independent of a function f .
(LSq )
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4.2. Sub-quadratic case Theorem 4.2. Suppose θ ∈ [1, 2] and let ς = 2(θ−1) . Then there exist constants C, D ∈ (0, ∞) θ such that f 2 ς 2 dμ C |∇f | dμ + D f 2 dμθ . (32) f 2 log 2 θ θ f dμθ −
β θ d
2 Proof. We note first that if θ ∈ [1, 2], then ς ∈ [0, 1]. Put g ≡ f eZ 1/2 . We have the following inequality
f 2 ς dμθ f 2 log 2 f dμθ ς g 2 θ + βd − log Z dλ = g log 2 g dλ θ ς g 2 ς 2 2 ς dλ + |log Z| g 2 dλ g log 2 dλ + g βd g dλ g 2 ς 2 ς 2 θς ς f d dμθ + |log Z| f 2 dμθ . = g log 2 dλ + β g dλ
(33)
Assume first that μθ f 2 = g 2 dλ = 1. Then we have
ς g2 ς g 2 log 2 dλ g 2 log+ g 2 dλ + Dς g dλ 2 ς 2+ε 2+ε ς 2+ε log+ dλ + Dς g ε
with Dς ≡ supx∈(0,1) x(log x1 )ς . Choosing suitable ε ∈ (0, 1), we can apply Sobolev inequality ¯ D¯ ∈ (0, ∞)) to get (with constants C,
ς g 2 ς 2+ε ς log+ C¯ |∇g|2 dλ + D¯ g 2 dλ + Dς g 2 log 2 ε g dλ C1 |∇g|2 dλ + D1
with C1 ≡ s
2+ε ε
ς
C¯
and
2+ε ς ¯ D1 ≡ s D + γς,s + Dς ε
W. Hebisch, B. Zegarli´nski / Journal of Functional Analysis 258 (2010) 814–851
827
where s ∈ (0, ∞) and γς,s ∈ (0, ∞) is a suitable constant. Using the definition of g, we have
|∇g| dλ 2 2
1 |∇f | dμθ + β 2 θ 2 2
2
f 2 d 2(θ−1) dμθ .
Now applying the U -bound of Theorem 2.6, we get
1 |∇g| dλ 2 + β 2 θ 2 Cθ 2
2
1 |∇f | dμθ + β 2 θ 2 Dθ 2
2
f 2 dμθ .
Thus we get (for the normalised function g)
g 2 ς g 2 log 2 C2 |∇f |2 dμθ + D2 g dλ
(34)
with some constants C2 , D2 ∈ (0, ∞). Now coming back to (33), we note that since θ ς = 2(θ − 1), we can use again the U -bound of Theorem 2.6 to bound the second term from the right-hand side of this relation. Combining this with (34), we arrive at the following bound
f 2 ς dμ C |∇f |2 dμθ + D f 2 log 2 θ f dμθ
(35)
with the constants C = C2 + β ς Cθ and D = D2 + β ς Dθ + |log Z|ς . At this stage we can remove the normalisation condition to arrive at the desired bound (32). 2 Using Theorem 4.2, we prove the following tight inequality. Theorem 4.3. For θ ∈ [1, 2] and ς =
2(θ−1) , θ
let
ς Φ(x) ≡ x log(1 + x) . Under the assumption of Theorem 4.2, if additionally μθ satisfies Poincaré inequality, there exists a constant cθ ∈ (0, ∞) such that (36) μθ Φ f 2 − Φ μθ f 2 cθ |∇f |2 dμθ . Proof. First we note that 1 + f 2 ς μθ Φ f 2 − Φ μθ f 2 μθ f 2 log 1 + μθ f 2
(37)
and 2 2 1 + f 2 ς 1 + f 2 ς 2 log μθ f 2 log = μ χ f μ f f θ θ 1 + μθ f 2 1 + μθ f 2 2 2 1 + f 2 ς 2 + μθ χ f μθ f f log . 1 + μθ f 2
(38)
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On the set {f μθ f 2 } we have
1+f 2 1+μθ f 2
f2 μθ f 2
and so
f 2 ς f 2 ς 2 log μθ χ f 2 μθ f 2 f 2 log μ f . θ μθ f 2 μθ f 2 On the other set {f μθ f 2 }, we have
1+μθ f 2 1+f 2
1+
μθ f 2 , f2
and therefore
2 1 + f 2 ς μθ χ f μθ f f log 2μθ f 2 . 1 + μθ f 2
2
2
Using these relations together with (38) we have 2 f 2 ς 2 2 + 2μθ f 2 μθ Φ f − Φ μθ f μθ f log μθ f 2
(39)
and thus, by Theorem 4.2, we obtain μθ Φ f 2 − Φ μθ f 2 Cμθ |∇f |2 + (D + 2)μθ f 2 .
(40)
Now according to Lemma A.1 of [28], one has the following analog of Rothaus lemma for a probability measure with Orlicz function Φ given in the theorem: ∃a, b ∈ (0, ∞) νΦ f 2 − Φ νf 2 a νΦ (f − νf )2 − Φ ν(f − νf )2 + bν(f − νf )2 .
(41)
Combining (40) and (41) with the Poincaré inequality for the measure μθ μθ (f − μθ f )2
1 μθ |∇f |2 M
we arrive at the following result
2 D+b 2 μθ |∇f |2 . μθ Φ f − Φ μθ f aC + M
2
Summarising, in the current section in essence our methods were based on the fact that the primary part of the interaction where a nice function of certain unbounded function d which length of the gradient |∇d| (with respect to a given set of fields) was bounded from above and stayed strictly away from zero. We also used number of times the Leibniz rule for the fields. 4.3. From coercive inequalities to U -bounds For a probability measure dμ ≡ e−U dλ/Z, we have shown that if for q ∈ (1, 2] the following bound is satisfied f q |∇U |q + U dμ C |∇f |q dμ + D |f |q dμ
W. Hebisch, B. Zegarli´nski / Journal of Functional Analysis 258 (2010) 814–851
829
together with q-Poincaré inequality Mμ|f − μf |q μ|∇f |q , then the following LSq inequality holds μ|f |q log
|f |q cμ|∇f |q . μ|f |q
We show that the following result in the converse direction is true as well. Theorem 4.4. Suppose q ∈ (1, 2] and for some constants a, b ∈ (0, ∞), we have |∇U |q aU + b and assume that the measure dμ ≡ e−U dλ/Z satisfies LSq . Then the following U -bound is true
|f | U dμ C q
|∇f | dμ + D q
|f |q dμ
with some constants C, D ∈ (0, ∞) independent of f . Proof. We note that by relative entropy inequality one has q 1 |f |q 1 q εU μ |f | U μ|f | log log μe μ μ|f |q . + ε μ|f |q ε Hence, if LSq is true, we get c μ |f |q U ε
1 εU μ|f |q . log μe |∇f | dμ + ε
q
Thus we will be finished if we show μeεU < ∞. This follows from the following result. Exp-bounds from LSq Theorem 4.5. Assume that a measure μ satisfies LSq with some q ∈ (1, 2]. Suppose that for some constants a, b ∈ (0, ∞), we have |∇f |q af + b. Then the following exp-bound is true μetf < ∞ for all t > 0 sufficiently small.
830
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Remark. For the case q = 2 see [1]. Proof of Theorem 4.5. By our assumption, we have μg q log
gq cμ|∇g|q . μg q
It is enough to prove the bound under additional assumption that f is bounded. Namely, given L ∈ (0, ∞), replace f by F ≡ χ(|f | L)f + Lχ(|f | > L). F satisfies our assumptions with the same constants. So we will get the claim letting L go to ∞. Since now f is bounded, exp tf is integrable and we have etf tf μ e log tf cq −q t q μ etf |∇f |q . μe By our assumption |∇f |q af + b, so we get etf μ etf log tf caq −q t q μ etf f + cbq −q t q μ etf μe which can be rearranged to get 1 − caq −q t q−1 μ
etf etf log μetf μetf
caq −q t q−1 log μ etf + cbq −q t q .
Taking into the account that
etf etf μ log μetf μetf
= t2
d 1 log μetf dt t
and setting G(t) ≡ 1t log μetf , after simple transformations we obtain the following differential inequality d G(t) βt q−2 G(t) + γ t q−2 dt with β(t) ≡
caq −q (1−caq −q t q−1 )
cbq −q which are well defined for caq −q t q−1 (1−caq −q t q−1 ) ∈ (1, 2], for caq −q t q−1 < ε < 1, after integration we get
and γ (t) ≡
Since G(t) → μf as t → 0 and q G(t) μf +
caq −q cbq −q t q−1 + (q − 1)(1 − ε) (1 − ε)
< 1.
t dτ τ q−2 G(τ ). 0
In our range of q ∈ (1, 2], this can be solved by iteration. Since G(t) is non-decreasing, in this interval one also has G(t) μf +
caq −q cbq −q t q−1 + t q−1 G(t) (q − 1)(1 − ε) (q − 1)(1 − ε)
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which for
caq −q q−1 (q−1)(1−ε) t
831
≡ δ < 1 yields the following bound μe(1−δ)tf exp tμf + Ct q
−q
cbq . One can check that our bound is independent of the cut off L in the given with C ≡ (q−1)(1−ε) interval of t. 2
By the above we have shown the equivalence of the LSq and U -bounds in particular in the cases of natural interactions dependent on the metric. Similar considerations can be provided in the subquadratic case for which the exponential bounds are known (see e.g. [25,6]). 5. Weighted U -bounds and coercive inequalities Let p 2 and suppose f is a smooth function supported away from the origin. Starting with the identity α
d − 2 (∇f )e−
βd p 2
βd p βd p α pβ p− α −1 d 2 (∇d)f e− 2 , = d − 2 ∇ f e− 2 + 2
squaring and integrating with the measure dλ, one obtains
d −α |∇f |2 e−βd dλ pβ p
+
βd p βd p d p−α−1 ∇ f e− 2 · (∇d)f e− 2 dλ
p2 β 2 4
d 2p−α−2 |∇d|2 f 2 e−βd dλ. p
Hence, after integration by parts in the first term on the right-hand side and simple rearrangements, one arrives at the following bound d
−α
2 −βd p
|∇f | e
p2 β 2 p dλ f 2 d 2p−α−2 |∇d|2 e−βd dλ 4
p(p − α − 1)β p−α−2 pβ p−α−1 p − f2 d d |∇d|2 + d e−βd dλ. 2 2
If we choose α = p − 2 and assume |∇d|
d −α |∇f |2 e−βd dλ p
1 σ
> 0, we obtain
p2 β 2 p f 2 d p e−βd dλ 2 4σ
p(p + 1)β pβ p − f2 |∇d|2 + dd e−βd dλ. 2 2 2 2
Finally assuming that there exist constants K ∈ (0, ∞) and δ ∈ (0, p4σβ2 ), such that p(p + 1)β pβ |∇d|2 + dd K + δd p 2 2
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we arrive at 2 2 p β p 2 p −βd p −α 2 −βd p − δ d e dλ d |∇f | e dλ + K f 2 e−βd dλ. f 4σ 2 α
By adjusting the constant on the right-hand side and replacing d −α by d −α ≡ (1 + d 2 )− 2 , we conclude with the following result. Theorem 5.1. Let dμ ≡ e−βd dλ/Z with p > 2. Suppose there are constants σ ∈ [1, ∞) and 2 2 K ∈ (0, ∞) and δ ∈ (0, p4σβ2 ) such that |∇d| σ1 and p
pβ pβ |∇d|2 + dd K + δd p . 2 2 Then there are constants C, D ∈ (0, ∞) such that μf 2 d p Cμ d 2−p |∇f |2 + Dμf 2 . Using this bound, by similar arguments as in the proof of Poincaré inequality (see Theorem 3.1), we now obtain Theorem 5.2. Under the assumptions of Theorem 4.4 there is a constant M ∈ (0, ∞) such that Mμ(f − μf )2 μ d 2−p |∇f |2 . Finally following our strategy from the beginning of Section 4 (see proof of Theorem 4.2), with appropriate amendments, we arrive at the following coercive inequality. Theorem 5.3. Under the assumptions of Theorem 4.4 there is a constant c ∈ (0, ∞) such that 2−p f2 2 cμ d μ f 2 log . |∇f | μf 2 5.1. Weighted U -bounds and coercive inequalities: Distributions with slow tails on Riemannian manifolds In this section we consider a non-compact smooth Riemannian manifold M of dimension 3 N < ∞. In this setup d(x) denotes the Riemannian distance of a point x from a given point x0 ∈ M called later on the origin. By ∇ and we denote the gradient and Laplace–Beltrami operators, respectively. The aim of this section is to discuss coercive inequalities involving probability measures dμ ≡ ρ dx with density (with respect to the corresponding Riemannian measure dλ on M) which is of the form ρ ≡ e−U (d) /Z with leading part of the function U given by a concave function (and therefore also defining a non-Riemannian distance on M). In particular we will consider the following cases: (i) U (d) = βd α , with α ∈ (0, ∞) and β > 0, (ii) U (d) = β log(1 + d) with β > 0.
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Before we go on we recall the following Laplacian comparison theorem (cf. [16,24] ([32,39, 40])). For a complete Riemannian manifold M with Ric (N − 1)K where K ∈ R: √ (∗) If K 0, then d (N − 1)d −1 + (N − 1) |K|. −1 (∗∗) If Ric 0, then d (N − 1)d . By similar computation as we have done in Section 2, for a smooth non-negative function f localised outside a ball Bε ≡ Bε (x0 ) centred at the origin we consider a field (∇f )e−U = ∇ f e−U + f (U ∇d)e−U
(42)
to which we will apply a functional α(v) ≡
W (∇d · v) dλ
(43)
defined with a positive weight function W ≡ W (d) to be specified later. Using the fact that |∇d| = 1 (for d = 0), together with arguments involving Hölder inequality and integration by parts one arrives at the following bound f Ve
−U
dλ
W |∇f |e−U dλ
(44)
with V ≡ χM\Bε W U − div(W ∇d) . Later on we will extend V to Bε in a convenient way by adding an arbitrary bounded continuous function. One can handle a function of arbitrary sign replacing f by |f | and using equality ∇|f | = sgn(f )∇f . To include f which are non-zero on a ball centred at the origin we write f = f0 + f1 where f0 = φf , f1 = (1 − φ)f and φ(x) = min(ε, max(2ε − d(x), 0)). Then
|f |V dμ =
|f |V dμ +
d(x)2ε
sup (V) {d2ε}
|f |V dμ
d(x)>2ε
φ|f | dμ +
|f |1 V dμ.
(45)
Next we have 1 |∇f1 | |∇f | + χ{εd<2ε} |f |, ε and therefore
|f1 |V dμ
W (1 − φ)|∇f | dμ +
(46)
sup ε −1 W
{εd<2ε}
εd<2ε
|f | dμ.
(47)
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Combining (42)–(47) we arrive at the following bound |f |V dμ W (1 − φ)|∇f | dμ + sup (V) φ|f | dμ +
sup ε −1 W
{εd<2ε}
{d2ε}
|f | dμ.
(48)
εd<2ε
Hence with B ≡ sup (V) + {d2ε}
sup
−1 ε W ,
{εd<2ε}
we have
|f |V dμ
W |∇f | dμ + B
|f | dμ.
(49)
Case (i) For U (d) = βd α , with α ∈ (0, ∞) and β > 0, choosing W (d) = α −1 d κ , with κ 1, we have V ≡ W U − div(W ∇d) = U − α −1 κd κ−1 − α −1 d κ d.
(50)
Thus if (∗) holds, we have V βd α−1+κ − χM\Bε α −1 κN d κ−1 + α −1 (N − 1) |K|d κ .
(51)
Hence we conclude with the following result Theorem 5.4. Let dμ ≡ e−U dλ/Z with U ≡ βd α where α ∈ (0, ∞). Suppose Ric (N − 1)K with K 0. • If α > 1, then for any κ 1, there exist constants c1 , b1 ∈ (0, ∞) such that |f |U dμ c1 d κ |∇f | dμ + b1 |f | dμ.
(52)
√ • If α = 1 and β > α −1 (N − 1) |K|, then for any κ 1, there exist constants c1 , b1 ∈ (0, ∞) such that (52) is true. • If α ∈ (0, 1) and Ric 0, then for any κ 1, there exist constants c1 , b1 ∈ (0, ∞) such that (52) is true. Moreover if (52) holds, then for any q ∈ (1, ∞), we have q(κ− pα ) |∇f |q dμ + b2 |f |q dμ |f |q U dμ c2 d q
with c2 ≡ c1 λq q−1 β p [1 − c1 /(pλ)]−1 and b2 ≡ b1 [1 − c1 /(pλ)]−1 .
(53)
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The second part follows from the first by substituting f q in place of f and using elementary arguments involving Young inequality. As a consequence, by similar arguments as earlier in this section, we obtain the following result on possible coercive inequalities. Theorem 5.5. Let dμ ≡ e−U dλ/Z with U ≡ βd α where α ∈ (0, ∞). Suppose Ric (N − 1)K with K 0. • If α > 1, then for any κ 1, there exists a constant c ∈ (0, ∞) such that |f |q μ|f | log c μ|f |q
q
d
q(κ− pα )
|∇f |q dμ.
(54)
√ • If α = 1 and β > α −1 (N − 1) |K|, then for any κ 1, there exists a constant c ∈ (0, ∞) such that (54) is true. • If α ∈ (0, 1) and Ric 0, then for any κ 1, there exists a constant c ∈ (0, ∞) such that (54) is true. As a consequence the following inequality holds Mμ|f − μf |q
d
q(κ− pα )
|∇f |q dμ
(55)
with some M ∈ (0, ∞). Case (ii) For U (d) = β log(1 + d) with β > 0, choosing W (d) = d log(1 + d) and setting V ≡ U + χM\Bε Wβ(1 + d)−1 − div(W ∇d) = U − χM\Bε 1 + log(1 + d) − χM\Bε d log(1 + d)d.
(56)
Thus if (∗) holds, we have V U − χM\Bε 1 + log(1 + d) − χM\Bε d log(1 + d) (N − 1)d −1 + (N − 1) |K| .
(57)
Hence we conclude with the following result Theorem 5.6. Let dμ ≡ (1 + d)−β dλ/Z with α ∈ (0, 1). Suppose Ric 0. If β > N , then
|f |U dμ c1
d log(1 + d)|∇f | dμ + b1
with c1 ≡ β · [β − N ]−1
|f | dμ
(58)
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and b1 ≡ β · [β − N ]−1 · N + sup (V) + {d2ε}
sup
−1 ε W .
{εd<2ε}
Hence, there exist cq , bq ∈ (0, ∞) such that
|f |q U dμ cq
d q log(1 + d)|∇f |q dμ + bq
|f |q dμ.
(59)
The second part follows from the first by substituting f q in place of f and using the following Young inequality q d ∇f q = q |f |q−1 · d|∇f | λq d q |∇f |q + λ−p |f |q p which implies
d log(1 + d)∇f q dμ =
d log(1 + d)q|f |q−1 |∇f | dμ
λq
d q log(1 + d)|∇f |q dμ +
q −p λ p
log(1 + d)q|f |q dμ.
From this and (58), choosing c1 pq λ−p < 1, one obtains
|f |q U dμ cq
d q log(1 + d)|∇f |q dμ + bq
|f |q dμ
with cq ≡ c1 λq (1 − c1 pq λ−p )−1 and bq ≡ b1 (1 − c1 pq λ−p )−1 . As a consequence of the above theorem, using arguments similar to those of Sections 4.1 and 4.2, we derive the following result on possible coercive inequalities. Theorem 5.7. Let dμ ≡ e−β log(1+d) dx/Z with β > N . Suppose Ric 0. Then for any q 1, there are constants Mq , cq ∈ (0, ∞), such that Mq μ|f − μf |q μ(1 + d)q log(e + d)|∇f |q
(60)
and μ|f |q log
|f |q cq μ(1 + d)q log(e + d)|∇f |q . μ|f |q
(61)
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5.1.1. Weighted inequalities at large β Let U ≡ β log(1 + d), with β > N ≡ dim(M). While the above results are true for any β > N , we will show that for sufficiently big β and Ric 0 due to the special nature of the interaction it is possible to improve the weight in the Poincaré and related Log-Sobolev inequalities. We start from noting that for a non-negative differentiable function supported outside a ball of radius r centred at the origin, one has
(1 + d)|∇f |e−U dx
=
(1 + d)∇d · ∇f e−U dx (1 + d) ∇d · ∇ f e−U + f ∇d · ∇U e−U dx
and so, taking into the account that |∇d|2 = 1, one gets
f β − 1 − (1 + d)d e−U dx
(1 + d)|∇f |e−U dx.
When Ric 0, we have d (N − 1)d −1 which implies the following bound Mβ
f e−U dx
(1 + d)|∇f |e−U dx
(62)
where Mβ ≡ [β − N − (N r−1) ]. Since |∇f | |∇|f ||, this inequality remains true for not necessarily positive function with f replaced by |f | on the right-hand side. Let now consider the following cutoff function ⎧ ⎨1 χ(t) ≡ 1 − ⎩ 0
(t−r) L
for 0 t 2r, for 2r t R, for t R,
with some R > 2r to be chosen later. Setting f˜1 ≡ (f − μf )χ and f˜2 ≡ (f − μf )χ , we have μ|f − μf | μ|f˜1 | + μ|f˜2 |. As f˜1 is compactly supported Lipschitz function, there is an m ≡ mR ∈ (0, ∞) independent of the function f , such that −1 ˜ μ|f˜1 | m−1 R μ|∇ f1 | mR μ |∇f |χ +
1 μ |f − μf |χ(2r < d < R) . mR (R − 2r)
The second term on the right-hand side can be treated with the help of (62) as follows. Setting χˆ to be a Lipschitz extension of χ(2r < d < R) supported outside the ball of radius r, we have μ |f − μf |χ(2r < d < R) μ |f − μf |χˆ Mβ−1 μ(1 + d)|∇f | + Mβ−1 sup |∇ χˆ |μ|f − μf |.
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Thus we obtain ˜ μ|f˜1 | m−1 R μ|∇ f1 |
1 −1 M + m−1 μ(1 + d)|∇f |χˆ R mR (R − 2r) β +
1 M −1 sup |∇ χˆ |μ|f − μf |. mR (R − 2r) β
(63)
On the other hand applying (62) to f˜2 we obtain μ|f˜2 | Mβ−1 μ(1 + d)|∇f |(1 − χ) + Mβ−1
1+R μ |f − μf |χ(r < d < R) . R−r
(64)
Combining (63) and (64) we arrive at μ|f − μf | c0 μ(1 + d)|∇f | + b0 μ|f − μf |
(65)
with + c0 ≡ m−1 R
1 + 1 Mβ−1 mR (R − 2r)
and b0 ≡ Mβ−1
1 1+R sup |∇ χˆ | + . mR (R − 2r) R−r
Since given R > 2r, one can choose β > N sufficiently large so that b0 < 1, we conclude with the following result Theorem 5.8. Suppose U = β log(1 + d), with β > N , and Ric 0. Then there exists β0 > N , such that for any β > β0 , one has Mμ|f − μf | μ(1 + d)|∇f |
(66)
with some constant M ∈ (0, ∞) independent of f . Consequently, we have Mq μ|f − μf |q μ(1 + d)q |∇f |q
(67)
with some constant Mq ∈ (0, ∞). The second part of the theorem follows by similar arguments as the ones used in the proof of Proposition 2.3 in [11]. Next we study the relative entropy estimate as follows. For a non-negative function f , setting f1 ≡ f χ and f2 ≡ f (1 − χ) with the same Lipschitz cutoff function χ , we have μf log
f f1 f2 μf1 log + μf2 log . μf μf1 μf2
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Since the function f1 is compactly supported and the density of the measure μ restricted to the ball BR (x0 ) bounded and bounded away from zero (via the arguments involving Sobolev inequality), we get μf1 log
f1 c1 μ|∇f1 | c1 μ |∇f |χ + b1 sup |∇χ|μf μf1
(68)
with some constants c1 , b1 ∈ (0, ∞) independent of f . Next we apply similar arguments based U on Sobolev inequality with the function F ≡ ff2eeU dx and the Riemannian measure dx to get 2
F log
F dx a F dx
|∇F | dx + b
F dx
with some constants a, b ∈ (0, ∞). Hence we have μf2 log
f2 aμ|∇f |(1 − χ) + μf (1 − χ) a|∇U | + b − log Z + μf2 U. μf2
(69)
In our current setup we have |∇U | β. Moreover, by simple relative entropy arguments, we have 1 eλU 1 μf2 U = μf2 log λU + log μeλU μf2 λ μe λ 1 f2 1 μf2 log + log μeλU μf2 λ μf2 λ which holds provided that β > N + λ. If we can choose λ > 1, this together with (69) implies μf2 log
f2 c2 μ|∇f |(1 − χ) + b2 μf (1 − χ) μf2
(70)
with −1 c2 ≡ a 1 − λ−1 and
1 −1 −1 λU aβ + b − log Z log μe . b2 ≡ 1 − λ λ Combining (70) and (68) we arrive at the following result Theorem 5.9. Suppose U = β log(1 + d), with β > N , and Ric 0. Then there exists β0 > N , such that for any β > β0 , one has μf log
f ¯ cμ(1 ¯ + d)|∇f | + bμf μf
(71)
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with some constants c, ¯ b¯ ∈ (0, ∞) independent of f . Consequently, if the weighted Poincaré inequality (67) is true for q > 1, we have μf q log
fq cq μ(1 + d)q |∇f |q μf q
(WLSq )
with some constant cq ∈ (0, ∞). We remark that (71) implies similar weighted LSq inequality with f replaced by |f |q and |∇f | by its qth power (which follows simply by substitution and use of Hölder inequality), while the tightening is obtained via Rothaus arguments (see e.g. [11]). 6. Optimal control distance on the Heisenberg group Heisenberg group Hl as a manifold is isomorphic to R2l+1 = R2l × R with the multiplication given by the formula 1 (x1 , z1 ) ◦ (x2 , z2 ) = x1 + x2 , z1 + z2 + S(x1 , x2 ) 2 where S(x, y) is standard symplectic form on R2l : S(x, y) =
l (xi yi+l − xi+l yi ). i=1
Vector fields spanning the corresponding Lie algebra are given as follows 1 Xi = ∂xi + xi+l ∂z , 2 1 Xi+l = ∂xi+l − xi ∂z , 2 Z = ∂z where i = 1, . . . , l. More generally, we say that a Lie algebra n is a stratified Lie algebra if it can be written as n=
m
ni ,
i
[ni , nj ] ⊂ ni+j and n is generated by n1 . Note that stratified Lie algebra is nilpotent. We say that Lie group N is stratified if it is connected, simply connected and its Lie algebra n is stratified. Since for stratified groups exponential mapping is a diffeomorphism from n to N , one can identify N with n. A Lie algebra is step two if it is stratified with m = 2. In other words it can be written in the form
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n=v⊕z where z is the centre (that is [n, z] = 0) and [v, v] ⊂ z. On a stratified Lie algebra n we define dilations by the formula δ(s)x = s i x for x ∈ ni (and extend linearly to the whole n). For s = 0, δ(s) is an automorphism of n. One can also define dilations on the corresponding group: δ(exp(X)) = exp(δ(X)). A Lie algebra n is of H-type (Heisenberg type) if it is step two and there exists an inner product ·,· on n such that z is an orthogonal complement to v, and the map JZ : v → v given by JZ X, Y = [X, Y ], Z for X, Y ∈ v and Z ∈ z satisfies JZ2 = −|Z|2 I for each Z ∈ z. Equivalently, for each v ∈ v of length 1 the mapping adv∗ given by adv∗ z, y = z, adv y = z, [v, y]
is an isometry from z∗ into v∗ . An H-type group is a connected and simply connected Lie group N whose Lie algebra is of H-type. We can identify H-type group N with its Lie algebra n defining multiplication on n by the formula: 1 (v1 , z1 ) · (v2 , z2 ) = v1 + v2 , z1 + z2 + [v1 , v2 ] 2 where v1 , v2 ∈ v and z1 , z2 ∈ z. It is easy to see that Heisenberg group is an H-type group. Also H-type group with onedimensional centre is isomorphic to the Heisenberg group, however there exist H-type groups with centre of arbitrary high dimension [23]. On H-type group we consider vector fields X1 , . . . , Xn which form an orthonormal basis of v and we introduce the following operators: Subelliptic gradient: ∇f = (X1 f, . . . , Xn f ). Kohn Laplacian: =
n
Xi2 .
i=1
On Heisenberg group Hl n = 2l and =
2l i=1
∂x2i + ∂z
l |x|2 2 ∂ . (xi+l ∂xi − xi ∂xi+l ) + 4 z i=1
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On general H-type group we similar, but more complicated expression: =
n i=1
∂v2i +
k
∂zi
Jα,i +
i=1
k |v|2 2 ∂zi 4 i=1
where Jα,i are vector fields corresponding to rotations. Length of a curve: smooth γ : [0, 1] → G is admissible if γ (s) = ni=1 ai (s)Xi (γ (s)). If 1 n γ is admissible, then |γ | = 0 ( i=1 ai2 (s))1/2 . Distance d(g) = inf |γ | where infimum is taken over all admissible γ such that γ (0) = e and γ (1) = g. d is homogeneous of degree 1 with respect to the dilations δ(s), namely for s > 0 d δ(s)g = sd(g). Lemma 6.1. On H-type group Z distance d((v, z)) depends only on |v| and |z|. Moreover if ¯ |z| = |¯z|, then d((v, z)) = d((v, ¯ z¯ )). v, ¯ z¯ ∈ H1 , |v| = |v|, Proof. Fix vectors V , Z ∈ N such that |V | = 1, |Z| = 1, v = |v|V , z = |z|Z. Put X = JZ (V ). Since JZ is antisymmetric and JZ2 = I , JZ is orthogonal, so |X| = 1. Also, for any S ∈ z of length 1, we have [X, Y ], S = JS X, Y |X||Y | so since
[V , X], Z = JZ V , X = X, X = |X|2 = 1
we have [V , X] = Z. Now, it is easy to see that the subgroup (in fact a subspace) of N generated by V , X, Z is isomorphic to H1 . Consequently, using images of curves from H1 to join with (v, z) we see that d((v, z)) d(((|v|, 0), z)) where on the right-hand we have distance in H1 . To get inequality in the opposite direction consider quotient group N/M where M = {t ∈ z: t, Z = 0}. It is easy to see that N/M is still an H-type group (note that since N/M has onedimensional centre it is enough to check the defining property just for JZ ). Hence, N/M is isomorphic to the Heisenberg group of appropriate dimension. For Heisenberg group our claim is well known. 2 It is known [30] that on Heisenberg group if g = (x, z) and x = 0, then d is smooth at g and |∇d| = 1, however when x = 0 than d is not differentiable at g. Lemma 6.2. Let A = (r, z) ∈ R2 : z > 0, r > −z. There are > 0 and a smooth function ψ(r, z) defined on A such that on each group N of H-type
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d((x, z)) = ψ |x|, |z| . Moreover, ∂r ψ < 0 when r = 0. Proof. First, by Lemma 6.1 without loss of generality we may assume that N = H1 . Also, if |x1 | = |x2 | and |z1 | = |z2 |, then d(x1 , z1 ) = d(x2 , z2 ), so ψ is uniquely defined for r 0. We need to show that it has smooth extension to A . Since d is homogeneous, it is enough to construct smooth extension in a neighbourhood of a single point g = (0, 1). There exists a smooth geodesic (length minimising curve) γ joining e = (0, 0) and g. We use length as a parametrisation of γ , so γ (d(g)) = g. For s < s0 = d(g) we have d(γ (s)) = s. Let γ (s) = (γx (s), γz (s)). Since square of Euclidean distance is smooth |γx |2 is smooth. We can write |γx |2 (s) = (s − s0 )2 ρ(s) where ρ is smooth and ρ(s0 ) = 1, so |γx |2 (s) has a square root φ(s) = (s0 − s)ρ 1/2 (s) which is smooth for s close to s0 . Since both φ and |γx | are positive square roots of |γx |2 for s0 − < s < s0 we have γx (s) = φ(s) for s0 − < s s0 . Put η(s, t) = tφ(s), t 2 γz (s) . Since γ is admissible |γz | (s0 ) = 0 so the Jacobi matrix at (s, t) = (s0 , 1) is
−1 0 0 2
and by the inverse function theorem η is invertible in a neighbourhood of (s0 , 1). So, there exist f1 , f2 such that (r, p) = η f1 (r, p), f2 (r, p) . We claim that ψ(r, p) = f1 (r, p)f2 (r, p) give us extension of ψ to a neighbourhood of g. Consider (x, z) close to g. Let (s, t) = (f1 (|x|, z), f2 (|x|, z)). We have |x| = tφ(s) = t γx (s) = δt γ (s) x , z = t 2 γz (s) = δt γ (s) z so d((x, z)) = d δt γ (s) = td γ (s) = ts = f1 (r, z)f2 (r, z) = ψ(r, z). Now it remains to find sign (∂r ψ)(0, z). Form equality (r, p) = η(f1 (r, p), f2 (r, p)) we see I = η · f . We substitute (r, p) = (0, 1) and note that this corresponds to (s0 , 1). So
1 0
0 1
=
−1 0 0 2
∂r f · ∂p f
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and using first row we get 1 = −(∂r f1 )(0, 1), 0 = −(∂r f2 )(0, 1) so (∂r ψ)(0, 1) = (∂r f1 )(0, 1)f2 (0, 1) + f1 (0, 1)(∂r f2 )(0, 1) = (∂r f1 )(0, 1) = −1.
2
Theorem 6.1. If N is an H-type group, then there is K such that if d(g) 1, then d K where is understood in the sense of distributions. Proof. Due to homogeneity, it is enough to prove the inequality only for g with d(g) = 1 (more precisely, in a small neighbourhood of each such g). Namely, if s = d(g) > 1, then d(g) = s −2 d δ(s)g = s −1 d(g). Next, d((x, z)) is smooth when x = 0, so it is enough to prove the inequality in a small neighbourhood of (0, z0 ) where z0 > is chosen so that d((0, z0 )) = 1. Below we give computation on Heisenberg group: xi ∂r ψ |x|, z , ∂xi d((x, z)) = ∂xi ψ |x|, z = |x| 2 x x2 xi 1 ∂x2i d((x, z)) = ∂xi ∂r ψ |x|, z = i 2 ∂r2 ψ |x|, z + ∂r ψ |x|, z − i 3 ∂r ψ |x|, z , |x| |x| |x| |x| 2n
2n − 1 ∂r ψ |x|, z + ∂r2 ψ |x|, z , |x| i=1 xi xi+n xi+n xi − ∂r ψ |x|, z = 0, (xi+n ∂xi − xi ∂xi+n )d((x, z)) = |x| |x| ∂x2i d((x, z)) =
d((x, z)) =
|x|2 2 2n − 1 ∂r ψ |x|, z + ∂r2 ψ |x|, z + ∂ ψ |x|, z . |x| 4 z
Since ψ is smooth the second term and third term is bounded in a neighbourhood of (0, z0 ). Since ∂r ψ(0, z0 ) < 0 the first term is unbounded, but negative in a neighbourhood of (0, z0 ), which gives the claim on Heisenberg group. On general H-type groups instead of xi+n ∂xi − xi ∂xi+n one must handle the Jα,i term. However, since Jα,i generates rotations in v space and d is rotationally invariant again Jα,i d = 0. 2 6.1. Counterexample for homogeneous norm On stratified groups N one may introduce a homogeneous norm, that is a continuous function φ : N → [0, ∞) such that φ(e) = 0, φ(x) > 0 for x = e and φ(δs (x)) = sφ(x) for s > 0. Homogeneous norms are equivalent to each other, if φ1 and φ2 are two homogeneous norms, then there is C such that C −1 φ1 φ2 Cφ1 .
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The optimal control distance d gives one example of homogeneous norm, but there are others. In particular, it is possible to choose homogeneous norm so that it is smooth for x = e (we will call such homogeneous norm smooth). Smooth homogeneous norms are convenient in many situations. For smooth homogeneous norm φ the condition (φ)(x) K for φ(x) 1 is automatically satisfied. However, we are going to prove that for such norm |∇φ|(x) = 0 for some x = e, and consequently Log-Sobolev inequality like the one for optimal control distance cannot hold. Theorem 6.2. Let N be a stratified group, and φ be a smooth homogeneous norm on N . There exists x = e such that |∇φ|(x) = 0. Proof. Let X1 , . . . , Xn be a basis of n1 . We claim that for (a1 , . . . , an ) ∈ R n − {0}, ai (Xi φ) exp ai Xi > 0. Namely, exp(t
(72)
ai Xi ) is a one parameter subgroup of N , so ∂t φ exp t ai Xi = ai (Xi φ) exp t ai Xi .
However, by homogeneity ai Xi = ∂t tφ exp = φ exp ∂t φ exp t ai Xi ai Xi > 0 so (72) holds. Using the X1 , . . . , Xn basis we identify n1 with R n . This identification gives us scalar product on n1 . We extend this scalar product to a scalar product on n such that ni is orthogonal to nj for i = j . ˜ be the unit sphere in n1 (in n respectively). Define mapping η : S → S by the Let S (S) n formula η(x) = (∇φ)(exp(x)) |∇φ|(exp(x)) (note that we use identification n1 = R here). By (72) on S, |∇φ|(exp(x)) > 0 so η is well defined. Also, η is homotopic with identity. Namely put χ( ai Xi ) = (a1 , . . . , an ). If ft is defined by the formula ft (x) = tη(x) + (1 − t)χ , then for x = ai Xi we have ft (x), x > 0, so ft takes values in R n − {0}. Consequently gt (x) = |fftt (x) (x)| gives homotopy of mappings from S to S. ˜ then η is homotopic to a constant. Namely, S˜ contains a homeoIf (∇φ)(exp(x)) = 0 on S, (∇φ)◦exp morphic copy of (n+1)-dimensional disc D having S as a boundary and |(∇φ)◦exp | gives required homotopy. However, it is well known that identity of the sphere is not homotopic to a constant — so we reach contradiction with assumption that (∇φ)(exp(x)) = 0. 2 Lemma 6.3. If f is smooth function on a stratified group N , d is optimal control metric on N , x0 ∈ N is fixed, then f (x) − f (x0 ) O d(x, x0 ) . If additionally (∇f )(x0 ) = 0, then f (x) − f (x0 ) O d 2 (x, x0 ) .
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Proof. Let γ : [0, 1] → N be an admissible curve joining x0 and x. We have γ (s) = ai (s)Xi (γ (s)), so f (x) − f (x0 ) =
1
(f ◦ γ ) =
0
1
1 ai (s)(Xi f ) ◦ γ 0
|γ |(∇f ) ◦ γ |γ | sup (∇f ) ◦ γ (s). s∈[0,1]
0
Put r = d(x, x0 ). If |γ | r + ε, then γ (s) ∈ B(x, r + ε) and f (x) − f (x0 ) (r + ε)
sup
(∇f )(y).
y∈B(x,r+ε)
Taking ε → 0 we get f (x) − f (x0 ) r
sup (∇f )(y). y∈B(x,r)
Since f is smooth the supremum is finite which gives the first claim of the lemma. If (∇f )(x0 ) = 0, then we can apply the first part to Xi f and get sup (∇f )(y) Cr y∈B(x,r)
f (x) − f (x0 ) Cr 2
sup (∇∇f )(y), y∈B(x,r)
sup (∇∇f )(y)
y∈B(x,r)
which gives the second claim.
2
Theorem 6.3. Let N be a stratified group and φ be a smooth homogeneous norm on N . For β > 0, p 1 put μβ,p = exp(−βφ p )/Z dλ, where Z is a normalising factor such that μβ,p is a probability measure. The measure μβ,p satisfies no LSq inequality with q ∈ (1, 2]. Proof. Fix β > 0, p 1, q ∈ (1, 2]. Suppose that μβ,p satisfies LSq . We are going to show that this leads to contradiction. Let x0 be such that (∇φ)(x0 ) = 0. For t > 0 put r = t (−p+1)/2 and f = max(min((2 − d(x, tx0 ))/r, 1), 0). By homogeneity and Lemma 6.3 we have |φ(x) − φ(tx0 )| C1 r 2 on B(tx0 , 2r) = {x: d(x, tx0 ) 2r}, so |φ(x)p − φ(tx0 )p | C2 . Consequently the exponential factor in μβ,p is comparable to a constant on support of f . Also |∇f | r −1 and μβ,p |f |q ≈ r Q exp −βφ(tx0 )p , log μβ,p |f |q ≈ −t p , μβ,p |∇f | ≈ r −q r Q exp −βφ(tx0 )p , μβ,p |f |q log |f |q μβ,p |f |q ≈ |f |q t p dμβ,p ≈ t p r Q exp −βφ(tx0 )p . B(tx0 ,r)
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Using LSq we get t p r Q exp −βφ(tx0 )p Mr −q r Q exp −βφ(tx0 )p for large t, so t p Mr −q = Mt −q(−p+1)/2 for large t, and p q(p − 1)/2. Since p 1 and q 2, this implies p p − 1 which is a contradiction. 2 7. Log Sobolev inequalities for heat kernel on the Heisenberg group The heat kernels bound of the following form 1 2 1 C 2 e−σ d (x)t p(x, t) e− σ d (x)t C|B(e, t 1/2 )| |B(e, t 1/2 )|
were well known since a few decades, see e.g. [17,38] and references therein. While the measures corresponding to the densities on the left and right have nice properties and in particular satisfy Poincaré and logarithmic Sobolev inequality, this kind of sandwich bound does not imply similar properties for the measure corresponding to the density in the middle. Namely on a stratified groups one can write C −1 p(x, t/σ )
1 exp −φ 2 (x)/t Cp(x, σ t) 1/2 |B(e, t )|
where C, σ 1 are constants and φ is a smooth homogeneous norm. In Theorem 6.3 we proved that the density in the middle does not satisfy logarithmic Sobolev inequality. We give another example in Appendix A. In [26] it was observed that asymptotics from [22] imply the following precise bound (extending [7]) on the heat kernel p (at time t = 1) on the three-dimensional Heisenberg group H1 : • (HK) There exists a constant L ∈ (0, ∞) such that for any x ≡ (x, z) ∈ H1 − 1 d 2 (x) − 1 d 2 (x) L−1 1 + xd(x) 2 e− 4 p(x) L 1 + xd(x) 2 e− 4 . Let dν0 ≡ ρ0 dλ ≡ e−
d 2 (x) 4
dλ/Z and set dμ = p dλ.
Theorem 7.1. There exist constants C1 , C2 , D1 , D2 ∈ (0, ∞) such that μ f 2 d 2 C2 μ|∇f |2 + D2 μf 2 and μ |f |d C1 μ|∇f | + D1 μ|f |.
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Proof. Put W =
−1 2
log(1 + εxd) for some ε ∈ (0, 1) to be chosen later. We have |∇W |2 = ε 2 ε2
|d∇x + x∇d|2 (1 + εxd)2 d 2 + x2 ε2 d 2 + 1 (1 + εxd)2
so, if ε is small enough W satisfies assumptions of Theorem 2.5. Now we observe that for ε ∈ (0, 1), we have − 1 − 1 − 1 1 1 + xd 2 1 + εxd 2 1 + xd 2 . ε This together with (HK) imply we can write μ = exp(−W − V )μ0 and apply Theorem 2.5 to get the first claim. We get the second claim using Theorem 2.2. 2 By similar arguments as in Section 3 we obtain the following result Theorem 7.2. Let dμ ≡ p dλ. There exists a constant M ∈ (0, ∞) such that Mμ(f − μf )2 μ|∇f |2 .
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We are now ready to prove the Log-Sobolev inequality for the heat kernel measure. Theorem 7.3. There exists a constant c ∈ (0, ∞) such that on Heisenberg group Hn we have f2 cμ|∇f |2 . μ f 2 log μf 2 Remark. The case of H1 is proven in [26]. While our proof uses heat kernel estimates from [26], in [26] large part is devoted to proof of estimate (1) for heat kernel measure on H1 — using our methods we could give different proof for this part, but instead we work directly with LogSobolev inequality. Proof of Theorem 7.3. First consider H1 . In the proof of Theorem 7.1 we wrote μ = e−W −V μ0 . Consider now μ1 = e−W μ = e−U dλ. μ1 satisfies Log-Sobolev inequality as a consequence of Theorem 4.1. The result for H1 follows, since μ is equivalent to μ1 . Now, write Hn = G/N , where G = ni=1 H1 , N = {((0, z1 ), . . . , (0, zn )): zi = 0} and let π be the canonical homomorphism from G to H . Since heat kernel on H is an image of product n n of heat kernels on G = ni=1 H1 , and since Log-Sobolev inequality holds on product, we have μHn f 2 log
f2 μ Hn f 2
(f ◦ π)2 = μG (f ◦ π)2 log μG (f ◦ π)2 2 2 cμG ∇(f ◦ π) = cμG (∇f ) ◦ π = cμHn |∇f |2 .
2
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Appendix A. Examples of no spectral gap In case of measures on real line the following necessary and sufficient condition for Poincaré inequality to hold was provided by Muckenhoupt [31] ([2]) which in the special case of a measure dμ ≡ ρ dx can be stated as follows: Given q ∈ [1, ∞) and q1 + p1 = 1 μ|f − μf |q Cμ|f |q
∃C ∈ (0, ∞)
⇔
B± ≡ sup B± (r) < ∞ r∈R±
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where 1 B± (r) ≡ μ [r, ±∞) q ·
ρ
− pq
1
p
.
[0,±r]
Consider ρ ≡ e−U dx/Z with U ≡ β|x|p (1 + ε cos x), defined ε ∈ (0, 1) and some β ∈ (0, ∞). Then, with r = 2nπ + π2 , we have
1
2nπ+ 83 π
B+ (r) >
e
−β|x|p (1− 2ε )
q
·
dx
2nπ+ 43 π
1
2nπ+ 23 π
e
+ pq β|x|p (1+ 2ε )
p
dx
2nπ− 23 π
4 π 3
1
1 p 4 π >e ·e 3 p p
p 1 β(2nπ) 4 4 ε ε = π exp 1 − 3n 1 + 2 − 1 + 3n 1 − 2 3 q
β(2nπ)p 1 4 ε+o → ∞ as n → ∞. ∼ π exp 3 q n −β q1 |2nπ+ 83 π|p (1− 2ε )
q
+ q1 β|2nπ− 23 π|p (1+ 2ε )
Alternatively one can study lower bound asymptotic for B± thinking of U = V + δV as a perturbation of V ≡ β|x|p as follows. We notice that by Jensen inequality r ∞
+V dx 1 r δV e−V dx p 0 δV e . B+ (r, U ) B+ (r, V ) exp − β ∞ −V + βε r +V q q dx dx 0 e r e Hence one can use a procedure based essentially on integration by parts to study the integrals in the exponential. For example in case p = 2 one gets the following an asymptotic lower bound B+ (r, U ) B+ (r, V ) exp −βεr cos r + O(1) . We summarise our considerations in the above as follows Proposition A.1. Suppose p 1. In any neighbourhood 1 p 1 −(1+δ)β|x|p e ρ Ce− 1+δ β|x| C
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with arbitrary δ ∈ (0, 1) and some C ∈ (1, ∞), of a measure dμ0 ≡ e Z dx satisfying the Poincaré inequality there is a measure dμ ≡ ρ dx for which this inequality fails. The example provided above illustrates similar phenomenon for other coercive inequalities. Note added in proof For the benefit of the reader we would like to mention the following two recent works [5] and [15] containing certain related results in Euclidean setup as well as some results concerning isoperimetry. We would like to thank Michel Ledoux and Patrick Cattiaux for this information. References [1] S. Aida, D.W. Stroock, Moment estimates derived from Poincaré and logarithmic Sobolev inequalities, Math. Res. Lett. 1 (1994) 75–86. [2] C. Ané, S. Blachère, D. Chafai, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmiques, Panor. Synthèses, vol. 10, S.M.F., Paris, 2000. [3] A.Val. Antonjuk, A.Vict. Antonjuk, Weighted spectral gap and logarithmic Sobolev inequalities and their applications, Preprint 93.33, Acad. Sci. of Ukraine, Kiev, 1993. [4] D. Bakry, L’hypercontractivité et son utilisation en théorie des semigroupes, in: Lectures on Probability Theory (École d’été de probabilités de St-Flour 1992), in: Lecture Notes in Math., vol. 1581, Springer, Berlin, 1994, pp. 1– 114. [5] D. Bakry, P. Cattiaux, A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal. 254 (2008) 727–759. [6] F. Barthe, P. Cattiaux, C. Roberto, Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry, Rev. Mat. Iberoamericana 22 (2006) 993–1067. [7] R. Beals, B. Gaveau, P.C. Greiner, Hamiltonian–Jacobi theory and heat kernel on Heisenberg groups, J. Math. Pures Appl. 79 (2000) 633–689. [8] S.G. Bobkov, F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal. 163 (1999) 1–28. [9] S. Bobkov, M. Ledoux, From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal. 10 (2000) 1028–1052. [10] S. Bobkov, M. Ledoux, Weighted Poincaré-type inequalities for Cauchy and other convex measures, Ann. Probab. 37 (2) (2009) 403–427. [11] S. Bobkov, B. Zegarli´nski, Entropy bounds and isoperimetry, Mem. Amer. Math. Soc. 176 (829) (2005), x+69 pp. [12] Th. Bodineau, B. Helffer, The log-Sobolev inequalities for unbounded spin systems, J. Funct. Anal. 166 (1999) 168–178. [13] H.J. Brascamp, E. Lieb, On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for Logconcave functions, and with an application to diffusion equation, J. Funct. Anal. 22 (1976) 366–389. [14] E.A. Carlen, D.W. Stroock, An application of the Bakry–Émery criterion to infinite dimensional diffusions, in: Séminaire de probabilités de Strasbourg, vol. 20, 1986, pp. 341–348. [15] P. Cattiaux, N. Gozlan, A. Guillin, C. Roberto, Functional inequalities for heavy tails distributions and application to isoperimetry, preprint, 2008. [16] B. Chow, Lu Peng, Ni Lei, Hamilton’s Ricci Flow, Grad. Stud. Math., Amer. Math. Soc., 2006. [17] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, 1989. [18] E.B. Davies, L. Gross, B. Simon, Hypercontractivity: A bibliographic review, in: Ideas and Methods in Quantum and Statistical Physics, Oslo, 1988, Cambridge Univ. Press, Cambridge, 1992, pp. 370–389. [19] B.K. Driver, T. Melcher, Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal. 221 (2005) 340–365. [20] I. Gentil, C. Roberto, Spectral gaps for spin systems: Some non-convex phase examples, J. Funct. Anal. 180 (2001) 66–84. [21] A. Guionnet, B. Zegarli´nski, Lectures on logarithmic Sobolev inequalities, in: Séminaire de Probabilités, XXXVI, in: Lecture Notes in Math., vol. 1801, Springer, Berlin, 2003, pp. 1–134.
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[22] H. Hueber, D. Müller, Asymptotics for some Green kernels on the Heisenberg group, Math. Ann. 283 (1989) 97– 119. [23] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980) 147–153. [24] A. Kasue, A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold, Japan J. Math. 8 (1982) 309–341. [25] R. Latała, K. Oleszkiewicz, Between Sobolev and Poincaré, Lecture Notes in Math., vol. 1745, 2000, pp. 147–168. [26] H.-Q. Li, Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg, J. Funct. Anal. 236 (2006) 369–394. [27] H.-Q. Li, Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg, C. R. Math. Acad. Sci. 344 (2007) 497–502. [28] P. Ługiewicz, B. Zegarli´nski, Coercive inequalities for Hörmander type generators in infinite dimensions, J. Funct. Anal. 247 (2007) 438–476. [29] V. Mazya, Sobolev Spaces, Springer, 1985. [30] R. Monti, Some properties of Carnot–Carathéodory balls in the Heisenberg group, Rend. Mat. Acc. Lincei 11 (2000) 155–167. [31] B. Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972) 31–32, collection of articles honoring the completion by Antoni Zygmunt of 50 years of scientific activity. [32] Ding Qing, A new Laplacian comparison theorem and the estimate of eigenvalues, Chinese Ann. Math. Ser. B 15 (1994) 35–42. [33] C. Roberto, B. Zegarli´nski, Orlicz–Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semigroups, J. Funct. Anal. 243 (2006) 28–66. [34] J. Rosen, Sobolev inequalities for weight spaces and supercontractivity, Trans. Amer. Math. Soc. 222 (1976) 367– 376. [35] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (3) (1982). [36] D.W. Stroock, B. Zegarli´nski, The logarithmic Sobolev inequality for continuous spin systems on a lattice, J. Funct. Anal. 104 (1992) 299–326. [37] D.W. Stroock, B. Zegarli´nski, The equivalence of the logarithmic Sobolev inequality and the Dobrushin–Shlosman mixing condition, Comm. Math. Phys. 144 (2) (1992) 303–323. [38] N.Th. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and Geometry on Groups, Cambridge Univ. Press, 1992. [39] F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997) 417–424. [40] F.-Y. Wang, Functional Inequalities, Markov Processes and Spectral Theory, Science Press, Beijing, 2005. [41] N. Yosida, The log-Sobolev inequality for weakly coupled lattice fields, Probab. Theory Related Fields 115 (1999) 1–40. [42] B. Zegarli´nski, Entropy bounds for Gibbs measures with non-Gaussian tails, J. Funct. Anal. 187 (2001) 368–395.
Journal of Functional Analysis 258 (2010) 852–868 www.elsevier.com/locate/jfa
A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit Olivier Glass Université Pierre et Marie Curie-Paris 6, UMR 7598 Laboratoire Jacques-Louis Lions, Paris F-75005, France Received 12 May 2009; accepted 30 June 2009 Available online 10 July 2009 Communicated by J. Coron
Abstract We revisit a result by Coron and Guerrero stating that the one-dimensional transport-diffusion equation ut + Mux − εuxx = 0 in (0, T ) × (0, L), controlled by the left Dirichlet boundary value is zero-controllable at a bounded cost as ε → 0+ , when T > 4.3L/M if M > 0 and when T > 57.2L/|M| if M < 0. By a completely different method, relying on complex analysis, we prove that this still holds when T > 4.2L/M if M > 0 and when T > 6.1L/|M| if M < 0. © 2009 Elsevier Inc. All rights reserved. Keywords: Null controllability; Vanishing viscosity limit
1. Introduction Let us fix L > 0 and M = 0. We consider the following transport-diffusion equation: ⎧ ⎨ ut + Mux − εuxx = 0 in (0, T ) × (0, L), u|t=0 = u0 in (0, L), ⎩ u|x=L = 0 in (0, T ). u|x=0 = v(t) in (0, T ), E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.035
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In the above equation v is a boundary control and ε is a small positive parameter, intended to tend to zero. The problem which we consider for this parabolic equation is connected to the zerocontrollability. We recall that the problem of zero-controllability is to determine whether it is possible given a time T > 0 and an initial data u0 in L2 (0, L), to find a control v ∈ L2 (0, T ) such that the corresponding solution of (1) satisfies u(T , x) = 0 for all x ∈ [0, L].
(2)
The controllability of parabolic equations in dimension 1, such as the one considered here for fixed ε > 0, was established by Fattorini and Russell [6]. The controllability of parabolic equation in higher dimensions was established independently by Fursikov and Imanuvilov (see [7]) and Lebeau and Robbiano (see [13]) in slightly different frameworks, and with different methods (both using the so-called Carleman estimates, though). In this paper, we investigate the cost of the control in the vanishing viscosity limit ε → 0+ , and in particular to determine in which situation it is possible to obtain a control which remains bounded as ε → 0+ . We will say that the system is uniformly zero-controllable if this property is satisfied. A motivation for studying the controllability of a transport equation in the vanishing viscosity limit, comes from the topic of the control of systems of conservation laws, in the context of weak entropy solutions, see for instance [1,2,4,8]. These solutions are discontinuous solutions (admitting shocks), which can be obtained via a vanishing viscosity limit. It is hence interesting in order to understand better the control properties of these equations, to know how the control behaves for small but not zero viscosity. Of course the linear model which we consider here is the simplest possible example of scalar conservation law. A first example of controllability result of a nonlinear conservation law in the vanishing viscosity limit was given in [9]. The problem under view was first introduced and studied by Coron and Guerrero [5]. Next Guerrero and Lebeau [10] extended some of the results of [5] in arbitrary dimension and with a variable vector field M. In these papers, it is proven that if the vector field M is such that the transport equation is not controllable (because there is a characteristic of M which stays in the domain without reaching the control zone ω) then the size of the control can grow as eC/ε . On the other side, if all the characteristics stay sufficiently long in the control zone ω or outside Ω, then the system uniformly zero-controllable. These results require that T is large enough, and in particular in [5] it is proven that in the one-dimensional case that (1) is uniformly zero-controllable when M > 0 provided that T > 4.3L/M, and when M < 0 provided that T > 57.2L/|M|. Clearly the transport equation (ε = 0) is controllable for T L/|M| (this time being optimal), so one could expect that in both cases the uniform controllability to hold for any time T > L/|M|. A very surprising result of [5] is that when M < 0, the control can blow up exponentially for any T < 2L/|M|, while this is shown only for times T < L/M when M > 0 (which is much more intuitive). What we establish in this paper is that we can improve the times 4.3L/M and 57.2L/|M| of Coron and Guerrero’s paper to T > 4.2L/M and T > 6.1L/|M|, respectively. Also (and perhaps more importantly), our proof is of completely different nature. Coron and Guerrero used a Carleman estimate to prove the observability inequality of the adjoint problem, and showed that the explosive nature of the constant coming from this Carleman estimate as ε → 0+ can be compensated by the constant of a dissipation estimate (the solution of (1) or its adjoint equation naturally decreases for T > 1/|M|, exponentially in −1/ε as ε → 0+ ), provided that T is
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large enough. Here, our method is closer to Russell’s harmonic analysis approach to some controllability problems (see in particular Fattorini, Russell [6] and Russell [18]). The observation inequality for the adjoint system is connected to a question concerning sum of exponentials. This requires the construction of some biorthogonal family to the family of exponentials, which relies on the Paley–Wiener theorem. Some analogous methods can be found for instance in [16,20– 22], but here the core of the proof is slightly different and relies on the construction of a complex “multiplier” due to Beurling and Malliavin [3]. Precisely, we show the following result. Theorem 1. Given M = 0 and T > 0, the system (1) is uniformly zero-controllable in the sense that there exist constants κ > 0 and K > 0 such that for any u0 ∈ L2 (0, L), any ε ∈ (0, 1), there exists v ∈ L2 (0, T ) such that the solution of (1) satisfies (2), and moreover κ vL2 (0,T ) K exp − u0 L2 (0,L) , ε
(3)
provided that: L if M > 0, M L T > 6.1 if M < 0. |M| T > 4.2
(4) (5)
Remark 1. The conjecture that the optimal times should be 1/M and 2/|M| is hence still open. We believe that the complex analytic technique could be a good approach to solve the problem, probably by finding a more accurate complex multiplier. 2. Notations and preliminaries 2.1. Observability inequality It is a standard fact (see Lions [15] and Russell [18]) that proving Theorem 1 is equivalent to establish an observability inequality for the adjoint equation with a constant as in (3). Precisely the adjoint equation is the following: ⎧ ⎪ ⎨ ϕt + Mϕx + εϕxx = 0 in (0, T ) × (0, L), ϕ = 0 on (0, T ) × {0, L}, ⎪ ⎩ ϕ(T , ·) = ϕT in (0, L).
(6)
It is then sufficient to show that for some κ > 0 and K > 0, one has for any ε ∈ (0, 1) and any ϕT ∈ L2 (0, L), one has ϕ(0, ·)
κ ∂x ϕ(·, 0) 2 K exp − . 2 L (0,L) L (0,T ) ε
(7)
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2 2.2. The operator −M∂x − ε∂xx
To diagonalize the operator 2 , P := −M∂x − ε∂xx
it suffices to remark that Mx
Mx M M2 2 2 ∂xx e 2ε u = e 2ε ∂xx u + ∂x u + 2 u , ε 4ε that is to say with the obvious notation for the multiplication operator 2 ◦e P = −εe− 2ε ◦ ∂xx Mx
Mx 2ε
+
M2 Id. 4ε
(8)
It follows that P is diagonalizable in L2 (0, L), with eigenvectors √ kπx Mx ek (x) := 2 exp − sin 2ε L
(9)
for k ∈ N \ {0} and corresponding eigenvalues λk := ε
k2π 2 M 2 , + 4ε L2
(10)
the family {ek , k ∈ N \ {0}} being a Hilbert basis of L2 (0, L) for the L2 ((0, L); exp( Mx ε ) dx) scalar product: L u, v :=
Mx u(x)v(x) dx. exp ε
(11)
0
3. Proof of Theorem 1 3.1. General strategy The strategy to prove Theorem 1 is connected to the method of moments, see for instance [6,16,18,20–22]. The idea is to construct a biorthogonal family in L2 (0, T ) to the family of exponentials
t → exp −λk (T − t) .
(12)
By the change of variables t → T − t, we can of course consider the family of exponentials t → exp(−λk t).
(13)
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To that purpose, as in the complex-analytic proof of the Müntz–Szász theorem (see for instance [17,19]) the idea is to construct a suitable family Jk (z) of entire functions of exponential type (see e.g. [12]), satisfying Jk (−iλj ) = δj k ,
(14)
where δj k is the Kronecker symbol. Then using the Paley–Wiener theorem we deduce our biorthogonal family ψk as the inverse Fourier transform of Jk (z) (up to a translation in time). The family Jk (z) is constructed from a single entire function having simple poles at (−iλk )k∈N\{0} . This function is naturally constructed as a Weierstrass product (which turns out to be explicit here), multiplied by a function (which we will designate as a “multiplier”) intended to make Jk of relevant exponential type and with suitable behaviour on the real axis. Such a method can be traced back to Paley and Wiener [17]. The construction of the multiplier which we employ here follows the work of Beurling and Malliavin [3]. Once the biorthogonal family is constructed with suitable estimates, obtaining the observability inequality (7) is rather straightforward. We develop these main steps in the following subsections. 3.2. The Weierstrass product Φ An entire function having the k 2 , k ∈ N \ {0} as its simple zeros is the following one: ∞ k=1
z 1− 2 k
√ sin(π z) = , √ π z
(15)
which is an entire function (despite the square roots). Now one can construct a function having simple zeros exactly at {−iλk , k ∈ N \ {0}} by
Φ(z) =
sin
L √ ε
L √ ε
iz −
iz −
M2 4ε
M2 4ε
.
(16)
It is elementary to see that Φ is of exponential type, and even satisfies Φ(z) C(M, ε) exp √L |z| as |z| → +∞. 2ε
(17)
A good candidate for Jk (z) would be Φ(z) , k )(z + iλk )
Φ (−iλ
(18)
but precisely because of (17), one could show by the Phragmen–Lindelöf method that such a function cannot be bounded on the real line, and hence it cannot be used directly to construct the family ψk by inverse Fourier transform. We must use a multiplier to “mollify” the function on the real line without perturbing too much the behavior at the above zeros.
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3.3. Beurling and Malliavin’s multiplier We follow Beurling and Malliavin’s construction [3] (see also Koosis [12, Chapter X]). We fix a :=
T , 2π
(19)
and L˜ := L + αε 1/4
and Lˆ := L + 2αε 1/4 ,
(20)
with α a positive real number independent of ε to be chosen later. Let us introduce s(t) = at −
L˜ √ t. √ π 2ε
(21)
Using that [3, p. 294] ∞
x 2 γ πγ log1 − 2 dt = |x|γ π cot 2 t
for 0 < γ < 2,
(22)
0
we see that ∞ 0
L˜ x 2 log1 − 2 ds(t) = − √ |x|. t 2ε
(23)
We notice that s is increasing for t larger than A :=
1 L˜ 2 . 2ε T
(24)
We also introduce B := 4A =
2 L˜ 2 , ε T
(25)
which satisfies s(B) = 0. Now one defines ν as the restriction of the measure ds(t) to the interval [B, +∞). Let us underline that this measure is positive. Next we introduce for z ∈ C: ∞ U (z) := 0
∞ 2 z z2 log1 − 2 dν(t) = log1 − 2 ds(t), t t B
(26)
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O. Glass / Journal of Functional Analysis 258 (2010) 852–868
and for z ∈ C \ R, ∞ g(z) :=
∞ z2 z2 log 1 − 2 dν(t) = log 1 − 2 ds(t). t t
(27)
B
0
By “atomizing” the measure dν in the above integral, we can define U˜ (z) :=
∞
z2 log1 − 2 d ν(t) , t
(28)
0
where [ · ] denotes the integer part and where t ν(t) =
dν.
(29)
z2 log 1 − 2 d[ν](t). t
(30)
0
In the same way as previously we introduce ∞ h(z) := 0
Of course,
U (z) = Re g(z)
and U˜ (z) = Re h(z) .
The main advantage of U˜ (and h) over U is that now exp(h(z)) is an entire function. Indeed, calling {μk , k ∈ N} the discrete set in R consisting of the discontinuities of the function t → [ν(t)], we have
z2 1− 2 . exp h(z) = μk k∈N
(31)
The convergence of this product is quite straightforward. Finally, the multiplier which we will use is the following:
f (z) := exp h(z − i) .
(32)
3.4. Estimates on the multiplier Before constructing the functions Jk themselves, let us prove some lemmas which will be useful to obtain properties on f .
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Lemma 1. For x ∈ R, one has L˜ U (x) − √ |x| + C1 aB, 2ε
(33)
where C1 is the following positive (and finite) constant 1 C1 := − min x∈R
√ x2 log1 − 2 d(t − t ) 2.34 < 2.35. t
(34)
0
Proof. Following (23), we have B x 2 L˜ U (x) + √ |x| = − log1 − 2 ds, t 2ε 0
which immediately gives (34) after the change of variable t → t/B. Now that the constant C1 is finite follows from explicit integration: 1 0
√ x + 1 √ x + 1 √ √ x2 − x ln √ log1 − 2 d(t − t ) = −π x + x ln x − 1 x − 1 t √ √ + 2 x arctan( x ).
2
(35)
Im(z)U (t) dt. |z − t|2
(36)
Lemma 2. For Im(z) < 0, we have 1 U (z) = −πa Im(z) − π
∞ −∞
Proof. This is essentially [12, Vol. I, Theorem G.1, p. 47] (see also [12, Vol. II, p. 161]). We recall this result for the reader’s convenience. Theorem 2. Let f (z) be analytic in Im(z) > 0 and at the points of the real axis. Suppose that
logf (z) O |z| , for Im(z) 0 and |z| large, and that +∞ −∞
log+ |f (x)| dx < ∞. 1 + x2
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O. Glass / Journal of Functional Analysis 258 (2010) 852–868
Then if f (z) has no zeros in Im(z) > 0, 1 logf (z) = A Im(z) + π
∞ −∞
Im(z) log|f (t)| dt, |z − t|2
there where A = lim sup y→+∞
log|f (iy)| . y
We notice that for any y ∈ R we have ∞ U (iy) =
y2 log1 + 2 dν, t
0
so that using ν(t) →a t
as t → +∞,
and integrating by parts we deduce lim sup y→+∞
U (±iy) = πa. ±y
(37)
Now applying Theorem 2 to exp(g(−z)) would yield the result, except that U is not analytic at the points of the real axis. But this is just a matter of considering exp(g(−z − iτ )) for small τ > 0 and passing to the limit by dominated convergence. 2 Lemma 3. For x ∈ R, one has L˜ U (x − i) πa + C1 aB − √ |x|. 2ε Proof. We apply (33) and (36); since ∞ −∞
1 dt = |x − i − t|2
∞
−∞
1 dt = π, 1 + |x − t|2
there is left to compute ∞ −∞
√ |t| dt. 1 + |x − t|2
(38)
O. Glass / Journal of Functional Analysis 258 (2010) 852–868
861
This can be cut into two are computed in a standard way via the respective √ integrals which √ changes of variable u = t and u = −t: ∞ 0
√ t π , dt = √ 1 + (x − t)2 2 1 + x 2 − 2x
and 0 −∞
√
−t π . dt = √ 1 + (x − t)2 2 1 + x 2 + 2x
By considering x > 0 and x < 0 we see that
2 1 + x 2 + 2x + 2 1 + x 2 − 2x 2 |x|, and the result follows.
2
Lemma 4. We have for z = x + iy ∈ C: ∞ 0
max(|x|, |y|) z2 |y| + . log1 − 2 d [ν](t) − ν(t) log 2|y| 2 max(|x|, |y|) t
Proof. This is [12, Vol. II, Lemma, p. 162].
(39)
2
Lemma 5. Denote 1 G(y) :=
√ y 2 log1 + 2 d(t − t ). t
(40)
0
For any y ∈ R one has y 2 log1 + 2 dt = πy, t
(41)
y 2 √ log1 + 2 d t = π 2|y|, t
(42)
y y 2 . log1 + 2 ds = aBG B t
(43)
∞ 0
∞ 0
B 0
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O. Glass / Journal of Functional Analysis 258 (2010) 852–868
Proof. These are easily obtained by integration by parts and change of variable, and noting s(B) = 0. 2 Lemma 6. For all y ∈ R one has ∞
∞ y2 y2 y2 log 1 + 2 d[s] log 1 + 2 ds − log 1 + 2 . t t B
B
(44)
B
Proof. By integrating by parts, recalling that s(B) = 0 and using 0 s(t) − [s(t)] 1, we obtain ∞
∞
y2 y2 s(t) − s(t) dt log 1 + 2 d [s] − s ∂t log 1 + 2 t t
B
B
∞ y2 y2 ∂t log 1 + 2 dt = − log 1 + 2 . t B
2
B
The conclusion of this paragraph is the following: Proposition 1. The function U˜ constructed above satisfies the following properties for some C > 0: ∀x ∈ R, ∀y ∈ R− ,
L˜ U˜ (x − i) − √ |x| + aBC1 + log+ |x| + πa, 2ε y L˜ y2 . U˜ (iy) πa|y| − √ |y| − log 1 + 2 − aBG B B ε
(45) (46)
Proof. Estimate (45) is a direct consequence of Lemmata 3 and 4, while estimate (46) follows from Lemmata 5 and 6 and the fact that y → U˜ (iy) is monotonous on R− . 2 3.5. The biorthogonal family ψk Now we introduce the function for any k ∈ N \ {0}: J˜k (z) :=
f (z) Φ(z) . Φ (−iλk )(z + iλk ) f (−iλk )
(47)
The construction of Section 3.3 was performed in order to get the following result. Proposition 2. For any k ∈ N \ {0}, the function J˜k is an entire function of exponential type πa. Moreover for ε > 0 small enough independent of k, it satisfies on the real line ˜2
−3/2 Lˆ J˜k (x) C exp L|M| + 1 (C1 − C2 ) L − T λk + √ λk 1 + |x| , 2ε π Tε 2 ε
(48)
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where C2 := −G(2) 1.97 > 1.95.
(49)
Proof. That J˜k is an entire function follows from the fact that Φ is entire and has only simple zeros at −iλk and that f is an entire function with f (−iλk ) = 0. From (18), we see that in order to prove that J˜k is of exponential type πa = T /2, it is sufficient to prove that f is of exponential type πa. That h satisfies |h(z)| C exp(πa|z|) is a consequence of Theorem 2 and (37) being valid for U˜ . It follows that f is also of exponential type T /2. Now turn to estimate (48). Using (16) and the fact that for y ∈ R− , x ∈ R → √ let us √ Im( ix + y − ix) is maximal at x = 0, we infer L|M| L √ exp( 2ε + √2ε |x| ) Φ(x) . M 4 1/4 Lε −1/2 |x 2 + 16ε 2|
Using (45), we infer √ ˜ L−L √ |x| + aBC1 + log+ (|x|) + πa) 2ε M 4 1/4 Lε −1/2 |x 2 + 16ε 2| L|M| 1/2 exp( 2ε + aBC1 )
L|M|
exp( 2ε − Φ(x) exp U˜ (x − i)
Cε
|x 2 +
M 4 1/4 | 16ε 2
,
√ provided that α 2 and with C independent of ε. Now a direct computation yields Φ (−iλk ) =
(−1)k . 2ελk
Finally, by (46) we get λ2 λk L˜ f (−iλk ) c exp πaλk − √ . λk − log 1 + k2 − aBG B B ε Using for instance log(1 + y 2 /24) of k and ε ∈ (0, 1) one has
√ |y|, we infer that for α large enough and independently
Lˆ λk f (−iλk ) c exp πaλk − √ . λk − aBG B ε Putting all these estimates together yields exp( J˜k (x) C
L|M| 2ε
+ aBC1 − πaλk + |x 2 +
M 2 1/4 2 |x 4ε |
Lˆ √ √ λk ε
− aBG( λBk ))
+ λ2k |1/2
.
(50)
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O. Glass / Journal of Functional Analysis 258 (2010) 852–868
Concerning the last term in the exponential, we use that in both cases T > 4L/|M| so that λk M2 T 2 2 B 8 L˜ 2
(51)
(at least for ε small so that T |M|/L˜ > 4) and the fact that G is a negative decreasing function. For larger ε it suffices to enhance a little bit the constant C in (50). 2 Remark 2. The constant C2 could be optimized a little bit further by making the optimization later (see Proposition 3). Now from Proposition 2 and the Paley–Wiener theorem, we deduce that J˜k is the Fourier– Laplace transform of some function ψ˜ k ∈ L2 (R), supported in [−T /2, T /2]. Now we define Jk (z) =
exp(−i T2 z) exp(− T2 λk )
J˜k (z).
(52)
We deduce that Jk is the Fourier–Laplace transform of the function ψk := TT /2 ψ˜ k , supported in [0, T ], where TT /2 is the translation at the source by T /2. From (48) and (52), we moreover deduce that for x ∈ R, ˜2 1 Lˆ Jk (x) C exp L|M| + 1 (C1 − C2 ) L + √ λk . 2ε π Tε (1 + |x|)3/2 ε
(53)
Moreover, due to (47) and (52), we have Jk (iλj ) = δj k .
(54)
1 Lˆ L|M| L˜ 2 + (C1 − C2 ) + √ λk , ψk L2 (R) C exp 2ε π Tε ε
(55)
Finally Parseval’s identity yields
and (54) translates into T ψk (t) exp(−λj t) dt = δj k .
(56)
0
As mentioned in Section 3.1, we will in fact consider t → ψk (T − t). We will still call the resulting function ψk . The new family (ψk ) still satisfies (55), and now (56) is replaced by T 0
ψk (t) exp −λj (T − t) dt = δj k .
(57)
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3.6. The constants The constants of the main statement appear in the next result. Proposition 3. We have for some κ > 0, L2 L L|M| 1 + (C1 − C2 ) − T λk + √ λk −κλk 2ε π Tε ε
for all k,
(58)
provided that L c+ T> |M|
with c+ := 2 +
4+
4 (C1 − C2 ) < 4.2, π
(59)
and we have for some κ > 0, L|M| 1 L2 L + (C1 − C2 ) − T λk + √ λk −κλk ε π Tε ε
for all k,
(60)
provided that L c− T> |M|
with c− := 3 +
9+
4 (C1 − C2 ) < 6.1. π
(61)
Proof. First we notice that L √ x → −T x + √ x ε is decreasing for values larger than
1 L2 4ε T 2
M2 4ε
λk
(in both cases). Next we only use that for all k, M2 , 4ε
(62)
hence we are led to decide when T is larger than the larger root of the polynomial 1 L2 M 2 2 L|M| (C1 − C2 ) − X +X , π ε 2ε ε for (58), respectively 1 L2 M 2 2 3L|M| (C1 − C2 ) − X +X , π ε 2ε 2ε for (60). Obvious computations give (59)–(61), and the estimates of c− and c+ come from (34) and (49). 2
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Remark 3. We do not use the “εk 2 π 2 /L2 ” part of λk , that is in some sense, we do not benefit from the high frequencies. Another possible strategy would be to use this part to absorb the 2 term π2 (C1 − C2 ) Lε , and to treat the low frequencies in another way, for instance by using the “spectral inequality” of Lebeau and Robbiano [13], Lebeau and Zuazua [14], Jerison and Lebeau [11] together with a dissipation estimate. But the constant appearing in this inequality is not explicit, so the constants c− and c+ would not be either. 3.7. Deducing the observability inequality Consider a solution ϕ of (6), where
ϕT (x) =
N
ck ek (x).
(63)
k=1
It is not restrictive to consider ϕT as the combination of a finite number of modes, since the inequalities which follow are independent of N . We see that
ϕ(t, x) =
N
ck exp −λk (T − t) ek (x),
(64)
k=1
and consequently √ π 2k ck = (∂x ϕ)(t, 0)ψk (t) dt. L T
0
Hence we deduce L ∂x ϕ|x=0 L2 (0,T ) ψk L2 (0,T ) . |ck | √ 2πk
(65)
And of course, ϕ(0, x) =
N
ck exp(−λk T )ek (x).
(66)
k=1
From (65) and (66) we deduce ϕ(0, x)
C∂x ϕL2 (0,T ) L2 (0,L)
N 1 k=1
k
exp(−λk T )ek (x)L2 (0,L) ψk L2 (0,T ) .
Now let us distinguish between the two cases M > 0 and M < 0.
(67)
O. Glass / Journal of Functional Analysis 258 (2010) 852–868
867
Case 1. If M > 0, then ek (x)
L2 (0,L)
1.
Hence using (55) and (67), we finally deduce ϕ(0, x) C
L2 (0,L)
N k=1
L|M| 1 1 L˜ 2 Lˆ exp + (C1 − C2 ) − T λk + √ λk ∂x ϕL2 (0,T ) . k 2ε π Tε ε
(68)
Using (58) we deduce ϕ(0, x)
L2 (0,L)
C∂x ϕL2 (0,T )
N k=1
1 κ κ Lˆ − L √ exp − λk . exp − λk + √ λk 2 k 2 ε
It is not difficult to see that for some constant C > 0 independent of ε one has κ κ Lˆ − L L˜ 2 − L2 κ M2 1 − λk + √ + (C1 − C2 ) C − λk C − , 2 π Tε 3 3 4ε ε and that N κ εκπ 2 2 1 exp − λk exp − k k 2 k 2L2
N 1 k=1
k=1
∞ k=1
εκπ 2 exp − k 2L2
C(T , L, M) . ε
This gives the desired result. Case 2. If M < 0, then ek (x)
L2 (0,L)
L|M| . exp 2ε
Hence using (55) and (67), we finally deduce ϕ(0, x)
L2 (0,L)
C
L|M| 1 L˜ 2 Lˆ exp + (C1 − C2 ) − T λk + √ λk ∂x ϕL2 (0,T ) , k ε π Tε ε
N 1 k=1
and we conclude as previously by using (60). This concludes the proof of Theorem 1.
(69)
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Acknowledgments The author would like to warmly thank the anonymous referee for valuable comments on a first version of this paper. He is supported by Grant JCJC06_137283 of the Agence Nationale de la Recherche. References [1] F. Ancona, G.M. Coclite, On the attainable set for Temple class systems with boundary controls, SIAM J. Control Optim. 43 (6) (2005) 2166–2190. [2] F. Ancona, A. Marson, On the attainable set for scalar nonlinear conservation laws with boundary control, SIAM J. Control Optim. 36 (1) (1998) 290–312. [3] A. Beurling, P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962) 291– 309. [4] A. Bressan, G.M. Coclite, On the boundary control of systems of conservation laws, SIAM J. Control Optim. 41 (2) (2002) 607–622. [5] J.-M. Coron, S. Guerrero, Singular optimal control: A linear 1-D parabolic–hyperbolic example, Asymptot. Anal. 44 (3,4) (2005) 237–257. [6] H.O. Fattorini, D.L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal. 43 (1971) 272–292. [7] A. Fursikov, O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser., vol. 34, Seoul National University, Korea, 1996. [8] O. Glass, On the controllability of the 1-D isentropic Euler equation, J. Eur. Math. Soc. (JEMS) 9 (3) (2007) 427– 486. [9] O. Glass, S. Guerrero, On the uniform controllability of the Burgers equation, SIAM J. Control Optim. 46 (4) (2007) 1211–1238. [10] S. Guerrero, G. Lebeau, Singular optimal control for a transport-diffusion equation, Comm. Partial Differential Equations 32 (10–12) (2007) 1813–1836. [11] D. Jerison, G. Lebeau, Nodal sets of sums of eigenfunctions, in: Harmonic Analysis and Partial Differential Equations, Chicago, IL, 1996, in: Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999, pp. 223–239. [12] P. Koosis, The Logarithmic Integral, vols. I, II, Cambridge Stud. Adv. Math., vol. 12, Cambridge University Press, Cambridge, 1988; Cambridge Stud. Adv. Math., vol. 21, Cambridge University Press, Cambridge, 1992. [13] G. Lebeau, L. Robbiano, Contrôle exact de l’équation de la chaleur, Comm. Partial Differential Equations 20 (1995) 335–356. [14] G. Lebeau, E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Ration. Mech. Anal. 141 (4) (1998) 297–329. [15] J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués, Tomes 1, 2, RMA, vols. 8, 9, Masson, Paris, 1988. [16] L. Miller, How violent are fast controls for Schrödinger and plate vibrations? Arch. Ration. Mech. Anal. 172 (3) (2004) 429–456. [17] R.C. Paley, N. Wiener, Fourier Transforms in the Complex Domain, Reprint of the 1934 original, Amer. Math. Soc. Colloq. Publ., vol. 19, American Mathematical Society, Providence, RI, 1987. [18] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev. 20 (1978) 639–739. [19] L. Schwartz, Étude des sommes d’exponentielles, 2ème éd, Publications de l’Institut de Mathématique de l’Université de Strasbourg, V. Actualités Sci. Ind., Hermann, Paris, 1959. [20] T.I. Seidman, Two results on exact boundary control of parabolic equations, Appl. Math. Optim. 11 (2) (1984) 145–152. [21] T.I. Seidman, S.A. Avdonin, S.A. Ivanov, The “window problem” for series of complex exponentials, J. Fourier Anal. Appl. 6 (3) (2000) 233–254. [22] T.I. Seidman, M.S. Gowda, Norm dependence of the coefficient map on the window size, Math. Scand. 73 (1994) 177–189.
Journal of Functional Analysis 258 (2010) 869–892 www.elsevier.com/locate/jfa
Classification of C∗-homomorphisms from C0(0, 1] to a C∗-algebra Leonel Robert ∗ , Luis Santiago The Fields Institute, 222 College Street, Toronto, Canada M5T 3J1 Received 12 May 2009; accepted 23 June 2009 Available online 7 August 2009 Communicated by D. Voiculescu
Abstract A class of C∗ -algebras is described for which the C∗ -homomorphisms from C0 (0, 1] to the algebra may be classified by means of the Cuntz semigroup functor. Examples are given of algebras—simple and nonsimple—for which this classification fails. It is shown that a suitable suspension of the Cuntz semigroup functor deals successfully with some of these counterexamples. © 2009 Elsevier Inc. All rights reserved. Keywords: Classification program; Cuntz semigroup; Positive elements
1. Introduction In this paper we consider the question of classifying the homomorphisms from C0 (0, 1] to a C*-algebra A. In [2], Ciuperca and Elliott showed that if A has stable rank 1 then this classification is possible—up to approximate unitary equivalence—by means of the Cuntz semigroup functor. They defined a pseudometric dW on the morphisms from Cu(C0 (0, 1]) to Cu(A), and showed if A has stable rank 1 then dW (Cu(φ), Cu(ψ)) = 0 for φ, ψ : C0 (0, 1] → A if and only if φ and ψ are approximately unitarily equivalent by unitaries in A∼ (the unitization of A). A classification result in the same spirit as Ciuperca and Elliott’s result is Thomsen’s [10, Theorem 1.2]. Thomsen showed that if X is a locally compact Hausdorff space such that
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E-mail addresses:
[email protected] (L. Robert),
[email protected] (L. Santiago). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.025
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dim X 2 and Hˇ 2 (X) = 0, then the approximate unitary equivalence class of a positive element in Mn (C0 (X)) is determined by its eigenvalue functions. Theorem 1 below applies to a class of C∗ -algebras that contains both the stable rank 1 C∗ -algebras and the C∗ -algebras considered by Thomsen. For this class of algebras the classification of homomorphisms by the functor Cu(·) must be rephrased in terms of stable approximate unitary equivalence. Given φ, ψ : C0 (0, 1] → A we say that φ and ψ are stably approximately unitarily equivalent if there are unitaries un ∈ (A ⊗ K)∼ , n = 1, 2, . . . , such that un φu∗n → ψ pointwise (where A is identified with the top left corner of A ⊗ K). If A is stable or has stable rank 1, then stable approximate unitary equivalence coincides with approximate unitary equivalence, but these relations might differ in general. The following theorem characterizes the C∗ -algebras for which the pseudometric dW (defined in the next section) determines the stable approximate unitary equivalence classes of homomorphism from C0 (0, 1] to the algebra. Theorem 1. Let A be a C∗ -algebra. The following propositions are equivalent. (I) For all x, e ∈ A, with e a positive contraction and ex = xe = x, we have that x ∗ x + e is stably approximately unitarily equivalent to xx ∗ + e. (II) If φ, ψ : C0 (0, 1] → A are such that dW (Cu(φ), Cu(ψ)) = 0 then φ is stably approximately unitarily equivalent to ψ. If (I) and (II) hold then dW (φ, ψ) dU (φ, ψ) 4dW (φ, ψ).
(1)
In (1) dU denotes the distance between the stable unitary orbits of φ(id) and ψ(id), where id ∈ C0 (0, 1] is the identity function. The inequalities (1) are derived in [2] for the stable rank 1 case, though their factor of 8 has now been improved to 4. By the bijective correspondence φ → φ(id) between homomorphisms φ : C0 (0, 1] → A and positive contractions of A the proposition (II) of the previous theorem may be restated as a classification of the stable unitary orbits of positive contractions in terms of the Cuntz equivalence relation of positive elements. The following theorem extends Ciuperca and Elliott’s classification result beyond the stable rank 1 case. Theorem 2. Suppose that (A ⊗ K)∼ has the property (I) of Theorem 1. Let hA ∈ A+ be strictly positive. Then for every α : Cu(C0 (0, 1]) → Cu(A), morphism in the category Cu, with α([id]) [hA ], there is φ : C0 (0, 1] → A, unique up to stable approximate unitary equivalence, such that Cu(φ) = α. The class of algebras that satisfy (I) is closed under the passage to quotients, hereditary subalgebras, and inductive limits (see Proposition 4 below). This class is strictly larger than the class of stable rank 1 C∗ -algebras. Any commutative C∗ -algebra satisfies (I). If X is a locally compact Hausdorff space with dim X 2 and Hˇ 2 (X) = 0 (the Cech cohomology with integer coefficients), then we deduce from [10, Theorem 1.2] that (C0 (X) ⊗ K)∼ satisfies (I) (and so Theorem 2 is applicable to C0 (X) ⊗ K). On the other hand, the C∗ -algebra M2 (C(S 2 )), with S 2 the 2-dimensional sphere, does not satisfy (I). In fact, there exists a pair of homomorphisms
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φ, ψ : C0 (0, 1] → M2 (C(S 2 )) such that Cu(φ) = Cu(ψ) but φ is not stably approximately unitarily equivalent to ψ (see Example 6 below). This phenomenon is not restricted to non-simple AH C∗ -algebras: by a slight variation—to suit our purposes—of the inductive limit systems constructed by Villadsen in [13], we construct a simple, stable, AH C∗ -algebra for which the Cuntz semigroup functor does not classify the homomorphism from C0 (0, 1] into the algebra (see Theorem 7). These counterexamples raise the question of what additional data is necessary to classify, up to stable approximate unitary equivalence, the homomorphisms from C0 (0, 1] to an arbitrary C∗ -algebra. In the last section of this paper we take a step in this direction by proving the following theorem. Theorem 3. Let A be an inductive limit of the form lim −→ C(Xi ) ⊗ K, with Xi compact metric spaces, and dim Xi 2 for all i = 1, 2, . . . . Let φ, ψ : C0 (0, 1] → A be homomorphisms such that Cu(φ ⊗ Id) = Cu(ψ ⊗ Id), where Id : C0 (0, 1] → C0 (0, 1] is the identity homomorphism. Then φ and ψ are approximately unitarily equivalent. 2. Preliminary definitions and results In this section we collect a number of definitions and results that will be used throughout the paper. 2.1. Relations on positive elements Let A be a C∗ -algebra and let a and b be positive elements of A. Let us say that (i) a is Murray–von Neumann equivalent to b if there is x ∈ A such that a = x ∗ x and b = xx ∗ —we denote this by a ∼ b, (ii) a is approximately Murray–von Neumann equivalent to b if there are xn ∈ A, n = 1, 2, . . . , such that xn∗ xn → a and xn xn∗ → b—we denote this by a ∼ap b, (iii) a is stably approximately unitarily equivalent to b if there are unitaries un ∈ (A ⊗ K)∼ , such that u∗n aun → b, where A is identified with the top left corner of A ⊗ K, (iv) a is Cuntz smaller than b if there are dn ∈ A, n = 1, 2, . . . , such that dn∗ bdn → a—we denote this by a Cu b, (v) a is Cuntz equivalent to b if a Cu b and b Cu a, and we denote this by a ∼Cu b. We have (i) ⇒ (ii) ⇒ (v). By [11, Remark 1.8], approximate Murray–von Neumann equivalence is the same as stable approximate unitary equivalence. We will make frequent use of this fact throughout the paper. The relations (i), (ii), and (iii) will also be applied to homomorphisms from C0 (0, 1] to A, via the bijection φ → φ(id) from these homomorphisms into the positive contractions of A. We will make frequent use of the following proposition. Proposition 1. Let a ∈ A+ and x ∈ A be such that a − x ∗ x < ε for some ε > 0. Then there is y such that (a − ε)+ = y ∗ y, yy ∗ xx ∗ , and y − x < Cε 1/2 a . The constant C is universal. Proof. The proof works along the same lines as the proof of [5, Lemma 2.2] (see also [7, Lemma 1]). We briefly sketch the argument here. We have a − ε1 x ∗ x, with ε1 such that a − x ∗ x < ε1 < ε. So (a − ε)+ ex ∗ xe, with e ∈ C ∗ (a) such that e(a − ε1 )e = (a − ε)+ .
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Set xe = x and let x = v| x | be its polar decomposition. Then y = v(a − ε)+ has the properties stated in the proposition. 2 It follows from the previous proposition (or from [5, Lemma 2.2]), that Cuntz comparison can be described in terms of Murray–von Neumann equivalence as follows: a Cu b if and only if for every ε > 0 there is b such that (a − ε)+ ∼ b ∈ Her(b). Here Her(b) denotes the hereditary subalgebra generated by b. We also have the following corollary of Proposition 1. Corollary 1. If a, b ∈ B + , where B is a hereditary subalgebra of A, then a ∼ap b in A if and only if a ∼ap b in B. Proof. If w ∗ w and ww ∗ belong to B for some w ∈ A, then w ∈ B. Thus, if a ∼ b in A then a ∼ b in B. Suppose that a ∼ap b in A. We may assume without loss of generality that a and b are contracand b − xx ∗ < ε. By Proposition 1 there tions. For ε > 0 let x ∈ A be such that a − x ∗ x < ε √ ∗ ∗ exists y such that (a − ε)+ = y y and yy − b C1 ε for some constant C1 . Applying√Proposition 1 again we get that there exists z ∈ A such that (yy ∗ − ε)+ = z∗ z, zz∗ − b C2 4 ε, and and b ∈ B. zz∗ b, for some constant C2 . Set zz∗ = b . We have (a − 2ε)+ ∼ (yy ∗ − ε)+ ∼ b √ ∗
∗
So there is w ∈ B such that (a − 2ε)+ = w w and b = ww . Since b − b C2 4 ε and ε is arbitrary, we get that a ∼ap b in B. 2 2.2. The Cuntz semigroup Let us briefly recall the definition of the (stabilized) Cuntz semigroup in terms of the positive elements of the stabilization of the algebra (see [3] and [8]). Let A be a C∗ -algebra. Given a ∈ (A ⊗ K)+ let us denote by [a] the Cuntz equivalence class of a. The Cuntz semigroup of A is defined as the set of Cuntz equivalence classes of positive elements of A⊗K. This set, denoted by Cu(A), is endowed with the order such that [a] [b] if a Cu b, and the addition operation [a] + [b] := [a + b ], where a and b are mutually orthogonal and Murray–von Neumann equivalent to a and b, respectively. If φ : A → B then Cu(φ) : Cu(A) → Cu(B) is defined by Cu(φ)([a]) := [φ(a)]. Coward, Elliott, and Ivanescu showed in [3] that Cu(·) is a functor from the category of C∗ -algebras to a certain category of ordered semigroups denoted by Cu. In order to describe this category let us first recall the definition of the far below relation. Let S be an ordered set such that the suprema of increasing sequences always exist in S. For x and y in S, let us say that x is far below y, and denote it by x y, if for every increasing sequence (yn ) such that y supn yn , we have x yk for some k. An ordered semigroup S is an object of the Cuntz category Cu if it has a 0 element and satisfies that (1) (2) (3) (4)
if (xn ) is an increasing sequence of elements of S then supn xn exists in S, if (xn ) and (yn ) are increasing sequences in S then supn (xn + yn ) = supn xn + supn yn , for every x ∈ S there is a sequence (xn ) with supremum x and such that xn xn+1 for all n, if x1 , x2 , y1 , y2 ∈ S satisfy x1 y1 and x2 y2 , then x1 + x2 y1 + y2 .
The morphisms of the category Cu are the order preserving semigroup maps that also preserve the suprema of increasing sequences, the far below relation, and the 0 element.
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2.3. The pseudometrics dU and dW Let us identify the C∗ -algebra A with the top left corner of A ⊗ K. Given positive elements a, b ∈ A let us denote by dU (a, b) the distance between the unitary orbits of a and b in A ⊗ K (with the unitaries taken in (A ⊗ K)∼ ). Following Ciuperca and Elliott (see [2]), let us define a pseudometric on the morphisms from Cu(C0 (0, 1]) to Cu(A) as follows: + α([et+r ]) β([et ]), + for all t ∈ R , dW (α, β) := inf r ∈ R β([et+r ]) α([et ]),
(2)
where α, β : Cu(C0 (0, 1]) → Cu(A) are morphisms in the Cuntz category and et is the function et (x) = max(x − t, 0), for x 0. It is easily shown that dW is a pseudometric. Notation convention. All throughout the paper we will use the notations (a − t)+ and et (a) interchangeably. They both mean the positive element obtained evaluating the function et (x) on a given selfadjoint element a. The pseudometric dW may be used to define a pseudometric—that we also denote by dW — on the positive elements of norm at most 1 by setting dW (a, b) := dW (Cu(φ), Cu(ψ)), where φ, ψ : C0 (0, 1] → A are such that φ(id) = a and ψ(id) = b. We have + et+r (a) Cu et (b), + dW (a, b) = inf r ∈ R for all t ∈ R . et+r (b) Cu et (a),
(3)
Notice that (3) makes sense for arbitrary positive elements a and b without assuming that they are contractions. We extend dW to all positive elements using (3). The following lemma relates the metrics dU and dW in a general C∗ -algebra (this is [2, Corollary 9.1]). Lemma 1. For all a, b ∈ A+ we have dW (a, b) dU (a, b) a − b . Proof. Let r be such that a − b < r and choose r1 such that a − b < r1 < r. Then for all t 0 we have a − t − r1 b − t. Multiplying this inequality on the left and the right by e1/2 , where e ∈ C ∗ (a) is such e(a − t − r1 ) = (a − t − r)+ = et+r (a), we get et+r (a) e1/2 (b − t)e1/2 e1/2 (b − t)+ e1/2 Cu et (b), for all t 0. Similarly we deduce that et+r (b) Cu et (a) for all t 0. It follows that dW (a, b) a − b . Since dW is invariant by stable unitary equivalence, dW (a, b) a − ubu∗ for any u unitary in (A ⊗ K)∼ . Hence dW (a, b) dU (a, b). 2 The question of whether dW —as defined in (2)—is a metric is linked to the property of weak cancellation in the Cuntz semigroup. Let us say that a semigroup in the category Cu has weak cancellation if x + z y + z implies x y for elements x, y, and z in the semigroup. It was proven in [2] that if Cu(A) has weak cancellation then dW is a metric on the morphisms from Cu(C0 (0, 1]) to Cu(A). Since this result is not explicitly stated in that paper, we reprove it here.
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Proposition 2. (See [2].) If Cu(A) has weak cancellation then dW is a metric on the Cuntz category morphisms from Cu(C0 (0, 1]) to Cu(A). Proof. By [7, Theorem 1], the map [f ] → (t → rank f (t)) is a well defined isomorphism from Cu(C0 (0, 1]) to the ordered semigroup of lower semicontinuous functions from (0, 1] to N∪{∞}. This isomorphism maps [et ] to 1(t,1] for all t ∈ [0, 1], with 1(t,1] the characteristic function of (t, 1]. Let us identify Cu(C0 (0, 1]) with the semigroup of lower semicontinuous functions from (0, 1] to N ∪ {∞} in this way. Then dW (α, β) = 0 says that α(1(t,1] ) = β(1(t,1] ) for all t. In order to show that α and β are equal it suffices to show that they agree on the functions 1(s,t) (their overall equality then follows by additivity and preservation of suprema of increasing sequences). Let ε > 0. We have α(1(s+ε,t−ε) ) + α(1(t−ε,1] ) α(1(s,1] ) = β(1(s,1] ) β(1(s,t) ) + β(1(t−ε,1] ) = β(1(s,t) ) + α(1(t−ε,1] ). Since A has weak cancellation α(1(s+ε,t−ε) ) β(1(s,t) ). Passing to the supremum over ε > 0 we get that α(1(s,t) ) β(1(s,t) ). By symmetry we also have β(1(s,t) ) α(1(s,t) ). Hence, α(1(s,t) ) = β(1(s,t) ). 2 Rørdam and Winter showed in [9, Theorem 4.3] that if A has stable rank 1 then Cu(A) has weak cancellation. In the next section we will extend this result to the case when the property (I) of Theorem 1 holds in (A ⊗ K)∼ . 3. Proofs of Theorems 1 and 2 3.1. Proof of Theorem 1 In this subsection we prove Theorem 1 of the introduction. For positive elements a, b ∈ A+ we use the notation a b to mean that b is a unit for a, that is to say, ab = ba = a. We start with a lemma. Lemma 2. Let A be a C∗ -algebra such that the property (I) of Theorem 1 holds in A. Let e, f, α, β ∈ A+ be such that e is a contraction, and α e,
α ∼ β f,
and f ∼ f e
for some f ∈ A+ .
Then for every δ > 0 there are α , e ∈ A+ such that α e e,
β + f ∼ α + e ,
and α − α < δ.
Proof. Since f ∼ f there exists x such that f = x ∗ x and xx ∗ = f . Let x = w|x| be the polar decomposition of x in the bidual of A. We have wf w ∗ = f . Set wβw ∗ = α1 . Then α1 ∼ α, α1 e, and α e. Hence α1 + e ∼ap α + e. By Proposition 1 this implies that for every δ > 0 there is z ∈ A such that
L. Robert, L. Santiago / Journal of Functional Analysis 258 (2010) 869–892
(α1 + e − δ )+ = z∗ z, zz∗ α + e, √ ∗ zz − (α + e) < C δ .
and
875
(4) (5)
Let z = w1 |z| be the polar decomposition of z in the bidual of A. Since e is a unit for α1 we have (α1 + e − δ )+ = α1 + (e − δ )+ (we assume δ < 1). It follows that the map c → w1 cw1∗ , sends the elements of Her((e − δ )+ ) into Her(e). By (5) if we let δ → 0 then (zz∗ − 1)+ can be made arbitrarily close to (α + e − 1)+ . Since (zz∗ − 1)+ = w1 (α1 − δ )+ w1∗ and (α + e − 1)+ = α, this means that we can choose δ small enough so that w1 α1 w1∗ − α < δ. Let α = w1 α1 w1∗ , e = w1 f w1∗ , and y = w1 w(β + f )1/2 . Then β + f = y ∗ y and yy ∗ = α + e . 2 Proof of Theorem 1. (II) ⇒ (I). Let φ, ψ : C0 (0, 1] → A be the homomorphism such that φ(id) =
1 (x ∗ x + e) x 2 + 1
and ψ(id) =
1 (xx ∗ + e). x 2 + 1
From the definition of the pseudometric dW we see that dW (Cu(φ), Cu(ψ)) = x 12 +1 dW (x ∗ x + e, xx ∗ + e). In order to prove that x ∗ x + e is stably approximately unitarily equivalent to xx ∗ + e it is enough to show that dW (x ∗ x + e, xx ∗ + e) = 0. That is, (x ∗ x + e − t)+ ∼Cu (xx ∗ + e − t)+ for all t ∈ R. Using that e is a unit for x ∗ x and xx ∗ we deduce that (x ∗ x + e − t)+ = x ∗ x + (e − t)+ ,
(xx ∗ + e − t)+ = xx ∗ + (e − t)+ ,
for 0 t < 1. Also, x ∗ x(e − t)+ = x ∗ x(1 − t) and xx ∗ (e − t)+ = xx ∗ (1 − t). It follows that x ∗ x and xx ∗ belong to the hereditary algebra generated by (e − t)+ . Therefore, (x ∗ x + e − t)+ ∼Cu (e − t)+ ∼Cu (xx ∗ + e − t)+ ,
for 0 t < 1.
If t 1 then (x ∗ x + e − t)+ = (x ∗ x + 1 − t)+ and (xx ∗ + e − t)+ = (xx ∗ + 1 − t)+ . Hence, (x ∗ x + e − t)+ ∼Cu (xx ∗ + e − t)+ for t 1. (I) ⇒ (II). Let us prove that (I) implies the inequalities (1). The proposition (II) clearly follows from this. Notice that the inequality dW dU was already established in Lemma 1. It rests to show that dU 4dW . Let φ, ψ : C0 (0, 1] → A be C∗ -homomorphisms. Set φ(id) = a and ψ(id) = b. Let r be such that dW (a, b) < r. We will show that dU (a, b) < 4r. Let m ∈ N be the number such that mr m+1 1 < (m + 1)r. Let the sequences (ai )m+1 i=1 , (bi )i=1 be defined as ai = ξm−i+1 (a), bi = ξm−i+1 (b) for i = 1, 2, . . . , m + 1, where ξk ∈ C0 (0, 1] is such that 1(kr+ε,1] ξk 1(kr,1] and ε > 0 is chosen small enough so that dW (a, b) + 2ε < r. m+1 The sequences (ai )m+1 i=1 and (bi )i=1 satisfy that ai ai+1 , ai ∼ di bi+1 ,
bi bi+1 ,
for i = 1, . . . , m,
bi ∼ ci ai+1 ,
and
for i = 1, . . . , m,
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for some positive elements ci and di . The first line follows trivially from the definition of the elements ai and bi . Let us prove the second line. From dW (a, b) < r − 2ε we get e(m−i+1)r−ε (a) Cu e(m−i)r+ε (b) bi+1 . By the definition of Cuntz comparison there exists d ∈ A+ such that e(m−i+1)r (a) ∼ d bi+1 . Since ai is expressible by functional calculus as a function of e(m−i+1)r (a), we get that there exists di ∈ A+ such that ai ∼ di bi+1 . We reason similarly to get the existence of ci . Let us now show by induction on n, for n = 1, 2, . . . , m, that there are sequences of elements (ai )ni=1 and (bi )ni=1 such that
ai ai+1 , bi bi+1 , for i = 1, 2, . . . , n − 1, ai − a < ε, for i odd, i n, i bi − b < ε, for i even, i n, i n i=1
ai ∼
n
bi ,
(6) (7) (8) (9)
i=1
and an = an , bn bn+1 if n is odd, and bn = bn , an an+1 if n is even. Since a1 ∼ d1 b2 , the induction hypothesis holds for n = 1 taking b1 = d1 . Suppose the induction holds for n and let us show that it also holds for n + 1. Let us consider the case that
n is odd (the case that n is even is dealt with similarly). We set bn+1 = bn+1 and leave the n
)n sequence (bi )i=1 unchanged. We are going to modify the sequence (a i i=1 in order to complete
the induction step. Set ni=1 ai = α, an+2 = e, ni=1 bi = β, and bn+1 = f . Then the conditions of the previous lemma apply. We thus have that for every δ > 0 there are α and e , such that α e an+2 ,
α − α < δ,
and β + f ∼ α + e .
. We remark that the elements a are It follows that β ∼ α , and so α = ni=1 ai
, with ai
ai+1 i
∗ all in the C -algebra generated by α and the elements ai are in the C∗ -algebra generated by α . In fact,
α − (n − i) + − α − (n − i + 1) + = ai ,
α − (n − i) + − α − (n − i + 1) + = ai
.
(10) (11)
Therefore, we may choose the number δ sufficiently small so that ai − ai
< ε for all i n.
We now rename the sequence (ai
)ni=1 as (ai )ni=1 and set an+1 = e . From β + f ∼ α + e we get n+1 n+1
that i=1 bi ∼ i=1 ai . This completes the induction.
m Continuing the induction up to n = m we find (ai )m i=1 and (bi )i=1 that satisfy (6)–(9). For the last part of the proof we split the analysis in to cases, m even and m odd. Suppose that m = 2k + 1. We have 2k+1
2k+1
ai
b ∼ i=1 i . 2k + 1 2k + 1 i=1
(12)
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Let a denote the sum on the left side of the last equation, and b the sum on the right. Let us
for all i and ai 1 for show that a − a < 2r + 2ε and b − b < 2r + 2ε. Since ai ai+1
for all i. Hence, all i, we have ai ai+1 2
k
i=1 a2i−1
a1 + 2 ki=1 a2i+1 ai
. 2k + 1 2k + 1 2k+1
+ a2k+1
i=1
2k + 1
Using that ai − ai < ε for i odd in the above inequalities we obtain 2
2k+1
a1 + 2 ki=1 a2i+1 + a2k+1 i=1 ai −ε + ε. 2k + 1 2k + 1 2k + 1
k
i=1 a2i−1
It follows now from the inequalities 2
k
i=1 ξ2i−1 (t) + ξ2k+1 (t)
2k + 1
t + 2r + ε,
t − 2r − ε
ξ1 (t) + 2 ki=1 ξ2i+1 (t) , 2k + 1
that 2k+1 a − 2r − 2ε
ai
a + 2r + 2ε. 2k + 1 i=1
Therefore a − a < 2r + 2ε.
for i = 1, 2, . . . , 2k, that b2k+1 Let us show that b − b < 2r + 2ε. Using that bi bi+1
b2k+2 , and that bi − bi < ε for all i even, we obtain the inequalities 2
2k+1
k
i=1 b2i
2k + 1
−ε
bi
2 2k + 1 i=1
k
+ b2k+2 + ε. 2k + 1
i=1 b2i
It follows from the estimates 2
k
1 ξ2i (t)
2k + 1
t − 2r − ε,
ξ0 (t) + 2 k1 ξ2i (t) t + 2r + ε, 2k + 1
that 2k+1 b − 2r − 2ε
bi
b + 2r + 2ε. 2k + 1 i=1
Hence b − b < 2r + 2ε. We have found a , b ∈ A+ such that a ∼ b , a − a < 2r + 2ε and b − b < 2r + 2ε. arbitrary the desired result follows. Therefore dU (a, b) 4r + 4ε. Since ε > 0 is 2k 2k 1
and b = 1
a For the case that m = 2k we take a = 2k i=1 i i=1 bi , and we reason simi2k
larly to how we did in the odd case to obtain that a − a < 2r + 2ε and b − b < 2r + 2ε. 2 Corollary 2. Let A be a C∗ -algebra with the property (I) of Theorem 1. The following propositions hold true:
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(i) If a and b are positive elements of A such that dW (a, b) < r, then for all ε > 0 there exists b ∈ A+ such that a − b < 4r and dU (b, b ) < ε. (ii) The set of positive elements of A is complete with respect to the pseudometric dU . Proof. (i) We may assume without loss of generality that a and b are contractions. We may also assume that A is σ -unital by passing to the subalgebra Her(a, b) if necessary (the property (I) holds for hereditary subalgebras by Proposition 4(iii)). Let c ∈ A+ be strictly positive. By the continuity of the pseudometrics dU and dW (see Lemma 1), it is enough to prove the desired proposition assuming that a and b belong to a dense subset of A+ . Thus, we may assume that a, b ∈ Her((c − δ)+ ) for some δ > 0. From dW (a, b) < r and the proof of Theorem 1 we get that there is x ∈ Her((c − δ)+ ) such that a − x ∗ x < 2r
and b − xx ∗ < 2r.
Let e ∈ A+ be a positive contraction that is a unit for the subalgebra Her((c − δ)+ ). Then x ∗ x + e ∼ap xx ∗ + e. This implies that for all ε > 0 there is a unitary u in (A ⊗ K)∼ such that u∗ eu − e < ε
and u∗ x ∗ xu − xx ∗ < ε.
Set eubu∗ e = b . If we take ε small enough such that a − x ∗ x < 2r − ε
and b − xx ∗ < 2r − ε,
then we have the following estimates: a − b a − ubu∗ < 4r − 2ε + uxx ∗ u∗ − x ∗ x < 4r, u∗ b u − b u∗ eubu∗ eu − ebu∗ eu + bu∗ eu − be < 2ε. This proves (i). (ii) Let (ci )∞ i=1 be a sequence of positive elements of A that is Cauchy with respect to the pseudometric dU . In order to show that (ci )∞ i=1 converges it is enough to show that it has a convergent subsequence. We may assume, by passing to a subsequence if necessary, that dU (ci , ci+1 ) < 21i for all i 1. Using mathematical induction we will construct a new sequence (ci )∞ i=1 such that
< 1 and d (c , c ) < 1 for all i. ci − ci+1 U i i 2i 2i−3 For n = 1 we set c1 = c1 . Suppose that we have constructed ci , for i = 1, 2, . . . , n, and let us
. We have dU (cn+1 , cn ) < 21n and dU (cn , cn ) < 21n (by the induction hypothesis). construct cn+1 1 1 , and so dW (cn , cn+1 ) < 2n−1 (by Lemma 1). Applying part (i) of the Hence dU (cn , cn+1 ) < 2n−1 1
and corollary to a = cn and b = cn+1 , we find a positive element d such that cn − d < 2n−3 1
dU (cn+1 , d) < 2n+1 . Setting cn+1 = d completes the induction.
< 1 for all i, the sequence (c )∞ is a Cauchy sequence with respect Since ci − ci+1 i i=1 2i−3 to the norm of A. Hence, it converges to an element c ∈ A+ . Since dU (ci , ci ) < 21i for all i, dU (ci , c) dU (ci , ci ) + dU (ci , c) → 0. That is, (ci )∞ i=1 converges to c in the pseudometric dU . Thus, A+ is complete with respect to dU . 2
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3.2. Approximate existence theorem Let A be a C∗ -algebra and hA a strictly positive element of A. The main result of this subsection, Theorem 4 below, states that every morphism α : Cu(C0 (0, 1]) → Cu(A) in the category Cu such that α([id]) [hA ], may be approximated in the pseudometric dW by a morphism of the form Cu(φ), with φ : C0 (0, 1] → A a C∗ -algebra homomorphism. Lemma 3. Let A be a C∗ -algebra. The following propositions hold true: (i) If a and b are two positive elements of A such that a Cu b, then for every ε > 0 there is b ∈ M2 (A)+ such that b ∼Cu b and a 0
0 0
− b < ε.
(ii) If a and b are two positive elements of A ⊗ K such that a Cu b then for every ε > 0 there exists b ∈ (A ⊗ K)+ such that b ∼Cu b and a − b < ε. Proof. (i) Let ε > 0 be given. Since a Cu b, by [5, Lemma 2.2] there exists d ∈ A such that 1 1 (a − ε/2)+ = d ∗ bd. Consider the vector c = (b 2 d, δb 2 ), where δ > 0. Then ∗
1 2
∗
1 2
cc = b dd b + δ b 2
∗
and c c =
(a − ε/2)+ δbd
δd ∗ b . δ2b
We may choose δ small enough such that a 0
0 0
− c∗ c < ε.
Since δ 2 b cc∗ (δ 2 + d 2 )b, we have cc∗ ∼Cu b. Thus, the desired result follows letting b = c∗ c. (ii) We may assume without loss of generality that A is stable. This implies that for every b ∈ (A ⊗ K)+ there is b ∈ A+ that is Murray–von Neumann equivalent to b, where A is being identified with the top corner of A ⊗ K. Thus, we may assume without loss of generality that b ∈ A+ . Every positive element a in (A ⊗ K)+ is approximated by the elements pn apn ∈ Mn (A) (with pn the unit of Mn (A∼ )). Therefore, we may also assume without loss of generality that a ∈ Mn (A) for some n. So we have a, b ∈ Mn (A)+ for some n. Now the existence of b ∈ M2n (A)+ with the desired properties is guaranteed by part (i) of the lemma. 2 Lemma 4. Let A be a C∗ -algebra and let (xk )nk=0 be elements of Cu(A) such that xk+1 xk for all k. Then there exists a ∈ (A ⊗ K)+ , with a 1, such that [a] = x0 and xk+1 [(a − k/n)+ ] xk for k = 1, . . . , n − 1. Proof. Let ε > 0. Let an ∈ (A ⊗ K)+ be such that [an ] = xn and an ε. Repeatedly applying
Lemma 3(ii), we can find positive elements (ak )n−1 i=0 such that [ak ] = xk and ak − ak+1 < ε for
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k = 1, . . . , n − 1. For all k we have a0 − ak < kε. It follows from Lemma 1 that dW (a0 , ak ) < kε. Hence,
ak − 2kε + Cu a0 − kε + Cu ak . Since xk+1 xk for all k, we can choose ε small enough such that
x0 = a0 x1 a0 − ε + x2 a0 − 2ε + · · ·
a0 − (n − 1)ε + xn . Set a0 /(nε) = a. Then [(a0 − kε)+ ] = [(a − k/n)+ ] for all k. The lemma now follows by noticing that an ε and a0 − an < (n − 1)ε imply that a 1. 2 Theorem 4. Let A be a C∗ -algebra and let hA be a strictly positive element of A. Let α : Cu(C0 (0, 1]) → Cu(A) be a morphism in Cu such that α([id]) [hA ]. Then for every ε > 0 there exists φ : C0 (0, 1] → A such that dW (Cu(φ), α) < ε. Proof. Let ε > 0 be given and let n be such that 1/2n−1 < ε. Set α([et ]) = xt for t ∈ [0, 1]. By Lemma 4, we can find a ∈ (A ⊗ K)+ such that a 1, [a] = x0 , and x(k+1)/2n
a − k/2n + xk/2n
(13)
for k = 1, . . . , 2n − 1. Let δ > 0 be such that (13) still holds after replacing a by (a − δ)+ . This is possible since
a − k/2n + = sup a − δ − k/2n + . δ>0
We have [a] = α([id]) [hA ]. By [5, Lemma 2.2], there exists d ∈ A ⊗ K such that (a − δ)+ = 1/2 1/2 dhA d ∗ . Set hA d ∗ dhA = a . Then a is in A+ and is Murray–von Neumann equivalent to (a − δ)+ . It follows that (a − t)+ is Murray–von Neumann equivalent to (a − δ − t)+ for all t ∈ [0, 1]. Therefore, [(a − k/2n )+ ] = [(a − δ − k/2n )+ ] for k = 1, . . . , 2n − 1. So we have found a positive element a in A+ such that x(k+1)/2n
a − k/2n + xk/2n
for k = 1, . . . , 2n − 1. Notice also that a = (a − δ)+ < 1. Let φ : C0 (0, 1] → A be such that φ(id) = a . Then
Cu(φ) [ek/2n ] α [ek/2n ]
and α [e(k+1)/2n ] Cu(φ) [ek/2n ] .
Any interval of length 1/2n−1 contains an interval of the form (k/2n , (k + 1)/2n ) for some k. Thus, for every t ∈ [0, 1] there exists k such that (k/2n , (k + 1)/2n ) ⊆ (t, t + 1/2n−1 ). It follows that
Cu(φ) [et+1/2n−1 ] Cu(φ) [ek/2n ] α [ek/2n ] α [et ]
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and
α [et+1/2n−1 ] α [e(k+1)/2n ] Cu(φ) [ek/2n ] Cu(φ) [et ] . These inequalities imply that dW (Cu(φ), α) 1/2n−1 < ε.
2
3.3. Weak cancellation in Cu(A) Proposition 3. Suppose that (A ⊗ K)∼ has the property (I) of Theorem 1. Then Cu(A) has weak cancellation. Proof. Suppose that [a] + [c] [b] + [c] for [a], [b], and [c] in Cu(A). Let us choose a, b, and c, such that ac = bc = 0. Taking supremum over δ > 0 in [(b − δ)+ ] + [(c − δ)+ ] we get that [a] + [c] [(b − δ)+ ] + [(c − δ)+ ] for some δ > 0. Hence, for every ε > 0 there are a1 and c1 in (A ⊗ K)+ such that
a1 + c1 ∈ Her (b − δ)+ + (c − δ)+ , a1 ∼ (a − ε)+ ,
c1 ∼ (c − ε)+ ,
and a1 c1 = 0.
We assume that ε < δ/2. Let us show that a1 is Cuntz smaller than b. Let g ∈ C0 (0, 1] be such that 0 g(t) 1, g(t) = 1 for t δ − ε and g(t) = 0 for t δ/2. Then g((c − ε)+ ) + g(b) is a unit for a1 and c1 . We have g(c1 ) ∼ g((c − ε)+ ). Let x be such that g(c1 ) = xx ∗ and g((c − ε)+ ) = x ∗ x. From (g(b) + x ∗ x)xx ∗ = xx ∗ we deduce that (1 − (g(b) + x ∗ x))x = 0. Let w ∈ (A ⊗ K)∼ be given by w=x+
1 − g(b) + x ∗ x .
We have w ∗ w = 1 − g(b). From a1 g(c1 ) = 0 and g(c1 ) = xx ∗ we get that a1 x = 0. Also ˜ Since a1 (1 − (g(b) + x ∗ x)) = 0. Hence a1 w = 0. Let b˜ ∈ (A ⊗ K)+ be given by ww ∗ = 1 − b. ∼ ∗ ∗ we have assumed that the property (I) holds in (A ⊗ K) , we have w w + 1 ∼ap ww + 1. From ˜ So b˜ ∼Cu g(b) Cu b. On the other hand, this we deduce 1 − w ∗ w ∼ap 1 − ww ∗ , i.e., g(b) ∼ap b. ˜ Hence a1 Cu b˜ b. ˜ from a1 w = 0 we deduce that a1 b = a1 , and so a1 a1 b. We have shown that [(a − ε)+ ] = [a1 ] [b] for all ε > 0. Letting ε → 0 we get [a] [b] as desired. 2 It would be desirable to relax the hypothesis of the previous proposition to the case that A ⊗ K has the property (I). However, we have not succeeded in proving this. The proof given above (and also the proof give in [9] for the stable rank 1 case) can be adapted to the following hypotheses: A ⊗ K has property (I) and contains a full projection. 3.4. Proof of Theorem 2 Proof of Theorem 2. The uniqueness of the homomorphism φ is clear by Theorem 1. Let us prove its existence. By Theorem 4, for every n there exists φn : C0 (0, 1] → A such that
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dW (Cu(φn ), α) < 1/2n+2 . It follows from Theorem 1 that
dU φn (id), φn+1 (id) 4dW Cu(φn ), Cu(φn+1 ) < 1/2n . This implies that (φn (id))n is a Cauchy sequence with respect to the pseudometric dU . By Corollary 2(ii), A+ is complete with respect to dU . Hence, there exists φ : C0 (0, 1] → A such that dU (φ(id), φn (id)) → 0. We have,
dW Cu(φ), α dW Cu(φ), Cu(φn ) + dW Cu(φn ), α
dU Cu(φ), Cu(φn ) + dW Cu(φn ), α → 0. So dW (Cu(φ), α) = 0. By Propositions 2 and 3, dW is a metric. Therefore Cu(φ) = α.
2
4. Examples and counterexamples 4.1. Algebras with the property (I) The following proposition provides us with examples of C∗ -algebras with the property (I) of Theorem 1. Proposition 4. The following propositions hold true. (i) If A is a C∗ -algebra of stable rank 1 then (I) holds in A. (ii) If X is a locally compact Hausdorff space such that dim X 2 and Hˇ 2 (X) = 0, then (I) holds in (C0 (X) ⊗ K)∼ . (iii) If (I) holds in A it also holds in every hereditary subalgebra and every quotient of A. ∗ (iv) If A ∼ = lim −→ Ai and (I) holds in the C -algebras Ai then it also holds in A. Proof. (i) Let x, e ∈ A be as in Theorem 1(I). Let B be the smallest hereditary subalgebra of A containing x ∗ x and xx ∗ . Then B has stable rank 1, and e is a unit for B. It is well known that in a C∗ -algebra of stable rank 1 Murray–von Neumann equivalent positive elements are approximately unitarily equivalent in the unitization of the algebra. Therefore, there are unitaries un ∈ B ∼ , n = 1, 2, . . . , such that u∗n x ∗ xun → xx ∗ . We also have u∗n eun = e for all n, since e is a unit for B. Hence u∗n (x ∗ x + e)un → xx ∗ + e, as desired. (ii) Let x, e ∈ (C0 (X) ⊗ K)∼ be as in Theorem 1(I). For every t ∈ X the operators ∗ x (t)x(t) + e(t) and x(t)x ∗ (t) + e(t), in K∼ , are approximately unitarily equivalent, since K∼ has stable rank 1. Let us denote by λ ∈ R the scalar such that x ∗ x + e − λ · 1 ∈ C0 (X) ⊗ K and xx ∗ + e − λ · 1 ∈ C0 (X) ⊗ K. Then the selfadjoint elements x ∗ x + e − λ · 1 and xx ∗ + e − λ · 1 have the same eigenvalues for any point t ∈ X, and so by Thomsen’s [10, Theorem 1.2] they are approximately unitarily equivalent in C0 (X) ⊗ K. (Thomsen’s result is stated for selfadjoint elements of C0 (X) ⊗ Mn , but it easily extends to selfadjoint elements of C0 (X) ⊗ K.) It follows that x ∗ x + e and xx ∗ + e are approximately unitarily equivalent in (C0 (X) ⊗ K)∼ . (iii) The property (I) passes to hereditary subalgebras because approximate Murray–von Neumann equivalence does (by Corollary 1). In order to consider quotients by closed two-sided ideals we first make the following claim: for every ε > 0 there is δ > 0 such that if x(1 − e) < δ and (1 − e)x < δ, with e a positive
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contraction, then dW (x ∗ x + e, xx ∗ + e) < ε. In order to prove this we notice that the inequality dW (x ∗ x + e, xx ∗ + e) < ε is implied by a finite set of relations of Cuntz comparison on positive elements obtained by functional calculus on x ∗ x + e and xx ∗ + e (see the proofs of Theorem 4 and Lemma 5(ii)). Using the continuity of the functional calculus, the argument used in the implication (II) ⇒ (I) of Theorem 1 can still be carried out, approximately, to obtain this finite set of Cuntz comparisons. Let us suppose that the algebra A has the property (I). Let x, e ∈ A/I be elements in a quotient of A such that ex = xe = x, and e is a positive contraction. Let x, ˜ e˜ ∈ A be lifts of x and e, with e˜ a positive contraction. Let (iλ ) be an approximate identity of I . Let e˜λ ∈ A be the positive ˜ 1/2 (1 − iλ )(1 − e) ˜ 1/2 . Then e˜λ is a lift of e for all λ, and contraction defined by 1 − e˜λ = (1 − e) ˜ x(1 ˜ − e˜λ ) → 0. Thus, we can find lifts x˜ and e˜λ of x and e, such that (1 − e˜λ )x ˜ <δ (1 − e˜λ )x, and x(1 ˜ − e˜λ ) < δ for any given δ > 0. By the claim made in the previous paragraph we can ˜ x˜ ∗ x˜ + e) ˜ < ε, for any given ε > 0. Since A has the property (I), choose δ such that dW (x˜ ∗ x˜ + e, ˜ x˜ ∗ x˜ + e) ˜ < 4ε. Passing to the quotient by I we get we have by Theorem 1 that dU (x˜ ∗ x˜ + e, ∗ ∗ dU (x x + e, x x + e) < 4ε, and since ε is arbitrary we are done. (iii) Let x, e ∈ A be as in Theorem 1(I). We may approximate these elements by the images of elements x , e ∈ An , with e a positive contraction, within an arbitrary degree of proximity. By possibly moving the elements x and e further along the inductive limit, we may assume that e is approximately a unit for x . We can then use the claim established in the proof of (ii) to get that dW ((x )∗ x + e , x (x )∗ + e ) can be made arbitrarily small (choosing x and e suitably). Since An has the property (I), we have that dU ((x )∗ x + e , x (x )∗ + e ) can be arbitrarily small. Going back to the limit algebra this implies that dU (x ∗ x + e, xx ∗ + e) is arbitrarily small, and so it is 0. 2 Example 5. Let D denote the unit disc in R2 and U its interior. Let B ⊆ M2 (D) be the hereditary subalgebra
C(D) C0 (U )
C0 (U ) . C0 (U )
By Propositions 4(ii) and (iv), (I) holds in B. Thus, the Cuntz semigroup functor classifies the homomorphisms from C0 (0, 1] to B up to stable approximate unitary equivalence. Let us show that, unlike the case of stable rank 1 algebras, stable approximate unitary equivalence and
approximate unitary equivalence do not agree in B. Let p ∈ B be the rank 1 projection 10 00 and let q ∈ B be a rank 1 projection that agrees with p on the boundary of D, and such that the projection induced by 1 − q in D/∼, the disc with the boundary points identified, is non-trivial. Then p and q are Murray–von Neumann equivalent projections, and so they are stably unitary equivalent. However, if there were u ∈ B ∼ unitary such that u∗ pu = q, then the partial isometry v = u∗ (1 − p) would be constant on T and such that v ∗ v = 1 − q and vv ∗ = 1 − p is trivial. This would contradict the non-triviality of 1 − q in D/∼. Examples of C∗ -algebras that do not have the property (I) are not hard to come by. If a unital A has (I), then for any two projections p and q in A such that p ∼ q, we have that p + 1 ∼ap q + 1 by (I). From this we deduce by functional calculus on p + 1 and q + 1 that 1 − p ∼ 1 − q. Thus, any unital C∗ -algebra with Murray–von Neumann equivalent projections that are not unitarily equivalent does not have (I). In particular, the algebra B ∼ , with B as in the previous example, does not have (I). C∗ -algebra
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4.2. The isometry question The following question was posed to us by Andrew Toms: if A has stable rank 1, is it true that dW = dU ? We formulate this question here for the algebras covered by Theorem 1. Question. Suppose that A has the property (I) of Theorem 1. Is it true that dW = dU ? We do not know the answer to this question, even in the case of stable rank 1 algebras. Proposition 5 below provides some evidence that the answer is yes. ∗ ∗ Lemma 5. Let A = lim −→(Ai , φi,j ) be the C -algebra inductive limit of the sequence of C + ∞ algebras (Ai )i=1 with connecting homomorphisms φi,j : Ai → Aj . Let a, b ∈ Ak for some k. Then A
(i) dU i (ai , bi ) → dUA (a∞ , b∞ ) as i → ∞, and Ai A (a , b ) as i → ∞, (ai , bi ) → dW (ii) dW ∞ ∞ where ai and bi denote the images of a and b by the homomorphism φk,i , for i = k + 1, k + 2, . . . , ∞. A
Proof. (i) We clearly have dUAn (an , bn ) dU n+1 (an+1 , bn+1 ) dUA (a∞ , b∞ ) for all n 1. Therefore, it is enough to show that for every ε > 0 there is n such that dUAn (an , bn ) dUA (a∞ , b∞ ) + ε. Let us denote dUA (a∞ , b∞ ) by r and let ε > 0. Let u ∈ (A ⊗ K)∼ be a unitary such that
∼ ua∞ u∗ − b∞ < ε + r. Since A ⊗ K = lim −→ Ai ⊗ K, there are n and a unitary u ∈ (An ⊗ K) such that u an (u )∗ − bn < ε + r. Hence dUAn (an , bn ) dUA (a∞ , b∞ ) + ε. An (an , bn ) (ii) We may assume without loss of generality that k = 1. As before, we have dW A A (a , b ) for all n 1. Thus, we need to show that for every ε > 0 there dWn+1 (an+1 , bn+1 ) dW ∞ ∞ An A (a , b ) + ε. (an , bn ) dW is n such that dW ∞ ∞ Let us denote dW (a∞ , b∞ ) by r and let ε > 0. Let us choose a grid of points {ti }m i=1 in (0, 1] such that ti < ti+1 and |ti − ti+1 | < ε for i = 1, . . . , m − 1 (e.g., choose m 1/ε and ti = i/m for i = 1, . . . , m). From the Cuntz inequality eti +r+ε/4 (a∞ ) Cu eti (b∞ ) and [5, Lemma 2.2], we deduce that there exists di ∈ A such that eti +r+ε/2 (a∞ ) = di et (b∞ )di∗ . Since A is the inductive limit of the C∗ -algebras An , we can find n and di ∈ An such that et +r+ε/2 (an ) − d et (bn ) d ∗ < ε/2. i i i i By [5, Lemma 2.2] applied in the algebra An , we have that eti +r+ε (an ) Cu eti (bn ) in An . Let us choose a value of n such that this inequality holds in An for all i = 1, 2, . . . , m − 1, and such that we also have eti +r+ε (bn ) Cu eti (an ) for all i = 1, 2, . . . , m − 1. Let t ∈ [0, 1]. Let i be the smallest integer such that t ti . Then [ti , ti +r +ε] ⊆ [t, t +r +2ε]. We have the following inequalities in An : et+r+2ε (an ) Cu eti +r+ε (an ) Cu eti (bn ) Cu et (bn ). An (an , bn ) r + 2ε. The same inequalities hold after interchanging an and bn . Thus, dW
2
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Proposition 5. Let A be such that A ⊗ K is an inductive limit of algebras of the form C(Xi ) ⊗ K, with dim Xi 2, Hˇ 2 (Xi ) = 0. Then the pseudometrics dU and dW agree on the positive elements of A. Proof. We may assume without loss of generality that A is stable. Let A = lim −→(C0 (Xi ) ⊗ K, φi,i+1 ). Since both dU and dW are continuous (by Lemma 1), it is enough to show that they are equal on a dense subset of A+ . Thus, we may assume that a and b belong to the image in A of some algebra C0 (Xi ) ⊗ K. Furthermore, in order to show that dU (a, b) = dW (a, b), it is enough to show, by Lemma 5, that this equality holds on all the algebras C0 (Xj ) ⊗ K, with j i. Thus, we may assume that the algebra A is itself of the form C0 (X) ⊗ K, with dim X 2 and Hˇ 2 (X) = 0. Finally, since ∞ n=1 Mn (C0 (X)) is dense in C0 (X) ⊗ K, we may assume that a, b ∈ Mn (C0 (X)) for some n ∈ N. So let a, b ∈ Mn (C0 (X)) be positive elements. Set dW (a, b) = r. Then for every x ∈ X we have dW (a(x), b(x)) r, where dW is now taken in the C∗ -algebra Mn (C). From the definition of dW we see that this means that for every t > 0, the number of eigenvalues of a(x) that are less than t is less than the number of eigenvalues of b(x) that are less than t + r, and vice-versa, the number of eigenvalues of b(x) less than t, is less than the number of eigenvalues of a(x) less than t + r. By the Marriage Lemma this means that the eigenvalues of a(x) and b(x) may be matched in such a way that the distance between the paired eigenvalues is always less than r. We then have that dU (a, b) < r by [10, Theorem 1.2]. 2 4.3. Counterexamples The counterexamples of this subsection are C∗ -algebras that not only do not have the property (I), but moreover the Cuntz semigroup functor does not distinguish the stable approximate unitary classes of homomorphisms from C0 (0, 1] to the algebra. Example 6. Let S 2 denote the 2-dimensional sphere. Let us show that there are homomorphisms φ, ψ : C0 (0, 1] → M2 (C(S 2 )) such that Cu(φ) = Cu(ψ) but φ is not stably approximately unitarily equivalent to ψ . Let λ1 and λ2 be continuous functions from S 2 to [0, 1] such that λ1 > λ2 , min λ2 = 0, and min λ1 max λ2 . Let P and E be rank one projections in M2 (C(S 2 )) such that E is trivial and P is non-trivial. Consider the positive elements a = λ1 P + λ2 (12 − P )
and b = λ1 E + λ2 (12 − E),
where 12 denotes the unit of M2 (C(S 2 )). Let us show that for every non-zero function f ∈ C0 (0, 1] we have f (a) ∼ f (b). In view of the computation of the Cuntz semigroup of S 2 obtained in [7, Theorem 2], it is enough to show that the rank functions of f (a) and f (b) are equal and non-constant. We have f (a) = f (λ1 )P +f (λ2 )(1−P ) and f (b) = f (λ1 )E +f (λ2 )(1−E). It is easily verified that the rank functions of f (a) and f (b) are both equal to 1U + 1V , where U = {x | f (λ1 (x)) = 0}, V = {x | f (λ2 (x)) = 0}, and 1U and 1V denote the characteristic functions of U and V . Since min λ2 = 0, the open set V is a proper subset of S 2 . So if V is non-empty, then the function 1U + 1V is non-constant. On the other hand, if V is empty, then f is 0 on the interval [0, max λ2 ]; in particular, f (min λ1 ) = 0. Thus, U is a proper subset of S 2 in this case, and so 1U + 1V is again non-constant.
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Let φ, ψ : C0 (0, 1] → M2 (C(S 2 )) be the homomorphisms such that φ(id) = a and ψ(id) = b. It follows from the discussion in the previous paragraph that Cu(φ) = Cu(ψ). Let us show that φ and ψ are not stably approximately unitarily equivalent. Let t = max( λλ12 ) and r > 0. Then et
a λ1
= (1 − t)P
and et+r
b λ1
= (1 − t − r)E.
In order that et+r (b/λ1 ) be Cuntz smaller than et (a/λ1 ) the value of r must be at least 1 − t. Thus, dW ( λa1 , λb1 ) 1 − t. Hence λa1 ap λb1 , and so a ap b. It follows that φ and ψ are not stably approximately unitarily equivalent. Next we construct a simple AH C∗ -algebra for which the Cuntz semigroup functor does not classify the homomorphisms from C0 (0, 1] into the algebra. Let us recall the definition given in [13] of a diagonal homomorphism from C(X) ⊗ K to C(Y ) ⊗ K (here X and Y are compact Hausdorff spaces). Let (pi )ni=1 be mutually orthogonal projections in C(Y ) ⊗ K and let λi : Y → X, i = 1, 2, . . . , n, be continuous maps. Let us define a homomorphism φ : C(X) → C(Y ) ⊗ K by φ(f ) =
n (f ◦ λi )pi . i=1
The homomorphism φ gives rise to a homomorphism φ˜ from C(X) ⊗ K to C(Y ) ⊗ K as follows: φ˜ is the composition of φ ⊗ id : C(X) ⊗ K → C(Y ) ⊗ K ⊗ K with id ⊗ α : C(Y ) ⊗ K ⊗ K → C(Y ) ⊗ K, where α is some isomorphism map from K ⊗ K to K. A homomorphism φ˜ obtained in this way is said to be a diagonal homomorphism arising from the data (pi , λi )ni=1 (the choice ˜ of α does not change the approximate unitary equivalence class of φ). Theorem 7. There exist a simple stable AH C∗ -algebra A, and homomorphisms φ, ψ : C0 (0, 1] → A, such that Cu(φ) = Cu(ψ) but φ and ψ are not approximately unitarily equivalent. Proof. Let us define the sequence of topological spaces (Xi )∞ i=1 by X1 = CP(1) and Xi+1 = Xi × CP(ni ), where ni = 2 · (i + 1)! and CP(n) denotes the complex projective space of dimension 2n. For every n let us denote by ηn the rank one projection in C(CP(n)) ⊗ K associated to the canonical line bundle of CP(n). For every i let πi : Xi+1 → Xi denote the projection map onto Xi . Let φ˜ i : C(Xi ) ⊗ K → C(Xi+1 ⊗ K) denote the diagonal homomorphism given by j j the data (1, πi ) ∪ (ηni , δy j )ij =1 , where (ηni )ij =1 are mutually orthogonal projections all Murray– i
j
von Neumann equivalent to ηni , and δy j : Xi+1 → Xi is the constant map equal to yi ∈ Xi for i
j
j = 1, 2, . . . , i. It is possible, and well known, to choose the points yi in such a way that the ∗ inductive limit A = lim −→(C(Xi ) ⊗ K, φi ) is a simple C -algebra (see [13]). Let us show that this inductive limit A provides us with the desired example. Let a, b ∈ C(X1 ) ⊗ K be the two positive elements constructed in Example 6 (notice that X1 is homeomorphic to S 2 ) and φa , ψb : C0 (0, 1] → C(X1 ) ⊗ K the homomorphisms associated to them. Set φ1,i (a) = ai and φ1,i (b) = bi for i = 2, 3, . . . , ∞. For i = 2, . . . , ∞, let us denote
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by φai and ψbi the homomorphisms from C0 (0, 1] to Ai associated to the positive elements ai and bi . Since Cu(φa ) = Cu(ψb ), we have Cu(φa∞ ) = Cu(ψb∞ ). Let us show that φa∞ and ψb∞ are not approximately unitarily equivalent. Equivalently, let us show that dU (a∞ , b∞ ) > 0. By Lemma 5(i) it suffices to show that dU (an , bn ) does not tend to 0 when n goes to infinity. Let us show that dU (an , bn ) (min λ1 )(1 − max(λ2 /λ1 )) for all n, where λ1 and λ2 are the functions used in the definition of a and b in Example 6. Let us denote by η˜ i ∈ C(Xi ) ⊗ K the projection e0 ⊗ 1 ⊗ · · · ⊗ ηi ⊗ · · · ⊗ 1, where ηi is placed in the i-th position of the tensor product. Here we view C(Xi ) ⊗ K as the tensor product
C CP(1) ⊗ K ⊗ C CP(n2 ) ⊗ · · · ⊗ C CP(ni ) . Let p be an arbitrary projection in C(X1 ) ⊗ K. It was observed in [13] that the image of p by φ1,i is Murray–von Neumann equivalent to the projection (p ⊗ 1 ⊗ · · · ⊗ 1) ⊕ k1 η˜ 1 ⊕ k2 η˜ 2 ⊕ · · · ⊕ ki η˜ i , where ki ∈ N. In this expression the multiplication by the coefficients ki indicates the orthogonal sum of ki copies of the projection η˜ i . In a similar manner, one can show that for every scalar function λ ∈ C(X1 ) the image of λp by φ1,i is Murray–von Neumann equivalent to λ(p ⊗ 1 ⊗ · · · ⊗ 1) ⊕
j
j
j
λ y1 η˜ 1 ⊕ λ y2 η˜ 2 ⊕ · · · ⊕ λ yi η˜ i . j
j
j
Since a and b have both the form λ1 p ⊕ λ2 q, for some projections p and q and scalar functions λ1 and λ2 , the formula above allows us to compute the images of a and b in C(Xi ) ⊗ K (i.e., the elements ai and bi ) up to Murray–von Neumann equivalence. We have that ai is Murray–von Neumann equivalent to j
j
j
λ1 y1 η˜ 1 ⊕ λ1 y2 η˜ 2 ⊕ · · · ⊕ λ1 yi η˜ i λ1 η˜ 1 ⊕ j
⊕ λ2 η˜ 1
⊕
j
j
j
j
j
λ2 y1 η˜ 1 ⊕ λ2 y2 η˜ 2 ⊕ · · · ⊕ λ2 yi η˜ i ,
j
j
j
where η˜ 1 = (12 − η1 ) ⊗ 1 ⊗ · · · ⊗ 1. A similar expression holds for bi . Let ai = ai /λ1 and bi = bi /λ1 . Let t = max(λ2 /λ1 ). Let us show that dW (ai , bi ) 1 − t. We have that (ai − t)+ is Murray–von Neumann equivalent to (1 − t)η˜ 1 ⊕ α1,j (y)η˜ 1 ⊕ α2,j (y)η˜ 2 ⊕ · · · ⊕ αi,j (y)η˜ i ⊕
j
where αk,j (y) = (
j
λ1 (yk ) λ1 (y)
j
β1,j (y)η˜ 1 ⊕
j
β2,j (y)η˜ 2 ⊕ · · · ⊕
j
− t)+ and βk,j (y) = (
j
βi,j (y)η˜ i ,
(14)
j j
λ2 (yk ) λ1 (y)
− t)+ for k, j = 1, . . . , i. It follows that
i
ai − t + [η˜ 1 ] + 2kj [η˜ j ] j =2
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in the Cuntz semigroup of C(Xi ) ⊗ K. For the element (bi − t)+ an expression identical to (14) may be found, except that the first summand of (14) is replaced with the term (1 − t)(1 ⊗ · · · ⊗ 1). It follows that for all r < 1 − t we have [1 ⊗ · · · ⊗ 1] [(bi − t − r)+ ]. Since we do not have [1 ⊗ · · · ⊗ 1] [η˜ 1 ] + ij =2 2kj [η˜ j ] (because the total Chern class of the projection on the right side is non-zero), we conclude that dW (ai , bi ) 1 − t. By Lemma 1 we have dU (ai , bi ) 1 − t. Hence
dU (ai , bi ) (min λ1 ) · dU ai , bi (min λ1 ) · 1 − max(λ2 /λ1 ) .
2
5. Classification by the functor Cu(· ⊗ Id) a the pseuLet A and B be C∗ -algebras. For a ∈ (A ⊗ K)+ a contraction, let us denote by dW dometric on the Cuntz category morphisms from Cu(A) to Cu(B) given by
a dW (α, β) := dW α ◦ Cu(φa ), β ◦ Cu(φa ) , where φa : C0 (0, 1] → A ⊗ K is such that φ(id) = a. We consider the set Mor(Cu(A), Cu(B)) a . A basis of entourages endowed with the uniform structure induced by all the pseudometrics dW for this uniform structure is given by the sets a UF,ε = (α, β) dW (α, β) < ε, a ∈ F , where ε > 0 and F runs through the finite subsets of positive contractions of A ⊗ K. We will prove the following theorem, of which Theorem 3 of the introduction is an obvious corollary. Theorem 8. For every ε > 0 there is a finite set F ⊂ C0 (0, 1] ⊗ C0 (0, 1], and δ > 0, such that
Cu(φ ⊗ Id), Cu(ψ ⊗ Id) ∈ UF,δ
⇒
dU φ(id), ψ(id) < ε,
for any pair of homomorphisms φ, ψ : C0 (0, 1] → A, where the C∗ -algebra A is an inductive limit of the form lim −→ C(Xi ) ⊗ K, with Xi compact metric spaces and dim Xi 2 for all i = 1, 2, . . . . Before proving Theorem 8 we need some preliminary definitions and results. We will consider the relation of Murray–von Neumann equivalence on projections in matrix algebras over possibly non-compact spaces. If P and Q are projections in the algebra Mn (Cb (X)) of continuous, bounded, matrix valued functions on X, we say that P and Q are Murray–von Neumann equivalent, and denote this by P ∼ Q, if there is v ∈ Mn (Cb (X)) such that P = vv ∗ and Q = vv ∗ . For a subset U of X, assumed either open or closed, we say that P is Murray–von Neumann equivalent to Q on the set U if the restrictions of P and Q to U are Murray–von Neumann equivalent in the algebra Mn (Cb (U )). Lemma 6. Let X be a finite CW-complex of dimension at most 2, and let C be a closed subset of X. If P and Q are projections in Mn (C(X)) such that P is Murray–von Neumann equivalent to Q on the set C, then there exists a finite subset F of X\C such that P is Murray–von Neumann equivalent to Q on X\F .
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889 m
0 Proof. Let X1 denote the 1-skeleton of X and ( i )m i=1 the 2-cells of X. Suppose that ( i )i=1 ˚ i \C for i m0 , and let F are the 2-cells intersected by the open set X\C. Choose points xi ∈ be the set of these points. Since X\F contracts to X 1 ∪ i>m0 i , it is enough to show that P is Murray–von Neumann equivalent to Q on X ∪ 1 i>m0 i (see [12, Theorem 1]). Let v be a vv ∗ and Q = v ∗ v on i>m0 i (v exists partial isometry defined on i>m0 i such that P = by hypothesis). Let us show that v extends to X1 ∪ i>m0 i . For this, it is enough to show that v extends from X1 ∩ i>m0 i to X1 . This is true by [6, Proposition 4.2(1)] (applied to 1-dimensional spaces). 2
Proposition 6. Let X be a finite CW-complex of dimension at most 2. Let ε > 0. Suppose that a, b ∈ Mn (C(X))+ are of the form a=
n
Pj λj ,
b=
j =1
n
Qj λj ,
(15)
j =1
where (Pj )nj=1 and (Qj )nj=1 are sequences of orthogonal projections of rank 1, (λj )nj=1 is a sequence of scalar functions such that λj λj +1 for j = 1, 2, . . . , n − 1, and i
Pj ∼
j =1
i
Qj
on the set x ∈ X λi (x) − λi+1 (x) ε ,
(16)
j =1
for i = 1, . . . , n (for i = n we take λi+1 = 0 in (16)). Then dU (a, b) < 2ε. Proof. Let ε > 0 and a and b be as in the statement of the lemma. Let us perturb the elements a and b by modifying the functions (λi )ni=1 in the following way: For i = 1, 2, . . . , n, let us denote by Ci the set {x ∈ X | λi (x) − λi+1 (x) ε}. By (16) and Lemma 6, there are finite sets Fi ⊆ X\Ci such that ij =1 Pj is Murray–von Neumann to ij =1 Qj on X\Fi for i = 1, 2, . . . , n. Let us choose the sets Fi so that they are disjoint for differenti’s (it is clear from the proof of Lemma 6 that this is possible). Furthermore, for every x ∈ ni=1 Fi let us choose an open neighborhood U (x) of x such that U (x) ∩ U (x ) = ∅ for x = x and U (x) ∩ Ci = ∅ for x ∈ Fi . Starting with i = 1, and proceeding to i = 2, . . . , n, let us perturb the function λi+1 on the set x∈Fi U (x) by an amount less than ε, and so that λi+1 (x) = λi (x) for x in some open set Vi such that Fi ⊂ Vi and Vi ⊆ x∈Fi U (x). Since the sets x∈Fi U (x) are disjoint for different values of i, the resulting perturbations of a and b are within a distance of ε of their original values. These perturbations, which we continue to denote by a and b, satisfy that a=
n
Pj λj ,
j =1 i j =1
Pj ∼
b=
n
Qj λj ,
(17)
j =1 i
Qj
on X\Vi , for i = 1, 2, . . . , n,
(18)
j =1
Vi ⊆ x λi (x) = λi+1 (x) ,
and Vi is open.
(19)
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The proposition will be proved once we show that, under the conditions (17)–(19), the elements a and b are Murray–von Neumann equivalent. This amounts to finding a sequence of orthogonal projections (Ri )ni=1 in Mn (C(X)) such that a = nj=1 Rj λj , and Ri ∼ Qi for i = 1, . . . , n. Let us show that this is possible. The sequence (Ri )ni=1 will be obtained by a series of modifications on the sequence (Pi )ni=1 . k −1 k −1 Let k0 be the smallest index such that Pk0 Qk0 . From j0=1 Pi ∼ j0=1 Qi and (18), we get that Pk0 ∼ Qk0 on X\Vk0 (since there is cancellation of projections over spaces of dimension at most 2). Let v be the partial isometry defined on X\Vk0 such that Pk0 = vv ∗ and Qk0 = v ∗ v on X\Vk0 . It is guaranteed by [6, Proposition 4.2(1)] that v can be extended to a partial isometry w on X such that w ∗ w = Qk0 and ww ∗ Pk0 + Pk0 +1 . Set ww ∗ = Pk 0 , with w being such an extension of v. Then Pk 0 is such that Pk 0 ∼ Qk0 , Pk 0 Pk0 + Pk0 +1 , and Pk 0 (x) = Pk0 (x) for all x ∈ X\Vk0 . Let Pk 0 +1 be the projection such that Pk 0 + Pk 0 +1 = Pk0 + Pk0 +1 . We have Pk0 λk0 + Pk0 +1 λk0 +1 = Pk 0 λk0 + Pk 0 +1 λk0 +1 . Thus, replacing Pk0 and Pk0 +1 by Pk 0 and Pk 0 +1 respectively, we obtain a new sequence of projections (Pi )ni=1 that satisfies (17) and (18), and also Pk ∼ Qk for k k0 . Continuing this process we obtain the desired sequence (Ri )ni=1 . 2 Proof of Theorem 8. Let ε > 0 (and assume ε < 1). Let gε ∈ C0 (0, 1] be a function such that gε (t) = εt for t ∈ [ε, 1], and 0 gε (t) 1 for t ∈ (0, 1]. Let F ⊆ C0 (0, 1] ⊗ C0 (0, 1] be the set F = {id ⊗ id, id ⊗ gε }. Let us prove that
Cu(φ ⊗ Id), Cu(ψ ⊗ Id) ∈ U
2 F, ε2
⇒
ε2 dU φ(id), ψ(id) < 2ε + , 2
where φ, ψ, and A are as in the statement of the theorem. Let us express what we wish to prove in terms of positive contractions (via the bijection φ → φ(id)). For a, b ∈ A positive contractions, we have id⊗id dW (a ⊗ id, b ⊗ id) = dW (a ⊗ id, b ⊗ id), id⊗gε
dW
(a ⊗ id, b ⊗ id) = dW (a ⊗ gε , b ⊗ gε ).
Thus, we want to show that ε2 , 2 ε2 dW (a ⊗ gε , b ⊗ gε ) < 2 dW (a ⊗ id, b ⊗ id) <
⇒
dU (a, b) < 2ε +
ε2 , 2
(20)
for a and b positive contractions. Let us first show that if we have (20) for the C∗ -algebras (Ai )∞ i=1 of a sequential inductive system, then we also have (20) for their inductive limit A. By the continuity of the pseudometrics dW and dU (see Lemma 1), it is enough to prove (20) assuming that a and b belong to a dense subset of the positive contractions of A. Thus, we may assume that a and b are the images in A of positive contractions in some C∗ -algebra Ai , i ∈ N. Suppose we have a , b ∈ Ai such that their
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images in A satisfy the inequalities of the left side of (20). By Lemma 5(ii), it is possible to move a and b along the inductive limit to a C∗ -algebra Aj , j i, so that these same inequalities hold A
in the C∗ -algebra Aj . We conclude that dU j (φi,j (a ), φi,j (b )) < ε. Moving a and b back to the limit we get the right side of (20). By the discussion of the previous paragraph it is enough to prove (20) for A = C(X) ⊗ K, with X a compact metric space of dimension at most 2. Moreover, since a compact metric space of dimension at most 2 is a sequential projective limit of finite CW-complexes of dimension at most 2 (see [4, Theorem 1.13.5]), we are reduced to proving (20) for the case that A = C(X) ⊗ K, where X is a finite CW-complex of dimension at most 2. Let us suppose A = C(X) ⊗ K, where X is a finite CW-complex of dimension at most 2. It is enough to prove (20) assuming that a, b ∈ Mn (C(X)) for some n ∈ N. Moreover, by Choi and Elliott’s [1, Theorem 1], we may assume that a(x) and b(x) have distinct eigenvalues (as matrices in Mn (C)) for all x ∈ X. (Choi and Elliott’s Theorem implies that such a set is dense in the set of positive contractions of Mn (C(X)) for dim X 2.) This implies (see the proof of [10, Theorem 1.2]) that a and b have the form a=
n
Pj λi
and b =
j =1
n
Qj μi ,
(21)
j =1
for some sequences of orthogonal projections of rank 1 (Pi )ni=1 and (Qi )ni=1 , and scalar eigenfunctions (λi )ni=1 and (μi )ni=1 , such that 1 λ1 (x) > λ2 (x) > · · · > 0 and 1 μ1 (x) > μ2 (x) > · · · > 0. From dW (a ⊗ id, b ⊗ id) < ε 2 /2 we deduce that dW (a, b) < ε 2 /2 (evaluating id at t = 1), 2
and so i − μi < ε /2 for all i (see the proof of Theorem 5). Let b ∈ Mn (C(X)) be given by λ n
2 b = i=1 Qi λi . Then dU (b, b ) < ε /2 and dW (a ⊗ gε , b ⊗ gε ) dW (a ⊗ gε , b ⊗ gε ) + dW (b ⊗ gε , b ⊗ gε ) < ε 2 . The implication (20) will be proven once we have shown that dW (a ⊗ gε , b ⊗ gε ) < ε 2
⇒
dU (a, b ) < 2ε.
In order to prove this, it is enough to show that the left side of this implication implies the condition (16) of Proposition 6 (applied to the elements a and b ). Let us choose ε > 0 such that dW (a ⊗ gε , b ⊗ gε ) < ε 2 − ε ε. By the definition of dW we have that
a ⊗ gε − (ε − ε ε) + Cu b ⊗ gε − ε − ε 2 + . Let us identify Mn (C(X))⊗C0 (0, 1] with Mn (C0 (X ×(0, 1])) and express the Cuntz comparison above in terms of the projections (Pi )ni=1 and (Qi )ni=1 , and the eigenfunctions (λi )ni=1 . We get n j =1
n
Pj (x) λj (x)gε (t) − ε + ε ε + Cu Qj (x) λj (x)gε (t) − ε + ε 2 + ,
(22)
j =1
for (x, t) ∈ X ×(0, 1]. Note: this Cuntz relation comparison is not to be understood as a pointwise relation, but rather as a relation in the C∗ -algebra Mn (C(X × (0, 1])).
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For i n let us define the set Ti = x ∈ X λi+1 (x)/λi (x) 1 − ε and λi (x) ε . Let Ci ⊆ X × (0, 1] be the closed set Ci = {(x, λi (x)) | x ∈ Ti }. Restricting the Cuntz comparison (22) to the set Ci , and using the definition of gε , we get that
λ1 λ2 − (1 − ε ) + P2 − (1 − ε ) + · · · + ε Pi λi λ i + + λ1 λ2 Cu Q1 − (1 − ε) + Q2 − (1 − ε) + · · · + εQi , λi λi + +
P1
on the closed set Ti . It follows that ij =1 Pj Cu ij =1 Qj on Ti . In the same way we prove that i i i i j =1 Qj Cu j =1 Pj on Ti , and so j =1 Pj ∼ j =1 Qj on Ti . If λi (x) − λi+1 (x) ε then λi+1 (x)/λi (x) 1 − ε and λi (x) ε. Hence, {x ∈ X | λi (x) − λi+1 (x) ε} ⊆ Ti . Therefore, the elements a and b satisfy the condition (16) of Proposition 6. This completes the proof of the theorem. 2 References [1] M.D. Choi, G.A. Elliott, Density of the selfadjoint elements with finite spectrum in an irrational rotation C∗ -algebra, Math. Scand. 67 (1) (1990) 73–86. [2] A. Ciuperca, G.A. Elliott, A remark on invariants for C∗ -algebras of stable rank one, Int. Math. Res. Not. IMRN 2008 (5) (2008), Art. ID rnm158, 33 pp. [3] K.T. Coward, G.A. Elliott, C. Ivanescu, The Cuntz semigroup as an invariant for C∗ -algebras, J. Reine Angew. Math. 623 (2008) 161–193. [4] R. Engelking, Dimension Theory, North-Holland Publishing Co., Amsterdam, 1978. [5] E. Kirchberg, M. Rørdam, Infinite non-simple C∗ -algebras: Absorbing the Cuntz algebras O∞ , Adv. Math. 167 (2) (2002) 195–264. [6] N.C. Phillips, Cancellation and stable rank for direct limits of recursive subhomogeneous algebras, Trans. Amer. Math. Soc. 359 (10) (2007) 4625–4652 (electronic). [7] L. Robert, The Cuntz semigroup of some spaces of dimension at most 2, preprint, arXiv:0711.4396v2. [8] M. Rørdam, The stable and the real rank of Z-absorbing C∗ -algebras, Internat. J. Math. 15 (10) (2004) 1065–1084. [9] M. Rørdam, W. Winter, The Jiang–Su algebra revisited, J. Reine Angew. Math., in press. [10] K. Thomsen, Homomorphisms between finite direct sums of circle algebras, Linear Multilinear Algebra 32 (1) (1992) 33–50. [11] K. Thomsen, Inductive limits of interval algebras: Unitary orbits of positive elements, Math. Ann. 293 (1) (1992) 47–63. [12] L.N. Vaserstein, Vector bundles and projective modules, Trans. Amer. Math. Soc. 294 (2) (1986) 749–755. [13] J. Villadsen, On the stable rank of simple C∗ -algebras, J. Amer. Math. Soc. 12 (4) (1999) 1091–1102.
Journal of Functional Analysis 258 (2010) 893–912 www.elsevier.com/locate/jfa
Asymptotics of Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin domains in Rd Denis Borisov a,1 , Pedro Freitas b,c,∗,2 a Department of Physics and Mathematics, Bashkir State Pedagogical University, October rev. st., 3a,
450000 Ufa, Russia b Department of Mathematics, Faculdade de Motricidade Humana (TU Lisbon), Portugal c Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2,
P-1649-003 Lisboa, Portugal Received 23 May 2009; accepted 25 July 2009 Available online 5 August 2009 Communicated by J.-M. Coron
Abstract We consider the Laplace operator with Dirichlet boundary conditions on a domain in Rd and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This generalizes our previous results in two dimensions and, as in that case, allows us to obtain an approximation for Dirichlet eigenvalues for a large class of domains, under very mild assumptions. As an application, we derive a three-term asymptotic expansion for the first eigenvalue of d-dimensional ellipsoids. © 2009 Elsevier Inc. All rights reserved. Keywords: Laplace spectrum; Thin domain asymptotics
* Corresponding author at: Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal. E-mail addresses:
[email protected] (D. Borisov),
[email protected] (P. Freitas). 1 The author was partially supported by RFBR and gratefully acknowledges the support from Deligne 2004 Balzan prize in mathematics and the grant of the President of Russia for young scientists and for Leading Scientific Schools (NSh-2215.2008.1). 2 The author was partially supported by POCTI/POCI2010, Portugal, and by project LC06002 of the Czech Ministry of Education, Youth and Sports. Part of this work was done while the author was visiting the Doppler Institute in Prague, and he would like to thank the people there, and in particular P. Exner and D. Krejˇciˇrík, for their hospitality.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.07.014
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1. Introduction In his 1967 paper [7] Joseph studied families of domains indexed by one parameter to obtain perturbation formulae approximating eigenvalues in a neighbourhood of a given domain. Within this context, he derived an elegant expression for the first eigenvalue of ellipses parametrized by their eccentricity e, namely, λ1 2 λ1 λ1 4 e − 3− e 2 16 2 λ1 6 λ1 3− e + O e8 , as e → 0, − 32 2
λ1 (e) = λ1 −
(1.1)
where λ1 = λ1 (0) is the first eigenvalue of the disk √ – to obtain the eigenvalue of ellipses of, say, area π , for instance, this should be divided by 1 − e2 and λ1 (0) be the corresponding value for the disk. The coefficient of order e6 in Joseph’s paper is actually incorrect – we are indebted to M. Ashbaugh for pointing this out to us, and also for mentioning Henry’s book [6] where this has been corrected. Although in principle quite general, the approach used by Joseph yields formulae which, in the case of domain perturbations, will allow us to obtain explicit asymptotic expansions only in very special cases such as that of ellipses above. The failure to obtain these expressions may be the case even when the eigenvalues and eigenfunctions of the original domain are known, as this does not necessarily mean that the coefficients appearing in the expansion may be computed in closed form. An example of this is the perturbation of a rectangle into a parallelogram, which Joseph considered as an example of what he called “pure shear.” With the purpose of obtaining approximations that can be computed explicitly, in a previous paper we considered instead the scaling of a given two-dimensional domain in one direction and studied the resulting singular perturbation as the domain approached a segment in the limit [1]. This approach may, of course, have the disadvantage that we might now be starting too far from the original domain. However, it allows for the explicit derivation of the coefficients in the expansion in terms of the functions defining the boundary of the domain. As was to be expected, and can be seen from the examples given in that paper, these four–term approximations are quite accurate close to the thin limit. A more interesting feature of this approach is that in some cases it also allows us to approximate eigenvalues quite well away from this limit, as may be seen from the following examples. The application of our formula to the ellipses considered above yields λ1 (ε) =
3 11 π π2 π + + + ε + O ε2 , + 2 2ε 4 8π 12 4ε
as ε → +0,
(1.2)
where we now considered ellipses of radii 1 and ε, ε being the stretch factor. The error in the approximation is comparable to that in Joseph’s formula, except that Eq. (1.2) is more accurate closer to the thin limit while (1.1) provides better approximations near the circle. This is also an advantage, since it is natural for numerical methods to perform better away from the thin limit, but to have more difficulties the closer they are to the singular case, suggesting that our formulae may also be useful for checking numerical methods close to the limit case. As another application we mention the case of the lemniscate 2 2 x1 + x22 = x12 − x22 .
D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
895
for which we have √ √ 97 593 3π 2π 2 2 3π + + ε + O ε2 , λ1 (ε) = 2 + + √ ε 24 4 ε 64 3π
as ε → +0,
yielding an error at ε equal to one which is in fact smaller than in the case of the disk above. For details, see [1]. In the present paper we extend the results in [1] to general dimension, in the sense that we now consider domains in Rd which are being scaled in one direction and approach a (d − 1)dimensional set in the limit as the stretch parameter goes to zero. Due to the increase in complexity in the corresponding formulae as a consequence of the fact that we are now considering arbitrary dimensions, we only obtained the first three non-zero coefficients in the asymptotic expansion of the principal eigenvalue. However, because of smoothness assumption near the point of global maximum, these include the coefficients of the two unbounded terms plus the constant term in the expansion – we know from the two-dimensional case that lack of smoothness at the point of maximum will yield other intermediate powers of ε [3,4]. As an example, we obtain an expansion for the first eigenvalue of the general d-dimensional ellipsoid 2 2 x1 xd E = (x1 , . . . , xd ) ∈ Rd : + ··· + 1 , a1 ad where the ai s are positive real numbers. If we choose as projecting hyperplane that which is orthogonal to the xd axis we obtain
d−1 d−1 d−1 d−1 π2 π 1 1 1 1 1 + + λ1 (Eε ) = 2 + 3 4 4ad ε 2 2ad ε i=1 ai a 2 2 i=1 j =i+1 ai aj i=1 i + O ε 1/2 ,
as ε → +0.
(1.3)
Besides the added complexity of the formulae, there are now extra technical difficulties related to the fact that there may exist multiple eigenvalues requiring a more careful approach. As in the two-dimensional case, the asymptotic expansions obtained depend on what happens locally at the point of global maximum width. Also as in that case, we cannot exclude the existence of a tail term approaching zero faster than any power of ε. However, we conjecture that if the boundary of the domain is analytic, then the expansions will actually correspond to the series developments of the corresponding eigenvalues. In the next section we establish the notation and state the main results of the paper, which are then proved in Sections 3 and 4. In the last section, and as an application, we derive the above expression for the first eigenvalue of the ellipsoid. 2. Statement of results Let x = (x , xd ), x = (x1 , . . . , xd−1 ) be Cartesian coordinates in Rd and Rd−1 , respectively, d 2, and ω ⊂ Rd−1 be a bounded domain having C 1 -boundary. By h± = h± (x ) ∈ C(ω) we
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denote two arbitrary functions such that H (x ) := h+ (x ) + h− (x ) 0 for x ∈ ω. We consider the thin domain defined by Ωε := x: −εh− (x ) < xd < εh+ (x ), x ∈ ω , where ε is a small positive parameter. We assume that the function H (x ) attains its global maximum at a single point x ∈ ω and that there exists a ball Bδ (x) := {x : |x − x| < δ} such that h± ∈ C ∞ (Bδ (x)). Let H0 := H (x) and the Taylor expansions for H and h− at x read as follows H (x ) = H0 +
∞
Hi (x − x),
h− (x ) = h0 +
i=2k
∞
hi (x − x),
(2.1)
i=1
where Hi and hi are homogeneous polynomials of order i, H2k (x − x) < 0 for x = x, and k 1. Our purpose is to study the asymptotic behaviour of the eigenvalues and eigenfunctions ∞ d−1 ) be a non-negative cutof the Dirichlet Laplacian −D Ωε in Ωε . Let χ = χ(x ) ∈ C (R off function equalling one as |x − x| < δ/3 and vanishing for |x − x| > δ/2. Denote Ωεδ := Ωε ∩ {x: |x − x| < δ}. Let Gn := −ξ −
2π 2 n2 H2k (ξ ) H03
be an operator in L2 (Rd−1 ). The spectrum of this operator consists of countably many isolated eigenvalues of finite multiplicity having only one accumulation point at infinity [5, Ch. IV, Sec. 46, Th. 1]. By Λn,1 < Λn,2 Λn,3 . . . we denote the eigenvalues of this operator arranged in non-decreasing order and taking the multiplicities into account. Denote by Ψn,m the associated eigenfunctions orthonormalized in L2 (Rd−1 ). It follows from [5, Ch. V, Sec. 43, Th. 2] that the functions Ψn,m decay exponentially at infinity. Our main results are the following. First, we obtain a two-parameter description for the eigenvalues. Theorem 1. Let Λ = Λn,M = Λn,M+1 = · · · = Λn,M+N −1 be a N -multiple eigenvalue of Gn for a given n ∈ N. Then there exist eigenvalues λn,m (ε) of −D Ωε , m = M, . . . , M + N − 1 taken counting multiplicities whose asymptotics as ε → +0 read as follows λn,m (ε) = ε −2 c0
(n,m)
+ ε −2
∞
(n,m) j
cj
η ,
η := ε α , α :=
j =2k (n,m)
c0
=
π 2 n2 , H02
(n,m)
c2k
= Λ,
1 , k+1
(2.2)
(2.3)
D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
897
(n,m)
and −c2k+1 are the eigenvalues of the matrix with the entries 2π 2 n2 H0−3 H2k+1 Ψn,m , Ψn,l L
2 (R
d−1 )
m, l = M, . . . , M + N − 1.
,
The remaining coefficients are determined by Lemmas 3.6 and 3.7. As in [1], for sufficiently small ε this allows us to derive the asymptotics for specific eigenvalues, and we give the explicit expansion for the first eigenvalue in terms of the functions H and h− in the case where H2 is negative for x = x. Theorem 2. For any N 1 there exists ε0 = ε0 (N ) such that for ε ε0 the first N eigenvalues of −D Ωε are λ1,m (ε), m = 1, . . . , N . If H2 (x ) = −
k = 1,
1 2 2 αi xi , 2 d−1
(2.4)
i=1
the lowest eigenvalue λ1,1 (ε) has the asymptotic expansion (1,1)
(1,1) c0 c2 (1,1) λ1,1 (ε) = 2 + + c4 + O ε 1/2 , ε ε
c0(1,1) (1,1)
c4
=
π2 = 2, H0
c2(1,1)
θj ,
θj :=
j =1
παj 3/2
,
H0
π 2 2 3H2 (ξ ) − 2H0 H4 (ξ ) Ψ0 , Ψ0 L (Rd−1 ) 4 2 H0 d−1 π 2 ∂h1 2 2π 2
1 , Ψ0 + 2 − 3 H3 (ξ )Ψ , L2 (Rd−1 ) H0 i=1 ∂xi H0
Ψ0 (ξ ) :=
1 (ξ ) := Ψ0 (ξ ) Ψ
=
d−1
ε → +0,
d−1 j =1
1/4
θj
π 1/4
d−1
3π 2 βppj ξj
p,j =1
2H03 θj (2θp + θj )
e
−
−
θj ξj2 2
(2.5)
(2.6)
,
d−1
π 2 βpqj ξp ξq ξj
p,q,j =1
H03 (θp + θq + θj )
,
(2.7)
where it is assumed that H3 (x ) is written as H3 (x ) =
d−1
βpqj ξp ξq ξj ,
p,q,j =1
and the constants βpqj are invariant under each permutation of the indices p, q, j : βpqj = βpj q = βqpj = βqjp = βjpq = βj qp .
(2.8)
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D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
Remark 2.1. The assumption (2.4) for H2 is not a restriction, since we can always achieve such form for H2 by an appropriate change of variables. 3. Proof of Theorem 1 In this section we construct the asymptotics for the eigenvalues and the eigenfunctions of the operator −D Ωε . This is first done formally, and justified rigorously afterwards. In the formal construction we employ the same approach as was used in [1, Sec. 3]. We are going to construct formally the asymptotic expansions for the eigenvalues λn,m (ε), (m) m = M, . . . , M + N − 1 which we relabel as λε , m = 1, . . . , N . We denote the associated (m) eigenfunctions by ψε . We construct their asymptotic expansions as the series λ(m) ε
= ε −2 μ(m) ε ,
μ(m) ε
(m) = c0
+
∞
(m)
ci η i ,
i=2k
ψε(m) (x) =
ε(m) (x), H (x )ψ
ε(m) (x) = ψ
∞
(m)
η i ψi
(ξ ),
i=0
ξ = (ξ , ξd ),
ξ :=
x − x , η
ξd :=
xd + εh− (x ) . εH (x )
(3.1)
We postulate the functions ψi (ξ ) to be exponentially decaying as ξ → +∞. It means that they are exponentially small outside Ωεδ (with respect to ε). In terms of the variables ξ the domain Ωεδ becomes {ξ : |ξ | < δη−1 , 0 < ξd < 1}. As η → 0, it “tends” to the layer Π := {ξ : 0 < ξd < 1} and this is why we shall construct the functions ψi as defined on Π . We rewrite the eigenvalue equation for ψε and λε in the variables ξ , (m)
d−1 ∂ ∂2 ∂ ∂ ∂ 2k+1 + η K + K i i ∂ξi ∂ξd ∂ξd ∂ξi ∂ξd2 i=1 d−1 ∂ 2k+2 2 ∂ 2k+2
ε(m) = μ(m)
(m) in Π, +η K +η K0 ψ ε ψε ∂ξd i ∂ξd
− η2k ξ + Kd
i=1
ε(m) ψ
= 0 on ∂Π,
where Ki = Ki (ξ, η), i = 0, . . . , d, 1 , + ηξ ) ∂h− ∂H 1 Ki (ξ, η) = (x + ηξ ) − ξd (x + ηξ ) , H (x + ηξ ) ∂xi ∂xi
Kd (ξ, η) =
H 2 (x
2 1 1 K0 (ξ, η) = H −1 (x + ηξ )x H (x + ηξ ) − H −2 (x + ηξ )∇x H (x + ηξ ) . 2 4
(3.2)
D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
899
√ (m) Remark 3.1. We have introduced the factor H (x ) in the series (3.1) for ψε in order to have a symmetric differential operator in the equation (3.2). We expand the functions Ki into the Taylor series w.r.t. η and employ (2.1) to obtain ∞
Kd (ξ, η) = H0−2 +
ηj Pj (ξ ), (d)
j =2k
Ki (ξ, η) =
∞
(i)
ηj Kj (ξ ),
j =0
Kj (ξ ) := Pj (ξ ) + ξd Qj (ξ ), (i)
(i)
(i)
K0 (ξ, η) =
∞
i = 1, . . . , d − 1,
ηi Pi (ξ ), (0)
(3.3)
i=0 (i)
(i)
where Pj , Qj are polynomials, and, in particular, P2k (ξ ) = − (d)
2H2k (ξ ) H03
P0 (ξ ) = (i)
,
P2k+1 (ξ ) = − (d)
1 ∂h1 (x), H0 ∂xi
2H2k+1 (ξ ) H03
,
Q0 (ξ ) = 0. (i)
(3.4)
We substitute (3.1), (3.3), (3.4) into (3.2) and equate the coefficients of like powers of η. This (m) leads us to the following boundary value problems for ψi ,
1 ∂2 (m) (m) ψj = 0 in Π, + c 0 H02 ∂ξd2 (m)
ψj = 0 on ∂Π, j = 0, . . . , 2k − 1, 1 ∂2 2H2k (ξ ) ∂ 2 (m) (m) (m) ψ ψ0(m) − + c = − + c ξ 0 2k 2k H02 ∂ξd2 H03 ∂ξd2
(3.5) in Π,
(m) = 0 on ∂Π, ψ2k 1 ∂2 2H2k (ξ ) ∂ 2 (m) (m) (m) (m) ψ ψj −2k − + c = − + c ξ j 0 2k H02 ∂ξd2 H03 ∂ξd2 (m) (m) + cj ψ0
+
j −2k−1
(m)
(m)
cj −q ψq(m) + Fj
(3.6)
in Π,
q=1 (m)
ψj
= 0 on ∂Π,
j 2k + 1,
(3.7)
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D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
(m)
Fj
:=
j −2k−1
Lj −q−2k ψq(m) ,
q=0
d−1 ∂ ∂ (i) ∂ (i) ∂ K + K Lj := ∂ξd j −1 ∂ξi ∂ξi j −1 ∂ξd i=1
j −2 d−1 2 ∂ ∂ (i) (d) ∂ (0) + Ks(i) Kj −s−2 + Pj +2k 2 + Pj −2 , ∂ξd ∂ξd ∂ξd
(3.8)
i=1 s=0
(0)
where P−1 = 0. Problems (3.5) can be solved explicitly with (m)
(m)
ψj (ξ ) = Ψj (m)
where j = 0, . . . , 2k − 1, and Ψj
(ξ ) sin πnξd ,
c0 =
π 2 n2 , H02
(3.9)
are the functions to be determined. The last identity proves
(n,m) . c0
formula (2.3) for We consider the problem (3.6) as posed on the interval (0, 1) and depending on ξ . It is solvable, if and only if the right-hand side is orthogonal to sin πnξd in L2 (0, 1). It implies the equation − + ξ
2π 2 n2 H2k (ξ )
(m)
(m)
(m)
(m)
= c2k Ψ0
Ψ0
H03
(m)
in Rd−1 .
(3.10)
(m)
Thus, c2k is an eigenvalue of the operator Gn , i.e., c2k = Λ. Then Ψ0 is one of the eigenfunctions associated with Λ. These eigenfunctions are assumed to be orthonormalized in L2 (Rd−1 ). We substitute Eq. (3.10) into (3.6) and see that formula (3.9) is valid also for j = 2k. The problems (3.7) are solvable, if and only if the right-hand sides are orthogonal to sin πnξd in L2 (0, 1). It gives rise to the equations
(m) (Gn − Λ)Ψj −2k
(m) = fj
+
j −2k−1
(m)
(m)
(m)
cj −q Ψq(m) + cj Ψ0 ,
q=1
(m) Ψj
(m) = Ψj (ξ ) := 2
1
(m)
(3.11)
(m)
(3.12)
ψj (ξ ) sin πnξd dξd , 0
(m) fj
(m) = fj (ξ ) := 2
1
Fj
(ξ ) sin πnξd dξd .
0
To solve problems (3.7), (3.11) we need some auxiliary lemmata. The first of these follows from standard results in the spectral theory of self-adjoint operators.
D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
901
Lemma 3.2. Let f ∈ L2 (Rd−1 ). The equation (Gn − Λ)u = f
(3.13)
is solvable, if and only if f, Ψ0(m) L
2 (R
d−1 )
= 0,
m = 1, . . . , N. (m)
The solution is unique up to a linear combination of the functions Ψ0
.
By gn we denote the sesquilinear form associated with Gn , gn [u, v] = (∇u, ∇v)L2 (Rd−1 ) − (H2k u, v)L2 (Rd−1 ) . The domain of this form is D(gn ) = W21 Rd−1 ∩ u: 1 + |ξ |k u ∈ L2 Rd−1 . By D(Gn ) we denote the domain of Gn . The set C0∞ (Rd−1 ) is dense in D(gn ) in the topology induced by gn [2, Ths. 1.8.1, 1.8.2]. Lemma 3.3. Let f ∈ L2 (Rd−1 ), u ∈ L2 (Rd−1 ) ∩ W21 (S) for each bounded domain S ⊂ Rd−1 and for each φ ∈ D(g) the identity H2k (ξ ) − Λ uφ dξ = ∇u · ∇φ dξ − f φ dξ (3.14) Rd−1
Rd−1
Rd−1
holds true. Then u ∈ D(Gn ) and Eq. (3.13) is valid. Proof. Let χ1 = χ1 (t) be a non-negative infinitely differentiable cut-off function taking values in [0, 1], equalling one as t < 1, and vanishing as t > 2. It is clear that for each t > 0 the function u(ξ )χ1 (|ξ |t) belongs to D(gn ). We substitute φ(ξ ) = u(ξ )χ1 (|ξ |t) into (3.14) and integrate by parts, χ1 ∇u 2L
2 (R
d−1 )
− (H2k χ1 u, χ1 u)L2 (Rd−1 )
= (χ1 f, χ1 u)L2 (Rd−1 ) +
1 uξ χ12 , u L (Rd−1 ) + Λ χ1 u 2L (Rd−1 ) . 2 2 2
(3.15)
Hence, ∇u 2L2 (B
t −1
(0))
− (H2k u, u)L2 (B −1 (0)) t
f L2 (Rd−1 ) u L2 (Rd−1 ) + Ct 2 u L2 (Rd−1 ) + Λ u 2L
2 (R
d−1 )
,
(3.16)
where the constant C is independent of t, and Br (a) := {ξ : |ξ − a| < r}. Passing to the limit as t → +0, we conclude that u ∈ D(gn ) and in view of (3.14) this function belongs to D(gn ) and solves Eq. (3.13). 2
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D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
Let V be the space of the functions f ∈ C ∞ (Rd−1 ) such that ∂τ f d−1 1 + |ξ |γ τ ∈ L2 R ∂ξ for each τ ∈ Zd+ , γ ∈ Z+ . Lemma 3.4. Let f ∈ V, and u be a solution to (3.13). Then u ∈ V. Proof. Since u ∈ D(Gn ), we have ∇u ∈ L2 (Rd−1 ), (1 + |ξ |k )u ∈ L2 (Rd−1 ), and due to standard smoothness improving theorems u ∈ C ∞ (Rd−1 ). The identity (3.15) is also valid with χ1 replaced by χ1 (|ξ |t)|ξ |β . Employing this identity and proceeding as in (3.16), we check that (1 + |ξ |β )∇u ∈ L2 (Rd−1 ), (1 + |ξ |k+β )u ∈ L2 (Rd−1 ), if (1 + |ξ |β )u ∈ L2 (Rd−1 ) for some β ∈ Z+ . Applying this fact by induction and using that (1 + |ξ |k )u ∈ L2 (Rd−1 ), we conclude that (1 + |ξ |γ )∇u ∈ L2 (Rd−1 ), (1 + |ξ |k+γ )u ∈ L2 (Rd−1 ) for each γ ∈ Z+ . We differentiate Eq. (3.13) w.r.t. ξi , (Gn − Λ)
∂u ∂f ∂H2k = + u. ∂ξi ∂ξi ∂ξi
The right-hand side belongs to L2 (Rd−1 ) and the function Lemma 3.3. Applying this lemma, we see that sure that
∂u ∂ξi
∂u ∂ξi
satisfies the hypothesis of
∈ D(Gn ). Proceeding as above, one can make
∂u 1 + |ξ |γ ∇ ∈ L2 Rd−1 ∂ξi for each γ ∈ Z+ . Repeating the described process, we complete the proof.
2
As it follows from Lemma 3.2, the solvability condition of Eq. (3.11) is (m) (l) fj , Ψ0 L
2 (R
d−1 )
+
j −2k−1
(m) (l) cj −q Ψq(m) , Ψ0 L
(m)
2 (R
d−1 )
+ cj δml = 0,
q=1
where m, l = 1, . . . , N , and δml is the Kronecker delta. Here we have supposed that the functions (m) Ψ0 are orthonormalized in L2 (Rd−1 ). In view of (3.12) these identities can be rewritten as (m) (l) 2 Fj , ψ 0 L
2
+2 (Π)
j −2k−1
(m) (l) cj −q ψq(m) , ψ0 L
2 (Π)
(m)
+ cj δml = 0,
(3.17)
q=1
where m, l = 1, . . . , N . Consider the problem (3.7) for j = 2k + 1. The solvability condition is Eq. (3.11) for the (m) (m) same j . Since Ψ0 ∈ V, the same is true for f2k+1 . By (3.17), this equation is solvable, if and only if (2k+1)
Tml
(m)
+ c2k+1 δml = 0, m, l = 1, . . . , N, (2k+1) (m) (l) Tml := 2 L1 ψ0 , ψ0 L (Π) . 2
(3.18)
D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
903
The definition of L1 and (3.4) yield (2k+1)
Tml
(m) (l) = 2π 2 n2 H0−3 H2k+1 Ψ0 , Ψ0 L
2 (R
d−1 )
(3.19)
.
(2k+1)
Hence, the matrix T (2k+1) with the entries Tml is symmetric. This matrix describes a (m) quadratic form on the space spanned over Ψ0 , m = 1, . . . , N . By the theorem on the simul(m) taneous diagonalization of two quadratic forms we conclude that the eigenfunctions Ψ0 can be d−1 (2k+1) is diagonal. chosen as orthonormalized in L2 (R ) and, in addition, so that the matrix T In what follows we assume that these functions are chosen in such a way. Then identities (3.18) imply (m)
c2k+1 = −τm(2k+1) ,
(3.20)
(2k+1)
are the eigenvalues of T (2k+1) . where τm By Lemma 3.2 the solution to (3.11) for j = 2k + 1 reads as follows (m)
Ψ1
(ξ ) = Φ1 (ξ ) + (m)
N
(m)
(p)
(3.21)
bp,1 Ψ0 ,
p=1 (m)
where Φ1
(l)
(m)
is orthogonal to all Ψ0 , l = 1, . . . , N , in L2 (Rd−1 ) and bp,1 are constants to be (m)
found. It follows from Lemma 3.3 that Φ1 ∈ V. The definition (3.8) of L1 and Eq. (3.11) for j = 2k + 1 imply that the right-hand side of the equation in (3.7) for j = 2k + 1 is zero. Hence, (m) the solution to the problem (3.7) for j = 2k + 1 is given by formula (3.9), where Ψ2k+1 is to be found. We substitute (3.9), (3.21) into the equation (3.11) for j = 2k + 2. In view of (3.17) and (3.20) the solvability condition for this equation is as follows (m) (2k+1) (m) (m) (m) (l) bl,1 τl − τm(2k+1) + c2k+2 δml + 2 L2 ψ0 + L1 Φ1 sin πnξd , ψ0 L
2 (Π)
= 0,
l = 1, . . . , N.
(3.22) (2k+1)
Assume that all the eigenvalues τm
(m)
(m)
bl,1 =
2(L2 ψ0
are different. In this case the last identities imply (m)
+ L 1 Φ1
(2k+1)
τm
(l)
sin πnξd , ψ0 )L2 (Π) (2k+1)
− τl
m = l,
,
(m) (m) (m) (m) c2k+2 = −2 L2 ψ0 + L1 Φ1 sin πnξd , ψ0 L
2 (Π)
,
(3.23)
(m)
and we can also let bm,1 = 0. (2k+1)
Now suppose that all the eigenvalues τm are equal. In this case the equations (3.22) do (m) not allow us to determine the constants bl,1 for m = l. Consider the matrix T (2k+2) with the entries (2k+2)
Tml
(m) (m) (l) := 2 L2 ψ0 + L1 Φ1 sin πnξd , ψ0 L
2 (Π)
.
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D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
Lemma 3.5. The matrix T (2k+2) is symmetric. Proof. Integrating by parts, we obtain (2k+2)
Tml
(m) (l) = 2 ψ0 , L 2 ψ0 L
2 (Π)
(m) (l) + 2 Φ1 sin πnξd , L1 ψ0 L
2 (Π)
.
Since by (3.7) (l)
L 1 ψ0 = −
1 ∂2 (m) (m) (m) (m) ψ2k+1 − c2k+1 ψ0 , + c 0 H02 ∂ξd2
in view of (3.9), (3.11), (3.21) we have (m) (l) 2 Φ1 sin πnξd , L1 ψ0 L
2 (Π)
(m) (l) = Φ1 , (Gn − Λ)Φ1 L (Rd−1 ) 2 (m) (l) = (Gn − Λ)Φ1 , Φ1 L (Rd−1 ) 2 (m) (m) = 2 L1 ψ0 , Φ1 sin πnξd L (Π) . 2
2
Since we supposed that all the eigenvalues of T (2k+1) are equal, we can make orthogonal (m) transformation in the space spanned over Ψ0 , m = 1, . . . , N , without destroying the orthonormality in L2 (Π) and diagonalization of T (2k+1) . We employ this freedom to diagonalize the matrix T (2k+2) which is possible due to Lemma 3.5. After such diagonalization we see that the (m) coefficients c2k+2 are determined by the eigenvalues of the matrix T (2k+2) : (m)
c2k+2 = −τm(2k+2) . (m)
If all these eigenvalues are distinct, we can determine the numbers bl,1 at the next step by formulae similar to (3.23). If all these eigenvalues are identical, at the next step we should consider the next matrix T (2k+3) and diagonalize it. There exists one more possibility. Namely, the matrix T (2k+1) can have different multiple eigenvalues. We do not treat this case here. The reason is that the formal construction of the asymptotics is rather complicated from the technical point of view and at the same time it does not require any new ideas in comparison with the cases discussed above. Thus, from now on, we consider two cases only. More precisely, in the first case we assume that the matrix T (2k+1) has (2k+1) N different eigenvalues τm , m = 1, . . . , N . In the second case we suppose that the matrix (2k+1) has only one eigenvalue τ (2k+1) with multiplicity N , while the matrix T (2k+2) has N T (2k+2) different eigenvalues τm , m = 1, . . . , N . (m)
Lemma 3.6. Assume that the matrix T (2k+1) has N different eigenvalues and choose Ψ0 being orthonormalized in L2 (Rd−1 ) and so that the matrix T (2k+1) is diagonal. Then problems (3.5), (3.6), (3.7) have solutions (m)
(m) (ξ ) + Ψ
(m) (ξ ) sin πnξd + ψj (ξ ) = ψ j j
N p=1
(m)
(p)
bj,p ψ0 (ξ ).
D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
Here the functions ψ j
(m)
905
are zero for j 2k + 1, while for other j they solve the problems
j −2k−1 (m) 1 ∂2 2H2k (ξ ) ∂ 2 (m) (m) (m)
q(m) + F (m)
− ψ − + c = + Λ ψ + cj −q ψ ξ j j 0 j −2k 2 2 3 2 H0 ∂ξd ∂ξd H0 q=2k+2
− 2 Fj(m) , sin πnξd L
2 (0,1)
(m) = 0 ψ j
sin πnξd
in Π,
on ∂Π,
and are represented as finite sums
(m) (ξ ) = ψ j
ψj,ς,1 (ξ )ψj,ς,2 (ξd ), (m)
(m)
ς
where ψj,ς,1 ∈ V, ψj,ς,2 ∈ C0∞ [0, 1], ψj,ς,2 (0) = ψj,ς,2 (1) = 0, and the functions ψj,ς,2 are
(m) ∈ V are solutions to Eqs. (3.11) and are orthogonal to sin πnξd in L2 (0, 1). The functions Ψ (m)
(m)
(m)
(m)
(m)
j
(l)
(m)
(m)
orthogonal to all Ψ0 , l = 1, . . . , N , in L2 (Rd−1 ). The constants bj,p and cj by the formulae (m)
(m)
b0,l = δml , (m) bj,l
=
(m) , ψ (l) ) + 2(F j +2k+1 0
bj,m = 0,
are determined
j 1,
j −1
(m) (m) q=1 cj +2k−q+1 bq,l , (m) (l) τ2k+1 − τ2k+1
(m)
m = l, j 1,
(m)
c2k = Λ, c2k+1 = −τm(2k+1) , (m) (m) (m)
,ψ c = −2 F , j 2k + 2, j
(m) = F j
j −2k−1
j
0
L2 (Π)
j −2k−2 N (m) (p) (m)
q + Ψ
q(m) sin πnξd + Lj −q−2k ψ bq,p Lj −q−2k ψ0 .
q=0
q=0
p=1
Lemma 3.7. Assume that all the eigenvalues of the matrix T (2k+1) are identical and that the ma(m) trix T (2k+2) has N different eigenvalues, and choose Ψ0 being orthonormalized in L2 (Rd−1 ) (2k+1) (2k+2) and T are diagonal. Then problems (3.5), (3.6), (3.7) have so that the matrices T solutions
(ξ ) + Ψ
ψj (ξ ) = ψ j j (m)
(m)
+
N p=1
(m)
(ξ ) sin πnξd
bj −1,p Φ1 (ξ ) sin πnξd + (m)
(p)
N
(m)
(p)
bj,p ψ0 (ξ ).
p=1
(m) are zero for j 2k + 1, while for other j they solve the problems Here the functions ψ j
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D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
−
j −2k−1 (m) 1 ∂2 2H2k (ξ ) ∂ 2 (m) (m) (m)
(m)
q(m) + F
ψ + c = − + Λ ψ + cj −q ψ ξ j j 0 j −2k 2 2 3 2 H0 ∂ξd ∂ξd H0 q=2k+2 (m)
, sin πnξd −2 F j L
2 (Π)
sin πnξd
in Π,
(m) = 0 on ∂Π, ψ j
(m) := F j
j −2k−1
N j −2k−2 (m) (m) (p)
q + Ψ
q(m) sin πnξd + Lj −q−2k ψ bq−1,p Lj −q−2k Φ1 sin πnξd
q=0
p=1
+
N j −2k−3 p=1
q=1
(p)
(m) bq,p Lj −q−2k ψ0 ,
q=0
and are represented as finite sums
(m) (ξ ) = ψ j
ψj,ς,1 (ξ )ψj,ς,2 (ξd ), (m)
(m)
ς
where ψj,ς,1 ∈ V, ψj,ς,2 ∈ C0∞ [0, 1], ψj,ς,2 (0) = ψj,ς,2 (1) = 0, and the functions ψj,ς,2 are
(m) ∈ V are solutions to the equations orthogonal to sin πnξd in L2 (0, 1). The functions Ψ (m)
(m)
(m)
(m)
(m)
j
(Gn − Λ)Ψ j
(m)
= f j +2k + (m)
j −1
q(m) + cj +2k−q Ψ (m)
q=1
+
j −2 N
j −3 N
(m)
(p)
(m) cj +2k−q bq,p Ψ0
q=1 p=1
(p) (m) (m) cj +2k−q bq−1,p Φ1
q=1 p=1
N (m) (p) f j +2k , Ψ0 L −
2 (R
(p)
d−1 )
Ψ0 ,
p=1
(m) and cj(m) are and are orthogonal to all Ψ0(l) , l = 1, . . . , N , in L2 (Rd−1 ). The constants bj,p determined by the formulae (m) bl,−1 = 0,
(m) (m) b0,l = δml , bj,m = 0, j 1, (m)
(m) , ψ (l) ) + j −1 c(m) 2(F q=1 j +2k−q+2 bq,l j +2k+2 0 (m) bj,l = , m= l, j 1, (m) (l) τ2k+1 − τ2k+1 (m)
c2k = Λ,
(m)
c2k+1 = −τ (2k+1) , (m) (m) (m)
,ψ , cj = −2 F j 0 L (Π) 2
(m)
c2k+2 = −τm(2k+2) , j 2k + 3.
These lemmata can be proven by induction. Remark 3.8. We observe that if Λ is simple, then N = 1 and the hypothesis of Lemma 3.6 is obviously true.
D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
907
We denote s (m) x − x xd + εh− (x ) (m) , , ψε,s (x) := χ(x ) H (x ) η j ψj η εH (x ) j =0
−2 λ(m) ε,s := ε c0
(m)
+ ε −2
s
(m)
s 2k.
η j cj ,
j =2k (m)
The next lemma follows from the construction of the functions ψj
(m)
and the constants cj .
(m)
Lemma 3.9. The functions ψε,s solve the boundary value problems (m) (m) (m) − D Ωε + λε,s ψε,s = gε,s ,
m = 1, . . . , N,
(3.24)
where the right-hand sides satisfy the estimate (m) g ε,s
L2 (Ωε )
3k−d = O ηs− 2 −2 ,
m = 1, . . . , N.
(3.25)
We rewrite problem (3.24) as (m) (m) ψε,s = Aε ψε,s +
1 (m)
1 + λε,s
(m) Aε gε,s ,
−1 where Aε := (−D Ωε + 1) . This operator is self-adjoint, compact and satisfies the estimate Aε 1. In view of this estimate and (3.25) we have
1
(m) Aε gε,s 1 + λ(m) L2 (Ωε ) ε,s
Cm,s ηs−
3k−d 2 +2k
,
m = 1, . . . , N,
where Cm,s are constants. We apply Lemma 1.1 to conclude that there exists an eigenvalue (m) s (ε) of Aε such that (m) (ε) − 1 + λ(m) −1 Cm,s ηs− 3k−d 2 +2k , s
ε,s
m = 1, . . . , N.
(m) −1 − 1 is an eigenvalue of the operator −D Hence, the number λ(m) s (ε) := s (ε) Ωε , which satisfies the inequality (m) 7k−d λ (ε) − λ(m) C
m,s ηs− 2 −4 , s ε,s
m = 1, . . . , N,
(3.26)
m,s are constants. where C (m)
m,s η C
m,s−1 as ε εs(m) . We choose the Let εs be a monotone sequence such that C (m) (m) (m) (m) (m) eigenvalue λε := λε,s as ε ∈ [εs , εs+1 ). Inequality (3.26) implies that the eigenvalue λε has the asymptotics (2.2). We employ Lemma 1.1 in [8, Ch. III, Sec. 1.1] once again with
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D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
√ 3k−d (m) α = Cm,s ηs− 2 , d = α. It yields that there exists a linear combination ψs (x, ε) of the (m) (m) eigenfunctions of −D Ωε associated with the eigenvalues lying in [λε − d, λε + d] such that (m) ψ (·, ε) − ψ (m) s
ε,s
L2 (Ωε )
2s−3k+d , =O η 4
m = 1, . . . , N.
(m)
(m)
Since the functions ψε,s are linearly independent for different m, the same is true for ψs (·, ε), (m) if s is large enough. Thus, the total multiplicity of the eigenvalues λε is at least N. The proof is complete. 4. Proof of Theorem 2 In order to prove Theorem 2 we need to ensure that, for sufficiently small ε, the asymptotic expansions for λ1,m , m = 1, . . . , N , provided by Theorem 1 do correspond to the first N eigenvalues of −D Ωε (counting multiplicities). In [1] this was done by means of adapting the proof of Theorem 1.1 in [4] from the situation where h− = 0 to our case. In the present context we need to show that, under the conditions for h± , this result may be extended to d dimensions. There are two important points that should be stressed here. On the one hand, we are assuming C ∞ regularity in a neighbourhood of the point of global maximum, and thus do not have to deal with what could now be more complex regularity issues at this point. On the other hand, since the proof of eigenvalue convergence given in [4] is based on convergence in the norm, it is not affected by details related to the possible higher multiplicities as was the case in the derivation of the formulae in the previous section. While still using the notation defined in Section 2, we also refer to the notation in [4]. In particular, the function h and the operator H defined there correspond to our width function H and operator Gn , respectively. We begin by assuming H to be strictly positive in ω. Let thus
π(xd + εh− (x )) 2 sin . ψ(x , xd ) = ψχ (x , xd ) = χ(x ) εH (x ) εH (x )
As in [4], we have ψχ (x , xd )
L2 (Ωε )
=
χ 2 (x ) dx ,
ω
while now
εh+ (x )
ω −εh− (x )
∇ψχ (x , xd )2 =
∇χ(x )2 +
π2 + v(x ) χ 2 (x ) dx , ε 2 H 2 (x )
ω
with 2 1 2 1 π2 1 1 + 2 ∇H (x ) . v(x ) = 2 ∇H (x ) − ∇h− (x ) + 4 3 π H (x ) 2
D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
909
In the notation of [4], the potential Wε appearing in the quadratic form qε [χ] (Eq. (1.3) on page 3 in that paper) is now defined by Wε (x ) =
1 1 π2 − + v(x ). ε 2 H 2 (x ) H 2 (x)
We consider the scaling x = eα t as before, which causes the domain ω to be scaled to ωε = eα ω. Then the proofs of Lemma 2.1 and Theorem 1.2 go through with minor changes (note that m = 2k, while I and Iε should be changed by ω and ωε , respectively). Similar remarks apply to the proofs in Section 4 of [4] leading to the proof of Theorem 1.3, except that due to regularity we do not need to worry about separating the domain into different parts as was necessary there for the intervals Iε . Finally, we relax the condition on the strict positivity of H mentioned above. This again follows in a similar fashion to what was done in Section 6.1 of [4]. We are now in conditions to proceed to the proof of (2.5). In the case considered the lowest eigenvalue of G1 is Λ = dj =1 θj , while the associated eigenfunction is given by (2.6). This (1,1)
proves the formula for c2 . In view of Remark 3.8, we can employ Lemma 3.6 to calculate (1,1) (1,1) c3 , c4 . Since Ψ0 is even w.r.t. each ξi , i = 1, . . . , d − 1, and H3 (−ξ ) = −H3 (ξ ), we (3) (1,1) conclude by (3.19) that T11 = 0. By Theorem 1 it yields that c3 = 0.
Eq. (3.11) for Ψ1 with j = 3 reads as follows
1 = (G1 − Λ)Ψ
2π 2 H03
(4.1)
H3 Ψ0 .
1 = RΨ0 , where R is a polynomial of the form We seek the solution as Ψ R(ξ ) :=
d−1
Cpqj ξp ξq ξj +
p,q,j =1
d−1
(4.2)
Cj ξ j ,
j =1
where Cpqj , Cj are constants to be found, and Cpqj are invariant under each permutation of
1 , Ψ0 )L (Rd−1 ) = 0. We the indices p, q, j . We also note that such a choice of R ensures that (Ψ 2
1 into (4.1) taking into account (2.8), substitute (4.2) and the formula for Ψ
2
d−1
(θp + θq + θj )Cpqj ξp ξq ξj + 2
p,q,j =1
=−
d−1
θ j Cj ξ j + 6
j =1
2π 2
d−1
H03
p,q,j =1
d−1
Cppj ξj
p,j =1
βpqj ξp ξq ξj .
It yields the formulae Cpqj = −
π 2 βpqj H03 (θp + θq + θj )
π 2 βppj 3 . 2 H03 θj (2θp + θj ) d−1
,
Cj =
p=1
(4.3)
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D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
It is easy to check that 1 ∂H2 1 (0) (ξ ), P0 = x H2 , H0 ∂xi 2H0 (d) P4 (ξ ) = H0−4 3H22 (ξ ) − 2H0 H4 (ξ ) .
Q1 (ξ ) = − (i)
(1,1)
Employing these identities, we write the formula for c4 (1,1)
c4
(4.4)
from Lemma 3.6
4 , ψ0 )L2 (Π) = −2(L2 ψ0 , ψ0 )L2 (Π) − 2(L1 Ψ
1 sin πξd , ψ0 )L2 (Π) = −2(F (d) (0) = π 2 P4 Ψ0 , Ψ0 L (Rd−1 ) − P0 Ψ0 , Ψ0 L (Rd−1 ) 2
+ 4π
d−1
Q(i) 1
i=1
−2
2
∂Ψ0 sin πξd , Ψ0 ξd cos πξd ∂ξi
L2 (Π)
d−1 (i) 2 ∂ 2 ψ0 (d)
1 , Ψ0 K0 , ψ + π 2 P3 Ψ 0 L2 (Rd−1 ) 2 ∂ξd L2 (Π) i=1
= π 2 P4(d) Ψ0 , Ψ0 L
2
d−1 d−1 ∂Ψ0 1 2 Q(i) αi − , Ψ 0 1 ∂ξ 2H0 i L2 (Rd−1 )
+ (Rd−1 )
i=1
+
i=1
2 d−1 (d) π 2 ∂h1
1 , Ψ0 (x) + π 2 P3 Ψ L2 (Rd−1 ) 2 H0 i=1 ∂xi
(d) (d) = π 2 P4 + P3 Ψ0 , Ψ0 L
2
+ (Rd−1 )
2 d−1 π 2 ∂h1 (x) . H02 i=1 ∂xi (1,1)
We substitute (4.2), (4.3), (4.4) into this identity and arrive at the desired formula for c4
.
5. The d-dimensional ellipsoid As an application of our results, we will derive the expression (1.3) for the asymptotic expansion for the first eigenvalue for a general ellipsoid. From the equation defining the boundary of E and assuming that, as mentioned in the Introduction, we are doing the scaling along the xd axis, we have 2 x1 xd−1 2 1/2 h± (x ) = ad 1 − − ··· − , a1 ad−1 while H (x ) = 2h± (x ). We thus have x located at the origin and H0 = 2ad . Expanding H around x we have
D. Borisov, P. Freitas / Journal of Functional Analysis 258 (2010) 893–912
911
4 x1 2 ad x1 xd−1 2 xd−1 4 − H (x ) = 2ad − ad + ··· + + ··· + a1 ad−1 4 a1 ad−1 2 2 2 x1 x 2 x1 x3 xd−2 xd−1 +2 +2 + ··· + 2 + ···, a1 a2 a1 a3 ad−2 ad−1
yielding Hk = hk = 0 for odd k and
H2 (x ) = −ad
d−1 2 xi i=1
H4 (x ) = −
ai
d−1 d−1 ad xi xj 2 . 4 ai aj i=1 j =1
Hence √ 2ad αi = , ai
θi =
π 2ai ad
and 1−d
1−d
2
2
x x 2 4 ad 4 − 4aπ ( a1 +···+ ad−1 ) d 1 d−1 . ψ0 (x ) = e (a1 . . . ad−1 )1/2
1 . It is now straightforward Note that since H3 is identically zero, there is no need to compute Ψ to obtain (1,1)
c0
=
π2 4ad2
(1,1)
and c2
=
d−1 π 1 . 2ad ai i=1
It remains to compute (1,1) c4
π2 = 16ad2
d−1 2 d−1 d−1 xi 2 xi xj 2 + 3 ψ0 (x ), ψ0 (x ) ai ai aj i=1
π2
= 2
d+3 2
d+3 2
ad
i=1 j =1
d−1 d−1 xi xj 2 e ai aj
L2 (Rd−1 )
x2 x2 − 2aπ ( a1 +···+ ad−1 ) d 1 d−1
dx
(a1 . . . ad−1 )1/2 i=1 j =1Rd−1
which, after some further simplifications, yields the desired result. References [1] D. Borisov, P. Freitas, Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains, Ann. Inst. H. Poincaré Anal. Non-Linéaire 26 (2009) 547–560. [2] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989.
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[3] P. Freitas, Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi, J. Funct. Anal. 251 (2007) 376–398, doi:10.1016/jfa.2007.04.012. [4] L. Friedlander, M. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip, Israel J. Math. 170 (2009) 337–354. [5] I.M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Oldbourne Press, London, 1965. [6] D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, with editorial assistance from J. Hale and A.L. Pereira, London Math. Soc. Lecture Note Ser., vol. 318, Cambridge Univ. Press, Cambridge, 2005. [7] D.D. Joseph, Parameter and domain dependence of eigenvalues of elliptic partial differential equations, Arch. Ration. Mech. Anal. 24 (1967) 325–351. [8] O.A. Olejnik, A.S. Shamaev, G.A. Yosifyan, Mathematical Problems in Elasticity and Homogenization, Stud. Math. Appl., vol. 26, North-Holland, Amsterdam, 1992.
Journal of Functional Analysis 258 (2010) 913–932 www.elsevier.com/locate/jfa
Asymptotic properties of Gabor frame operators as sampling density tends to infinity ✩ Wenchang Sun Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China Received 31 May 2009; accepted 23 September 2009 Available online 1 October 2009 Communicated by N. Kalton
Abstract We study the asymptotic properties of Gabor frame operators defined by the Riemannian sums of inverse windowed Fourier transforms. When the analysis and the synthesis window functions are the same, we give necessary and sufficient conditions for the Riemannian sums to be convergent as the sampling density tends to infinity. Moreover, we show that Gabor frame operators converge to the identity operator in operator norm whenever they are generated with locally Riemann integrable window functions in the Wiener space. © 2009 Elsevier Inc. All rights reserved. Keywords: Gabor frame; Windowed Fourier transform; Frame operator; Sampling density; Walnut’s representation
1. Introduction and the main result Given t, ω ∈ Rd , we define the time-frequency shift τ (t, ω) for functions g on Rd by τ (t, ω)g (x) = g(x − t)ei2πx,ω . The windowed Fourier transform of f ∈ L2 (Rd ) with respect to g ∈ L2 (Rd ) is defined by (Fg f )(t, ω) = f, τ (t, ω)g .
(1.1)
✩ This work was supported partially by the National Natural Science Foundation of China (10971105 and 10990012) and the Natural Science Foundation of Tianjin (09JCYBJC01000). E-mail address:
[email protected].
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.018
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And we can reconstruct f from (Fg f )(t, ω) with the following formula, f=
1 g, ˜ g
(Fg f )(t, ω)τ (t, ω)g˜ dt dω,
(1.2)
R2d
where g˜ ∈ L2 (Rd ) satisfies g, ˜ g = 0 and the integral is convergent in L2 (Rd ) norm (see Proposition 2.4 below). Let a, b > 0 be constants. Whenever {τ (na, mb)g: m, n ∈ Zd } is a frame for L2 (Rd ), that is, there exist positive numbers A and B, called the lower and upper frame bounds, respectively, such that Af 22
f, τ (na, mb)g 2 Bf 2 , 2
∀f ∈ L2 Rd ,
n,m∈Zd
˜ m, n ∈ Zd } and {τ (na, mb)g: m, n ∈ Zd } then there is some g˜ ∈ L2 (Rd ) such that {τ (na, mb)g: 2 d are a pair of dual frames and for any f ∈ L (R ), f=
f, τ (na, mb)g τ (na, mb)g. ˜
(1.3)
n,m∈Zd
In this case, we can recover f from the frame coefficients {f, τ (na, mb)g: m, n ∈ Zd } if a dual frame {τ (na, mb)g: ˜ m, n ∈ Zd } is known. If {τ (na, mb)g: m, n ∈ Zd } is a tight frame, i.e., A = B, things become simple. We can choose g˜ = g/A and then (1.3) holds for any f ∈ L2 (Rd ). However, it is not always an easy thing in general, or not even possible, to find a function g˜ for (1.3) to hold. In fact, much work has been done on the construction of dual Gabor frames, e.g., see [1,3–8,13–16,18–21] for an overview. Define Sa,b;g,g˜ f =
(ab)d f, τ (na, mb)g τ (na, mb)g. ˜ g, ˜ g d
(1.4)
n,m∈Z
Then Sa,b;g,g˜ f can be regarded as a Riemannian sum of the integral in (1.2). We may expect that it is well defined for every f ∈ L2 (Rd ) and that it converges to f whenever a and b tend to zero. Weisz [22] proved that this is the case if both g and g˜ are in S0 (Rd ) := {g: Fg g ∈ L1 (R2d )}, and he proved the convergence in various norms. When we consider Sa,b;g,g˜ as a frame operator, the convergence of Sa,b;g,g˜ f implies that {τ (na, mb)g: m, n ∈ Zd } is asymptotically close to a tight frame. Note that Weisz’s result [22] involves window functions in S0 (Rd ). For example, when we consider window functions g = g˜ = χ[0,1]d , it is not clear whether Sa,b;g,g˜ f is convergent to f as (a, b) tends to (0, 0) since neither g nor g˜ is in S0 (Rd ). For the case of g = g, ˜ we give necessary and sufficient conditions on g for Sa,b;g,g f to be well defined for every f ∈ L2 (Rd ) and convergent as (a, b) tends to (0, 0). And for the general
W. Sun / Journal of Functional Analysis 258 (2010) 913–932
915
case, we give a weaker condition on g, g˜ for Sa,b;g,g˜ f to be convergent. As a consequence, we find that whenever both g and g˜ are in the Wiener space
g · χn+[0,1)d ∞ < ∞ , W Rd := g: g is measurable and gW := n∈Zd
then Sa,b;g,g˜ f is convergent to f as (a, b) tends to (0, 0). Since S0 (Rd ) is a proper subspace of W (Rd ) [10, Theorem 3.2.13], our result improves the one of Weisz’s. Moreover, we also consider the convergence of Sa,b;g,g˜ f for f ∈ Lp (Rd ). With the help of Walnut’s representation, we show that Sa,b;g,g˜ is convergent to the identity operator in operator norm for any 1 p ∞ provided g, g˜ ∈ W (Rd ) and g · g˜ is locally Riemann integrable, i.e., it is Riemann integrable on [−A, A]d for any A > 0. Specifically, we have the following. Theorem 1.1. Let g, g˜ ∈ W (Rd ). Then we have: (i) For any f ∈ Lp (Rd ), 1 p < ∞, lim
(a,b)→(0,0)
Sa,b;g,g˜ f − f p = 0
(1.5)
and the conclusion fails if p = ∞. (ii) Moreover, if g · g˜ is locally Riemann integrable, then for any 1 p ∞, lim
(a,b)→(0,0)
Sa,b;g,g˜ − I Lp →Lp = 0.
(1.6)
Note that Theorem 1.1(ii) implies that {τ (na, mb)g: n, m ∈ Z} is frame for L2 (Rd ) whenever a and b are small enough. Moreover, the condition that g · g˜ is locally Riemann integrable is not redundant. A counterexample is given in Section 3. The paper is organized as following. In Section 2, we consider the Gabor frame operators on L2 (Rd ). We give necessary and sufficient conditions for Sa,b;g,g to converge to the identity operator in weak∗ sense. And in Section 3, we prove the strong and weak∗ convergence result for Gabor frame operators on Lp (Rd ), 1 p ∞. At the end of this paper, we give an example. Rd ) 2. Gabor frame operators on L2 (R In this section, we study the convergence of Sa,b;g,g f for f ∈ L2 (Rd ). First, we consider the special case of g = g. ˜ In this case, we write simply Sa,b instead of Sa,b;g,g , i.e., Sa,b f =
(ab)d f, τ (na, mb)g τ (na, mb)g. 2 g2 d
(2.1)
n,m∈Z
Recall that a sequence {ϕn : n ∈ Z} in a Hilbert space H is said to be a Bessel sequence if f, ϕn 2 < ∞, n∈Z
∀f ∈ H.
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We refer to [2,4,11,23] for an introduction to frames and Bessel sequences. In particular, it was shown that {ϕn : n ∈ Z} is a Bessel sequence if and only if there exists some constant M < +∞ such that f, ϕn 2 < Mf 2 ,
∀f ∈ H.
n∈Z
The constant M is called the upper frame bound for the Bessel sequence. The main result in this section is the following. Theorem 2.1. Let g ∈ L2 (Rd ) and Sa,b be defined as in (2.1), where a, b > 0. Then the following assertions are equivalent. (i) For any f ∈ L2 (Rd ), there exist constants af , bf > 0 such that Sa,b f is well defined whenever 0 < a < af and 0 < b < bf and the limit lim(a,b)→(0,0) Sa,b f exists in the L2 (Rd ) sense. (ii) Sa,b is well defined on L2 (Rd ) for a, b ∈ (0, 1] and there exists some constant M < +∞ such that Sa,b M,
∀a, b ∈ (0, 1].
d (iii) There is some constant M < +∞ such that {(ab)d/2 g−1 2 τ (na, mb)g: n, m ∈ Z } is a Bessel sequence with upper frame bound M, ∀a, b ∈ (0, 1].
Remark 2.1. It was shown in [8,11] that there exist functions g ∈ L2 (Rd ) such that {τ (na, mb)g: n, m ∈ Zd } is not a Bessel sequence even if both a and b are small enough. In this case, we see from the above theorem that Sa,b f does not converge for all f ∈ L2 (Rd ). For the limit lima,b→0 Sa,b f to make sense, Sa,b f must be well defined whenever a and b are small enough. The following lemma gives a necessary and sufficient condition for Sa,b f to be well defined. Lemma 2.2. Let g ∈ L2 (Rd ) and Sa,b be defined as in (2.1), where a, b > 0. Then the following assertions are equivalent. (i) For any f ∈ L2 (Rd ), there exist constants af , bf > 0 such that Sa,b f is well defined whenever 0 < a < af and 0 < b < bf . (ii) Sa,b f is well defined for any f ∈ L2 (Rd ) and any a, b > 0. (iii) {τ (na, mb)g: m, n ∈ Zd } is a Bessel sequence in L2 (Rd ) for any a, b > 0. Proof. First, we assume that (i) holds. Fix some f ∈ L2 (Rd ). Then we have 2 f, τ (na, mb)g 2 = g2 Sa,b f, f < ∞, (ab)d d
n,m∈Z
∀0 < a < af , 0 < b < bf .
W. Sun / Journal of Functional Analysis 258 (2010) 913–932
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For any a, b > 0, we can find some positive integer K such that a/K < af and b/K < bf . Hence f, τ (na, mb)g 2 f, τ (na/K, mb/K)g 2 < ∞, n,m∈Zd
∀a, b > 0.
n,m∈Zd
Since f is arbitrary, the above inequality implies that {τ (na, mb)g: m, n ∈ Zd } is a Bessel sequence. Hence (iii) holds. On the other hand, it is easy to see that (iii) ⇒ (ii) and (ii) ⇒ (i) are obvious. This completes the proof. 2 Next we consider the convergence of the partial sum of the series (2.1). Define Sa,b;g,g,N ˜ f =
(ab)d g, ˜ g
f, τ (na, mb)g τ (na, mb)g. ˜
(2.2)
n,m∈Zd na∞ ,mb∞ N
We write Sa,b;N instead of Sa,b;g,g,N whenever g = g. ˜ ˜ The following result appeared in the proof of [22, Theorem 4]. To keep the paper selfcontained, we include a proof. Lemma 2.3. Suppose that g, g˜ ∈ L2 (Rd ) and g, ˜ g = 0. Then for any f ∈ L2 (Rd ) and N 1, 1 Sa,b;g,g,N lim ˜ f − (a,b)→(0,0) g, ˜ g
(Fg f )(t, ω)τ (t, ω)g˜ dt dω = 0. 2
(t,ω)∈[−N,N ]2d
Proof. For a, b > 0, define
Em,n = (t, ω): t − na∞ a/2, ω − mb∞ b/2 ,
n, m ∈ Zd .
Then we have g, ˜ f − ˜ gSa,b;g,g,N
(Fg f )(t, ω)τ (t, ω)g˜ dt dω
(t,ω)∈[−N,N ]2d
2
f, τ (t, ω)g τ (t, ω)g˜ − f, τ (na, mb)g τ (na, mb)g˜ dt dω 2
n,m∈Zd Em,n na∞ ,mb∞ N
+
f, τ (t, ω)g τ (t, ω)g˜ dt dω 2
Da,b;N
n,m∈Zd Em,n na∞ ,mb∞ N
f, τ (t, ω)g − f, τ (na, mb)g τ (t, ω)g˜ dt dω 2
(2.3)
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+
f, τ (na, mb)g τ (t, ω)g˜ − τ (na, mb)g˜ dt dω 2
n,m∈Zd Em,n na∞ ,mb∞ N
f, τ (t, ω)g τ (t, ω)g˜ dt dω 2
+ Da,b;N
= I + II + III, where
I=
f, τ (t, ω)g − f, τ (na, mb)g τ (t, ω)g˜ dt dω, 2
n,m∈Zd Em,n na∞ ,mb∞ N
II =
f, τ (na, mb)g τ (t, ω)g˜ − τ (na, mb)g˜ dt dω, 2
n,m∈Zd Em,n na∞ ,mb∞ N
III =
f, τ (t, ω)g τ (t, ω)g˜ dt dω, 2
Da,b;N
a b Da,b;N = (t, ω): t∞ − N < , ω∞ − N < . 2 2 Since f, τ (t, ω)g is uniformly continuous on [−N, N ]2d , we have lim
(a,b)→(0,0)
I
lim
(a,b)→(0,0)
22d (N + a)d (N + b)d g ˜ 2·
sup
t−na∞
f, τ (t, ω)g − f, τ (na, mb)g
= 0. Observe that |f, τ (na, mb)g| f 2 g2 and that τ (t, ω)g is a uniformly continuous map from R2d to L2 (Rd ). We also have lim
(a,b)→(0,0)
II = 0.
Finally, since f, τ (t, ω)gτ (t, ω)g ˜ 2 f 2 g2 g ˜ 2 and the measure of Da,b;N tends to zero as (a, b) tends to (0, 0), we get lim
(a,b)→(0,0)
Now the conclusion follows.
2
III = 0.
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The following proposition is known in Fourier analysis, e.g., see [9] or [11, p. 48] for a proof. ˜ g = 0. Then for any ε > 0 and f ∈ Proposition 2.4. Suppose that g, g˜ ∈ L2 (Rd ) and g, L2 (Rd ), there exists some N > 0 such that f − 1 (Fg f )(t, ω)τ (t, ω)g˜ dt dω < ε. g, ˜ g 2 (t,ω)∈[−N,N ]2d
To prove the main result in this section, we also need the following lemma. Lemma 2.5. Suppose that g ∈ L2 (Rd ), f ∈ Cc∞ (Rd ) and Λ ⊂ Z2d . Then we have 1/2 2 1/2 2 d (F (F (ab) f )(na, mb) − f )(t, ω) dt dω g g (m,n)∈Λ
(m,n)∈Λ E
m,n
f ;a,b g2 , where Em,n is defined by (2.3) and f ;a,b :=
α,β∈Id , |α|+|β|>0
2|α|+2|β| a |α| b|β| X β D α f . 2 |α| π
(2.4)
Here we use the following set of multi-indexes: Id := {(i1 , . . . , id ): ik = 0 or 1}, α = (α1 , . . . , αd ), β β |α| = α1 + · · · + αd , x α = x1α1 · · · xdαd , (X β f )(x) = x β f (x) = x1 1 · · · xd d f (x), and D α f (x) = ∂ |α| α α ∂x11 ···∂xdd
f (x) stands for classical partial derivatives.
Proof. It is easy to see that for any f, g ∈ L2 (Rd ), (Fg f )(t, ω) = e−i2πt,ω (Ff g)(−t, −ω).
(2.5)
Now, suppose that f ∈ Cc∞ (Rd ). We see from [8, Eqs. (3.15) and (3.18)] that (Ff g)(t, ω)ei2πt,ω − (Ff g)(na, mb)ei2πt,mb 2 dt dω 2
2 f ;a,b g2 .
m,n∈Zd Em,n
It follows that (Fg f )(t, ω) − ei2πna−t,mb (Fg f )(na, mb)2 dt dω (m,n)∈Λ E
m,n
(Ff g)(−t, −ω)ei2πt,ω − (Ff g)(−na, −mb)ei2πt,mb 2 dt dω = (m,n)∈Λ E
m,n
using (2.5)
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(Ff g)(−t, −ω)ei2πt,ω − (Ff g)(−na, −mb)ei2πt,mb 2 dt dω
(m,n)∈Zd Em,n
=
(Ff g)(t, ω)ei2πt,ω − (Ff g)(na, mb)ei2πt,mb 2 dt dω
(m,n)∈Zd Em,n
2f ;a,b g22 .
(2.6)
By the triangle inequalities in both 2 (Z2d ) and L2 (Em,n ), we get
d
(ab)
2 f, τ (na, mb)g
1/2
(m,n)∈Λ
1/2
(Fg f )(na, mb) 2 2 =
L (Em,n )
(m,n)∈Λ
=
(Fg f )(t, ω) − (Fg f )(t, ω) − ei2πna−t,mb (Fg f )(na, mb) 2 2
1/2
L (Em,n )
(m,n)∈Λ
(Fg f )(t, ω)
L2 (Em,n )
(m,n)∈Λ
− (Fg f )(t, ω) − ei2πna−t,mb (Fg f )(na, mb) L2 (E (Fg f )(t, ω) 2 2
m,n )
2
1/2
1/2
L (Em,n )
(m,n)∈Λ
−
(Fg f )(t, ω) − ei2πna−t,mb (Fg f )(na, mb) 2 2
1/2
L (Em,n )
(m,n)∈Λ
1/2 2 (Fg f )(t, ω) dt dω − f ;a,b g2 , (m,n)∈Λ E
m,n
where we used (2.6) in the last step. Similarly we can prove that
2 (ab)d f, τ (na, mb)g
(m,n)∈Λ
Now the conclusion follows.
1/2
1/2 (Fg f )(t, ω)2 dt dω + f ;a,b g2 . (m,n)∈Λ E
m,n
2
We are now ready to give the proof of Theorem 2.1.
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d Proof of Theorem 2.1. (i) ⇒ (iii). By Lemma 2.2, {(ab)d/2 g−1 2 τ (na, mb)g: m, n ∈ Z } is 2 d 2 d 2 a Bessel sequence in L (R ) for any a, b > 0. Define the operator Ta,b : L (R ) → as the following:
d Ta,b f := (ab)d/2 g−1 2 f, τ (na, mb)g : m, n ∈ Z . Fix some f ∈ L2 (Rd ). Suppose that lim(a,b)→(0,0) Sa,b f = h. Then there are constants > 0 such that
af , bf
Sa,b f − h2 1,
∀0 < a < af , 0 < b < bf .
Hence Sa,b f 2 1 + h2 . Therefore, Ta,b f 22 = Sa,b f, f 1 + h2 f 2 ,
∀0 < a < af , 0 < b < bf .
(2.7)
On the other hand, for any a, b ∈ (0, 1], we can find some integer K < max{1/af , 1/bf } such that a/K < af and b/K < bf , where we use the symbol x to denote the minimal integer which is greater than or equal to x. It follows that Ta,b f 22 =
2 (ab)d f, τ (na, mb)g g22 d m,n∈Z
K 2d ·
(ab)d f, τ (na/K, mb/K)g 2 2 2d g2 K d m,n∈Z
= K 2d Sa/K,b/K f, f K 2d 1 + h2 f 2 ,
0 < a, b 1.
sup Ta,b f 22 < ∞,
∀f ∈ L2 Rd .
(2.8)
Hence
0
By the Banach–Steinhaus theorem [17], we have M :=
sup Ta,b 2 < ∞. 0
d Hence {(ab)d/2 g−1 2 τ (na, mb)g: m, n ∈ Z } is a Bessel sequence with upper frame bound M.
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(iii) ⇔ (ii). Since Sa,b f, f =
(ab)d f, τ (na, mb)g 2 , g22 m,n∈Z
the equivalence of (ii) and (iii) is obvious. (ii) ⇒ (i). This is a special case of Theorem 2.6 (see below).
2
For the case of g = g, ˜ we have the following result. Theorem 2.6. Let g, g˜ ∈ L2 (Rd ) be such that g, ˜ g = 0. Let Sa,b;g,g˜ be defined as in (1.4), where a, b > 0. Suppose that there is some constant M < +∞ such that Sa,b;g,g M
and Sa,b;g, ˜ g˜ M,
∀0 < a, b 1.
Then Sa,b;g,g˜ f is well defined for any f ∈ L2 (Rd ) and lim
(a,b)→(0,0)
Sa,b;g,g˜ f = f,
∀f ∈ L2 Rd .
Proof. First, we assume that f ∈ Cc∞ (Rd ). Let N 1 to be determined later. Denote ΛN = [−N, N ]2d . We have Sa,b;g,g˜ f − f = Sa,b;g,g˜ f − Sa,b;g,g,N ˜ f + Sa,b;g,g,N ˜ f 1 − (Fg f )(t, ω)τ (t, ω)g˜ dt dω g, ˜ g (t,ω)∈ΛN
+
1 g, ˜ g
(Fg f )(t, ω)τ (t, ω)g˜ dt dω − f.
(2.9)
(t,ω)∈ΛN
Take some ε > 0. By Proposition 2.4, we can find some N > 1 such that 1 g, ˜ g
(Fg f )(t, ω)τ (t, ω)g˜ dt dω − f <ε
(t,ω)∈ΛN
(2.10)
2
and
(Fg f )(t, ω)2 dt dω < ε 2 g2 . 2
(t,ω)∈[−N / +1,N −1]2d
(2.11)
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923
On the other hand, we see from Lemma 2.3 that there exist constants a0 , b0 > 0 such that Sa,b;g,g,N ˜ f −
1 g, ˜ g
(Fg f )(t, ω)τ (t, ω)g˜ dt dω < ε,
2
(t,ω)∈ΛN
0 < a < a0 , 0 < b < b0 .
(2.12)
Without loss of generality, we assume that a0 , b0 1. Combining (2.9), (2.10) and (2.12), we get Sa,b;g,g˜ f − f 2 Sa,b;g,g˜ f − Sa,b;g,g,N ˜ f 2 + 2ε.
(2.13)
Next we estimate Sa,b;g,g˜ f − Sa,b;g,g,N ˜ f 2 . We have Sa,b;g,g˜ f − Sa,b;g,g,N ˜ f 2 = sup Sa,b;g,g˜ f − Sa,b;g,g,N ˜ f, h h2 =1
= sup
h2 =1 (na,mb)∈Λ / N
1 sup | g, ˜ g| h2 =1 ·
(ab)d (Fg f )(na, mb)(Fg˜ h)(na, mb) g, ˜ g
2 (ab) (Fg f )(na, mb)
d
(na,mb)∈Λ / N
2 (ab)d (Fg˜ h)(na, mb)
1/2
1/2 .
(2.14)
n,m∈Zd
Since lim(a,b)→(0,0) f ;a,b = 0, by choosing a0 and b0 small enough, we can also suppose that f ;a,b < ε,
0 < a < a0 , 0 < b < b0 .
(2.15)
It follows from Lemma 2.5 that
2 (ab)d (Fg f )(na, mb)
1/2
(na,mb)∈Λ / N
(na,mb)∈Λ / NE
(Fg f )(t, ω)2 dt dω
1/2 + f ;a,b g2
m,n
(Fg f )(t, ω)2 dt dω
1/2 + f ;a,b g2
(t,ω)∈[−N / +1,N −1]2d
g2 + 1 ε,
0 < a < a0 , 0 < b < b0 ,
where we used (2.11) in the last step. Observe that
(2.16)
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2 (ab)d (Fg˜ h)(na, mb) = g ˜ 22 Sa,b;g, ˜ g˜ h, h
n,m∈Zd 2 g ˜ 22 Sa,b;g, ˜ g˜ · h2
Mg ˜ 22 h22 .
(2.17)
Now we see from (2.14), (2.16) and (2.17) that Sa,b;g,g˜ f − Sa,b;g,g,N ˜ f 2
g ˜ 22 M 1/2 g2 + 1 ε. |g, ˜ g|
By (2.13), we get g ˜ 22 M 1/2 g2 + 1 ε + 2ε, |g, ˜ g|
Sa,b;g,g˜ f − f 2
0 < a < a0 , 0 < b < b0 .
Hence lim
(a,b)→(0,0)
Sa,b;g,g˜ f − f 2 = 0,
∀f ∈ Cc∞ Rd .
For arbitrary f ∈ L2 (Rd ) and ε > 0, since Cc (Rd ) is dense in L2 (Rd ), we can find some f0 ∈ Cc∞ (Rd ) such that f − f0 2 < ε. Using the result already proved, we can find some a0 , b0 > 0 such that Sa,b;g,g˜ f0 − f0 2 < ε,
0 < a < a0 , 0 < b < b0 .
Observe that Sa,b;g,g˜ h1 , h2 = h1 , τ (na, mb)g · τ (na, mb)g, ˜ h2 n,m∈Zd 1/2 Sa,b;g,g h1 , h1 1/2 · Sa,b;g, ˜ g˜ h2 , h2 Mh1 2 h2 2 , ∀h1 , h2 ∈ L2 Rd .
We have Sa,b;g,g˜ M,
0 < a, b 1.
Hence Sa,b;g,g˜ f − f 2 Sa,b;g,g˜ (f − f0 ) 2 + Sa,b;g,g˜ f0 − f0 2 + f0 − f 2 (M + 2)ε,
0 < a < a0 , 0 < b < b0 .
Therefore, Sa,b;g,g˜ f converges to f in L2 (Rd ) as (a, b) tends to (0, 0).
2
W. Sun / Journal of Functional Analysis 258 (2010) 913–932
925
3. Gabor frame operators on Lp (Rd ) In this section, we consider the convergence of the operators Sa,b;g,g˜ on Lp (Rd ). It was shown in [11,12,21] that Sa,b;g,g˜ is a bounded operator from Lp (Rd ) to Lp (Rd ) for any 1 p ∞ whenever both g and g˜ are in W (Rd ). We begin with a property on the Wiener space W (Rd ). Proposition 3.1. (See [11, Lemma 6.1.2].) If g ∈ W (Rd ) and a > 0, then d g(x − ak) 1 + 1 gW , a
a.e.
k∈Z
The Walnut’s representation [21] is a fundamental result for Gabor frame operators. Here we cite a version from [11]. For g, g˜ ∈ W (Rd ) and a, b > 0, define Ga,b;n (x) =
n ˜ − ak), g x − − ak g(x b d
n ∈ Zd .
k∈Z
Proposition 3.2 (Walnut’s representation). Let g, g˜ ∈ W (Rd ). Then for any a, b > 0 and 1 p ∞, Sa,b;g,g˜ is a bounded linear operator from Lp (Rd ) to Lp (Rd ). Moreover, 1 d n , a Ga,b;n (x)f x − (Sa,b;g,g˜ f )(x) = g, ˜ g b d
∀f ∈ Lp Rd .
n∈Z
The following lemma is a combination of [11, Lemma 6.3.1] and [21, Lemma 2.1]. Lemma 3.3. For any g, g˜ ∈ W (Rd ), we have n∈Z
1 d Ga,b;n ∞ 1 + (2 + 2b)d gW g ˜ W, a
∀a, b > 0,
(3.1)
and
lim
(a,b)→(0,0)
a d Ga,b;n ∞ = 0.
n∈Zd \{0}
Proof. Put Ek = k + [0, 1]d , k ∈ Zd . By Proposition 3.1, we have
1 Ga,b;n ∞ 1 + a
d g · − n · g˜ . b W
(3.2)
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W. Sun / Journal of Functional Analysis 258 (2010) 913–932
It follows that n∈Zd \{0}
d
d
1 Ga,b;n ∞ 1 + a 1 = 1+ a
g · − n · g˜ b W d
n∈Z \{0}
g ·− n ·χ Ek · g˜ · χEk b ∞ d
n∈Zd \{0} k∈Z
1 d n · g˜ · χEk ∞ . · χEk 1+ g ·− a b ∞ d d
k∈Z
(3.3)
n∈Z \{0}
Fix some integer K 0. Observe that g · − n · χ Ek b ∞ d d
n∈Z
n∈Z
=
l∈Zd
g · χEl ∞
l∈Zd
El ∩(Ek −n/b)=∅
g · χEl ∞
n∈Zd
El ∩(Ek −n/b)=∅
(2 + 2b)d
g · χEl ∞
l∈Zd
= (2 + 2b)d gW . We have g · − n · χ · g˜ · χ ∞ (2 + 2b)d gW · Ek Ek b d
k∞ K
n∈Z
k∞ K
g˜ · χEk ∞ . (3.4)
By setting K = 0, we see from (3.3) and (3.4) that (3.1) holds. On the other hand, for k, l, n ∈ Zd with k∞ K, n = 0 and El ∩ (Ek − n/b) = ∅, we have l∞ 1/b − K − 2. Hence for k ∈ Zd with k∞ K, g · − n · χ Ek b ∞ d d
n∈Z \{0}
l∈Z
g · χEl ∞
n∈Zd \{0}
El ∩(Ek −n/b)=∅
(2 + 2b)d
l∞ 1/b−K−2
g · χEl ∞ .
W. Sun / Journal of Functional Analysis 258 (2010) 913–932
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Consequently, g · − n · χ · g˜ · χEk ∞ Ek b ∞ d
k∞ K
n∈Z \{0}
(2 + 2b)d
g · χEl ∞
l∞ 1/b−K−2
(2 + 2b)d
g˜ · χEk ∞
k∞ K
g · χEl ∞ g ˜ W.
(3.5)
l∞ 1/b−K−2
Putting (3.3), (3.4) and (3.5) together, we get
a d Ga,b;n ∞
n∈Zd \{0}
(1 + a)d
g · − n · χ · g˜ · χEk ∞ Ek b ∞ d d
k∈Z
n∈Z \{0}
(1 + a) (2 + 2b) d
d
g˜ · χEk ∞ gW +
k∞ K
g · χEl ∞ g ˜ W .
l∞ 1/b−K−2
Now we can make n∈Zd \{0} a d Ga,b;n ∞ arbitrarily small by choosing 0 < a < 1 and K and 1/b − K large enough. This completes the proof. 2 Note that Ga,b;0 is independent of b. Let
Ga (x) :=
ad ad Ga,b;0 (x) = g(x − ak)g(x ˜ − ak), g, ˜ g g, ˜ g d
x ∈ Rd .
k∈Z
By Proposition 3.1, we have 1 (1 + a)d g · g ˜ W < ∞. | g, ˜ g| 0
M0 := sup Ga − 1∞ sup 0
The following lemma is the key to Theorem 1.1. Lemma 3.4. Let g, g˜ ∈ W (Rd ) and 1 p ∞. Then we have: (i) for any f ∈ Lp (Rd ), lim(a,b)→(0,0) (Sa,b;g,g˜ f − f p − (Ga − 1)f p ) = 0; (ii) lim(a,b)→(0,0) (Sa,b;g,g˜ − I Lp →Lp − Ga − 1∞ ) = 0.
(3.6)
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W. Sun / Journal of Functional Analysis 258 (2010) 913–932
Proof. Define operators Ta;g,g˜ and Ra,b;g,g˜ on Lp (Rd ) by Ta;g,g˜ f = (Ga − 1)f, 1 n Ra,b;g,g˜ f = a d Ga,b;n · f · − , g, ˜ g b d
f ∈ Lp R d .
n∈Z \{0}
Now we can rewrite the Walnut’s representation as ∀f ∈ Lp Rd .
Sa,b;g,g˜ f − f = Ta;g,g˜ f + Ra,b;g,g˜ f,
(3.7)
By Lemma 3.3, we have
lim
(a,b)→(0,0)
n∈Zd \{0}
ad Ga,b;n ∞ = 0. |g, ˜ g|
(3.8)
Hence lim
(a,b)→(0,0)
Ra,b;g,g˜ Lp →Lp
lim
(a,b)→(0,0)
n∈Zd \{0}
ad Ga,b;n ∞ = 0. |g, ˜ g|
(3.9)
On the other hand, it is easy to see that Ta;g,g˜ Lp →Lp = Ga − 1∞ . Now the conclusion follows.
2
Lemma 3.5. Suppose that f ∈ W (Rd ) is locally Riemann integrable. Then we have d a f (y + na) − f (x) dx = 0. lim sup a→0 d y∈R n∈Zd
(3.10)
Rd
Proof. Let Ek = k + [0, 1]d , k ∈ Zd . Since f ∈ W (Rd ), for any ε > 0, we can find some K > 0 such that
f (x) dx < ε
and
f · χEk ∞ < ε.
k∞ K−1
x∞ K
It follows that for any y ∈ Rd , n∈Zd
y+na∞ >K
a d f (y + na)
n∈Zd
k∈Zd
y+na∞ >K y+na∈Ek
a d f · χEk ∞
W. Sun / Journal of Functional Analysis 258 (2010) 913–932
929
a d f · χEk ∞
k∈Zd n∈Zd k∞ K−1 y+na∈Ek
a
d
1 2+ a
d
f · χEk ∞
k∈Zd k∞ K−1
(2a + 1)d ε.
(3.11)
On the other hand, since f is Riemann integrable on any bounded domain, there is some a0 ∈ (0, 1) such that for any y ∈ Rd and 0 < a < a0 ,
a f (y + na) − d
n∈Zd y+na∞ K
f (x) dx < ε.
x∞ K
It follows that d a f (y + na) − f (x) dx n∈Zd
Rd
a f (y + na) − d
n∈Zd y+na∞ K
f (x) dx +
x∞ K
+
a d f (y + na)
n∈Zd y+na∞ >K
f (x) dx
x∞ K
2ε + (2a + 1)d ε 2 + 3d ε, ∀y ∈ Rd , 0 < a < a0 . Now the conclusion follows.
2
We are now ready to prove the main result. Proof of Theorem 1.1. First, we consider the case of p = 2. By Walnut’s representation and (3.1), it is easy to see that Sa,b;g,g
1 8d d d 2 (1 + a) (2 + 2b) g g2W , W g22 g22
0 < a, b 1
Sa,b;g, ˜ g˜
1 8d d d 2 (1 + a) (2 + 2b) g ˜ g ˜ 2W , W g ˜ 22 g ˜ 22
0 < a, b 1.
and
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W. Sun / Journal of Functional Analysis 258 (2010) 913–932
Now we see from Theorem 2.6 that lim
(a,b)→(0,0)
∀f ∈ L2 Rd .
Sa,b;g,g˜ f − f 2 = 0,
(3.12)
By Lemma 3.4, we have lim (Ga − 1)f 2 = 0,
a→0
∀f ∈ L2 Rd .
(3.13)
Take some f ∈ Lp (Rd ), 1 p < ∞. For any ε > 0, there is some A > 0 such that
f (x)p dx < ε p .
x∞ >A
And there is some δ > 0 such that for any measurable set E ⊂ Rd with |E| < δ,
f (x)p dx < ε p .
E
Here we use | · | to denote the Lebesgue measure of a measurable set. By (3.13), we have
1 lim x ∈ [−A, A]d : Ga (x) − 1 ε lim 2 a→0 a→0 ε
Ga (x) − 12 · χ
[−A,A]d
2 (x) dx = 0.
Rd
Hence, we can find some 0 < a0 < 1 such that for any 0 < a < a0 ,
x ∈ [−A, A]d : Ga (x) − 1 ε < δ. It follows that
(Ga − 1)f p = p
Ga (x) − 1 f (x)p dx
{x∈[−A,A]d : |Ga (x)−1|ε}
Ga (x) − 1 f (x)p dx
+ {x∈[−A,A]d : |Ga (x)−1|<ε}
+
Ga (x) − 1 f (x)p dx
d x ∈[−A,A] /
p
p
p
M0 ε p + ε p f p + M0 ε p , where M0 is defined in (3.6). Hence lim (Ga − 1)f p = 0.
a→0
0 < a < a0 ,
W. Sun / Journal of Functional Analysis 258 (2010) 913–932
931
Now we see from Lemma 3.4 that for any f ∈ Lp (Rd ), 1 p < ∞, lim
(a,b)→(0,0)
Sa,b;g,g˜ f − f p = 0.
This proves (1.5) for 1 p < ∞. Next we show that (1.5) is not true for p = ∞. For simplicity, we consider only the case of d = 1. Take some E ⊂ [0, 1] such that E is nowhere dense and is of positive measure. Let g = g˜ = χE . For any a > 0, we have
(na + E) ∩ [0, 1] = x ∈ [0, 1]: Ga (x) > 0 =
(na + E) ∩ [0, 1].
n∞ 1/a
n∈Z
Since each of na + E is nowhere dense, so is Ga (x) > 0} is nowhere dense. Hence
n∞ 1/a (na
+ E). Therefore, {x ∈ [0, 1]:
x ∈ [0, 1]: Ga (x) − 1 = 1 x ∈ [0, 1]: Ga (x) = 0 > 0. Let f0 = χ[0,1] . Then we have (Ga − 1)f0
∞
1,
∀a > 0.
By Lemma 3.4, lim(a,b)→(0,0) Sa,b;g,g˜ f0 − f0 ∞ 1. That is, (1.5) fails for p = ∞. At last, we consider the strong convergence. Since g · g˜ is locally Riemann integrable, we see from Lemma 3.5 that lim
(a,b)→(0,0)
Ga − 1∞ = 0.
(3.14)
Using Lemma 3.4 again, we get lim
(a,b)→(0,0)
Sa,b;g,g˜ − I Lp →Lp = 0.
2
Example 3.1. Consider the example given by Feichtinger and Janssen [5]. Take some 0 < ε < 1. Let g = g˜ = χE , where E = (0, 1) \ G,
G=
l ε l ε . − , + 3k 32k 3k 32k
k1 l∈Z
It is easy to see that E is nowhere dense and of positive measure no less that 1 − ε. It was shown in [5] that {τ (na, mb)g: n, m ∈ Z} has no positive lower frame bound for any a, b > 0. Hence, Sa,b;g,g − I L2 →L2 cannot converge to 0. In other words, as an operator on L2 (Rd ), Sa,b;g,g does not converge to the identity operator strongly. Nevertheless, Sa,b;g,g converges to the identity operator on Lp (Rd ), 1 p < ∞, in weak∗ sense since g ∈ W (Rd ).
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W. Sun / Journal of Functional Analysis 258 (2010) 913–932
Note that in this example, the window function g, which is equal to g · g, ˜ is not locally Riemann integrable since E = {x: g is not continuous at x} is of positive measure. Therefore, this example illustrates that the Riemann integrability in Theorem 1.1(ii) is not redundant. Acknowledgment The author thanks the referee for valuable suggestions which helped to improve the paper. References [1] P.G. Casazza, O. Christensen, A.J.E.M. Janssen, Weyl–Heisenberg frames, translation invariant systems and the Walnut representation, J. Funct. Anal. 180 (2001) 85–147. [2] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003. [3] O. Christensen, Pairs of dual Gabor frame generators with compact support and desired frequency localization, Appl. Comput. Harmon. Anal. 20 (2006) 403–410. [4] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. [5] H.G. Feichtinger, A.J.E.M. Janssen, Validity of W H -frame bound conditions depends on lattice parameters, Appl. Comput. Harmon. Anal. 8 (2000) 104–112. [6] H.G. Feichtinger, T. Strohmer (Eds.), Gabor Analysis: Theory and Applications, Birkhäuser, Boston, 1998. [7] H.G. Feichtinger, W. Sun, Stability of Gabor frames with arbitrary sampling points, Acta Math. Hungar. 113 (2006) 187–212. [8] H.G. Feichtinger, W. Sun, Sufficient conditions for irregular Gabor frames, Adv. Comput. Math. 26 (2007) 403–430. [9] H.G. Feichtinger, F. Weisz, Inversion formulas for the short-time Fourier transform, J. Geom. Anal. 16 (2006) 507–521. [10] H.G. Feichtinger, G. Zimmermann, A Banach space of test functions for Gabor analysis, in [6], pp. 123–170. [11] K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001. [12] K. Gröchenig, C. Heil, Gabor meets Littlewood–Paley: Gabor expansions in Lp (Rd ), Studia Math. 146 (2001) 15–33. [13] Q. Gu, D. Han, When a characteristic function generates a Gabor frame, Appl. Comput. Harmon. Anal. 24 (2008) 290–309. [14] A.J.E.M. Janssen, The duality condition for Weyl–Heisenberg frames, in: H.G. Feichtinger, T. Strohmer (Eds.), Gabor Analysis: Theory and Applications, Birkhäuser, Boston, 1998. [15] A.J.E.M. Janssen, Zak transforms with few zeros and the tie, in: H.G. Feichtinger, T. Strohmer (Eds.), Advances in Gabor Analysis, Birkhäuser, Boston, 2003. [16] A. Ron, Z. Shen, Weyl–Heisenberg frames and Riesz bases in L2 (Rd ), Duke Math. J. 89 (1997) 148–153. [17] W. Rudin, Real and Complex Analysis, McGraw–Hill, 1997. [18] W. Sun, X. Zhou, On the stability of Gabor frames, Adv. in Appl. Math. 26 (2001) 181–191. [19] W. Sun, X. Zhou, Irregular wavelet/Gabor frames, Appl. Comput. Harmon. Anal. 13 (2002) 63–76. [20] W. Sun, X. Zhou, Irregular Gabor frames and their stability, Proc. Amer. Math. Soc. 131 (2003) 2883–2893. [21] D.F. Walnut, Continuity properties of the Gabor frame operator, J. Math. Anal. Appl. 165 (1992) 479–504. [22] F. Weisz, Inversion of the short-time Fourier transform using Riemannian sums, J. Fourier Anal. Appl. 13 (2007) 357–368. [23] R. Young, An Introduction to Non-Harmonic Fourier Series, revised first edition, Acad. Press, San Diego, CA, 2001.
Journal of Functional Analysis 258 (2010) 933–955 www.elsevier.com/locate/jfa
High-frequency propagation for the Schrödinger equation on the torus Fabricio Macià 1 Universidad Politécnica de Madrid, DEBIN, ETSI Navales, Avda. Arco de la Victoria s/n, 28040 Madrid, Spain Received 1 June 2009; accepted 24 September 2009
Communicated by J. Bourgain
Abstract The main objective of this paper is understanding the propagation laws obeyed by high-frequency limits of Wigner distributions associated to solutions to the Schrödinger equation on the standard d-dimensional torus Td . From the point of view of semiclassical analysis, our setting corresponds to performing the semiclassical limit at times of order 1/ h, as the characteristic wave-length h of the initial data tends to zero. It turns out that, in spite that for fixed h every Wigner distribution satisfies a Liouville equation, their limits are no longer uniquely determined by those of the Wigner distributions of the initial data. We characterize them in terms of a new object, the resonant Wigner distribution, which describes high-frequency effects associated to the fraction of the energy of the sequence of initial data that concentrates around the set of resonant frequencies in phase-space T ∗ Td . This construction is related to that of the so-called two-microlocal semiclassical measures. We prove that any limit μ of the Wigner distributions corresponding to solutions to the Schrödinger equation on the torus is completely determined by the limits of both the Wigner distribution and the resonant Wigner distribution of the initial data; moreover, μ follows a propagation law described by a family of density-matrix Schrödinger equations on the periodic geodesics of Td . Finally, we present some connections with the study of the dispersive behavior of the Schrödinger flow (in particular, with Strichartz estimates). Among these, we show that the limits of sequences of position densities of solutions to the Schrödinger equation on T2 are absolutely continuous with respect to the Lebesgue measure. © 2009 Elsevier Inc. All rights reserved. Keywords: Semiclassical (Wigner) measures; Schrödinger equation on the torus; Quantum limits; Two-microlocal Wigner measures; Resonances; Strichartz estimates
E-mail address: [email protected]. 1 This research has been supported by grants MTM2007-61755 (MEC) and Santander-Complutense 34/07-15844.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.020
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F. Macià / Journal of Functional Analysis 258 (2010) 933–955
1. Introduction In this article, we shall consider solutions to Schrödinger’s equation on the standard flat torus Td := Rd /(2πZd ), 1 i∂t u(t, x) + x u(t, x) = 0, 2
(t, x) ∈ R × Td .
(1)
We are interested in understanding the propagation of high-frequency effects associated to solutions to (1). More precisely, given a sequence (uh ) of initial data which oscillates at frequencies of the order of 1/ h (see condition (5) below) we would like to describe in a quantitative manner the propagation of these oscillation effects under the action of the Schrödinger group eitx /2 . Some understanding in this direction can be obtained by analysing the structure of weak ∗ limits of sequences of measures of the form: it /2 e x uh (x)2 dx,
(2)
where (uh ) is a bounded sequence in L2 (Td ) and dx denotes the Lebesgue measure on Td . These limiting measures give information about the regions on which the energy of (eitx /2 uh ) concentrates; a natural question in this context is to understand their dependence on t; and in particular, their dependence on the initial data (uh ). However, it is usually difficult to deal directly with (2). This is due to the presence of the modulus in (2) which prevents us from keeping track of the characteristic directions of oscillation of the functions uh . It is preferable instead to consider their Wigner distributions, which are phase-space densities that take into account simultaneously concentration on physical and Fourier space, and which project onto (2). The main issue addressed in this article is to study the propagation laws obeyed by Wigner distributions of solutions to the Schrödinger equation (1). As a consequence of our results, we shall prove that for d = 2 a limit of a sequence of densities (2) (corresponding to sequence (uh ) satisfying a standard oscillation condition) is absolutely continuous with respect to the Lebesgue measure. At the end of this introduction we discuss the connections of this result with Strichartz estimates. vectors of the standard orthonormal basis of Let ψk (x) := (2π)−d/2 eik·x , k ∈ Zd , denote the 2 u(k)ψk in L2 (Td ) is defined for h > 0 L (Td ). The Wigner distribution of a function u = k∈Zd as: wuh (x, ξ ) :=
u(k) u(j )ψk (x)ψj (x)δ h (k+j ) (ξ ), 2
k,j ∈Zd
(3)
where δp stands for the Dirac delta centered at the point p. The distribution wuh is in fact a measure on T ∗ Td ∼ = Td × Rd ; one easily checks that: Rd
2 wuh (x, dξ ) = u(x) ,
wuh (dx, ξ ) = Td
2 u(k) δhk (ξ ). k∈Zd
Therefore, wuh may be viewed as a microlocal lift of the density |u(x)|2 to phase space T ∗ Td : it allows to describe simultaneously the distribution of energy of u on physical and Fourier space. The distributions wuh are not positive, although their limits as h → 0+ are indeed finite positive
F. Macià / Journal of Functional Analysis 258 (2010) 933–955
935
Radon measures.2 Let us note that the definition of the Wigner distribution in a general compact Riemannian manifold usually depends on various choices (coordinate charts, partitions of unity) that however have no effect on their asymptotic behavior as h → 0+ (see, for instance, [13,17], and the references therein). Our formula (3) corresponds to identifying elements in L2 (Td ) to those functions in L2loc (Rd ) that are 2πZd -periodic and then consider their canonical Euclidean Wigner distributions. One easily checks that this definition is consistent with the generally accepted one. The reader can refer to the book [10] for a comprehensive discussion on Wigner distributions. We shall assume in the following that (uh ) is bounded in L2 (Td ) and write wuhh (t, ·) := wehitx /2 u . h
Then, after possibly extracting a subsequence, the associated Wigner distributions at t = 0 converge (see, for instance, [11–13,16]): wuhh (0, ·) μ0 ∈ M+ T ∗ Td ,
as h → 0+ in D T ∗ Td .
(4)
The measure μ0 is usually called the semiclassical or Wigner measure of (uh ). In addition, one can recover the asymptotic behavior of |uh |2 from μ0 . More precisely: |uh | dx 2
μ0 (·, dξ ),
vaguely as h → 0+ ,
Rd
provided the densities |uh |2 converge and the sequence (uh ) verifies the h-oscillation property3 : lim sup
uh (k)2 → 0,
h→0+ |k|>R/ h
as R → ∞.
(5)
Wigner distributions wuhh (t, ·) associated to solutions to the Schrödinger equation are completely determined by those of their initial data wuhh (0, ·) as they solve the classical Liouville equation: ∂t wuhh (t, x, ξ ) +
ξ · ∇x wuhh (t, x, ξ ) = 0. h
(6)
As a consequence of this, it is possible to show that the rescaled Wigner distributions wuhh (ht, ·) converge (after possibly extracting a subsequence) locally uniformly in t to the measure (φ−t )∗ μ0 , where (φt )∗ denotes the push-forward operator on measures induced by the geodesic flow φt (x, ξ ) := (x + tξ, ξ ) on T ∗ Td . This result, sometimes known as the classical limit, holds in a much more general setting, see, for instance, [12,14,16]. Eq. (6) provides interesting consequences when (un ) is a sequence of (normalised) eigenfunctions −x . Suppose that −x un = λn un and that limn→∞ λn = ∞. Since wuhn is quadratic in un , 2 From now on, we denote the set of such measures by M (T ∗ Td ). + 3 This condition expresses that no positive fraction of the energy of the sequence (u ) concentrates at frequencies h
asymptotically larger than 1/ h.
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F. Macià / Journal of Functional Analysis 258 (2010) 933–955
√ we have wuhn (t, ·) = wuhn (0, ·), for every t ∈ R. Setting h := 1/ λn (so that (un ) is h-oscillating) Eq. (6) implies that any semiclassical measure μ0 of this sequence is invariant by the geodesic flow, i.e. (φs )∗ μ0 = μ0 . Moreover, one can show that μ0 is a probability measure concentrated on the unit cosphere bundle S ∗ Td ; in this context, semiclassical measures are usually called quantum limits. The problem of classifying all possible quantum limits in Td is very hard. Their structure has been clarified by Jakobson [15] for d = 2. For arbitrary d 1, Bourgain has proved (see again [15]) that the projection of a quantum limit onto Td (which is an accumulation point of the measures (|un |2 )) is absolutely continuous with respect to Lebesgue measure. In particular, the sequence (un ) cannot concentrate on sets of dimension lower than d. The article [15] also provides partial results when d 3 (see also [2]). When (un ) is not formed by eigenfunctions, it is not anymore clear that wuhn (t, ·) converges pointwise in t. However, we have the following result, proved in [17] for a general compact manifold: Existence of limits. Given a bounded h-oscillating sequence (uh ) in L2 (Td ), there exists a subsequence such that, for every a ∈ Cc∞ (T ∗ Td ) and every ϕ ∈ L1 (R), lim
h →0+ R×T ∗ Td
ϕ(t)a(x, ξ )wuhh (t, dx, dξ ) dt =
ϕ(t)a(x, ξ )μ(t, dx, dξ ) dt,
(7)
ϕ(t)m(x)μ(t, dx, dξ ) dt,
(8)
R×T ∗ Td
where the limit μ is in L∞ (R; M+ (T ∗ Td )).4 Moreover, lim
h →0+
2 ϕ(t)m(x)eitx /2 uh (x) dx dt =
R Td
R T ∗ Td
for every ϕ ∈ L1 (R) and m ∈ C(Td ). In addition, as one can check by taking limits in Eq. (6), the invariance property satisfied by semiclassical measures corresponding to sequences of eigenfunctions also holds in this more general setting. Invariance. For a.e. t ∈ R, the measure μ(t, ·) is invariant by the geodesic flow on T ∗ Td : (φs )∗ μ(t, ·) = μ(t, ·),
for every s ∈ R.
(9)
See also [17] for a proof in a more general context. In that reference, a characterization of the propagation law for these measures in the class of compact manifolds with periodic geodesic flow (the so-called Zoll manifolds) was given. In fact, a formula relating μ and μ0 exists: μ(t, ·) equals the average of μ0 along the geodesic flow for a.e. t ∈ R (which is well-defined due to the periodicity of the geodesic flow); note, in particular, that μ is constant in time. This fits our setting when d = 1; but when d 2, the dynamics of the geodesic flow in the torus are more complex than those in Zoll manifolds. In both cases, the geodesic flow is completely integrable. However, the torus possesses geodesics of arbitrary large minimal periods, as well as non-periodic, dense, geodesics. 4 Note that, in contrast with the semiclassical limit, it is not true that (w h (t, ·)) (or any subsequence) converges to uh μ(t, ·), even almost everywhere.
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It will turn out that this added complexity will have an effect on our problem. However, there is still a class of sequences of initial data for which the measures μ and μ0 are related by an averaging process. More precisely, μ is obtained by averaging μ0 when the initial data do not see the set of resonant frequencies:
Ω := ξ ∈ Rd : k · ξ = 0 for some k ∈ Zd \ {0} . Proposition 1 (Non-resonant case). (See [17, Proposition 10].) Suppose μ and μ0 are given respectively by (7) and (4) for some sequence (uh ) bounded in L2 (Td ) and verifying (5). If μ0 (Td × Ω) = 0 then, for a.e. t ∈ R, μ(t, x, ξ ) =
1 (2π)d
μ0 (dy, ξ ). Td
Note that in this case, any limit (8) of the densities |eitx /2 uh (x)|2 is a constant function in t and x.5 The role of resonances. Therefore, all the difficulties in analysing the structure of μ rely on understanding its behavior when μ0 actually sees the set of resonant frequencies Ω. Given ξ0 ∈ Ω, in [17], Proposition 11, sequences of initial data (uh ) and (vn ) are constructed such that both have |ρ(x)|2 dx δξ0 (ξ ) as a semiclassical measure. However, when ρ ∈ L2 (Td ) is invariant in the ξ0 -direction, the corresponding limits (7) of the evolved Wigner distributions are, respectively, it /2 e x ρ(x)2 dx δξ (ξ ) 0
and
1 (2π)d
ρ(y)2 dyδξ (ξ ). 0
Td
Two conclusions can be extracted from this fact: (i) The measures μ(t, ·) may have a non-trivial dependence on t, which is not directly related to the dynamics of the geodesic flow and, most importantly, (ii) the measure μ0 corresponding to the initial data no longer determines uniquely the measures μ(t, ·) corresponding to the evolution. The structure of μ is therefore much more complex when μ0 sees the set of resonant frequencies; before stating our main result, let us introduce some notation. We define set W of simple resonant directions in Rd as follows. Consider the subset Ω1 ⊂ Ω consisting of simple resonances (that is, Ω1 is formed by the ξ ∈ Ω such that λξ ∈ Zd for some real λ = 0). Consider the equivalence relation ∼ on Ω1 \ {0} defined by x ∼ y if and only if x, y ∈ Ω1 \ {0} lie on a line through the origin. Define W as the set of equivalence classes of ∼. In other words, W is the subset of the real projective space RPd−1 obtained by projecting Ω1 \ {0} ⊂ Rd using the canonical covering projection. For each ω ∈ W define, γω := {tνω : t ∈ R}/2πZd ⊂ Td ,
where νω ∈ ω.
5 Note also that for d = 1, the condition μ (Td × Ω) = 0 reduces to μ ({ξ = 0}) = 0. This has to be interpreted 0 0 as the requirement that no positive fraction of the energy of the sequence of initial data concentrates at frequencies asymptotically smaller than 1/ h.
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This is the (closed) geodesic of Td issued from the point 0 in the direction νω . There is a bijection between W and the set of closed geodesics in Td that pass through 0. In what follows, L2 (γω ) will denote the space of (equivalence classes of) square integrable functions on γω with respect to arc-length measure. Given ω ∈ W, we shall denote by Iω ⊂ Rd the hyperplane through the origin orthogonal to ω. The structure of the measures μ(t, ·) is given by the next result (in Theorem 9 in Section 3, we precise the nature of the propagation law for μ(t, ·)). Theorem 2. Let (uh ) be a bounded, h-oscillating sequence in L2 (Td ) with semiclassical measure μ0 . Suppose that (7) holds for some measure μ. Then, for a.e. t ∈ R, μ(t, x, ξ ) =
ρωt (x, ξ ) +
ω∈W
1 (2π)d
μ0 (dy, ξ ),
(10)
Td
where ρωt , for t ∈ R and ω ∈ W, is a signed measure concentrated on Td × Iω whose projection on the first component is absolutely continuous with respect to the Lebesgue measure. Moreover, the measure ρωt follows a propagation law related to the Schrödinger flow on L2 (γω ), and can be computed solely in terms of ρω0 , which is in turn completely determined by the initial data (uh ). Note that the sum ω∈W ρωt is concentrated on Td × Ω, and, as we shall prove in Theorem 9, ρωt is related to the trace of a density matrix in L2 (γω ) evolving according to the Schrödinger equation on γω . The measure ρω0 will be obtained as the limit of a new object, the resonant Wigner distribution of the initial data uh , which describes the concentration of energy of the sequence (uh ) over Td × Ω at scales of order one. We introduce its definition, along with a description of the properties that are relevant to our analysis in the next section. Let us just mention that the resonant Wigner distribution may be viewed as a two-microlocal object, in the spirit of the 2-microlocal semiclassical measures introduced by Fermanian-Kammerer [7,8], Gérard and Fermanian-Kammerer [9], Miller [18], and Nier [19]. In particular, ρω0 might vanish even if μ0 (Td × Iω ) > 0. The condition for ρω0 to be zero is the following (see Proposition 12 in Section 4). Suppose that ρωt are given by formula (10). If (uh ) satisfies: lim
h→0+
uh (k)2 = 0,
for every N > 0,
|k·νω |
where νω ∈ ω is a unit vector, then ρωt = 0 for every t ∈ R. Using this characterization, we are able to describe the propagation of wave-packet type solutions, see Proposition 13 in Section 4. We also give there an example of sequence (uh ) for which some of the ρωt are non-zero. As a consequence of formula (10) we prove in Section 3 the following result for the position densities (2). Corollary 3. Let d = 2 and (uh ) be a bounded, h-oscillating sequence in L2 (T2 ) with a semiclassical measure μ0 . If μ0 ({ξ = 0}) = 0 then, up to some subsequence, for every ϕ ∈ L1 (R) and m ∈ C(T2 ),
F. Macià / Journal of Functional Analysis 258 (2010) 933–955
lim
h →0+
R
2 ϕ(t)m(x)eitx /2 uh (x) dx dt =
ϕ(t)m(x)ν(t, dx) dt, R
T2
939
T2
and the measure ν ∈ L∞ (R; M+ (T2 )) is absolutely continuous with respect to the Lebesgue measure in T2 . This result is somehow related to the analysis of dispersion (Strichartz) estimates for the Schrödinger equation. For instance, when d = 1 we have the following inequality due to Zygmund [20]: it/2 e u
L4 (Tt ×Tx )
C u L2 (T) ,
(11)
for some constant C > 0 (note that eit/2 is 2πZ-periodic in t). This estimate implies that for d = 1 any limit of averages in time of the densities (2) is absolutely continuous with respect to Lebesgue measure (and is in fact an L2 (T)-function). However, as shown by Bourgain [1], estimate (11) fails when d = 2; although a version of (11) with a loss of derivatives does hold.6 Therefore, the result given in Corollary 3 supports in some sense the possibility that an inequality such as (11) holds on Tt × T2x in some Lp -space with 2 < p < 4. The present analysis can be extended to more general tori and Schrödinger-type equations arising as the quantization of completely integrable Hamiltonian systems. These issues will be addressed elsewhere. 2. The resonant Wigner distribution 2.1. Preliminaries and definition Let ω ∈ W be a simple resonant direction; as before, denote by Iω ⊂ Rd the linear hyperplane orthogonal to ω. Then there exists a unique pω ∈ ω ∩ Zd such that: (i) the (non-zero) components of pω are coprime; (ii) the first non-zero component of pω is positive. Clearly, ω ∩ Zd = {npω : n ∈ Z}; therefore, the component in the direction pω of any k ∈ Zd is of the form n νω , |pω |
where n = k · pω ∈ Z and νω :=
pω . |pw |
Moreover, since Bezout’s theorem ensures the existence of c ∈ Zd satisfying pω · c = 1, we have that for any given n ∈ Z there exists at least one k ∈ Zd such that k · pω = n. In other words, the set of orthogonal projections onto ω of points in Zd consists of the vectors n/|pω |νω for n ∈ Z. Note that, in particular, the sets 6 References [1,3] describes also positive results. In [4–6] Strichartz estimates in general compact manifolds are established, together with a detailed analysis of the loss of derivatives phenomenon in specific geometries.
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ωn⊥ := r ∈ Iω :
n νω + r ∈ Zd ⊂ Iω , |pω |
are non-empty. ⊥ = ∅ if and only if m ≡ n (mod |p |2 ), in which case It is not difficult to see that ωn⊥ ∩ ωm ω ⊥ ⊥ ⊥ ωn = ωm . This implies that the set ω := n∈Z ωn⊥ consisting of the orthogonal projections on Iω of vectors in Zd is a subgroup of Rd . The results discussed so far imply the following. Proposition 4. For p ∈ Z, denote by Zp := Z/pZ the group of congruence classes modulo p. The map: [c]∈Z|p
ω |2
n νω + r [c] × ωc⊥ → Zd : (n, r) → |pω |
is well-defined and bijective. Moreover, the map h : ω⊥ → Z|pω |2 defined by h(r) := [n] if r ∈ ωn⊥ is a well-defined group homomorphism whose kernel is ω0⊥ ⊂ Zd . Therefore, the quotient group ω⊥ /ω0⊥ is isomorphic to Z|pω |2 and consists of the cosets ωn⊥ = r + ω0⊥ where r is any element of ωn⊥ . The geodesic γω , passing through 0 and pointing in the direction ω, has length 2π|pω |. Therefore, it can be identified to Tω := R/(2π|pω |Z) in such a way that arc-length measure on γω corresponds to a (suitably normalized) Haar measure on Tω . The functions: i
m
s
e |pω | ω (s) := √ φm , 2π|pω |
m ∈ Z,
ω ⊗ φ ω the operator on for an orthonormal basis of L2 (γω ). For n, m ∈ Z, we shall denote by φm n 2 L (γω ) given by:
ω φm
⊗ φnω
ω φk =
ω φm 0
if k = n, otherwise.
We shall denote by L(L2 (γω )), K(L2 (γω )) and L1 (L2 (γω )), the space of linear bounded, compact and trace-class operators on L2 (γω ), respectively. We write: J :=
{ω} × Iω ;
ω∈W
consider on each {ω} × Iω the topology induced by Rd and endow J with the disjoint union topology. To every (ω, ξ ) ∈ J we associate the vector spaces L(L2 (γω )) and K(L2 (γω )); this defines vector bundles over J : πL :
(ω,ξ )∈J
L L2 (γω ) → J ,
πK :
(ω,ξ )∈J
K L2 (γω ) → J .
F. Macià / Journal of Functional Analysis 258 (2010) 933–955
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Let us denote respectively by X (J ) and X0 (J ) the spaces of continuous, compactly supported, sections of the bundles πL and πK . That is, k ∈ X (J ) whenever k(ω, ξ ) ∈ L(L2 (γω )) for every (ω, ξ ) ∈ J and k(ω, ·) is continuous, compactly supported, and non-zero for at most a finite number of ω. Similar considerations hold for the elements of X0 (J ). The dual of X (J ) (resp. X0 (J )) will be denoted by X (J ) (resp. X0 (J )). Given μ ∈ X (J ) (resp. X0 (J )), μ(ω, ·) can be identified to a measure on Iω taking values on L(L2 (γω )) (resp. L1 (L2 (γω ))). (J ) will stand for the cone of positive elements of X (J ); hence, μ ∈ X (J ) if Finally, X0,+ 0 0 1 2 Iω b(ξ )μ(ω, dξ ) ∈ L (L (γω )) is positive and Hermitian whenever b ∈ Cc (Iω ) is positive. Appendix A provides background material, additional details and references on operator-valued measures. The resonant Wigner distribution of u ∈ L2 (Td ), is defined as:
Rhu (ω, ξ ) :=
[c]∈Z|p
ω |2
m,n∈[c] r∈ωc⊥
m n ω νω + r νω + r δhr (ξ )φm u u ⊗ φnω . |pω | |pω |
(12)
Clearly, Rhu ∈ X (J ). 2.2. Boundedness and convergence (J ). Our next result shows, in particular, that Rhu ∈ X0,+
Proposition 5. Let u ∈ L2 (Td ). Then for every ω ∈ W and b ∈ Cc (Iω ), b(ξ )Rhu (ω, dξ ) is a Hermitian, trace-class operator of L2 (γω ), Iω
which is positive if b is non-negative. In addition, we have the following bound: h tr b(ξ )Ru (ω, dξ ) u 2L2 (Td ) sup b(r). r∈Iω
Iω (J ). In particular, Rhu ∈ X0,+
Proof. Define u 2m,ω :=
2 m ; u ν + r |p | ω ω
⊥ r∈ωm
because of Proposition 4, one has u 2L2 (Td ) =
2 m∈Z u m,ω .
Kb :=
b(ξ )Rhu (ω, dξ ); Iω
Take b ∈ Cc (Iω ) and set
(13)
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then Kb =
ω m≡n (mod |pω |2 ) kb (m, n)φm
⊗ φnω with:
m n νω + r νω + r . b(hr) u u |pω | |pω | ⊥
kb (m, n) =
r∈ωc
The operator Kb is bounded, since:
kb (m, n)2 b 2 ∞
u 2m,ω u 2n,ω
L (Iω )
m≡n (mod |pω |2 )
m≡n (mod |pω |2 )
b 2L∞ (Iω )
2
u 2n,ω
= b 2L∞ (Iω ) u 4L2 (Td ) .
n∈Z
Moreover, Kb is Hermitian as soon as b is real valued, since kb (m, n) = kb (n, m); therefore, Kb is a Hilbert–Schmidt operator on L2 (γ ω ). ω . Then Now, given v ∈ L2 (γω ) write v = m∈Z vm φm (Kb v|v)L2 (γω ) =
2 n νω + r . b(hr) vn u |pω | ⊥
[c]∈Z|p
r∈ωc
ω |2
n∈[c]
This quantity is positive whenever b 0 and b ≡ 0. Therefore, for such b the operator Kb is Hilbert–Schmidt (and hence compact), Hermitian and positive. Thus, it will be trace-class as soon as its trace is finite. This is clearly the case, since tr Kb =
2 n u νω + r sup b(r) u 2L2 (Td ) . b(hr) |pω | r∈Iω ⊥
n∈Z r∈ωn
For a general b non-necessarily positive, the result follows by expressing b = b+ − b− and applying the above estimate to each term separately. 2 If (uh ) is a bounded family in L2 (Td ), estimate (13) then shows that Rhuh (ω, ·) is a uniformly (J ). bounded family in X0,+ Proposition 6. Let (uh ) be a bounded sequence in L2 (Td ). Then, there exist a subsequence (uh ) (J ), such that, for every ω ∈ W and b ∈ X (J ): and a finite measure μR ∈ X0,+ 0 lim tr
h →0+
b(ω, ξ )Rhuh (ω, dξ ) = tr
b(ω, ξ )μR (ω, dξ ).
Rd
Iω
Moreover, the total mass of μR (ω, ·) satisfies: μR (ω, dξ ) lim inf uh L2 (Td ) ;
tr
h →0+
Iω
(14)
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and, due to the structure of Rhuh ,
ω b(ξ )μR (ω, dξ )φnω φm
Iω
L2 (γω )
= 0,
if m ≡ n mod |pω |2 .
Proof. Estimate (13) implies that each Rhuh (ω, ·) is uniformly bounded in the space7 M+ (Iω ; L1 (L2 (γω ))) of positive measures on Iω with values in L1 (L2 (γω )) by a constant C > 0 independent of ω ∈ W. Since M+ (Iω ; L1 (L2 (γω ))) may be identified to the cone of positive elements of the dual of Cc (Iω ; K(L2 (γω ))), statement (14) follows from the Banach–Alaoglu theorem and a standard diagonal argument. Finally, the bound on the total mass of μR (ω, ·) is a consequence of estimate (13), 2 (J ) obtained as a limit (14) as a In what follows, we shall refer to a measure μR ∈ X0,+ resonant Wigner measure of the sequence (uh ). As we mentioned in the introduction, resonant Wigner measures are closely related to the two-microlocal semiclassical measures introduced in [7–9,18,19]. However, our definition of the resonant Wigner distribution gives rise to a global object (see the discussion in [7, p. 518]); moreover, resonant Wigner measures describe the energy concentration (at scales of order one) of the sequence (uh ) on the non-smooth set Td × Ω in phase space.
2.3. Additional properties Our next result is a manifestation of the two-microlocal character of resonant Wigner measures. It characterizes the sequences (uh ) for which μR is identically zero. Proposition 7. Let (uh ) be h-oscillatory and suppose that (14) holds for the sequence (uh ). Given any ω ∈ W, one has μR (ω, ·) = 0 if and only if : lim
h→0+
uh (k)2 = 0,
for every N > 0.
(15)
|k·pω |
Proof. Let N ∈ N and denote by πN the projection in L2 (γω ) onto the subspace spanned by (φjω )0|j |N . Then πN is compact and 2 n uh δhr (ξ ). ν + r tr πN Rhuh (ω, ξ ) = ω |pω | ⊥ |n|N r∈ωn
Therefore, for every ϕ ∈ Cc (Iω ), lim
h→0+
2 n νω + r = tr πN ϕ(ξ )μR (ω, dξ ) . ϕ(hr)uh |pω | ⊥
|n|N r∈ωn
Iω
7 We refer the reader to Appendix A for precise definitions of spaces of operator-valued measures.
(16)
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Since (uh ) is h-oscillatory, we can suppose without loss of generality that there exists R > 0 such that uh (k) = 0 if |hk| > R. Now suppose μR (ω, ·) = 0; by taking ϕ(ξ ) = 1 for |ξ | R in (16) we conclude (15). Now suppose that (15) holds. Then tr(πN Iω ϕ(ξ )Rhuh (ω, dξ )) = 0 for every ϕ ∈ Cc (Iω ) and every N > 0. By letting N tend to infinity we conclude that tr Iω ϕ(ξ )μR (ω, dξ ) = 0, which implies, since ϕ is arbitrary, μR (ω, ·) = 0. 2 Since the energy of sequence (uh ) may concentrate on Iω at scales larger than one, typically the restriction to Iω of the semiclassical measure of (uh ) is larger than tr μR (ω, ξ ). This is the content of our next result. Proposition 8. Let (uh ) have a semiclassical measure μ0 and satisfy (14). Then, for every nonnegative ϕ ∈ Cc (Rd ) and every ω ∈ W,
ϕ(ξ )μR (ω, dξ )
tr Iω
ϕ(ξ )μ0 (dx, dξ ).
(17)
T ∗ Td
Proof. Consider the projector πN defined in the proof of Proposition 7. For every ϕ ∈ Cc1 (Rd ) and every N > 0 one has: tr πN ϕ(ξ )Rh (ω, dξ ) − uh Iω
|pω ·k|N
2 ϕ(hk) uh (k)
hN ∇ξ ϕ L∞ (Iω ) uh 2L2 (Td ) . Therefore, if in addition ϕ is non-negative, 2 tr πN ϕ(ξ )Rhuh (ω, dξ ) ϕ(hk)uh (k) + O(h); k∈Zd
Iω
taking limits as h → 0+ we get, for every N > 0: tr πN ϕ(ξ )μR (ω, dξ ) Iω
ϕ(ξ )μ0 (dx, dξ ).
T ∗ Td
Letting N → ∞ and using the density of Cc1 (Rd ) in Cc (Rd ) we conclude the proof of the proposition. 2 3. Resonant Wigner measures and the Schrödinger flow Before stating our main result, we need some more notation. Given ω ∈ W, denote by Lω the length of γω , equal to 2π|pω |. There is a well-defined extension operator Eω from the space of Lω /|pω |2 Z-periodic functions in L1 (γω ) to the space L1 (Td ). If f ∈ L1 (γω ) is Lω /|pω |2 Z-
F. Macià / Journal of Functional Analysis 258 (2010) 933–955
945
periodic then put Eω f (x) :=
|pω | f (x · νω ), (2π)d−1
x ∈ Td .
With that normalization, it is not difficult to check that Eω f L1 (Td ) = f L1 (γω ) . Moreover, Eω f is invariant by translations along vectors orthogonal to ω. The following is a reformulation of Proposition 15 from Appendix A in our setting. Let μR ∈ X0,+ (J ); for fω ∈ L∞ (γω ) denote by mfω the operator in L2 (γω ) defined by multiplication by fω . The measures ρ˜ω ∈ M(γω × Iω ), ω ∈ W, defined by:
fω (s)ϕ(ξ )ρ˜ω (ds, dξ ) := tr mfω ϕ(ξ )μR (ω, dξ ) ,
γω ×Iω
Iω
and ρ˜ω (·, ξ ) is absolutely continuous with respect to for fω ∈ C(γω ) and ϕ ∈ Cc (Iω ) are positive arc-length measure ds in γω . Clearly, γω ρ˜ω (ds, ·) = tr μR (ω, ·). We shall say that ρ˜ω is the trace density of μR (ω, ·). Let ∂ω2 denote the Laplacian in γω (with respect to the arc-length metric, that is, when γω is identified to R/(2π|pω |Z)). The next result complements Theorem 2. Theorem 9. Let (uh ) satisfy the hypotheses of Theorem 2. Then the measures ρωt ∈ M(Td × Iω ), ω ∈ W, t ∈ R, appearing in formula (10) are uniquely determined by the initial data (uh ) as follows. Let μ0R ∈ X0,+ (J ) be a resonant Wigner measure corresponding to (uh ). Let μtR ∈ X0,+ (J ), t ∈ R, solve the density matrix Schrödinger equation: ⎧ ⎪ ⎨ i∂t μt (ω, ξ ) = − 1 ∂ 2 , μt (ω, ξ ) , R 2 ω R ⎪ ⎩ μt (ω, ξ ) = μ0 (ω, ξ ). R t=0 R
(18)
Let ρ˜ωt be the trace density of μtR (ω, ·). Then ρωt (·, ξ ) = Eω ρ˜ωt (·, ξ ) − tr μtR (ω, ξ ) . In particular, each ρωt (·, ξ ) is absolutely continuous with respect to Lebesgue measure in Td , has zero total mass and is invariant under the geodesic flow. Before giving the proof of Theorems 2 and 9, we shall first need some preparatory lemmas. Let a ∈ Cc∞ (T ∗ Td ) and write it as a(x, ξ ) = k∈Zd ak (ξ )ψk (x). Given ϕ ∈ S(R) define for every ω ∈ W the operator-valued function: ka,ϕ (ω, ξ ) :=
1 (2π)d/2
[c]∈Z|p
ω |2
n,m∈[c] m =n
2 n − m2 ω ϕ ⊗ φnω . a m−n νω (ξ )φm |pω | 2|pω |2
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Lemma 10. For every (ω, ξ ) ∈ J , the operator ka,ϕ (ω, ξ ) is a Hilbert–Schmidt operator on L2 (γω ). Moreover, 2 1 n 2 ϕ a(·, ξ ) − a0 (ξ )ψ0 L2 (Td ) . L (L2 (γω )) (2π)d 2 d
ka,ϕ (ω, ξ ) 2 2 ω∈W
(20)
n∈Z
In particular, ka,ϕ ∈ X0 (J ). Proof. A direct computation of the Hilbert–Schmidt norm gives: 2 2 l|pω |2 1 ϕ l alpω (ξ ) + n = ω )) (2π)d 2
ka,ϕ (ω, ξ ) 2 2 2 L (L (γ
l∈Z\{0}
n∈Z
n 2 1 alp (ξ )2 ω ϕ 2 , (2π)d l∈Z\{0}
n∈Z
since, for every fixed l = 0 the map that associates n ∈ Z to l(l|pω |2 + 2n) ∈ Z is injective. Therefore, summing in ω gives the estimate (20). 2 Lemma 11. Let (uh ) be a bounded sequence in L2 (Td ) such that (7) holds. Then, given a ∈ Cc∞ (T ∗ Td ) such that Td a(x, ·) dx = 0 and ϕ ∈ S(R), we have: h ϕ(t) w h (t, ·), a dt − tr ka,ϕ (ω, ξ )Ruh (ω, dξ ) Ca,ϕ h, uh ω∈W
R
Iω
for some constant Ca,ϕ > 0. Proof. Let ϕ ∈ S(R) and take a ∈ Cc∞ (T ∗ Td ) such that a0 ≡ 0; from formula (3) we deduce:
ϕ(t) wuhh (t, ·), a dt =
|k|2 − |j |2 k+j 1 h a uh (k)uh (j ). ϕ j −k 2 2 (2π)d/2 d k,j ∈Z
R
This expression can be written as: 1 d/2 (2π)
ω∈W k,j ∈Zd k−j ∈ω
2 k+j |k| − |j |2 aj −k h uh (k)uh (j ), ϕ 2 2
(21)
since, recall a0 ≡ 0, and, by definition, the lines ω ∈ W do not contain the origin. Now, for ω ∈ W fixed, the sum in k − j ∈ ω in (21) may be rewritten in terms of the parametrization introduced
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in Proposition 4 to give: [c]∈Z|p |2 ω (m,n,r)∈Cω,[c]
2 m+n m n m − n2 a h ϕ + hr u + r u + r , ν ν ν (n−m) ω h ω h ω |pω | νω 2|pω | |pω | |pω | 2|pω |2 (22)
where, for ω ∈ W and [c] ∈ Z|pω |2 we have set:
Cω,[c] := (m, n, r): m, n ∈ [c], m = n, r ∈ ωc⊥ . Notice that the reason for (22) to hold is that the condition k − j ∈ ω results inthe fact that k, j can be written as k = |pmω | νω + r and j = |pnω | νω + r for a unique (m, n, r) ∈ [c]∈Z 2 Cω,[c] . |pω | Comparing (22) with the expression (12) defining the resonant Wigner distribution of uh we find that:
ϕ(t) wuhh (t, ·), a dt = tr ka,ϕ (ω, ξ )Rhuh (ω, dξ ) ω∈W
R
Iω
tr ra,ϕ (ω, ξ )Rhuh (ω, dξ ) ,
+
ω∈W
Iω
where: 1 ra,ϕ (ω, ξ ) := (2π)d/2
[c]∈Z|p |2 ω m,n∈[c]
2 n − m2 ω l(m, n, ω, ξ )φm ϕ ⊗ φnω , 2|pω |2
with m+n l(m, n, ω, ξ ) := a m−n νω h νω + ξ − a m−n νω (ξ ). |pω | |pω | 2|pω | Let us estimate the remainder term. First note that: h m + n sup ∇ξ a n−m νω (ξ ). sup l(m, n, ω, ξ ) |pω | |pω | 2 ξ ∈Iω ξ ∈Iω Proceeding as in the proof of Lemma 10, we use (23) to estimate: ra,ϕ (ω, ξ ) 2 2
L (L2 (γω
))
2 supξ ∈Iω |∇ξ alpω (ξ )|2 n h2 n ϕ . 2 (2π)d 2 l 2 |pω |2 n∈Z
l∈Z\{0}
(23)
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Since a ∈ Cc∞ (T ∗ Td ) and a0 ≡ 0, ω∈W
l
−2
−2
|pω |
l∈Z\{0}
2 sup ∇ξ alpω (ξ )
1/2 is finite.
ξ ∈Iω
Therefore: h tr ra,ϕ (ω, ξ )Ruh (ω, dξ ) Ca,ϕ uh 2L2 (Td ) h, ω∈W
Iω
2
and the result follows.
Proof of Theorems 2 and 9. Suppose that (Rhuh ) converges along some subsequence (uh ) to the resonant Wigner measure μ0R (this is the case, by Proposition 6). Let a ∈ Cc∞ (T ∗ Td ) and ϕ ∈ S(R); in view of estimate (20) we have that: lim
h →0+
tr ka,ϕ (ω, ξ )Rhuh (ω, dξ ) = tr ka,ϕ (ω, ξ )μ0R (ω, dξ ) . ω∈W
ω∈W
Iω
Iω
Applying Lemma 11 we obtain:
R
T ∗ Td
(24)
ϕ(t)a(x, ξ )μ(t, dx, dξ )
=
ω∈W
ka,ϕ (ω, ξ )μ0R (ω, dξ ) +
tr
a(ξ )μ0 (dx, dξ ),
T ∗ Td
Iω
where a(ξ ) := (2π)−d Td a(x, ξ ) dx. Therefore, it only remains to identify the term involving the resonant Wigner measure μ0R . Simple inspection gives:
ka,ϕ (ω, ξ )μR (ω, dξ ) = 0
tr
ϕ(t) tr
R
Iω
pa (ω, ξ )eit∂ω /2 μ0R (ω, dξ )e−it∂ω /2 , 2
2
(25)
Iω
where ∂ω2 denotes the Laplacian on L2 (γω ) and pa (ω, ξ ) :=
1 (2π)d/2
[c]∈Z|pω | n,m∈[c] m =n
ω a m−n νω (ξ )φm ⊗ φnω . |pω |
Note that μtR (ω, ξ ) := eit∂ω /2 μ0R (ω, ξ )e−it∂ω /2 2
2
(26)
F. Macià / Journal of Functional Analysis 258 (2010) 933–955
949
solves (18). For m ≡ n (mod |pω |2 ), define the measures μtR (ω, ξ )(m, n):
ϕ(ξ )μtR (ω, dξ )(m, n) := Iω
ω ϕ(ξ )μtR (ω, dξ )φnω φm
Iω
, L2 (γω )
ϕ ∈ Cc (Iω ),
(27)
so
μtR (ω, ξ ) =
ω μtR (ω, ξ )(m, n)φm ⊗ φnω .
m≡n (mod |pω |2 )
Given k ∈ Zd , let ω ∈ W be the unique resonant direction such that k ∈ ω. In view of (24)–(26), we have that, for a.e. t ∈ R the kth Fourier coefficient of μ(t, ·) is given by: ψk (x)μ(t, dx, ξ ) =
1 (2π)d/2
Td
μtR (ω, ξ )(m, n).
m−n=k·pω
The trace density of μtR (ω, ξ ) is precisely the measure 1 ρ˜ωt (·, ξ ) := √ 2π|pω | d
ω μtR (ω, ξ )(m, n)φk·p + ω
k∈ω∩Z m−n=k·pω
1 tr μtR (ω, ξ ). 2π|pω |
ω As, for k ∈ ω ∩ Zd one has that φk·p is Lω /|pω |2 -periodic and ω
ω = Eω φk·p ω
|pω | ψk , (2π)d−1
we find that 1 d/2 (2π) d
μtR (ω, ξ )(m, n)ψk = Eω ρ˜ωt (·, ξ ) − tr μtR (ω, ξ ) = ρωt (·, ξ ),
k∈ω∩Z m−n=k·pω
and identity (10) follows.
2
The proof of Corollary 3 is an easy consequence of formula (10) and the properties of resonant Wigner distributions. Proof of Corollary 3. The weak form of Egorov’s theorem proved in [17], Theorem 2(ii), gives in this case: b(ξ )μ(t, dx, dξ ) = b(ξ )μ0 (dx, dξ ), (28) T ∗ T2
T ∗ T2
for every b ∈ Cc (R2 ) and a.e. t ∈ R (this can also be directly deduced from Eq. (6)). Identity (28), together with our hypothesis μ0 ({ξ = 0}) = 0 implies μ(t, {ξ = 0}) = 0 for a.e. t ∈ R. Notice
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F. Macià / Journal of Functional Analysis 258 (2010) 933–955
that since d = 2 the lines Iω only intersect at the origin. As a consequence of this, we deduce that μ(t, ·) =
μ(t, ·)T2 ×Iω + μ(t, ·)T2 ×(R2 \Ω) .
(29)
ω∈W
Let ν0 := (2π)−2
T2
μ0 (dy, ·), then we obtain from formula (10) the following expressions:
μ(t, ·)T2 ×Iω = ρωt + ν0 Iω ,
μ(t, ·)T2 ×(R2 \Ω) = ν0 Rd \Ω .
Recall that all the measures involved in the right-hand side of Eq. (29) are mutually disjoint. In particular, the fact that μ(t, ·) 0 for a.e. t ∈ R implies that ρωt + ν0 Iω 0 as well. Theorem 2 shows that the projection on T2 of every ρωt + ν0 Iω is absolutely continuous with respect to the Lebesgue measure. Therefore, the monotone convergence theorem ensures that R2 μ(t, · ,dξ ) is also absolutely continuous with respect to the Lebesgue measure for a.e. t ∈ R. The result then follows applying identity (8). 2 4. Additional properties and examples The following is a direct consequence of Proposition 7 and Theorem 9. Proposition 12. Let (uh ) be an h-oscillating sequence such that (7) holds. If in addition, one has lim
h→0+
uh (k)2 = 0,
for every N > 0,
|k·pω |
for some ω ∈ W then the corresponding term ρωt in (10) vanishes identically. As an example, we shall apply Proposition 12 to analyse the propagation of wave-packet type solutions to the Schrödinger equation in this context. Given (x0 , ξ0 ) ∈ T ∗ Td and ρ ∈ h ∈ L2 (Td ) to be the 2πZd -periodization of the function: Cc∞ ((−π, π)d ) define c(x 0 ,ξ0 ) uh (x) :=
1
ρ hd/4
x − x0 iξ0 / h·x e . √ h
h The Poisson summation formula ensures that the Fourier coefficients of c(x are: 0 ,ξ0 )
√ d/2 d/4 h c h ρ h(k − ξ0 / h) e−i(k−ξ0 / h)·x0 . (x0 ,ξ0 ) (k) = (2π) It is not difficult to prove that wchh
(x0 ,ξ0 )
(0, ·) ρ 2L2 (Rd ) δx0 ⊗ δξ0 ,
as h → 0+ .
Proposition 13. Let μ be the semiclassical measure given by the limit (7) corresponding to the h . Then, for almost every t ∈ R, initial data c(x 0 ,ξ0 ) μ(t, x, ξ ) = ρ 2L2 (Rd ) dx δξ0 (ξ ).
F. Macià / Journal of Functional Analysis 258 (2010) 933–955
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Proof. Let ω ∈ W; if ξ0 ∈ / Iω then ρωt = 0; therefore we may suppose that ξ0 ∈ Iω . The conclusion will follow as soon as we show that ρωt = 0 also holds in this case. Take a function χ ∈ Cc∞ ((−2, 2)), identically equal to one in [−1, 1] and taking values between 0 and 1. Let N > 0 and write χN (ξ ) := χ(pω · ξ/N). Then √ √ 2 2 d d/2 h c χN (k) ρ ( hk − ξ0 / h ) . (x0 ,ξ0 ) (k) (2π) h
|k·pω |
k∈Zd
Applying the Poisson summation formula, we find that the right-hand side of the above inequality equals vh 2L2 (Td ) , where vh stands for the 2πZd -periodization of χN (Dx )uh . Since this function
is in S(Rd ), we have, for every s > d an estimate: vh 2L2 (Td ) Cs
χN (Dx )uh (x)2 1 + |x|2 s/2 dx.
Rd
Applying Plancherel’s identity, we get, after changing variables and taking into account that ξ0 · pω = 0,
χN (Dx )uh (x)2 1 + |x|2 s/2 dx =
Rd
(1 − hξ )s/4 wh (ξ )2 dξ , (2π)d
Rd
√ s/4 where wh (ξ ) := χN (ξ/ h) ρ (ξ ). The functions (1 − h √ξ ) wh are uniformly bounded in d S(R ), and supported on the strips Sh := {ξ : |ξ · pω | 2 hN }. Taking, for instance, s/4 ∈ N, we find that, for every ε > 0 there exists Rε > 0 such that:
(1 − hξ )s/4 wh (ξ )2 dξ C Sh ∩ B(0; Rε ) + ε. d (2π)
Rd
This expression tends to ε as h → 0+ . Therefore, as ε is arbitrary, Proposition 12 ensures that ρωt = 0, and the conclusion follows. 2 If instead we consider the purely oscillating profiles uh (x) defined as the periodizations of: ρ(x)eiξ0 / h·x , with ξ0 ∈ Ω a simple resonance (i.e. such that λξ0 ∈ Zd for some λ ∈ R \ {0}) we find that some of the terms ρωt are non-zero. More precisely, write ρper to denote the periodization of ρ and set, for f ∈ L1 (Td ): |ξ0 | f ξ0 (x) := L
L/|ξ 0|
f (x + sξ0 ) ds, 0
where L denotes the length of the periodic geodesic issued from (x, ξ0 ). We have the following.
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F. Macià / Journal of Functional Analysis 258 (2010) 933–955
Proposition 14. Let ξ0 ∈ Ω \ {0} be a simple resonance. The semiclassical measure μ given by (7) corresponding to the initial data uhn , where for simplicity we have taken ξ0 / hn ∈ Zd , is given by: 2 μ(t, x, ξ ) = eitx /2 ρper ξ (x) dx δξ0 (ξ ). 0
Proof. It is easy to check that the Wigner measure corresponding to the sequence of initial data (uhn ) is precisely: 2 μ0 (x, ξ ) = ρper (x) dx δξ0 (ξ ). Therefore, in view of (17), the resonant Wigner measure of (uhn ) satisfies μR (ω, ·) ≡ 0, when/ Iω . Let us compute the measures μR (ω, ·) when ξ0 ∈ Iω . Start noticing that the Poisson ever ξ0 ∈ summation formula gives: uh =
ρ (k − ξ0 / hn )ψk ;
k∈Zd
hence, for b ∈ C ∞ (Iω ): b(ξ )Rhuhnn (ω, dξ ) Iω
=
[c]∈Z|p
ω |2
m,n∈[c] r∈ωc⊥
m n ξ0 ξ0 ω ρ ρ ⊗ φnω . b(hr)φm νω + r − νω + r − |pω | hn |pω | hn
Since ξ0 / hn ∈ Iω ∩ Zd , each of the summands in m and n can be rewritten as:
ω ρ (k) ρ (j )b hk ⊥ + ξ0 φm ⊗ φnω ,
k,j ∈Zd ,k−j =λpω k·pω =m,j ·pω =n
for some λ ∈ Z and where k ⊥ denotes the projection of k onto Iω . As hn → 0 this converges to:
ω ρ (k) ρ (j )b(ξ0 )φm ⊗ φnω .
k,j ∈Zd ,k−j =λpω k·pω =m,j ·pω =n
Therefore, μR (ω, ξ ) =
ω ρ (k) ρ (j )φm ⊗ φnω δξ0 (ξ ).
λ∈Z k,j ∈Zd ,k−j =λpω k·pω =m,j ·pω =n
The conclusion then follows using the formula defining ρωt .
2
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953
Appendix A. Operator-valued measures Let H be a separable Hilbert space, we denote by L(H ), K(H ), and L1 (H ) the spaces of bounded, compact and trace-class operators on H , respectively. If A ∈ L1 (H ), tr A denotes the trace of A; A L1 (H ) := tr |A| defines a norm on L1 (H ). With this norm, L1 (H ) is the dual of K(H ), the duality being tr(AB). When is X a locally compact, σ -compact, Hausdorff metric space, the space M(X; L1 (H )) of trace-operator-valued Radon measures on X consists of linear operators μ : Cc (X) → L1 (H ) bounded in the following sense: given K ⊂ X compact there exists CK > 0 such that for every ϕ ∈ Cc (K), μ, ϕ
L1 (H )
CK sup ϕ(x). x∈K
Note that M(X; L1 (H )) is the dual of Cc (X; K(H )), the space of compactly supported functions from X into K(H ). An element μ ∈ M(X; L1 (H )) is positive if for every non-negative ϕ ∈ Cc (X) the operator μ, ϕ is Hermitian and positive. The set of such positive elements is denoted by M+ (X; L1 (H )). Given a positive measure μ on defines the scalar valued positive measure tr μ as tr μ, ϕ := trμ, ϕ,
for ϕ ∈ Cc Rd .
We refer the reader to [11] for a clear presentation of operator-valued measures, as well as a proof of a Radon–Nykodim theorem in this context. When H = L2 (T , ν), where T is locally compact, σ -compact, Hausdorff metric space equipped with a Radon measure ν, then the operators μ, b may be represented by their integral kernels kb ∈ L2 (T × T ). Thus, μ can be viewed as an L2 (T × T )-valued measure. Given f ∈ Cc (T × X), we denote by mf (x) the operator acting on L2 (T , ν) by multiplication by f (·, x). Clearly, mf ∈ Cc (X; L(L2 (T ))). The following construction is used in the proof of Theorems 2 and 9. Proposition 15. Let H = L2 (T , ν) and μ ∈ M+ (X; L1 (H )). The linear operator ρμ : Cc (T × X) → R : g → trμ, mg extends to a positive Radon measure on T × X, which is finite if μ is. Moreover, for every ϕ ∈ Cc (X), ϕ(x)ρμ (·, dx) ∈ L1 (T , ν). X
In addition ρμ = 0
if and only if μ = 0.
(30)
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F. Macià / Journal of Functional Analysis 258 (2010) 933–955
Proof. Given compact sets S ⊂ T and K ⊂ X and functions f ∈ Cc (S) and ϕ ∈ Cc (K) we have, as L1 (H ) is an ideal in L(H ): tr mf μ, ϕ mf L(H ) trμ, ϕ CK f L∞ (T ,ν) sup ϕ(x) x∈K
for some constant CK > 0. Therefore, as Cc (T ) ⊗ Cc (X) is dense in Cc (T × X), the functional ρμ extends to a measure on T × X. The positivity of ρμ follows easily from the properties of the trace of linear operators. Let ϕ ∈ Cc (X) and write μ, ϕ =
∞
λj φj ⊗ φj
j =1
where λj are the eigenvalues of μ, ϕ, the φj , j ∈ N, form an orthonormal basis of H consisting of eigenvectors, and φj ⊗ φj is the projection on the linear span of φj . Now,
f (t)ϕ(x)ρμ (dt, dx) = tr mf
λj φj ⊗ φj
=
j =1
T ×X
Since μ, ϕ is trace class,
∞
∞
j =0 λj
j =0
f |φj |2 dν.
λj
(31)
T
is absolutely convergent; therefore,
ϕ(x)ρμ (·, dx) = X
∞
∞
λj |φj |2
j =0
is in L1 (T ) as we wanted to show. Finally, to see that ρμ = 0 implies μ = 0 simply take f and ϕ non-negative in formula (31). As the left-hand side is zero, all λj must vanish, that is μ, ϕ = 0. Since ϕ is arbitrary, we must have μ = 0. 2 Acknowledgments Much of this research has been done as the author was visiting the Département de Mathématiques at Université de Paris-Sud and the Mathematics Department at University of Texas at Austin. He wishes to thank these institutions for their kind hospitality. He also wants to express his gratitude to Patrick Gérard for his interest and the numerous fruitful discussions they had regarding this work. References [1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I, Schrödinger equations, Geom. Funct. Anal. 3 (2) (1993) 107–156. [2] J. Bourgain, Eigenfunction bounds for the Laplacian on the n-torus, Int. Math. Res. Not. 3 (1993) 61–66. [3] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Amer. Math. Soc. Colloq. Publ., vol. 46, American Mathematical Society, Providence, RI, 1999. [4] N. Burq, P. Gérard, N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (3) (2004) 569–605.
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[5] N. Burq, P. Gérard, N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (1) (2005) 187–223. [6] N. Burq, P. Gérard, N. Tzvetkov, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. École Norm. Sup. (4) 38 (2) (2005) 255–301. [7] C. Fermanian-Kammerer, Mesures semi-classiques 2-microlocales, C. R. Math. Acad. Sci. Paris Sér. I 331 (7) (2000) 515–518. [8] C. Fermanian-Kammerer, Propagation and absorption of concentration effects near shock hypersurfaces for the heat equation, Asymptot. Anal. 24 (2) (2000) 107–141. [9] C. Fermanian-Kammerer, P. Gérard, Mesures semi-classiques et croisement de modes, Bull. Soc. Math. France 130 (1) (2002) 123–168. [10] G.B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud., vol. 122, Princeton University Press, Princeton, NJ, 1989. [11] P. Gérard, Microlocal defect measures, Comm. Partial Differential Equations 16 (11) (1991) 1761–1794. [12] P. Gérard, Mesures semi-classiques et ondes de Bloch, Séminaire sur les Équations aux Dérivées Partielles, 1990– 1991, Exp. No. XVI, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 1991. [13] P. Gérard, E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J. 71 (2) (1993) 559–607. [14] P. Gérard, P.A. Markowich, N.J. Mauser, F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math. 50 (4) (1997) 323–379. [15] D. Jakobson, Quantum limits on flat tori, Ann. of Math. 145 (2) (1997) 235–266. [16] P.-L. Lions, T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana 9 (3) (1993) 553–618. [17] F. Macià, Semiclassical measures and the Schrödinger flow on Riemannian manifolds, Nonlinearity 22 (2009) 1003– 1020. [18] L. Miller, Propagation d’ondes semi-classiques à travers une interface et mesures 2-microlocales, PhD thesis, École Polytechnique, 1996. [19] F. Nier, A semi-classical picture of quantum scattering, Ann. Sci. École Norm. Sup. (4) 29 (2) (1996) 149–183. [20] A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974) 189–201.
Journal of Functional Analysis 258 (2010) 956–977 www.elsevier.com/locate/jfa
An abstract form of a theorem of Helson and applications to sets of synthesis and sets of uniqueness ✩ A. Ülger Koç University, Department of Mathematics, 34450 Sariyer, Istanbul, Turkey Received 7 June 2009; accepted 30 September 2009 Available online 8 October 2009 Communicated by D. Voiculescu
Abstract Let E be a compact perfect subset of the real line R such that the restriction of the Fourier transform a → a |E from L1 (R) into C(E) is onto. Helson proved that then, for μ ∈ M(E), lim|y|→∞ | μ(y)| = 0 is possible only if μ = 0. In this paper we present an abstract version of this theorem of Helson and provide some applications of it to the study of sets of spectral synthesis and sets of uniqueness. © 2009 Elsevier Inc. All rights reserved. Keywords: Helson set; Set of synthesis; Set of uniqueness; Group algebra; Multipliers; Arens product; Weakly almost periodic functions
0. Introduction Let G be an infinite locally compact commutative group equipped with its Haar measure, the dual L1 (G) its group algebra and M(G) its measure algebra [42]. We shall denote by G 1 group of G. The group G being infinite, the Fourier transform φ : L (G) → C0 (G) is never onto for which the restriction φE : L1 (G) → C0 (E) of the Fourier but there are closed subsets E of G a |E , is onto. Such a set, because of Helson’s Theorem [14] cited transform to the set E, φE (a) = in the abstract, is said to be a Helson set. The kernel of the homomorphism φE is the closed ideal a = 0 on E k(E) = a ∈ L1 (G): ✩ This work of the author is partially supported by Turkish Academy of Sciences and Tubitak Isbap-Project No. 107T896. E-mail address: [email protected].
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.021
A. Ülger / Journal of Functional Analysis 258 (2010) 956–977
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so that, for a Helson set E, the Banach algebras L1 (G)/k(E) and C0 (E) are isomorphic. Rudin in [42, Theorem 5.6.7], extending Helson’s Theorem to all locally compact commutative the intersection k(E)⊥ ∩ C0 (G) is trivial. That is, groups, proved that, for a Helson set E ⊆ G, ⊥ ⊥ k(E) ∩ C0 (G) = {0}. Here k(E) is the annihilator of the ideal k(E) in the dual space L∞ (G) of L1 (G). The reader can find a shorter proof of this theorem in Doss’ paper [8] and another proof in Hewitt and Ross [18, Section 41.18]. As a search in MathSciNet shows, Helson’s Theorem in the year following its publication has attracted a great deal of attention and a considerable number of papers related in one way or other to this theorem of Helson has appeared. As far as we were able to check, almost all these papers study one of the following problems: Constructing Helson sets [21,24,25,29,36,48], comparing Helson sets with other thin sets [27,28,33,41] and studying the stability properties of Helson sets [16,44,49,50]. These references are by no means exhaustive. We do not know any paper whose main theme is the following question. the intersection k(E)⊥ ∩ C0 (G) is trivial? Question: What makes that, for a Helson set E ⊆ G, This is this question around which this paper is centred. It is clear that this question is closely related to the problem of characterizing the sets of uniqueness for the trigonometric series [20,26,51]. We prove an abstract and fairly general result (Theorem 2.4) which, when specialized to the group algebra L1 (G) of a noncompact locally compact σ -compact commutative group G, says that the only fact that makes that the intersection k(E)⊥ ∩ C0 (G) is trivial is the fact that the space k(E)⊥ (which is isomorphic to M(E) = C0 (E)∗ ) is weakly sequentially complete. Actually, if X is any weakly sequentially complete norm closed L1 (G)-submodule of L∞ (G) (i.e. a ∗ f is in X for each a ∈ L1 (G) and f ∈ X) then X ∩ C0 (G) = {0}. Our result applies, beside the group algebra L1 (G), to the Fourier algebra A(G) of a nondiscrete locally compact amenable first countable group G [13], to the Herz–Figa–Talamanca algebra Ap (G) of a nondiscrete locally compact amenable second countable group G [17], to the Beurling algebra L1 (G, ω) of a noncompact locally compact σ -compact commutative group G [4,40] and to several other Banach algebras. The paper also contains a certain number of applications of the main result to the study of sets of synthesis and sets of uniqueness in the setting of the group algebra L1 (G) of a locally compact σ -compact commutative group G. In Section 1 we have gathered the preliminary results needed in the subsequent sections. Section 2 contains the main result of the paper (Theorem 2.4) and some of its corollaries. In Section 3, as applications of our main result, we present a series of results related to the spectrum a set σ (f ) of the functions f ∈ L∞ (G) and to the question of “when is a given subset E of G of spectral synthesis”. Section 4 is devoted to the problem of sets of uniqueness. The main result is a set of uniqueness if, for each of that section (Theorem 4.7) says that a closed subset E of G ∞ f ∈ L (G) whose spectrum is contained in the set E, the space Zf = {a ∗ f : a ∈ L1 (G)} is weakly sequentially complete. Our proofs are essentially functional analytic. The main ingredients of the proofs are: a) The notion of narrow spectrum introduced by Beurling for the group algebra L1 (R) [1] and extended by Domar [7] and Lindahl [32] to abstract Banach algebras; and b) a lemma (proved below) there is an element eγ in L1 (G)∗∗ \ {0} such that, for a ∈ L1 (G), saying that, for any γ ∈ G, a (γ )eγ . Here the product aeγ denotes the Arens product (defined in the next section) of aeγ = the elements a and eγ in the algebra L1 (G)∗∗ .
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1. Preliminaries and notation Our notation and terminology are standard. Concerning harmonic analysis, they are essentially those of Rudin’s book [42]; and, concerning Banach algebras, they are those of books [3] and [23]. To be fixed, for any Banach space X, we always consider X as naturally embedded into its second dual X ∗∗ . The natural duality between X and X ∗ is denoted as f, x (and also as
x, f ). If we need to be more precise, instead of x, f , we write x, f X,X∗ . Let now A be a commutative Banach algebra. We denote by (A) the Gelfand spectrum of A and consider each element of (A) as a multiplicative functional on A. One can make the second dual A∗∗ of A into a Banach algebra by equipping it with one of the two Arens multiplications. We shall equip A∗∗ with the first Arens multiplication and use only this multiplication. This multiplication is defined in three steps as follows: For a, b in A, f in A∗ and n, m in A∗∗ , the functionals b.f , n.f and the product nm are defined by the equalities
a, b.f = ab, f ,
a, n.f = a.f, n ,
mn, f = m, n.f . Equipped with this multiplication, A∗∗ is a Banach algebra and A is a subalgebra of it. In general A∗∗ is not commutative but, A being commutative, for a ∈ A and m ∈ A∗∗ , am = ma. This multiplication is in general not separately weak∗ to weak∗ continuous on A∗∗ but, for n ∈ A∗∗ fixed, the mapping m → mn is continuous in the weak∗ topology of A∗∗ . If (eα )α∈I is a BAI (= bounded approximate identity) in A then each weak∗ cluster point e of this net in A∗∗ is a “right identity” in A∗∗ . That is, for each m ∈ A∗∗ , me = m. In general we do not have em = m. Conversely any right identity in A∗∗ is a cluster point in (A∗∗ , weak∗ ) of some BAI of A. We remark that, as one can see easily from the definition of the Arens multiplication, for m, n ∈ A∗∗ and γ ∈ (A), we also have mn, γ = n, γ . m, γ . For these results we refer the reader to the book [3, Section 2.9]. For a locally compact commutative group G equipped with its Haar measure (denoted dt), let A = L1 (G) be the group algebra of G. The multiplication on L1 (G) is of course the convolution. We define the duality between L1 (G) and L∞ (G) as a, f = G a(−t)f (t) dt. On L1 (G)∗∗ we put the first Arens multiplication as defined above. For f ∈ L∞ (G) and a ∈ L1 (G), as one can easily check, a.f is just a ∗ f . To illustrate the abstract results and be able to give examples, we need some concrete examples of Banach algebras. A large class of Banach algebras that contain the group algebra L1 (G) of a locally compact commutative group G and also the Fourier algebra A(G) of an arbitrary locally compact group G defined by Eymard [13] as special cases is the Herz–Figa–Talamanca algebra Ap (G). To introduce this algebra, let G be an arbitrary locally compact group equipped with its left Haar measure and 1 < p < ∞. The space Ap (G) is the space of the continuous functions of the form a=
∞
vn ⊗ u∨ n,
n=1
where vn ∈ Lq (G), un ∈ Lp (G) ( p1 + The norm of a is defined as
1 q
−1 = 1), u∨ n (t) = un (t ) and
∞
n=1 vn q .un p
< ∞.
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aAp (G) = inf
∞
959
vn q .un p ,
n=1
where the infimum is taking on all the representation of a as above. The space Ap (G), equipped with the above norm and the pointwise multiplication, is a commutative, semisimple, regular Tauberian Banach algebra and Ap (G) ⊆ C0 (G) [17]. The Gelfand spectrum of Ap (G) (via Dirac measures) is G. The Banach algebra Ap (G) has a BAI iff the group G is amenable. The dual space of Ap (G), which is a subalgebra of the operator algebra B(Lp (G)), is denoted as P Mp (G). This is the space of the p-pseudomeasures on G. For a ∈ L1 (G), let ρ(a) : Lp (G) → Lp (G) be the convolution operator defined by ρ(a)(b) = a ∗ b. The norm closure of the space {ρ(a): a ∈ L1 G)} in B(Lp (G)) is denoted as P Fp (G). This is the space of the p-pseudofunctions on G. The pointwise multiplier algebra of the algebra Ap (G) is denoted as Bp (G). When G is amenable P Fp (G) ⊆ P Mp (G) and Bp (G) = P Fp (G)∗ . For ample information on these algebras we refer the reader to the paper [17] and the book [38]. For p = 2, Ap (G) = A(G), the Fourier algebra of G; Bp (G) = B(G), the Fourier–Stieltjes algebra of G; P Mp (G) = VN(G), the von Neumann algebra of G and P Fp (G) = C ∗ (G), the group C ∗ -algebra of G [13]. Finally, when G is commutative, via Fourier transform, A(G) = B(G) = M(G), VN(G) = L∞ (G) and C ∗ (G) = C0 (G) (see [13] or [38]). L1 (G), We shall use these algebras only to illustrate some of the results presented in the paper. The other notions and notation will be introduced as needed. 2. An abstract form of Helson’s Theorem In this section A will be an arbitrary commutative Banach algebra. Our aim is to present an abstract version of Helson’s Theorem mentioned in the introduction, which has appeared in [14]. Our main ingredient is the fact that, when A is semisimple and regular, for any f ∈ A∗ , the narrow spectrum of f and its Beurling (or weak∗ ) spectrum are the same. We start by recalling these notions. Let f ∈ A∗ , f = 0, be a given functional. The following subsets σ∗ (f ) and σ (f ) of (A) ∗
σ∗ (f ) = {a.f : a ∈ A} w ∩ (A)
∗
and σ (f ) = {a.f : a ∈ A} ∩ A1 w ∩ (A)
are called, respectively, the weak∗ -spectrum and the narrow spectrum of f . Here A1 denotes the closed unit ball of A. The reader can find the proof of the following theorem in Lindahl’s paper [32, Theorem 4]. Theorem 2.0. Suppose that A is semisimple and regular. Then σ∗ (f ) = σ (f ) for each f in A∗ . Let Ac = {a ∈ A: the support of a , Supp( a ), is compact}. We recall that the algebra A is said to be “Tauberian” if the space Ac is dense in A. This condition, when the algebra A is semisimple and regular, guarantees that the set σ∗ (f ) is not empty whenever f = 0. Our main ingredient in this section is the following version of the above theorem. We recall that the algebra A is said to be weakly compactly generated if it is generated by a weakly compact set. If this is the case (e.g. A is separable) then the closed unit ball A∗1 of A∗ under the weak∗ topology σ (A∗ , A) is sequentially compact [6, p. 228]. We shall need such a condition since we shall work with the weakly sequentially complete subspaces of A∗ .
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Theorem 2.1. Suppose that the algebra A is semisimple, regular, Tauberian and weakly compactly generated. Let f ∈ A∗ and γ ∈ (A) be two functionals. Then γ ∈ σ (f ) iff there exists a sequence (an )n0 in A such that an .f 1 for all n 0 and an .f → γ in the weak∗ topology σ (A∗ , A) of A∗ . An essential question for us is the question when the sequence (an .f )n0 in the preceding theorem is weakly Cauchy? (i.e., When does, for each m ∈ A∗∗ , the sequence ( m, an .f )n0 converge?). To explain the motivation behind the next definition, let for a moment A be the group algebra L1 (G) of a locally compact commutative group G. Let (fn )n0 be a sequence in C0 (G) that converges in the weak∗ topology of the space L∞ (G) to some element f of L∞ (G). Then, for any a ∈ L1 (G), the sequence (a ∗ fn )n0 is weakly Cauchy. Indeed, this sequence, which lies in the space C0 (G), by the uniform boundedness principle, is uniformly bounded and it converges pointwise on G. To see this last point it is enough to observe that, for t ∈ G, (a ∗ fn )(t) = at , fn → at , f , where at is the translate of a by t. The conclusion now follows from the Lebesgue Dominated Convergence Theorem. Definition 2.2. Let Y be a norm closed subspace of A∗ . We shall say that the subspace Y has the wCp (= weak Cauchy property) if for any sequence (fn )n0 in Y that converges in the weak∗ topology of A∗ to some f ∈ A∗ , the sequence (a.fn )n0 is weakly Cauchy for each a ∈ A. Such a subspace Y of A∗ will play the role that is played by the space C0 (G) in Helson’s Theorem. Based on the Lebesgue Dominated Convergence Theorem, we have just seen that the subspace C0 (G) of L∞ (G) has the wCp. There is another more fundamental reason for this that allows itself to be generalized: The dual space of the space C0 (G) is the multiplier algebra of the Banach algebra L1 (G) [31]. It is this fact that is behind the examples given below. A norm closed subspace X of A∗ will be said “invariant” if a.f is in X for each f ∈ X and a ∈ A. That is, X is an A-module for the action (a, f ) → a.f . This canonical module action is the only module action that we shall use on the subspaces of A∗ , whatever the algebra A is. We remark that, for any closed ideal I of A, the subspace X = I ⊥ of A∗ , the annihilator of I in A∗ , is invariant in this sense. In the following examples, the basic pattern is this: We have a closed invariant subspace Y of A∗ whose dual Y ∗ identifies “naturally” with the multiplier algebra M(A) of A. Here the term “naturally” means this: For f ∈ Y , a ∈ A and T ∈ M(A), we have: T , a.f M(A),Y =
T (a), f A,A∗ . Examples 2.3. a) Let A = A(G) be the Fourier algebra of a locally compact amenable group G [13]. As recalled in the preceding section, the multiplier algebra of A(G) is the Fourier– Stieltjes algebra B(G) of G and the dual space of A(G) is the von Neumann algebra VN(G) of G. Let Y = C ∗ (G) be the group C ∗ -algebra of G. The dual space of C ∗ (G) is the algebra B(G) and, since G is amenable, C ∗ (G) is a closed subspace of VN(G). Let now (fn )n0 be a sequence in C ∗ (G) that converges in the weak∗ -topology of the space VN(G) to some element f of the space VN(G) and a ∈ A(G) a given element. For any u ∈ B(G), the product ua is in A(G) since A(G) is an ideal in B(G). Moreover, for g ∈ C ∗ (G), the functional a.g is in C ∗ (G) and
u, a.g B(G),C ∗ (G) = ua, g A(G),VN(G) .
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These facts combined with the Hahn Banach Theorem imply that the sequence (a.fn )n0 is a weakly Cauchy sequence in the space VN(G). Hence the invariant subspace Y = C ∗ (G) of VN(G) has the wCp. b) Let A = Ap (G) (1 < p < ∞) be the Herz–Figa–Talamanca algebra of a locally compact amenable group G [17]. The multiplier algebra of Ap (G) is the space Bp (G) of p-Fourier– Stieltjes algebra of G. The dual space of Ap (G) is the space p-pseudomeasures P Mp (G) on G. The multiplier algebra Bp (G) of the algebra Ap (G) is the dual space of the space P Fp (G) of the p-pseudofunctions on G, which is an invariant subspace of P Mp (G). Exactly for the same reasons as in the preceding example, the invariant subspace Y = P Fp (G) of P Mp (G) has the wCp. c) Let A = L1 (G, ω) be the Beurling algebra associated with a locally compact commutative group G and a continuous weight function ω : G → [0, ∞[ such that w(t) 1 for t ∈ G (see [4, Chapter 7] or [40, Section 6.3]). The multiplier algebra of L1 (G, ω) is the weighted measure algebra M(G, ω). The subspace C0 (G, 1/w) = {f : G → C: the function ω1 f is in C0 (G)} of L∞ (G, 1/ω) is a closed invariant subspace of this space and M(G, ω) is the dual space of C0 (G, 1/ω). Exactly for the same reasons as in a), the invariant subspace Y = C0 (G, 1/ω) of L∞ (G, 1/ω) has the wCp. d) Suppose that the algebra A is an ideal in its second dual and has a BAI. Then, since A has a BAI, the space AA∗ = {a.f : a ∈ A and f ∈ A∗ } is a norm closed subspace of A∗ [3, Corollary 2.9.26] or [18, 32.22]. Since A is an ideal in its second dual, for each a ∈ A, the multiplication operator τa : A → A, τa (b) = ab, is weakly compact and τa∗ (f ) = a.f . This implies that the invariant subspace Y = AA∗ of A∗ has the wCp. Actually, since A is an ideal in its second dual, the dual space of AA∗ identifies with the multiplier algebra of A. e) Let B be another commutative Banach algebra and h : A → B an onto bounded Banach algebra homomorphism. Then, since h is onto, as one can see it easily, for any closed invariant subspace Y0 of B ∗ , the subspace Y = h∗ (Y0 ) of A∗ is invariant. Moreover, since h∗ is an isomorphism, Y has the wCp iff the space Y0 has the wCp. To the above list of examples, one can also add, for instance, some of the Segal algebras, the semigroup algebras (e.g. L1 (R+ )) or the hypergroup algebras . . . etc., that they display the same pattern. The main result of this paper is the following theorem. This theorem is an abstract form and a far reaching generalization of Helson’s Theorem mentioned in the introduction. The corollaries that follow the proof will justify, we hope, this affirmation. Compared with the known proofs of Helson’s Theorem [8,14], [18, Section 41.18.], [42, Theorem 5.6.7], as the reader will notice, our proof is much shorter and simpler, though it requires the algebra to be weakly compactly generated. This hypothesis is not an important restriction for the purpose of this paper. Theorem 2.4. Suppose that the Banach algebra A is semisimple, regular, Tauberian and weakly compactly generated. Let X and Y be two norm closed invariant subspaces of A∗ such that a) X is weakly sequentially complete. b) Y has the wCp. c) X ∩ Y ∩ (A) = ∅. Then X ∩ Y = {0}.
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Proof. For a contradiction, suppose that X ∩ Y = {0}. Let f ∈ X ∩ Y , f = 0 be a functional in this intersection. Then, since the algebra A is semisimple, regular and Tauberian, the spectrum of the functional f is not empty. Let γ be in the spectrum σ (f ) of f . By Theorem 2.1, there is a sequence (an )n0 in A such that the sequence (an .f )n0 converges in the weak∗ topology a (γ ) = 1. Since the sequence fn = an .f lies in the space Y of A∗ to γ . Choose a ∈ A such that and since this space has the wCp, the sequence (aan .f )n0 is weakly Cauchy. Since the space X is invariant and f ∈ X too, the sequence (aan .f )n0 is also in the space X. As this space is weakly sequentially complete, the sequence (aan .f )n0 converges weakly to some element of the space X. Since the sequence (an .f )n0 converges in the weak∗ topology of A∗ to γ and a.γ = a (γ )γ = γ , necessarily aan .f → γ in the weak topology of A∗ . Since the space Y is closed in A∗ and the sequence (aan .f )n0 lies in Y , γ ∈ Y too. Hence γ ∈ X ∩ Y ∩ (A). This contradicts hypothesis c). Thus X ∩ Y = {0}. 2 be the Fourier transform, E ⊆ G a Helson set and Let now φ : L1 (G) → C0 (G) 1 a|E , the restriction of the Fourier transform to E. Since the hoφE : L (G) → C0 (E), φE (a) = momorphism φE is surjective, the Banach algebras C0 (E) and L1 (G)/k(E) are isomorphic. The μ:G→C adjoint φE∗ of φE sends each measure μ ∈ M(E) to its Fourier-Stieltjes transform μ: μ ∈ M(E)}. This is a weakly sequentially complete invariant subspace so that k(E)⊥ = { of L∞ (G). We recall that Helson’s Theorem states that k(E)⊥ ∩ C0 (G) = {0} [14]. The next corollary is considerable stronger than Helson’s Theorem since the space X in this corollary is neither related to the Fourier transform or is assumed to be weak∗ closed. The group algebra L1 (G) is weakly compactly generated if the group G is σ -compact [5, p. 143]. Since G = ∅ so that this result follows directly from the preceding theorem. is not compact, C0 (G) ∩ G Corollary 2.5. Let G be a noncompact locally compact σ -compact commutative group and A = L1 (G). Then, for any weakly sequentially complete norm closed invariant subspace X of L∞ (G), the intersection X ∩ C0 (G) is trivial. Let now G be a locally compact amenable group and A = A(G) the Fourier algebra of G. Helson’s Theorem has been extended to Fourier algebra A(G) of a first countable (or compact) group G by Dunkl and Ramirez [9] by quite a complicated method. When G is first countable, the algebra A(G) is weakly compactly generated [22, Theorem 3.2] so that the compact space (A(G)∗1 , w ∗ ) is sequentially compact. We give here, as an immediate corollary of Theorem 2.4 above, the exact analogue of the preceding corollary for the algebra Ap (G), so for the algebra A(G) as well. Since the subspace P Fp (G) of P Mp (G) has wCp (Example 2.3(b)) and (Ap (G)) ∩ P Fp (G) = ∅. The following corollary too is immediate from Theorem 2.4. Corollary 2.6. Let G be a nondiscrete locally compact amenable group such that the space (Ap (G)∗1 , w ∗ ) (1 < p < ∞) is sequentially compact. Then, for any weakly sequentially complete norm closed invariant subspace X of P Mp (G), the intersection X ∩ P Fp (G) is trivial. Now let G be a noncompact locally compact σ -compact commutative group and ω : G → [0, ∞[ is a continuous weight function chosen so that the algebra L1 (G, ω) is regular and has a BAI (see [4, Chapter 7] and [40, Section 6.3]). We recall that this algebra is always
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semisimple. In general, the Gelfand spectrum of the algebra L1 (G, ω) is quite different from that of the algebra L1 (G). However, whatever the spectrum of the algebra L1 (G, ω) is, since the group G is not compact, the spectrum of the algebra L1 (G, ω) and the space C0 (G, 1/ω) are disjoint. Indeed, by definition of the space C0 (G, 1/ω), for f ∈ C0 (G, 1/ω), the function f/ω vanishes at infinity whereas, for γ ∈ (L1 (G, 1/ω)), |γ /ω| = 1 on G. The following result also follows directly from Theorem 2.4. Corollary 2.7. Under the above hypothesis on G and ω, for any weakly sequentially complete norm closed invariant subspace X of L∞ (G, 1/ω), the intersection X ∩ C0 (G, ω) is trivial. So far we have considered only weakly compactly generated Banach algebras. To a certain extent we can weaken this hypothesis. Let X and Y be two invariant norm closed subspaces of A∗ . Suppose that X is weakly sequentially complete and Y has wCp. Now let I be a closed ideal of the algebra A and B = A/I . The subspace X0 = X ∩ I ⊥ of B ∗ is weakly sequentially complete and the subspace Y0 = I ⊥ ∩ Y has the wCp. The proof of following proposition can easily be deduced from the proof of Theorem 2.4. For that reason we omit the proof. Proposition 2.8. Let I be a closed ideal of the algebra A such that the quotient algebra B = A/I is semisimple, regular, Tauberian and weakly compactly generated. Let X and Y be two invariant norm closed subspaces of A∗ such that X is weakly sequentially complete and Y has the wCp. If X ∩ Y ∩ I ⊥ ∩ (B) = ∅ then X ∩ Y ∩ I ⊥ = {0}. If, in addition, X ⊆ I ⊥ then X ∩ Y = {0}. a closed set which is a set of For instance, if A = L1 (G) (G is commutative) and E ⊆ G synthesis (see the next section) then the algebra B = L1 (G)/k(E) is semisimple, regular and Tauberian. So, if the algebra B is also separable for instance, the preceding proposition applies to it. Remarks 2.9. a) For f ∈ A∗ , put Zf = {a.f : a ∈ A}, the norm closure of the space {a.f : a ∈ A} in A∗ . The space Zf is an invariant subspace of A∗ . Let X and Y be two norm closed (not necessarily invariant) subspaces of A∗ . As another version of Theorem 2.4, we can state the following: The algebra A being as in Theorem 2.4, suppose that a) For each f ∈ Y , the space Zf has the wCp. b) For each g ∈ X, the space Zg is weakly sequentially complete; and c) Zf ∩ Zg ∩ (A) = ∅ for each f ∈ Y and g ∈ X. Then Zf ∩ Zg = {0} for each f ∈ Y and g ∈ X. Moreover, if f ∈ Zf and g ∈ Zg for each f ∈ X and g ∈ Y , then X ∩ Y = {0}. b) A celebrated Banach space theorem due to Rosenthal [6, Chapter XI] says this: A Banach space Y does not contain an isomorphic copy of the sequence space 1 iff every bounded sequence (yn )n0 in Y has a subsequence which is weakly Cauchy. c) So, with the notation of Remark a), in Theorem 2.4, instead of assuming that the space Y has the wCP, it is enough to assume that, for each f ∈ Y , the space Zf does not contains an isomorphic copy of 1 . d) Let A be as in Theorem 2.4 and Y a norm closed invariant subspace of A∗ . Put Y0 = {f ∈ ∗ A : a.f ∈ Y for each a ∈ A}. This is also a norm closed invariant subspace of A∗ and Y ⊆ Y0 . It is easy to see that Y0 has the wCp iff Y does. If A = L1 (G), Y = C0 (G) and G is not discrete then
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Y0 is strictly larger than C0 (G) since every bounded Borel measurable function f : G → C with compact support (and also the uniform limits of such functions) are in Y0 . In the results given above and also below where C0 (G) is involved, if one uses this space Y0 = {f ∈ L∞ (G): a ∗ f ∈ C0 (G), ∀a ∈ L1 (G)} instead of C0 (G), the results remain valid. However, for the sake of clarity and concreteness we preferred to work with the space C0 (G) rather than with the space Y0 . In the rest of the paper we shall work with the group algebra L1 (G) of a locally compact commutative group but all the results presented below are valid, mutatis mutandis, for the Herz– Figa–Talamanca algebra Ap (G) of a locally compact group G whose discrete version Gd is amenable, for instance. 3. Applications to the study of sets of spectral synthesis In this section G will be a noncompact locally compact commutative group and A = L1 (G) the group algebra of G. On L1 (G)∗∗ we put the first Arens multiplication as defined in Section 1. Theorem 2.4 above has some applications to the problem of when is a given closed subset E of a set of synthesis. In this section we shall present these applications. We begin by recalling the G relevant definitions. As usual, to E we associate two ideals, k(E) (already Let E be a nonempty closed subset of G. defined) and j (E): a has compact support disjoint from E . j (E) = a ∈ L1 (G): Among the closed ideals of L1 (G) whose hull is E, k(E) is the largest one and j (E) is the smallest one. If k(E) = j (E) then the set E is said to be a set of spectral synthesis. In the books [18, Chapter X], [23, Chapter 5], [42, Chapter 7], the reader can find ample information ∗ on this notion. The annihilator of the ideal k(E) in the space L∞ (G) is the subspace Span(E)w of L∞ (G), the weak∗ closure of the linear span of the set E in the space L∞ (G). The annihilator of the closed ideal j (E) in L∞ (G) is the subspace ∞ L∞ E (G) = f ∈ L (G): σ (f ) ⊆ E . ∗
∗
∞ w ⊆ Both spaces Span(E)w and L∞ E (G) are invariant subspaces of L (G) and Span(E) ∞ ∞ LE (G). These two subspaces of L (G) are equal iff the set E is a set of synthesis. We shall mostly use the following characterization of sets of synthesis. For the sake of completeness, we include a proof. and In the rest of this section, E will denote a nonempty closed subset of the dual group G 1 ∗∗ e ∈ L (G) will be a fixed right identity. We also recall that, in order to have b ∗ a, f =
b, a ∗ f (rather than b ∗ a, f = b, a ∨ ∗ f , where a ∨ (t) = a(−t)), in Section 1, we have defined the (L1 (G), L∞ (G))-duality as a, f = G a(−t)f (t) dt.
Lemma 3.1. The set E is a set of synthesis iff a ∗ f = 0 for each a ∈ k(E) and f ∈ L∞ E (G). Proof. Suppose first that E is a set of synthesis. Then k(E)⊥ = L∞ E (G). Since for a ∈ k(E) and b ∈ L1 (G), the product b ∗ a is in k(E), for f ∈ k(E)⊥ ,
b, a ∗ f = b ∗ a, f = 0. This being true for all b ∈ L1 (G), a ∗ f = 0.
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Conversely, suppose that for each a ∈ k(E) and f ∈ L∞ E (G), a ∗ f = 0. Then
a, f = ea, f = e, a ∗ f = 0. ⊥ This being true for each a ∈ k(E), we conclude that f ∈ k(E)⊥ so that L∞ E (G) = k(E) . Hence E is a set of synthesis. 2
Our main tool to study the sets of synthesis is the following lemma. We recall that every maximal ideal Ker(γ ) of L1 (G) has a bounded approximate identity. A justification (valid for all the Banach algebras considered in this paper) of this affirmation can be found in [47, Lemma 4.3]. considered as a multiplicative functional on L1 (G) and for m We also recall that, for any γ ∈ G 1 ∗∗ and n in L (G) , we have γ , mn = γ , m . γ , n . That is, γ is also multiplicative on L1 (G)∗∗ . In particular the set {m ∈ L1 (G)∗∗ : γ , m = 0}, which is, by bipolar theorem, the second dual of the ideal Ker(γ ), is a maximal ideal of the Banach algebra L1 (G)∗∗ . there is an element eγ in L1 (G)∗∗ such that γ , eγ = 1 and, for Lemma 3.2. For each γ ∈ G, 1 each a ∈ L (G), aeγ = a (γ )eγ . Proof. Let uγ be a right identity in Ker(γ )∗∗ . Let us see that the element eγ = e − euγ of L1 (G)∗∗ will do the job. As uγ ∈ Ker(γ )∗∗ , uγ , γ = 0 so that
γ , eγ = γ , e − euγ = γ , e = 1. a (γ )eγ . Let a ∈ L1 (G) and decompose it in Let us now verify that for each a ∈ L1 (G), aeγ = 1 ∗∗ ∗∗ the direct sum L (G) = Ker(γ ) ⊕ Ce as a = n + λe. Since n ∈ Ker(γ )∗∗ and Ker(γ )∗∗ = a (γ ). Multiplying the equality a = n + a (γ )e from the {m ∈ L1 (G)∗∗ : γ , m = 0}, one has λ = right by uγ and using the fact that neγ = n, we get a (γ )euγ . auγ = n + Then subtracting this from a = n + a (γ )e we get a (γ ).(e − euγ ). a − auγ = a (γ )eγ . Since a − auγ = a(e − euγ ), we see that we have aeγ =
2
As a first use of this lemma we present the following result. If eγ , f = 0 then γ ∈ σ (f ). Proposition 3.3. Let f ∈ L∞ (G) and γ ∈ G. Proof. Since, for all a ∈ L1 (G), a (γ ). eγ , f ,
a, eγ .f = aeγ , f = we see that eγ , f = 0 iff eγ .f = 0. The preceding line also shows that eγ .f = eγ , f .γ .
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This being observed, suppose now that eγ , f = 0. Then eγ .f = eγ , f .γ = 0. Let (ai )i∈I be a bounded net in L1 (G) that converges to eγ in the weak∗ topology of L1 (G)∗∗ . Then ai ∗ f → eγ .f in the weak∗ topology of L∞ (G). Since eγ .f = eγ , f .γ and eγ , f = 0, dividing ai by
eγ , f , we see that γ ∈ σ∗ (f ). Since, by Theorem 2.0, σ∗ (f ) = σ (f ), γ ∈ σ (f ). 2 The family ( eγ , f )γ ∈G can be thought as the “Fourier coefficients of f ” in a generalized sense. However this family, unless f is almost periodic, does not characterize f in any way. For any f ∈ L∞ (G) and a ∈ L1 (G), as is well known and easy to see [32, Lemma 5], σ (a ∗ f ) ⊆ σ (f ) ∩ Supp( a ). Corollary 3.4. Let a ∈ k(E) and f ∈ L∞ E (G). Then eγ , a ∗ f = 0. a) For all γ ∈ G, b) Either a ∗ f = 0 or the set σ (a ∗ f ) is perfect. c) The function a ∗ f always lies in the space L∞ ∂E (G), where ∂E is the topological boundary of the set E in the space G. Since Proof. a) Let γ ∈ G. a (γ ). eγ , f ,
eγ , a ∗ f = eγ a, f = aeγ , f = a (γ ). eγ , f = 0. If γ ∈ / E then eγ , f = 0 too. Indeed, if we we see that if γ ∈ E, eγ , a ∗ f = had eγ , f = 0, by the preceding proposition, we would have γ ∈ σ (f ), which is not possible eγ , a ∗ f = 0. since σ (f ) ⊆ E. Hence, whatever the set E is, for all γ ∈ G, b) Suppose that γ is an isolated point of the set σ (a ∗ f ). Then, for some open neighborhood b(γ ) = 1 and Supp( b) ⊆ V . V of γ , V ∩ σ (a ∗ f ) = {γ }. Choose an element b ∈ L1 (G) such that Since σ (b ∗ a ∗ f ) ⊆ Supp( b) ∩ σ (a ∗ f ) ⊆ V ∩ σ (a ∗ f ) = {γ }, we conclude for instance by [40, Proposition 7.1.17] that b ∗ a ∗ f = λγ , where λ is a constant. The constant λ cannot be zero. Indeed, since γ ∈ σ (a ∗ f ), by definition of the set σ (a ∗ f ), there is a net (bi )i∈I in L1 (G) such that, in the weak∗ topology of L∞ (G), bi ∗ a ∗ f → γ . Since b(γ ) = 1 and b ∗ a ∗ f = λγ , we conclude that, on the one hand, in the weak∗ topology ∞ of L (G), b(γ )γ = γ ; b ∗ bi ∗ a ∗ f → and, on the other hand, b ∗ bi ∗ a ∗ f = λbi (γ )γ so that λbi (γ )γ → γ . Hence λ = 0. Applying now eγ to the equality b ∗ a ∗ f = λγ and using that b(γ ) = 1, we get a (γ ). b, eγ .f = a (γ ). eγ , f . λ = eγ , b ∗ a ∗ f = eγ a, b ∗ f =
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As λ = 0, this is not possible by assertion a). Hence, for a ∈ k(E) and f ∈ L∞ E (G), whenever a ∗ f = 0, the set σ (a ∗ f ) is perfect. c) Since σ (a ∗ f ) ⊆ σ (f ) ∩ Supp( a ) ⊆ E ∩ Supp( a) and a = 0 on E, E ∩ Supp( a ) ⊆ ∂E so that a ∗ f ∈ L∞ ∂E (G).
2
As a consequence of the preceding result we have the following result. eγ , f = 0. In particular, for each f ∈ C0 (G), Corollary 3.5. For each f ∈ C0 (G) and γ ∈ G, f = 0, the set σ (f ) is perfect. Since eγ .f = eγ , f γ , if we had eγ , f = 0, by assertion a) Proof. Let f ∈ C0 (G) and γ ∈ G. of the preceding corollary, we would have eγ .f = eγ , f .γ = 0, / C0 (G). Indeed, the group which is not possible since the function eγ .f is in C0 (G) and γ ∈ G is not compact and for all m ∈ L1 (G)∗∗ and f ∈ C0 (G), the function m.f is in the space C0 (G). This is because each function in C0 (G) is weakly almost periodic. Thus, eγ , f = 0 for and f ∈ C0 (G). If a γ ∈ σ (f ) were an isolated point of the set σ (f ) then, as seen all γ ∈ G in the proof of Corollary 3.4(b), we would have, for some b ∈ L1 (G), b ∗ f = γ . This is not / C0 (G). Hence, for any f ∈ C0 (G), f = 0, the set σ (f ) possible since b ∗ f is in C0 (G) and γ ∈ is perfect. 2 By WAP(G) and AP (G) we denote, respectively, the spaces of the weakly almost periodic and the almost periodic functions on the group G [2]. These are norm closed L1 (G)-submodules of L∞ (G), C0 (G) ⊆ WAP(G) and AP (G) ∩ C0 (G) = {0}. We now introduce the following subset of L∞ (G), . X(G) = f ∈ L∞ (G): either f = 0 or eγ , f = 0 for at least one γ ∈ G The next result shows that this set X(G) is fairly large. Proposition 3.6. a) The inclusion AP (G) ⊆ X(G) holds. b) If f ∈ L∞ (G) and the set σ (f ) has an isolated point then f ∈ X(G). c) If f ∈ L∞ (G) and for some net (ai )i∈I in L1 (G), the net (ai ∗ f )i∈I converges in the weak then f ∈ X(G). topology of L∞ (G) to some γ ∈ G d) X(G) ∩ C0 (G) = {0}. Proof. a) Let f ∈ AP (G), f = 0, be a given almost periodic function. Then the norm spectrum of f , that is, the intersection a ∗ f : a ∈ L1 (G) ∩ G
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is nonempty [12]. Let γ be an element in this intersection. Then, for some net (ai )i∈I in L1 (G), ai ∗ f → γ in the norm topology of L∞ (G). Then 1 = eγ , γ = lim eγ , ai ∗ f = lim ai (γ ). eγ , f . i
i
This proves that eγ , f = 0. So f ∈ X(G). The same argument (replacing the norm convergence by the weak convergence) also proves assertion c). To prove assertion b), if γ is an isolated point of σ (f ) then, as seen in the proof of Corollary 3.4(b), there is some b ∈ L1 (G) such that b ∗ f = γ . This implies that b(γ ). eγ , f . 1 = eγ , γ = eγ , b ∗ f = This shows that eγ , f = 0. So f ∈ X(G). and f ∈ C0 (G), eγ , f = 0, the interd) Since, by the preceding corollary, for each γ ∈ G section X(G) ∩ C0 (G) is trivial. 2 Next we present a simple but general result. We put Z(E) = k(E) ∗ L∞ E (G). That is, Z(E) = a ∗ f : a ∈ k(E) and f ∈ L∞ E (G) . We recall that, by Corollary 3.4, Z(E) ⊆ L∞ ∂E (G). Proposition 3.7. a) The set E is a set of synthesis iff Z(E) ⊆ X(G). b) The set E is a set of synthesis iff, for each a ∈ k(E) and f ∈ L∞ E (G), the space Za∗f = {b ∗ a ∗ f : b ∈ L1 (G)} is reflexive. Proof. a) Suppose that for each a ∈ k(E) and f ∈ L∞ E (G), the function a ∗ f is in the set X(G). eγ , a ∗ f = 0. By Then, by definition of the set X(G), either a ∗ f = 0 or, for some γ ∈ G, Corollary 3.4(a), this latter case is not possible. Hence a ∗f = 0, and the set E is a set of synthesis by Lemma 3.1. The converse implication, by Lemma 3.1, is always true. b) Suppose first that, for each a ∈ k(E) and f ∈ L∞ E (G), the space Za∗f is reflexive. Let (G) and let us see that a ∗ f = 0. If a ∗ f = 0 then σ (a ∗ f ) = ∅. Let a ∈ k(E) and f ∈ L∞ E γ ∈ σ (a ∗ f ). By Theorem 2.0, there is a net (bi )i∈I in the space L1 (G) such that bi ∗ a ∗ f 1 for all i ∈ I and the net (bi ∗ a ∗ f )i∈I converges in the weak∗ -topology of the space L∞ (G) to γ . Since the net (bi ∗ a ∗ f )i∈I is in the space Za∗f and since this space is reflexive, the net (bi ∗ a ∗ f )i∈I converges to γ in the weak topology of the space L∞ (G). But then, 1 = eγ , γ = lim eγ , bi ∗ a ∗ f = lim bi (γ ). eγ , a ∗ f = 0 i
i
by Corollary 3.4. This contradiction proves that a ∗ f = 0, and the set E is a set of synthesis by Lemma 3.1. The converse is trivial again by Lemma 3.1. 2
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The preceding proposition shows that, in the study of sets of synthesis, the most important subsets of L∞ (G) are the set Z(E) and the space L∞ ∂E (G). The first main result of this section is the following theorem. As seen above (Proposition 3.6), the set X(G) + C0 (G) is considerably larger than the space AP (G) + C0 (G). Theorem 3.8. Suppose that the group G is σ -compact and that a) The set Z(E) = k(E) ∗ L∞ E (G) is contained in the sum X(G) + C0 (G). b) For each γ ∈ ∂E, there is a measure μ ∈ M(G) such that μ(γ ) = 1 and, with Fμ = (G) is weakly sequentially complete. Supp( μ) ∩ ∂E, the space L∞ Fμ Then the set E is a set of synthesis. Proof. Let a ∈ k(E) and f ∈ L∞ E (G). We want to prove that a ∗ f = 0. By hypothesis a), by a ∗ f decomposes as a ∗ f = g + h, where g ∈ X(G) and h ∈ C0 (G). As, for all γ ∈ G, Corollaries 3.4 and 3.5, both eγ , a ∗ f = 0 and eγ , h = 0, we have 0 = eγ , a ∗ f = eγ , g + eγ , h = eγ , g eγ , g = 0. Hence, by definition of the set X(G), g = 0. Thus a ∗ f = h so that, for all γ ∈ G, so that the function a ∗ f is in the space C0 (G). Let us see that this is possible only if a ∗ f = 0. Indeed, for a contradiction, suppose that a ∗ f = 0, and let γ ∈ σ (a ∗ f ). By Corollary 3.4, γ ∈ ∂E. Hence, by hypothesis b), there is a measure μ ∈ M(G) such that μ(γ ) = 1 and the μ) ∩ ∂E is such that the space L∞ (G) is weakly sequentially complete. Next set Fμ = Supp( Fμ observe that the function μ ∗ a ∗ f , which is in the space C0 (G), is also in the space L∞ Fμ (G) since σ (μ ∗ a ∗ f ) ⊆ Supp( μ) ∩ ∂E = Fμ . Hence, since by Corollary 2.5, the intersection (G) ∩ C (G) is trivial, μ ∗ a ∗ f = 0. However this is not possible since γ ∈ σ (a ∗ f ) L∞ 0 Fμ and μ(γ ) = 1. To explain this last point, let γ = lim an ∗ a ∗ f n→∞
in the weak∗ -topology of L∞ (G)
for some sequence (an )n0 in L1 (G). Since L1 (G) is an ideal in M(G), μ ∗ γ = lim μ ∗ an ∗ a ∗ f n→∞
in the weak∗ -topology of L∞ (G).
As μ ∗ γ = μ(γ )γ = γ , this contradicts the equality μ ∗ a ∗ f = 0. This contradiction proves that σ (a ∗ f ) = ∅. Hence a ∗ f = 0, and E is a set of synthesis by Lemma 3.1. 2 Remark 3.9. If, in the preceding theorem, the set Fμ is contained in some set F which is a “HS-set”, that is, it is both a Helson set and a set of synthesis, then the space L∞ Fμ (G) is weakly
∞ ⊥ ⊥ sequentially complete. Indeed since then L∞ Fμ (G) ⊆ LF (G) = k(F ) and the space k(F ) is weakly sequentially complete. In particular, if ∂E is a HS-set then condition b) holds automatically. The reader can find a characterization of compact HS-sets in Saeki [45].
The next result also gives two conditions sufficient for a set E to be a set of synthesis. We recall that the space C0 (G) is weak∗ -dense in the space L∞ (G).
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Theorem 3.10. Suppose that the group G is σ -compact and that ∞ ∗ a) The intersection C0 (G) ∩ L∞ E (G) is weak dense in the space LE (G). ∞ b) The space L∂E (G) is weakly sequentially complete.
Then E is a set of synthesis. Proof. Let a ∈ k(E) and f ∈ L∞ E (G). We want again to prove that a ∗ f = 0. First let g be in the (G). Then the function a ∗ g is in C0 (G) ∩ L∞ intersection C0 (G) ∩ L∞ E ∂E (G). This intersection being trivial by Corollary 2.5, we see that a ∗ g = 0 for each g in C0 (G) ∩ L∞ E (G). This, by a), implies that a ∗ f = 0. Hence E is a set of synthesis. 2 An outstanding open problem in the theory of the spectral synthesis is the problem whether the union of two sets of synthesis is a set of synthesis. The next result, which is the second main result of this section, is related to this problem. Let F be another nonempty closed subset of the We put dual group G. D = ∂E ∩ ∂F ∩ ∂(E ∪ F ). The set X(G) (defined just before Proposition 3.6) has the same meaning as above. We also recall that Z(E) = k(E) ∗ L∞ E (G). Theorem 3.11. Suppose that the group G is σ -compact, both sets E and F are set of synthesis and that a) Z(E ∪ F ) ∩ WAP(G) ⊆ X(G) + C0 (G). b) For each γ ∈ D there is a measure μ ∈ M(G) with μ(γ ) = 1 and such that the set Fμ = D ∩ Supp( μ) is contained in some HS-set V . Then E ∪ F is a set of synthesis. Proof. Let a ∈ k(E ∪ F ) and f ∈ L∞ E∪F (G). We want to prove that a ∗ f = 0. For a contradiction, suppose that a ∗ f = 0. Then σ (a ∗ f ) = ∅. Take γ ∈ σ (a ∗ f ). One can easily see that γ ∈ D (see [43, Lemma 3]). By hypothesis b), there is a measure μ ∈ M(G) with μ(γ ) = 1 and μ) is contained in some HS-set V . Then the function μ ∗ a ∗ f such that the set Fμ = D ∩ Supp( ∞ is both in the spaces Z(E ∪ F ) and L∞ V (G). Since V is a HS-set, LV (G) ⊆ WAP(G) (see Corollary 4.3 below). It follows that the function μ ∗ a ∗ f is in the intersection Z(E ∪ F ) ∩ WAP(G). Hence, by hypothesis a), μ ∗ a ∗ f = g + h for some g ∈ X(G) and h ∈ C0 (G). Since, by Corol laries 3.4(a) and 3.5, for all γ ∈ G,
eγ , μ ∗ a ∗ f = 0 and eγ , h = 0, eγ , g = 0. This, by definition of the set X(G), is possible only if we see that, for all γ ∈ G, g = 0 so that μ ∗ a ∗ f = h, and the function μ ∗ a ∗ f is in the space C0 (G). Since μ ∗ a ∗ f ∞ is also in the space L∞ V (G) and since, by Corollary 2.5, LV (G) ∩ C0 (G) = {0}, necessarily μ ∗ a ∗ f = 0. Since γ ∈ σ (a ∗ f ) and μ(γ ) = 1, the function μ ∗ a ∗ f cannot be the zero function. From this contradiction we deduce that the function a ∗ f must be the zero function.
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Thus, Z(E ∪ F ) = k(E ∪ F ) ∗ L∞ E∪F (G) = {0}, and the union E ∪ F is a set of synthesis by Lemma 3.1. 2 We remark that if the set D is contained in some HS-set then condition b) of the preceding theorem holds automatically and Z(E ∪ F ) ⊆ L∞ D (G) ⊆ WAP(G) (Corollary 4.3 below). If (G) ∩ WAP(G) ⊆ X(G) + C (G) then condition a) holds automatically. L∞ 0 D 4. Helson sets, Arens regularity and sets of uniqueness As in the preceding section, in this section too, G will be a noncompact locally compact commutative group. It happens that the notion of Helson set is somewhat related to the notion of Arens regularity. In the first part of this section we shall study the connections between these two notions. In the second part, as applications of Theorem 2.4, we present some sufficient conditions to be a set of uniqueness. We start by recalling Arens regularity for a given closed subset E of G notion. Let A be a commutative Banach algebra. The second dual A∗∗ of A equipped with the Arens multiplication as defined in Section 1 is a Banach algebra. The Banach algebra A∗∗ is in general not commutative. When the algebra A∗∗ is commutative the algebra A is said to be Arens regular. a) Every commutative C ∗ -algebra is Arens regular. b) The closed subalgebras (so the uniform algebras) and the quotient algebras of Arens regular algebras are Arens regular. c) The group algebra L1 (G), unless G is finite, is not Arens regular. The reader can find ample information on this much studied notion in the books [3,37] and the memoir [4] among many other sources. One of the most important characterizations of Arens regularity is this: The algebra A is Arens regular iff each functional f ∈ A∗ is weakly almost periodic. (i.e. the set H (f ) = {a.f : a ∈ A1 } is relatively weakly compact). We denote by WAP(A) the set of the weakly almost periodic functionals on A. This is a norm closed invariant subspace of A∗ . For the algebra A = L1 (G), one has WAP(A) = WAP(G) [10, Theorem 1.1]. We shall need the following result. Lemma 4.1. Let B be another commutative Banach algebra and φ : A → B a bounded onto homomorphism. Then φ ∗ (WAP(B)) = WAP(A) ∩ ker(φ)⊥ . Proof. We first observe that, φ being onto, by the Open Mapping Theorem, there is a constant c > 0 such that φ(A1 ) ⊇ c.B1 . Then, since φ is a homomorphism, we observe that, for a ∈ A and g ∈ B∗, φ(a).g = a.φ ∗ (g). These two observations imply that φ ∗ (WAP(B)) ⊆ WAP(A). Since always φ ∗ (WAP(B)) ⊆ ker(φ)⊥ , the inclusion φ ∗ (WAP(B)) ⊆ WAP(A) ∩ ker(φ)⊥ is established. To prove the reverse inclusion, let f ∈ WAP(A) ∩ ker(φ)⊥ be a given functional. Since φ is onto and ker(φ) ⊆ ker(f ), by Sard’s quotient theorem [19, p. 176], there is a functional g ∈ B ∗ such that g ◦ φ = f . That is, φ ∗ (g) = f . It remains to prove that g ∈ WAP(B). To see this we first recall that the set {a.f : a ∈ A1 } is relatively weakly compact since f ∈ WAP(A). As φ ∗ (g) = f , φ(a).g: a ∈ A1 = a.φ ∗ (g): a ∈ A1 = {a.f : a ∈ A1 },
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and the inclusion 1 {b.g: a ∈ B1 } ⊆ .{φ(a).g: a ∈ A1 } c holds, we see that the functional g is weakly almost periodic on B. Hence φ ∗ (WAP(B)) = WAP(A) ∩ ker(φ)⊥ . 2 As a consequence of this lemma we have the following result. Proposition 4.2. Let I be a closed ideal of A. Then the quotient algebra A/I is Arens regular iff I ⊥ ⊆ WAP(G). Proof. Let π : A → A/I be the quotient homomorphism. Since ker(π) = I , by the above lemma, WAP (A/I ) = I ⊥ ∩ WAP(A). Hence, by what we have noted prior to Lemma 4.1, the algebra A/I is Arens regular iff (A/I )∗ = WAP(A/I ). That is, the algebra A/I is Arens regular iff I ⊥ = I ⊥ ∩ WAP(G), which is possible iff I ⊥ ⊆ WAP(A). 2 Since every commutative C ∗ -algebra is Arens regular and Arens regularity is invariant under Banach algebra isomorphisms, we have the following corollary. At this point we recall that, the inclusion L∞ (G) ⊆ as proved by Loomis in [34], for a compact scattered subset E of G, E is a set of synthesis so that AP (G) holds. We also recall that every compact scattered set E ⊆ G ⊥ L∞ E (E) = k(E) . The next result is an analogue of this result of Loomis for Helson sets. This result is immediate from the preceding proposition. the inclusion k(E)⊥ ⊆ WAP(G) holds. Corollary 4.3. For any closed Helson set E ⊆ G, Contrary to closed scattered sets, since not every Helson set is a set of synthesis [30,46], from the inclusion k(E)⊥ ⊆ WAP(G) we cannot conclude that the inclusion L∞ E (G) ⊆ WAP(G) holds, unless E is a set of synthesis. As is well-known [2, Chapter 2], there is a unique invariant mean M on the space WAP(G). This mean induces the decomposition WAP(G) = AP (G) ⊕ W0 (G), where W0 (G) = {f ∈ WAP(G): M(|f |) = 0}. The space C0 (G) is strictly contained in the space W0 (G) [2, p. 31]. Corollary 4.3 and Corollary 2.5 imply that, for any closed Helson set E, the inclusion k(E)⊥ ⊆ AP (G) + W0 (G) \ C0 (G) holds.
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Remarks 4.4. a) The converse of the preceding corollary is not true. Indeed there exist closed such that L∞ (G) ⊆ AP (G), in particular k(E)⊥ ⊆ WAP(G), which are not Helson sets E ⊆ G E sets [39]. We remark that any closed set E for which the inclusion L∞ E (G) ⊆ AP (G) holds is a ⊥. set of synthesis (Proposition 3.7) so that L∞ (G) = k(E) E b) Since every positive definite function on the group G is weakly almost periodic [11, Theo ∧ is contained in the space WAP(G). From this one can deduce another rem 11.2], B(G) = M(G) proof of Corollary 4.3 but Proposition 4.2 is of independent interest. c) Corollary 4.3 proves incidentally that the space W0 (G), which is a norm closed invariant subspace of L∞ (G), does not have the wCp. Indeed, since for every nonempty closed Helson we have k(E)⊥ ⊆ AP (G) + W0 (G) \ C0 (G), if the space W0 (G) had the wCp, by set E ⊆ G, Theorem 2.4, we would have k(E)⊥ ∩ W0 (G) = {0} so that we would always have k(E)⊥ ⊆ AP (G), which is not true for instance if E contains a nonempty perfect subset [34]. In the rest of this section we are going to present a couple of results related to the set of uniqueness. We start by recalling the definition of this notion. Let Z be the group of the integer numbers and T = Z, the unit circle group. A subset E of T is said to be a set of uniqueness for the trigonometric series if the only int satisfying the condition trigonometric series ∞ n=−∞ cn e ∀t ∈ T \ E,
∞ n=−∞
cn eint = lim
n→∞
n
cn eikt = 0
k=−n
is the identically null series (i.e. cn = 0 for all n ∈ Z). The reader can find in the books [20,26] and [51] ample information about historical development of this notion. We shall not consider the finer subclasses of the sets of uniqueness. For these we refer the reader to the book [26] and Lyons’ paper [35]. Some of the important results obtained in the past about sets of uniqueness are: a) Every countable subset (closed or not) of T is a set of uniqueness (Cantor, Lebesgue, W.H. Young). b) Every closed scattered subset of T is a set of uniqueness (W.H. Young). c) The union of countably many closed sets of uniqueness is a set of uniqueness (Bary). d) There exist perfect sets of uniqueness (Bary, Rajchman, Salem-Zygmund). f) There exist sets of multiplicity (= nonuniqueness) of Lebesgue measure zero (Menshov). g) There exist closed Helson sets which are not sets of uniqueness (Th.W. Körner). For precise references about these historical results we refer the reader to the books just mentioned, especially [26]. For a general locally compact commutative group G, the set of uniqueness can be defined as follows. This is the approach introduced by Piatetski–Shapiro (see [15, p. 191] and [26, II.4.1 and V.4.1]). is said to be a set of uniqueness if L∞ (G) ∩ C0 (G) = {0}. Definition 4.5. A closed subset E of G E We are going to work with this definition. From this definition it is clear that this notion is closely related to Corollary 2.5. In the theorem given below we have collected some consequences of Corollary 2.5 related to sets of uniqueness. It follows from Helson’s Theorem that every HS-set is a set of uniqueness. Körner [29] and subsequently Kaufman [25] settling an outstanding open problem proved that there exist closed Helson sets in T which are not sets of
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uniqueness. For such a set E, the space k(E)⊥ is weakly sequentially complete but the space L∞ E (Z) is not. As in Section 2, for each f ∈ L∞ (G), we put Zf = {a ∗ f : a ∈ L1 (G)}. The bar denotes the norm closure in L∞ (G). The space Zf is apparently a much simpler space than the space L∞ E (G). The next lemma displays the role that the space Zf can play in the study of sets of uniqueness. Then E is a set of uniqueness iff, for each f ∈ Lemma 4.6. Let E be a closed subset of G. L∞ (G), Z ∩ C (G) = {0}. f 0 E ∞ Proof. Since, for each f ∈ L∞ E (G), the inclusion Zf ⊆ LE (G) holds, the necessity of the condition is obvious. To prove its sufficiency, suppose that, for each f ∈ L∞ E (G), Zf ∩ C0 (G) = {0}. (G) ∩ C (G) = {0}. For a contradiction, suppose that there is an We have to prove that L∞ 0 E (G)∩C (G), f = 0. Since f ∈ C (G), f is weakly almost periodic. So, for any bounded f ∈ L∞ 0 0 E approximate identity (ei )i∈I in L1 (G), ei ∗ f → f weakly. This shows that f ∈ Zf . Since f ∈ C0 (G) too, we see that f ∈ Zf ∩ C0 (G). This contradiction completes the proof. 2
The next theorem, which is the main result of this section, offers some conditions implying that a given set E is a set of uniqueness. Theorem 4.7. Suppose that the group G is σ -compact and that E is a nonempty closed subset of Then each of the following conditions implies that E is a set of uniqueness. the dual group G. The space L∞ E (G) is weakly sequentially complete. For each f ∈ L∞ E (G), the space Zf is weakly sequentially complete. For each f ∈ L∞ E (G), Zf ∩ C0 (G) = {0}. μ) ∩ For each γ ∈ E, there is a measure μ ∈ M(G) such that μ(γ ) = 1 and, with Fμ = Supp( E, the space L∞ Fμ (G) is weakly sequentially complete. e) For each γ ∈ E, there is a measure μ ∈ M(G) such that μ(γ ) = 1 and the set E ∩ Supp( μ) is contained in some HS-set Vμ . f) For each γ ∈ E, there is a measure μ ∈ M(G) such that μ(γ ) = 1 and the set Fμ = E ∩ Supp( μ) is a set of uniqueness.
a) b) c) d)
Proof. It is clear that condition a) is stronger than condition b). Condition b), by Corollary 2.5, implies condition c); and, by Lemma 4.6, condition c) implies that E is a set of uniqueness. Hence each of the conditions a), b) and c) implies that E is a set of uniqueness. Now suppose that condition d) holds. Let, if there is any, f ∈ L∞ E (G) ∩ C0 (G), f = 0. Since f = 0, σ (f ) = ∅. Take a γ ∈ σ (f ). As σ (f ) ⊆ E, γ ∈ E. So, there is a measure μ ∈ M(G) such μ) ∩ E, the space L∞ that μ(γ ) = 1 and, with Fμ = Supp( Fμ (G) is weakly sequentially complete. As σ (μ ∗ f ) ⊆ Supp( μ) ∩ E = Fμ , the function μ ∗ f is in the space L∞ Fμ (G). On the other hand, since f ∈ C0 (G), the function μ ∗ f is in C0 (G). Hence the function μ ∗ f is in the intersection L∞ Fμ (G) ∩ C0 (G). This intersection, by Corollary 2.5 is trivial. So μ ∗ f = 0. However this is not possible since γ ∈ σ (f ) and μ(γ ) = 1. This contradiction proves that the set E is a set of uniqueness. The proof that condition e) implies that E is a set of uniqueness is very similar. So we omit it. To finish the proof, suppose that condition f) holds. Let, if there is any, f ∈ L∞ E (G) ∩ C0 (G),
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f = 0. Since f = 0, σ (f ) = ∅. Take a γ ∈ σ (f ). By condition f), there is a measure μ ∈ M(G) μ) is a set of uniqueness. Proceeding exactly as such that μ(γ ) = 1 and the set Fμ = E ∩ Supp( in the proof of d), we get that the function μ ∗ f is in the intersection L∞ Fμ (G) ∩ C0 (G). The set Fμ being a set of uniqueness, this intersection is trivial. So μ ∗ f = 0, and as above, we conclude that E is a set of uniqueness. 2 The final result of the paper is the following proposition, which is another version of Theorem 3.10 above. The proof, being very similar, is omitted. a nonempty closed set. If the interProposition 4.8. Suppose that G is σ -compact and E ⊆ G ∞ ∞ ∗ section LE (G) ∩ C0 (G) is weak -dense in LE (G) and the set ∂E is a set of uniqueness then E is a set of synthesis. We finish the paper with some questions and remarks. 5. Questions and remarks The results presented Suppose that G is σ -compact and E a nonempty closed subset of G. in the last two sections arise a certain number of questions. We present here some of them. As above, for f ∈ L∞ (G), we put Zf = {a ∗ f : a ∈ L1 (G)}. Q1). Characterize those f ∈ L∞ (G) for which the space Zf has the wCp. Q2). Characterize those f ∈ L∞ (G) for which the space Zf is weakly sequentially complete. Q3). Characterize those f ∈ L∞ (G) for which one has Zf ∩ C0 (G) = {0}. Q4). If L∞ E (G) ⊆ WAP(G), is then E a set of synthesis? The characterizations sought after may be in terms of topological properties of the set σ (f ) or in terms of geometric properties of the Banach space Zf . R1). The weak sequential completeness is not the only geometric property on L∞ E (G) that implies that E is a set of uniqueness or set of synthesis (see the next remark). For instance, if ∞ ∗ on the unit sphere S = {f ∈ L∞ E (G): f = 1} of the space LE (G) the weak and the weak topologies agree then E is a set of uniqueness. To prove this, let, if there is any, f ∈ L∞ E (G) ∩ C0 (G), f = 0 be a function and γ ∈ σ (f ). By Theorem 2.1, there is some sequence (an )n0 in L1 (G) such that an ∗ f 1 for all n 0 and an ∗ f → γ in the weak∗ topology of L∞ (G). Then 1 = γ lim inf an ∗ f lim sup an ∗ f 1 n
n
so that an ∗ f → γ . Hence an ∗ f → γ weakly. But then, since an ∗ f ∈ C0 (G) for all n 0, γ ∈ C0 (G). This contradiction proves that L∞ E (G) ∩ C0 (G) = {0}. R2). Let this time S be the unit sphere of L∞ ∂E (G). Suppose that, whenever fn ∈ S, f ∈ S and fn → f in the weak∗ topology of L∞ (G), a ∗ fn → a ∗ f weakly, for each a ∈ L1 (G). Then, as in the preceding remark, one can see that the set E is a set of synthesis. is separable (this is the case if G is also first countable). Then, whatever R3). Suppose that G the set E is, one can find a function f ∈ AP (G) such that σ (f ) = E. To see this, let (γn )n1 1 be a dense sequence in E. Let f = ∞ n=1 n2 γn . Then f ∈ AP (G) and σ (f ) = E. Indeed, as
eγn , f = n12 γn , the sequence (γn )n1 is in σ (f ) (Proposition 3.3) so that E ⊆ σ (f ). The reverse inclusion being trivial, σ (f ) = E. However, for instance, if E is a perfect set of uniqueness,
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it is not possible to find a function f ∈ C0 (G) such that σ (f ) = E. The perfect sets that can be realized as σ (f ) for some f ∈ C0 (G) should have some special feature that this author does not completely comprehend. R4). Let f ∈ L∞ (G) and put If = {a ∈ L1 (G): a ∗ f = 0}. It is easy to see that ⊥ If ∩ C0 (G) = {0} iff If is σ (M(G), C0 (G))-dense in M(G). We do not know whether there is a similar condition equivalent to the condition that Zf ∩ C0 (G) = {0}. Acknowledgments The author expresses his sincere thanks to the referee for his careful reading of this paper and his suggestions. References [1] A. Beurling, Un théorème sur les fonctions bornées et uniformément continues sur l’axe réel, Acta Math. 77 (1945) 127–136. [2] R.B. Burckel, Weakly Almost Periodic Functions on Semigroups, Gordon and Breach, Science Publishers, Inc., New York, 1970. [3] H.G. Dales, Banach Algebras and Automatic Continuity, Oxford University Press, Oxford, 2000. [4] H.G. Dales, A.T. Lau, The Second Duals of the Beurling Algebras, Mem. Amer. Math. Soc., vol. 836, Amer. Math. Soc., Providence, RI, 2005. [5] J. Diestel, Geometry of Banach Spaces-Selected Topics, Lecture Notes in Math., vol. 485, Springer-Verlag, Berlin, 1975. [6] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York, 1984. [7] Y. Domar, Some results on narrow spectral analysis, Math. Scand. 20 (1967) 5–18. [8] R. Doss, Elementary proof of a theorem of Helson, Proc. Amer. Math. Soc. 27 (1971) 418–420. [9] Ch.F. Dunkl, D.E. Ramirez, Helson sets in compact and locally compact groups, Michigan Math. J. 19 (1972) 65–69. [10] Ch.F. Dunkl, D.E. Ramirez, Weakly almost periodic functionals on the Fourier algebra, Trans. Amer. Math. Soc. 185 (1973) 501–514. [11] W.E. Eberlein, Abstract ergodic theorems and weakly almost periodic functions, Trans. Amer. Math. Soc. 67 (1949) 217–240. [12] W.E. Eberlein, The point spectrum of weakly almost periodic functions, Michigan Math. J. 3 (1955-56) 137–139. [13] P. Eymard, L’Algebre de Fourier d’un group locallement compact, Bull. Soc. Math. France 92 (1964) 181–236. [14] H. Helson, Fourier transforms on perfect sets, Studia Math. 14 (1954) 209–213. [15] C. Herz, The spectral theory of bounded functions, Trans. Amer. Math. Soc. 94 (1960) 181–232. [16] C. Herz, Drury’s lemma and Helson sets, Studia Math. 42 (1972) 205–219. [17] C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (3) (1973) 91–123. [18] E. Hewitt, K. Ross, Abstract Harmonic Analysis (I and II), Springer, Berlin, 2002. [19] R. Holmes, Geometric Functional Analysis and Its Applications, Springer-Verlag, New York, 1975. [20] J.P. Kahane, R. Salem, Ensembles Parfaits et Series Trigonometriques, Herman, Paris, 1963. [21] J.P. Kahane, Sur les rearrangements des fonctions de la classe A, Studia Math. 31 (1968) 287–293. [22] E. Kaniuth, A.T. Lau, G. Schlichting, Weakly compactly generated Banach algebras associated to locally compact groups, J. Operator Theory 40 (1998) 323–337. [23] E. Kaniuth, A Course in Commutative Banach Algebras, Grad. Texts Math., Springer, 2008. [24] R. Kaufman, Examples in Helson sets, Bull. Amer. Math. Soc. 72 (1966) 139–140. [25] R. Kaufman, M-sets and distributions, pseudofunctions and Helson sets, Astérisque 5 (1973) 225–230. [26] A.S. Kechris, A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness, Cambridge Univ. Press, Cambridge, 1987. [27] Th.W. Körner, Some results on Kronecker, Dirichlet and Helson sets, Ann. Inst. Fourier (Grenoble) 20 (2) (1970) 219–324. [28] Th.W. Körner, Relations between Dirichlet, Kronecker and Helson sets, in: L.-A. Lindahl, F. Poulsen (Eds.), Thin Sets in Harmonic Analysis, Marcel Dekker Inc., New York, 1971 (Chapter XIII). [29] Th.W. Körner, A pseudofunction on a Helson set (I and II), Astérisque 5 (1973) 3–224, and 231–239.
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Journal of Functional Analysis 258 (2010) 978–998 www.elsevier.com/locate/jfa
Hankel operators and the Stieltjes moment problem ✩ Hélène Bommier-Hato, El Hassan Youssfi ∗ LATP, U.M.R. C.N.R.S. 6632, CMI, Université de Provence, 39 Rue F-Joliot-Curie, 13453 Marseille Cedex 13, France Received 19 June 2009; accepted 9 August 2009 Available online 1 September 2009 Communicated by N. Kalton
Abstract Let s be a non-vanishing Stieltjes moment sequence and let μ be a√representing measure of it. We denote by μn the image measure in Cn of μ ⊗ σn under the map (t, ξ ) → tξ , where σn is the rotation invariant probability measure on the unit sphere. We show that the closure of holomorphic polynomials in L2 (μn ) is a reproducing kernel Hilbert space of analytic functions and describe various spectral properties of the corresponding Hankel operators with anti-holomorphic symbols. In particular, if n = 1, we prove that there are nontrivial Hilbert–Schmidt Hankel operators with anti-holomorphic symbols if and only if s is exponentially bounded. In this case, the space of symbols of such operators is shown to be the classical Dirichlet space. We mention that the classical weighted Bergman spaces, the Hardy space and Fock type spaces fall in this setting. © 2009 Elsevier Inc. All rights reserved. Keywords: Hankel operator; Fock space; Bergman kernel
1. Introduction ¯ In this paper we consider Hankel operators and the ∂-canonical solution operator in a Hilbert space of analytic functions related to a Stieltjes moment sequence. We recall that a sequence s = (sd ), d ∈ N0 , is said to be a Stieltjes moment sequence if it has the form
✩
This research is partially supported by the French ANR DYNOP, Blanc07-198398.
* Corresponding author.
E-mail addresses: [email protected] (H. Bommier-Hato), [email protected] (E.H. Youssfi). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.08.004
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+∞ sd = t d dμ(t), 0
where μ is a non-negative measure on [0, +∞[, called a representing measure for s. These sequences have been characterized by Stieltjes [22] in terms of some positive definiteness conditions. We denote by S the set of such sequences. It follows from the above integral representation that each s ∈ S is either non-vanishing, that is, sd > 0 for all d, or else sd = δ0d for all d. We denote by S∗ the set of all non-vanishing elements of S. Fix an element s = (sd ) ∈ S∗ . By Cauchy–Schwarz inequality we see that the sequence sd+1 sd is non-decreasing and hence converges as d → +∞ to the radius of convergence of the entire series
Fs (λ) :=
+∞
λd
d=n−1
sd+1 sd
sd+1−n
,
λ ∈ C.
1
Set Rs := limd→+∞ = limd→+∞ sd d . The sequences s for which the radius Rs is finite are called exponentially bounded [5]. Denote by Ωs the ball in Cn centered at the origin with radius Rs with the understanding that n when R = +∞. We denote by A2 (s) the Hilbert space of those holomorphic functions Ωs = C s f (z) = α∈Nn aα zα on Ωs that satisfy 0
α∈Nn0
α!s|α| |aα |2 < +∞ (|α| + n − 1)!
equipped with the natural inner product f, g :=
α∈Nn0
α!s|α| aα b¯α (|α| + n − 1)!
if f (z) = α∈Nn aα zα and g(z) = α∈Nn bα zα are two elements of A2 (s). 0 0 Now let σ = σn be the rotation invariant probability measure on the unit sphere Sn in Cn and n let μ be a representing n the image measure in C of μ ⊗ σn under √ measure of s. We denote by μ n the map (t, ξ ) → tξ from [0, +∞[ × Sn onto C . We consider the Hilbert space L2 (μn ) of square integrable complex-valued functions in Cn with respect to the measure μn . Our first result is the following: Theorem A. The measure μn is supported by the closure of the domain Ωs . In addition, for each set compact K ⊂ Ωs there exists C = C(K) > 0 such that sup f (z) C f L2 (μn )
z∈K
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for all holomorphic polynomials f in Cn . Furthermore, the space A2 (s) coincides with the closure of the holomorphic polynomials in L2 (μn ) and its reproducing kernel is given by Ks (z, w) =
1 Fs(n−1) z, w , (n − 1)!
z, w ∈ Ωs .
The classical weighted Bergman spaces, weighted Fock spaces and Hardy spaces are of the form A2 (s); each of these space is associated to an appropriate choice of the sequence s, see [2,20,26]. To state further results we consider the orthogonal projection Ps associated to A2 (μn ). It is given for all g ∈ L2 (μn ) by (Ps g)(z) = Ks (z, w)g(w) dμn (w), z ∈ Ωs . Ωs
This integral operator can be extended in a natural way to functions g that satisfy Ks (z, ·)g ∈ L1 (μn ) for all z ∈ Ωs . This extension allows us to define Hankel operators. To do so, denote by T(s) the class of all f ∈ A2 (s) such that f ϕKs (z, ·) ∈ L1 (μn ) for all holomorphic polynomials ϕ and z ∈ Ωs and the function
Hf¯ (ϕ)(z) := Ks (z, w)ϕ(w) f¯(z) − f¯(w) dμn (w), z ∈ Ωs , Cn
is the restriction to Ωs of a function in L2 (μn ). This is a densely defined operator from A2 (s) into L2 (μn ) which will be called the Hankel operator Hf¯ with symbol f¯. It can be written in the form Hf¯ (ϕ) = (I − Ps )(f¯ϕ) for all holomorphic polynomials ϕ. It is not hard to see that the class T(s) contains all holomorphic polynomials. Finally, if f ∈ k T(s), we denote by Spec(f ) the set of all multi-indices k ∈ Nn0 such that ∂∂zfk (0) = 0. Our second result is the following Theorem B. Suppose that f is a holomorphic polynomial. Then (1) Hf¯ is bounded if and only if sup d∈N0
sd+2|k| sd+|k| n − 1 sd+|k| − + sd+|k| sd d sd
< +∞
(1.1)
for all k ∈ Spec(f ). (2) Hf¯ is compact if and only if lim
d→+∞
for all k ∈ Spec(f ).
sd+2|k| sd+|k| n − 1 sd+|k| − + sd+|k| sd d sd
=0
(1.2)
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(3) If p > 0, then Hf¯ is in the Schatten class Sp (A2 (s), L2 (μn )) if and only if
d
n−1
d∈N
sd+2|k| sd+|k| − sd+|k| sd
p 2
+ (n − 1)d
n−1− p2
sd+|k| sd
p 2
< +∞
for all k ∈ Spec(f ). We point out that if the sequence s is exponentially bounded then (1.1) and (1.2) hold. The last assertion of Theorem B shows that if n 2, and the Schatten class Sp (A2 (s), L2 (μn )) contains nontrivial Hankel operators with anti-holomorphic symbols, then p > 2n. The converse to this statement is not true as shown by the authors in [11]. In particular, in higher dimensions there are no nontrivial Hilbert–Schmidt Hankel operators with anti-holomorphic symbols. The situation in the one-dimensional case is completely different. More precisely: Theorem C. Suppose that n = 1 and f is a nonconstant holomorphic function in f ∈ T(s). Then Hf¯ is in the Hilbert–Schmidt class S2 (A2 (s), L2 (μn )) if and only if s is exponentially bounded and f is in the classical Dirichlet space D(Ωs ). In addition, the trace Tr(Hf∗¯ Hf¯ ) of Hf∗¯ Hf¯ is given by 1 Tr Hf∗¯ Hf¯ = π =
2 f (z) dA(z)
Ωs
f (z) − f (w)2 Ks (z, w)2 dA(z) dA(w)
Ωs
where dA(z) is the Lebesgue measure in C. The first equality shows the characterization in the latter theorem depends only on the limit limd→+∞ sd+1 sd . The above result has been proved by separate methods in the two simple particular cases of Hardy and Bergman spaces [26]. Now we shall characterize the boundedness, the compactness and the membership in a Schatten class of S the canonical solution operator of the ∂¯ on the space H(0,1) (Ωs ) consisting of ¯ (0, 1)-forms with holomorphic coefficients in L2 (μn ) defined by ∂(Sf ) = f and Sf is orthogo2 nal to holomorphic elements of L (μn ). The spectral properties of this operator were studied by Haslinger [12,13], Haslinger and Helfer [14] and Lovera and Youssfi [17]. Corollary 1.1. Consider the canonical solution operator S to the ∂¯ from H(0,1) (Ωs ) to L2 (μn ). Then the following are equivalent: (1) S is bounded on H(0,1) (Ωs ). (2) For all j = 1, . . . , n, the Hankel operator Hz¯ j is bounded from A2 (s) into L2 (μn ). (3) There is j = 1, . . . , n, such that the Hankel operator Hz¯ j is bounded from A2 (s) into L2 (μn ).
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(4) There is a positive constant C > 0 such that
sd+n sd+n−1
−
sd+n−1 sd+n−2
+
n − 1 sd+n C d sd+n−1
for all positive integers d. Corollary 1.2. Consider the canonical solution operator S to the ∂¯ from H(0,1) (Ωs ) to L2 (μn ). Then the following are equivalent: (1) (2) (3) (4)
S is compact on H(0,1) (Ωs ). For all j = 1, . . . , n, the Hankel operator Hz¯ j is compact from A2 (s) into L2 (μn ). There is j = 1, . . . , n, such that the Hankel operator Hz¯ j is compact A2 (s) into L2 (μn ). We have lim
d→+∞
sd+n sd+n−1
−
sd+n−1 n − 1 sd+n + sd+n−2 d sd+n−1
= 0.
In each of the two preceding corollaries, the equivalence between the two assertions (1) and (4) was established in Lovera and Youssfi [17] and later by Haslinger and Lamel [15]. Corollary 1.3. Consider the canonical solution operator S to the ∂¯ from H(0,1) (Ωs ) to L2 (μn ) and let p > 0. Then the following are equivalent: (1) S is in the Schatten class Sp (H(0,1) (Ωs )L2 (μn )). (2) For all j = 1, . . . , n, the Hankel operator Hz¯ j is in the Schatten class Sp (A2 (s), L2 (μn )). (3) There is j = 1, . . . , n, such that the Hankel operator Hz¯ j is in the Schatten class Sp (A2 (s), L2 (μn )). (4) There is a positive constant C such that d∈N
d
n−1
sd+n sd+n−1
sd+n−1 − sd+n−2
p 2
+ (n − 1)d
n−1− p2
sd+n sd+n−1
p 2
C
for all positive integers d. In the latter corollary, the equivalence between the two assertions (1) and (4) was established in Lovera and Youssfi [17] in the case p 2 and later by Haslinger and Lamel [15] in the general case. To state another result, we let M(s) be the subspace of T(s) consisting of those functions f for which the Hankel operator Hf¯ is bounded on A2 (s). We equip M(s) with norm f M(s) := Hf¯ + f (0). The subspace of M(s) consisting of functions f such that Hf¯ is a compact operator will be denoted by M∞ (s). Then it is not hard to see that M∞ (s) is a closed subspace of M(s).
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If p > 0, we denote by Mp (s) the subspace of those functions f ∈ M(s) such that the Hankel operator Hf¯ is the Schatten class Sp (A2 (s), L2 (μn )). We equip Mp (s) with quasi-norm f Mp (s) := Hf¯ Sp + f (0). Then we have the following Theorem D. Let X ∈ {M(s), M∞ (s), Mp (s)} and let U be a rotation in Cn . Then the following assertions hold. (1) If f ∈ X, then f ◦ U ∈ X and f ◦ U X = f X . (2) If f ∈ X, then zk ∈ X for all k ∈ Spec(f ). (3) If the sequence s is either exponentially bounded or satisfies lim
d→+∞
sd+l sd2
1 d
=0
for all l ∈ N0 ,
(1.3)
then the spaces M(s), M∞ (s) and Mp (s), p 1, are Banach spaces and the space Mp (s), 0 < p < 1, is a quasi-Banach space. We point out that there are examples of Stieltjes moment sequences that do not satisfy (1.3) as shown by Boas type sequences [10]. There is a sequence of positive real numbers s satisfying s0 1 and sn+1 (nsn )n+1 . It is not hard to see by Theorem B that the spaces M(s), M∞ (s) and Mp (s) corresponding to such sequences are trivial, namely, they consist only of constant functions. Another type of Stieltjes moment sequences for which Theorem B applies to show that the corresponding spaces M(s), M∞ (s) and Mp (s) are trivial are the Stieltjes sequences s that satisfy s0 1 and sd2 δsd+1 sd−1
for all d 1
(1.4)
for some 0 < δ < 1. Arbitrary sequences satisfying (1.4) were studied by Bisgaard and Sasvári [9] and Bisgaard [8]. They were shown in [8] to be Stieltjes moment sequences as long 2 as d1 δ d 14 . 2. The Hilbert space A2 (s) and related operators We first fix some notations. Let Nn0 denote the set of all n-tuples with components in the set N0 of all non-negative integers. If α = (α1 , . . . , αn ) ∈ Nn0 , we let |α| := α1 + · · · + αn denote the length of α. If β = (β1 , . . . , βn ) ∈ Nn0 satisfies αj βj for all j = 1, . . . , n, then we write α β. Otherwise, set α β. Finally, if A and B are two quantities, we use the symbol A ≈ B whenever A C1 B and B C2 A, where C1 and C2 are positive constants independent of the varying parameters. Proof of Theorem A. We first observe that if a positive real number r satisfies μ(]r, +∞[) = 0, 1 then for all non-negative integers d, we have sd r d μ(]0, +∞[) and hence lim supd sd d r.
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This shows that the radius of convergence of the series Fs is smaller than or equal to the infimum of all such real numbers r. Conversely, suppose that r > 0 satisfies μ(]r, +∞[) > 0. Then r d μ ]r, +∞[ sd for all non-negative integers d. Therefore, 1
1
r lim inf sd d lim sup sd d . d
d
Since
sup r: μ ]r, +∞[ > 0 = inf r: μ ]r, +∞[ = 0 , 1
we see that Rs2 = limd→+∞ sd d . Therefore, the measure μn is supported by the closure Ω s . (n−1) Since both series Fs and Fs have the same radius of convergence it follows that for each z ∈ Ωs , the series +∞
Ks (z, w) =
(d + n − 1)! 1 z, w d , (n − 1)! d!sd
w ∈ Ωs ,
0
converges on Ω s . Moreover, by Fatou’s lemma and orthogonality of the holomorphic monomials with respect to μn we have
Ks (z, w)2 dμn (w)
1 (n − 1)!
2
2 N (d + n − 1)! d lim inf z, w dμn (w) N →+∞ d!sd
Ωs
Ωs
=
1 (n − 1)!
2
d=0
2 N (d + n − 1)! d z, w
dμn (w) N →+∞ d!s lim inf
Ωs
d
d=0
= Ks (z, z). Hence for any fixed z ∈ Ωs , the series Ks (z, w) converges in L2 (μn ). In addition, a little computing shows that for any α ∈ Nn0 , we have
+∞
(d + n − 1)! 1 w Ks (z, w) dμn (w) = (n − 1)! d!sd
w α z, w d dμn (w)
α
d=0
Ωs
=
1 (|α| + n − 1)! (n − 1)! |α|!s|α|
Ωs
=z . α
Ωs
w α z, w |α| dμn (w)
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985
This shows that the kernel Ks (z, w) reproduces holomorphic polynomials. Moreover, it satisfies sup f (z) sup Ks (z, z) f L2 (μn )
z∈K
z∈K
for all holomorphic polynomials and each set compact K ⊂ Ω. The remaining part of the proof follows by standard arguments. 2 We point out that Rs is always strictly positive. We recall from the previous work of the authors [11] the expression of the operators Hz¯ k and Hz¯ k Hz¯ l on holomorphic homogeneous polynomials. Lemma 2.1. Suppose that k and l are in Nn0 . Then the domain Dom(Hz¯∗k ) of Hz¯∗k contains all polynomials in w and w. ¯ Moreover, if f is a holomorphic homogeneous polynomial of degree d, then Hz¯∗l Hz¯ k f =
sd+|l| Γ (n + d + |l| − |k|) ∂ |k| l sd Γ (d + n − |k|) l ∂ |k| z k f. z f − sd+|l|−|k| Γ (n + d + |l|) ∂zk sd−|k| Γ (d + n) ∂z
In particular, Hz¯∗l Hz¯ k f is a holomorphic homogeneous polynomial of degree d + |l| − |k|. In particular, for each α in Nn0 , the monomial zα is an eigenvector for the operator Hz¯∗k Hz¯ k and the corresponding eigenvalue λα is given by λα =
s|α| Γ (|α| + n − |k|) α! s|α|+|k| Γ (n + |α|) (α + k)! − s|α| Γ (n + |α| + |k|) α! s|α|−|k| Γ (|α| + n) (α − k)!
if α k and λα =
s|α|+|k| Γ (n + |α|) (α + k)! , s|α| Γ (n + |α| + |k|) α!
otherwise. For simplicity reasons, we introduce some notations. We set fn (t1 , . . . , tn ) := −|k|2 t k +
n j =1
kj2
tk , tj kj
k
with the understanding that kj2 ttj = 0 as long as kj = 0 and t (α) :=
1 + α1 1 + αn , ,..., |α| + n |α| + n
tj tj
t ∈ Rn ,
k −1
= tj j
(2.1)
for kj 1. We also let
α ∈ Nn0 .
Lemma 2.2. The function fn given by (2.1) satisfies fn (t1 , . . . , tn ) 0 for all non-negative real numbers t1 , . . . , tn that satisfy t1 + · · · + tn = 1. In particular, fn (t (α)) 0 for all α ∈ Nn0 .
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Proof. Setting rj =
kj |k| ,
the lemma follows from the inequality n r2 j j =1
tj
1,
which holds for all t1 , . . . , tn , r1 , . . . , rn ∈ ]0, +∞[ that satisfy t1 + · · · + tn = r1 + · · · + rn = 1. This inequality, in turn, can be proved by induction on n.
2
Lemma 2.3. Suppose that α and k are in Nn0 . If n = 1, set γα,k := 0 and if n > 1, set γα,k :=
Γ (n + |α|) (α + k)! Γ (|α| + n − |k|) α! 1 − . n − 1 Γ (n + |α| + |k|) α! Γ (|α| + n) (α − k)!
Then γα,k 0, for all α ∈ Nn0 that satisfy α k. In addition, if n 2, then 1 1 1 fn t (α) + O , γα,k = n−1d +n d for all k, α ∈ Nn0 , satisfying α k, where d := |α|. Proof. We consider the particular case of the constant Stieltjes moment sequence sd = 1, d ∈ N0 , represented by the Dirac measure μ = δ1 . If α ∈ Nn0 , then (n − 1)γα,k is the eigenvalue of Hz¯∗k Hz¯ k corresponding to the eigenvector zα . Applying the previous lemma we see that (n − 1)γα,k 0 and hence the first part of the lemma follows. Next, we prove the second part of lemma. From the property of the Gamma function [18] (y − z)(y + z − 1) 1 Γ (x + y) as x → +∞, = x y−z 1 + +O 2 Γ (x + z) 2x x where y and z are real numbers, we get Γ (d + n) = (d + n)−|k| 1 − Γ (d + n + |k|) Γ (d + n − |k|) −|k| 1+ = (d + n) Γ (d + n)
1 |k|(|k| − 1) +O 2 as d → +∞, 2(d + n) d |k|(|k| + 1) 1 as d → +∞. +O 2 2(d + n) d
By the proof of Lemma 3.3 in [11], we have, when α k, n n kj (kj − 1) (α + k)! = (1 + αj )kj −1 (1 + αl )kl + q(α) (1 + αj )kj + α! 2 l =j j =1 j =1 q(α) h(t) − g(t) + = (d + n)|k| t k + d +n (d + n)|k|
H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 258 (2010) 978–998
where h(t) :=
n k2t k j j =1
2tj
,
g(t) :=
n kj t k j =1
2tj
.
Using a similar argument, we also have kj (kj + 1) α! = (1 + αj )kj −1 (1 + αl )kl + r(α) (1 + αj )kj − (α − k)! 2 n
j =1
n
= (d + n)|k| t k −
j =1
r(α) h(t) + g(t) + d +n (d + n)|k|
l =j
where q and r are polynomials of degree at most |k| − 2. Γ (d + n) (α + k)! Γ (d + n − |k|) α! − Γ (d + n + |k|) α! Γ (d + n) (α − k)! |k|(|k| − 1) 1 −|k| (α + k)! 1− +O 2 = (d + n) α! 2(d + n) d |k|(|k| + 1) 1 α! 1+ +O 2 − (d + n)−|k| (α − k)! 2(d + n) d 1 1 h(t) − g(t) |k|(|k| − 1) tk + +O 2 +O 2 = 1− 2(d + n) d +n d d 1 1 h(t) + g(t) |k|(|k| + 1) k t − +O 2 +O 2 − 1+ 2(d + n) d +n d d 1 1 = −|k|2 t k + 2h(t) + O . d +n d The lemma now follows since fn (t) = −|k|2 t k + 2h(t).
2
Lemma 2.4. If α ∈ Nn0 , then the eigenvalue λα of the operator Hz¯∗k Hz¯ k satisfies λα =
s|α|+|k| s|α| − s|α| s|α|−|k|
k n − 1 s|α| 1 1 t (α) + O fn t (α) + O + d d + n s|α|−|k| d
if α k and λα = otherwise.
s|α|+|k| 1 O , s|α| d
987
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Proof. By Lemma 2.1 and the definition of γα,k , we have λα =
s|α| s|α|+|k| s|α| Γ (d + n) (α + k)! + (n − 1)γn,k − . s|α| s|α|−|k| Γ (d + n + |k|) α! s|α|−|k|
By the estimates in the proof of Lemma 2.3 we deduce that Γ (d + n) (α + k)! 1 t (α) . +O = t Γ (d + n + |k|) α! d The latter equation, combined with Lemma 2.3, completes the proof of the first part of the lemma. To prove the remaining part of the lemma, suppose that for some j0 = 1, . . . , n we have that kj0 1 and αj0 < kj0 . Then by Lemma 2.1 we have λα =
s|α|+|k| Γ (d + n) (α + k)! . s|α| Γ (d + n + |k|) α!
Set α = (α1 , . . . , αj0 −1 , 0, αj0 +1 , . . . , αn ) and k = (k1 , . . . , kj0 −1 , 0, kj0 +1 , . . . , kn ). Arguing like in Lemma 3.4 in [11], we get (α + k)! (2kj0 )! α!
n
(1 + αj )kj +
j =1, j =j0
n j =1, j =j0
kj (kj − 1) (1 + αj )kj −1 (1 + αs )kl 2
l =j,j0
+ q(α ), where q(α ) is a polynomial of degree at most |k | − 2. This inequality, combined with the estimate Γ (d + n) 1 =O Γ (d + n + |k|) (d + n)|k| gives the second part of the lemma.
2
Theorem 2.5. Fix k ∈ Nn0 and consider the Hankel operator Hz¯ k from the dense subspace of A2 (s) consisting of holomorphic polynomials into L2 (μn ). Then: (1) Hz¯ k is bounded if and only if sup d∈N0
sd+2|k| sd+|k| n − 1 sd+|k| − + sd+|k| sd d sd
< +∞.
(2.2)
(2) Hz¯ k is compact if and only if lim
d→+∞
sd+2|k| sd+|k| n − 1 sd+|k| − + sd+|k| sd d sd
= 0.
(2.3)
H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 258 (2010) 978–998
989
Proof. We consider the sequence (λα )α of eigenvalues of the Hz¯∗k Hz¯ k . Let Σn be the simplex consisting of those t = (t1 , . . . , tn ) ∈ Rn such that tj 0 and t1 + · · · + tn = 1. Since the set α1 + 1 d +n
d∈N0
,...,
αn + 1 , |α| = d d +n
is dense in Σn , it follows that sup fn
|α|=d
α1 + 1 αn + 1 ,..., d +n d +n
≈ sup fn (t) t∈Σn
and sup t (α)k ≈ sup t k
|α|=d
t∈Σn
as d tends to +∞. These estimates, combined with Lemma 2.4, implies that (λα )α is bounded if and only if (2.2) holds and lim|α|→+∞ λα = 0 if and only if (2.3) holds. The theorem now follows since Hz¯ k is bounded if and only if Hz¯∗k Hz¯ k is bounded and compactness of Hz¯ k is equivalent to that of Hz¯∗k Hz¯ k . 2 Next, let p > 0. We shall study the membership of the operator Hz¯ k in a Schatten class Sp . p Recall that Hz¯ k is in Sp if and only if Hz¯∗k Hz¯ k is in S p , that is to say the series λα2 is convergent. 2 p Let d be an integer. We shall estimate the sum Sd = |α|=d λα , when d → +∞. The calculations above lead to study the cases α k and its opposite separately. Let Bd := {α ∈ Nn0 , |α| = d}. We partition Bd = Bd ∪ Bd , where Bd = {α ∈ Bd , α k} and Bd = Bd \ Bd . Thus p p Sd can be written in the form Sd = Sd + Sd , where Sd = α∈B λα and Sd = α∈B λα . We d d shall use the following lemmas (see [11]). Lemma 2.6. If n 2, then we have the estimates Bd ≈ Bd ≈ d → +∞.
d n−1 (n−1)!
and Bd ≈ d n−2 as
Lemma 2.7. Suppose that n 2 and g is a continuous function on Rn−1 . Consider the open set n−1 n−1 D := {(t1 , . . . , tn−1 ) ∈ R+ , n−1 j =1 tj < 1}. For a multi-index β = (β1 , . . . , βn−1 ) in N0 , set βn−1 + 1 β1 + 1 cβ,d := ,..., , d d n−1 βj βj + 1 n−1 , ⊂D . Jd := β ∈ N0 : d d
j =1
Then limd→+∞
1 d n−1
β∈Jd
g(cβ,d ) =
D g(t) dt.
The above results enable us to estimate Sd when d = |α| → +∞.
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Lemma 2.8. If p > 0, then Sd ≈ d n−1
sd+|k| sd − sd sd−|k|
p
+ (n − 1)d n−1−p
p
sd sd−|k|
.
p p Proof. Recall that Sd = Sd + Sd , where Sd = α∈B λα and Sd = α∈B λα . First we shall d d estimate Sd . By Lemma 2.4, we know that this sum has the following expansion when d = |α| → +∞ Sd
≈
sd+|k| sd − sd sd−|k|
p p k 1 t (α) + O d α∈Bd
p p 1 n − 1 sd fn t (α) + O + . d + n sd−|k| d
α∈Bd
Using the properties of the function x → x p and Lemma 2.7 we see that there exists a constant M > 0, such that pkn p n k 1 pk pk t (α) + O ≈ d n−1 t1 1 · · · tn−1n−1 1 − tj dt, d
α∈Bd
α∈Bd
j =1
D
p p n 1 n−1 fn t (α) + O ≈d tj dt. fn t1 , . . . , tn−1 , 1 − d j =1
D
Therefore, Sd
≈d
n−1
≈ d n−1
p n − 1 sd +d d + n sd−|k| p p sd sd − + (n − 1)d n−1−p sd−|k| sd−|k|
sd+|k| sd − s|α| sd−|k| sd+|k| s|α|
p
n−1
as d → +∞. To estimate Sd we observe that if n = 1, then . Sd S|k|
On the other hand, if n 2, by Lemma 2.4 we see that for α ∈ Bd we have λpα
sd+|k| p −p . = (n − 1) O d sd
Since Bd ≈ d n−2 , we see that Sd = O( Sd + Sd . 2
Sd+|k| d p ).
The lemma follows from the relation Sd =
We then characterize the Schatten class membership of Hz¯ k .
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991
Theorem 2.9. Let k ∈ Nn0 . Then the Hankel operator Hz¯ k is in the Schatten class Sp (A2 (s), L2 (μn )) if and only if d
n−1
sd+2|k| sd+|k| − sd+|k| sd
p 2
+ (n − 1)d
n−1− p2
sd+|k| sd
p 2
< +∞.
(2.4)
Proof. We use that the operator Hz¯ k is in the Schatten class Sp (A2 (s), L2 (μn )) if and only if p Hz¯∗k Hz¯ k is in S 2 (A2 (s)). Therefore, the theorem follows from Lemma 2.8. 2 Lemma 2.10. If U is a unitary transformation in Cn , the operator Uf := f ◦ U is a unitary isometry from L2 (μn ) onto itself and from A2 (s) onto itself. Moreover the following assertions hold. (1) If f ∈ M(s), then Uf ∈ M(s) and Uf M(s) = f M(s) . (2) If f ∈ M∞ (s), then Uf ∈ M∞ (s). (3) If f ∈ Mp (s), then Uf ∈ Mp (s) and Uf Mp (s) = f Mp (s) . Proof. Let U be a unitary transformation in Cn and denote U ∗ its adjoint, which is also its inverse. It is clear that the operator U is a unitary isometry from L2 (μn ) onto itself and from A2 (s) onto itself. Let f be in M(s). If g is a holomorphic polynomial, then by a change of variable we see that
HUf (g)(z) = Ks (U z, w)g(U ∗ w) Uf (z) − f¯(w) dμm (w) Cn
=
Ks (U z, w)(U ∗ g)(w) f¯(U z) − f¯(w) dμn (w)
Cn
= Hf¯ (U ∗ g)(U z) = (U Hf¯ U ∗ )(g)(z). Therefore, HUf = U Hf¯ U ∗
(2.5)
and thus HUf = Hf¯ , showing that Uf M(s) = f M(s) . This proves part (1) of the lemma. The proof of parts (2) and (3) of the lemma are similar.
2
Let Tn := {ζ = (ζ1 , . . . , ζn ) ∈ Cn : |ζj | = 1, j = 1, . . . n} and for ζ = (ζ1 , . . . , ζn ) ∈ Tn , let Uζ be the unitary linear transformation in Cn defined by Uζ (z) = (ζ1 z1 , . . . , ζn zn ), for all z = (z1 , . . . , zn ) ∈ Cn . Like in [1] we have the following
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Lemma 2.11. If f ∈ T(s) and g ∈ A2 (s), the mappings ζ → Uζ g and ζ → HUζ f (g) are continuous from Tn to L2 (μn ). Proof. Let g ∈ A2 (μn ) and write g(z) =
α∈Nn0 ak z
α.
If ζ, η ∈ Tn , then
(Uζ − Uη )g 2 2 = g ◦ Uζ − g ◦ Uη 2L2 (μ ) L (μn ) n 2 = |aα |2 (Uζ z)α − (Uη z)α L2 (μ α∈Nn0
=
n)
2 |aα |2 cα ζ α − ηα ,
α∈Nn0
where cα =
α 2 z dμn (z),
α ∈ Nn0 .
Cn
Since
2 |aα |2 cα < +∞ and ζ α − ηα 4,
α∈Nn0
the dominated convergence theorem leads to lim (Uζ − Uη )g L2 (μ ) = 0,
ζ →η
n
showing that the mapping ζ → Uζ g is continuous from Tn to L2 (μn ). This, combined with the fact that Uζ is unitary and the equalities HUζ f − HUη f = Uζ Hf¯ Uζ¯ − Uη Hf¯ Uη¯ = Uζ Hf¯ Uζ¯ − Uζ Hf¯ Uη¯ + Uζ Hf¯ Uη¯ − Uη Hf¯ Uη¯ = Uζ Hf¯ (Uζ¯ − Uη¯ ) + (Uζ − Uη )Hf¯ Uη¯ , shows that the mapping ζ → HUζ f (g) is also continuous from Tn to L2 (μn ).
2
Lemma 2.12. Assume that f ∈ T(s). (1) If f ∈ M(s), then for any multi-index k ∈ Spec(f ), the monomial zk is in M(s). (2) If f ∈ M∞ (s), then for any multi-index k ∈ Spec(f ), the monomial zk is in M∞ (s). (3) If p > 0 and f ∈ Mp (s), then for any multi-index k ∈ Spec(f ), the monomial zk is in Mp (s).
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Proof. To prove (1), suppose that f ∈ M(s) and write f (z) = mula we have
k∈Nn0 ak z
k.
993
By the Cauchy for-
f (Uζ z)ζ¯ k dmn (ζ ),
ak z k = Tn
where dmn (ζ ) is the normalized Lebesgue measure on Tn . If g is a holomorphic polynomial and h ∈ L2 (μn ), an application of Fubini’s theorem leads to
HUζ f (g), h ζ¯ k dmn (ζ ) = Ha
(g), h .
(2.6)
dmn (ζ ).
(2.7)
kz
k
Tn
By Lemmas 2.11, 2.12 we see that H
ak
(g) L2 (μ ) zk
H
Uζ f (g)
n
Tn
Since HUζ f (g) Hf¯ g L2 (μn ) for all ζ in Tn , it follows that Ha
kz
k
k
is bounded and ak zk is
in M(s). Therefore, zk ∈ M(s) as long as ∂∂zfk (0) = 0. This proves part (1) of the lemma. Suppose now that f ∈ M∞ (s) and let (gq ) be a sequence in A2 (s) which converges weakly to 0. lim HUζ f (gq ) L2 (μ ) = 0, n
q→+∞
for all ζ ∈ Tn ,
so that by (2.7) and the dominated convergence theorem we see that lim Ha
q→+∞
k kz
(gq ) L2 (μ ) = 0 n
k
and hence zk ∈ M∞ (s) whenever ∂∂zfk (0) = 0. Therefore part (2) of the lemma holds. To establish the remaining part of the lemma, we recall that if T is a compact operator from A2 (s) to L2 (μn ) then its singular numbers νq (T ), q ∈ N0 , are given by νq (T ) := inf T − A A∈Rq
where Rq is the space of all operators from A2 (s) to L2 (μn ) with finite rank at most q. Assume that f ∈ Mp (s). Then the sequence (νq (Hf¯ ))q is in l p . Moreover, there are an orthonormal system (uq )q in A2 (s) and an orthonormal system (vq )q in L2 (μn ) such that Hf¯ =
+∞ q=0
νq (Hf¯ ) ·, uq vq ,
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where the series converges in the operator norm. See Proposition 16.3 in [19]. If q is a positive integer, consider the operator with rank at most q given by Aq :=
q−1
νj (Hf¯ ) ·, ukj vkj
j =0
where for each integer j = 0, . . . , q − 1 and z ∈ Cn the functions ukj and vkj are defined by ukj (z) := (Uζ uj )(z)ζ¯ k dmn (ζ ) and vkj (z) := (Uζ hj )(z)ζ¯ k dmn (ζ ). Tn
Tn
The dominated convergence theorem, combined with (2.6) and (2.5), yields
(Ha
kz
k
− Aq )(g), h =
+∞
νj (Hf¯ ) Uζ¯ g, uj vj Uζ¯ h ζ¯ k dmn (ζ ).
Tn j =q
Due to the facts that the sequence (νj (Hf¯ ))j is non-increasing and the systems (uj )j and (vj )j are orthonormal it follows that +∞ νq (Hf¯ ) Uζ¯ g, uj vj Uζ¯ h νq (Hf¯ ) g A2 (s) h L2 (μn ) j =q
for all ζ ∈ Tn , g ∈ A2 (s) and h ∈ L2 (μn ). Hence Ha
kz
k
− Aq νq (Hf¯ ).
This implies that νq (Ha
kz
k
) νq (Hf¯ )
showing that ak zk ∈ Mp (s). Consequently, zk ∈ Mp (s) if and only if the lemma is now complete. 2
∂kf ∂zk
(0) = 0. The proof of
Proof of Theorem B. Let f ∈ A2 (s). Suppose that Hf¯ is bounded and let k ∈ Spec(f ). By Lemma 2.12 we see that the monomial zk is in M(s). Now Theorem 2.5 implies that f satisfies (1.1). If Hf¯ is compact (resp. in the Schatten class) then a similar argument shows that f satisfies part (2) (resp. part (3)) of Theorem B. 2 Proof of Corollaries 1.1, 1.2 and 1.3. As mentioned before, the equivalence between (1) and (4) in the statement of each of the corollaries was in established in [17] and [15]. The double equivalence (2) ⇐⇒ (3) ⇐⇒ (4) in the statement of each of the corollaries follows from Theorem B. 2 Lemma 2.13. Suppose that Rs = +∞ and the sequence s satisfies (1.3). Then the function w → g(w)Ks (z, w) is in L2 (μn ) for all holomorphic polynomials g and z ∈ Cn .
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995
Proof. We first observe that +∞
(d + n − 1)! 1 z, w d , Ks (z, w) = (n − 1)! d!sd
z ∈ Cn , w ∈ Cn .
0
Therefore, for any α ∈ Nn0 and z ∈ Cn ,
α w Ks (z, w)2 dμn (w) =
Cn
=
1 (n − 1)! 1 (n − 1)! 1 (n − 1)!
2 +∞ 0
2 +∞ 0
(d + n − 1)! d!sd (d + n − 1)! d!sd
2
α w z, w d 2 dμn (w)
Cn
2
Rs |z|
2d
t |α|+d dμ(t)
0
2 +∞ 0
(d + n − 1)! d!sd
2 s|α|+d |z|2d .
Now assumption (1.3) ensures that the latter series converges for all z ∈ Cn .
2
Lemma 2.14. Assume that s satisfies (1.2). Then the spaces M(s) and Mp (s), p 1, are Banach spaces and Mp (s), 0 < p < 1, is a quasi-Banach space. Proof. We prove the lemma for M(s). Let (fq )q∈N0 be a Cauchy sequence in M(s). Without loss of generality we may assume that fq (0) = 0 for all n. The sequence (Hf¯q )q∈N0 is a Cauchy sequence of bounded operators on A2 (s). Therefore, there is an operator T in A2 (s) such that (Hf¯q )q∈N0 converges to T in the norm operator. Let f := T (1) be the conjugate of the image T (1) of the constant function 1 under T . Since H ¯ (1) = f¯q , it follows that fq
fq − f L2 (μn ) = f¯q − T (1) L2 (μ ) n = Hf¯q (1) − T (1) L2 (μ
n)
Hf¯q − T 1 L2 (μm ) showing that lim fq − f L2 (μn ) = 0.
(2.8)
q→∞
Thus f ∈ A2 (s). We shall show that the Hankel operator Hf¯ with symbol f¯ is bounded. We shall prove that f ∈ T(s) and Hf¯ coincides with T on holomorphic polynomials. Let g be a holomorphic polynomial. We first observe by Lemma 2.13 that for all z ∈ Cn we have Ps (f¯ − fq )g (z) fq − f 2 L (μn ) gKs (z, ·) L2 (μ
n)
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H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 258 (2010) 978–998
so that by (2.8) we see that limq→+∞ Ps ((f¯ − fq )g)(z) = 0. Since again by (2.8) we have that limq→+∞ (f¯ − fq )g(z) = 0, it follows that lim (Hfq − Hf¯ )(g)(z) = 0.
q→+∞
This proves that T g = Hf¯ (g) and hence f ∈ T(s) and T = Hf¯ . Therefore M(s) is a Banach space. The proof of that Mp (s) is a Banach space for p 1, and a quasi-Banach space for 0 < p < 1 is similar. 2 Proof of Theorem C. Suppose that n = 1 and f is as in the hypothesis of Theorem C. A straightforward calculation appealing to Lemma 2.1 shows that for all non-negative integers j , k we have Tr Hz¯∗k Hz¯ j = 0, as long as j = k and +∞ k−1 ∗ sd+2k sd+k sd+k Tr Hz¯ k Hz¯ k = + − sd+|k| sd sd d=k
d=0
= kRs2 . Writing f =
k∈N ak z
k
yields k|ak |2 Tr Hf∗¯ Hf¯ = Rs2 k∈N
=
1 π
2 f (z) dA(z).
Ωs
This proves the first equality of the theorem. Next we prove the second equality. Writing 2 ¯ Ks (z, w) = ∞ k=0 fk (z)fk (w), where (fk ) is an orthonormal basis of A (s), we observe by a standard argument that for any positive operator T on A2 (s) we have Tr(T ) =
∞ Tfk , fk A2 (s) k=0
T Ks (·, z), Ks (·, z) A2 (s) dA(z).
= Ωs
Applying this equality to T = Hf∗¯ Hf¯ and using the reproducing property of the kernel Ks implies that 2 2 ∗ Tr Hf¯ Hf¯ = f (z) − f (w) K(z, w) dA(z) Ωs
and hence completes the proof of the theorem.
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Proof of Theorem D. Follows from Lemmas 2.10, 2.12 and 2.14.
997
2
3. Concluding remarks It was proved by the authors [11] that if s := (sd )d is the Stieltjes moment sequence given by s=
2s + 2n 1 Γ m m
where m is positive real number, then the space M(s) is finite-dimensional and consists of polynomials of degree at most m2 . See also [16] and [21] for the one-dimensional case. It would be of interest to characterize Stieltjes moment sequences having this property. Such sequences must not be exponentially bounded since, in view of Theorem B, the space M(s) contains all holomorphic polynomials whenever the corresponding sequence s is exponentially bounded. Similar discussion can be invoked relatively to Mp (s). This issue has been treated in the case of Bergman space on the unit ball. See [27] and [25]. Fix a Stieltjes moment sequence and for a linear operator T on A2 (s), the Berezin transform of T is the function T defined on Ωs by T (z) := T κz , κz , where κz is the normalized reproducing kernel κz (w) :=
Ks (w, z) 1
.
Ks (z, z) 2
∗ H is equivaIf k ∈ Nn0 , we do not know whether the boundedness of the Berezin transform H k z¯ k z¯ lent to that of the operator Hz¯ k . Finally, it would be of interest to consider further study of the general setting of Hankel operators with arbitrary symbols but reasonably defined as was done in the classical Fock space. See [3,4,6,7,23,24] and the references therein.
References [1] P. Ahern, E.H. Youssfi, K. Zhu, Compactness of Hankel operators on Hardy–Sobolev spaces of the polydisk, J. Operator Theory 61 (2009) 301–312. [2] J. Arazy, S.D. Fisher, J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988) 989– 1053. [3] W. Bauer, Mean oscillation and Hankel operators on the Segal–Bargmann space, Integral Equations Operator Theory 52 (2005) 1–15. [4] W. Bauer, Hilbert–Schmidt Hankel operators on the Segal–Bargmann space, Proc. Amer. Math. Soc. 132 (2005) 2989–2996. [5] C. Berg, J.P.R. Christensen, P. Ressel, Harmonic Analysis on Semigroups, Grad. Texts in Math., Springer-Verlag, 1984. [6] C.A. Berger, L. Coburn, Toeplitz operators on the Segal–Bargmann space, Trans. Amer. Math. Soc. 301 (1987) 813–829. [7] C.A. Berger, L. Coburn, K. Zhu, Toeplitz Operators and Function Theory in n-Dimensions, Lecture Notes in Math., vol. 1256, Springer, 1987. [8] T.M. Bisgaard, Stieltjes moment sequences and positive definite matrix sequences, Proc. Amer. Math. Soc. 126 (1998) 3227–3237.
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[9] T.M. Bisgaard, Z. Sasvári, Stieltjes moment sequences and positive definite matrix sequences, Math. Nachr. 186 (1997) 81–99. [10] R.P. Boas, The Stieltjes moment problem for functions of bounded variation, Bull. Amer. Math. Soc. 45 (1939) 399–404. [11] H. Bommier-Hato, E.H. Youssfi, Hankel operators on weighted Fock spaces, Integral Equations Operator Theory 59 (2007) 1–17. [12] F. Haslinger, The canonical solution operator to ∂¯ restricted to Bergman spaces and spaces of entire functions, Ann. Fac. Sci. Toulouse Math. (6) 11 (2002) 57–70. ¯ J. Math. Kyoto [13] F. Haslinger, Schrödinger operators with magnetic fields and the canonical solution operator to ∂, Univ. 46 (2006) 249–257. [14] F. Haslinger, B. Helfer, Compactness of the solution operator to ∂¯ in weighted L2 -spaces, J. Funct. Anal. 243 (2007) 679–697. ¯ J. Funct. Anal. 255 (2008) 13–24. [15] F. Haslinger, B. Lamel, Spectral properties of the canonical solution operator to ∂, [16] W. Knirsch, G. Schneider, Continuity and Schatten–von Neumann p-class membership of Hankel operators with antiholomorphic symbols on (generalized) Fock spaces, J. Math. Anal. Appl. 320 (2006) 403–414. ¯ [17] S. Lovera, E.H. Youssfi, Spectral properties of the ∂-canonical solution operator, J. Funct. Anal. 208 (2004) 360– 376. [18] W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, 1966. [19] R. Meise, D. Vogt, Introduction to Functional Analysis, Clarendon Press, Oxford, 1997. [20] W. Rudin, Function Theory on the Open Unit Ball in Cn , Springer-Verlag, 1980. [21] G. Schneider, Hankel operators with antiholomorphic symbols on the Fock space, Proc. Amer. Math. Soc. 132 (2004) 2399–2409. [22] T. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse 8 (1894) 1–122; Ann. Fac. Sci. Toulouse 9 (1895) 5–47. [23] K. Stroethoff, Hankel operators in the Fock space, Michigan Math. J. 39 (1992) 3–16. [24] J. Xia, D. Zheng, Standard deviation and Schatten class Hankel operators on the Segal–Bargmann space, Indiana Univ. Math. J. 53 (2004) 1381–1399. [25] K. Zhu, Hilbert–Schmidt Hankel operators on the Bergman space, Proc. Amer. Math. Soc. 109 (1990) 721–730. [26] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990. [27] K. Zhu, Schatten class Hankel operators on the Bergman space of the unit ball, Amer. J. Math. 113 (1991) 147–167.
Journal of Functional Analysis 258 (2010) 999–1059 www.elsevier.com/locate/jfa
Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian ✩ Olivier Druet a , Emmanuel Hebey b,∗ , Jérôme Vétois c a Ecole Normale Supérieure de Lyon, Département de Mathématiques, UMPA, 46 allée d’Italie,
69364 Lyon cedex 07, France b Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvin,
95302 Cergy-Pontoise cedex, France c Université de Nice, Département de Mathématiques, Laboratoire J.A. Dieudonné, Parc Valrose,
06108 Nice cedex 2, France Received 21 June 2009; accepted 7 July 2009 Available online 17 July 2009 Communicated by Paul Malliavin To the memory of T. Aubin
Abstract We prove bounded stability for strongly coupled critical elliptic systems in the inhomogeneous context of a compact Riemannian manifold when the potential of the operator is less, in the sense of bilinear forms, than the geometric threshold potential of the conformal Laplacian. © 2009 Elsevier Inc. All rights reserved. Keywords: Critical equations; Elliptic systems; Riemannian manifolds; Stability; Strong coupling
Let (M, g) be a smooth compact Riemannian manifold of dimension n 3. For p 1 an integer, let also Mps (R) denote the vector space of symmetrical p × p real matrices, and A be a C 1 map from M to Mps (R). We write that A = (Aij )i,j , where the Aij ’s are C 1 real-valued ✩
The authors were partially supported by the ANR grant ANR-08-BLAN-0335-01.
* Corresponding author.
E-mail addresses: [email protected] (O. Druet), [email protected] (E. Hebey), [email protected] (J. Vétois). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.07.004
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functions in M. Let g = − divg ∇ be the Laplace–Beltrami operator on M, and H 1 (M) be the Sobolev space of functions in L2 (M) with one derivative in L2 (M). The Hartree–Fock coupled systems of nonlinear Schrödinger equations we consider in this paper are written as g ui +
p
−2
Aij (x)uj = |U|2
ui
(0.1)
j =1
p 2n in M for all i, where |U|2 = i=1 u2i , and 2 = n−2 is the critical Sobolev exponent for the 1 embeddings of the Sobolev space H (M) into Lebesgue’s spaces. The systems (0.1) are weakly coupled by the linear matrix A, and strongly coupled by the Gross–Pitaevskii type nonlinearity in the right-hand side of (0.1). As is easily seen, (0.1) is critical for Sobolev embeddings. Coupled systems of nonlinear Schrödinger equations like (0.1) are now parts of several important branches of mathematical physics. They appear in the Hartree–Fock theory for Bose– Einstein double condensates, in fiber-optic theory, in the theory of Langmuir waves in plasma physics, and in the behavior of deep water waves and freak waves in the ocean. A general reference in book form on such systems and their role in physics is by Ablowitz, Prinari and Trubatch [1]. The systems (0.1) we investigate in this paper involve coupled Gross–Pitaevskii type equations. Such equations are strongly related to two branches of mathematical physics. They arise, see Burke, Bohn, Esry and Greene [9], in the Hartree–Fock theory for double condensates, a binary mixture of Bose–Einstein condensates in two different hyperfine states. They also arise in the study of incoherent solitons in nonlinear optics, as described in Akhmediev and Ankiewicz [2], Christodoulides, Coskum, Mitchell and Segev [13], Hioe [24], Hioe and Salter [25], and Kanna and Lakshmanan [26]. A strong solution U of (0.1) is a p-map with components in H 1 satisfying (0.1). By elliptic regularity strong solutions are of class C 2,θ , θ ∈ (0, 1). In the sequel a p-map U = (u1 , . . . , up ) from M to Rp is said to be nonnegative if ui 0 in M for all i. We aim in this paper in discussing bounded stability for our systems (0.1). With respect to the notion of analytic stability, as defined and investigated in Druet and Hebey [18], no bound on the energy of the solution is required in the stronger notion of bounded stability. This prevents, see Section 2, the existence of standing waves with arbitrarily large amplitude for the corresponding critical vector-valued Klein–Gordon and Schrödinger equations. Let SA be the set consisting of the nonnegative strong solutions of (0.1). Bounded stability is defined as follows. Definition. The system (0.1) is bounded and stable if there exist C > 0 and δ > 0 such that for any A ∈ C 1 (M, Mps (R)) satisfying A − AC 1 < δ, and for any U ∈ SA , there holds that UC 2,θ C for θ ∈ (0, 1). An equivalent definition is that for any sequence (Aα )α of C 1 -maps from M to Mps (R), α ∈ N, and for any sequence of nonnegative nontrivial strong solutions Uα of the associated systems, if Aα → A in C 1 as α → +∞, then, up to a subsequence, Uα → U in C 2 as α → +∞ for some nonnegative solution U of (0.1). Moreover, see Druet and Hebey [18], we can assert that U is automatically nontrivial if g + A is coercive, or, more generally, if g + A does not possess nonnegative nontrivial maps in its kernel. The question we address in this paper is to find conditions on the vector-valued operator g + A which guarantee the bounded stability of (0.1). We answer the question in the theorem below when the potential of the operator is less, in the sense of bilinear forms, than the geometric
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threshold potential of the conformal Laplacian. As one can check, there is a slight difference between the case n = 3, where the Green’s matrix of g + A and the positive mass theorem come into play, and the case n 4. Following standard terminology we say that g + A is coercive if the energy of the operator controls the H 1 -norm, and we say that −A is cooperative if the nondiagonal components Aij of A, i = j , are nonpositive in M. When −A is cooperative, see Hebey [23], the existence of U = (u1 , . . . , up ) such that U solves (0.1) and ui > 0 in M for all i, implies the coercivity of g + A. In the sequel we let Sg be the scalar curvature of g and let Idp be the identity matrix in Mps (R). The theorem we prove is stated as follows. Theorem. Let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, and A : M → Mps (R) be a C 1 -map satisfying that A<
n−2 Sg Idp 4(n − 1)
(0.2)
in M in the sense of bilinear forms. When n = 3 assume also that g + A is coercive and that −A is cooperative. Then the associated system (0.1) is bounded and stable. A closely related notion to stability, which has been intensively investigated, is that of compactness. Among possible references we refer to Brendle [6,7], Brendle and Marques [8], Druet [14,15], Druet and Hebey [17], Gidas and Spruck [21], Khuri, Marques and Schoen [27], Li and Zhang [29,30], Li and Zhu [32], Marques [33], Schoen [37,38], and Vétois [42]. A system like (0.1) is said to be compact if sequences of nonnegative solutions of (0.1) converge, up to a subsequence, in the C 2 -topology. A direct consequence of our theorem is as follows. Corollary. Let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, and A : M → Mps (R) be a C 1 -map satisfying (0.2). When n = 3 assume also that g + A is coercive and that −A is cooperative. Then (0.1) is compact. Another consequence of our theorem is in terms of standing waves and phase stability for vector-valued Schrödinger and Klein–Gordon equations. Roughly speaking, we refer to Section 2 for more details, it follows from our result that fast oscillating standing waves for Schrödinger and Klein–Gordon equations cannot have arbitrarily large amplitude. The same phenomenon holds true for slow oscillating standing waves if the potential matrix A is sufficiently small. Instability comes in the intermediate regime. Condition (0.2) in the theorem is the global vector-valued extension of the seminal condition introduced by Aubin [3]. Aubin proved in [3] that (0.2), when satisfied at one point in the manifold, and when A and U are functions, implies the existence of a minimizing solution of (0.1). Our theorem establishes that (0.2) does not only provide the existence of minimal energy solution to the equations, but also provides the stability of the equations in all dimensions. The condition turns out to be sharp. Assuming that (0.2) is an equality, then, see Druet and Hebey [16,18], we can construct various examples of unstable systems like (0.1) in any dimension n 6. These include the existence of clusters (multi peaks solutions with fewer geometrical blow-up points) and the existence of sequences (Uα )α of solutions with unbounded energy (namely such that Uα H 1 → +∞ as α → +∞). By the analysis in Brendle [6] and Brendle and Marques [8] we even get examples of noncompact systems in any dimension n 25. Of course, the
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sphere, because of the noncompactness of its conformal group, is another example where noncompactness holds true (however, in this case, in all dimensions). Conversely, when we avoid large dimensions, avoid the sphere, and restrict the discussion to compactness, it follows from the analysis developed in this paper that for any smooth compact Riemannian 3-manifold (M, g), assumed not to be conformally diffeomorphic to the unit 3-sphere, for any p 1, and any C 1 -map A : M → Mps (R), if the inequality in (0.2) is large, g + A is coercive, and −A is cooperative, then the associated system (0.1) is compact. Our paper is organized as follows. In Section 1 we provide a complete classification of nonnegative solutions of the strongly coupled critical Euclidean limit system associated with (0.1) and thus obtain the shape of the blow-up singularities associated to our problem. We briefly discuss the dynamical notion of phase stability in Section 2. In Section 3 we prove strong pointwise control estimates for blowing-up sequences of solutions of perturbed equations. These estimates hold true without assuming (0.2). In Section 4 we prove sharp asymptotic estimates for sequences of solutions of perturbed equations when we assume (0.2) and get that rescalings of such sequences locally converge to the Green’s function plus a globally well-defined harmonic function with no mass. We construct parametrix for vector-valued Schrödinger operators when n = 3 in Section 5 and get an extension of the positive mass theorem of Schoen and Yau [39] to the vectorvalued case we consider here. This is the only place in the paper where we use the 3-dimensional assumptions that g + A is coercive and that −A is cooperative. We prove the theorem in Section 6 by showing that there should be a mass in the rescaled expansions of blowing-up sequences of solutions of perturbed equations. 1. Nonnegative solutions of the limit system Of importance in blow-up theory, when discussing critical equations, is the classification of the solutions of the critical limit Euclidean system we get after blowing up the equations. In our case, we need to classify the nonnegative solutions of the limit system −2
ui = |U|2
(1.1)
ui ,
p where |U|2 = i=1 u2i , and = − ni=1 ∂ 2 /∂xi2 is the Euclidean Laplace–Beltrami operator. The result we prove here provides full classification of nonnegative solutions of (1.1). It is stated as follows. Proposition 1.1. Let p 1 and U be a nonnegative C 2 -solution of (1.1). Then there exist a ∈ Rn , p−1 λ > 0, and Λ ∈ S+ , such that U(x) =
λ λ2 +
|x−a|2 n(n−2)
n−2 2
Λ
(1.2)
p−1
for all x ∈ Rn , where S+ consists of the elements (Λ1 , . . . , Λp ) in S p−1 , the unit sphere in Rp , which are such that Λi 0 for all i. We prove Proposition 1.1 by using the moving sphere method and the result in Druet and Hebey [18] where the classification of nonnegative H 1 -solutions of (1.1) is achieved by variational arguments. The method of moving sphere, a variant of the method of moving planes,
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has been intensively investigated in recent years. Among possible references we refer to Chen and Li [11], Chou and Chu [12], Li and Zhang [28], Li and Zhu [31] and Padilla [34]. Proposition 1.1 in the special case p = 1 was known for long time and goes back to Caffarelli, Gidas and Spruck [10]. The novelty in Proposition 1.1 is that p is arbitrary. For any a ∈ Rn , and any λ > 0, we define the Kelvin transform Ua,λ = Ka,λ (U) of a map U : Rn → Rp as the p-map defined in Rn \ {a} by Ua,λ (x) = Ka,λ (x)n−2 U a + Ka,λ (x)2 (x − a) λ for all x ∈ Rn \ {a}, where Ka,λ is given by Ka,λ (x) = |x−a| . As one can check, for any u ∈ 2 n n n C (R , R), for any a ∈ R , for any λ > 0, and for any x ∈ R \ {a},
ua,λ (x) = Ka,λ (x)n+2 u a + Ka,λ (x)2 (x − a) .
(1.3)
In particular, if U is a nonnegative solution of (1.1), so is Ua,λ in Rn \ {a} for all a ∈ Rn and all λ > 0. Writing that Ua,λ = ((u1 )a,λ , . . . , (up )a,λ ), it follows that −2
(ui )a,λ = |Ua,λ |2
(1.4)
(ui )a,λ
in Rn \ {a} for all a ∈ Rn , all λ > 0, and all i = 1, . . . , p. Before proving Proposition 1.1 we establish three lemmas. Our approach is based on the analysis developed in Li and Zhang [28]. Lemma 1.1. Let U be a nonnegative C 2 -solution of (1.1). For any point a in Rn , there exists a positive real number λ0 (a) such that for any λ in (0, λ0 (a)), there holds (ui )a,λ ui in Rn \ Ba (λ) for i = 1, . . . , p. Proof. Without loss of generality, we may take a = 0. We denote (ui )0,λ = (ui )λ for i = 1, . . . , p. By the superharmonicity of the function ui and by the strong maximum principle, for i = 1, . . . , p, there holds either ui ≡ 0 or ui > 0 in Rn . In case ui > 0, as is easily seen, there exists a positive real number r0 such that for any r ∈ (0, r0 ) and for any point θ ∈ S n−1 , there holds d n−2 r 2 ui (rθ ) > 0, dr for i = 1, . . . , p. It follows that for any λ ∈ (0, r0 ], there holds (ui )λ ui
(1.5)
in B0 (r0 ) \ B0 (λ). On the other hand, by the superharmonicity of the function ui and by the Hadamard Three-Sphere theorem as stated, for instance, in Protter and Weinberger [35], for any real number r > r0 and for any point x ∈ B0 (r) \ B0 (r0 ), we get 2−n r0 − r 2−n ui (x) |x|2−n − r 2−n min ui + r02−n − |x|2−n min ui ∂B0 (r0 )
|x|2−n − r 2−n min ui ∂B0 (r0 )
∂B0 (r)
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for i = 1, . . . , p. Letting r → +∞ gives ui (x)
r0 |x|
n−2 min ui
(1.6)
∂B0 (r0 )
for i = 1, . . . , p. We take 1 min∂B0 (r0 ) ui n−2 λ0 = r0 min , i∈I0 maxB (r ) ui 0 0 where I0 = i ∈ {1, . . . , p} s.t. ui ≡ 0 in Rn . For any real number λ ∈ (0, λ0 ) and for any point x ∈ Rn \ B0 (r0 ), there holds (ui )λ (x)
λ0 |x|
n−2
max ui B0 (r0 )
r0 |x|
n−2 min ui
∂B0 (r0 )
(1.7)
for i = 1, . . . , p. It follows from (1.5)–(1.7) that for any λ in (0, λ0 ), there holds (ui )λ ui in Rn \ B0 (λ) for i = 1, . . . , p. This ends the proof of Lemma 1.1. 2 By Lemma 1.1, for any point a in Rn , we can now define λ(a) = sup λ > 0 s.t. (ui )a,λ ui in Rn \ Ba (λ) for i = 1, . . . , p . The next lemma in the proof of Proposition 1.1 is as follows. Lemma 1.2. Let U be a nonnegative C 2 -solution of (1.1). If there holds that λ(a) < +∞ for some point a in Rn , then there holds |Ua,λ(a) | ≡ |U| in Rn \ {a}. Proof. Without loss of generality, we may take a = 0. We denote λ(0) = λ and (ui )0,λ = (ui )λ for i = 1, . . . , p. By definition of λ, in case λ < +∞, we get that for any λ ∈ (0, λ], there holds (ui )λ ui
(1.8)
in Rn \ B0 (λ) for i = 1, . . . , p, and that there exist an index i0 and a sequence of real numbers (λα )α in (λ, +∞) converging to λ such that property (1.8) does not hold true for i = i0 and λ = λα . For any positive real number λ, we let vλ be the function defined on Rn \ {0} by vλ = ui0 − (ui0 )λ . By (1.1), (1.4) and (1.8), we get ∗ −2
−vλ = |U|2
∗ −2
ui0 − |Uλ |2
(ui0 )λ 0
(1.9)
in Rn \ B0 (λ). We clearly have that min vλ = min vλ = 0.
Rn \B0 (λ)
∂B0 (λ)
(1.10)
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We claim that there holds vλ ≡ 0 in Rn \ B0 (λ). In order to prove this claim, we proceed by contradiction and assume that vλ ≡ 0 in Rn \ B0 (λ). By (1.10) and by the Hopf lemma, it follows that the outward normal derivative of the function vλ on ∂B0 (λ) is positive. By the continuity of ∇ui0 , we then get that there exists a real number r0 > λ such that for any λ ∈ [λ, r0 ), there holds vλ > 0
(1.11)
in B0 (r0 ) \ B0 (λ). Using the Hadamard Three-Sphere theorem as in Lemma 1.1, we also get that for any point x ∈ Rn \ B0 (r0 ), there holds vλ (x)
r0 |x|
n−2 min vλ .
∂B0 (r0 )
(1.12)
On the other hand, by the uniform continuity of the function ui0 on B0 (r0 ), there exists a positive real number ε such that for any λ ∈ [λ, λ + ε] and for any point x ∈ Rn \ B0 (r0 ), there holds n−2
v (x) − vλ (x) = (ui )λ (x) − (ui ) (x) 1 r0 min vλ . 0 0 λ λ ∂B0 (r0 ) 2 |x|
(1.13)
It follows from (1.11)–(1.13) that for any λ ∈ [λ, λ + ε], there holds vλ 0 in Rn \ B0 (λ). This contradicts the definition of λ, and this ends the proof of our claim, namely that there holds vλ ≡ 0 in Rn \ B0 (λ). Taking into account that vλ (x) = −
λ |x|
n−2 vλ
λ |x|
2 x
for all points x in Rn \ {0}, we even get that there holds vλ ≡ 0 in Rn \ {0}. Moreover, the function ui0 cannot be identically zero without contradicting the definition of λ, and thus, by the maximum principle, ui0 is nowhere vanishing. By (1.9), it follows that there holds |Uλ | ≡ |U| in Rn \ {0}. This ends the proof of Lemma 1.2. 2 The third and last lemma in the proof of Proposition 1.1 states as follows. Lemma 1.3. Let U be a nonnegative C 2 -solution of (1.1). If there holds that λ(a) = +∞ for some point a in Rn , then the p-map U is identically zero. Proof. By definition of λ(a), in case λ(a) = +∞, we get that for any positive real number λ, there holds (ui )a,λ ui in Rn \ Ba (λ) for i = 1, . . . , p. Without loss of generality we may here again assume that a = 0. In particular, we get λn−2 ui (0) lim inf |x|n−2 ui (x) . |x|→+∞
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Letting λ → +∞, it follows that for i = 1, . . . , p, either ui (0) = 0 or |x|n−2 ui (x) → +∞ as |x| → +∞. If there holds ui (0) = 0 for some i = 1, . . . , p, then by the superharmonicity of the function ui and by the strong maximum principle, ui is identically zero. Therefore, we may now assume that there holds |x|n−2 ui (x) → +∞ as |x| → +∞ for all i = 1, . . . , p such that ui ≡ 0. We then claim that there holds λ(y) = +∞ for all points y in Rn . Indeed, if not the case, namely if there holds λ(y) < +∞ for some point y in Rn , then by Lemma 1.2, we get
|x|n−2 U(x) = |x|n−2 Uy,λ(y) (x) → λ(y)n−2 U(y)
as |x| → +∞, which is a contradiction. By Lemma 11.2 in Li and Zhang [28] if there holds λ(y) = +∞ for all points y in Rn , then we get that the p-map U is constant. Taking into account that U satisfies (1.1), it follows that U is identically zero. 2 We are now in position to end the proof of Proposition 1.1. Proof of Proposition 1.1. By Lemma 1.3, we may assume that for any point y ∈ Rn , there holds λ(y) < +∞. By Lemma 1.2, it follows that for any point y in Rn , there holds |Uy,λ(y) | ≡ |U| in Rn \ {y}. By Lemma 11.1 in Li and Zhang [28], we then get that there exist a point a ∈ Rn and two positive real numbers λ and λ such that
U(x) =
λ λ + |x − a|2
n−2 2
(1.14)
for all points x in Rn . For any positive real number R, we define the function ηR in R+ by ηR (x) = η(x/R), where η is a smooth cutoff function in R+ satisfying η ≡ 1 in [0, 1], 0 η 1 in [1, 2], and η ≡ 0 in [2, +∞). For any positive real number R, multiplying (1.1) by ηR ui , summing over i and integrating by parts in Rn gives 1 2 2 |∇U| ηR dx + |U| ηR dx = |U|2 ηR dx. (1.15) 2 Rn
Rn
Rn
By (1.14), we get
ηC 0 (Rn )
|U|2 ηR dx
R2 Rn
|U|2 dx = O R 2−n
(1.16)
B0 (2R)\B0 (R)
as R → +∞. Passing to the limit into (1.15) as R → +∞, it follows from (1.16) that 2 |∇U| dx = |U|2 dx < +∞. Rn
Rn
By Proposition 3.1 in Druet and Hebey [18] we then get that the p-map U is of the form (1.2). This ends the proof of Proposition 1.1. 2
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2. Phase stability We very briefly discuss the implications that the stationary notion of bounded stability introduced in the introduction has in terms of dynamics. For this we define a notion of phase stability, see below, and discuss standing waves of critical nonlinear Klein–Gordon and Schrödinger equations associated with (0.1). The critical nonlinear vector-valued Schrödinger equations we consider in this section are written as ∂ui − g ui − Aij (x)uj + |U|2 −2 ui = 0 i ∂t p
(2.1)
j =1
in M for all i. The critical nonlinear vector-valued Klein–Gordon equations we consider are written as ∂ 2 ui + u + Aij (x)uj − |U|2 −2 ui = 0 g i ∂t 2 p
(2.2)
j =1
in M for all i. In the above equations A ∈ C 1 (M, Mps (R)). Vector-valued Schrödinger equations traditionally arise as a limiting case of the Zakharov system associated with plasma physics. In this framework equation (2.1) is a special case of the traditional vector nonlinear Schrödinger equation corresponding to the addition of a matrix potential in the linear part of the equation, and to the choice α = 1 of the thermal velocity parameter in the original equations. Let Ue−iωt be the standing waves model for (2.1) and (2.2), where the amplitude U : M → Rp is assumed to be nonnegative. It is easily checked that Ue−iωt is a standing wave for (2.1) if and only if U solves g ui +
p
Aij (x) − ωδ ˜ ij uj = |U|2 −2 ui
(2.3)
j =1
in M for all i, where ω˜ = ω, and that it is a standing wave for (2.2) if and only if U solves (2.3) with ω˜ = ω2 . In other words, Ue−iωt is a standing wave for (2.1) and (2.2) if and only if U solves (0.1) with the phase translated matrix A − ω Idp and A − ω2 Idp . In what follows, we define phase stability by the property that a convergence of the phase implies a convergence of the amplitude. When phase stability holds true, the corresponding standing wave sequence converges to another standing wave and phase stability clearly prevents the existence of standing waves with arbitrarily large amplitude in L∞ -norm. Definition. A phase ω is stable if for any sequence of standing waves with amplitudes Uα and phases ωα , the convergence ωα → ω in R as α → +∞ implies that, up to a subsequence, Uα → U in C 2 as α → +∞. An easy consequence of our theorem and of (2.3) is that large phases are always stable (with extra assumptions on A when n = 3). In particular, fast oscillating standing waves (|ω| 1) for the critical nonlinear vector-valued Klein–Gordon and Schrödinger equations cannot have arbitrarily large amplitude. We also get that small phases are stable, and thus that slow oscillating standing waves (|ω| 1) cannot have arbitrarily large amplitude as well, if the potential A is sufficiently small. Recall that standing waves here are like Ue−iωt , where U 0.
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Corollary. Large phases, required to be positive for (2.1), are generically stable. In particular, fast oscillating standing waves cannot have arbitrarily large amplitude. Small phases are also stable, and slow oscillating standing waves cannot have arbitrarily large amplitude as well, if the potential A is sufficiently small. To be more precise, assume that −A is cooperative, that ω˜ = ω (resp. ω˜ = ω2 ) is such that g + (A − ω˜ Idp ) is coercive, and that A<
n−2 Sg + ω˜ Idp . 4(n − 1)
(2.4)
Classical minimization arguments give that standing waves with nonnegative amplitude and phase ω exist for the critical nonlinear vector-valued Klein–Gordon and Schrödinger equations. Our theorem provides the stability of such standing waves with respect to ω. As is easily 1 checked, (2.4) is satisfied pby large phases. Let (aij )i,j be a symmetrical matrix of C functions aij : M → R such that j =1 aij (x) = 1 for all i = 1, . . . , p and all x ∈ M, and let A(g) be the
n−2 C 1 maps from M to Mps (R) given by A(g)ij = 4(n−1) Sg aij for all i, j = 1, . . . , p. By Druet and Hebey [16,18], the system (0.1) associated with A(g) is unstable when posed on spherical space forms in any dimension n 6. By the noncompactness of the conformal group on the sphere the system is noncompact when posed on the sphere in any dimension n 3, and by the constructions in Brendle [6] and Brendle and Marques [8], there are examples of nonconformally flat manifolds for any n 25 such that the system (0.1) associated with A(g) is noncompact, and thus also unstable. If A − ω Idp = A(g), or A − ω2 Idp = A(g), we then get instability of the phase ω for (2.1) and (2.2). However, if A is sufficiently small such that (0.2) is satisfied, then (2.4) is still satisfied with |ω| 1 sufficiently small, and our theorem provides the stability of such ω’s. In particular, small phases are also stable, and thus slow oscillating standing waves cannot have arbitrarily large amplitude as well, if the potential A is sufficiently small. Instability comes in the intermediate regime.
3. Pointwise controls in blow-up theory We let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, and (Aα )α be a sequence in C 1 (M, Mps (R)), α ∈ N. We consider the sequence of approximated equations g ui +
p
−2
Aαij (x)uj = |U|2
ui ,
(3.1)
j =1
where Aα = (Aαij )i,j , and we assume that Aα → A
(3.2)
in C 1 (M, Mps (R)) as α → +∞ for some A ∈ C 1 (M, Mps (R)). We let (Uα )α be a sequence of nonnegative solutions of (3.1) and we assume that max |Uα | → +∞ M
(3.3)
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as α → +∞. For U ∈ C 1 (M, Rp ) we define |U|Σ by |U|Σ =
p
(3.4)
ui ,
i=1
where U = (u1 , . . . , up ). If U 0 solves an equation like (0.1), summing the equations in (0.1), we get that g |U|Σ + Λ|U|Σ 0, where, for example, Λ = pA∞ and A∞ = maxM maxij |Aij (x)|. In particular, |U|Σ satisfies the maximum principle and we get that either |U|Σ ≡ 0 or |U|Σ > 0 everywhere in M. As a consequence, either U ≡ 0 or |U| > 0 everywhere in M, and we get that |U| is of class C 2,θ , θ ∈ (0, 1), exactly like U is. In what follows we let (xα )α be a sequence of points in M and (ρα )α , 0 < ρα < ig /7, be a sequence of positive real numbers, where ig is the injectivity radius of g. We assume that the xα ’s and ρα ’s are such that ∇|Uα |(xα ) = 0 and
n−2
dg (xα , x) 2 Uα (x) C
(3.5)
for all α, all x ∈ Bxα (7ρα ), and some C > 0 independent of α and x. We define μα =
1
(3.6)
2
|Uα (xα )| n−2
for all α, and aim in getting pointwise control estimates on the Uα ’s around the xα ’s. We start with a general Harnack type inequality in Lemma 3.1 and then get our control estimates in Lemmas 3.2, 3.4 and 3.5 under the additional assumption that n−2
lim ρα 2
α→+∞
sup |Uα | = +∞.
(3.7)
Bxα (6ρα )
Lemma 3.3 is used as an intermediate state between the asymptotic description in Lemma 3.2 and the sharp pointwise control in Lemma 3.4. Lemma 3.1. Let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, (Aα )α be a sequence in C 1 (M, Mps (R)), and (Uα )α be a sequence of nonnegative solutions of (3.1) such that (3.2) and (3.3) hold true. Let (xα )α and (ρα )α be such that (3.5) holds true, and let R 6 be given. There exists C > 1 such that for any sequence (sα )α of positive real numbers satisfying that sα > 0 and Rsα 6ρα for all α, there holds sα ∇Uα L∞ (Ωα ) C sup |Uα | C 2 inf |Uα |, Ωα
Ωα
where Ωα is given by Ωα = Bxα (Rsα ) \ Bxα ( R1 sα ) and, for U = (u1 , . . . , up ), ∇UL∞ = maxi ∇ui L∞ .
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Proof. Let R 6 be given and (sα )α be a sequence of positive real numbers such that sα > 0 and Rsα 6ρα for all α. We set for x ∈ B0 ( 7R 6 ), n−2 Uˆα (x) = sα 2 Uα expxα (sα x) , Aˆ α (x) = Aα expxα (sα x) , and gˆ α (x) = expxα g (sα x).
2 (B ( 7R )) as α → +∞, where gˆ is some Riemannian metric Up to a subsequence, gˆ α → gˆ in Cloc 0 6 7R in B0 ( 6 ), and gˆ = ξ as soon as sα → 0, where ξ is the Euclidean metric. We know thanks to (3.5) that
Uˆα (x) C|x|1− n2
(3.8)
in B0 ( 7R 6 ) \ {0}. Thanks to Eq. (3.1), we also get that gˆα (uˆ α )i + sα2
p
Aˆ αij (x)(uˆ α )j = |Uˆα |2 −2 (uˆ α )i
(3.9)
j =1
ˆ in B0 ( 7R ˆ α )1 , . . . , (uˆ α )p ). It follows from (3.8) and (3.9) that 6 ) for all i, where Uα = ((u
gˆ (uˆ α )i C 2 −2 |x|−2 + ps 2 Aα ∞ α α
sup
|Uˆα |
12 B0 ( 13R 12 )\B0 ( 13R )
12 in B0 ( 13R 12 ) \ B0 ( 13R ) for all i = 1, . . . , p. Sobolev embeddings lead then to the existence of some D > 0 such that
sup
∇(uˆ α )i D
B0 (R)\B0 ( R1 )
sup
|Uˆα |
(3.10)
12 B0 ( 13R 12 )\B0 ( 13R )
for all i = 1, . . . , p. Let i ∈ {1, . . . , p} be given and let uˆ α = |Uˆα |Σ , where | · |Σ is as in (3.4). By the maximum principle, uˆ α > 0. Summing the equations in (3.9) we have that gˆα uˆ α = Fα uˆ α
(3.11)
in B0 ( 7R 6 ), where p Fα = |Uˆα |
2 −2
ˆ α ˆ α )j i,j =1 Aij (u
− sα2
uˆ α
.
(3.12)
Combining (3.8) and (3.12) we get that |Fα |
7R 6
2
−2
C2
+ sα2 Aα ∞
(3.13)
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6 in B0 ( 7R ˆ α of (3.11), 6 ) \ B0 ( 7R ). Thanks to the Harnack inequality that we apply to the solutions u see for instance Theorem 4.17 of [22], we get the existence of some D > 0 independent of α, K and x such that
sup uˆ α D
Bx (2K)
inf uˆ α + KFα Ln (Bx (2K)) sup uˆ α
Bx (K)
Bx (2K)
for all α and all balls Bx (2K) ⊂ B0 ( 7R 6 ). Using (3.13) and choosing K small enough clearly leads to the existence of some D > 0 such that uˆ α
sup B0 (R)\B0 ( R1 )
D
for all α. It remains to note that (3.14). 2
uˆ α
sup 12 B0 ( 13R 12 )\B0 ( 13R )
1 2 ˆα pu
inf
B0 (R)\B( R1 )
uˆ α D
uˆ α
sup
(3.14)
B0 (R)\B0 ( R1 )
|Uα |2 uˆ 2α to conclude the lemma with (3.10) and
Lemmas 3.2 to 3.5 below are involved with getting pointwise control estimates on the Uα ’s. Lemma 3.2. Let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, (Aα )α be a sequence in C 1 (M, Mps (R)), and (Uα )α be a sequence of nonnegative solutions of (3.1) such that (3.2) and (3.3) hold true. Let (xα )α and (ρα )α be such that (3.5) and (3.7) hold true. After passing to a subsequence, n−2 μα 2 Uα expxα (μα x) →
1 1+
|x|2
n−2 2
Λ
(3.15)
n(n−2) p−1
p−1
1 (Rn ) as α → +∞, where μ is as in (3.6), Λ ∈ S in Cloc is the set of vectors in α + , and S+ p R with nonnegative components and such that |Λ| = 1. Moreover, μραα → +∞ as α → +∞. In particular, μα → 0 as α → +∞.
Proof. Let yα ∈ Bxα (6ρα ) and να > 0 be such that
Uα (yα ) =
sup |Uα |
1− n and Uα (yα ) = να 2
Bxα (6ρα )
for all α. By (3.7), να → 0 and ρα να−1 → +∞ as α → +∞. By (3.5), dg (xα , yα ) Cνα for all α. Let Ωα = B0 (ρα να−1 ), Ωα ⊂ Rn . For x ∈ Ωα we set n−2 U˜α (x) = να 2 Uα expxα (να x)
(3.16)
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2 (Rn ) as α → +∞, where and gα (x) = (expxα g)(να x). Since να → 0 we get that gα → δ in Cloc ξ is the Euclidean metric. As is easily checked,
gα (u˜ α )i + να2
p
A˜ αij (x)(u˜ α )j = |U˜α |2 −2 (u˜ α )i
(3.17)
j =1
for all i, where U˜α = ((u˜ α )1 , . . . , (u˜ α )p ) and A˜ αij (x) = Aαij expxα (να x) for all α and all i, j . Since |U˜α | 1 in Ωα , and since ρα να−1 → +∞ so that Ωα → Rn , we get 1 (Rn ) as α → +∞, where U˜ 0 from (3.17) and standard elliptic theory that U˜α → U˜ in Cloc solves (1.1). Let y˜α be given by y˜α =
1 exp−1 xα (yα ). να
By (3.16) we have that |y˜α | C for all α and we may thus assume that, up to a subsequence, y˜α → y˜0 as α → +∞. Since |U˜α (y˜α )| = 1, we get that |U˜ (y˜0 )| = 1 and y˜0 is a point where |U˜ | ˜ since xα is a critical point attains its maximum. Also we have that 0 is a critical point of |U| ˜ of |Uα |, and we have that
U˜ (0) = lim
α→+∞
να μα
n−2 2
(3.18)
.
˜ attains its maximum 1 at y˜0 , we get that By Proposition 1.1, since |U| ˜ U(x) =
n−2
1 1+
2
Λ,
|x−y˜0 |2 n(n−2)
p−1 ˜ we get that y˜0 = 0, and by for all x ∈ Rn , where Λ ∈ S+ . Since 0 is a critical point of |U|, (3.18) we get that να = μα (1 + o(1)). This proves Lemma 3.2. 2
At this point we define ϕα : (0, ρα ) → R+ by 1 ϕα (r) = |∂Bxα (r)|g
|Uα |Σ dσg ,
(3.19)
∂Bxα (r)
where |∂Bxα (r)|g is the volume of the sphere of center xα and radius r for the induced metric and | · |Σ is as in (3.4). As a consequence of Lemma 3.2 we have that (μα r)
n−2 2
ϕα (μα r) →
r 1+
r2 n(n−2)
n−2 2
|Λ|Σ
(3.20)
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1013
1 ([0, +∞)) as α → +∞. We define r ∈ [2R μ , ρ ] by in Cloc α 0 α α
n−2 rα = sup r ∈ [2R0 μα , ρα ] s.t. s 2 ϕα (s) 0 in [2R0 μα , r]
(3.21)
where R02 = n(n − 2). Thanks to (3.20) we have that rα → +∞ μα
(3.22)
as α → +∞, while the definition of rα gives that r
n−2 2
ϕα is non-increasing in [2R0 μα , rα ]
(3.23)
and that n−2 r 2 ϕα (r) (rα ) = 0
if rα < ρα .
(3.24)
|Uα |.
(3.25)
Given R > 0 we define ηR,α =
sup Bxα (Rrα )\Bxα ( R1 rα )
Now we can prove the following estimate. Lemma 3.3. Let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, (Aα )α be a sequence in C 1 (M, Mps (R)), and (Uα )α be a sequence of nonnegative solutions of (3.1) such that (3.2) and (3.3) hold true. Let (xα )α and (ρα )α be such that (3.5) and (3.7) hold true, and let R 6 be such that Rrα 6ρα for all α 1. For any ε > 0 there exists Cε > 0 such that, after passing to a subsequence, n−2
Uα (x) Cε μα 2 (1−2ε) dg (xα , x)(2−n)(1−ε) + ηR,α
rα dg (xα , x)
(n−2)ε (3.26)
for all x ∈ Bxα (Rrα ) \ {xα } and all α, where ηR,α is as in (3.25), μα is as in (3.6), and rα is as in (3.21). Proof. By Lemma 3.1 there exists C > 1 such that 1 C
|Uα | ϕα (sα ) C
sup Bxα (Rsα )\Bxα ( R1 sα )
inf
Bxα (Rsα )\Bxα ( R1 sα )
|Uα |
(3.27)
for all 0 < sα rα and all α. By (3.23) and (3.27) we then get that for D 1 sufficiently large, sup x∈Bxα (Rrα )\Bxα (Dμα )
dg (xα , x)
n−2 2
Uα (x) C
sup
Dμα rrα
C(Dμα )
n−2 2
r
n−2 2
ϕα (r)
ϕα (Dμα )
(3.28)
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and it follows from (3.20) and (3.28) that lim
lim
sup
D→+∞ α→+∞ x∈B
dg (xα , x)
n−2 2
Uα (x) = 0.
(3.29)
xα (Rrα )\Bxα (Dμα )
In particular, by (3.22) and (3.29), n−2
rα 2 ηR,α → 0
(3.30)
as α → +∞. Let G be the Green’s function of g in M, where we choose G such that G 1. Then, see for instance Aubin [4,5],
1
τ dg (x, y)
dg (x, y)n−2 G(x, y) − (3.31)
(n − 2)ωn−1 and
dg (x, y)n−1 ∇G(x, y) − 1 τ dg (x, y)
ωn−1 for some continuous function τ : R+ → R+ satisfying τ (0) = 0. We fix 0 < ε < n−2
Φαε (x) = μα 2
(1−2ε)
(3.32) 1 2
and set
G(xα , x)1−ε + ηR,α rα(n−2)ε G(xα , x)ε .
By (3.31) it suffices, in order to get Lemma 3.3, to prove that |Uα |Σ = O(1). ε Bxα (Rrα )\{xα } Φα sup
(3.33)
We have Φαε (x) → +∞ as x → xα . Let yα ∈ Bxα (Rrα ) \ {xα } be such that |Uα |Σ |Uα (yα )|Σ . = ε Φαε (yα ) Bxα (Rrα )\{xα } Φα sup
(3.34)
First we assume that dg (xα , yα ) 0 as α → +∞. Then rα 0 since there holds dg (xα , yα ) Rrα and we get that Φαε (yα ) CηR,α for some C > 0 independent of α. By Lemma 3.1 we can also write that |Uα (yα )| CηR,α for some C > 0 independent of α. This proves (3.33) when dg (xα , yα ) 0 as α → +∞. From now on we assume that dg (xα , yα ) → 0 as α → +∞ and we distinguish three different cases: d (x ,y )
Case 1. g μαα α → D as α → +∞; Case 2. yα ∈ ∂Bxα (Rrα ) for all α; d (x ,y ) Case 3. yα ∈ Bxα (Rrα ) and g μαα α → +∞ as α → +∞.
(3.35)
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Assume first that we are in case 1. Then, by Lemma 3.2,
n−2 2
μα Uα (yα ) →
n−2
1 1+
2
(3.36)
Λ
D2 n(n−2)
p−1
as α → +∞, where Λ ∈ S+ . By (3.22), (3.30) and (3.31), n−2 2
μα
Φαε (yα ) =
1 (n − 2)ωn−1
1−ε
μα dg (xα , yα )
(n−2)(1−ε) + o(1)
n−2 + O ηR,α μα 2 rα(n−2)ε dg (xα , yα )(2−n)ε 1−ε 1 + o(1) = (n − 2)ωn−1 D n−2 n−2 (1−2ε) + O ηR,α rα(n−2)ε μα 2 1−ε 1 + o(1) = (n − 2)ωn−1 D n−2
− n−2 (1−2ε) n−2 (1−2ε) + o rα 2 μα 2 1−ε 1 + o(1) = (n − 2)ωn−1 D n−2 if D = 0, and if D = 0, noting that by (3.31), n−2
μα 2 Φαε (yα ) Cμα(n−2)(1−ε) dg (xα , yα )−(n−2)(1−ε) , we get that n−2
lim μα 2 Φαε (yα ) = +∞.
α→+∞
It follows that in case 1, for D = 0 or D > 0, using (3.36), n−2 2 1 |Uα (yα )| n−2 1−ε → (n − 2)ω D n−1 2 D Φαε (yα ) 1 + n(n−2)
(3.37)
as α → +∞, and (3.33) follows from (3.37). Now we assume we are in case 2. Then, by the definition of ηR,α , we have that |Uα (yα )| ηR,α and since by (3.31),
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Φαε (yα ) ηR,α rα(n−2)ε G(xα , yα )ε ε 1 ηR,α rα(n−2)ε + o(1) dg (xα , yα )−(n−2)ε (n − 2)ωn−1 ε 1 = ηR,α + o(1) (n − 2)ωn−1 R n−2 we get that, here again, (3.33) holds true. At this point it remains to discuss case 3. Since yα ∈ Bxα (Rrα ) in case 3, it follows from (3.34) and (3.42) below that g |Uα |Σ (yα ) g Φαε (yα ) . |Uα |Σ (yα ) Φαε (yα )
(3.38)
Since 2 −1 g |Uα |Σ C1 |Uα |Σ + C2 |Uα |Σ ,
where C1 , C2 > 0 are independent of α, we get by (3.35) and (3.29) that g |Uα |Σ (yα ) = 0. |Uα |Σ (yα )
(3.39)
|∇Gxα |2 ε Φα G2xα
(3.40)
g Φαε (yα ) = ε(1 − ε)(n − 2)2 . Φαε (yα )
(3.41)
lim dg (xα , yα )2
α→+∞
On the other hand, we compute g Φαε = ε(1 − ε) and by (3.31), (3.32) and (3.40) we get that lim dg (xα , yα )2
α→+∞
Combining (3.38), (3.39) and (3.41) we get a contradiction so that only cases 1 and 2 can occur. This ends the proof of Lemma 3.3. 2 In the above process we used that if Ω is an open subset of M, u, v are C 2 -positive functions in Ω, and x0 ∈ Ω is a point where uv achieves its supremum in Ω, then g v(x0 ) g u(x0 ) . v(x0 ) u(x0 ) Indeed, ∇( uv ) =
u∇v−v∇u u2
(3.42)
so that u(x0 )∇v(x0 ) = v(x0 )∇u(x0 ). Then,
u(x0 )g v(x0 ) − v(x0 )g u(x0 ) v g (x0 ) = u u2 (x0 ) and we get (3.42) by writing that g ( uv )(x0 ) 0. At this point, thanks to Lemma 3.3, we can prove the following sharp estimate.
O. Druet et al. / Journal of Functional Analysis 258 (2010) 999–1059
1017
Lemma 3.4. Let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, (Aα )α be a sequence in C 1 (M, Mps (R)), and (Uα )α be a sequence of nonnegative solutions of (3.1) such that (3.2) and (3.3) hold true. Let (xα )α and (ρα )α be such that (3.5) and (3.7) hold true, and let R 6 be such that Rrα 6ρα for all α 1. There exists C > 0 such that, after passing to a subsequence, n−2
Uα (x) + dg (xα , x)∇Uα (x) Cμα 2 dg (xα , x)2−n
(3.43)
for all x ∈ Bxα ( R2 rα ) \ {xα } and all α, where, for U = (u1 , . . . , up ) and x ∈ M, ∇U(x) = maxi |∇ui (x)|, where μα is as in (3.6), and where rα is as in (3.21). Proof. We prove that there exist C, C > 0 such that
n−2
Uα (x) C μα 2 dg (xα , x)2−n + ηR,α
(3.44)
for all x ∈ Bxα ( R2 rα ) \ {xα } and all α, and n−2
ηR,α C μα 2 rα2−n
(3.45)
for all α. Lemma 3.4 follows from Lemma 3.1, (3.44) and (3.45). In particular, it suffices to prove (3.44) and (3.45). We start with the proof of (3.45) assuming (3.44). By (3.23), for any η ∈ (0, 1), (ηrα )
n−2 2
n−2
ϕα (ηrα ) rα 2 ϕα (rα )
for all α 1. By (3.27) we then get that n−2 1 n−2 rα 2 ηR,α (ηrα ) 2 C
|Uα |.
sup ∂Bxα (ηrα )
Assuming (3.44) it follows that n−2 n−2 1 ηR,α η 2 μα 2 (ηrα )2−n + ηR,α C and if we choose η ∈ (0, 1) sufficiently small such that Cη
n−2 2
12 , we obtain that
n−2
ηR,α η2−n μα 2 rα2−n . This proves (3.45) when we assume (3.44). Now it remains to prove (3.44). For this it suffices to prove that for any sequence (yα )α such that yα ∈ Bxα
R rα \ {xα } 2
(3.46)
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for all α, there exists C > 0 such that, up to a subsequence,
n−2
Uα (yα ) C μα 2 dg (xα , yα )2−n + ηR,α .
(3.47)
Let (yα )α be such that yα satisfies (3.46) for all α. As a preliminary remark one can note that (3.47) directly follows from Lemma 3.2 if dg (xα , yα ) = O(μα ). In a similar way, (3.47) follows from Lemma 3.1 if rα−1 dg (xα , yα ) 0 as α → +∞. From now on we assume that lim
α→+∞
1 dg (xα , yα ) = +∞ and μα
lim
1
α→+∞ rα
dg (xα , yα ) = 0.
(3.48)
/ Sp(g ), where Sp(g ) is the spectrum of g , and let G be the Let λ > 1 be such that λpA∞ ∈ Green’s function of g − λpA∞ . There exist, see for instance Robert [36], positive constants C1 > 1 and C2 , C3 > 0 such that 1 dg (x, y)2−n − C2 G(x, y) C1 dg (x, y)2−n , C1
∇G(x, y) C3 dg (x, y)1−n
and (3.49)
for all x = y. By (3.49) there exists δ > 0 such that G 0 in Bxα (δrα ) for all α. By (3.48), yα ∈ Bxα ( 2δ rα ) for α 1. By the Green’s representation formula,
G(yα , x) g |Uα |Σ − λpA∞ |Uα |Σ (x) dvg (x)
|Uα |Σ (yα ) = Bxα (δrα )
G(yα , x) ∂ν |Uα |Σ (x) dσg (x)
+ ∂Bxα (δrα )
∂ν G(yα , x) |Uα |Σ (x) dσg (x),
−
(3.50)
∂Bxα (δrα )
where ν is the unit outward normal to ∂Bxα (δrα ). Since λ > 1, g |Uα |Σ − λpA∞ |Uα |Σ |Uα |2 −2 |Uα |Σ √ p|Uα |2 −1
and since G 0 in Bxα (δrα ) we get with (3.49) that
G(yα , x) g |Uα |Σ − λpA∞ |Uα |Σ (x) dvg (x)
Bxα (δrα )
C Bxα (δrα )
2 −1
dg (yα , x)2−n Uα (x)
dvg (x).
(3.51)
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Independently, by (3.49) and Lemma 3.1,
G(yα , x) ∂ν |Uα |Σ (x) dσg (x) CηR,α ,
1019
and
∂Bxα (δrα )
∂ν G(yα , x) |Uα |Σ (x) dσg (x) CηR,α
(3.52)
∂Bxα (δrα )
for some C > 0. Combining (3.50)–(3.52), we get that
1 |Uα |Σ (yα ) C
2 −1
dg (yα , x)2−n Uα (x)
dvg (x) + ηR,α .
(3.53)
Bxα (δrα )
We fix ε =
2 n+2 .
By Lemmas 3.2 and 3.3, and by (3.48), we can write that
2 −1
dg (yα , x)2−n Uα (x)
dvg (x)
Bxα (δrα )
n−2 = O μα 2 dg (xα , yα )2−n n+2 2 (1−2ε) + O μα
2−n
dg (yα , x)
Bxα (δrα )\Bxα (μα )
2 −1 (n+2)ε + O ηR,α rα
−(n+2)(1−ε)
dg (xα , x)
dvg (x)
dg (yα , x)2−n dg (xα , x)−(n+2)ε dvg (x)
Bxα (δrα )\Bxα (μα )
2 −1 2 n−2 rα = O μα 2 dg (xα , yα )2−n + O ηR,α and we thus get from (3.5), that
2 −1
dg (yα , x)2−n Uα (x)
dvg (x)
Bxα (δrα )
n−2 = O μα 2 dg (xα , yα )2−n + O(ηR,α ).
(3.54)
2 −1 2 −2 Indeed rα2 ηR,α = (rα2 ηR,α )ηR,α , and by (3.5),
n−2 2 2 −2 1 rα ηR,α 2 −2 = rα 2
C
sup
|Uα |
Bxα (Rrα )\Bxα ( R1 rα )
sup
dg (xα , x)
n−2 2
Uα (x) C.
x∈Bxα (Rrα )
Then (3.47) follows from (3.53) and (3.54). This ends the proof of Lemma 3.4.
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At this point we define Bα by Bα (x) =
μα μ2α +
n−2 2
dg (xα ,x)2 n(n−2)
(3.55)
for all α, where x ∈ M. As a last estimate in this section we prove Lemma 3.5 below. Lemma 3.5. Let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, (Aα )α be a sequence in C 1 (M, Mps (R)), and (Uα )α be a sequence of nonnegative solutions of (3.1) such that (3.2) and (3.3) hold true. Let (xα )α and (ρα )α be such that (3.5) and (3.7) hold true. There exist C > 0 and (εα )α such that, up to a subsequence, n−2 |Uα − Bα Λ| Cμα 2 rα2−n + Sα + εα Bα
(3.56)
p−1
in Bxα (2rα ) \ {xα } for all α, where Λ ∈ S+ is as in Lemma 3.2, εα → 0 as α → +∞, Sα (x) = dg (xα , x)3−n for all x, μα is as in (3.6), and rα is as in (3.21). Proof. Let G be the Green’s function of g + 1 in M. Let (yα )α be any sequence of points in Bxα (2rα ) \ {xα }. By the Green’s representation formula, for any i = 1, . . . , p,
G(yα , x) g (uα )i + (uα )i (x) dvg (x)
(uα )i = Bxα (2rα )
+
G(yα , x) ∂ν (uα )i (x) dσg (x)
∂Bxα (2rα )
−
∂ν G(yα , x) (uα )i (x) dσg (x),
(3.57)
∂Bxα (2rα )
where ν is the unit outward normal to Bxα (2rα ) and Uα = ((uα )1 , . . . , (uα )p ). We have, see, for instance, Druet, Hebey and Robert [20], that G 0 and that there exist positive constants C1 , C2 > 0 such that
1
C1 dg (x, y), and
dg (x, y)n−2 G(x, y) −
(n − 2)ωn−1
∇G(x, y) C2 dg (x, y)1−n (3.58) for all x = y. By (3.58) and Lemma 3.1,
G(yα , x) ∂ν (uα )i (x) dσg (x)
Cη6,α ,
∂Bxα (2rα )
∂Bxα (2rα )
∂ν G(yα , x) (uα )i (x) dσg (x)
Cη6,α
(3.59)
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and by (3.45), n−2
η6,α Cμα 2 rα2−n .
(3.60)
By Lemma 3.4 and (3.58),
n−2
G(yα , x)|Uα | dvg (x) Cμα 2 Bxα (2rα )
dg (yα , x)2−n dg (xα , x)2−n dvg (x) Bxα (2rα )
and by Giraud’s lemma we get that
1
G(yα , x)|Uα | dvg (x) Cμα2
if n = 3,
Bxα (2rα )
G(yα , x)|Uα | dvg (x) Cμα 1 + ln dg (xα , yα )
if n = 4,
and
Bxα (2rα )
n−2
G(yα , x)|Uα | dvg (x) Cμα 2 dg (xα , yα )4−n
if n 5.
(3.61)
Bxα (2rα )
Now we let Rα : M → Rp be given by Rα (x) =
2 −2 G(x, y) Uα (y)
Uα (y) dvg (y)
(3.62)
Bxα (2rα )
for x ∈ M, and let f : M → R be given by f (x) = (n − 2)ωn−1 dg (x0 , x)n−2 G(x0 , x) if x = x0 and f (x0 ) = 1, where, up to a subsequence, xα → x0 as α → +∞. By (3.58), f is continuous at x0 and
f (x) − 1 Cdg (x0 , x).
(3.63)
Rα (yα ) = 0. lim − f (y )Λ α α→+∞ B (y )
(3.64)
We claim that
α
α
As is easily checked, Lemma 3.5 follows from (3.64). Indeed, by (3.64), since (yα )α is arbitrary in Bxα (2rα ) \ {xα }, for any x ∈ M
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Rα (x) − Bα (x)Λ
Rp
Rα (x) − f (x)Bα (x)ΛRp + f (x) − 1 Bα (x) Rα
− f Λ Bα (x) + f (x) − 1 Bα (x) Bα L∞
εα Bα (x) + f (x) − 1 Bα (x),
(3.65)
where εα → 0 as α → +∞, and by (3.63) we can write that
f (x) − 1 Cdg (x0 , x) εα + Cdg (xα , x), where εα = Cdg (x0 , xα ) is such that εα → 0 as α → +∞. Moreover, n−2
dg (xα , x)Bα (x) μα 2 dg (xα , x)3−n
(3.66)
and we thus get (3.56) by combining (3.57), (3.59), (3.60), (3.61), (3.65) and (3.66). Summarizing, at this point, it remains to prove (3.64). Up to passing to a subsequence we may assume that yα → y0 as α → +∞. Suppose first that y0 = x0 . By Lemmas 3.2, 3.4, and the Lebesgue’s dominated convergence theorem, writing that
n−2 2
2 −2
U˜α (x) dvg˜α (x), G yα , expxα (μα x) U˜α (x)
Rα (yα ) = μα
B0 (2 μrαα )
where n−2 U˜α (x) = μα 2 Uα expxα (μα x) g˜ α (x) = expxα g (μα x),
and (3.67)
we get that Rα (yα ) = α→+∞ Bα (yα )
lim
dg (x0 , y0 )2 n(n − 2)
n−2 2
−1
u02
dx G(x0 , y0 )Λ,
Rn
where u0 (x) =
1 1+
|x|2
n−2 2
.
n(n−2)
Since
−1
u02 Rn
n−2 dx = (n − 2)ωn−1 n(n − 2) 2
(3.68)
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1023
we get that if y0 = x0 , then lim
α→+∞
Rα (yα ) = (n − 2)ωn−1 dg (x0 , y0 )n−2 G(x0 , y0 )Λ Bα (yα ) = f (y0 )Λ.
This proves (3.64) when y0 = x0 . Now we assume that y0 = x0 . In addition, as a first case to consider, we assume also that dg (xα , yα ) →D μα
(3.69)
as α → +∞ for some D 0. Let zα be such that yα = expxα (μα zα ). Then n−2 2 |zα |2 Rα (yα ) = 1+ μn−2 α Bα (yα ) n(n − 2)
α Bxα ( 2r μα
˜ α |U˜α |2 −2 U˜α dvg˜ , G α
(3.70)
)
where U˜α and g˜ α are as in (3.67), and ˜ α (x) = G expx (μα zα ), expx (μα x) . G α α By (3.58), ˜ α (x) → dg expxα (μα zα ), expxα (μα x) G
1 (n − 2)ωn−1
(3.71)
as α → +∞ for all x, and we also have that dg expxα (μα zα ), expxα (μα x) = μα dg˜α (zα , x).
(3.72)
Combining (3.70), (3.71) and (3.72), by Lemmas 3.2 and 3.4, and by the Lebesgue’s dominated convergence theorem we get that n−2 2 |z0 |2 Rα (yα ) u0 (x)2 −1 dx Λ, lim = 1+ α→+∞ Bα (yα ) n(n − 2) (n − 2)ωn−1 |x − z0 |n−2 Rn
−1
where zα → z0 as α → +∞, and u0 is as in (3.68). We have that u0 = u02 G0 (x, y) =
1 (n − 2)ωn−1 |y − x|n−2
is the Green’s function of , we get from (3.73) that Rα (yα ) = Λ. α→+∞ Bα (yα ) lim
, and since
(3.73)
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This proves (3.64) when y0 = x0 and we assume (3.69). Now it remains to consider the case where y0 = x0 and dg (xα , yα ) → +∞ μα
(3.74)
as α → +∞. Then Rα (yα ) = Bα (yα )
n−2 2 1 − n−2 + o(1) dg (xα , yα )n−2 μα 2 Iα , n(n − 2)
(3.75)
where
2 −2 G(yα , x) Uα (x)
Uα (x) dvg (x).
Iα = Bxα (2rα )
We write that
2 −2 G(yα , x) Uα (x)
Uα (x) dvg (x)
Iα = Ωα
+
2 −2 G(yα , x) Uα (x)
Uα (x) dvg (x),
Ωαc
where 1 Ωα = x ∈ Bxα (2rα ) s.t. dg (yα , x) dg (xα , yα ) 2 and Ωαc = Bxα (2rα ) \ Ωα . We have that
− n−2 μα 2
Ωα
2 −2 G(yα , x) Uα (x)
Uα (x) dvg (x)
= 1 μα
2 −2
U˜α (x) dvg˜α (x), G yα , expxα (μα x) U˜α (x)
exp−1 xα (Ωα )
where U˜α and g˜ α are as in (3.67). Let zα = expxα (μα x). For x ∈ dg (yα , zα ) → +∞ μα
1 μα
exp−1 xα (Ωα ),
(3.76)
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as α → +∞, and since dg (yα , zα ) − dg (xα , zα ) dg (xα , yα ) dg (yα , zα ) + dg (xα , zα ) and dg (xα , zα ) = μα |x|, we get that lim
α→+∞
dg (xα , yα ) = 1. dg (yα , zα )
(3.77)
By (3.58) and (3.77), lim dg (xα , yα )n−2 Gα yα , expxα (μα x) =
α→+∞
1 . (n − 2)ωn−1
By Lemmas 3.2 and 3.4, and by the Lebesgue’s dominated convergence theorem, we then get that
− n−2 2
2 −2 G(yα , x) Uα (x)
Uα (x) dvg (x)
lim dg (xα , yα )n−2 μα
α→+∞
=
1 (n − 2)ωn−1
Ωα
−1
u02
dx Λ
Rn
n−2 = n(n − 2) 2 Λ.
(3.78)
Independently, by (3.58) and by Lemma 3.4, − n−2 2
2 −2 G(yα , x) Uα (x)
Uα (x) dvg (x)
dg (xα , yα )n−2 μα
Ωαc
Cdg (xα , yα )−4 μ2α
dg (yα , x)2−n dvg (x)
Ωαc
C
μα dg (xα , yα )
2 = o(1)
(3.79)
since 1 dg (xα , x) dg (xα , yα ) − dg (yα , x) dg (xα , yα ) 2 for x ∈ Ωαc . Noting that (3.64) follows from (3.75), (3.76), (3.78) and (3.79), we get that (3.64) holds true when y0 = x0 and we assume (3.74). This ends the proof of Lemma 3.5. 2
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4. Sharp pointwise blow-up estimates In this section we prove sharp blow-up estimates for sequences of solutions of perturbed equations like (3.1) when we assume (0.2). The main result of this section is Lemma 4.3. Lemmas 4.1 and 4.2 are preliminary lemmas for the proof of Lemma 4.3. In what follows we let Xα be the 1-form given by Xα (x) = 1 −
1 Rcg (x) ∇fα (x), ∇fα (x) ∇fα (x), 6(n − 1)
(4.1)
where fα (x) = 12 dg (xα , x)2 and, in local coordinates, (Rcg )ij = g iμ g j ν Rμν , where the Rij ’s are the components of the Ricci curvature Rcg of g. Lemma 4.1. Let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, (Aα )α be a sequence in C 1 (M, Mps (R)), and (Uα )α be a sequence of nonnegative solutions of (3.1) such that (3.2) and (3.3) hold true. Let (xα )α and (ρα )α be such that (3.5) and (3.7) hold true. Let R1,α be given by R1,α =
p i=1B (r ) xα α
1 ∇Xα − (divg Xα )g ∇(uα )i , ∇(uα )i dvg , n
(4.2)
where Uα = ((uα )1 , . . . , (uα )p ), Xα is as in (4.1), and A is the musical isomorphism of A. Then R1,α = μα rα if n = 3, 1 2−n + o μn−2 if n = 4, R1,α = o μ2α ln α rα μα 2−n if n 5, R1,α = o μ2α + o μn−2 α rα
(4.3)
where μα is as in (3.6) and rα is as in (3.21). Proof. Thanks to the expression of Xα , 1 (∇Xα )ij − (divg Xα )gij = O dg (xα , x)2 n for all i, j . Assuming n = 3 we can write by Lemma 3.4 that
2 dg (xα , x)2 ∇Uα (x) dvg (x)
|R1,α | C Bxα (rα )
Cμα Bxα (rα )
Cμα rα .
dg (xα , x)−2 dvg (x)
(4.4)
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This proves (4.3) when n = 3. From now on we assume that n 4. We have that
1 ∇Xα − (divg Xα )g n
(∇Bα , ∇Bα ) = O dg (xα , ·)3 |∇Bα |2 ,
(4.5)
where Bα is as in (3.55). Thanks to (4.4) and (4.5) we can write that
2
dg (xα , x)3 ∇Bα (x) dvg (x)
R1,α = O Bxα (rα )
+O Bxα (rα )
+O
dg (xα , x) ∇Bα (x) × ∇(Uα − Bα Λ)(x) dvg (x)
2
2 dg (xα , x) ∇(Uα − Bα Λ)(x) dvg (x) .
2
(4.6)
Bxα (rα )
We have that
2
1 if n = 4, dg (xα , x)3 ∇Bα (x) dvg (x) = o μ2α ln μα
Bxα (rα )
2
dg (xα , x)3 ∇Bα (x) dvg (x) = o μ2α
if n 5.
Bxα (rα )
Moreover, given i ∈ {1, . . . , p}, integrating by parts,
2 dg (xα , x)2 ∇ (uα )i − Bα Λi (x) dvg (x)
Bxα (rα )
(uα )i − Bα Λi (x) dg (xα , x)2 ∇ (uα )i − Bα Λi (x) dσg (x)
=O ∂Bxα (rα )
2
dg (xα , x) (uα )i − Bα Λi (x) dσg (x)
+O ∂Bxα (rα )
+O
(uα )i − Bα Λi (x) 2 dvg (x)
Bxα (rα )
+ Bxα (rα )
dg (xα , x)2 (uα )i − Bα Λi (x) g (uα )i − Bα Λi (x) dvg (x),
(4.7)
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and we get by Lemma 3.4 that
2 dg (xα , x)2 ∇ (uα )i − Bα Λi (x) dvg (x)
Bxα (rα )
dg (xα , x)2 (uα )i − Bα Λi (x) g (uα )i − Bα Λi (x) dvg (x)
= Bxα (rα )
4−n + O μn−2 +O α rα
(uα )i − Bα Λi 2 dvg .
(4.8)
Bxα (rα )
We have that Bα2 dvg = 64ω3 μ2α ln Bxα (rα )
rα 1 if n = 4, + o μ2α ln μα μα
u20 dx μ2α + o μ2α
Bα2 dvg =
and
if n 5,
(4.9)
Rn
Bxα (rα )
where u0 is as in (3.68). Moreover,
(uα )i − Bα Λi 2 dvg = o μ2 α
(4.10)
Bxα (μα )
by Lemma 3.2, while if Sα is as in Lemma 3.5, we can write that
4−n + o μ2α . Sα2 dvg = O μn−2 α rα
μn−2 α
(4.11)
Bxα (rα )\Bxα (μα )
By (3.22), 4−n = o μ2α μn−2 α rα
(4.12)
if n 5. By Lemma 3.5 and (4.9)–(4.12) we then get that
(uα )i − Bα Λi 2 dvg = o μ2 ln 1 if n = 4, α μα
and
Bxα (rα )
(uα )i − Bα Λi 2 dvg = o μ2 α
Bxα (rα )
and coming back to (4.8) we get that
if n 5,
(4.13)
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2 dg (xα , x)2 ∇ (uα )i − Bα Λi (x) dvg (x)
Bxα (rα )
dg (xα , x)2 (uα )i − Bα Λi (x) g (uα )i − Bα Λi (x) dvg (x)
= Bxα (rα )
1 if n = 4, and + o μ2α ln μα = dg (xα , x)2 (uα )i − Bα Λi (x) g (uα )i − Bα Λi (x) dvg (x) Bxα (rα )
+ o μ2α if n 5.
(4.14)
Thanks to Eqs. (3.1) satisfied by the Uα ’s, and thanks to the expression of g in geodesic polar coordinates,
dg (xα , x)2 (uα )i − Bα Λi (x) g (uα )i − Bα Λi (x) dvg (x)
Bxα (rα )
2 −1
dg (xα , x)2 (uα )i − Bα Λi (x) × Uα (x)
dvg (x)
=O Bxα (rα )
dg (xα , x)2 (uα )i − Bα Λi (x) Bα (x)2 −1 dvg (x)
+O Bxα (rα )
+O
Bxα (rα )
+O
dg (xα , x)2 (uα )i − Bα Λi (x) × Uα (x) dvg (x)
dg (xα , x)3 (uα )i − Bα Λi (x) × ∇Bα (x) dvg (x) .
(4.15)
Bxα (rα ) −1
By Lemmas 3.2, 3.4 and 3.5, letting Fα = |Uα |2
−1
+ Bα2
, we can write that
dg (xα , x)2 (uα )i − Bα Λi (x) Fα (x) dvg (x)
Bxα (rα )
= Bxα (μα )
dg (xα , x)2 (uα )i − Bα Λi (x) Fα (x) dvg (x)
+
dg (xα , x)2 (uα )i − Bα Λi (x) Fα (x) dvg (x)
Bxα (rα )\Bxα (μα )
2−n . = o μ2α + o μn−2 α rα
(4.16)
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In a similar way, by Lemmas 3.4 and 3.5,
dg (xα , x)2 (uα )i − Bα Λi (x) × Uα (x) dvg (x)
Bxα (rα )
dg (xα , x)2 (uα )i − Bα Λi (x) × Uα (x) dvg (x)
= Bxα (μα )
dg (xα , x)2 (uα )i − Bα Λi (x) × Uα (x) dvg (x)
+ Bxα (rα )\Bxα (μα )
4−n , = o μ2α + O μn−2 α rα
(4.17)
and since n−2
∇Bα (x) Cμα 2 dg (xα , x)1−n ,
we also have that
dg (xα , x)3 (uα )i − Bα Λi (x) × ∇Bα (x) dvg (x)
Bxα (rα )
4−n . = o μ2α + O μn−2 α rα
(4.18)
Plugging (4.15)–(4.18) into (4.14), we get that
2 dg (xα , x)2 ∇ (uα )i − Bα Λi (x) dvg (x)
Bxα (rα )
1 2−n 2 if n = 4, = o μn−2 + o μ r ln α α α μα 2−n = o μn−2 + o μ2α if n 5. α rα
and
Noting that
2
1 if n = 4, dg (xα , x)2 ∇Bα (x) dvg (x) = O μ2α ln μα
Bxα (rα )
Bxα (rα )
and since
2
dg (xα , x)2 ∇Bα (x) dvg (x) = O μ2α if n 5,
(4.19)
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dg (xα , x)2 ∇Bα (x) × ∇(Uα − Bα Λ)(x) dvg (x)
Bxα (rα )
1 2
2
dg (xα , x) ∇Bα (x) dvg (x)
2
Bxα (rα )
1 2 2 dg (xα , x) ∇(Uα − Bα Λ)(x) dvg (x) ,
2
× Bxα (rα )
we get (4.3) by plugging (4.19) into (4.6). This ends the proof of Lemma 4.1.
2
Another lemma we need for the proof of Lemma 4.3 is as follows. Lemma 4.2. Let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, (Aα )α be a sequence in C 1 (M, Mps (R)), and (Uα )α be a sequence of nonnegative solutions of (3.1) such that (3.2) and (3.3) hold true. Let (xα )α and (ρα )α be such that (3.5) and (3.7) hold true. Let R2,α be given by R2,α =
Aα Uα , Xα (∇Uα ) Rp dvg
Bxα (rα )
n−2 + 4n
(g divg Xα )|Uα |2 dvg Bxα (rα )
n−2 + 2n
(divg Xα )Aα Uα , Uα Rp dvg ,
(4.20)
Bxα (rα )
where Uα = ((uα )1 , . . . , (uα )p ), ·,·Rp is the scalar product in Rp , Xα (∇Uα )i = (Xα , ∇(uα )i ), and Xα is as in (4.1). Then R2,α = O(μα rα )
if n = 3,
rα 1 if n = 4, + o μ2α ln μα μα R2,α = C(n)LA,Λ (x0 )μ2α + o μ2α if n 5,
R2,α = C(4)LA,Λ (x0 )μ2α ln
(4.21)
where n−2 LA,Λ (x) = A(x)Λ, Λ Rp − Sg (x), 4(n − 1) μα is as in (3.6), rα is as in (3.21), Λ is as in Lemma 3.2, C(4) = −64ω3 , C(n) = − when n 5, u0 is as in (3.68), and xα → x0 as α → +∞.
Rn
u20 dx
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Proof. By the expression of Xα ,
Xα (x) = O dg (xα , x) , divg Xα (x) = n + O dg (xα , x)2 , and n Sg (xα ) + O dg (xα , x) . g (divg Xα )(x) = n−1
(4.22)
Assume first that n = 3. By (4.22),
Aα Uα , Xα (∇Uα ) dvg = O
Bxα (rα )
Uα (x) × ∇Uα (x)dg (xα , x) dvg (x)
Bxα (rα )
and by Lemma 3.4 we get that
Aα Uα , Xα (∇Uα ) dvg
Bxα (rα )
dg (xα , x)−2 dvg (x) Cμα rα .
Cμα
(4.23)
Bxα (rα )
Similarly, it follows from (4.22) and Lemma 3.4 that
2
( div X )|U | dv g g α α g Cμα rα
and that
Bxα (rα )
(divg Xα )Aα Uα , Uα Rp dvg
Cμα rα .
(4.24)
Bxα (rα )
It follows from (4.23) and (4.24) that (4.21) holds true when n = 3. From now on we assume that n 4. We write that Aα (x) = Aα (xα ) + O dg (xα , x) . Then, by (4.22),
Aα Uα , Xα (∇Uα ) dvg
Bxα (rα )
=
p
Aαij (xα )
i,j =1
(uα )i Xα (∇Uα )j dvg
Bxα (rα )
+O Bxα (rα )
dg (xα , x) Uα (x) × ∇Uα (x) dvg (x) .
2
(4.25)
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By the Cauchy–Schwarz inequality,
dg (xα , x)2 Uα (x) × ∇Uα (x) dvg (x)
Bxα (rα )
1 2
2
dg (xα , x) Uα (x) dvg (x)
Bxα (rα )
1 2 2 dg (xα , x) ∇Uα (x) dvg (x) .
3
×
(4.26)
Bxα (rα )
By Lemmas 3.2 and 3.4,
2 dg (xα , x) Uα (x) dvg (x)
Bxα (rα )
2 dg (xα , x) Uα (x) dvg (x)
= Bxα (μα )
2 dg (xα , x) Uα (x) dvg (x)
+ Bxα (rα )\Bxα (μα )
4−n = o μ2α + O μn−2 . α rα
(4.27)
Independently, thanks to Eqs. (3.1) satisfied by the Uα ’s, integrating by parts,
2
dg (xα , x)3 ∇(uα )i (x) dvg (x)
Bxα (rα )
dg (xα , x) (uα )i (x) × ∇(uα )i (x) dσg (x)
3
=O ∂Bxα (rα )
2
dg (xα , x)2 (uα )i (x) dσg (x)
+O ∂Bxα (rα )
+O
Bxα (rα )
+O
2
dg (xα , x)3 Uα (x) dvg (x)
2 dg (xα , x) Uα (x) dvg (x)
Bxα (rα )
for all i = 1, . . . , p. By Lemma 3.4,
(4.28)
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4−n , dg (xα , x)3 Uα (x) × ∇Uα (x) dσg (x) = O μn−2 α rα
∂Bxα (rα )
and
2
4−n . dg (xα , x)2 Uα (x) dσg (x) = O μn−2 α rα
(4.29)
∂Bxα (rα )
By Lemmas 3.2 and 3.4,
2
dg (xα , x)3 Uα (x) dvg (x)
Bxα (rα )
2
dg (xα , x)3 Uα (x) dvg (x)
= Bxα (μα )
2
dg (xα , x)3 Uα (x) dvg (x)
+ Bxα (rα )\Bxα (μα )
4−n = o μ2α + O μn−2 . α rα
(4.30)
In particular, we get from (4.25)–(4.30) that
Aα Uα , Xα (∇Uα ) dvg
Bxα (rα )
p
=
4−n . (uα )i Xα (∇Uα )j dvg + o μ2α + O μn−2 α rα
Aαij (xα )
i,j =1
(4.31)
Bxα (rα )
Integrating by parts, by (4.22) and (4.27), p
Aαij (xα )
i,j =1
(uα )i Xα (∇Uα )j dvg
Bxα (rα )
=−
p n α Aij (xα ) 2 i,j =1
4−n . (uα )i (uα )j dvg + o μ2α + O μn−2 α rα
(4.32)
Bxα (rα )
By (4.22) and (4.27) we also have that
g (divg Xα ) |Uα |2 dvg
Bxα (rα )
nSg (xα ) = n−1
Bxα (rα )
4−n , |Uα |2 dvg + o μ2α + O μn−2 α rα
(4.33)
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1035
and that (divg Xα )Aα Uα , Uα Rp dvg Bxα (rα )
=n
p
4−n . (uα )i (uα )j dvg + o μ2α + O μn−2 α rα
Aαij (xα )
i,j =1
(4.34)
Bxα (rα )
By (4.31)–(4.34),
p
R2,α = −
Aαij (xα )
i,j =1
+
(uα )i (uα )j dvg
Bxα (rα )
n−2 Sg (xα ) 4(n − 1)
4−n . |Uα |2 dvg + o μ2α + O μn−2 α rα
(4.35)
Bxα (rα )
Let Sα be as in Lemma 3.5. We can write that n−2 2
μα
4−n , Bα dvg = O μn−2 α rα
rα2−n
and
Bxα (rα )
n−2
μα 2
4−n . Bα Sα dvg = o μ2α + O μn−2 α rα
(4.36)
Bxα (rα )
By (4.9) and (4.13), and by Lemma 3.5, we get (4.21) from (4.35) and (4.36). This ends the proof of Lemma 4.2. 2 Now, at this point, we can state the main result of this section. This is the subject of the following lemma. We assume (0.2) in the lemma. Lemma 4.3. Let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, (Aα )α be a sequence in C 1 (M, Mps (R)), and (Uα )α be a sequence of nonnegative solutions of (3.1) such that (0.2), (3.2) and (3.3) hold true. Let (xα )α and (ρα )α be such that (3.5) and (3.7) hold true. Assume rα → 0 as α → +∞, where rα is as in (3.21). Then ρα = O(rα ) and 1− n rαn−2 μα 2 Uα expxα (rα x) →
n−2 2
(n(n − 2)) |x|n−2
Λ
+ H(x)
(4.37)
2 (B (2) \ {0}) as α → +∞, where μ is as in (3.6), Λ is as in Lemma 3.2, and H is a in Cloc 0 α harmonic function in B0 (2) which satisfies that Λ, H(0)Rp 0 with equality if and only if H(0) = 0. Moreover, assuming n 4, it is necessarily the case that rα → 0 as α → +∞.
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Proof. Let R 6 be such that Rrα 6ρα for α 1. We assume first that rα → 0 as α → +∞. Then we set, for x ∈ B0 (3), 1− n Wα (x) = rαn−2 μα 2 Uα expxα (rα x) , gα (x) = expxα g (rα x), and A˜ α (x) = Aα expxα (rα x) . 2 (Rn ) as α → +∞, where ξ is the Since rα → 0 as α → +∞, we have that g˜ α → ξ in Cloc Euclidean metric. Thanks to Lemma 3.4 we also have that
Wα (x) C|x|2−n
(4.38)
in B0 ( R2 ) \ {0}. By (3.1), gα (wα )i + rα2
p
A˜ αij (wα )j =
j =1
μα rα
2
−2
|Wα |2
(wα )i
(4.39)
in B0 ( R2 ), for all i, where Wα = ((wα )1 , . . . , (wα )p ) and A˜ α = (A˜ αij )i,j . Thanks to (3.22) and by standard elliptic theory, we then deduce that, after passing to a subsequence, Wα → W
(4.40)
2 (B ( R ) \ {0}) as α → +∞, where W satisfies in Cloc 0 2
W = 0
(4.41)
in B0 ( R2 ) \ {0}. Moreover, thanks to (4.38), we know that
W(x) C|x|2−n
(4.42)
in B0 ( R2 ) \ {0}. Thus we can write that W(x) =
Λ˜ + H(x) |x|n−2
(4.43)
where Λ˜ ∈ Rp has nonnegative components and H satisfies H = 0 in B0 ( R2 ). In order to see that Λ˜ = (n(n − 2))(n−2)/2 Λ, it is sufficient to integrate (4.39) in B0 (1) to get that −
∂ν Wα dσgα
∂B0 (1)
=
μα rα
2
−2
|Wα |2 B0 (1)
Wα dvgα − rα2 B0 (1)
A˜ α Wα dvgα .
(4.44)
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1037
By (4.38), |Wα | dvg˜α C
(4.45)
B0 (1)
and by changing x into
μα rα x,
we can write that −2
|Wα |2
Wα dvgα = rα2 μ−2 α
|U˜α |2
−2
U˜α dvg˜α ,
B0 ( μrαα )
B0 (1)
where U˜α and g˜ α are as in (3.67). By Lemmas 3.2 and 3.4, we then get that lim
α→+∞
μα rα
2
Wα dvgα
B0 (1)
−1
u02
=
−2
|Wα |2 dx Λ
Rn
n−2 = (n − 2)ωn−1 n(n − 2) 2 Λ,
(4.46)
where u0 is as in (3.68). Noting that by (4.40) and (4.43), lim
α→+∞ ∂B0 (1)
˜ ∂ν Wα dσgα = −(n − 2)ωn−1 Λ,
(4.47)
we get that n−2 Λ˜ = n(n − 2) 2 Λ
(4.48)
thanks to (4.45)–(4.47) by passing into the limit in (4.44) as α → +∞. Now we prove that Λ, H(0) 0 and that rα → 0 if n 4. For that purpose, we let Xα be the vector field given by (4.1) and we apply the Pohozaev identity in Druet and Hebey [18] to Uα in Bxα (rα ). We get that
n−2 Aα Uα , Xα (∇Uα ) dvg + 4n
Bxα (rα )
+
n−2 2n
(g divg Xα )|Uα |2 dvg Bxα (rα )
(divg Xα )Aα Uα , Uα dvg Bxα (rα )
= Q1,α + Q2,α + Q3,α , where
(4.49)
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O. Druet et al. / Journal of Functional Analysis 258 (2010) 999–1059
n−2 Q1,α = 2n
(divg Xα )∂ν Uα , Uα dσg ∂Bxα (rα )
−
1 2 Xα (ν)|∇Uα | − Xα (∇Uα ), ∂ν Uα dσg , 2
∂Bxα (rα )
Q2,α = −
p i=1B (r ) xα α
Q3,α =
1 (∇Uα )i , (∇Uα )i dvg , ∇Xα − (divg Xα )g n
n−2 2n
Xα (ν)|Uα |2 dσg ∂Bxα (rα )
−
n−2 4n
∂ν (divg Xα ) |Uα |2 dσg ,
∂Bxα (rα )
and ν is the unit outward normal derivative to Bxα (rα ). We have that
Xα (x) = O dg (xα , x)
and ∇(divg Xα )(x) = O dg (xα , x) .
It follows that 4−n . Q3,α = O μnα rα−n + O μn−2 α rα
(4.50)
By Lemmas 4.1 and 4.2, by (4.49) and (4.50), we can write that Q1,α = O μ3α rα−3 + O(μα rα ) if n = 3, 1 rα Q1,α = C(4) A(x0 )Λ, Λ Rp − Sg (x0 ) μ2α ln 6 μα 1 2 + o μ2α rα−2 if n = 4, + o μα ln μα n−2 Sg (x0 ) μ2α Q1,α = C(n) A(x0 )Λ, Λ Rp − 4(n − 1) + o μ2α + o μn−2 if n 5, α rα 2 − n where the constants C(4) and C(n) are as in Lemma 4.2. We wrote here that 2−n μnα rα−n = μn−2 α rα
μα rα
2
2−n . = o μn−2 α rα
By Lemma 3.4, (4.22), and the expression of Q1,α , we have that 2−n . Q1,α = O μn−2 α rα
(4.51)
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1039
Thanks to (0.2), this clearly implies that rα → 0 as α → +∞ if n 4. Now, assuming that rα → 0 as α → +∞, it is easily checked thanks to (4.40), (4.42) and (4.43), that Q1,α = −
nn−2 (n − 2)n 2−n ˜ H(0) p + o(1) μn−2 ωn−1 Λ, α rα . R 2
(4.52)
Coming back to (4.51), it follows from (4.52) that nn−2 (n − 2)n ˜ H(0) p = 0 if n = 3, ωn−1 Λ, R 2 nn−2 (n − 2)n r ˜ H(0) p = −C4 (x0 ) lim rα2 ln α ωn−1 Λ, R α→+∞ 2 μα
if n = 4,
nn−2 (n − 2)n ˜ H(0) p = −Cn (x0 ) lim μα4−n rαn−2 if n 5, ωn−1 Λ, R α→+∞ 2
(4.53)
where n−2 Sg (x0 ) , Cn (x0 ) = C(n) A(x0 )Λ, Λ Rp − 4(n − 1) and C(n) is as in Lemma 4.2. Since Cn (x0 ) > 0 by (0.2), we get with (4.48) and (4.53) that
Λ, H(0) Rp 0
(4.54)
in all dimensions. In what follows we still assume that rα → 0 as α → +∞. We multiply line i of the system (3.1) by (uα )j and integrate over Bxα (rα ). We obtain that
(uα )j g (uα )i dvg + p
Bxα (rα )
Aαik (uα )j (uα )k dvg
k=1B (r ) xα α
−2
=
|Uα |2
(uα )j (uα )i dvg .
Bxα (rα )
Inverting i and j and substracting one to the other, we get that
(uα )j g (uα )i − (uα )i g (uα )j dvg
Bxα (rα )
=
p k=1B (r ) xα α
(uα )i Aαjk − (uα )j Aαik (uα )k dvg .
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Integrating by parts, this leads to
(uα )i ∂ν (uα )j − (uα )j ∂ν (uα )i dσg
∂Bxα (rα )
=
p
(uα )i Aαjk − (uα )j Aαik (uα )k dvg .
k=1B (r ) xα α
Thanks to (4.40) and (4.43), since H is harmonic in B0 (r) with r > 1, it is easily checked that
(uα )i ∂ν (uα )j − (uα )j ∂ν (uα )i dσg
∂Bxα (rα )
(Wi ∂ν Wj − Wj ∂ν Wi ) dσ + o(1)
2−n = μn−2 α rα ∂B0 (1)
2−n = (n − 2)ωn−1 Λ˜ i Hj (0) − Λ˜ j Hi (0) + o(1) μn−2 α rα .
(4.55)
Suppose first that n = 3. Then, by Lemma 3.4, (uα )i (uα )j dvg = O(μα rα ). Bxα (rα )
Suppose now that n 4. By Lemma 3.5, by (4.9), and by (4.13), with similar computations to those developed in the proof of Lemma 4.1, we have that
(uα )i (uα )j dvg = Bxα (rα )
Bα2 dvg
1 Λi Λj + o(1) + o μ2α ln μα
Bxα (rα )
rα 1 = 64ω3 Λi Λj + o(1) μ2α ln + o μ2α ln μα μα
if n = 4, while
(uα )i (uα )j dvg = Bxα (rα )
Bα2 dvg
4−n Λi Λj + o(1) + o μ2α + O μn−2 α rα
Bxα (rα )
=
u20 dx Λi Λj + o(1) μ2α
Rn
if n 5, where u0 is as in (3.68). In particular, we get that
O. Druet et al. / Journal of Functional Analysis 258 (2010) 999–1059
p
1041
(uα )i Aαjk − (uα )j Aαik (uα )k dvg
k=1B (r ) xα α
= O(μα rα ) if n = 3 rα if n = 4 = 64ω3 Aj k (x0 )Λi Λk − Aik (x0 )Λj Λk + o(1) μ2α ln μα = Aj k (x0 )Λi Λk − Aik (x0 )Λj Λk + o(1) μ2α u20 dx if n 5.
(4.56)
Rn
Assuming that Λ, H(0)Rp = 0, we get from (4.53) that lim rα2 ln
α→+∞
rα = 0 if n = 4, μα
and
lim μα4−n rαn−2 = 0 if n = 4.
α→+∞
Coming back to (4.55) and (4.56) we get that Λi Hj (0) = Λj Hi (0) for all i, j ∈ {1, . . . , p}. Multiplying by Λi and summing over i, we then deduce that Hj (0) = Λ, H(0) Λj , which proves that H(0) = 0 if Λ, H(0)Rp = 0. At this point it remains to prove that ρα = O(rα ). We still assume that rα → 0 as α → +∞ and we proceed by contradiction so that we also assume that rα →0 ρα
(4.57)
as α → ∞. Then (4.43) holds true in B0 (R) for all R. Since H is harmonic we then get from (4.43) that (n(n − 2)) R n−2
n−2 2
Λ+
1 |∂B0 (R)|
H dσ ∂B0 (R)
n−2 2
(n(n − 2)) Λ + H(0) R n−2 1 W dσ = |∂B0 (R)| =
∂B0 (R)
and hence, since W 0, and since |Λ| = 1, we get that (n(n − 2)) R n−2
n−2 2
+ Λ, H(0) Rp 0.
(4.58)
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Passing into the limit in (4.58) as R → +∞ we get that Λ, H(0)Rp 0. By (4.54) we also have that Λ, H(0)Rp 0. It follows that Λ, H(0)Rp = 0. Hence, H(0) = 0. However, since rα < ρα by (4.57), we get with (3.24) that there holds ((r (n−2)/2 ϕ(r)) (1) = 0, where
1 ϕ(r) = ωn−1 r n−1
|W|Σ dσ ∂B0 (r)
=
(n(n − 2)) r n−2
n−2 2
|Λ|Σ + H(0) Σ .
Hence,
n−2
H(0) = n(n − 2) 2 |Λ|Σ Σ and since H(0) = 0, we get a contradiction with the fact that |Λ| = 1. In particular, (4.57) is false, and thus, ρα = O(rα ). This ends the proof of the lemma. 2 5. Construction of a parametrix for g + A when n = 3 Let (M, g) be a smooth compact Riemannian manifold of dimension n = 3, p 1 be an integer, and A be a map in C 1 (M, Mps (R)). We prove the existence of a parametrix for multivalued Schrödinger operators like g + A and get a positive mass theorem for such parametrix from the positive mass theorem of Schoen and Yau [39] (see also Witten [43]). We assume here that g + A is coercive and −A is cooperative
(5.1)
and we also assume that Sg Idp 8
A<
(5.2)
in the sense of bilinear forms. Let η ∈ C ∞ (M × M), 0 η 1, be such that η(x, y) = 1 if dg (x, y) δ and η(x, y) = 0 if dg (x, y) 2δ, where δ > 0 is small. For x = y we define H (x, y) =
η(x, y) , ω2 dg (x, y)
(5.3)
where ω2 is the volume of the unit 2-sphere. The result we prove in this section is as follows. We refer to the end of the section for a remark on how to get the Green’s matrix from Proposition 5.1. Proposition 5.1. Let (M, g) be a smooth compact Riemannian 3-manifold, p 1 be an integer, and A : M → Mps (R) be a C 1 -map satisfying (5.1). Let Λ 0 be a nonnegative vector in Rp . There exists G : M × M \ D → Rp , G 0, such that for any x ∈ M, and any i = 1, . . . , p, g (Gx )i +
p j =1
Aij (Gx )j = Λi δx ,
(5.4)
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where D is the diagonal in M × M, Gx (y) = G(x, y), G = (G1 , . . . , Gp ), δx is the Dirac mass at x, and G can be written as G(x, y) = H (x, y)Λ + R(x, y)
(5.5)
for all x, y ∈ M × M \ D, where R : M × M → Rp is continuous in M × M. Moreover, there exists C > 0 such that R(x, x) CΛ for all x ∈ M if we also assume that A satisfies (5.2). In particular, R(x, x)i > 0 for at least one i if Λ ≡ 0 and (5.1)–(5.2) hold true. Proof. (i) First we construct G such that (5.4) holds true. We have that, see, for instance, Aubin [4,5],
g,y H (x, y) + H (x, y)
C dg (x, y)
(5.6)
g,y,dist. H (x, y) = δx + g,y H (x, y)
(5.7)
and
in the sense of distributions, where δx is the Dirac mass at x. We define the maps Γ1 , Γ2 : M × M → Rp by Γ1 (x, y)i = − g,y H (x, y) Λi − H (x, y) Aij (y)Λj , p
j =1
Γ2 (x, y)i = −
Γ1 (x, z)i g,y H (z, y) dvg (z) M
−
p
Aij (y)
j =1
Γ1 (x, z)j H (z, y) dvg (z),
(5.8)
M
for all (x, y) ∈ M × M \ D and all i = 1, . . . , p. By Giraud’s lemma and (5.6), Γ2 is continuous in M × M. Given x ∈ M, we let Sx : M → Rp be the solution of the linear system g (Sx )i +
p
Aij (Sx )j = (Γ2,x )i
(5.9)
j =1
for all i = 1, . . . , p, where Γ2,x (·) = Γ2 (x, ·). The existence of Sx easily follows from the variational theory and the coercivity of g + A. In particular, Sx ∈ H 2,q for all q. We define G : M × M \ D → Rp by H (z, y)Γ1 (x, z) dvg (z) + S(x, y),
G(x, y) = H (x, y)Λ + M
(5.10)
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where S(x, y) = Sx (y). By Giraud’s lemma and (5.6), (x, y) →
H (z, y)Γ1 (x, z) dvg (z)
(5.11)
M
is continuous in M × M. Let ϕ ∈ C 2 (M), x ∈ M, and i ∈ {1, . . . , p}. Thanks to (5.7)–(5.9) we get that G(x, y)i g ϕ(y) dvg (y) +
p
Aij (y)G(x, y)j ϕ(y) dvg (y) = ϕ(x)Λi ,
j =1 M
M
where G is as in (5.10). This proves (5.4). (ii) We prove (5.5) and that G 0. Let x, x ∈ M. By (5.9), g (Sx )i − (Sx )i + Aij (Sx )j − (Sx )j = (Γ2,x )i − (Γ2,x )i . p
(5.12)
j =1
Multiplying (5.12) by (Sx )i − (Sx )i , integrating over M, and summing over i, it follows that M
∇(Sx − Sx ) 2 dvg +
A(Sx − Sx , Sx − Sx ) dvg M
Γ2,x − Γ2,x L2 Sx − Sx C 0 and by the coercivity of g +A we get that Sx −Sx L2 CΓ2,x −Γ2,x L2 . Then, by standard elliptic theory, we obtain that Sx − Sx C 0 CΓ2,x − Γ2,x L2 .
(5.13)
In a similar way, we get by (5.9) that Sx L2 CΓ2,x C 0 C and then, by standard elliptic theory, we can write that Sx C 1 C. Writing that
S(x , y ) − S(x, y) S(x , y ) − S(x , y) + S(x , y) − S(x, y)
Sx − Sx C 0 + ∇Sx C 0 dg (y, y ) we get from (5.13), the above estimate on Sx , and the continuity of Γ2 , that S is continuous in M × M. Together with (5.10), and the above remark that the map in (5.11) is continuous, this proves (5.5). Now we prove that G 0. Given u : M → R a continuous function, we let u+ = max(u, 0) and u− = min(u, 0) so that u = u+ + u− . By (5.5), there exists δ > 0 such that for any i, if Λi > 0 then (Gx )− i has its support in M \ Bx (δ). On the other hand, if Λi = 0 then, by (5.4), (5.5), standard elliptic theory, and the Sobolev embedding theorem, we can write that g (Gx )i ∈ Lq for all q < 3, then that (Gx )i ∈ H 2,q for all such q, and at last that (Gx )i ∈ H 1,s
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for all s. In both cases we can multiply (5.4) by (Gx )− i , integrate over M, sum over i, and get that
∇G− 2 dvg +
x
M
− A G− x , Gx dvg +
M
− A G+ x , Gx dvg = 0.
(5.14)
M
0 − + + − Noting that for any that −A is u, v +∈ C −, u v + u v 0 in M, it follows from the fact cooperative that M A(Gx , Gx ) dvg 0. Coming back to (5.14), we get that G− x ≡ 0 for all x, and thus that G 0. (iii) We prove the last part of the proposition that there exists C > 0 such that R(x, x) CΛ for all x ∈ M if we also assume that A satisfies (5.2). By (5.2), and since g + A is coercive, S ˜ i be the Green’s the Schrödinger operators g + Aii and g + 8g are also coercive. We let G
function of g + Aii and Gg be the Green’s function of g + any i ∈ {1, . . . , p},
Sg 8 .
By (5.4), for any x ∈ M and
˜ i )x Λi + Aii (Gx )i − (G ˜ i )x Λi = − g (Gx )i − (G Aij (Gx )j 0
(5.15)
j =i
˜ i )x Λi is continuous in M. Then, by the since G 0 and −A is cooperative. By (5.5), (Gx )i − (G maximum principle, we get with (5.15) that ˜ i Λi Gi G
(5.16)
for all i. By (5.2), given i ∈ {1, . . . , p}, and x ∈ M, there exists hi > 0 smooth and such that hi
Sg − Aii (Gg )x 8
(5.17)
in M. By the coercivity of g + Aii there exists θi ∈ C 2 , θi > 0, such that g θi + Aii θi = hi .
(5.18)
Noting that by (5.17) and (5.18), ˜ i )x − (Gg )x − θi + Aii (G ˜ i )x − (Gg )x − θi 0, g (G ˜ i )x − (Gg )x − θi is continuous in M, we get that and that (G ˜ i )x (Gg )x + θi (G
(5.19)
for all i and all x. Combining (5.16) and (5.19) it follows from the positive mass theorem of Schoen and Yau [39–41] that there exists C > 0 such that R(x, x) CΛ for all x ∈ M. This ends the proof of Proposition 5.1. 2
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Fix x ∈ M. As a remark there holds that there exists C > 0 such that
dg (x, y) ∇Rx (y) C
(5.20)
for all y ∈ M \ {x}, where Rx (y) = R(x, y). By (5.4) and (5.6) we get that there exists C > 0 such that dg (x, y)|g Rx (y)| C for all y ∈ M \ {x}. In order to get (5.20) it suffices to prove that for any sequence (yα )α in M \ {x} such that yα → x as α → +∞,
dg (x, yα ) ∇Rx (yα ) = O(1).
(5.21)
Let sα = dg (x, yα ) and set Rα (y) = Rx (expx (sα y)). Let also gα be given by gα (y) = (expx g)(sα y), and y˜α be such that yα = expx (sα y˜α ). We can write that |gα Rα (y)| Csα |y|−1 1 (R3 ) since s → 0 as α → +∞. Moreover while Rα is bounded and gα → ξ as α → +∞ in Cloc α |y˜α | = 1 for all α. Let y0 be such that y˜α → y0 as α → +∞. Since |y0 | = 1, we can write by the above estimates and standard elliptic theory that Rα is bounded in the C 1 -topology in the Euclidean ball of center y0 and radius 1/4. This proves (5.21) and thus (5.20). It also follows from the proof that sα
max ∇Rx (y) = o(1)
y∈∂Bx (sα )
(5.22)
for all sequences (sα )α of positive real numbers such that sα → 0 as α → +∞. Indeed, there holds that gα Rα → 0 uniformly in compact subsets of R3 \ {0} as α → +∞. Hence Rα → R 1 (R3 \ {0}), where R is a bounded harmonic map in R3 \ {0}. By Liouville’s theorem we in Cloc get that R is constant and (5.22) follows. Given j ∈ {1, . . . , p}, let Λj ∈ Rp be defined by (Λj )i = δij for all i = 1, . . . , p, where the δij ’s are the Kronecker symbols. Also let Gj be the parametrix given by Proposition 5.1 when Λ = Λj and G = (Gij )i,j be the matrix given by Gij = (Gj )i for all i, j = 1, . . . , p. Then
g,y
p
Giα (x, y)fα (x) dvg (x) +
α=1 M
p α,j =1
Gj α (x, y)fα (x) dvg (x) = fi (y)
Aij (y) M
for all f ∈ C ∞ (M, Rp ), all i ∈ {1, . . . , p}, and all y ∈ M. In other words, G is the Green’s matrix of g + A. 6. Proof of the theorem We prove our theorem in what follows. Let (M, g) be a smooth compact Riemannian manifold of dimension n 3, p 1 be an integer, (Aα )α be a sequence in C 1 (M, Mps (R)), and (Uα )α be a sequence nonnegative solutions of (3.1) such that (0.2), (3.2) and (3.3) hold true. As a preliminary remark we claim that there exists C > 0 such that for any α the following holds true. Namely that there exist Nα ∈ N and Nα critical points of |Uα |, denoted by (x1,α , x2,α , . . . , xNα ,α ), such that dg (xi,α , xj,α )
n−2 2
Uα (xi,α ) 1
(6.1)
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for all i, j ∈ {1, . . . , Nα }, i = j , and min
i=1,...,Nα
n−2
2
Uα (x) C dg (xi,α , x)
(6.2)
for all x ∈ M and all α. We prove (6.1) and (6.2). Clearly |Uα |Σ satisfies the maximum principle since, summing the equations in (3.1), g |Uα |Σ + pAα ∞ |Uα |Σ 0, where |Uα |Σ is given by (3.4). Hence, |Uα |Σ > 0 and we also get that |Uα | > 0 in M. In particular, we can use Lemma 1.1 of Druet and Hebey [19] and we get the existence of Nα ∈ N and of (x1,α , x2,α , . . . , xNα ,α ) a family of critical points of |Uα | such that (6.1) holds true for all i, j ∈ {1, . . . , Nα }, i = j , and min
i=1,...,Nα
n−2
2
Uα (x) 1 dg (xi,α , x)
(6.3)
for all critical points of |Uα |. We claim now that there exists C > 0 such that (6.2) holds true for all x ∈ M and all α. We proceed by contradiction and assume that min
n−2
2
Uα (xα ) → +∞ dg (xi,α , xα )
min
n−2
2
Uα (xα )
dg (xi,α , xα )
i=1,...,Nα
(6.4)
as α → +∞, where i=1,...,Nα
= sup M
min
i=1,...,Nα
n−2
2
Uα (x) . dg (xi,α , x)
(6.5)
1− n
We set |Uα (xα )| = μα 2 . Thanks to (6.4) and (6.5), since M is compact so that the distance between two points in M is always bounded, μα → 0 as α → +∞. We let Sα be the above set of critical points xi,α of |Uα |. By (6.4), dg (xα , Sα ) → +∞ μα as α → +∞. We set, for x ∈ Ωα = B0 ( μδα ), where 0 < δ < 12 ig is fixed, n−2 Vα (x) = μα 2 Uα expxα (μα x) , gα (x) = expxα g (μα x), and A˜ α (x) = Aα expxα (μα x) .
(6.6)
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2 (Rn ) as α → +∞, where ξ is the Euclidean metric, since μ → 0 We have that gα → ξ in Cloc α as α → +∞. Thanks to (3.1),
gα (vα )i + μ2α
p
A˜ αij (vα )j = |Vα |2 −2 (vα )i
(6.7)
j =1
in Ωα , for all i, where Vα = ((vα )1 , . . . , (vα )p ), and A˜ α = (A˜ αij )i,j . We have that |Vα (0)| = 1 and also that, thanks to (6.5) and (6.6), for any R > 0, lim sup sup |Vα | = 1.
(6.8)
α→+∞ B0 (R)
Indeed, for any x ∈ Bxα (Rμα ), for any i = 1, . . . , Nα , dg (xi,α , x) dg (xi,α , xα ) − Rμα dg (xα , Sα ) − Rμα Rμα . dg (xα , Sα ) 1 − dg (xα , Sα ) By standard elliptic theory we then get by (6.7) that, after passing to a subsequence, Vα → U
(6.9)
1 (Rn ) as α → +∞, where U has nonnegative components and satisfies in Cloc −2
U = |U|2
U
in Rn with |U| |U(0)| = 1. It follows from Proposition 1.1 that U = 1+
|x|2 n(n − 2)
1− n 2
Λ
for some Λ ∈ Rp with nonnegative components such that |Λ| = 1. In particular, |U| has a strict local maximum at 0 which proves that |Uα | has a local maximum, and hence a critical point, (n−2)/2 |Uα (yα )| → 1 as α → +∞. This clearly violates (6.3) yα with dg (xα , yα ) = o(μα ) and μα thanks to (6.6) since for any i = 1, . . . , Nα , dg (xi,α , yα ) dg (xi,α , xα ) − dg (xα , yα ) dg (xα , Sα ) + o(μα ) Rα μα + o(μα ) = Rα μα 1 + o(1) where Rα → +∞ as α → +∞ by (6.6). Thus we have contradicted (6.4). This concludes the proof of (6.1) and (6.2).
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Now we consider the family (x1,α , . . . , xNα ,α ) given by (6.1) and (6.2) and we define dα by dα =
min
1i<j Nα
(6.10)
dg (xi,α , xj,α ).
If Nα = 1, we set dα = 14 ig , where ig is the injectivity radius of (M, g). We claim that dα 0
(6.11)
as α → +∞. In order to prove this claim, we proceed by contradiction. Assuming on the contrary that dα → 0 as α → +∞, we see that Nα 2 for α large, and we can thus assume that the concentration points are ordered in such a way that dα = dg (x1,α , x2,α ) dg (x1,α , x3,α ) · · · dg (x1,α , xNα ,α ).
(6.12)
We set, for x ∈ B0 (δdα−1 ), 0 < δ < 12 ig fixed, n−2 Uˆα (x) = dα 2 Uα expx1,α (dα x) , Aˆ α (x) = Aα expx1,α (dα x) , and gˆ α (x) = expx1,α g (dα x).
2 (Rn ) as α → +∞ since d → 0 as α → +∞. Thanks to (3.1) we It is clear that gˆ α → ξ in Cloc α have that
gˆα (uˆ α )i + dα2
p
Aˆ αij (uˆ α )j = |Uˆα |2 −2 (uˆ α )i
(6.13)
j =1
in B0 (δdα−1 ), for all i, where Uˆα = ((uˆ α )1 , . . . , (uˆ α )p ), and Aˆ α = (Aˆ αij )i,j . For any R > 0, we also let 1 NR,α Nα be such that dg (x1,α , xi,α ) Rdα
for 1 i NR,α ,
dg (x1,α , xi,α ) > Rdα
for NR,α + 1 i Nα .
and
Such an NR,α does exist thanks to (6.12). We also have that NR,α 2 for all R > 1 and that (NR,α )α is uniformly bounded for all R > 0 thanks to (6.10). Indeed, suppose there are kα points xi,α , i = 1, . . . , kα , such that dg (x1,α , xi,α ) Rdα for all i = 1, . . . , kα . By (6.10), Bxi,α
dα 2
∩ Bxj,α
dα 2
=∅
for all i = j . Then, kα 3R dα Volg Bx1,α dα Volg Bx1,α 2 2 i=1
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and we get an upper bound for kα depending only on R. In the sequel, we set xˆi,α = dα−1 exp−1 x1,α (xi,α ) for all 1 i Nα such that dg (x1,α , xi,α ) 12 ig . Thanks to (6.2), for any R > 1, there exists CR > 0 such that |Uˆα | CR .
sup
N2R,α
B0 (R)\
i=1
(6.14)
Bxˆi,α ( R1 )
Mimicking the proof of Lemma 3.1, one easily gets that, for any R > 1, there exists DR > 1 such that ∇ Uˆα L∞ (ΩR,α ) DR sup |Uˆα | DR2 inf |Uˆα | ΩR,α
ΩR,α
(6.15)
where N2R,α
ΩR,α = B0 (R) \
i=1
1 . Bxˆi,α R
Assume first that, for some R > 0, there exists 1 i NR,α such that
Uˆα (xˆi,α ) = O(1).
(6.16)
Since (3.5) is satisfied by the sequences xα = xi,α and ρα = 18 dα , it follows from Lemma 3.2 that (3.7) cannot hold and thus that (|Uˆα |)α is uniformly bounded in Bxˆi,α ( 34 ). In particular, by standard elliptic theory, and thanks to (6.13), (Uˆα )α is uniformly bounded in C 1 (Bxˆi,α ( 12 )). Since, by (6.1), we have that |xˆi,α |
n−2 2
Uˆα (xˆi,α ) 1,
we get the existence of some δi > 0 such that n n 1 1 |Uˆα | |xˆi,α |1− 2 R 1− 2 2 2
in Bxˆi,α (δi ). Assume now that, for some R > 0, there exists 1 i NR,α such that
Uˆα (xˆi,α ) → +∞
(6.17)
as α → +∞. Since (3.5) and (3.7) are satisfied by the sequences xα = xi,α and ρα = 18 dα , it follows from Lemma 4.3 that the sequence (|Uˆα (xˆi,α )| × |Uˆα |)α is uniformly bounded in Ωˆ α = Bxˆi,α (δ˜i ) \ Bxˆi,α
δ˜i 2
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for some δ˜i > 0. Thus, using (6.15), we can deduce that these two situations are mutually exclusive in the sense that either (6.16) holds true for all i or (6.17) holds true for all i. Now we split the conclusion of the proof into two cases. In the first case we assume that there exist R > 0 and 1 i NR,α such that |Uˆα (xˆi,α )| = O(1). Then, thanks to the above discussion, we get that
Uˆα (xˆj,α ) = O(1) for all 1 j NR,α and all R > 0. Now, as above, we get that (|Uˆα |)α is uniformly bounded in 1 (Rn ). Thus, by standard elliptic theory, there exists a subsequence of (Uˆ ) which converges Cloc α α 1 (Rn ) to some Uˆ solution of in Cloc Uˆ = |Uˆ |2 −2 Uˆ
in Rn . Still thanks to the above discussion, we know that U ≡ 0 and has nonnegative components. Moreover, |U| possesses at least two critical points, namely 0 and xˆ2 , the limit of xˆ2,α . This is absurd thanks to the classification of Proposition 1.1. In the second case we assume that there exist R > 0 and 1 i NR,α such that |Uˆα (xˆi,α )| → +∞ as α → +∞. Then, thanks to the above discussion,
Uˆα (xˆj,α ) → +∞ as α → +∞, for all 1 j NR,α and all R > 0. By (6.13) we have that gˆα (vˆα )i + dα2
p
Aˆ αij (vˆα )j =
j =1
1 |Vˆ α |2 −2 (vˆα )i , −2 2 ˆ |Uα (0)|
where Vˆ α = |Uˆα (0)|Uˆα and Vˆ α = ((vˆα )1 , . . . , (vˆα )p ). Applying Lemma 4.3 and standard elliptic theory, and thanks to (6.15) and to the above discussion, one easily checks that, after passing to a subsequence,
Uˆα (0) Uˆα → G ˆ 1 (Rn \ {xˆ } in Cloc i i∈I ) as α → +∞, where
I = 1, . . . , lim
lim NR,α
R→+∞ α→+∞
and, for any R > 0,
ˆ G(x) =
N˜ R i=1
Λ˜ i + Hˆ R (x) |x − xˆi |n−2
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in B0 (R), where 2 N˜ R N2R is such that |xˆN˜ R | R and |xˆN˜ R +1 | > R, and where N2R,α → N2R as α → +∞. In this expression, the Λ˜ i ’s are nonzero vectors with nonnegative components and Hˆ R is a harmonic function in B0 (R). We have that N˜
Hˆ R1 (x) − Hˆ R2 (x) =
R2
i=N˜ R1 +1
Λ˜ i |x − xˆi |n−2
for all 0 < R1 < R2 . We can write that Λ˜ 1 ˆ G(x) = n−2 + X(x) |x| in B0 ( 12 ) where, for any R > 1,
X(x) =
N˜ R i=2
Λ˜ i + Hˆ R (x). |x − xˆi |n−2
ˆ = (G ˆ 1, . . . , G ˆ p ), X = (X1 , . . . , Xp ), and Λ˜ 1 = ((Λ˜ 1 )1 , . . . , (Λ˜ 1 )p ). We have that G ˆi 0 Let G 2−n ˜ for all for all 1 i p. Hence, by the maximum principle, we get that Xi (0) −(Λ1 )i R R > 1, so that Xi (0) 0 for all 1 i p. By Lemma 4.3 we now have that Λ˜ 1 , X(0)Rp 0 with equality if and only if X(0) = 0. Since all the components of X(0) and of Λ˜ 1 are nonnegative, we are actually in the case of equality so that X(0) = 0. Let i be such that (Λ˜ 2 )i > 0. By the maximum principle, Xi (0) (Λ˜ 2 )i −
(Λ˜ 1 )i (Λ˜ 2 )i − . R n−2 (R − 1)n−2
Choosing R 1 sufficiently large we get that Xi (0) > 0 and this is in contradiction with X(0) = 0. By the above discussion we get that (6.11) holds true. Clearly, this implies that (Nα )α is uniformly bounded. Now we let (xα )α be a sequence of maximal points of |Uα |. Thanks to (3.3) and to (6.11), we clearly have that (3.5) and (3.7) hold for the sequences (xα )α and ρα = δ for some δ > 0 fixed. This clearly contradicts Lemma 4.3 in dimensions n 4 and thus concludes the proof of the theorem in dimensions n 4. Suppose now that n = 3. In addition to (0.2), (3.2) and (3.3) we assume that g + A is coercive and that −A is cooperative. Up to a subsequence, since (Nα )α is bounded, there holds that Nα = N for all α. Let xi = lim xi,α α→+∞
(6.18)
for all i = 1, . . . , N . Let also μi,α be given by (3.6) with xi,α instead of xα . By the above discussion, μi,α → 0 for all i = 1, . . . , N . Up to a subsequence we can assume that μ1,α = maxi μi,α
O. Druet et al. / Journal of Functional Analysis 258 (2010) 999–1059
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for all α. Still up to a subsequence we define μi 0 by μi,α . α→+∞ μ1,α
μi = lim
(6.19)
By Lemma 3.4, there exist C, δ > 0 such that
Uα (x) Cμ1/2 dg (xi,α , x)−1
(6.20)
i,α
in Bxi,α (2δ) for all i. By (6.20) and Harnack’s inequality we thus get that 1/2
|Uα | Cμ1,α in M \
N
i=1 Bxi,α (δ).
(6.21)
−1/2 Let U˜α be given by U˜α = μ1,α Uα . Then
g (u˜ α )i +
p
Aαij (x)(u˜ α )j = μ21,α |U˜α |2
−2
(u˜ α )i
(6.22)
j =1
for all i, where the (u˜ α )i ’s are the components of U˜α . By (6.21), (6.22), and standard elliptic theory, we then get that, up to a subsequence, −1/2
μ1,α Uα → Z
(6.23)
1 (M \ S) as α → +∞, where S is the finite set consisting of the x ’s defined in (6.18). Let in Cloc i Φ ∈ C ∞ (M, Rp ) be given. By (3.1),
Uα , (g Φ + AΦ) dvg =
M
|Uα | Uα , Φ dvg + o
|Uα | dvg .
4
M
(6.24)
M
For any R > 0,
N Φ(xi ), |Uα | Uα , Φ dvg =
|Uα | Uα dvg
4
M
i=1
4
Bxi,α (Rμi,α )
+o
N
|Uα | dvg 5
i=1B xi,α (Rμi,α )
|Uα |4 Uα , Φ dvg .
+ M\
N
i=1 Bxi,α (Rμi,α )
(6.25)
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O. Druet et al. / Journal of Functional Analysis 258 (2010) 999–1059
By Lemma 3.2,
lim
−1/2
lim μ1,α
R→+∞ α→+∞
N
|Uα |4 Uα dvg
Φ(xi ),
i=1
Bxi,α (Rμi,α )
N √ 1/2 = 3ω2 μi Λi , Φ(xi ) ,
(6.26)
i=1 p−1
where the Λi ’s are vectors in S+
given by (3.15), and
−1/2 μ1,α
N
|Uα |5 dvg C
(6.27)
i=1B xi,α (Rμi,α )
for some C > 0 independent of α and R. By (6.20) and (6.21) we can also write that lim
−1/2
|Uα |4 Uα , Φ dvg = 0
lim μ1,α
R→+∞ α→+∞
(6.28)
N
M\
i=1 Bxi,α (Rμi,α )
and that
1/2 |Uα | dvg = O μ1,α .
(6.29)
M
Plugging (6.25)–(6.29) into (6.24) it follows that −1/2
μ1,α
N √ 1/2 Uα , (g Φ + AΦ) dvg = 3ω2 μi Λi , Φ(xi ) + o(1).
(6.30)
i=1
M
Since Φ ∈ C ∞ (M, Rp ) is arbitrary, it follows from (6.23) and (6.30) that g Z + AZ =
N √ 1/2 3ω2 μi Λi δxi .
(6.31)
i=1
Since g + A is coercive, by Proposition 5.1 and (6.31), there holds that Z(x) =
√
3ω2
N
1/2
μi
H (xi , x)Λi + Ri (xi , x) ,
(6.32)
i=1
where H is as in (5.3), and Ri is a continuous function in M × M such that Ri (xi , xi ) CΛi for some C > 0. Let i = 1, . . . , N be arbitrary and Xα be the vector field given by Xα = ∇fα ,
O. Druet et al. / Journal of Functional Analysis 258 (2010) 999–1059
1055
where fα (x) = 12 dg (xi,α x)2 . We apply the Pohozaev identity in Druet and Hebey [18] to Uα in Bxi,α (r) for r > 0 small. We get that
1 Aα Uα , Xα (∇Uα ) dvg + 12
Bxi,α (r)
+
1 6
(g divg Xα )|Uα |2 dvg Bxi,α (r)
(divg Xα )Aα Uα , Uα dvg = Q1,α + Q2,α + Q3,α ,
(6.33)
Bxi,α (r)
where Xα (∇Uα ) is as in Lemma 4.2, Q1,α =
1 6
(divg Xα )∂ν Uα , Uα dσg ∂Bxi,α (r)
−
1 2 Xα (ν)|∇Uα | − Xα (∇Uα ), ∂ν Uα dσg , 2
∂Bxi,α (r)
Q2,α = −
p j =1B
Q3,α =
1 6
1 ∇Xα − (divg Xα )g 3
xi,α (r)
Xα (ν)|Uα |2 dσg − ∂Bxi,α (r)
(∇Uα )j , (∇Uα )j dvg ,
1 12
∂ν (divg Xα ) |Uα |2 dσg ,
∂Bxi,α (r)
and ν is the unit outward normal derivative to Bxi,α (r). By (6.23), lim μ−1 1,α (Q1,α + Q3,α )
α→+∞
1 = 6
(divg X)∂ν Z, Z dσg ∂Bxi (r)
−
1 X(ν)|∇Z|2 − X(∇Z), ∂ν Z dσg 2
∂Bxi (r)
−
1 12
∂ν (divg X) |Z|2 dσg ,
∂Bxi (r)
where X = ∇f and f (x) = 12 dg (xi , x)2 . We have that divg X = 3 + O dg (xi , x)2
and |∇ divg X| = O dg (xi , x)
(6.34)
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while, by (6.20), there also holds that |Z| Cdg (xi , x)−1 in a neighbourhood of xi . From (5.20) we have in addition that dg (xi , x)|∇Ri (x)| C for all x = xi . It follows that lim
r→0 ∂Bxi (r)
∂ν (divg X) |Z|2 dσg = 0
(6.35)
and that
1 6
(divg X)∂ν Z, Z dσg = ∂Bxi (r)
1 2
∂ν Z, Z dσg + o(1)
(6.36)
∂Bxi (r)
as r → 0. We choose δ > 0 in the definition of η in (5.3) such that dg (xj , xk ) 4δ for all j, k = 1, . . . , N such that xj = xk . Since the parametrix in Proposition 5.1 are nonnegative, it follows from our choice of δ that Rj (xj , xi ) 0 for all j = i. In a neighbourhood of xi we get from (6.32) that Z(x) =
N √ √ 1/2 1/2 3dg (xi , x)−1 μi Λi + 3ω2 μj Rj (xj , x).
(6.37)
j =1
By (5.22) and (6.37) we compute 1 2
∂ν Z, Z dσg − ∂Bxi (r)
1 X(ν)|∇Z|2 − X(∇Z), ∂ν Z dσg 2
∂Bxi (r)
N 3ω2 1/2 1/2 =− μj Rj (xj , xi ) + o(1). μi Λ i , 2
(6.38)
j =1
Combining (6.36) and (6.38) it follows that 1 6
(divg X)∂ν Z, Z dσg ∂Bxi (r)
−
1 X(ν)|∇Z|2 − X(∇Z), ∂ν Z dσg 2
∂Bxi (r)
N 3ω2 1/2 =− Rj (xj , xi ) + o(1). μi Λ i , 2 j =1
Noting that 1 (∇Xα )μν − (divg Xα )gμν = O dg (xi,α , x)2 3
(6.39)
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1057
for all i and all μ, ν, we can write with Lemma 3.4 that |Q2,α | Cμ1,α r. It follows that lim lim μ−1 1,α Q2,α = 0.
(6.40)
r→0 α→+∞
Still by Lemma 3.4, we also have that lim lim μ−1 1,α
r→0 α→+∞
Aα Uα , Xα (∇Uα ) dvg = 0,
Bxi,α (r)
lim lim μ−1 1,α
(g divg Xα )|Uα |2 dvg = 0,
r→0 α→+∞
and
Bxi,α (r)
lim lim μ−1 1,α
(divg Xα )Aα Uα , Uα dvg = 0.
r→0 α→+∞
(6.41)
Bxi,α (r)
Multiplying (6.33) by μ−1 1,α , passing to the limit as α → +∞, and then as r → 0, we get with (6.34), (6.35), (6.39), (6.40) and (6.41), that 1/2 μi Λi ,
N
1/2 μj Rj (xj , xi )
=0
(6.42)
j =1
for all i. We fix i = 1. Then μ1 = 1. As already mentioned, according to our choice of δ in the definition of η in (5.3), we get that Rj (xj , x1 ) 0 for all j = 1. By Proposition 5.1 we also have that R1 (x1 , x1 ) CΛ1 for some C > 0. Since the Λj ’s are nonnegative vectors, it follows that Λ1 , ω2
N
1/2 μj Rj (xj , x1 )
Λ1 , R1 (x1 , x1 ) C|Λ1 |2
(6.43)
j =1
and we get a contradiction by combining (6.42) and (6.43) since the Λi ’s are nonzero vectors. This concludes the proof of the theorem when n = 3. References [1] M.J. Ablowitz, B. Prinari, A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, Cambridge, 2004. [2] N. Akhmediev, A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett. 82 (1998) 2661– 2664. [3] T. Aubin, Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976) 269–296. [4] T. Aubin, Nonlinear Analysis on Manifolds. Monge–Ampère Equations, Grundlehren Math. Wiss., vol. 252, 1982. [5] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monogr. Math., Springer, 1998. [6] S. Brendle, Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc. 21 (2008) 951–979. [7] S. Brendle, On the conformal scalar curvature equation and related problems, Surv. Differ. Geom. XII (2008) 1–19. [8] S. Brendle, F.C. Marques, Blow-up phenomena for the Yamabe equation II, J. Differential Geom. 81 (2009) 225– 250.
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[9] J.P. Burke, J.L. Bohn, B.D. Esry, C.H. Greene, Hartree–Fock theory for double condensates, Phys. Rev. Lett. 78 (1997) 3594–3597. [10] L.A. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989) 271–297. [11] W. Chen, C. Li, A necessary and sufficient condition for the Nirenberg problem, Comm. Pure Appl. Math. 48 (1995) 657–667. [12] K.S. Chou, C.W. Chu, On the best constant for a weighted Sobolev–Hardy inequality, J. London Math. Soc. 48 (1993) 137–151. [13] D.N. Christodoulides, T.H. Coskun, M. Mitchell, M. Segev, Theory of incoherent self-focusing in biased photorefractive media, Phys. Rev. Lett. 78 (1997) 646–649. [14] O. Druet, From one bubble to several bubbles: The low-dimensional case, J. Differential Geom. 63 (2003) 399–473. [15] O. Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not. 23 (2004) 1143–1191. [16] O. Druet, E. Hebey, Blow-up examples for second order elliptic PDEs of critical Sobolev growth, Trans. Amer. Math. Soc. 357 (2004) 1915–1929. [17] O. Druet, E. Hebey, Elliptic equations of Yamabe type, Int. Math. Res. Surv. 1 (2005) 1–113. [18] O. Druet, E. Hebey, Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, preprint, 2008. [19] O. Druet, E. Hebey, Stability and instability for Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds, Math. Z., in press. [20] O. Druet, E. Hebey, F. Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, Math. Notes, vol. 45, Princeton University Press, 2004. [21] B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981) 883–901. [22] Q. Han, F. Lin, Elliptic Partial Differential Equations, CIMS Lecture Notes, vol. 1, Courant Institute of Mathematical Sciences, 1997; second edition published by the American Mathematical Society. [23] E. Hebey, Critical elliptic systems in potential form, Adv. Differential Equations 11 (2006) 511–600. [24] F.T. Hioe, Solitary waves for N coupled nonlinear Schrödinger equations, Phys. Rev. Lett. 82 (1999) 1152–1155. [25] F.T. Hioe, T.S. Salter, Special set and solutions of coupled nonlinear Schrödinger equations, J. Phys. A 35 (2002) 8913–8928. [26] T. Kanna, M. Lakshmanan, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations, Phys. Rev. Lett. 86 (2001) 5043–5046. [27] M. Khuri, F.C. Marques, R. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom. 81 (2009) 143–196. [28] Y.Y. Li, L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math. 90 (2003) 27–87. [29] Y.Y. Li, L. Zhang, A Harnack type inequality for the Yamabe equations in low dimensions, Calc. Var. Partial Differential Equations 20 (2004) 133–151. [30] Y.Y. Li, L. Zhang, Compactness of solutions to the Yamabe problem II, Calc. Var. Partial Differential Equations 24 (2005) 185–237. [31] Y.Y. Li, M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995) 383–417. [32] Y.Y. Li, M. Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math. 1 (1999) 1–50. [33] F.C. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom. 71 (2005) 315–346. [34] P. Padilla, Symmetry properties of positive solutions of elliptic equations on symmetric domains, Appl. Anal. 64 (1997) 153–169. [35] M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, 1967. [36] F. Robert, Green’s functions estimates for elliptic type operators, personal notes, 2006. [37] R.M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in: Topics in Calculus of Variations, Montecatini Terme, 1987, in: Lecture Notes in Math., vol. 1365, SpringerVerlag, Berlin, 1989, pp. 120–154. [38] R.M. Schoen, On the number of constant scalar curvature metrics in a conformal class, in: Differential Geometry: A Symposium in Honor of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro, 1988, in: Pitman Monogr. Surv. Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 311–320. [39] R.M. Schoen, S.T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979) 45–76.
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[40] R.M. Schoen, S.T. Yau, Proof of the positive action conjecture in quantum relativity, Phys. Rev. Lett. 42 (1979) 547–548. [41] R.M. Schoen, S.T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988) 47–71. [42] J. Vétois, Multiple solutions for nonlinear elliptic equations on compact Riemannian manifolds, Internat. J. Math. 18 (2007) 1071–1111. [43] E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981) 381–402.
Journal of Functional Analysis 258 (2010) 1060–1065 www.elsevier.com/locate/jfa
Corrigendum
Corrigendum to “Semiclassical non-concentration near hyperbolic orbits” [J. Funct. Anal. 246 (2) (2007) 145–195] Hans Christianson 1 Department of Mathematics, University of California, Berkeley, CA 94720, USA Received 2 June 2009; accepted 2 June 2009 Available online 24 June 2009 Communicated by Paul Malliavin
1. The corrected statements This erratum fixes several mistakes in the named paper. The statement and proof of Proposition 6.2 is incorrect, however the statement of Theorem 1 is correct. The first assertion of Theorem 2 is correct, however we give a modified version below. There is a typo in the statement of the Main Theorem, however the statement about quasimodes is correct. Specifically, Eq. (1.1) should be replaced with u C
log(1/ h) P (h)u + C log(1/ h)(I − A)u, h
that is, (log 1/ h)1/2 in the first term on the right-hand side should be replaced by log(1/ h). We will give a modified, global version of Theorem 1 below which we then use to correct the proof of Theorem 2 and the Main Theorem. As a consequence of the mistake in the proof of Proposition 6.2, the application to the damped wave equation in Theorem 5 is also incorrect. In this erratum, we fix these mistakes (see [1] for an in-depth discussion of the optimality of the corrected Theorem 5 below). We begin by stating the correct results, and then indicate where the mistakes in the proof occur, and how to remedy them. First let us recall the notation used in [3]. We are ultimately interested in a semiclassical pseudodifferential operator P (h) on a comDOI of original article: 10.1016/j.jfa.2006.09.012. E-mail address: [email protected]. 1 Present address: Massachusetts Institute of Technology, Department of Mathematics, 77 Mass. Ave., Cambridge, MA 02139-4307, USA. 0022-1236/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.003
H. Christianson / Journal of Functional Analysis 258 (2010) 1060–1065
1061
pact manifold X whose classical flow generates a closed hyperbolic orbit γ in the energy level {p = 0} ⊂ T ∗ X where p is the principal symbol of P (h). In [3] we define the perturbed family of operators Q(z) = P (h) − z − iha w , for a ∈ C ∞ (T ∗ X). The mistake in this work comes from the fact that the complex absorbing potential ha w is not large enough for our purposes. In fact, the size necessary is h log(1/ h), which is no longer “lower order”. Hence in this erratum, for simplicity, we take an absorbing potential of size 1, and define = P (h) − z − iW w , Q(z) where W ≡ 0 in a small neighbourhood of γ and W ≡ 1 away from γ . Theorem 1. There exist constants c0 > 0, h0 > 0, and N0 such that for u ∈ L2 (X), z ∈ [−1, 1] + i(−c0 h, +∞), and 0 < h < h0 we have Q(z)u
L2 (X)
C −1 hN0 uL2 (X)
(1.1)
for some constant C, provided W w is elliptic outside a sufficiently small neighbourhood of γ . With this theorem and the semiclassical adaptation of the “three-lines” theorem from complex analysis, we obtain the following theorem, which is the same as Theorem 2 in [3] with Q(z) replaced with Q(z). is as above, and z ∈ [−1/2, 1/2]. Then there is h0 > 0 and Theorem 2. Suppose Q(z) 0 < C < ∞ such that for 0 < h < h0 , Q(z) −1
L2 (X)→L2 (X)
C
log(1/ h) . h
(1.2)
If ϕ ∈ Cc∞ (X) is supported away from γ , then log(1/ h) . C L2 (X)→L2 (X) h
Q(z) −1 ϕ
(1.3)
The point is that with Theorem 2 and the control theory arguments from [2] we recover the first estimate in Theorem 2 from [3], and the same estimate then clearly holds in the complex neighbourhood z ∈ [−1/2, 1/2] + i −c0 h/ log(1/ h), c0 h/ log(1/ h) , by perturbation. From this, and the rather abstract Theorem 3 in [4] we can correct the statement of Theorem 5 in [3] from exponential decay to sub-exponential decay: Theorem 5. Suppose u solves the damped wave equation on X with initial data (0, f ) and damping a 0. Assume a controls X geometrically outside a small neighbourhood of γ . Then for any > 0, there is a constant C > 0 such that ut 2L2 (X) + ∇u2L2 (X) Ce−t
1/2 /C
f 2H .
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H. Christianson / Journal of Functional Analysis 258 (2010) 1060–1065
Fig. 1. Various cutoffs.
2. The corrected proofs In this section we begin by proving Theorems 1 and 2 above. The idea of the proof of Theorem 1 in [3] is to conjugate Q(z) by microlocally defined weights and cut off to a neighbourhood by weights and cutting off, or for that matter of γ . Of course this is the same as conjugating Q(z) P (h) − z, since the perturbations are all supported away from γ . For Theorem 1 , we conju by globally defined weights, which have the cutoff built in, and then use the complex gate Q(z) absorbing term iW w to control the interactions. The first step is to observe that the quantum normal form for this operator is only microlocal in a neighbourhood of γ . Let Vt be the effective Hamiltonian for a symplectic deformation family κt for the symplectic transformation κ constructed in Proposition 4.3. Observe Vt is defined only locally near γ , say in a neighbourhood of size 4 > 0. Let ψ0 ∈ C ∞ (T ∗ X) be equal to 1 in a 2 t = ψ0 Vt , which is now defined neighbourhood and 0 outside the 4 neighbourhood, and let V globally on T ∗ X. We can now find a globally defined symplectic deformation family κt such that κ0 = id, κ1 = κ near γ and id away, and an h-FIO F quantizing κ1 similar to Proposition 3.2, t . For the remainder of this section, we now using the globally defined effective Hamiltonian V assume P (h) has been conjugated by F so that the principal symbol p takes the form of that in Proposition 4.3 near γ . Now assume W ≡ 1 outside a neighbourhood of size /2 of γ , and let ψ1 ∈ C ∞ (T ∗ X) be 1 on an neighbourhood of γ and 0 outside the set where ψ0 = 1. Choose also χ1 , χ2 ∈ C ∞ (T ∗ X) so that χ1 ≡ 1 on {W = 1} with support in {ψ1 = 1} and χ1 ≡ 1 on supp ψ1 with support in {ψ0 = 1} (see Fig. 1 for a schematic drawing). Let Th,h˜ be the rescaling operator defined in (2.6) 1 (z) be the rescaled operator in [3], and let Q w −1 1 (z) = T ˜ Q(z)χ Q 2 T ˜. h,h h,h
w = Let G be as defined in Section 5 in [3] (defined in the rescaled coordinates), and let G Th,h˜ ψ1w T −1˜ Gw . We define the conjugated operators similar to before with s > 0 sufficiently h,h small, only now everything is defined globally: w w 2 (s, z) = e−s G Q1 (z)es G , Q
H. Christianson / Journal of Functional Analysis 258 (2010) 1060–1065
1063
˜ where now the w superscript indicates the h-Weyl quantization. Now as in (5.18), on the set where ψ1 ≡ 1, we have for Im z −c0 h for some c0 > 0, ˜ 2 (s, z)U, U hh U 2 . − Im Q C
(2.1)
On {ψ1 = 1}, we have 2 (s, z)U, U C h log(1/ h)U 2 , Q h˜
so if W ≡ 1 outside, say, an /2 neighbourhood of γ , the estimate (2.1) holds everywhere. More precisely, there is a zero-order pseudodifferential operator Γ w which is bounded below by hh˜ so that 2 (s, z)U, U = Γ w T ˜ χ1w T −1 + T ˜ W w χ2w T −1 U, U . − Im Q h,h h,h ˜ ˜ h,h
h,h
w w is equal to zero outside a compact set, the operators e±s G Now, since G are identity outside a compact set, and hence the rescaled operators w
T −1˜ e±s G Th,h˜ h,h
are globally defined. But then w w −1 s G Th,h˜ u, u − Im T −1˜ e−s G Th,h˜ Q(z)T ˜e h,h
= − Im
h,h
w w w −1 s G T ˜e Th,h˜ u, u T −1˜ e−s G Th,h˜ Q(z)χ 2 h,h h,h
w w − Im T −1˜ e−s G Th,h˜ Q(z) 1 − χ2w T −1˜ es G Th,h˜ u, u h,h h,h −1 w w −1 w w −1 = T ˜ Γ Th,h˜ χ1 T ˜ + Th,h˜ W χ2 T ˜ Th,h˜ u, u h,h
h,h
h,h
w w T −1˜ e−s G Th,h˜ W w 1 − χ2w T −1˜ es G Th,h˜ u, u h,h h,h
+ = T −1˜ Γ w Th,h˜ χ1w + W w u, u h,h
hh˜ u2 . C Conjugating back and using the fact that all operators used are tempered yields Theorem 1 . Theorem 2 then follows immediately using [3, Lemma 6.3]. To get the Main Theorem from [3], Theorem 2 from [3], and indeed the improvement to a neighbourhood | Im z| c0 h/ log(1/ h) which gives Theorem 5 , we now use the control theory arguments from [2]. That is, if B1 ∈ Ψ 0 , B1 ≡ 1 near γ , has wavefront set sufficiently close to γ , ϕ ∈ Ψ 0 has wavefront set away from gamma and ϕ ≡ 1 on supp ∇B1 , and a ∈ C ∞ (T ∗ X) controls T ∗ X geometrically outside a neighbourhood of γ , we have for z ∈ [−1/2, 1/2]
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∞ −1 Q(z)B u B1 u C Q(z) 1u + O h
−1 B1 Q(z) + Q(z), B1 u + O h∞ u = C Q(z)
−1 ϕ Q(z), B1 u + O h∞ u −1 B1 Q(z)u + C Q(z) C Q(z) log(1/ h) Q(z)u + C log1/2 (1/ h)ϕu C ˜ + O h∞ u. h
(2.2)
Here we have used that Q(z)B 1 = Q(z)B1 , WF[Q(z), B1 ] is away from γ , Q(z) is elliptic away from {p = 0} ⊃ γ , and we take ϕ˜ ∈ Ψ 0 satisfying ϕ˜ ≡ 1 on WF[Q(z), B1 ] but WF ϕ˜ ∩ γ = ∅. Next, we assume a w = A∗ A for some non-negative definite A ∈ Ψ 0 , so that Au2 = au, u, and [3, Lemma 6.1] implies (1 − B1 )u C Q(z)u + CAu + O h∞ u. h Then, again using [3, Lemma 6.1] on the term with ϕu ˜ in (2.2) all told we have the estimate u C
log(1/ h) Q(z)u + C log1/2 (1/ h)Au + O h∞ u. h
Since Q(z)B 1 u = Q(z)B1 u = P (h) − z B1 u modulo O(h∞ )u, the preceding calculations hold true with P − z replacing Q(z), which is the Main Theorem from [3]. Finally, to get the first estimate in [3, Theorem 2] (and the improvement to a complex neighbourhood), we calculate for z ∈ [−1/2, 1/2], Au2 = au, u 1 = Im Q(z)u, u h 1 Q(z)uu, h so that we have for any > 0, 1/2 1 log1/2 (1/ h)Au log1/2 (1/ h) Q(z)uu h log(1/ h) Q(z)u + u. 2h Combining this with (2.2) and taking > 0 sufficiently small yields Theorem 2 from [3]. The improvement to | Im z| c0 h/ log(1/ h) follows from taking c0 > 0 sufficiently small, since then the order of the perturbation is the same order as the estimate.
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References [1] Nicolas Burq, Hans Christiansen, Imperfect geometric control and overdamping for the damped wave equation, in preparation. [2] Nicolas Burq, Maciej Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17 (2) (2004) 443–471 (electronic). [3] Hans Christianson, Semiclassical non-concentration near hyperbolic orbits, J. Fund. Anal. 246 (2) (2007) 145–195. [4] Hans Christianson, Applications of cutof resolvent estimates to the wave equation, Math. Res. Lett., in press.
Journal of Functional Analysis 258 (2010) 1066–1067 www.elsevier.com/locate/jfa
Corrigendum
Corrigendum to “Carathéodory interpolation on the non-commutative polydisk” [J. Funct. Anal. 229 (2) (2005) 241–276] Dmitry S. Kaliuzhnyi-Verbovetskyi Department of Mathematics, Drexel University, 3141 Chestnut St., Philadelphia, PA 19104, United States Received 2 June 2009; accepted 2 June 2009 Available online 16 June 2009 Communicated by G. Pisier
In the proof of the main result, Theorem 4.11, the argument for the case where c∅ > 0 was using an implicit assumption that the operator c∅ has a bounded inverse. Thus, unfortunately, the possibility that λ = 0 is a point of the (continuous) spectrum of c∅ was not covered. The following argument should replace the one given in the paper (the second paragraph on page 269). Consider the case where c∅ > 0. Let S be the N -tuple of operators on the finite-dimensional Hilbert space HΛ as defined in Example 4.6. Since S consists of Λ-jointly nilpotent contractions, Re p(S) 0 by the assumption on the non-commutative polynomial p. Let w ∈ Λ \ {∅}, y1 ∈ Y, y2 ∈ Y be arbitrary. Then, identifying the words in Λ with the corresponding orthonormal basis vectors in HΛ , we have 0 2 Re p(S)(y1 ⊗ ∅ + y2 ⊗ w), y1 ⊗ ∅ + y2 ⊗ w = 2 Re p(S)(y1 ⊗ ∅ + y2 ⊗ w), y1 ⊗ ∅ + y2 ⊗ w c∅ ⊗ IHΛ v + = 2 Re cv ⊗ S (y1 ⊗ ∅ + y2 ⊗ w), y1 ⊗ ∅ + y2 ⊗ w 2 v∈Λ\∅
= (c∅ ⊗ IHΛ )(y1 ⊗ ∅), y1 ⊗ ∅ + 2 Re cw ⊗ Sw (y2 ⊗ w), y1 ⊗ ∅ + (c∅ ⊗ IHΛ )(y2 ⊗ w), y2 ⊗ w
= c∅ y1 , y1 + 2 Recw y2 , y1 + c∅ y2 , y2 . DOI of original article: 10.1016/j.jfa.2005.03.007. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.004
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c∅ cw In other words, for every w ∈ Λ \ ∅ the operator block matrix cw∗ c∅ is positive semidefinite. By the classical result of Schur (see, e.g., Theorem XVI.1.1 in [2] or Lemma 1.2 in [1]) there 1/2 1/2 exist contractions tw ∈ L(Y) such that cw = c∅ tw c∅ , w ∈ Λ \ ∅. Define the non-commutative
I polynomial q(z) := 2Y + w∈Λ\∅ tw zw . Since for every N -tuple T of Λ-jointly nilpotent contractions on a Hilbert space H one has
1/2
1/2 Re p(T) = c∅ ⊗ IH Re q(T) c∅ ⊗ IH 0 1/2
and the operator c∅ has a dense range in Y, we obtain that Re q(T) 0. As we have shown above, Problem 4.2 for the data tw , w ∈ Λ, is solvable, that is, there exists g ∈ HAnc N (Y) such that 1/2 1/2 IY nc g∅ = 2 and gw = tw , w ∈ Λ \ {∅}. Then f (z) := c∅ g(z)c∅ ∈ HAN (Y) solves Problem 4.1 for the data cw , w ∈ Λ. Note that the remainder of the proof of Theorem 4.11 (the general case c∅ 0) now only needs the replacement of ran c∅ by its closure, ran c∅ . Acknowledgment The author is thankful to Mark Nudelman for pointing out the gap in the proof of Theorem 4.11. References [1] M.A. Dritschel, On factorization of trigonometric polynomials, Integral Equations Operator Theory 49 (1) (2004) 11–42. [2] C. Foia¸s, A.E. Frazho, The Commutant Lifting Approach to Interpolation Problems, Oper. Theory Adv. Appl., vol. 44, Birkhäuser-Verlag, Basel, 1990, xxiv+632 pp.
Journal of Functional Analysis 258 (2010) 1068–1069 www.elsevier.com/locate/jfa
Corrigendum
Corrigendum to “E0-dilation of strongly commuting CP0-semigroups” [J. Funct. Anal. 255 (2008) 46–89] Orr Moshe Shalit Department of Mathematics, Technion, Haifa 32000, Israel Received 2 June 2009; accepted 2 June 2009 Available online 13 June 2009 Communicated by D. Voiculescu
In the paper “E0 -dilation of strongly commuting CP0 -semigroups” [1], I proved the existence of an E0 -dilation to every pair of commuting CP0 -semigroups, under the assumption that these semigroups satisfy an additional assumption of strong commutativity. The proof given there contains a gap. The gap, a way to fill it, and some consequences will be outlined below. As explained in [1, Lemma 4.3], one of the equivalent ways to define strong commutativity of CP maps is as follows: Definition 1. Two CP maps R and S on a von Neumann algebra M are said to commute strongly if there is an isomorphism of W∗ -correspondences v : M ⊗R M ⊗S M → M ⊗S M ⊗R M such that v(I ⊗ I ⊗ I ) = I ⊗ I ⊗ I . Two semigroups R = {Rt }t0 and S = {St }t0 acting on M are then said to commute strongly if Rs and St commute strongly for each s, t 0. It turns out that this definition of strong commutativity is not enough to make the proof of [1, Proposition 4.9] go through. The problem in that proof is precisely the passage from Eq. (14) to Eq. (15). All the results in the paper that depend Proposition 4.9 – and the main result concerning the E0 -dilation of a strongly commuting pair of CP0 -semigroups (Theorem 6.6) is one of these results – are therefore not adequately established. DOI of original article: 10.1016/j.jfa.2008.04.003. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.002
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However, if one replaces the above definition of strong commutativity with a stronger one, then almost all of the results become valid with the proofs unchanged. The new definition of strong commutativity that makes the proofs and most results of [1] hold is the following. Definition 2. Two semigroups R = {Rt }t0 and S = {St }t0 acting on M are said to commute strongly if there is a family {vs,t }s,t0 of isomorphisms vs,t : M ⊗Rs M ⊗St M → M ⊗St M ⊗Rs M with vs,t (I ⊗ I ⊗ I ) = I ⊗ I ⊗ I , such that for all s, s , t, t 0 the following diagrams commute M ⊗Rs+s M ⊗St M ⏐ ⏐
vs+s ,t
−−−−→
M ⊗St M ⊗Rs+s M ⏐ ⏐
(vs,t ⊗I )(I ⊗vs ,t )
M ⊗Rs M ⊗Rs M ⊗St M −−−−−−−−−→ M ⊗St M ⊗Rs M ⊗Rs M and M ⊗Rs M ⊗St+t M ⏐ ⏐
vs,t+t
−−−−→
M ⊗St+t M ⊗Rs M ⏐ ⏐
(I ⊗vs,t )(vs,t ⊗I )
M ⊗Rs M ⊗St M ⊗St M −−−−−−−−−→ M ⊗St M ⊗St M ⊗Rs M where the vertical arrows are given by a ⊗Rs+s b ⊗St c → a ⊗Rs I ⊗Rs b ⊗St c, etc. To put it shortly, the new definition of strong commutativity requires that for all s, t 0 the maps Rs and St strongly commute, and that this “pointwise” strong commutativity is compatible with the semigroup structure. If Definition 2 is taken as the definition of strongly commuting semigroups, then all the results and proofs in [1] are correct, except the following: – The examples of strongly commuting semigroups given in the appendix are shown to satisfy only the old definition of strong commutativity. In particular, it is not established that all commuting CP-semigroups on Mn (C) strongly commute. – As a consequence of the last sentence, Corollary 6.7 does not follow from Theorem 6.6, and should therefore be erased (until somebody proves it). References [1] O.M. Shalit, E0 -dilation of strongly commuting CP0 -semigroups, J. Funct. Anal. 255 (1) (2008) 46–89.
Journal of Functional Analysis 258 (2010) 1070–1072 www.elsevier.com/locate/jfa
Addendum
Addendum to “Contractive projections on Banach algebras” [J. Funct. Anal. 254 (10) (2008) 2513–2533] ✩ Anthony To-Ming Lau a , Richard J. Loy b,∗ a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada b Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
Received 12 June 2009; accepted 3 July 2009 Available online 18 July 2009 Communicated by N. Kalton
Abstract The purpose of this addendum is to clarify two shortcomings in the paper of the title. © 2008 Elsevier Inc. All rights reserved. Keywords: Banach algebra; Conditional expectation; Norm one projection; Locally compact group; Group algebra
It has been pointed out in [1] that there are difficulties with two of our original arguments in [3]. Firstly, in the proof of Theorem 3.1, take E to be a subset X of non-zero finite measure. Let f = 1E , so f ∗ ∈ L1 (X) by hypothesis. Set h = f ∗ f . Then h is a bounded right uniformly continuous function, lying in L1 (X) by hypothesis. Further, for x ∈ G, h(x) = λ(E ∩ Ex −1 ), so take U = {x ∈ G: h(x) > 0} and argue as before. Secondly, the appeal to [4] and [5] in Theorem 5.1 to get an expansion for the projection satisfying (i) and (ii) overlooks the fact that these results preclude the case p = 2. In the case p = 2, the orthogonal projection onto B certainly satisfies (ii) for any choice of an orthonormal basis (yn ) in B. But (i) need not hold. ✩ We thank Matthew Daws for kindly communicating to us his concerns before the review, and the referee for providing the elegant proof of Proposition 0.4. DOI of original article: 10.1016/j.jfa.2008.02.008. * Corresponding author. E-mail addresses: [email protected] (A.T.-M. Lau), [email protected] (R.J. Loy).
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.07.007
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Also, the line of inequalities just before Eq. (5.1) shows equality holds in Hölder’s inequality, so that a|yn |p = b|yn∗ |q or some constants a, b, which gives the supports are the same. The remark about singleton supports is spurious and should be ignored. As we show below, Theorem 5.1 remains true in the case p = 2. We need some properties of closed subalgebras of 2 (Λ). The following gives much more than we require, but is of some interest in its own right. 0.1. Proposition. Let 1 p < ∞, and let B be a closed subalgebra of p (Λ). Define M to the collection of subsets S of Λ minimal subject to the condition that 1S ∈ B. Then J = {1S : S ∈ M} is a family of orthogonal idempotents with span dense in B, and {T : T ∈ M} . B = x ∈ p (Λ): x is constant on each S ∈ M and zero off In particular, any closed ideal in B is the kernel of its hull. Proof. Let x ∈ B be non-zero. Since x ∈ p , |x| attains its maximum on some finite set Sx ; by scaling assume this maximum is 1. Take ε > 0, and consider {xλ : λ ∈ Sx } as a subset of the unit circle. By [2, Theorem 26.14] there is a non-zero integer k such that |xλk − 1| < ε for each λ ∈ Sx . Since |xλ−k − 1| = |xλk − 1| < ε, we may suppose that k > 0. It follows that there is a sequence (kj ) in N such that k xλj
→
0, 1,
λ∈ / Sx , λ ∈ Sx ,
as j → ∞. This convergence is dominated by |x|, and so it follows that 1Sx ∈ B. Thus M is a well-defined collection of finite subsets of Λ, and each Sx is the union of finitely many elements of M. Set J = {1S : S ∈ M}. By minimality, S ∩ S = ∅ for S, S ∈ M,S = S , so that distinct elements of J have disjoint supq port. Take x ∈ B non-zero. Then Sx = j =1 S (j ) for some S (1) , . . . , S (q) ∈ M. Take λ1 ∈ S (1) and set y = x1S (1) − xλ1 1S (1) ∈ B. If x in not constant on S (1) , then y ∈ B is non-zero, with Sy = S (1) \ {λ1 }, contradicting the minimality of S (1) . It follows that x is constant on each of the sets S (j ) . Thus by subtracting an element of span J , x becomes zero on its previous maximum set. Repeating this process, it follows that x lies in the closure of span J . Thus J is a family of orthogonal idempotents with span dense in B. In fact, for x ∈ B, setting αS (x) to be the (constant) value x takes on the set S ∈ M, we have x=
αS (x)1S ,
S∈M
the series being absolutely convergent in norm. The final statements of the proposition are now clear (in the case of an ideal, the elements of M must be singletons). 2 0.2. Corollary. Any closed subalgebra of p (Λ) is closed under conjugation.
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0.3. Corollary. Let B be a closed subalgebra of 2 (Λ). Then there is an orthonormal basis (eν ) of B such that each supp eν is finite, and for ν = μ, supp eμ ∩ supp eν = ∅. Proof. Normalizing the elements of J gives the required orthonormal basis.
2
In the case that Λ is countable, then J is countable, and Corollary 0.3 gives an expression for the orthogonal projection onto B which satisfies the conditions (i) and (ii) of Theorem 5.1. The argument is completed by the following elementary result that must be well known, but we have been unable to find a proof in the literature. 0.4. Proposition. Let K be a closed subspace of a Hilbert space H . Then the only norm one projection of H onto K is the orthogonal one. Proof. Let Q : H → K be a norm one projection. Let x ∈ (ker Q)⊥ , so that x, x − Qx = 0. That is, x 2 = x, Qx x 2 so that x = Qx by Cauchy–Schwarz inequality. Thus (ker Q)⊥ ⊆ K, and hence K ⊆ ker Q. It follows immediately that Q is the orthogonal projection onto K, as required. 2 References [1] M.D. Daws, Math. Rev. 2406685, 2009. [2] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis I, Grundlehren Math. Wiss., vol. 115, Springer-Verlag, New York, 1979. [3] A.T.-M. Lau, R.J. Loy, Contractive projections on Banach algebras, J. Funct. Anal. 254 (2008) 2513–2533. [4] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I and II, Classics Math., Springer-Verlag, Berlin, 1996. [5] B. Randrianantoanina, 1-complemented subspaces of spaces with 1-unconditional bases, Canad. J. Math. 49 (1997) 1242–1264.
Journal of Functional Analysis 258 (2010) 1073–1120 www.elsevier.com/locate/jfa
A noncommutative extended de Finetti theorem Claus Köstler ∗ University of Illinois at Urbana-Champaign, Department of Mathematics, Altgeld Hall, 1409 West Green Street, Urbana, 61801, USA Received 25 June 2008; accepted 27 October 2009 Available online 5 November 2009 Communicated by D. Voiculescu
Abstract The extended de Finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is a noncommutative version of this theorem. In contrast to the classical result of Ryll-Nardzewski, exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa’s notion of commuting squares and Kümmerer’s generalized Bernoulli shifts. Our main result is applicable to classical probability, quantum probability, in particular free probability, braid group representations and Jones subfactors. © 2009 Elsevier Inc. All rights reserved. Keywords: Noncommutative de Finetti theorem; Distributional symmetries; Exchangeability; Spreadability; Noncommutative conditional independence; Mean ergodic theorem; Noncommutative Kolmogorov zero–one law; Noncommutative Bernoulli shifts
Contents 0. 1. 2.
Introduction and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074 Preliminaries and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 Noncommutative stationary processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083
* Address for correspondence: Aberystwyth University, Institute of Mathematics and Physics, Aberystwyth, Ceredigion, SY23 3BZ, Wales, United Kingdom. E-mail address: [email protected].
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3. 4. 5. 6. 7. 8. 9.
Conditional independence and conditional factorizability . . . . . . . . . . . . Stationarity and conditional independence/factorizability . . . . . . . . . . . . Noncommutative i.i.d. sequences may be non-stationary . . . . . . . . . . . . Stationarity with strong mixing and noncommutative Bernoulli shifts . . . . Spreadability implies conditional order independence . . . . . . . . . . . . . . Spreadability implies conditional full independence . . . . . . . . . . . . . . . . Some applications and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. A glimpse on braidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. The prototype of a noncommutative conditioned central limit law . 9.3. Noncommutative Lp -inequalities for spreadable random sequences Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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0. Introduction and main result The characterization of random objects with distributional symmetries is of major interest in modern probability theory and Kallenberg’s recent monograph [35] provides an impressive account on this subject. Already in the early 1930s, de Finetti showed that infinite exchangeable random sequences are conditionally i.i.d. or, more intuitively formulated, mixtures of i.i.d. random variables [18,13]. An early version of his celebrated characterization is that for every infinite sequence of exchangeable {0, 1}-valued random variables X ≡ (X1 , X2 , X3 , . . .), there exists a probability measure ν on [0, 1] such that the law L(X) is given by L(X) =
m(p) dν(p).
[0,1]
Here m(p) denotes the infinite product of the measure with Bernoulli distribution (p, 1 − p). An extension of this result from the set {0, 1} to any compact Hausdorff space Ω goes back to Hewitt and Savage [26] and soon after it was realized by Ryll-Nardzewski [62] that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability for infinite random sequences. A further extension to standard Borel spaces is provided by Aldous in his monograph on exchangeability [6]. Let us mention that ‘spreadability’ is also known as ‘contractability’ in probability theory and shares common ground with ‘subsymmetric sequences’ in Banach space theory. Our main result is a noncommutative version of the following extended de Finetti theorem. We have adapted its formulation in [35, Theorem 1.1] to the purposes of this paper: Theorem 0.1. Let X ≡ (Xn )n∈N : (Ω, Σ, μ) → (Ω0 , Σ0 ) be a sequence of random variables, where (Ω, Σ) and (Ω0 , Σ0 ) are standard Borel spaces and μ is a probability measure. Then the following conditions are equivalent: (a) X is exchangeable; (b) X is spreadable; (c) X is conditionally i.i.d.
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Here the conditioning is with respect to the tail σ -field of the random sequence X. Three different proofs of this result can be found in [35] and it is worthwhile to point out that the two implications (a) ⇒ (b) and (c) ⇒ (a) are fairly clear; the main work rests on proving the implication (b) ⇒ (c). An early noncommutative version of de Finetti’s theorem was given by Størmer for exchangeable states on the infinite tensor product of C∗ -algebras [69]. His pioneering work stimulated further results in quantum statistical physics and quantum probability, with focus on bosonic systems [30,29,17]. A quite general noncommutative analogue of de Finetti’s theorem is obtained by Accardi and Lu in a C∗ -algebraic setting, where only the tail algebra (generated by the exchangeable infinite noncommutative random sequence) is required to be commutative [5]. Quite recently, inspired by Good’s formula and Speicher’s free cumulants [67], a combinatorial approach by Lehner unifies cumulant techniques in a ∗-algebraic setting of exchangeability systems [50,49,51,52]. He shows that exchangeability entails properties of cumulants, as they are known in classical probability to be characterizing for (conditional) independence. Presently, no results on noncommutative versions of de Finetti’s theorem seem to be available in the literature beyond the case of commutative tail algebras and Lehner’s combinatorial results for exchangeability systems; and no results at all are present in the noncommutative realm for the extended de Finetti theorem, Theorem 0.1. Our framework towards a noncommutative version of the extended de Finetti theorem needs to be capable to efficiently handle tail events. This suggests to deal right from the beginning with W∗ -algebraic probability spaces. We will work with noncommutative probability spaces (M, ψ) which consist of a von Neumann algebra M (with separable predual) and a faithful normal state ψ on M. A noncommutative random variable ι from (A0 , ϕ0 ) to (M, ψ) is given by an injective ∗-homomorphism ι : A0 → M such that ϕ0 = ψ ◦ ι. Furthermore, we require that the ψ -preserving conditional expectation from M onto ι(A0 ) exists (see Section 1 for further details). Here we will constrain our investigations to an infinite sequence I of identically distributed random variables ι ≡ (ιn )n∈N0 from (A0 , ϕ0 ) to (M, ψ), called for brevity: random sequence I . The assumption of identical distributions improves the transparency of our approach, since it allows us to realize ι as injective mappings from the single probability space (A0 , ϕ0 ). A treatment beyond identically distributed random variables is possible and of course of interest; it would start with a probability space (M, ψ) and a sequence of (not necessarily injective) normal ∗homomorphisms from a von Neumann algebra A into M. Since the distributional symmetries considered herein will lead anyway to stationarity (which implies identical distributions), we omit this primary technical generalization. We recall that M is of the form L∞ (Ω, Σ, μ) forsome standard Borel space (Ω, Σ, μ) as soon as M is commutative; and then one has ψ = Ω · dμ. In this case a random variable X : (Ω, Σ, μ) → (Ω0 , Σ0 ) reappears as an injective ∗-homomorphism ι : L∞ (Ω0 , Σ0 , μX ) → L∞ (Ω, Σ, μ) with ι(f ) := f ◦ X (the measure μX is the distribution of X). Given a sequence of random variables (Xn )n∈N0 , the constraint of identically distributed Xn ’s ensures that we can identify all image measures μXn with the single measure μX0 . Throughout we will work with a noncommutative notion of conditional independence which, by our main result, can actually be seen to emerge out of the transfer of the extended de Finetti theorem to noncommutative probability. It encompasses Popa’s notion of ‘commuting squares’ in subfactor theory [58,22] as well as Voiculescu’s freeness with amalgamation [76], aside of tensor independence and many other examples coming from generalized Gaussian random variables [10,24].
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vonNeumann Consider the random sequence I which generates the subalgebras MI := tail := ι (A ) for subsets I ⊂ N and the tail algebra M 0 0 i∈I i n∈N0 kn ιk (A0 ). Let EMtail tail denote the ψ -preserving conditional expectation from M onto M . Then we say that I is full Mtail -independent if EMtail (xy) = EMtail (x)EMtail (y) for all x ∈ Mtail ∨ MI and y ∈ Mtail ∨ MJ with I ∩ J = ∅. We will also meet a weaker notion of independence, called order Mtail -independence, which requires the (ordered) sets I and J to satisfy I < J or I > J , instead of disjointness. These two notions of conditional independence do not require Mtail ⊂ MI and allow a noncommutative dual formulation of random measures as they are necessary in the context of de Finetti’s theorem. Interesting on its own, the paradigm of an infinite sequence X of exchangeable {0, 1}-valued random variables clearly illustrates that, in its algebraic reformulation, stipulating the inclusion Mtail ⊂ MI implies the triviality Mtail C and thus forces X to be i.i.d. Thus it is crucial to allow Mtail ⊂ MI if one is interested in transferring results on distributional symmetries to a noncommutative setting. In order to formulate our main result, we informally introduce the relevant distributional symmetries. Given the two random sequences I and I with random variables ι resp. ι˜, both from (A0 , ϕ0 ) to (M, ψ), we write distr
(ι0 , ι1 , ι2 , . . .) = (˜ι0 , ι˜1 , ι˜2 , . . .) if I and I have the same distribution: ψ ιi(1) (a1 )ιi(2) (a2 ) · · · ιi(n) (an ) = ψ ι˜i(1) (a1 )˜ιi(2) (a2 ) · · · ι˜i(n) (an ) for all n-tuples i : {1, 2, . . . , n} → N0 (a1 , . . . , an ) ∈ An0 and n ∈ N. Now a random sequence I is said to be exchangeable if its distribution is invariant under permutations: distr
(ι0 , ι1 , ι2 , . . .) = (ιπ(0) , ιπ(1) , ιπ(2) , . . .) for any finite permutation π ∈ S∞ of N0 . We say that a random sequence I is spreadable if every subsequence has the same distribution: distr
(ι0 , ι1 , ι2 , . . .) = (ιn0 , ιn1 , ιn2 , . . .) for any (strictly increasing) subsequence (n0 , n1 , n2 , . . .) of (0, 1, 2, . . .). Finally, I is stationary if the distribution is shift-invariant: distr
(ι0 , ι1 , ι2 , . . .) = (ιk , ιk+1 , ιk+2 , . . .) for all k ∈ N. We are ready to state our main result, a noncommutative dual version of Theorem 0.1.
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Theorem 0.2. Let I be a random sequence with (identically distributed) random variables ι ≡ (ιi )i∈N0 : (A0 , ϕ0 ) → (M, ψ) and consider the following conditions: (a) I is exchangeable; (b) I is spreadable; (c) I is stationary and full Mtail -independent; (d) I is full Mtail -independent; (co ) I is stationary and order Mtail -independent; (do ) I is order Mtail -independent. Then we have the implications: (a)
⇒
(b)
⇒
(c) ⇓
⇒
(d) ⇓
(co )
⇒
(do ).
Moreover, there are examples such that (a) (b) (c) (d) and (co ) (do ). Similar to the classical case, the hard part of the proof is that spreadability implies conditional full independence. This is done by means from noncommutative ergodic theory. One might object that a noncommutative version of the extended de Finetti theorem should provide an equivalence of these conditions. But our investigations show that such a common folklore understanding would be conceptually misleading in the noncommutative world. The crucial implications from distributional symmetries to conditional (full/order) independence are still valid. All listed converse implications fail due to deep structural reasons, and the others are presently open in the generality of our setting. The failure of the implication ‘(b) ⇒ (a)’ relies on the fact that, in the noncommutative realm, spreadability of infinite random sequences goes beyond the representation theory of the symmetric group. As developed in [21], braid group representations with infinitely many generators lead to braidability, a new symmetry intermediate to exchangeability and spreadability. This braidability extends exchangeability and provides a rich source of spreadable noncommutative random sequences such that the reverse implication ‘(b) ⇒ (a)’ fails. Some of these ‘counter-examples’ are known in subfactor theory as vertex models from quantum statistical physics. The inequivalence of exchangeability and spreadability is a familiar phenomenon for random arrays [35]. Since this phenomenon already occurs for infinite sequences in the noncommutative setting, it provides another facet of the common folklore result that (d + 1)-dimensional classical models correspond to d-dimensional quantum models [16]. Examples for the failure of the implication ‘(c) ⇒ (b)’ are also available in the context of braid group representations. It is shown in [21] that an appropriate cocycle perturbation of the unilateral shift of a stationary random sequence may obstruct spreadability without effecting the structure of conditional full independence. Again, related ‘counter-examples’ arise in the most natural manner. For example, the symbolic shift on the Artin generators of the braid group B∞ induces an endomorphism of the braid group von Neumann algebra L(B∞ ) such that, when
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acting on the subalgebra L(B2 ), the resulting stationary random sequence exhibits the failure of ‘(c) ⇒ (b)’. Such phenomena are impossible in the classical case by Theorem 0.1. Finally, one cannot expect in the noncommutative realm that i.i.d. random sequences are automatically stationary. The failure of the implication ‘(d) ⇒ (c)’, and thus of ‘(do ) ⇒ (co )’, is closely related to the fact that our notion of noncommutative conditional independence is more general than (conditioned versions of) tensor independence or free independence. The latter two notions of independence enjoy universality properties [66,8,53] which immediately entail stationarity of an i.i.d. random sequence. In particular, they are rigid with respect to certain ‘local perturbations’ of noncommutative random sequences. But we will see that, starting with a stationary (conditionally full/order) independent random sequence, our more general notion of independence is non-rigid with respect to such ‘local perturbations’. Related examples arise again in the context of braid group representations or noncommutative Gaussian random variables. Thus stationarity plays are more distinguished role in the quantum setting and cannot simply be deduced from independence properties as it is the case for classical probability or Voiculescu’s free probability. A closer look at Theorem 0.2 reveals that it is ‘dual’ to the usual formulations of de Finetti’s theorem. In terms of quantum physics, our theorem is formulated in the Heisenberg picture, whereas the usual formulations use the Schrödinger picture. Or equivalently phrased: our result is on the level of the von Neumann algebra, whereas the latter identify the geometry of exchangeable states in the predual of the von Neumann algebra. Using the theory of noncommutative L1 -spaces it would be of interest to examine in detail the geometry of exchangeable, spreadable or ‘conditionally independent’ subspaces. We summarize the content of this paper. Section 1 introduces our setting of noncommutative probability spaces, random sequences and distributional symmetries. It closes with the proof of some of the elementary implications of Theorem 0.2. Section 2 provides the needed background results on noncommutative stationary processes and their endomorphisms. Since spreadability immediately implies stationarity, most parts of the proof of Theorem 0.2 will be carried out in an equivalent framework of stationary processes. We introduce in Section 3 two noncommutative versions of classical conditional independence, called ‘conditional independence’ (CI) and ‘conditional factorizability’ (CF). Both notions are equivalent if the conditioning is trivial or appropriate additional algebraic structure is supposed. But (CF) is a priori weaker than (CI) and more easily to control in applications. Their definition reflects that the conditioning is with respect to a von Neumann algebra which may not be contained in the image of two random variables. Further we relate ‘conditional independence’ to Popa’s ‘commuting squares’ of von Neumann algebras. The main result of Section 4 is that (CI) and (CF) are equivalent for a stationary random sequence if the conditioning is with respect to a subalgebra of the fixed point algebra of the corresponding endomorphism, see Theorem 4.2. Moreover, we introduce the notions of ‘conditional order independence’ (CIo ) and ‘conditional order factorizability’ (CFo ). These two notions are apparently weaker and reflect that the index set N0 of the random sequence is considered as an ordered set. Already (CFo ), the weakest of the four properties, will suffice to establish mixing properties of stationary processes. Finally, we illustrate (CI) and (CF) by the algebraic reformulation of de Finetti’s original example, an infinite sequence of exchangeable {0, 1}-valued random variables. Section 5 focuses on appropriate ‘local perturbations’ of C-independent stationary random sequences. Our main result is that a noncommutative i.i.d. random sequence may be non-stationary.
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We provide related examples and disprove the implications ‘(d) ⇒ (c)’ and ‘(do ) ⇒ (co )’ of Theorem 0.2. Section 6 provides a noncommutative generalization of Kolmogorov’s zero–one-law for a random sequence with (CFo ). Further we show in Theorem 6.4 that (CFo ) and stationarity imply strong mixing over the tail algebra and fixed point characterization results. We coin in this section also the notion of a noncommutative Bernoulli shift, as it is suggested by our results on distributional symmetries and inspired by Kümmerer’s notion of a generalized Bernoulli shift. These shifts can be recognized as the unilateral ‘discrete’ version of noncommutative continuous Bernoulli shifts from [25]. Section 7 is devoted to an integral part of the noncommutative extended de Finetti theorem, the proof that spreadability of a random sequence yields conditional order independence (CIo ). Here the conditioning is shown to be with respect to the tail algebra of the random sequence. Section 8 upgrades the results of the previous section. Our main result is Theorem 8.1 which provides the proof of the crucial part of Theorem 0.2: spreadability implies conditional full independence (CI) of a random sequence. An important tool within its proof is a local version of the mean ergodic theorem, Theorem 8.4. It will allow us to perform mean ergodic approximations in a spreadability preserving manner. Applications and an outlook are contained in Section 9. We cite results from [21] on braidability and on the failure of the implications ‘(a) ⇐ (b)’ and ‘(b) ⇐ (c)’ of the noncommutative extended de Finetti theorem, Theorem 0.2. Moreover, we present a general central limit theorem for spreadable random sequences which can be regarded to be the noncommutative prototype of a conditioned central limit theorem. We also discuss briefly its potential connections to interacting Fock spaces. Finally, we give immediate applications of Theorem 0.2 to inequalities in noncommutative L1 -spaces, as they appear in the work of Junge and Xu. 1. Preliminaries and terminology Noncommutative notions of probability spaces have in common that they consist of an algebra which is equipped with a linear functional. Here we shall work with the W∗ -algebraic version of such spaces, since they allow us to capture probabilistic tail events of random sequences. We refer the reader to [4,42,48,76,53] for further information on noncommutative probability spaces, in particular ∗-algebraic or C∗ -algebraic settings. Definition 1.1. A probability space (M, ψ) consists of a von Neumann algebra M with separable predual and a faithful normal state ψ on M. A von Neumann subalgebra M0 of M is said to be ψ-conditioned if the ψ-preserving conditional expectation EM0 from M onto M0 exists. Two probability spaces (M1 , ψ1 ) and (M2 , ψ2 ) are said to be isomorphic if there exists an isomorphism Π : M1 → M2 such that ψ1 = ψ2 ◦ Π . The ψ-preserving automorphisms of M will be denoted by Aut(M, ψ). By Takesaki’s theorem, the ψ-preserving conditional expectation EM0 exists if and only if ψ for all t ∈ R [71, IX, Theorem 4.2]. Here σt denotes the modular automorphism group associated to (M, ψ). Thus the existence of such a conditional expectation is automatic if ψ is a trace, i.e. ψ(xy) = ψ(yx) for all x, y ∈ M. ψ σt (M0 ) = M0
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The noncommutative generalization of random variables is cast in terms of ∗-homomorphisms [4]. For the purpose of this paper the following definition of a random variable will be sufficient. Definition 1.2. Let (A0 , ϕ0 ) and (M, ψ) be two probability spaces. A random variable is an injective ∗-homomorphism ι : A0 → M satisfying two additional properties: (i) ϕ0 = ψ ◦ ι; (ii) ι(A0 ) is ψ -conditioned. A random variable will also be addressed as the mapping ι : (A0 , ϕ0 ) → (M, ψ). Every classical random variable in the context of standard measure spaces yields by algebraisation a random variable in the sense of Definition 1.2. Conversely, if the von Neumann algebra M is commutative then the usual notion of a random variable on standard probability spaces can be recovered from Definition 1.2. Note that our assumption of injectivity is no restriction if a single random variable is considered. Remark 1.3. Assertion (ii) in the above definition is superfluous if ψ is a trace. Note also that this assertion has equivalent formulations: (iii) ι intertwines the modular automorphism groups of (A0 , ϕ0 ) and (M, ψ); (iv) There exists a (unique) unital completely positive map ι+ : M → A0 satisfying ψ(xι(a)) = ϕ0 (ι+ (x)a) for all x ∈ M and a ∈ A0 . The map ι+ is also called the adjoint of ι. We refer the reader to [2,23,25,7] for further details and background results on the equivalences of (ii) to (iv). Remark 1.4. Commonly (selfadjoint) operators in the von Neumann algebra M (or, more generally, its noncommutative Lp -spaces) are also denoted as ‘noncommutative random variables’ in the literature. Of course, we can easily produce a random variable in the operator sense from our setting by considering ι(x) for some fixed x ∈ A0 . We are interested in sequences of random variables. Notation 1.5. We write I < J for two subsets I, J ⊂ N0 if i < j for all i ∈ I and j ∈ J . The cardinality of I is denoted by |I |. For N ∈ N, we denote by I + N the shifted set {i + N | i ∈ I }. Definition 1.6. An (identically distributed) random sequence I is a sequence of random variables ι ≡ (ιi )i∈N0 : (A0 , ϕ0 ) → (M, ψ). The family (AI )I ⊂N0 , with von Neumann subalgebras AI =
i∈I
ιi (A0 ),
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is called the canonical filtration (generated by I ) and I is said to be minimal if AN0 = M. The von Neumann subalgebra Atail :=
ιk (A0 )
n∈N0 kn
is called the tail algebra of I . Suppose that a second random sequence I is defined by the random variables ψ ), ι˜ ≡ (˜ιi )i∈N0 : (A0 , ϕ0 ) → (M, and that I and I are minimal. Then I and I are isomorphic if there exists an isomorphism → M such that ψ ◦ Π = ψ and Π ◦ ι˜n = ιn for all n ∈ N0 . Π :M Whenever it is convenient, we may turn a random sequence into a minimal one by restricting the probability space (M, ψ) to (AN0 , ψ|AN0 ). We have already introduced distributional symmetries in the introduction. Here we present equivalent definitions which are less intuitive, but more convenient within our proofs. Notation 1.7. The group S∞ is the inductive limit of the symmetric groups Sn , n 2, where Sn is generated on N0 by the transpositions πi : (i − 1, i) → (i, i − 1) with 1 i < n. By [n] we denote the linearly ordered set {1, 2, . . . , n}. Definition 1.8. Let i, j : [n] → N0 be two n-tuples. (i) i and j are translation equivalent, in symbols: i ∼θ j, if there exists k ∈ N0 such that i = θ k ◦ j or
θ k ◦ i = j.
Here denotes θ the right translation m → m + 1 on N0 . (ii) i and j are order equivalent, in symbols: i ∼o j, if there exists a permutation π ∈ S∞ such that i=π ◦j
and π|j([n]) is order preserving.
(iii) i and j are symmetric equivalent, in symbols: i ∼π j, if there exists a permutation π ∈ S∞ such that i = π ◦ j. We have the implications (i ∼θ j) ⇒ (i ∼o j) ⇒ (i ∼π j).
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Remark 1.9. Order equivalence of two n-tuples i and j can also be expressed with the help of the partial shifts (θN )N 0 : N0 → N0 , where θN (n) =
n if n < N; n + 1 if n N .
Each θN is order-preserving and it is easy to see that i ∼o j if and only if there exist partial shifts θN1 , θN2 , . . . , θNk such that θN1 ◦ θN2 ◦ · · · ◦ θNk ◦ i = j. Note also that any subsequence (n0 , n1 , n2 , . . .) of the infinite sequence (0, 1, 2, 3, . . .) can be approximated via actions of the subshifts (θN )N 0 . Remark 1.10. Order equivalence is used in the context of a general limit theorem in [39] and our present formulation is an equivalent one. For the notation of mixed higher moments of random variables, it is convenient to use Speicher’s notation of multilinear maps. Notation 1.11. Let the random sequence I be given by ι ≡ (ιi )i∈N0 : (A0 , ϕ0 ) → (M, ψ). We put, for i : [n] → N0 , a = (a1 , . . . , an ) ∈ An0 and n ∈ N, ι[i; a] := ιi(1) (a1 )ιi(2) (a2 ) · · · ιi(n) (an ), ψι [i; a] := ψ ι[i; a] . We are ready to introduce distributional symmetries in terms of the mixed moments of a random sequence. Definition 1.12. A random sequence I is (i) exchangeable if, for any n ∈ N, ψι [i; ·] = ψι [j; ·]
whenever i ∼π j;
(ii) spreadable if, for any n ∈ N, ψι [i; ·] = ψι [j; ·]
whenever i ∼o j;
(iii) stationary if, for any n ∈ N, ψι [i; ·] = ψι [j; ·] whenever i ∼θ j. We close this section with the proof of the obvious implications in the noncommutative extended de Finetti theorem.
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Proof of Theorem 0.2, elementary parts. It is evident from Definition 1.8 and Definition 1.12 that exchangeability implies spreadability, and that spreadability implies stationarity. This shows the implication ‘(a) ⇒ (b)’ and the elementary parts on stationarity of the implications ‘(b) ⇒ (c)’ and ‘(b) ⇒ (co )’. The implications ‘(c) ⇒ (d)’ and ‘(co ) ⇒ (do )’ are trivial. 2 2. Noncommutative stationary processes Exchangeable or spreadable random sequences are stationary and can thus be expressed as stationary processes. Since the remaining sections of this paper will rest on this well-known connection, we provide more in detail some of their specific properties in this section. We will introduce stationary processes such that they are in a canonical correspondence to stationary random sequences (see Definition 1.12). Their notion is very closely related to Kümmerer’s approach in [45,47] (see also [20, Section 2.1]). Moreover, we present a result from [43] which shows that a unilateral stationary process (as introduced next) extends to a bilateral stationary process. Definition 2.1. A (unilateral) stationary process M consists of a probability space (M, ψ), a ψ-conditioned subalgebra M0 ⊂ M and an endomorphism α of M satisfying (i) unitality: α(1) = 1; (ii) stationarity: ψ ◦ α = ψ ; ψ (iii) conditioning: α and the modular automorphism group σt commute for all t ∈ R. The stationary process M is also denoted by the quadruple (M, ψ, α, M0 ) and ια ≡ ιαi i∈N : (M0 , ψ0 ) → (M, ψ), 0
ιαi := α i M , 0
is called the random sequence associated to M , for brevity also denoted by I α . The family of von Neumann subalgebras (MI )I ⊂N0 , with MI :=
α i (M0 ),
i∈I
is called the canonical filtration (generated by M ). The von Neumann subalgebra Mtail =
α n (M)
n∈N0
is called the tail algebra of M . We denote by Mα the fixed point algebra of the endomorphism α. are isomorphic if there exists an isoFinally, two minimal stationary processes M and M → M such that morphism Π : M , ψ ◦Π =ψ
Π ◦ α = α ◦ Π,
0 ) = M0 . Π(M
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The modular condition (iii) is needed for a non-tracial state ψ to ensure: Von Neumann algebras generated by the α n (M0 )’s and their intersections are ψ -conditioned (see Remark 2.2). Stationary processes and stationary random sequences are in correspondence (see Lemma 2.5). A unilateral stationary process extends to a bilateral stationary process (see Theorem 2.7). Remark 2.2. Condition (iii) of Definition 2.1 entails that the ψ-preserving conditional expectaψ tion EMI from M onto MI exists: M0 is globally σt -invariant and now condition (iii) implies ψ that α(M0 ) and, more generally, MI are globally σt -invariant. Thus Takesaki’s theorem on the existence of ψ -preserving conditional expectations applies. Of course, the condition (iii) can be dropped if ψ is a trace. We are indebted to Kümmerer for simple examples on hyperfinite IIIλ ψ factors such that α(A0 ) fails to be globally σt -invariant without condition (iii) [41]. To avoid reiterations throughout this paper we shall use the following convention for properties of a stationary process. Definition 2.3. The stationary process M is said to have property ‘A’ if its associated random sequence I α has property ‘A’. For example, M is minimal if its associated random sequence I α is minimal. The canonical filtrations of a stationary process M and its associated random sequence I α always coincide. But the tail algebra M tail of M may be larger than the tail algebra of I α , M I tail =
ι(α) k (M0 ) =
n∈N0 kn
α k (M0 ).
n∈N0 kn
Lemma 2.4. If M is minimal, then MI tail = Mtail . Proof. This is easily concluded from kn
and minimality.
(α)
ιk (M0 ) =
kn
α k (M0 ) = α n
α k (M0 ) ⊆ α n (M)
k0
2
We continue with the correspondence between stationary processes and stationary random sequences under the condition of minimality. We include this well-known result for reasons of transparency since the proof of the noncommutative version of the extended de Finetti theorem makes heavily use of it. Lemma 2.5. There is a one-to-one correspondence between (equivalence classes of ) (a) minimal stationary processes M = (M, ψ, α, M0 );
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(b) minimal stationary random sequences I with random variables (ιn )n0 : (A0 , ϕ0 ) → (M, ψ). The correspondence from (a) to (b) is given by (A0 , ϕ0 ) := (M0 , ψ|M0 )
and ιn := α n M . 0
The correspondence from (b) to (a) is established via M0 := ι0 (A0 )
and α ι[i; a] := ι[θ ◦ i; a]
for all n ∈ N, n-tuples i : [n] → N0 and a ∈ An0 . (Here θ is the shift from Definition 1.8.) Proof. We omit all fairly clear parts of the proof and only show that the properties of I imply ψ ψ the modular condition ασt = σt α. Since the von Neumann algebras ιn (A0 ) are ψ -conditioned, ψ ϕ the random variables ιn intertwine σt 0 and σt , the modular automorphism groups of (A0 , ϕ0 ) and (M, ψ) (see Remark 1.3 and [7, Lemma 2.5]). Thus ϕ ψ ψ ϕ σt ◦ α ι[i; a] = σt ι[θ ◦ i; a] = ι θ ◦ i; σt 0 (a) = α ι i; σt 0 (a) ψ = α ◦ σt ι[i; a] establishes ασt = σt α on a weak∗ -total subset of M. Here σt 0 (a) denotes the n-tuple ϕ ϕ (σt 0 (a1 ), . . . , σt 0 (an )). 2 ψ
ψ
ϕ
We will need the next theorem for approximations in the proof of Theorem 4.2. ˆ ψ, ˆ 0 ) is said to be bilateral if the enˆ α, Definition 2.6. A stationary process Mˆ = (M, ˆ M ˆ domorphism αˆ is an automorphism of M. A bilateral stationary process Mˆ is minimal if ˆ 0 ). ˆ M = n∈Z αˆ n (M Theorem 2.7. A unilateral stationary process M = (M, ψ, α, M0 ) extends to a bilateral ˆ ψ, ˆ 0 ). In other words, there exists a random variable ˆ α, stationary process Mˆ = (M, ˆ M ˆ ψ) ˆ such that j : (M, ψ) → (M, ˆ0 j (M0 ) = M
and j α n = αˆ n j
(n ∈ N0 ).
ˆ αˆ = j (Mα ). If Mˆ is minimal, then M This theorem is immediate from Kümmerer’s work on state-preserving Markov dilations. We provide some results from [43] which are essential for its proof. Let (A, ϕ) and (B, ψ) be two probability spaces. A morphism T : (A, ϕ) → (B, ψ) is a unital completely positive map T : A → B satisfying ϕ = ψ ◦ T . The morphisms from (A, ϕ) into itself are denoted by Mor(A, ϕ).
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Definition 2.8. A morphism T ∈ Mor(A, ϕ) admits a state-preserving dilation if there exists a ˆ ϕ), ˆ ϕ), ˆ ϕ) probability space (A, ˆ an automorphism Tˆ ∈ Aut(A, ˆ two morphisms j : (A, ϕ) → (A, ˆ n = QTˆ n j for all n ∈ N . A state-preserving dilation is ˆ ϕ) and Q : (A, ˆ → (A, ϕ) such that T 0 minimal if Aˆ = n∈Z Tˆ n j (A). Note in above definition that T n = QTˆ n j reads as idA = Qj for n = 0. This implies that j is ˆ ϕ) a random variable from (A, ϕ) to (A, ˆ and the composition j Q is the ϕ-preserving ˆ conditional ˆ expectation from A onto j (A). We refer the reader to [42] for further details. Proposition 2.9. (See [43].) Let (A, ϕ) be a probability space and suppose α is a ϕ-preserving unital endomorphism of A. Then the following are equivalent: (a) α admits a state-preserving dilation. ϕ (b) α commutes with the modular automorphism group σt . We include the proof from [43] for the convenience of the reader. It uses inductive limits of C∗ -algebras (see for example [63, Subsection 1.23]). Note also for the proof that a ϕ-preserving endomorphism α of A is injective. Indeed, α(x ∗ x) = 0 implies ϕ ◦ α(x ∗ x) = ϕ(x ∗ x) = 0 for x ∈ A. But ϕ is faithful and thus x = 0. Proof. The implication ‘(a) ⇒ (b)’ is shown in [42, 2.1.8]. So it remains to prove the converse. For n ∈ N0 , let An := A with C∗ -isomorphisms jn : A → An , where j0 is defined by identifying A with A0 . We identify next jn (x) and jn+1 (α(x)) for x ∈ A. This turns An into a subalgebra of An+1 . Consequently we obtain the infinite chain of embeddings A0 → A1 → A2 → · · · . Let A∞ denote the C∗ -inductive limit of this chain with norm · on A∞ . We may consider each An as a C∗ -subalgebra of A∞ such that A∞ =
An · .
n∈N0
Now the state ϕ∞ on the C∗ -algebra A∞ is introduced as the inductive limit of the states ϕn := ϕ ◦ jn−1 on An (see [63, 1.23.10]). Moreover the map α∞ jn (x) := jn α(x) ,
x ∈ A,
extends by continuity to a ϕ∞ -preserving automorphism of the C∗ -algebra A∞ , denoted by the same symbol. (Note that jn (x) is the image of jn+1 (x) under α, since we have identified jn (x) and jn+1 (α(x)).) ϕ Since α commutes with the modular automorphism group σt , the subalgebra α n (A) ⊂ A is ϕ globally σt -invariant and the ϕ-preserving conditional expectation from A onto α n (A) exists for all n ∈ N (see [71, Chapter IX, Theorem 4.2]). Correspondingly, for each n ∈ N, we find a completely positive map Qn : An → A such that, for m n,
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Qn ◦ j0 = idA ,
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Qn |Am = Qm .
By continuity this leads to a completely positive map Q∞ : A∞ → A such that, for n 1, ϕ∞ = ϕ ◦ Q∞ ,
Q∞ ◦ j0 = idA ,
Q∞ |An = Qn .
ϕ
ϕ
Let σt m denote the modular automorphism group associated to (Am , ϕm ). It follows σt m (x) = ϕ σt n (x) for x ∈ An , n m. Therefore the modular groups on Am extend to a group σt on A∞ such that ϕ∞ satisfies the KMS condition with respect to σt (see [54, 8.12.3]). Hence ϕ∞ extends to a faithful normal state ϕˆ on Aˆ := Πϕ∞ (A∞ ) (compare [54, 8.14.4]). ˆ the map Now it is routine to show that α∞ extends to the ϕ-preserving ˆ automorphism αˆ of A, Q∞ to the completely positive map Q : Aˆ → A satisfying ϕ ◦ Q = ϕ, ˆ and the injection j0 to an injective ∗-homomorphism j : A → Aˆ such that ϕ = ϕˆ ◦ j and j (A) = Πϕ∞ (A0 ) . Finally, α n = Qαˆ n j is immediately verified for n ∈ N0 . 2 Proof of Theorem 2.7. The endomorphism α of M satisfies the condition (b) of Proposiˆ ψ), ˆ ˆ a ψ-preserving ˆ tion 2.9. Thus there exists a probability space (M, automorphism αˆ of M, ˆ ψ) ˆ such that j α n = αˆ n j for all n ∈ N0 . Clearly and a random variable j : (M, ψ) → (M, ˆ 0 := j (M0 ) is a ψ-conditioned ˆ Finally, M ˆ αˆ = j (Mα ) is the content of ˆ M subalgebra of M. [42, Corollary 3.1.4]. 2 3. Conditional independence and conditional factorizability From our investigations of distributional symmetries emerge two closely related noncommutative generalizations of classical conditional independence. Here we concentrate on the case of two random variables; the more general setting of random sequences is covered in the consecutive section where we will meet a further ramification of these two notions. Definition 3.1. Let (M, ψ) be a probability space with three ψ -conditioned von Neumann subalgebras M0 , M1 and M2 . Then M1 and M2 are said to be (i) M0 -independent or conditionally independent if EM0 (xy) = EM0 (x)EM0 (y) for all x ∈ M1 ∨ M0 and y ∈ M2 ∨ M0 ; (ii) M0 -factorizable or conditionally factorizable if EM0 (xy) = EM0 (x)EM0 (y) for all x ∈ M1 and y ∈ M2 . This definition does not assume the inclusion M0 ⊂ M1 ∩ M2 . It is open if conditional factorizability implies conditional independence and thus the equivalence of these two notions. But this is of course the case if M0 C, and we will state in Lemma 3.6 further conditions under which M0 -factorizability implies M0 -independence.
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Remark 3.2. An alternative formulation of M0 -independence is sometimes easier to verify in applications. Under the assertions of Definition 3.1, the following are equivalent: (a) M1 and M2 are M0 -independent; 1 and M 2 of M such that M0 ⊂ (b) there exist M0 -independent von Neumann subalgebras M 1 ∩ M 2 and Mi ⊂ M i (i = 1, 2). M Since this equivalence is fairly clear, we omit its proof. Remark 3.3. If M0 C, we will also write C-independence instead of M0 -independence. Note that M1 and M2 are C-independent if and only if ψ(xy) = ψ(x)ψ(y) for all x ∈ M1 and y ∈ M2 [44]. The failure of the inclusion M0 ⊂ M1 ∩ M2 happens frequently in the context of distributional symmetries and is, in classical probability, intimately related to random probability measures. We illustrate this by the most simple example which may be taken from classical probability (just choose A C2 ⊗ C2 in Example 3.4). Example 3.4. Let A1 and A2 be two C-independent von Neumann subalgebras of the probability space (A, ϕ). We define the probability space (M, ψ) by M := A ⊕ A and ψ := 12 (ϕ ⊕ ϕ). For i = 1, 2, the embeddings Ai x → x ⊕ x ∈ M define the von Neumann subalgebras M1 and M2 , respectively. Furthermore, we put M0 = C1A ⊕ C1A C2 . One has Mi ∨ M0 = Ai ⊕ Ai for i = 1, 2 and calculates EM0 (xy) = EM0 (x)EM0 (y) for all x ∈ M1 ∨ M0 and y ∈ M2 ∨ M0 . Thus M1 and M2 are M0 -independent. But M1 ∩ M2 C, so M0 ⊂ M1 ∩ M2 . Remark 3.5. Another calculation shows in the above example that M1 and M2 are Cindependent. But this is rather an accident because we have chosen identical states on each component of the direct sum. Evidently, M0 -independence implies M0 -factorizability; but the converse is open. Frequently this can be concluded if additional algebraic structures are available (see also Theorem 4.2). All presently known examples (within our setting) satisfy at least one of the following conditions. Lemma 3.6. M0 -factorizability and M0 -independence are equivalent under each of the following additional assertions: (i) (ii) (iii) (iv) (v)
(trivial conditioning) M0 C; (central conditioning) M0 ⊂ M ∩ M ; (classical probability) M = M ; (relative commutants) M0 ⊂ M1 ∩ M2 ; (commuting squares) M0 ⊂ M1 ∩ M2 .
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Proof. Each of the assertions (i) to (iv) implies that the vector spaces {ax | a ∈ M0 , x ∈ M1 } and {yb | b ∈ M0 , y ∈ M2 } are weak∗ total in M0 ∨ M1 and M0 ∨ M2 , respectively. Thus the module property of conditional expectations and M0 -factorizability imply EM0 (axyb) = aEM0 (xy)b = aEM0 (x)EM0 (y)b = EM0 (ax)EM0 (yb). This equalities extend bilinearly and an approximation argument completes the proof in the cases (i) to (iv). The proof under the assertion (v) is trivial. 2 Our notion of conditional independence is in close contact with Popa’s notion of commuting squares [57,58,56]. Detailed information on their role in subfactor theory is provided in [32,22]. We will make frequent use of some of their properties. Note that these assertions do not apply for conditional factorizability. Proposition 3.7. Suppose M0 ⊂ M1 ∩ M2 , in addition to the assertions of Definition 3.1. Then the following are equivalent: (i) (ii) (iii) (iv)
M1 and M2 are M0 -independent; EM1 (M2 ) = M0 ; E M1 E M2 = E M0 ; EM1 EM2 = EM2 EM1 and M1 ∩ M2 = M0 .
In particular, it holds M0 = M1 ∩ M2 if one and thus all of the four assertions are satisfied. Proof. The tracial case for ψ is proved in [22, Proposition 4.2.1]. The non-tracial case follows from this, after some minor modifications of the arguments therein. 2 We close this section with some remarks on examples and references which are closely related to conditional independence in our noncommutative setting. The author is presently not aware of published examples in the quantum setting beyond the assertions stated in Lemma 3.6. It would be of interest to find examples of von Neumann algebras which are conditionally factorizable, but not conditionally independent, if possible at all. Remarks 3.8. (1) C-independence emerged from investigations of Kümmerer on the structure of stationary quantum Markov processes [42–45]. Its generalization to commuting squares is explored further from the perspective of noncommutative probability in [61,37,38,25,39]. (2) Examples for C-independence are classical independence, tensor independence and free independence. Further examples originate from pioneering work of Bo˙zejko and Speicher [11,12] and are given by generalized or noncommutative Gaussian random variables [10,24,40]. The most well-known among them are q-Gaussian random variables. Crucial for the appearance of C-independence are the presence of white noise functors [46,24] and a vacuum vector of the underlying deformed Fock space which is separating for the considered von Neumann algebras.
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(3) Sources of examples for M0 -independence are, of course, conditional independence in probability theory and random probability measures on standard Borel probability spaces. Further examples, satisfying the inclusion M0 ⊂ M1 ∩ M2 , arise from amplifications of examples for C-independence by tensor product constructions. Freeness with amalgamation as well as commuting squares from subfactor theory are further sources of M0 -independence (with M0 ⊂ M1 ∩ M2 ). We refer to [25] for a more detailed treatment of some of these examples. (4) M0 -independence appears, also under the assumption M0 ⊂ M1 ∩ M2 , in the work of Junge and Xu on noncommutative Rosenthal inequalities [34] and within Junge’s quantum probabilistic approach to embedding Pisier’s operator Hilbert space OH into the predual of the hyperfinite III1 -factor [33]. 4. Stationarity and conditional independence/factorizability This section is devoted to show in Theorem 4.2 that conditional factorizability implies conditional independence in the context of stationarity and under a certain conditioning. We close with an illustration of conditional independence and conditional factorizability by an algebraic treatment of an infinite sequence of exchangeable {0, 1}-valued random variables. Due to the noncommutativity of our setting, there are (at least) two natural ways to extend the notions of conditional independence and conditional factorizability (see Definition 3.1) from two random variables to random sequences indexed by N0 . One may regard N0 as a set, or as an ordered set (with its natural order). Definition 4.1. The (identically distributed) random sequence I , given by ι ≡ (ιi )i∈N0 : (A0 , ϕ0 ) → (M, ψ), with canonical filtration (AI )I ⊂N0 , is said to be (CI) full N -independent or conditionally full independent, if AI and AJ are N -independent for all I, J ⊂ N0 with I ∩ J = ∅; (CIo ) order N -independent or conditionally order independent, if AI and AJ are N -independent for all I, J ⊂ N0 with I < J or I > J . We say that I is (CF) full N -factorizable or conditionally full factorizable, if AI and AJ are N -factorizable for all I, J ∈⊂ N0 with I ∩ J = ∅; (CFo ) order N -factorizable or conditionally order factorizable, if AI and AJ are N -factorizable for all I, J ⊂ N0 with I < J or I > J . Note above that N is a ψ -conditioned von Neumann subalgebra of M, as required in Definition 3.1. We will deliberately drop the attributes ‘full’ or ‘order’ if we want to address conditional independence or conditional factorizability only on the informal level or if it is clear from the context whether the index set N0 is regarded with order structure or without it. Obviously we have the following implications:
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(CI) ⇓ (CIo )
⇒ ⇒
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(CF) ⇓ (CFo ).
We record that this gives the following implications in the noncommutative extended de Finetti theorem. Proof of Theorem 0.2, (c) ⇒ (co ) and (d) ⇒ (do ). This is obvious for N = Mtail from Definition 4.1 and above diagram. 2 A natural question is to ask if the converse implications in above diagram are also valid. Actually, we do not know an answer in this generality. But an affirmative answer is available for the equivalence of conditional independence and conditional factorizability if the random sequence I is stationary and N contained in the fixed point algebra of the corresponding stationary process (see Lemma 2.5 for this correspondence). Theorem 4.2. Let M be a minimal stationary process and suppose the ψ -conditioned von Neumann subalgebra N satisfies N ⊂ Mα . Then the following are equivalent: (CI) M is full N -independent; (CF) M is full N -factorizable. Furthermore, the following are equivalent under the same assertions: (CIo ) M is order N -independent; (CFo ) M is order N -factorizable. We will see in Section 6 that conditional order factorizability (CFo ), the weakest of the four properties, already suffices to identify N as the fixed point algebra of the endomorphism α which equals moreover the tail algebra. There it will suffice, due to Theorem 4.2, to establish these fixed point characterization results of Kolmogorov type on the level of conditional factorizability. Moreover we will benefit from this simplification in Section 7 and Section 8 when showing that spreadability implies conditional independence. We prepare the proof of Theorem 4.2 by two lemmas on approximations. Lemma 4.3. Let x1 , . . . , xp ∈ B1 (M), the unit ball of M. Suppose further that each xi is approximated by a sequence (xi,n )n∈N ⊂ B1 (M) in the strong operator topology. Then x1 x2 · · · xp = SOT- lim x1,n x2,n · · · xp,n . n→∞
A proof of this standard fact is omitted. Lemma 4.4. Suppose M is a minimal bilateral stationary process and let the function N : N → Z be given. Then every a ∈ Mα is approximated by a sequence (an )n∈N ⊂ M in the strong operator topology such that an ∈ M{0,1,...,n−1}+N (n)
and an a.
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Proof. We assume without loss of generality that all a is in the unit ball B1 (M) ∩ Mα . The alg ∗-algebra MN0 is weak∗ -dense in M. Thus, by Kaplansky’s density theorem, a ∈ N is approxalg
imated by a sequence (bn )n ⊂ MN0 ∩ B1 (M) in the strong operator topology. Put an := α N (n) EM[0,n−1] (bn ) ∈ M{0,1,...,n−1}+N (n) and note that α is an automorphism of M, since we are working in the bilateral setting. We claim that SOT- lim an n
= a.
(4.1)
Indeed, the sequence (EM[0,n−1] )n is norm bounded and converges to idM in the pointwise alg strong operator topology; this is clear on MN0 and an 2ε -argument gives the general case. Thus (EM[0,n−1] (bn ))n converges to a in the strong operator topology. We use next the ψ-topology which is induced by the maps M x → ψ(x ∗ x)1/2 . Since the strong operator topology and the ψ-topology coincide on bounded sets, an − aψ = α N (n) EM[0,n−1] (bn ) − a ψ = EM[0,n−1] (bn ) − a ψ completes the proof.
2
Proof of Theorem 4.2. Only the implications ‘(CFo ) ⇒ (CIo )’ and ‘(CF) ⇒ (CI)’ require a proof, since their reverse implications are trivial. We can assume by Theorem 2.7 that M = (M, ψ, α, M0 ) is a minimal bilateral stationary process. This will allow us to approximate elements of N in an appropriate manner. Note that full (resp. order) N -factorizability of the family (MI )I ⊂N0 implies immediately full (resp. order) N -factorizability of (MI )I ⊂Z by stationarity; this is clear for finite sets I and the general case is done by approximation. We need to show that full (resp. order) N -factorizability of (MI )I ⊂N implies EN (xy) = EN (x)EN (y) for all x ∈ MI ∨ N and y ∈ MJ ∨ N with I ∩ J = ∅ (resp. I < J ). For this purpose, we start with bounded sets I, J ⊂ N0 and consider monomials of the form x = z1 a1 · · · zp ap
and y = zp+1 ap+1 · · · z2p a2p ,
with zi ∈ MI , zp+i ∈ MJ and ai , ai+p ∈ N (i = 1, . . . , p). We approximate all ai ’s in the strong operator topology and can assume without loss of generality that all zi ’s and ai ’s are in the unit ball B1 (M). Let Ni : N → Z be given function which will be specified later. By Lemma 4.4, there exist sequences (ai,n )n∈N ⊂ B1 (M) satisfying ai = SOT- lim ai,n , n→∞
ai,n ∈ M{0,1,...,n−1}+N (n) .
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Let xn := z1 a1,n z2 a2,n · · · zp ap,n , yn := zp+1 ap+1,n zp+2 ap+2,n · · · z2p a2p,n . We specify next the choice of the functions Ni . Let Ni (n) := −n and Np+i (n) := N for i = 1, . . . , p, where N > max I ∪ J . Note that the sets In := I ∪ {−n, n + 1, . . . , −1}, Jn := J ∪ {N, N + 1, . . . , N + n − 1} are disjoint if I and J are disjoint; and that In < Jn if I < J . Since xn ∈ MIn and yn ∈ MJn we conclude from order (resp. full) N -factorizability that EN (xn yn ) = EN (xn )EN (yn ), which entails EN (xy) − EN (x)EN (y) = EN (xy − xn yn ) + EN (xn )EN (yn − y) + EN (xn − x)EN (y). We infer from Lemma 4.3 and the SOT – SOT-continuity of conditional expectations that the righthand side of this equation vanishes for n → ∞ in the strong operator topology. Thus full (resp. order) N -factorizability implies, for each p ∈ N, EN (z1 a1 · · · zp ap zp+1 ap+1 · · · z2p a2p ) = EN (z1 a1 · · · zp ap )EN (zp+1 ap+1 · · · z2p a2p )
(4.2)
for any z1 , . . . , zp ∈ MI , zp+1 , . . . , z2p ∈ MJ and a1 , . . . , a2p ∈ N whenever I and J are disjoint (resp. ordered) and bounded. This equality extends by bilinearity to the ∗-algebras MI ∪ N and MJ ∪ N . (By filling in additional factors 1M if necessary we can always achieve that monomials have the same number of factors.) Since MI ∪ N and MJ ∪ N are weak∗ dense in MI ∨ N and MJ ∨ N , the equality (4.2) extends further to the weak∗ closure, using Kaplansky’s density theorem and arguments similar to that in the proof of Theorem 6.4. Finally, another density argument extends the validity of (4.2) from bounded disjoint sets I and J to possibly unbounded disjoint sets. 2 Remark 4.5. At the time of this writing and in the generality of our setting, we have no information about the validity of the remaining implications (CIo ) ⇒ (CI) and (CFo ) ⇒ (CF), even under the assumptions of stationarity and N C. In particular, we do not know if an infinite stationary random sequence exists which is conditionally order independent, but fails to be conditionally full independent.
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We continue with an illustration of above concepts of conditional independence and conditional factorizability for stationary random sequences. The example is the von Neumann algebraic reformulation of infinite sequence of zero–one-valued random variables, as they have been the subject of de Finetti’s pioneering investigations on exchangeability [18]. We will observe in this example why it is too restrictive to assume that N is contained in the image of random variables. Example 4.6. Let (A0 , ϕ0 ) be given by A0 = C2
and ϕ0 = trp
with trp ((a1 , a2 )) = pa1 + (1 − p)a2 for some fixed p ∈ (0, 1). We realize the probability space (M, ϕ) as a mixture of infinite coin tosses with respect to some probability measure ν on the standard measurable space ([0, 1], Σ), assuming ν({0}) = ν({1}) = 0 and ν({p}) < 1 for any p ∈ (0, 1): ⊕ M(p) dν(p),
M =
M(p) =
C2 ,
n∈N0
[0,1]
⊕ ψ =
ψ(p) dν(p),
ψ(p) =
trp .
n∈N0
[0,1]
Here M(p) denotes the infinite von Neumann algebraic tensor product of C2 with respect to the infinite tensor product state on ψ(p) construc which are obtained by passing through theGNS tion starting from the ∗-algebra k∈N nk=0 C2 equipped with the product state k∈N nk=0 trp . We refer the reader to [70] for further information on direct integrals of von Neumann algebras and states. The random variable ιi : (A0 , ϕ0 ) → (M, ψ), with i ∈ N0 , is defined by the constant embedding of a ∈ C2 into the i-th factor of each fiber of the direct integral: ⊕ ιi (a) =
1A0 ⊗ · · · ⊗ 1A0 ⊗ a ⊗ 1A0 ⊗ · · · dν(p).
[0,1]
i factors
Finally, we put ⊕ N :=
C1M(p) dν(p) L∞ [0, 1], ν .
[0,1]
Note that our assumptions on the measure ν imply N C. The canonical filtration (AI )I ⊂N0 generated by the random sequence ι ≡ (ιi )i∈N0 is defined by
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AI =
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ιi (A0 ).
i∈I
The random sequence ι is minimal, i.e. we have M=
ιi (A0 ).
n∈N0
This follows if we can ensure that n∈N0 ιi (A0 ) contains N . Indeed, Kakutani’s theorem entails that the family of infinite product states {ψ(p)}p∈(0,1) is mutually disjoint [27]. We conclude ∞ from this that every element x ∈ N L ([0, 1], ν) can be approximated by a bounded sequence (xn )n∈N ⊂ i∈N0 ιi (A0 ) in the weak operator topology. This implies the minimality of the random sequence. |I | An elementary computation shows AI C2 for any finite set I ⊂ N0 . In the case of an infinite set I , we restrict the family of infinite product states {ψ(p)}p∈(0,1) to AI and conclude again by the Kakutani theorem [27] that these restricted states are mutually disjoint. This implies that the von Neumann algebra AI contains a copy of N whenever |I | = ∞. Now it is straightforward to verify the conditional full factorizability (CF) EN (xy) = EN (x)EN (y) for all x ∈ AI and y ∈ AJ with disjoint subsets I, J ⊂ N0 . Since all von Neumann algebras are commutative, it is immediate from the module property of conditional expectation that (CF) upgrades to (CI), i.e. EN (xy) = EN (x)EN (y) for all x ∈ AI ∨ N and y ∈ AJ ∨ N with disjoint subsets I, J ⊂ N0 . Thus the random sequence (ιi )i∈N0 is full N -independent. But N ⊂ AI ∩ AJ if one of the sets I or J is finite. Remark 4.7. There are ∗-algebraic, C∗ -algebraic and W∗ -algebraic approaches to noncommutative probability and it is instructive to compare them at the hand of Example 4.6. Of course, the alg ∗-algebras AI := i∈I ιi (A0 ) as well as its norm-closure are contained in the von Neumann alg algebra AI . The latter contains a copy of L∞ ([0, 1], ν) if |I | is infinite, but AI and its norm closure do not. 5. Noncommutative i.i.d. sequences may be non-stationary It is folklore in classical probability and free probability that independence resp. freeness of an identically distributed random sequence implies stationarity. But this implication fails in our broader context of noncommutative independence. Theorem 5.1. There exist full C-independent identically distributed random sequences I which fail to be stationary.
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Proof, in particular of Theorem 0.2, (c) (d) and (co ) (do ). See Example 5.2 or Example 5.4 below. Since full C-independence implies order C-independence we have also shown (co ) (do ). 2 Let us first outline our strategy to produce such examples. Recall from the introduction that an infinite random sequence I with random variables (ιn )n0 : (A0 , ϕ0 ) → (M, ψ) is identically distributed by definition. Suppose now that I is C-independent and stationary. Our goal is to ‘perturbate’ the random variables ιn such that C-independence is preserved, but stationarity is obstructed. This can be done in two ways, for the domain or the codomain of each random variable ιn . Example 5.2 (Perturbation of codomain). Consider (R, tr), the hyperfinite II1 -factor equipped with its normalized trace. Let (Mm , trm ) be the complex m × m-matrices equipped with the normalized trace. The canonical embeddings M2 x → ιn (x) := 1M2 ⊗ · · · ⊗ 1M2 ⊗ x
⊗ 1M2 ⊗ · · ·
n-th position
define the random sequence I with random variables (ιn )n0 : (M2 , tr2 ) → (R, tr). It is easily verified that I is C-independent and stationary. We will deform this random sequence to obtain a non-stationary random sequences as follows. Under the canonical identification of M2 ⊗ M2 and M4 , the unitary matrix ⎡
1 ⎢0 Uω = ⎣ 0 0
0 0 1 0
0 1 0 0
⎤ 0 0⎥ ⎦, 0 ω
|ω| = 1,
defines the trace-preserving automorphism x → Uω xUω∗ of M2 ⊗ M2 . It is well known in subfactor theory that the inclusions Uω (M2 ⊗ 1M2 )Uω∗ ∪ C1M2 ⊗M2
⊂
M2 ⊗ M2 ∪ ⊂ M2 ⊗ 1M2
form a commuting square [31,61,32]. We canonically amplify this automorphism to the automorphism γω ∈ Aut(R, tr) which acts trivial on all higher tensor product factors. Consider now the random sequence I (ω) with random variables (ι(ω) n )n0 defined by ι(ω) n :=
ιn
if n = 1,
γω ι0
if n = 1.
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Note that I (1) is the random sequence I . Clearly I (ω) is identically distributed for any uni(ω) modular ω ∈ C. We note that the von Neumann algebras ιn (M2 ) mutually commute for n = 1. (ω) (ω) So do ι1 (M2 ) and ιn (M2 ) for n 2. We conclude from this that I (ω) is full C-independent. But we calculate for a = 01 01 that (ω) 1 (ω) (ω) (ω) tr ι0 (a)ι1 (a)ι0 (a)ι1 (a) = (ω + ω), 2 and (ω) (ω) (ω) (ω) tr ι2 (a)ι3 (a)ι2 (a)ι3 (a) = 1. This leads us to the conclusion that I (ω) is stationary if and only if ω = 1. Remark 5.3. Example 5.2 illustrates that the distribution of two C-independent (identically distributed) random variables does not determine their joint distribution. This is in contrast to two distinguished examples for C-independence, tensor independence and free independence. See [66,8] for further information on the related universality properties. We sketch next how local perturbations of random variables on their domain are capable to produce such effects. Suppose the minimal stationary random sequence I with random variables (ιn )n∈N0 : (A0 , ϕ0 ) → (M, ψ) is C-independent (in the ordered or full sense). Furthermore, let γ ≡ (γn )n0 ⊂ Aut(A0 , ϕ0 ) be a sequence of ‘local perturbations’. Then we can associate to each sequence γ a random sequence I (γ ) by putting (γ )
ιn := ιn ◦ γn . The random sequence I (γ ) is again minimal and C-independent. Suppose that there is a sequence γ with distr (γ ) (γ ) (γ ) (ι0 , ι1 , . . . , ιn−1 , ιn , ιn+1 , . . .) = ι0 , ι1 , . . . , ιn−1 , ιn , ιn+1 , . . .
for some n ∈ N. We conclude immediately that the random sequence on the right-hand side fails to be stationary, but it is still identically distributed and enjoys C-independence. Example 5.4 (Perturbation of domain). Let 2 (N) be the real Hilbert space of square-summable sequences and consider the q-Gaussian field Γq (2 (N)) for some fixed 0 < q < 1. These fields are the von Neumann algebra generated by q-Gaussian field operators ωq (f ), f ∈ 2 (N), acting on the q-deformed Fock space Fq (2 (N)) (see [10,24] for further details). Γq (2 (N)) is a nonhyperfinite II1 -factor and we denote its normalized trace by trq . The second quantization of the canonical unilateral shift on 2 (N) provides us with a unital trq -preserving endomorphism α of Γq (2 (N)). Identify R with the subspace generated by the first coordinate of 2 (N). Doing so we
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obtain the abelian von Neumann subalgebra Γq (R) ⊂ Γq (2 (N)) and we denote the restriction of trq to this subalgebra by the same symbol. Now it is straightforward to see that
ιn := α n Γ
q (R)
defines a full C-independent random sequence I with random variables (ιn0 ) : Γq (R), trq → Γq 2 (N) , trq , which is of course stationary. Let γ ∈ Aut(Γq (R), trq ) be fixed and consider the random sequence Iγ which is obtained from perturbating the first random variable of I : γ
ιn :=
ιn ι0 ◦ γ
if n = 1, if n = 0.
The central result by van Leeuwen and Maassen in [72] on the obstruction for q-deformation of the convolution product can be reformulated as: Theorem 5.5. Let 0 < q < 1. There is a ‘perturbation’ γ ∈ Aut(Γq (R), trq ) such that 4 4 trq ωq (f ) + α ωq (f ) = trq γ ωq (f ) + α ωq (f ) for 0 = f ∈ R. Note that ω(f ), γ (ω(f )) and α(ω(f )) have identical distributions and each of the first two random variables is C-independent from the third one. Thus the knowledge of the individual distributions of C-independent random variables does not completely determine their joint distributions; this depends on the concrete realization of the random variables. The ‘perturbation’ γ is constructed in [72] starting from a μ-preserving point transformation on the spectrum of the (selfadjoint) q-Gaussian field operator ωq (f ), for some fixed f ∈ R, where μ is induced by the spectral measure of ωq (f ) with respect to trq . Corollary 5.6. Iγ is full C-independent and non-stationary. Proof. It is immediate from its construction that I is full C-independent. The perturbation γ of the domain of the first random variable does not effect its range. Thus Iγ is also full Cindependent. Let a := ωq (f ) for notational convenience. A straightforward computation yields for the lefthand side of the inequality in Theorem 5.5 that 4 trq a + α(a) = 2 trq a 4 + 4 trq a 2 trq a 2 + 2 trq aα(a)aα(a) . (Expand the product; use traciality, C-independence, trq ◦ α = trq and the centredness of a.) Similarly, the right-hand side of this inequality simplifies to 4 trq γ (a) + α(a) = 2 trq a 4 + 4 trq a 2 trq a 2 + 2 trq γ (a)α(a)γ (a)α(a) .
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Since trq γ (a)α(a)γ (a)α(a) = trq aα(a)aα(a) by Theorem 5.5, we have trq γ (a)α(a)γ (a)α(a) = trq α(a)α 2 (a)α(a)α 2 (a) distr
and consequently (ι0 ◦ γ , ι1 , ι2 , . . .) = (ι1 , ι2 , ι3 , . . .).
2
The invariance of all finite joint distributions of an identically distributed random sequence under all local automorphisms seems to be a very strong condition. If the von Neumann algebra M is abelian and γ ∈ Aut(A0 , ϕ0 ) ergodic, such a local invariance property implies the Cindependence of the random sequence by an application of the mean ergodic theorem. In the noncommutative context, this observation invites to introduce ‘top-order C-independence’ for a random sequence I , i.e., the von Neumann algebras k
(ι0 , ι1 , ι2 , . . .) = (ι0 ◦ γ0 , ι1 ◦ γ1 , ι2 ◦ γ2 , . . .). Does the ergodicity of Aut(A0 , ϕ0 ) imply that I is full C-independent? And if so, can one show that this C-independence must be either tensor independence or free independence? 6. Stationarity with strong mixing and noncommutative Bernoulli shifts We provide a noncommutative generalization of the Kolmogorov zero–one law. Furthermore we show that conditional factorizability implies strong mixing in the context of stationarity. This leads us to a noncommutative generalization of classical Bernoulli shifts. Theorem 6.1. Let I be an order N -factorizable random sequence where N is a ψ-conditioned von Neumann subalgebra of Mtail . Then it holds N = Mtail . In particular, an order Cindependent random sequence has a trivial tail algebra.
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The last assertion is a noncommutative Kolmogorov zero–one law. Note also that order N factorizability (CFo ) is implied by (CIo ), (CF) or (CI). Proof. Assume without loss of generality that I is minimal. We show first that M and Mtail alg are N -independent. Let a ∈ Mtail and x ∈ MN0 . Thus there exists some bounded subset J ⊂ N0 such that x ∈ MJ . Because Mtail ⊂ M[n,∞) for all n ∈ N0 , we find some n such that J < [n, ∞). Consequently, the order N -factorizability implies EN (ax) = EN (a)EN (x). Now let x ∈ M. By minimality and Kaplansky’s density theorem, there exists a bounded sealg quence (xk )k∈N in MN0 of M such that x = WOT- limk xk . Note that, for all k, we have xk ∈ MJk with some bounded subset Jk . We conclude that, for any y ∈ M, ψ yEN (ax) = lim ψ yEN (axk ) k
= lim ψ yEN (a)EN (xk ) k
= ψ yEN (a)EN (x) . This gives the factorization EN (ax) = EN (a)EN (x)
(6.1)
for all a ∈ Mtail and x ∈ M. This factorization implies the N -independence of Mtail and M. Indeed, the ψ-preserving conditional expectation EMtail from M onto Mtail exist since Mtail is ψ globally σt -invariant. The latter is easily concluded from the fact that the ranges of the random variables ιn are ψ -conditioned and the definition of Mtail . We are left to verify that (6.1) extends to elements a ∈ Mtail ∨ N and x ∈ M ∨ N . But this is evident, because N ⊂ M and N ⊂ Mtail . Thus M and Mtail are N -independent. To prove N = Mtail , we are left to show the inclusion Mtail ⊂ N . We infer from the N independence of M and Mtail that Mtail and Mtail are N -independent. We use the module property of conditional expectations and N -independence to get, for every x ∈ Mtail , ∗ EN x − EN (x) x − EN (x) = EN x ∗ x − EN x ∗ EN (x) = 0. Now the faithfulness of EN implies x = EN (x) and thus Mtail ⊂ N . The last assertion is clear since order C-factorizability and order C-independence are equivalent (see Definition 3.1). 2 Remark 6.2. The assumptions in Theorem 6.1 can be further weakened since an inspection of its proof shows that only the ranges of the random variables matter. It suffices that the probability space (M, ψ) is equipped with an order N -factorizable family of ψ-conditioned von Neumann subalgebras (Mk )k∈N .
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It is well known that the Kolmogorov zero–one law implies strong mixing properties of an independent stationary random sequence. Here we are interested in a conditioned noncommutative version of this classical result. It is convenient to formulate it in terms of the minimal stationary process M associated to a stationary random sequence I . Definition 6.3. A stationary process M or its endomorphism α is said to be strongly mixing over N if, for any x ∈ M, WOT-
lim α n (x) = EN (x).
n→∞
Here N is a ψ-conditioned von Neumann subalgebra of M. Theorem 6.4. Let the minimal stationary process M be order N -factorizable for the ψ-conditioned subalgebra N of Mα . Then α is strongly mixing over N . Moreover we have N = Mα = Mtail . In particular, these three subalgebras are trivial if M is order C-independent. The condition N ⊂ Mα is non-trivial if Mtail C (see Remark 6.5). Proof. Since Mα ⊂ Mtail , we conclude N = Mα = Mtail from Theorem 6.1. We are left to prove the mixing properties. Suppose x ∈ MI and y ∈ MJ for bounded sets I, J ⊂ N0 . One calculates lim ψ y ∗ α n (x) = lim ψ EN y ∗ α n (x) n→∞ = lim ψ EN y ∗ EN α n (x) n→∞ = ψ EN y ∗ EN (x) = ψ y ∗ EN (x) .
n→∞
Here we used that J < (I + n) for n sufficiently large and applied order N -factorizability to obtain the second equality. The third equality uses that N ⊂ Mα implies EN ◦ α = EN . To extend these equations to arbitrary x, y ∈ M, we use the minimality of the stationary process and approximate x and y by bounded sequences (xi )i and, respectively, (yi )i from the alg ∗-algebra MN0 in the strong operator topology. Since ψ y ∗ α n (x) = ψ (y − yi )∗ α n (x) + ψ yi∗ α n (x − xi ) + ψ yi∗ α n (xi ) and since the estimates
ψ (y − yi )∗ α n (x) ψ |y − yi |2 1/2 ψ |x|2 1/2 ,
∗ n
y α (x − xi ) ψ |yi |2 1/2 ψ |x − xi |2 1/2 i
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are uniform in n, we conclude the convergence of ψ(y ∗ α n (x)) to ψ(y ∗ EMtail (x)) by an ε/3argument. Now the claimed mixing property follows from the norm density of the functionals {ψ(y ·) | y ∈ M} in M∗ and the boundedness of the set {α n (x) | n ∈ N0 }. 2 Remark 6.5. The condition N ⊂ Mα in Theorem 6.4 is non-trivial. Consider a minimal stationary process M with N = Mtail = M C. Then M is N -factorizable and EN is the identity map on M. Furthermore, α is easily seen to be an automorphism. It follows from Definition 6.3 that α is strongly mixing over N if and only if α is the identity. Remark 6.6. Conditional order factorizability (CFo ) is the weakest form of independence or factorizability introduced in Definition 4.1; thus Theorem 6.1 and Theorem 6.4 are also valid if (CFo ) is replaced by (CF), (CIo ) or (CI). An important class of stationary processes in classical probability are Bernoulli shifts; and a noncommutative notion of such shifts emerges in [43] from the study of stationary quantum Markov processes. Here we are interested in their amalgamated version, as studied in [61] and, in a bilateral continuous ‘time’ formulation, in [25]. Definition 6.7. An (ordered/full) Bernoulli shift (over N ) is a minimal stationary process B = (B, χ, β, B0 ) with the following properties: (i) N ⊂ B α ∩ B0 is a χ -conditioned von Neumann subalgebra; (ii) the canonical filtration (BI )I ⊂N0 is (order/full) N -independent. The endomorphism β is also called a Bernoulli shift over N with generator B0 . Note that this definition of a Bernoulli shift contains a subtle redundancy: one could drop the modular condition on the endomorphism β and conclude it from the fact that its ranges β n (B0 ) must be χ -conditioned, as required by our definition of independence. This entails that χ β commutes with σt , the modular automorphism group of (B, χ). Corollary 6.8. Let M = (M, ψ, α, M0 ) be a minimal stationary process. Further suppose N ⊂ Mα is a ψ -conditioned von Neumann subalgebra and B = (M, ψ, α, M0 ∨ N ). Then the following are equivalent: (a) M is (order/full) N -factorizable; (b) M is (order/full) N -independent; (c) B is an (ordered/full) Bernoulli shift over N . In particular, it holds N = Mα = Mtail . Proof. We already know the equivalence of (a) and (b) from Theorem 4.2. The equivalence of (b) and (c) is also clear since the family (MI )I ⊂N0 is (order/full) N -independent if and only if the family (MI ∨ N )I ⊂N0 is so. We are left to show N = Mα = Mtail . But this is content of Theorem 6.4. 2
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We provide next a result which is useful for applications where one wants to identify a given process as a Bernoulli shift. Suppose M = (M, ψ, α, M0 ) is an (order/full) N -factorizable minimal stationary process for some ψ -conditioned von Neumann subalgebra N ⊂ Mα . Furthermore let C0 be a ψ -conditioned von Neumann subalgebra of M0 . Put B :=
α n (C0 ∨ N ),
χ := ψ|B ,
β := α|B ,
B0 := C0 ∨ N .
n0
This defines the minimal stationary process B = (B, χ, β, B0 ) which is subject of the next result. Corollary 6.9. B is an (ordered/full) Bernoulli shift over N and N = B β = B tail . Proof. Theorem 4.2 implies the (order/full) N -independence of M . Since B0 ⊂ M0 ∨ N (order/full) N -independence is inherited by the minimal stationary process B. Now an application of Theorem 6.1 to the random sequence associated to B ensures N = B tail . We are left to prove N ⊂ B0 ∩ B β . Clearly N ⊂ B0 . Thus it suffices to show N = B β . Since N = Mα by Theorem 6.4 and N ⊂ B0 , we have Mα ⊂ B0 and consequently Mα ⊂ B. But this implies B β = Mα and consequently N = B β . 2 Remark 6.10. Our notion of a Bernoulli shift is motivated from Kümmerer’s work on noncommutative stationary Markov processes in [42,44,43,45,46]. An ordered Bernoulli shift here is the unilateral discrete version of noncommutative continuous Bernoulli shifts introduced in [25]. Note that Definition 6.7 of a Bernoulli shift is not restricted to tensor independence; it is casted in the broader context of conditional independence. 7. Spreadability implies conditional order independence The main result of this section is Theorem 7.1 which is an integral part of the noncommutative extended de Finetti theorem, Theorem 0.2. Theorem 7.1. Suppose I is a spreadable random sequence. Then I is stationary and order Mtail -independent. It is immediate from Definition 1.12 that spreadability implies the stationarity of a random sequence. Thus we can reformulate Theorem 7.1 in terms of stationary processes, as done in Theorem 7.2. Throughout this section, we consider the minimal stationary process M ≡ (M, ψ, α, M0 ) and, replacing its generator M0 by M0 ∨ Mα , the minimal stationary process B ≡ M, ψ, α, M0 ∨ Mα . Theorem 7.2. Suppose the stationary process M is spreadable and minimal. Then M is order Mtail -independent and Mtail = Mα . In particular, B is an ordered Bernoulli shift.
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The proof of Theorem 7.2 needs some preparation and is postponed to the end of this section. It entails of course the proofs of Theorem 7.1 and Theorem 0.2 (b) ⇒ (co ) through the correspondence stated in Lemma 2.5. Proposition 7.3. Suppose the minimal stationary process M is spreadable. Then there exists the ψ -preserving conditional expectation EMtail of M onto Mtail and WOT- lim α n
n
(x) = EMtail (x),
x ∈ M.
Moreover, we have Mtail = Mα . Proof. Let MI := n∈I α n (M0 ) for I ⊂ N0 . Let x, y ∈ |I |<∞ MI . Consequently we can assume x ∈ MI and y ∈ MJ such that there exists N ∈ N with I ∩ (J + N ) = ∅. We infer from spreadability that ψ(yα n (x)) = ψ(yα n+1 (x)) for all n N . Due to minimality this establishes the limit lim ψ yα n (x)
n→∞
on the WOT -dense ∗-algebra |I |<∞ MI . A standard approximation argument ensures now the existence of this limit for x, y ∈ M, using the norm density of the functionals {ψ(y·) | y ∈ M} and the boundedness of the set {α n (x) | n ∈ N}. We conclude from this that the pointwise WOT limit of the sequence (α n )n defines a linear map Q : M → M such that Q(M) ⊂ Mtail . It is easily seen that the linear map Q enjoys ψ =ψ ◦Q
and Q(x) x for x ∈ M.
Thus Q is a conditional expectation from M onto Mtail , if we can insure that Q(x) = x for all tail tail x ∈ M . To this end let x ∈ M and y ∈ |I |<∞ MI . We infer from Mtail ⊂ α N (M) and M[N,∞) ⊂ α N (M) for all N ∈ N that there exists some N ∈ N such that x ∈ α N (M) and y ∈ M[0,N −1] . We approximate x ∈ M in the WOT-sense by a sequence (xk )k ⊂ |I |<∞ α N (MI ) and conclude further from the definition of Q and from spreadability that ψ yQ(x) = lim ψ yQ(xk ) = lim lim ψ yα n (xk ) k
k
n
= lim ψ(yxk ) = ψ(yx). k
This shows that Q(x) = x for all x ∈ Mtail . Thus Q is the conditional expectation of M onto Mtail with respect to ψ (see [71, Chapter IX, Definition 4.1]), which we denote from now on by EMtail . We need to identify the tail algebra as the fixed point algebra. Proposition 7.3 gives pointwise EMtail EMα = WOT- limn α n EMα = EMα and thus Mα ⊂ Mtail . The inclusion Mtail ⊂ Mα follows from αEMtail = limn αα n = EMtail in the pointwise WOT-sense. 2 Remark 7.4. The proof of Proposition 7.3 shows that the ψ-preserving conditional expectation onto the tail algebra Mtail and the fixed point algebra Mα of the endomorphism α exist under
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weaker assertions. One does not need that α and the modular automorphism group σt commute (this compatibility condition is required in Definition 2.1). It is convenient to use Speicher’s notion of multilinear maps also for the endomorphism α. We put α[i; a] := α i(1) (a1 )α i(2) (a2 ) · · · α i(n) (an ) for n-tuples i : [n] → N0 and a = (a1 , a2 , . . . , an ) ∈ M0 . Definition 7.5. A stationary process M = (M, ψ, α, M0 ) or its endomorphism α is N spreadable if there exists a ψ-conditioned von Neumann subalgebra N of M such that EN α[i; a] = EN α[j; a] for any n ∈ N, i, j : [n] → N0 with i ∼o j and a ∈ Mn0 . Lemma 7.6. The following are equivalent for a minimal stationary process M : (a) M is spreadable; (b) M is Mtail -spreadable; (c) M is Mα -spreadable. Proof. (b) and (c) are equivalent since Mtail = Mα by Proposition 7.3. Obviously (b) implies (a) and we are left to prove the converse. Let us first treat the case Mtail ⊂ M0 . We already know Mtail = Mα from Proposition 7.3. Consider the n-tuple (ax1 , x2 , . . . , xn ) ∈ Mn0 with a ∈ Mα . We conclude from this that, for i, j : [n] → N0 with i ∼o j, ψ aα[i; x1 , x2 , . . . , xn ] = ψ α[i; ax1 , x2 , . . . , xn ] = ψ α[j; ax1 , x2 , . . . , xn ] = ψ aα[j; x1 , x2 , . . . , xn ] . Using ψ = ψ ◦ EMtail and the module property of EMtail , we conclude that α is conditionally Mtail -spreadable by standard arguments. The more general case Mtail ⊂ M0 is treated similar. We approximate a ∈ Mtail by a sequence (ak )k0 ⊂ M such that ak ∈
lk
α l (M0 )
and a = SOT- lim ak . k→∞
Thus we can assume that each ak is a linear combination of monomials α[ik ; ak ], for some nk tuple ik : [nk ] → {k, k + 1, . . .} and a ∈ Mn0 k . Now we compute as before that, for i, j : [n] → N0 with i ∼o j and sufficiently large k, ψ α[ik ; ak ]α[i; x1 , x2 , . . . , xn ] = ψ α[ik ; ak ]α[j; x1 , x2 , . . . , xn ] . This equality extends by linearity and weak∗ density arguments to
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ψ aα[i; x1 , x2 , . . . , xn ] = ψ aα[j; x1 , x2 , . . . , xn ] for every a ∈ Mtail . We conclude from this the Mtail -spreadability of the stationary process.
2
Lemma 7.7. Suppose M be a minimal stationary process. If M is spreadable, then M is order Mtail -factorizable. Proof. We need to show that the canonical filtration (MI )I ⊂N0 satisfies the factorization rule EMtail (xy) = EMtail (x)EMtail (y) alg
alg
for all x ∈ MI and y ∈ MJ whenever I < J or I > J . Let x ∈ MI and y ∈ MJ . Then, for all n ∈ N0 , EMtail (xy) = EMtail xα n (y) , since spreadability implies Mtail -spreadability (Lemma 7.6). We use the mixing properties of α (Proposition 7.3) to conclude EMtail (xy) = WOT- lim EMtail xα n (y) = EMtail (x)EMtail (y). n→∞ This establishes the order Mtail -factorizability of a spreadable stationary process.
2
Proof of Theorem 7.2. Lemma 7.7 shows that M is order Mtail -factorizable and Proposition 7.3 insures Mtail = Mα . Thus Theorem 4.2 applies for N = Mtail and ensures that M is conditionally Mtail -independent. Finally, Corollary 6.8 entails that B is an ordered Bernoulli shift over Mtail . 2 8. Spreadability implies conditional full independence We have already shown in the previous section that spreadability implies conditional order independence. Here this result will be strengthened to conditional full independence. Theorem 8.1. Let I be a spreadable random sequence. Then I is stationary and full Mtail independent. Theorem 8.1 establishes the implication (b) ⇒ (c) of Theorem 0.2, the noncommutative extended de Finetti theorem. We will prove it in terms of the corresponding stationary process M = (M, ψ, α, M0 ) and, replacing the generator M0 by M0 ∨ Mα , denote by B the stationary process (M, ψ, α, M0 ∨ Mα ). Theorem 8.2. Suppose the stationary process M is spreadable and minimal. Then M is full Mtail -independent and Mtail = Mα . In particular, B is a full Bernoulli shift. The proofs of Theorem 8.1 and Theorem 8.2 require a certain refined version of the mean ergodic theorem. Let us start with its usual formulation and include for the convenience of the reader how its proof reduces to the usual result for contractions on Hilbert spaces.
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Theorem 8.3. Let (M , ψ) be a probability space and α a ψ-preserving endomorphism of M. Then we have, for each x ∈ M, 1 k α (x) = EMα (x). n→∞ n n−1
SOT-
lim
k=0
Proof. The strong operator topology and the ψ -topology generated by the maps x → ψ(x ∗ x)1/2 , x ∈ M, coincide on norm bounded sets in M. Thus this mean ergodic theorem is an immediate consequence of the usual mean ergodic theorem in Hilbert spaces (see [55, Theorem 1.2] for example). 2 This mean ergodic theorem would allow us to give an alternative proof of that spreadability implies conditional order independence (CIo ), after having identified the tail algebra as the fixed point algebra of the stationary process in Proposition 7.3 and established conditional spreadability in Lemma 7.6. We illustrate this by an example. Given the stationary process (M, ψ, α, M0 ), let a, b ∈ M0 and consider M{1,2} x = α(a)α 2 (a)α(a)α 2 (a), M{3,4} y = α 4 (b)α 3 (b)α 4 (b)α 3 (b)α 4 (b). We have {1, 2} < {3, 4} and thus spreadability implies ! n−1 n 1 k α (y) EMα (xy) = EMα xα (y) = EMα x n k=0
for all n 1. Thus Theorem 8.3 implies EMα (xy) = EMα (x)EMα (y). But such an argument falls short of establishing the apparently stronger version, conditional full independence (CI). For example, consider the two elements x = α(a)α 3 (a)α(a)α 3 (a), y = α 4 (b)α 2 (b)α 4 (b)α 2 (b)α 4 (b). Thus we have x ∈ MI and y ∈ MJ with I = {1, 3} and J ∈ {2, 4}. Since the tuples (1, 3, 1, 3, 4, 2, 4, 2, 4) and (1, 3, 1, 3, 4 + n, 2 + n, 4 + n, 2 + n, 4 + n) are order equivalent if and only if n = 0, the previous arguments fails. We observe that spreadability implies, in particular, EMα (xy) = EMα xα 4+n (b)α 2 (b)α 4+n (b)α 2 (b)α 4+n (b) = E Mα
! n−1 1 4+k α (b)α 2 (b)α 4+k (b)α 2 (b)α 4+k (b) , x n k=0
but a direct application of the mean ergodic theorem is still out of reach.
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To overcome such difficulties we need to provide a more elaborated version of Theorem 8.3 which allows us to preserve relative localisation properties of the canonical filtration (MI )I while performing mean ergodic averages. Since this result is of interest in its own, we formulate it in greater generality than necessary for our purposes. Theorem 8.4. Let (M, ψ) be a probability space and suppose {αN }N ∈N0 is a family of ψpreserving completely positive linear maps of M satisfying ⊂ MαN+1 for all N ∈ N0 ; (i) MαN (ii) M = N ∈N0 MαN . Furthermore let 1 k αN n n−1
(n)
MN :=
and TN :=
k=0
" N
(N ) α lN Ml . l=0 l
Then we have SOT-
lim TN (x) = EMα0 (x)
N →∞
for any x ∈ M. Proof. Since the family {TN | N ∈ N0 } is bounded, its pointwise SOT -convergence follows by a standard approximation argument if we can establish this convergence on the weak∗ -dense α N ∗-subalgebra N ∈N0 M of M. (n)
Let x ∈ MαN0 for some N0 ∈ N and N N0 . Since αN (x) = x and thus MN (x) = x, the ordered product has at most N0 non-trivially acting factors: TN (x) =
$
# "N
(N ) α lN Ml l=0 l
(x) =
# "N −1 0
(N ) αllN Ml l=0
$ (x).
The assertions on the fixed point algebras Mαk imply that, for any k N and n ∈ N, EMαk αN = EMαk
(n)
and EMαk MN = EMαk .
Thus we can rewrite TN (x) as a finite telescope sum, assuming N N0 : (N )
(N )
(N )
TN (x) = M0 α1N M1 α22N M2
2
(N )
N · · · αN MN (x) (N −1)N
) = M0(N ) α1N M1(N ) α22N M2(N ) · · · αN00−1 MN(N0 −1 (x) $ # "N −1 (N ) 0 αllN Ml − EMαl EMαN0 (x) = l=0
+
# "N −2 0 l=0
$ (N ) αllN Ml − EMαl EMαN0 −1 (x)
C. Köstler / Journal of Functional Analysis 258 (2010) 1073–1120
+
# "N −3 0 l=0
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$ (N ) αllN Ml − EMαl EMαN0 −2 (x)
+ ··· (N ) (N ) + M0 − EMα0 α1N M1 − EMα1 EMα2 (x) (N ) + M0 − EMα0 EMα1 (x) + EMα0 (x). The strong operator topology and the ψ -topology generated by x → x2ψ := ψ(x ∗ x) coincide on bounded sets of M. Thus # "k−1 $ (N ) (N ) lN αl Ml − EMαl EMαk (x) 2k−1 Mk−1 − EMαk−1 EMαk (x)ψ , l=0 ψ
and the usual mean ergodic theorem, Theorem 8.3, entail that all terms of above telescope sum, except EMα0 (x), vanish in the limit N → ∞. 2 We will connect this refined mean ergodic theorem to partial shifts which canonically emerge from a spreadable endomorphism. Recall for this purpose the notion of partial shifts θN of N0 and their relation to order invariance of tuples (see Remark 1.9): θN (n) =
n if n < N; n + 1 if n N .
Clearly θN is an order preserving map of N0 into itself and so are the compositions of such maps with N ∈ N0 . Here we are interested in compositions of the type θN,l N :=
" N
θ iN +li i=0 i
(N −1)N +lN−1 N 2 +lN θN ,
= θ00 θ1N +l1 θ22N +l2 · · · θN −1 l
where l N = (l0 , l1 , . . . , lN ) ∈ {0, 1, . . . , N − 1}N +1 . Note that the θi ’s in the ordered product do not commute for different i’s. We record two simple, but crucial properties of this composition. Lemma 8.5. For any (N + 1)-tuples l N , k N ∈ {0, 1, . . . , N − 1}N +1 , it holds θN,l N (i) < θN,k N (j )
whenever i < j < N
and θN,l N (I ) ∩ θN,k N (J ) = ∅ whenever I ∩ J = ∅ and max I ∪ J < N.
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Proof. Since all θN ’s are order preserving, it suffices to consider j = i + 1. One calculates θN,l N (i + 1) − θN,l N (i) = 1 +
i (kj − lj ) + (i + 1)N + ki+1 > 0. j =0
Moreover this ensures that the images of disjoint sets I, J (bounded by N ) are disjoint.
2
Suppose the stationary process M ≡ (M, ψ, α, M0 ) is minimal and let, for N ∈ N, MN −1 :=
α k (M0 ).
0k
Spreadability of M allows us to promote the partial shifts θN of N0 to endomorphisms of M. Let α[i; a] := α i(1) (a1 )α i(2) (a2 ) · · · α i(n) (an ) for n-tuples i : [n] → N0 and a = (a1 , a2 , . . . , an ) ∈ Mn0 . Lemma 8.6. Suppose the endomorphism α of M is spreadable and let N ∈ N0 . Then the complex linear extension of the map α[i; a] → α[θN ◦ i; a] defines a ψ-preserving unital endomorphism αN of M, such that MN ⊂ MαN+1 . In particular, MN := (M, ψ, αN , MN ) is a minimal stationary process. alg
Proof. The map αN is well defined on the ∗-algebra MN0 , the C-linear span of monomials α[i; a]. Indeed, the faithfulness of ψ and spreadability give
α[θN ◦ ik ; ak ] = 0
⇔
k
⇔
2 $
2 $ # #
ψ α[θN ◦ ik ; ak ] = ψ α[ik ; ak ]
= 0
k
k
α[ik ; ak ] = 0.
k alg
Thus αN is well defined on MN0 . Now it is routine to check that αN extends to a ψ -preserving unital endomorphism of M, denoted by the same symbol. The inclusion MN −1 ⊂ MαN is alg immediately concluded by approximation from the definition of αN on MN0 . It is also clear that αN commutes with the modular automorphism group of (M, ψ) since α does so. Thus (M, ψ, αN , MN ) is a stationary process which is easily seen to be minimal. 2
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1111
Corollary 8.7. The minimal stationary processes MN and their endomorphisms αN are spreadable. Moreover, it holds for N ∈ N0 : (i) αN +1 |αN (M) = αN |αN (M) ; ⊂ MαN+1 ; (ii) MαN (iii) M = N ∈N0 MαN . Proof. The spreadability of MN is immediate from definition of αN in Lemma 8.6 and the spreadability of α. (i) Clearly, θN +1 |θN (N0 ) = θN |θN (N0 ) . Thus αN +1 and αN coincide on the C-linear span of all monomials of the form α[θN ◦ i; a] = αN (α[i; a)]). Now the assertion follows from the weak∗ density of this span in αN (M). (ii) MαN is contained in αN (M). By (i), αN and αN +1 coincide on αN (M). Thus MαN ⊂ MαN+1 . n αN by (iii) This is evident from the minimality of M since 0n
and y = α[j; b], q
for p-tuples i : [p] → I , a ∈ M0 and q-tuples j : [q] → J , b ∈ M0 . Recall that Mα is Mα -spreadable by Lemma 7.6 and so EMα0 (xy) = EMα α[i; a]α[j; b] = EMα α[θN,k N ◦ i; a]α[θN,k N ◦ j, b] for any k N ∈ {0, 1, . . . , N − 1}N +1 and N > max I ∪ J . By Lemma 8.5, the maps θN,k N are order preserving on N0 and I ∩ J = ∅ implies θN,k N (I ) ∩ θN,l N (J ) = ∅ for any (N + 1)-tuples k N , l N ∈ {0, 1, . . . , N − 1}N +1 . Thus EMα α[i; a]α[j; b] = EMα α[θN,k N ◦ i; a]α[θN,l N ◦ j, b] for all k N , l N . Consequently we can pass on the right side of this equation to the mean ergodic averages by summing over the variables k0 , k1 , . . . , kN and l0 , l1 , . . . , lN . Doing so we find EMα α[i; a]α[j; b] = EMα TN α[i; a] TN α[j; b]
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for all N > max I ∪ J , where TN :=
" N
1 k αN . n n−1
(N )
l=0
(n)
with MN :=
αllN Ml
k=0
Since Corollary 8.7 ensures that all assumptions of the refined mean ergodic theorem Theorem 8.4 are satisfied, the pointwise SOT -convergence of TN to EMα0 (= EMα ) for N → ∞ establishes EMα α[i; a]α[j; b] = EMα α[i; a] EMα α[j; b] for any i and j with disjoint ranges. This generalizes to the C-linear span of monomials α[in ; an ] and α[jn ; bn ], provided the range of the tuples in is contained in I and the range of the tuples jn is contained in J . Now a density argument establishes the factorization EMα (xy) = EMα (x)EMα (y) for all x ∈ MI and y ∈ MJ whenever I and J are finite disjoint subsets of N0 . Finally, another approximation removes the assumption of the finiteness of I and J . Thus we have established that the spreadability of a minimal stationary process M implies its full Mα factorizability. By Theorem 4.2, full Mα -factorizability and full Mα -independence are equivalent. In particular, we know already Mα = Mtail from Theorem 7.2. Finally, Corollary 6.8 entails that B is a full Bernoulli shift. 2 Remark 8.9. The refined version of the mean ergodic theorem, Theorem 8.4, is motivated in parts from product representations of endomorphisms as their study is started in [20] and as they are applied to braid group representations in [21]. Suppose the probability space (M, ψ) is equipped with a tower M0 ⊂ M1 ⊂ M2 ⊂ · · · of ψ-expected subalgebras such that M = (γk )k∈N ⊂ Aut(M, ψ) satisfying
n0 Mn
γk (Mn ) = Mn
and consider a family of automorphisms
if k n,
γk |Mn−1 = id |Mn−1
if k n + 1.
Then αN := lim γN +1 · · · γn n→∞
exists in the pointwise strong operator topology and defines a family of ψ-preserving endomorphisms {αN }N ∈N0 of M such that MN ⊂ MαN ⊂ MαN+1 for all N ∈ N0 .
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Suppose now in addition that αN |α k (M) = α0 |α k (M) 0
0
if k N .
Then it can be seen that the refined mean ergodic theorem preserves localization properties with respect to the filtration (AI )I ⊂N0 , where AI := i∈I α0i (M0 ). To be more precise, suppose x ∈ AI and y ∈ AJ with I ∩ J = ∅. Then for every N , there exist sets IN , JN with IN ∩ JN = ∅ such that TN (x) ∈ AIN and TN (y) ∈ AJN . Such a feature turned out to be crucial for the proof that spreadability implies conditional full independence. 9. Some applications and outlook We briefly address some further developments and applications of Theorem 0.2. 9.1. A glimpse on braidability The Artin’s braid group B∞ is presented by the generators σ1 , σ2 , . . . , subject to the relations σi σj σi = σj σi σj σi σj = σj σi
if | i − j |= 1, if | i − j |> 1.
Bn is an important extension of the symmetric group Sn and we introduce in [21] ‘braidability’ as a notion which extends exchangeability. Definition 9.1. A random sequence I with random variables ι ≡ (ιn )n0 : (A0 , ϕ0 ) → (M, ψ) is ρ-braidable if there exists a representation ρ : B∞ → Aut(M, ψ) satisfying: ιn = ρ(σn σn−1 · · · σ1 )ι0 ι0 = ρ(σn )ι0
for all n 1;
if n 2.
Note that the representation ρ may be non-faithful and comprises representations of S∞ . More precisely, it is shown in [21] that the following are equivalent: (i) I is exchangeable; (ii) I is ρ-braidable and ρ(σn2 ) = id for all n ∈ N. So exchangeability clearly implies braidability. A main result of [21] is that braidability is intermediate between two distributional symmetries and thus provides a refinement of the noncommutative extended de Finetti theorem, Theorem 0.2: Theorem 9.2. (See [21].) Let I be an infinite random sequence and consider the following statements:
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(a) (ab) (b) (c)
C. Köstler / Journal of Functional Analysis 258 (2010) 1073–1120
I I I I
is exchangeable; is braidable; is spreadable; is stationary and full Mtail -independent.
Then we have the implications: (a)
⇒
(ab)
⇒
(b)
⇒
(c).
Starting from braid group representations, this result implies a rich structure of triangular arrays of commuting squares, similar as they emerge from the Jones fundamental construction in subfactor theory. We refer the interested reader to [21] for further details and developments. We need another result from [21] to complete the proof of Theorem 0.2. Theorem 9.3. (See [21].) There exist examples of infinite random sequences such that the implications ‘(a) ⇐ (b)’ and ‘(b) ⇐ (c)’ fail in Theorem 0.2 resp. Theorem 9.2. Proof. See Theorem 5.6, Theorem 5.9, Example 6.1 and Example 6.4 in [21].
2
9.2. The prototype of a noncommutative conditioned central limit law Another immediate application of Theorem 0.2 is given by noncommutative central limit theorems. They are an integral component of quantum probability [14,28,19,77,59] and free probability [73,74,64,75,76]. Unified general versions of them are obtained in the setting of ∗-algebraic probability spaces in [65,68] and the related algebraic techniques are of growing interest in operator algebras. Especially Speicher’s interpolation technique for q-commutation relations [65] is successfully applied for results on hypercontractivity in [9,36] and the embedding of Pisier’s operator Hilbert space OH into the predual of the hyperfinite III1 factor due to Junge [33]. To control the existence of a limit distribution in a ∗-algebraic setting, general limit theorems need to stipulate three more or less technical conditions on mixed moments of the random variables: a singleton condition, a growth condition and some appropriate form of order-invariance condition on second order correlations [68]. These three conditions have been replaced by two conditions in [39] when working with tracial W∗ -algebraic probability spaces: a growth condition and order-invariance (which equals ‘spreadability’ herein). This leads to precise formulas for the higher moments of additive flows with stationary independent increments whenever they are spreadable. An application of Theorem 0.2 allows us to show that additive flows with spreadable increments have automatically independent stationary increments. In particular, one obtains for such additive flows a noncommutative generalization of [35, Theorem 1.15], the continuous version of the extended de Finetti theorem. Related results will be published elsewhere. Let us present here only a simple version of the central limit theorem for spreadable random sequences, the ‘discrete time’ analogue of spreadable additive flows. We need to introduce some notation for its formulation. Let O(p) denote the set of equivalence classes [i] for p-tuples i : {1, 2, . . . , p} → N0 under the following equivalence relation: two p-tuples i and j are order equivalent if
C. Köstler / Journal of Functional Analysis 258 (2010) 1073–1120
i(k) i(l)
⇔
j(k) j(l)
1115
for all k, l = 1, . . . , p.
Furthermore, let
& % O2 (p) := [i] ∈ O(p) i−1 (k) ∈ {0, 2}, k ∈ N0 , the set of all equivalence classes of p-tuples with pair partitions as pre-image and let P2 (p) denote the set of all pair partitions of {1, 2, . . . , p}. Note that P2 (p) has the cardinality p!! = (p − 1)(p − 3) · · · 5 · 3 · 1 for p even and p!! = 0 for p odd and that |O2 (p)|, the cardinality of O2 (p), satisfies p!! =
|O2 (p)| . (p/2)!
The following result can be easily deduced from [39, Theorem 4.4], since condition (d) of Theorem 0.2 implies the vanishing of so-called ‘singletons’. Theorem 9.4. Let the spreadable random sequence I be given by the random variables (ιn )n0 : (M0 , ψ0 ) → (M, ψ) and consider N −1 1 ιn (x) SN (x) := √ N n=0
for some fixed x ∈ M0 with EMtail (x) = 0. Then lim ψ SN (x)p = p!! · ap (x) N →∞
with the average ' ap (x) :=
1 |O2 (p)|
(
[i]∈O2 (p) ψ(ιi(1) (x)ιi(2) (x) · · · ιi(p) (x))
0
for even p, for odd p.
This result can be regarded as the prototype of a noncommutative version of conditional central limit theorems in classical probability. We refer the reader to [15] for more information on this matter. Note also that above theorem can be promoted to an operator equation: SOT-
lim EMtail SN (x)p = p!! · Ap (x)
N →∞
with the average ' Ap (x) :=
1 |O2 (p)|
0
(
[i]∈O2 (p) EMtail (ιi(1) (x)ιi(2) (x) · · · ιi(p) (x))
for even p, for odd p.
Let us discuss in greater detail the example that the ιk (x)’s mutually commute for fixed x. Then the averages a2p (x) and A2p (x) can be easily computed by Theorem 0.2 and the module property of conditional expectations:
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p a2p (x) = ψ EMtail x 2 , p A2p (x) = EMtail x 2 . If the tail algebra Mtail is trivial, we obtain the normal distribution as central limit law, since then a2p (x) = ψ(x 2 )p = a2 (x)p and thus lim ψ SN (x)2p = (2p)!! · a2 (x)p .
N →∞
But if Mtail is non-trivial, the limit law is different from the normal distribution; it is a mixture of them. There seems to be an interesting connection to interacting Fock space models (as introduced in [1,3]) in the conditional case. Given x ∗ = x ∈ M0 with EMtail (x) = 0 and EMtail (x 2 ) = 0 in the setting of above example, there exists a monotone increasing sequence (λ2p )p with λ2 = 1 such that, for all p, a2p (x) = λ2p a2 (x)p . Here the properties of (λ2p )p are deduced from the fact that Lp (Mtail , ψ|Mtail ) isomorphic to a classical Lp -space (w.r.t. some probability measure). Now λ2p+2 λ2p is concluded from the monotony of the Lp -norms. Already this simple class of examples hints at that non-trivial tail algebras lead to interesting examples of interacting Fock space models through central limit techniques, such that the limit object ‘limN →∞ SN (x)’ reappears as the sum of creation and annihilation operator on an appropriately chosen interacting Fock space. Moreover, it is worthwhile to mention that the central limit law is Wigner’s semicircle law if the averages ap (x) are connected to the second order moment ψ(x 2 ) by the formula a2p (x) =
p Cp ψ x2 , (2p)!!
whenever EMtail (x) = 0 and ψ(x 2 ) = 0. Here Cp denotes the p-th Catalan number. The amazing analogy of results in classical probability and free probability prompts of course the question if the condition A2p (x) =
p Cp E tail x 2 (2p)!! M
can be better understood in the context of freeness with amalgamation. At this stage of our knowledge we regard it to be of major interest to identify concrete central limit laws which can emerge from spreadable random sequences. This line of research is continued in [21], where we will investigate central limit laws in the context of braid group representations as stated in Theorem 9.2. At the time of this writing we have strong numerical evidence that the spectral distributions of q-Gaussian random variables are among the central limit laws for random sequences constructed on simple examples of Jones towers on the hyperfinite II1 factor.
C. Köstler / Journal of Functional Analysis 258 (2010) 1073–1120
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9.3. Noncommutative Lp -inequalities for spreadable random sequences As a third application we address Junge’s L1 -inequality for systems of independent, conditioned top-subsymmetric copies of a von Neumann algebra [33, Theorem 1.1]. Top-subsymmetry is a slight generalization of subsymmetry or, in our formulation, spreadability. By Theorem 0.2, the assertion of independence is automatically satisfied in the context of spreadability. Given the random sequence I , we identify the probability space (A0 , ψ0 ) with its embedding (M0 , ψ0 ) := (ι0 (A0 ), ψ|ι0 (A0 ) ) and thus have ι0 (x) = x for all x ∈ M0 . The endomorphisms ιk extend to isometric embeddings from L1 (M0 ) into L1 (M), the Haagerup L1 -spaces, and are denoted by the same symbol. Similarly, the state-preserving conditional expectation from M onto Mtail extends to a projection from L1 (M) onto L1 (Mtail ), in the following just denoted by E. We refer the reader for further information on the technical details to [33] and the references cited therein. The main inequality of [33] can now be reformulated as follows. We are indebted to Junge who pointed out to the author this immediate reformulation. Theorem 9.5. Suppose I is a spreadable random sequence with above identification and let x ∈ L1 (M0 ) with E(x) = 0. Then, for all n ∈ N, n−1 1/2 √ √ + nE x x ∗ 1/2 . ιk (x) ∼ inf nx1 1 + nE x2∗ x2 3 3 1 1 x=x1 +x2 +x3 k=0
1
Here a ∼ b means that there exists an absolute constant c > 0 such that c−1 a b ca. This constant is independent of n and x in the above stated theorem. A corollary of this inequality is the following estimate: n−1 1 1/2 + E x x ∗ 1/2 . lim √ ιk (x) ∼ inf E x2∗ x2 3 3 1 1 n→∞ n x=x2 +x3 k=0
1
Of course, a further immediate application is given by noncommutative Rosenthal inequalities of Junge and Xu [34]. They established the noncommutative version of inequalities for the p-norm of independent mean-zero random variables found by Rosenthal [60]. With Theorem 0.2 at our hands, spreadable random sequences produce a rich class of new examples. The noncommutative Rosenthal inequalities are even of interest for independent copies of a single random variable since we have still a very incomplete picture on the resulting central limit laws. Theorem 9.6. Let 2 p < ∞. Suppose I is a spreadable random sequence and let (xn )n0 ⊂ Lp (M0 ) with E(xn ) = 0 for all n. Then there exist universal constants δp and ηp such that, n−1
δp−1 sp,n (x)
where
k=0
ιk (xk ) ηp sp,n (x), p
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! 1 n−1 !1 ) ' n−1 ! 1 n−1 2 2
p p
ιk (xk ) sp,n (x) = max E xk∗ xk E xk xk∗ , , . k=0
p
k=0
p
k=0
p
We note that a similar inequality is valid for 1 < p < 2 (see [34, Theorem 6.1]). In the special case of constant selfadjoint sequences, i.e. xn = x, the above inequality yields n−1 1 % 1/2 ∗ 1/2 & , E xx . lim √ ιk (x) ∼p max E x ∗ x p p n→∞ n k=0
p
Here a ∼p b means that there exists a constant cp such that cp−1 a b cp a. Acknowledgments The present paper took its origin from work with Rolf Gohm on one of the most simple examples coming from the Jones fundamental construction [21], and joint work with Roland Speicher on the structure of noncommutative white noises [39]. At both occasions we found ‘spreadability’ without being aware of it. We are indebted to Marius Junge and Wojciech Jaworski who independently pointed out possible connections to distributional symmetries and initiated the author’s investigations resulting in the present paper. We are thankful to several helpful discussions with Benoit Collins, Rolf Gohm, Marius Junge, James Mingo and Roland Speicher in the course of writing this paper. References [1] L. Accardi, M. Bo˙zejko, Interacting Fock spaces and Gaussianization of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (4) (1998) 663–670. [2] L. Accardi, C. Cecchini, Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal. 45 (1982) 245–273. [3] L. Accardi, V. Crismale, Y. Lu, Constructive universal central limit theorems based on interacting Fock spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (4) (2005) 631–650. [4] L. Accardi, A. Frigerio, J. Lewis, Quantum stochastic processes, Publ. Res. Inst. Math. Sci. 18 (1982) 97–133. [5] L. Accardi, Y. Lu, A continuous version of de Finetti’s theorem, Ann. Probab. 21 (1993) 1478–1493. [6] D. Aldous, Exchangeability and related topics, in: École d’été de probabilités de Saint-Flour, XIII – 1983, in: Lecture Notes in Math., vol. 1117, Springer, Berlin, 1985. [7] C. Anantharaman-Delaroche, On ergodic theorems for free group actions on noncommutative spaces, Probab. Theory Related Fields 135 (4) (2006) 520–546. [8] A. Ben Ghorbal, M. Schürmann, Non-commutative notions of stochastic independence, Math. Proc. Cambridge Philos. Soc. 133 (3) (2002) 531–561. [9] P. Biane, Free hypercontractivity, Comm. Math. Phys. 184 (2) (1997) 457–474. [10] M. Bo˙zejko, B. Kümmerer, R. Speicher, q-Gaussian processes: Non-commutative and classical aspects, Comm. Math. Phys. 185 (1997) 129–154. [11] M. Bo˙zejko, R. Speicher, An example of generalized Brownian motion, Comm. Math. Phys. 137 (1991) 519–531. [12] M. Bo˙zejko, R. Speicher, Completely positive maps on Coxeter groups, deformed commutation relations and operator spaces, Math. Ann. 300 (1994) 97–120. [13] D. Cifarelli, E. Regazzini, De Finetti’s contribution to probability and statistics, Statist. Sci. 11 (4) (1996) 253–282. [14] C. Cushen, R. Hudson, A quantum-mechanical central limit theorem, J. Appl. Probab. 8 (3) (1971) 454–469. [15] J. Dedecker, F. Merlevède, Necessary and sufficient conditions for the conditional central limit theorem, Ann. Probab. 30 (3) (2002) 1044–1081. [16] D. Evans, Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford Math. Monogr., Oxford Science Publications, Oxford University Press, New York, 1998.
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[53] A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability, London Math. Soc. Lecture Note Ser., vol. 335, Cambridge University Press, 2006. [54] G. Pedersen, C∗ -Algebras and Their Automorphism Groups, Academic Press, 1979. [55] K. Petersen, Ergodic Theory, Cambridge Stud. Adv. Math., Cambridge University Press, Cambridge, 1983. [56] M. Pimsner, S. Popa, Entropy and index for subfactors, Ann. Sci. Ecole Norm. Sup. 19 (1986) 57–106. [57] S. Popa, Maximal injective subalgebras in factors associated with free groups, Adv. Math. 50 (1983) 27–48. [58] S. Popa, Orthogonal pairs of ∗-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983) 253–268. [59] J. Quaegebeur, A noncommutative central limit theorem for CCR-algebras, J. Funct. Anal. 57 (1) (1984) 1–20. [60] H. Rosenthal, On the subspaces of Lp , p > 2 spanned by sequences of independent random variables, Israel J. Math. 8 (1970) 273–303. [61] C. Rupp, Non-commutative Bernoulli shifts on towers of von Neumann algebras, PhD thesis, University Tübingen, dissertation, 1995. [62] C. Ryll-Nardzewski, On stationary sequences of random variables and the de Finetti’s equivalence, Colloq. Math. 4 (1957) 149–156. [63] S. Sakai, C∗ -Algebras and W∗ -Algebras, Springer-Verlag, 1971. [64] R. Speicher, A new example of ‘independence’ and ‘white noise’, Probab. Theory Related Fields 84 (1990) 141– 159. [65] R. Speicher, A noncommutative central limit theorem, Math. Z. 209 (1) (1992) 55–66. [66] R. Speicher, On universal products, in: Free Probability Theory, Waterloo, ON, 1995, in: Fields Inst. Commun., American Mathematical Society, Providence, RI, 1997. [67] R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (627) (1998). [68] R. Speicher, W. von Waldenfels, A general central limit theorem and invariance principles, in: Quantum Probability and Related Topics, vol. IX, 1994. [69] E. Størmer, Symmetric states of infinite tensor products of C∗ -algebras, J. Funct. Anal. 3 (1969) 48–68. [70] M. Takesaki, Theory of Operator Algebras I, Springer, 1979. [71] M. Takesaki, Theory of Operator Algebras II, Encyclopaedia Math. Sci., Springer, 2003. [72] H. van Leeuwen, H. Maassen, An obstruction for q-deformation of the convolution product, J. Phys. A 29 (15) (1996) 4741–4748. [73] D. Voiculescu, Symmetries of some reduced free product C∗ -algebras, in: S.S.H. Araki, C.C. Moore, D. Voiculescu (Eds.), Operator Algebras and Their Connection with Topology and Ergodic Theory, in: Lecture Notes in Math., vol. 1132, Springer, 1985. [74] D. Voiculescu, Addition of certain non-commuting variables, J. Funct. Anal. 66 (1986) 323–346. [75] D. Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991) 201–220. [76] D. Voiculescu, K. Dykema, A. Nica, Free Random Variables, CRM Monogr. Ser., vol. 1, American Mathematical Society, 1992. [77] W. von Waldenfels, An algebraic central limit theorem in the anti-commuting case, Z. Wahrsch. Verw. Gebiete 42 (1978) 135–140.
Journal of Functional Analysis 258 (2010) 1121–1139 www.elsevier.com/locate/jfa
Lp spectral theory and heat dynamics of locally symmetric spaces Lizhen Ji a,b,1 , Andreas Weber c,∗ a Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China b Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA c Institut für Algebra und Geometrie, Universität Karlsruhe (TH), Englerstr. 2, 76128 Karlsruhe, Germany
Received 26 September 2008; accepted 11 November 2009 Available online 18 November 2009 Communicated by C. Villani
Abstract In this paper we first derive several results concerning the Lp spectrum of locally symmetric spaces with rank one. In particular, we show that there is an open subset of C consisting of eigenvalues of the Lp Laplacian if p < 2 and that corresponding eigenfunctions are given by certain Eisenstein series. On the other hand, if p > 2 there is at most a discrete set of real eigenvalues of the Lp Laplacian. These results are used in the second part of this paper in order to show that the dynamics of the Lp heat semigroups for p < 2 is very different from the dynamics of the Lp heat semigroups if p 2. © 2009 Elsevier Inc. All rights reserved. Keywords: Locally symmetric spaces; Lp heat semigroups; Eisenstein series; Lp spectrum; Chaotic semigroups
1. Introduction The purpose of this paper is twofold. We are first concerned with the Lp spectrum of the Laplace–Beltrami operator on locally symmetric spaces with rank one and then we will use the obtained results about the Lp spectrum in order to show that the dynamics of the Lp heat semigroups for p < 2 is very different from the dynamics of the Lp heat semigroups if p 2. * Corresponding author.
E-mail addresses: [email protected] (L. Ji), [email protected] (A. Weber). 1 Partially supported by NSF grant DMS 0604878.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.011
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In contrast to the L2 spectrum of the Laplace–Beltrami operator on locally symmetric spaces the Lp spectrum (as set), p ∈ [1, ∞), is only known in special situations. But there are several reasons to study also the Lp spectrum. Firstly, from a physical point of view, the natural space to study heat diffusion is L1 : If the function u(t, ·) 0 denotes the heat distribution at the time t, the total amount of heat in some region Ω is given by the L1 norm of u(t, ·)|Ω and hence, the L1 norm has a physical meaning. But, on the other hand, L1 is more difficult to handle than the reflexive Lp spaces (p > 1) as the heat semigroup on Lp (p > 1) is always bounded analytic whereas this is in general not true for the heat semigroup on L1 . Secondly, there are already many results for differential operators on domains of Euclidean space concerning various aspects of Lp spectral theory, see e.g. [1,11–13,24–26,31–33,45]. Another point to mention is the fact that the question whether the Lp spectrum of the Laplace– Beltrami operator on a Riemannian manifold depends non-trivially on p is related to the geometry, in particular the volume growth, of the respective manifold [42]. But even if one is only interested in the L2 case, it could be worth studying the Lp spectra, as the knowledge of the Lp spectra yields in some cases further information on the decay of L2 eigenfunctions of the Laplace–Beltrami operator (see e.g. [14,27,43]). Furthermore, in [46] J. Wang was interested in the L2 spectrum of the Laplacian on complete Riemannian manifolds with non-negative Ricci curvature. In order to calculate this spectrum he first calculated the L1 spectrum and then used a result of Sturm [42] to show that the Lp spectrum does not depend on p. And finally, in the theory of (locally) symmetric spaces there are new phenomena that cannot occur in the L2 case. One example for this is the fact that the Lp spectrum (p > 2) of a symmetric space of non-compact type contains an uncountable number of eigenvalues of the Laplace–Beltrami operator on Lp . More precisely, we have the following result by Taylor: Theorem 1.1. (Cf. [43].) Let X denote a symmetric space of non-compact type. Then for any p ∈ [1, ∞) we have σ (X,p ) = PX,p , where 2 2 2 PX,p = ρ − z : z ∈ C, |Re z| ρ · − 1 . p Furthermore, if p > 2 any point in the interior of the parabolic region PX,p is an eigenvalue for X,p and eigenfunctions corresponding to these eigenvalues are given by spherical functions. For p = 2 we have PX,2 = [ρ2 , ∞) and X,2 has no eigenvalues (for a definition of ρ we refer to Section 2.2). This follows easily by using the Helgason–Fourier transform which turns X,2 into a multiplication operator. The Lp spectrum of certain locally symmetric spaces was examined in various articles: In [14] Davies, Simon and Taylor determined the Lp spectrum of real hyperbolic spaces and certain geometrically finite quotients Γ \ Hn . For symmetric spaces of non-compact type with rank one the results by Lohoué and Rychener in [34] should be mentioned. They derived estimates of the resolvent ( − z)−1 of the Laplacian on Lp spaces. More precisely, in the rank one case, all the complex numbers z are determined such that the respective resolvent is a bounded operator
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on Lp . More general locally symmetric spaces are treated in [43,47] but a precise identification of the Lp spectrum (as set) was only obtained if the universal covering is a rank one symmetric space. Related results are also contained in the classical paper [9] by Clerc and Stein. One of the aims in this paper is to derive similar results as in Theorem 1.1 for locally symmetric spaces M with rank one. For this, we first establish some Lp estimates of Eisenstein series which are generalized eigenfunctions for the Laplacian and therefore, they can be regarded as analog to the spherical functions on symmetric spaces of non-compact type. Our estimates show that plenty of the Eisenstein series are contained in Lp (M) if 1 p < 2. From this result in turn, it will follow that these Eisenstein series are actually (honest) eigenfunctions for the Lp Laplacian, 1 < p < 2. A point to mention here is that, compared to symmetric spaces, the roles p < 2 and p > 2 are interchanged. This means that the interior of a similarly defined parabolic region PM,p consists (besides a discrete set) of eigenvalues if 1 < p < 2. If p 2 there is only a discrete set of real eigenvalues possible. In the second part of this paper we investigate the dynamics of the Lp heat semigroups e−tM,p : Lp (M) → Lp (M) on locally symmetric spaces M with rank one and on products of rank one spaces. As all these spaces M have finite volume, it follows from Hölder’s inequality Lq (M) → Lp (M) if p q and hence, the semigroup e−tM,p can be regarded as an extension of the semigroup e−tM,q (see Section 2.1 for further details). In this part, we will make use of the results concerning the Lp spectrum in order to show that the Lp heat semigroups on locally symmetric spaces with rank one have a certain chaotic behavior if 1 < p < 2. This contrasts the fact that such a behavior is not possible for the Lp heat semigroups if p 2. One reason for this is that the spaces Lp (M) become larger and larger if p ↓ 1 and hence, there is more space for potential chaotic behavior available. In particular, if p < 2 there are suddenly plenty of Eisenstein series contained in Lp (M). If the (Q-)rank of a locally symmetric space with finite volume is greater than one the theory of Eisenstein series is more complicated than in the case of rank one but similar results hold in this case, too. A first step into this direction is Proposition 4.9 where Riemannian products of locally symmetric spaces with rank one are treated. The general case is more difficult to work out but as the asymptotic behavior of an automorphic form at infinity is controlled by its constant term a similar result as Proposition 3.5 can be proven. We plan to treat the higher Q-rank case in a future publication. Analogous results for symmetric spaces of non-compact type have been obtained in [28]. 2. Preliminaries 2.1. The heat semigroup on Lp spaces In this section we denote by M an arbitrary complete Riemannian manifold and by = − div(grad) the Laplace–Beltrami operator acting on differentiable functions of M. If we denote by M the Laplacian with domain Cc∞ (M) (the set of differentiable functions with compact support), this is an essentially self-adjoint operator and hence, its closure M,2 is a self-adjoint operator on the Hilbert space L2 (M). Since M,2 is positive, −M,2 generates a bounded analytic semigroup e−tM,2 on L2 (M) which can be defined by the spectral theorem for unbounded
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self-adjoint operators. The semigroup e−tM,2 is a submarkovian semigroup (i.e., e−tM,2 is positive and a contraction on L∞ (M) for any t 0) and we therefore have the following: (1) The semigroup e−tM,2 leaves the set L1 (M) ∩ L∞ (M) ⊂ L2 (M) invariant and hence, e−tM,2 |L1 ∩L∞ may be extended to a positive contraction semigroup Tp (t) on Lp (M) for any p ∈ [1, ∞]. These semigroups are strongly continuous if p ∈ [1, ∞) and consistent in the sense that Tp (t)|Lp ∩Lq = Tq (t)|Lp ∩Lq . (2) Furthermore, if p ∈ (1, ∞), the semigroup Tp (t) is a bounded analytic semigroup with angle √ of analyticity θp π2 − arctan 2|p−2| . p−1 For a proof of (1) we refer to [11, Theorem 1.4.1]. For (2) see [33]. In general, the semigroup T1 (t) needs not be analytic. However, if M has bounded geometry T1 (t) is analytic in some sector (cf. [44,10]). In the following, we denote by −M,p the generator of Tp (t) and by σ (M,p ) the spectrum of M,p . Furthermore, we will write e−tM,p for the semigroup Tp (t). Because of (2) from above, the Lp -spectrum σ (M,p ) has to be contained in the sector π z ∈ C \ {0}: arg(z) − θp ∪ {0} 2 |p − 2| ⊂ z ∈ C \ {0}: arg(z) arctan √ ∪ {0}. 2 p−1
If we identify as usual the dual space of Lp (M), 1 p < ∞, with Lp (M), p1 + p1 = 1, the dual operator of M,p equals M,p and therefore we always have σ (M,p ) = σ (M,p ). It should also be mentioned that the family M,p , p 1, is consistent, which means that the restrictions of M,p and M,q to the intersection of their domains coincide: Lemma 2.1. If p, q ∈ [1, ∞), the operators M,p and M,q are consistent, i.e. M,p f = M,q f
for any f ∈ dom(M,p ) ∩ dom(M,q ).
Proof. Since the semigroups e−tM,p and e−tM,q are consistent, we have for f ∈ dom(M,p ) ∩ dom(M,q ): ·Lp 1 −tM,p e f − f −−− → −M,p f t
(t ↓ 0)
and 1 ·Lq 1 −tM,p e f − f = e−tM,q f − f −−− → −M,q f t t Furthermore, M,p f − M,q f ∈ Lp (M) + Lq (M)
(t ↓ 0).
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and Lp (M) + Lq (M) = {h1 + h2 : h1 ∈ Lp (M), h2 ∈ Lq (M)} is a Banach space for the norm gLp +Lq := inf h1 Lp + h2 Lq : h1 ∈ Lp (M), h2 ∈ Lq (M) with g = h1 + h2 . In particular, we obtain 1 −t M,p f − f + M,p f − M,q f Lp +Lq e f M,p t Lp 1 −t M,q + e f − f + M,q f q → 0 (t ↓ 0). t L
2
If M is a Riemannian manifold with finite volume we have by Hölder’s inequality Lq (M) → for 1 p q ∞. Hence, by consistency, the semigroup e−tM,p (resp. M,p ) can be regarded as extension of the semigroup e−tM,q (resp. M,q ), p q. We conclude this subsection with a general result that will be needed later. For this we first recall a uniqueness result of Lp solutions of the heat equation by Strichartz, cf. [41, Theorem 3.9].
Lp (M)
Theorem 2.2. Let v : (0, ∞) × M → R denote a differentiable solution of the heat equation ∂ p Dt for some constants C p ∂t u = −u with v(t, ·) ∈ L (M) for each t > 0 and v(t, ·)L Ce p and D, p ∈ (1, ∞). Then there is a function f ∈ L (M) with v(t, x) = e−tM,p f (x). Corollary 2.3. Let p ∈ (1, ∞) and f : M → R denote a differentiable function such that f ∈ Lp (M) and f = μf for some μ ∈ R. Then f ∈ dom(M,p ) and M,p f = μf . Proof. We put v(t, x) = e−μt f (x). Then the conditions of Theorem 2.2 are obviously fulfilled and hence, e−tM,p f (x) = e−μt f (x). From this it follows 1 −t M,p f − f + μf e t p → 0 (t → 0) L and therefore M,p f = μf .
2
Note, that it is not completely obvious that a differentiable Lp function f that satisfies the eigenequation f = μf is contained in the domain of M,p . The purpose of Corollary 2.3 was to show exactly this. We do not know whether Theorem 2.2 and Corollary 2.3 are true in the case p = 1, too. Therefore, we need to restrict ourselves in some situations below to the case p ∈ (1, ∞). Note also, that if p = ∞ a uniqueness result as Theorem 2.2 cannot hold as there are Riemannian manifolds on which non-constant solutions v(t, x) of the heat equation exist such that v(0, x) = 1, cf. [2, Proposition 7.9].
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2.2. Locally symmetric spaces Let X denote always a symmetric space of non-compact type. Then G := Isom0 (X) is a noncompact, semi-simple Lie group with trivial center that acts transitively on X and X = G/K, where K ⊂ G is a maximal compact subgroup of G. We denote the respective Lie algebras by g and k. Given a corresponding Cartan involution θ : g → g we obtain the Cartan decomposition g = k ⊕ p of g into the eigenspaces of θ . The subspace p of g can be identified with the tangent space TeK X. We assume, that the Riemannian metric ·,· of X in p ∼ = TeK X coincides with the restriction of the Killing form B(Y, Z) := tr(ad Y ◦ ad Z), Y, Z ∈ g, to p. For any maximal abelian subspace a ⊂ p we refer to Σ = Σ(g, a) as the set of restricted roots for the pair (g, a), i.e. Σ contains all α ∈ a∗ \ {0} such that hα := Y ∈ g: ad(H )(Y ) = α(H )Y for all H ∈ a = {0}. These subspaces hα = {0} are called root spaces. Once a positive Weyl chamber a+ in a is chosen, we denote by Σ + the subset of positive 1
roots and by ρ := 2 α∈Σ + (dim hα )α half the sum of the positive roots (counted according to their multiplicity). We say that a locally symmetric space M = Γ \ X, where Γ ∼ = π1 (M) is a non-uniform lattice in G that acts without fixed points on X, has rank one if it admits a decomposition M = M0 ∪ Z1 ∪ · · · ∪ Zk ,
(1)
into a compact Riemannian manifold M0 with boundary and finitely many ends Zi , i = 1, . . . k, associated to rank one Γ cuspidal parabolic subgroups Pi ⊂ G, i = 1, . . . , k, in particular, each Zi is a fibered cusp and M = Γ \ X is a non-compact locally symmetric space with finite volume, cf. also [39]. If rank(X) 2 it follows by Margulis’ arithmeticity result that Γ is in particular arithmetic [35,50], if rank(X) = 1 this needs however not be true [20,35]. We will now recall some basic facts about the geometry and L2 spectral theory of locally symmetric spaces with rank one in order to fix notation. More details can be found e.g. in [22,29,38,19]. For a more general class of Riemannian manifolds (manifolds with cusps of rank one) which are basically defined via the decomposition (1) above, we refer to W. Müller’s book [37]. 2.2.1. Langlands decomposition and reduction theory By P = NP AP MP we denote the Langlands decomposition of a rank one Γ cuspidal parabolic subgroup P ⊂ G into a unipotent subgroup NP , a one-dimensional abelian subgroup AP , and a reductive subgroup MP . In the case where X denotes a higher rank symmetric space, this decomposition can be found in [8, III.1.11]. In the case where X has rank one, we refer to [8, I.1.9]. See also the remark concerning the comparison of the real and rational Langlands decomposition in [8, III.1.12].
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If we denote by XP = MP /K ∩ MP the boundary symmetric space, we have the horocyclic decomposition of X: X∼ = NP × AP × XP . More precisely, if we denote by τ : MP → XP the canonical projection, we have an analytic diffeomorphism n, a, τ (m) → nam · x0 ,
μ : NP × AP × XP → X,
(2)
for some x0 ∈ X. Note, that the boundary symmetric space XP is a Riemannian product of a symmetric space of non-compact type by a Euclidean space. We denote in the following by g, aP , and nP the Lie algebras of the (real) Lie groups G, AP , and NP defined above. Associated with the pair (g, aP ) there is a root system Φ(g, aP ). If we define for α ∈ Φ(g, aP ) the root spaces gα := Z ∈ g: ad(H )(Y ) = α(H )(Y ) for all H ∈ aP , we have the root space decomposition
g = g0
gα .
α∈Φ(g,aP )
Furthermore, the Γ cuspidal parabolic subgroup P defines an ordering of Φ(g, aP ) such that nP =
gα .
α∈Φ + (g,a
P)
The root spaces gα , gβ to distinct positive roots α, β ∈ Φ + (g, aP ) are orthogonal with respect to the Killing form: B(gα , gβ ) = {0}. We also define ρP :=
α∈Φ + (g,a
(dim gα )α. P)
If rank(X) = 1, we have ρ = ρP . If X is a higher rank symmetric space this needs not always be true. But as we assume that M = Γ \ X is a rank one locally symmetric space the root systems Φ(g, aPi ) are canonically isomorphic (cf. [4, 11.9]) and moreover, we can conclude ρP1 = · · · = ρPk . Furthermore, we denote by Φ ++ (g, aP ) the set of simple positive roots. Recall, that we call a positive root α ∈ Φ + (g, aP ) simple if 12 α is not a root.
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Let us now define for any t ∈ aP the set AP ,t = eH ∈ AP : α(H ) > α(t) for all α ∈ Φ ++ (g, aP ) . Then any end Zi in the decomposition of the rank one locally symmetric space M can be identified with a so-called Siegel set Ui × APi ,ti × Vi ⊂ NPi × APi × XPi , more precisely, Zi = π(Ui × APi ,ti × Vi ), where π : X → Γ \ X denotes the canonical projection and Ui ⊂ NPi , Vi ⊂ XPi are sufficiently large bounded subsets. 2.2.2. L2 spectral theory One knows that the L2 spectrum σ (M,2 ) of the Laplace–Beltrami operator M,2 on a rank one (non-compact) locally symmetric space M is the union of a point spectrum and an absolutely continuous spectrum. The point spectrum consists of a (possibly infinite) sequence of eigenvalues 0 = λ0 < λ1 λ2 · · · with finite multiplicities such that below any finite number there are only finitely many eigenvalues. The absolutely continuous spectrum equals [b2 , ∞) where b = ρP1 = · · · = ρPk =: ρP . In what follows, we denote by L2dis (M) the subspace spanned by all eigenfunctions of M,2 and by L2con (M) the orthogonal complement of L2dis (M) in L2 (M). The absolutely continuous part of the L2 spectrum is parametrized by generalized eigenfunctions of Γ \ X which are given by Eisenstein series. Therefore, we recall several basic facts about Eisenstein series. Our main reference here is [22]. Definition 2.4. Let f be a measurable, locally integrable function on Γ \ X. The constant term fP of f along some rank one Γ cuspidal parabolic subgroup P of G is defined as
fP (x) =
f (nx) dn,
(ΓP ∩NP )\NP
where the measure dn is normalized such that the volume of (ΓP ∩ NP ) \ NP equals one and ΓP = Γ ∩ P . A function f on Γ \ X with the property fP = 0 for all rank one Γ cuspidal parabolic subgroups P of G is called cuspidal and the subspace of cuspidal functions in L2 (Γ \ X) is denoted by L2cus (Γ \ X). It is known that L2cus (M) ⊂ L2dis (M) and this inclusion is in general strict as the non-zero constant functions are not contained in L2cus (M) if M is non-compact.
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Let P be rank one Γ cuspidal parabolic subgroup of G and ΓMP the image of ΓP = Γ ∩ P under the projection NP AP MP → MP . Then, ΓMP acts discretely on the boundary symmetric space XP and the respective quotient ΓMP \ XP , called boundary locally symmetric space, has finite volume. Furthermore, we denote by a∗P the dual of aP and put ∗ ++ a∗+ (g, aP ) . P = λ ∈ aP : λ, α > 0 for all α ∈ Φ For any ϕ ∈ L2cus (ΓMP \ XP ) and Λ ∈ a∗P ⊗ C with Re(Λ) ∈ ρP + a∗+ P we define the Eisenstein series E(P |ϕ, Λ) as follows: E(P |ϕ, Λ : x) =
e(ρP +Λ)(HP (γ x)) ϕ zP (γ x) ,
(3)
γ ∈ΓP \Γ
where μ(nP (x), eHP (x) , zP (x)) = x (cf. (2)). This series converges uniformly for x in compact subsets of X and is holomorphic in Λ. Furthermore, E(P |ϕ, Λ) can meromorphically be continued (as a function of Λ) to a∗P ⊗ C. By definition, the Eisenstein series are Γ invariant and hence, they define functions on M = Γ \ X. We have the following lemma. Lemma 2.5. Let ϕ ∈ L2cus (ΓMP \ XP ) be an eigenfunction of ΓMP \XP ,2 with respect to some eigenvalue ν. Then we have for any Λ ∈ a∗P ⊗ C that is not a pole of E(P |ϕ, Λ) the following: E(P |ϕ, Λ) = ν + ρP 2 − Λ, Λ E(P |ϕ, Λ). Proof. In rational horocyclic coordinates we have for a function f that is constant along NP the formula f = ρP 2 f + eρP AP e−ρP f + XP f, where AP , and XP denote the Laplacians on AP and XP . This follows from an analogous calculation as in the proof of [23, Proposition 3.8] or [30, Theorem 15.4.1]. We therefore obtain e(ρP +Λ)(HP ) ϕ(zP ) = ν + ρP 2 − Λ, Λ e(ρP +Λ)(HP ) ϕ(zP ). As is G invariant, the other terms in the Eisenstein series E(P |ϕ, Λ) satisfy this equation too, and hence, the Eisenstein series E(P |ϕ, Λ) itself satisfies this equation in the region of absolute convergence. By meromorphic continuation, the claim follows. 2 We conclude this section with the description of the constant term of an Eisenstein series on a rank one locally symmetric space. The boundary locally symmetric space ΓMP \ XP is compact for rank one Γ cuspidal parabolic subgroups P and thus any L2 function on ΓMP \ XP is cuspidal as the cuspidal condition is empty. Let μ be an eigenvalue of some boundary locally symmetric space ΓMPi \ XPi and choose an orthonormal basis of the μ-eigenspace of ΓMPj \ XPj for any j = 1, . . . , k. The union of these
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μ
μ
μ
bases is denoted by {ϕ1 , . . . , ϕl(μ) }. Each ϕm is associated to a unique Pj (m) , i.e. ϕm is an eigenμ function on ΓMPj (m) \ XPj (m) and defines an Eisenstein series E(Pj (m) |ϕm , Λ). The poles of the Eisenstein series in the half plane Re(Λ) 0 are contained in the interval (0, ρP ]. Furthermore, if Λ is a pole, ρP 2 − Λ, Λ is an L2 eigenvalue of the Laplacian and an eigenfunction is given by the residue of the corresponding Eisenstein series [22,38]. μ μ For the constant term EPj (Pj (m) |ϕm , Λ) of E(Pj (m) |ϕm , Λ) along Pj we have (for more details cf. [29,38]) μ HP (ρ +Λ)(HPj ) μ , Λ e j z = δj,j (m) e Pj ϕm (z) EPj Pj (m) ϕm +
l(μ)
e
(ρPj −Λ)(HPj )
μ cmi (Λ)ϕi (z),
(4)
i=1
where cmi (Λ) are the entries of the scattering matrix cPj |Pj (m) (w : Λ). The scattering matrix cP2 |P1 (w : Λ) is a bounded (linear) operator L2cus (ΓMP1 \ XP1 ) → L2cus (ΓMP2 \ XP2 ) and [cP2 |P1 (w : Λ)ϕ] and ϕ are eigenfunctions for the same eigenvalue. 3. Lp spectral theory of locally symmetric spaces Let us denote in the following by M = Γ \ X a locally symmetric space with rank one, by λ0 , . . . , λr the eigenvalues that are strictly smaller than ρP 2 , and by P1 , . . . , Pk representatives of rank one Γ cuspidal parabolic subgroups of G. For any p ∈ [1, ∞) we define the parabolic region 2 2 2 PM,p = ρP − z : z ∈ C, |Re z| ρP · − 1 ⊂ C. p
(5)
Our main concern in this section is to prove (for the notation see the preceding section) Theorem 3.1. Let M = Γ \ X denote a locally symmetric space with rank one, p ∈ (1, 2), and Λ ∈ aPj ⊗ C with |Re(Λ)(H 0 )| < 2−p p ρP . Then the Eisenstein series μ E Pj (m) ϕm ,Λ are eigenfunctions of M,p with eigenvalue μ + ρP 2 − (Λ(H 0 ))2 if Λ is not a pole. The proof of Theorem 3.1 follows from Proposition 3.5 below, Lemma 2.5, and Corollary 2.3. Theorem 3.1 contrasts the following fact. Proposition 3.2. Let M = Γ \ X denote a locally symmetric space with rank one and p 2. Then there is at most a discrete set of eigenvalues for M,p .
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Proof. In the case p = 2 this is well known and was stated in the previous section. Let now p > 2 and assume that ϕ is some eigenfunction for M,p . As Lp (M) → L2 (M) and because of Lemma 2.1 the function ϕ is also an eigenfunction for M,2 . This shows the claim. 2 In what follows, H 0 ∈ a+ Pj denotes the unique element with norm one (note, that dim APj = 1), i.e. we have ρPj (H 0 ) = ρP . Lemma 3.3. Let Sj = Uj × APj ,tj × Vj denote a Siegel set associated with Pj , p ∈ [1, 2), and let Λ ∈ aPj ⊗ C with |Re(Λ)(H 0 )| < 2−p p ρP . Then we have μ , Λ ∈ Lp (Sj ) EPj Pj (m) ϕm if Λ is not a pole. Proof. The volume form of the symmetric space X = NPj × APj × XPj with respect to rational horocyclic coordinates is given by dvolX = h(z)e−2ρP y dz dy where h > 0 is smooth on NPj × XPj and log(a) = yH 0 for any a ∈ APj , cf. [5,6]. The integrals
∞ (ρ ±Λ(H 0 ))y p −2ρ y P e P e dy tj
are readily seen to be finite if |Re(Λ)(H 0 )| <
2−p p ρP ,
and the claim follows.
2
Lemma 3.4. The functions μ μ E Pj (m) ϕm , Λ − EPj Pj (m) ϕm ,Λ are rapidly decreasing in the Siegel set Sj if Λ is not a pole. Proof. Recall, that we call a Γ invariant function f on X rapidly decreasing on a Siegel set S associated to a rank one Γ cuspidal parabolic subgroup P if for all Λ ∈ a∗ we have supx∈S |f (x)|eΛ(HP (x)) < ∞, see e.g. [36, I.2.12]. The proof now follows from [22, p. 13]. In the case G = SL(2, R) it can also be found in [7, 7.6]. 2 From the preceding lemmas it follows immediately Proposition 3.5. Let M = Γ \ X denote a locally symmetric space with rank one, p ∈ [1, 2), and Λ ∈ aPj ⊗ C with |Re(Λ)(H 0 )| < 2−p p ρP . Then we have μ , Λ ∈ Lp (M) E Pj (m) ϕm if Λ is not a pole.
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From Theorem 3.1 we immediately obtain Corollary 3.6. There is a discrete set B ⊂ PM,p ∩ {z ∈ C: Im(z) > 0} such that each point in the interior of PM,p \ B is an eigenvalue of M,p if p ∈ (1, 2) and PM,p ⊂ σ (M,p ) for all p ∈ (1, ∞). Proof. Let B˜ denote the (discrete) set of points Λ ∈ aPj ⊗ C with − 2−p p ρP < Re(Λ) < 0 ˜ such that each Λ ∈ B is a pole for all Eisenstein series. We define B = z ∈ C: ∃Λ ∈ B˜ such that z = ρP 2 − Λ, Λ . Then the first statement follows clearly when we choose μ = 0. Note, that this is possible as the boundary locally symmetric space is compact. Note also that the poles with positive real part correspond to L2 eigenvalues (see Section 2.2.2) and hence to Lp eigenvalues if 1 < p 2. The second statement follows by duality and from the fact that the spectrum is a closed subset of C. 2 As by Hölder’s inequality L2 (M) → Lp (M) for any p ∈ (1, 2], it follows that each L2 eigenvalue λj is also an Lp eigenvalue (cf. Corollary 2.3). Hence, by duality, {λj : j ∈ N} ⊂ σ (M,p ) for all p ∈ (1, ∞). We therefore have {λ0 , . . . , λr } ∪ PM,p ⊂ σ (M,p ) for all p ∈ (1, ∞). From Taylor’s results in [43] it follows also an “upper bound” for the Lp spectrum, i.e. σ (M,p ) ⊂ {λ0 , . . . , λr } ∪ PM,p ,
where PM,p
2 2 2 = ρP − z : z ∈ C, |Re z| ρ · − 1 p
and hence, PM,p ⊂ PM,p . Note, that we have equality here if and only if ρ = ρP and this condition is obviously fulfilled if X is a rank one symmetric space as in this case dim a = dim aP . But the condition ρ = ρP holds also for an important class of (Q-)rank one locally symmetric spaces M = Γ \ X – the so-called Hilbert modular varieties. For these spaces X can be a higher rank symmetric space. To introduce this class, let k be a totally real number field, and Ok be the ring of integers of k. Then SL(2, Ok ) is called a Hilbert modular group. It is an arithmetic subgroup of G = SL(2, R) × · · · × SL(2, R) = SL(2, R)r , where there is one factor for each embedding of k into R. The group SL(2, Ok ) acts properly on the product (H2 )r = H2 × · · · × H2 with a finite volume quotient, which is called a Hilbert modular variety. More generally, for any finite index subgroup Γ of SL(2, Ok ), the quotient of Γ \ H2 × · · · × H2 is often also called a Hilbert modular variety. The Q-rank of M = Γ \ X is equal to 1 whereas the rank of its associated symmetric space, i.e., the universal covering of the Hilbert modular variety, is equal to r. Unless the number field k = Q, the symmetric space has rank strictly greater than 1. Let P∞ be the parabolic subgroup of upper triangular matrices of SL(2, R). Then there exists a minimal rational parabolic subgroup r . The R-split component A P whose real locus P is equal to the product P∞ × · · · × P∞ = P∞ P
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in the real Langlands decomposition of P is equal to the product A∞ × · · · × A∞ , where A∞ =
a 0
0
a −1
: a>0 ,
and the Q-split component AP of the rational Langlands decomposition of P is the onedimensional diagonal subgroup AP = (g, . . . , g): g ∈ A∞ ⊂ A∞ × · · · × A∞ . Under the identification of the dual space aP with itself, half the sum of positive roots ρ is given by ρ = ( 12 , . . . , 12 ). The orthogonal projection of ρ onto the subspace aP is equal to half the sum ρP of the rational roots. This implies that ρ = ρP . More information about Hilbert modular varieties can be found in [18]. From these remarks we obtain Corollary 3.7. Let M denote a locally symmetric space with Q-rank one whose universal covering is a symmetric space of rank one or let M denote a Hilbert modular variety. Then we have σ (M,p ) = {λ0 , . . . , λr } ∪ PM,p for p ∈ (1, ∞). 4. Heat dynamics 4.1. Chaotic semigroups There are many different definitions of chaos. We will use the following one which is basically an adaption of Devaney’s definition [17] to the setting of strongly continuous semigroups, cf. [16]. Definition 4.1. A strongly continuous semigroup T (t) on a Banach space B is called chaotic if the following two conditions hold: (i) There exists an f ∈ B such that its orbit {T (t)f : t 0} is dense in B. (ii) The set of periodic points {f ∈ B: ∃t > 0 such that T (t)f = f } is dense in B. Remark 4.2. (1) As with {T (t)f : t 0} also the set {T (q)f : q ∈ Q0 } is dense, B is necessarily separable. (2) The orbit of any point T (t)f in a dense orbit {T (t)f : t 0} is again dense in B. Hence, the set of points with a dense orbit is a dense subset of B or empty. (3) For a separable Banach space B condition (i) in the definition above is equivalent to topological transitivity of the semigroup T (t), which means that for any pair of non-empty open subsets U, V ⊂ B there is a t > 0 with T (t)U ∩ V = ∅, cf. [16].
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(4) If both subsets B0 = f ∈ B: T (t)f → 0 (t → ∞) and B∞ = f ∈ B: ∀ε > 0 ∃g ∈ B, t > 0 such that g < ε, T (t)g − f < ε are dense in B, the semigroup T (t) has dense orbits. However, this condition is not necessary, cf. [16]. (5) Chaotic semigroups exist only on infinite dimensional Banach spaces. When looking at the Jordan canonical form of a bounded operator on a (real or complex) finite dimensional Banach space a proof of this is straightforward, see e.g. [21, Proposition 11]. A sufficient condition for a strongly continuous semigroup to be chaotic in terms of spectral properties of its generator was given by Desch, Schappacher, and Webb: Theorem 4.3. (See [16].) Let T (t) denote a strongly continuous semigroup on a separable Banach space B with generator A and let Ω denote an open, connected subset of C with Ω ⊂ σpt (A) (the point spectrum of A). Assume that there is a function F : Ω → B such that (i) Ω ∩ iR = ∅. (ii) F (λ) ∈ ker(A − λ) for all λ ∈ Ω. (iii) For all φ ∈ B in the dual space of B, the mapping Fφ : Ω → C, λ → φ ◦ F is analytic. Furthermore, if for some φ ∈ B we have Fφ = 0 then already φ = 0 holds. Then the semigroup T (t) is chaotic. In [16] it was also required that the elements F (λ), λ ∈ Ω, are non-zero but as remarked in [3] this assumption is redundant. In order to make this paper more comprehensive, we include the idea of the proof. Proof of Theorem 4.3. A major role in the proof is played by the following observation: let U ⊂ Ω be any subset that contains an accumulation point. Then it follows that the subset BU = span{F (λ): λ ∈ U } is dense in B. Indeed, if we suppose the contrary, by the Hahn–Banach Theorem there exists some φ ∈ B , φ = 0, such that φ ◦ F (λ) = 0 for all λ ∈ U . As U contains an accumulation point, it follows from the identity theorem for complex analytic functions that Fφ = 0. But this is a contradiction. For the subsets U0 = {λ ∈ Ω: Re(λ) < 0}, U∞ = {λ ∈ Ω: Re(λ) > 0}, and Uper = Ω ∩ iQ it follows now BU0 ⊂ B0 , BU∞ ⊂ B∞ , and BUper ⊂ {f ∈ B: ∃t > 0 such that T (t)f = f }. As all these sets are dense in B, the proof is complete. 2 In the theory of dynamical systems chaotic semigroups are highly unwanted because of their difficult dynamics. Not much more appreciated are so-called subspace chaotic semigroups: Definition 4.4. A strongly continuous semigroup T (t) on a Banach space B is called subspace chaotic if there is a closed, T (t) invariant subspace V = {0} of B such that the restriction T (t)|V is a chaotic semigroup on V.
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Because of Remark 4.2 such a subspace is always infinite dimensional. Banasiak and Moszy´nski showed that a subset of the conditions in Theorem 4.3 yield a sufficient condition for subspace chaos: Theorem 4.5. (See [3, Criterion 3.3].) Let T (t) denote a strongly continuous semigroup on a separable Banach space B with generator A. Assume, there is an open, connected subset Ω ⊂ C and a function F : Ω → B, F = 0, such that (i) Ω ∩ iR = ∅. (ii) F (λ) ∈ ker(A − λ) for all λ ∈ Ω. (iii) For all φ ∈ B , the mapping Fφ : Ω → C, λ → φ ◦ F is analytic. Then the semigroup T (t) is subspace chaotic. Furthermore, the restriction of T (t) to the T (t) invariant subspace V = span F (Ω) is chaotic. The proof of this result is similar to the proof of Theorem 4.3. Note, that it is not required Ω ⊂ σpt (A) here, i.e. either F (λ) is an eigenvector or F (λ) = 0. But, as explained in [3], the assumption Ω ⊂ C is not really weaker. 4.2. Lp heat dynamics on locally symmetric spaces Theorem 4.6. Let M = Γ \ X denote a locally symmetric space with rank one. (a) If p ∈ (1, 2) there is a constant cp > 0 such that for any c > cp the semigroup e−t (M,p −c) : Lp (M) → Lp (M) is subspace chaotic. (b) If p 2 and c ∈ R the semigroup e−t (M,p −c) is not subspace chaotic. Proof. For the proof of part (a), we will check the conditions of Theorem 4.5. If p < 2, the interior of PM,p ∩ {z ∈ C: Im(z) < 0} consists completely of eigenvalues, cf. Corollary 3.6, and the apex of PM,p is at the point cp = ρP 2 − ρP 2 ·
2 −1 p
2 =
1 4ρP 2 1− . p p
Hence, the point spectrum of (M,p − c) intersects the imaginary axis for any c > cp . We assume in the following c > cp and denote by Ω the interior of the set (PM,p − c) ∩ z ∈ C: Im(z) < 0 . Then, if the usual analytic branch of the square root is chosen, Ω is mapped (analytically) by −1 h(z) = iρP z + c − ρP 2 onto the strip 2 −1 . z ∈ C: Im(z) > 0, 0 < Re(z) < p
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If we now define F : Ω → Lp (M),
z → E Pj (1) ϕ10 , h(z)ρPj (1)
the map Ff : Ω → C, z → M F (z)(x)f (x) dx is analytic as a composition of analytic mappings for all f ∈ Lp (M). Note, that the integral is always finite as the Eisenstein series F (z) are contained in Lp (M). Furthermore, it follows from Theorem 3.1 that each F (z) is an eigenfunction of (M,p − c) for the eigenvalue z and the proof of part (a) is complete. If p 2, the point spectrum of M,p , and hence of (M,p − c), is a discrete subset of R. On the other hand, the intersection of the point spectrum of the generator of a chaotic semigroup with the imaginary axis is always infinite, cf. [15] and its erratum. 2 From the proof of Theorem 4.6 we immediately obtain Corollary 4.7. Let M = Γ \ X denote a locally symmetric space with rank one. If p ∈ (1, 2) and c > cp , the restriction of e−t (M,p −c) to V is chaotic for any of the subspaces 0 , h(z)ρPj (m) : z ∈ Ω . V = span E Pj (m) ϕm Remark 4.8. Let μ Vj (m),μ = span E Pj (m) ϕm , h(z)ρPj (m) : z ∈ Ω . μ
As the Eisenstein series for the eigenfunctions ϕm lead to Lp eigenvalues in the interior of PM,p + μ it can be shown similarly that the semigroups e−t (M,p −c) V
j (m),μ
are chaotic, if c > cp + μ. Proposition 4.9. Let Mi = Γi \ Xi , i = 1, . . . , k, denote locally symmetric spaces with rank one and M = M1 × · · · × Mk their Riemannian product. If p ∈ (1, 2) there are a constant cp > 0 and a closed e−tM,p -invariant subspace V ⊂ Lp (M) such that for all c > cp the semigroup e−t (M,p −c) |V has dense orbits. Proof. We restrict ourselves to the case k = 2. By Lp (M1 ) ⊗ Lp (M2 ) we denote the tensor product of the spaces Lp (M1 ) and Lp (M2 ). For the uniform cross norm gp on this tensor product ˜ gp Lp (M2 ) of the normed space (Lp (M1 ) ⊗ Lp (M2 ), gp ) as in [48] the completion Lp (M1 ) ⊗ p coincides with L (M1 × M2 ). Furthermore, we have e−tM1 ×M2 ,p = e−tM1 ,p ⊗ e−tM2 ,p , cf. [40,48]. By Corollary 4.7 the semigroups Ti (t) = e−t (Mi ,p −ci ) V
i
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are chaotic if ci > cp,i and the subspaces Vi are chosen accordingly (i = 1, 2). Let now cp = cp,1 + cp,2 , c > cp and choose ci , with ci > cp,i , i = 1, 2, and c = c1 + c2 . Then it follows from ˜ gp Lp (M2 ) [49, Corollary 2.2] that the tensor product T1 (t)⊗T2 (t) on Lp (M1 ×M2 ) = Lp (M1 ) ⊗ is a strongly continuous semigroup that has dense orbits (it is even recurrent hypercyclic). Hence, the semigroup e−t (M1 ×M2 ,p −c) = e−t (M1 ,p −c1 ) ⊗ e−t (M2 ,p −c2 ) restricted to the subspace V = V1 ⊗ V2 ⊂ Lp (M1 × M2 ) has dense orbits. 2 Acknowledgment We want to thank the referee for many valuable comments, in particular, for pointing out that our results hold true also in the case of general rank one locally symmetric spaces with finite volume. References [1] Wolfgang Arendt, Gaussian estimates and interpolation of the spectrum in Lp , Differential Integral Equations 7 (5–6) (1994) 1153–1168, MR MR1269649 (95e:47066). [2] Robert Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974) 193–240, MR MR0356254 (50 #8725). [3] Jacek Banasiak, Marcin Moszy´nski, A generalization of Desch–Schappacher–Webb criteria for chaos, Discrete Contin. Dyn. Syst. 12 (5) (2005) 959–972, MR MR2128736 (2005k:37027). [4] Armand Borel, Introduction aux groupes arithmétiques, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969, MR MR0244260 (39 #5577). [5] Armand Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Differential Geom. 6 (1972) 543–560, MR MR0338456 (49 #3220). [6] Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974) 235–272, MR MR0387496 (52 #8338). [7] Armand Borel, Automorphic Forms on SL2 (R), Cambridge Tracts in Math., vol. 130, Cambridge University Press, Cambridge, 1997, MR MR1482800 (98j:11028). [8] Armand Borel, Lizhen Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Math. Theory Appl., Birkhäuser Boston Inc., Boston, MA, 2006, MR MR2189882. [9] J.L. Clerc, E.M. Stein, Lp -multipliers for noncompact symmetric spaces, Proc. Natl. Acad. Sci. USA 71 (1974) 3911–3912, MR MR0367561 (51 #3803). [10] E. Brian Davies, Pointwise bounds on the space and time derivatives of heat kernels, J. Operator Theory 21 (2) (1989) 367–378, MR MR1023321 (90k:58214). [11] E. Brian Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92, Cambridge University Press, 1990, MR MR1103113 (92a:35035). [12] E. Brian Davies, Lp spectral independence and L1 analyticity, J. London Math. Soc. (2) 52 (1) (1995) 177–184, MR MR1345724 (96e:47034). [13] E. Brian Davies, Lp spectral theory of higher-order elliptic differential operators, Bull. London Math. Soc. 29 (5) (1997) 513–546, MR MR1458713 (98d:35164). [14] E. Brian Davies, Barry Simon, Michael E. Taylor, Lp spectral theory of Kleinian groups, J. Funct. Anal. 78 (1) (1988) 116–136, MR MR937635 (89m:58205). [15] R. deLaubenfels, H. Emamirad, Chaos for functions of discrete and continuous weighted shift operators, Ergodic Theory Dynam. Systems 21 (5) (2001) 1411–1427, MR MR1855839 (2002j:47030). [16] Wolfgang Desch, Wilhelm Schappacher, Glenn F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems 17 (4) (1997) 793–819, MR MR1468101 (98j:47083). [17] Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, second ed., Addison–Wesley Studies in Nonlinearity, Addison–Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989, MR MR1046376 (91a:58114). [18] Eberhard Freitag, Hilbert Modular Forms, Springer-Verlag, Berlin, 1990, MR MR1050763 (91c:11025). [19] H. Garland, M.S. Raghunathan, Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups, Ann. of Math. (2) 92 (1970) 279–326, MR MR0267041 (42 #1943).
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Journal of Functional Analysis 258 (2010) 1140–1166 www.elsevier.com/locate/jfa
Products of longitudinal pseudodifferential operators on flag varieties Robert Yuncken University of Victoria, Department of Mathematics and Statistics, PO Box 3060 STN CSC, Victoria, BC, Canada Received 21 November 2008; accepted 12 October 2009 Available online 7 November 2009 Communicated by S. Vaes
Abstract Associated to each set S of simple roots of SL(n, C) is an equivariant fibration X → XS of the complete flag variety X of Cn . To each such fibration we associate an algebra JS of operators on L2 (X ), or more generally on L2 -sections of vector bundles over X . This ideal contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. Together, they form a lattice of operator ideals whose common intersection is the compact operators. Thus, for instance, the product of negative order pseudodifferential operators along the fibres of two such fibrations, X → XS and X → XT , is a compact operator if S ∪ T is the full set of simple roots. The construction of the ideals uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU(n), which may be described as ‘essential orthogonality of subrepresentations’. © 2009 Elsevier Inc. All rights reserved. Keywords: Semisimple Lie groups; Pseudodifferential operators; Noncommutative harmonic analysis; Operator algebras
1. Introduction Let X = X1 × X2 be a product of compact manifolds. If A1 and A2 are longitudinal smoothing operators along the respective product fibrations, then their product A1 A2 is a smoothing operator on X . More generally, if A1 and A2 are longitudinal pseudodifferential operators of negative order then their product, whilst not being a classical pseudodifferential operator, is certainly a compact operator on L2 (X ). In this article we extend, and generalize, the latter fact to a class of highly nontrivial multiply-fibred manifolds—the complete flag varieties for Cn . E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.007
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The motivation for studying longitudinal pseudodifferential operators on flag varieties comes from the representation theory of semisimple groups, where they appear frequently. For instance, the Kunze–Stein intertwining operators between principle series representations of SL(n, C) are of this form (see, e.g., [8]). In [2], Bernstein proposed a longitudinal Sobolev theory related to such operators. However, as far as the present author is aware, certain desirable properties of this Sobolev theory seem to fail (see [12, Chapter 5]). In this light, the results presented here constitute a weaker analytic construction which, while far less powerful than a full Sobolev theory, is sufficient for certain applications to index theory. The author’s specific motivation for this work is the Baum–Connes Conjecture, an important conjecture in equivariant index theory which remains unsolved for discrete subgroups of higher rank ( 2) semisimple Lie groups. (See, e.g., [4] for an overview of the Baum–Connes Conjecture and its many consequences.) For the interested reader, we provide here a sketch of the motivating application, the full details of which can be found in the preprint [13]. This material is not necessary for an understanding of what follows. By work of Kasparov, the Baum–Connes Conjecture follows if one can prove that a particular canonical element of equivariant bivariant K-theory—the γ element—is trivial, in the weak sense that it acts trivially upon the K-theory of the reduced group C ∗ -algebra. While there are many possible choices of Kasparov cycle representing γ , successful models for rank-one groups in [7,6,5]1 can in hindsight be seen to be based upon the Bernstein–Gelfand–Gelfand (‘BGG’) complex ([3], see also [1]). For more discussion of why this complex is a likely starting point for a construction, see [13]. For G = SL(3, C), the BGG complex is constructed from longitudinal differential operators along the two equivariant fibrations of the complete flag variety. The conclusion of [13] is that a Kasparov cycle representing γ can indeed be constructed from this BGG complex. The analysis involved in the construction relies upon the results presented here. To get a heuristic idea of the construction, one should use as analogy the construction of the Kasparov product of two Khomology cycles defined from elliptic differential operators on closed manifolds: underlying this is a pair of longitudinal operators on the coordinate fibrations of the product manifold. If a similar construction can be made for general semisimple groups, as is expected, it should rely upon analysis of the present kind. Remark 1.1. For rank-one groups, the triviality of γ in the above sense is proven by a homotopy argument. This part of the proof of the Baum–Connes Conjecture suffers from its own analytic difficulties when the group has property T. This will certainly be a significant obstacle in the higher rank cases. We will not speculate here upon possible resolutions of that problem. We now state the main results, beginning with some important notation. Let K be a compact semisimple Lie group with Lie algebra k. Let G be the complex semisimple group associated to the complexification g = kC of k. Fix a maximal torus T ⊆ K with Lie algebra t, so h = tC is the Cartan subalgebra of g. The set of roots of K will be denoted by R. We fix a choice of positive roots R + and denote the set of simple roots by Σ . Given a set S ⊆ Σ of simple roots, let S ⊆ R + denote the set of roots which are nonnegative linear combinations of elements of S. To any subset S ⊆ Σ one associates a parabolic subgroup 1 The first proof for discrete subgroups of Sp(n, 1) was due to Lafforgue [9] using a different model for γ . The unpublished proof of Julg [5] referenced here is more in line with the present discussion.
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PS ⊆ G by defining its Lie algebra to be
pS =
gα ⊕ h ⊕
α∈R +
g−α
α∈S
(gα denoting the α-root space of g). We put KS = PS ∩ K. Note that {KS }S⊆Σ is a lattice of subgroups of K, i.e., KS ⊆ KT if and only if S ⊆ T . Example 1.2. Let K = SU(n + 1) with simple roots Σ = {α1 , . . . , αn }, where αi is the weight of the matrix Ei,i+1 ∈ sl(n + 1, C) with 1 in the (i, i + 1)-entry, and zeros elsewhere. The subgroups KS (S ⊆ Σ) are block-diagonal subgroups of SU(n + 1). In SU(5), for instance, ⎧⎛ ⎪ ⎪ ⎨⎜ K{α1 ,α2 } = ⎜ ⎝ ⎪ ⎪ ⎩
⎫ ⎪ ⎪ ⎟ A ∈ U(3), ω1 , ω2 ∈ U(1), ⎬ ⎟: . ⎠ det(A).ω1 .ω2 = 1 ⎪ ⎪ ⎭ ⎞
A ω1 ω2
Note that the semisimple part of KS is a product of special unitary groups. For S ⊆ Σ, the flag variety K/KS will be denoted by XS . Thus, associated to each S ⊆ Σ , we have an equivariant fibration qS : X → X S of the complete flag variety X = K/T. Note that K∅ = T, so that X∅ = X . Theorem 1.3. Let K = SU(n), so that X is the complete flag variety for Cn , and let E be a vector pseubundle over X . Let S1 , . . . , SN ⊆ Σ , and for each i = 1, . . . , N let Ai be a longitudinal . If S = Σ, dodifferential operator on E of negative order tangent to the fibration X → X S i i i then the product i Ai is a compact operator on L2 (X , E). Remark 1.4. This suggests an obvious question about longitudinal pseudodifferential operators on multiply foliated manifolds in general. Suppose X is a compact manifold which admits two foliations F1 and F2 with compact leaves. Suppose further that the tangent bundles to the foliations, T F1 and T F2 , span a distribution in T X which is totally non-integrable. If Ai is a longitudinal pseudodifferential operator of negative order along the leaves of Fi (for i = 1, 2), is A1 A2 a compact operator on X ? We suspect the answer is yes. However, this level of generality is greater than is necessary for the representation theoretic applications we have in mind. Furthermore, the symmetry present in flag varieties allows us to take a noncommutative harmonic analysis approach to these questions, as we now describe. Although this paper is nominally about pseudodifferential operators, the results actually apply to a much larger class of operators, defined in terms of harmonic analysis. Moreover, the proofs are facilitated by passing to this larger class of operators. Throughout, we will use Kˆ to denote the set of irreducible unitary representations of a compact group K (considered up to unitary equivalence), which are also referred to as K-types.
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Definition 1.5. Let S ⊆ Σ . Let H be any unitary representation space of K, that is, a Hilbert space equipped with a unitary representation U : K → B(H). For σ ∈ Kˆ S , pσ will denote the orthogonal projection onto the σ -isotypical subspace of H (with the representation U restricted to KS ). If F ⊆ Kˆ S is a collection of irreducibles, we write pF for the projection σ ∈F pσ onto the closed span of the corresponding isotypical subspaces. In fact, we need to generalize slightly the type of spaces H on which these projections are defined. For any S ⊆ Σ, the KS -isotypical projections pσ on a unitary representation space H commute with all weight-space projections (see Lemma 2.1). Thus, pσ is also well defined on each weight-space. Definition 1.6. By a harmonic K-space we shall mean any direct sum of weight spaces H = i (Hi )μi , where μi are weights and (Hi )μi denotes the μi -weight space of a unitary K-representation space Hi . The key example of a harmonic K-space is the L2 -section space of a homogeneous line bundle (see Section 7). Occasionally, we will need to qualify that a harmonic K-space is ‘not too large’, in the following sense. Definition 1.7. A harmonic K-space H will be called finite multiplicity if, for each π ∈ Kˆ , pπ H is finite dimensional. For instance, the regular representation L2 (K) is finite multiplicity by the Peter–Weyl Theorem. So too is the L2 -section space of any K-equivariant vector bundle over X (Remark 7.1). The central concept of this paper is the following definition. Definition 1.8. Let S ⊆ Σ. Let H1 and H2 be harmonic K-spaces and A : H1 → H2 be a bounded linear map between them. For each σ, τ ∈ Kˆ S , put Aσ τ = pσ Apτ , so that (Aσ τ )σ,τ ∈Kˆ S is the matrix of A with respect to the KS -harmonic decomposition. We say A is (i) KS -harmonically finite if all but finitely many matrix entries Aσ τ are zero, (ii) KS -harmonically proper if the matrix (Aσ τ ) is row- and column-finite, i.e., for each fixed σ there are only finitely many τ with Aσ τ or Aτ σ nonzero. If H1 = H2 = H, the set of KS -harmonically proper operators is an algebra, and the KS harmonically finite operators form an ideal in that algebra. Closing these in operator-norm, we obtain a C ∗ -algebra and ideal. Definition 1.9. For any S ⊆ Σ , let KS (H1 , H2 ) (resp. AS (H1 , H2 )) denote the operator-norm closure of the KS -harmonically finite operators (resp. KS -harmonically proper operators) from H1 to H2 . If H1 = H2 = H, we shall write KS (H) for KS (H, H) and AS (H) for AS (H, H). It is notationally convenient to think of AS and KS as C ∗ -categories whose objects are harmonic K-spaces and whose morphism sets are given by Definition 1.9 above. However, it is worth remarking that we shall need none of the technicalities of C ∗ -categories. This framework simply
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allows us to write A ∈ AS or A ∈ KS , with the domain and target spaces H1 and H2 implied by the operator A. It is not true in general that the C ∗ -categories KS form a lattice of ideals (see Example 3.6). We make the following adjustment. Definition 1.10. With A =
T ⊆Σ
AT , define JS = KS ∩ A.
The main result is that the C ∗ -categories JS form a lattice of ideals. More precisely, we have the following: Theorem 1.11. (i) If S ⊆ T ⊆ Σ then JT is an ideal in JS . (ii) For any S, T ⊆ Σ, JS ∩ JT = JS∪T . (iii) If H1 and H2 are finite multiplicity harmonic K-spaces, then JΣ (H1 , H2 ) is the space of compact operators from H1 to H2 . (See Lemma 3.7 for (i) and Lemma 3.8 for (iii). Property (ii) is proven in Section 5.) We now wish to relate this harmonic analysis back to longitudinal pseudodifferential operators. For S ⊆ Σ, let FS denote the foliation of X given by the fibration X → XS , and let −p ΨFS (E) denote the set of longitudinal pseudodifferential operators of order −p tangent to FS on a K-homogeneous bundle E over X . Recall that the L2 -section space L2 (X ; E) is a finite multiplicity harmonic K-space. Proposition 1.12. Let K be a product of special unitary groups. Let E be an equivariant vector −p bundle over X . For S ⊆ Σ, we have ΨF0 S (E) ⊆ AS (L2 (X ; E)) and ΨFS (E) ⊆ JS (L2 (X ; E)) for any −∞ −p < 0. For the proof, see Section 9. Theorem 1.3 is an immediate corollary of Proposition 1.12 and Theorem 1.11. −p We also note that the norm closure of ΨFS (E) (for any −∞ −p < 0) has an interpretation as compact Hilbert module operators or as a groupoid C ∗ -algebra. See Section 6. −p
Remark 1.13. Excluding the case S = Σ , the inclusion ΨFS (E) ⊆ JS (L2 (X ; E)) above is far from an equality, even after passing to the norm closure of the left-hand side. Operators A ∈ −p ΨFS (E) preserve XS -supports, in the sense that if a section s ∈ L2 (X ; E) is zero on a given fibre of X → XS , then so is As. On the other hand, the operator of left translation by any k ∈ K belongs to AS (L2 (X ; E)) (see Remark 7.2), so operators in the ideal JS (L2 (X ; E)) need not preserve XS -supports. Similar remarks hold for ΨF0 S (E) ⊆ AS (L2 (X ; E)). −p
For S = Σ, the norm closure of ΨFΣ (E) is equal to JΣ (L2 (X ; E)) = K(L2 (X ; E)) (the compact operators), while AΣ (L2 (X ; E)) = B(L2 (X ; E)).
The paper is organized as follows. Sections 2 to 5 deal with the harmonic analysis. Section 2 provides some basics on harmonic decompositions, Section 3 elaborates upon the definition of the C ∗ -categories AS and KS , and Section 4 proves some useful lemmas about tensor operators.
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Section 5 proves the main result—property (ii) of Theorem 1.11. The remaining sections describe how this harmonic analysis can be applied to longitudinal pseudodifferential operators on the complete flag variety of Cn . Some comments on the proof of Theorem 1.11(ii) may be illuminating. At the heart of the proof one finds the following property of a pair of subgroups of a compact group. Definition 1.14. Two closed subgroups K1 and K2 of a compact group K will be called essentially orthotypical if for any irreducible representations σ1 of K1 and σ2 of K2 and any > 0, there are only finitely many irreducible representations π of K which contain unit vectors ξ1 , ξ2 , where ξi is of type σi for Ki (i = 1, 2) and |ξ1 , ξ2 | > . An equivalent formulation is that the product of the isotypical projections pσ1 pσ2 is compact on any finite multiplicity unitary representation of K. (Cf. Lemma 5.1.) We prove, for K = SU(n), that if S ∪ T = Σ then the subgroups KS and KT defined above are essentially orthotypical. We expect the analogous result to be true for arbitrary compact semisimple groups K. Indeed one might ask the following more general question. Question 1.15. Is it true for any compact group K that subgroups K1 and K2 are essentially orthotypical whenever they generate K? Remark 1.16. Essential orthotypicality for generating subgroups can be viewed as a strong version of Kazhdan’s property T. (Compact groups satisfy property T trivially, of course.) Taking K1 ∪ K2 as a generating set for K, the ‘almost invariant vectors’ definition of property T implies that there exists a constant 0 < c < 1 with the following property: Let π be an irreducible representation of K on V π which contains unit vectors ξi fixed by Ki (i = 1, 2). If |ξ1 , ξ2 | > c, then π is the trivial representation of K. On the other hand, essential orthotypicality says that for any > 0, the condition |ξ1 , ξ2 | >
implies that π belongs to some finite set of irreducibles of K. 2. Harmonic decompositions Let K be any compact group, and H a closed subgroup. Let U be a unitary representation of ˆ we let pσ denote K on a Hilbert space H. As above, for any irreducible representation σ ∈ H the projection onto the σ -isotypical subspace of H (with representation restricted to H). By orthogonality of characters, this can be written explicitly as pσ = dim σ.
χσ (h)U (h) dh, H
where χσ is the character of σ . Lemma 2.1. Let H1 , H2 be closed subgroups of K, and let σ ∈ Hˆ 1 , τ ∈ Hˆ 2 . (i) If H1 and H2 commute, then pσ and pτ commute. (ii) If H1 H2 , then pσ and pτ commute.
(2.1)
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Proof. Making the change of variables h2 → h1 h2 h−1 1 in the following integral, we get
pσ pτ = dim σ. dim τ.
χσ (h1 )χτ (h2 )U (h1 h2 ) dh1 dh2
h1 ∈H1 h2 ∈H2
= dim σ. dim τ.
χσ (h1 )χτ h1 h2 h−1 1 U (h2 h1 ) dh1 dh2 .
h1 ∈H1 h2 ∈H2
In both cases (i) and (ii), χτ (h1 h2 h−1 1 ) = χτ (h2 ), so the latter integral equals pτ pσ .
2
Now we specialize to the case where K is compact semisimple and H = KS , for some S ⊆ Σ , as defined in the introduction. Consider first the case S = ∅, for which K∅ = T. The irreducible representations of T correspond to the weights μ of K, via the exponential map. The corresponding harmonic projections— which we will denote by pμ rather than the cumbersome peμ —are the projections onto the weight spaces of a K-representation. More generally, for any S ⊆ Σ, the family of projections {pσ | σ ∈ Kˆ S } gives an orthogonal decomposition of any unitary representation space of K. Note that, by Lemma 2.1, the weightspace projections pμ commute with all of the projections pσ for σ ∈ Kˆ S . Thus, {pσ | σ ∈ Kˆ S } also gives an orthogonal decomposition of any harmonic K-space, as defined in Definition 1.6. 3. C ∗ -algebras associated to the fibrations Recall that in Definition 1.9, we defined the C ∗ -categories KS and AS as the norm-closure of the KS -harmonically finite and KS -harmonically proper operators, respectively. In this section, we prove some basic properties of KS and AS . Recall that for any set F ⊆ Kˆ S of KS -types, we define pF = σ ∈F pσ . An operator A between harmonic K-spaces is KS -harmonically finite if and only if A = pF ApF for some finite set F ⊂ Kˆ S . In particular, the KS -isotypical projections pσ themselves are KS -harmonically finite, as is pF for any finite F ⊂ Kˆ S . If S = T ⊆ Σ , the KT -isotypical projections turn out to belong to AS , although this is not easy to prove yet. (See Lemma 5.7.) For now, we content ourselves with the easy case where T is a subset or superset of S. Lemma 3.1. Let S ⊆ Σ . Suppose A : H1 → H2 is a bounded operator between harmonic ˆ S ). Then A is KS -harmonically proper, K-spaces which commutes with all projections pσ (σ ∈ K and hence belongs to AS . In particular, (i) if T ⊆ S or T ⊇ S then pF ∈ AS for any F ⊆ Kˆ T . (ii) if A is an intertwiner between unitary K-representations, then A ∈ AS . Proof. The main assertion is immediate since pσ Apσ = pσ pσ A = 0 for all σ = σ in Kˆ S . Consequently, (i) follows from Lemma 2.1 and (ii) is trivial. 2 Lemma 3.2. If S ⊆ T ⊆ Σ , then KT ⊆ KS .
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Proof. Suppose A is a KT -harmonically finite operator, so that A = pF ApF for some finite set F ⊂ Kˆ T . Each irreducible representation for KT decomposes into finitely many irreducibles for KS , so there is a finite set F ⊂ Kˆ S of KS -types which occur in the members of F . Then pF = pF pF = pF pF , and it follows that A = pF ApF . Thus A is KS -harmonically finite. Taking completions gives KT ⊆ KS . 2 For the purpose of the next few lemmas, we will fix an enumeration of the irreducible representations of KS as {σ0 , σ1 , σ2 , . . .}. Also, let Fj = {σ0 , . . . , σj } ⊆ Kˆ S . Recall that a harmonic K-space H is called finite multiplicity if the projections pπ (π ∈ Kˆ ) are all finite rank on H. Lemma 3.3. If H1 , H2 are finite multiplicity harmonic K-spaces, then KΣ (H1 , H2 ) = K(H1 , H2 ), the algebra of compact operators from H1 to H2 . Proof. Since pF = π∈F pπ is finite rank on H1 and H2 for any finite set F ⊆ Kˆ = Kˆ Σ , KΣ harmonically finite operators are finite-rank. Thus KΣ (H1 , H2 ) ⊆ K(H1 , H2 ). On the other j hand, the projections pFj = i=1 pσj converge to 1 in the strong operator topology. Therefore, for any finite rank operator A : H1 → H2 , the KΣ -harmonically finite operators pFj ApFj converge to A as j → ∞. Thus K(H1 H2 ) ⊆ KΣ (H1 H2 ). 2 The next two lemmas give equivalent characterizations of the C ∗ -categories KS and AS . Lemma 3.4. Let K : H1 → H2 be a bounded linear map between harmonic K-spaces. The following are equivalent: (i) K ∈ KS . (ii) pF⊥j K → 0 and KpF⊥j → 0 in norm as j → ∞. (iii) pFj KpFj → K in norm as j → ∞. Proof. For (i) ⇒ (ii), note that the both pF⊥j K and KpF⊥j are eventually zero if K is KS -harmonically finite. Thus, (ii) follows for all K ∈ KS by density. For (ii) ⇒ (iii), write K − pFj KpFj = KpF⊥j + pF⊥j KpFj . For (iii) ⇒ (i), note that pFj KpFj is KS -harmonically finite. 2 Lemma 3.5. Let A : H1 → H2 be a bounded linear map between harmonic K-spaces. The following are equivalent: (i) A ∈ AS . (ii) For any σ ∈ Kˆ S , pF⊥j Apσ → 0 and pσ APF⊥j → 0 in norm as j → ∞. (iii) For any σ ∈ Kˆ S , Apσ and pσ A are in KS . (iv) A is a two-sided multiplier of KS , i.e., AK ∈ KS for all right-composable K ∈ KS and KA ∈ KS for all left-composable K ∈ KS . Here, left- and right-composable mean that the appropriate domain and target spaces agree. Proof. (i) ⇒ (ii): If A is KS -harmonically proper the two sequences in (ii) are eventually zero. By density, (ii) holds for all A ∈ AS .
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(ii) ⇒ (iii): Immediate from Lemma 3.4(ii). (iii) ⇒ (iv): Let K ∈ KS be right-composable with A. By Lemma 3.4(iii), AK = limj →∞ ApFj KpFj . If A satisfies (iii) then ApFj ∈ KS for all j , so AK ∈ KS . Similarly for right-composable K ∈ KS . (iv) ⇒ (i): Let A be a two-sided multiplier of KS . Let > 0. Starting with A0 = A, we will construct a sequence (Ak )k∈N of two-sided multipliers for KS such that Ak+1 − Ak < .2−k−1 ,
(3.1)
as well as a sequence a0 , a1 , a2 , . . . ∈ N such that pF⊥a Ak pFj = 0 for all 0 j < k,
(3.2)
pFj Ak pF⊥a = 0 for all 0 j < k.
(3.3)
j
j
The norm-limit A∞ of this sequence will be an -approximation of A by (3.1), and it will be
ˆ S , if σ ∈ KS -harmonically proper as follows. For any σ = σj ∈ K / Faj ,
pσ A∞ pσ = lim pσ Ak pσ = lim pσ pF⊥a Ak (pFj pσj ) = 0, k→∞
j
k→∞
and pσ A∞ pσ = 0 similarly. Since is arbitrary, we will conclude that A ∈ AS . To inductively define Ak+1 , assume that we have Ak as above. Since Ak is a two-sided multiplier, Ak pFk ∈ KS and pFk Ak ∈ KS . By Lemma 3.4 there is an integer ak (without loss of generality, larger than k) such that the operators Bk = pF⊥a Ak pFk k
and Ck = pFk Ak pF⊥a
k
have norm less than .2−k−2 . Note that Bk , Ck ∈ KS , so they are trivially two-sided multipliers for KS . Now put Ak+1 = Ak − Bk − Ck . It is clear that (3.1) is satisfied. Since all isotypical projections for KS commute, (3.2) and (3.3) hold for Ak+1 with 0 j < k. We need to prove them for j = k. Since ak k we have pF⊥a Ak+1 pFk = pF⊥a Ak pFk − pF⊥a Bk pFk − pF⊥a Ck pFk k
k
k
= pF⊥a Ak pFk k
− pF⊥a Ak pFk k
= 0, and pF⊥a Ak+1 pFk = 0 similarly. k
2
k
−0
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Note, in particular, that KS is an ideal in AS . Recall that KT ⊆ KS for any S ⊆ T ⊆ Σ . The following example shows that KT is not generally an ideal in KS . Example 3.6. Let H be the zero weight space of the right regular representation on L2 (K). This is the space of T-invariant functions, so H ∼ = L2 (X ). By the Peter–Weyl Theorem, H is a finite multiplicity harmonic K-space, so KΣ (H) = K(H) by Lemma 3.3. At the other extreme, since the zero weight projection on H is the identity operator, every bounded operator on H is K∅ harmonically finite, i.e., K∅ (H) = B(H). If ∅ T Σ then H contains infinitely many KT -types, namely, any KT -type with nontrivial 0-weight space. Thus 1 ∈ / KT (H). On the other hand, H contains the KT -fixed vectors of L2 (K), which is an infinite dimensional subspace, so the projection onto the trivial KT -type is not in K(H). We get K∅ (H) KT (H) KΣ (H ), so KT (H) cannot be an ideal in K∅ (H) = B(H). To remedy this, as mentioned in Definition 1.10, we let A = KS ∩ A.
T ⊆Σ
AT , and define JS =
Lemma 3.7. If S ⊆ T ⊆ Σ , then JT JS . Proof. By Lemma 3.2, JT ⊆ JS . Since JS ⊆ A ⊆ AT by definition, Lemma 3.5(iv) shows that JS multiplies JT . 2 Note also that by Lemma 3.2, KΣ ⊆ KS ⊆ AS for all S ⊆ Σ , and thus JΣ = KΣ . Lemma 3.3 gives the following. Lemma 3.8. If H1 , H2 are finite multiplicity harmonic K-spaces, JΣ (H1 , H2 ) is the space of compact operators from H1 to H2 . 4. Tensor operators Notation 4.1. Throughout, we use 1S to denote the trivial representation of KS for any S ⊆ Σ . Also, for any σ ∈ Kˆ S , we use V σ to denote the finite dimensional Hilbert space on which it acts. The dual representation will be denoted σ † . Definition 4.2. Let H and H be Hilbert spaces. For v ∈ H , we denote by Θv the operator Θv : H → H ⊗ H;
ξ → v ⊗ ξ.
The adjoint of Θv is the operator Θv∗ : w ⊗ ξ → v, wξ . Lemma 4.3. Let U , U be unitary representations of K on Hilbert spaces H and H , respectively. For any v ∈ H , the operator Θv : H → H ⊗ H; belongs to A =
S⊆Σ
AS .
ξ → v ⊗ ξ,
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Proof. Fix S ⊆ Σ. To begin with, suppose that v is KS -isotypical, say v ∈ pσ H . Let σ1 ∈ Kˆ S . The representation σ ⊗ σ1 , being finite dimensional, contains only finitely many KS -types, so pσ2 Θv pσ1 = 0 for all but finitely many σ2 ∈ Kˆ S . Moreover, by orthogonality of characters, the multiplicity of σ1 in σ ⊗ σ2 is the same as the multiplicity of σ2 in σ † ⊗ σ1 , which is again zero for all but finitely many σ2 . Hence pσ1 Θv pσ2 = 0 for all but finitely many σ2 ∈ Kˆ S . Therefore Θv is KS -harmonically proper. Since the KS -isotypical vectors span a dense subspace of H , the general case follows from an approximation argument. 2 Using such tensor operators, we can relate any KS -isotypical projection pσ to the projection p1S onto KS -fixed vectors as follows. † Fix σ ∈ Kˆ S . Let m = dim V σ , let v1 , . . . , vm be an orthonormal basis for V σ and v1† , . . . , vm the dual basis of V σ † . Lemma 4.4. With the above notation, on any Hilbert space H with a unitary representation of KS , we have pσ = m.
m
Θ ∗† p1S Θv † . vi
i=1
(4.1)
i
Note that the projection p1S here is acting on the representation V σ † ⊗ H.
Proof. To begin with, suppose H is itself an irreducible KS -representation, say H = V σ , for σ ∈ Kˆ S . By orthogonality of characters, the representation σ † ⊗ σ contains a trivial subrepre sentation if and only if σ = σ . Thus, both sides of (4.1) are zero on V σ for σ = σ . On the other † hand, if σ = σ , the trivial subrepresentation of V σ † ⊗ V σ is spanned by m k=1 vk ⊗ vk . For each i, j , we get Θ
† vi
vj −
→ vi†
Θ∗
† vi 1 † 1 ⊗ vj −→ δij vk ⊗ vk −→ δij vi . m m
m
p1S
k=1
Hence m.
m
∗ i=1 Θv † p1S Θvi† vj
= vj , for each j = 1, . . . , m. Eq. (4.1) for H = V σ follows.
i
Now let Φ : V σ → H be an intertwining operator from any σ ∈ Kˆ S to a unitary KS representation. In the diagram
Θ
V
σ
† vi
V σ† ⊗ V
σ
1⊗Φ
Φ Θ
H
p1S
† vi
V σ† ⊗ H
Θ ∗†
V σ† ⊗ V
σ
1⊗Φ p1S
V σ† ⊗ H
vi
Θ ∗† vi
Vσ
Φ
H
the left and right squares obviously commute, while the middle square commutes because 1 ⊗ Φ is an intertwiner (Lemma 3.1(ii)). Since the top row equals pσ the bottom row equals pσ on the image of Φ. Such images span a dense subspace. 2
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In order to work with harmonic K-spaces, rather than KS -spaces, we need to rewrite this lemma as follows. Let ι : V σ † → V π be an inclusion of σ † as a KS -subrepresentation of some K-representation π . (Such a representation π always exists—for instance the induced representation IndKKS σ † contains σ † as a KS -subrepresentation, by Frobenius Reciprocity.) Let
wi = ι(vi† ) ∈ V π for i = 1, . . . , m. The commuting diagram Θ
H
† vi
p1S
V σ† ⊗ H ι⊗1
H
Θwi
Θ ∗† vi
V σ† ⊗ H
H
ι⊗1 p1S
Vπ ⊗H
Θw∗
i
Vπ ⊗H
H
immediately yields the following. Lemma 4.5. On any Hilbert space H with a unitary K-representation, pσ = m.
m
Θw∗ i p1S Θwi .
(4.2)
i=1
5. Lattice of ideals The purpose of this section is to prove property (ii) of Theorem 1.11, namely that JS ∩ JT = JS∪T . We begin with a lemma which expands upon the notion of ‘essential orthotypicality’ from the introduction. Lemma 5.1. Let K be as above and S, T ⊆ Σ. The following are equivalent. (i) On any harmonic K-space H, pτ pσ ∈ KS∪T (H) for all σ ∈ Kˆ S , τ ∈ Kˆ T . (ii) KS and KT are essentially orthotypical as subgroups of KS∪T , i.e., for any σ ∈ Kˆ S , τ ∈ Kˆ T and any > 0, there exist only finitely many irreducible representations π ∈ Kˆ S∪T having unit vectors ξ ∈ pσ V π , η ∈ pτ V π with |η, ξ | > . (iii) For any σ ∈ Kˆ S and any > 0, there exist only finitely many irreducible representations π ∈ Kˆ S∪T having a unit vector ξ ∈ pσ V π and a unit vector η fixed by KT with |η, ξ | > . Proof. (i) ⇒ (ii): Let σ ∈ Kˆ S , τ ∈ Kˆ T and π ∈ Kˆ S∪T . Let H = L2 (K) with the right regular representation. Note that every π ∈ Kˆ S∪T occurs with nonzero multiplicity in L2 (K). By assumption, pσ pτ ∈ KS∪T , so by Lemma 3.4(ii), for any > 0 there exists a finite set F ⊆ Kˆ S∪T such that pF⊥ pσ pτ < . Note that pF⊥ commutes with pτ and pσ (Lemma 2.1). If π∈ / F , then for any unit vectors ξ ∈ pσ pπ H and η ∈ pτ pπ H, η, ξ = p ⊥ η, ξ = p ⊥ pτ η, pσ ξ = p ⊥ pσ pτ η, ξ < . F
F
F
This proves (ii). (ii) ⇒ (iii): Immediate, by letting τ be the trivial representation of KT .
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(iii) ⇒ (i): By subsequently cutting down by weight projections, it suffices to prove (i) in the case where H is a unitary representation space for K. Fix > 0, and let F be the finite set of irreducible KS∪T representations which contain vectors ξ, η as in (iii) above. Then |p1T pF⊥ v, pσ pF⊥ w| < for any v, w ∈ H, so ⊥ p p1 ∗ p ⊥ pσ = p ⊥ (p1 pσ ) = (p1 pσ )p ⊥ < . T T T F F F F Therefore, by Lemma 3.4, pσ p1T ∈ KS∪T .
(5.1)
We want to generalize (5.1) from the trivial representation 1T of KT to arbitrary τ ∈ Kˆ T . From Lemma 4.5, we a representation π of K and unit vectors w1 , . . . , wm ∈ can choose ∗ p Θ . The operators Θ V π , such that pτ = m. m Θ 1 wi wi are in AS by Lemma 4.3, so by T i=1 wi ˆ Lemma 3.5, there exist finite sets Fi ⊆ KS of KS -types such that pσ Θ ∗ p ⊥ < /m2 . wi Fi
(5.2)
Now, pσ pτ = m.
m
pσ Θw∗ i p1T Θwi
i=1
= m.
m
pσ Θw∗ i pFi p1T Θwi + m.
i=1
m
pσ Θw∗ i pF⊥i p1T Θwi .
(5.3)
i=1
Note that (5.1) implies pFi p1T ∈ KS∪T , and since pσ , Θwi ∈ AS∪T , the first term of (5.3) is in KS∪T . By (5.2), the second term has norm less than , and since was arbitrary, pσ pτ ∈ KS∪T . 2 We next describe a structural reduction that can be made in proving the equivalent properties of Lemma 5.1. We follow the notation of the introduction, namely, g is the complexification of k, h is its Cartan subalgebra, and pS is the parabolic subalgebra of g associated to S ⊆ Σ, for which kS = pS ∩ k. Let S, T ⊆ Σ , and put Σ = S ∪ T . The Lie algebra kΣ of KΣ is the intersection of k with the reductive part
gλ ⊕ h ⊕
λ∈Σ
g−λ .
(5.4)
λ∈Σ
of pΣ . Next, let h be the subalgebra of h spanned by the commutators [gλ , g−λ ] with λ ∈ Σ , so that the semisimple part of (5.4) is g :=
λ∈Σ
gλ ⊕ h ⊕
λ∈Σ
g−λ .
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Let k = g ∩ k. The orthocomplement of h in h (with respect to the Killing form) is the centre z of kΣ , and we get the decomposition kΣ = k ⊕ z. Let KΣ = K .Z be the corresponding decomposition at the group level. (The subgroups K and Z may have nontrivial but finite intersection.) Since K is itself semisimple, with simple root set Σ = S ∪ T , we may associate to the subsets S, T ⊆ Σ subgroups KS , KT of K , just as we did for K. Thanks to the decomposition above, their Lie algebras are related to those of KS and KT by kS = kS ⊕ z, kT = kT ⊕ z. Thus we have decompositions KΣ = K .Z,
KS = KS .Z,
KT = KT .Z.
Lemma 5.2. With notation as above, if KS and KT are essentially orthotypical subgroups of K then KS and KT are essentially orthotypical subgroups of KΣ . Remark 5.3. The converse is also true, although we shall not need it. Proof. By Schur’s Lemma, any irreducible representation π of KΣ is scalar on the central subgroup Z, and π remains irreducible upon restriction to K . Likewise, irreducible representations σ ∈ Kˆ S , τ ∈ Kˆ T are scalar on Z and irreducible upon restriction to KS and KT , respectively. Suppose KS and KT are essentially orthotypical subgroups of K . Fix σ ∈ Kˆ S and τ ∈ Kˆ T , and let > 0. If π ∈ Kˆ Σ contains the KS -type σ and the KT -type τ nontrivially, then the restriction π|Z is equal to σ |Z (and also, for that matter, τ |Z ). If moreover there exist unit vectors ξ ∈ pσ V π and η ∈ pτ V π with |η, ξ | > , then the KS -type of ξ and KT -type of η are prescribed, and hence there are only finitely many possibilities for the restriction π|K . Since KΣ = K .Z, this gives only finitely many possibilities for π . 2 Throughout the remainder of the paper, we make the standing assumption that K is a product of special unitary groups, N i=1 SU(ni ) (ni 2). It is worth remarking, however, that we expect the results to be true for arbitrary compact semisimple groups. Proposition 5.4. The equivalent properties of Lemma 5.1 are true for any S, T ⊆ Σ (with K a product of special unitary groups). Proof. We work inductively on the size of S ∪ T . If #(S ∪ T ) = 0 or 1, the result is immediate from Lemma 3.2. So let #(S ∪ T ) = n 2, and suppose we have proven the proposition for any lesser cardinalities. By Lemma 5.2, it suffices to prove that KS and KT are essentially orthotypical subgroups of the group K . Replacing K by K , which is itself a product of special unitary groups (cf. Example 1.2), we may therefore assume without loss of generality that S ∪ T = Σ. We start with the case K = SU(n + 1). Let the simple weights of K be Σ = {α1 , . . . , αn }, as in Example 1.2. The heart of the proof is the case S = Σ \ {αn },
T = Σ \ {α1 }.
(5.5)
This case is proven by a technical computation, which for clarity we separate out as Lemma 5.5 below. Let us assume for the moment that we have proven this case and proceed to the other cases. Still with K = SU(n + 1), let S, T ⊆ Σ be arbitrary, subject to S ∪ T = Σ . Without loss of generality, α1 ∈ S (otherwise interchange S and T ). Let S = S \ {α1 }, T = T \ {α1 }. Let σ |S
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denote the finite set of irreducible representations in Kˆ S which occur in the restriction of σ to KS , and similarly define τ |T ⊆ Kˆ T . Then pσ pτ = pσ pσ |S pτ |T pτ . Now #(S ∪ T ) = #(Σ \ {α1 }) = n − 1, so pσ |S pτ |T ∈ KΣ\{α1 } by the inductive hypothesis. Thus, for any > 0, there is a finite set F1 ⊆ Kˆ Σ\{α1 } such that pσ pτ − pσ pF1 pσ |S pτ |T pτ < .
(5.6)
Next consider the product pσ pF1 . Let S = S \ {αn } and T = Σ \ {α1 , αn }. As above, we let σ |S denote the finite set of irreducible representations occurring in the restriction of σ to KS , and let F1 |T denote the finite set of irreducible representations of KT which occur in the restriction of any ρ ∈ F1 to T . Then pσ pF1 = pσ pσ |S pF1 |T pF1 . Again, the inductive assumption implies pσ |S pF1 |T ∈ KS ∪T = KΣ\{αn } , so for some finite set F2 ⊆ Kˆ Σ\{αn } , we have pσ pF1 − pσ pσ |S pF1 |T pF2 pF1 < .
(5.7)
Combining the approximations (5.6) and (5.7) yields pσ pτ − pσ pσ |S pF |T (pF pF )pσ |S pτ |T pτ < 2 . 2 1 1 Here, pF2 pF1 ∈ KΣ by the assumed case (5.5). All the other projections are in AΣ by Lemma 3.1. Since was arbitrary, we conclude that pτ pσ ∈ KΣ . (i) We now deal with the case where K is a product of special unitary groups, K = N i=1 K with (i) ˆ (i) K = SU(ni ). Here, irreducible representations of K are of the form i πi , where πi ∈ K . (i) Let Σi ⊆ Σ denote the set of roots of K which come from roots of K . Put Si = S ∩ Σi , (i) (i) Ti = T ∩ Σi . Then KS = i KSi , where KSi is the subgroup of K(i) associated to the set of simple roots Si ⊆ Σi . For σ ∈ Kˆ S , we have a corresponding decomposition σ = i σi , with ˆ σi ∈ Kˆ (i) i pσi . Similarly, for τ ∈ KT we have pτ = i pτi , with all the Si . We also have pσ = analogous notation. Since pσi and pτj commute for i = j , pσ p τ =
pσi pτi .
i
By the preceding cases, pσi pτi ∈ KΣi for each i, so for any > 0, we can find finite sets Fi ⊆ Kˆ (i) such that pσi pτi − pσi pτi pFi < /N . Therefore, N pF i pσ pτ − pσ pτ i=1
N N pF i (pσ pτ − pσ pτ pFi ) i=1
j =i+1
< .
ˆ Now N i=1 pFi = pF , where F is the finite set { i πi | πi ∈ Fi } ⊆ KΣ . Thus, pσ pτ can be arbitrarily well approximated by elements of KΣ . 2
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It remains to prove the fundamental case (5.5). For this we make a computation using Gelfand–Tsetlin bases for the irreducible representations of SU(n). We provide a quick review of Gelfand–Tsetlin bases here, which we take from the expository article of Molev [10]. The irreducible unitary representations of SU(n) are in correspondence with irreducible Clinear representations of its complexified Lie algebra sl(n, C). One begins by considering irreducible representations of gl(n, C). The weights of gl(n, C) are indexed by n-tuples of integers λ = (λ1 , . . . , λn ), which act on the Cartan (diagonal) subalgebra by the formula ⎛ λ:⎝
⎞
t1 ..
⎠ →
. tn
λ i ti .
i
A weight is dominant if the entries are descending: λ1 · · · λn . These are the highest weights of irreducible gl(n, C)-representations. Let πλ denote the irreducible representation of gl(n, C) with highest weight λ. One now considers the successive restrictions of this representation to the ‘upper-left’ subalgebras gn ⊇ gn−1 ⊇ · · · ⊇ g1 , where ! gk =
gl(k, C) 0
" 0 . I
∼ gl(n − 1, C) which occur in πλ are It is known that the irreducible representations of gn−1 = precisely those with highest weights μ = (μ1 , . . . , μn−1 ) satisfying the interlacing conditions λi μi λi+1
(i = 1, . . . , n − 1).
(5.8)
Moreover, these representations all occur with multiplicity one. Thus, a nested sequence of irreducible subrepresentations of gn−1 , . . . , g1 is specified uniquely by the rows of a Gelfand–Tsetlin pattern ⎞ ⎛ λn,1 λn,2 ········· λn,n−1 λn,n ⎜ λn−1,1 λn−1,2 · · · λn−1,n−2 λn−1,n−1 ⎟ ⎟ ⎜ ⎟ ⎜ . . ⎟, ⎜ . . Λ=⎜ . . ⎟ ⎟ ⎜ ⎠ ⎝ λ2,1 λ2,2 λ1,1 satisfying the interlacing conditions λk+1,i λk,i λk+1,i+1
(i = 1, . . . , k; k = 1, . . . n − 1),
(5.9)
where, the kth row (λk,1 , . . . , λk,k ) is the highest weight of the gk -subrepresentation. The resulting irreducible representations of g1 ∼ = gl(1, C) are one-dimensional, so choosing a nonzero vector from each defines an orthogonal basis for the representation space of πλ . In the most elegant choice (due to Želobenko), these orthogonal basis vectors are not of unit length. If we denote the basis vector corresponding to the Gelfand–Tsetlin pattern Λ by ξΛ , and
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set lk,i = λk,i − i + 1, then the norm of ξΛ is ξΛ 2 =
n
k=2 1ij
(lk,i − lk−1,j )! (lk−1,i − lk−1,j )!
1i<j k
(lk,i − lk,j − 1)! . (lk−1,i − lk,j − 1)!
(5.10)
(We also follow the notational convention that if Λ is an inadmissible pattern, that is it does not satisfy the interlacing conditions (5.9), then ξΛ = 0.) The advantage of Želobenko’s choice is the simplicity of the formulae for the representation πλ in this basis. Let Ep,q denote the n × n-matrix with all entries zero except for a 1 in the (p, q)-position. Then πλ (Ek,k )ξΛ =
k
λk,i −
i=1
πλ (Ek,k+1 )ξΛ = −
k i=1
(5.11)
λk−1,i ξΛ ,
i=1
k (lk,i − lk+1,1 ) · · · (lk,i − lk+1,k+1 ) i=1
πλ (Ek+1,k )ξΛ =
k−1
(lk,i − lk,1 ) · · · ∧ · · · (lk,i − lk,k )
ξΛ+δk,i ,
(lk,i − lk−1,1 ) · · · (lk,i − lk−1,k−1 ) ξΛ−δk,i , (lk,i − lk,1 ) · · · ∧ · · · (lk,i − lk,k )
(5.12)
(5.13)
where Λ ± δk,i is the Gelfand–Tsetlin pattern obtained by adding ±1 to the entry λk,i of Λ, and the symbol ∧ in the denominator indicates that the zero term should be omitted. In particular, Eq. (5.11) shows that each Gelfand–Tsetlin vector ξΛ is a weight vector with weight (s1 − s0 , s2 − s1 , . . . , sn − sn−1 ),
(5.14)
where sk = ki=1 λk,i is the sum of the entries in the kth row (and s0 = 0 by convention). Now restrict the representations πλ from gl(n, C) to sl(n, C). Note that two representations πλ and πλ restrict to the same representation of sl(n, C) if, and only if, they differ by a multiple of the trace, i.e., πλ (X) − πλ (X) = Tr(X) for all X ∈ gl(n, C). On diagonal elements the trace is precisely the weight (1, 1, . . . , 1). Hence two Gelfand–Tsetlin patterns define the same basis vector for the sl(n, C)-representation πλ = πλ if and only if they differ in each entry by a fixed overall constant. Lemma 5.5. Inside K = SU(n) (n 3), let ⎧⎛ ⎪ ⎪ ⎨ ⎜ KS = ⎜ ⎝ ⎪ ⎪ ⎩
⎫ ⎞ 0 ⎪ ⎪ .. ⎟ ⎬ A . ⎟ : A ∈ U(n − 1), z ∈ S 1 , z(det A) = 1 , ⎪ 0⎠ ⎪ ⎭ 0 ··· 0 z ⎧⎛ ⎫ ⎞ z 0 ··· 0 ⎪ ⎪ ⎪ ⎪ ⎨⎜ 0 ⎬ ⎟ 1 ⎟ ⎜ : A ∈ U(n − 1), z ∈ S . KT = ⎝ .. , z(det A) = 1 ⎠ ⎪ ⎪ A ⎪ ⎪ ⎩ . ⎭ 0
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Let σ ∈ Kˆ S and > 0. There are only finitely many irreducible representations π of SU(n) which contain unit vectors ξ ∈ pσ V π and η ∈ p1T V π with |η, ξ | > . Proof. Suppose the representation π ∈ Kˆ contains a unit vector η fixed by the subgroup KT . If we put η = π(w)η, where ⎛ w=⎝
.
..
1
⎞ ⎠,
1 then η is fixed by w KT w −1 = KS , and hence is annihilated by the complexified Lie algebra (kS )C . Note that (kS )C contains the upper-left subalgebra gn−1 , so η is a multiple of the Gelfand– Tsetlin vector ξΛ0 with all entries of the pattern Λ0 below the top row being zero (modulo addition of a constant in each entry). In view of the interlacing conditions (5.9), we must have ⎛ ⎜ ⎜ Λ0 = ⎜ ⎜ ⎝
··· 0 ··· 0
0
m 0
..
0 . ..
. 0
−m
0
0
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
0 for some m, m 0. Moreover, η is of weight zero, since (kS )C contains the Cartan subalgebra h, so by (5.14), m = m . Thus, we conclude that the representation π of K necessarily has highest weight of the form λ = λm = (m, 0, . . . , 0, −m). Let πm = πλm , and let η = ηm be a KT -fixed unit vector in V πm . Then ηm has weight zero and is annihilated by πm (Ek,k+1 ) for each k = 2, . . . , n − 1. By the weight formula (5.14), the zero-weight space of πm is spanned by the Gelfand–Tsetlin vectors whose patterns have zero row-sums, i.e., ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
0
m
0
mn−1 ..
··· ···
0 −mn−1
0
.
.
..
−m2
m2
−m
⎞ ⎟ ⎟ ⎟. ⎟ ⎠
(5.15)
0 We will denote such a Gelfand–Tsetlin pattern by Λ(M), where M is the n-tuple (m = mn , mn−1 , . . . , m2 , m1 = 0) with m mn−1 · · · m2 0. Let xΛ = ξξΛΛ , the normalized Gelfand–Tsetlin vectors. We claim that ηm , xΛ(M) =
n−1
1
+ k − 1) 2 . 1 m+n−2
k=2 (2mk
(n − 2)! 2
(5.16)
n−2
This will prove the lemma, for the following reason. Let σ ∈ Kˆ S . If πm contains the KS type σ nontrivially, then by the interlacing conditions (5.8), the highest weight of σ as an sl(n − 1, C)-representation must be of the form (q, 0, . . . , 0, −q ) for some 0 q, q m. Moreover, if q = q , the Gelfand–Tsetlin patterns occurring in this sl(n − 1, C)-subrepresentation have
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nonzero (n − 1)th row-sum, and thus all Gelfand–Tsetlin vectors have nonzero weights by (5.14). Thus the only KS -types in which there can exist ξ with ηm , ξ = 0 are those with highest weight (q, 0, . . . , 0, −q) for some q ∈ N. Denote this KS -type by σq . πm Let > 0. Let ξ ∈ V be a unit vector of KS -type σq . Write ξ in the Gelfand–Tsetlin basis: ξ = Λ bΛ ξΛ . Since the weight zero Gelfand–Tsetlin vectors of type σq in πm are the ξΛ(M) with mn−1 = q, we get ηm , ξ |bΛ(M) |.ηm , xΛ(M) , M
where the sum is over M = (m, q, mn−2 , mn−3 , . . . , m2 , 0). The number of such n-tuples M is bounded by q n−3 , and since ξ = 1, each |bΛ(M) | 1. Thus, the formula (5.16) gives 1
2 (n−2) C ηm , ξ q n−3 · (2q + n −1) = m+n−2 , 1 m+n−2 (n − 2)! 2 n−2 n−2 for some constant C independent of m. It follows that there are only finitely many m ∈ N— and hence finitely many representations πm —for which this inner product can be greater than , which proves the lemma. Therefore, let us prove Eq. (5.16). For 1 < k < n, let M ± ek denote the n-tuple (mn , . . . , mk ± 1, . . . , m1 ). Recall also that we use Λ ± δk,i to denote the pattern obtained by adding ±1 to the (k, i)-entry of Λ. Note that Λ(M) + δk,i does not satisfy the interlacing conditions (5.9) unless i = 1 or k. Note also that Λ(M) + δk,k = Λ(M − ek ) + δk,1 . Recall that ηm satisfies πm (Ek,k+1 )ξΛ(M) = 0 for k = 2, . . . , n − 1. Write ηm in the unnormalized Gelfand–Tsetlin basis for πλ : ηm =
aM ξΛ(M) .
M
By the Gelfand–Tsetlin formula (5.12), πm (Ek,k+1 )ξΛ(M) (mk − mk+1 )( k−1 j =1 (mk + j ))(mk + mk+1 + k) ξΛ(M)+δk,1 =− k−2 ( j =1 (mk + j ))(2mk + k − 1) (−mk − mk+1 − k + 1)( k−2 j =0 (−mk − j ))(−mk + mk+1 + 1) ξΛ(M)+δk,k − (−2mk − k + 1)( k−2 j =1 (−mk − j )) =
(mk+1 − mk )(mk+1 + mk + k)(mk + k − 1) ξΛ(M)+δk,1 (2mk + k − 1) +
(mk+1 − mk + 1)(mk+1 + mk + k − 1)mk ξΛ(M−ek )+δk,1 , (2mk + k − 1)
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for k = 2, . . . , n − 1. Comparing the coefficients of ξΛ(M)+δk,1 in the equation πm (Ek,k+1 )ηm =
aM πm (Ek,k+1 )ξΛ(M) = 0,
M
we see that (mk+1 − mk )(mk+1 + mk + k)(mk + k − 1) aM (2mk + k − 1) +
(mk+1 − mk )(mk+1 + mk + k)(mk + 1) aM+ek = 0, (2mk + k + 1)
so that aM+ek = −
(mk + k − 1) (2mk + k + 1) · aM , (mk + 1) (2mk + k − 1)
(5.17)
for k = 2, . . . , n − 1. For each fixed k = 2, . . . , n − 1, Eq. (5.17) gives a recurrence relation which can be used to reduce the parameter mk . Thus reducing m2 , . . . , mn−1 to zero in turn, we get n−1 m −1 k (i + k − 1) (2i + k + 1) a(m,mn−1 ,...,m2 ,0) = ± · a(m,0,...,0) (i + 1) (2i + k − 1) k=2 i=0
n−1 (mk + k − 2)! (2mk + k − 1) · =± a(m,0,...,0) mk !(k − 2)! (k − 1) k=2
±a(m,0,...,0) = (n − 2)!
n−1 (mk + k − 2)! k=2
mk !(k − 2)!
· (2mk + k − 1) .
(5.18)
This computes the coefficients aM of ηm up to an overall scalar factor of a(m,0,...,0) . In order to determine this scalar (up to phase), we use that the norm of ηm is 1. We first compute ξΛ(M) by 2 )! Eq. (5.10). This is straightforward but tedious. The k = 2 term in (5.10) is m2 ! (2m m2 ! = (2m2 )!. For 3 k n, the terms with i = 1 give (mk − mk−1 )!( k−3 j =1 (mk + j )!)(mk + mk−1 + k − 2)! k−3 0!( j =1 (mk−1 + j )!)(2mk−1 + k − 2)! mk !( k−3 j =1 (mk + j )!)(2mk + k − 2)! × k−3 mk−1 !( j =1 (mk−1 + j )!)(mk + mk−1 + k − 2)! k−3 2 (mk + j )! mk−1 ! (2mk + k − 2)! = (mk − mk−1 )! · · ; · (mk−1 + j )! mk ! (2mk−1 + k − 2)! j =0
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the terms with 1 < i < k − 1 are all 1; and the terms with i = k − 1 give mk−1 !mk ! . (mk − mk−1 )! Thus, ξΛ(M) = 2
n
#
k=3
k (mk + j − 3)! (mk−1 + j − 3)!
j =3
2
$ (2mk + k − 2)! · mk−1 ! · (2m2 )! (2mk−1 + k − 2)! 2
2 n 2 n n (mk + j − 3)! = · mk−1 ! (mk−1 + j − 3)! j =3 k=j k=3 n (2mk + k − 2)! 1 × (2m2 )! (2mk−1 + k − 3)! (2mk−1 + k − 2)
k=3
2 n 2 n (mn + j − 3)! = mk−1 ! (mj −1 + j − 3)! j =3 k=3 n (2mn + n − 2)! 1 × (2m2 )! (2m2 )! (2mk−1 + k − 2) k=3 n (m + k − 3)!2 mk−1 !2 1 · = (2m + n − 2)! (2mk−1 + k − 2) (mk−1 + k − 3)!2 k=3 n−1 (m + k − 2)!2 mk !2 1 · = (2m + n − 2)! . 2 (2mk + k − 1) (mk + k − 2)!
(5.19)
k=2
Combining (5.18) and (5.19), we have (2m + n − 2)! (n − 2)!2 n−1 2 n−1 (m + k − 2)! × (2mk + k − 1) . (k − 2)!
aM ξΛ(M) 2 = |a(m,0,...,0) |2 .
k=2
(5.20)
k=2
Since ηm is a unit vector, the sum of the quantities (5.20) over all descending n-tuples M = (m, mn−1 , . . . , m2 , 0) is 1. Note that only the final term in (5.20) depends on the variables m2 , . . . , mn−1 . For that term the sum is given by a combinatorial identity mmn−1 ··· ···m3 m2 0
n−1 k=2
! " m+n−2 2 (2mk + k − 1) = (n − 2)! , n−2
which we prove in Lemma 5.6 below. Thus,
R. Yuncken / Journal of Functional Analysis 258 (2010) 1140–1166
1 = |a(m,0,...,0) |2 . ×
(2m + n − 2)! (n − 2)!2
n−1 (m + k − 2)! k=2
1161
2
(k − 2)!
! " m+n−2 2 (n − 2)! . n−2
(5.21)
Dividing (5.20) by (5.21) gives n−1 aM ξΛ(M) = 2
k=2 (2mk + k − 1) 2 . (n − 2)! m+n−2 n−2
Since |ηm , xΛ(M) | = |aM ξΛ(M) , xΛ(M) | = aM ξΛ(M) , this proves Eq. (5.16).
2
We needed the following combinatorial identity. Lemma 5.6. Let n 3. For any m ∈ N,
n−1
mmn−1 ··· k=2 ···m3 m2 0
! " m+n−2 2 (2mk + k − 1) = (n − 2)! . n−2
(5.22)
Proof. The identity ! " ! " m i +p−1 2 m+p 2 (2i + p) =p p−1 p i=0
can be easily proven by induction on m, with the key inductive computation being p
! " ! " (m − 1) + p 2 m+p−1 2 + (2m + p) p p−1 ! "2 % & 2 m+p p2 m = + (2m + p) p p (m + p)2 (m + p)2 & " % ! m + p 2 m2 p + 2mp 2 + p 3 = p (m + p)2 ! " m+p 2 =p . p
Now Eq. (5.22) is proven by induction on n, as follows. If n = 3, then (5.22) is m
(2m2 + 1) = (m + 1)2 ,
m2 =0
which is just (5.23) with p = 1. For n > 3, write the left-hand side of (5.22) as
(5.23)
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m
(2mn−1 + n − 2) ·
mn−1 =0
=
m
n−2
(2mk + k − 1)
mn−1 mn−2 ··· k=2 ···m3 m2 0
" ! mn−1 + n − 3 2 (2mn−1 + n − 2)(n − 3)! , n−3
mn−1 =0
by the inductive hypothesis. Applying (5.23) with p = n − 2, gives the result.
2
This completes the proof of Proposition 5.4. The proof is unquestionably very computational. It would be extremely satisfying to have a proof of Proposition 5.4 which is more geometric in nature, especially given the expected wide generality of the result, as suggested in Question 1.15. The next result generalizes Lemma 3.1(i). Corollary 5.7. Let K be a product of special unitary groups, let H be a harmonic K-space, and let T ⊆ Σbe arbitrary. If τ ∈ Kˆ T , the projection pτ is in AS (H) for any S ⊆ Σ. Hence, pτ ∈ A(H) = S⊆Σ AS (H). Proof. Using Proposition 5.4, for any σ ∈ Kˆ S , pτ pσ ∈ KS∪T ⊆ KS , and pσ pτ ∈ KS similarly. Apply Theorem 3.5(iii). 2 Since pσ ∈ KS and JS = KS ∩ A for σ ∈ Kˆ S , we have the following. Corollary 5.8. Let K be a product of special unitary groups and H a harmonic K-space, and let S ⊆ Σ . For any σ ∈ Kˆ S , pσ ∈ JS (H). We can now complete the proof of the main theorem: showing that JS ∩ JT = JS∪T for all S, T ⊆ Σ . Proof of Theorem 1.11(ii). By Lemma 3.7, JS∪T ⊆ JS ∩ JT . For the reverse inclusion, we prove that JS .JT ⊆ JS∪T . Suppose A ∈ JS and B ∈ JT . Lemma 3.4 gives arbitrarily close norm-approximations ApF of A and pF B of B, for some finite sets F ⊂ Kˆ S , F ⊂ Kˆ T . By Proposition 5.4, pF pF ∈ KS∪T . Moreover, Corollary 5.7 gives pF , pF ∈ A, so in fact pF pF ∈ JS∪T . By Lemma 3.7, JS∪T is an ideal in JS and JT , so ApF pF B ∈ JS∪T . Since the latter is an arbitrarily close norm-approximation of AB, we have AB ∈ JS∪T . 2 6. Longitudinal pseudodifferential operators on a fibre bundle In the remainder of the paper, we apply the above harmonic analysis to the titular longitudinal pseudodifferential operators on the complete flag variety of Cn . In particular, we will prove Theorem 1.3. We begin with some generalities on longitudinal pseudodifferential operators. Most of this background can be found in [11]. Let X be a compact manifold and F a foliation of X . Let E be a vector bundle over X . The set of longitudinal pseudodifferential operators of order p on E, tangent to F , will be denoted p by ΨF (E). Put a Riemannian metric on X and Hermitian metric on E, so that we can define the L2 -sections of E. The order zero longitudinal pseudodifferential operators are bounded on
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L2 (X ; E). For any ∞ −p < 0, the set of order −p longitudinal pseudodifferential operators −p
−p
ΨF (E) is an ideal in ΨF0 (E). The operator-norm closure ΨF (E) of this ideal is independent of −∞ −p < 0. Remark 6.1. Here is a fuller explanation, which goes beyond what we shall use. Let S ∗ F be the cosphere bundle of the foliation. The tangential principal symbol map Symb0 : ΨF0 (E) → C S ∗ F , End(E) extends continuously to the norm closure of ΨF0 (E). The kernel of this map contains any longi-
tudinal pseudodifferential operator of negative order, and can be shown to be equal to ΨF−∞ (E). In fact, the kernel is equal to the groupoid C ∗ -algebra Cr∗ (GF ; E) of the foliation.
The ideal ΨF−∞ (E) is much simplified in the case where the foliation comes from a smooth q
fibre bundle X → Y. One can define a C(Y)-valued inner product on the continuous sections of E by L2 -integration along the fibres: s1 , s2 C(Y ) (y) =
s1 (x), s2 (x) x dVolq −1 (y) (x),
(6.1)
q −1 (y)
for s1 , s2 ∈ C(X ; E). Using this, C(X ; E) completes to a Hilbert C(Y)-module, which we denote by EY (X ; E). The following fact is certainly well known, although we are not aware of a specific reference. We therefore provide a very brief proof. Proposition 6.2. The algebra ΨF−∞ (E) is isomorphic to the algebra of compact Hilbert module operators K(EY (X ; E)). Proof (sketch). Since X is compact, the choice of metrics on X and E will not affect the algebras. If the fibration is trivial (X = Y × V) and the bundle E is the pullback of a bundle on the fibre V then the result is a bundle version of the standard fact that the completion of the smoothing operators on a compact manifold is the compact operators. To generalize this, observe that the bundle E → X is locally of the above product form. Use a partition of unity subordinate to a finite trivializing cover of Y to show that the two algebras of the proposition are each included in the other with bounded change in norm. 2 7. Homogeneous vector bundles over the flag variety We now restrict our attention to the complete flag variety X = K/T. For this we will use the harmonic analysis of the regular representation on L2 (K). When working with harmonic projections p σ on L2 (K), we will always take them to be defined with respect to the right regular representation of K. Also, as earlier, we assume K is a product of special unitary groups N i=1 SU(ni ) (ni 2). Let μ be a weight of K. We use Eμ = K ×T V μ to denote the K-homogeneous vector bundle over X induced from μ. Thus, the continuous sections of Eμ are identified with
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( ' C(X ; Eμ ) = s ∈ C(K) s(kt) = eμ t −1 s(k) for all k ∈ K, t ∈ T = p−μ C(K).
(7.1)
Hence L2 (X ; Eμ ) = p−μ L2 (K) is a harmonic K-space. More generally, any K-homogeneous vector bundle E over X decomposes equivariantly into homogeneous line bundles, so that L2 (X ; E) is a harmonic K-space. Remark 7.1. Since L2 (K) is a finite multiplicity harmonic K-space, so too are the section spaces L2 (X ; Eμ ), and hence L2 (X ; E) for any homogeneous vector bundle E over X . Remark 7.2. The left regular representation of K descends to an action of K on L2 (X ; Eμ ). Since this action commutes with the right-isotypical projections pσ , it follows that the left-translation operators belong to AS , for every S ⊆ Σ . (Cf. Remark 1.13.) If s1 , s2 ∈ C(X ; Eμ ) then by the equivariance property (7.1), s1 (k)s2 (k) is constant on right T-cosets. This defines a C(X )-valued (ie, pointwise) inner product of sections. It corresponds to the natural K-invariant Hermitian metric on Eμ . qS
Recall that associated to each S ⊆ Σ there is a fibration X = K/T → K/KS = XS of the complete flag variety. The fibre of X → XS is KS /T, which has a KS -invariant measure induced from Haar measure on KS . This extends to a family of smooth measures on the fibres of X → XS by left translation by K. From the previous section, there is a C(XS )-valued inner product on C(X ; Eμ ) given by s1 , s2 C(XS ) (k) =
s1 (kh)s2 (kh) dh.
h∈KS
Comparing with Eq. (2.1), we can rewrite this as s1 , s2 C(XS ) = p1S (s1 s2 ),
(7.2)
where 1S is the trivial representation of KS . We will denote the resulting Hilbert C(XS )module completion by ES (X ; Eμ ), instead of EXS (X ; Eμ ). Note the extreme cases E∅ (X ; Eμ ) = C(X ; Eμ ) and EΣ (X ; Eμ ) = L2 (X ; Eμ ). 8. Multiplication operators Recall the Peter–Weyl isomorphism, L2 (K) ∼ =
V π† ⊗ V π,
π∈Kˆ
1 dim V π 2 . w † , π(·)v ↔ w † ⊗ v, which intertwines the right regular representation with
1 ⊗ π.
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Lemma 8.1. Let f ∈ C(K). The operator Mf of multiplication by f on L2 (K) belongs to A = A S⊆Σ S . Proof. Fix S ⊆ Σ. The matrix units, f (k) = w † , π(k)v
(8.1)
with π ∈ Kˆ and v ∈ V π , w † ∈ V π† , span a dense subset of C(K). We may further suppose that v is isotypical for KS —say, v ∈ pσ V π for some σ ∈ Kˆ S —since such vectors span V π . Fix such an f . Let σ ∈ Kˆ S . Let s ∈ L2 (K). If s is itself a matrix unit, s(k) = η† , ρ(k)ξ , for some ρ ∈ Kˆ and ξ ∈ V ρ , η† ∈ V ρ† , then the right isotypical projection pσ acts on s as pσ s(k) = η† , ρ(k)pσ ξ .
(8.2)
Multiplying Eqs. (8.1) and (8.2), we get Mf pσ s(k) = w † ⊗ η† , (π ⊗ ρ)(k)(v ⊗ pσ ξ ) . Since v ∈ pσ V π by assumption, the vector v ⊗ pσ ξ lies in a KS -subrepresentation of π ⊗ ρ isomorphic to σ ⊗ σ , which decomposes into a finite set F of KS -types. Thus, for any s ∈ L2 (K), pF⊥ Mf pσ s = 0. The adjoint of multiplication by f is multiplication by f , which is itself a KS -isotypical matrix unit. (Specifically, if we denote by v → v † the canonical anti-linear isomorphism from V π to V π† , then f (k) = (w † , π(k)v) = (w, π † (k)v † ).) It follows that pσ Mf pF⊥ = (pF⊥ Mf pσ )∗ = 0 for some finite F ⊂ Kˆ S . Thus Mf is KS -harmonically proper. Since Mf = f ∞ , the result for general f ∈ C(K) follows from the density of the span of the matrix units. 2 Suppose μ and ν are weights for K. If f ∈ C(X ; Eμ ) and s ∈ L2 (X ; Eν ), then the product f.s is in L2 (X ; Eμ+ν ), as can be readily verified from the defining equivariance property of (7.1). By Lemma 8.1, the multiplication operator Mf also descends to an operator in A(L2 (X ; Eν ), L2 (X ; Eν+μ )). 9. Longitudinal pseudodifferential operators on the flag variety To prove Theorem 1.3, it only remains to prove Proposition 1.12, which states that longitudinal pseudodifferential operators of negative order tangential to the fibration X → XS are in JS . Proof of Proposition 1.12. Let E be a K-homogeneous vector bundle over X . From Section 6, −p elements of ΨFS (E) are compact operators on the Hilbert module ES (X ; E). By decomposing E into a direct sum of K-homogeneous line bundles, we are reduced to considering compact Hilbert module operators from ES (X ; Eμ ) to ES (X ; Eν ) for some weights μ, ν. These are densely
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spanned by the rank-one operators, which have the form A = t2 .t1 , ·C(XS ) , for some t1 ∈ ES (X ; Eμ ), t2 ∈ ES (X ; Eν ). By Eq. (7.2), A = Mt2 p1S Mt1 . Since p1S ∈ JS (Lemma 5.8) and multiplication operators are in A (Lemma 8.1), we have A ∈ JS . As for the order 0 operators, since the character χσ of any finite dimensional KS -representation is smooth, Eq. (2.1) describes the KS -isospectral projections on L2 (X ; Eμ ) as smooth convolu(E) for all σ ∈ Kˆ S . Thus, if A ∈ ΨF0 S (E), tion operators along the fibres of FS . That is, pσ ∈ ΨF−∞ S −∞ then Apσ and pσ A are in ΨFS (E) ⊆ JS . By Lemma 3.5, A ∈ AS . 2 References [1] R. Baston, M. Eastwood, The Penrose Transform. Its Interaction with Representation Theory, Oxford Math. Monogr., Oxford Sci. Publ., The Clarendon Press, Oxford University Press, New York, 1989. [2] J. Bernstein, Analytic structures on representation spaces of reductive groups, in: Proceedings of the International Congress of Mathematicians, vol. II, Berlin, 1998, Doc. Math. (1998) 519–525. [3] I. Bernstein, I. Gel’fand, S. Gel’fand, Differential operators on the base affine space and a study of g-modules, in: Lie Groups and Their Representations, Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971, Halsted, New York, 1975, pp. 21–64. [4] Nigel Higson, The Baum–Connes conjecture, in: Proceedings of the International Congress of Mathematicians, vol. II, Berlin, 1998, Doc. Math. (1998) 637–646. [5] Pierre Julg, La conjecture de Baum–Connes à coefficients pour le groupe Sp(n, 1), C. R. Math. Acad. Sci. Paris 334 (7) (2002) 533–538. [6] P. Julg, G. Kasparov, Operator K-theory for the group SU(n, 1), J. Reine Angew. Math. 463 (1995) 99–152. [7] G. Kasparov, Lorentz groups: K-theory of unitary representations and crossed products, Dokl. Akad. Nauk SSSR 275 (3) (1984) 541–545. [8] A. Knapp, Representation Theory of Semisimple Groups, Princeton Math. Ser., vol. 36, Princeton University Press, Princeton, NJ, 1986. [9] V. Lafforgue, Banach KK-theory and the Baum–Connes conjecture, in: Proceedings of the International Congress of Mathematicians, vol. II, Beijing, 2002, Higher Ed. Press, Beijing, 2002, pp. 795–812. [10] A.I. Molev, Gelfand–Tsetlin bases for classical Lie algebras, in: M. Hazewinkel (Ed.), Handbook of Algebra, vol. 4, Elsevier, 2006, pp. 109–170. [11] Calvin C. Moore, Claude L. Schochet, Global Analysis on Foliated Spaces, second ed., Math. Sci. Res. Inst. Publ., vol. 9, Cambridge University Press, New York, 2006. [12] R. Yuncken, Analytic structures for the index theory of SL(3, C), PhD thesis, Penn State University, 2006. [13] R. Yuncken, The Bernstein–Gelfand–Gelfand complex and Kasparov theory for SL(3, C), preprint, http://arxiv.org/ abs/0802.2094.
Journal of Functional Analysis 258 (2010) 1167–1224 www.elsevier.com/locate/jfa
New Orlicz–Hardy spaces associated with divergence form elliptic operators Renjin Jiang 1 , Dachun Yang ∗ School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China Received 10 December 2008; accepted 23 October 2009 Available online 4 November 2009 Communicated by C. Kenig Dedicated to Professor Lizhong Peng in celebration of his 66th birthday
Abstract Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0, ∞) of strictly critical lower type pω ∈ (0, 1] and ρ(t) = t −1 /ω−1 (t −1 ) for t ∈ (0, ∞). In this paper, the authors study the Orlicz–Hardy space Hω,L (Rn ) and its dual space BMOρ,L∗ (Rn ), where L∗ denotes the adjoint operator of L in L2 (Rn ). Several characterizations of Hω,L (Rn ), including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The ρ-Carleson measure characterization and the John– Nirenberg inequality for the space BMOρ,L (Rn ) are also given. As applications, the authors show that the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL map Hω,L (Rn ) continuously into L(ω). The authors further show that the Riesz transform ∇L−1/2 maps Hω,L (Rn ) into the classical Orlicz–Hardy n , 1] and the corresponding fractional integral L−γ for certain γ > 0 maps space Hω (Rn ) for pω ∈ ( n+1 n n ω is determined by ω and γ , and satisfies the same property Hω,L (R ) continuously into H ω,L (R ), where as ω. All these results are new even when ω(t) = t p for all t ∈ (0, ∞) and p ∈ (0, 1). © 2009 Elsevier Inc. All rights reserved. Keywords: Divergence form elliptic operator; Gaffney estimate; Orlicz–Hardy space; Lusin-area function; Maximal function; Molecule; Carleson measure; John–Nirenberg inequality; Dual; BMO; Riesz transform; Fractional integral
* Corresponding author.
E-mail addresses: [email protected] (R. Jiang), [email protected] (D. Yang). 1 Present address: Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), 40014,
Finland. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.018
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1. Introduction Ever since Lebesgue’s theory of integration has taken a center stage in concrete problems of analysis, the need for more inclusive classes of function spaces than the Lp (Rn )-families naturally arose. It is well known that the Hardy spaces H p (Rn ) when p ∈ (0, 1] is a good substitute of Lp (Rn ) when studying the boundedness of operators, for example, the Riesz operator is bounded on H p (Rn ), but not on Lp (Rn ) when p ∈ (0, 1]. The theory of Hardy spaces H p on the Euclidean space Rn was initially developed by Stein and Weiss [36]. Later, Fefferman and Stein [15] systematically developed a real-variable theory for the Hardy spaces H p (Rn ) with p ∈ (0, 1], which now plays an important role in various fields of analysis and partial differential equations; see, for example, [35,10,17,29,33]. A key feature of the classical Hardy spaces is their atomic decomposition characterizations, which were obtained by Coifman [9] when n = 1 and Latter [27] when n > 1. On the other hand, as another generalization of Lp (Rn ), the Orlicz space was introduced by Birnbaum–Orlicz in [7] and Orlicz in [30], since then, the theory of the Orlicz spaces themselves has been well developed and the spaces have been widely used in probability, statistics, potential theory, partial differential equations, as well as harmonic analysis and some other fields of analysis; see, for example, [31,32,8,28,1,23]. Moreover, the Orlicz–Hardy spaces are also good substitutes of the Orlicz spaces in dealing with many problems of analysis, say, the boundedness of operators. In particular, Strömberg [37] and Janson [24] introduced generalized Hardy spaces Hω (Rn ), via replacing the norm · Lp (Rn ) by the Orlicz-norm · L(ω) in the definition of H p (Rn ), where ω is an Orlicz function on [0, ∞) satisfying some control conditions. Viviani [39] further characterized these spaces Hω on spaces of homogeneous type via atoms. The dual spaces of these spaces were also studied in [37,24,39,22]. All theories of these spaces are intimately connected with properties of harmonic analysis and of the Laplacian operator on Rn . In recent years, function spaces, especially Hardy spaces and BMO spaces, associated with different operators inspire great interests; see, for example, [3,5,6,12–14,19,40,25,18] and their references. In particular, Auscher, Duong and McIntosh [3] first introduced the Hardy space HL1 (Rn ) associated with an operator L whose heat kernel satisfies a pointwise Poisson type upper bound by means of a corresponding variant of the Lusin-area function, and established its molecular characterization. Duong and Yan [13,14] introduced its dual space BMOL (Rn ) and established the dual relation between HL1 (Rn ) and BMOL (Rn ). Yan [40] further generalized these p results to the Hardy spaces HL (Rn ) with certain p 1 and their dual spaces. Also, Auscher and Russ [6] studied the Hardy space HL1 on strongly Lipschitz domains associated with a divergence form elliptic operator L whose heat kernels have the Gaussian upper bounds and regularity. Very recently, Auscher, McIntosh and Russ [5] treated the Hardy space H p with p ∈ [1, ∞] associated to Hodge Laplacian on a Riemannian manifold with doubling measure, and Hofmann and Mayboroda [19] further studied the Hardy space HL1 (Rn ) and its dual space adapted to a second order divergence form elliptic operator L on Rn with bounded complex coefficients and these operators may not have the pointwise heat kernel bounds. Motivated by [19,24,39], in this paper, we study Orlicz–Hardy spaces Hω,L (Rn ) associated to the divergence form elliptic operator L in [19] and their dual space BMOρ,L∗ (Rn ), where L∗ denotes the adjoint operator of L in L2 (Rn ), the positive function ω on (0, ∞) is concave and of strictly critical lower type pω ∈ (0, 1] and ρ(t) = t −1 /ω−1 (t −1 ) for all t ∈ (0, ∞). A typical example of such Orlicz functions is ω(t) = t p for all t ∈ (0, ∞) and p ∈ (0, 1]. As applications, we obtain the boundedness of the Riesz transform, the Littlewood–Paley g-function and the fractional integral associated with L on Hω,L (Rn ), which may not be bounded on the classical
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Orlicz–Hardy space Hω (Rn ) or the Orlicz space L(ω). Thus, it is necessary to introduce and study the Orlicz–Hardy space Hω,L (Rn ). Recall that the classical BMO(Rn ) was originally introduced and studied by John and Nirenberg [26] in the context of partial differential equations, which has been identified as the dual space of H 1 (Rn ) in the work by Fefferman and Stein [15]. Also, the generalized space BMOρ (Rn ) was introduced and studied in [37,24,39,22] and it was proved therein to be the dual space of Hω (Rn ). To state the main content of this paper, we first recall some notation and known facts on second order divergence form elliptic operators on Rn with bounded complex coefficients from [2,19]. Let A be an n × n matrix with entries {aj,k }nj, k=1 ⊂ L∞ (Rn , C) satisfying the ellipticity conditions, namely, there exist constants 0 < λA ΛA < ∞ such that for all ξ, ζ ∈ Cn and almost every x ∈ Rn , λA |ξ |2 Re A(x)ξ, ξ and A(x)ξ, ζ ΛA |ξ ||ζ |. (1.1) Then the second order divergence form operator is given by Lf ≡ div(A∇f ),
(1.2)
interpreted in the weak sense via a sesquilinear form. Following [19], set pL ≡ inf p 1: supe−tL Lp (Rn )→Lp (Rn ) < ∞ t>0
and p L ≡ sup p ∞: supe−tL Lp (Rn )→Lp (Rn ) < ∞ . t>0
It was proved by Auscher [2] that if n = 1, 2, then pL = 1 and p L = ∞, and if n 3, then pL < 2n/(n + 2) and p L > 2n/(n − 2). Moreover, thanks to a counterexample given by Frehse [16], this range is also sharp, which was pointed out to us by Professor Pascal Auscher. For all f ∈ L2 (Rn ) and x ∈ Rn , define
SL f (x) ≡
2 dy dt 2 −t 2 L t Le f (y) n+1 t
1/2 (1.3)
,
Γ (x)
where and in what follows, Γ (x) ≡ {(y, t) ∈ Rn × (0, ∞): |y − x| < t}. The space Hω,L (Rn ) is defined to be the completion of the set {f ∈ L2 (Rn ): SL f ∈ L(ω)} with respect to the quasinorm
SL f (x) dx 1 . ω f Hω,L (Rn ) ≡ SL f L(ω) = inf λ > 0: λ Rn
p
If p 1 and ω(t) = t p for all t ∈ (0, ∞), we then denote the Hardy space Hω,L (Rn ) by HL (Rn ). The Hardy space HL1 (Rn ) was studied by Hofmann and Mayboroda in [19] (see also [20] for a corrected version).
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In this paper, we first obtain the molecular decomposition of the Orlicz–Hardy space Hω,L (Rn ). Using this molecular decomposition, we then establish the dual relation between the spaces Hω,L (Rn ) and BMOρ,L∗ (Rn ), and the molecular characterization of Hω,L (Rn ). Characterizations via the Lusin-area function associated to the Poisson semigroup and the maximal functions are also obtained. We also establish the ρ-Carleson measure characterization and the John–Nirenberg inequality for the space BMOρ,L (Rn ). As applications, we show that the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL map Hω,L (Rn ) continuously into L(ω); in particular, ∇L−1/2 maps Hω,L (Rn ) into the classical Orlicz–Hardy space Hω (Rn ) n for pω ∈ ( n+1 , 1]. Moreover, we show that the corresponding fractional integral L−γ for all n ω is determined by ω γ ∈ (0, n2 ( p1L − p1 )) maps Hω,L (Rn ) continuously into H ω,L (R ), where L and γ , and satisfies the same property as ω. All these results are new even when ω(t) = t p for all t ∈ (0, ∞) and p ∈ (0, 1). When p = 1 and ω(t) = t for all t ∈ (0, ∞), some of results are also new. The key step of the above approach is to establish a molecular characterization of the Orlicz– Hardy space Hω,L (Rn ). To this end, a main difficulty encountered is the convergence problem of the summation of molecules, i.e., in what sense does the molecular characterization hold? In Theorem 5.1 below, we prove that our molecular characterization holds in the dual of BMOρ,L∗ (Rn ). This is quite different from the cases for the Hardy space HL1 (Rn ) in [19] and the Hardy space H 1 (ΛT ∗ M) in [5], which only need that the molecular characterizations hold pointwise; see [19, (1.11)] (or its corrected version in [20]) and [5, Definition 6.1]. Recall that M denotes a complete Riemannian manifold and
ΛT ∗ M ≡
Λk T ∗ M
0kdim M
the bundle over M whose fibre at each x ∈ M is given by ΛTx∗ M, the complex exterior algebra over the cotangent space Tx∗ M; see [5, p. 194]. In this paper, to obtain the molecular characterization of Hω,L (Rn ), we first need to show that the dual space of Hω,L (Rn ) is BMOρ,L∗ (Rn ) in Theorem 4.1 below. The key ingredients used in the proof of Theorem 4.1 is the Calderón reproducing formula (Lemma 4.3 below) and the atomic decomposition of the tent space Tω (Rn+1 + ) (Theorem 3.1 below). We point out that the dual space of HL1 (Rn ) was already obtained in [19, Theorems 8.2 and 8.6] by a different, but more complicated, approach, without invoking the atomic decomposition of the tent space. Also, the dual space of H 1 (ΛT ∗ M) was obtained in [5] as a direct corollary of the dual theorem on the corresponding tent space; see [5, Theorem 5.8]. Another key tool used in this paper to obtain the maximal function characterizations of Hω,L (Rn ) and their applications in boundedness of operators is Lemma 5.1 below, which gives a sufficient condition for the boundedness of linear or non-negative sublinear operators from Hω,L (Rn ) to L(ω). Such a condition for the molecular Hardy space in HL1 (Rn ) case was also given in [19, Lemma 3.3], which is a direct corollary of the definition of the molecular Hardy space; see its corrected version in [20]. To obtain Lemma 5.1, we need the following impor2 n+1 tant observation that for all f ∈ Hω,L (Rn ) ∩ L2 (Rn ), since t 2 Le−t L f ∈ T22 (Rn+1 + ) ∩ Tω (R+ ), 2 by Proposition 3.1 below, the atomic decomposition of t 2 Le−t L f holds in both Tω (Rn+1 + ) and
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p
T2 (Rn+1 + ) for all p ∈ [1, 2]. Then by the fact that the operator πL,M , which is introduced in [14] and initially defined on F ∈ L2 (Rn+1 + ) with compact support by
∞ 2 M+1 −t 2 L dt t L e F (·, t) , πL,M F ≡ CM t
(1.4)
0 p
p n L ) (see Proposition 4.1 below), we further is bounded from T2 (Rn+1 + ) to L (R ) for p ∈ (pL , p p n obtain the L (R )-convergence with p ∈ (pL , 2] and the Hω,L (Rn )-convergence of the corresponding molecular decomposition for functions in Hω,L (Rn )∩L2 (Rn ) in Proposition 4.2 below. These convergences are necessary and play a fundamental role in the whole paper, which is totally different from the δ-representation used in [19,20,18]. Here and in what follows, M ∈ N and
∞ CM
t 2(M+2) e−2t
2
dt = 1. t
0
We remark that the convergence of the atomic decomposition of the tent spaces was also already carefully dealt with in [5] (We thank Professor Pascal Auscher to point out this to us). To be precise, in [5, pp. 209–210], Auscher, McIntosh and Russ proved that for any functions F in the intersection of the tent spaces T 1,2 (ΛT ∗ M) and T 2,2 (ΛT ∗ M) with the support M × [ , ∞) for some > 0, Fn ≡ F χB(x0 ,n)×(1/n,n) for any x0 ∈ M has an atomic decomposition which converges in both T 1,2 (ΛT ∗ M) and T 2,2 (ΛT ∗ M); see [5, (4.5)]. Observe that the compact support of Fn plays an important role in establishing the convergence of its atomic decomposition in [5]. p n+1 However, Proposition 3.1 below is true for all functions in Tω (Rn+1 + ) ∩ T2 (R+ ) without assuming the compact supports. To obtain this proposition, we need to subtly use the construction of the supports of atoms in the atomic decomposition of tent spaces Tω (Rn+1 + ) in Theorem 3.1 below and the Lebesgue dominated convergence theorem. This paper is organized as follows. In Section 2, we recall some notions and known results concerning operators associated with L and describe some basic assumptions on the Orlicz function ω considered in this paper. We point out that throughout the whole paper, we always assume that ω on (0, ∞) is concave and of strictly critical lower type pω ∈ (0, 1]. These restrictions are necessary for the Orlicz–Hardy space Hω,L (Rn ) to have the molecular characterization; see Theorem 5.1 below. Thus, under these restrictions, the Orlicz–Hardy space Hω,L (Rn ) behaves more closely like the classical Hardy space. We leave the study on the Orlicz–Hardy space with a Young function in a forthcoming paper, which may have some properties similar to those of the spaces H p (ΛT ∗ M) with p ∈ (1, ∞] as in [5]. In Section 3, we introduce the tent spaces Tω (Rn+1 + ) associated to ω and establish its atomic characterization; see Theorem 3.1 below. By the proof of Theorem 3.1, we observe that if a p n+1 function F ∈ Tω (Rn+1 + ) ∩ T2 (R+ ), p ∈ (0, ∞), then there exists an atomic decomposition of F p n+1 which converges in both Tω (Rn+1 + ) and T2 (R+ ); see Proposition 3.1 below. As a consequence, n+1 n+1 2 we prove that if F ∈ Tω (R+ ) ∩ T2 (R+ ), then there exists an atomic decomposition of F p n+1 which converges in both Tω (Rn+1 + ) and T2 (R+ ) for all p ∈ [1, 2]; see Corollary 3.1 below. In Section 4, we first introduce the Orlicz–Hardy space Hω,L (Rn ), and then prove that the opp p n L ) erator πL,M in (1.4) maps the tent space T2 (Rn+1 + ) continuously into L (R ) for p ∈ (pL , p
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n and Tω (Rn+1 + ) continuously into Hω,L (R ) (see Proposition 4.1 below). Combined this with Corollary 3.1, we obtain a molecular decomposition for elements in Hω,L (Rn ) ∩ L2 (Rn ) which converges in Lp (Rn ) for p ∈ (pL , 2]; see Proposition 4.2 below. Via this molecular decomposition of Hω,L (Rn ), we further obtain the duality between Hω,L (Rn ) and BMOρ,L∗ (Rn ) (see Theorem 4.1 below). We also remark that the proof of Theorem 4.1 is much simpler than the proof of [19, Theorem 8.2]. In Section 5, we introduce the molecular Hardy space, where the summation of molecules converges in the space (BMOρ,L∗ (Rn ))∗ , the dual space of BMOρ,L∗ (Rn ). Then we show that the molecular Hardy space is equivalent to the Orlicz–Hardy space Hω,L (Rn ) with equivalent norms; see Theorem 5.1 below. Furthermore, we characterize Hω,L (Rn ) via the Lusin-area function associated to the Poisson semigroup, and the maximal functions; see Theorem 5.2 below. We also point out that a sufficient condition for the boundedness of linear or non-negative sublinear operators from Hω,L (Rn ) to L(ω) is also given in Lemma 5.1 below, which plays a key role in the proof of Theorem 5.2 and is very useful in applications (see Section 7 of this paper). This condition is also necessary if ω(t) = t p for all t ∈ (0, ∞) and p ∈ (0, 1]. Section 6 is devoted to establish the ρ-Carleson measure characterization (see Theorem 6.1 below) and the John–Nirenberg inequality (see Theorem 6.2 below) for the space BMOρ,L (Rn ). In Section 7, as applications, we give some sufficient conditions which guarantee the boundedness of linear or non-negative sublinear operators from Hω,L (Rn ) to L(ω); in particular, we show that the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL map Hω,L (Rn ) continuously into L(ω); see Theorem 7.1 below. A fractional variant of Theorem 7.1 is also given in this section; see Theorem 7.2 below. Using Theorem 7.2, we prove that the fractional n ω integral L−γ for all γ ∈ (0, n2 ( p1L − p1 )) maps Hω,L (Rn ) continuously into H ω,L (R ), where L is determined by ω and γ and satisfies the same property as ω; see Theorem 7.3 below. In parp q ticular, L−γ maps HL (Rn ) continuously into HL (Rn ) for 0 < p < q 1 and n/p − n/q = 2γ ; see Remark 7.3 below. Applying Theorems 7.1 and 7.3, we further show that ∇L−1/2 maps p n , 1], and in particular, HL (Rn ) into the Hω,L (Rn ) continuously into Hω (Rn ) for pω ∈ ( n+1 n , 1]; see Theorem 7.4 below. Moreover, we show classical Hardy space H p (Rn ) for p ∈ ( n+1 n n n that Hω,L (R ) ⊂ Hω (R ) for all pω ∈ ( n+1 , 1] in Remark 7.4 below. It was also pointed out by Hofmann and Mayboroda in [19] that HL1 (Rn ) is a proper subspace of H 1 (Rn ) for certain L as in (1.2). We remark that if L = − + V with V ∈ L1loc (Rn ) is the Schrödinger operator on Rn , then it was proved in [18] that ∇L−1/2 maps HL1 (Rn ) into the classical Hardy space H 1 (Rn ). We point out that this paper is strongly motivated by Hofmann and Mayboroda [19], and we also directly use some estimates from [19] which simplify the proofs of some theorems of this paper. Finally, we make some conventions. Throughout the whole paper, L always denotes the second order divergence form operator as in (1.2). We denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. The symbol X Y means that there exists a positive constant C such that X CY ; the symbol α for α ∈ R denotes the maximal integer no more than α; B(zB , rB ) denotes an open ball with center zB and radius rB and CB(zB , rB ) ≡ B(zB , CrB ). Set N ≡ {1, 2, . . .} and Z+ ≡ N ∪ {0}. For any subset E of Rn , we denote by E the set Rn \ E.
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2. Preliminaries In this section, we recall some notions and notation on the divergence form elliptic operator, and present some basic properties on Orlicz functions and also describe some basic assumptions on them. 2.1. Some notions on the divergence form elliptic operator L In this subsection, we present some known facts about the operator L considered in this paper. A family {St }t>0 of operators is said to satisfy the L2 off-diagonal estimates, which is also called the Gaffney estimates (see [19]), if there exist positive constants c, C and β such that for arbitrary closed sets E, F ⊂ Rn , St f L2 (F ) Ce−(
dist(E,F )2 β ) ct
f L2 (E)
for every t > 0 and every f ∈ L2 (Rn ) supported in E. Here and in what follows, for any p ∈ (0, ∞] and E ⊂ Rn , f Lp (E) ≡ f χE Lp (Rn ) ; for any sets E, F ⊂ Rn , dist(E, F ) ≡ inf{|x − y|: x ∈ E, y ∈ F }. The following results were obtained in [2,4,19,21]. Lemma 2.1. (See [21].) If two families of operators, {St }t>0 and {Tt }t>0 , satisfy Gaffney estimates, then so does {St Tt }t>0 . Moreover, there exist positive constants c, C, and β such that for arbitrary closed sets E, F ⊂ Rn , dist(E,F )2 β
Ss Tt f L2 (F ) Ce−( c max{s,t} ) f L2 (E) for every s, t > 0 and every f ∈ L2 (Rn ) supported in E. Lemma 2.2. (See [4,21].) The families, −tL , tLe−tL t>0 , e t>0
1/2 −tL t ∇e , t>0
(2.1)
as well as
(I + tL)−1
t>0
,
1/2 t ∇(I + tL)−1 t>0 ,
(2.2)
are bounded on L2 (Rn ) uniformly in t and satisfy the Gaffney estimates with positive constants c, C depending on n, λA , ΛA as in (1.1) only. For the operators in (2.1), β = 1, while in (2.2), β = 1/2. 2n 2n Lemma 2.3. (See [2,19].) There exist pL ∈ [1, n+2 ), p L ∈ ( n−2 , ∞] and c, C ∈ (0, ∞) such that
L , the families {e−tL }t>0 and {tLe−tL }t>0 satisfy (i) for every p and q with pL < p q < p p q L − L off-diagonal estimates, i.e., for arbitrary closed sets E, F ⊂ Rn , −tL e f
Lq (F )
n 1 1 dist(E,F )2 ( − ) + tLe−tL f Lq (F ) Ct 2 q p e− ct f Lp (E)
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for every t > 0 and every f ∈ Lp (Rn ) supported in E. The operators {e−tL }t>0 and n 1
( −1)
{tLe−tL }t>0 are bounded from Lp (Rn ) to Lq (Rn ) with the norm Ct 2 q p ; (ii) for every p ∈ (pL , p L ), the family {(I + tL)−1 }t>0 satisfies Lp − Lp off-diagonal estimates, i.e., for arbitrary closed sets E, F ⊂ Rn , dist(E,F ) n 1 1 ( − ) − (I + tL)−1 f q Ct 2 q p e ct 1/2 f Lp (E) L (F )
for every t > 0 and every f ∈ Lp (Rn ) supported in E. Lemma 2.4. (See [19].) Let k ∈ N and p ∈ (pL , p L ). Then the operator given by for any f ∈ Lp (Rn ) and x ∈ Rn ,
SLk f (x) ≡
2 dy dt 2 k −t 2 L t L e f (y) n+1 t
1/2 ,
Γ (x)
is bounded on Lp (Rn ). 2.2. Orlicz functions Let ω be a positive function defined on R+ ≡ (0, ∞). The function ω is said to be of upper type p (resp. lower type p) for certain p ∈ [0, ∞), if there exists a positive constant C such that for all t 1 (resp. t ∈ (0, 1]) and s ∈ (0, ∞), ω(st) Ct p ω(s).
(2.3)
Obviously, if ω is of lower type p for certain p > 0, then limt→0+ ω(t) = 0. So for the sake of convenience, if it is necessary, we may assume that ω(0) = 0. If ω is of both upper type p1 and lower type p0 , then ω is said to be of type (p0 , p1 ). Let pω+ ≡ inf p > 0: there exists C > 0 such that (2.3) holds for all t ∈ [1, ∞), s ∈ (0, ∞) , and pω− ≡ sup p > 0: there exists C > 0 such that (2.3) holds for all t ∈ (0, 1], s ∈ (0, ∞) . The function ω is said to be of strictly lower type p if for all t ∈ (0, 1) and s ∈ (0, ∞), ω(st) t p ω(s), and define pω ≡ sup p > 0: ω(st) t p ω(s) holds for all s ∈ (0, ∞) and t ∈ (0, 1) . It is easy to see that pω pω− pω+ for all ω. In what follows, pω , pω− and pω+ are called to be the strictly critical lower type index, the critical lower type index and the critical upper type index of ω, respectively.
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Remark 2.1. We claim that if pω is defined as above, then ω is also of strictly lower type pω . In other words, pω is attainable. In fact, if this is not the case, then there exist certain s ∈ (0, ∞) and t ∈ (0, 1) such that ω(st) > t pω ω(s). Hence there exists ∈ (0, pω ) small enough such that ω(st) > t pω − ω(s), which is contrary to the definition of pω . Thus, ω is of strictly lower type pω . Throughout the whole paper, we always assume that ω satisfies the following assumption. Assumption (A). Let pω be defined as above. Suppose that ω is a positive Orlicz function on R+ with pω ∈ (0, 1], which is continuous, strictly increasing and concave. Notice that if ω satisfies Assumption (A), then ω(0) = 0 and ω is obviously of upper type 1. Since ω is concave, it is subadditive. In fact, let 0 < s < t, then s+t st ω(t) ω(t) + ω(s) = ω(s) + ω(t). t ts t For any concave function ω of strictly lower type p, if we set ω(t) ≡ 0 ω(s)/s ds for t ∈ [0, ∞), then by [39, Proposition 3.1], ω is equivalent to ω, namely, there exists a positive constant C such ω(t) Cω(t) for all t ∈ [0, ∞); moreover, ω is strictly increasing, concave and that C −1 ω(t) continuous function of strictly lower type p. Since all our results are invariant on equivalent functions, we always assume that ω satisfies Assumption (A); otherwise, we may replace ω by ω. ω(s + t)
Convention (C). From Assumption (A), it follows that 0 < pω pω− pω+ 1. In what folω ≡ pω+ ; otherwise pω+ < 1 and we lows, if (2.3) holds for pω+ with t ∈ [1, ∞), then we choose p + + choose p ω ∈ (pω , 1) to be close enough to pω . For example, if ω(t) = t p with p ∈ (0, 1], then pω = pω+ = p ω = p; if ω(t) = t 1/2 ln(e4 + t), + ω < 1. then pω = pω = 1/2, but 1/2 < p Let ω satisfy Assumption (A). A measurable function f on Rn is said to be in the Lebesgue type space L(ω) if
ω f (x) dx < ∞. Rn
Moreover, for any f ∈ L(ω), define
|f (x)| f L(ω) ≡ inf λ > 0: ω dx 1 . λ Rn
Since ω is strictly increasing, we define the function ρ(t) on R+ by setting, for all t ∈ (0, ∞), ρ(t) ≡
t −1 , ω−1 (t −1 )
(2.4)
where and in what follows, ω−1 denotes the inverse function of ω. Then the types of ω and ρ have the following relation; see [39] for its proof.
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Proposition 2.1. Let 0 < p0 p1 1 and ω be an increasing function. Then ω is of type (p0 , p1 ) if and only if ρ is of type (p1−1 − 1, p0−1 − 1). 3. Tent spaces associated to Orlicz functions In this section, we study the tent spaces associated to Orlicz functions. We first recall some notions. n For any ν > 0 and x ∈ Rn , let Rn+1 + ≡ R × (0, ∞) and Γν (x) ≡ (y, t) ∈ Rn+1 + : |x − y| < νt n denoting the cone of aperture ν with vertex x ∈ Rn . For any closed set F of R , denote by Rν F the union of all cones with vertices in F , i.e., Rν F ≡ x∈F Γν (x); and for any open set O in Rn , denote the tent over O by Tν (O), which is defined as Tν (O) ≡ [Rν (O )] . Notice that Tν (O) = (x, t) ∈ Rn × (0, ∞): dist x, O νt .
respectively. In what follows, we denote Γ1 (x), R1 (F ) and T1 (O) simply by Γ (x), R(F ) and O, n Let F be a closed subset of R and O ≡ F . Assume that |O| < ∞. For any fixed γ ∈ (0, 1), we say that x ∈ Rn has the global γ -density with respect to F if |B(x, r) ∩ F | γ |B(x, r)| for all r > 0. Denote by F ∗ the set of all such x. Obviously, F ∗ is a closed subset of F . Let O ∗ ≡ (F ∗ ) . Then it is easy to see that O ⊂ O ∗ . In fact, we have O ∗ = x ∈ Rn : M(χO )(x) > 1 − γ , where M denotes the Hardy–Littlewood maximal function on Rn . As a consequence, by the weak type (1, 1) of M, we have |O ∗ | C(γ )|O|, where and in what follows, C(γ ) denotes a positive constant depending on γ . The proof of the following lemma is similar to that of [11, Lemma 2]; we omit the details. Lemma 3.1. Let ν, η ∈ (0, ∞). Then there exist positive constants γ ∈ (0, 1) and C(γ , ν, η) such that for any closed subset F of Rn whose complement has finite measure and any non-negative measurable function H on Rn+1 + ,
n H (y, t)t dy dt C(γ , ν, η) H (y, t) dy dt dx, Rν (F ∗ )
F
Γη (x)
where F ∗ denotes the set of points in Rn with global γ -density with respect to F . n Let ν ∈ (0, ∞). For all measurable functions g on Rn+1 + and all x ∈ R , let
2 dy dt 1/2 g(y, t) n+1 Aν (g)(x) ≡ , t Γν (x)
and denote A1 (g) simply by A(g).
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p
Coifman, Meyer and Stein [11] introduced the tent space T2 (Rn+1 + ) for p ∈ (0, ∞), which is defined as the space of all measurable functions g such that gT p (Rn+1 ) ≡ A(g)Lp (Rn ) < ∞. + 2 On the other hand, let ω satisfy Assumption (A). Harboure, Salinas and Viviani [22] defined the tent space Tω (Rn+1 + ) associated to the function ω as the space of measurable functions g on Rn+1 such that A(g) ∈ L(ω) with the norm defined by + gTω (Rn+1 ) +
≡ A(g)
A(g)(x) dx 1 . = inf λ > 0: ω L(ω) λ Rn
Lemma 3.2. Let η, ν ∈ (0, ∞). Then there exists a positive constant C, depending on η and ν, such that for all measurable functions H on Rn+1 + , C
−1
ω Aη (H )(x) dx
Rn
ω Aν (H )(x) dx C
Rn
ω Aη (H )(x) dx.
(3.1)
Rn
Proof. By the symmetry, we only need to establish the first inequality in (3.1). To this end, let λ ∈ (0, ∞) and Oλ ≡ {x ∈ Rn : Aν (H )(x) > λ}. If |Oλ | = ∞, then Rn ω(Aν (H )(x)) dx = ∞ and the inequality automatically holds. Now, assume that |Oλ | < ∞. Applying Lemma 3.1 with Fλ ≡ (Oλ ) , we have
Rη (Fλ∗ )
H (y, t)2 dy dt t
H (y, t)2 dy dt dx t n+1
Fλ Γν (x)
2 Aν (H )(x) dx.
Fλ
Here and in what follows, we denote (Fλ )∗ and (Oλ )∗ = ((Fλ )∗ ) simply by Fλ∗ and Oλ∗ , respectively. Observe that
Fλ∗ Γη (x)
H (y, t)2 dy dt dx t n+1
Rη (Fλ∗ )
H (y, t)2 dy dt , t
which implies that
2 Aη (H )(x) dx
Fλ∗
2 Aν (H )(x) dx.
Fλ
Here and in what follows, for a measurable function g on Rn and λ > 0, let σg (λ) denote the distribution of g, namely, σg (λ) = |{x ∈ Rn : |g(x)| > λ}|. Hence, we have 1 σAη (H ) (λ) Oλ∗ + 2 λ
(Oλ )
2 1 Aν (H )(x) dx |Oλ | + 2 λ
λ tσAν (H ) (t) dt. 0
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Since ω is of upper type 1 and lower type pω ∈ (0, 1], we have
t ω(t) ∼
ω(u) du u
(3.2)
0
for each t ∈ (0, ∞), which further implies that
ω Aη (H )(x) dx ∼
Rn
Aη (H )(x) Rn
ω(t) dt dx ∼ t
0
∞
ω(t) dt + σAν (H ) (t) t ω(t) dt + σAν (H ) (t) t
0
∞
ω(t) t3
0
∞
σAη (H ) (t)
ω(t) dt t
0
0
∞
t sσAν (H ) (s) ds dt 0
∞
∞ sσAν (H ) (s)
0
ω(t) dt ds t3
s
ω Aν (H )(x) dx.
Rn
This proves the first inequality in (3.1), and hence, finishes the proof of Lemma 3.2.
2
We next give the atomic characterization of the tent space Tω (Rn+1 + ). Let p ∈ (1, ∞). A function a on Rn+1 is called an (ω, p)-atom if + (i) there exists a ball B ⊂ Rn such that supp a ⊂ B; (ii) aT p (Rn+1 ) |B|1/p−1 [ρ(|B|)]−1 . 2
+
Since ω is concave, by the Jensen inequality, it is easy to see that for all (ω, p)-atoms a, we have aTω (Rn+1 ) 1. + Furthermore, if a is an (ω, p)-atom for all p ∈ (1, ∞), we then call a an (ω, ∞)-atom. Theorem 3.1. Let ω satisfy Assumption (A). Then for any f ∈ Tω (Rn+1 + ), there exist (ω, ∞)∞ ⊂ C such that for almost every (x, t) ∈ Rn+1 , atoms {aj }∞ and numbers {λ } j j =1 + j =1 f (x, t) =
∞
λj aj (x, t).
(3.3)
j =1
Moreover, there exists a positive constant C such that for all f ∈ Tω (Rn+1 + ), ∞ |λj | 1 Cf Tω (Rn+1 ) , |Bj |ω Λ({λj aj }j ) ≡ inf λ > 0: + λ|Bj |ρ(|Bj |) j =1
j appears as the support of aj . where B
(3.4)
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Proof. We prove this theorem by borrowing some ideas from the proof of Theorem 1 in Coifman, n k Meyer and Stein [11]. Let f ∈ Tω (Rn+1 + ). For any k ∈ Z, let Ok ≡ {x ∈ R : A(f )(x) > 2 } and n+1 Fk ≡ (Ok ) . Since f ∈ Tω (R+ ), for each k, Ok is an open set and |Ok | < ∞. Since ω is of upper type 1, by Lemma 3.1, for k ∈ Z and k 0, we have
R(Fk∗ )
f (y, t)2 dy dt t
f (y, t)2 dy dt dx t n+1
Fk Γ (x)
2 A(f )(x) dx
Fk
ω A(f )(x) dx → 0,
Fk
as k → −∞, which implies that f = 0 almost everywhere in k∈Z R(Fk∗ ), and hence, supp f ⊂ dx dt ∗ ∪ E}, where E ⊂ Rn+1 and { k∈Z O + k E t = 0. Thus, for each k, by applying the Whitney decomposition to the set Ok∗ , we obtain a set Ik of indices and a family {Qk,j }j ∈Ik of disjoint cubes such that (i) j ∈Ik Qk,j = Ok∗ , and if i = j , then Qk,j ∩ Qk,i = ∅, √ √ (ii) n(Qk,j ) dist(Qk,j , (Ok∗ ) ) 4 n(Qk,j ), where (Qk,j ) denotes the side-length of Qk,j . Next, for each j ∈ Ik , we choose a ball Bk,j with the same center as Qk,j and with radius ∗ ∗ (Qk,j ). Let Ak,j ≡ B k,j ∩ (Qk,j × (0, ∞)) ∩ (Ok \ Ok+1 ),
11 √ n-times 2
−1 ak,j ≡ 2−k |Bk,j |−1 ρ |Bk,j | f χAk,j ∗ ∗ and λk,j ≡ 2k |Bk,j |ρ(|B k,j |). Notice that {(Qk,j × (0, ∞)) ∩ (Ok \ Ok+1 )} ⊂ Bk,j . From this, we conclude that f = k∈Z j ∈Ik λk,j ak,j almost everywhere. k,j . Let Let us show that for each k ∈ Z and j ∈ Ik , ak,j is an (ω, ∞)-atom supported in B q p ∈ (1, ∞), q ≡ p be the conjugate index of p, i.e., 1/q + 1/p = 1, and h ∈ T2 (Rn+1 + ) with ∗ ∗ ) = R(F ), by Lemma 3.1 and the Hölder inequality, h q n+1 1. Since Ak,j ⊂ (O k+1
T2 (R+ )
k+1
we have
ak,j , h
(ak,j χA )(y, t)h(y, t) dy dt k,j t
Rn+1 +
ak,j (y, t)h(y, t) dy dt dx t n+1
−1 |Bk,j |1/p−1 ρ |Bk,j | ,
A(ak,j )(x)A(h)(x) dx
(Ok+1 )
Fk+1 Γ (x)
−1 2−k |Bk,j |−1 ρ |Bk,j |
Bk,j ∩Ok+1
p A(f )(x) dx
1/p hT q (Rn+1 ) 2
+
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
k,j for all p ∈ (1, ∞), hence, an (ω, ∞)which implies that ak,j is an (ω, p)-atom supported in B atom. By (3.2), for any λ > 0, we further obtain
|Bk,j |ω
k∈Z j ∈Ik
|Qk,j |ω
k∈Z j ∈Ik
ω
k∈ZO k
2k λ
2k λ
2A(f )(x) λ
Rn
0
|Ok∗ |ω
k∈Z
dx
k
Rn 2 k+1
Rn k
|λk,j | λ|Bk,j |ρ(|Bk,j |)
ω
2k λ
ω
|Ok |ω
k∈Z
2k λ
2k λ
dx
t dt dx λ t
dt ω(t) dx t
Rn
A(f )(x) dx, ω λ
which implies that (3.4) holds, and hence, completes the proof of Theorem 3.1.
(3.5)
2
Remark 3.1. (i) Notice that the definition Λ({λj aj }j ) in (3.4) is different from [22,39]. In fact, if p ∈ (0, 1] and ω(t) = t p for all t ∈ (0, ∞), then Λ({λj aj }j ) here coincides with ( j |λj |p )1/p , which seems to be natural. (ii) Let {λi }i,j ⊂ C and {aji }i,j be (ω, p)-atoms for certain p ∈ (1, ∞), where i = 1, 2. If 1 1 j 2 2 n+1 j λj aj , j λj aj ∈ Tω (R+ ), then by the fact that ω is subadditive and of strictly lower type pω , we have 2 i i pω i i pω Λ λj aj i,j Λ λj a j j . i=1
(iii) Since ω is concave, it is of upper type 1. Then, with the same notation as in Theorem 3.1, we have ∞ j =1 |λj | CΛ({λj aj }j ) Cf Tω (Rn+1 ) . +
Let p ∈ (0, 1] and q ∈ (p, ∞) ∩ [1, ∞). Recall that a function a on Rn+1 + is called a (p, q)atom if (i) there exists a ball B ⊂ Rn such that supp a ⊂ B; (ii) aT q (Rn+1 ) |B|1/q−1/p . 2
+
We have the following convergence result.
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1181 p
n+1 Proposition 3.1. Let ω satisfy Assumption (A) and p ∈ (0, ∞). If f ∈ (Tω (Rn+1 + ) ∩ T2 (R+ )), p n+1 then the decomposition (3.3) holds in both Tω (Rn+1 + ) and T2 (R+ ).
Proof. We use the same notation as in the proof of Theorem 3.1. We first show that (3.3) holds −1 is convex, by the Jensen inequality and the in Tω (Rn+1 + ). In fact, since ω is concave and ω Hölder inequality, for each k ∈ Z and j ∈ Ik , we have ω−1
1 |Bk,j |
Rn
|λk,j | ω A(λk,j ak,j )(x) dx A(λk,j ak,j )(x) dx |Bk,j | Rn
|λk,j | |λk,j | . ak,j T 2 (Rn+1 ) 1/2 + 2 |Bk,j |ρ(|Bk,j |) |Bk,j |
From this and the continuity of ω together with the subadditive property of ω and A, it follows that
ω A f− λk,j ak,j (x) dx |k|+|j |N
Rn
ω A(λk,j ak,j )(x) dx
|k|+|j |>NRn
|Bk,j |ω
|k|+|j |>N
|λk,j | |Bk,j |ρ(|Bk,j |)
→ 0, (3.6)
as N → ∞, by (3.5). Now for any > 0, by the fact that ω is of upper type 1 and (3.6), there exists N0 ∈ N such that when N > N0 ,
1 ω A f− λk,j ak,j (x) dx 1, |k|+|j |N
Rn
which implies that when N > N0 , f − Tω (Rn+1 + ).
|k|+|j |N
λk,j ak,j Tω (Rn+1 ) . Thus, (3.3) holds in +
p
We now prove that (3.3) holds in T2 (Rn+1 + ). For the case p ∈ (0, 1], notice that {Ak,j }k∈Z,j ∈Ik are independent of ω. In this case, letting ak,j ≡ 2−k |Bk,j |−1/p f χAk,j and λk,j ≡ 2k |Bk,j |1/p , we then have that {ak,j }k∈Z,j ∈Ik are (p, q)-atoms, where q ∈ (p, ∞) ∩ [1, ∞), and p p ak,j imk∈Z j ∈Ik |λk,j | f p n+1 , which combined with the fact that λk,j ak,j = λk,j T2 (R+ ) p T2 (Rn+1 + ) in
this case. plies that (3.3) holds in p Let us now consider the case p ∈ (1, ∞). To prove that (3.2) holds in T2 (Rn+1 + ), it suffices to show that for any β > 0, there exists N0 ∈ N such that if N > N0 , then λk,j ak,j = f χAk,j < β. (3.7) |k|+|j |>N
p
T2 (Rn+1 + )
|k|+|j |>N
p
T2 (Rn+1 + )
To see this, noticing that {Ak,j }k∈Z, j ∈Ik are disjoint, hence, we have k∈Z j ∈Ik
|f χAk,j | = |f |.
(3.8)
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
Write HN,1 ≡ k<−N, j ∈Ik f χAk,j and HN,2 ≡ k>N, j ∈Ik f χAk,j . To estimate the term HN,1 , q let q be the conjugate index of p and h ∈ T2 (Rn+1 + ) with hT q (Rn+1 ) 1. Notice that for 2
+
∗ ∗ ∗ each k < −N , Ak,j ⊂ (O −N ) , and hence supp HN,1 ⊂ (O−N ) = R(F−N ). From this, (3.8), Lemma 3.1 and the Hölder inequality, we deduce that
dy dt HN,1 , h (f χ )(y, t)h(y, t) Ak,j t ∗ ) k<−N, j ∈Ik R(F−N
F−N Γ (x)
k<−N, j ∈Ik
dy dt (f χAk,j )(y, t)h(y, t) n+1 dx t
A(f )(x)A(h)(x) dx
F−N
p A(f )(x) dx
1/p ,
F−N
which implies that
p A(f )(x) dx
HN,1 T p (Rn+1 ) +
2
1/p .
F−N
Then by the Lebesgue dominated convergence theorem, we have lim HN,1 T p (Rn+1 ) = 0,
N →∞
+
2
which implies that there exists N1 ∈ N such that if N N1 , then HN,1 T p (Rn+1 ) < β/3. + 2 ∗ and hence, supp H ∗ For the term HN,2 , notice that for each k > N , Ak,j ⊂ O N,2 ⊂ ON , which N together with (3.8) implies that
p p p A A(f )(x) dx. HN,2 p n+1 = f χAk,j (x) dx T2 (R+ )
k>N, j ∈Ik
Rn
∗ ON
∗ | |O | → 0 as N → ∞, by the continuity of Lebesgue integrals (or the Lebesgue Since |ON N dominated convergence theorem in measures), we have
lim HN,2 T p (Rn+1 ) = 0,
N →∞
+
2
which implies that there exists N2 ∈ N such that if N N2 , then HN,2 T 2 (Rn+1 ) < β/3. + 2 Now let HN,3 ≡ −N1 kN2 , |k|+|j |>N f χAk,j . Since Ak,j ⊂ B k,j , by (3.8), we obtain HN,3
p p T2 (Rn+1 + )
=
A Rn
−N1 kN2 , |k|+|j |>N
p A(f )(x) dx.
f χAk,j (x)
−N1 kN2 , |k|+|j |>N
Bk,j
p
dx
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1183
From the Whitney decomposition, it follows that for each fixed k, j ∈Ik
and hence, limN →∞ lim N →∞
|Bk,j |
|Qk,j | Ok∗ |Ok | < ∞,
j ∈Ik
{j ∈Ik : |j |>N } |Bk,j | = 0,
! −N1 kN2 , |k|+|j |>N
which implies that
Bk,j lim
N →∞
|Bk,j | = 0.
−N1 kN2 |j |+|k|>N
Applying the continuity of Lebesgue integrals (or the Lebesgue dominated convergence theorem in measures) again, we obtain lim HN,3 T p (Rn+1 ) = 0,
N →∞
2
+
which implies that there exists N3 ∈ N such that if N N3 , then HN,3 T p (Rn+1 ) < β/3. + 2 Letting N0 ≡ max{N1 , N2 , N3 } and noticing that when N > N0 ,
|k|+|j |>N
f χAk,j
p
T2 (Rn+1 + )
=
|k|+|j |>N
|f χAk,j |
p
T2 (Rn+1 + )
3
HNi ,i T p (Rn+1 ) < β,
i=1
we then obtain (3.7), which completes the proof of Proposition 3.1.
2
+
2
As a consequence of Proposition 3.1, we have the following corollary which plays an important role in this paper. p
n+1 n+1 2 Corollary 3.1. Let ω satisfy Assumption (A). If f ∈ Tω (Rn+1 + ) ∩ T2 (R+ ), then f ∈ T2 (R+ ) p n+1 for all p ∈ [1, 2], and hence, the decomposition (3.3) holds in T2 (R+ ).
Proof. Observing that ω is of upper type 1, we have
Rn
p A(f )(x) dx
A(f )(x) dx +
{x∈Rn : A(f )(x)<1}
{x∈Rn :
{x∈Rn : A(f )(x)1}
ω A(f )(x) dx + f 2 2
2 A(f )(x) dx
A(f )(x)<1}
T2 (Rn+1 + )
< ∞,
p
which implies that f ∈ T2 (Rn+1 + ). Then by Proposition 3.1, we have that the decomposition p n+1 (3.3) holds in T2 (R+ ), which completes the proof of Corollary 3.1. 2 p,c
n+1 n+1 In what follows, let Tωc (Rn+1 + ) and T2 (R+ ) denote the set of all functions in Tω (R+ ) p and T2 (Rn+1 + ) with compact supports, respectively, where p ∈ (0, ∞).
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
Lemma 3.3. p,c
p,c
2,c n+1 n+1 (i) For all p ∈ (0, ∞), T2 (Rn+1 + ) ⊂ T2 (R+ ). In particular, if p ∈ (0, 2], then T2 (R+ ) coincides with T22,c (Rn+1 + ). 2,c n+1 (ii) Let ω satisfy Assumption (A). Then Tωc (Rn+1 + ) coincides with T2 (R+ ). p,c
2,c n+1 Proof. By (1.3) in [11, p. 306], we have T2 (Rn+1 + ) ⊂ T2 (R+ ) for all p ∈ (0, ∞). If p ∈ p,c 2,c n+1 (0, 2], then from the Hölder inequality, it is easy to follow that T2 (Rn+1 + ) ⊂ T2 (R+ ). Thus, (i) holds. 2,c n+1 Let us prove (ii). To prove Tωc (Rn+1 + ) ⊂ T2 (R+ ), by (i), it suffices to show that p,c n+1 n+1 c Tω (R+ ) ⊂ T2 (R+ ) for certain p ∈ (0, ∞). Suppose that f ∈ Tωc (Rn+1 + ) and supp f ⊂ K, n such that K ⊂ B. where K is a compact set in Rn+1 . Let B be a ball in R Then supp A(f ) ⊂ B. + This, together with the lower type property of ω, yields that
p A(f )(x) ω dx =
p A(f )(x) ω dx +
{x∈Rn : A(f )(x)<1}
Rn
|B| +
···
{x∈Rn : A(f )(x)1}
ω A(f )(x) dx < ∞.
Rn p ,c
2,c n+1 That is, f ∈ T2 ω (Rn+1 + ) ⊂ T2 (R+ ). n+1 Conversely, let f ∈ T21,c (Rn+1 + ) supporting in a compact set K in R+ . Then there exists a ball B such that K ⊂ B and supp A(f ) ⊂ B. This, together with the upper type property of ω, yields that
Rn
ω A(f )(x) dx {x∈Rn :
ω(1) dx +
A(f )(x)<1}
{x∈Rn :
A(f )(x) dx A(f )(x)1}
|B| + f T 1 (Rn+1 ) < ∞, 2
+
which implies that f ∈ Tωc (Rn+1 + ), and hence, completes the proof of Lemma 3.3.
2
4. Orlicz–Hardy spaces and their dual spaces In this section, we always assume that the Orlicz function ω satisfies Assumption (A). We introduce the Orlicz–Hardy space associated to L via the Lusin-area function and establish its duality. Let us begin with some notions and notation. Let SL be the same as in (1.3). It follows from Lemma 2.4 that the operator SL is bounded on Lp (Rn ) for p ∈ (pL , p L ). Hofmann and Mayboroda [19] introduced the Hardy space HL1 (Rn ) associated to L as the completion of {f ∈ L2 (Rn ): SL f ∈ L1 (Rn )} with respect to the norm f H 1 (Rn ) ≡ SL f L1 (Rn ) . L Using some ideas from [14,19], we now introduce the Orlicz–Hardy space Hω,L (Rn ) associated to L and ω as follows.
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1185
ω,L (Rn ) Definition 4.1. Let ω satisfy Assumption (A). A function f ∈ L2 (Rn ) is said to be in H if SL f ∈ L(ω); moreover, define
SL f (x) f Hω,L (Rn ) ≡ SL f L(ω) = inf λ > 0: dx 1 . ω λ Rn
ω,L (Rn ) in the norm The Orlicz–Hardy space Hω,L (Rn ) is defined to be the completion of H · Hω,L (Rn ) . In what follows, for a ball B ≡ B(xB , rB ), we let U0 (B) ≡ B, and for j ∈ N, Uj (B) ≡ B(xB , 2j rB ) \ B(xB , 2j −1 rB ). Definition 4.2. Let q ∈ (pL , p L ), M ∈ N and ∈ (0, ∞). A function α ∈ Lq (Rn ) is called an (ω, q, M, )-molecule adapted to B if there exists a ball B such that (i) αLq (Uj (B)) 2−j |2j B|1/q−1 ρ(|2j B|)−1 , j ∈ Z+ ; (ii) for every k = 1, . . . , M and j ∈ Z+ , there holds −2 −1 k r L α B
Lq (Uj (B))
1/q−1 j −1 ρ 2 B 2−j 2j B .
L ), then α is called an (ω, ∞, M, )Finally, if α is an (ω, q, M, )-molecule for all q ∈ (pL , p molecule. Remark 4.1. (i) Since ω is of strictly lower type pω , we have that for all f1 , f2 ∈ Hω,L (Rn ), p
p
p
f1 + f2 Hωω,L (Rn ) f1 Hωω,L (Rn ) + f2 Hωω,L (Rn ) . p
p
In fact, if letting λ1 ≡ f1 Hωω,L (Rn ) and λ2 ≡ f2 Hωω,L (Rn ) , by the subadditivity, the continuity and the lower type pω of ω, we have
ω Rn
2
SL (f1 + f2 )(x) SL (fi )(x) dx dx ω (λ1 + λ2 )1/pω (λ1 + λ2 )1/pω i=1Rn
2 i=1
p
λi λ1 + λ2
ω
SL (fi )(x) 1/pω
λ1
Rn p
dx 1,
which implies f1 + f2 Hω,L (Rn ) (f1 Hωω,L (Rn ) + f2 Hωω,L (Rn ) )1/pω , and hence, the desired conclusion. ω,L (Rn ) is (ii) From the theorem of completion of Yosida [41, p. 56], it follows that H n n dense in Hω,L (R ), namely, for any f ∈ Hω,L (R ), there exists a Cauchy sequence {fk }∞ k=1 ⊂ ω,L (Rn ) such that limk→∞ fk − f Hω,L (Rn ) = 0. Moreover, if {fk }∞ is a Cauchy sequence H k=1 ω,L (Rn ), then there uniquely exists f ∈ Hω,L (Rn ) such that limk→∞ fk − f Hω,L (Rn ) = 0. in H (iii) If ω(t) = t, then the space Hω,L (Rn ) is just the space HL1 (Rn ) introduced by Hofmann and Mayboroda [19]. Furthermore, when ω(t) ≡ t p for all t ∈ (0, ∞) with p ∈ (0, 1], we then p denote the space Hω,L (Rn ) simply by HL (Rn ).
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
4.1. Molecular decompositions of Hω,L (Rn ) n+1 2 In what follows, let L2c (Rn+1 + ) denote the set of all functions in L (R+ ) with compact supports. Recall that ω is a concave function of strictly lower type pω , where pω ∈ (0, 1].
Proposition 4.1. Let ω satisfy Assumption (A), M ∈ N and M > n2 ( p1ω − 12 ), and πL,M be as in (1.4). p,c
(i) The operator πL,M , initially defined on T2 (Rn+1 + ), extends to a bounded linear operator p n+1 p n from T2 (R+ ) to L (R ), where p ∈ (pL , p L ). (ii) The operator πL,M , initially defined on Tωc (Rn+1 + ), extends to a bounded linear operator n ). from Tω (Rn+1 ) to H (R ω,L + Proof. Let k ∈ N. By Lemma 2.4 and a duality argument, we know that the operator SLk ∗ is bounded on Lp (Rn ) for p ∈ (pL∗ , p L∗ ), where p1 ∗ + 1 = 1 = p1L + 1 . p L
L
p,c
p L∗
Let f ∈ T2 (Rn+1 L ). For any g ∈ Lq (Rn ) ∩ L2 (Rn ), where + ), where p ∈ (pL , p by the Hölder inequality, we have
1 p
+
1 q
= 1,
2 ∗ M+1 −t 2 L∗ dy dt πL,M (f )(x)g(x) dx f (y, t) t L e g(y) t Rn
Rn+1 +
A(f )(x)SLM+1 g(x) dx A(f )Lp (Rn ) SLM+1 g Lq (Rn ) ∗ ∗
Rn
f T p (Rn+1 ) gLq (Rn ) , 2
+
p,c
p n which implies that πL,M maps T2 (Rn+1 + ) continuously into L (R ). Then by a density argup p n ment, we obtain that πL,M is bounded from T2 (Rn+1 + ) to L (R ). This proves (i). n+1 c Let us prove (ii). Assume that f ∈ Tω (R+ ). By Theorem 3.1, we have f = ∞ j =1 λj aj ∞ are as in Theorem 3.1 and Λ({λ a } ) f pointwise, where {λj }∞ and {a } j j j j j =1 j =1 Tω (Rn+1 ) . +
From Lemma 3.3(ii), it follows that f ∈ T22,c (Rn+1 + ), which together with (i) and Corollary 3.1 further implies that πL,M (f ) =
∞
λj πL,M (aj ) ≡
j =1
∞
λj αj
j =1
in Lp (Rn ) for p ∈ (pL , 2]. On the other hand, notice that the operator SL is bounded on Lp (Rn ), which together with the subadditivity and the continuity of ω yields that
Rn
∞ ω SL πL,M (f ) (x) dx
j =1Rn
ω |λj |SL (αj )(x) dx.
(4.1)
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1187
We claim that for any fixed ∈ (0, ∞), αj = πL,M (aj ) is a multiple of an (ω, ∞, M, )molecule adapted to Bj for each j . In fact, assume that a is an (ω, ∞)-atom supported in the ball B ≡ B(xB , rB ) and q ∈ (pL , p L ). Since for q ∈ (pL , 2), each (ω, 2, M, )-molecule is also an (ω, q, M, )-molecule, to prove the above claim, it suffices to show that α ≡ πL,M (a) is a multiple of an (ω, q, M, )molecule adapted to B with q ∈ [2, p L ). By (i), for i = 0, 1, 2, we have −1 . αLq (Ui (B)) = πL,M (a)Lq (U (B)) aT q (Rn+1 ) |B|1/q−1 ρ |B| i
2
+
For i 3, let q ∈ (1, 2] being the conjugate number of q and h ∈ Lq (Rn ) satisfying hLq (Rn ) 1 and supp h ⊂ Ui (B). By the Hölder inequality and Lemmas 2.1 and 2.3, we have πL,M (a), h
rB
a(x, t) t 2 L∗ M+1 e−t 2 L∗ (h)(x) dx dt t
0 B
A(a)
Lq (Rn )
2 ∗ M+1 −t 2 L∗ A χ t L e (h) 1/q −1/2
aT q (Rn+1 ) |B| 2
+
B 1/q −1/2
2
2 dx dt 2 ∗ M+1 −t 2 L∗ t L e (h)(x, t) t
" rB
aT q (Rn+1 ) |B|
t
+
Lq (Rn )
B
n(1/2−1/q )
1/2
dist(B, Ui (B))2 exp − ct 2
2
dt t
#1/2
0 −1/2
|B| 2
−1 ρ |B|
" rB t
n(1−2/q )
t i 2 rB
2( +n/pω −n/q)
dt t
#1/2
0
1/q−1 i −1 2 B ρ 2 B ,
−i i
(4.2)
which implies that α satisfies Definition 4.2(i). We now show that α also satisfies Definition 4.2(ii). Let k ∈ {1, . . . , M}. If i = 0, 1, 2, let h be the same as in the proof of (4.2); similarly to the proof of (4.1), we have −2 −1 k r L πL,M (a), h
rB
B
t rB
2k
a(x, t) t 2 L∗ M+1−k e−t 2 L∗ (h)(x) dx dt t
0 B
A(a)
Lq (Rn )
M+1−k S ∗ (h) L
Lq (Rn )
−1 aT q (Rn+1 ) |B|1/q−1 ρ |B| , 2
+
which is the desired estimate, where we used the Hölder inequality and Lemma 2.4 by noticing that q ∈ (pL∗ , 2]. If i 3, an argument similar to that used in the estimate of (4.2) also yields
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
the desired estimate. Thus, α = πL,M (a) is a multiple of an (ω, q, M, )-molecule adapted to B with q ∈ [2, p L ), and the claim is proved. Let q ∈ (pL , p L ) and > n( p1ω − 1 ), where p ω is as in Convention (C). We now claim that p ω for all (ω, q, M, )-molecules α adapted to the ball B ≡ B(xB , rB ) and λ ∈ C,
ω |λ|SL (α)(x) dx |B|ω
|λ| . |B|ρ(|B|)
Rn
(4.3)
Once this is proved, then we have αHω,L (Rn ) 1, which together with (4.1) further implies that for all f ∈ Tωc (Rn+1 + ),
∞ ω SL πL,M (f ) (x) dx |Bj |ω j =1
Rn
|λj | . |Bj |ρ(|Bj |)
Thus, for all f ∈ Tωc (Rn+1 + ), we have πL,M (f )
Hω,L (Rn )
Λ {λj aj }j f Tω (Rn+1 ) , +
which combined with a density argument implies (ii). Now, let us prove the claim (4.3). Observe that if q > 2, then an (ω, q, M, )-molecule is also an (ω, 2, M, )-molecule. Thus, to prove the claim (4.3), it suffices to show (4.3) for q ∈ (pL , 2]. To this end, write
ω |λ|SL (α)(x) dx
Rn
2 M ω |λ|SL I − e−rB L α (x) dx +
Rn
2 M ω |λ|SL I − e−rB L (αχUj (B) ) (x) dx
j =0Rn ∞
sup
j =0 1kMRn
≡
2 M α (x) dx ω |λ|SL I − I − e−rB L
Rn
∞
+
∞ j =0
Hj +
∞
k 2 − k r2 L r Le M B ω |λ|SL M B
M χUj (B) rB−2 L−1 α (x) dx
M
Ij .
j =0
For each j 0, let Bj ≡ 2j B. Since ω is concave, by the Jensen inequality and the Hölder inequality, we obtain
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
Hj
∞
1189
2 M ω |λ|SL I − e−rB L (αχUj (B) ) (x) dx
k=0U (B ) k j
∼
∞
2 M ω |λ|χUk (Bj ) (x)SL I − e−rB L (αχUj (B) ) (x) dx
k=0 k 2 Bj
∞ k |λ| 2 Bj ω |2k Bj | k=0
2 M SL I − e−rB L (αχUj (B) ) (x) dx
Uk (Bj )
∞ k 2 Bj ω k=0
|λ| SL I − e−rB2 L M (αχU (B) ) q . j L (Uk (Bj )) |2k Bj |1/q
By the proof of [19, Lemma 4.2] (see [19, (4.22) and (4.27)]), we have that for k = 0, 1, 2, SL I − e−rB2 L M (αχU (B) ) q j L (U
k (Bj ))
αLq (Uj (B)) ,
and for k 3, SL I − e−rB2 L M (αχU (B) ) 2 q j L (U
k (Bj ))
k
1
4M+2n(1/2−1/q) α2Lq (Uj (B)) ,
2k+j
which, together with Definition 4.2, 2Mpω > n(1 − pω /2) and Assumption (A), implies that √ ∞ k |λ| k2−(2M+n/2−n/q)(j +k)−j |λ|2−j 2 Bj ω + Hj |Bj |ω |Bj |ρ(|Bj |) |2k Bj |1/q |Bj |1−1/q ρ(|Bj |) k=3 ∞ √ |λ| −jpω kn(1−pω /q) −pω (2M+n/2−n/q)(j +k) 2 k2 2 1+ |Bj |ω |Bj |ρ(|Bj |) k=3 |λ| . 2−jpω |Bj |ω |Bj |ρ(|Bj |)
Since ρ is of lower type 1/ pω − 1 and > n(1/pω − 1/ pω ), we further have ∞
Hj
j =0
∞ j =0
∞
|B|ρ(|B|) pω |λ| 2−jpω |Bj | ω |Bj |ρ(|Bj |) |B|ρ(|B|) 2
2
−jpω j n(1−pω / pω )
j =0
∞ j =0
pω |B| pω / |λ| |Bj | ω |Bj | |B|ρ(|B|)
−jpω
2
|λ| |B|ω |B|ρ(|B|)
|λ| |B|ω . |B|ρ(|B|)
(4.4)
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
Similarly, we have ∞
Ij |B|ω
j =0
|λ| , |B|ρ(|B|)
which completes the proof of (4.3), and hence, the proof of Proposition 4.1.
2
Proposition 4.2. Let ω satisfy Assumption (A), > n(1/pω − 1/pω+ ) and M > n2 ( p1ω − 12 ). If f ∈ Hω,L (Rn ) ∩ L2 (Rn ), then f ∈ Lp (Rn ) for all p ∈ (pL , 2] and there exist (ω, ∞, M, )∞ molecules {αj }∞ j =1 and numbers {λj }j =1 ⊂ C such that f=
∞
(4.5)
λj αj
j =1
in both Hω,L (Rn ) and Lp (Rn ) for all p ∈ (pL , 2]. Moreover, there exists a positive constant C independent of f such that for all f ∈ Hω,L (Rn ) ∩ L2 (Rn ), ∞ |λj | Λ {λj αj }j ≡ inf λ > 0: 1 Cf Hω,L (Rn ) , |Bj |ω (4.6) λ|Bj |ρ(|Bj |) j =1
where for each j , αj is adapted to the ball Bj . Proof. Let f ∈ Hω,L (Rn ) ∩ L2 (Rn ). For each N ∈ N, define ON ≡ {(x, t) ∈ Rn+1 + : |x| < N, 1/N < t < N}. Then by the L2 (Rn )-functional calculi for L, we have
∞ 2 M+2 −2t 2 L dt 2 = lim πL,M χON t 2 Le−t L f t L e f f = CM N →∞ t 0
in L2 (Rn ), where M ∈ N, πL,M and CM are as in (1.4). 2 On the other hand, by Definition 4.1 and Lemma 2.4, we have that t 2 Le−t L f ∈ T22 (Rn+1 + )∩ 2 p n+1 n+1 2 −t L Tω (R+ ). An application of Corollary 3.1 shows that t Le f ∈ T2 (R+ ), which to2L 2 −t gether with Proposition 4.1(i) implies that {πL,M (χON (t Le f ))}N is a Cauchy sequence in Lp (Rn ). Then via taking subsequence, we have 2 f = lim πL,M χON t 2 Le−t L f N →∞
in Lp (Rn ). 2 Now applying Theorem 3.1 and Proposition 3.1 to t 2 Le−t L f , we obtain (ω, ∞)-atoms 2 p ∞ n+1 ∞ 2 −t L f = {aj }∞ j =1 λj aj in T2 (R+ ) and j =1 and numbers {λj }j =1 ⊂ C such that t Le Λ({λj aj }j ) t 2 Le−t L f Tω (Rn+1 ) , which combined with Proposition 4.1(i) further yields that 2
+
∞ ∞ 2 f = πL,M t 2 Le−t L f = λj πL,M (aj ) ≡ λj αj j =1
j =1
(4.7)
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1191
in Lp (Rn ) for p ∈ (pL , 2]. By the proof of Proposition 4.1, we know that αj is a multiple of an (ω, ∞, M, )-molecule for any > 0, and M ∈ N and M > n2 ( p1ω − 12 ). Notice that Λ({λj αj }j ) = Λ({λj aj }j ). We therefore obtain (4.6). To finish the proof of Proposition 4.2, it remains to show that (4.5) holds in Hω,L (Rn ). In fact, by Lemma 2.4, (3.5), (4.3) and (4.7) together with the continuity and the subadditivity of ω, we have # #
" " N ∞
ω SL f − λj αj (x) dx ω SL (λj αj )(x) dx Rn
j =1
j =N +1Rn
∞ j =N +1
|Bj |ω
|λj | |Bj |ρ(|Bj |)
→ 0,
as N → ∞. We point out that here, in the last inequality, to use (4.3), we need to choose p ω as in Convention (C) such that > n(1/pω − 1/ pω ), which is guaranteed by the assumption + > n(1/pω − 1/p ∞ω ). This combined nwith an argument similar to the proof of Proposition 3.1 yields that f = j =1 λj αj in Hω,L (R ), which completes the proof of Proposition 4.2. 2 L ) and M > Corollary 4.1. Let ω satisfy Assumption (A), > n(1/pω − 1/pω+ ), q ∈ (pL , p n 1 1 n ), there exist (ω, q, M, )-molecules {α }∞ and num( − ). Then for every f ∈ H (R ω,L j j =1 2 pω 2 ∞ n ). Furthermore, if letting Λ({λ α } ) bers {λj }∞ ⊂ C such that f = λ α in H (R j j ω,L j j j j =1 j =1 be as in (4.6), then there exists a positive constant C independent of f such that Λ({λj αj }j ) Cf Hω,L (Rn ) . Proof. If f ∈ Hω,L (Rn ) ∩ L2 (Rn ), then it immediately follows from Proposition 4.2 that all results hold. n 2 n Otherwise, there exist {fk }∞ k=1 ⊂ (Hω,L (R ) ∩ L (R )) such that for all k ∈ N, f − fk Hω,L (Rn ) 2−k f Hω,L (Rn ) . (f − fk−1 ) in Hω,L (Rn ). By Proposition 4.2, we have that for all Set f0 ≡ 0. Then f = ∞ ∞ k=1k kk k ∈ N, fk − fk−1 = j =1 λj αj in Hω,L (Rn ) and Λ({λkj ajk }j ) fk − fk−1 Hω,L (Rn ) , where for k k n all j and k, αjk is an (ω, q, M, )-molecule. Thus, f = ∞ k,j =1 λj αj in Hω,L (R ), and it further follows from Remark 3.1(ii) that ∞ ∞ k k pω k k pω p p Λ λj αj k,j Λ λj a j j fk − fk−1 Hωω,L (Rn ) f Hωω,L (Rn ) , k=1
which completes the proof of Corollary 4.1.
k=1
2
q,M,
Let Hω,fin (Rn ) denote the set of all finite combinations of (ω, q, M, )-molecules. From Corollary 4.1, we immediately deduce the following density result. Corollary 4.2. Let ω satisfy Assumption (A), > n(1/pω − 1/pω+ ) and M > n2 ( p1ω − 12 ). Then q,M,
the space Hω,fin (Rn ) is dense in the space Hω,L (Rn ).
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
4.2. Dual spaces of Hω,L (Rn ) In this subsection, we study the dual space of the Orlicz–Hardy space Hω,L (Rn ). We begin with some notions. Following [19], for > 0 and M ∈ N, we introduce the space n 2 MM, <∞ , ω (L) ≡ μ ∈ L R : μMM, (L) ω where
M −k 1/2 L μ 2 μMM, (L) ≡ sup 2 B 0, 2j ρ B 0, 2j L (U j
ω
j 0
j (B(0,1)))
.
k=0
Notice that if φ ∈ MM, ω (L) with norm 1, then φ is an (ω, 2, M, )-molecule adapted to B(0, 1). Conversely, if α is an (ω, 2, M, )-molecule adapted to certain ball, then α ∈ MM, ω (L). 2L 2 −1 −t M, ∗ M, Let At denote either (I + t L) or e and f ∈ (Mω (L)) , the dual of Mω (L). We claim that (I − A∗t )M f ∈ L2loc (Rn ) in the sense of distributions. In fact, for any ball B, if ψ ∈ L2 (B), then it follows from the Gaffney estimates via Lemmas 2.1 and 2.2 that (I − At )M ψ ∈ MM, ω (L) for all > 0 and any fixed t ∈ (0, ∞). Thus, I − A∗ M f, ψ ≡ f, (I − At )M ψ C t, rB , dist(B, 0) f M, ∗ ψ 2 , L (B) t (M (L)) ω
which implies that (I − A∗t )M f ∈ L2loc (Rn ) in the sense of distributions. Finally, for any M ∈ N, define n MM ω,L∗ R ≡
$
M, ∗ Mω (L) .
>n(1/pω −1/pω+ )
Definition 4.3. Let q ∈ (pL , p L ), ω satisfy Assumption (A), ρ be as in (2.4) and M > n2 ( p1ω − 12 ). q,M
n n A functional f ∈ MM ω,L (R ) is said to be in BMOρ,L (R ) if
f BMOq,M (Rn ) ≡ sup ρ,L
B⊂Rn
1 1 I − e−rB2 L M f (x)q dx ρ(|B|) |B|
1/q
< ∞,
B
where the supremum is taken over all balls B of Rn . q,M
n In what follows, when q = 2, we denote BMOρ,L (Rn ) simply by BMOM ρ,L (R ). The proofs of following Lemmas 4.1 and 4.2 are similar to those of Lemmas 8.1 and 8.3 of [19], respectively; we omit the details.
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1193 q,M
Lemma 4.1. Let ω, ρ, q and M be as in Definition 4.3. A functional f ∈ BMOρ,L (Rn ) if and n only if f ∈ MM ω,L (R ) and
1 1 I − I + r 2 L −1 M f (x)q dx sup B n ρ(|B|) |B| B⊂R
1/q
< ∞.
B
Moreover, the quantity appeared in the left-hand side of the above formula is equivalent to f BMOq,M (Rn ) . ρ,L
Lemma 4.2. Let ω, ρ and M be as in Definition 4.3. Then there exists a positive constant C such n that for all f ∈ BMOM ρ,L (R ),
2 dx dt 2 M −t 2 L 1 1 t L e sup f (x) t B⊂Rn ρ(|B|) |B|
1/2
Cf BMOM
ρ,L (R
n)
.
B
The following lemma is a slight variant of Lemma 8.4 and Remark of Section 9 in [19]. > M + 1 + n . Lemma 4.3. Let ω, ρ and M be as in Definition 4.3, q ∈ (pL∗ , 2], , 1 > 0 and M 4 M n Suppose that f ∈ Mω,L∗ (R ) satisfies
Rn
|(I − (I + L∗ )−1 )M f (x)|q dx < ∞. 1 + |x|n+ 1
(4.8)
)-molecule α, Then for every (ω, q , M, M f, α = C
2 ∗ M −t 2 L∗ dx dt 2 , t L e f (x)t 2 Le−t L α(x) t
Rn+1 +
M is a positive constant satisfying where q ∈ [2, ∞) satisfying 1/q + 1/q = 1 and C ∞ 2(M+1) −2t 2 dt M t e = 1. C 0
t
Proof. If we let ω(t) ≡ t for all t ∈ (0, ∞), then this lemma are just Lemma 8.4 and Remark of Section 9 in [19]. )-molecule adapted to a ball B. Then from Definition 4.2, Otherwise, let α be an (ω, q , M, )-molecule, where it is easy to see that ρ(|B|)α is an ( ω, q , M, ω(t) ≡ t for all t ∈ (0, ∞), and hence Lemma 4.3 holds for ρ(|B|)α, which implies the desired conclusion and hence, completes the proof of Lemma 4.3. 2 q,M
From Lemma 4.1, it is easy to follow that all f ∈ BMOρ,L (Rn ) satisfy (4.8) for all 1 ∈ q,M
(0, ∞), and hence, Lemma 4.3 holds for all f ∈ BMOρ,L (Rn ). Now, let us give the main result of this section.
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
Theorem 4.1. Let ω satisfy Assumption (A), ρ be as in (2.4), > n(1/pω − 1/pω+ ), M > n2 × > M + n . Then (Hω,L (Rn ))∗ , the dual space of Hω,L (Rn ), coincides with ( p1ω − 12 ) and M 4 n BMOM ρ,L∗ (R ) in the following sense:
2,M, n n (i) Let g ∈ BMOM ρ,L∗ (R ). Then the linear functional , which is initially defined on Hω,fin (R ) by
(f ) ≡ g, f ,
(4.9)
has a unique extension to Hω,L (Rn ) with (Hω,L (Rn ))∗ CgBMOs ∗ (Rn ) , where C is a ρ,L positive constant independent of g. n (ii) Conversely, for any ∈ (Hω,L (Rn ))∗ , then ∈ BMOM ρ,L∗ (R ), (4.9) holds for all f ∈ 2,M, (Rn ) and BMOM Hω,fin
(R ρ,L∗
n)
C(Hω,L (Rn ))∗ , where C is a positive constant inde-
pendent of .
2,M, n n n 2 n Proof. Let g ∈ BMOM ρ,L∗ (R ). For any f ∈ Hω,fin (R ) ⊂ Hω,L (R ), we have that f ∈ L (R ) n+1 2 and hence, t 2 Le−t L f ∈ (Tω (Rn+1 + ) ∩ T2 (R+ )) by Lemma 2.4. By Theorem 3.1, there exist ∞ ∞ j }∞ such that (3.4) holds. Notice that {λj }j =1 ⊂ C and (ω, ∞)-atoms {aj }j =1 supported in {B j =1 g satisfies (4.8) with q = 2 (by Lemma 4.1), which, together with Lemmas 4.2 and 4.3, the Hölder inequality and Remark 3.1(iii), yields that 2
2 ∗ M −t 2 L∗ dx dt 2L 2 −t g, f = C t L e g(x)t Le f (x) M t Rn+1 +
∞ j =1
∞ j =1
∞ j =1
|λj |
dx dt 2 ∗ M −t 2 L∗ t L e g(x)aj (x, t) t
Rn+1 +
|λj |aj T 2 (Rn+1 ) 2
+
j B
|λj |gBMOM
(R ρ,L∗
n)
∼ f Hω,L (Rn ) gBMOM
2 dx dt 2 ∗ M −t 2 L∗ t L e g(x) t
2 t 2 Le−t L f T
(R ρ,L∗
n+1 ω (R+ )
n)
.
1/2
gBMOM
(R ρ,L∗
n)
(4.10)
Then by a density argument via Corollary 4.2, we obtain (i). Conversely, let ∈ (Hω,L (Rn ))∗ . For any (ω, 2, M, )-molecule α, it follows from 4.3 that n αHω,L (Rn ) 1. Thus, |(α)| (Hω,L (Rn ))∗ , which implies that ∈ MM ω,L∗ (R ). n To finish the proof of (ii), we still need to show that ∈ BMOM ρ,L∗ (R ). To this end, for any 1 ball B, let φ ∈ L2 (B) with φL2 (B) |B|1/2 ρ(|B|) and α ≡ (I − [I + rB2 L]−1 )M φ. Then from Lemma 2.3, we deduce that for each j ∈ Z+ and k = 0, 1, . . . , M ,
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
2 −k r L α
L2 (Uj (B))
B
1195
−1 M−k −k I + rB2 L φ L2 (U (B)) = I − I + rB2 L j
dist(B, Uj (B)) φL2 (B) exp − crB 2−2j (M+ ) 2j n(1/pω −1/2)
1 |2j B|1/2 ρ(|2j B|)
2−2j
1 |2j B|1/2 ρ(|2j B|)
,
where c is as in Lemma 2.3 and 2M > n(1/pω − 1/2). Thus, α is a multiple of an (ω, 2, M, )molecule. Since (I − ([I + t 2 L]−1 )∗ )M is well defined and belongs to L2loc (Rn ) for any fixed t > 0 we have I − I + r 2 L −1 ∗ M , φ = , I − I + r 2 L −1 M φ = , α (Hω,L (Rn ))∗ , B B which further implies that 1/2
1 1 I − I + r 2 L −1 ∗ M (x)2 dx B ρ(|B|) |B| B
=
sup
% & φ (H (Rn ))∗ . , I − I + r 2 L −1 M ω,L B |B|1/2 ρ(|B|)
φL2 (B) 1
n Thus, ∈ BMOM ρ,L∗ (R ) and BMOM
Theorem 4.1.
2
(R ρ,L∗
n)
(Hω,L (Rn ))∗ , which completes the proof of
n n Remark 4.2. It follows from Theorem 4.1 that the spaces BMOM ρ,L (R ) for all M > 2 × n ( p1ω − 12 ) coincide with equivalent norms. Thus, in what follows, we denote BMOM ρ,L (R ) simply n by BMOρ,L (R ).
Rn ) 5. Several equivalent characterizations of Hω,L (R In this section, we establish several equivalent characterizations of the Orlicz–Hardy spaces. Let us begin with some notions. Definition 5.1. Let q ∈ (pL , p L ), ω satisfy Assumption (A), M > n2 ( p1ω − 12 ) and > n × (1/pω − 1/pω+ ). A distribution f ∈ (BMOρ,L∗ (Rn ))∗ is said to be in the space Hω,L (Rn ) ∞ ∞ if there exist {λj }∞ j =1 λj αj in j =1 ⊂ C and (ω, q, M, )-molecules {αj }j =1 such that f = (BMOρ,L∗ (Rn ))∗ and q,M,
∞ |Bj |ω Λ {λj αj }j = inf λ > 0: j =1
where for each j , αj is adapted to the ball Bj .
|λj | λ|Bj |ρ(|Bj |)
1 < ∞,
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224 q,M,
If f ∈ Hω,L (Rn ), then define its norm by
f H q,M, (Rn ) ≡ inf Λ {λj αj }j , ω,L
where the infimum is taken over all possible decompositions of f as above. For any f ∈ L2 (Rn ) and x ∈ Rn , define the Lusin-area function associated to the Poisson semigroup as follows, 1/2
√ t∇e−t L f (y)2 dy dt , (5.1) SP f (x) ≡ t Γ (x)
where Γ (x) is as in (1.3). Let β ∈ (0, ∞). Following [19], we define nontangential the maximal operators by setting, for all f ∈ L2 (Rn ) and x ∈ Rn , 1/2
2 −t 2 L 1 β dz e Nh g(x) ≡ sup g(z) (5.2) n (y,t)∈Γβ (x) (βt) B(y,βt)
and
β NP g(x) ≡
sup (y,t)∈Γβ (x)
1 (βt)n
2 −t √L e g(z) dz
1/2 ,
(5.3)
B(y,βt)
where and in what follows, Γβ (x) ≡ {(y, t) ∈ Rn × (0, ∞): |x − y| < βt}. In what follows, we denote Nh1 and NP1 simply by Nh and NP . We also define the radial maximal functions by setting, for all f ∈ L2 (Rn ) and x ∈ Rn , 1/2
2 −t 2 L 1 e f (y) dy (5.4) Rh f (x) ≡ sup n t>0 t B(x,t)
and
RP f (x) ≡ sup t>0
1 tn
2 −t √L e f (y) dy
1/2 .
(5.5)
B(x,t)
Similarly to Definition 4.1, we define the space Hω,SP (Rn ) as follows. ω,S (Rn ) Definition 5.2. Let ω satisfy Assumption (A). A function f ∈ L2 (Rn ) is said to be in H P if SP (f ) ∈ L(ω); moreover, define
SP (f )(x) dx 1 . ω f Hω,SP (Rn ) ≡ SP (f ) L(ω) = inf λ > 0: λ Rn
ω,S (Rn ) in the norm The Orlicz–Hardy space Hω,SP (Rn ) is defined to be the completion of H P · Hω,SP (Rn ) .
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1197
The spaces Hω,Nh (Rn ), Hω,NP (Rn ), Hω,Rh (Rn ) and Hω,RP (Rn ) are defined in a simiq,M, lar way. We now show that all the spaces Hω,L (Rn ), Hω,L (Rn ), Hω,SP (Rn ), Hω,Nh (Rn ), n n n Hω,NP (R ), Hω,Rh (R ) and Hω,RP (R ) coincide with equivalent norms. 5.1. The molecular characterization In this subsection, we establish the molecular characterization of the Orlicz–Hardy spaces, which gives some understanding of the “distributions” in Hω,L (Rn ) as elements of the dual of BMOρ,L∗ (Rn ). We start with the following auxiliary result. ≡ B(x0 , R) for some Proposition 5.1. Let ω satisfy Assumption (A). Fix t ∈ (0, ∞) and B x0 ∈ Rn and R > 0. Then there exists a positive constant C(t, R) such that for all φ ∈ L2 (Rn ) t 2 Le−t 2 L φ ∈ BMOρ,L (Rn ) and supported in B, 2 −t 2 L t Le φ
BMOρ,L (Rn )
C(t, R)φL2 (B) .
Proof. Let M > n2 ( p1ω − 12 ). For any ball B ≡ B(xB , rB ), let 1/2
2 1 1 −rB2 L M 2 −t 2 L I −e t Le φ(x) dx . H≡ ρ(|B|) |B| B
For the case when rB R, from the L2 (Rn )-boundedness of the operators e−rB L and 2 t 2 Le−t L (Lemma 2.2), it follows that 2
H
1 1/2 ρ(|B|) |B|
φL2 (B) .
Let us consider the case when rB < R. It follows from the upper type property that 1/2 ρ |B| |B|
R rB
n(1/pω −1/2)
|B|1/2 ρ |B| .
(5.6)
r2 2 On the other hand, noticing that I − e−rB L = 0 B Le−rL dr, thus, by the Minkowski inequality 2 and the L2 (Rn )-boundedness of t 2 Le−t L (Lemma 2.2), we have
I − e−rB2 L M t 2 Le−t 2 L φ(x)2 dx
1/2
B
2 #1/2 " rB2 rB2 2 M+1 −(r1 +···+rM +t 2 )L = e φ(x) dr1 · · · drM dx ··· t L B rB2
rB2
0
···
0
0
0
t2 φL2 (B) dr1 · · · drs (r1 + · · · + rM + t 2 )M+1
rB t
2M φL2 (B) .
(5.7)
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
By the fact that M > n2 ( p1ω − 12 ) and the estimates (5.6) and (5.7), we obtain 2M φL2 (B) R H . 1/2 t |B| ρ(|B|) Thus, t 2 Le−t L φBMOρ,L (Rn ) C(t, R)φL2 (B) , which completes the proof of Proposition 5.1. 2 2
Theorem 5.1. Let q ∈ (pL , p L ), ω satisfy Assumption (A), M > n2 ( p1ω − 12 ) and > n ×
(1/pω − 1/pω+ ). Then the spaces Hω,L (Rn ) and Hω,L (Rn ) coincide with equivalent norms. q,M,
Proof. By Corollary 4.1, for all f ∈ Hω,L (Rn ), there exist (ω, q, M, )-molecules {αj }∞ j =1 ∞ ∞ n adapted to balls {Bj }∞ j =1 λj αj in Hω,L (R ) j =1 and numbers {λj }j =1 ⊂ C such that f = and Λ({λj αj }j ) f Hω,L (Rn ) . Then Theorem 4.1 implies that the decomposition also holds in q,M, (BMOρ,L∗ (Rn ))∗ , and hence, Hω,L (Rn ) ⊂ Hω,L (Rn ).
Conversely, let f ∈ Hω,L (Rn ). Then there exist {λj }∞ j =1 ⊂ C and (ω, q, M, )-molecules ∞ ∗ (Rn ))∗ and {αj }∞ such that f = λ α in (BMO j j ρ,L j =1 j =1 q,M,
∞ |Bj |ω Λ {λj αj }j = inf λ > 0: j =1
|λj | λ|Bj |ρ(|Bj |)
1 < ∞,
where for each j , αj is adapted to the ball Bj . For all x ∈ Rn , by Proposition 5.1, we have ∞ 2 2 −t 2 L t Le SL f (x) = (f ) 2
dt
1/2
L (B(x,t)) t n+1
0
∞" =
φL
0
∞ ∞ ) j =1
' ∞ (#2 1/2 dt 2 ∗ −t 2 L∗ λj αj , t L e φ sup t n+1 1 2
0
j =1
(B(x,t))
2 −t 2 L *2 dt t Le sup λj αj , φ n+1 t 1 2
φL
1/2
∞
SL (λj αj )(x).
j =1
(B(x,t))
Then from (4.3) together with the continuity and the subadditivity of ω, it follows that
∞ ω SL f (x) dx
∞ ω SL (λj αj )(x) dx |Bj |ω
j =1Rn
Rn
j =1
|λj | , |Bj |ρ(|Bj |)
which implies that f Hω,L (Rn ) Λ({λj αj }j ). By taking the infimum over all decompositions of f as above, we obtain that f Hω,L (Rn ) f H q,M, (Rn ) , which completes the proof of Theorem 5.1.
2
ω,L
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1199
5.2. Characterizations by the maximal functions In this subsection, we characterize the Orlicz–Hardy space via the Lusin-area function SP and the maximal functions Nh , NP , Rh and RP . Let us begin with the following very useful auxiliary result on the boundedness of linear or non-negative sublinear operators from Hω,L (Rn ) to L(ω). Lemma 5.1. Let q ∈ (pL , 2], ω satisfy Assumption (A), M > n2 ( p1ω − 12 ) and > n × (1/pω − 1/pω+ ). Suppose that T is a non-negative sublinear (resp. linear) operator which maps Lq (Rn ) continuously into weak-Lq (Rn ). If there exists a positive constant C such that for all (ω, ∞, M, )-molecules α adapted to balls B and λ ∈ C,
|λ| , (5.8) ω T (λα)(x) dx C|B|ω |B|ρ(|B|) Rn
then T extends to a bounded sublinear (resp. linear) operator from Hω,L (Rn ) to L(ω); moreover, Hω,L (Rn ) . such that for all f ∈ Hω,L (Rn ), Tf L(ω) Cf there exists a positive constant C Proof. It follows from Proposition 4.2 that for every f ∈ Hω,L (Rn ) ∩ L2 (Rn ), f ∈ Lq (Rn ) ∞ with q ∈ (pL , 2] and there exist {λj }∞ j =1 ⊂ C and (ω, ∞, M, )-molecules {αj }j =1 such that ∞ n q n f = j =1 λj αj in both Hω,L (R ) and L (R ); moreover, Λ({λj αj }j ) f Hω,L (Rn ) . Thus, if T is linear, then it follows from the fact that T is of weak type (q, q) that T (f ) = ∞ j =1 T (λj αj ) almost everywhere. If T is a non-negative sublinear operator, then sup t 1/q x ∈ Rn : t>0
" N # N λj αj (x) > t f − λj αj T (f )(x) − T j =1
j =1
→ 0,
Lq (Rn )
as N → ∞. Thus, there exists a subsequence {Nk }k ⊂ N such that " T
Nk
# λj αj
→ T (f )
j =1
almost everywhere, as k → ∞, which together with the non-negativity and the sublinearity of T further implies that T (f ) −
∞
" T (λj αj ) = T (f ) − T
j =1
" T (f ) − T
Nk j =1 Nk
#
" +T
λj αj #
Nk j =1
# λj αj
−
∞
T (λj αj )
j =1
λj αj .
j =1
By letting k → ∞, we see that T (f ) ∞ j =1 T (λj αj ) almost everywhere. Thus, by the subadditivity and the continuity of ω and (5.8), we finally obtain
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
∞ ω T (f )(x) dx
∞ ω T (λj αj )(x) dx |Bj |ω
j =1Rn
Rn
j =1
|λj | , |Bj |ρ(|Bj |)
which implies that T (f )L(ω) Λ({λj αj }j ) f Hω,L (Rn ) . This, combined with the density of Hω,L (Rn ) ∩ L2 (Rn ) in Hω,L (Rn ), then finishes the proof of Lemma 5.1. 2 Remark 5.1. Let p ∈ (0, 1]. We point out that the condition (5.8) is also necessary, if ω(t) ≡ t p for all t ∈ (0, ∞). However, for a general ω as in Lemma 5.1, it is still unclear whether (5.8) is necessary or not. Theorem 5.2. Let ω satisfy Assumption (A). Then the spaces Hω,L (Rn ), Hω,SP (Rn ), Hω,Nh (Rn ) and Hω,NP (Rn ) coincide with equivalent norms. Before we prove Theorem 5.2, we recall some auxiliary operators introduced in [19]. Let β ∈ (0, ∞). For any g ∈ L2 (Rn ) and x ∈ Rn , let
2 dy dt 1/2 2 −t √L β t Le SP g(x) ≡ g(y) n+1 t Γβ (x)
and β Sh g(x) ≡
t∇e−t 2 L g(y)2 dy dt t n+1
1/2 .
Γβ (x)
We denote SP1 g and Sh1 g simply by SP g and Sh g, respectively. The proof of the following lemma is similar to that of [19, Lemma 5.4]. We omit the details. Lemma 5.2. There exists a positive constant C such that for all g ∈ L2 (Rn ) and x ∈ Rn , SP g(x) CSP g(x)
(5.9)
and SL g(x) C Sh g(x). Rn ) and Hω,SP (R Rn ). Let > n( p1ω − 1+ ) and M > n2 ( p1ω − 12 ). Suppose Equivalence of Hω,L (R pω that f ∈ Hω,S (Rn ) ∩ L2 (Rn ). It follows from (5.9) that SP f L(ω) f H (Rn ) . Moreover, P
since SP is bounded on L2 (Rn ) (see [19, (5.15)]), by (5.9), we have
ω,SP
SP f L2 (Rn ) SP f L2 (Rn ) f L2 (Rn ) . √
n+1 2 Thus, we obtain t 2 Le−t L f ∈ (Tω (Rn+1 + ) ∩ T2 (R+ )). Let C be a positive constant such that ∞ 2(s+1) −t 2 2 −t dt 2 n C e t e t = 1. Then by the L (R )-functional calculi, we have 0 t
f=
√ C πL,M t 2 Le−t L f CM
in L2 (Rn ), where CM is the same as in (1.4).
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
Since t 2 Le−t
√ Lf
1201
n ∈ Tω (Rn+1 + ), by Proposition 4.1, we obtain that f ∈ Hω,L (R ) and
√ f Hω,L (Rn ) t 2 Le−t L f T
n+1 ω (R+ )
∼ SP f L(ω) f Hω,SP (Rn ) .
Then a density argument yields that Hω,SP (Rn ) ⊂ Hω,L (Rn ). Conversely, similarly to the proof of (4.3), by using the estimates in the proof of [19, Theorem 5.3], we have
|λ| , ω |λ|SP (α)(x) dx |B|ω |B|ρ(|B|) Rn
where α is an (ω, 2, M, )-molecule adapted to the ball B and λ ∈ C. By the L2 (Rn )boundedness of SP and Lemma 5.1, we have f Hω,SP (Rn ) = SP f L(ω) f Hω,L (Rn ) , which implies that Hω,L (Rn ) ⊂ Hω,SP (Rn ). Thus, Hω,L (Rn ) and Hω,SP (Rn ) coincide with equivalent norms. β
β
In what follows, the operators Nh and NP are as in (5.2) and (5.3), respectively. Lemma 5.3. Let 0 < β < γ < ∞. Then there exists a positive constant C, depending on β and γ , such that for all g ∈ L2 (Rn ), γ β β C −1 Nh g L(ω) Nh g L(ω) C Nh g L(ω)
(5.10)
γ β β C −1 NP g L(ω) NP g L(ω) C NP g L(ω) .
(5.11)
and
Proof. We only prove (5.10); the proof of (5.11) is similar. γ β Since β < γ , for any x ∈ Rn , it is easy to see that Nh g(x) ( γβ )n Nh g(x), which implies the first inequality. To show the second inequality in (5.10), without loss of generality, we may assume that β Nh gL(ω) < ∞. Let σ ∈ (0, ∞), n β β n ∗ n . (5.12) Eσ ≡ x ∈ R : Nh g(x) > σ and Eσ ≡ x ∈ R : M(χEσ )(x) > 3γ Suppose that x ∈ / Eσ∗ . Thus, for any (y, t) ∈ Γ2γ (x), we have B(y, βt) Eσ ; otherwise, M(χEσ )(x) >
|B(y, βt)| = |B(x, 3γ t)|
β 3γ
n ,
which contradicts with x ∈ / Eσ∗ . Thus, there exists z ∈ (B(y, βt) ∩ (Eσ ) ), which further implies that 1/2
2 −t 2 L 1 β e g(u) du Nh g(z) σ. (5.13) (βt)n B(y,βt)
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
For every (w, t) ∈ Γγ (x), we cover the ball B(w, γ t) by no more than N (n, β, γ ) balls N (n,β,γ ) {B(yi , βt)}i=1 , where (yi , t) ∈ Γ2γ (x) and N (n, β, γ ) depends only on n, β, γ . Thus, by (5.13), we obtain
1 (γ t)n
2 −t 2 L e g(z) dz
1/2
n/2 N (n,β,γ ) 1 β γ (βt)n i=1
B(w,γ t)
2 −t 2 L e g(z) dz
1/2
B(yi ,βt)
C(n, β, γ )σ, where C(n, β, γ ) is a positive constant depending on n, β, γ . From this, it follows that for all γ σ > 0, {x ∈ Rn : Nh g(x) > C(n, β, γ )σ } ⊂ Eσ∗ . This combined (3.2) yields that
γ ω Nh g(x) dx ∼
Rn
γ
N h g(x)
Rn
∞ ∼
ω(t) dt dx ∼ t
0
∞
ω(t) γ x ∈ Rn : Nh g(x) > t dt t
0
ω(t) γ x ∈ Rn : Nh g(x) > C(n, β, γ )t dt t
0
∞
ω(t) ∗ Et dt t
0
∞
ω(t) |Et | dt ∼ t
0 γ
β ω Nh g(x) dx,
Rn
β
which further implies that Nh gL(ω) Nh gL(ω) , and hence, completes the proof of Lemma 5.3. 2 Rn ) and Hω,Nh (R Rn ). By (3.2) and Lemmas 5.2 and 3.2, we have Equivalence of Hω,L (R 1/2 SL f L(ω) Sh f L(ω) Sh f L(ω) .
(5.14)
Recall that σg denote the distribution function of a function g. The estimate (6.36) of [19] says that for any λ ∈ (0, ∞), 1 σS1/2 f (λ) 2 h λ
λ tσN β f (t) dt + σN β f (λ), h
h
0
where β ∈ (0, ∞) is large enough. Since ω is of upper type 1, by (5.14), (3.2), (5.15) and Lemma 5.3, we obtain that
Rn
ω SL f (x) dx
1/2 ω S f (x) dx ∼
1/2
Sh f (x)
h
Rn
Rn
0
ω(u) du dx u
(5.15)
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
∞ ∼
1203
ω(u) σ 1/2 (u) du u Sh f
0
∞
, + u ω(u) 1 tσN β f (t) dt + σN β f (u) du h h u u2
0
0
∞
∞ tσN β f (t) h
t
0
ω(t) du dt + u2 t
β ω Nh f (x) dx
Rn
β ω Nh f (x) dx
Rn
ω Nh f (x) dx,
Rn
which implies that f Hω,L (Rn ) f Hω,Nh (Rn ) , and hence, Hω,Nh (Rn ) ⊂ Hω,L (Rn ). Conversely, let Rh be as in (5.4). For all g ∈ L2 (Rn ) and x ∈ Rn , we also define RM h g(x) ≡ sup t>0
1 tn
2 2 M −t 2 L t L e g(y) dy
1/2 .
B(x,t)
By the proof of [19, Theorem 6.3], we know that the operators Rh and RM h are bounded on L2 (Rn ). Since Lemma 5.3 implies that for all f ∈ L2 (Rn ) ∩ Hω,L (Rn ), 1/2 Nh f L(ω) Nh f L(ω) Rh f L(ω) , by Lemma 5.1 and a density argument, to show Hω,L (Rn ) ⊂ Hω,Nh (Rn ), it suffices to prove that for all (ω, 2, M, )-molecules α adapted to balls B and λ ∈ C,
ω Rh (λα)(x) dx |B|ω
Rn
|λ| . |B|ρ(|B|)
Since ω is concave, by the Jensen inequality and the Hölder inequality, we obtain
Rn
∞ ω Rh (λα)(x) dx
ω Rh (λα)(x) dx
j =0U (B) j
∞ j Rh (λα)L2 (Uj (B)) 2 Bω . |2j B|1/2 j =0
(5.16)
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
For j ∈ Z+ and j 10, by the L2 (Rn )-boundedness of the operator Rh and the definition of the molecule, we have 10 j Rh (λα)L2 (Uj (B)) |λ| 2 B ω |B|ω . |B|ρ(|B|) |2j B|1/2 j =0
Since M > n2 ( p1ω − 12 ), we let a ∈ (0, 1) such that a(2M + n/2) > n/pω . For j ∈ N and j > 10, write Rh (λα)(x)
sup t2aj −2 rB
1 tn
1/2
B(x,t)
+
2 −t 2 L e (λα)(y) dy
sup t>2aj −2 rB
1 tn
2 −t 2 L e (λα)(y) dy
1/2 ≡ Hj + Ij .
B(x,t)
For the case t 2aj −2 rB , let Vj (B) ≡ 2j +3 B \ 2j −3 B,
Rj (B) ≡ 2j +5 B \ 2j −5 B
and Ej (B) ≡ Rj (B) . (5.17)
If x ∈ Uj (B) and |x − y| < t, then we have y ∈ Vj (B) and dist(Vj (B), Ej (B)) ∼ 2j rB , which together with Lemma 2.3 yields that Hj L2 (Uj (B))
sup t2aj −2 rB
+
1 tn
sup
1/2
L2 (Uj (B)
B(·,t)
t2aj −2 rB
2 −t 2 L e (λαχRj (B) )(y) dy
1 tn
2 −t 2 L e (λαχEj (B) )(y) dy
1/2
L2 (Uj (B)
B(·,t)
1/2 Rh (λαχRj (B) )L2 (Rn ) + Uj (B) 1/2 λαL2 (Rj (B)) + Uj (B)
sup t2aj −2 rB
t −n/2 e
sup t2aj −2 rB
t −n/2
t 2j rB
−
(2j rB )2 ct 2
λαL2 (Ej (B))
N λαL2 (Rn )
−1 j −1/2 −1 −1/2 2 B |λ|2−j ρ 2j B + |λ|2j (1−a)(n/2−N ) ρ |B| |B| , where c is a positive constant as in Lemma 2.3 and N ∈ N is large enough such that (1 − a) × (N − n/2)pω > n(1 − pω /2). Then by an argument similar to the proof of (4.4) and the fact that ω is of lower type pω , we have ∞ j Hj L2 (Uj (B)) 2 B ω |2j B|1/2
j =11
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
∞ j 2 Bω j =11
1205
∞ j |λ|2−j |λ|2j (1−a)(n/2−N ) 2 Bω + |2j B|ρ(|2j B|) |2j B|1/2 ρ(|B|)|B|1/2 j =11
|B|ω |B|ω
|λ| + |B|ρ(|B|)
∞
|B|2j n(1−pω /2) 2j (1−a)(n/2−N )pω ω
j =11
|λ| ρ(|B|)|B|
|λ| , |B|ρ(|B|)
(5.18)
which is a desired estimate. For the term Ij , by the L2 (Rn )-boundedness of the operator RM h , we have Ij L2 (Uj (B)) sup
t>2aj −2 rB
1 tn
rB t
4M
2 M −t 2 L 2 −M 2 t L e λ r L α (y) dy B
1/2
L2 (Uj (B))
B(·,t)
2 −M 2−2aMj RM α L2 (U h λ rB L
j (B))
−1 −1/2 |λ|2−2aMj ρ |B| |B| ,
which together with the fact that apω (2M + n/2) > n implies that ∞ ∞ j Ij L2 (Uj (B)) j |λ|2−2aMj 2 B ω 2 B ω |2j B|1/2 |2j B|1/2 ρ(|B|)|B|1/2
j =11
j =11
∞ j =11
2
j n(1−pω /2) −2aMjpω
2
|λ| , |B|ω |B|ρ(|B|)
|λ| |B|ω |B|ρ(|B|)
(5.19)
which is a desired estimate. Combining the estimates (5.18) and (5.19) yields (5.16), and hence, completes the proof of that Hω,L (Rn ) ⊂ Hω,Nh (Rn ). Therefore, Hω,L (Rn ) and Hω,Nh (Rn ) coincide with equivalent norms. Rn ) and Hω,NP (R Rn ). The proof of the equivalence of Hω,L (Rn ) and Equivalence of Hω,L (R Hω,NP (Rn ) is similar to that of the equivalence of Hω,L (Rn ) and Hω,Nh (Rn ); we omit the details. This finishes the proof of Theorem 5.2. From Theorem 5.2, it is easy to deduce the following radial maximal function characterizations of Hω,L (Rn ). Recall that Rh and RP are defined in (5.4) and (5.5), respectively. Corollary 5.1. Let ω satisfy Assumption (A). Then the spaces Hω,L (Rn ), Hω,Rh (Rn ) and Hω,RP (Rn ) coincide with equivalent norms.
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
Proof. We only give the proof of the equivalence between Hω,Rh (Rn ) and Hω,L (Rn ), since the proof of the equivalence between Hω,RP (Rn ) and Hω,L (Rn ) is similar. For any f ∈ (Hω,L (Rn ) ∩ L2 (Rn )), by (5.2) and (5.4), we obviously have Rh f Nh f , which implies that Hω,Nh (Rn ) ⊂ Hω,Rh (Rn ). 1/2 1/2 Conversely, since for all f ∈ L2 (Rn ), we have Nh f Rh f, where Nh is as in (5.2). 2 n n Then by Lemma 5.3, we obtain that for all f ∈ L (R ) ∩ Hω,Rh (R ), 1/2 Nh f L(ω) Nh f L(ω) Rh f L(ω) , which implies that Hω,Rh (Rn ) ⊂ Hω,Nh (Rn ), and hence, completes the proof of Corollary 5.1. 2 6. The Carleson measure and the John–Nirenberg inequality In this section, we characterize the space BMOρ,L∗ (Rn ) via the ρ-Carleson measure and establish the John–Nirenberg inequality for elements in BMOρ,L∗ (Rn ), where L∗ denotes the conjugate operator of L in L2 (Rn ). Recall that a measure dμ on Rn+1 + is called a ρ-Carleson measure if dμρ ≡ sup
B⊂Rn
1 |B|[ρ(|B|)]2
| dμ|1/2 < ∞, B
denotes the tent over B; see [22]. where the supremum is taken over all balls B of Rn and B Theorem 6.1. Let ω satisfy Assumption (A), ρ be as in (2.4) and M > n2 ( p1ω − 12 ). (i) If f ∈ BMOρ,L∗ (Rn ), then dμf is a ρ-Carleson measure and there exists a positive constant C independent of f such that dμf ρ Cf 2BMO ∗ (Rn ) , where ρ,L
2 dx dt M 2 ∗ . dμf ≡ t 2 L∗ e−t L f (x) t
(6.1)
n ∗ (ii) Conversely, if f ∈ MM ω,L∗ (R ) satisfies (4.8) with certain q ∈ (pL , 2] and 1 > 0, and dμf is a ρ-Carleson measure, then f ∈ BMOρ,L∗ (Rn ) and there exists a positive constant C independent of f such that f 2BMO ∗ (Rn ) Cdμf ρ , where dμf is as in (6.1). ρ,L
Proof. It follows from Lemma 4.2 that (i) holds. > M + 1 + n and > n( 1 − To show (ii), let M 4 pω M f, g = C
Rn+1 +
1 ). pω+
By Lemma 4.3, we have
2 ∗ M −t 2 L∗ dx dt 2 t L e f (x)t 2 Le−t L g(x) , t
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
)-molecules and q = where g is a finite combination of (ω, q , M, obtain that
q q−1 .
1207
Then by (4.10), we
f, g dμf 1/2 gH (Rn ) . ρ ω,L q ,M,
Since Hω,fin (Rn ) is dense in Hω,L (Rn ), we obtain that f ∈ (Hω,L (Rn ))∗ , which combined with Theorem 4.1 implies that f ∈ BMOρ,L∗ (Rn ) and f 2BMO ∗ (Rn ) duf ρ . This finishes ρ,L the proof of Theorem 6.1. 2 Recall that for every cube Q, (Q) denotes its side-length. Lemma 6.1. Let F ∈ L2loc (Rn+1 + ). Suppose that there exist β ∈ (0, 1) and N ∈ (0, ∞) such that √
for certain a ∈ ( 5 2 n , ∞) and all cubes Q ⊂ Rn , " (Q)
x ∈ Q:
F (y, t)2 dy dt t n+1
#1/2
> Nρ |Q| β|Q|.
0 B(x,3at)
Then
sup cubes Q⊂Rn
1 |Q|[ρ(|Q|)]p
" (Q)
Q
F (y, t)2 dy dt t n+1
0 B(x,at)
#p/2 dx
2N p 1−β
(6.2)
for all p ∈ (1, ∞). (Q) 2 dy dt 1/2 > Nρ(|Q|)}. Applying the Whitney Proof. Let Ω ≡ {x ∈ Q: ( 0 B(x,3at) |F (y, t)| t n+1 ) decomposition to Ω, we obtain a family {Qj }j of disjoint cubes such that ( j Qj ) = Ω and √ √ dist(Qj , Q\Ω) ∈ ( n(Qj ), 4 n(Qj )); see the proof of [19, Lemma 10.1]. For δ ∈ (0, (Q)), define 1 M(δ) ≡ sup | Q| cubes Q⊂Q
Q)
" ( Q
δ
B(x,a(t−δ))
where B(x, a(t − δ)) ≡ ∅ if δ t. Now, observe that
" (Q) δ
Q
#p/2 F (y, t) 2 dy dt dx ρ(|Q|) t n+1
B(x,a(t−δ))
" (Q) Q\Ω
0 B(x,3at)
#p/2 F (y, t) 2 dy dt dx ρ(|Q|) t n+1
#p/2 F (y, t) 2 dy dt dx, ρ(|Q|) t n+1
1208
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
" (Q
j)
+
{j : (Qj )>δ}Q
+
j Q j
"
δ
j
#p/2 F (y, t) 2 dy dt dx ρ(|Q|) t n+1
B(x,a(t−δ))
(Q)
#p/2 F (y, t) 2 dy dt dx ρ(|Q|) t n+1
max{(Qj ),δ} B(x,a(t−δ))
N |Q| + β|Q|M(δ) + p
"
j Q j
(Q)
#p/2 F (y, t) 2 dy dt dx ρ(|Q|) t n+1
max{(Qj ),δ} B(x,a(t−δ))
≡ N p |Q| + β|Q|M(δ) + I. √ √ Since dist(Qj , Q \ Ω) ∈ ( n(Qj ), 4 n(Qj )), there exists x ∈ (Q \ Ω) such that for all x ∈ Qj , √ |x − x | |x − xQj | + |xQj − x | 5 n(Qj ). √ Then by the fact that a 5 n/2, we obtain
(y, t): y ∈ B x, a(t − δ) , max (Qj ), δ < t < (Q) ⊂ (y, t): y ∈ B( x , 3at), t < (Q) ,
which implies that
I
" (Q)
#p/2 F (y, t) 2 dy dt dx N p |Q|. ρ(|Q|) t n+1
sup
x ∈Q\Ω j Q j
0 B( x ,3at)
⊂ Q, let Ω ≡ {x ∈ Q: For every cube Q
(Q)
" (
Q)
|Ω| x ∈ Q:
0
2 dy dt 1/2 B(x,3at) |F (y, t)| t n+1
F (y, t)2 dy dt t n+1
#1/2
> Nρ(|Q|)}. Then
β|Q|. > Nρ |Q|
0 B(x,3at)
Repeating the above estimates, we obtain
" (
Q) Q
δ
#p/2 F (y, t) 2 dy dt + β|Q|M(δ), dx 2N p |Q| ρ(|Q|) t n+1
B(x,a(t−δ))
implies that (1 − β)M(δ) 2N p . Letting δ → 0, we finally which via taking the supremum on Q obtain
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1 |Q|[ρ(|Q|)]p
" (Q)
Q
F (y, t)2 dy dt t n+1
#p/2 dx lim M(δ) δ→0
0 B(x,at)
1209
2N p , 1−β
2
which implies (6.2), and hence, completes the proof of Lemma 6.1.
Theorem 6.2. Let ω satisfy Assumption (A), ρ be as in (2.4) and M > n2 ( p1ω − 12 ). Then the q,M
spaces BMOρ,L∗ (Rn ) for all q ∈ (pL∗ , p L∗ ) coincide with equivalent norms. Proof. It follows from the Hölder inequality that f BMOp,M (Rn ) f BMO2,M
(R ρ,L∗
ρ,L∗
where pL∗ < p < 2 < q < p L∗ . Let us now show that f BMO2,M
(R ρ,L∗
+ (Q)
√ 0 B(x,9 nt)
Q
+ (Q)
n)
(R ρ,L∗
n)
,
f BMOp,M (Rn ) for all p ∈ (pL∗ , 2). Write ρ,L∗
2 dy dt 2 ∗ M −t 2 L∗ t L e f (y) n+1 t
f BMOq,M
n)
1/p
,p/2 dx
2 ∗ M −t 2 L∗ t L e
√ 0 B(x,9 nt)
Q
2 dy dt √ 2 −1 M × I − I + 9 n(Q) L∗ f (y) n+1 t
+ (Q)
dx
2 ∗ M −t 2 L∗ t L e
+ Q
1/p
,p/2
√ 0 B(x,9 nt)
2 dy dt √ 2 −1 M f (y) n+1 × I − I − I + 9 n(Q) L∗ t
1/p
,p/2
≡ H + I.
dx
√ √ Let 9 nQ denote the cube with the same center as Q and side-length 9 n times (Q). Then by the Hölder inequality, Lemmas 2.1 and 2.3, we have
H
+ (Q)
2 j =0
×
Q
2 ∗ M −t 2 L∗ t L e
√ 0 B(x,9 nt)
χUj (9√nQ)
≡ H1 + H2 .
√ 2 −1 M 2 dy dt I − I + 9 n(Q) L∗ f (y) n+1 t
1/p
,p/2 dx
+
∞ j =3
···
1210
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
By Lemma 2.4 and Proposition 4 in [11], we obtain
H1
2 A
√ 9 n
j =0
2 ∗ M −t 2 L∗ √ 2 −1 M t L χUj (9√nQ) I − I + 9 n(Q) L∗ e f Lp (Rn )
2 2 ∗ M −t 2 L∗ A t L χ e
√ Uj (9 nQ)
j =0
√ 2 −1 M I − I + 9 n(Q) L∗ f Lp (Rn )
2 √ I − I + 9 n(Q) 2 L∗ −1 M f p L (χ
√ Uj (9 nQ) )
j =0
ρ |Q| |Q|1/p f BMOp,M (Rn ) . ρ,L∗
√ √ For the term H2 , noticing that Uj (9 nQ) can be covered by 2j n cubes of side-length 9 n(Q), which together with Lemma 2.3 and the Hölder inequality implies that
H2
(Q)
∞
|Q|
1/p−1/2
2 ∗ M −t 2 L∗ t L e
√ 0 10 nQ
j =3
√ 2 −1 M 2 dx dt × χUj (9√nQ) I − I + 9 n(Q) L∗ f (x) t |Q|
1/p−1/2
1/2
(Q)
∞ j 2 − (2 n(Q)) ct 2 e j =3
0
√ 2 −1 M 2 × I − I + 9 n(Q) L∗ f Lp (U
dt
1/2
√ j (9 nQ)) t 1+n/p−n/2
ρ |Q| |Q|1/p f BMOp,M (Rn ) ρ,L∗
× |Q|
1/p−1/2
+ (Q)
∞ j =3
t 2j (Q)
N 2
2j n
dt
,1/2
t 1+n/p−n/2
0
ρ |Q| |Q|1/p f BMOp,M (Rn ) , ρ,L∗
where c is the positive constant as in Lemma 2.3 and N ∈ N is large enough. Thus, H H1 + H2 ρ |Q| |Q|1/p f BMOp,M (Rn ) . ρ,L∗
Applying the formula that
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1211
−1 M √ I − I − I + (9 n(Q))2 L∗ f =
M
√ √ 2 −k 2 −1 M k 9 n(Q) L∗ I − I + 9 n(Q) L∗ CM f,
k=1 k denotes the combinatorial number, by a way similar to the estimate of H, we also have where CM I ρ(|Q|)|Q|1/p f BMOp,M (Rn ) . ρ,L∗
Combining the estimates of H and I yields that + (Q)
Q
√ 0 B(x,9 nt)
2 dy dt 2 ∗ M −t 2 L∗ t L e f (y) n+1 t
1/p
,p/2 dx
ρ |Q| |Q|1/p f BMOp,M (Rn ) , ρ,L∗
which together with Lemma 6.1 implies that sup balls B⊂Rn
∼
2 dx dt 2 ∗ M −t 2 L∗ 1 1 t L e f (x) ρ(|B|) |B| t B
sup cubes Q⊂Rn
2 dx dt 2 ∗ M −t 2 L∗ 1 1 t L e f (x) ρ(|Q|) |Q| t
1/2
Q
"
1/2
sup cubes Q⊂Rn
1 |Q|[ρ(|Q|)]2
(Q)
Q
√ 0 B(x,3 nt)
2 dy dt 2 ∗ M −t 2 L∗ t L e f (y) n+1 dx t
#1/2
f BMOp,M (Rn ) . ρ,L∗
Then by Theorem 6.1, we obtain f BMO2,M Finally, let us show that f BMOq,M
ρ,L∗
(R ρ,L∗
n)
f BMOp,M (Rn ) .
f BMO2,M (Rn )
ρ,L∗
(R ρ,L∗ q
n)
for all q ∈ (2, p L∗ ). Let q be the
conjugate index q. For any ball B, let h ∈ L2 (B) ⊂ L (B) such that hLq (B) From Lemma 2.3, similarly to the proof of Theorem 4.1, it is easy to follow that is a multiple of an (ω, q , M, )-molecule, and hence, I − e−rB2 L M h H
ω,L (R
n)
1 . |B|1−1/q ρ(|B|) 2 (I − e−rB L )M h
1.
n n ∗ Now let f ∈ BMO2,M ρ,L∗ (R ). By Theorem 4.1, f ∈ (Hω,L (R )) , and hence,
I − e−rB2 L∗ M f, h = f, I − e−rB2 L M h f BMO2,M
(R ρ,L∗
Taking supremum over all such h yields that f BMOq,M pletes the proof of Theorem 6.2.
2
(R ρ,L∗
n)
f BMO2,M
n)
.
(R ρ,L∗
n)
, which com-
1212
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
7. Some applications In this section, we establish the boundedness on Orlicz–Hardy spaces of the Riesz transform and the fractional integral associated with the operator L as in (1.2). Recall that the Littlewood–Paley g-function gL is defined by setting, for all f ∈ L2 (Rn ) and x ∈ Rn , " ∞ #1/2 2 dt 2 −t 2 L t Le gL f (x) ≡ f (x) . t 0
By the proof of Theorem 3.4 in [19], we know that gL is bounded on L2 (Rn ). Similarly to Theorems 3.2 and 3.4 in [19], we have the following conclusion. Theorem 7.1. Let ω satisfy Assumption (A) and p ∈ (pL , 2]. Suppose that the non-negative sublinear operator or linear operator T is bounded on Lp (Rn ) and there exist C > 0, M ∈ N and M > n2 ( p1ω − 12 ) such that for all closed sets E, F in Rn with dist(E, F ) > 0 and all f ∈ Lp (Rn ) supported in E, T I − e−tL M f p C L (F )
t dist(E, F )2
M f Lp (E)
(7.1)
and T tLe−tL M f
t C Lp (F ) dist(E, F )2
M f Lp (E)
(7.2)
for all t > 0. Then T extends to a bounded sublinear or linear operator from Hω,L (Rn ) to L(ω). In particular, the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL is bounded from Hω,L (Rn ) to L(ω). Proof. Let > n(1/pω − 1/ pω ), where p ω is as in Convention (C). Since T is bounded on Lp (Rn ), by Lemma 5.1, to show that T is bounded from Hω,L (Rn ) to L(ω), it suffices to show that for all λ ∈ C and (ω, ∞, M, )-molecules α adapted to balls B,
ω T (λα)(x) dx |B|ω
Rn
|λ| . |B|ρ(|B|)
To prove (7.3), we write
ω T (λα)(x) dx
Rn
Rn
2 M ω |λ|T I − e−rB L α (x) dx +
Rn
2 M α (x) dx ω |λ|T I − I − e−rB L
(7.3)
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
∞
2 M ω |λ|T I − e−rB L (αχUj (B) ) (x) dx
j =0 Rn
+
∞
sup
j =0 1kM Rn
≡
∞
1213
Hj +
j =0
∞
k 2 − k r2 L r Le M B ω |λ|T M B
−2 −1 M χUj (B) rB L α (x) dx
M
Ij .
j =0
For each j 0, let Bj ≡ 2j B. Since ω is concave, by the Jensen inequality and the Hölder inequality, we obtain Hj
∞
2 M ω |λ|T I − e−rB L (αχUj (B) ) (x) dx
k=0 U (B ) k j
∼
∞
k=0
2 M ω |λ|χUk (Bj ) (x)T I − e−rB L (αχUj (B) ) (x) dx
2k Bj
∞ k |λ| 2 Bj ω |2k Bj | k=0
2 M T I − e−rB L (αχUj (B) ) (x) dx
Uk (Bj )
∞
k 2 Bj ω
k=0
|λ| T I − e−rB2 L M (αχU (B) ) p . j L (Uk (Bj )) |2k Bj |1/p
By the Lp (Rn )-boundedness of T , Lemma 2.3 and (7.1), we have that for k = 0, 1, 2, T I − e−rB2 L M (αχU (B) ) p j L (U
k (Bj ))
αLp (Uj (B)) ,
and that for k 3, T I − e−rB2 L M (αχU (B) ) p j L (U
k (Bj
))
1 2k+j
2M α2Lp (Uj (B)) ,
which, together with Definition 4.2, 2Mpω > n(1 − pω /2) and Assumption (A), implies that ∞ k |λ|2−(2M)(j +k)−j |λ|2−j 2 Bj ω + Hj |Bj |ω |Bj |ρ(|Bj |) |2k Bj |1/p |Bj |1−1/p ρ(|Bj |) k=3 ∞ |λ| −jpω kn(1−pω /p) −2Mpω (j +k) 2 2 2 1+ |Bj |ω |Bj |ρ(|Bj |) k=3 |λ| . 2−jpω |Bj |ω |Bj |ρ(|Bj |)
1214
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
Since ρ is of lower type 1/ pω − 1 and > n(1/pω − 1/ pω ), we further have ∞
Hj
j =0
∞
2
−jpω
|B|ρ(|B|) pω |λ| |Bj | ω |Bj |ρ(|Bj |) |B|ρ(|B|)
2
−jpω
pω |B| pω / |λ| |Bj | ω |Bj | |B|ρ(|B|)
2
−jpω j n(1−pω / pω )
j =0
∞ j =0
∞
2
j =0
|λ| |B|ω |B|ρ(|B|)
|λ| |B|ω . |B|ρ(|B|)
Similarly, we have ∞ j =0
|λ| . Ij |B|ω |B|ρ(|B|)
Thus, (7.3) holds, and hence, T is bounded from Hω,L (Rn ) to L(ω). It was proved in [19, Theorem 3.4] that operators gL and ∇L−1/2 satisfy (7.1) and (7.2); thus, gL and ∇L−1/2 are bounded from Hω,L (Rn ) to L(ω), which completes the proof of Theorem 7.1. 2 We now give a fractional variant of Theorem 7.1. To this end, we first make some assumptions. Let ω and pω satisfy Assumption (A) and q ∈ [pω , 1]. In what follows, for all t ∈ (0, ∞), define v(t) ≡ ω−1 (t)t 1/q−1/pω .
(7.4)
Assumption (B). Let ω satisfy Assumption (A), q ∈ [pω , 1] and pL < r1 min{2, r2 } r2 < p L satisfying that 1/pω − 1/q = 1/r1 − 1/r2 . Suppose that v as in (7.4) is convex and t −1 v(0) ≡ limt→0+ v(t) = 0. Then for all t ∈ (0, ∞), let ω(t) ≡ v −1 (t) and ρ (t) ≡ . ω−1 (t −1 ) Remark 7.1. (i) It is easy to see that if ω is as in Assumption (B), then ω also satisfies Assump+ 1 = . Moreover, tion (A) with p ω = q and p + ω 1/p +1/q−1/p ω
ρ (t) ≡
t −1 ω−1 (t −1 )
=
ω
t −1 ω−1 (t −1 )t 1/pω −1/q
= ρ(t)t −1/pω +1/q .
(ii) Let p ∈ (0, 1] and ω(t) ≡ t p for all t ∈ (0, ∞). In this case, pω = p = pω+ , q ∈ [p, 1] and ω(t) ≡ t q for all t ∈ (0, ∞). ω satisfy Assumption (B). Suppose that the linear operator Theorem 7.2. Let q, r1 , r2 , ω and T is bounded from Lr1 (Rn ) to Lr2 (Rn ) and there exist C > 0, M ∈ N and M > n2 ( p1ω − 12 ) +
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1215
n( p1ω − p1+ ) satisfying that for all closed sets E, F in Rn with dist(E, F ) > 0, all f ∈ Lr1 (Rn ) ω supported in E, and all t > 0, T I − e−tL M f r C L 2 (F )
t dist(E, F )2
M f Lr1 (E)
(7.5)
f Lr1 (E) .
(7.6)
and T tLe−tL M f
C Lr2 (F )
t dist(E, F )2
M
If T is commutative with L, then T extends to a bounded linear operator from Hω,L (Rn ) to n H ω,L (R ). Proof. Let ∈ (n( p1ω −
1 ), M − n2 ( p1ω − 12 )). pω+ L2 (Rn )), f ∈ Lr1 (Rn ) and
It follows from Proposition 4.2 that for ev-
∩ there exist {λj }∞ ery f ∈ (Hω,L j =1 ⊂ C and (ω, ∞, 2M, ) ∞ ∞ n molecules {αj }j =1 adapted to balls {Bj }j =1 such that f = ∞ λ j =1 j αj holds in both Hω,L (R ) and Lr1 (Rn ); moreover, Λ({λj αj }j ) f Hω,L (Rn ) . We first show that T maps each (ω, ∞, 2M, )-molecule into a multiple of an ( ω, r2 , M, )molecule. To this end, assume that α is an (ω, ∞, 2M, )-molecule adapted to a ball B ≡ B(xB , rB ). By the boundedness of T from Lr1 (Rn ) to Lr2 (Rn ) and Remark 7.1, we have T α ∈ Lr2 (Rn ) and for any k ∈ {0, · · · , M} and j ∈ {0, . . . , 10}, (Rn )
2 −k r L T α B
Lr2 (Uj (B))
1/r −1 −1 −k rB2 L α Lr1 (Rn ) 2−j 2j B 1 ρ 2j B 1/r −1 j −1 2 B ∼ 2−j 2j B 2 ρ .
For j 11, let Wj (B) ≡ (2j +3 B \ 2j −3 B) and Ej (B) ≡ (Wj (B)) . Thus, 2 −k r L T α B
Lr2 (Uj (B))
2 M 2 −k rB L α Lr2 (U T I − e−rB L
j (B))
2 M 2 −k rB L α Lr2 (U + T I − I − e−rB L
j (B))
≡ H + I.
By the boundedness from Lr1 (Rn ) to Lr2 (Rn ) of T , Lemma 2.3, (7.5), the choice of and Remark 7.1, we have −k 2 M H T I − e−rB L χWj (B) rB2 L α Lr2 (U
j (B))
−k 2 M χEj (B) rB2 L α Lr2 (U + T I − e−rB L −k rB2 L α Lr1 (W
+ (B))
j (B))
M
rB2
2 −k r L α
B dist(Uj (B), Ej 1/r −1 −1 −1 2−j 2j B 1 ρ 2j B + 2−2j M |B|1/r1 −1 ρ |B| 1/r −1 j −1 2 B 2−j 2j B 2 ρ . j
(B))2
Lr1 (Ej (B))
1216
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
Similarly, by (7.6), we have 2 krB − krB2 L I sup T M Le M
M
−k−M χWj (B) rB2 L α Lr2 (U
1kM
2 krB − krB2 L + sup T M Le M
2 −k−M χEj (B) rB L α
M
Lr2 (Uj (B))
1kM
−k−M rB2 L α Lr1 (W
j
j (B))
+ (B))
rB2 dist(Uj (B), Ej (B))2
M
2 −k−M r L α
Lr1 (Ej (B))
B
1/r −1 j −1 2 B 2−j 2j B 2 ρ .
Combining the above estimates, we finally obtain that T α is a multiple of an ( ω, r2 , M, )molecule. r2 n Since T is bounded from Lr1 (Rn ) to Lr2 (Rn ), we have Tf = ∞ j =1 λj T (αj ) in L (R ). To finish the proof, it remains to show that Tf Hω,L (Rn ) f Hω,L (Rn ) . To this end, by Lemma 2.4, the subadditivity and the continuity of ω and (4.3) with ω and ρ replaced respectively by ω and ρ , we obtain
∞ ω SL (Tf )(x) dx
∞ ω |λj |SL (T αj )(x) dx |Bj | ω
j =1Rn
Rn
j =1
|λj | . (7.7) |Bj | ρ (|Bj |)
Choose γ ∈ (Λ({λj αj }j ), 2Λ({λj αj }j )]. Then for each j ∈ N, we have γ |λj |; otherwise, there exists i ∈ N such that γ < |λi |, which together with the strictly increasing property of ω further implies that ∞
|Bj |ω
j =1
|λj | γ |Bj |ρ(|Bj |)
|Bi |ω
|λi | γ |Bi |ρ(|Bi |)
> |Bi |ω
1 |Bi |ρ(|Bi |)
= 1.
This contradicts to the assumption λ > Λ({λj αj }j ). Thus, the claim is true. Therefore, by this claim and the strictly increasing property of ω, for each j ∈ N, we have |Bj |ω
|λj | γ |Bj |ρ(|Bj |)
1/pω −1/q
1/pω −1/q |Bj |ω ω−1 |Bj |−1 1,
which implies that |λj | −1 |λj | −1 |λj | ω |Bj |−1 = ω |Bj |−1 |Bj |1/pω −1/q = −1 γ γ γ |Bj | ρ (|Bj | ) 1/q−1/pω |λj | −1 |λj | ω |Bj |−1 ω γ γ |Bj |ρ(|Bj |) |λj | −1 ω . = ω γ |Bj |ρ(|Bj |)
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
1217
Since ω satisfies Assumption (B), we further obtain ω
|λj | γ |Bj | ρ (|Bj |)
ω
|λj | , γ |Bj |ρ(|Bj |)
and hence, by (7.7),
Rn
∞ ∞ |λj | |λj | SL (Tf )(x) dx 1. ω |Bj | ω |Bj |ω γ γ |Bj | ρ (|Bj |) γ |Bj |ρ(|Bj |) j =1
j =1
Thus, Tf Hω,L (Rn ) γ f Hω,L (Rn ) , which together with a standard density argument completes the proof of Theorem 7.2. 2 Remark 7.2. If we let p ∈ (0, 1] and ω(t) ≡ t p for all t ∈ (0, ∞), by Remark 7.1(ii) and Theorem 7.2, we know that the operator T of Theorem 7.2 in this case extends to a bounded linear p q operator from HL (Rn ) to HL (Rn ). In what follows, let γ ∈ (0, n2 ( p1L − p1L )). Recall that the generalized fractional integral L−γ is given by setting, for all f ∈ L2 (Rn ) and x ∈ Rn , L
−γ
1 f (x) ≡ Γ (γ )
∞
t γ e−tL f (x)
dt . t
0 n Applying Theorem 7.2, we obtain the boundedness of L−γ from Hω,L (Rn ) to H ω,L (R ) as follows.
Theorem 7.3. Let q, r1 , r2 , ω and ω satisfy Assumption (B) and γ ∈ (0, n2 ( p1L − p1L )) satisfying that n(1/pω − 1/q) = 2γ . Then the operator L−γ satisfies (7.5) and (7.6) and hence, is bounded n from Hω,L (Rn ) to H ω,L (R ). Proof. Let M ∈ N and M > n2 ( p1ω − 12 ) + n( p1ω − p1+ ). By [2, Proposition 5.3], the operator L−γ ω is bounded from Lr1 (Rn ) to Lr2 (Rn ). Thus, by Theorem 7.2, to show Theorem 7.3, we only need to prove that L−γ satisfies (7.5) and (7.6). We only give the proof of the former one, since (7.6) can be proved in a similar way. Let E, F be closed sets in Rn with dist(E, F ) > 0 and f ∈ Lr1 (Rn ) supported in E. Write −γ M L I − e−tL f Lr2 (Rn ) =
∞ 1 M s γ −1 e−sL I − e−tL f ds Γ (γ ) 0
Lr2 (Rn )
t M s γ −1 e−sL I − e−tL f ds 0
≡ H1 + H2 .
Lr2 (Rn )
∞ + · · · t
Lr2 (Rn )
1218
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
It follows from Lemma 2.3 that
t H1
γ −1 −sL s e f
t Lr2 (Rn )
ds + sup 1kM
0
t
s
γ −1
s
n 1 1 2 ( r2 − r1 )
0
t dist(E, F )2
e
) − dist(E,F cs
2
γ −1 −sL −ktL s e e f
ds
0
t ds +
Lr2 (Rn )
s
γ −1
t
n 1 1 2 ( r2 − r1 )
e
) − dist(E,F ct
2
ds f Lr1 (Rn )
0
M f Lr1 (Rn ) ,
here and in what follows, c is the positive t constant as in Lemma 2.3. For the term H2 , since I − e−tL = 0 Le−rL dr, by Lemmas 2.1 and 2.3, and the Minkowski inequality, we obtain #M " t
∞ γ −1 −sL −rL H2 s e Le dr f t
∞
0
s γ −1 sM
t
∞
t
t ···
0
ds Lr2 (Rn )
(sL)M e−sL e−(r1 +···+rM )L f
Lr2 (Rn )
dr1 · · · drM ds
0
s γ −1 n2 ( r1 − r1 ) M − dist(E,F )2 cs s 2 1 t e dsf Lr1 (Rn ) sM
t
∞
rt dist(E, F )2
M
e−r
dr f Lr1 (Rn ) r
t dist(E, F )2
M f Lr1 (Rn ) ,
0
which implies that (7.5) holds for the operator L−γ , and hence, completes the proof of Theorem 7.3. 2 Remark 7.3. Similarly to Remark 7.2, as a special case of Theorem 7.3, we know that the opp p erator L−γ maps HL (Rn ) continuously into HL (Rn ), where γ , p, q satisfy 0 < p q 1 and n(1/p − 1/q) = 2γ . Using Theorems 7.1 and 7.3, we further obtain the following boundedness of the Riesz transform ∇L−1/2 from Hω,L (Rn ) to Hω (Rn ). We first recall some notions; see [39,34,22]. In what follows, let S(Rn ) denote the space of all Schwartz functions and S (Rn ) the space of all Schwartz distributions. n , 1]. A function a is called a (ρ, 2)Definition 7.1. Let ω satisfy Assumption (A) and pω ∈ ( n+1 atom if
(i) supp a ⊆ B, where B is a ball of Rn ; (ii) aL2 (Rn ) |B|−1/2 [ρ(|B|)]−1 ; (iii) Rn a(x) dx = 0.
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n Definition 7.2. Let ω satisfy Assumption (A) and pω ∈ ( n+1 , 1]. The Orlicz–Hardy space n ) is defined to be the set of all distributions f ∈ S (Rn ) that can be written as f = (R H ω∞ ∞ n j =1 bj in S (R ), where {bj }j =1 is a sequence of multiples of (ρ, 2)-atoms such that ∞
|Bj |ω
j =1
bj L2 (Rn )
|Bj |1/2
< ∞,
where supp bj ⊂ Bj . Moreover, define f Hω (Rn ) ≡ inf λ > 0:
∞ j =1
|Bj |ω
bj L2 (Rn ) λ|Bj |1/2
1 ,
where the infimum is taken over all decompositions of f as above. It is well known that the classical Orlicz–Hardy space defined by using grand maximal functions is equivalent to the above atomic Orlicz–Hardy space Hω (Rn ) as in Definition 7.2; see [39,34]. Based on this fact, in what follows, we denote both spaces by the same notation. Recall that Hω (Rn ) is complete. n Theorem 7.4. Let ω satisfy Assumption (A) and pω ∈ ( n+1 , 1]. Then the Riesz transform ∇L−1/2 p is bounded from Hω,L (Rn ) to Hω (Rn ). In particular, ∇L−1/2 is bounded from HL (Rn ) to n , 1] H p (Rn ) for all p ∈ ( n+1
Proof. Let > 1 + n( p1ω − p1 ) and M > n2 ( p1ω − 12 ) + , where p ω is as in Convention (C). ω Suppose that α is an (ω, ∞, M, )-molecule associated to a ball B ≡ B(xB , rB ). We first show that Rn ∇L−1/2 α(x) dx = 0. From the L2 (Rn )-boundedness of ∇L−1/2 (see [2, Theorem 4.1]), it follows that for j = 0, 1, . . . , 10, −1/2 ∇L α L2 (U
j (B))
−1 ∇L−1/2 α L2 (Rn ) αL2 (Rn ) |B|−1/2 ρ |B| .
(7.8)
For j 11, let Wj (B) ≡ (2j +3 B \ 2j −3 B) and Ej (B) ≡ (Wj (B)) . By the fact that ∇L−1/2 satisfies (7.1) and (7.2) (see Theorem 3.4 in [19]) together with the L2 (Rn )-boundedness of ∇L−1/2 and Lemma 2.2, we have −1/2 ∇L α
L2 (Uj (B))
2 M 2 M α L2 (U (B)) ∇L−1/2 I − e−rB L α L2 (U (B)) + ∇L−1/2 I − I − e−rB L j j −1/2 2 M I − e−rB L (χWj (B) + χEj (B) )α L2 (U (B)) ∇L j 2 kr 2 L M −1/2 krB L − B 2 −M e M (χWj (B) + χEj (B) ) rB L + sup ∇L α 2 M 1kM L (Uj (B)) αL2 (Wj (B)) + 2−2j M αL2 (Rn )
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R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
−M + rB2 L α L2 (W
−M + 2−2j M rB2 L α L2 (Rn ) −1/2 j −1 ρ 2 B 2−j + 2−j (2M−n/pω +n/2) 2j B . j (B))
(7.9)
Combining the above estimates and using the Hölder inequality, we see that ∇L−1/2 α ∈ L1 (Rn ). We now choose pL < s min{t, 2} t < p L such that 1/s − 1/t = 1/n. Since an (ω, ∞, M, )-molecule is also an (ω, s, M, )-molecule, by the fact that L−1/2 is bounded from Ls (Rn ) to Lt (Rn ) (see [2, Proposition 5.3]) and the Hölder inequality, we have that for j = 0, 1, . . . , 10, −1/2 L α
L1 (Uj (B))
1−1/t −1/2 n(1−1/t) L Uj (B) α Lt (Rn ) 2j rB αLs (Rn ) −1 −1 |B|1−1/t+1/s−1 ρ |B| ∼ |B|1/n ρ |B| .
For j 11, let Wj (B) ≡ (2j +3 B \ 2j −3 B) and Ej (B) ≡ (Wj (B)) . By Theorem 7.3, we have that L−1/2 satisfies (7.5) and (7.6), which together with Lemma 2.3 and the Hölder inequality yields that −1/2 L α
L1 (Uj (B))
1−1/t −1/2 2 M 2 M L I − e−rB L α Lt (U (B)) + L−1/2 I − I − e−rB L α Lt (U (B)) Uj (B) j j 1−1/t −1/2 2 M L Uj (B) I − e−rB L (χWj (B) + χEj (B) )α Lt (U (B)) j
−1/2 krB2 L − krB2 L e M + sup L M 1kM
2 −M (χWj (B) + χEj (B) ) rB L α
M
1−1/t Uj (B) αLs (Wj (B)) + 2−2j M αLs (Rn ) −M −M + rB2 L α Ls (W (B)) + 2−2j M rB2 L α Ls (Rn )
Lt (Uj (B))
j
−1 2−j ( −1) + 2−j (2M−n[1−1/t]) |B|1/n ρ |B| . Since > 1 + n(1/pω − p ω ) and M > 2pnω , we obtain that L−1/2 α ∈ L1 (Rn ). ∞ n Now we choose {ϕj }∞ j =0 ⊂ C0 (R ) such that ∞ n (i) j =0 ϕj (x) = 1 for almost every x ∈ R ; (ii) for each j ∈ Z+ , supp ϕj ⊂ 2Bj , ϕj = 1 on Bj and 0 ϕj 1; all j ∈ Z+ and x ∈ Rn , |ϕj (x)| + |∇ϕj (x)| Cϕ ; (iii) there exists Cϕ > 0 such that for ∞ (iv) there exists Nϕ ∈ N such that j =0 χ2Bj Nϕ . −1/2 α, ∇L−1/2 α ∈ L1 (Rn ), we obtain Using the properties of {ϕj }∞ j =0 and the facts that L
Rn
∇L−1/2 α(x) dx =
∞
j =0Rn
∇ ϕj L−1/2 α (x) dx.
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
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For each j , let ηj ∈ C0∞ (Rn ) such that ηj = 1 on 2Bj and supp ηj ⊂ 4Bj . Then for each i = 1, 2, . . . , n, we have
∂ ∂ ϕj L−1/2 α (x) dx = ηj (x) ϕj L−1/2 α (x) dx ∂xi ∂xi Rn
Rn
∂ ϕj L−1/2 α (x) ηj (x) dx = 0, ∂xi
=− Rn
which implies that Rn ∇L−1/2 α(x) dx = 0. k ≡ To finish the proof, we borrow some ideas from [38]. For k ∈ Z+ , let χk ≡ χUk (B) , χ k . Then we have |Uk (B)|−1 χk , mk ≡ Uk (B) ∇L−1/2 α(x) dx and Mk ≡ ∇L−1/2 αχk − mk χ ∇L−1/2 α =
∞
Mk +
k=0
in L2 (Rn ). Now let Nj ≡
∞
k=j
mk . Since
∇L−1/2 α =
∞
Rn
mk χ k
k=0
∇L−1/2 α(x) dx = 0, we have
Mk +
k=0
∞
∞
Nk+1 ( χk+1 − χ k ).
(7.10)
k=0
Obviously, for all k ∈ Z+ , Rn Mk (x) dx = 0. Furthermore, by (7.8), (7.9) and the Hölder inequality, we have that for all k ∈ Z+ , Mk L2 (Rn ) ∇L−1/2 α L2 (U
k (B))
−1/2 k −1 ρ 2 B 2−k 2k B ,
which implies that {2k Mk }k∈Z+ is a family of (ρ, 2)-atoms up to a harmless constant. k | |2k B|−1 and the Hölder To deal with the second sum in (7.10), by (7.8), (7.9), | χk+1 − χ inequality, we have that for all k ∈ Z+ , Nk+1 ( χk+1 − χ k )
L2 (Rn )
∞ |2j B|1/2 ∇L−1/2 α 2 L (Uj (B)) k 1/2 |2 B| j =k
−1/2 k −1 ρ 2 B 2−k 2k B . This, together with Rn [ χk+1 − χ k ] dx = 0, implies that for each k ∈ Z+ , 2k Nk+1 ( χk+1 − χ k ) is a (ρ, 2)-atom up to a harmless constant. By Assumption (A) and Convention (C), ρ is of lower type 1/ pω − 1, which implies that ∞ ∞ j Mj L2 (Rn ) j Nj +1 ( χj +1 − χ j )L2 (Rn ) 2 B ω 2 B ω + λ|2j B|1/2 λ|2j B|1/2 j =0
j =0
∞ j =0
2
−jpω +j n(1−pω / pω )
1 |B|ω λ|B|ρ(|B|)
1 ∼ |B|ω . λ|B|ρ(|B|)
(7.11)
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Now, suppose that f ∈ Hω,L (Rn ) ∩ L2 (Rn ). By Proposition 4.2, there exist (ω, ∞, M, )∞ ∞ and numbers {λ }∞ ⊂ C such that f = adapted to balls {B } molecules {αk }∞ k k=1 k k=1 k=1 λk αk k=1 n 2 n in both Hω,L (R ) and L (R ) with Λ({λk αk }k ) f Hω,L (Rn ) . For each (ω, ∞, M, )-molecule αk , by the above argument, we decompose ∇L−1/2 αk into 2 n a summation of multiples of (ρ, 2)-atoms with ∞harmless constants, which converges in L (R ). −1/2 αk = j =1 bk,j , where bk,j is a multiple of a (ρ, 2)-atom For simplicity, we write it as ∇L supported in Bk,j with a harmless constant. Thus, by (7.11), we obtain ∞ ∞ −1/2 |λk |bk,j L2 (Rn ) ∇L 1 f H (Rn ) = inf λ > 0: |Bk,j |ω ω λ|Bk,j |1/2 k=1 j =1 ∞ |λk | inf λ > 0: 1 |Bk |ω λ|Bk |ρ(|Bk |) k=1 ∼ Λ {λk αk }k f Hω,L (Rn ) . Then, by a standard density argument, we see that ∇L−1/2 extends to a bounded linear operator from Hω,L (Rn ) to Hω (Rn ). This finishes the proof of Theorem 7.4. 2 n Remark 7.4. Let ω satisfy Assumption (A) and pω ∈ ( n+1 , 1]. We claim that the Orlicz–Hardy p n n n n , 1]. spaces Hω,L (R ) ⊂ Hω (R ). In particular, HL (R ) ⊂ H p (Rn ) for all p ∈ ( n+1 n 1 1 + Let ∈ (n(1/pω − 1/pω ), ∞), M ∈ N and M > 2 ( pω − 2 ). For all (ω, ∞, M, )-molecules α, we claim that Rn α(x) dx = 0. To show this, write
−1 2 −1 −1 α = div A∇L−1 α = rB div A rB ∇ I + rB2 L rB L α + rB ∇ I + rB2 L α ,
where α is adapted to the ball B ≡ B(xB , rB ). From the Hölder inequality, Lemma 2.2 and Definition 4.2, it follows that for j = 0, 1, . . . , 10, 1/2 −1 2 −1 rB ∇ I + r 2 L −1 r 2 L −1 α 1 rB L α L2 (Rn ) Uj (B) rB ∇ I + rB2 L B B L (Uj (B)) −1 −1 |B|1/2 rB2 L α L2 (Rn ) ρ |B| . For j 11, let Wj (B) ≡ (2j +3 B \ 2j −3 B) and Ej (B) ≡ (Wj (B)) . By Lemma 2.2 and the Hölder inequality, we have rB ∇ I + r 2 L −1 r 2 L −1 α 1 B B L (Uj (B)) 1/2 −1 −1 (χWj (B) + χEj (B) ) rB2 L α L2 (U (B)) Uj (B) rB ∇ I + rB2 L j
1/2 2 −1 dist(U (B), E (B)) j j r L α 2 s n α Uj (B) + exp − L (R ) B L (Wj (B)) crB −1 −1 rB n/2+ −1 2−j ρ 2j B ρ |B| + 2j n/2 j 2−j ρ |B| . 2 rB The above two estimates imply that rB ∇(I + rB2 L)−1 (rB2 L)−1 α ∈ L1 (Rn ).
R. Jiang, D. Yang / Journal of Functional Analysis 258 (2010) 1167–1224
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Similarly, we have that rB ∇(I + rB2 L)−1 α ∈ L1 (Rn ), and hence, ∇L−1 α ∈ L1 (Rn ). ∞ Let {ϕj }∞ j =0 be as in the proof of Theorem 7.4. Using the properties of {ϕj }j =0 and the facts −1 1 n that α, ∇L α ∈ L (R ) together with the divergence theorem, we obtain
α(x) dx =
∞
div ϕj A∇L−1 α (x) dx
j =0Rn
Rn
=
∞
ϕj (x)N2Bj (x), A∇L−1 α(x) dσ2Bj x = 0,
j =0∂(2B ) j
where N 2Bj denotes the outward unit norm vector to 2Bj and σ2Bj the surface measure over ∂(2Bj ). Then following the proof of Theorem 7.4, we obtain that for all f ∈ Hω,L (Rn ) ∩ L2 (Rn ), f Hω (Rn ) f Hω,L (Rn ) . By a density argument, we obtain that Hω,L (Rn ) ⊂ Hω (Rn ), which completes the proof of the above claim. Acknowledgments Dachun Yang would like to thank Professor Steve Hofmann, Professor Pascal Auscher and Professor Lixin Yan very much for some helpful discussions on the subject of this paper. The authors would also like to thank the referee very much for his many valuable remarks which made this article more readable. Dachun Yang is supported by the National Natural Science Foundation (Grant No. 10871025) of China. References [1] K. Astala, T. Iwaniec, P. Koskela, G. Martin, Mappings of BMO-bounded distortion, Math. Ann. 317 (2000) 703– 726. [2] P. Auscher, On necessary and sufficient conditions for Lp -estimates of Riesz transforms associated to elliptic operators on Rn and related estimates, Mem. Amer. Math. Soc. 186 (2007) 1–75. [3] P. Auscher, X.T. Duong, A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, unpublished manuscript, 2005. [4] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on Rn , Ann. of Math. (2) 156 (2002) 633–654. [5] P. Auscher, A. McIntosh, E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008) 192–248. [6] P. Auscher, E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of Rn , J. Funct. Anal. 201 (2003) 148–184. [7] Z. Birnbaum, W. Orlicz, Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen, Studia Math. 3 (1931) 1–67. [8] S.-S. Byun, F. Yao, S. Zhou, Gradient estimates in Orlicz space for nonlinear elliptic equations, J. Funct. Anal. 255 (2008) 1851–1873. [9] R.R. Coifman, A real variable characterization of H p , Studia Math. 51 (1974) 269–274. [10] R.R. Coifman, P.-L. Lions, Y. Meyer, P. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993) 247–286. [11] R.R. Coifman, Y. Meyer, E.M. Stein, Some new functions and their applications to harmonic analysis, J. Funct. Anal. 62 (1985) 304–335. [12] X.T. Duong, J. Xiao, L. Yan, Old and new Morrey spaces with heat kernel bounds, J. Fourier Anal. Appl. 13 (2007) 87–111.
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[13] X.T. Duong, L. Yan, New function spaces of BMO type, the John–Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005) 1375–1420. [14] X.T. Duong, L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005) 943–973. [15] C. Fefferman, E.M. Stein, H p spaces of several variables, Acta Math. 129 (1972) 137–193. [16] J. Frehse, An irregular complex valued solution to a scalar uniformly elliptic equation, Calc. Var. Partial Differential Equations 33 (2008) 263–266. [17] L. Grafakos, Modern Fourier Analysis, second ed., Grad. Texts in Math., vol. 250, Springer, New York, 2008. [18] S. Hofmann, G. Lu, D. Mitrea, M. Mitrea, L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies–Gaffney estimates, submitted for publication. [19] S. Hofmann, S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009) 37–116. [20] S. Hofmann, S. Mayboroda, Correction to “Hardy and BMO spaces associated to divergence form elliptic operators”, arXiv:0907.0129. [21] S. Hofmann, J.M. Martell, Lp bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003) 497–515. [22] E. Harboure, O. Salinas, B. Viviani, A look at BMOϕ (ω) through Carleson measures, J. Fourier Anal. Appl. 13 (2007) 267–284. [23] T. Iwaniec, J. Onninen, H 1 -estimates of Jacobians by subdeterminants, Math. Ann. 324 (2002) 341–358. [24] S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47 (1980) 959–982. [25] R. Jiang, D. Yang, Y. Zhou, Orlicz–Hardy spaces associated with operators, Sci. China Ser. A 52 (2009) 1042–1080. [26] F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961) 415–426. [27] R.H. Latter, A characterization of H p (Rn ) in terms of atoms, Studia Math. 62 (1978) 93–101. [28] S. Martínez, N. Wolanski, A minimum problem with free boundary in Orlicz spaces, Adv. Math. 218 (2008) 1914– 1971. [29] S. Müller, Hardy space methods for nonlinear partial differential equations, Tatra Mt. Math. Publ. 4 (1994) 159–168. [30] W. Orlicz, Über eine gewisse Klasse von Räumen vom Typus B, Bull. Inst. Acad. Pol. Ser. A 8 (1932) 207–220. [31] M. Rao, Z. Ren, Theory of Orlicz Spaces, Dekker, New York, 1991. [32] M. Rao, Z. Ren, Applications of Orlicz Spaces, Dekker, New York, 2000. [33] S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations 19 (1994) 277–319. [34] C.F. Serra, Molecular characterization of Hardy–Orlicz spaces, Rev. Un. Mat. Argentina 40 (1996) 203–217. [35] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993. [36] E.M. Stein, G. Weiss, On the theory of harmonic functions of several variables. I. The theory of H p -spaces, Acta Math. 103 (1960) 25–62. [37] J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979) 511–544. [38] M.H. Taibleson, G. Weiss, The molecular characterization of certain Hardy spaces, Astérisque 77 (1980) 67–149. [39] B.E. Viviani, An atomic decomposition of the predual of BMO(ρ), Rev. Mat. Iberoamericana 3 (1987) 401–425. [40] L. Yan, Classes of Hardy spaces associated with operators, duality theorem and applications, Trans. Amer. Math. Soc. 360 (2008) 4383–4408. [41] K. Yosida, Functional Analysis, sixth ed., Springer-Verlag, Berlin, 1978.
Journal of Functional Analysis 258 (2010) 1225–1246 www.elsevier.com/locate/jfa
Heat kernel bounds, ancient κ solutions and the Poincaré conjecture Qi S. Zhang Department of Mathematics, University of California, Riverside, CA 92521, USA Received 13 January 2009; accepted 2 November 2009
Communicated by L. Gross
Abstract We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3-dimensional ancient κ solutions to the Ricci flow. As an application, using the W entropy associated with the heat kernel, we give a different and much shorter proof of Perelman’s classification of backward limits of these ancient solutions. The method is partly motivated by Cao (2007) [1] and Sesum (2006) [27]. The current paper or Chow and Lu (2004) [6] combined with Chen and Zhu (2006) [4] and Zhang (2009) [31] lead to a simplified proof of the Poincaré conjecture without using reduced distance and reduced volume. © 2009 Elsevier Inc. All rights reserved. Keywords: Heat kernel bound; Ancient solutions; Ricci flow
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Proof of Theorem 1.1: the heat kernel bounds . . . . . . . . . . . 3. Applications to ancient solutions and the Poincaré conjecture Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.002
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1. Introduction The main goal of the paper is to establish certain Gaussian type upper bound for the heat kernel (fundamental solutions) of the conjugate heat equation associated with 3-dimensional ancient κ solutions to the Ricci flow. Heat kernel estimates have been an active area of research. When coupled with Ricci flow, various estimates can be found in [11], [24, Section 9], [22], and [30]. For example, in Section 9 of [24] Perelman proved a lower bound for the fundamental solution of the conjugate heat equation for general Ricci flow. So far an upper bound corresponding to this lower bound has been missing. Our result is a progress in this direction when the Ricci flow is a 3-dimensional ancient κ solution. One motivation of the work is that it induces a simpler proof of the Poincaré conjecture. The most difficult analytical parts of the proof can now be treated by one unifying theme: Perelman’s W entropy and related (log) Sobolev inequalities and heat kernel estimates. Let us explain the point in more detail. From Perelman’s original papers [24–26] and the works by Cao and Zhu [3], Kleiner and Lott [15] and Morgan and Tian [20], and Tao [29,28], it is clear that the bulk of the proof of the Poincaré conjecture is consisted of two items. One is the proof of local noncollapsing with or without surgeries, and the other is the classification of backward limits of ancient κ solutions. After these are done, one can show that regions where the Ricci flow is close to forming singularity have simple topological structure, i.e. canonical neighborhoods. Then one proceeds to prove that the singular region can be removed by finite number of surgeries in finite time. When the initial manifold is simply connected, the Ricci flow becomes extinct in finite time [26] (see also [8]). Thus the manifold is diffeomorphic to S3 , as conjectured by Poincaré. Besides the results and techniques by R. Hamilton, the main new tools Perelman used in carrying out the proof are several monotone quantities along Ricci flow. These include the W entropy, reduced volume and the associated reduced distance. In [24], Perelman first used his W entropy to prove local non-collapsing for smooth Ricci flows. However he then turned to the reduced volume (distance) to prove the classification and non-collapsing with surgeries. The W entropy is not used anymore. The reduced distance, not being smooth or positive in general, is one of the causes of the complexity of the original proof. It turns out that the W entropy is just the formula in a log Sobolev inequality (cf. [10] in the fixed metric case) and the monotonicity of the W entropy implies certain uniform Sobolev inequalities along the Ricci flow. Using this idea and being inspired by the last section of [25] and [15], we proved in [31] a stronger local non-collapsing result for Ricci flow with surgeries. The proof, without using reduced distance or volume, is short and seems more accessible. It also strengthens and clarifies the original result by doing analysis at one time level each time, thus avoiding the complication associated with surgeries. In the wake of this development, it would be desirable that the classification mentioned above can also be done by using the W entropy alone. Such a view was also expressed in [28] e.g. As one application of the main result of the paper, using the W entropy associated with the heat kernel, we give a different and much shorter proof of Perelman’s classification of backward limits of these ancient solutions. Thus, the current paper together with [31] and [4] (see explanation 4 paragraphs below) lead to a simplified proof of the Poincaré conjecture. Of course we still follow the framework by Perelman. However, much of the highly intensive analysis involving reduced distance and volume is now replaced by the study of the W entropy and the related uniform Sobolev inequalities and heat kernel estimates. Sobolev inequalities and heat kernels are familiar to many mathematicians. Therefore the current proof is more accessible to a wider audience. Besides, due to the relative simplicity and less assumptions on curvature,
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we hope the current technique can lead to better understanding of other problems for Ricci flow. We should mention that the reduced distance and volume are still needed for the proof of the geometrization conjecture. Specifically, they are needed, but only in the proof of Perelman’s no local collapsing Theorem II with surgeries. Let us outline the proof. In the next section we prove Theorem 1.1 concerning the bounds for the heat kernel of the conjugate heat equation. The proof follows the framework in Section 5 of [30]. There an upper bound in the case of Ricci flow with non-negative Ricci curvature was given. In the current situation, the ancient κ solutions provide better control on curvature and volume. These allow us to find a better Gaussian upper bound for the heat kernel. These bounds can be regarded as generalization of the heat kernel bounds of Li and Yau [19] in the fixed metric case. Using this heat kernel bound, in Section 3 we show that the W entropy associated with the heat kernel is uniformly bounded from below after certain scaling. After this done, we use Perelman’s monotonicity formula for the W entropy to prove the backward limit is a shrinking gradient Ricci soliton. This part of the arguments resembles that in the paper [1] and [27] where forward convergence results for normalized Ricci flow were proved. Finally one needs to prove universal non-collapsing for ancient κ solutions without reduced distance or volume. But this is already done in [4], even in certain more general 4-dimensional situation. We will just describe their proof. Now let us introduce the definitions and notations in order to present our result precisely. M denotes a complete compact, or non-compact Riemannian manifold, unless stated otherwise; g, Rij (or Ric) will be the metric and Ricci curvature; ∇, denote the corresponding gradient and Laplace–Beltrami operator; c with or without index denote generic positive constant that may change from line to line. If the metric g(t) evolves with time, then d(x, y, t) will denote the corresponding distance function; dg(x, t) or dg(t) denote the volume element under g(t); We will use B(x, r; t) to denote the geodesic ball centered at x with radius r under the metric g(t); |B(x, r; t)|s to denote the volume of B(x, r; t) under the metric g(s). We will still use ∇, to denote the corresponding gradient and Laplace–Beltrami operator for g(t), without mentioning the time t, when no confusion arises. We use the following concept of ancient κ solutions according to Perelman. Definition 1.1. A solution to the Ricci flow ∂t g = −2Ric is an ancient κ solution if it satisfies the following properties. 1. It is complete (compact or non-compact) and defined on an ancient time interval (−∞, T0 ], T0 0. 2. It has non-negative curvature operator and bounded curvature at each time level. 3. It is κ non-collapsed on all scales for some positive constant κ, i.e. Suppose that x0 ∈ M, t0 ∈ (−∞, T0 ]. Let P (x0 , t0 , r, −r 2 ) be the parabolic ball (x, t) d(x, x0 , t) < r, t0 − r 2 < t < t0 . Then M is κ non-collapsed at (x0 , t0 ) at scale r if |Rm| r −2 on P (x0 , t0 , r, −r 2 ) and vol(B(x0 , t0 , r)) κr 3 .
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For convenience, we take the final time T0 of the ancient solution to be 0 throughout the paper. The conjugate heat equation is u − Ru − ∂τ u = 0.
(1.1)
Here and always τ = −t. and R are the Laplace–Beltrami operator and the scalar curvature with respect to g(t). This equation, coupled with the initial value uτ =0 = u0 is well posed if M is compact or the curvature is bounded, and if u0 is bounded [11]. We use G = G(x, τ ; x0 , τ0 ) to denote the heat kernel (fundamental solution) of (1.1). Here τ > τ0 and x, x0 ∈ M. Existence of G was established in [11]. The main result of the paper is Theorem 1.1. (i) Let (M, g(t)) be an n-dimensional ancient κ solution of the Ricci flow. Suppose also that D0 for some D0 > 0 and for t ∈ [−T , 0]. Here T is any positive number or R(x, t) 1+|t| T = ∞. Then there exist positive numbers a and b depending only on n, κ and D0 such that the following holds. For all x, x0 ∈ M, there hold G(x, τ ; x0 , τ0 ) (τ −τa )n/2 and 0
G(x, τ ; x0 , τ0 )
a 2 e−bd (x,x0 ,t0 )/(τ −τ0 ) , √ |B(x, τ − τ0 , t0 )|t0
where τ = −t, τ0 = −t0 , τ > τ0 and t ∈ [−T , 0]. D0 for all t 0, namely (M, g(t)) is a type I ancient solution, (ii) In particular, if R(x, t) 1+|t| there exist positive numbers a1 and b1 depending only on κ and D0 such that the following holds. For all x, x0 ∈ M, and all τ = −t > 0, 1 a1 2 2 e−d (x,x0 ,t)/(b1 τ ) G(x, τ ; x0 , τ/2) n/2 e−b1 d (x,x0 ,t)/τ . n/2 a1 τ τ Remark. The full Gaussian lower bound in part (ii) of the theorem is not needed for √ the application in Section 3. One only needs the lower bound for one point in the ball B(x0 , b|t|, t) for some b > 1, which is a simple consequence of the upper bound. The Gaussian upper and lower bounds seem to be of interest that is independent of the Poicaré conjecture. For instance, Perelman [24] used heat kernel bounds to prove his pseudo locality theorem. In Section 9 of [24], a lower bound for the heat kernel was proved. However the upper bound is missing. In this sense, this paper is not just a reproof of a known result. 2. Proof of Theorem 1.1: the heat kernel bounds We divide the proof into three steps. The first two are for part (i) of the theorem. We always assume that all the time variables involved are not smaller than −T , so that the condition D0 holds. As mentioned in the introduction, the proof follows the framework of TheR(·.t) 1+|t| orem 5.2 in [30], where certain upper bound for G under Ricci flow with non-negative Ricci curvature was derived. Comparing with that case, we have two new ingredients coming from ancient κ solutions. One is the non-collapsing condition on all scales. The other is the bound
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on the scalar curvature. These allow us to prove a better bound. During the proof, there will be overlaps with [30]. They are here so that the paper is self contained. Without loss of generality we assume τ0 = 0 in G(x, τ ; x0 , τ0 ). It is convenient to work with the reversed time τ . Note that the Ricci flow is a backward flow with respect to τ and the conjugate heat equation is a forward heat equation with a potential term. Step 1. Since Ricci 0, it is well known (see Theorem 3.7 [14] e.g.) the following Sobolev inequality holds: Let B(x, r, t) be a proper subdomain for (M, g(t)). For all v ∈ W 1,2 (B(x, r, t)), there exists cn > 0 depending only on the dimension n such that
(n−2)/n v 2n/(n−2) dg(t)
cn r 2 2/n |B(x, r, t)|t
|∇v|2 + r −2 v 2 dg(t).
(2.1)
√ For our purpose, we only need to take r = c |t|, for c 1. By the assumption that R(x, t) D0 1+|t| and the κ non-collapsing property, we have
B(x, |t|, t) κD −n |t|n/2 . 0 t Therefore the above Sobolev inequality becomes v
2n/(n−2)
(n−2)/n cn D 2 dg(t) 2/n0 |∇v|2 + |t|−1 v 2 dg(t) κ
(2.2)
√ for all v ∈ W 1,2 (B(x, |t|, t)). Before moving forward, we would like to clarify a technical point in the definition of Perelman’s κ non-collapsing for ancient solutions, as given in Definition 1.1. The issue is whether the metric balls B(x, r, t) in the definition are required to be a proper subdomain of the manifold M. When M is non-compact, B(x, r, t) is always a proper subdomain so this issue is mute. Now one assumes that M is compact. Without requiring B(x, r, t) being a proper subdomain, if r is larger than the diameter of M, then B(x, r, t) is the whole manifold. In this case |B(x, r, t)|t cannot be greater than κr n for large r. So to be κ non-collapsed in such a large ball B(x, r, t), the curvature condition of κ non-collapsing must be violated, i.e. at some point in the parabolic ball |Rm| is greater than 1/r 2 . On the other hand, assume |Rm| 1/r 2 in the parabolic ball. If |B(x, r, t)| κr n , and if the Ricci curvature is non-negative, then by standard volume comparison theorem, the diameter of the manifold at time t is at least cr. Thus the κ non-collapsing condition implies certain restrictions on diameter and curvature of the compact manifold. In this paper, we take this explanation for Perelman’s κ non-collapsing, i.e. B(x, r, t) in the definition of κ ancient solutions is not required to be a proper subdomain. This seems to be the prevailing view √ in the literature. That is why the Sobolev imbedding (2.2) holds without requiring that B(x, |t|, t) is a proper subdomain of M. A natural question is: what happens when B(x, r, t) is implicitly assumed as a proper subdomain in the definition of κ solutions? Then we have to make this extra assumption throughout. This also an issue in Perelman’s original proof using reduced distance and volume. However either way does not affect the application for the Poincaré conjecture in the next section. The reason is the case of compact ancient solutions can be taken care of by other ways. See the proof of Theorem 3.1 Case 4.
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Another explanation is that, for a proof of Poincaré conjecture, one only needs to deal with ancient solutions resulting from blowing up finite time singularity. For these ancient solutions, ˜ g(s)) there exists desired lower bound for diameters. Indeed, suppose a Ricci flow (M, ˜ develops singularity at finite time S. By [31], for s ∈ [0, S), there is a uniform Sobolev inequality: for smooth function u |u|
n/(n−2)
dμ g(s) ˜
(n−2)/n
A
˜ +B 4|∇u|2 + Ru2 dμ g(s)
u2 dμ g(s) ˜ .
Here A and B are positive constants depending only on S and the initial manifold. Suppose, after blowing up the singularity, one obtains a compact ancient solution (M, g(t)), t 0. Then the constant B will drop out by the scaling. So, for any smooth function v on M, it holds
|v|n/(n−2) dμ g(t)
(n−2)/n
A
4|∇v|2 + Rv 2 dμ g(t) .
If R(x, t) D0 /|t|, we can take v = 1 to deduce |M|g(t) c|t|n/2 . Since the curvature is non-negative, the√classical volume comparison theorem shows that the diameter of M at time t is greater than c |t|. This is a desired lower bound. Next we show that, under the assumptions of the theorem, (M, g(t)) possesses a space time doubling property: the distance between two points at times t1 and t2 are comparable if t1 and t2 are comparable. The proof is very simple. Given x1 , x2 ∈ M, let r be a shortest geodesic connecting the two. Then ∂t d(x1 , x2 , t) = −
Ric(∂r , ∂r ) ds. r
Since the sectional curvature is non-negative, it holds Ric(x, t) R(x, t)
D0 . 1 + |t|
Therefore −
D0 d(x1 , x2 , t) ∂t d(x1 , x2 , t) 0. 1 + |t|
After integration, we arrive at: D |t1 /|t2 | 0 d(x1 , x2 , t1 )/d(x1 , x2 , t2 ) 1
(2.3)
for all t2 < t1 < 0. Note that the above inequality is of local nature. If the distance is not smooth, then one can just shift one point, say x1 , slightly and then obtain the same integral inequality by taking limits.
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Similarly, we have dg(t) = − 0 ∂t √ B(x, |t1 |,t1 )
√ B(x, |t1 |,t1 )
D0 R(y, t) dg(t) − 1 + |t|
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dg(t). √ B(x, |t1 |,t1 )
Upon integration, we know that the volume of the balls
B x, |t3 |, t4
t5
(2.4)
is all comparable for t3 , t4 , t5 ∈ [t2 , t1 ], provided that t1 and t2 are comparable. Let u be a positive solution to (1.1) in the region Qσ r (x, τ ) ≡ (y, s) y ∈ M, τ − (σ r)2 s τ, d(y, x, −s) σ r . Here r =
√
|t|/8 > 0, 2 σ 1. Given any p 1, it is clear that up − pRup − ∂τ up 0.
(2.5)
Let φ : [0, ∞) → [0, 1] be a smooth function such that |φ | 2/((σ − 1)r), φ 0, φ(ρ) = 1 when 0 ρ r, φ(ρ) = 0 when ρ σ r. Let η : [0, ∞) → [0, 1] be a smooth function such that |η | 2/((σ −1)r)2 , η 0, η 0, φ(s) = 1 when τ −r 2 s τ , φ(s) = 0 when s τ −(σ r)2 . Define a cut-off function ψ = φ(d(x, y, −s))η(s). Writing w = up and using wψ 2 as a test function on (2.5), we deduce 2 ∇ wψ ∇w dg(y, −s) ds + p Rw 2 ψ 2 dg(y, −s) ds − (∂s w)wψ 2 dg(y, −s) ds. (2.6) By direct calculation 2 ∇ wψ 2 ∇w dg(y, −s) ds = ∇(wψ) dg(y, −s) ds − |∇ψ|2 w 2 dg(y, −s) ds. Next we estimate the right-hand side of (2.6). − (∂s w)wψ 2 dg(y, −s) ds 1 1 = w 2 ψ∂s ψ dg(y, −s) ds + (wψ)2 R dg(y, −s) ds − (wψ)2 dg(y, −τ ). 2 2 Observe that ∂s ψ = η(s)φ d(y, x, −s) ∂s d(y, x, −s) + φ d(y, x, −s) η (s) φ d(y, x, −s) η (s). This is so because φ 0 and ∂s d(y, x, −s) 0 under the Ricci flow with non-negative Ricci curvature. Hence
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−
(∂s w)wψ 2 dg(y, −s) ds
1 w 2 ψφ d(y, x, −s) η (s) dg(y, −s) ds + 2 1 − (wψ)2 dg(y, −τ ). 2
(wψ)2 R dg(y, −s) ds (2.7)
Combing (2.6) with (2.7), we obtain, in view of p 1 and R 0,
∇(wψ)2 dg(y, −s) ds + 1 (wψ)2 dg(y, −τ ) 2 c w 2 dg(y, −s) ds. (σ − 1)2 r 2
(2.8)
Qσ r(x,τ )
By Hölder’s inequality (ψw)2(1+(2/n)) dg(y, −s)
(n−2)/n 2/n . (ψw)2 dg(y, −s) (ψw)2n/(n−2) dg(y, −s)
(2.9)
√ By the κ non-collapsing assumption, |B(x, |t|, t)|t κc2 r n . Since M has non-negative Ricci curvature, by the standard volume comparison theorem, the diameter of M at time t satisfies n ωn diam M, g(t) /2 B x, diam M, g(t) /2, t t κc2 r n √ where ωn is the volume of√the unit ball in Rn . Recall that r = |t|/8. Hence the diameter is a least constant multiple of c |t| for some c = cn > 0. Therefore by the distance doubling property (2.3), B(x, σ r, −s) is a proper subdomain of M, s ∈ [τ − (σ r)2 , τ ]. Here we just take the number 8 for simplicity. √ If it is not large enough, we just replace it by a sufficiently large number D and consider r = |t|/D instead. By the Sobolev inequality (2.2), it holds 2n/(n−2)
(ψw)
(n−2)/n ∇(ψw)2 + r −2 (ψw)2 dg(y, −s), dg(y, −s) c(κ, D0 )
for s ∈ [t − (σ r)2 , t]. Substituting this and (2.8) to (2.9), we arrive at the estimate w Qr (x,τ )
2θ
dg(y, −s) ds c(κ, D0 )
1 (σ − 1)2 r 2
θ
w dg(y, −s) ds 2
Qσ r (x,τ )
,
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with θ = 1 + (2/n). Now we apply the above inequality repeatedly with the parameters σ0 = 2, σi = 2 − ij =1 2−j and p = θ i . This shows an L2 mean value inequality c(κ, D0 ) sup u r n+2 Qr/2 (x,τ )
u2 dg(y, −s) ds.
2
(2.10)
Qr (x,τ )
This inequality clearly also holds if one replaces r by any positive number r < r since |B(x, r , t)| kcn |B(x, r, t)|(r /r)n cr n by the doubling condition for manifolds with nonnegative Ricci curvature. Then one can just rerun the above Moser’s iteration. From here, by a generic trick of Li and Schoen [18], applicable here since it uses only the doubling property of the metric balls, we arrive at the L1 mean value inequality sup Qr/2 (x,τ )
u
c(κ, D0 ) r n+2
u dg(z, −s) ds. Qr (x,τ )
We remark that the doubling constant is uniform since the metrics have non-negative Ricci curvature. √ Now we take u(x, τ ) = G(x, τ ; x0 , 0). Note that M u(z, s) dg(z, −s) = 1 and r = |t|. G(x, τ ; x0 , 0)
c(κ, D0 ) . |t|n/2
(2.11)
Step 2 (Proof of the Gaussian upper bound). We begin by using a modified version of the exponential weight method due to Davies [9]. Pick a point x0 ∈ M, a number λ < 0 and a function f ∈ C0∞ (M, g(0)). Consider the functions F and u defined by F (x, τ ) ≡ eλd(x,x0 ,t) u(x, τ ) ≡ eλd(x,x0 ,t)
G(x, τ ; y, 0)e−λd(y,x0 ,0) f (y) dg(y, 0).
(2.12)
Here and always τ = −t. It is clear that u is a solution of (1.1). By direct computation, ∂τ
F 2 (x, τ ) dg(x, t) = ∂τ
e2λd(x,x0 ,t) u2 (x, τ ) dg(x, t)
= 2λ
e2λd(x,x0 ,t) ∂τ d(x, x0 , t)u2 (x, τ ) dg(x, t) +
+2
e2λd(x,x0 ,t) u2 (x, τ )R(x, t) dg(x, t)
e2λd(x,x0 ,t) u − R(x, t)u(x, τ ) u(x, τ ) dg(x, t).
By the assumption that Ricci 0 and λ < 0, the above shows
∂τ
F 2 (x, τ ) dg(x, t) 2
e2λd(x,x0 ,t) uu(x, τ ) dg(x, t).
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Using integration by parts, we turn the above inequality into F 2 (x, τ ) dg(x, t)
∂τ
−4λ
e2λd(x,x0 ,t) u∇d(x, x0 , t)∇u dg(x, t) − 2
e2λd(x,x0 ,t) |∇u|2 dg(x, t).
Observe also
∇F (x, τ )2 dg(x, t) =
λd(x,x ,t) 2 0 ∇ e u(x, τ ) dg(x, t)
=
e
2λd(x,x0 ,t)
|∇u| dg(x, t) + 2λ 2
e2λd(x,x0 ,t) u∇d(x, x0 , t)∇u dg(x, t)
+ λ2
e2λd(x,x0 ,t) |∇d|2 u2 dg(x, t).
Combining the last two expressions, we deduce
∂τ
F (x, τ ) dg(x, t) −2 2
∇F (x, τ )2 dg(x, t) + λ2
e2λd(x,x0 ,t) |∇d|2 u2 dg(x, t).
By the definition of F and u, this shows
F (x, τ ) dg(x, t) λ 2
∂τ
2
F (x, τ )2 dg(x, t).
Upon integration, we derive the following L2 estimate
2τ
F 2 (x, τ ) dg(x, t) eλ
2τ
F 2 (x, 0) dg(x, 0) = eλ
f (x)2 dg(x, 0).
(2.13)
Recall that u is a solution to (1.1). Therefore, by the mean value inequality (2.10), the following holds c(κ, D0 ) u(x, τ ) 1+n/2 τ
τ
u2 (z, s) dg(z, −s) ds.
2
√ τ/2 B(x, |t|/2,−s)
By the definition of F and u, it follows that c(κ, D0 ) u(x, τ ) 1+n/2 τ
τ
2
√ τ/2 B(x, |t|/2,−s)
e−2λd(z,x0 ,−s) F 2 (z, s) dg(z, −s) ds.
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√ In particular, this √ holds for x = x0 . In this case, for z ∈ B(x0 , |t|/2, −s), there holds d(z, x0 , −s) |t|/2. Therefore, by the assumption that λ < 0, √ c(κ, D0 ) u(x0 , τ ) 1+n/2 e−λ 2|t| τ
τ
F 2 (z, s) dg(z, −s) ds.
2
√ τ/2 B(x0 , |t|/2,−s)
This combined with (2.13) shows that u(x0 , τ )2
c(κ, D0 ) λ2 τ −λ√2|t| e τ n/2
f (y)2 dg(y, 0),
i.e.
2 c(κ, D0 ) λ2 τ −λ√2|t| G(x0 , τ ; z, 0)e−λd(z,x0 ,0) f (z) dg(z, 0) e f (y)2 dg(y, 0). τ n/2
Now, we fix y0 such that d(y0 , x0 , 0)2 4t. Then it is clear that, by λ < 0 and the triangle inequality, λ −λd(z, x0 , 0) − d(x0 , y0 , 0) 2 when d(z, y0 , 0)
√ B(y0 , |t|,0)
√ |t|. In this case, the above integral inequality implies
√ 2 2 c(κ, D0 )eλd(x0 ,y0 ,0)+λ τ −λ 2|t| G(x0 , τ ; z, 0)f (z) dg(z, 0) f (y)2 dg(y, 0). τ n/2
Note that this inequality hold for all −T t < 0 and λ < 0. For an arbitrarily fixed t ∈ [−T , 0], we take λ=−
d(x0 , y0 , 0) βτ
with β > 0 sufficiently large. Since f is arbitrary, this shows, for some b > 0,
c(κ, D0 )e−bd(x0 ,y0 ,0) G (x0 , τ ; z, 0) dg(z, 0) τ n/2 2
√ B(y0 , |t|,0)
2 /τ
√ Hence, there exists z0 ∈ B(y0 , |t|, 0) such that G2 (x0 , τ ; z0 , 0)
c(κ, D0 ) 2 e−bd(x0 ,y0 ,0) /τ . √ 0 , |t|, 0)|0
τ n/2 |B(x
.
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In order to get the upper bound for all points, let us consider the function v = v(z, l) ≡ G(x0 , τ ; z, l). This is a solution to the conjugate of the conjugate equation (1.1), i.e. z G(x, τ ; z; l) + ∂l G(x, τ ; z, l) = 0,
∂l g = 2Ric.
Therefore, we can use Theorem 3.3 in [30], after a reversal in time. Note this theorem was stated only for compact manifolds. However, as remarked there, it is valid in the non-compact case whenever the maximum principle for the heat equation holds. This is the case since the curvature is uniformly bounded for κ solutions. Since the proof is quite short, we will present it in Appendix A. It is just a simple generalization of Hamilton’s first result in [12] to the Ricci flow case. Consequently, for δ > 0, C > 0, G(x0 , τ ; y0 , 0) CG1/(1+δ) (x0 , τ, z0 , 0)M δ/(1+δ) , where M = supM×[0,τ/2] G(x0 , τ, ·, ·). By Step 1, there exists a constant c(κ, D0 ) > 0, such that M
c(κ, D0 ) . τ n/2
Consequently G2 (x0 , τ ; y0 , 0)
c(κ, D0 ) c(κ, D0 ) 2 2 e−bd(x0 ,y0 ,0) /t e−bd(x0 ,y0 ,0) /t . √ √ 2 τ n/2 |B(x0 , |t|, 0)|0 |B(x0 , |t|, 0)|0
The last step holds since the Ricci curvature is non-negative. Since x0 and y0 are arbitrary, the proof of part (i) is done. Step 3. In this step, we prove the upper and lower bound for G(x, τ ; x0 , τ/2) in the case of type I ancient solution. The upper bound is already √ proved in view the distance and volume comparison result (2.3), (2.4) and the fact that |B(x, |t|, t)|t c(κ, D0 )|t|n/2 . So we just need to prove the lower bound. For a number β > 0 to be fixed later, the upper bound implies G2 (x, τ ; x0 , τ/2) dg(x, t) √ B(x0 , β|t|,t)
1 √ |B(x0 , β|t|, t)|t
2 G(x, τ ; x0 , τ/2) dg(x, t)
√ B(x0 , β|t|,t)
1 1− = √ |B(x0 , β|t|, t)|t
√ B(x0 , β|t|,t)c
2 G(x, τ ; x0 , τ/2) dg(x, t)
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1 1− √ |B(x0 , β|t|, t)|t
√ B(x0 , β|t|,t)c
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2 c(κ, D0 ) −bd(x0 ,y0 ,t)2 /t e dg(x, t) . τ n/2
Since the Ricci curvature is non-negative, one can use the volume doubling property to compute that √ B(x0 , β|t|,t)c
c(κ, D0 ) −bd(x0 ,y0 ,t)2 /t e dg(x, t) 1/2 τ n/2
provided √ that β is sufficiently large. Here we stress that all constants are independent of t. Since |B(x0 , β|t|, t)|t cn (β|t|)n/2 by standard volume comparison theorem, this shows G2 (x, τ ; x0 , τ/2) dg(x, t) √ B(x0 , β|t|,t)
c(κ, D0 ) . |t|n/2
√ Hence there exists x1 ∈ B(x0 , β|t|, t) such that G(x1 , τ ; x0 , τ/2)
c(κ, D0 ) . |t|n/2
For applications in Section 3, this lower bound is already sufficient. An √ inspection of the proof shows that actually for any λ ∈ [3/4, 4], it holds, for some xλ ∈ B(x0 , β|t|, t), G(xλ , λτ ; x0 , τ/2)
c(κ, D0 ) . |t|n/2
It is well known that such a lower bound implies the full Gaussian lower bound if one has a suitable Harnack inequality. Such Harnack inequality already exists. For the heat kernel, it is in Section 9 of [24]. For all positive solutions it is in Corollary 2.1(a) in [16] and [2]. Applying Corollary 2.1(a) in [16], we get 1 n [4|γ (s)|2 + (τ/4)2 R] ds 3 τ G x3/4 , τ ; x0 , τ/2 G(x, τ ; x0 , τ/2) , exp 0 4 τ 3/4 2(τ/4) where γ is a smooth curve on M such that γ (0) = x3/4 and γ (1) = x. Also |γ (s)|2 = g−l (γ (s), γ (s)), and l = 3τ/4 + sτ/4. This inequality together with the decay property of R and compatibility of distances conclude G(x, τ ; x0 , τ/2) This finishes the proof of the theorem.
c(κ, D0 ) −b1 d(x,x0 ,t)2 /τ e . |t|n/2
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3. Applications to ancient solutions and the Poincaré conjecture In this section we use Theorem 1.1 to give a different proof for Perelman’s classification result of backward limits of ancient κ solutions. Theorem 3.1 (Perelman). Let g(·, t) with t ∈ (−∞, 0] be a non-flat, 3-dimensional ancient κ solution for some κ > 0. Then there exist sequences of points {qk } ⊂ M and times tk → −∞, k = 1, 2, . . . , such that the scaled metrics gk (x, s) ≡ R(qk , tk )g(x, tk + sR −1 (qk , tk )) around qk ∞ topology. converge to a non-flat gradient shrinking soliton in Cloc Proof. We divide the proof into several cases. Case 1 is when the section curvature is zero somewhere and M is non-compact. Then Hamilton’s strong maximum principle for tensors show that M = M2 × R1 where M2 is a 2dimensional, non-flat ancient κ solution. According to Hamilton, M2 is either S 2 or RP 2 . So the theorem is already proved in this case. This case can also be covered in Case 4 below together. Case 2 is when the section curvature is zero somewhere and M is compact. Then, again using maximum principle, Hamilton (see Theorem 6.64 in [7] e.g.) showed that M is the metric quotient of R3 with the flat metric or that of S 2 × R1 . So the theorem is also proved in this case. Case 3 is when the sectional curvature is positive everywhere and M is a type II ancient solution, i.e. supt<0 |t|R(·, t) = ∞. In this case Hamilton [13] showed by a scaling argument and his matrix maximum principle that the backward limit is a steady gradient soliton. See also Theorem 9.29 in [7], in which a proof is given for the non-compact case. However the compact case can be proved in the same way with the κ non-collapsing assumption. So one can take a scaling limit to a shrinking gradient soliton. See Theorem 9.66 in [7] e.g. If the ancient solution arises from the blow-up of finite time type II singularity, then Hamilton [13] even proved that M is a steady gradient soliton. If M is compact, then it is well known that M is an Einstein manifold. Since the curvature is positive, M has to be S 3 . So there is only one case left. Case 4: M has positive sectional curvature and is of type I ancient solution. In the special case that M is compact and the ancient solution arises from blowing up finite time singularity, N. Sesum already proved the theorem in this case [27]. Actually Sesum proved a stronger result, namely, M is a shrinking gradient soliton. See also [3, p. 302] and the work of X.D. Cao [1]. For the non-compact case, a similar result was proved by A. Naber [21]. However it is not clear if the non-compact gradient soliton is flat. So we will assume that M is non-compact and of type I for the rest of the proof. In fact our proof works in both compact and non-compact cases by considering M × R. D0 By the k non-collapsing assumption and the bound R(·, t) 1+|t| , we can find a sequence τk → ∞ such that the following holds: The pointed manifolds (M, gk , yk ) with the metric gk ≡ τk−1 g(·, −sτk ) converge, in C0∞ sense, to a pointed manifold (M∞ , g∞ (·, s), y∞ ). Here s > 0.
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We aim to prove that g∞ is a gradient, shrinking Ricci soliton. Note that we are scaling by τk−1 . By the upper and lower bound on the scalar curvature, this scaling is equivalent to scaling by the scalar curvatures. We define, for x ∈ M and s 1, the functions n/2
uk = uk (x, s) ≡ τk G(x, sτk ; x0 , 0). Here G is the heat kernel of the conjugate heat equation and x0 is a fixed point. We choose yk = x0 in the scaled metrics above. By Theorem 1.1 (actually (2.11)), we know that uk (x, s) U0 uniformly for all k = 1, 2, . . . , x ∈ M and s in a compact interval. Here U0 is a positive constant. Note that uk is a positive solution of the conjugate heat equation under the metric on (M, gk (s)) i.e. gk uk − Rgk uk − ∂s uk = 0. We have seen that uk and Rgk are uniformly bounded on compact intervals of s in (0, ∞), and also the Ricci curvature is non-negative and the curvature tensors are uniformly bounded. The standard parabolic theory shows that uk is Hölder continuous uniformly with respect to gk . Hence α sense, modulo diffeowe can extract a subsequence, still called {uk }, which converges in Cloc α morphism, to a Cloc function u∞ on (M∞ , g∞ (s), y∞ ). Using integration by parts, it is easy to see that u∞ is a weak solution of the conjugate heat equation on (M∞ , g∞ (s)), i.e. (u∞ φ − R∞ u∞ φ + u∞ ∂s φ) dg∞ (s) ds = 0 for all φ ∈ C0∞ (M∞ × (−∞, 0]). Here R∞ is the scalar curvature of the limit manifold. By standard parabolic theory, the function u∞ , being bounded on compact time intervals, is a smooth solution of the conjugate heat equation on (M∞ , g∞ (s), y∞ ). We need to show that u∞ is not zero. One can even show that it is actually the fundamental solution of the conjugate heat equation with pole at y∞ (the image of the same x0 in the limiting manifold). Let u = u(x, τ ) = G(x, τ ; x0 , 0). We claim that for a constant a > 0 and all τ 1, u(x0 , τ )
a . τ n/2
Here is the proof. Define f by (4πτ )−n/2 e−f = u. By Corollary 9.4 in [24], which is a consequence of his differential Harnack inequality for fundamental solutions, we have, for τ = −t, 1 1 −∂t f (x0 , t) R(x0 , t) − f (x0 , t). 2 2τ Since R(x0 , t) c/τ , we can integrate the above from τ = 1 to get f (x0 , τ ) c +
f (x0 , 1) C. τ
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Here we have use the fact that f (x0 , 1) is bounded, by the standard short time bounds for G = G(x0 , 1; x0 , 0). This proves the claim. By definition of uk as a scaling of u, we know that uk (x0 , s) b > 0 for s ∈ [1, 4]. Here b is independent of k. Therefore u∞ (x0 , s) b > 0. The maximum principle shows u∞ is positive everywhere. A detailed proof of this corollary of Perelman’s can be found in [23]. In the proof, short time behavior of Perelman’s reduced distance is used at one stage. However, an inspection of the proof shows that one can use asymptotic formula for the heat kernel in the same paper instead. See the paper [17] e.g. Let us recall that Perelman’s W entropy for each uk is Wk (s) = W (gk , uk , s) =
s |∇fk |2 + Rk + fk − n uk dgk (s)
where fk is determined by the relation (4πs)−n/2 e−fk = uk ; and Rk is the scalar curvature under gk . By the uniform upper bound for uk , we know that there exists c0 > 0 such that fk = − ln uk −
n ln(4πs) −c0 2
for all k = 1, 2, . . . and s ∈ [1, 3]. Here the choice of this interval for s is just for convenience. Any finite time interval also works. Since M is non-compact, one needs to justify the integral in Wk (s) is finite. For fixed k, uk has a generic Gaussian upper and lower bound with coefficients depending on τk and curvature tensor and their derivatives, as shown in [11]. The manifold has non-negative Ricci curvature and bounded curvature. So the term fk uk which is essentially −uk ln uk is integrable. The term |∇fk |2 uk = |∇uk |2 /uk is integrable by Theorem 3.3 in [30], given in Appendix A. These together imply that Wk (s) is well defined. Since M uk dgk = 1, we know that Wk (s) −c0 − n
(3.1)
for all k = 1, 2, . . . and s ∈ [1, 3]. There is an alternative proof of the lower bound for Wk . Actually Wk (s) is uniformly bounded from below if uk is replaced by any v ∈ W 1,2 such that v 2 = 1. This can be seen since (M, gk (s), yk ), s ∈ [1, 3], has uniformly bounded curvature operator and is κ non-collapsed. Therefore, a uniform Sobolev inequality holds, which implies the lower bound of Wk (s). The later is nothing but a lower bound on the best constants of log Sobolev inequalities. By scaling it is easy to see that Wk (s) = W (g, u, sτk ), where u = u(x, l) = G(x, l, x0 , 0). According to [24], dWk (s) = −2s ds
2 Ricg + Hessg fk − 1 gk uk dgk (s) 0. k k 2s
(3.2)
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Note that the integral on the right-hand side is finite by a similar argument as in the case of Wk (s). So, for a fixed s, Wk (s) = W (g, u, sτk ) is a non-increasing function of k. Using the lower bound on Wk (s) (3.1), we can find a function W∞ (s) such that lim Wk (s) = lim W (g, u, sτk ) = W∞ (s).
k→∞
k→∞
Now we pick s0 ∈ [1, 2]. Clearly we can find a subsequence {τnk }, tending to infinity, such that W (g, u, s0 τnk ) W g, u, (s0 + 1)τnk W (g, u, s0 τnk+1 ). Since lim W (g, u, s0 τnk ) = lim W (g, u, s0 τnk+1 ) = W∞ (s0 ),
k→∞
k→∞
we know that lim W (g, u, s0 τnk ) − W g, u, (s0 + 1)τnk = 0.
k→∞
That is lim Wnk (s0 ) − Wnk (s0 + 1) = 0.
k→∞
Integrating (3.2) from s0 to s0 + 1, we use the above to conclude that s0 +1
lim
k→∞
2 1 s Ricgnk + Hessgnk fnk − gnk unk dgnk (s) ds = 0. 2s
s0
Therefore Ric∞ + Hess∞ f∞ −
1 g∞ = 0. 2s
So the backward limit is a gradient shrinking Ricci soliton. Finally we need to show the soliton is non-flat. We can assume the original ancient solution is not a gradient shrinking soliton. Otherwise there is nothing to prove. Hence, we know that Wk (s) < Wk (0) = W0 = 0 where W0 is the Euclidean W entropy with respect to the standard Gaussian. By the upper bound of uk , we know that the integrand in Wk (s), s ∈ [1, 3], is bounded from below by a negative constant. Replace the −n in the definition of Wk (s) by a sufficiently large constant a0 , if necessary. Applying Fatuo’s lemma on a sequence of exhausting domains, we find that W (g∞ , u∞ , s) Wk (s) < W0 = 0.
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If the gradient shrinking soliton g∞ were flat, it is known to be R3 . Note M∞ u∞ 1 by Fatuo’s lemma again. It is known that W (g∞ , u∞ , s) 0 on the Euclidean space for such a u∞ . We have reached a contradiction, which implies g∞ is not flat. 2 Remark. Case 4 with positive curvature tensor seems can also be dealt with by the method in [6]. There Chow and Lu actually constructed an embedded region of the flow, which is close to S 2 × R. They even do not need to assume the soliton is κ non-collapsed on all scales. Also the on diagonal lower bound of the fundamental solution G in the middle of the proof can be extended to full lower bound by the theorem in Appendix A. But we do not need it here. In the last part of the section, we discuss the ramification of the above method to the proof of the Poincaré conjecture. After the classification of the backward limits and κ non-collapsing with surgeries, the only part of Perelman’s proof of the Poincaré conjecture that requires the reduced distance and volume is the universal non-collapsing of ancient κ solutions. Interestingly, a different proof of this fact already exists in Section 3.2 of the paper of Chen and Zhu [4], where certain more general 4-dimensional result is proved (see the paragraph after the proof of Proposition 3.4 there). In the 3-dimensional case, the proof looks longer than Perelman’s original proof. However it is basically a reshuffling of certain arguments suggested by Perelman, all which are needed to prove the canonical neighborhood property for ancient κ solutions. In this sense, the proof of the universal non-collapsing is a by product of canonical neighborhood property for ancient κ solutions. Indeed, the canonical neighborhood property for ancient κ solutions can be proved exactly the same way without the universal non-collapsing property, except that the constants in the property depend on the non-collapsing constant κ. But this is enough to show that after a conformal change of metric using the scalar curvature function, the ancient solution is close to model manifolds which are universal non-collapsed. Therefore the former is also universal non-collapsed. Let us state the result and sketch the proof. The details can be found in [4]. See also [32, Chapter 7]. Proposition 3.1 (Perelman). There exists a positive constant κ0 with the following property. Suppose we have a non-flat, 3-dimensional ancient κ solution arising from finite time singularity of a Ricci flow, for some κ > 0. Then either the solution is κ0 non-collapsed on all scales or it is a metric quotient of the round 3 sphere. Proof. (Sketched as a special case of Chen and Zhu’s proof in Section 3.2, the statement after Proposition 3.4 [4].) Note we use an extra assumption that κ solution is arising from finite time singularity of a Ricci flow. This will make the proof more transparent since type II κ solution in this case is just steady gradient Ricci soliton as proved by Hamilton [13]. If the 3-dimensional M is compact, then they are explicitly known to be gradient solitons as mentioned in Cases 1–4 in the proof of the previous theorem. Anyway they are not needed in singularity analysis leading to the Poincaré conjecture. So we just need to prove that noncompact 3-dimensional κ ancient solutions are universal non-collapsed on all scales. The proof is divided into 3 steps. Step 1 (One proves the compactness of ancient κ solutions with any fixed κ > 0, i.e.). The set of non-flat 3-dimensional ancient κ solutions, for any fixed κ > 0, is compact modulo scaling in the following sense: for any sequence of such solutions and marking points in space time (xk , 0)
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∞ converging subsequence whose limit is also an ancient with R(xk , 0) = 1, one can extract a Cloc κ solution.
The proof is identical to that in [24], the theorem in Section 11.7. Note that no universal non-collapsing is needed here. This actually is the original order of proof by Perelman. Step 2 (One proves certain elliptic type estimates for the scalar curvature). There exist a positive constant η and a positive increasing function w : [0, ∞) → (0, ∞) with the following property. Let (M, gij (t)), −∞ < t 0, is a 3-dimensional ancient κ solution for a fixed κ > 0. Then (i) for every x, y ∈ M and t ∈ (−∞, 0], there holds R(x, t) R(y, t)w R(y, t)d 2 (x, y, t) ; (ii) for all x ∈ M and t ∈ (−∞, 0], there hold |∇R| ηR 3/2 (x, t),
|∂t R|(x, t) ηR 2 (x, t);
(iii) suppose for some (y, t0 ) in space time and a constant ζ > 0 there holds |B(y, R(y, t0 )−1/2 , t0 )|t0 ζ. R(y, t0 )−3/2 Then there exist a positive functions w depending only on ζ such that, for all x ∈ M, R(x, t0 ) R(y, t0 )w R(y, t0 )d 2 (x, y, t0 ) . The proof of statements (i) and (ii) is almost a carbon copy of Theorem 6.4.3 in [3] (3D case) or Proposition 3.3 (4D case) in [4], or the corresponding results in [15] and [20]. The one difference is that one uses κ non-collapsing assumption instead the universal non-collapsing that is being proved. Therefore the constant η and the function w may depend on κ. Part (iii) is the remark after Proposition 3.3 (4D case) in [4], which includes the 3 dimension case as a special situation. Its proof is a moderate refinement of that of statement (i), by keeping a careful track of constants. Step 3. For any point (x, t), one shows that either it is a center of the neck, or it lies in a compact manifold with boundary, called M . After scaling by scalar curvature at one of its boundary points, this manifold is close to a compact manifold of finite diameter and whose scalar curvature is bounded between two positive constants which are independent of the non-collapsing constant κ. This step follows Proposition 3.4 in [4] which is a 4-dimensional result that includes the 3 dimension one as a special case. They use a blow-up argument, taking advantage of the property that a boundary point of M is the centered of a ball which is 2 close to that of S 2 × R after scaling. Then they use (iii) in Step 2 to obtain the bounds on scalar curvature. The bounds depend only on the non-collapsing constant of S 2 × R. This means that after scaling by scalar curvature, every point on the ancient solution has a ball of fixed diameter that is close to a model manifold which is universal non-collapsed. Therefore ancient κ solution is also universal non-collapsed. 2
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Let us close by presenting the flow chart of a simplified proof of the Poincaré conjecture without reduced distance or volume. The details will appear in [32, Chapters 7–9]. Step 1. W entropy and its monotonicity [24]. See also [5,3,15,20]. Step 2. Local non-collapsing result via Step 1 [24]. See also [5,3,15,20]. Step 3. Getting ancient κ solutions by blowing up of singularity using Step 2 and Hamilton’s compactness theorem [24]. See also [5,3,15,20]. Step 4. (i) Showing the backward limits of ancient κ solutions are gradient shrinking solitons. Earlier work of Hamilton [13] for type II case and [6] or this paper for type I case. (ii) Universal non-collapsing of ancient κ solutions. Section 3.2 of [4]. (iii) Curvature and volume estimates for ancient solutions [24]. See also [5,3,15,20]. Step 5. Classification of gradient shrinking solitons [24]. See also [5,3,15,20]. Step 6. Canonical neighborhood property [24]. That is: regions of high scalar curvature resemble the ancient solution after appropriate scaling. See also [5,3,15,20]. Step 7. Surgery procedure, including properties of the standard solution [25]. See also [3,15, 20]. Step 8. Local κ non-collapsing with surgeries [31]. Step 9. Canonical neighborhood property with surgeries [25]. See also [3,15,20]. Step 10. Existence of Ricci flow with surgeries, i.e. proving there are finitely many surgeries within finite time [25]. See also [3,15,20] and [32, Chapter 9]. Step 11. Finite time extinction of Ricci flow on simply connected manifolds [26]. See also [8] and [20]. Acknowledgments I wish to thank Professors Huaidong Cao, Xiaodong Cao, Bennett Chow, and Xiping Zhu for their helpful suggestions, and the anonymous referee for helpful comments that improve the paper. Appendix A Here we state and prove Theorem 3.3 in [30], which was used at the end of Step 2 in the proof of Theorem 1.1. See also [2]. Theorem A.1. Let M be a compact or complete non-compact Riemannian manifold with bounded curvature and equipped with a family of Riemannian metric evolving under the forward Ricci flow ∂t g = −2Ric with t ∈ [0, T ]. Suppose u is any positive solution to u − ∂t u = 0 in M × [0, T ]. Then, it holds |∇u(x, t)| u(x, t)
1 M log t u(x, t)
for M = supM×[0,T ] u and (x, t) ∈ M × [0, T ].
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Moreover, the following interpolation inequality holds for any δ > 0, x, y ∈ M and 0 < t T : u(y, t) c1 u(x, t)1/(1+δ) M δ/(1+δ) ec2 d(x,y,t)
2 /t
.
Here c1 , c2 are positive constants depending only on δ. Proof. This is almost the same as that of Theorem 1.1 in [12]. By direct calculation
M M |∇u|2 u log − ∂t u log =− , u u u ∂i u∂j u 2 2 |∇u|2 = ∂i ∂j u − ( − ∂t ) 0. u u u The first inequality follows immediately from the maximum principle since t
M |∇u|2 − u log u u
is a subsolution of the heat equation. We remark that the maximum principle is valid since the curvature is uniformly bounded for κ solutions. One can find detailed justification of this in [5, vol. II, Chapter 14] e.g. To prove the second inequality, we set l(x, t) = log M/u(x, t) . Then the first inequality implies
√ ∇ l(x, t) 1/ t. Fixing two points x and y, we can integrate along a geodesic to reach
d(x, y, t) log M/u(x, t) log M/u(y, t) + . √ t
The result follows by squaring both sides.
2
References [1] Xiaodong Cao, Dimension reduction under the Ricci flow on manifolds with nonnegative curvature operator, Pacific J. Math. 232 (2) (2007) 263–268. [2] Xiaodong Cao, Richard Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials, arXiv:0807.0568, Geom. Funct. Anal., in press. [3] Huai-Dong Cao, Xi-Ping Zhu, A complete proof of the Poincaré and geometrization conjectures—Application of the Hamilton–Perelman theory of the Ricci flow, Asian J. Math. 10 (2) (2006) 165–492. [4] Bing-Long Chen, Xi-Ping Zhu, Ricci flow with surgery on four-manifolds with positive isotropic curvature, J. Differential Geom. 74 (2) (2006) 177–264. [5] Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, Jim Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, Lei Ni, The Ricci Flow: Techniques and Applications. I, II, Amer. Math. Soc., 2007.
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[6] Bennett Chow, Peng Lu, On the asymptotic scalar curvature ratio of complete type I-like ancient solutions to the Ricci flow on noncompact 3-manifolds, Comm. Anal. Geom. 12 (1–2) (2004) 59–91. [7] Bennett Chow, Peng Lu, Lei Ni, Hamilton’s Ricci Flow, Grad. Stud. Math., vol. 77, Amer. Math. Soc./Science Press, Providence, RI/New York, 2006. [8] Tobias H. Colding, William P. Minicozzi II, Estimates for the extinction time for the Ricci flow on certain 3manifolds and a question of Perelman, J. Amer. Math. Soc. 18 (3) (2005) 561–569. [9] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92, Cambridge University Press, Cambridge, 1990. [10] Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (4) (1975) 1061–1083. [11] C. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal. 12 (2002) 425– 436. [12] Richard S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1) (1993) 113–126. [13] Richard S. Hamilton, Eternal solutions to the Ricci flow, J. Differential Geom. 38 (1) (1993) 1–11. [14] Emmanuel Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lect. Notes, Amer. Math. Soc., 2000. [15] Bruce Kleiner, John Lott, Notes on Perelman’s papers, arXiv:math.DG/0605667v1, 2006. [16] Shilong Kuang, Qi S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal. 255 (4) (2008) 1008–1023. [17] Junfang Li, Xiangjin Xu, Notes on Perelman’s differential Harnack inequality, preprint. [18] Peter Li, Richard Schoen, Lp and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math. 153 (3–4) (1984) 279–301. [19] Peter Li, S.T. Yau, On the parabolic kernel of the Schödinger operator, Acta Math. 156 (1986) 153–201. [20] John W. Morgan, Gang Tian, Ricci Flow and the Poincaré Conjecture, Amer. Math. Soc. and Clay Institute, 2006. [21] Aaron Naber, Noncompact shrinking 4-solitons with nonnegative curvature, arXiv:0710.5579. [22] Lei Ni, Ricci flow and nonnegativity of sectional curvature, Math. Res. Lett. 11 (5–6) (2004) 883–904. [23] Lei Ni, A note on Perelman’s LYH inequality, Comm. Anal. Geom. 14 (2006) 883–905. [24] Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159. [25] Grisha Perelman, Ricci flow with surgery on three manifolds, arXiv:math.DG/0303109. [26] Grisha Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math/0307245. [27] Natasa Sesum, Convergence of the Ricci flow towards Ricci soliton, Comm. Anal. Geom. 14 (2006). [28] Terence Tao, Course blog at UCLA, http://terrytao.wordpress.com/2008. [29] Terence Tao, Perelman’s proof of the Poincaré conjecture: A nonlinear PDE perspective, arXiv:math/0610903. [30] Qi S. Zhang, Some gradient estimates for the heat equation on domains and for an equation by Perelman, Int. Math. Res. Not. IMRN 2006 (2006), Article ID 92314, 39 pp. [31] Qi S. Zhang, Strong noncollapsing and uniform Sobolev inequalities for Ricci flow with surgeries, Pacific J. Math. 239 (1) (2009) 179–200; Qi S. Zhang, Announcement: A uniform Sobolev inequality for Ricci flow with surgeries and applications, C. R. Math. Acad. Sci. Paris 346 (9–10) (2008) 549–552. [32] Qi S. Zhang, Sobolev Inequalities, Heat Kernels under Ricci Flow and the Poincaré Conjecture, CRC Press (in English), Science Press Beijing (in Chinese), in press.
Journal of Functional Analysis 258 (2010) 1247–1272 www.elsevier.com/locate/jfa
Strong regularizing effect of a gradient term in the heat equation with the Hardy potential ✩ Boumediene Abdellaoui a , Ireneo Peral b,∗ , Ana Primo b a Département de Mathématiques, Université Aboubekr Belkaïd, Tlemcen, Tlemcen 13000, Algeria b Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Received 14 January 2009; accepted 10 November 2009 Available online 18 November 2009 Communicated by C. Kenig To Lucio Boccardo on his 60th birthday with our gratitude and friendship
Abstract We deal with the following parabolic problem, ⎧ u p ⎪ ⎨ ut − u + |∇u| = λ 2 + f, u > 0 in Ω × (0, T ), |x| on ∂Ω × (0, T ), ⎪ ⎩ u(x, t) = 0 x ∈ Ω, u(x, 0) = u0 (x), where Ω ⊂ RN , N 3, is a bounded regular domain such that 0 ∈ Ω or Ω = RN , 1 < p 2, λ > 0 and f 0, u0 0 are in a suitable class of functions. For p > p∗ ≡ NN−1 , we will show that the above problem
has a solution for all λ > 0, f ∈ L1 (ΩT ) and u0 ∈ L1 (Ω). We prove also that p ∗ is optimal for the existence result. These results prove the strong regularizing effect of a gradient term in the problem studied in Baras and Goldstein (1984) [3]. The Cauchy problem is also studied. © 2009 Elsevier Inc. All rights reserved. Keywords: Semilinear heat equations; Optimal power for existence and nonexistence; Complete and instantaneous blow-up; Fujita type exponent
✩
Work partially supported by projects MTM2007-65018, MICINN, Spain and A-8174-07, AECI, M.A.E., Spain.
* Corresponding author.
E-mail addresses: [email protected] (B. Abdellaoui), [email protected] (I. Peral), [email protected] (A. Primo). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.008
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1. Introduction In the celebrated paper by P. Baras and J.A. Goldstein [3], it could be found an answer to some questions formulated by H. Brezis and J.L. Lions about the problem,
(BG)
⎧ λ ⎪ ⎪ ⎨ ut − u = 2 u + f |x| u(x, 0) = u0 (x) ⎪ ⎪ ⎩ u(x, t) = 0
if x ∈ Ω ⊂ RN , N 3, t > 0, λ ∈ R, if x ∈ Ω, u0 ∈ L2 (Ω), if x ∈ ∂Ω, t > 0,
where Ω is a bounded domain such that 0 ∈ Ω. We could summarize the results by Baras–Goldstein as follows. • If λ ΛN ≡ ( N 2−2 )2 , the problem (BG) has a unique global solution if
−α1
|x|
T u0 (x) dx < ∞,
Ω
|x|−α1 f dx dt < ∞,
0 Ω
with α the smallest root of α 2 − (N − 2)α + λ = 0. • If λ > ΛN , the problem (BG) has no (even local) solution for u0 0. Notice that ΛN = ( N 2−2 )2 is the optimal constant in the Hardy inequality, ΛN RN
u2 dx |x|2
|∇u|2 dx.
RN
As a consequence of the nonexistence result for λ > ΛN , a complete and instantaneous blowup result for solutions to the approximated problems is also obtained. Notice that the effect of the Hardy potential is related in a strong way to the value of λ, then we could say that is a spectral complete and instantaneous blow-up. On the other hand, the following parabolic problem
(P )
⎧ u p ⎪ ⎨ ut − u = |∇u| + λ 2 + f, u > 0 in Ω × (0, T ), |x| u(x, t) = 0 on ∂Ω × (0, T ), ⎪ ⎩ u(x, 0) = u0 (x), x ∈ Ω,
was studied in [1], where Ω ⊂ RN , N 3, is a bounded regular domain such that 0 ∈ Ω or Ω = RN , p > 1, λ > 0 and f 0, u0 0 are in a suitable class of functions. The main results obtained in [1] are the following: • There exists a critical exponent p+ (λ) such that for p p+ (λ), there is no solution for nontrivial data and for any λ, even 0 < λ ΛN . • p+ (λ) is optimal in the sense that, if p < p+ (λ) there exists solution for suitable data and for 0 < λ ΛN .
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This means that the effect of the reaction term, which is, in this case, a large power of the modulus of the gradient (the power depends on λ), reinforces the Hardy potential in such a way that the blow-up occurs even in the subcritical interval of λ. The Cauchy problem is also studied in [1] and, as a new feature for problem (P ), a Fujita exponent F (λ) < p+ (λ) is found. This is a major difference with the case λ = 0. In this work we will analyze the effect of a power of the gradient as an absorption term in the heat equation with a Hardy potential. Precisely we will consider the following problem, ⎧ u p ⎪ ⎨ ut − u + |∇u| = λ 2 + f |x| u(x, t) = 0 ⎪ ⎩ u(x, 0) = u0 (x)
in ΩT ≡ Ω × (0, T ), on ∂Ω × (0, T ), if x ∈ Ω,
(1.1)
where Ω is either an open bounded domain in RN such that 0 ∈ Ω, or Ω = RN , N 3, 1 < p 2 and λ > 0. We assume that f and u0 are nonnegative measurable functions under some summability hypotheses that will be precised. The behavior of the problem (1.1) is completely different to the previous ones. The main goals of this paper are to show these differences. We summarize here the main results obtained in the paper. • We find that p ∗ = NN−1 is the threshold to regularize the blow-up in the problem studied by Baras and Goldstein in [3], i.e., if p > p ∗ there exists a solution to problem (1.1) for all λ > 0, even for L1 (Ω) data, while if p p ∗ , problem (1.1) has no local positive solution for λ > ΛN independently of the data. The critical value p ∗ appears in the elliptic case, see [2]. • We will prove the regularizing effect of the gradient term in (1.1) even if λ ΛN , i.e., for p > p ∗ problem (1.1) has a solution for f ∈ L1 (ΩT ) and u0 ∈ L1 (Ω), without the integrability restriction as in [3]. • We also analyze the Cauchy problem. The paper is organized as follows. In Section 2 we recall some well known comparison results and a weak maximum principle obtained in [1]. For the reader convenience, we have included the proof of the latter in Appendix A. Also in Section 2, we derive some local estimates for a positive solution to (1.1). In Section 3 we prove the existence result for p > p ∗ and λ > 0. Notice that in this case no condition is imposed on the data f and u0 , apart from f ∈ L1 (ΩT ) and u0 ∈ L1 (Ω). In the same section, we consider also the case p < p ∗ and λ ΛN . We realize that in order to get the existence result, the summability condition on the data are the same as in [3]. Section 4 deals with the nonexistence result for problem (1.1), namely we show that problem (1.1) has no local positive solution if p p ∗ and λ > ΛN . The proof is based on some local a priori estimates and the Hardy inequality. These results show the optimality of the power p ∗ . Section 5 deals with the Cauchy problem, namely Ω = RN . In the case where p > p ∗ we will show that the Cauchy problem has a solution for any f ∈ L1loc (RN × (0, T )) and u0 ∈ L1loc (RN ). However in the case p p ∗ , we use some extra regularity conditions on f and u0 to ensure the existence of a solution. In Appendix A we will explain the proof of the pointwise convergence of the gradients and the proof of the mentioned comparison principle.
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2. Preliminaries In this paper we will use the following weak version of the maximum principle proved in [1]. The interest of this version is the minimal conditions imposed on the sub and the supersolution. For the reader convenience, we include the proof in Appendix A. Lemma 2.1 (Weak maximum principle). Let h(x, t) be a measurable function such that |h| ∈ N L2r0 ([0, T ]; L2p0 (Ω)) where p0 , r0 > 1 and 2p + r10 < 1. Assume that w(x, t) 0 verifies, 0 1,p1
(i) w ∈ C((0, T ); L1 (Ω)) ∩ Lr1 ([0, T ]; W0 (ii) w is a subsolution to the problem
(Ω)), where r1 , p1 1 such that
wt − w |h||∇w| w(x, t) = 0 w(x, 0) = 0
N 2p1
+ r11 >
in ΩT , on ∂Ω × (0, T ), in Ω.
N +1 2 ,
(2.1)
Then w ≡ 0. As a consequence, we get the following comparison result. 1,p
Corollary 2.2 (Comparison principle). Let u, v ∈ C((0, T ); L1 (Ω)) ∩ Lp ((0, T ); W0 (Ω)), for some p > 1, with |ut − u| ∈ L1 (ΩT ) and |vt − v| ∈ L1 (ΩT ). Consider a Caratheodory function H (x, t, s) such that, H (x, t, ·) ∈ C 1 (RN ) for all (x, t) ∈ ΩT with sups∈RN |∇s H (x, t, s)| = N h(x, t) ∈ L2r ([0, T ]; L2p (Ω)), for p, r > 1 and 2p + 1r < 1. Assume that u and v verify
ut − u H (x, t, ∇u) + f u(x, 0) = u0 (x)
in Ω, in Ω,
vt − v H (x, t, ∇v) + f v(x, 0) = v0 (x)
in Ω, in Ω,
(2.2)
where f ∈ L1 (ΩT ), u0 , v0 ∈ L1 (Ω) and v0 (x) u0 (x) in Ω. Then v u in ΩT . From the above comparison result we obtain that the approximated problem ⎧ ⎪ ⎪ ⎨ ut − u ±
|∇u|p =f 1 + α|∇u|p
u(x, t) = 0 ⎪ ⎪ ⎩ u(x, 0) = u0 (x)
in ΩT , on ∂Ω × (0, T ), in Ω,
(2.3)
where α > 0, f ∈ L1 (ΩT ) and u0 ∈ L1 (Ω), has a unique positive solution for 1 < p 2. Notice that if problem (1.1) has a supersolution, then using again the above comparison principle and following the compactness argument used in [7], we get the existence of a minimal solution (see details in [1]). +2 1 Corollary 2.3. Assume that 1 p < N N +1 and f ∈ L (ΩT ). Let u, v be two functions such that 1,p 1 u, v ∈ W (Ω), ut − u, vt − v ∈ L (ΩT ) and
B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
ut − u |∇u|p + f u(x, 0) = u0 (x)
in ΩT , in Ω,
vt − v |∇v|p + f v(x, 0) = v0 (x)
1251
in ΩT , in Ω,
(2.4)
where v u on ∂Ω × (0, T ), u0 , v0 ∈ L1 (Ω) and v0 (x) u0 (x) in Ω. Then v u in ΩT . Proof. Consider w = v − u. It is clear that w ∈ W 1,p (Ω), w 0 on ∂Ω × (0, T ), w0 (x) 0 in Ω and wt − w ∈ L1 (ΩT ). In order to conclude, it is sufficient to prove that w+ = 0. We have wt − w |∇v|p − |∇u|p in ΩT . Since 1 p <
N +2 N +1
< 2, it follows that ⎧ ⎨ wt − w a(p, x, t)|∇w| in ΩT , w 0 on ∂Ω × (0, T ), w0 (x) 0 in Ω, ⎩ w ∈ W 1,p (Ω), wt − w ∈ L1 (ΩT ),
with a(p, x, t) p|∇u|p−1 if p > 1 and a(p, x, t) = 1 if p = 1. Therefore, applying Kato’s inequality (see [11]) there results ⎧ (w ) − w a(p, x, t)|∇w |, + + ⎨ + t w+ (x, 0) = 0 in Ω, ⎩ 1,p w+ ∈ W0 (Ω). +2 2r 2q Since p < N N +1 , then a(p, x, t) ∈ L ([0, T ]; L (Ω)), for q, r > 1 and the previous comparison lemma, we conclude that w+ = 0. 2
N 2q
+
1 r
< 1. By using
We will use also the following well known strong maximum principle, in particular to get local behavior of our solutions. Lemma 2.4 (Strong maximum principle). Suppose that 1 < p 2 and let w ∈ C(0, T ; L2 (Ω)) ∩ L2 (0, T ; W01,2 (Ω)) be a solution to ⎧ ⎪ ⎪ ⎨ wt − w +
|∇w|p = f (x, t) a + |∇w|p
w(x, t) = 0 ⎪ ⎪ ⎩ w(x, 0) = w0 (x)
in ΩT ≡ Ω × (0, T ), on ∂Ω × (0, T ), if x ∈ Ω,
(2.5)
where f ∈ L∞ (ΩT ), w0 ∈ L∞ (Ω) with f 0 or w0 0 and a 0. Then w(x, t) > 0 in ΩT . Proof. Without loss of generality we can assume that a > 0. It is clear that 0 is a strict subsolution to (2.5), hence from the above comparison principle we obtain that w 0 and w ≡ 0. Notice that w solves wt − w + B(x, t)∇w = h(x, t) 0,
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∇w where B(x, t) = |∇w| a+|∇w|p . Since |B(x, t)| C for all (x, t) ∈ ΩT , then the Harnack inequality obtained in Theorem 1.2 from [13], allows us to conclude that w > 0. 2 p−2
As a direct application of the strong maximum principle, we obtain the following result that describes the local behavior of a solution close to x = 0 and t > 0. 1,p
Proposition 2.5. Let 1 < p 2 and u ∈ Lp (0, T ; W0 (Ω)) a supersolution to ⎧ u ⎪ ⎨ ut − u + |∇u|p λ 2 |x| ⎪ ⎩ u(x, t) = 0 u(x, 0) = u0 (x) 0
in Ω × (0, T ), on ∂Ω × (0, T ), in Ω,
then u ∈ / L∞ (ΩT ), namely for all t > 0 fixed, limx→0 u(x, t) = ∞. Proof. Fixed t1 > 0, by Lemma 2.4, there exist c, η > 0 and t1 < t2 < T such that u > c η in Bη (0) × (t1 , t2 ). Let β > 1 and we consider v(r, t) = α(t − t1 )β log |x| in Bη (0), with suit-
λc λu able α > 0 in such a way that vt − v + |∇v|p |x| 2 |x|2 in Bη (0) × (t1 , t2 ). In order to conclude, it is sufficient to prove that v < u in Bη (0) × (t1 , t2 ). Since
⎧ ⎨ vt − v + |∇v|p ut − u + |∇u|p (v − u) 0 ⎩ (v − u)(t1 ) = 0
in Bη (0) × (t1 , t2 ), on ∂Bη (0) × (t1 , t2 ), in Bη (0),
we consider w = v − u that satisfies ⎧ ⎨ wt − w + |∇v|p − |∇u|p 0 in Bη (0) × (t1 , t2 ), w0 on ∂Bη (0) × (t1 , t2 ), ⎩ in Bη (0). w(t1 ) = 0 Since 1 < p 2, it follows that wt − w p|∇v|p−2 ∇v∇w p|∇v|p−1 |∇w| in Bη (0) × (t1 , t2 ). Applying Kato’s inequality (see [11]), we conclude that (w+ )t − w+ p|∇v|p−1 |∇w+ | in Ω × (t1 , t2 ). Since |∇v|p−1 ∈ L∞ ((t1 , t2 ); Lq (Bη (0))) for q > N2 , then from Corollary 2.2, we conclude that w+ = 0. 2 We recall the following Hardy inequality for 1 < p < N that we will use in some proofs, ΛN,p RN
up dx |x|p
|∇u|p dx,
∀u ∈ C0∞ RN
(2.6)
RN
where ΛN,p = ( N p−p )p is the optimal constant which is not attained. See for instance [9] for details.
B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
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3. Existence result for λ > 0 and p > p ∗ We will consider the usual k-truncation function Tk (σ ), σ, |σ | k, Tk (σ ) = k sign(σ ), |σ | > k, and Gk (σ ) = σ − Tk (σ ).
(3.1)
The following estimates will be used in a systematic way. Define ⎧ q+1 s ⎨ q s q 1 q+1 q Ψk,q (s) = Tk (t) dt that is, Ψk,q (s) = 1 ⎩ q k q+1 q + (s − k)k q q+1
0
if s k,
(3.2)
if s > k.
We call σ Θk (s) = Ψk,1 (s) =
Tk (σ ) dσ,
then
sTk (s) Θk (s) and Θk (s) ks. 2
0
Next we prove the main result on the regularizing effect of the gradient term. Theorem 3.1. Assume that p > NN−1 , then problem (1.1) has a positive solution for all λ > 0, for all f ∈ L1 (ΩT ) and for all u0 ∈ L1 (Ω). Proof. Consider a sequence fn ∈ L∞ (ΩT ), un0 ∈ L∞ (Ω) such that fn ↑ f in L1 (ΩT ) and un0 ↑ u0 in L1 (Ω). Since 1 < p 2, then using existence result of [7] and the comparison principle in Corollary 2.2, there exists a minimal bounded positive solution un to the problems ⎧ un p ⎪ ⎪ ⎨ unt − un + |∇un | = λ |x|2 +
1 n
+ fn (x)
in ΩT , (3.3)
un > 0 in ΩT , un = 0 on ∂Ω × (0, T ), ⎪ ⎪
⎩ un ∈ L2 0, T ; W01,2 (Ω) ∩ L∞ (ΩT ). Using Tk (un ) as a test function in (3.3), it follows that
Θk un (x, t) dx +
Ω
T
T |∇Tk un | dx dt +
0 Ω
T =λ 0 Ω
Hence, since p >
N N −1 ,
un Tk un |x|2 +
1 n
|∇un |p Tk un dx dt
2
0 Ω
T dx dt +
fn (x)Tk un dx dt +
0 Ω
Θk un0 (x) .
Ω
applying Hölder, Young and Hardy inequalities, we get
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B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
Θk un (x, t) dx +
Ω
T
T |∇Tk un | dx +
0 Ω
T 0 Ω
εkλ ΛN,p
0 Ω
|un |p dx dt + λC() |x|p
εkλ
T
|∇Ψk,p un,p |p dx
2
T 0 Ω
1
dx dt + k f L1 (ΩT ) + u0 L1 (Ω) |x|p
|∇un |p dx dt + λkT C1 (ε) + k f L1 (ΩT ) + u0 L1 (Ω) .
(3.4)
0 Ω
Therefore, by (3.2),
Θk un (x, t) dx +
T
Ω
|∇un |p dx dt 0 Ω
Θk un (x, t) dx +
Ω
T
T |∇Tk un |2 dx dt + k 0 {un k}
0 Ω
Θk un (x, t) dx +
Ω
T
T |∇Tk un | dx +
T
|∇Ψk,p un,p |p dx + Cq |Ω|T
2
0 Ω
εkλ C ΛN
|∇un |p dx dt + Cq |Ω|T
0 Ω
|∇un |p dx dt + λkT C(ε) + k f L1 (ΩT ) + u0 L1 (Ω) .
0 Ω
Then, we conclude that: 1,p
(1) un u weakly in Lp (0, T ; W0 (Ω)), (2) Tk un Tk u weakly in L2 (0, T ; W01,2 (Ω)). Let Gk (s) = s − Tk (s) and define ψk−1 (s) = T1 (Gk−1 (s)), then ψk−1 (un )|∇un |p |∇un |p χ{un k} . As in [6], using ψk−1 (un ) as a test function in (3.3), there results that
1 2
ψk−1 (un ) 2 dx +
Ω
T
∇ψk−1 (un ) 2 dx dt +
0 Ω
T =
λ 0 Ω
un |x|2 +
1 n
+ fn (x) ψk−1 (un ) dx dt.
T ψk−1 (un )|∇un |q dx dt 0 Ω
B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
1255
∗
Since {un } is uniformly bounded in Lp (0, T ; Lp (Ω)), it follows that (x, t) ∈ ΩT , such that k − 1 < un (x, t) < k → 0 and (x, t) ∈ Ω, such that un (x, t) > k → 0 as k → ∞, uniformly in n. Moreover T λ 0 Ω
un |x|2 +
1 n
un |x|2 + n1
→
u |x|2
strongly in L1 (ΩT ), therefore we conclude that
+ fn (x) ψk−1 (un ) dx dt → 0 as k → ∞ uniformly in n.
Hence lim
k→∞ {ΩT ∩un k}
|∇un |p dx dt = 0,
uniformly in n.
(3.5)
Using Theorem A.1 in Appendix A we obtain that
Tk un → Tk u strongly in L2 0, T ; W01,2 (Ω) .
(3.6)
Finally, since
|∇un |p dx dt = ΩT
∇Tk (un ) p dx dt +
|∇un |p dx dt,
ΩT ∩{un k}
ΩT
then using (3.5), (3.6) and by Vitali’s lemma, it follows that |∇un |p → |∇u|p Then we have proved the existence result.
strongly in L1 (ΩT ).
2
Corollary 3.2. The same existence result holds if f is substituted by a regular measure with respect to the L2 (0, T ; W01,2 (Ω)) capacity, namely the existence result holds for all
f ∈ L1 (ΩT ) + L2 0, T ; W −1,2 (Ω) . We refer to [12] (see also [8]) for more details about the parabolic capacity. If p < NN−1 and λ ΛN , then as it is proved in [3] the existence result requires the data with some extra summability. More precisely, we have the next theorem.
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B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
Theorem 3.3. Assume that 1 < p NN−1 and 0 < λ ΛN . Let f and u0 be positive functions T √ such that 0 Ω f |x|−α1 dx dt < ∞, Ω u0 (x)|x|−α1 dx < ∞ where α1 = N 2−2 − ΛN − λ. 1,q +2 Then there exists u ∈ C(0, T ; L1 (Ω)) ∩ Lq (0, T ; W0 (Ω)), for all q < max{p, N N +1 }, a positive solution to problem (1.1). Proof. Define fn (x, t) = min{f (x, t), n} ∈ L∞ (ΩT ) and u0n (x) = min{u0 (x), n} ∈ L∞ (Ω). Consider λ ΛN . It is easy to check that there exists a unique positive solution to the problem −ϕ = λ
1 ϕ + |x|2 |x|2
in Ω, ϕ|∂Ω = 0.
It is clear that ϕ is a regular function in Ω\{0} and that ϕ(x) C|x|−α1 in a neighborhood of the origin. Notice that in the case λ < ΛN , ϕ ∈ W01,2 (Ω) and in the case λ = ΛN , α1 = N 2−2 and φ2 ϕ ∈ H (Ω), the completion of C0∞ (Ω) with respect to the norm φ2H = Ω (|∇φ|2 − ΛN |x| 2 ) dx. Let wn be the unique positive solution to
⎧ 1 ⎪ ⎪ ⎨ wt − w = λTn |x|2 w + fn (x) w(x, 0) = u0n ⎪ ⎪
⎩ w ∈ L2 0, T ; W01,2 (Ω) .
in ΩT , in Ω,
It is clear that wn is a supersolution to the problem ⎧
1 ⎪ p ⎪ un + fn (x) ⎪ unt − un + |∇un | = λTn ⎨ |x|2
in Ω × (0, T ), (3.7)
un (x, 0) = un0 (x), ⎪ ⎪ ⎪
⎩ 1,p un ∈ C 0, T ; L1 (Ω) ∩ Lp 0, T ; W0 (Ω) .
Since 0 is a strict subsolution, then we claim that problem (3.7) has a minimal solution un such that un un+1 . To prove the claim we use an iteration argument. Fixed n 1, we consider the next approximated problem ⎧
(n) ⎪ |∇vk |p 1 ⎪ (n) (n) (n) ⎪ ⎪ vkt − vk + vk + fn (x) = λTn ⎪ 2 (n) p 1 ⎨ |x| 1 + k |∇vk | (n) ⎪ v (x, 0) = un0 (x), ⎪ k ⎪ ⎪
⎪ (n) ⎩ vk ∈ C 0, T ; L2 (Ω) ∩ L2 0, T ; W01,2 (Ω) .
in Ω × (0, T ),
It is clear that vk ∈ L∞ (ΩT ), therefore using the comparison principle in Corollary 2.2, it fol(n) (n) (n) (n+1) (n) . It is not difficult to see that vk L2 (0,T ;W 1,2 (Ω)) lows that vk vk+1 wn and vk vk (n)
(n)
0
C(n), thus vk ↑ un in L1 (ΩT ) as k → ∞. Hence using the compactness argument as in [7], (n) (n) the fact that p < 2 and the monotonicity of the sequence {vk }, we get vk → un strongly in
B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
1257
1,p
Lp (0, T ; W0 (Ω)). It is clear by construction that un is the minimal solution of (3.7) and that un un+1 . Thus the claim follows. Using the main hypotheses on f and u0 , we can take ϕ as a test function in (3.7), then
T fn (x, t)ϕ dx dt < C,
uniformly in n.
un0 (x)ϕ(x) dx < C,
0 Ω
Ω
Therefore, T
T
un (x, T )ϕ dx + Ω
|∇un | ϕ dx dt + λ p
0 Ω
T
0 Ω
un dx dt |x|2
f (x, t)ϕ dx dt + 0 Ω
u0 ϕ(x) dx C. Ω
Hence, in particular, we conclude that T 0 Ω
un dx dt C, |x|2
uniformly in n.
Taking Tk un as a test function in (3.7), we have that
Θk un (x, T ) dx +
Ω
T
T |∇Tk un | dx dt +
0 Ω
T λk 0 Ω
|∇un |p Tk un dx dt
2
0 Ω
un dx dt + k |x|2
T f (x) dx dt + 0 Ω
Θk u0n (x) dx.
Ω
T Thus k1 0 Ω |∇Tk un |2 dx dt C and Tk un Tk u weakly in L2 (0, T ; W01,2 (Ω)). In particular, for k = 1, we obtain that T
T |∇un | dx dt
0 Ω
T |∇T1 un | dx dt + C(p, Ω, T ) +
|∇un |p T1 un dx dt C,
2
p
0 Ω
0 Ω 1,p
therefore un u weakly in Lp (0, T ; W0 (Ω)). un 1 Since {un } is an increasing sequence and |x| 2 is bounded in L (ΩT ), it follows that
un |x|2
→
u |x|2
strongly in L1 (ΩT ). To finish we have just to prove the strong convergence of |∇un |p to |∇u|p in L1 (ΩT ). This follows using the same computation as in the proof of Theorem 3.1 and the convergence result of Appendix A. 2
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4. Nonexistence and blow-up results: optimality of p ∗ ≡
N N −1 .
In this section we will use the concept of very weak solution which, roughly speaking, is the more general setting for which the equation has a meaning in the distributional sense. Definition 4.1. We say that u ∈ C((0, T ); L1loc (Ω)) is a very weak supersolution (subsolution) |u| 1 1 1 ∞ p to problem (1.1) if |x| 2 ∈ Lloc (ΩT ), |∇u| ∈ Lloc (ΩT ), f ∈ Lloc (ΩT ) and for all φ ∈ C0 (Ω × (0, T )) such that φ 0, we have that T
T
T (−φt − φ)u dx dt +
0 Ω
|∇u| φ dx dt () p
0 Ω
0 Ω
u λ 2 + cf φ dx dt. |x|
(4.1)
If u is a very weak super and subsolution, then we say that u is a very weak solution. Notice that the concept of very weak solution is local in nature. The goal of this section is to prove a nonexistence result in the very weak sense given by Definition 4.1. According with such very weak sense, we will use in the proof local arguments. The main theorem in this section is the following. Theorem 4.2. Assume that p p ∗ ≡ positive very weak solution.
N N −1 .
If λ > ΛN =
(N −2)2 , 4
then problem (1.1) has no
Proof. Without loss of generality, we will assume that u˜ 0 = 0 and f is a nonnegative function such that f ∈ L∞ (ΩT ). We argue by contradiction. Suppose that problem (1.1) has a nonnegative very weak solution u˜ for some λ > ΛN . By the comparison principle in Lemma 2.1 and Corollary 2.2, we can construct a solution u in Ω1 × (0, T ). Thanks to the regularity result in [5], 1,q u ∈ Lr (0, T ; Wloc (Ω)), with 1 r 2, 1 q NN−1 and Nr + Nq > N + 1. Since 1 < p 2, then using the strong maximum principle proved above and by Proposition 2.5, there exists Br (0) × (t1 , t2 ) ⊂ Ω1 × (0, T1 ) such that u > c 1 in Br (0) × (t1 , t2 ) ⊂ 2 Ω1 × (0, T1 ). For fixed η r to be chosen later, consider φ ∈ C0∞ (Bη (0)). Using φu as a test function in (1.1) and integrating in Bη (0) × (t1 , t2 ), it follows that
log u(x, t2 ) |φ|2 dx −
t2 t1 Bη (0)
Bη (0)
t2
|∇u|2 φ 2 dx dt + 2 u2
t2
φ∇φ ∇u dx dt u
t1 Bη (0)
|∇u|p φ 2 dx dt u
+ t1 Bη (0)
t2
λ t1 Bη (0)
φ2 dx dt. |x|2
Let us analyze the left-hand side of previous inequality (4.2) term by term.
(4.2)
B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
log u(x, t2 ) |φ|2 dx
2∗
|φ| dx
Bη (0)
2∗ 2
Bη (0)
S
−1
N log u(x, t1 ) 2 dx
1259
2
N
Bη (0)
N log u(x, t2 ) 2 dx
|∇φ| dx 2
Bη (0)
2
N
,
Bη (0)
where S is the optimal constant in the Sobolev inequality. N Since u ∈ C([0, T1 ]; L1 (Ω1 )), then Bη (0) (log u(x, t2 )) 2 dx → 0 as η → 0. Hence for all ε > 0, there exists η > 0 such that
log u(x, t2 ) |φ|2 dx ε |∇φ|2 dx. Bη (0)
Bη (0)
Moreover, t2
|∇u|p φ 2 dx dt = u
t1 Bη (0)
t2
|∇u| |∇u|p−1 φ 2 dx dt u
t1 Bη (0)
1 ε0−2 2
t2
|∇u|2 2 1 φ dx dt + ε02 2 u2
t1 Bη (0)
t2 |∇u|2(p−1) φ 2 dx dt, t1 Bη (0)
where ε0 is a positive number that we will choose later. On the other hand we have t2
φ∇φ ∇u dx dt ε12 u
2 t1 Bη (0)
t2
φ 2 |∇u|2 dx dt + ε1−2 u2
t1 Bη (0)
t2 |∇φ|2 dx dt. t1 Bη (0)
Hence it follows that t2 λ t1 Bη (0)
T ∇u 2 2 φ2 1 −2 2 dx dt − 1 − ε1 − ε0 u φ dx dt 2 |x|2 0 Bη (0)
1 + ε02 2
t1 |∇u|
2(p−1) 2
t1 Bη (0)
+ε
φ dx dt
+ ε1−2 T
|∇φ|2 dx dt Bη (0)
|∇φ|2 dx.
Bη (0)
For 1 > ε1 > 0 verifying (ε1−2 T + ε)−1 λ > ΛN , we pick ε0 small enough such that (1 − ε12 − 1 −2 2 ε0 ) 0. Therefore we conclude that
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B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
−2
−1 ε1 T + S −1 ε λ
Bη (0)
−1 φ2 1 dx e02 ε1−2 t2 + S −1 ε 2 2 |x| +
t2 |∇u|2(p−1) φ 2 dx dt t1 Bη (0)
|∇φ|2 dx.
Bη (0)
We deal now with the mixed term,
t2
2∗
|∇u|2(p−1) φ 2 dx
|φ| dx
t1 Bη (0)
2∗ t2 t1
Bη (0)
S
−1
|∇u|N (p−1) dx
N
N
dt
Bη (0)
t2
2 N
2q |∇u| dx q
t1
2
2
2
1 N q
2 |∇φ|2 dx
dt
Bη (0)
N
,
Bη (0)
where S is the classical Sobolev constant and q = N (p − 1). • If p < NN−1 then N (p − 1) < NN−1 . Moreover, since p 2, we have 2q N 2. In order to 2N N +3 N +3 satisfy q > N + 1, it is sufficient that q < N +1 . Since for N 3, min{ N +1 , NN−1 } = NN−1 ,
it follows that q < NN−1 , and then classical regularity of entropy solution allows (see for instance [4]) us to conclude that t2
2q |∇u| dx q
t1
• For p =
N N −1 ,
1 N q
dt → 0 as η → 0.
Bη (0)
the summability of the solution directly gives us, t2
2q |∇u| dx q
t1
1 N q
dt
as η → 0.
Bη (0)
Hence we can choose ε0 and ε1 such that
−1 1 2 −2 e0 ε1 T + S −1 ε S −1 2
T |∇u| dx 0
2
2q q
1 N q
N
dt
Bη (0)
Then there exists η > 0 small enough and ε0 , ε1 < 1 such that φ2 dx |∇φ|2 dx, λ1 |x|2 Bη (0)
Bη (0)
with λ1 > ΛN , a contradiction with the Hardy inequality.
2
→ 0 as η → 0.
B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
1261
5. Cauchy problem In this section we consider the case of Cauchy problem, namely we assume that Ω ≡ RN . Consider the following problem ut − u + |∇u|p = λ
u +f |x|2
in RN , t > 0, λ > 0,
u(x, 0) = u0 (x)
in RN .
(5.1)
The main goal of this section is to prove the existence of a local solution under minima conditions of the data. These conditions will depend on the value of p. We begin by the more general case, namely we will assume that p > NN−1 . Hence we have the next theorem. Theorem 5.1. Let p > NN−1 , u0 (x) ∈ L1loc (RN ) and f ∈ L1loc (RN × (0, T )). Assume that λ > 0, then for all T > 0 there exists a solution u ∈ C(0, T ; L1 (RN )) ∩ Lp (0, T ; W 1,p (RN )) to the Cauchy problem (5.1). Proof. Let Bn be the ball in RN with radius n and centered at the origin. We consider
1,p un ∈ C 0, T ; L1 (Bn ) ∩ Lp (0, T ), W0 (Bn ) , the solutions to the following problems, ⎧ u p ⎪ ⎪ ⎨ unt − un + |∇un | = λ |x|2 + f
∀T > 0,
in Bn , t > 0,
un (x, 0) = u0 (x) ⎪ ⎪ ⎩ un (x, t) = 0
in Bn , t > 0, on ∂Bn , t > 0,
(5.2)
such that un un+1 . Let φ ∈ C0∞ (RN ) with φ 0, then we fix n large enough such that D ≡
p Supp φ Bn (0). Using Tk un φ p as a test function in (5.2), with p = p−1 , it follows that ∂ ∂t
p
φ Θk (un ) dx + Bn
φ |∇Tk un | dx + a
Tk (un )φ a−1 ∇un ∇φ dx
2
Bn
+
p
Bn
p
Tk (un )φ |∇un |p dx Bn
λ
Bn
un Tk (un )φ p dx + k |x|2
f φ p dx.
Bn
Notice that Tk (un )φ p −1 ∇un ∇φ dx ε φ p |∇un |p Tk (un ) dx + C(ε)k |∇φ|p dx. Bn
Hence we conclude that
Bn
RN
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B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
∂ ∂t
p
φ Θk (un ) dx + Bn
p
Bn
λ Bn
un Tk (un |x|2
)φ p
p
φ p ∇Ψk,p (un ) dx
φ |∇Tk un | dx + C 2
Bn
dx + k(f, T , φ),
where Ψk,p is defined in (3.2). Notice that
Bn
φ p |∇Tk un |2 dx =
φ p |∇Tk un |2 dx D
p
φ p ∇Ψk,p (un ) dx =
Bn
D
2
φ p ∇ φ p Tk un dx − C(k, D, φ),
D
p
φ p ∇Ψk,p (un ) dx
C
p φ ∇ φ p Ψk,p (un ) dx − C(k, D, φ) p
D
φ p Θk un (x, t) dx.
D
Then ∂ ∂t
φ p Θk (un ) dx + C1 D
λ D
p
∇ φ Tk un 2 dx + C2
D
un Tk (un |x|2
)φ p
p
∇ φ Ψk,p (un ) p dx
D
φ p Θk un (x, t) dx + C(k, f, T , φ, D).
dx + C D
Using the definition of Ψ and by Hardy inequality we obtain
p
∇ φ Ψk,p (un ) p dx Λp
D
Since p >
Bn
N N −1 ,
(φ p Ψk,p (un ))p dx C |x|p
D
(φ p un )p dx − C(k, φ, D). |x|p
it follows that D
Tk (un )φ p un dx ε |x|p
D
ε D
(φ p un )p dx + C(ε, k) |x|p
D
1
dx |x|p
p
(φ un )p dx + C(ε, k, D). |x|p
Thus ∂ ∂t
φ p Θk (un ) dx C D
D
φ p Θk un (x, t) dx + C(k, f, T , φ, D).
B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
Denote Yn (T ) =
T 0
Dφ
p Θ (u ) dx dt. k n
1263
Integrating in [0, T ], we have
Yn (t) CYn (t) + C. Therefore, T Y (T ) e
CT
e−CT C(T ) dx + eCT Y (0),
with Y (0) =
φ p Θk (u0 ) dx < ∞.
RN
0
Hence we conclude that Θk (un ) is bounded in L1loc (RN ) uniformly in n. Hence the above computation allows us to conclude that T
uniformly in n,
∇Tk (un ) 2 φ p dx dt C
uniformly in n,
0 D
T
un φ p dx C |x|2
0 Ω
T
|∇un |p φ p dx dt C
uniformly in n.
0 Ω 1,p
Thus there exists u ∈ C(0, T ; L1loc (RN )) ∩ Lp (0, T , Wloc (RN )) such that 1,2 L2 (0, T , Wloc (RN ))
un |x|2
→
u |x|2
strongly
and un → u strongly in L1loc (RN × (0, T )), Tk (un ) → Tk (u) strongly in 1,p in Lp (0, T , Wloc (RN )). To get the existence result we have just to show that |∇un |p → |∇u|p strongly in L1loc (RN × (0, T )), which follows using the same argument as in the proof of Theorem 3.1 and the local convergence result Theorem A.1 in Appendix A. Hence we conclude. 2 Now in the case where p
N N −1 ,
Theorem 5.2. Assume that p (1) (2)
we have the following existence result local in time.
N N −1 ,
and let u0 , f be nonnegative functions such that:
−α1 (λ) u (x) dx < ∞ and T1 −α1 (λ) f (x, t) dx dt < ∞ for some r > 0. 0 B0 (r) |x| 0 B0 (r) |x| 2 2 |u0 (x)|e−c0 |x| C0 and |f (x, t)|e−c1 |x| C1 as |x| → ∞ with c0 , c1 > 0, C0 , C1 > 0.
Then there exists T > 0 and
1,p u ∈ C 0, T ; L1loc RN ∩ Lp 0, T ; Wloc RN a local positive solution to (5.1).
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Proof. Define N
vλ (x, t) = t α1 (λ)− 2 |x|−α1 (λ) e−
|x|2 4t
,
(5.3)
from [1] we know that vλ solves (vλ )t − vλ − λ
vλ =0 |x|2
in RN × (0, ∞).
Fixed T > 0 to be chosen later, we define wλ (x, t) = vλ (x, t +T ). Then wλ ∈ C(0, ∞; L2 (RN ))∩ L2 (0, ∞; W 1,2 (RN )) and wλ solves ⎧ w ⎨ (wλ )t − wλ = λ λ in RN × (0, ∞), |x|2 ⎩ |x|2 N wλ (x, 0) = w0 (x) ≡ T α1 (λ)− 2 |x|−α1 (λ) e− 4T . Let wn be the solution to the problem ⎧ wn ⎪ ⎪ ⎨ (wn )t − wn = λ |x|2
in Bn (0) × (0, ∞),
wn (x, t) = 0 ⎪ ⎪ ⎩ wn (x, 0) = w0 (x)
on ∂Bn (0) × (0, ∞),
(5.4)
in R . N
It is clear that {wn } is an increasing sequence in n and wn ↑ wλ in L2 (0, ∞; W 1,2 (RN )). For t < T , we set w˜ n (x, t) = wn (x, T − t), then using w˜ n as a test function in (5.2), we get
T1 un (x, T1 )w˜ n (x, T1 ) dx +
Bn
|∇un |p w˜ n dx dt 0 Bn
T1 f w˜ n dx dt +
= 0 Bn
u0 w˜ n (x, 0) dx, Bn
where T1 < T . Hence it follows that T1
un (x, T1 )wn (x, T − T1 ) dx + Bn
|∇un |p wn (x, T − t) dx dt 0 Bn
T1 f wn (x, T − t) dx dt +
= 0 Bn
u0 wn (x, T ) dx. Bn
B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
1265
Then for T < min{ 2c10 , 2c11 }, using the hypotheses on f and u0 and by monotonicity in n, we obtain that
T1 f w(x, T − t) dx dt + 0 RN
u0 w(x, T ) dx < ∞.
RN
Thus we conclude that ⎧ ⎪ ⎪ un (x, T1 )wn (x, T − T1 ) dx C ⎪ ⎪ ⎪ ⎪ ⎨Bn T1 ⎪ ⎪ ⎪ ⎪ |∇un |p wn (x, T − t) dx dt C ⎪ ⎪ ⎩
uniformly in n, (5.5) uniformly in n.
0 Bn
Therefore there exists a nonnegative function u such that un ↑ u a.e. in RN × (0, T ), T1 |∇u|p w(x, T − t) dx dt C 0 RN
and un (·, T1 )wn (·, T − T1 ) → u(·, T1 )wλ (·, T − T1 ) strongly in L1 (RN ) for all T1 < T . To esun timate the term |x| 2 , we fix 0 < μ < λ and consider the function vμ (x, t) as in (5.3) where λ is substituted by μ. Using the same computation as above, taking advantage of estimates (5.5), it follows that T1 0 Bn
un wn,μ (x, T − t) dx dt C |x|2
uniformly in n,
where wn,μ is the solution to problem (5.4), with λ substituted by μ. Hence, as n → ∞, it follows that T1 0
RN
|x|2 u α1 (λ)− N2 −α1 (μ) − 4(T −t) (T − t) |x| e dx dt C |x|2
It is clear from the above computation that
un |x|2+α1 (μ)
→
u |x|2+α1 (μ)
for all T1 < T .
strongly in L1loc (RN × (0, T )).
To obtain that u solves (5.1) we have just to prove the strong convergence of {|∇un |p } to |∇u|p in L1loc (RN × (0, T )) which follows using Vitali’s lemma and the same computation as in the proof of Theorem 3.1 and the local convergence result of Theorem A.1 in Appendix A. 2
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Appendix A A.1. Pointwise convergence of the gradients In this part of Appendix A we prove the pointwise convergence of the gradients used in the proof of Theorem (3.1). Since in general we need the local strong convergence of the gradient, then the case of the Cauchy problem follows using the same argument. Therefore, for fixed n ∈ N and k 1, we consider the truncated problems ⎧ un p ⎪ + fn ⎪ ⎨ unt − un + |∇un | = λ 1 2 n + |x| un (x, t) = 0 ⎪ ⎪ ⎩ un (x, 0) = u0n
in Ω1 × (0, T ), on ∂Ω1 × (0, T ), if x ∈ Ω1 ,
(A.1)
where Ω1 is any bounded domain with Ω1 Ω, λ 0, u0n ↑ u0 in L1 (Ω) and fn ↑ f in L1 (ΩT ). From the computation of the proof of Theorem 3.1 we get the existence of 1,2 un u 2 u ∈ C([0, T ); L1 (Ω)) such that |x| 2 → |x|2 , Tk (un ) Tk (u) weakly in L (0, T ; W0 (Ω)) and 1,p
un u weakly in Lp (0, T ; W0 (Ω)). It is clear that T |∇un |q dxdt < C,
q<
0 Ω1
N +2 uniformly in n. N +1
Theorem A.1. Assume that {un } is defined by (A.1), then ∇un → ∇u a.e. (x, t) ∈ Ω1 × (0, T ). Proof. To prove the result is sufficient to show that for some θ > 0, we have T
∇(un − u) θ φ dx dt → 0 as k → ∞.
0 Ω1
As in [5], we denote by ω(ν, m, ε) any quantity that satisfies lim lim sup lim sup ω(ν, m, ε) = 0,
ε→0+ m→∞
ν→∞
and by ων,ε (m) any quantity that goes to zero as m → ∞ for a fixed ν and ε. We perform the following time regularization. For v ∈ L2 (0, T ; W01,2 (Ω)), we define t vν (x, t) = −∞
v(x, s)eν(s−t) ds,
B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
1267
where v(x, t) =
v(x, t) 0
if t ∈ [0, T ], if t ∈ / [0, T ].
It is clear that vν → v strongly in L2 (0, T ; W01,2 (Ω)), (vν ) = ν(v − vν ) in the weak sense, i.e., (vν ) , w = ν
(v − vν )w dx dt
for all w ∈ L2 0, T ; W01,2 (Ω) .
ΩT
Consider 0 < 2θ < 1 and the following split,
∇(un − u) 2θ dx dt =
∇(un − u) 2θ dx dt +
∇(un − u) 2θ dx dt
ΩT ∩u<m
ΩT ∩um
ΩT
≡ Im + Jm . We estimate the first integral,
∇(un − u) 2θ dx dt c
ΩT ∩um
T
q |∇un | + |∇u| dx dt
2θ q
1− 2θ q . meas u(x, t) m
0 Ω
Since meas{|u(x, t)| m} → 0 as m → ∞, it follows that Im → 0 as m → ∞.
(A.2)
We deal now with Jm . Since un u, then T Jm =
∇ Tm (un ) − Tm (u) 2θ dx dt.
0 Ω
Using the definition of (Tm (u))ν and the monotonicity of un it follows that lim χ{|un −(Tm (u))ν |>ε} = χ{|u−(Tm (u))ν |>ε}
n→∞
lim χ{|u−(Tm (u))ν |>ε} = χ{|u−Tm (u)|>ε}
ν→∞
for all ν and m ∈ N, for all m ∈ N.
Hence we have
∇ Tm (un ) − Tm (u) 2θ χ{|u −(T (u)) |>ε} dx dt n m ν
Jm ΩT
+ ΩT
∇ Tm (un ) − Tm (u) 2θ χ{|u −(T (u)) |ε} dx dt. n m ν
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B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272 q
Since |∇(Tm (un ) − Tm (u))|2θ is bounded in L 2θ (ΩT ) and since χ{|u−(Tm (u))ν |>ε} goes to zero almost everywhere, we conclude that
∇ Tm (un ) − Tm (u) 2θ χ{|u
n −(Tm (u))ν |>ε}
dx dt = ω(ν, m, ε).
(A.3)
ΩT
On the other hand, we have
∇ Tm (un ) − Tm (u) 2θ χ{|u
ΩT
1−θ C meas(ΩT )
n −(Tm (u))ν |ε}
dx dt
∇ Tm (un ) − Tm (u) 2 φχ{|u −(T (u)) |ε} dx dt n m ν
θ .
ΩT
Notice that
∇ Tm (un ) − Tm (u) 2 χ{|u
ΩT
n −(Tm (u))ν |ε}
∇ un − Tm (u) 2 χ{|u
n −(Tm (u))ν |ε}
dx dt
dx dt
ΩT
∇un ∇ un − Tm (u) χ{|un −(Tm (u))ν |ε} dx dt
ΩT
−
∇Tm (u)∇ un − Tm (u) φχ{|un −(Tm (u))ν |ε} dx dt.
ΩT
Using the weak convergence of Tm (un ) and the monotonicity of {un } we get easily that
∇Tm (u)∇ un − Tm (u) φχ{|un −(Tm (u))ν |ε} dx dt = ων,ε (m).
ΩT
We deal now with the first integral of Jm . Since (Tm (u))ν → Tm (u) strongly in L2 (0, T ; W01,2 (Ω)) it follows that
∇ un − Tm (u) 2 χ{|u
ΩT
=
n −(Tm (u))ν |ε}
dx dt
∇un ∇ un − Tm (u) ν χ{|un −(Tm (u))ν |ε} dx dt + ωε (m, ν).
ΩT
Using Tε (un − (Tm (u))ν ) as a test function in (A.1) we get
(A.4)
B. Abdellaoui et al. / Journal of Functional Analysis 258 (2010) 1247–1272
(un )t , Tε un − Tm (u) ν + + = ΩT
1269
∇un ∇Tε un − Tm (u) ν dx dt
ΩT
Tε un − Tm (u) ν |∇un |p
ΩT
un Tε un − Tm (u) ν λ 1 + fn dx dt. 2 n + |x| un 2 n +|x|
By the estimate obtained in Theorem 3.1 we know that λ 1
, fn and |∇un |p are uniformly
bounded in L1 (ΩT ). Moreover, since |Tε (un − (Tm (u))ν )| ε, then
(un )t , Tε un − Tm (u) ν + ∇un ∇Tε un − Tm (u) ν dx dt Cε. ΩT
A simple variation of Lemma 3.1 in [5] allows us to prove that (un )t , Tε (un − (Tm (u))ν ) ων,ε (m). Thus
∇un ∇Tε un − Tm (u) ν dx dt ω(ν, m, ε).
(A.5)
ΩT
Putting together (A.3), (A.4) and (A.5) we get Jm = ω(ν, m, ε). Therefore, from (A.2) and (A.6) the result follows.
(A.6)
2
Lemma A.2 (Maximum principle). Consider h = (h1 , h2 , . . . , hN ), a measurable function in N + r10 < 1. Consider (ΩT )N such that |h| ∈ L2r0 ([0, T ]; L2p0 (Ω)) where p0 , r0 > 1 and 2p 0 w(x, t) 0 such that, 1,p1
(i) w ∈ C((0, T ); L1 (Ω)) ∩ Lr1 ([0, T ]; W0 (ii) w is a subsolution to problem
(Ω)), where r1 , p1 1 satisfy
⎧ ⎨ wt − w |h||∇w| w(x, t) = 0 ⎩ w(x, 0) = 0
in ΩT , on ∂Ω × (0, T ), in Ω.
N 2p1
+
1 r1
>
N +1 2 ,
(A.7)
Then w ≡ 0. N + r11 > N 2+1 , then usProof. Since w ∈ Lr1 ([0, T ]; W0 1 (Ω)) with r1 , p1 1 such that 2p 1 r p ing Gagliardo–Nirenberg inequality it follows that w ∈ L 2 ([0, T ]; L 2 (Ω)) with r2 , q2 1 and N 1 N N 1 N 1 N 2p2 + r2 > 2 . Since 2p0 + r0 < 1, then 2p + r > 2 . Hence, we can choose β > 0 such that 1,p
1 N β+1 ( 2p0
+
1 ) > N2 . r0
0
0
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Let show that T w β−1 |∇w|2 dx dt < ∞.
(A.8)
0 Ω s For ε > 0, consider m (s) ≡ ( 1+εs )β and define Dε (s) = test function in (2.1), we get
T
Dε w(x, T ) dx + β
Ω
0 Ω
T
w 1 + εw
0 Ω
w 1 + εw
β−1
s 0
m (y) dy. Taking mε (w) as a
|∇w|2 dx dt (1 + εw)2
β |h||∇w| dx dt.
Using Young inequality there results T 0 Ω
w 1 + εw T
η1 0 Ω
β |h||∇w| dx dt
w 1 + εw
β−1
T |∇w| dx dt + η2
|h|2 (x, t)w β+1 dx dt.
2
0 Ω
On the other hand, using Hölder inequality, T
T |h| w 2
β+1
dx dt
0 Ω
|h|
2p0
dx
1 p0
w
Ω
0
dx
1 p0
dt
Ω
T
|h|
2p0
0
p0 (β+1)
dx
r0 p0
1 T r0 dt
Ω
w 0
r0 dx
p0
Ω
By the hypothesis on h and the choice of β, we obtain T |h| 0
1 T r0
r0 2p0
dx
p0
dt
Ω
w 0
1
p0 (β+1)
r0 dx
p0
dt
r0
Ω
Letting ε → 0, then by Fatou’s lemma, we find
T w
Ω
and (A.8) follows.
β+1
(x, T ) dx + β
w β−1 |∇w|2 dx dt < ∞, 0 Ω
1
p0 (β+1)
< ∞.
dt
r0
.
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Using the assumption on h, there exists q > 2 such that |h|q ∈ Lr2 ([0, T ], Lp2 (Ω)) and 1 r2
N 2p2
+
< 1. Consider ψ the solution to the problem q ⎧ ⎨ ψt − ψ = h(x, T − t) ψ ψ(x, t) = 0 ⎩ ψ(x, 0) = 1
in ΩT , on ∂Ω × (0, T ), in Ω.
(A.9)
Using regularity result of [10], we know that ψ ∈ L∞ (ΩT ). Consider ψ1 (x, t) = ψ(x, T − t), then ψ1 solves ⎧ q dψ1 ⎪ ⎪ − ψ1 = h(x, t) ψ1 ⎨− dt ψ1 (x, t) = 0 ⎪ ⎪ ⎩ ψ1 (x, T ) = 1
in ΩT , on ∂Ω × (0, T ), in Ω.
(A.10)
w Using ( 1+εw )β ψ1 as a test function in (2.1), passing to the limit as ε → 0 and using the estimate (A.8), we get
1 β +1
t
w β+1 (x, t)ψ1 (x, t) dx + β Ω
0 Ω
1 + β +1 t
w β−1 |∇w|2 ψ1 dx ds
t
w β+1 (−ψ1 )t − ψ1 dx ds
0 Ω
|h||∇w|w β ψ1 dx ds 0 Ω
t
|h||∇w|w θ1 +θ2 +θ3 ψ1 dx ds,
= 0 Ω
(q−2)(β+1) β+1 with θ1 = β−1 (it is clear that θ1 + θ2 + θ3 = β). Using Young 2 , θ2 = q and θ3 = 2q inequality with θ1 , θ2 , θ3 and by definition of ψ1 , there results
1 β +1
t
w
β+1
(x, t)ψ1 (x, t) dx + β
Ω
+
w β−1 |∇w|2 ψ1 dx ds 0 Ω
1 β +1
t
w β+1 |h|q ψ1 dx ds
0 Ω
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t η1
t w
β−1
|∇w| ψ1 dx ds + η2
w β+1 |h|q ψ1 dx ds
2
0 Ω
0 Ω
t + η3
w β+1 (x, s)ψ1 (x, s) dx ds. 0 Ω
Choosing η1 and η2 small, it follows that 1 β +1
t
w Ω
β+1
(x, t)ψ1 (x, t) dx η3
w β+1 (x, s)ψ1 (x, s) dx ds. 0 Ω
Since w 0 and ψ1 > 0 in Ω × (0, t) for t < T , then by Gronwall’s inequality we obtain that w ≡ 0 and then we conclude. 2 References [1] B. Abdellaoui, I. Peral, A. Primo, Optimal results for parabolic problems arising in some physical models with critical growth in the gradient respect to a Hardy potential, submitted for publication. [2] B. Abdellaoui, I. Peral, A. Primo, Breaking of resonance and regularizing effect of a first order quasi-linear term in some elliptic equations, Ann. Inst. H. Poincaré Anal. Non Lineaire 25 (2008) 969–985. [3] P. Baras, J.A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1) (1984) 121– 139. [4] D. Blanchard, F. Murat, Renormalised solutions of nonlinear parabolic problems with L1 data: Existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A 127 (6) (1997) 1137–1152. [5] L. Boccardo, A. Dall’Aglio, T. Gallouët, L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal. 147 (1) (1997) 237–258. [6] L. Boccardo, T. Gallouët, L. Orsina, Existence and nonexistence of solutions for some nonlinear elliptic equations, J. Anal. Math. 73 (1997) 203–223. [7] A. Dall’Aglio, D. Giachetti, J.P. Puel, Nonlinear parabolic equations with natural growth in general domains, Differential Integral Equations 20 (4) (2007) 361–396. [8] J. Droniou, A. Porretta, A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal. 19 (2) (2003) 99–161. [9] J. Garcia Azorero, I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998) 441–476. [10] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasi-Linear Equations of Parabolic Type, Transl. Math. Monogr., vol. 23, Amer. Math. Soc., Providence, 1968. [11] L. Oswald, Isolated positive singularities for a non linear heat equation, Houston J. Math. 14 (4) (1988) 543–572. [12] M. Pierre, Parabolic capacity and Sobolev spaces, SIAM J. Math. Anal. 14 (3) (1983) 522–533. [13] N. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968) 205– 226.
Journal of Functional Analysis 258 (2010) 1273–1309 www.elsevier.com/locate/jfa
Nonlinear mobility continuity equations and generalized displacement convexity J.A. Carrillo a,b , S. Lisini c,∗ , G. Savaré c , D. Slepˇcev d a ICREA (Institució Catalana de Recerca i Estudis Avançats), Spain b Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain c Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Italy d Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, USA
Received 24 January 2009; accepted 19 October 2009 Available online 5 November 2009 Communicated by C. Villani
Abstract We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex. © 2009 Elsevier Inc. All rights reserved. Keywords: Gradient flows; Displacement convexity; Nonlinear diffusion equations; Parabolic equations; Wasserstein distance; Nonlinear mobility
* Corresponding author.
E-mail addresses: [email protected] (J.A. Carrillo), [email protected] (S. Lisini), [email protected] (G. Savaré), [email protected] (D. Slepˇcev). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.016
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1. Introduction 1.1. Displacement convexity and Wasserstein distance In [33], McCann introduced the notion of displacement convexity for integral functionals of the form: given U : [0, +∞) → R, U (μ) :=
U ρ(x) dx
for μ = ρL d ,
Ω
defined on the set Pac (Ω) of the Borel probability measures in a convex open domain Ω ⊂ Rd , which are absolutely continuous with respect to the Lebesgue measure L d . Displacement convexity of U means convexity along a particular class of curves, given by displacement interpolation between two given measures. These curves turned out to be the geodesics of the space Pac (Ω) endowed with the euclidean Wasserstein distance. We recall that the Wasserstein distance W between two Borel probability measures μ0 and μ1 on Ω is defined by the following optimal transportation problem (Kantorovich relaxed version) (see [42,43]) W (μ0 , μ1 ) := min 2
|x − y| dγ (x, y): γ ∈ Γ (μ0 , μ1 ) , 2
Ω×Ω
where Γ (μ0 , μ1 ) is the set of admissible plans/couplings between μ0 and μ1 , that is the set of all Borel probability measures on Ω × Ω with first marginal μ0 and second marginal μ1 . We introduce the “pressure” function P , defined by P (r) := rU (r) − U (r) − U (0) =
r
sU (s) ds
so that
0
P (r) = rU (r),
P (0) = 0.
(1.1)
The main result of [33] states that under the assumption P (r)r (1 − 1/d)P (r) 0 ∀r ∈ (0, +∞),
(1.2a)
or, equivalently, r →
P (r) r 1−1/d
is nonnegative and nondecreasing on (0, +∞),
(1.2b)
the functional U is convex along the constant speed geodesics induced by W , i.e. for every curve (μs )s∈[0,1] ⊂ Pac (Ω) satisfying W (μs1 , μs2 ) = |s1 − s2 |W (μ0 , μ1 )
∀s1 , s2 ∈ [0, 1],
(1.3)
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the map s → U (μs ) is convex in [0, 1]. This class of curves can be, equivalently, defined by displacement interpolation, using the Brenier’s optimal transportation map pushing μ0 onto μ1 (see [42], for example). Note that (1.2a) and (1.1) imply the convexity of U . For power-like functions U, P U (ρ) =
if β = 1, if β = 1,
1 β β−1 ρ
ρ log ρ
P (ρ) = ρ β ,
(1.2a) is equivalent to β 1 − 1/d.
(1.4)
1.2. The link with a nonlinear diffusion equation Among the various applications of this property, a remarkable one concerns a wide class of nonlinear diffusion equations. The seminal work of Otto [34] contributed the key idea that a solution of the nonlinear diffusion equation ∂t ρ − ∇ · ρ∇U (ρ) = 0 in (0, +∞) × Ω,
(1.5)
with homogeneous Neumann boundary condition on ∂Ω can be interpreted as the trajectory of the gradient flow of U with respect to the Wasserstein distance. This means that the equation is formally the gradient flow of U with respect to the local metric which for a tangent vector s has the form −∇ · (ρ∇p) = s in Ω, s, s ρ = ρ|∇p|2 dx where ∇p · n = 0 on ∂Ω Ω
where n is a unit normal vector to ∂Ω. Let us note here that Eq. (1.5) corresponds via (1.1) to ∂t ρ − P (ρ) = 0.
(1.6)
In particular, the heat equation, for P (ρ) = ρ, is the gradient flow of the logarithmic entropy U (ρ) = Ω ρ log ρ dx. Let us also note that the metric above satisfies 2 s, s ρ = inf ρ|v| dx: s + ∇ · (ρv) = 0 in Ω and v · n = 0 on ∂Ω . Ω
The key property of this metric is that the length of the minimal geodesic between two given measures is nothing but the Wasserstein distance. More precisely 1 W (μ0 , μ1 ) = inf 2
v s (x) 2 ρs (x) dx ds: ∂s ρ + ∇ · (ρv) = 0 in (0, 1) × Rd ,
0 Rd
¯ ρ0 L = μ0 , ρ1 L = μ1 . supp(ρs ) ⊂ Ω, d
d
This dynamical formulation of the Wasserstein distance was rigorously established by Benamou and Brenier in [5] and extended to more general situations in [2] and [30].
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As for the classical gradient flows of convex functions in euclidean spaces, the flow associated with (1.5) is a contraction with respect to the Wasserstein distance. In [2] the authors showed that one of the possible ways to rigorously express the link between the functional U , the distance W , and the solution of the diffusion equation (1.5) is given by the evolution variational inequality satisfied by the measures μt = ρ(t, ·)L d associated with (1.5): 1 d+ 2 W (μt , ν) U (ν) − U (μt ) 2 dt
∀ν ∈ Pac (Ω).
(1.7)
1.3. A new class of “dynamical” distances In a number of problems from mathematical biology [8,9,14,17,18,26,27,36] mainly in chemotaxis with prevention of overcrowding, mathematical physics [10,13,21,22,28,29,40], studies of phase segregation [23,39], and studies of thin liquid films [6], the mobility of “particles” depends on the density ρ itself. For instance, a typical choice of the mobility to avoid overcrowding in chemotaxis is to assume a saturation of the density, see [26], after normalization this leads to m(ρ) = ρ(1 − ρ). The equation for the population density is ∂t ρ = ∇ · m(ρ)∇(ρ ∗ W ) + ∇ρ = ∇ · m(ρ)∇ ρ ∗ W + U (ρ) with U (ρ) = ρ log ρ + (1 − ρ) log(1 − ρ) and W the fundamental solution of −S + δS with δ 0. Another source of models comes from mathematical physics. The same equation has been derived as a hydrodynamical limit of interacting particles system with Kawasaki exchange dynamics in studies of phase segregation [23]. Moreover, these type of equations appear in the study of relaxation towards equilibrium of fermionic or bosonic particles based on kinetic models [28,29]. In their diffusive approximation, they lead to equations of the form ∂t ρ = ∇ · m(ρ)x + ∇ρ = ∇ · m(ρ)∇ V + U (ρ) with m(ρ) = ρ(1 ± ρ), V (x) = |x|2 and U (ρ) = ρ log ρ ± (1 ∓ ρ) log(1 ∓ ρ), with + corresponding to bosons and − to fermions. More precisely the local metric in the configuration space is formally given as follows: For a tangent vector s (euclidean variation) 2
s, s ρ =
m(ρ)|∇p|2 dx
where
−∇ · (m(ρ)∇p) = s ∇p · n = 0
in Ω, on ∂Ω
Ω
where m : [0, +∞) → [0, +∞) is the mobility function. The global distance generated by the local metric is given by 1 Wm,Ω (μ0 , μ1 ) := inf 2
v s (x) 2 m ρs (x) dx ds: ∂s ρ + ∇ · m(ρ)v = 0
0 Rd
¯ ρ0 L = μ0 , ρ1 L = μ1 . in (0, 1) × R , supp(ρs ) ⊂ Ω, d
d
d
(1.8)
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This distance was recently introduced and studied in [19] in the case when m is concave and nondecreasing. Similarly to the case m(r) = r, it is easy to check formally that the trajectory of the gradient flow of U with respect to the modified distance Wm,Ω solves ∂t ρ − ∇ · m(ρ)∇U (ρ) = 0
in (0, +∞) × Ω
(1.9)
with homogeneous Neumann boundary conditions on ∂Ω. Assuming that U m and U mm are locally integrable, we can define in this case the function P and the auxiliary function H by r P (r) :=
r
H (r) :=
U (z)m(z) dz, 0
r
U (z)m(z)m (z) dz = 0
P (z)m (z) dz,
0
so that P = mU ,
H = m P ,
P (0) = H (0) = 0,
and, at least for smooth solutions, the problem (1.9) is equivalent to (1.6). By means of a formal computation, detailed in Section 2, the second derivative of the internal energy functional U along a geodesic curve (μs )s∈[0,1] satisfying as in (1.3) Wm,Ω (μs1 , μs2 ) = |s1 − s2 |Wm,Ω (μ0 , μ1 ) is nonnegative, i.e. (1.2b) holds
d2 U ds 2
∀s1 , s2 ∈ [0, 1],
(μs ) 0, if the following generalization of McCann condition (1.2a),
P (r)m(r) (1 − 1/d)H (r) 0 ∀r ∈ (0, +∞).
(1.10a)
It can also be expressed by requiring that r →
H (r) m1−1/d (r)
is nonnegative and nondecreasing in (0, +∞).
(1.10b)
Note that P (r) 0 by (1.10a) implies the convexity of U . As in the case of the Wasserstein distance, in dimension d = 1 the condition (1.10a) reduces to the usual convexity of U . In dimension d 2, still considering the relevant example of powerlike functions U, P , m as in (1.4), we get U (ρ) =
1 β β−1 ρ
ρ log ρ
if β = 1, , if β = 1
m(ρ) = ρ α ,
P (ρ) =
and condition (1.10a) is equivalent to α ∈ (0, 1],
γ 1 − α/d.
β γ ρ , γ
γ := α + β − 1
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In this case the heat equation corresponds to γ = α + β − 1 = 1 and it is therefore the gradient flow of the functional 1 U (ρ) = ρ 2−α dx (1.11) (2 − α)(1 − α) Ω
with respect to the distance Wm,Ω induced by the mobility function m(ρ) = ρ α . Understanding diffusion equations as gradient flows with respect to these new metrics has a functional analysis interest, since it naturally leads to new functional inequalities and to a new interpretation of classical ones. As in the Wasserstein framework [1,15], geodesic/displacement convexity of integral functionals usually plays a crucial role; in particular, in the case of Fokker– Planck equations with a log-concave invariant measure γ = e−V L d , one can recover the family of the classical Beckner inequalities (interpolating between Log–Sobolev and Poincaré inequalities, see [4]) by exploiting the geodesic convexity of the functional obtained integrating ρ 2−α with respect to γ as in (1.11). In this case (considered in [20]) also the definition of the transport distance involves the reference measure γ [19]. Analogous inequalities can be expected in the nonlinear diffusion case. Convexity and dissipation inequalities for second order diffusion equations are also a crucial tool (see the “metric” techniques developed by [25,32] in the linear mobility case m(ρ) = ρ) for the study of nonnegative solutions to a certain class of fourth-order nonlinear diffusion equations ∂t ρ + ∇ · m(ρ)∇ ρ β−1 ρ β = 0.
(1.12)
Particularly interesting cases correspond to the values β = 1 (equation of thin-film type for power-like mobilities or Cahn–Hiliard when m(ρ) = ρ(1 − ρ)) and β = 1/2 (the so called Derrida–Lebowitz–Speer–Spohn equation also arising in quantum drift-diffusion models). They are formally the gradient flows of the first-order functionals U (ρ) :=
1 2β
β 2 ∇ ρ dx
(1.13)
Ω
with respect to Wm,Ω . In view of applications to Cahn–Hiliard models, another interesting example, still leading to the heat equation, is represented by the functional U (ρ) =
ρ log ρ + (1 − ρ) log(1 − ρ) dx,
0 ρ 1,
L d -a.e. in Ω,
Ω
and the distance induced by m(ρ) = ρ(1 − ρ), ρ ∈ [0, 1]. Notice that in this case the positivity domain of the mobility m is the finite interval [0, 1], a case that has not been explicitly considered in [19], but that can be still covered by a careful analysis (see [31]). 1.4. Geodesic convexity and contraction properties Our aim is to prove rigorously the geodesic convexity of the integral functional U under conditions (1.10a), (1.10b) and the metric characterization of the nonlinear diffusion equation
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(1.9) as the gradient flow of U with respect to the distance Wm,Ω (1.8). If one tries to follow the same strategy which has been developed in the more familiar Wasserstein framework, one immediately finds a serious technical difficulty, due to the lackness of an “explicit” representation of the geodesics for Wm,Ω . In fact, the McCann’s proof of the displacement convexity of the functionals U is strictly related to the canonical representation of the Wasserstein geodesics in terms of optimal transport maps. Existence of a minimal geodesic connecting two measures at a finite Wm,Ω distance has been proved by [19]. However, an explicit representation is no longer available. On the other hand in [16], following the eulerian approach introduced in [35], the authors presented a new proof of McCann’s convexity result for integral functionals defined on a compact manifold without the use of the representation of geodesics. Here, following the same approach of [16], we reverse the usual strategy which derives the existence and the contraction property of the gradient flow of a functional from its geodesic convexity. On the contrary, we show that under the assumption (1.10a) smooth solutions of (1.9) satisfy the following Evolution Variational Inequality analogous to (1.7), 1 d+ 2 (μt , ν) U (ν) − U (μt ) W 2 dt m,Ω ∀t ∈ [0, +∞), ∀ν ∈ P(Ω): Wm (ν, μ0 ) < +∞.
(1.14)
This is sufficient to construct a nice gradient flow generated by U and metrically characterized by (1.14), as showed in [2] and [3]. The remarkable fact proved by [16] is that whenever a functional U admits a flow, defined at least in a dense subset of D(U ), satisfying (1.14), the functional itself is convex along the geodesics induced by the distance Wm,Ω . As a by-product we obtain stability, uniqueness, and regularization results for the solutions of the problem (1.9) in a suitable subspace of P(Ω) metrized by Wm,Ω . Concerning the assumptions on m, its concavity is a necessary and sufficient condition to write the definition of Wm,Ω with a jointly convex integrand [19], which is crucial in many properties of the distance, in particular for its lower semicontinuity with respect to the usual weak convergence of measures. Since m 0 on [0, ∞) the concavity implies that the mobility must be nondecreasing. This is the case considered in [19]. However we are also able to treat the case when the mobility is defined on an interval [0, M) where it is nonnegative and concave. It that case the configuration space is restricted to absolutely continuous measures with densities bounded from above by M. Such mobilities are of particular interest in applications as mentioned before. 1.5. Plan of the paper In next section, we show the heuristic computations for the convexity of functionals with respect to Wm,Ω . Section 3 is devoted to introduce the notation and to review the needed concepts on Wm,Ω from [19]. Moreover, we prove a key technical regularization lemma: Lemma 3.6. Subsection 3.4 addresses the question of finiteness of Wm,Ω (μ0 , μ1 ), providing new sufficient conditions on m and μ0 , μ1 in order to ensure that Wm,Ω (μ0 , μ1 ) < +∞. After a brief review of some basic properties of the diffusion equation (1.6), in Section 4 we try to get some insight on the features of the generalized McCann condition (1.10a), (1.10b), we recall some basic facts on the metric characterization of contracting gradient flows and their relationships with geodesic convexity borrowed from [2,16], and we state our main results Theorems 4.10 and 4.12. The core
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of our argument in smooth settings is collected in Section 5, whereas the last Section concludes the proofs of the main results. At the end of the paper we collect some final remarks and open problems. 2. Heuristics We first discuss, in a formal way, the conditions for the displacement convexity of the internal, the potential and the interaction energy, with respect to the geodesics corresponding to the distance (1.8). For simplicity, we assume that Ω = Rd and that densities are smooth and decaying fast enough at infinity so that all computations are justified. 2.1. Geodesics We first obtain the optimality condition for the geodesic equations in the fluid dynamical formulation of the new distance (1.8). As in [7], we insert the nonlinear mobility continuity equation (1.8) ∂s ρ + ∇ · m(ρ)v = 0 in (0, 1) × Rd .
(2.1)
inside the minimization problem as a Lagrange multiplier. As a result, we get the unconstrained minimization problem 1
2 1 v s (x) m ρs (x) dx ds 2
W2m (μ0 , μ1 ) = inf sup (ρ,v ) ψ
1 −
0 Rd
ρs (x)∂s ψ(s, x) + m ρs (x) v s (x) · ∇ψ(s, x) dx ds
0 Rd
+
ρ1 (x)ψ(1, x) dx −
Rd
ρ0 (x)ψ(0, x) dx .
Rd
Applying a formal minimax principle and thus taking first an infimum with respect to v we obtain the optimality condition v = ∇ψ, and the following formal characterization of the distance
1 Wm (μ0 , μ1 ) = sup inf − ρ 2 ψ
1
1 |∇ψ| m(ρ) dx ds −
2
2
0 Rd
+
ρ1 (x)ψ(1, x) dx −
Rd
ρ∂s ψ dx ds 0 Rd
ρ0 (x)ψ(0, x) dx ,
Rd
which provides the further optimality condition 1 ∂s ψ + m ρs (x) |∇ψ|2 = 0. 2
(2.2)
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We thus end up with a coupled system of differential equations in (0, 1) × Rd [19, Remark 5.19], ⎧ ⎨ ∂s ρ + ∇ · m(ρ)∇ψ = 0, 1 ⎩ ∂s ψ + m (ρ)|∇ψ|2 = 0. 2
(2.3)
2.2. Internal energy We use the formal equations (2.3) for the geodesics associated to the distance (1.8) to compute the conditions under which the internal energy functional is displacement convex. If therefore (ρs , ψs ) is a smooth solution of (2.3), which decays sufficiently at infinity, we proceed as usual [11,35,42] to obtain the following formulas: d U (μ) = − P (ρ)ψ dx, ds Rd
where ρ denotes, as usual in this paper, the density of μ with respect to the Lebesgue measure, and
d2 U (μ) = ds 2
P (ρ)m(ρ) − H (ρ) (ψ)2 dx +
Rd
−
1 2
1 H (ρ) −∇ψ · ∇ψ + |∇ψ|2 dx 2
Rd
P (ρ)m (ρ)|∇ρ|2 |∇ψ|2 dx.
Rd
As usual, the Bochner formula 1 1 −∇ψ · ∇ψ + |∇ψ|2 = | Hess ψ|2 (ψ)2 , 2 d and the fact that H (ρ) 0, allow us to estimate it as d2 U (μ) ds 2
1 P (ρ)m(ρ) − (1 − 1/d)H (ρ) (ψ)2 dx − 2
Rd
P (ρ)m (ρ)|∇ρ|2 |∇ψ|2 dx.
Rd
Therefore, under conditions of concavity of the mobility m(ρ) and the generalized displacement McCann’s condition (1.10a), the functional U is convex along the geodesics of the distance Wm . 2.3. Potential energy Similar heuristic formulas can be obtained for the potential and the interaction energy, as in [11,42]. We consider the potential energy functional V (μ) :=
V (x) dμ, Rd
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with V a given smooth potential. The equation that we formally obtain as the gradient flow of V with respect to the distance Wm is the conservation law ∂t ρ = ∇ · m(ρ)∇V ,
(2.4)
which is an equation of hyperbolic type. As before, it is easy to check that the second derivative of V along a geodesic satisfying (2.3) is d2 V (μ) = ds 2
m(ρ)m (ρ) (Hess V ∇ψ) · ∇ψ dx
Rd
+
1 2 m(ρ)m (ρ) (∇ρ · ∇ψ)(∇V · ∇ψ) − (∇ρ · ∇V )|∇ψ| dx. 2
Rd
This formula allows us to show that this functional cannot be convex along geodesics if m is not linear. Technically, the reason is the presence of the terms linearly depending on ∇ρ. We present a simple example: Example. Let us first construct the example in one dimension. The expression for the second derivative of the functional above reduces to d2 V (μ) = ds 2
m(ρ)m (ρ)Vxx ψx2 dx R
1 + 2
m(ρ)m (ρ)ρx Vx ψx2 dx =: I + II.
R
Consider the case that V is nontrivial. Then Vx = 0 on some interval. For notational simplicity, we assume that Vx > 0 on [−2, 2]. Since the mobility m we are considering is not a linear function of ρ there exists z > 0 such that m (z) = 0. Again for notational simplicity, let us assume that m (z) < 0
on
1 3 , . 2 2
The fact that we chose Vx to be positive and m negative is irrelevant because the sign of term II can be controlled by the sign of ρx . Let η be a piecewise linear function on R: ⎧3 ⎪ ⎨2 η(x) = 1 − x ⎪ ⎩1 2
if x < − 12 , if x ∈ [− 12 , 12 ], if x > 12 .
The fact that the function is Lipschitz, but not smooth is irrelevant; smooth approximations of the given η, can also be used in the construction. Let ηε (x) = η( xε ). Let σ ∈ C0∞ (R, [0, 1]),
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Fig. 1. A profile at which the potential energy is not convex.
supported in [−1, 1], such that σ = 1 on [− 14 , 14 ] and R σ (x)dx = 1. Let ρε = σ ηε . Note that R ρε dx = 1. A typical profile of ρε is given in Fig. 1. The test velocity (tangent vector at s = 0) we consider also needs to be localized near zero. A simple choice is ψε (0) = ηε . Let ρε (s) be the corresponding geodesics given by (2.1) and (2.2). Let us observe how, at s = 0, the terms I and II scale with ε: Iε max m(z)m (z) max Vxx (x) z∈[0,2]
x∈[−1,1]
1 1 ε∼ , 2 ε ε
1 1 1 1 min Vx ε ∼ − 2 . II ε − min m(z) m (z) 2 z∈[ 12 , 32 ] ε x∈[−1,1] ε 2 ε d2 Thus, for ε small enough, ds 2 s=0 V (ρε (s)) < 0. Furthermore note that the square of the length of the tangent vector dtd ρε (0) is
m ρε (0) (∂x ψε )2 dx ∼ R
1 . ε
Thus for any λ ∈ R there exists ε > 0 such that d 2 V ρε (s) + λ m ρε (0) (∂x ψε )2 dx < 0 ds 2 s=0 R
which implies that V is not λ-convex for any λ ∈ R. Let us conclude the example by remarking that it can be extended to multidimensional domains. In particular it suffices to extend the 1-D profile to d-D to be constant in every other direction and then use a cut-off. We only sketch the elements of the construction. We can assume that ∇V (0) = ed . Let ρ˜ε (x) = ρε (xd ). Let xˆ = (x1 , . . . , xd−1 ). To cut-off in the directions perpendicular to ed we use the length scales 1 l δ ε. Let θl,δ be smooth cut-off function equal to 1 on [−l, l] and equal to 0 outside of [−l − δ, l + δ]; with |∇θl,δ | < Cδ and |D 2 θl,δ | < δC2 . Let ρl,δ,ε (x) = ρ˜ε (xd )θl,δ (|x|). ˆ Let ψl,δ,ε (x) = ηε (xd )θl,δ (|x|). ˆ Checking the scaling of appropriate terms is straightforward.
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2.4. Interaction energy Consider the interaction energy functional W (μ) :=
1 2
W (x − y)ρ(x)ρ(y) dx dy, Rd Rd
with W a given smooth potential. The equation that we formally obtain as the gradient flow of W with respect to the distance Wm is the interaction equation ∂t ρ = ∇ · m(ρ)(ρ ∗ ∇W ) .
(2.5)
As before, it is easy to check that d2 W (μ) = ds 2
m ρ(x) m ρ(x) ρ(y)∇ψ(x) · Hess W (x − y)∇ψ(x) dx dy
Rd Rd
m ρ(x) m ρ(y) ∇ψ(y) · Hess W (x − y)∇ψ(x) dx dy
− Rd Rd
+
m ρ(x) m ρ(x) ρ(y) ∇ρ(x) · ∇ψ(x) ∇W (x − y) · ∇ψ(x) dx dy
Rd Rd
−
1 2
2 m ρ(x) m ρ(x) ρ(y) ∇ρ(x) · ∇W (x − y) ∇ψ(x) dx dy.
Rd Rd
It can be demonstrated that if m is nonlinear then the interaction energy is not geodesically convex. As for the potential energy, the reason lies in the presence of derivatives of ρ in the expression above. More precisely, in one dimension the second derivative of W (ρ) reduces to d2 W (ρ) = ds 2
m ρ(x) m ρ(x) ρ(y)ψx2 (x)Wxx (x − y) dx dy
R R
−
m ρ(x) m ρ(y) ψy (y)Wxx (x − y)ψx (x) dx dy R R
+
1 2
m ρ(x) m ρ(x) ρ(y)ρx (x)ψx2 (x) Wx (x − y) dx dy.
R R
It turns out that the example for the lack of (semi)convexity provided for the potential energy is also an example (with V replaced by W ) for the interaction energy. The estimates of the terms are similar, so we leave the details to the reader.
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3. Notation and preliminaries In this section, following [19], we shall recall the main properties of the distance Wm,Ω introduced in (1.8). For the sake of simplicity, we only consider here the case of a bounded open domain Ω, so that it is not restrictive to assume that all the measures (Radon, i.e. locally finite, in the general approach of [19]) involved in the various definitions have finite total variation. Since we deal with arbitrary mobility functions m, these distances do not exhibit nice homogeneity properties as in the Wasserstein case; therefore we deal with finite Borel measures without assuming that their total mass is 1. 3.1. Measures and continuity equation d d We denote by M+ (Rd ) (resp. M+ c (R )) the space of finite positive Borel measures on R d d d (resp. with compact support) and by M(R ; R ) the space of R -valued Borel measures on Rd with finite total variation. By Riesz representation theorem, the space M(Rd ; Rd ) can be identified with the dual space of Cc0 (Rd ; Rd ) and it is endowed with the corresponding weak∗ topology. We denote by |ν| ∈ M+ (Rd ) the total variation of the vector measure ν ∈ M(Rd ; Rd ). ν admits the polar decomposition ν = w|ν| with w ∈ L1 (|ν|; Rd ). If B is a Borel subset of Rd (typically an open or closed set) we denote by M+ (B) (resp. M+ (B; Rd )) the subset of M+ (Rd ) (resp. M(Rd ; Rd )) whose measure μ are concentrated on B, i.e. μ(Rd \ B) = 0 (resp. |μ|(Rd \ B) = 0). Notice that if B is a compact subset of Rd then the convex set in M+ (B) of measures with a fixed total mass m is compact with respect to the weak∗ topology. If m > 0, M+ (B, m) is the convex subset of M+ (B) whose measures have fixed total mass μ(B) = m. ¯ we denote by CEΩ (μ0 → μ1 ) Let Ω be a bounded open subset of Rd . Given μ0 , μ1 ∈ M+ (Ω) + ¯ and (ν s )s∈(0,1) ∈ M(Ω; ¯ Rd ) the collection of time dependent measures (μs )s∈[0,1] ⊂ M (Ω) such that
1. s → μs is weakly∗ continuous in M+ (Rd ) with μ|s=0 = μ0 and μ|s=1 = μ1 ; 1 ¯ ds < +∞; 2. (ν s )s∈(0,1) is a Borel family with 0 |ν s |(Ω) 3. (μ, ν) is a distributional solution of ∂s μs + ∇ · ν s = 0
in (0, 1) × Rd .
¯ = If (μ, ν) ∈ CEΩ (μ0 → μ1 ) then it is immediate to check that the total mass μs (Rd ) = μs (Ω) m is a constant, independent of s. In particular, μ0 (Rd ) = μ1 (Rd ). 3.2. Mobility and action functional We fix a right threshold M ∈ (0, +∞] and a concave mobility function m ∈ C 0 [0, M) strictly positive in (0, M). We denote by m(M) the left limit of m(r) as r ↑ M. We can also introduce the maximal left interval of monotonicity of m whose right extreme is M↑ := sup m ∈ [0, M): m|[0,m] is nondecreasing . We distinguish two situation:
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Case A. M = +∞, so that m is nondecreasing and M↑ = M = +∞; typically m(0) = 0 and the main example is provided by m(r) = r α , α ∈ [0, 1]. This is the case considered in [19]. When m (+∞) := limr↑+∞ r −1 m(r) = limr↑+∞ m (r) = 0 we are in the sublinear growth case. A linear growth of m corresponds to m (+∞) > 0. Case B. M < +∞, so that 0 M↑ M and m is nonincreasing in the right interval [M↑ , M] (but we also allow m to be constant or even decreasing in [0, M) with M↑ = 0). Typically m(0) = m(M) = 0 (in this case 0 < M↑ < M) and the main example is m(r) = r(M − r), or, more generally, m(r) = r α0 (M − r)α1 , α0 , α1 ∈ (0, 1]. Remark 3.1. Many properties proved in the Case A can be extended to the Case B, but there are important exceptions. The most important one concerns the upper bound on the two measures μ0 , μ1 in order to satisfy Wm,Ω (μ0 , μ1 ) < +∞ in Case B: they should be absolutely continuous with respect to L d with essentially bounded densities ρ i M, i = 0, 1. Another important difference concerns the subadditivity property [19, Theorem 5.12], Wm,Ω μ0 + σ 0 , μ1 + σ 1 Wm,Ω μ0 , μ1 + Wm,Ω σ 0 , σ 1 which does not hold in Case B. We refer to [31] for further technical details. Using the conventions a/b = 0
if a = b = 0,
a/b = +∞
if a > 0 = b,
(3.1)
the corresponding action density function φm : R × Rd → [0, +∞] is defined by φm (ρ, w) =
|w |2 m(ρ)
+∞
if ρ ∈ [0, M], if ρ < 0 or ρ > M.
It is not difficult to check that, under the convention (3.1), the function φm is (jointly) convex and lower semicontinuous. Given that m is concave and φm is convex, when M = +∞ we can define the recession ∞ : Rd → [0, +∞] (recall (3.1)), function ϕm ∞ ϕm (w) := lim rφm (1, w/r) = r↑+∞
|w|2 , m (∞)
m (∞) := lim m (r) = lim r→+∞
r→+∞
m(r)
r
0.
We introduce now the action functional Φm,Ω : M+ Rd × M Rd ; Rd → [0, +∞], defined on couples of measures μ ∈ M+ (Rd ), ν ∈ M(Rd ; Rd ). In order to define it we consider the usual Lebesgue decomposition μ = ρL d + μ⊥ , ν = wL d + ν ⊥ and distinguish the following cases:
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1. If the support of μ or ν is not contained in Ω¯ then Φm,Ω (μ, ν) = +∞; 2. When M < +∞ (Case B), we set Φm,Ω (μ, ν) :=
φm (ρ, w) dx +∞ Ω
if μ⊥ = 0, ν ⊥ = 0, otherwise;
notice that if Φm,Ω (μ, ν) < +∞ then ρ ∈ L∞ (Ω) with 0 ρ M, L d -a.e. in Ω and w ∈ L2 (Ω; Rd ). 3. When M = +∞ and m (∞) = 0 (Case A, sublinear growth) then Φm,Ω (μ, ν) :=
φm (ρ, w) dx +∞ Ω
if ν ⊥ = 0, otherwise;
4. Finally, when M = +∞ and m (∞) > 0 (Case A, linear growth) then we set Φm,Ω (μ, ν) :=
φm (ρ, w) dx + +∞ Ω
Ω¯
∞ (w ⊥ ) dμ⊥ ϕm
if ν ⊥ = w ⊥ μ⊥ μ⊥ , otherwise.
3.3. The modified Wasserstein distance ¯ we define Let Ω be a bounded open set. Given μ0 , μ1 ∈ M+ (Ω) 1
1/2 0 1 0 1 Φm,Ω (μs , ν s ) ds : (μ, ν) ∈ CEΩ μ → μ Wm,Ω μ , μ := inf
(3.2)
0
1 = inf
0 1/2 1 Φm,Ω (μs , ν s ) . ds: (μ, ν) ∈ CEΩ μ → μ
(3.3)
0
We refer to [19, Theorem 5.4] for the equivalence between (3.2) and (3.3). Wm,Ω (μ0 , μ1 ) = +∞ if the set of connecting curves CEΩ (μ0 → μ1 ) is empty. The following three propositions are proved in [19], see Theorems 5.5–5.7, 5.15, and Proposition 5.14. ¯ endowed with the distance Wm,Ω is a complete pseudoProposition 3.2. The space M+ (Ω) metric space (the distance can assume the value +∞), inducing as strong as, or stronger topology than the weak∗ one. ¯ the space M+ [σ ] := {μ ∈ M+ (Ω): ¯ Wm,Ω (μ, σ ) < +∞} is Given a measure σ ∈ M+ (Ω), m,Ω a complete metric space whose measures have the same total mass of σ . ¯ such that Wm,Ω (μ0 , μ1 ) < +∞ there exists a minMoreover, for every μ0 , μ1 ∈ M+ (Ω) imizing couple (μ, ν) in (3.2) (unique, if m is strictly concave and sublinear) and the curve (μs )s∈[0,1] is a constant speed geodesic for Wm,Ω , thus satisfying Wm,Ω (μt , μs ) = |t − s|Wm,Ω (μ0 , μ1 )
∀s, t ∈ [0, 1].
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Proposition 3.3 (Lower semicontinuity). If Ωn , Ω are bounded open sets such that L d |Ωn weakly∗ converges to L d |Ω , Mn ∈ (0, +∞] is a nonincreasing sequence converging to M, mn is a sequence of nonnegative concave functions in the intervals (0, Mn ) such that mn (r) mn (r)
∀r ∈ (0, Mn )
if n n ,
lim mn (r) = m(r)
n→∞
∀r ∈ (0, M),
and μn0 , μn1 are sequences of measures weakly∗ convergent to μ0 and μ1 respectively, then lim inf Wmn ,Ωn μn0 , μn1 Wm,Ω (μ0 , μ1 ).
n→+∞
(3.4)
¯ Then the following in˜ m, μ0 , μ1 ∈ M+ (Ω). Proposition 3.4 (Monotonicity). Let Ω˜ ⊃ Ω, m equality holds Wm˜ ,Ω˜ (μ0 , μ1 ) Wm,Ω (μ0 , μ1 ). Proposition 3.5. Let k ∈ Cc∞ (Rd ) be a nonnegative convolution kernel, with Rd k(x) dx = 1 ¯ and ν ∈ and supp(k) = B¯ 1 (0), and let kε (x) := ε −d k(x/ε). For every μ, μ0 , μ1 ∈ M+ (Ω) d ¯ M(Ω; R ) we have Φm,Ωε (μ ∗ kε , ν ∗ kε ) Φm,Ω (μ, ν)
∀ε > 0,
Wm,Ωε (μ0 ∗ kε , μ1 ∗ kε ) Wm,Ω (μ0 , μ1 )
∀ε > 0,
lim Wm,Ωε (μ0 ∗ kε , μ1 ∗ kε ) = Wm,Ω (μ0 , μ1 ),
ε→0
(3.5) (3.6)
where Ωε := Ω + Bε (0). Proof. If Φm,Ω (μ, ν) < +∞ then μ, ν are supported in Ω¯ and [19, Theorem 2.3] yields Φm,Ω (μ, ν) = Φm,Rd (μ, ν) Φm,Rd (μ ∗ kε , ν ∗ kε ) = Φm,Ωε (μ ∗ kε , ν ∗ kε ), being μ ∗ kε , ν ∗ kε supported in Ω¯ ε . Notice that only the concavity of m (and not its monotonicity) plays a role here. A similar argument and [19, Theorem 5.15] yields (3.5). The limit (3.6) is an immediate consequence of (3.4) and (3.5). 2 The next technical lemma provides a crucial approximation result for curves with finite Φm,Ω energy. It allows for measures to be approximated by ones with smooth, positive densities. Lemma 3.6. Let Ω be an open bounded convex set and let (μ, ν) ∈ CEΩ (μ0 → μ1 ) with given 1 constant mass m and finite energy 0 Φm,Ω (μs , ν s ) ds < +∞. For every ε > 0, δ ∈ [0, 1] there exist a decreasing family of smooth convex sets Ω ε ↓ Ω and a family of curves (με,δ , ν ε,δ ) ∈ ε,δ CEΩ ε (με,δ 0 → μ1 ) with the following properties
J.A. Carrillo et al. / Journal of Functional Analysis 258 (2010) 1273–1309 ε με,δ i = (1 − δ)μi ∗ kε + δλ , ε,δ d με,δ s = ρs L |Ω ε ,
λε :=
ε με,δ s Ω = m,
(3.7)
ρ ε,δ , w ε,δ ∈ C ∞ [0, 1] × Ω¯ ε ,
(3.8)
m L d (Ω ε )
ε,δ d ν ε,δ s = w s L |Ω ε ,
L d |Ω ε ,
m ρ ε,δ δ d > 0, L (Ω)
ε ∂s ρsε,δ + ∇ · wε,δ s = 0 in (0, 1) × Ω ,
1 cε2
1 Φ
m,Ω ε
ε,δ ε,δ μs , ν s ds
0
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1
1 Φm,Ω (μs , ν s ) ds = lim
ε,δ↓0
0
ε,δ ds, Φm,Ω ε με,δ s , νs
0
where cε := 1 + 2ε. Proof. Let us extend (μs , ν s ) outside the unit interval by setting ν s ≡ 0 and μs ≡ μ0 if s < 0, μs ≡ μ1 if s > 1; it is immediate to check that (μ, ν) still satisfy the continuity equation. We then consider a family of smooth and convex open sets Ω ε satisfying Ω + B2ε (0) ⊂ Ω ε ⊂ Ω + B3ε (0) ˜ εs and are concentrated and define μ˜ εs := μ ∗ kε , ν˜ εs := ν ∗ kε which have smooth densities ρ˜sε , w in Ω¯ + Bε (0). We perform a further time convolution with respect to a 1-dimensional family of nonnegative smooth mollifiers hε (z) := ε −1 h(z/ε) with support in [−ε, ε] and integral 1, μ¯ εs :=
μ˜ εz hε (s − z) dz,
ν˜ εz hε (s − z) dz,
ν¯ εs :=
R
R
with corresponding densities ρsε , w εs . Notice that μ¯ ε−ε = με,0 ¯ ε1+ε = με,0 0 ,μ 1 and, by the convexity of φm and Jensen’s inequality, we have φm ρsε , w εs
˜ εz hε (s − z) dz, φm ρ˜zε , w
Φm,Ω ε μ¯ εs , ν¯ εs
R
Φm,Ω ε μ˜ εs , ν˜ εs hε (s − z) dz
R
so that, being ν¯ εs = 0 if s < −ε or s > 1 + ε, 1+ε 1 ε ε ε ε ε ε Φm,Ω ε μ¯ s , ν¯ s ds = Φm,Ω ε μ¯ s , ν¯ s ds Φm,Ω ε μ˜ s , ν˜ s ds Φm,Ω (μs , ν s ) ds. −ε
R
R
0
We eventually set μεs := μ¯ εcε s−ε ,
ν εs := cε ν¯ εcε s−ε ,
cε := 1 + 2ε
and ε ε με,δ s := (1 − δ)μs + δλ ,
ε ν ε,δ s := ν s .
It is then easy to check that all the requirements are satisfied.
2
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3.4. Couple of measures at finite Wm,Ω distance We discuss now some cases when it is possible to prove that the distance between two measures is finite. We already know [19, Cor. 5.25] (in the Case A, but the same argument can be easily adapted to cover the case M < +∞) that when Ω is convex and bounded if μi = ρi L d with ρi L∞ (Rd ) < M
then Wm,Ω (μ0 , μ1 ) < ∞.
(3.9)
We focus on the Case A, M = +∞, and exploit some ideas of [37]. In order to refine the condition (3.9), we first introduce the functions −1/2 km,d (r) := r 1+2/d m(r) ,
1 Km,d (r) := d
+∞ km,d (z) dz,
r > 0.
r
Observe that Km,d is either everywhere finite or identically +∞. In particular, in the case m(r) = r α , Km,d is finite if and only if α > 1 − 2/d. Theorem 3.7. Let Ω be a bounded, open convex set of Rd . Suppose that M = +∞, m > 0, and that Km,d is finite (in particular α > 1 − 1/2d in the homogeneous case m(r) = r α ). Then any ¯ m) have finite distance Wm,Ω (μ0 , μ1 ) < +∞ and the topology two measures μ0 , μ1 ∈ M+ (Ω, ¯ m) coincides with the usual weak∗ topology. In particuinduced by Wm,Ω on the space M+ (Ω, + ¯ m), Wm,Ω ) is compact and separable. lar, the metric space (M (Ω, Proof. We fix an open set B with compact closure in Ω and a reference measure λ = ρL ¯ d |B d with λ(Ω) = m and 0 < ρ(x) ¯ b for L -a.e. x in B. Since Wm,Ω satisfies the triangular inequality, the first part of the theorem follows if we show that Wm,Ω (λ, μ) < +∞ for every ¯ m). μ ∈ M+ (Ω, Let r : B → Ω¯ be the Brenier map pushing λ onto μ: we know that r is cyclically monotone. We set r s := (1 − s)i + sr with image Bs ⊂ (1 − s)B¯ + s Ω¯ ⊂ Ω¯ and inverse s s = r −1 s : Bs → B, and v s := (r − i) ◦ r −1 s = i − s s . It is well known that s s is a Lipschitz map with Lipschitz constant bounded by (1 − s)−1 and that the curve μs := (r s )# λ belongs to CEΩ (λ → μ) with μs = ρ¯s L d |Bs ,
ρ¯s = χ Bs ρ(s ¯ s )Js ,
Js := det Ds s ,
ν s = ρ¯s v s L d .
Since the map r → r/m(r) is nondecreasing and Js (1 − s)−d , it follows that Φm,Ω (μs , ν s ) = Bs
ρ¯s2 |v s |2 dx = m(ρ¯s )
b(1 − s)−d m(b(1 − s)−d )
B
2 ρ(y)J ¯ s (r s (y)) r(y) − y ρ(y) ¯ dy m(ρ(y)J ¯ s (r s (y)))
r(y) − y 2 ρ(y) ¯ dy =
B
Taking the square root and applying (3.3), since
b(1 − s)−d W 2 (λ, μ). m(b(1 − s)−d ) 2
(3.10)
(3.11)
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1 0
b(1 − s)−d m(b(1 − s)−d )
1/2
b1/d ds = d
+∞
z m(z)
1/2
1291
z−1−1/d dz = b1/d Km,d (b)
b
we get the estimate Wm,Ω (λ, μ) b1/d Km,d (b) W2 (λ, μ).
(3.12)
A completely analogous calculation with μ := μ0 (resp. μ := μ1 ) and μs = μ0,s (resp. μs = μ1,s ) shows that Wm,Ω (μi,1−ε , μi ) b1/d Km,d bε −d W2 (λ, μi ) ∀ε > 0, i = 0, 1. On the other hand, taking into account that the density of μi,1−ε is bounded by bε −d , we can apply (3.12) with μ0,1−ε instead of λ, obtaining Wm,Ω (μ0,1−ε , μ1,1−ε ) b1/d ε −1 Km,d bε −d W2 (μ0,1−ε , μ1,1−ε ). Therefore, the triangular inequality yields Wm,Ω (μ0 , μ1 ) b1/d Km,d bε −d W2 (μ0 , λ) + W2 (μ1 , λ) + ε −1 W2 (μ0,1−ε , μ1,1−ε ) . Applying this estimate to a sequence μn weakly∗ converging to μ (and therefore converging also with respect to W2 ), since the corresponding geodesic interpolants with λ μn,1−ε converge to μ1−ε as n → ∞ with respect to W2 , we easily obtain lim sup Wm,Ω (μn , μ) 2b1/d Km,d bε −d W2 (μ, λ). n→∞
Since limε↓0 Km,b (bε −d ) = 0, taking ε arbitrarily small, we conclude.
2
In the next result we do not assume any particular condition on m, but we ask that μi L d with densities satisfying some extra integrability assumptions. Theorem 3.8. Let Ω be a bounded, open convex set of Rd and assume that M = +∞, m > 0. If the measures μi = ρi L d |Ω ∈ M+ (Ω, m), i = 0, 1, satisfy Ω
ρi (x)2 dx < +∞, m(ρi (x))
i = 0, 1,
(3.13)
then Wm,Ω (μ0 , μ1 ) < +∞. Proof. We argue as in the previous proof, keeping the same notation and observing that for 0 s 1/2, (3.11) yields Φm,Ω (μs , ν s )
b 2d W 2 (λ, μ). m(b 2d ) 2
(3.14)
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When 1/2 s 1 we invert the role of λ and μ = ρL d in (3.10) obtaining Φm,Ω (μs , ν s ) = Ω
2 ρ(y)J˜s (˜s s (y)) s˜ (y) − y ρ(y) dy ˜ m(ρ(y)Js (˜s s (y)))
(3.15)
where s˜ s = (1 − s)s + si is the optimal map pushing μ onto μs and J˜s = det D˜s −1 s satisfies J˜s s −d . (3.15) then yields for 1/2 s 1, Φm,Ω (μs , ν s ) 2d+1
2 ρ(y)2 s˜ (y) + |y|2 dy. m(ρ(y))
(3.16)
Ω
Since the range of s˜ (y) is μ-essentially bounded, the integral in (3.16) is finite thanks to (3.13). Integrating (3.14) in (0, 1/2) and (3.16) in (1/2, 1) we conclude that Wm,Ω (λ, μ) is finite. 2 4. Geodesic convexity of integral functionals and their gradient flows 4.1. Nonlinear diffusion equations: weak and limit solutions We consider a 2,1 (0, M) with mU ∈ L1loc [0, M) convex density function U ∈ Wloc
(4.1a)
and a pressure function P : [0, M) → R defined by r P (r) :=
m(z)U (z) dz.
(4.1b)
0 1,1 Let us observe that P ∈ Wloc ([0, M)) is nondecreasing, continuous, and P (0) = 0. When U has a superlinear growth at +∞ the corresponding internal energy functional U : D(U ) ⊂ d M+ c (R ) → (−∞, +∞] is defined as
U (μ) :=
U ρ(x) dx,
d d 1 . D(U ) := μ = ρL d ∈ M+ c R : U (ρ) ∈ L R
(4.2)
Rd
Since U is bounded from below by a linear function and μ has compact support, the integral in (4.2) is always well defined. U is lower semicontinuous with respect to weak convergence in d M+ c (R ) if and only if U (+∞) := lim
r↑+∞
U (r) = lim U (r) = +∞. r↑+∞ r
When U (+∞) < +∞ we define the functional U as
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U (μ) :=
U (ρ) dx + U (+∞)μ⊥ Rd ,
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μ = ρL d + μ⊥ ,
Rd
where μ⊥ is the singular part of μ in the usual Lebesgue decomposition. Let Ω ⊂ Rd be a bounded, open, and connected set with Lipschitz boundary ∂Ω and exterior unit normal n. We will often suppose that Ω is convex in the sequel. We consider the homogeneous Neumann boundary value problem for the nonlinear diffusion equation ∂t ρ − P (ρ) = 0 in (0, +∞) × Ω,
∂n P (ρ) = 0
on (0, +∞) × ∂Ω,
(4.3)
with nonnegative initial condition ρ(0, ·) = ρ0 . We also introduce the dissipation rate of U along the flow by D(ρ) =
|∇P (ρ)|2 dx = m(ρ)
Ω
φm ρ, ∇P (ρ) dx
∀0 ρ ∈ L1 (Ω), P (ρ) ∈ W 1,1 (Ω).
(4.4)
Ω
We collect in the following result some well established facts [41] on weak and classical solutions to (4.3). Theorem 4.1 (Very weak and classical solutions). Let us suppose that Ω is bounded and ρ0 ∈ L∞ (Ω). There exists a unique solution ρ ∈ L∞ ((0, +∞) × Ω) ∩ C 0 ([0, +∞); L1 (Ω)) with P (ρ) ∈ L∞ ((0, +∞) × Ω) ∩ L2 ((0, +∞); W 1,2 (Ω)) to (4.3) satisfying the following weak formulation +∞ 0
ρ∂t ζ − ∇P (ρ) · ∇ζ dx dt = 0 ∀ζ ∈ Cc∞ (0, +∞) × Rd ,
(4.5)
Ω
and the initial condition ρ(0, ·) = ρ0 . The energy U is decreasing along the flow and satisfies the identity Ω
U ρ(T , x) dx +
T 0 Ω
|∇P (ρ)|2 dx dt = m(ρ)
U ρ0 (x) dx
∀T > 0.
(4.6)
Ω
The map ρ0 → St ρ0 := ρ(t, ·) can be extended to a C 0 contraction semigroup S = S(P , Ω) in the positive cone of L1 (Ω), whose curves St ρ0 are also called “limit L1 -solutions” of (4.3), and it satisfies ess infΩ ρ0 St ρ0 ess supΩ ρ0 . If moreover U, m ∈ C ∞ (0, M), U is uniformly convex, Ω is smooth and infΩ ρ0 > 0, then ρ ∈ ¯ and is a classical solution to (4.3). C ∞ ((0, +∞) × Ω) Let us briefly discuss here two useful lemma, whose proof follows from a standard variational argument.
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Lemma 4.2. If ρ0 , U (ρ0 ) ∈ L1 (Ω) then the limit L1 -solution ρ = S(ρ0 ) satisfies P (ρ) ∈ L1loc ([0, +∞); W 1,1 (Ω)), the weak formulation (4.5), and the energy inequality
U ρ(T , x) dx +
Ω
T
|∇P (ρ)|2 dx dt m(ρ)
0 Ω
U ρ0 (x) dx.
(4.7)
Ω
d Proof. Let us first show that we can find a constant C depending only on P , ω := L (Ω), m = Ω ρ dx, and the constant cp in the Poincaré inequality for Ω such that
P (ρ)
L1 (Ω)
C 1 + ∇P (ρ)L1 (Ω)
ρ dx = m,
∀ρ ∈ L1 (Ω),
P (ρ) ∈ W 1,1 (Ω).
(4.8)
Ω
In fact, setting p := inequality yield
Ω
P (ρ) dx and := L d ({x ∈ Ω: P (ρ) p/2}) Poincaré and Chebyshev
1 p(ω − ) 2
P (ρ) − p dx cp
Ω
∇P (ρ) dx,
1 p P (m/), 2
Ω
so that if ω/2 we get p 2P (2m/ω), whereas if ω/2 we obtain p 4ω−1 cp
∇P (ρ) dx.
Ω
If now ρt = St ρ0 is the L1 (Ω)-limit of a sequence ρn,t = St ρn,0 of bounded solutions with U (ρn,0 ) → U (ρ0 ) in L1 (Ω) as n ↑ +∞, from the uniform bound (4.6) we obtain for every bounded Borel set T ⊂ (0, +∞), every B ⊂ Ω, and every nonnegative constants a, b such that m(r) a + br,
∇P (ρn ) dx dt m(ρn )1/2 L1 (T×B)
T B
T×B
Ca
|∇P (ρn )|2 dx dt m(ρn )
1/2
1/2 + bρn L1 (T×B) .
Taking T = (0, T ), B = Ω and applying (4.8), we obtain a uniform bound of the sequence P (ρn ) in L1 (0, T ; W 1,1 (Ω)); since ρn converges to ρ in L1 ((0, T ) × Ω), we obtain that ∇P (ρn ) is uniformly integrable and therefore it converges weakly to ∇P (ρ) in L1 ((0, T ) × Ω). It follows that P (ρ) ∈ L1 (0, T ; W 1,1 (Ω)) and we can then pass to the limit in the weak formulation (4.5) written for ρn , obtaining the same identity for ρ. The inequality (4.7) eventually follows by the same limit procedure, recalling that the dissipation functional (4.4) is lower semicontinuous with respect to weak convergence in L1 (Ω). 2
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The following stability result is used in the sequel; its proof is an easy adaption of [41, Prop. 6.10]. Proposition 4.3. Let Ω n ⊂ Rd be a decreasing sequence of open, bounded, convex sets converging to Ω and let S n = S(P , Ω n ), S(P , Ω) be the associated semigroups provided by Theorem 4.1. If (after a trivial extension to 0 outside Ω n ) ρ0n ∈ L1 (Ω n ) is converging strongly in L1 (Rd ) to ρ0 ∈ L1 (Ω), then Stn (ρ0n ) → St (ρ0 ) in the same L1 sense, as n ↑ +∞ for every t > 0. 4.2. The generalized McCann condition Let us assume that P m ∈ L1loc ([0, M)) and let us introduce a primitive function H of h := = U mm ,
P m
r H (r) := H0 +
P (z)m (z) dz
for some H0 0.
(4.9)
0
Notice that in the most common case when m (0+ ) = limr↓0 r −1 m(r) > 0, the local integrability of h in a right neighborhood of 0 implies the local integrability of m U (which we already required in (4.1a)) and the fact that P is bounded from below. These restrictions can be removed in the 1-dimensional case, see Remark 4.15. Definition 4.4 (Generalized McCann condition). Let U, P , H and m be defined in the interval (0, M) according to (4.1a), (4.1b) and (4.9). We say that the energy density U and the corresponding pressure function P satisfy the d-dimensional generalized McCann condition for the mobility m, denoted by GMC(m, d), if for a suitable choice of H0 U (r)m2 (r) = P (r)m(r) (1 − 1/d)H (r) 0 ∀r ∈ (0, M),
(4.10a)
or, equivalently, r →
H (r) m1−1/d (r)
is nonnegative and nondecreasing in (0, +∞).
(4.10b)
Before analyzing some properties related to GMC(m, d) let us consider in more detail the nonegativity condition of (4.10a) in dimension d = 1 and in the two distinct Cases A–B we introduced in Section 3.2. Dimension d = 1. In the 1-dimensional the generalized McCann condition GMC(m, 1) reduces to the usual convexity of U : we will also comment on this issue in the next Remark 4.15. Case A. M = +∞ and d > 1. The minimal admissible choice for H corresponds to H0 = 0 in (4.9). Notice that the existence of a nonnegative primitive of h = P m in (0, +∞) is in fact equivalent to its local integrability in a right neighborhood of 0. Case B. M < +∞ and d > 1. In this case we have to assume h = P m ∈ L1 (0, M) and we can choose
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M H0 :=
−
< +∞.
P m dx
(4.11)
0
If moreover P is locally Lipschitz near 0 and M, and m(0) = m(M) = 0, then imposing the first inequality of (4.10a) at r = 0+ yields H0 = 0 and at r = M− yields the compatibility condition M 0 P m dr = 0. We collect in the following remarks some simple properties related to this definition. Remark 4.5 (Elementary properties). 1. (Linear mobility) (4.10a) is consistent with the usual McCann condition (1.2a) in the linear case of m(r) = r. 2. (Dimension d = 1) As in the case of McCann condition, in space dimension d = 1 (4.10a) is equivalent to the convexity of U or to the monotonicity of P . 3. (Local boundedness of U when d > 1) In dimension d > 1 the energy density function U is bounded in a right neighborhood of 0 (and in a left neighborhood of M, in the case M < +∞). Since U = P /m the property is immediate if m(0) > 0. If m(0) = 0 then m (0) > 0 and therefore P is bounded around 0 and the formula r0
U (r) = U (r0 ) + U (r0 )(r − r0 ) +
(z − r)+ P (z) dz, m(z)
r ∈ (0, r0 ],
0
shows that limr↓0 U (r) < +∞. 4. (Constant mobility) When m(r) ≡ c > 0 (4.10a) is still equivalent to the convexity of U . 5. (The power-like case) In the case of P (r) = r γ (γ = α + β − 1 if U (r) = r β ) and m(r) = r α , (4.10a) is satisfied if and only if γ 1−
α . d
(4.12)
6. (The case P (r) = r) It is immediate to check that the couple (r, m) always satisfies (4.10a): it corresponds to the entropy function Um whose second derivative is m−1 . After fixing some r0 ∈ (0, M) (the choice r0 = 0 is admissible if m−1 is integrable in a right neighborhood of 0), we obtain r Um (r) :=
r −z dz, m(z)
Pm (r) = r − r0 .
r0
7. (The case of the logarithmic entropy) U (r) = r log r satisfies GMC(r α , d) if and only if γ = α d/(d + 1). 8. (Linearity) If P1 and P2 satisfy GMC(m, d) then also α1 P1 + α2 P2 satisfies GMC(m, d), for every α1 , α2 0. Analogously, if P satisfies GMC(m1 , d) and GMC(m2 , d) then P satisfies GMC(α1 m1 + α2 m2 , d). In particular, if P satisfies GMC(m, d) then P (r) + αr satisfies GMC(m + β, d) for every α, β 0.
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9. (Shift) If M = +∞ and P satisfies GMC(m, d) then P satisfies GMC(m(· + α), d) and P (· − α) satisfies GMC(m, d), for every α 0. The next two properties are more technical and require a detailed proof. Lemma 4.6 (Smoothing). Let us assume that P satisfies GMC(m, d) and let us fix two constants 0 < M < M < M. Then there exists a family Pη , mη , η > 0, with smooth restriction to [M , M ] such that Pη P is strictly increasing, mη m is concave, Pη satisfies GMC(mη , d) (in [0, M ]), and Pη , mη converge uniformly to P , m in [M , M ] as η ↓ 0. Moreover, if P is locally integrable in a right neighborhood of 0, then we can choose M = 0. Proof. When M > 0 it is not restrictive (up to choosing a smaller M ) to assume that M is a ˜ η (r) := m(r) + η, Lebesgue point of the derivative of P . Let H be as in (4.9) and let us set m P˜η (r) = P (r) + ηr, H˜ η (r) = H0 + ηm(0) + η +
r
2
˜ η (r) dr = H (r) + ηm(r) + η2 η2 > 0. P˜η (r)m
0
˜ η , d) and moreover By the previous remark (points 6 and 8) P˜η satisfies GMC(m η η ˜ η − (1 − 1/d)H˜ η = P m − (1 − 1/d)H + (η + m) (m + η) η2 /d. P˜η m d d
(4.13)
By choosing a family of mollifiers hδ , δ > 0, with support in [0, δ], we introduce the functions P˜η,δ (r) := ˜ η,δ (r) := m
P˜η (M ) + P˜η (r)
r
M
P˜η ∗ hδ ds
if r M , if r < M ,
r
˜ η ∗ hδ ds ˜ η (M ) + M m m ˜ η (r) m
if r M , if r < M ,
which are smooth in [M , M ] and satisfy the requested monotonicity/concavity conditions. converges to P˜ in L1 (0, M ] and m ˜ η,δ is uniformly bounded and converges pointSince P˜η,δ η loc ˜ η as δ → 0, we conclude that the corresponding continuous functions H˜ η,δ converge wise a.e. to m uniformly to H˜ η as δ ↓ 0. By (4.13), we can find a sufficiently small δ = δη depending on η such that ˜ (1 − 1/d)H˜ η,δη 0. m P˜η,δ η η,δη
A standard diagonal argument concludes the proof.
2
Lemma 4.7 (Minimal asymptotic behaviour). When d > 1 and M = +∞, the function r Pmin (r) := 0 m(z)−1/d dz satisfies GMC(m, d) and provides an (asymptotic) lower bound for every any other P , since for every r0 > 0 there exists a constant c0 > 0 such that (r) = c0 m(r)−1/d , P (r) c0 Pmin
U (r) c0 m(r)−1−1/d
for a.e. r r0 .
(4.14)
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Proof. In fact f (r) := P (r)m(r) satisfies
r
f (r) (1 − 1/d) H (r0 ) +
f (r)m (r)/m(r) dr . r0
Gronwall Lemma then yields (4.14) with c0 := (1 − 1/d)H (r0 )m(r0 )1/d−1 .
2
Notice that in the case m(r) = r α we obtain the functions Pmin (r) = c r γ0 with exponent γ0 = 1 − α/d, which is consistent with (4.12). The corresponding energy density functions are then Umin (r) = cr 2−α(1+1/d) : in particular, when α < d/(d + 1), all the energy functions have a superlinear growth as r ↑ ∞. Remark 4.8 (A sufficient condition). It is possible to give a simpler sufficient condition than (4.10a), at least when mU is integrable in a right neighborhood of 0 and M = +∞: if the map r → m1/d (r)P (r) = m1+1/d (r)U (r) is nondecreasing in (0, +∞)
(4.15)
then (4.10a) is satisfied. In fact, assuming U smooth for simplicity, (4.15) is equivalent to 0 m1/d P + 1/d m1/d−1 m P . Multiplying the inequality by m1−1/d and integrating from 0 to r we get (4.10a). Condition (4.15) gives the same sharp bound (4.12) in the power case. 4.3. The metric approach to gradient flows We recall here some basic facts about the metric notion of gradient flows, referring to [2] for further details. Let (D, W) be a metric space, not assumed to be complete, and let V : D(V ) → (−∞, +∞] be a lower semicontinuous functional. A family of continuous maps St : D → D, t 0, is a C 0 -(metric) contraction gradient flow of V with respect to W if
St+h (u) = Sh St (u) ,
lim St (u) = S0 (u) = u t↓0
1 1 2 W St (u), v − W2 (u, v) t V (v) − V St (u) 2 2
∀u ∈ D, t, h 0,
(4.16a)
∀t > 0, u ∈ D, v ∈ D(V ). (4.16b)
Thanks to [16, Prop. 3.1], conditions (4.16a), (4.16b) imply
St (D) ⊂ D(V ) ∀t > 0 and the map t → V St (u) is not increasing in (0, +∞),
1 d+ 2 W St (u), v + V St (u) V (v) ∀u ∈ D, v ∈ D(V ), t 0, 2 dt 1 V St (u) V (v) + W2 (u, v) ∀u ∈ D, v ∈ D(V ), t > 0, 2t W2 St1 (u), St0 (u) 2(t1 − t0 ) V (St0 u) − Vinf ∀u ∈ D(V ), 0 t0 t1 , W St (u), St (v) W(u, v) ∀u, v ∈ D, t 0.
(4.17)
(4.18)
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In (4.17) we used the usual notation d+ ζ (t + h) − ζ (t) ζ (t) = lim sup dt h h→0+ for every real function ζ : [0, +∞) → R. The following approximated convexity estimate [16, Theorem 3.2] plays an important role in the sequel. Theorem 4.9 (Approximated convexity). Let us suppose that S is metric contraction gradient flow of V with respect to W according to (4.16a), (4.16b) and let s → us ∈ D, s ∈ [0, 1], be a Lipschitz (“almost” geodesic) curve such that u0 , u1 ∈ D(V ) and W(ur , us ) L|r − s|
∀r, s ∈ [0, 1],
L2 W2 (u0 , u1 ) + δ 2 .
(4.19)
Then for every s ∈ [0, 1] and t > 0, we have s(1 − s) 2 δ . V St (us ) (1 − s)V (u0 ) + sV (u1 ) + 2t In particular, if us is a minimal geodesic, i.e. (4.19) holds with δ = 0, then V (us ) (1 − s)V (u0 ) + sV (u1 )
∀s ∈ [0, 1].
4.4. Main results We state our main result about the generation of a contractive gradient flow of U with respect to Wm,Ω . Theorem 4.10 (Contractive gradient flow). Let us assume that Ω is a bounded, convex open set, and the functions U, P , H satisfy the generalized McCann condition GMC(m, d). For every reference measure σ ∈ M+ (Ω) with finite energy U (σ ) < +∞ the functional U generates a unique metric contraction gradient flow S = S(U , m, Ω) in the space D := μ ∈ M+ (Ω): μ L d |Ω , Wm,Ω (μ, σ ) < +∞, U (μ) < +∞ endowed with the distance Wm,Ω . Moreover S is characterized by the formula St μ0 = ρt L d |Ω , where ρt = St ρ0 is a limit L1 -solution of (4.3). When m satisfies the finiteness condition of Theorem 3.7 (in particular m(r) = r α with α > 1 − 2/d) we obtain a much more refined result, which in particular shows the continuous dependence of S on the weak∗ topology. Corollary 4.11. Under the same assumptions on Ω, U, P of the previous theorem, if moreover M = +∞ and m satisfies the finiteness condition of Theorem 3.7, then the semigroup S can be ¯ m), which is continuuniquely extended to a contraction semigroup on every convex set M+ (Ω, ∗ ous with respect to the weak convergence of the initial data. If U has a superlinear growth, then St (μ0 ) = ρt L d L d |Ω for every t > 0 and ρt is a weak solution of (4.3) according to (4.5).
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We conclude this section with our main convexity result. Theorem 4.12 (Convexity). Let us assume that Ω is a bounded convex open set, and the functions U, P , H satisfy the generalized McCann condition GMC(m, d). For every μ0 , μ1 ∈ M+ (Ω) ∩ D(U ) with finite distance Wm,Ω (μ0 , μ1 ) < +∞ there exists a constant speed minimizing geodesic for Wm,Ω , μ : [0, 1] → M+ (Ω) connecting μ0 to μ1 such that U (μs ) sU (μ1 ) + (1 − s)U (μ0 )
∀s ∈ [0, 1].
(4.20)
The proof of Theorems 4.10 and 4.12 will be developed in the next two sections. Remark 4.13 (Weak and strong convexity). When (4.20) holds for all the (constant speed, minimizing) geodesics, the functional U is called strongly geodesically convex. When m is strictly concave and has a sublinear growth (or M < +∞) then every two measures with finite Wm,Ω distance can be connected by a unique geodesic [19, Theorem 5.11], so that there is no difference between strong or weak convexity and (4.20) yields that the map s → U (μs ) is convex in [0, 1]. Remark 4.14 (Absolutely continuous measures). Even when geodesics are not unique, the proof of Theorem 4.12 shows in fact that (4.20) is satisfied by any geodesic μs with μs L d for every s ∈ Rd , which surely exist if U has a superlinear growth. Along this class of geodesics we still obtain that the map s → U (μs ) is convex in [0, 1]. Remark 4.15 (The one-dimensional case). When the space dimension d = 1, then the generalized McCann condition GMC(m, 1) reduces to the usual convexity of U . In this case, a simple approximation argument shows that we can cover also the case of functions U which are not bounded in a right neighborhood of 0 (and in a left neighborhood of M, if M < +∞) and the integrability assumptions on U m of (4.1a) and on U mm of (4.11) can be dropped. 5. Action inequalities in the smooth case This Section contains the proof of the action inequality in the smooth case. This is a core of the proof of the main result. Indeed, in the next Section, the proof of Theorem 4.10 will be obtained by using the approximation results of Lemma 3.6 and 4.6, and passing to the limit on the inequality (5.13). In this section we assume that Ω is a smooth and bounded open set. We consider a smooth curve μs := ρs L d |Ω ,
ρ ∈ C ∞ [0, 1] × Ω¯ , 0 < m0 ρ m1 < M, μs (Ω) ≡ m,
s ∈ [0, 1].
(5.1)
We also assume that P and m are of class C ∞ in [m0 , m1 ]. We consider the semigroup S = S(P , Ω) defined by Theorem 4.1 and we set μs,t := ρs,t L d |Ω ,
ρs,t (·) = ρ(s, t, ·) := Sst ρs ,
s ∈ [0, 1], t 0.
(5.2)
Classical theory of quasilinear parabolic equation shows that ρ ∈ C ∞ ([0, 1] × [0, ∞) × Ω) ∩ ¯ C ∞ ([0, 1] × (0, +∞) × Ω).
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Since the semigroup St preserve the lower and upper bounds on ρ and Ω ∂s ρ dx = 0, for ¯ every (s, t) ∈ [0, 1] × [0, +∞) we can introduce the unique solution ζs,t = ζ (s, t, ·) ∈ C ∞ (Ω), of the uniformly elliptic Neumann boundary value problem ⎧ −∇ · m(ρ)∇ζ = ∂s ρ ⎪ ⎪ ⎪ ⎨ ∇ζ · n = 0 ⎪ ⎪ ζ (x) dx = 0. ⎪ ⎩
in Ω, on ∂Ω,
(5.3)
Ω
It is easy to check that ζ depends smoothly on s and t. Notice that (5.3) is equivalent to
m(ρ) ∇ζ · ∇η dx = Ω
∂s ρ η dx
¯ ∀η ∈ C 1 (Ω).
(5.4)
Ω
By construction, for every t 0 the curve s → (μs,t , ν s,t ) with ν s,t := m(ρs,t )∇ζs,t L d |Ω belongs to CEΩ (μ0 → μ1,t ) and its energy can be evaluated by integrating the action As,t := Φm,Ω (μs,t , ν s,t ) =
m(ρs,t )|∇ζs,t |2 dx Ω
with respect to s in the interval [0, 1]. The integral provides an upper bound of the Wm,Ω -distance between μ0 and μ1,t = ρ1,t L d , which corresponds to the solution of the nonlinear diffusion equation (4.3) with initial datum ρ1 . As it was shown in [16], evaluating the time derivative of the action As,t is a crucial step to prove that (4.3) satisfies the EVI formulation (4.16b). Next lemma, which does not require any convexity assumption on Ω, collects the main calculations. Lemma 5.1. Let ρs , ρs,t , and ζs,t be as in (5.1), (5.2), and (5.3). Then for every (s, t) ∈ [0, 1] × (0, +∞) we have 1 ∂ ∂ 1 As,t = |∇ζs,t |2 m(ρs,t ) dx 2 ∂t ∂t 2 =−
Ω
∇P (ρs,t ) · ∇ζs,t dx Ω
−s
2 P (ρs,t )m(ρs,t ) − H (ρs,t ) (ζs,t )2 + H (ρs,t ) D 2 ζs,t dx
Ω
+s Ω
1 + s 2
P (ρs,t )m (ρs,t )|∇ρs,t |2 |∇ζs,t |2 dx
∂Ω
where H is defined in (4.9).
H (ρs,t )∇|∇ζs,t |2 · n dH d−1 ,
(5.5)
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Proof. For keep the notation simple, we omit the explicit dependence of ρ, ζ on s, t. By the definition of ρ we easily get ∂t ρ = sP (ρ)
∇P (ρ) · ∇η dx = −
and s Ω
∂t ρ η dx
¯ ∀η ∈ C 1 (Ω).
(5.6)
Ω
Further differentiation with respect to s yields
∇P (ρ) · ∇η dx + s
− Ω
∂s P (ρ)η dx =
Ω
∂st ρη dx
(5.7)
Ω
¯ with ∇η · n = 0 on ∂Ω. On the other hand, differentiating (5.4) with respect for all η ∈ C 2 (Ω) to t we obtain ¯ m(ρ)∂t ∇ζ · ∇η dx = ∂st ρ η dx − ∂t m(ρ)∇ζ · ∇η dx ∀η ∈ C 1 (Ω). (5.8) Ω
Ω
Ω
The time derivative of the action functional is 1 ∂ 1 2 2 |∇ζ | m(ρ) dx = ∂t m(ρ)|∇ζ | dx + ∂t ∇ζ · ∇ζ m(ρ) dx ∂t 2 2 Ω (5.8)
=
Ω (5.7)
Ω
1 ∂st ρζ dx − 2
Ω
∂t m(ρ)|∇ζ |2 dx Ω
= −
∇P (ρ) · ∇ζ dx + s Ω
∂s P (ρ)ζ dx − Ω
1 2
∂t m(ρ)|∇ζ |2 dx Ω
= I + II + III.
(5.9)
We evaluate separately the various contributions: concerning the second integral II we introduce the auxiliary function G G(r) := P (r)m(r) − H (r),
so that G (r) = P (r)m(r),
and we get II = s
∂s P (ρ)ζ dx Ω
= s
P (ρ)∂s ρζ dx
Ω (5.4)
= s Ω
P (ρ)m(ρ)∇ζ · ∇ζ dx + s
Ω
ζ P (ρ)m(ρ)∇ρ · ∇ζ dx
(5.10)
J.A. Carrillo et al. / Journal of Functional Analysis 258 (2010) 1273–1309
(5.10)
= s
P (ρ)m(ρ)∇ζ · ∇ζ dx + s Ω
H (ρ)∇ζ · ∇ζ dx − s Ω
ζ ∇G(ρ) · ∇ζ dx Ω
= s
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G(ρ)(ζ )2 dx. Ω
The third integral of (5.9) is 1 2
III = −
(5.6) s = 2
m (ρ)∂t ρ|∇ζ |2 dx
Ω
P (ρ)m (ρ)|∇ρ|2 |∇ζ |2 dx +
s 2
Ω
(4.9) s = 2
P (ρ)m (ρ)∇|∇ζ |2 · ∇ρ dx
Ω
s P (ρ)m (ρ)|∇ρ|2 |∇ζ |2 dx + 2
Ω
∇|∇ζ |2 · ∇H (ρ) dx.
(5.11)
Ω
A further integration by parts in the last integral and the Bochner formula 1 ∇ζ · ∇ζ − |∇ζ |2 = −|D 2 ζ |2 2 yield (5.5).
2
Corollary 5.2. Under the same notation and assumptions of Lemma 5.1, if Ω is convex and U satisfies the generalized McCann condition GMC(m, d) (4.10a), (4.10b), then 1 ∂ ∂ 1 As,t = 2 ∂t ∂t 2
|∇ζs,t |2 m(ρs,t ) dx −
∂ ∂ U (μs,t ) = − ∂s ∂s
Ω
U (ρs,t ) dx,
(5.12)
Ω
and 1 2 1 Wm,Ω (μ1,t , μ0 ) + tU (μ1,t ) 2 2
1 As,t ds + 0
1 2
t U (μ1,τ ) dτ 0
1 As,0 ds + tU (μ0 ).
(5.13)
0
Proof. We determine the sign of the terms in the right-hand side of (5.5) thanks to (4.10a), (4.10b) and the convexity of Ω. Recalling that |D 2 ζ | d1 (ζ )2 and H 0 we obtain that the second integral in the right-hand side of (5.5) is nonpositive − Ω
2 (4.10a) P (ρ)m(ρ) − H (ρ) (ζ )2 + H (ρ) D 2 ζ dx 0.
(5.14)
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Since P is increasing and m is concave, we have P (ρ)m (ρ) 0 which yields
P (ρ)m (ρ)|∇ρ|2 |∇ζ |2 dx 0.
(5.15)
Ω
Since H is nonnegative, the smoothness and convexity of Ω and the smoothness of ζ yields, see [24,25,34], 2 H (ρ)∇|∇ζ |2 · n dH d−1 0. (5.16) ∇|∇ζ | · n 0, on ∂Ω, ∂Ω
Combining (5.14), (5.15) and (5.16), (5.5) yields the inequality ∂ 1 ∂t 2
|∇ζ | m(ρ) dx −
∇ζ · ∇P (ρ) dx.
2
Ω
(5.17)
Ω
On the other hand ∂ (5.4) U (ρ) dx = U (ρ)∂s ρ dx = m(ρ)∇U (ρ) · ∇ζ dx = ∇P (ρ) · ∇ζ dx, ∂s Ω
Ω
Ω
(5.18)
Ω
so that (5.12) follows by (5.17) and (5.18). Integrating (5.12) with respect to s and t, we obtain the second inequality in (5.13). The first inequality in (5.13) follows from the definition of Wm,Ω and the monotonicity of τ → U (μ1,τ ) (see the energy identity (4.6)). 2 6. Proof of the main theorems 6.1. The generation result Recall that St (μ0 ) = St (ρ0 )L d |Ω when μ0 = ρ0 L d |Ω ; Theorem 4.10 is an immediate consequence of the following result. Theorem 6.1. Let Ω be a bounded, convex open set of Rd and let us assume that μi ∈ ¯ i = 0, 1, have finite distance Wm,Ω (μ0 , μ1 ) < +∞ and satisfy U (μ0 ) < +∞ and M+ (Ω), μ1 = ρ1 L d |Ω L d . If U satisfies the generalized McCann condition GMC(m, d) (4.10a), (4.10b) then 1 2 1 W (St μ1 , μ0 ) + tU (St μ1 ) W2m,Ω (μ1 , μ0 ) + tU (μ0 ) 2 m,Ω 2
∀t 0.
Proof. Since Wm,Ω (μ0 , μ1 ) < +∞ there exists a geodesic curve (μ, ν) ∈ CEΩ (μ0 → μ1 ) such that 1 Φm,Ω (μs , ν s ) ≡
Φm,Ω (μs , ν s ) ds = W2m,Ω (μ0 , μ1 ). 0
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Applying Lemma 3.6, we find a family of approximating curves (με,δ , ν ε,δ ) and smooth convex open sets Ωε satisfying (3.7), (3.8) and 1
Φm,Ωε με,δ , ν ε,δ ds cε2 W2m,Ω (μ0 , μ1 ).
0
Let mη , Pη as in Lemma 4.6, for constants 0 < M < M < M such that M ρ ε,δ M in Ωε , ε,δ,η ∈ C ∞ (Rd ) obtained by solving (5.3) with respect to mη and ∂s ρ ε,δ in Ωε . Since and let ζs mη m, by Theorem 5.21 of [19] we easily have
ε,δ,η 2 ε,δ ∇ζ mη ρ dx = Φm
η ,Ωε
s
ε,δ ε,δ μ ,ν Φm,Ωε με,δ , ν ε,δ ∀η > 0, s ∈ [0, 1].
Ωε
If Sε,η = S(Uη , mη , Ωε ) is the semigroup associated with S(Pη , Ωε ) and the corresponding integral functional Uη , (5.13) gives ε,η ε,η 1 1 2 + tUη St με,δ , με,δ Wmη ,Ωε St με,δ 1 0 1 2 2
1
ε,δ,η 2 ε,δ ∇ζ mη ρ dx ds + tUη με,δ s
0
0 Ωε
cε2 2 . Wm,Ω (μ1 , μ0 ) + tUη με,δ 0 2
Passing to the limit as η ↓ 0 (notice that the functions St ρ ε,δ take their values in [M , M ]), we get ε,η
cε2 2 1 2 ε ε,δ ε,δ Wm,Ωε St μ1 , μ0 + tU Sεt με,δ Wm,Ω (μ1 , μ0 ) + tU με,δ 1 0 2 2 where Sε = S(U , m, Ωε ) is associated with S(P , Ωε ). We can then pass to the limit as δ ↓ 0: since ρ ε,δ → ρ ε in L∞ (Ωε ) we immediately have c2 1 2 ε ε ε Wm,Ωε St μ1 , μ0 + tU Sεt με1 ε W2m,Ω (μ1 , μ0 ) + tU με0 . 2 2 Finally as ε ↓ 0 we conclude, recalling Proposition 4.3.
2
6.2. Geodesic convexity The proof of Theorem 4.12 follows immediately from the generation result Theorem 6.1 and Theorem 4.9 if every measure μs of the geodesic curve is absolutely continuous with respect to L d |Ω (see also Remark 4.14). On the other hand, this property is not known a priori, so we need a more refined argument. Proof. As in Proposition 3.5 we set μεs := μs ∗ kε , ν εs := ν s ∗ kε and we denote by Sε = S(U , m, Ωε ). By (3.5) and the contraction property given by Theorem 4.10 in Ωε we have
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Wm,Ωε Sεt μεs1 , Sεt μεs2 Wm,Ωε μεs1 , μεs2 Wm,Ω (μs1 , μs2 ) = |s1 − s2 |Wm,Ω (μ0 , μ1 )
(6.1)
and by (3.6), Theorem 4.9 and (4.18) we have δε2 := W2Ω,m (μ0 , μ1 ) − W2m,Ωε με0 , με1 → 0 as ε ↓ 0, δ2 U Sεt μεs (1 − s)U με0 + sU με1 + ε s(1 − s), 2t ε ε ε 2 Wm,Ωε St μi , μi t U μi − inf U t U (μi ) − inf U ,
(6.2) (6.3)
where the second inequality in (6.3) follows from Jensen’s inequality. We choose now a countable set C dense in [0, 1] and containing 0 and 1, a vanishing sequence (tk )k∈N and another vanishing sequence (εk )k∈N so that limk↑+∞ tk−1 δε2k = 0. By compactness and a standard diagonal argument, up to extracting a further subsequence, we can find limit points μ˜ s for s ∈ C such that ε
Stkk μεs k μ˜ s
weakly as k ↑ +∞.
By (6.1) and Proposition 3.3 we get Wm,Ω (μ˜ s1 , μ˜ s2 ) |s1 − s2 |Wm,Ω (μ0 , μ1 )
∀s1 , s2 ∈ C .
(6.4)
˜ (6.3) yields μ˜ 0 = μ0 , μ˜ 1 = μ1 so that we can extend μ˜ to a continuous curve (still denoted by μ) connecting μ0 and μ1 still satisfying (6.4) for every s1 , s2 ∈ [0, 1]. The triangular inequality shows that (6.4) is in fact an equality and the curve μ˜ is a constant speed minimizing geodesic. On the other hand, the lower semicontinuity of U with respect to weak convergence and (6.2) yields U (μ˜ s ) (1 − s)U (μ0 ) + sU (μ1 )
∀s ∈ C .
(6.5)
A further lower semicontinuity and density argument shows that (6.5) holds for every s ∈ [0, 1]. 2 7. Final remarks and open problems This paper is a first step towards the investigation of the geometry of spaces of measures metrized by Wm,Ω , the induced convexity notions for integral functionals and the corresponding generation of gradient flows with applications to various nonlinear evolutionary PDE’s. Since a sufficiently general theory is far from being developed and understood, it is in some sense surprising that one can reproduce in this setting the celebrated McCann convexity result. On the other hand, many interesting and basic problems remain open: here is just a provisional list.
J.A. Carrillo et al. / Journal of Functional Analysis 258 (2010) 1273–1309
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– At the level of the distance Wm,Ω only partial results on some basic properties (such as density of regular measures or necessary and sufficient conditions ensuring the finiteness of the distance), are known and a complete and accurate picture is still missing. – The situation is even less clear in the case of unbounded domains: in this paper we restricted our attention to the bounded case only. – The study of other integral functionals is completely open, as well as applications to different types of evolution equations, like scalar conservation laws (2.4), interaction equation (2.5) or nonlinear fourth order equation (1.12). – It would be interesting to study other metric quantities (e.g. the metric slope) and the pseudo-Riemannian structure (tangent space, Alexandrov curvature, etc.) connected with the distance and the energy functionals, see [2,12]. – The regularization properties and asymptotic behaviour of the gradient flow U and its perturbation can be studied as well: in the Wasserstein case the geodesic convexity of a functional yields many interesting estimates. – The convergence of the so called “Minimizing movement” or JKO-scheme could be exploited in this and other situations: in the case of geodesically convex energies, further information on the Alexandrov curvature of the distance Wm,Ω would be crucial, see [2,38]. Acknowledgments J.A.C. acknowledges the support from DGI-MICINN (Spain) project MTM2008-06349-C0303. G.S. and S.L. have been partially supported by MIUR-PRIN’06 grant for the project “Variational methods in optimal mass transportation and in geometric measure theory”. D.S. was partially supported by NSF grant DMS-0638481. D.S. would also like to thank the Center for Nonlinear Analysis (NSF grants DMS-0405343 and DMS-0635983) for its support during the preparation of this paper. We thank the referee for the careful reading of the manuscript and a number of useful suggestions. References [1] M. Agueh, N. Ghoussoub, X. Kang, Geometric inequalities via a general comparison principle for interacting gases, Geom. Funct. Anal. 14 (14) (2004) 215–244. [2] L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures Math. ETH Zürich, Birkhäuser Verlag, Basel, 2005. [3] L. Ambrosio, G. Savaré, Gradient flows of probability measures, in: C.M. Dafermos, E. Feireisl (Eds.), Handbook of Differential Equations: Evolutionary Equations, vol. 3, Elsevier, 2006, pp. 1–136. [4] W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc. 105 (1989) 397–400. [5] J.-D. Benamou, Y. Brenier, A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem, Numer. Math. 84 (2000) 375–393. [6] A. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc. 45 (1998) 689–697. [7] Y. Brenier, Extended Monge–Kantorovich theory, in: Optimal Transportation and Applications, Martina Franca, 2001, in: Lecture Notes in Math., vol. 1813, Springer, Berlin, 2003, pp. 91–121. [8] M. Burger, M. di Francesco, Y. Dolak, The Keller–Segel model for chemotaxis with prevention of overcrowding: Linear vs. nonlinear diffusion, SIAM J. Math. Anal. 38 (2006) 1288–1315. [9] M. Burger, M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media 3 (2008) 749–785. [10] J.A. Carrillo, P. Laurençot, J. Rosado, Fermi–Dirac–Fokker–Planck equation: Well-posedness & long-time asymptotics, J. Differential Equations 247 (2009) 2209–2234.
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[11] J.A. Carrillo, R.J. McCann, C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana 19 (2003) 1–48. [12] J.A. Carrillo, R.J. McCann, C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal. 179 (2006) 217–263. [13] J.A. Carrillo, J. Rosado, F. Salvarani, 1D nonlinear Fokker–Planck equations for fermions and bosons, Appl. Math. Lett. 21 (2008) 148–154. [14] F.A.C.C. Chalub, J.F. Rodrigues, A class of kinetic models for chemotaxis with threshold to prevent overcrowding, Port. Math. (N.S.) 63 (2006) 227–250. [15] D. Cordero-Erausquin, B. Nazaret, C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo– Nirenberg inequalities, Adv. Math. 182 (2) (2004) 307–332. [16] S. Daneri, G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal. 3 (2008) 1104–1122. [17] Y. Dolak, C. Schmeiser, The Keller–Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math. 66 (2005) 286–308. [18] M. Di Francesco, J. Rosado, Fully parabolic Keller–Segel model for chemotaxis with prevention of overcrowding, Nonlinearity 21 (11) (2008) 2715–2730. [19] J. Dolbeault, B. Nazaret, G. Savaré, A new class of dynamic transport distances between measures, Calc. Var. Partial Differential Equations 34 (2009) 193–231. [20] J. Dolbeault, B. Nazaret, G. Savaré, Beckner inequalities and weighted transport distances, in preparation. [21] T.D. Frank, Classical Langevin equations for the free electron gas and blackbody radiation, J. Phys. A 37 (2004) 3561–3567. [22] T.D. Frank, Nonlinear Fokker–Planck Equations, Springer Ser. Synergetics, Springer, 2005. [23] G. Giacomin, J. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits, J. Stat. Phys. 87 (1997) 37–61. [24] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math., vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. [25] U. Gianazza, G. Savaré, G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal. 194 (2009) 133–220. [26] T. Hillen, K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding., Adv. in Appl. Math. 26 (4) (2001) 280–301. [27] D. Horstmann, From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (2003) 103–165. [28] G. Kaniadakis, Generalized Boltzmann equation describing the dynamics of bosons and fermions, Phys. Lett. A 203 (1995) 229–234. [29] G. Kaniadakis, P. Quarati, Kinetic equation for classical particles obeying an exclusion principle, Phys. Rev. E 48 (1993) 4263–4270. [30] S. Lisini, Characterization of absolutely continuous curves in Wasserstein spaces, Calc. Var. Partial Differential Equations 28 (2007) 85–120. [31] S. Lisini, A. Marigonda, On a class of modified Wasserstein distance induced by concave mobility functions defined on bounded intervals, preprint available on http://cvgmt.sns.it/people/lisini/. [32] D. Matthes, R.J. McCann, G. Savaré, A family of nonlinear fourth order equations of gradient flow type, arXiv: 0901.0540v1, 2009. [33] R.J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997) 153–179. [34] F. Otto, The geometry of dissipative evolution equation: the porous medium equation, Comm. Partial Differential Equations 26 (2001) 101–174. [35] F. Otto, M. Westdickenberg, Eulerian calculus for the contraction in the Wasserstein distance, SIAM J. Math. Anal. 37 (2005) 1227–1255. [36] B. Perthame, A.L. Dalibard, Existence of solutions of the hyperbolic Keller–Segel model, Trans. Amer. Math. Soc. 361 (2009) 2319–2335. [37] F. Santambrogio, Absolute continuity and summability of transport densities: Simpler proofs and new estimates, Calc. Var. Partial Differential Equations 36 (2009) 343–354. [38] G. Savaré, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, C. R. Math. Acad. Sci. Paris 345 (2007) 151–154. [39] D. Slepˇcev, Coarsening in nonlocal interfacial systems, SIAM J. Math. Anal. 40 (2008) 1029–1048. [40] J. Sopik, C. Sire, P.H. Chavanis, Dynamics of the Bose–Einstein condensation: Analogy with the collapse dynamics of a classical self-gravitating Brownian gas, Phys. Rev. E 74 (2006) 011112.
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Journal of Functional Analysis 258 (2010) 1310–1360 www.elsevier.com/locate/jfa
Spectral asymptotics for Laplacians on self-similar sets Naotaka Kajino 1 Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan Received 4 February 2009; accepted 2 November 2009 Available online 13 November 2009 Communicated by L. Gross
Abstract Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a ‘geometric counting function’ defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace–Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function. © 2009 Elsevier Inc. All rights reserved. Keywords: Self-similar sets; Dirichlet forms; Eigenvalue counting function; Partition function; Short time asymptotics; Sub-Gaussian heat kernel estimate; Sierpinski carpets
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics on self-similar sets . . . . . . . . . . . . . . . . . . . . . 2.1. Scales on the shift space . . . . . . . . . . . . . . . . . 2.2. Self-similar structures and measures . . . . . . . . . 2.3. Systems of neighborhoods associated with scales
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E-mail address: [email protected]. URL: http://www-an.acs.i.kyoto-u.ac.jp/~kajino.n/. 1 JSPS Research Fellows DC (20·6088): Supported by Japan Society for the Promotion of Science. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.001
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2.4. Qdistances adapted to scales and cell-counting dimension . Framework: Self-similar Dirichlet spaces . . . . . . . . . . . . . . . . . Spectral and geometric counting functions . . . . . . . . . . . . . . . . Short time asymptotics of the partition function . . . . . . . . . . . . Rational boundary and cell-counting dimension . . . . . . . . . . . . Sharpness of the key estimate . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Intersection type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Sharpness of the key estimate . . . . . . . . . . . . . . . . . . . . 7.3. Positivity of capacity for subsets of the boundary . . . . . . 8. Examples: Sierpinski carpets . . . . . . . . . . . . . . . . . . . . . . . . . 9. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Miscellaneous lemmas for Section 7 . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. 4. 5. 6. 7.
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1. Introduction Mathematical analysis on fractal spaces began when Goldstein [19] and Kusuoka [31] had constructed the Brownian motion on the Sierpinski gasket (Fig. 1.1 below), whose transition density (heat kernel) has proved to be subject to the two-sided sub-Gaussian estimate by the result of Barlow and Perkins [10]. Since then many results have been obtained concerning the spectra of Laplacians on self-similar sets. For example, let {λSG n }n∈N be the non-decreasing enumeration of the eigenvalues of the Laplacian associated with the Brownian motion on the Sierpinski gasket, where each eigenvalue is repeated according to its multiplicity. The corresponding eigenvalue counting function is defined by NSG (x) := # n ∈ N λSG n x
(1.1)
for each x ∈ [0, ∞), where #A denotes the number of all the elements of a set A. By the results of Fukushima and Shima [18], Kigami and Lapidus [30] and Barlow and Kigami [8], there exists a log 5-periodic right-continuous discontinuous function G : R → (0, ∞) with 0 < infR G < supR G < ∞, such that NSG (x) = x dS /2 G(log x) + O(1)
(1.2)
as x → ∞, where dS := log 9/ log 5. This result is in remarkable contrast to Weyl’s theorem [35,36] for the Dirichlet Laplacian on bounded open subsets of Euclidean spaces in two important points, as suggested in the early 1980s by Physicists, e.g. Rammal and Toulouse [34] and Rammal [33]. First, the ratio x −dS /2 NSG (x) is bounded away from 0 and ∞ but does not converge as x → ∞. Secondly, the number dS , called the spectral dimension of the Sierpinski gasket, is different from its boxcounting dimension (and its Hausdorff dimension) df = log 3/ log 2 with respect to the Euclidean distance; dS < df . By [30,8], the same kind of result is known to be valid for nested fractals, a class of finitely ramified self-similar sets. The purpose of this paper is twofold. First, we give a geometric characterization of the spectral dimension dS based on a framework due to Kigami [28]. Secondly, we prove the same kind
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Fig. 1.1. The Sierpinski gasket.
Fig. 1.2. The Sierpinski carpet.
Fig. 1.3. The similitudes {Fi }i∈S .
of asymptotic behavior as in (1.2) of the partition function, the Laplace–Stieltjes transform of the eigenvalue counting function, for the case of infinitely ramified self-similar sets such as the Sierpinski carpet (Fig. 1.2). All our results are applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets (see [6,7]), but in this introduction we illustrate the main results by treating the case of the Sierpinski carpet as a particular example. Let {Fi }i∈S , S := {1, . . . , 8}, be a family of similitudes on R2 as described in Fig. 1.3, where the whole square denotes [0, 1]2 . The Sierpinski carpet K is defined as the self-similar set associated with {Fi }i∈S , that is, the unique non-empty compact subset of R2 such that K = i∈S Fi (K). Let V0 := [0, 1]2 \ (0, 1)2 , which should be regarded as the boundary of K:
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In fact, V0 is the smallest subset of K that satisfies Fi (K) ∩ Fj (K) = Fi (V0 ) ∩ Fj (V0 ) for any distinct i, j ∈ S. As #V0 = ∞, K is infinitely ramified. Let ν be the self-similar measure with weight (1/8, . . . , 1/8). By the results of Barlow and Bass [1–4] and Kusuoka and Zhou [32, Section 8], there exists a regular Dirichlet form (E, F ) on L2 (K, ν) satisfying F ⊂ {u | u: K → R, u is continuous} (=: C(K)) and such that E(u, v) =
1 i∈S
r
E(u ◦ Fi , v ◦ Fi ),
u, v ∈ F ,
(1.3)
for some r ∈ (0, 1) (note also the recent result [7] on uniqueness of such (E, F )). Moreover, by looking at [32, Theorems 4.5, 5.4, 6.9 and 7.2], we easily verify that (E, F ) is a resistance form on K whose associated resistance metric is compatible with the original (Euclidean) topology of K. (See [27, Chapter 2] and [29, Part I] for basic theory of resistance forms.) Let μ be a Borel probability measure on K which is elliptic, i.e. there exists γ ∈ (0, ∞) such that μ(Kwi ) γ μ(Kw ) for any w ∈ m∈N∪{0} S m (=: W∗ ) and any i ∈ S, where Fw := Fw1 ◦ · · · ◦ Fwm and Kw := Fw (K) for w = w1 · · · wm ∈ W∗ . Then by [29, Corollary 5.4 and Theorem 8.4], (E, F ) is a regular Dirichlet form on L2 (K, μ). Also, (1.3) implies the strong locality of (E, F ). This Dirichlet space (L := (K, S, {Fi }i∈S ), μ, E, F , r) is the framework of our study. To explain our first main result, let us define several notions concerning the description of the geometryof the space (L, μ, E, F , r). Let |w| := m for w = w1 · · · wm ∈ S m , m ∈ N ∪ {0}. Set g(w) := r |w| μ(Kw ) for w ∈ W∗ and define (1.4) Λs := w1 · · · wm ∈ W∗ g(w1 · · · wm−1 ) > s g(w1 · · · wm ) for s ∈ (0, 1], with the convention that g(w1 · · · wm−1 ) = 2 when m = 0. g is called the gauge function and the collection S := {Λs }s∈(0,1] is called the scale, respectively, associated with the Dirichlet space (L, μ, E, F , r). We regard each Kw , w ∈ Λs (or strictly speaking, the union K (0) (Λs , Kw ) := {Kv | v ∈ Λs , Kv ∩ Kw = ∅}) as a ball of radius s. There may not be an associated distance, but under certain conditions we can associate a qdistance d adapted to S (see Subsection 2.4 below and [28, Section 2.3]) so that, for some c1 , c2 ∈ (0, ∞), , is comparable to metric balls with respect to d of each K (0) (Λs , Kw ), s ∈ (0, 1], w ∈ Λs radii c1 s and c2 s. It is clear that K = w∈Λs Kw . Also for distinct w, v ∈ Λs , we see that Kw ∩ Kv = Fw (V0 ) ∩ Fv (V0 ), that is, Kw and Kv intersect only on their boundaries. In this sense, {Kw | w ∈ Λs } may be thought of as a covering of K by ‘balls of radius s’ with small overlaps. Now our first main theorem (Theorem 4.3) together with Proposition 4.4 yields the following statement. Let F0 := {u ∈ F | u|V0 = 0} and let HN (resp. HD ) be the non-negative self-adjoint operator on L2 (K, μ) associated with (E, F ) (resp. (E|F0 ×F0 , F0 )). Theorem 1.1. Let NN (resp. ND ) be the eigenvalue counting function of HN (resp. HD ). Then there exist c1 , c2 ∈ (0, ∞) and δ ∈ [1, ∞) such that for any x ∈ [δ, ∞), c1 #Λx −1/2 ND (x) NN (x) c2 #Λx −1/2 .
(1.5)
Note that HN and HD have compact resolvents by [29, Lemma 8.6] (we will give a direct proof of this fact in Section 4). Hence NN and ND can be defined in the present situation. The important point about Theorem 1.1 is the generality of the measure μ: The only assumption on μ is that it is elliptic, and in particular μ need not be a self-similar measure. With such
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Fig. 1.4. The weight (μi )i∈S .
a weak assumption, we have a geometric description (1.5) of the asymptotic order of NN (x) and ND (x) as x → ∞. On the other hand, if μ is a self-similar measure on K with weight (μi )i∈S , then we can easily show the following estimate of #Λs : s −dS #Λs Γ s −dS ,
s ∈ (0, 1], (1.6) d/2 = 1 and Γ := where dS (∈ (0, ∞)) is the unique d ∈ R that satisfies i∈S (rμi ) −d S (mini∈S γi ) . By (1.5) and (1.6), we may call dS the spectral dimension of the Dirichlet space (L, μ, E, F , r), and we have a geometric characterization (1.6) of dS . Next we turn to the second purpose of this paper. In the rest of this introduction, μ is assumed to be a self-similar measure on K with weight (μi )i∈S ∈ (0, 1)S , i∈S μi = 1. Unfortunately, it seems extremely difficult to verify directly an asymptotic behavior similar to (1.2) of Nb for b ∈ {N, D} in the present case, as K is infinitely ramified. But since it may be possible to make use of arguments on the corresponding diffusion process and heat kernel estimates, there is some hope of proving a result similar to (1.2) for the associated partition function Zb : (0, ∞) → (0, ∞) defined by −tH −tλb b n = e = e−ts dNb (s), (1.7) Zb (t) := Tr e n∈N
[0,∞)
where {λbn }n∈N is the non-decreasing enumeration of the eigenvalues of Hb , b ∈ {N, D}. In fact, our second main result (Theorem 5.2) and its corollary (Corollary 5.4) lead us to the following √ theorem. Let γi := rμi for i ∈ S and let dS be as in (1.6). Theorem 1.2. Assume the following condition on (μi )i∈S (see Fig. 1.4 above): μ1 = μ3 = μ5 = μ7 ,
μ2 = μ6
and μ4 = μ8 .
(1.8)
Then we have the following statements. (1) Non-lattice case: If i∈S Z log γi is a dense additive subgroup of R, then for b ∈ {N, D}, t dS /2 Zb (t) converges as t ↓ 0, so does x −dS /2 Nb (x) as x → ∞ and lim t dS /2 ZN (t) = lim t dS /2 ZD (t) ∈ (0, ∞),
(1.9)
NN (x) ND (x) = lim d /2 ∈ (0, ∞). d /2 S x→∞ x x→∞ x S
(1.10)
t↓0
lim
t↓0
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(2) Lattice case: If i∈S Z log γi is a discrete additive subgroup of R with generator T ∈ (0, ∞), then there exists a continuous T -periodic function G : R → (0, ∞) such that, for b ∈ {N, D},
1 1 log = 0. (1.11) lim t dS /2 Zb (t) − G t↓0 2 t This theorem is an extension of Hambly’s recent result [21, Theorem 1.1], which concentrates on the case where μi = 1/8 for any i ∈ S. The reason for the condition (1.8) is that, by [28, Theorems 3.2.3 and 3.4.5], it is equivalent to the following (sub-)Gaussian heat kernel upper bound (UHK): With some β ∈ (1, ∞) and a distance d on K which is ‘adapted to the scale S’, for any (t, x, y) ∈ (0, 1] × K × K, ptN (x, y)
1 d(x, y)β β−1 c1 , exp −c2 μ(Bt 1/β (x, d)) t
(UHK)
where {ptN }t∈(0,∞) is the (unique) jointly continuous heat kernel of {e−tHN }t∈(0,∞) and Br (x, d) := {y ∈ K | d(y, x) < r}. (See [29, Theorem 9.4] for existence and continuity of the heat kernel, and Definition 5.1 for the precise statement of (UHK).) Note that in (UHK) we allow the cases with strong spatial inhomogeneity: Unless μi = 1/8 for any i ∈ S, lim supt↓0 (log μ(Bt 1/β (x, d)))/ log t −1 and lim inft↓0 (log μ(Bt 1/β (x, d)))/ log t −1 depend highly on x ∈ K. The key part of the proof of Theorem 1.2 is to prove that the difference ZN − ZD is sufficiently smaller, compared with ZN and ZD . In fact, we have the following estimate. Theorem 1.3. Assume (1.8). Choose d∂ ∈ (0, ∞) so that 2γ1d∂ + (max{γ2 , γ4 })d∂ = 1. Then there exist c3 , c4 ∈ (0, ∞) such that for any t ∈ (0, 1], c3 t −d∂ /2 ZN (t) − ZD (t) c4 t −d∂ /2 . Note that d∂ admits the following estimate; there exists c5 , c6 ∈ (0, ∞) such that
c5 s −d∂ # {w ∈ Λs | Kw ∩ V0 = ∅} c6 s −d∂ , s ∈ (0, 1].
(1.12)
(1.13)
In this sense we will call d∂ the cell-counting dimension of V0 with respect to the scale S. Since we have a trivial lower bound ZN (t) − ZD (t) 0, t ∈ (0, ∞), the upper inequality of (1.12) suffices for the proof of Theorem 1.2, and it is a special case of Theorem 5.11. Note that the lower bound in (1.12) is new even when μi = 1/8 for any i ∈ S, and essentially as its corollary, the following sharp remainder estimate also follows. Theorem 1.4. Suppose μi = 1/8 for any i ∈ S and let G : R → (0, ∞) be as in Theorem 1.2(2). Then there exist c7 , c8 ∈ (0, ∞) such that for any t ∈ (0, 1],
1 1 −d∂ /2 −dS /2 log − ZD (t) c8 t −d∂ /2 . c7 t t G (1.14) 2 t Theorem 1.3 is a special case of Theorem 7.7, which may be seen as the third main result of this article. In fact, Theorem 7.7 treats the similar lower bound for the case with Dirichlet (killing) condition on a general self-similar subset of positive capacity.
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Finally, we remark that almost all the arguments illustrated so far apply also to any generalized Sierpinski carpet, which has been defined in [6,7]. See Section 8 for details. The organization of this paper is as follows. In Section 2, we introduce a number of notions, including that of scales and gauge functions, to describe geometry of self-similar sets. In Section 3, we introduce the notion of self-similar Dirichlet spaces as the framework of our spectral analysis. We show our first main result (Theorem 4.3) in Section 4. Section 5 is devoted to the statement and the proof of our second main theorem (Theorem 5.2) on an asymptotic expansion of the partition function. The key for Theorem 5.2 is Theorem 5.11, where the sub-Gaussian heat kernel upper bound plays a crucial role. As a complement to the results of Section 5, in Section 6 we provide a practical method of calculating the cell-counting dimension of the boundary of selfsimilar sets. In Section 7, we state and prove our ‘third main theorem’ Theorem 7.7, asserting the sharpness as in (1.12) of the order estimate of the partition functions given in Theorem 5.11. In Section 8, we apply the results of the previous sections to generalized Sierpinski carpets. Then the paper is concluded by mentioning related open problems. Finally, Appendix A provides a few easy but important facts playing essential roles in Section 7, which are not suitable to be included in the main text. Notation. Throughout this paper, we follow the following notations and conventions. (1) N = {1, 2, 3, . . .}, i.e. 0 ∈ / N. (2) Given a topological space E, let B(E) denote the Borel σ -field of E. A measure μ defined on the measurable space (E, B(E)) is called a Borel measure on E. For f : E → R, we write f ∞ := supx∈E |f (x)| and suppE [f ] := {x ∈ E | f (x) = 0}. We also write C(E) := {f | f : E → R, f is continuous}, Cb (E) := {f | f ∈ C(E), f ∞ < ∞} and C∞ (E) := {f | f ∈ C(E), {x ∈ E | |f (x)| δ} is compact for any δ ∈ (0, ∞)}. Moreover, for A ⊂ E, intE A denotes the interior of A in E. 2. Basics on self-similar sets In this section, we review basic notions on self-similar sets. See Kigami [28, Sections 1.1–1.3 and 2.3] for details and proofs. 2.1. Scales on the shift space First we define the notion of scales on the shift space and state their basic properties. Definition 2.1 (Words and shift space). Let S be a non-empty finite set. (1) We define Wm (S) := S m := {w1 · · · wm | wi ∈ S for i = 1, . . . , m} for m ∈ N, and W0 (S) := {∅}, where ∅ is an element called the empty word. We also set W# (S) := m∈N Wm (S) and W∗ (S) := W# (S) ∪ {∅}. For w ∈ W∗ (S), the length of w, which is denoted by |w|, is defined to be the unique m ∈ N ∪ {0} satisfying w ∈ Wm (S). (2) For w = w1 · · · wm ∈ W∗ (S), v = v1 · · · vn ∈ W∗ (S), we set wv := w1 · · · wm v1 · · · vn . Also for w 1 , w 2 ∈ W∗ (S), we define w1 w2 1
w <w
2
if and only if w 1 = w 2 v if and only if w w 1
2
for some v ∈ W∗ (S), and w = w . 1
2
(3) For w = w1 · · · wm ∈ W# (S), we write w[−1] := w1 · · · wm−1 .
and
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(4) The (one-sided) shift space with symbols S is defined by Σ(S) := S N := {ω = ω1 ω2 ω3 · · · | ωi ∈ S for any i ∈ N}. For each i ∈ S, we define σi : Σ(S) → Σ(S) by σi (ω1 ω2 ω3 · · ·) := iω1 ω2 ω3 · · · . We also define σ : Σ(S) → Σ(S) by σ (ω1 ω2 ω3 · · ·) := ω2 ω3 ω4 · · · . For w = w1 · · · wm ∈ W∗ (S), we write σw := σw1 ◦ · · · ◦ σwm and Σw (S) := σw (Σ(S)). Note that is a partial order on W∗ (S). We fix a non-empty finite set S in the rest of this subsection. We will write Wm , W∗ , Σ and so forth instead of Wm (S), W∗ (S) and Σ(S) when no confusion can occur. We consider Σ to be a topological space with the product topology inherited from the discrete topology of S. With this topology, Σ is a compact metrizable space. Definition 2.2 (Partitions). (1) Let Λ be a finite subset ofW∗ . We call Λ a partition of Σ if and only if Σw ∩ Σv = ∅ for w, v ∈ Λ with w = v, and Σ = w∈Λ Σw . (2) Let Λ1 and Λ2 be two partitions of Σ. Then we say that Λ1 is a refinement of Λ2 , and write Λ1 Λ2 , if and only if each w 1 ∈ Λ1 admits an element w 2 ∈ Λ2 such that w 1 w 2 . Note that the relation , which is defined on the collection of all partitions of Σ, is a partial order. Note also that, for w, v ∈ W∗ , Σw ∩ Σv = ∅ if and only if either w v or v w. Let Λ1 and Λ2 be partitions of Σ with Λ1 Λ2 . Then for any w 1 ∈ Λ1 , there exists a unique w 2 ∈ Λ2 such that w 1 w 2 . Therefore we can naturally define a mapping Λ1 → Λ2 by w 1 → w 2 , with w 1 and w 2 as above. This mapping is surjective, hence #Λ1 #Λ2 , where #A denotes the number of the elements of a set A. Definition 2.3 (Scales). Let Λs be a partition of Σ for any s ∈ (0, 1]. Then the family S := {Λs }s∈(0,1] of partitions of Σ is called a scale on Σ if and only if S satisfies the following three properties: (S1) Λ1 = W0 . Λs1 Λs2 for any s1 , s2 ∈ (0, 1] with s1 s2 . (S2) min{|w| | w ∈ Λs } → ∞ as s ↓ 0. (Sr) For any s ∈ (0, 1) there exists ε ∈ (0, 1 − s] such that Λs = Λs for any s ∈ (s, s + ε). Remark. In Kigami [28], a family S = {Λs }s∈(0,1] of partitions satisfying (S1) and (S2) is called a scale on Σ , and S is called right-continuous if S satisfies (Sr) in addition. But since we use only right-continuous scales (in the sense of [28]), we simply call them scales. Definition 2.4 (Gauge functions). A function g : W∗ → (0, 1] is called a gauge function on W∗ if and only if g has the following two properties: (G1) g(wi) g(w) for any w ∈ W∗ and any i ∈ S. (G2) max{g(w) | w ∈ Wm } → 0 as m → ∞. There is a natural bijection between the collection of all scales on Σ and that of all gauge functions on W∗ , as in the following theorem.
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Theorem 2.5. (1) Let g be a gauge function on W∗ . For each s ∈ (0, 1], define Λs (g) := w ∈ W∗ g(w[−1] ) > s g(w) ,
(2.1)
with the convention that g(w[−1] ) = 2 when w = ∅. We also set S(g) := {Λs (g)}s∈(0,1] . Then S(g) is a scale on Σ . We call S(g) the scale induced by the gauge function g. (2) Let S = {Λs }s∈(0,1] be a scale on Σ . Then there exists a unique gauge function lS on W∗ such that S = S(lS ). We call lS the gauge function of the scale S. By this theorem, we can identify a scale on Σ with its gauge function. Next we define some regularity conditions for scales. Definition 2.6 (Elliptic scales). Let S = {Λs }s∈(0,1] be a scale on Σ and l be its gauge function. We consider the following two conditions on S: (EL1) There exists β1 ∈ (0, 1) such that l(wi) β1 l(w) for any w ∈ W∗ and any i ∈ S. (EL2) There exist β2 ∈ (0, 1) and k ∈ N such that l(wv) β2 l(w) for any w ∈ W∗ and any v ∈ Wk . S is called elliptic if and only if its gauge function l satisfies both (EL1) and (EL2). The following proposition, which asserts a doubling property of the function (0, 1] s → #Λs for a scale {Λs }s∈(0,1] , is fundamental for the results in Section 4. Proposition 2.7. Let S = {Λs }s∈(0,1] be a scale on Σ whose gauge function l satisfies (EL2) and let β2 ∈ (0, 1) and k ∈ N be as in (EL2). Then #Λβ2 s (#S)k #Λs and #Λs (#Λβ2 )s −α for any s ∈ (0, 1], where α := −(k log #S)/ log β2 (∈ [0, ∞)). Proof. Let s ∈ (0, 1]. For any w ∈ Λs and any v ∈ Wk , we have l(wv) β2 l(w) β2 s by (EL2) and Theorem 2.5. Therefore there is a unique τ ∈ Λβ2 s such that wv τ . Thus we can define a mapping η : Λs × Wk → Λβ2 s by η(w, v) := τ , with w, v, τ as above. Let τ ∈ Λβ2 s . Since Λβ2 s Λs we can choose w ∈ Λs and v ∈ W∗ so that τ = wv. If |v| k + 1, then l(τ[−1] ) = l(wv[−1] ) β2 l(w) β2 s, which contradicts τ ∈ Λβ2 s . Hence |v| k. This shows that η is surjective, and #Λβ2 s (#S)k #Λs follows. j Let := max{j ∈ N ∪ {0} | s β2 }. Then β2 < β2− s 1. Therefore (log s)/ log β2 and #Λs (#S)k #Λβ − s (#S)(k log s)/ log β2 #Λβ2 = (#Λβ2 )s −α . 2 2
Finally we define the notion of self-similar scales and prove a basic asymptotic property of these scales. Definition 2.8 (Self-similar scales). Let α = (αi )i∈S ∈ (0, 1)S . Define a gauge function gα on W∗ by gα (w) := αw , where αw1 ···wm := αw1 · · · αwm for w1 · · · wm ∈ W∗ . Also let S(α) = {Λs (α)}s∈(0,1] be the scale induced by gα . We call S(α) the self-similar scale with weight α. Clearly, any self-similar scale is elliptic.
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S Proposition 2.9. Let α = (αi )i∈S ∈ (0, 1) and let d(α) (∈ [0, ∞)) be the unique d ∈ R that satisfies i∈S αid = 1. Set α := mini∈S αi . Then for any s ∈ (0, 1],
s −d(α) #Λs (α) α −d(α) · s −d(α) .
(2.2)
Proof. We will write Λs and d instead of Λs (α) and d(α) in this proof. Let μ be the Bernoulli measure on Σ = S N with weight (αid )i∈S . Let s ∈ (0, 1]. By Theorem 2.5, αw[−1] > s αw , hence αs < αw s, for any w ∈ Λs . Since Σ = w∈Λs Σw (disjoint), (αs) #Λs = d
w∈Λs
(αs) d
d αw
w∈Λs
and (2.2) is immediate from this.
=
w∈Λs
μ(Σw ) = μ(Σ) = 1
s d = s d #Λs
w∈Λs
2
2.2. Self-similar structures and measures In this subsection we introduce the notion of self-similar structures and recall related definitions and results. Definition 2.10 (Self-similar structures). (1) Let K be a compact metrizable space, S be a non-empty finite set and Fi : K → K be a continuous injection for each i ∈ S. The triple (K, S, {Fi }i∈S ) is called a self-similar structure if and only if there exists a continuous surjection π : Σ = Σ(S) → K such that π ◦ σi = Fi ◦ π for each i ∈ S. (2) Let L = (K, S, {Fi }i∈S ) be a self-similar structure. For w = w1 · · · wm ∈ W∗ , we set Fw := Fw1 ◦ · · · ◦ Fwm and Kw := Fw (K), where F∅ := L and idK for w = ∅. We define the critical set C m (C ), the post critical set PL of L by CL := π −1 ( i,j ∈S, i =j (Ki ∩ Kj )) and PL := ∞ σ L m=1 respectively. We also set V0 := V0 (L) := π(PL ). Note that PL ∈ B(Σ) and V0 ∈ B(K). (3) We say that L is strongly finite if and only if supx∈K #(π −1 (x)) < ∞, and that L is post critically finite (or simply p.c.f.) if and only if #PL < ∞. Given a self-similar structure L = (K, S, {Fi }i∈S ), we always assume #K 2, and hence #S 2, to exclude the trivial case where K is just a one-point set. The set V0 is regarded as the ‘boundary’ of K. In fact, by [27, Proposition 1.3.5(2)], if w, v ∈ W∗ and Σw ∩ Σv = ∅ then Kw ∩ Kv = Fw (V0 ) ∩ Fv (V0 ). We fix a self-similar structure L = (K, S, {Fi }i∈S ) in the rest of this subsection. The following easy lemma is fundamental for our study. Lemma 2.11. Assume K = V0 . Set K I := K \ V0 and KwI := Fw (K I ) for each w ∈ W∗ . Then KwI is an open subset of K and KwI ⊂ K I for any w ∈ W∗ . Moreover, let Λ be a partition of Σ I I = and set KΛ := w∈Λ KwI . Then K \ KΛ w∈Λ Fw (V0 ). Proof. The first two statements follow from Kigami [28, Proof of Theorem 1.2.7], but we include I the Fw (V0 ) ∪ proof for easeof the reading. Let w I∈ W∗ and set m := |w|. Since K \ Kw = I . Therefore K ⊃ F (V ) ⊃ V , K is an open subset of K and V ⊂ K \ K 0 0 w w v∈Wm \{w} v v∈Wm v 0 KwI ⊂ K \ V0 = K I .
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I , hence F (V ) = F (V ) ⊂ K \ K I = Next let w ∈ Λ. Then clearly Fw (V0 ) ⊂ K \ KΛ w 0 w 0 Λ I I . The converse inclusion K \ KΛ by the compactness of K. Therefore w∈Λ Fw (V0 ) ⊂ K \ KΛ I ∪ follows from K = w∈Λ Kw = w∈Λ (KwI ∪ Fw (V0 )) = KΛ w∈Λ Fw (V0 ). 2
The following easy lemma is used (only) in Subsection 7.2. Lemma 2.12. Assume that K = V0 . Let Λ be a partition of Σ and Γ ⊂ Λ. Then for any w ∈ Λ \ Γ , Kw ∩ intK ( v∈Γ Kv ) = ∅. Proof. Let w ∈ Λ \ Γ and suppose Kw ∩ intK K(Γ ) = ∅. Then U := Fw−1 (intK K(Γ )) = open subset of K. We have U ⊂ V0 since Kw ∩ Fw−1 (Kw ∩ int K K(Γ )) is a non-empty intK K(Γ ) ⊂ v∈Γ (Kw ∩ Kv ) = v∈Γ (Fw (V0 ) ∩ Fv (V0 )) ⊂ Fw (V0 ). Therefore intK V0 = ∅, which contradicts K = V0 by [27, Theorem 1.3.8]. Hence Kw ∩ intK K(Γ ) = ∅. 2 Next we consider some classes of Borel probability measures on K. Definition 2.13. (1) We define a collection M(K) of Borel probability measures by
M(K) := μ μ is a Borel probability measure on K, μ {x} = 0 for any x ∈ K,
μ(Kw ) > 0 and μ Fw (V0 ) = 0 for any w ∈ W∗ . (2.3) (2) A Borel probability measure μ on K is called elliptic if and only if the following holds: (ELm) There exists γ ∈ (0, ∞) such that μ(Kwi ) γ μ(Kw ) for any (w, i) ∈ W∗ × S. By [28, Theorem 1.2.4], if K = V0 then every elliptic Borel probability measure on K belongs to M(K). Definition 2.14 (Self-similar measures). Let (μi )i∈S ∈ (0, 1)S satisfy i∈S μi = 1. A Borel probability measure μ on K is called a self-similar measure with weight (μi )i∈S if and only if the following equality (of Borel measures on K) holds: μ=
μi μ ◦ Fi−1 .
(2.4)
i∈S
Let (μi )i∈S ∈ (0, 1)S satisfy i∈S μi = 1. If ν is the Bernoulli measure on Σ with weight (μi )i∈S , then ν ◦ π −1 is a self-similar measure on K with the same weight. Therefore there does exist a self-similar measure with the given weight. See [27, Section 1.4] for details. Let μ be a self-similar measure with weight (μi )i∈S . If K = V0 , then by [28, Theorem 1.2.7 and its proof], μ(Kw ) = μw and μ(Fw (V0 )) = 0 for any w ∈ W∗ . In particular, a self-similar measure with given weight is unique and elliptic in this case. 2.3. Systems of neighborhoods associated with scales Let L = (K, S, {Fi }i∈S ) be a self-similar structure. In this subsection, we define a fundamental (n) system of neighborhoods {Us (x, S)}s∈(0,1] of x ∈ K associated with a scale S = {Λs }s∈(0,1] . (n) (n) Intuitively, Us (x, S) is a union of Kw ’s over w ∈ Λs which are around x. Us (x, S) is regarded as a ‘ball of radius s’, although there may not be an associated distance. See [28, Chapter 2] for
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existence of such distances. We then introduce the notion of the volume doubling property with respect to a scale defined in [28, Section 1.3]. This property is closely related with (sub-)Gaussian heat kernel estimate, and will be mentioned again in Section 5. In the rest of this subsection, we fix a self-similar structure L = (K, S, {Fi }i∈S ) and a scale S = {Λs }s∈(0,1] on Σ. Definition 2.15. Let Γ ⊂ W∗ and A ⊂ K. (1) We set W (Γ, A) := {w ∈ Γ | Kw ∩ A = ∅} and K(Γ ) := w∈Γ Kw . (2) Define W (0) (Γ, A) := W (Γ, A), and inductively, K (n) (Γ, A) := K(W (n) (Γ, A)) and (n+1) W (Γ, A) := W (Γ, K (n) (Γ, A)) for n = 0, 1, 2, . . . . The following lemma is immediate by the above definitions. Lemma 2.16. Let A ⊂ K. (1) Let Λ be a partition of Σ. Then A ⊂ intK (K (0) (Λ, A)), and for any n ∈ N ∪ {0}, K (n) (Λ, A) ⊂ intK (K (n+1) (Λ, A)) and W (n) (Λ, A) ⊂ W (n+1) (Λ, A). (2) Let Λi , i = 1, 2, be partitions of Σ with Λ1 Λ2 . Then for any n ∈ N ∪ {0}, K (n) (Λ1 , A) ⊂ K (n) (Λ2 , A). Definition 2.17. For x ∈ K, s ∈ (0, 1] and n ∈ N ∪ {0}, we define Λns,x := W (n) (Λs , {x}) and (n) (0) Us (x, S) := K (n) (Λs , {x}). We write Λs,x := Λ0s,x , Ks (x, S) := Us (x, S) and Us (x, S) := Us(1) (x, S). We also set Λs,w := W (Λs , Kw ) for s ∈ (0, 1] and w ∈ W∗ . (n)
Clearly, {Us (x, S)}s∈(0,1] is decreasing as s ↓ 0 and forms a fundamental system of neighborhoods of x in K. Definition 2.18 (Locally finite scales). We say that S is locally finite with respect to L, or simply (L, S) is locally finite, if and only if sup{#(Λs,w ) | s ∈ (0, 1], w ∈ Λs } < ∞. Definition 2.19 (Volume doubling property). Let μ ∈ M(K). For n ∈ N ∪ {0}, (L, S, μ) is said (n) to satisfy (VD)n if and only if there exist α ∈ (0, 1) and cV ∈ (0, ∞) such that μ(Us (x)) (n) cV μ(Uαs (x)) for any (s, x) ∈ (0, 1] × K. We say that μ is volume doubling with respect to S, or simply (L, S, μ) satisfies (VD), if and only if (L, S, μ) satisfies (VD)n for some n ∈ N. 2.4. Qdistances adapted to scales and cell-counting dimension Next we introduce the notions of qdistances and cell-counting dimension. We continue to fix a self-similar structure L = (K, S, {Fi }i∈S ) and a scale S = {Λs }s∈(0,1] on Σ . Definition 2.20 (Qdistances). Let E be a set, α ∈ (0, ∞) and d : E × E → [0, ∞). Then d is said to be an α-qdistance on E if and only if d α := d(·,·)α is a distance on E. Also d is called a qdistance on E if d is an α-qdistance for some α ∈ (0, ∞). If d is an α-qdistance on E, then E is regarded as being equipped with the topology given by the distance d α .
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Notation. Let d : K × K → [0, ∞). Then we set Br (x, d) := {y ∈ K | d(x, y) < r} for any x ∈ K and any r ∈ (0, ∞). We also set diamd A := supy,z∈A d(y, z) and distd (x, A) := infy∈A d(x, y) for any x ∈ K and any non-empty A ⊂ K. Definition 2.21. A qdistance d on K is said to be adapted to S if and only if there exist β1 , β2 ∈ (0, ∞) and n ∈ N such that for any (s, x) ∈ (0, 1] × K, Bβ1 s (x, d) ⊂ Us(n) (x, S) ⊂ Bβ2 s (x, d).
(2.5)
If d is adapted to S, then {Us(n) (x, S)}s∈(0,1], x∈K may be thought of as real balls. Since is a fundamental system of neighborhoods of x, the topology determined by d is the same as the original one of K in this case. (n) {Us (x, S)}s∈(0,1]
Lemma 2.22. Let μ ∈ M(K), let d be a qdistance on K adapted to S and let n ∈ N be as in Definition 2.21. Then (L, S, μ) satisfies (VD)n if and only if there exists cV ∈ (0, ∞) such that for any (r, x) ∈ (0, ∞) × K,
μ B2r (x, d) cV μ Br (x, d) .
(2.6)
Proof. Note that infx∈K μ(Br (x, d)) > 0 for a fixed r ∈ (0, ∞), since x → μ(Br (x, d)) is a (0, ∞)-valued lower semicontinuous function on a compact space K. Now the statement is straightforward from (2.5). 2 Definition 2.23 (Cell-counting dimension). Let η ∈ [0, ∞) and A ⊂ K. We say that the cellcounting dimension of A with respect to S is bounded from above (resp. below) by η, and write dimS A η (resp. dimS A η), if and only if sups∈(0,1] s η #W (Λs , A) < ∞ (resp. infs∈(0,1] s η #W (Λs , A) > 0). We call η the cell-counting dimension of A with respect to S, and write dimS A = η, if and only if both dimS A η and dimS A η hold. Note that η ∈ [0, ∞) satisfying dimS A = η, if exists, is unique. The notion of cell-counting dimension corresponds to that of box-counting dimension in the settings of metric spaces. In fact, we have the following proposition. Proposition 2.24. Let d be a qdistance on K adapted to S, let A ⊂ K and η ∈ [0, ∞). For r ∈ (0, ∞), let Nr (A) be the smallest number N of balls {Br (xi , d)}N i=1 of radius r that can cover A. Suppose that (L, S) is locally finite. Then dimS A η (resp. dimS A η) if and only if supr∈(0,1] r η Nr (A) < ∞ (resp. infr∈(0,1] r η Nr (A) > 0). Proof. Take β1 , β2 > 0 and n ∈ N so that (2.5) holds. We may assume that β1 1 β2 . Let s ∈ (0, 1]. We choose xw ∈ Kw for each w ∈ W (Λs , A). Then A⊂
w∈W (Λs ,A)
Kw ⊂
w∈W (Λs ,A)
so Nβ2 s (A) #W (Λs , A). Therefore
Us(n) (xw , S) ⊂
w∈W (Λs ,A)
Bβ2 s (xw , d),
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β2
1323
inf r η Nr (A) inf s η #W (Λs , A),
(2.7)
η
(2.8)
r∈(0,β2 ]
s∈(0,1]
sup r η Nr (A) β2 sup s η #W (Λs , A).
r∈(0,β2 ]
s∈(0,1]
By (2.8), dimS A η implies supr∈(0,1] r η Nr (A) < ∞. Suppose infr∈(0,1] r η Nr (A) > 0. Then A = ∅ and Nr (A) 1 for any r > 0. Therefore infr∈[1,β2 ] r η Nr (A) 1 and infr∈(0,β2 ] r η Nr (A) > 0. Now this and (2.7) imply dimS A η. For the converse implications, let M := sup{#Λn+1 s,x | s ∈ (0, 1], x ∈ K}. Since (L, S) is locally finite, M < ∞ by [28, Lemma 1.3.6]. Let s ∈ (0, 1] and N := Nβ1 s (A) and choose {xi }N i=1 ⊂ K N N (n) (n) so that A ⊂ i=1 Bβ1 s (xi , d) (⊂ i=1 Us (xi , S)). If w ∈ W (Λs , A), Us (xi , S) ∩ Kw = ∅, N n+1 hence w ∈ Λn+1 s,xi for some i ∈ {1, . . . , N}. Therefore W (Λs , A) ⊂ i=1 Λs,xi and #W (Λs , A) N n+1 i=1 #Λs,xi MN = MNβ1 s (A). This yields M −1 β1 inf s η #W (Λs , A) η
s∈(0,1]
inf r η Nr (A),
r∈(0,β1 ] −η
sup s η #W (Λs , A) Mβ1
s∈(0,1]
sup r η Nr (A).
r∈(0,β1 ]
(2.9) (2.10)
η
If dimS A η, then A = ∅ and infr∈[β1 ,1] r η Nr (A) β1 , which together with (2.9) implies infr∈(0,1] r η Nr (A) > 0. On the other hand, by (2.10), supr∈(0,1] r η Nr (A) < ∞ implies dimS A η. This completes the proof. 2 3. Framework: Self-similar Dirichlet spaces In this section, we introduce our framework of spectral analysis on self-similar structures, which we call self-similar Dirichlet spaces. See Fukushima, Oshima and Takeda [17] for basic notions concerning Dirichlet forms on locally compact separable metrizable spaces. The following lemma is immediate from the results of Subsection 2.2. Lemma 3.1. Let (K, S, {Fi }i∈S ) be a self-similar structure, μ ∈ M(K) and w ∈ W∗ . (1) The Borel probability measure μw on K defined by μw := μ(Kw )−1 μ ◦ Fw belongs to M(K), and K u ◦ Fw dμw = μ(Kw )−1 Kw u dμ for any u : K → [0, ∞] Borel measurable. In particular, if we set ρw u := u ◦ Fw for u : K → [−∞, ∞], then ρw defines a bounded linear operator ρw : L2 (K, μ) → L2 (K, μw ). (2) If μ is a self-similar measure and K = V0 , then μw = μ. Definition 3.2. For u : K → R, w ∈ W∗ , define uw : K → R by u := w
u ◦ Fw−1 0
on Kw , on K \ Kw .
Clearly, if u is Borel measurable then so is uw for any w ∈ W∗ .
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Now we introduce the notion of self-similar Dirichlet spaces. Note that under the situation of the next definition, we can regard F ∩ C(K) as a subspace of C(K), hence u ◦ Fi (∈ C(K)) as an element of L2 (K, μ) for u ∈ F ∩ C(K). Definition 3.3 (Self-similar Dirichlet spaces). Let L = (K, S, {Fi }i∈S ) be a self-similar structure satisfying K = V0 and let μ be an elliptic Borel probability measure on K. A (symmetric) regular Dirichlet form (E, F ) on L2 (K, μ) is called self-similar with resistance scaling ratio r = (ri )i∈S ∈ (0, ∞)S if and only if the following four conditions are satisfied: (SSDF1) u ◦ Fi ∈ F ∩ C(K) for any u ∈ F ∩ C(K) and any i ∈ S. (SSDF2) For any u, v ∈ F ∩ C(K),
E(u, v) =
1 E(u ◦ Fi , v ◦ Fi ). ri
(3.1)
i∈S
(SSDF3) ui ∈ F ∩ C(K) for any i ∈ S and any u ∈ F ∩ C(K) with suppK [u] ⊂ K I (:= K \ V0 , recall Lemma 2.11), where ui is as in Definition 3.2. √ (SSDF4) The function g : W∗ → (0, ∞) defined by g(w) := rw μ(Kw ) is a gauge function on W∗ and the scale induced by g is elliptic. If (E, F ) is a self-similar regular Dirichlet form on L2 (K, μ) with resistance scaling ratio r = (ri )i∈S , then we call (L, μ, E, F , r) a self-similar Dirichlet space. Remark. (1) For a self-similar Dirichlet space (L = (K, S, {Fi }i∈S ), μ, E, F , r = (ri )i∈S ), (i) μ ∈ M(K) (by [28, Theorem 1.2.4]). (ii) 1 ∈ F (by the compactness of K and the regularity of (E, F )). (2) If μ is a self-similar measure with weight (μi )i∈S , then (SSDF4) is equivalent to the condition that ri μi < 1 for any i ∈ S. In the rest of this section, (L = (K, S, {Fi }i∈S ), μ, E, F , r = (ri )i∈S ) is assumed to be a selfsimilar Dirichlet space. √ Notation. Set g(w) := rw μ(Kw ) for w ∈ W∗ and let S = {Λs }s∈(0,1] be the scale on Σ induced by the gauge function g. We write E1 (u, v) := E(u, v) + K uv dμ for u, v ∈ F . Also for A ∈ B(K), we write μ|A := μ|B(A) . We state several preliminary results on (L, μ, E, F , r) needed in the following sections. Lemma 3.4. (E, F ) is a local Dirichlet form.
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Proof. Let u, v ∈ F ∩ C(K) with u, v = 0 and suppK [u] ∩ suppK [v] = ∅. Since suppK [u] and suppK [v] are compact, we can choose m ∈ N so that for each w ∈ Wm , either Kw ∩ suppK [u] = ∅ or Kw ∩ suppK [v] = ∅ holds. Then by (3.1) we have E(u, v) =
1 E(u ◦ Fw , v ◦ Fw ) = 0. rw
w∈Wm
Now the local property of (E, F ) follows by [17, Problem 1.4.1 and Theorem 3.1.2].
2
Definition 3.5. Let U be a non-empty open subset of K. Define CU := u ∈ F ∩ C(K) suppK [u] ⊂ U
and FU := CU ,
(3.2)
where the closure is taken in the Hilbert space (F , E1 ). We also set E U := E|FU ×FU . We call (E U , FU ) the part of the Dirichlet form (E, F ) on U . Since u = 0 μ-a.e. on K \ U for any u ∈ FU , we can regard FU as a subspace of L2 (U, μ|U ) in the natural way. Then by [17, Theorem 1.4.2(v) and Lemma 1.4.2(ii)], we easily see that (E U , FU ) is a local regular Dirichlet form on L2 (U, μ|U ). Lemma 3.6. Let w ∈ W∗ . Then uw ∈ CKwI for any u ∈ CK I and ρw (CKwI ) = CK I . Proof. ρw (CKwI ) ⊂ CK I is clear by (SSDF1). Conversely if u ∈ CK I , then using (SSDF3) repeatedly, we have uw ∈ CKwI . Hence u = uw ◦ Fw ∈ ρw (CKwI ). 2 The following lemma is used (only) in Subsection 7.2. Lemma 3.7. There exist c, α ∈ (0, ∞) such that cμ(Kw ) s α for any s ∈ (0, 1], w ∈ Λs . Proof. Since the scale S = {Λs }s∈(0,1] is assumed to be elliptic by (SSDF4), we easily see that there exists β1 ∈ (0, 1) such that g(w) β1 s for any s ∈ (0, 1] and any w ∈ Λs . It is also easy |w| to show that there exist c1 ∈ (0, ∞) and β2 ∈ (0, 1) such that g(w) c1 β2 for any w ∈ W∗ . Since μ is also assumed to be elliptic, we can choose γ ∈ (0, 1) so that μ(Kwi ) γ μ(Kw ) for any w ∈ W∗ and any i ∈ S. Then μ(Kw ) γ |w| for any w ∈ W∗ . Now set α := (log γ )/ log β2 (∈ (0, ∞)) and let s ∈ (0, 1] and w ∈ Λs . Then |w|
β1 s g(w) c1 β2 = c1 γ |w|/α c1 μ(Kw )1/α . Thus (c1 /β1 )α μ(Kw ) s α .
2
4. Spectral and geometric counting functions Now we start to study spectral properties of self-similar Dirichlet forms. In this section, we state and prove our first main result (Theorem 4.3). Throughout this section, let (L = Dirichlet space and S = {Λs }s∈(0,1] be (K, S, {Fi }i∈S ), μ, E, F , r = (ri )i∈S ) be a self-similar √ the scale induced by the gauge function g : w → rw μ(Kw ).
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First we define the eigenvalue counting and partition functions of a non-negative selfadjoint operator on a Hilbert space. Note that, in the present setting, L2 (U, μ|U ) is an infinitedimensional separable Hilbert space for any U ⊂ K non-empty open. Definition 4.1 (Eigenvalue counting and partition functions). Let H be a non-negative selfadjoint operator on an infinite-dimensional separable Hilbert space H. (1) The partition function ZH of H (or of the contraction semigroup {e−tH }t∈(0,∞) or of the corresponding closed form on H) is defined by ZH (t) := Tr(e−tH ), t ∈ (0, ∞). (2) Suppose that H has compact resolvent and let {λH n }n∈N be the non-decreasing enumeration of the eigenvalues of H , where each eigenvalue is repeated according to its multiplicity. The eigenvalue counting function NH of H is defined by
NH (x) := # n ∈ N λH (4.1) n x , x ∈ [0, ∞), and then we have the following equalities for ZH :
−tλH ZH (t) = Tr e−tH = e n = n∈N
e−ts dNH (s),
t ∈ (0, ∞).
(4.2)
[0,∞)
Note that NH (x) < ∞ for any x ∈ [0, ∞) since limn→∞ λH n = ∞, and that ZH is (0, ∞)-valued, strictly decreasing and continuous provided ZH (t) < ∞ for any t ∈ (0, ∞). Notation. Let HN (resp. HD ) be the non-negative self-adjoint operator associated with the closed I form (E, F ) on L2 (K, μ) (resp. (E K , FK I ) on L2 (K I , μ|K I )). For b ∈ {N, D}, if Hb has comb pact resolvent, then we write λbn := λH n and Nb := NHb . Definition 4.2 (Uniform Poincaré inequality). We say that (E, F ) satisfies the uniform Poincaré inequality, (PI) for short, if and only if there exists CPI ∈ (0, ∞) such that
w 2 E(u, u) CPI u − uμ dμw , u ∈ ρw F ∩ C(K) (PI) K
for any w ∈ W∗ , where uν :=
K
u dν for a Borel probability measure ν on K.
The uniform Poincaré inequality yields the following estimate for the eigenvalue counting functions NN and ND , which is the main theorem of this section. Theorem 4.3. Assume that (E, F ) is conservative, i.e. E(1, 1) = 0, and satisfies (PI). Then there exist c1 , c2 ∈ (0, ∞) and δ ∈ [1, ∞) such that for any x ∈ [δ, ∞), c1 #Λx −1/2 ND (x) NN (x) c2 #Λx −1/2 .
(4.3)
Remark. In the arguments below, we will prove that HN and HD have compact resolvents under the situation of Theorem 4.3. We provide a few simple sufficient conditions for (PI) before proving Theorem 4.3.
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Proposition 4.4. (PI) holds for each of the following two cases. (1) F ⊂ C(K), (E, F ) is a resistance form on K and its associated resistance metric R is compatible with the original topology of K. (2) μ is a self-similar measure and there exists CPI ∈ (0, ∞) such that 2 E(u, u) CPI u − uμ dμ, u ∈ F ∩ C(K). (4.4) K
Proof. (1) This is immediate by [28, Proof of Lemma B.2]. (See Kigami [27, Chapter 2] and [29, Part I] for the definition and basic properties of resistance forms.) (2) trivially yields (PI) since ρw (F ∩ C(K)) ⊂ F ∩ C(K) and μw = μ for any w ∈ W∗ . 2 The rest of this section is devoted to the proof of Theorem 4.3. The proof is split into several lemmas and is based on the so-called minimax principle or the variational formula for the eigenvalues of non-negative self-adjoint operators. See Davies [15, Chapter 4] for details about the minimax principle. We first show the upper inequality of (4.3). Lemma 4.5. Suppose that (E, F ) is conservative and satisfies (PI). Define 2 λ(L) := sup E(u, u) u ∈ L, |u| dμ = 1 , L ⊂ F ∩ C(K) subspace,
(4.5)
K
λn := inf λ(L) L is an n-dimensional subspace of F ∩ C(K) .
(4.6)
Let Λ be a partition of Σ . Then −1 λ#Λ+1 CPI max rw μ(Kw ) . w∈Λ
(4.7)
In particular, HN has compact resolvent, so does HD and λn = λN n for any n ∈ N. Proof. The statements of the final sentence follow from (4.7) in view of the minimax principle, CK I ⊂ F ∩ C(K) and (SSDF4), so it suffices to show (4.7). Note that we may regard ρw (F ∩ C(K)) as a subspace of L2 (K, μw ) for w ∈ W∗ . Also, regarded as subspaces of C(K), ρw (F ∩ C(K)) ⊂ F ∩ C(K) by (SSDF1). Under these identifications, we define
FN,Λ := u ∈ L2 (K, μ) u ◦ Fw ∈ ρw F ∩ C(K) for any w ∈ Λ , 1 E N,Λ (u, v) := E(u ◦ Fw , v ◦ Fw ), u, v ∈ FN,Λ . rw
(4.8)
w∈Λ
Similarly to (4.5) and (4.6), we set |u|2 dμ = 1 , λ(L) := sup E N,Λ (u, u) u ∈ L,
L ⊂ FN,Λ subspace,
(4.9)
K
λΛ n
:= inf λ(L) L is an n-dimensional subspace of FN,Λ .
(4.10)
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F ∩ C(K) ⊂ FN,Λ by definition, and E N,Λ coincides with E on (F ∩ C(K)) × (F ∩ C(K)) Λ by (SSDF2). Hence λn λn for any n ∈ N. Let L0 := { w∈Λ aw 1Kw | aw ∈ R for each w ∈ Λ}. Note that L0 is a #Λ-dimensional subspace of FN,Λ and E N,Λ |L0 ×L0 ≡ 0. Let L ⊂ FN,Λ be a (#Λ + 1)-dimensional subspace and is naturally associated with a non-negative := L + L0 . Then the bilinear form E N,Λ on L set L N,Λ By the theory (u, v) = K Au · v dμ, u, v ∈ L. self-adjoint operator A on L by the equality E of finite-dimensional real symmetric matrices, the (#Λ + 1)-th smallest eigenvalue λA of A is given by , λA = inf λ L L is a (#Λ + 1)-dimensional subspace of L corresponding to λA is orthogonal where λ(L ) isas in (4.9). Moreover, the eigenfunction u∈L to L0 , that is, K u ◦ Fw dμw = μ(Kw )−1 Kw u dμ = 0 for any w ∈ Λ. We can normalize u so that K |u|2 dμ = 1. Then by (PI), λ(L) λA = E N,Λ (u, u) =
1 1 E(u ◦ Fw , u ◦ Fw ) CPI |u ◦ Fw |2 dμw rw rw
w∈Λ
= CPI
w∈Λ
1 rw μ(Kw )
w∈Λ
u2 dμ
K
CPI . maxw∈Λ rw μ(Kw )
Kw
Taking the infimum over L yields (4.7).
2
Lemma 4.6. Assume that (E, F ) is conservative and satisfies (PI). Then there exists c2 ∈ (0, ∞) such that for any x ∈ [1, ∞), NN (x) c2 #Λx −1/2 .
(4.11)
−1 C s −2 , hence Proof. Let s ∈ (0, 1]. By (4.7), λN PI #Λs +1 CPI (maxw∈Λs rw μ(Kw )) −2 NN (CPI s /2) #Λs . We may assume 1 CPI /2 (=: α). Let x ∈ [1, ∞) and set s 2 := α/x (∈ (0, 1]). Then NN (x) #Λ√αx −1/2 . Proposition 2.7 implies that there exists c2 > 0 such that #Λ√αt c2 #Λt for any t ∈ (0, 1]. Thus the result follows. 2
Next we prove the lower bound of (4.3). Lemma 4.7. There exists CD ∈ (0, ∞) such that for any w ∈ W∗ ,
λ1 KwI :=
inf
CD E(u, u) . |u|2 dμ rw μ(Kw )
u∈CK I , u ≡0 K I w w
(4.12)
Proof. Take v ∈ W∗ so that Kv ⊂ K I . By the regularity of (E, F ) and [17, Problem 1.4.1], there exists u ∈ CK I such that u 0 on K and u = 1 on Kv . Let w ∈ W∗ . Then Lemma 3.6 implies that uw ∈ CKwI . By (SSDF2) and the ellipticity of μ,
E(u, u) 1 E(u, u) E(u, u) E(uw , uw ) , = λ1 KwI |v| w 2 w 2 γ rw μ(Kw ) rw K |u | dμ rw μ(Kwv ) K |u | dμ
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where γ is the constant given in (ELm) (Definition 2.13(2)). Since u ∈ CK I and v ∈ W∗ is independent of w ∈ W∗ , (4.12) has been proved. 2 Lemma 4.8. Assume that HD has compact resolvent. For each w ∈ W∗ , let Hw be the nonI negative self-adjoint operator on L2 (KwI , μ|KwI ) associated with (E Kw , FKwI ). Let Λ be a parI , μ| tition of Σ and let HΛ be the non-negative self-adjoint operator on L2 (KΛ K I ) associated Λ
I
with (E KΛ , FK I ) (recall Lemma 2.11). Then Hw and HΛ have compact resolvents. Moreover, if Λ we set NKwI := NHw and NK I := NHΛ , then for any x ∈ [0, ∞), Λ
NKwI (x) = NK I (x) ND (x). Λ
(4.13)
w∈Λ
Proof. If w ∈ Λ, then by FKwI ⊂ FK I ⊂ FK I and the minimax principle, Hw and HΛ have Λ compact resolvents and the inequality in (4.13) holds. So we show the equality in (4.13). The self-similarity of (E, F ) implies that E(u1 , u2 ) = 0 for any wi ∈ Λ, i = 1, 2, with w1 = w2 and any ui ∈ FKwI , i = 1, 2. i Let w ∈ Λ and u ∈ CK I . Since K \ KwI = Fw (V0 ) ∪ τ ∈Λ\{w} Kτ , (Lw :=) Kw ∩ suppK [u] ⊂ Λ
KwI . Therefore u · 1KwI ∈ C(K) and suppK [u · 1KwI ] ⊂ Lw ⊂ KwI . Since Lw is compact and KwI is open in K, we may take ϕw ∈ F ∩ C(K) such that ϕw 0, ϕw |Lw = 1 and ϕw |K\KwI = 0 by [17, Problem 1.4.1]. Then u · 1KwI = u · ϕw ∈ F by [17, Theorem 1.4.2(ii)], hence u · 1KwI ∈ CKwI . It follows that CK I = w∈Λ CKwI , where CKwI , w ∈ Λ, are orthogonal to each other with respect to Λ
taking the closure of both sides in the Hilbert both E and the inner product of L2 (K, μ). Therefore space (F , E1 ) leads to the equality FK I = w∈Λ FKwI and again FKwI , w ∈ Λ are orthogonal Λ
to each other with respect to both E and the inner product of L2 (K, μ). This fact immediately implies that each eigenspace of HΛ is the direct sum over w ∈ Λ of those of Hw with the same eigenvalue. Now the desired equality is obvious. 2 Lemma 4.9. Suppose that HD has compact resolvent. Then there exist c1 ∈ (0, ∞) and δ ∈ [1, ∞) such that for any x ∈ [δ, ∞), c1 #Λx −1/2 ND (x).
(4.14)
Proof. Since the gauge function g of S = {Λs }s∈(0,1] is assumed to satisfy (EL1), we may choose β ∈ (0, 1) so that g(w) βs for any s ∈ (0, 1] and any w ∈ Λs . Let s ∈ (0, 1]. Then by Lemma 4.7, we have
λ1 KwI
CD CD CD = 2 2 2 rw μ(Kw ) g(w) β s
for any w ∈ Λs . Note that under the assumption of this lemma, λ1 (KwI ) is the smallest eigenvalue of Hw for any w ∈ W∗ . Now let δ := max{CD β −2 , 1}, x ∈ [δ, ∞) and s 2 := δ/x. Since x CD /β 2 s 2 , λ1 (KwI ) x and Nw (x) 1 for any w ∈ Λs . Hence by Lemma 4.8, ND (x)
w∈Λs
Nw (x) #Λs = #Λ√δx −1/2 .
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By Proposition 2.7, there exists c1 > 0 such that c1 #Λδ −1/2 t #Λt for any t ∈ (0, 1]. Thus the result follows. 2 Proof of Theorem 4.3. HN and HD have compact resolvents by Lemma 4.5. Since FK I ⊂ F , the minimax principle shows that ND (x) NN (x) for any x ∈ [0, ∞). Now the statement is immediate from Lemmas 4.6 and 4.9. 2 5. Short time asymptotics of the partition function In this section we assume that (L = (K, S, {Fi }i∈S ), μ, E, F , r = (ri )i∈S ) is a self-similar Dirichlet space and that S = {Λs }s∈(0,1] is the scale induced by the gauge function g : w → √ rw μ(Kw ). We also assume throughout this section that μ is a self-similar measure with weight √ (μi )i∈S . In particular, S is a self-similar scale with weight γ = (γi )i∈S , where γi := ri μi . We set dS := d(γ ), where d(γ ) is as in Proposition 2.9 with α = γ . We have dS > 0 since #S 2, and dimS K = dS by (2.2). Notation. Let {TtN }t∈(0,∞) and {TtD }t∈(0,∞) be the strongly continuous contraction semigroups I associated with the closed forms (E, F ) on L2 (K, μ) and (E K , FK I ) on L2 (K I , μ|K I ), respectively. For b ∈ {N, D}, let Zb denote the partition function associated with {Ttb }t∈(0,∞) (recall Definition 4.1). Note that if {Ttb }t∈(0,∞) is ultracontractive (see Definition A.1(1)) then by [14, Theorem 2.1.4] Hb has compact resolvent and Zb (t) ∈ (0, ∞) for any t ∈ (0, ∞). In our case, the (sub-)Gaussian heat kernel upper bound is formulated as follows. Definition 5.1 (UHK). We say that the (sub-)Gaussian heat kernel upper bound holds for (L, μ, E, F , r), or simply (UHK) holds, if and only if the following conditions are valid: The semigroup {TtN }t∈(0,∞) has a heat kernel {ptN }t∈(0,∞) , and there exist β ∈ (1, ∞), a (2/β)qdistance d adapted to S and c1 , c2 ∈ (0, ∞) such that for each t ∈ (0, 1], ptN (x, y)
1 d(x, y)2 β−1 c1 exp −c μ × μ-a.e. (x, y) ∈ K × K. 2 μ(B√t (x, d)) t
(5.1)
If dimS V0 d∂ for some d∂ ∈ [0, dS ), then (UHK) leads us to the following asymptotic behavior of Zb , which is the main theorem of this section. Theorem 5.2 (Short time asymptotics of the partition function). Let d∂ ∈ [0, dS ) and suppose that dimS V0 d∂ and (UHK) hold. Then we have the following statements. (1) Non-lattice case: If i∈S Z log γi is a dense additive subgroup of R, then t dS /2 ZN (t) and t dS /2 ZD (t) converge as t ↓ 0 and lim t dS /2 ZN (t) = lim t dS /2 ZD (t) ∈ (0, ∞). t↓0
t↓0
(5.2)
(2) Lattice case: If i∈S Z log γi is a discrete additive subgroup of R, let T ∈ (0, ∞) be its gend erator. Define mi := − log γi /T (∈ N) and pi := γi S for each i ∈ S and let Q be the polym i nomial defined by Q(z) := (1 − i∈S pi z )/(1 − z). Set q := min{|z| | z ∈ C, Q(z) = 0}
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(q := ∞ if Q = 1), m := max{the order of zero of Q at w | w ∈ C, |w| = q, Q(w) = 0} and dM := dS − T −1 log q. Then there exists a continuous T -periodic function G : R → (0, ∞) such that, for any b ∈ {N, D}, as t ↓ 0, ⎧
⎨ O(t −d∂ /2 ) 1 1 Zb (t) − t −dS /2 G log = O(t −d∂ /2 (log(t −1 ))m ) ⎩ 2 t O(t −dM /2 (log(t −1 ))m−1 )
if e(dS −d∂ )T < q, if e(dS −d∂ )T = q, if e(dS −d∂ )T > q.
(5.3)
Remark.In the lattice case we have q > dM ∈ (d∂ , dS ) if e(dS −d∂ )T > q. In fact, 1, and therefore m Q(1) = i∈S mi pi i∈S pi = 1. If i∈S pi z i = 1 for z ∈ C with |z| = 1, then the triangle mi = zmj for any i, j ∈ S. Hence z = 1. Also clearly | mi inequality implies that z i∈S pi z | i∈S pi |z| = |z| < 1 if z ∈ C and |z| < 1. Thus q > 1. As a special case of the above theorem, we have the following. Corollary 5.3. Let d∂ ∈ [0, dS ) and suppose that dimS V0 d∂ and (UHK) hold. If γi = γ for any i ∈ S for some γ ∈ (0, 1), then there exists a continuous log(γ −1 )-periodic function G : R → (0, ∞) such that, for any b ∈ {N, D}, as t ↓ 0,
1 1 log = O t −d∂ /2 . (5.4) Zb (t) − t −dS /2 G 2 t Proof. Since i∈S Z log γi = Z log(γ −1 ), we are in the lattice case of Theorem 5.2 and Q = 1 in the notation there. As q = ∞ > e−(dS −d∂ ) log γ the corollary follows by (5.3). 2 In the non-lattice case, we have the similar asymptotic behavior of NN and ND . Corollary 5.4. Let d∂ ∈ [0, dS ) and suppose that dimS V0 d∂ and (UHK) hold. If i∈S Z log γi is a dense additive subgroup of R, then x −dS /2 NN (x) and x −dS /2 ND (x) converge as x → ∞ and NN (x) ND (x) = lim d /2 ∈ (0, ∞). d /2 S x→∞ x x→∞ x S lim
(5.5)
Proof. This is immediate from Theorem 5.2(1) and Karamata’s Tauberian theorem (see Feller [16, p. 445, Theorem 2]). 2 The rest of this section is devoted to the proof of Theorem 5.2. The proof is split into several propositions and lemmas. We first give an easy lemma on the structure of (E, F ). Lemma 5.5. (SSDF1) and (SSDF2) are valid with F in place of F ∩ C(K). Moreover, if w ∈ W∗ then uw ∈ FKwI for any u ∈ FK I and ρw (FKwI ) = FK I . Remark. If u, v : K → R are Borel measurable and u = v μ-a.e., then for any w ∈ W∗ , it easily follows from μw = μ that uw = v w μ-a.e. Proof. Let w ∈ W∗ . Since μw = μ, ρw defines a bounded linear operator on L2 (K, μ), and also on (F ∩ C(K), E1 ) by (SSDF1) and (SSDF2). Let u ∈ F and choose {un }n∈N ⊂ F ∩ C(K) so
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that un → u in (F , E1 ). Then un ◦ Fw → u ◦ Fw in L2 (K, μ). Also in (F , E1 ), {un ◦ Fw }n∈N is a Cauchy sequence and converges to some f ∈ F . Hence u ◦ Fw = f ∈ F and un ◦ Fw → u ◦ Fw in (F , E1 ), which also immediately yields (3.1) for u, v ∈ F. By the equalities ρw (CKwI ) = CK I (by Lemma 3.6), K I |u|2 dμ = μw K I |u ◦ Fw |2 dμ w
for u ∈ L2 (KwI , μ|KwI ) and E(u ◦ Fw , u ◦ Fw ) = rw E(u, u) for u ∈ CKwI , we easily see that ρw (FKwI ) = FK I . Finally, let u ∈ FK I and choose {un }n∈N ⊂ CK I so that un → u in (F , E1 ). w w 2 2 Then {uw n }n∈N ⊂ CKwI by Lemma 3.6 and K |u − un | dμ = μw K |u − un | dμ → 0 as w w w w n → ∞. Since rw E(um − un , um − un ) = E(um − un , um − un ) for any m, n ∈ N, {uw n }n∈N KI
converges to some g ∈ FKwI in (FKwI , E1 w ) and then uw = g ∈ FKwI .
2
Lemma 5.6. Suppose that HN has compact resolvent and let Λ be a partition of Σ . Then
ND γw2 x ND (x) NN (x) NN γw2 x (5.6) NK I (x) = Λ
w∈Λ
w∈Λ
for any x ∈ [0, ∞). Moreover, there exist c1 , c2 ∈ (0, ∞) and δ ∈ [1, ∞) such that for any x ∈ [δ, ∞), c1 x dS /2 ND (x) NN (x) c2 x dS /2 .
(5.7)
Proof. Noting Proposition 2.9, Lemma 4.8 and that μ(Kw ∩ Kv ) = 0 for w, v ∈ Λ with w = v, the same arguments as in [30, Sections 2 and 6] immediately show the lemma. 2 Now we turn to estimates of partition functions. We need the following notations. Notation. (1) We set Ac := K \ A for A ⊂ K. (2) Let U ⊂ K be non-empty open. The contraction semigroup on L2 (U, μ|U ) associated with (E U , FU ) is denoted by {TtU }t∈(0,∞) . Suppose {TtU }t∈(0,∞) is ultracontractive. Then its heat kernel, which exists by [14, Theorem 2.1.4] and is unique up to μ × μ-a.e., is denoted U . We always set ptU := 0 on K × K \ U × U . Also, ZU (t) := Tr(TtU ) = by {pt }Ut∈(0,∞) 2 K×K (pt/2 ) d(μ × μ) (∈ (0, ∞)) denotes the associated partition function. Lemma 5.7. Suppose that {TtN }t∈(0,∞) is ultracontractive and let Λ be a partition of Σ . Then ZK I (t) =
Λ
w∈Λ
ZD
t γw2
ZD (t) ZN (t)
w∈Λ
ZN
t γw2
(5.8)
for any t ∈ (0, ∞). Moreover, there exist c1 , c2 ∈ (0, ∞) such that for any t ∈ (0, 1], c1 t −dS /2 ZD (t) ZN (t) c2 t −dS /2 . KI
(5.9)
Proof. By [28, Proposition C.1], {TtD }t∈(0,∞) and {Tt Λ }t∈(0,∞) are also ultracontractive. Therefore (5.8) is an immediate consequence of (5.6) and (4.2). For t ∈ (0, 1], using Proposition 2.9, letting Λ := Λ√t in (5.8) immediately leads to (5.9). 2 In the propositions below we establish important consequences of (UHK).
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Remark. In the following Proposition 5.8, Lemma 5.9, Proposition 5.10 and Theorem 5.11 and their proofs, we do not use the assumption that μ is a self-similar measure. Proposition 5.8. Suppose that (UHK) holds. Then: (1) (2) (3) (4)
The semigroup {TtN }t∈(0,∞) is ultracontractive. (L, S, μ) satisfies (VD). (L, S) is locally finite. Let d be the qdistance as in Definition 5.1. Then there exists cV > 0 such that cV μ(Bs (x, d)) μ(Us (x, S)) for any (s, x) ∈ (0, 1] × K.
Proof. (1) Let t ∈ (0, 1]. Since x → μ(B√t (x, d)) is a (0, 1]-valued lower semicontinuous function on a compact space K, η(t) := infx∈K μ(B√t (x, d)) ∈ (0, 1] By (UHK), ptN c1 η(t)−1 μ × μ-a.e. on K × K, hence we easily see that TtN 2→∞ c1 η(t)−1 for t ∈ (0, 1]. Also N N N N for t ∈ (1, ∞), TtN 2→∞ = T1N Tt−1 2→∞ T1 2→∞ Tt−1 2→2 T1 2→∞ . Hence the semigroup {TtN }t∈(0,∞) is ultracontractive. (2) This is proved in exactly the same way as [28, Proofs of Lemma 3.5.5 and Theorem C.3], based on Lemma 4.7 and with a few slight modifications. (3) Since S is (assumed to be) elliptic, (2) and [28, Theorem 1.3.5] imply the statement. (4) We may choose n ∈ N, β1 ∈ (0, 1] and β2 ∈ [1, ∞) so that (2.5) holds. Then (2) and [28, Theorem 1.3.5] imply (VD)n . Therefore there exists cV > 0 such that
(n)
cV μ Bs (x, d) cV μ U −1 (x, S) μ Us(n) (x, S) μ Us (x, S) β2 s
for any (s, x) ∈ (0, 1] × K. This completes the proof.
2
Lemma 5.9. Suppose that (UHK) holds and let β ∈ (1, ∞) and a (2/β)-qdistance d be as in Definition 5.1. Let F and L be closed subsets of K such that F L K. Then there exist c1 , c2 ∈ (0, ∞) such that, with
1 distd (x, L \ F )2 β−1 c1 , exp −c2 Φ(t, x) := μ(U√t (x, S)) t
(t, x) ∈ (0, 1] × K,
(5.10)
for any t ∈ (0, 1], c
c
0 ptF (x, y) − ptL (x, y) Φ(t, x) + Φ(t, y) 0 ZF c (t) − ZLc (t) Φ(t, x) dμ(x).
μ × μ-a.e. (x, y) ∈ F c × F c , (5.11) (5.12)
Fc c
c
Proof. By Proposition 5.8(1) and [28, Proposition C.1], {TtF }t∈(0,∞) and {TtL }t∈(0,∞) are ulc c tracontractive. Therefore the heat kernels {ptF }t∈(0,∞) and {ptL }t∈(0,∞) exist and ZF c and ZLc c c are (0, ∞)-valued and continuous on (0, ∞). Note that 0 ptL ptF ptN μ × μ-a.e. for any t ∈ (0, ∞), which follows by [28, (C.2)] and a monotone class argument. Let δ > 0 and set Uδ := {x ∈ F c | distd (x, L \ F ) < δ}. Then Uδ is an open subset of F c satisfying L \ F ⊂ Uδ . Note that L \ F includes the (topological) boundary of Lc in F c . Since
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c
(E F , FF c ) is a local regular Dirichlet form by Lemma 3.4, Grigor’yan’s result [20, Theorem 10.4] implies that for each t ∈ (0, ∞), for μ × μ-a.e. (x, y) ∈ Lc × Lc , c
c
ptF (x, y) − ptL (x, y)
c
sup t/2st s∈Q∪{t/2,t}
+
μ-esssup psF (x, u)
sup t/2st s∈Q∪{t/2,t}
u∈Uδ
c
μ-esssup psF (v, y). v∈Uδ
(5.13)
(In fact, [20, Theorem 10.4] may not be true when the right-hand side of [20, (10.12)] is essentially unbounded on some compact subset. It is, however, actually valid in the present setting, c since the function t → μ × μ-esssupK×K ptF is [0, ∞)-valued and non-increasing by [20, Lemmas 3.1 and 3.2].) Moreover, (UHK), Proposition 5.8 and [28, Theorem 1.3.5] imply that there exist cV , cVD ∈ [1, ∞) such that cV μ(Bs (x, d)) μ(Us (x, S)) and cVD μ(Us/2 (x, S)) μ(Us (x, S)) for any (s, x) ∈ (0, 1] × K. Let t ∈ (0, 1] and s ∈ [t/2, t]. By (UHK), with c1 , c2 ∈ (0, ∞) as in Definition 5.1, for μ × μ-a.e. (x, u) ∈ Lc × Uδ , c 0 psF (x, u) psN (x, u)
1 d(x, u)2 β−1 c1 exp −c2 μ(B√s (x, d)) s
1
distd (x, Uδ )2 β−1 c1 cV cVD =: Φ(t, x, δ) , exp −c2 √ μ(U t (x, S)) t c
which yields μ-esssupu∈Uδ psF (x, u) Φ(t, x, δ) for μ-a.e. x ∈ Lc . Thus we conclude that c
sup t/2st, s∈Q∪{t/2,t} c
μ-esssup psF (x, u) Φ(t, x, δ) u∈Uδ c
μ-a.e. x ∈ Lc .
c
Also, by the symmetry of psF , i.e. psF (x, y) = psF (y, x) for μ × μ-a.e. (x, y) ∈ K × K, sup t/2st, s∈Q∪{t/2,t}
c
μ-esssup psF (v, y) Φ(t, y, δ) v∈Uδ
c
μ-a.e. y ∈ Lc .
c
These estimates together with (5.13) imply ptF (x, y) − ptL (x, y) Φ(t, x, δ) + Φ(t, y, δ) for μ × μ-a.e. (x, y) ∈ Lc × Lc . Now we define Φ(t, x) by (5.10) with c1 replaced by c1 cV cVD . Then limδ↓0 Φ(t, x, δ) = Φ(t, x) for any x ∈ K. Therefore setting δ := n−1 with n ∈ N and letting n → ∞, we see that (5.11) holds for μ × μ-a.e. (x, y) ∈ Lc × Lc . On the other hand, for μ × μ-a.e. (x, y) ∈ F c × (L \ F ), c
c
c
0 ptF (x, y) − ptL (x, y) = ptF (x, y)
1 d(x, y)2 β−1 c1 exp −c 2 μ(B√t (x, d)) t
1 distd (x, L \ F )2 β−1 c1 cV cVD = Φ(t, x) Φ(t, x) + Φ(t, y). exp −c2 μ(U√t (x, S)) t
N. Kajino / Journal of Functional Analysis 258 (2010) 1310–1360 c
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c
Therefore by the symmetry of ptF and ptL , (5.11) follows also for μ × μ-a.e. (x, y) ∈ F c × F c yield F c \ Lc × Lc . Moreover, (5.11) and the symmetry of pt/2
Fc
Lc pt/2 (x, y)2 − pt/2 (x, y)2 d(μ × μ)(x, y)
0 ZF c (t) − ZLc (t) = F c ×F c
=
F c
Fc Lc Lc pt/2 (x, y) + pt/2 (x, y) pt/2 (x, y) − pt/2 (x, y) d(μ × μ)(x, y)
F c ×F c
2
Fc pt/2 (x, y) Φ(t/2, x) + Φ(t/2, y) d(μ × μ)(x, y)
F c ×F c
=4
c
F pt/2 (x, y)Φ(t/2, x) dμ(y) dμ(x) 4
Fc Fc
Φ(t/2, x) dμ(x),
Fc
F c (·, y) dμ(y) 1 μ-a.e. on F c . Now μ(U√ (x, S)) where we used the fact that F c pt/2 t cVD μ(U√t/2 (x, S)) leads to (5.12). 2 Proposition 5.10. Assume that (L, S) is locally finite. Let d∂ ∈ [0, ∞), β ∈ (1, ∞) and d be a (2/β)-qdistance adapted to S. Let A ⊂ K be non-empty and suppose dimS A d∂ . Let c1 , c2 ∈ (0, ∞). Then there exists c ∈ (0, ∞) such that for any t ∈ (0, 1], K
1 distd (x, A)2 β−1 c1 dμ(x) ct −d∂ /2 . exp −c 2 μ(U√t (x, S)) t
(5.14)
Combining Proposition 5.10 with Proposition 5.8(3) and (5.12), we have the following estimate, which is the key for the proof of Theorem 5.2. Theorem 5.11 (Key estimate). Suppose that (UHK) holds. Let F and L be closed subsets of K such that F ⊂ L K. Let d∂ ∈ [0, ∞) and suppose dimS (L \ F ) d∂ . Then there exists c ∈ (0, ∞) such that for any t ∈ (0, 1], 0 ZF c (t) − ZLc (t) ct −d∂ /2 .
(5.15)
Proof of Proposition 5.10. First let s ∈ (0, 1] and w ∈ Λs . Choose x0 ∈ Kw \ F w (V0 ) ( = ∅). Then for any x ∈ Kw \ Fw (V0 ), Ks (x, S) = Kw = Ks (x0 , S) and Us (x, S) = v∈Λs,w Kv = Us (x0 , S). Since μ(Fw (V0 )) = 0, we have Kw
1 dμ(x) = μ(Us (x, S))
Kw \Fw (V0 )
μ(Ks (x0 , S)) 1 dμ(x) = 1. μ(Us (x0 , S)) μ(Us (x0 , S))
(5.16)
Choose n ∈ N, β1 ∈ (0, 1] and β2 ∈ [1, ∞) so that (2.5) holds. Let s ∈ (0, 1] and set cA := sups∈(0,1] s d∂ #W (Λs , A) (< ∞ by dimS A d∂ ) and M := sup{#Λs,w | s ∈ (0, 1], w ∈ Λs }
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(< ∞ by the local finiteness of (L, S)). For 0 k n, we set Λks,A := W (k) (Λs , A) (recall Def inition 2.15(2)). Then for 0 k n − 1, since Λk+1 w∈Λk Λs,w , s,A = W (Λs , w∈Λk Kw ) = s,A
s,A
k we have #Λk+1 s,A M#Λs,A . Therefore
#Λns,A M n #Λ0s,A = M n #W (Λs , A) cA M n s −d∂ . Let Ks (A) :=
w∈Λns,A
Kw (=
(n) x∈A Us (x, S)).
(5.17)
If x ∈ K and distd (x, A) < β1 s, then (n)
d(x, y) < β1 s for some y ∈ A. Hence x ∈ Bβ1 s (y, d) ⊂ Us (y, S) ⊂ Ks (A). Therefore distd (x, A) β1 s,
x ∈ K \ Ks (A).
(5.18)
Recall that S is (assumed to be) elliptic. Therefore we may choose c3 ∈ (1, ∞) so that g(w) s c3 g(w) for any s ∈ (0, 1] and any w ∈ Λs . We also easily see that there exist c4 , γ ∈ (1, ∞) such that g(wv) c4 γ −|v| g(w) for any w, v ∈ W∗ . Moreover, by Proposition 2.7 there exist cS , α ∈ (0, ∞) such that #Λs cS s −α ,
s ∈ (0, 1].
(5.19)
Let N := N(s) := max{k ∈ N ∪ {0} | 2k s 1}, and for 0 k N let ψsk : Λs → Λ2k s be the natural surjection, so that w ψsk (w) for any w ∈ Λs . Let 0 k N and w ∈ Λ2k s . To estimate #((ψsk )−1 (w)), let v ∈ (ψsk )−1 (w). Then v w, g(w) 2k s c3 g(w), g(v) s c3 g(v) and g(v) c4 γ −(|v|−|w|) g(w). Therefore γ |v|−|w| c4 g(w)/g(v) c4 2k sc3 s −1 = c3 c4 2k , hence |v| − |w| (k log 2 + log c3 c4 )/ log γ (=: k ), where a := max{j ∈ Z | j a} for a ∈ R. Hence by setting c5 := (#S)1+(log c3 c4 )/ log γ /(#S − 1) and Γ := 2(log #S)/ log γ we have #((ψsk )−1 (w)) (#S)k +1 /(#S − 1) c5 Γ k for any w ∈ Λ2k s . Then by (5.17),
−1 n
Λ2k s,A = # ψsk
−1
# ψsk (w) c5 Γ k #Λn2k s,A
w∈Λnk
2 s,A
−d
k c5 Γ k cA M n 2k s ∂ = c5 cA M n 2−d∂ Γ s −d∂ .
(5.20)
Note also that
K2k s (A) =
Kw =
w∈Λnk
2 s,A
Kv =
v∈(ψsk )−1 (w) 2 s,A
w∈Λnk
Kw .
(5.21)
w∈(ψsk )−1 (Λnk ) 2 s,A
√ Now let t ∈ (0, 1], N := N ( √ t) and let Φ(t, x), (t, x) ∈ (0, 1]×K, be the integrand in the lefthand side of (5.14). Since 2N +1 t > 1, the observations (5.16)–(5.18), (5.20), (5.21) and (5.19) yield the following estimate: Φ(t, x) dμ(x) = Φ(t, x) dμ(x) + Φ(t, x) dμ(x) K
K√t (A)
0
√ √ 2k t (A)\K2k−1 t (A)
+ K\K2N √t (A)
Φ(t, x) dμ(x)
N. Kajino / Journal of Functional Analysis 258 (2010) 1310–1360
K√t (A)
c1 dμ(x) t (x, S))
μ(U√
+
2
0
√ √ 2k t (A)\K2k−1 t (A)
μ(U√
K\K2N √t (A)
w∈Λn√
t,A
Kw
c1 √ μ(U t (x, S))
+
0
2
+
w∈Λ√t K w
c1 #Λn√t,A +
k−1
c1 exp(−c2 β1β−1 4 β−1 ) dμ(x) μ(U√t (x, S))
1 −1
c1 exp −c2 β12 /4 β−1 t β−1 dμ(x) t (x, S))
+
1337
t,A
)
Kw
dμ(x) 2
k−1
c1 exp(−c2 β1β−1 4 β−1 ) dμ(x) μ(U√t (x, S))
1 −1
c1 exp −c2 β12 /4 β−1 t β−1 dμ(x) t (x, S))
μ(U√
2 k−1
n
k −1 Λ2k √t,A c1 exp −c2 β1β−1 4 β−1 # ψ√ t
0
1 −1
+ c1 exp −c2 β12 /4 β−1 t β−1 #Λ√t c1 cA M n t −d∂ /2 +
2 k−1
k c1 c5 cA M n 2−d∂ Γ exp −c2 β1β−1 4 β−1 t −d∂ /2
0
1 −1
+ c1 cS t −α/2 exp −c2 β12 /4 β−1 t β−1 ct −d∂ /2 , where c ∈ (0, ∞) is a constant determined solely by the constants given in the assumptions. Thus the proof is complete. 2 Proof of Theorem 5.2. Since ZN = ZK = Z∅c and ZD = ZK I = Z(V0 )c , Theorem 5.11 implies that there exists c0 ∈ (0, ∞) such that for any t ∈ (0, 1], 0 ZN (t) − ZD (t) ct −d∂ /2 . Let γ := mini∈S γi . By (5.8), for any t ∈ (0, γ 2 ], 0 ZD (t) −
i∈S
ZD
t γi2
t t ZN − Z c0 (#S)t −d∂ /2 . D 2 2 γ γ i i i∈S
(5.22)
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On the other hand, 0 ZD (t)− i∈S ZD (tγi−2 ) ZD (t) ZD (γ 2 ) for any t ∈ [γ 2 , ∞). Therefore if we set cZ := max{c0 (#S), ZD (γ 2 )}, then 0 ZD (t) −
ZD
i∈S
t γi2
cZ t −d∂ /2 ,
t ∈ (0, 1].
(5.23)
Define ΨD (x) := max{0, ZD (x −1 ) − ZD (1)} for each x ∈ (0, ∞). Then ΨD (x) = 0 for any x ∈ (0, 1]. Moreover, by (5.23) we easily see that 0 ΨD (x) −
ΨD γi2 x cx d∂ /2
(5.24)
i∈S
for any x ∈ (0, ∞), with a different constant c ∈ (0, ∞). We closely follow [27, Proof of Theorem 4.1.5] in the rest of this proof. Define f (t) := e−dS t ΨD (e2t ) and u(t) := e−dS t (ΨD (e2t ) − i∈S ΨD (γi2 e2t )) for t ∈ R. f and u are bounded d and continuous. Letting pi := γi S for i ∈ S, so that i∈S pi = 1, we have the following renewal equation f (t) =
pi f t − log γi−1 + u(t),
t ∈ R.
(5.25)
i∈S
We have f (t) = u(t) = 0 for any t ∈ (−∞, 0], and (5.24) yields 0 u(t) ce−(dS −d∂ )t for any t ∈ [0, ∞). Since we assume that dS − d∂ > 0, all the conditions required for the renewal theorem [27, Theorems B.4.2 and B.4.3] are satisfied (see also Feller [16, Chapter XI] for the renewal theorem). Thus, for the non-lattice case, we have
lim f (t) =
t→∞
i∈S
d γi S
∞ −1 −1 log γi u(t) dt (∈ R), 0
and this means that t dS /2 ZD (t) converges as t ↓ 0. limt↓0 t dS /2 ZD (t) ∈ (0, ∞) by (5.9). (5.22) implies that t dS /2 ZN (t) also converges to the same limit as t ↓ 0. For the lattice case, it is clear that the series j ∈Z u(· + j T ) is uniformly absolutely convergent on every compact subset of R, hence the function G on R defined by G(t) := j ∈Z u(t + j T ), t ∈ R, where (M) −1 := i∈S mi pi , is T -periodic and continuous. By [27, M Theorem B.4.3], limt→∞ |G(t) − f (t)| = 0, and this is clearly equivalent to limt↓0 |t dS /2 ZD (t) − G(2−1 log(t −1 ))| = 0. Then (5.9) implies that G is (0, ∞)-valued. Moreover, [27, Theorem B.4.3] leads also to the following estimate of |f (t) − G(t)|:
as t → ∞,
⎧ −(dS −d∂ )t ) ⎨ O(e f (t) − G(t) = O(t m e−(dS −d∂ )t ) ⎩ O(t m−1 q −t/T )
if e(dS −d∂ )T < q, if e(dS −d∂ )T = q, if e(dS −d∂ )T > q.
Now all the statements for the lattice case are obvious from (5.22) and (5.26).
2
(5.26)
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6. Rational boundary and cell-counting dimension This and the next sections are devoted to giving some complementary statements concerning the main result of the previous section (Theorem 5.2) and its proof. In this section, we provide a practical method of calculating the cell-counting dimension of self-similar subsets with respect to a self-similar scale. We also see that the inequality dimS V0 < dS is valid for all typical examples. Let S be a non-empty finite set. Definition 6.1. Let X be a non-empty finite subset of W# (= W∗ \ {∅}). (1) We write w = (w)1 · · · (w)|w| for any w ∈ W# . We define ιX : Σ(X) → Σ = Σ(S) and ιW X : W∗ (X) → W∗ = W∗ (S) to be the natural identifications, that is, ιX (x1 x2 · · ·) := (x1 )1 · · · (x1 )|x1 | (x2 )1 · · · (x2 )|x2 | · · · , ιW X (x1 · · · xm ) := (x1 )1 · · · (x1 )|x1 | · · · (xm )1 · · · (xm )|xm | . (2) We set Σ[X] := ιX (Σ(X)) and Σw [X] := σw (Σ[X]). (3) X is called independent if and only if ιX is injective. Clearly, if X is independent then ιW X is also injective. Accordingly, when X is independent, we will often identify x1 · · · xm ∈ W∗ (X) with ιW X (x1 · · · xm ) ∈ W∗ and Σ(X) with Σ[X] through ιX . Note that, for X ⊂ W# non-empty finite, Σ[X] is compact since ιX is continuous. Below we collect basic facts on Σx [X], where X ⊂ W# is non-empty finite and x ∈ W∗ . Definition 6.2. (1) Let Σ0 ⊂ Σ be non-empty and x ∈ W∗ . For each ω ∈ Σ , we define OΣ0 ,x (ω) := #({n ∈ N ∪ {0} | σ n ω ∈ σx (Σ0 )}), where we allow ∞ as a value of OΣ0 ,x (ω). (2) Let X ⊂ W# be non-empty finite and let x ∈ W# . We define AX,x (w) := (z, x0 , x1 , . . . , xm ) m ∈ N ∪ {0}, z ∈ W∗ , x0 = x, x1 , . . . , xm ∈ X, zx0 x1 · · · xm w < zx0 x1 · · · xm−1 for each w ∈ W∗ , with the convention that zx0 x1 · · · xm−1 =: z when m = 0. Definition 6.3. A subset X of W# is called separated if and only if it is non-empty, finite and independent and satisfies OΣ[X],y (ω) < ∞ for any ω ∈ Σ for some y ∈ W# . By [28, Lemma 1.6.3], supw∈W∗ #(AX,y (w)) < ∞ in this case. The following lemma is useful for concrete examples, and is easily proved. Lemma 6.4. Let S1 S be non-empty, let X ⊂ W# (S1 ) be non-empty finite and x ∈ W# (S \ S1 ). Then supω∈Σ OΣ[X],x (ω) = 1. Lemma 6.5. Let S = {Λs }s∈(0,1] be a scale on Σ with gauge function l, let X ⊂ W# be separated with y ∈ W# as in Definition 6.3 and set M := supw∈W∗ #(AX,y (w)) (< ∞). For s ∈ (0, 1], define Λs [X] := w ∈ Λs Σw ∩ Σ[X] = ∅ , Λs (X) := x1 · · · xm ∈ W∗ (X) l(x1 · · · xm−1 ) > s l(x1 · · · xm )
(6.1)
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with the convention that l(x1 · · · xm−1 ) = 2 when m = 0. Then for any s ∈ (0, 1], #Λs [X] #Λs (X) M#Λs [X].
(6.2)
Proof. Let s ∈ (0, 1] and let x = x1 · · · xm ∈ Λs (X). Since l(x) s, there exists a unique ϕ(x) ∈ Λs such that x ϕ(x). Clearly ϕ(x) ∈ Λs [X], and we have a map ϕ : Λs (X) → Λs [X]. Let w ∈ Λs [X]. Choose x1 x2 · · · ∈ Σw ∩ Σ[X] and let m0 := min{m | m 0, |w| |x1 · · · xm |}. Then we see that x1 · · · xm0 ∈ Λs (X) and ϕ(x1 · · · xm0 ) = w. Hence ϕ is surjective and #Λs [X] #Λs (X). Next let x1 · · · xm ∈ ϕ −1 (w). Then x1 · · · xm w and l(w) s < l(x1 · · · xm−1 ). Hence w < x1 · · · xm−1 and yx1 · · · xm yw < yx1 · · · xm−1 , namely (∅, y, x1 , . . . , xm ) ∈ AX,y (yw). by x1 · · · xm → (∅, y, x1 , . . . , xm ). Thus we have an injection ϕ −1 (w) → AX,y (yw) defined Hence #ϕ −1 (w) #(AX,y (yw)) M and #Λs (X) = w∈Λs [X] #ϕ −1 (w) M#Λs [X]. 2 Proposition 6.6. Let X ⊂ W# beseparated, α = (αi )i∈S ∈ (0, 1)S and let d(α, X) (∈ [0, ∞)) be the unique d ∈ R that satisfies x∈X αxd = 1. For each s ∈ (0, 1] let Λs [X] be as in (6.1) with Λs := Λs (α) (recall Definition 2.8). Then there exist c1 , c2 ∈ (0, ∞) such that for any s ∈ (0, 1], c1 s −d(α,X) #Λs [X] c2 s −d(α,X) .
(6.3)
Proof. Let l := gα (recall Definition 2.8), and for s ∈ (0, 1] let Λs (X) be as in (6.1). Since {Λs (X)}s∈(0,1] is a self-similar scale on Σ(X) with weight (αx )x∈X , Proposition 2.9 implies the existence of c2 ∈ [1, ∞) such that s −d(α,X) #Λs (X) c2 s −d(α,X) for any s ∈ (0, 1]. Then Lemma 6.5 implies the assertion. 2 Let L = (K, S, {Fi }i∈S ) be a self-similar structure in the rest of this section. Notation. Let π : Σ = Σ(S) → K denote the canonical projection associated with L. For nonempty finite X ⊂ W# and w ∈ W∗ , we set K[X] := π(Σ[X]) and Kw [X] := π(Σw [X]). Proposition Xk ⊂ W# be separated and wk ∈ W∗ for each k = 1, . . . , N . 6.7. Let N ∈ N and let N S Set Γ := N k=1 Σwk [Xk ] and L := k=1 Kwk [Xk ] (= π(Γ )). Let α = (αi )i∈S ∈ (0, 1) and set dk (α) := d(α, Xk ) for k = 1, . . . , N and dΓ (α) := maxk∈{1,...,N } dk (α). (1) If either (L, S(α)) is locally finite or π −1 (L) = Γ , then dimS(α) L = dΓ (α). d(α) (2) Let να be the Bernoulli measure on Σ with weight (αi )i∈S . Then dΓ (α) < d(α) if and only if να (Γ ) = 0. In most typical cases, V0 = π(PL ) is written in the form of Γ in Proposition 6.7. Considering such situations, we set the following definition. Definition 6.8 (Rational boundary). We say that L is of rational boundary, or simply (RB) holds, if and only if there exist N ∈ N and a separated set Xk ⊂ W# and wk ∈ W∗ for each k = 1, . . . , N , such that PL = N k=1 Σwk [Xk ]. Roughly speaking, (RB) says that the boundary V0 is a finite union of self-similar sets. (RB) implies that V0 = N i=1 Kwk [Xk ], which is clearly compact, hence that V0 = V0 . When (RB) holds, we can explicitly calculate dimS(α) V0 as in the following theorem.
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Theorem 6.9 (Cell-counting dimension for rational boundaries). Assume (RB). Let α = (αi )i∈S ∈ (0, 1)S and d∂ (α) := max1kN d(α, Xk ) with N , Xk as in Definition 6.8. Then dimS(α) V0 = d∂ (α). Moreover, d∂ (α) < d(α) if and only if K = V0 . Kigami [28, Definition 1.5.10] has introduced the notion of rationally ramified self-similar structures as a class of self-similar sets with sufficiently good ramification structure, in order to argue the volume doubling property and the (sub-)Gaussian estimate of heat kernels on self-similar sets in a general framework. For example, any p.c.f. self-similar structure and any generalized Sierpinski carpet ([6,7], see also [28, Section 3.4] and [25, §2]) are rationally ramified. By [28, Proof of Proposition 1.5.13(1)], any rationally ramified self-similar structure satisfies (RB). See [28, Sections 1.5 and 1.6 and Chapter 2] for details about rationally ramified self-similar structures. Proof of Theorem 6.9. π(PL ) = V0 by definition, and π −1 (V0 ) = PL by [27, Proposition 1.3.5(1)]. Therefore dimS(α) V0 = d∂ (α) by Proposition 6.7(1). If K = V0 then by Proposition 2.9, d∂ (α) = dimS(α) V0 = dimS(α) K = d(α). Conversely, assume K = V0 (= V0 ). Let να d(α) be the Bernoulli measure on Σ with weight (αi )i∈S . Then να ◦ π −1 is a self-similar measure on K with the same weight. [28, Theorem 1.2.7] implies 0 = να ◦ π −1 (V0 ) = να (PL ). Now Proposition 6.7(2) yields d∂ (α) < d(α). 2 Proof of Proposition 6.7. We write S, Λs , d, dk , dΓ , ν and μ instead of S(α), Λs (α), d(α), dk (α), dΓ (α), να and μα in this proof. Set α := mini∈S αi , αW := min1kN αwk and Λs [Xk ] := {w ∈ Λs | Σw ∩ Σ[Xk ] = ∅} for k = 1, . . . , N and s ∈ (0, 1], as in Lemma 6.5. Then Proposition 6.6 implies that there exist c1 , c2 ∈ (0, ∞) such that for any k ∈ {1, . . . , N}, c1 s −dk #Λs [Xk ] c2 s −dk ,
s ∈ (0, 1].
(6.4)
−dΓ −d and 1 #W (Λs , L) #Λs α −d s −d α −d αW for s ∈ [αW , 1], (1) Since 1 s −dΓ αW we may assume that s ∈ (0, αW ]. Let s ∈ (0, αW ]. Then
W (Λs , L) ⊃ {w ∈ Λs | Σw ∩ Γ = ∅} =
N
N w ∈ Λs Σw ∩ Σwk [Xk ] = ∅ = wk v v ∈ Λsαw−1 [Xk ] .
k=1
k=1
(6.5)
k
Choose J ∈ {1, . . . , N} so that dJ = dΓ . By (6.4) and dΓ = dJ , dJ −dΓ s . #W (Λs , L) # wJ v v ∈ Λsαw−1 [XJ ] = #Λsαw−1 [XJ ] c1 αw J J
(6.6)
J
To estimate #W (Λs , L) from above, let M := 1 if π −1 (L) = Γ and otherwise let M := sup{#Λt,w | t ∈ (0, 1], w ∈ Λt } (< ∞ by the assumption). Then we have #W (Λs , L) M#{w ∈ Λs | Σw ∩ Γ = ∅}. Indeed, if π −1 (L) = Γ then clearly W (Λs , L) = {w ∈ Λs | Σw ∩ Γ = ∅} and #W (Λs , L) = #({w ∈ Λs | Σw ∩ Γ = ∅}). If π −1 (L) = Γ , let v ∈ W (Λs , L). Choose x ∈ Kv ∩ L, ω ∈ Γ ∩ π −1 (x) and w ∈ Λs so that ω ∈ Σw . Then x ∈ Kw ∩ Kv = ∅, hence
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v ∈ Λs,w , and ω ∈ Σw ∩ Γ = ∅. Therefore W (Λs , L) ⊂ {Λs,w | w ∈ Λs , Σw ∩ Γ = ∅} and #W (Λs , L) M#({w ∈ Λs | Σw ∩ Γ = ∅}). Now by (6.4), (6.5), N N N #W (Λs , L) dk −dk # wk v v ∈ Λsαw−1 [Xk ] = #Λsαw−1 [Xk ] c2 αw s cs −dΓ , k k k M k=1
k=1
k=1
dk where c := N k=1 c2 αwk , and dimS L = dΓ follows from this and (6.6). (2) Let k ∈ {1, . . . , N}. Note that Σs [Xk ] := w∈Λs [Xk ] Σw , s ∈ (0, 1], is decreasing as s ↓ 0 −d ν(Σ [X ]). Now since and that s∈(0,1] Σs [Xk ] = Σ[Xk ]. Also, ν(Σ[Xk ]) = αw wk k k c1 α d s d−dk α d s d #Λs [Xk ]
d αw = ν Σs [Xk ] s d #Λs [Xk ] c2 s d−dk ,
w∈Λs [Xk ]
d d−dk
we have lim sup c1 α s s↓0
−d ν Σ[Xk ] = αw ν Σwk [Xk ] lim inf c2 s d−dk . k
Hence dk < d if and only if ν(Σwk [Xk ]) = 0, and the statement follows.
s↓0
2
7. Sharpness of the key estimate In this section, we prove a better lower bound for (5.15) in Theorem 5.11 in terms of the cell-counting dimension of L \ F , under the condition that L \ F includes a self-similar subset of positive capacity. This shows a sharpness of the upper bound in (5.15). For this purpose, we need the notion of intersection type introduced by Kigami [28, Section 2.2]. Subsection 7.1 is devoted to a brief description of basic facts on intersection type. The statement and the proof of sharpness of the key estimate is provided in Subsection 7.2 (Theorem 7.7). The proof of Theorem 7.7 relies heavily on strict positivity of heat kernels and of hitting probabilities, which is separately argued in Appendix A in the framework of a general regular Dirichlet form. In Subsection 7.3 we establish a reasonable sufficient condition for the positivity of capacity (Theorem 7.18), which plays an essential role in applying Theorem 7.7 to generalized Sierpinski carpets in Section 8. 7.1. Intersection type Throughout this subsection, we fix a self-similar structure L = (K, S, {Fi }i∈S ) and a scale S = {Λs }s∈(0,1] on Σ = Σ(S). We state basic definitions on intersection type only briefly. See Kigami [28, Section 2.2] for basic facts about intersection type. Definition 7.1. (1) Define IP(L) := {(w, v) | w, v ∈ W# , Kw ∩ Kv = ∅, Σw ∩ Σv = ∅}. Each (w, v) ∈ IP(L) is called an intersection pair of L. (2) Set A := {(A, B, ϕ) | A, B ⊂ V0 non-empty compact, ϕ : A → B homeomorphism}. For each (w, v) ∈ IP(L), we define ΦIT ((w, v)) ∈ A by
ΦIT (w, v) := Fw−1 (Kw ∩ Kv ), Fv−1 (Kw ∩ Kv ), Fv−1 ◦ Fw |Fw−1 (Kw ∩Kv ) . Definition 7.2 (Intersection type). (1) We set IT (L) := ΦIT (IP(L)). Each element of IT (L) is called an intersection type of L.
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(2) We define IP(L, S) := {(w, v) | w, v ∈ Λs for some s ∈ (0, 1] and (w, v) ∈ IP(L)} and IT (L, S) := ΦIT (IP(L, S)). We say that S is intersection type finite with respect to L, or simply (L, S) is intersection type finite, if and only if #IT (L, S) < ∞. Definition 7.3. (1) A non-empty finite subset Γ of W∗ is said to be a sub-partition of Σ if and only if Σw ∩ Σv = ∅ for any w, v ∈ Γ with w = v. (2) Let Γ1 , Γ2 ⊂ W∗ be sub-partitions of Σ . A bijection ϕ : Γ1 → Γ2 is called an Lisomorphism if and only if ϕ possesses the following two properties: (i) For w, v ∈ Γ1 , (w, v) ∈ IP(L) if and only if (ϕ(w), ϕ(v)) ∈ IP(L). (ii) ΦIT ((w, v)) = ΦIT ((ϕ(w), ϕ(v))) for any w, v ∈ Γ1 with (w, v) ∈ IP(L). (3) Let ϕ : Γ1 → Γ2 be an L-isomorphism between sub-partitions Γ1 , Γ2 of Σ . We define Fϕ : K(Γ1 ) → K(Γ2 ) (recall Definition 2.15(1)) by Fϕ |Kw := Fϕ(w) ◦ Fw−1 for any w ∈ Γ1 . Fϕ is a well-defined homeomorphism. We call Fϕ the L-similitude associated with ϕ. Moreover, if μ is a self-similar measure on K and K = V0 , define a bounded linear operator ρϕ : L2 (K(Γ2 ), μ|K(Γ2 ) ) → L2 (K(Γ1 ), μ|K(Γ1 ) ) by ρϕ u := u ◦ Fϕ . Also for u : K → R, we define uϕ : K → R by uϕ :=
u ◦ Fϕ−1 0
on K(Γ2 ), on K \ K(Γ2 ). n
Definition 7.4. Let n ∈ N ∪ {0}. For (s1 , x1 ), (s2 , x2 ) ∈ (0, 1] × K, we write (s1 , x1 ) ∼ (s2 , x2 ) L,S
if and only if there exists an L-isomorphism ϕ : Λns1 ,x1 → Λns2 ,x2 such that ϕ(Λks1 ,x1 ) = Λks2 ,x2 for any k = 0, . . . , n. Such ϕ is called an (n, L, S)-isomorphism between (s1 , x1 ) and (s2 , x2 ). n,ϕ n Clearly, ∼ is an equivalence relation on (0, 1] × K. Moreover, we write (s1 , x1 ) ∼ (s2 , x2 ) if L,S
L,S
and only if ϕ : Λns1 ,x1 → Λns2 ,x2 is an (n, L, S)-isomorphism between (s1 , x1 ) and (s2 , x2 ). The following lemma is used in the next subsection. (n)
(n)
Notation. For n ∈ N ∪ {0} and (s, x) ∈ (0, 1] × K, we set Vs (x, S) := intK (Us (x, S)). n+1,ϕ
Lemma 7.5. Let n ∈ N ∪ {0} and (s1 , x1 ), (s2 , x2 ) ∈ (0, 1] × K. If (s1 , x1 ) ∼ (s2 , x2 ), then L,S
(n) (n) Fϕ (Vs1 (x1 , S)) = Vs2 (x2 , S),
where Fϕ is the L-similitude associated with ϕ. (n+1)
(n)
Proof. For i = 1, 2, let Ui := Usi (xi , S). Then Usi (xi , S) ⊂ intK Ui by Lemma 2.16(1). (n) (n) Therefore Vsi (xi , S) = intUi (Usi (xi , S)). Since Fϕ : U1 → U2 is a homeomorphism and (n) (n) Fϕ (Us1 (x1 , S)) = Us2 (x2 , S), the assertion is now immediate. 2 7.2. Sharpness of the key estimate Throughout this subsection, (L = (K, S, {Fi }i∈S ), μ, E, F , r = (ri )i∈S ) is a self-similar Dirichlet space with μ a self-similar measure with weight (μi )i∈S , S = {Λs }s∈(0,1] is the self-
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√ similar scale with weight γ := (γi )i∈S , γi := ri μi , and dS := d(γ ) (= dimS K > 0). We follow the notations introduced in Section 5. The following conditions are required to verify a sharp lower bound for (5.15). Definition 7.6. (1) We say that (L, μ, E, F , r) satisfies the strong domain self-similarity (SSDF3S), or simply (SSDF3S) holds, if and only if F has the following property: (SSDF3S) For any sub-partitions Γ1 , Γ2 of Σ , any L-isomorphism ϕ : Γ1 → Γ2 and any u ∈ F ∩ C(K) with suppK [u] ⊂ intK K(Γ1 ), if uϕ ∈ C(K) and suppK [uϕ ] ⊂ intK K(Γ2 ) then uϕ ∈ F ∩ C(K), where uϕ is as in Definition 7.3(3). (2) We say that (L, μ, E, F , r) is local weight type finite, or simply (LWTF) holds, if and only if {rw /rv | (w, v) ∈ IP(L, S)} and {μw /μv | (w, v) ∈ IP(L, S)} are finite. Clearly, (SSDF3S) is stronger than (SSDF3) (let Γ1 = {∅} and Γ2 = {i}, i ∈ S). The following is the main theorem of this section. See Definition A.1(3) for the condition (CHK), and Definition A.4 for the definition of CapE . Theorem 7.7 (Sharpness of the key estimate). Assume that K is connected and that (E, F ) is conservative. Suppose that (L, S) is intersection type finite and that (LWTF), (SSDF3S), (CHK) and (UHK) hold. Let F ⊂ K be a closed subset of K, let w ∈ W∗ and let X ⊂ W# be separated and satisfy CapE (K[X]) > 0. Set L := F ∪ Kw [X] and d∂ := d(γ , X) (recall Proposition 6.6) and suppose F L K. Then there exist c1 , c2 ∈ (0, ∞) such that for any t ∈ (0, 1], c1 t −d∂ /2 ZF c (t) − ZLc (t) c2 t −d∂ /2 .
(7.1)
Remark. (1) If K is a generalized Sierpinski carpet, then we can construct a conservative selfsimilar Dirichlet space satisfying (SSDF3S) and (CHK). In this case, (UHK) implies (LWTF) and that (L, S) is intersection type finite. See Section 8 for details. (2) We have dimS (L \ F ) = d∂ in the situation of Theorem 7.7. In fact, since (L, S) is locally finite by (UHK) and Proposition 5.8(3), Proposition 6.7(1) implies that for any v ∈ W∗ , dimS Kv [X] = d∂ . As Kw [X] ⊂ F , we can choose x ∈ W∗ (X) so that Kwx [X] ∩ F = ∅. Then Kwx [X] ⊂ L \ F ⊂ Kw [X]. Hence dimS (L \ F ) = d∂ follows. (3) The lower bound in (7.1) is the essence of Theorem 7.7. In fact, since dimS (L \ F ) = d∂ , the upper bound in (7.1) follows from (UHK) and Theorem 5.11. As a corollary of Theorem 7.7, we have a sharp estimate for the reminder term in (5.4) under the condition (RB), as follows. Recall that (RB) implies V0 = V0 ( = K). Corollary 7.8. Assume that K is connected and that (E, F ) is conservative. Suppose that (L, S) is intersection type finite and that (LWTF), (SSDF3S), (CHK) and (UHK) hold. Suppose also that γi = γ for any i ∈ S for some γ ∈ (0, 1) and that L satisfies (RB) with N ∈ N and Xk ⊂ W# for k ∈ {1, . . . , N} as in Definition 6.8. Let d∂ := max1kN d(γ , Xk ) (= dimS V0 ∈ [0, dS ) by Theorem 6.9) and let G be the continuous log(γ −1 )-periodic function given in Corollary 5.3. If CapE (K[XJ ]) > 0 for some J ∈ {1, . . . , N} satisfying d(γ , XJ ) = d∂ , then there exist c1 , c2 ∈ (0, ∞) such that for any t ∈ (0, 1],
1 1 −d∂ /2 −dS /2 log − ZD (t) c2 t −d∂ /2 . t G (7.2) c1 t 2 t
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The rest of this subsection is devoted to the proofs of Theorem 7.7 and Corollary 7.8. First we prepare easy consequences of the assumptions. In the proofs below, {ptN }t∈(0,∞) always denotes the jointly continuous heat kernel of {TtN }t∈(0,∞) when (CHK) holds. Remark. In the following Lemmas 7.9–7.12 and their proofs, we do not use the assumption that μ is a self-similar measure. Lemma 7.9. Suppose that (CHK) and (UHK) hold and let β, d, c1 and c2 be as in Definition 5.1. Then (5.1) is valid for any (t, x, y) ∈ (0, 1] × K × K. Proof. This is immediate by the lower semicontinuity of x → μ(B√t (x, d)) on K.
2
See Definition A.2 for the definitions of Feller and strong Feller properties. Lemma N 7.10. Suppose that (E, F ) is conservative and that (CHK) holds. Set Pt (x, A) := A pt (x, y) dμ(y) for (t, x) ∈ (0, ∞) × K and A ∈ B(K). Then {Pt }t∈(0,∞) is a μ-symmetric conservative strong Feller Markovian transition function on (K, B(K)) whose associated Markovian semigroup on L2 (K, μ) is {TtN }t∈(0,∞) . Moreover, if (UHK) holds in addition then {Pt }t∈(0,∞) is Feller. Proof. Let t ∈ (0, ∞). Then Pt (·, K) = TtN 1 = 1 μ-a.e. since (E, F ) is conservative, and Pt (·, K) = K ptN (·, y) dμ(y) ∈ C(K). Therefore Pt (x, K) = 1 for any x ∈ K. Now since {ptN }t∈(0,∞) ⊂ C(K × K) and it is a heat kernel of {TtN }t∈(0,∞) , it is clear that {Pt }t∈(0,∞) is a μ-symmetric conservative strong Feller Markovian transition function on (K, B(K)) whose associated Markovian semigroup on L2 (K, μ) is {TtN }t∈(0,∞) . Next, suppose that (UHK) holds in addition. Let c, α ∈ (0, ∞) be as in Lemma 3.7 and let β, d, c1 and c2 be as in Definition 5.1. By Proposition 5.8(4), there exists cV ∈ (0, ∞) such that cV μ(B√t (x, d)) μ(U√t (x, S)), hence ccV μ(B√t (x, d)) t α/2 , for any (t, x) ∈ (0, 1] × K. Therefore by (UHK) and Lemma 7.9 we see that
1
d(x, y)2 β−1 0 ptN (x, y) cc1 cV t −α/2 exp −c2 , (t, x, y) ∈ (0, 1] × K × K. (7.3) t Now for f ∈ C(K), by K ptN (·, y) dμ(y) = 1 on K, (7.3) and the uniform continuity of f we easily see that limt↓0 Pt f − f ∞ = 0, proving the Feller property of {Pt }t∈(0,∞) . 2 Notation. As in Appendix A, for a non-empty open subset U of K, let U := U ∪ {U } denote the one-point compactification of U . Lemma 7.11. Suppose that (E, F ) is conservative and that (CHK) and (UHK) hold. (1) Let {Pt }t∈(0,∞) be as in Lemma 7.10. Then there exists a conservative diffusion X = (Ω, M, {Xt }t∈[0,∞] , {Px }x∈K ) on K whose transition function is {Pt }t∈(0,∞) . (2) For A ∈ B(K ) and ω ∈ Ω, define σA (ω) := inf{t ∈ [0, ∞) | Xt (ω) ∈ A} (inf ∅ := ∞) and τA (ω) := σK \A (ω). Let U be a non-empty open subset of K and define Xt (ω) if t < τU (ω), XtU (ω) := if t τU (ω) U
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for t ∈ [0, ∞] and ω ∈ Ω. Also set PU := PK . Then the process X U := (Ω, M, {XtU }t∈[0,∞] , {Px }x∈U ) is a diffusion on U with μ|U -symmetric strong Feller transi2 tion function {PU t }t∈(0,∞) whose associated Markovian semigroup on L (U, μ|U ) is U U {Tt }t∈(0,∞) . Moreover, (E , FU ) satisfies (CHK) with jointly continuous heat kernel {ptU }t∈(0,∞) ⊂ Cb (U × U ), PU A) = A ptU (x, y) dμ(y) for any (t, x) ∈ (0, ∞) × U t (x, and any A ∈ B(U ), and ZU (t) = U ptU (x, x) dμ(x) for any t ∈ (0, ∞). Proof. (1) By [11, Chapter I, Theorem 9.4] and the Feller property of {Pt }t∈(0,∞) , there exists a Hunt process X = (Ω, M, {Xt }t∈[0,∞] , {Px }x∈K ) on K with transition function {Pt }t∈(0,∞) . Since (E, F ) is local by Lemma 3.4, [17, Theorem 4.5.4(ii)] implies that X is a diffusion, and it is conservative since Px [Xt ∈ K] = Pt (x, K) = 1 for (t, x) ∈ (0, ∞) × K. (2) By [17, Theorems 4.4.2 and 4.4.3], X U is a Hunt process on U with μ|U -symmetric 2 transition function {PU t }t∈(0,∞) whose associated Markovian semigroup on L (U, μ|U ) is U U U {Tt }t∈(0,∞) . The definition of X immediately implies that X is a diffusion. Since the transition function {Pt }t∈(0,∞) of X is both Feller and strong Feller, [13, p. 69, Section 1, Proof of Theorem] implies that {PU t }t∈(0,∞) is strong Feller. By Proposition 5.8(1) and [28, Proposition C.1], {TtU }t∈(0,∞) is ultracontractive. Therefore Proposition A.3(1) implies that (E U , FU ) satisfies (CHK) with jointly continuous heat kernel {ptU }t∈(0,∞) ⊂ Cb (U × U ) and that PU A) = A ptU (x, y) dμ(y) for any (t, x) ∈ (0, ∞) × U , A ∈ B(U ). Then we have t (x, U U (x, y)2 dμ(y) dμ(x) = ZU (t) = U ×U pt/2 U pt (x, x) dμ(x) for t ∈ (0, ∞). 2 Convention. In the situation of Lemma 7.11(2), we set ptU (x, y) := 0 for t ∈ (0, ∞) and (x, y) ∈ K ×K \U ×U , as stated in Notation before Lemma 5.7. Note that, with this convention, ptU may not be continuous on K × K, although it is continuous on U × U . We also set pt∅ (x, y) := 0 for any (t, x, y) ∈ (0, ∞) × K × K. Lemma 7.12. Suppose that (E, F ) is conservative and that (CHK) and (UHK) hold. Let U , V be non-empty open subsets of K. Then for any (t, x, y) ∈ (0, ∞) × K × K, ptN (x, y) − ptU (x, y) ptV (x, y) − ptU ∩V (x, y).
(7.4)
Proof. Let t ∈ (0, ∞). By Lemma 7.11, ptU , ptV and ptU ∩V are continuous on U × U , V × V and (U ∩ V ) × (U ∩ V ), respectively. Since ptN ptU and ptN ptV on K × K, (7.4) is trivial if either x ∈ / U ∩ V or y ∈ / U ∩ V . Let x, y ∈ U ∩ V . Then ptN (x, y) − ptU (x, y) − ptV (x, y) + ptU ∩V (x, y) N U V U ∩V (x, z)) dμ(z) U (y,S) (pt (x, z) − pt (x, z) − pt (x, z) + pt = lim s s↓0 μ(Us (y, S)) = lim
Px [Xt ∈ Us (y, S), τU t] − Px [Xt ∈ Us (y, S), τU ∩V t < τV ] μ(Us (y, S))
= lim
Px [Xt ∈ Us (y, S), τU t] − Px [Xt ∈ Us (y, S), τU t < τV ] μ(Us (y, S))
s↓0
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= lim s↓0
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Px [Xt ∈ Us (y, S), τU t, τV t] 0. μ(Us (y, S))
Thus the result follows.
2
The following lemma is the key for the proof of the lower bound of (7.1). wy0
Lemma 7.13. Under the assumption of Theorem 7.7, let y0 ∈ W∗ (X) and set Λs Λs | Σv ∩ Σwy0 [X] = ∅} for s ∈ (0, 1]. Then (recall Definitions 2.15 and 2.17) inf
N
K\K [X] wy0 2 √ [X] > 0. pt (x, x) − pt w (x, x) dμ(x) t ∈ 0, γwy , v ∈ Λ 0 t
K(Λ√
[X] := {v ∈
(7.5)
t,v )
The proof of Lemma 7.13 is given later. We first complete the proof of Theorem 7.7 using Lemma 7.13. Proof of Theorem 7.7. We follow the notations in Lemmas 7.10 and 7.11 above. Let β ∈ (1, ∞) and a (2/β)-qdistance d be as in Definition 5.1. Since F L = F ∪ Kw [X], Kw [X] ⊂ F and we may choose y ∈ W∗ (X) so that Kwy ∩ F = ∅. If F = ∅, let c1 , c2 ∈ (0, ∞) and Φ(t, x) be as in Lemma 5.9 with F and L replaced with ∅ and F , respectively, let c, α ∈ (0, ∞) be as in Lemma 3.7 and let δ := 2−1 infx∈Kwy distd (x, F ) (∈ (0, ∞)). Similarly to Lemma 7.9, by (5.11) c and Lemma 7.11(2), ptN (x, x) − ptF (x, x) 2Φ(t, x) for any (t, x) ∈ (0, 1] × K. Therefore, 2
with c3 := 2cc1 and c4 := c2 δ β−1 , for any t ∈ (0, 1] and any x ∈ K satisfying distd (x, Kwy ) (22/β − 1)β/2 δ, c − 1
ptN (x, x) − ptF (x, x) 2Φ(t, x) c3 t −α/2 exp −c4 t β−1
(7.6)
c
since distd (x, F ) δ. If F = ∅ then ptN (x, x) = ptF (x, x) for any (t, x) ∈ (0, ∞) × K and (7.6) is trivially valid with some α, δ, c3 , c4 ∈ (0, ∞). We set δβ := (22/β − 1)β/2 δ. For each s ∈ (0, 1], set Λs [X] := {v ∈ Λs | Σv ∩ Σ[X] = ∅}, and, as in Lemma 7.13, wy Λs [X] := {v ∈ Λs | Σv ∩ Σwy [X] = ∅}. By Proposition 6.6, there exists cX ∈ (0, ∞) such wy that #Λs [X] cX s −d∂ for any s ∈ (0, 1]. Let s ∈ (0, γwy ]. Then we easily see that Λs [X] = wy d∂ −d∂ s . {wyv | v ∈ Λsγ −1 [X]}. Therefore #Λs [X] = #Λsγ −1 [X] cX γwy wy
wy
2 ] so that diam K 2−β/2 δ for any v ∈ Λ√ , and let t ∈ (0, t ]. We Choose t∗ ∈ (0, γwy d v β ∗ t∗ Lc p F c p N and √ easily see that distd (x, Kwy ) δβ for any x ∈ v∈Λwy √ [X] K(Λ t,v ). Since pt t t
c ptL
K\K [X] pt w
t
ptN
on K × K, using (7.6) and Lemma 7.13 we see that
ZF c (t) − ZLc (t) Fc
c = pt (x, x) − ptL (x, x) dμ(x) K
{distd (·,Kwy )δβ }
c c ptF (x, x) − ptL (x, x) dμ(x)
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N
c − 1
pt (x, x) − ptL (x, x) dμ(x) − c3 t −α/2 exp −c4 t β−1
{distd (·,Kwy )δβ }
wy {Kτ |τ ∈Λ√t,v , ∃ v∈Λ√ [X]}
1 M
N
K\K [X] − 1
pt (x, x) − pt w (x, x) dμ(x) − c3 t −α/2 exp −c4 t β−1
t
N
K\K [X] − 1
pt (x, x) − pt w (x, x) dμ(x) − c3 t −α/2 exp −c4 t β−1
wy
v∈Λ√ [X] K(Λ√ ) t t,v
wy CX wy − 1
− 1
#Λ√t [X] − c3 t −α/2 exp −c4 t β−1 2c5 t −d∂ /2 − c3 t −α/2 exp −c4 t β−1 , M
where M := sup{#Λs,v | s ∈ (0, 1], v ∈ Λs } (< ∞ since (L, S) is locally finite by Propowy wy d∂ sition 5.8(3)), CX ∈ (0, ∞) is the infimum in (7.5) and c5 := CX cX γwy /(2M). Choose −
1
t0 ∈ (0, t∗ ] so that c3 t −α/2 exp(−c4 t β−1 ) c5 t −d∂ /2 for any t ∈ (0, t0 ]. Then we have ZF c (t) − ZLc (t) c5 t −d∂ /2 for any t ∈ (0, t0 ]. c c To consider the case t ∈ [t0 , 1], let {λFn }n∈N (resp. {λL }n∈N ) be the eigenvalues of the n c c non-negative self-adjoint operator associated with (E F , FF c ) (resp. (E L , FLc )), similarly to c c c c F L Definition 4.1(2). Then λn λn for any n ∈ N by the minimax principle, and λFn < λL n for c c F L some n ∈ N since n∈N e−λn t = ZF c (t) > ZLc (t) = n∈N e−λn t for t ∈ (0, t0 ]. Therefore ZF c (t) > ZLc (t) for any t ∈ (0, ∞), and ZF c − ZLc is a (0, ∞)-valued continuous function on (0, ∞). Hence we can choose c6 ∈ (0, ∞) so that ZF c (t) − ZLc (t) c6 t −d∂ /2 for any t ∈ [t0 , 1]. 2 Therefore it suffices for us to prove Lemma 7.13. We need to prepare a few easy lemmas. The following lemma is stated in [24, p. 600] and is easily proved by using Lemma 5.5. Lemma 7.14. CapE (Fw (A)) min{rw−1 , μw } CapE (A) for any w ∈ W∗ and any A ⊂ K. (n)
(n)
Notation. For n ∈ N ∪ {0} and (s, x) ∈ (0, 1] × K, we set Cs,x := CV (n) (x,S) and Fs,x := n
n,ϕ
n
s
n,ϕ
FV (n) (x,S) . We also abbreviate ∼ to ∼ and ∼ to ∼ in the rest of this subsection. L,S
s
L,S
Lemma 7.15. Suppose that (SSDF3S) holds. Let n ∈ N ∪ {0} and (s1 , x1 ), (s2 , x2 ) ∈ (0, 1] × K n+1,ϕ
(n)
(n)
(n)
(n)
satisfy (s1 , x1 ) ∼ (s2 , x2 ). Then ρϕ (Cs2 ,x2 ) = Cs1 ,x1 and ρϕ (Fs2 ,x2 ) = Fs1 ,x1 , where we regard (n) (n) (n) Csi ,xi and Fsi ,xi as subspaces of L2 (Vsi (xi , S), μ|V (n) (x ,S) ) for i = 1, 2. si
i
Proof. Recall that the L-similitude Fϕ associated with ϕ induces a homeomorphism Fϕ : (n) (n) (n) Vs1 (x1 , S) → Vs2 (x2 , S) by Lemma 7.5. Let u ∈ Cs1 ,x1 . Then we easily see that uϕ ∈ C(K) (n) (n) and suppK [uϕ ] ⊂ Vs2 (x2 , S). (SSDF3S) implies uϕ ∈ F ∩ C(K), hence uϕ ∈ Cs2 ,x2 . There(n) (n) (n) (n) (n) fore u = ρϕ uϕ ∈ ρϕ (Cs2 ,x2 ), and it follows that Cs1 ,x1 ⊂ ρϕ (Cs2 ,x2 ). By Cs2 ,x2 ⊂ ρϕ −1 (Cs1 ,x1 ) and ρϕ −1 = ρϕ−1 , we conclude that ρϕ (Cs2 ,x2 ) = Cs1 ,x1 . Since Kw ∩ Vsi (xi , S) = ∅ for w ∈ (n)
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Λsi \ Λnsi ,xi by Lemma 2.12, i = 1, 2, it easily follows from (SSDF2) and the self-similarity of μ that there exist c1 , c2 ∈ (0, ∞) such that c1 E1 (ρϕ u, ρϕ u) E1 (u, u) c2 E1 (ρϕ u, ρϕ u), (n)
u ∈ Cs(n) . 2 ,x2
(7.7)
(n)
Now ρϕ (Fs2 ,x2 ) = Fs1 ,x1 follows from (7.7) by taking the closures in the Hilbert space (F , E1 ) (n) (n) for the equality ρϕ (Cs2 ,x2 ) = Cs1 ,x1 . 2 n,ϕ
Definition 7.16. Let n ∈ N ∪ {0}. For (s1 , x1 ), (s2 , x2 ) ∈ (0, 1] × K, we write (s1 , x1 ) ∼ (s2 , x2 ) ∗
if and only if ϕ : Λns1 ,x1 → Λns2 ,x2 is an (n, L, S)-isomorphism between (s1 , x1 ) and (s2 , x2 ) such that rw /rv = rϕ(w) /rϕ(v) and μw /μv = μϕ(w) /μϕ(v) for any w, v ∈ Λns1 ,x1 . We also write n
n,ϕ
∗
∗
(s1 , x1 ) ∼ (s2 , x2 ) if and only if (s1 , x1 ) ∼ (s2 , x2 ) for some (n, L, S)-isomorphism ϕ : Λns1 ,x1 → Λns2 ,x2
n
between (s1 , x1 ) and (s2 , x2 ). Clearly, ∼ is an equivalence relation on (0, 1] × K. ∗
Lemma 7.17. Suppose that (L, S) is both locally finite and intersection type finite and that n (LWTF) holds. Then for any n ∈ N ∪ {0}, ((0, 1] × K)/∼ is a finite set. ∗
Proof. Let n ∈ N ∪ {0}. As (L, S) is locally finite, L is strongly finite (recall Definition 2.10(3)) by [28, Lemma 1.3.6]. Since (L, S) is also intersection type finite, [28, Theorem 2.2.13] imn plies that the quotient ((0, 1] × K)/∼ is a finite set. Therefore there exist J ∈ N and (si , xi ) ∈ (0, 1] × K, i = 1, . . . , J , such that for any (s, x) ∈ (0, 1] we can choose i ∈ {1, . . . , J } so that n (s, x) ∼ (si , xi ). Let Mr := #{rw /rv | (w, v) ∈ IP(L, S)} and Mμ := #{μw /μv | (w, v) ∈ IP(L, S)}. Mr , Mμ ∈ N by (LWTF). Let i ∈ {1, . . . , J }, wi ∈ Λ0si ,xi and let (s, x) ∈ (0, 1] × K satisfy n,ϕ
(si , xi ) ∼ (s, x). Then for w ∈ Λnsi ,xi , there are at most Mrn+1 (resp. Mμn+1 ) possibilities for the value rϕ(w) /rϕ(wi ) (resp. μϕ(w) /μϕ(wi ) ). Therefore the cardinality of the set
(rϕ(w) /rϕ(wi ) , μϕ(w) /μϕ(wi ) )w∈Λns ,x ϕ is an (n, L, S)-isomorphism i i between (si , xi ) and some (s, x) ∈ (0, 1] × K (n+1)#Λn
n
si ,xi is bounded from above by Mi := (Mr Mμ ) . Hence each equivalence class of ∼ n n contains at most Mi equivalence classes of ∼, which implies that #(((0, 1] × K)/(∼)) ∗ ∗ J M < ∞. 2 i i=1
Proof of Lemma 7.13. We follow the notations in Lemmas 7.10, 7.11 and 7.15 and Definition 7.16 above. We fix n ∈ N \ {1} throughout this proof. By Proposition 5.8(3) and Lemma 7.17, there exist J ∈ N and (si , xi ) ∈ (0, 1] × K, i = 1, . . . , J , such that for any (s, x) ∈ (0, 1] × K we n+1
can choose i ∈ {1, . . . , J } so that (s, x) ∼ (si , xi ). For i ∈ {1, . . . , J }, fix wi ∈ Λsi ,xi and set (n)
∗
Ui := Usi (xi , S) and Vi := Vsi (xi , S). As n 2, Kw√i [X] ⊂ Kwi ⊂ K(Λsi ,wi ) ⊂ Ui ⊂ Vi . 2 ] and v ∈ Λwy √ 0 [X]. Then γv t γ −1 γv , where γ := mini∈S γi . Since Let t ∈ (0, γwy 0 t Σv ∩ Σwy0 [X] = ∅, we may choose y1 y2 · · · ∈ Σ(X) so that wy0 y1 y2 · · · ∈ Σv . Set j := min{k |
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k ∈ N ∪ {0}, wy0 · · · yk v}, yv := y0 · · · yj and sv := γwyv . Fix xv ∈ Kwyv \ Fwyv (V0 ) and (n) set Uv := Usv (xv , S) and Vv := Vsv (xv , S). We easily see that wyv ∈ Λsv and 1 sv−2 t −2(M +1) X γ , where MX := maxz∈X |z|. As n 2, we have Kwyv [X] ⊂ Kwyv ⊂ K(Λsv ,wyv ) = Uv ⊂ Vv . n+1,ϕ
n+1 Choose i ∈ {1, . . . , J } and ϕ : Λn+1 (si , xi ). Then since sv ,xv → Λsi ,xi so that (sv , xv ) ∼ ∗ {ϕ(wyv )} = ϕ(Λsv ,xv ) = Λsi ,xi wi , we see that Λsi ,xi = {wi } and ϕ(wyv ) = wi . By (SSDF3S), n+1,ϕ
Lemmas 7.5 and 7.15 and (sv , xv ) ∼ (si , xi ), Fϕ : Vv → Vi is a homeomorphism, ∗
ρϕ L2 (Vi , μ|Vi ) = L2 (Vv , μ|Vv ) and ρϕ (FVi ) = FVv , 2 −1 μ−1 |ρ u| dμ = μ |u|2 dμ, u ∈ L2 (Vi , μ|Vi ), ϕ wyv wi Vv
Vi
rwyv E(ρϕ u, ρϕ u) = rwi E(u, u),
u ∈ FVi .
(7.8)
−1 , it follows that Since ϕ|Λ1s ,x : Λ1sv ,xv → Λ1si ,xi is an L-isomorphism and Fϕ |Kwyv = Fwi ◦ Fwy v v v Fϕ (Usv (xv , S)) = Usi (xi , S) and that Fϕ (Vv \ Kwyv [X]) = Vi \ Kwi [X]. Therefore (7.8) together with (CHK) of (E U , FU ) for a non-empty open subset U of K implies that for any (s, x, y) ∈ (0, ∞) × Vv × Vv ,
Vi Vv μwyv pss 2 (x, y) = μwi psγ 2 Fϕ (x), Fϕ (y) , v
V \Kwyv [X]
μwyv pssv2 v
wi
wi
Since wyv v and sv = γwyv Lemma 7.12 and (7.9) imply that
Vi \Kwi [X]
(x, y) = μwi psγ 2
Fϕ (x), Fϕ (y) .
(7.9)
√ t we have Uv = K(Λsv ,wyv ) ⊂ K(Λ√t,v ). Therefore
N
K\K [X] pt (x, x) − pt w (x, x) dμ(x)
K(Λ√t,v )
Uv
= μwi μ−1 wyv =
Uv
Vi p 2
i
Vv
V \K [X] pt (x, x) − pt v wyv (x, x) dμ(x)
Uv
Vi p 2
γw sv−2 t
γw sv−2 t
Ui
N
K\K [X] pt (x, x) − pt wyv (x, x) dμ(x)
Vi \Kw [X] Fϕ (x), Fϕ (x) − p 2 −2i Fϕ (x), Fϕ (x) dμ(x) γw sv t
i
i
(x, x) − p
Vi \Kwi [X]
γw2 sv−2 t
(x, x) dμ(x).
i
Recall that 1 sv−2 t βX , where βX := γ −2(MX +1) , hence γw2i sv−2 t ∈ [γw2i , βX γw2i ]. Therefore for the proof of Lemma 7.13 it suffices to prove that for any a, b ∈ (0, ∞) with a < b and for any i ∈ {1, . . . , J } satisfying Λsi ,xi = {wi },
N. Kajino / Journal of Functional Analysis 258 (2010) 1310–1360
inf
t∈[a,b] Ui
1351
Vi
Vi \Kwi [X] pt (x, x) − pt (x, x) dμ(x) > 0.
(7.10)
Let a, b ∈ (0, ∞), a < b and let i ∈ {1, . . . , J } satisfy Λsi ,xi = {wi }. Since K is assumed to be connected, it is also arcwise connected by [27, Theorem 1.6.2], and any non-empty open subset of K is locally arcwise connected. Let Vc,i be the connected component of Vi containing xi . Then Vc,i is an arcwise connected open subset of K. By Lemma 7.11(2) and Proposition A.3(2), V V pt i (x, y) pt c,i (x, y) > 0 for any (t, x, y) ∈ (0, ∞) × Vc,i × Vc,i . On the other hand, we have Ui ⊂ Vc,i since Ui is connected and xi ∈ Kwi ⊂ Ui ⊂ Vi . Hence by (CHK) of (E Vi , FVi ), qi := inf ptVi (x, y) (t, x, y) ∈ [a/2, b] × Ui × Ui > 0.
(7.11)
V
We write V,i := (Vi ) and define σAi (ω) := inf{t ∈ [0, ∞) | Xt i (ω) ∈ A} (inf ∅ := ∞) and τAi (ω) := σV,i \A (ω) for A ∈ B(V,i ) and ω ∈ Ω. CapE (K[X]) > 0 and Lemma 7.14 imply CapE (Kwi [X]) > 0. Since Kwi [X] ⊂ Vi and Kwi [X] is compact, [17, Theorem 4.4.3(ii)] implies that CapE Vi (Kwi [X]) ∈ (0, ∞). Therefore by Theorem A.5, there exist a μ|Vi -regular closed subset Fi of Vi , zi ∈ Kwi [X] ∩ Fi and an open neighborhood Gi of zi in Vi such that hi :=
inf
x∈Gi ∩Fi
Px σKi w
i [X]
a/2 > 0.
(7.12)
Let Ai := Fi ∩ Gi ∩ intK Ui . Note that Gi ∩ intK Ui is an open neighborhood of zi in Vi since zi ∈ Kwi [X] ⊂ Kwi ⊂ intK Ui . Therefore μ(Ai ) > 0 by the μ|Vi -regularity of Fi . Now let t ∈ [a, b] and x ∈ Ai . Since (E U , FU ) satisfies (CHK) for U = Vi \ Kwi [X], Vi , Vi \Kwi [X] Vi (x, y)) dμ(y) Vi \Kwi [X] Us (x,S) (pt (x, y) − pt Vi (x, x) = lim pt (x, x) − pt s↓0 μ(Us (x, S)) = lim
Px [XtVi ∈ Us (x, S)] − Px [XtVi ∈ Us (x, S), t < σKi w
i [X]
μ(Us (x, S))
s↓0
= lim s↓0
]
Px [XtVi
∈ Us (x, S), σKi w
i [X]
μ(Us (x, S))
t]
.
(7.13)
As x ∈ intK Ui , we may choose δ ∈ (0, 1] so that Us (x, S) ⊂ intK Ui for any s ∈ (0, δ]. Let s ∈ (0, δ]. In the calculation below, we write X i (t, ω) := XtVi (ω) for each (t, ω) ∈ [0, ∞]×Ω and σi := σKi w [X] . Since X i (σi (ω), ω) ∈ Kwi [X] for ω ∈ {σi < ∞}, by the strong Markov property i
of the diffusion X Vi (see [26, Corollary 2.6.18], for example), (7.11) and (7.12), V Px Xt i ∈ Us (x, S), σi t Px XtVi ∈ Us (x, S), σi a/2 Vi PXi (σi (ω),ω) Xt−σ ∈ Us (x, S) dPx (ω) = i (ω) {σi a/2}
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= {σi a/2}
Us (x,S)
{σi a/2}
i
Vi X σ pt−σ (ω), ω , y dμ(y) dPx (ω) i i (ω)
qi dμ(y) dPx (ω) = qi Px [σi a/2]μ Us (x, S) qi hi μ Us (x, S) .
Us (x,S)
Hence the limit in (7.13) is bounded from below by qi hi , that is, Vi \Kwi [X]
ptVi (x, x) − pt
(x, x) qi hi ,
(t, x) ∈ [a, b] × Ai .
(7.14)
Therefore for any t ∈ [a, b], since Ai ⊂ Ui ,
Vi
Vi \Kwi [X] pt (x, x) − pt (x, x) dμ(x)
Ui
V
Vi \Kwi [X]
pt i (x, x) − pt
(x, x) dμ(x)
Ai
qi hi μ(Ai ) (> 0), proving (7.10). This completes the proofs of Lemma 7.13 and of Theorem 7.7.
2
Proof of Corollary 7.8. As ΣwJ [XJ ] ⊂ PL for some wJ ∈ W∗ by (RB), we also have Σ[XJ ] ⊂ PL and hence K[XJ ] ⊂ V0 .By K = V0 (= V0 ) we may choose w ∈ W# so that I = V c ⊂ K I = V c by Kw ∩ V0 = ∅. Let := |w| and V := v∈W Fv (V0 ). Then (∅ =) KW 0 Lemma 2.11, hence V0 ⊂ V K. Since (L, S) is locally finite, Proposition 6.7(1) implies that dimS V = d∂ = dimS Kw [XJ ]. Therefore dimS (V \ V0 ) = d∂ by Kw [X] ⊂ V \ V0 . By Theorem 7.7 there exists c3 ∈ (0, ∞) such that ZK I (t) − ZK I \Kw [XJ ] (t) c3 t −d∂ /2 for any t ∈ (0, 1]. Also Theorem 5.11 implies that there exists c4 ∈ (0, ∞) such that 0 ZK I (t) − ZK I (t) W
c4 t −d∂ /2 for any t ∈ (0, 1]. Note that ZK I = ZVc ZK I \Kw [XJ ] , that ZK I = ZD and that W
ZK I (t) = (#S) ZD (tγ −2 ) ZD (t) for any t ∈ (0, ∞) by Lemma 5.7. Hence we conclude W
that
c3 t
−d∂ /2
ZD (t) − (#S) ZD
t γ 2
c4 t −d∂ /2 ,
t ∈ (0, 1].
(7.15)
Since CapE (V0 ) > 0 by V0 ⊃ K[XJ ] and CapE (K[XJ ]) > 0 (or by Theorem 7.18, which is presented in the next subsection), [17, Corollary 2.3.1] implies that 1 ∈ / FK I . By [27, Theorem 1.6.2], Lemma 7.11(1) and Proposition A.3(2), (E, F ) is irreducible. [12, Theorem 2.1.11] I implies that E(u, u) > 0 for any u ∈ FK I \ {0}. Hence λD 1 := λ1 (K ) > 0 (recall (4.12)). Let D 1 (t) := eλ1 t Z (t), t ∈ (0, ∞). Then Z 1 is clearly (0, ∞)-valued and non-increasing. ThereZD D D fore
t D D 1 1 ZD (t) = ZD (t)e−λ1 t ZD (1)e−λ1 t , t ∈ [1, ∞). (7.16) 0 ZD (t) − (#S) ZD γ 2 Now by (7.15) and (7.16), we can follow the arguments of [27, Proofs of Theorems 4.1.5 and B.4.3] to prove that there exists a continuous log(γ − )-periodic function G : R → (0, ∞)
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and c1 , c2 ∈ (0, ∞) such that (7.2) holds for any t ∈ (0, 1], with G replaced by G . But then G = G since G and G are both log(γ − )-periodic. 2 7.3. Positivity of capacity for subsets of the boundary As in the previous subsection, in this subsection (L = (K, S, {Fi }i∈S ), μ, E, F , r = (ri )i∈S ) is assumed to be a self-similar Dirichlet space with μ a self-similar measure with weight (μi )i∈S √ and S = {Λs }s∈(0,1] to be the self-similar scale with weight γ := (γi )i∈S , γi := ri μi . As usual, let π : Σ → K denote the canonical projection. The purpose of this subsection is to state and prove the following Theorem 7.18, which asserts that every subset of V0 with non-empty interior in V0 has positive capacity. This kind of statement is indispensable when we apply Theorem 7.7 to concrete examples. Notation. For each u ∈ F , its quasi-continuous modification, which exists and is unique up to E-q.e., is denoted by u. ˜ Note that FU = {u ∈ F | u˜ = 0 E-q.e. on K \ U } for any non-empty open subset U of K by [17, Corollary 2.3.1]. See [17, Chapter 2] for details. Theorem 7.18. Assume that K is connected, that (E, F ) is conservative and that (CHK) and (UHK) hold. Then CapE (G) > 0 for any non-empty open subset G of V0 . Remark. Since #S 2, the connectivity of K implies that V0 = ∅. Proof. Let G be a non-empty open subset of V0 . Then we may choose an open subset O of K so that G = O ∩ V0 . Also there exists x ∈ O ∩ V0 . Let ω ∈ π −1 (x). Then ω ∈ PL since π −1 (V0 ) = PL by [27, Proposition 1.3.5(1)]. Therefore there exist w = w1 · · · wm ∈ W# and j ∈ S \ {w1 } such that Fw (x) ∈ Kw ∩ Kj (recall Definition 2.10(2)). Since Fw : K → Kw is a homeomorphism, Fw (O) is an open subset of Kw and we can choose an open subset Ow of K so that Fw (O) = Ow ∩ Kw . Then Fw (x) ∈ Fw (O ∩ V0 ) = Ow ∩ Fw (V0 ). Let Uw be the connected component of Ow containing Fw (x) and set U := Fw−1 (Uw ) ( x). Then Fw (x) ∈ Uw ∩ Kw ∩ Kj = ∅, and as in the proof of (7.10), Uw is an arcwise connected open subset of K. Also, Fw (U ∩ V0 ) = Uw ∩ Fw (V0 ) ⊂ Ow ∩ Fw (V0 ) = Fw (O) ∩ Fw (V0 ) = Fw (G) and therefore U ∩ V0 ⊂ G. Since intK V0 = ∅ by [28, Theorem 1.2.2], U ∩ K I = U \ V0 = ∅ and Uw ∩ KwI = Fw (U ∩ K I ) = ∅. Similarly, since Fj−1 (Uw ) is also a non-empty open subset of K, Fj−1 (Uw ) ∩ K I = Fj−1 (Uw ) \ V0 = ∅ and Uw ∩ KjI = Fj (Fj−1 (Uw ) ∩ K I ) = ∅. Now suppose CapE (G) = 0. Then CapE (U \ (U ∩ K I )) = CapE (U ∩ V0 ) = 0 and therefore FU = FU ∩K I . Let u ∈ FUw . Then u˜ = 0 E-q.e. on K \ Uw . Using Lemma 7.14, we see that u˜ ◦ Fw is a quasi-continuous modification of u ◦ Fw ∈ F . As Fw ({y ∈ K \ U | u˜ ◦ Fw (y) = 0}) ⊂ ˜
= 0}, Lemma 7.14 yields {y ∈ K \ Uw | u(y)
min rw−1 , μw CapE y ∈ K \ U u˜ ◦ Fw (y) = 0
CapE Fw y ∈ K \ U u˜ ◦ Fw (y) = 0 CapE y ∈ K \ Uw u(y) ˜
= 0 = 0. Therefore u˜ ◦ Fw = 0 E-q.e. on K \ U , hence u ◦ Fw ∈ FU = FU ∩K I ⊂ FK I . By Lemma 5.5, I uw := u·1Kw = (u◦Fw )w ∈ FKwI , which implies u w = 0 E-q.e. on K \Kw . Since u w = u·1Kw = I I u = u˜ μ-a.e. on Kw , [17, Lemma 2.1.4] yields u ˜ E-q.e. on Kw . Also, u˜ = 0 E-q.e. on w =u ˜ = 0 E-q.e. on KwI \ Uw . Thus u K \ Uw by u ∈ FUw and therefore u w =u w = 0 E -q.e. on
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K \ (Uw ∩ KwI ), hence u · 1Uw ∩Kw = u · 1Kw = uw ∈ FUw ∩KwI (⊂ FUw ) and u · 1Uw \Kw = u − u · 1Uw ∩Kw ∈ FUw . Recalling |w| = m, it follows that, for any u, v ∈ FUw , E(u · 1Uw ∩Kw , v · 1Uw \Kw ) =
1
E (u · 1Uw ∩Kw ) ◦ Fτ , (v · 1Uw \Kw ) ◦ Fτ = 0, rτ
τ ∈Wm
hence E(u, v) = E(u · 1Uw ∩Kw , v · 1Uw ∩Kw ) + E(u · 1Uw \Kw , v · 1Uw \Kw ).
(7.17)
Since μ(Uw ∩ Kw ) μ(Uw ∩ KwI ) > 0 and μ(Uw \ Kw ) μ(Uw ∩ KjI ) > 0, (7.17) together with [17, Theorem 1.6.1] contradicts the fact that (E Uw , FUw ) is irreducible by Proposition A.3(2) and the arcwise connectivity of Uw . Thus CapE (G) > 0 follows. 2 8. Examples: Sierpinski carpets In this section, we illustrate the results of the previous sections by applying them to a class of infinitely ramified self-similar sets called generalized Sierpinski carpets, whose definition was originally given by Barlow and Bass [6, Section 2] but has recently been modified by Barlow, Bass, Kumagai and Teplyaev [7], Hino [23] and Kigami [28, Section 3.4]. We follow the formulation of Hino [23] in the argument below, but their formulations of generalized Sierpinski carpets are all equivalent, as stated in Kajino [25, Section 2]. Definition 8.1 (Generalized Sierpinski carpets). Let d ∈ N and set Q0 := [0, 1]d . Let L ∈ N, d L 2 and set Q1 := { i=1 [(ki − 1)L−1 , ki L−1 ] | k1 , . . . , kd ∈ {1, . . . , L}}. Let S ⊂ Q1 be nonFq (x) := L−1 x + zq , where zq ∈ Rd is empty, and for each q ∈ S we define Fq : Rd → Rd by S chosen so that Fq (Q0 ) = q (⊂ Q0 ). We also set Q1 := q∈S q. Let GSC(d, L, S) be the self-similar set associated with {Fq }q∈S , that is, the unique nonempty compact subset K of Rd that satisfies K = q∈S Fq (K). We call GSC(d, L, S) a generalized Sierpinski carpet if and only if S satisfies the following four conditions: (GSC1) (Symmetry) QS1 is preserved by all the isometries of Q0 . (GSC2) (Connectedness) QS1 is connected. (GSC3) (Non-diagonality) If B is a d-dimensional rectangle with each side length L−1 or 2L−1 which is the union of elements of Q1 , intRd (B ∩ QS1 ) is either empty or connected. (GSC4) (Borders included) {(x1 , 0, . . . , 0) | x1 ∈ [0, 1]} ⊂ QS1 . In particular, we call GSC(2, 3, SSC ) the Sierpinski carpet (see Fig. 1.2), where SSC := {[(k1 − 1)/3, k1 /3] × [(k2 − 1)/3, k2 /3] | (k1 , k2 ) ∈ {1, 2, 3}2 \ {(2, 2)}}. In the rest of this section, we fix a generalized Sierpinski carpet GSC(d, L, S). Let K := GSC(d, L, S) and L := (K, S, {Fq }q∈S ) be the self-similar structure associated with {Fq }q∈S . The following proposition is immediate by the assumptions. Proposition 8.2. (1) K is connected (by (GSCj ), j = 1, 2, 4 and [27, Theorem 1.6.2]). (2) Let k ∈ {1, . . . , d}. Set Hk,s := {(x1 , . . . , xd ) ∈ Rd | xk = s} and Sk,s := {q ∈ S | q ∩ Hk,s = ∅} for s ∈ [0, 1] and let Rk : Rd → Rd be the reflection in the hyperplane Hk,1/2 . Then Rk induces natural bijections Sk,0 → Sk,1 and Sk,1 → Sk,0 given by q → Rk (q). (3) L satisfies (RB) with PL = dk=1 (Σ[Sk,0 ] ∪ Σ[Sk,1 ]), and V0 = K \ (0, 1)d = K.
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Next we discuss self-similar Dirichlet forms on K. We follow the arguments in [28, Section 3.4]. Let ν be the self-similar measure on K with weight ((#S)−1 , . . . , (#S)−1 ). By combining the arguments of Barlow and Bass [5,6] and Kusuoka and Zhou [32], as in Hambly, Kumagai, Kusuoka and Zhou [22] (note also the recent result [7] on uniqueness of the Dirichlet form on generalized Sierpinski carpets), we have a conservative self-similar Dirichlet space (in the sense of Definition 3.3) (L, ν, E, F , r = (rq )q∈S ) satisfying (SSDF3S), (CHK) and (UHK) and with rq = r for any q ∈ S for some r ∈ (0, ∞). Now let μ be a self-similar measure on K with weight (μq )q∈S satisfying rμq < 1 for any q ∈ S. By a result of Barlow and Kumagai [9, Lemma 2.5], μ is smooth with respect to (E, F ), that is, μ(A) = 0 for any A ∈ B(K) with CapE (A) = 0. By [17, Theorem 6.2.1], we can construct the time changed Dirichlet space (E μ , Fμ ) of (E, F ) with respect to μ, which is a regular Dirichlet form on L2 (K, μ). Since the whole space K is a quasi-support of μ by [9, Proposition 2.6] and [17, (5.1.22) and Theorem 5.1.5], [17, Theorem 1.5.2(iii) and (6.2.22)] yield Fμ ∩ C(K) = F ∩ C(K) and E μ (u, v) = E(u, v) for any u, v ∈ F ∩ C(K). Therefore (L, μ, E μ , Fμ , r) is a conservative self-similar Dirichlet space satisfying (SSDF3S). Moreover, by the discussions of [9] (see also [28, Section 3.4]), we can verify (CHK) and the assumptions √ μ μ of [28, Theorem 3.2.3] for (L, μ, E μ , Fμ , r). Finally, let γq := rμq for q ∈ S, γ μ := (γq )q∈S μ μ μ and S = {Λs }s∈(0,1] be the self-similar scale with weight γ . Then by [28, Theorems 3.2.3, 3.4.5 and Proof of Lemma 3.5.16], we have the following criterion for (UHK), (LWTF) and (L, Sμ ) being intersection type finite (see also [25, Proposition 3.3 and Theorem 3.5] for a short self-contained treatment of (VD)). Proposition 8.3. The following four conditions are equivalent. (0) (1) (2) (3)
(μq )q∈S is weakly symmetric, i.e. μq = μRk (q) for any k ∈ {1, . . . , d} and any q ∈ Sk,0 . (L, Sμ ) is locally finite. (L, Sμ , μ) satisfies (VD). (UHK) holds for (L, μ, E μ , Fμ , r).
Moreover, if any one of these four conditions holds, then (L, Sμ ) is intersection type finite and (L, μ, E μ , Fμ , r) satisfies (LWTF). Hence we conclude that if (μq )q∈S is weakly symmetric then all the statements of Theμ orem 5.2 are valid for (L, μ, E μ , Fμ , r) with d∂ = d∂ := max{d(γ μ , Sk,0 ) | k ∈ {1, . . . , d}} (= dimSμ V0 < dimSμ K in view of Theorem 6.9). Moreover, suppose that (μq )q∈S is weakly symmetric. Then Theorem 7.18 implies that CapE μ (K[Sk,j ]) > 0 for any k ∈ {1, . . . , d} and any j ∈ {0, 1}. Therefore Theorem 7.7 implies the following reminder estimate. For U ⊂ K non-empty open, let ZU,μ denote the partition function associated with ((E μ )U , (Fμ )U ), ZN,μ := ZK,μ and ZD,μ := ZK I ,μ . μ
Theorem 8.4. Assume that (μq )q∈S is weakly symmetric. Let k ∈ {1, . . . , d}, j ∈ {0, 1} and dk := d(γ μ , Sk,0 ). Then there exist c1 , c2 ∈ (0, ∞) such that for any t ∈ (0, 1], μ
μ
c1 t −dk /2 ZN,μ (t) − ZK\K[Sk,j ],μ (t) c2 t −dk /2 .
(8.1)
On the other hand, if μq = (#S)−1 for any q ∈ S, i.e. μ = ν, then dEuc dw /2 is a (2/dw )qdistance adapted to Sν , where dEuc is the Euclidean distance and dw := logL (#S/r). Hence
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by Proposition 2.24, dimSν K = 2df /dw and dimSν V0 = 2db /dw in this case, where df := logL (#S) and db := logL (#S1,0 ). Therefore Corollary 7.8 implies the following sharp reminder estimate for ZD,ν . Theorem 8.5. Let G be the log(#S/r)1/2 -periodic function given by Corollary 5.3 for (L, ν, E, F , r). Then there exist c3 , c4 ∈ (0, ∞) such that for any t ∈ (0, 1],
1 1 log − ZD,ν (t) c4 t −db /dw . (8.2) c3 t −db /dw t −df /dw G 2 t 9. Concluding remarks We conclude the present paper with a brief discussion of open problems. Consider the situation of Theorem 5.2. In the non-lattice case, we have shown an asymptotic behavior of the eigenvalue counting functions (Corollary 5.4) by virtue of Karamata’s Tauberian theorem. Unfortunately, in the lattice case we do not have any similar result for the eigenvalue counting functions. The main difficulty here is that the T -periodic function G given in Theorem 5.2 may be non-constant. In this case, it seems hopeless to verify the so-called ‘Tauberian conditions’ on G. It also seems extremely difficult to apply the renewal theorem directly to the eigenvalue counting function, since we cannot use probabilistic arguments to estimate NN (x) − ND (x). This is why Hambly [21] and this article have treated the partition function mainly and not the eigenvalue counting function. Acknowledgments Most parts of this paper originate from the author’s Master thesis in Kyoto University. I would like to express my deepest gratitude toward Professor Jun Kigami for having taken good care of my study during the Undergraduate and Master courses, and having led me to the world of analysis on fractals. I would like to thank Professors Takashi Kumagai and Masanori Hino for valuable comments and fruitful discussions and all the members of Sub-department of Applied Analysis for their indispensable cares and advices. I would like to thank also Dr. David Croydon for very careful reading of an earlier manuscript. Appendix A. Miscellaneous lemmas for Section 7 In this appendix, we present basic results on continuity and positivity of heat kernels and positivity of hitting probabilities for regular Dirichlet forms. Those results play essential roles in the proof of Theorem 7.7. Let E be a locally compact separable metrizable space and let E := E ∪ {E } denote its one-point compactification. Throughout this appendix, we assume that μ is a Borel measure on E satisfying μ(F ) < ∞ for any compact F ⊂ E and μ(O) > 0 for any non-empty open O ⊂ E, that (E, F ) is a (symmetric) regular Dirichlet form on L2 (E, μ) and that H and {Tt }t∈(0,∞) are the non-negative self-adjoint operator with domain D[H ] and the strongly continuous contraction semigroup, respectively, associated with the closed form (E, F ) on L2 (E, μ). The following definition is just a reminder for the readers. Definition A.1. (1) {Tt }t∈(0,∞) is called ultracontractive if and only if Tt (L2 (E, μ)) ⊂ L∞ (E, μ) and Tt : L2 (E, μ) → L∞ (E, μ) is a bounded linear operator for any t ∈ (0, ∞).
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(2) A family {pt }t∈(0,∞) of R-valued B(E × E)-measurable functions on E × E is called a heat kernel of {Tt }t∈(0,∞) if and only if for each t ∈ (0, ∞) and for any f ∈ L2 (E, μ), (A.1) Tt f = pt (·, y)f (y) dμ(y) μ-a.e. on E. E
Clearly, for t ∈ (0, ∞), such an integral kernel pt of Tt , if exists, is unique up to μ × μ-a.e. and satisfies pt (x, y) = pt (y, x) 0 μ × μ-a.e. on K × K. See [20, Section 2] for details. (3) We say that (E, F ) satisfies (CHK), or simply (CHK) holds, if and only if {Tt }t∈(0,∞) admits a heat kernel {pt }t∈(0,∞) which is jointly continuous, i.e. such that p = pt (x, y) : (0, ∞) × E × E → R is continuous. Clearly, such {pt }t∈(0,∞) , if exists, is unique. By [14, Theorem 2.1.4], if μ(E) < ∞ then the ultracontractivity of {Tt }t∈(0,∞) implies the existence of a heat kernel {pt }t∈(0,∞) ⊂ L∞ (E × E, μ × μ). Next let us recall the following definitions. See [17, Section 1.4] for the definitions of (sub-)Markovian transition functions, their μ-symmetry and the Markovian semigroup on L2 (E, μ) associated with a μ-symmetric (sub-)Markovian transition function. Definition A.2. Let {Pt }t∈(0,∞) be a (sub-)Markovian transition function on (E, B(E)). (1) {Pt }t∈(0,∞) is called conservative if and only if Pt (x, E) = 1 for any (t, x) ∈ (0, ∞) × E. (2) We say that {Pt }t∈(0,∞) has the Feller property, or simply it is Feller, if and only if Pt (C∞ (E)) ⊂ C∞ (E) for any t ∈ (0, ∞) and limt↓0 Pt u − u∞ = 0 for any u ∈ C∞ (E). (3) We say that {Pt }t∈(0,∞) has the strong Feller property, or simply it is strong Feller, if and only if Pt u ∈ Cb (E) for any bounded Borel measurable u : E → R. The following proposition provides a sufficient condition for (CHK) and for strict positivity of the jointly continuous heat kernel. Proposition A.3. Assume μ(E) < ∞ and suppose that {Tt }t∈(0,∞) is ultracontractive. Let {Pt }t∈(0,∞) be a μ-symmetric strong Feller (sub-)Markovian transition function on (E, B(E)) whose associated Markovian semigroup on L2 (E, μ) is {Tt }t∈(0,∞) . Then: (1) (CHK) holds with jointly continuous heat kernel {pt }t∈(0,∞) ⊂ Cb (E × E), and Pt (x, A) = A pt (x, y) dμ(y) for any (t, x) ∈ (0, ∞) × E and any A ∈ B(E). (2) Suppose that E is arcwise connected and that there exists a Hunt process X = (Ω, M, {Xt }t∈[0,∞] , {Px }x∈E ) on E whose transition function is {Pt }t∈(0,∞) . Then pt (x, y) ∈ (0, ∞) for any (t, x, y) ∈ (0, ∞) × E × E. In particular, (E, F ) is irreducible. Proof. (1) Let ϕ ∈ D[H ] and λ ∈ [0, ∞) satisfy H ϕ = λϕ. By the ultracontractivity of {Tt }t∈(0,∞) , T1 ϕ = e−λ ϕ ∈ L∞ (E, μ), and we may choose a bounded Borel measurable version of ϕ. Since ϕ = eλ T1 ϕ = eλ P1 ϕ μ-a.e. on E and P1 ϕ ∈ Cb (E) by the strong Feller property of {Pt }t∈(0,∞) , we may assume that ϕ ∈ Cb (E). Now as in [14, Proof of Theorem 2.1.4], for any T ∈ (0, ∞), the eigenfunction expansion [14, (2.1.4)] of the heat kernel defines an absolutely norm-convergent series in the Banach space Cb ([T , ∞) × E × E). Hence the heat kernel {pt }t∈(0,∞) defined by [14,(2.1.4)] is jointly continuous, proving (CHK). Moreover, if t ∈ (0, ∞) and A ∈ B(E) then Pt 1A , A pt (·, y) dμ(y) ∈ Cb (E) and both of them are equal to Tt 1A μ-a.e. on E, hence they are equal at every point of E.
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(2) Suppose pt (x, x) = 0 for some (t, x) ∈ (0, ∞) × E. Then pt/2 (x, y) = 0 for any y ∈ E since 0 = pt (x, x) = E pt/2 (x, z)2 dμ(z). Inductively, for each n ∈ N, pt/2n (x, y) = 0 for any y ∈ E and hence Px [Xt/2n ∈ E] = Pt/2n (x, E) = E pt/2n (x, y) dμ(y) = 0. Therefore Xt/2n = n→∞ −−− → E for any n ∈ N Px -a.s., which then implies that X0 = E Px -a.s. since Xt/2n (ω) − X0 (ω) in E for any ω ∈ Ω. This contradicts Px [X0 = x] = 1. Therefore pt (x, x) ∈ (0, ∞) for any (t, x) ∈ (0, ∞) × E. Now based on the arcwise connectivity of E, the positivity of pt follows in exactly the same way as in [28, Proof of Theorem A.4]. Finally, for A ∈ B(E), ( (E\A)×A pt d(μ × μ) =) E\A Tt 1A dμ = 0 implies μ × μ((E \ A) × A) = μ(E \ A)μ(A) = 0, which is sufficient for (E, F ) to be irreducible, by [17, pp. 46–48, Section 1.6]. 2 In the theorem below, we deduce a uniform positivity of short time hitting probabilities by assuming the positivity of capacity. Recall the following definitions. Definition A.4. (1) A closed subset F of E is called μ-regular if and only if, for any open subset U of E, either μ(U ∩ F ) > 0 or U ∩ F = ∅. (2) We define, with the convention that inf ∅ := ∞, for U ⊂ E open, capE (U ) := inf E1 (u, u) u ∈ F , u 1 μ-a.e. on U for A ⊂ E. CapE (A) := inf capE (U ) U ⊂ E open, A ⊂ U
(A.2) (A.3)
CapE is clearly an extension of capE . Moreover, let A ⊂ E and let S (x) be a statement on x for each x ∈ A. Then we say that S holds E -q.e. on A if and only if CapE ({x ∈ A | S (x) fails}) = 0. When A = E we simply say ‘S holds E-q.e.’ instead. Theorem A.5. Let X = (Ω, M, {Xt }t∈[0,∞] , {Px }x∈E ) be a μ-symmetric Hunt process on E whose Dirichlet form on L2 (E, μ) is (E, F ). For A ∈ B(E ) and ω ∈ Ω, define σA (ω) := inf t ∈ [0, ∞) Xt (ω) ∈ A
(inf ∅ := ∞).
(A.4)
If A ∈ B(E) and CapE (A) ∈ (0, ∞), then there exists a μ-regular closed subset F of E with the following properties: A ∩ F = ∅, and for any x0 ∈ A ∩ F , any t ∈ (0, ∞) and any s ∈ (0, 1) there exists an open neighborhood U of x0 in E such that inf Px [σA t] s.
x∈U ∩F
(A.5)
Proof. Let σA+ (ω) := inf{t ∈ (0, ∞) | Xt (ω) ∈ A} (inf ∅ := ∞) for ω ∈ Ω, and set NA := {x ∈ E | Px [σA = σA+ ] = 1}. Then CapE (NA ) = 0 by [17, Theorems 4.1.3, 4.2.1(ii) and A.2.6(i)]. Let 1 (x) := E [e−σA ] and p 1+ (x) := E [e−σA+ ] for x ∈ E. Then p 1 = p 1+ on E \ N . Since p 1+ pA x x A A A A A 1 is also quasi-continuous. is quasi-continuous by [17, Theorem 4.2.5] and CapE (NA ) = 0, pA By [17, Theorem 2.1.2(i)] there exists a μ-regular closed subset F of E such that CapE (A) > 1 )| is continuous. CapE (E \ F ) and (pA F This F possesses the required properties. Indeed, A ∩ F = ∅ follows from CapE (A) > CapE (E \ F ). Let x0 ∈ A ∩ F , t ∈ (0, ∞) and s ∈ (0, 1) and set Ms,t := s + (1 − s)e−t (< 1). 1 )| is continuous and p 1 (x ) = 1, we may choose an open neighborhood U of x in E Since (pA F 0 A 0 1 (x) M so that pA s,t for any x ∈ U ∩ F . Now let x ∈ U ∩ F . Then since
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1 Ms,t pA (x) = Ex e−σA = Ex e−σA 1{σA t} + Ex e−σA 1{σA >t}
Px [σA t] + e−t Px [σA > t] = Px [σA t] + e−t 1 − Px [σA t] , we conclude that Px [σA t] (Ms,t − e−t )/(1 − e−t ) = s. Therefore (A.5) follows.
2
Remark. The author has been taught the idea of using Ex [e−σA ] to deduce lower bounds for Px [σA t] by Prof. Masanori Hino. References [1] M.T. Barlow, R.F. Bass, The construction of Brownian motion on the Sierpinski carpet, Ann. Inst. H. Poincaré 25 (1989) 225–257. [2] M.T. Barlow, R.F. Bass, Local times for Brownian motion on the Sierpinski carpet, Probab. Theory Related Fields 85 (1990) 91–104. [3] M.T. Barlow, R.F. Bass, On the resistance of the Sierpi´nski carpet, Proc. R. Soc. Lond. Ser. A 431 (1990) 354–360. [4] M.T. Barlow, R.F. Bass, Transition densities for Brownian motion on the Sierpinski carpet, Probab. Theory Related Fields 91 (1992) 307–330. [5] M.T. Barlow, R.F. Bass, Coupling and Harnack inequalities for Sierpinski carpets, Bull. Amer. Math. Soc. 29 (1993) 208–212. [6] M.T. Barlow, R.F. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math. 51 (1999) 673–744. [7] M.T. Barlow, R.F. Bass, T. Kumagai, A. Teplyaev, Uniqueness of Brownian motion on Sierpinski carpets, J. Eur. Math. Soc. (JEMS), in press. [8] M.T. Barlow, J. Kigami, Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets, J. London Math. Soc. 56 (1997) 320–332. [9] M.T. Barlow, T. Kumagai, Transition density asymptotics for some diffusion processes with multi-fractal structures, Electron. J. Probab. 6 (2001) 1–23. [10] M.T. Barlow, E.A. Perkins, Brownian motion on the Sierpinski gasket, Probab. Theory Related Fields 79 (1988) 543–623. [11] R.M. Blumenthal, R.K. Getoor, Markov Processes and Potential Theory, Academic Press, New York, 1968; republished by Dover Publications, Inc., New York, 2007. [12] Z.-Q. Chen, M. Fukushima, Symmetric Markov Processes, Time Change and Boundary Theory, book manuscript, 2009. [13] K.L. Chung, Doubly-Feller process with multiplicative functional, in: Seminar on Stochastic Processes, 1985, in: Progr. Probab. Statist., vol. 12, Birkhäuser, Boston, 1986, pp. 63–78. [14] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92, Cambridge University Press, 1989. [15] E.B. Davies, Spectral Theory and Differential Operators, Cambridge Stud. Adv. Math., vol. 42, Cambridge University Press, 1995. [16] W. Feller, An Introduction to Probability Theory and Its Applications, vol. II, second ed., Wiley, New York, 1971. [17] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Stud. Math., vol. 19, Walter de Gruyter, Berlin, 1994. [18] M. Fukushima, T. Shima, On a spectral analysis for the Sierpinski gasket, Potential Anal. 1 (1992) 1–35. [19] S. Goldstein, Random walks and diffusions on fractals, in: H. Kesten (Ed.), Percolation Theory and Ergodic Theory of Infinite Particle Systems, in: IMA Math. Appl., vol. 8, Springer, New York, 1987, pp. 121–129. [20] A. Grigor’yan, Heat kernel upper bounds on fractal spaces, preprint, 2004. [21] B. M. Hambly, Asymptotics for functions associated with heat flow on the Sierpinski carpet, Canad. J. Math., in press. [22] B.M. Hambly, T. Kumagai, S. Kusuoka, X.Y. Zhou, Transition density estimates for diffusion processes on homogeneous random Sierpinski carpets, J. Math. Soc. Japan 52 (2000) 373–408. [23] M. Hino, personal communication, May 14, 2008. [24] M. Hino, T. Kumagai, A trace theorem for Dirichlet forms on fractals, J. Funct. Anal. 238 (2006) 578–611. [25] N. Kajino, Remarks on non-diagonality conditions for Sierpinski carpets, in: Proceedings of the 1st MSJ-SI, “Probabilistic Approach to Geometry”, Adv. Stud. Pure Math., in press.
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[26] I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, second ed., Grad. Texts in Math., vol. 113, Springer-Verlag, New York, 1991. [27] J. Kigami, Analysis on Fractals, Cambridge Tracts in Math., vol. 143, Cambridge University Press, 2001. [28] J. Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem. Amer. Math. Soc. 199 (932) (2009). [29] J. Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, preprint, 2009. [30] J. Kigami, M.L. Lapidus, Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys. 158 (1993) 93–125. [31] S. Kusuoka, A diffusion process on a fractal, in: K. Ito, N. Ikeda (Eds.), Probabilistic Methods on Mathematical Physics, Proceedings of Taniguchi International Symposium, Katata & Kyoto, 1985, Kinokuniya, Tokyo, 1987, pp. 251–274. [32] S. Kusuoka, X.Y. Zhou, Dirichlet forms on fractals: Poincaré constant and resistance, Probab. Theory Related Fields 93 (1992) 169–196. [33] R. Rammal, Spectrum of harmonic excitations on fractals, J. Physique 45 (1984) 191–206. [34] R. Rammal, G. Toulouse, Random walks on fractal structures and percolation clusters, J. Physique Lettres 44 (1983) L13–L22. [35] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71 (1912) 441–479. [36] H. Weyl, Über die Abhängigkeit der Eigenschwingungen einer Membran von deren Begrenzung, J. Angew. Math. 141 (1912) 1–11.
Journal of Functional Analysis 258 (2010) 1361–1425 www.elsevier.com/locate/jfa
Stochastic Volterra equations in Banach spaces and stochastic partial differential equation Xicheng Zhang a,b,∗ a Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China b School of Mathematics and Statistics, The University of New South Wales, Sydney, 2052, Australia
Received 12 March 2009; accepted 10 November 2009 Available online 18 November 2009 Communicated by Paul Malliavin
Abstract In this paper, we study the existence-uniqueness and large deviation estimate for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then we apply them to a large class of semilinear stochastic partial differential equations (SPDE), and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Moreover, stochastic Navier–Stokes equations are also investigated. © 2009 Elsevier Inc. All rights reserved. Keywords: Stochastic Volterra equation; Large deviation; Stochastic Navier–Stokes equation
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Gronwall’s inequality of Volterra type . . 2.2. Itô’s integral in 2-smooth Banach spaces 2.3. A criterion for Laplace principles . . . . .
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* Address for correspondence: Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China. E-mail address: [email protected].
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.006
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Abstract stochastic Volterra integral equations . . . . . . . . . . . . 3.1. Global existence and uniqueness . . . . . . . . . . . . . . . . . 3.2. Path continuity of solutions . . . . . . . . . . . . . . . . . . . . 3.3. Local existence and uniqueness . . . . . . . . . . . . . . . . . 3.4. Continuous dependence of solutions with respect to data 4. Large deviation for stochastic Volterra equations . . . . . . . . . . . 5. Semilinear stochastic evolutionary integral equations . . . . . . . . 6. Semilinear stochastic partial differential equations . . . . . . . . . . 7. Application to stochastic Navier–Stokes equations . . . . . . . . . 7.1. Unique maximal strong solution for SNSEs . . . . . . . . . 7.2. Non-explosion and large deviation for 2D SNSEs . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The aims of this paper are three folds: First of all, we prove the existence and uniqueness of solutions with continuous paths for stochastic Volterra integral equations with singular kernels in a 2-smooth Banach space. Secondly, the large deviation principles (abbrev. LDP) of Freidlin– Wentzell’s type for stochastic Volterra equations are established under small perturbations of multiplicative noises. Thirdly, we apply them to several classes of semilinear stochastic partial differential equations (abbrev. SPDE). Compared with the well-known results on SPDEs, the main contributions of the present paper are that we can prove the existence and uniqueness of strong solutions (in the sense of SDE and PDE) for SPDEs, and give a unified treatment for the LDPs to a large class of SPDEs. In finite-dimensional space, stochastic Volterra integral equations with regular kernels and driven by Brownian motions were first studied by Berger and Mizel [3]. Later, Protter [52] studied stochastic Volterra equations driven by general semimartingales. Using the Skorohod integral, Pardoux and Protter [47] also investigated stochastic Volterra equations with anticipating coefficients. The study of stochastic Volterra equations with singular kernels can be found in [14,16, 65,36,44], etc. Recently, the present author [68] studied the approximation of Euler’s type and the LDP of Freidlin–Wentzell’s type for stochastic Volterra equations with singular kernels. In particular, the kernels in [68] can be used to deal with fractional Brownian motion kernels as well as fractional order integral kernels. The study of LDP for stochastic Volterra equations is also referred to [44,36]. Since the work of Freidlin and Wentzell [21], the theory of small perturbation large deviations for stochastic differential equations (abbrev. SDE) has been studied extensively (cf. [2,62], etc.). In the classical method, to establish such an LDP for SDE, one usually needs to discretize the time variable and then prove various necessary exponential continuity and tightness for approximation equations in different spaces by using comparison principle. However, such verifications would become rather complicated and even impossible in some cases, e.g., stochastic evolution equations with multiplicative noises. Recently, Dupuis and Ellis [19] systematically developed a weak convergence approach to the theory of large deviation. The central idea is to prove some variational representation formula for the Laplace transform of bounded continuous functionals, which will lead to proving a Laplace principle which is equivalent to the LDP. In particular, for Brownian functionals, an elegant vari-
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ational representation formula has been established by Boué and Dupuis [5] and Budhiraja and Dupuis [10]. A simplified proof was given by the present author [67]. This variational representation has already been proved to be very effective for various finite and infinite-dimensional stochastic dynamical systems even with irregular coefficients (cf. [54,55,11,68,56], etc.). One of the main advantages of this argument is that one only needs to make some simple moment estimates (see Section 4 below). On the other hand, it is well known that in the deterministic case, many PDE problems of parabolic and hyperbolic types can be written as Volterra type integral equations in Banach spaces by using the corresponding semigroup and the variation-of-constants formula (cf. [22, 28,48]). An obvious merit of this procedure is that the unbounded operators in PDEs no longer appear and the analysis is entirely analogous to the ODE case. Thus, one naturally expects to take the same advantages for SPDEs in Banach spaces. However, it is not all Banach spaces in which stochastic integrals are well defined. One can only work in a class of 2-smooth Banach spaces. The definition of stochastic integrals in 2-smooth Banach spaces and related properties such as Burkholder–Davis–Gundy’s (abbrev. BDG) inequality, Girsanov’s theorem, stochastic Fubini’s theorem and the distribution of stochastic integrals can be found in [43,6,7,45], etc. Thus, similar to the deterministic case, we can develop a parallel theory in 2-smooth Banach spaces for SPDEs. It should be emphasized that besides the usual SPDEs driven by multiplicative Brownian noises, a class of stochastic evolutionary integral equations appearing in viscoelasticity and heat conduction with memory (cf. [53]) can also be written as abstract stochastic Volterra equations in Banach spaces. In the past three decades, the theory of general SPDEs has been developed extensively by numerous authors mainly based on two different approaches: semigroup method based on the variation-of-constants formula (cf. [64,15,6–8,66], etc.) and variation method based on Galerkin’s finite-dimensional approximation (cf. [46,35,58,34,41,51,69,26], etc.). A new regularization method is given in [71]. An overview for the classification and applications of SPDEs are referred to the recent book of Kotelenez [33]. In the author’s knowledge, most of the wellknown results are primarily concentrated on the mild or weak solutions, even measure-valued solutions. Such notions of solutions naturally appear in the study of SPDEs driven by the space– time white noises, and in this case one cannot obtain any differentiability of solutions with respect to the spatial variable. Nevertheless, when one considers an SPDE driven by the spatial regular and time white noises, it is reasonable to require the existence of spatial regular solutions or classical solutions in the sense of PDE. For linear SPDEs, such regular solutions are relatively easy and well known (cf. [35,58,20], etc.). However, for non-linear SPDEs, there seems to be few results (cf. [34,39,67, 71]). A major difficulty to prove the spatial regularity of solutions is that one cannot use the usual bootstrap method in the theory of PDE since there is no differentiability of solutions with respect to the time variable. The present author [67] solved this problem by using a non-linear interpolation result due to Tartar [63]. Obviously, for the regularity theory of SPDEs, by using Sobolev’s embedding theorem (cf. [1]), it is natural to consider the Lp -solutions of SPDEs. This is also why we need to work in 2-smooth Banach spaces. It should be noticed that the Lp -theory for SPDEs has been established in [6–8,34,17,18,66], etc. But, there are few results to deal with the Lp -strong solution in the sense of PDE. In the present paper, we shall prove a general result about the existence of strong solutions in the sense of both SDE and PDE (see Theorem 6.6 below). We now describe the structure of this paper: In Section 2, we prepare some preliminaries for later use, and divide it into three subsections. In Section 2.1, we prove a Gronwall’s lemma of
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Volterra type under rather weak assumptions on kernel functions. Moreover, two simple examples are provided to show this lemma. In Section 2.2, we recall the Itô integral in 2-smooth Banach spaces and Burkholder–Davies–Gundy’s inequality as well as Kolmogorov’s continuity criterion of random fields in random intervals. In Section 2.3, we recall a criterion of Laplace principle established by Budhiraja and Dupuis [5,10] (see also [70]). In Section 3, using the Gronwall inequality of Volterra type in Section 2.1, we first prove the existence and uniqueness of solutions for stochastic Volterra equations in 2-smooth Banach spaces under global Lipschitz conditions and singular kernels. Next, in Section 3.2, we study the regularity of solutions under slightly stronger assumptions on kernels. Moreover, a BDG type of inequality for stochastic Volterra type integral is also proved. In Section 3.3, employing the usual localizing method, we prove the existence of a unique maximal solution for stochastic Volterra equation under local Lipschitz conditions. Lastly, in Section 3.4, we discuss the continuous dependence of solutions with respect to the coefficients. In Section 4, using the weak convergence method, we prove the Freidlin–Wentzell large deviation principle for the small perturbations of stochastic Volterra equations under a compactness assumption and some uniform non-explosion conditions for the controlled equations. We also refer to [38,56] for the application of weak convergence approach in the LDPs of stochastic evolution equations (the case of evolution triple). In the proof of Section 4, we need to use the Yamada–Watanabe Theorem in infinite-dimensional space, which has been established by Ondreját [45] (see also [57] for the case of evolution triple). We want to say that although Ondreját only considered the case of convolution semigroup, their proofs are also adapted to more general stochastic Volterra equations. Moreover, since we are considering the path continuous solution, the proof in [45] can be simplified. In Section 5, a simple application in a class of semilinear stochastic evolutionary integral equations is presented, which has been studied in [13,4,31], etc., for additive noises. Such type of stochastic evolution equations appears in viscoelasticity, heat conduction in materials with memory, and electrodynamics with memory [53]. In Section 6, we apply our general results to a large class of semilinear stochastic evolution equations driven by multiplicative Brownian noises. A basic result in semigroup theory states that if f is a Hölder continuous function in Banach space X, then t Tt−s f (s) ds is continuous in D(L),
t → 0
where Tt is an analytic semigroup and L is the generator of Tt . We will use this result to prove the existence of strong solutions (in the sense of PDE) for semilinear SPDEs. The corresponding LDPs are also obtained (see also [61,49,11,56,38], etc., for the study of LDPs of stochastic evolution equations). More applications can be found in an uncompressed version [72]. In Section 7, we prove the existence and uniqueness of local Lp -strong solutions for stochastic Navier–Stokes equations (SNSE) in any dimensional case. In the two-dimensional case, we also obtain the non-explosion of solutions. Moreover, the LDPs for two-dimensional SNSEs are established in the case of both Dirichlet boundary and periodic boundary. We remark that the Lp -solutions for SNSEs have been studied by Brze´zniak and Peszat [9] (bounded domain) and Mikulevicius and Rozovskii [40] (the whole space). The large deviation result for two-dimensional SNSEs with additive noise was proved by Chang [12] using Girsanov’s transformation. In [60], the authors also used the weak convergence method to prove the large
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
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deviation estimate for two-dimensional SNSEs with multiplicative noises. But, they worked in the L2 -space and only for weak solutions. Here we can do it for strong solutions in Lp space. We conclude this introduction by making the following C ONVENTION: Throughout this paper, the letter C with or without subscripts will denote a positive constant, whose value may change from one place to another. Moreover, we also use the notation E1 E2 to denote E1 C · E2 , where C > 0 is an unimportant constant. 2. Preliminaries 2.1. Gronwall’s inequality of Volterra type Let := {(t, s) ∈ R2+ : s t}. We first recall the following result due to Gripenberg [27, Theorem 1 and p. 88]. Lemma 2.1. Let κ : → R+ be a measurable function. Assume that for any T > 0, t t →
κ(t, s) ds ∈ L∞ (0, T )
0
and ·+ lim sup κ(· + , s) ds ↓0 ·
< 1. L∞ (0,T )
Define t r1 (t, s) := κ(t, s),
rn+1 (t, s) :=
κ(t, u)rn (u, s) du,
n ∈ N.
(2.1)
s
Then for any T > 0, there exist constants CT > 0 and γ ∈ (0, 1) such that · rn (·, s) ds 0
CT nγ n ,
∀n ∈ N.
(2.2)
L∞ (0,T )
In particular, the series r(t, s) :=
∞ n=1
converges for almost all (t, s) ∈ , and
rn (t, s)
(2.3)
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t r(t, s) − k(t, s) =
t k(t, u)r(u, s) du =
s
r(t, u)k(u, s) du
(2.4)
s
and for any T > 0, t t →
r(t, s) ds ∈ L∞ (0, T ).
(2.5)
0
The function r defined by (2.3) is called the resolvent of κ. All the functions κ in Lemma 2.1 will be denoted by K . In what follows, we shall denote by K0 the subclass of K with the property that ·+ lim sup κ(· + , s) ds ↓0 ·
= 0. L∞ (0,T )
We also denote by K>1 the set of all nonnegative measurable functions κ on with the property that for any T > 0 and some β = β(T ) > 1, t t →
κ β (t, s) ds ∈ L∞ (0, T ).
(2.6)
0
It is clear that K>1 ⊂ K0 ⊂ K and for any κ1 , κ2 ∈ K0 (resp. K>1 ) and C1 , C2 0, C1 κ1 + C2 κ2 ∈ K0
(resp. K>1 ).
Let 0 h ∈ L1loc (R+ ). If κ(t, s) = h(s), then κ ∈ K0 and t r(t, s) = h(s) exp
h(u) du ;
s
if κ(t, s) = h(t − s), then κ ∈ K0 and r(t, s) = a(t − s) :=
∞
an (t − s),
n=1
where t a1 (t) = h(t),
an+1 (t) :=
h(t − s)an (s) ds. 0
(2.7)
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When 0 h ∈ L1 (R+ ), a classical result due to Paley and Wiener (cf. [42, p. 207, Theorem 5.2]) says that ∞ a ∈ L (R+ ) 1
if and only if
h(t) dt < 1.
(2.8)
0
ˆ ˆ In this case, a(s) ˆ = h(s)/(1 − h(s)), where the hat denotes the Laplace transform, i.e.: ˆ := h(s)
∞
e−st h(t) dt,
s 0.
0
We want to say that (2.8) is useful in the study of large time asymptotic behavior of solutions for Volterra equations. An important extension to nonintegrable convolution kernel can be found in [59,29] (see also [27]). A simple example is provided in Example 3.2 below. We now prove the following Gronwall’s lemma of Volterra type (see also [28, Lemma 7.1.1] for a case of special convolution kernel). Lemma 2.2. Let κ ∈ K and rn and r be defined respectively by (2.1) and (2.3). Let f, g : R+ → R+ be two measurable functions satisfying that for any T > 0 and some n ∈ N, t t →
rn (t, s)f (s) ds ∈ L∞ (0, T )
(2.9)
0
and for almost all t ∈ (0, ∞), t r(t, s)g(s) ds < +∞.
(2.10)
0
If for almost all t ∈ (0, ∞), t f (t) g(t) +
κ(t, s)f (s) ds,
(2.11)
r(t, s)g(s) ds.
(2.12)
0
then for almost all t ∈ (0, ∞), t f (t) g(t) + 0
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Proof. First of all, if we define t h(t) := g(t) +
r(t, s)g(s) ds, 0
then by (2.4) and (2.10) t h(t) = g(t) +
κ(t, s)h(s) ds
for a.a. t ∈ (0, ∞).
0
Thus, by (2.11) we have t f (t) − h(t)
κ(t, s) f (s) − h(s) ds
for a.a. t ∈ (0, ∞).
(2.13)
0
Set f˜(t) := f (t) − h(t) and define f˜∗ (t) := ess sup f˜(s),
t > 0,
s∈[0,t]
and
τ0 := inf t > 0: f˜∗ (t) > 0 . Clearly, t → f˜∗ (t) is non-decreasing and f˜(t) 0 for a.a. t ∈ [0, τ0 ).
(2.14)
We want to prove that τ0 = +∞. Iterating inequality (2.13), we have f˜(t)
t 0
rn (t, s)f˜(s) ds
t rn (t, s)f (s) ds,
∀n ∈ N.
0
By (2.9), one knows that f˜∗ (T ) < +∞ for any T > 0. Moreover, for almost all t > 0, (2.14)
f˜(t)
t τ0
˜∗
rn (t, s)f˜(s) ds f (t)
t rn (t, s) ds, τ0
∀n ∈ N.
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Suppose τ0 < +∞. Then, for any T > τ0 , we have · 0 < f˜∗ (T ) f˜∗ (T ) · rn (·, s) ds τ0
as n → ∞, which is impossible. So, τ0 = +∞.
(2.2)
−→ 0 L∞ (τ0 ,T )
2
The following two examples show that (2.12) is sensitive to κ ∈ K . Example 2.3. For C0 > 0, set C0 , κC0 (t, s) := √ 2 t − s2
s < t.
It is clear that t
κC0 (t, u) du = C0 (π/2) − arcsin(s/t) .
s
From this, one sees that
κC0 ∈ /K, if C0 2/π; c κC0 ∈ K ∩ K0 , if 0 < C0 < 2/π .
Consider the following Volterra equation t x(t) =
κC0 (t, s)x(s) ds,
t 0.
0
If C0 = 1, there are at least two solutions x(t) ≡ 0 and x(t) = t; if C0 = π2 , there are infinitely many solutions x(t) ≡ constant; if 0 < C0 < 2/π , by Lemma 2.2 there is only one solution x(t) ≡ 0 in L∞ loc (R+ ). Example 2.4. For C0 > 0 and α, β ∈ [0, 1), set α,β
κC0 (t, s) :=
C0 , (t − s)α s β
s < t.
It is clear that t u
α,β κC0 (t, s) ds
1 = C0 t
1−α−β u/t
1 ds. (1 − s)α s β
(2.15)
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From this, one sees that ⎧ α,β ⎪ /K, ⎪ ⎪ κC0 ∈ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ κCα,β ∈ /K, ⎪ ⎪ ⎨ 0
if α + β > 1 and C0 > 0; 1 if α + β = 1 and C0 0
⎪ ⎪ ⎪ α,β ⎪ ⎪ κC0 ∈ K ∩ K0c , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ κ α,β ∈ K , >1 C0
1 if α + β = 1 and C0 < 0
1 ds; (1 − s)α s β 1 ds; (1 − s)α s β
if α + β < 1 and C0 > 0.
Consider the following Volterra equation t x(t) =
α,β
kC0 (t, s)x(s) ds,
t 0.
0
If α + β < 1, by Lemma 2.2 there is only one solution x(t) ≡ 0 in L∞ loc (R+ ); if α = β = C0 = √ 1/2, there are at least two solutions x(t) ≡ 0 and x(t) = t. 2.2. Itô’s integral in 2-smooth Banach spaces Throughout this paper, we shall fix a stochastic basis (Ω, F , P ; (Ft )t0 ), i.e., a complete probability space with a family of right-continuous filterations. In what follows, without special declarations, all expectations E are taken with respect to the probability measure P . Let {W k (t): t 0, k ∈ N} be a sequence of independent one-dimensional standard Brownian motions on (Ω, F , P ; (Ft )t0 ). Let l 2 be the usual Hilbert space of all square summable real number sequences, {ek , k ∈ N} the usual orthonormal basis of l 2 . Let X be a separable Banach space, and L(l 2 ; X) the set of all bounded linear operators from l 2 to X. For an operator B ∈ L(l 2 ; X), we also write B = (B1 , B2 , . . .) ∈ XN ,
Bk = Bek .
Definition 2.5. An operator B ∈ L(l 2 ; X) is called radonifying if the series
Bek · W k (1) converges in L2 (Ω; X).
k
We shall denote by L2 (l 2 ; X) the space of all radonifying operators, and write for B ∈ L2 (l 2 ; X), 2 1/2
B L2 (l 2 ;X) := EBek · W k (1)X .
(2.16)
Here and below, we use the convention that the repeated indices will be summed. The following proposition is well known, and a detailed proof was given in [45, Proposition 2.5].
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Proposition 2.6. The space L2 (l 2 ; X) with norm (2.16) is a separable Banach space. In order to introduce the stochastic integral of an X-valued measurable (Ft )-adapted process with respect to W , in the sequel, we assume that X is 2-smooth (cf. [50]), i.e., there exists a constant CX 2 such that for all x, y ∈ X,
x + y 2X + x − y 2X 2 x 2X + CX y 2X . Let now s → B(s) be an L2 (l 2 ; X)-valued measurable and (Ft )-adapted process with T
B(s)2
L2 (l 2 ;X)
ds < +∞ a.s., ∀T > 0.
0
One can define the Itô stochastic integral (cf. [45, Section 3]) t t → It (B) :=
t B(s) dW (s) =
0
Bk (s) dW k (s) ∈ X 0
such that t → It (B) is an X-valued continuous local (Ft )-martingale. Moreover, let τ be any (Ft )-stopping time, then t∧τ
t B(s) dW (s) =
0
1{s<τ } · B(s) dW (s). 0
The following BDG inequality for It (B) holds (cf. [45, Section 5]). Theorem 2.7. For any p > 0, there exists a constant Cp > 0 depending only on p such that t p T p/2 2 B(s) 2 ds E sup B(s) dW (s) . Cp E L2 (l ;X) t∈[0,T ]
0
X
(2.17)
0
The following two typical examples of 2-smooth Banach spaces are usually met in applications. Example 2.8. Let X be a separable Hilbert space. Clearly, X is 2-smooth. In this case, L2 (l 2 ; X) consists of all Hilbert–Schmidt operators of mapping l 2 into X, and
B L2 (l 2 ;X) =
∞ k=1
1/2
Bek 2X
.
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Example 2.9. Let (E, E, μ) be a measure space, H a separable Hilbert space. For p 2, let Lp (E, μ; H) be the usual H-valued Lp -space over (E, E, μ). Then X = Lp (E, μ; H) is 2smooth (cf. [50,6]). In this case, by BDG’s inequality for Hilbert valued martingale we have
B 2L
2
2/p Bk (x) · W k (1)p μ(dx)
=E (l 2 ;X)
H
E
2/p p EBk (x) · W k (1)H μ(dx)
E
Cp
∞ Bk (x)2
H
E
2/p
p/2
μ(dx)
k=1
= Cp B 2Lp (E,μ;l 2 ⊗H) .
(2.18)
Hence, Lp E, μ; l 2 ⊗ H → L2 l 2 ; X = L2 l 2 ; Lp (E, μ; H) . We also recall the following Kolmogorov’s continuity criterion, which can be derived directly by Garsia’s inequality (cf. [64]). Theorem 2.10. Let {X(t), t 0} be an X-valued stochastic process, and τ a bounded random time. Suppose that for some C0 , p > 0 and δ > 1, p E X(t) − X(s) · 1{s,t∈[0,τ ]} X C0 |t − s|δ . Then there exist constants C1 > 0 and a ∈ (0, (δ − 1)/p) independent of C0 and a continuous version X˜ of X such that E
˜ − X(s) ˜
X(t) X ap |t − s| s=t∈[0,τ ] p
sup
C1 · C0 .
2.3. A criterion for Laplace principles It is well known that there exists a Hilbert space so that l 2 ⊂ U is Hilbert–Schmidt with embedding operator J and {W k (t), k ∈ N} is a Brownian motion with values in U, whose covariance operator is given by Q = J ◦ J ∗ . For example, one can take U as the completion of l 2 with respect to the norm generated by scalar product
h, h
:= U
∞ hk h
k
k=1
k2
1 2
,
h, h ∈ l 2 .
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For T > 0 and a Banach space B, we denote by B(B) the Borel σ -field, and by CT (B) the space of all continuous functions from [0, T ] to B, which is endowed with the uniform norm. Define 2T
·
:= h =
˙h(s) ds: h˙ ∈ L2 0, T ; l 2
(2.19)
0
with the norm T
h 2 :=
h(s) ˙ 22 ds
1/2
l
T
,
0
where the dot denotes the generalized derivative. Let μ be the law of the Brownian motion W in CT (U). Then CT (U), 2T , μ forms an abstract Wiener space. For T , N > 0, set
DN := h ∈ 2T : h 2 N T
and ATN :=
h : [0, T ] → l 2 is a continuous and (Ft )-adapted . process, and for almost all ω, h(·, ω) ∈ DN
(2.20)
It is well known that with respect to the weak convergence topology in 2T (cf. [32]), DN is metrizable as a compact Polish space.
(2.21)
Let S be a Polish space. A function I : S → [0, ∞] is given. Definition 2.11. The function I is called a rate function if for every a < ∞, the set {f ∈ S: I (f ) a} is compact in S. Let {Z : CT (U) → S, ∈ (0, 1)} be a family of measurable mappings. Assume that there is a measurable map Z0 : 2T → S such that: (LD)1 For any N > 0, if a family {h , ∈ (0, 1)} ⊂ ATN (as random variables in DN ) converges k in distribution to h ∈ ATN , then for some subsequence k , Zk (· + h√(·) ) converges in k distribution to Z0 (h) in S. (LD)2 For any N > 0, if {hn , n ∈ N} ⊂ DN weakly converges to h ∈ 2T , then for some subsequence hnk , Z0 (hnk ) converges to Z0 (h) in S.
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For each f ∈ S, define I (f ) :=
1 inf
h 22 , 2 {h∈2T : f =Z0 (h)} T
(2.22)
where inf ∅ = ∞ by convention. Then under (LD)2 , I (f ) is a rate function. In fact, assume that I (fn ) a. By the definition of I (fn ), there exists a sequence hn ∈ 2 such that Z0 (hn ) = fn and 1 1
hn 22 a + . 2 n T By the weak compactness of D2a+2 , there exist a subsequence nk (still denoted by n) and h ∈ 2T such that hn weakly converges to h and
h 22 lim hn 22 2a. T
n→∞
T
Hence, by (LD)2 we have lim Z0 (hnk ) − Z0 (h)S = 0
k→∞
and I Z0 (h) a. We recall the following result due to [5,10] (see also [67, Theorem 4.4]). Theorem 2.12. Under (LD)1 and (LD)2 , {Z , ∈ (0, 1)} satisfies the Laplace principle with the rate function I (f ) given by (2.22). More precisely, for each real bounded continuous function g on S:
g(Z ) = − inf g(f ) + I (f ) . lim log Eμ exp − →0 f ∈S
(2.23)
In particular, the family of {Z , ∈ (0, 1)} satisfies the large deviation principle in (S, B(S)) with the rate function I (f ). More precisely, let ν be the law of Z in (S, B(S)), then for any A ∈ B(S): − inf o I (f ) lim inf log ν (A) lim sup log ν (A) − inf I (f ), f ∈A
→0
→0
f ∈A¯
where the closure and the interior are taken in S, and I (f ) is defined by (2.22).
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3. Abstract stochastic Volterra integral equations In this section, we consider the following stochastic Volterra integral equation in 2-smooth Banach space X: t X(t) = g(t) +
A t, s, X(s) ds +
0
t
B t, s, X(s) dW (s),
(3.1)
0
where g(t) is an X-valued measurable and (Ft )-adapted process, and A : × Ω × X → X ∈ M × B(X)/B(X) and B : × Ω × X → L2 l 2 ; X ∈ M × B(X)/B L2 l 2 ; X . Here and below, := {(t, s) ∈ R2+ : s t}, and M denotes the progressively measurable σ field on × Ω generated by the sets E ∈ B() × F with properties: 1E (t, s, ·) ∈ Fs for all (t, s) ∈ , and s → 1E (t, s, ω) is right continuous for any t ∈ R+ and ω ∈ Ω. We start with the global existence and uniqueness of solutions for Eq. (3.1) under global Lipschitz conditions and singular kernels. 3.1. Global existence and uniqueness In this subsection, we make the following global Lipschitz and linear growth conditions on the coefficients: (H1) For some p 2 and any T > 0, t ess sup
t∈[0,T ]
p κ1 (t, s) + κ2 (t, s) · Eg(s)X ds < +∞,
0
where κ1 and κ2 are from (H2) and (H3) below. (H2) There exists κ1 ∈ K0 such that for all (t, s) ∈ , ω ∈ Ω and x ∈ X, A(t, s, ω, x) κ1 (t, s) · x X + 1 X and B(t, s, ω, x)2
L2 (l 2 ;X)
κ1 (t, s) · x 2X + 1 .
(H3) There exists κ2 ∈ K0 such that for all (t, s) ∈ , ω ∈ Ω and x, y ∈ X, A(t, s, ω, x) − A(t, s, ω, y) κ2 (t, s) · x − y X X
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and B(t, s, ω, x) − B(t, s, ω, y)2
L2 (l 2 ;X)
κ2 (t, s) · x − y 2X .
We now prove the following basic existence and uniqueness result. Theorem 3.1. Assume that (H1)–(H3) hold. Then there exists a unique measurable (Ft )-adapted process X(t) such that for almost all t 0, t X(t) = g(t) +
A t, s, X(s) ds +
t
B t, s, X(s) dW (s),
P -a.s.,
(3.2)
t p p p E X(t) X CT ,p,κ1 E g(t) X + ess sup κ1 (t, s) · E g(s) X ds
(3.3)
0
0
and for any T > 0 and some CT ,p,κ1 > 0,
t∈[0,T ]
0
for almost all t ∈ [0, T ], where p is from (H1). Moreover, if t t →
κ1 (t, s) ds ∈ L∞ (R+ ),
(3.4)
0
then for almost all t 0, t p p p E X(t) X Cp,κ1 E g(t) X + κ˜ 1 (t, s) · Eg(s)X ds 0
u
t +
rκ˜ 1 (t, u) · 0
p κ˜ 1 (u, s) · E g(s) X ds du ,
(3.5)
0
where κ˜ 1 = C˜ p,κ1 · κ1 , rκ˜ 1 is defined by (2.3) in terms of κ˜ 1 , and Cp,κ1 , C˜ p,κ1 are constants only depending on p, κ1 . Proof. We use Picard’s iteration to prove the existence. Let X1 (t) := g(t) and define recursively for n ∈ N, t Xn+1 (t) = g(t) + 0
A t, s, Xn (s) ds +
t 0
B t, s, Xn (s) dW (s).
(3.6)
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Fix T > 0 below. By (H2), BDG’s inequality (2.17) and Hölder’s inequality we have p p EXn+1 (t)X Eg(t)X + E
t
A t, s, Xn (s) ds X
p
0
t p + E B t, s, Xn (s) dW (s)
X
0
p Eg(t)X + E
t
κ1 (t, s) · Xn (s)X + 1 ds
p
0
t +E
B t, s, Xn (s) 2
L2
p 2
ds (l 2 ;X)
0
p Eg(t)X +
t
p κ1 (t, s) · E Xn (s)X + 1 ds ·
0
t +
t
p−1 κ1 (t, s) ds
0
p κ1 (t, s) · E Xn (s)X + 1 ds ·
t
0
p −1 2
κ1 (t, s) ds 0
p Eg(t)X + CT ,p · CT + CT ,p
t
p κ1 (t, s) · EXn (s)X ds,
0
where CT := ess supt∈[0,T ] | Set
t 0
p−1
κ1 (t, s) ds| and CT ,p := CT fm (t) :=
(p−2)/2
+ CT
.
p sup EXn (t)X .
n=1,...,m
Then p fm (t) CT ,p,κ1 Eg(t)X + 1 +
t κ˜ 1 (t, s) · fm (s) ds, 0
where κ˜ 1 = CT ,p,κ1 · κ1 and the constant CT ,p,κ1 is independent of m. Let rκ˜ 1 be defined by (2.3) in terms of κ˜ 1 . Note that by (2.4) t 0
p rκ˜ 1 (t, s) · Eg(s)X ds −
t 0
p κ˜ 1 (t, s) · Eg(s)X ds
(3.7)
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t t
p rκ˜ 1 (t, u)κ˜ 1 (u, s) du · Eg(s)X ds
= s
0
u
t =
rκ˜ 1 (t, u) 0
p κ˜ 1 (u, s) · E g(s) X ds du.
0
Hence, by Lemma 2.2 and (H1), we obtain that for almost all t ∈ [0, T ], p sup EXn (t)X =
n∈N
lim fm (t) CT ,p,κ1
m→∞
p Eg(t)X +
t
p rκ˜ 1 (t, s) · Eg(s)X ds
0
CT ,p,κ1
p Eg(t)X +
t
p κ˜ 1 (t, s) · Eg(s)X ds
0
u
t +
rκ˜ 1 (t, u) 0
p κ˜ 1 (u, s) · Eg(s)X ds du
(3.8)
0
(2.5)
CT ,p,κ1
p Eg(t)X + ess sup
t∈[0,T ]
t
p κ1 (t, s) · E g(s) X ds .
0
On the other hand, set Zn,m (t) := Xn (t) − Xm (t) and 2 f (t) := lim sup EZn,m (t)X . n,m→∞
As the above calculations, by (H3) we have t 2 2 A t, s, Xn (s) − A t, s, Xm (s) ds E Zn+1,m+1 (t) X E
X
0
t 2 B t, s, Xn (s) − B t, s, Xm (s) dW (s) + E 0
t 0
2 κ2 (t, s) · EZn,m (s)X ds.
X
(3.9)
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1379
By (3.9), (H1) and using Fatou’s lemma, we get t f (t)
κ2 (t, s) · f (s) ds. 0
By Lemma 2.2 again, we have for almost all t ∈ [0, T ], 2 f (t) = lim sup EZn,m (t)X = 0. n,m→∞
Hence, there exists an X-valued (Ft )-adapted process X(t) such that for almost all t ∈ [0, T ], 2 lim EXn (t) − X(t)X = 0.
n→∞
Taking limits for (3.6), one finds that (3.2) holds. Moreover, estimate (3.3) follows from (3.9). Note that when (3.4) is satisfied, the constant CT ,p in (3.7) is independent of T . Hence, estimate (3.5) is direct from (3.8). The uniqueness follows by similar calculations as above. 2 Example 3.2. Let for δ > 0, h(s) :=
e−δs s log2 s
,
t > s 0.
It is easy to see that h ∈ L1 (R+ ). Consider the following stochastic Volterra equation: t t X(t) = x0 log(t ∧ 1) + h(t − s)A X(s) ds + 0
h(t − s)B X(s) dW (s),
0
where A : X → X and B : X → L2 (l 2 ; X) are global Lipschitz continuous functions. By elementary calculations, one finds that t sup t0
0
e−δ(t−s) |log(s ∧ 1)| (t − s) log2 (t − s)
ds < +∞.
So, (H1)–(H3) are satisfied with p = 2. Moreover, by (2.8) and (3.5), one finds that if δ is large enough, then for any T > 0, 2 sup EX(t)X < +∞.
tT
We remark that in this example, X(0) = ∞.
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
3.2. Path continuity of solutions In this subsection, in addition to (H2) and (H3), we also assume that: (H1) The process t → g(t) is continuous and (Ft )-adapted, and for any p 2 and T > 0, E
!
p " sup g(t)X < +∞.
t∈[0,T ]
(H4) For all s < t < t , ω ∈ Ω and x ∈ X, A t , s, ω, x − A(t, s, ω, x) λ t , t, s · x X + 1 X and B t , s, ω, x − B(t, s, ω, x)2 L
2 (l
2 ;X)
λ t , t, s · x 2X + 1 ,
where λ is a positive measurable function satisfying that for any T > 0 and some γ = γ (T ), C = C(T ) > 0, t
γ λ t , t, s ds C t − t ,
0 t < t T .
(3.10)
0
Theorem 3.3. Assume that (H1) and (H2)–(H4) hold, and the kernel function κ1 in (H2) belongs to K>1 . Then there exists a unique X-valued continuous (Ft )-adapted process X(t) such that P -a.s., for all t 0, t X(t) = g(t) +
A t, s, X(s) ds +
0
t
B t, s, X(s) dW (s)
(3.11)
0
and for any p 2 and T > 0, E
!
p " sup X(t)X < +∞.
t∈[0,T ]
(3.12)
Moreover, if for some δ > 0 and any p 2, T > 0, p δp Eg t − g(t)X CT ,p t − t , then, t → X(t) admits a Hölder continuous modification and for any p 2, T > 0 and some a > 0,
X(t ) − X(t) X E sup |t − t|ap t=t ∈[0,T ] p
CT ,p,a .
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1381
Proof. First of all, for any p 2 and T > 0, by (H1) and (3.3) we have p ess sup EX(t)X < +∞.
(3.13)
t∈[0,T ]
Set t J (t) :=
B t, s, X(s) dW (s)
0
and write for 0 t < t T , J t − J (t) =
t
B t , s, X(s) − B t, s, X(s) dW (s)
0
t +
B t , s, X(s) dW (s) =: J1 t , t + J2 t , t .
t
In view of κ1 ∈ K>1 , (2.6) holds for some β > 1. Fix p 2β ∗ (β ∗ := β/(β − 1)). By BDG inequality (2.17), (H2) and Hölder’s inequality, we have p E J 2 t , t X E
t
2 κ1 t , s · X(s)X + 1 ds
p 2
t
t
β k1 t , s ds
p
t
2β
E
t
∗ X(s)2β + 1 ds
X
t
(2.6)
p −1 t − t 2β ∗
t
p p (3.13) EX(s)X + 1 ds t − t 2β ∗ ,
t
and by (H4) and Minkowski’s inequality, p E J 1 t , t X E
t
2 λ t , t, s · X(s)X + 1 ds
p 2
0
t
p 2 λ t , t, s · EX(s)X p + 1 ds
0 (3.13)
p 2β ∗
t
0
λ t , t, s ds
p 2
(3.10)
γp t − t 2 .
p 2
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
Hence, for all 0 t < t T , p γp p EJ t − J (t)X t − t 2 + t − t 2β ∗ . Similarly, we may prove that for all 0 t < t T and p β ∗ , t p t γp p E A t , s, X(s) ds − A t, s, X(s) ds t − t + t − t β ∗ . 0
X
0
The desired conclusions follow from Theorem 2.10.
2
We conclude this subsection by proving a lemma, which will be used frequently later. We put it here since the proof is similar to Theorem 3.3. Lemma 3.4. Let τ be an (Ft )-stopping time and G : × Ω → L2 l 2 ; X ∈ M /B L2 l 2 ; X . Assume that for all 0 s < t < t and ω ∈ Ω, G(t, s, ω)2
κ(t, s) · f 2 (s, ω), G t , s, ω − G(t, s, ω)2 2 λ t , t, s · f 2 (s, ω), L (l ;X) L2 (l 2 ;X) 2
(3.14) (3.15)
where κ ∈ K>1 and for any T > 0 and some α > 1 and γ > 0, t
γ λα t , t, s ds CT t − t ,
∀0 t < t T ,
0
and (s, ω) → f (s, ω) is a positive measurable process with T∧τ E
f p (s) ds < +∞,
∀p 2.
0
t Then t → J (t) := 0 G(t, s) dW (s) ∈ X admits a continuous modification on [0, τ ), and for any T > 0 and p large enough t p T∧τ p E sup G(t, s) dW (s) f (s) ds , CT E t∈[0,T ∧τ ]
0
X
where the constant CT is independent of f and τ .
0
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1383
Proof. Fix T > 0 and write for 0 t < t T , J t − J (t) =
t
G t , s dW (s) +
t
=: J1 t , t + J2 t , t .
t
G t , s − G(t, s) dW (s)
0
In view of κ ∈ K>1 and (2.6), by BDG’s inequality (2.17) and Hölder’s inequality we have, for some β > 1 and p 2β ∗ (β ∗ = β/(β − 1)), t ∧τ p p EJ1 t , t · 1{t ,t∈[0,τ )} X E G t , s dW (s)
X
t∧τ
p/2
t ∧τ
E
2 G t , s
L2 (l 2 ;X)
ds
t∧τ
t ∧τ
(3.14)
E
κ t , s · f 2 (s) ds
p/2
t∧τ
t
κ β t , s ds
p
t ∧τ
2β
·E
t
f
2β ∗
p 2β ∗
(s) ds
t∧τ
p −1 t − t 2β ∗ · E
T∧τ
f p (s) ds
0
and for p 2α ∗ (α ∗ = α/(α − 1)), p EJ1 t , t · 1{t ,t∈[0,τ )} X E
t∧τ G t , s − G(t, s)2
p/2
L2 (l 2 ;X)
ds
0 (3.15)
t∧τ
E
λ t , t, s · f 2 (s) ds
p/2
0
t
α
p
λ t , t, s ds
·E
0
γp t − t 2α · E
t∧τ
2α
f 0
T∧τ
p
f (s) ds . 0
2α ∗
(s) ds
p 2α ∗
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
Hence, for any p 2(α ∗ ∨ β ∗ ) and 0 t < t T , ( p −1)∧ γp p 2α · E E J t − J (t) · 1{t ,t∈[0,τ )} X t − t 2β ∗
T∧τ
p
f (s) ds . 0
2
The desired result now follows by Theorem 2.10. 3.3. Local existence and uniqueness In this subsection, we assume that:
(H2) For any R > 0, there exists κ1,R ∈ K>1 such that for all (t, s) ∈ , ω ∈ Ω and x ∈ X with
x X R, A(t, s, ω, x) + B(t, s, ω, x)2 L X
2 (l
2 ;X)
κ1,R (t, s).
(H3) For any R > 0, there exists κ2,R ∈ K0 such that for all (t, s) ∈ , ω ∈ Ω and x, y ∈ X with x X , y X R, A(t, s, ω, x) − A(t, s, ω, y) κ2,R (t, s) · x − y X X and B(t, s, ω, x) − B(t, s, ω, y)2
L2 (l 2 ;X)
κ2,R (t, s) · x − y 2X .
(H4) For any R > 0, there exists a measurable function λR satisfying that for any T > 0 and some γ , C > 0, t
γ λR t , t, s ds C t − t ,
0 t < t T ,
0
such that for all s < t < t , ω ∈ Ω and x ∈ X with x X R, A t , s, ω, x − A(t, s, ω, x) + B t , s, ω, x − B(t, s, ω, x)2 X L
2 (l
2 ;X)
λR t , t, s .
We first introduce the following notion of local solutions. Definition 3.5. Let τ be an (Ft )-stopping time, and {X(t); t ∈ [0, τ )} an X-valued continuous (Ft )-adapted process. The pair of (X, τ ) is called a local solution of Eq. (3.1) if P -a.s., for all t ∈ [0, τ ), t X(t) = g(t) + 0
A t, s, X(s) ds +
t 0
B t, s, X(s) dW (s);
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
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(X, τ ) is called a maximal solution of Eq. (3.1) if lim X(t, ω)X = +∞
t↑τ (ω)
on ω: τ (ω) < +∞ , P -a.s.
We call (X, τ ) a non-explosion solution of Eq. (3.1) if
P ω: τ (ω) < +∞ = 0. Remark 3.6. The stochastic integral in the above definition is defined on [0, τ ) by t
B t, s, X(s) dW (s) = lim
t∧τ n
B t, s, X(s) dW (s),
n→∞
0
t < τ,
0
where τn := inf{t > 0: X(t) X > n} τ as n → ∞. We now prove the following main result in this section. Theorem 3.7. Under (H1) –(H4) , there exists a unique maximal solution (X, τ ) for Eq. (3.1) in the sense of Definition 3.5. Proof. For n ∈ N, let χn be a positive smooth function on R+ with χn (s) = 1, s n, and χn (s) = 0, s n + 1. Define An (t, s, ω, x) := A(t, s, ω, x) · χn x X , Bn (t, s, ω, x) := B(t, s, ω, x) · χn x X . It is easy to see that for An and Bn , (H2) holds with κ1,n+1 , (H4) holds with λn+1 , and (H3) holds with some κ3,n ∈ K0 . Thus, by Theorem 3.3 there exists a unique continuous (Ft )-adapted process Xn (t) such that for any p 2 and T > 0, E
!
p " sup Xn (t)X CT ,p,n
t∈[0,T ]
and t Xn (t) = g(t) + 0
An t, s, Xn (s) ds +
t
Bn t, s, Xn (s) dW (s).
(3.16)
0
We have the following claim: Let τ be any stopping time. The uniqueness holds for (3.16) on [0, τ ). We remark that when τ = T is non-random, it follows from Theorem 3.1. Let Xi (t), i = 1, 2, be two X-valued continuous (Ft )-adapted processes and satisfy on [0, τ ),
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
t Xi (t) = g(t) +
An t, s, Xi (s) ds +
t
0
Bn t, s, Xi (s) dW (s),
i = 1, 2.
0
Set Z(t) := X1 (t) − X2 (t). Since κ3,n ∈ K0 , as the calculations in (3.7), by BDG’s inequality (2.17) and (H3) for An and Bn , we have p EZ(t) · 1{t<τ } X t∧τ
κ3,n (t, s) · Z(s)X ds
E
p
t∧τ +E
0
t =E
p 2
0
κ3,n (t, s) · 1{s<τ } · Z(s)X ds
0
t
2 κ3,n (t, s) · Z(s)X ds
p
t +E
2 κ3,n (t, s) · 1{s<τ } · Z(s)X ds
p 2
0
p κ3,n (t, s) · EZ(s) · 1{s<τ } X ds.
0
By Lemma 2.2, we get p EZ(t) · 1{t<τ } X = 0 for almost all t ∈ [0, T ], which implies by the arbitrariness of T and the continuities of Xi (t), i = 1, 2, X1 (·)|[0,τ ) = X2 (·)|[0,τ ) . The claim is proved. Now, for n ∈ N, define the stopping times
τn := inf t > 0: Xn (t)X > n and
σn := inf t > 0: Xn+1 (t)X > n . By the above claim, we have Xn (·)|[0,τn ∧σn ) = Xn+1 (·)|[0,τn ∧σn ) ,
(3.17)
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1387
which implies τn σn τn+1 ,
a.s.
Hence, we may define τ (ω) := lim τn (ω) n→∞
and for all t < τ (ω), X(t, ω) := Xn (t, ω),
if t < τn (ω).
Clearly, (X, τ ) is a maximal solution of Eq. (3.1) in the sense of Definition 3.5. ˜ τ˜ ) be another maximal solution of Eq. (3.1) in the We next prove the uniqueness. Let (X, sense of Definition 3.5. Define the stopping times
˜ >n τ˜n := inf t > 0: X(t) X and τˆn := τn ∧ τ˜n ,
τˆ := τ ∧ τ˜ .
It is clear that τˆn τˆ
a.s. as n → ∞,
and ˜ = 1[0,τˆn ) (t) · g(t) + 1[0,τˆn ) (t) · 1[0,τˆn ) (t) · X(t)
t
˜ A t, s, X(s) ds
0
t + 1[0,τˆn ) (t) ·
˜ B t, s, X(s) dW (s)
0
t = 1[0,τˆn ) (t) · g(t) + 1[0,τˆn ) (t) ·
˜ ds An t, s, X(s)
0
t + 1[0,τˆn ) (t) ·
˜ dW (s). Bn t, s, X(s)
0
By the above claim again, we have ˜ [0,τˆ ) . X(·)|[0,τˆn ) = X(·)| n
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
So ˜ [0,τˆ ) . X(·)|[0,τˆ ) = X(·)| By the definition of maximal solution we must have τˆ = τ = τ˜ .
2
We have the following simple criterion of non-explosion. Theorem 3.8. Assume that (H1) , (H2) and (H4) hold, and κ1 in (H2) belongs to K>1 . Then there is no explosion for Eq. (3.1). Proof. Let (X, τ ) be a maximal solution of Eq. (3.1). Define
τn := inf t > 0: X(t)X n . By BDG’s inequality (2.17) and Hölder’s inequality, and using the same method as estimating (3.17), we have, for any T > 0 and some β > 1 and p 2β ∗ (β ∗ = β/(β − 1)), p p EX(t) · 1{tτn } X Eg(t)X + E
t∧τ p n A t, s, X(s) ds X 0
t∧τ p n + E B t, s, X(s) dW (s)
X
0
p Eg(t)X + E
t∧τ p n κ1 (t, s) · X(s) X + 1 ds 0
t∧τ n B t, s, X(s) 2 +E
L2
p 2
(l 2 ;X)
ds
0
p Eg(t)X + E
t∧τ p∗ n β β ∗ X(s) + 1 ds X
0
t∧τ p∗ n 2β 2β ∗ X(s) + 1 ds +E X
0
CT ,p
p Eg(s)X + 1 +
t 0
where the constant CT ,p is independent of n.
p EX(s) · 1{sτn } X ds ,
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1389
By Gronwall’s inequality, we obtain p sup EX(t) · 1{tτn } X CT ,p .
t∈[0,T ]
Using this estimate, as in the proofs of Theorem 3.3 and Lemma 3.4, we can prove that for any T > 0 and p 2, sup E
! sup
" X(t)p CT ,p . X
t∈[0,T ∧τn ]
n∈N
Hence, lim P {τn T } = lim P
n→∞
#
n→∞
lim E n→∞
!
sup
$ X(t) n X
sup
" X(t)p /np
t∈[0,T ∧τn ]
X
t∈[0,T ∧τn ]
lim CT ,p /np = 0, n→∞
which produces the non-explosion, i.e., P {τ < ∞} = 0.
2
Remark 3.9. One cannot directly prove p sup EX(t ∧ τn )X < +∞,
∀t 0,
n∈N
to obtain the non-explosion, because it does not in general make sense to write t∧τ n
B t ∧ τn , s, X(s) dW (s).
0
3.4. Continuous dependence of solutions with respect to data In this subsection, we study the continuous dependence of solutions for Eq. (3.1) with respect to the coefficients. Let {(gm , Am , Bm ), m ∈ N} be a sequence of coefficients associated to Eq. (3.1). Assume that for each m ∈ N, (gm , Am , Bm ) satisfies (H1) –(H4) with the same κ1,R , κ2,R and λR as (g, A, B), and for each p 2, lim
p sup Egm (t) − g(t)X = 0
m→∞ t∈[0,T ]
and for each T , R > 0,
(3.18)
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
t lim
sup
m→∞ t∈[0,T ], x R X
t lim
(3.19)
0
Bm (t, s, x) − B(t, s, x)2
sup
m→∞ t∈[0,T ], x R X
Am (t, s, x) − A(t, s, x) ds = 0, X
L2 (l 2 ;X)
ds = 0.
(3.20)
0
Let (Xm , τm ) (resp. (X, τ )) be the unique maximal solution associated with (gm , Am , Bm ) (resp. (g, A, B)). For each R > 0 and m ∈ N, define
τmR := inf t > 0: X(t)X , Xm (t)X > R . Suppose that for each t > 0,
lim sup P τmR < t = 0.
R→∞ m
Then we have: Theorem 3.10. For each t > 0 and > 0,
lim P Xm (t) − X(t)X = 0.
m→∞
Proof. For R > 0 and m ∈ N, set R Zm (t) := Xm (t) − X(t) · 1{tτmR } . Then R R R R R R Zm (t) = J1,m (t) + J2,m (t) + J3,m (t) + J4,m (t) + J5,m (t),
where R J1,m (t) := 1{tτmR ] · gm (t) − g(t) , R t∧τ m
R (t) := 1{tτmR ] J2,m
Am t, s, Xn (s) − Am t, s, X(s) ds,
· 0
R t∧τ m
Am t, s, X(s) − A t, X(s) ds,
R J3,m (t) := 1{tτmR ] · 0
R t∧τ m
Bm t, s, Xm (s) − Bm t, s, X(s) dW (s),
R J4,m (t) := 1{tτmR ] · 0
(3.21)
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1391
R t∧τ m
Bm t, s, X(s) − B t, s, X(s) dW (s).
R J5,m (t) := 1{tτmR ] · 0
Fix T > 0. Clearly, for any p 2 and t ∈ [0, T ], R p p EJ1,m (t)X sup Egm (t) − g(t)X =: J1,m . t∈[0,T ]
R (t), by (H3) and Hölder’s inequality we have, for p large enough (κ For J2,m 2,R ∈ K>1 )
R p EJ2,m (t)X E
R t∧τ m
κ2,R (t, s) · Xm (s) − X(s)X ds
p
0
t
p
t
β
β κ2,R (t, s) ds
·E
0
t C
R β ∗ Z (s) ds
p∗ β
X
m
0
R p EZm (s)X ds.
0 R (t), we have For J3,m
R p EJ3,m (t)X E
R t∧τ m
Am (t, s, x) − A(t, s, x) ds X
sup
x X R
0
t sup
p
sup
t∈[0,T ] x X R
Am (t, s, x) − A(t, s, x) ds X
p
0
R =: J3,m .
Similarly, by BDG’s inequality (2.17) we have, for p large enough R p EJ4,m (t)X C
t
R p EZm (s)X ds
0
and p R (t)X Cp EJ5,m
t sup
sup
t∈[0,T ] x X R
Bm (t, s, x) − B(t, s, x)2
L2
0
p 2
(l 2 ;X)
ds
R =: J5,m .
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
Combining the above calculations, we get R p R R (t)X J1,m + J3,m + J5,m +C EZm
t
R p EZm (s)X ds.
0
By Gronwall’s inequality and (3.18)–(3.20) we get, for any R > 0 and p large enough R p lim EZm (t)X = 0.
m→∞
Hence
P Xm (t) − X(t)X P Xm (t) − X(t)X · 1{tτmR } + P τmR < t R p p
EZm (t)X / + P τmR < t . First letting m → ∞ and then R → ∞, we then get the desired limit by (3.21).
2
4. Large deviation for stochastic Volterra equations In this section, we study the large deviation of small perturbations for stochastic Volterra equations and work in the finite time interval [0, T ]. In what follows, we fix a densely defined closed linear operator L on X for which
Sφ := λ ∈ C: 0 < φ |arg λ| π ⊂ ρ(L),
(4.1)
and for some C 1, (λ − L)−1
L(X)
C , 1 + |λ|
λ ∈ Sφ ,
where ρ(L) denotes the resolvent set of L. The above operator L is also called sectorial (cf. [28, p. 18]). It is well known that L generates an analytic semigroup Tt = e−Lt ,
t 0.
Moreover, we also assume that L−1 is a bounded linear operator on X, i.e., 0 ∈ ρ(L). Thus, for any α ∈ R, the fractional power Lα is well defined (cf. [28,48]). For α > 0, we define the fractional Sobolev space Xα by Xα := D Lα with the norm
x Xα := Lα x X .
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1393
For α < 0, Xα is defined as the completion of X with respect to the above norm. It is clear that Xα is still 2-smooth, and B ∈ L2 (l 2 ; Xα ) if and only if Lα B ∈ L2 (l 2 ; X), i.e.,
B L2 (l 2 ;Xα ) = Lα B L
2 (l
2 ;X)
(4.2)
.
We recall the following well-known properties about Tt for later use (cf. [28, pp. 24–27] or [48, p. 74]). Proposition 4.1. (i) Tt : X → Xα for each t > 0 and α > 0. (ii) For each t > 0, α ∈ R and every x ∈ Xα , Tt Lα x = Lα Tt x. (iii) For some δ > 0 and each t, α > 0, the operator Lα Tt is bounded in X and α L Tt x Cα t −α e−δt x X , X
∀x ∈ X.
(iv) Let α ∈ (0, 1] and x ∈ Xα , then
Tt x − x X Cα t α x Xα . (v) For any 0 β < α, β
β
x Xβ Cα,β x 1− α x Xα α ,
∀x ∈ Xα .
In addition to (H2) , (H3) and (H4) , in this section we assume that g and A, B are nonrandom, and: (H1) For some δ > 0, g(t) − g t CT t − t δ , X
t, t ∈ [0, T ],
and for some α > 0, sup g(t)X < +∞.
t∈[0,T ]
α
(H2) For the same α as in (H1) and any R > 0, there exists a kernel function κα,R ∈ K0 such that for all (t, s) ∈ and x ∈ X with x X R, A(t, s, x)
Xα
2 + B(t, s, x)L
2 (l
2 ;X α 2
)
κα,R (t, s).
Remark 4.2. If the κα,R in (H2) belongs to K>1 , then (H2) implies (H2) in view of Xα → X.
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Consider the following small perturbation of stochastic Volterra equation (3.1) t X (t) = g(t) +
√ A t, s, X (s) ds +
0
t
B t, s, X (s) dW (s),
(4.3)
0
where ∈ (0, 1). By Theorem 3.7, there exists a unique maximal solution (X , τ ) for Eq. (4.3). Below, we fix T > 0 and work in the finite time interval [0, T ], and assume that for each ∈ (0, 1), τ > T ,
a.s.
By Yamada–Watanabe’s theorem (cf. [45,57]), there exists a measurable mapping Φ : CT (U) → CT (X) such that X (t, ω) = Φ W (·, ω) (t). It should be noticed that although the equation considered in [45] is a little different from Eq. (3.1), the proof is still adapted to our more general equation. We now fix a family of processes {h , ∈ (0, 1)} in ATN (see (2.20) for the definition of ATN ), and put h (·, ω) (t). X (t, ω) := Φ W (·, ω) + √ Here, we have used a little confused notations X and X , but they are clearly different. By Girsanov’s theorem (cf. [45, Section 7]), X (t) solves the following stochastic Volterra equation (also called control equation): t X (t) = g(t) +
A t, s, X (s) ds +
0
+
√
t
t
B t, s, X (s) h˙ (s) ds
0
B t, s, X (s) dW (s).
(4.4)
0
˙ = 0 for t > T so Although h is defined only on [0, T ], we can extend it to R+ by setting h(t) that Eq. (4.4) can be considered on R+ . We shall always use this extension below. Let τ be the explosion time of Eq. (4.4). For n ∈ N, define
τn := inf t 0: X (t)X > n . Then τn τ as n → ∞. We have:
(4.5)
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Lemma 4.3. For any α0 ∈ (0, α), there is an a > 0 such that for p sufficiently large p
X (t ) − X (t) Xα
sup E
sup
0
|t − t|ap
t=t ∈[0,T ∧τn ]
∈(0,1)
CN,n,T ,p,κα,n ,α0 .
Proof. Note that X (t) · 1{tτ } n
g(t)
Xα t∧τ n
Xα
A t, s, X (s)
+
Xα
ds
0 t∧τn
+
B t, s, X (s) h˙ (s)
t∧τ n √ ds + B t, s, X (s) dW (s) Xα
0
0
=: J1 (t) + J2 (t) + J3 (t) + J4 (t). By (H2) and (4.5) we have p E J2 (t) Cn E
t∧τ n
p Cn,T ,p,κα,n
κα,n (t, s) ds 0
and by Hölder’s inequality p E J3 (t) E
t∧τ n B t, s, X (s) h˙ (s)
p Xα
ds
0 t∧τ n B t, s, X (s) E
L2 (l 2 ;Xα )
· h˙ (s) 2 ds
p
l
0 t∧τ n B t, s, X (s) 2 N E L p 2
2
p 2
(l 2 ;X
α
ds )
0
CN,n,T ,p,κα,n , where we have used that h ∈ ATN . Similarly, by BDG’s inequality (2.17) and (H2) we have p E J4 (t) Cp E
t∧τ n B t, s, X (s) 2
L2 (l 2 ;Xα
0
p 2
ds )
Cn,T ,p,κα,n .
Xα
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Combining the above calculations, we get sup
p sup EX (t) · 1{tτn } X CN,n,T ,p,κα,n , α
∈(0,1) t∈[0,T ]
p 2.
(4.6)
Moreover, as in the proofs of Theorem 3.3 and Lemma 3.4, by (H1) , (H2) and (H4) , for some β3 > 1 and p 2β3∗ (β3∗ := β3 /(β3 − 1)), we have that for any 0 t < t T , p δp γp p∗ sup E X t − X (t) · 1{t ,tτn } X CT ,p,n t − t + t − t 2 + t − t 2β3 .
∈(0,1)
Thus, by (v) of Proposition 4.1 and (4.6), for any α0 ∈ (0, α) and p large enough we have p sup E X t − X (t) · 1{t ,tT ∧τn } X
∈(0,1)
α0
δp γp p 1− α0 α . CN,n,T ,p,κα,n ,α0 t − t + t − t 2 + t − t 2β ∗ The desired estimate now follows by Theorem 2.10.
2
In order to obtain the tightness of the laws of {X , ∈ (0, 1)} in CT (X), we assume that: (C1) L−1 is a compact operator on X. (C2) limn→∞ sup∈(0,1) P {ω: τn (ω) < T } = 0. Note that (C2) implies
P ω: τ (ω) > T = 1. We now prove the following key lemma for the large deviation principle of Eq. (4.3). ˜ F˜ , P˜ ) Lemma 4.4. Under (C1) and (C2), there exist subsequence k ↓ 0, a probability space (Ω, k k k h ˜ ˜ ˜ ˜ and a sequence {(h , X , W )}k∈N as well as (h, X , W ) defined on this probability space and taking values in DN × CT (X) × CT (U) such that: (i) (h˜ k , X˜ k , W˜ k ) has the same law as (hk , X k , W ) for each k ∈ N; (ii) (h˜ k , X˜ k , W˜ k ) → (h, X h , W˜ ) in DN × CT (X) × CT (U), P˜ -a.s. as k → ∞; (iii) (h, X h ) uniquely solves the following Volterra equation: t X (t) = g(t) + h
A t, s, X h (s) ds +
0
In particular, (LD)1 in Section 2.3 holds.
t 0
˙ ds. B t, s, X h (s) h(s)
(4.7)
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Proof. Let α0 ∈ (0, α) and a > 0 be as in Lemma 4.3. For R > 0, set KR := x ∈ CT (X): sup x(t)X + t∈[0,T ]
x(t) − x(s) Xα0
sup
|t − s|a
s=t∈[0,T ]
R .
By (C1), Xα0 → X is compact (cf. [28, p. 29, Theorem 1.4.8]). Thus, by Ascoli–Arzelà’s theorem (cf. [30]), the set KR is compact in CT (X). For any δ > 0, by (C2) we can choose n sufficiently large such that
sup P ω: τn (ω) < T δ.
∈(0,1)
By Lemma 4.3 and Chebyschev’s inequality, for any R > n we have
P X (·) ∈ / KR = P X (·) ∈ / KR , τn T + P X (·) ∈ / KR , τn < T
X (t) − X (s) Xα0
P sup R − n + P τn < T a |t − s| s=t∈[0,T ∧τn ] E
sup
s=t∈[0,T ∧τn ]
p
X (t) − X (s) Xα 0
|t − s|ap
/(R − n)p + δ
CN,n,T ,p,κα,n ,α0 /(R − n)p + . Therefore, for R large enough we have
sup P X (·) ∈ / KR 2δ. ∈(0,1)
Thus, by the compactness of DN (see (2.21)), the laws of (h , X , W ) in DN × CT (X) × CT (U) is tight. By Skorohod’s embedding theorem (cf. [30]), the conclusions (i) and (ii) hold. We now prove (iii). Note that by (i) (cf. [45, Section 8]) t
˜k
X (t) = g(t) + 0
+
√ k
t
A t, s, X˜ k (s) ds +
t
B t, s, X˜ k (s) h˙˜ k (s) ds
0
B t, s, X˜ k (s) dW˜ k (s)
0
=: g(t) + J1k (t) + J2k (t) + J3k (t),
P˜ -a.s.
Set
τ˜nk := inf t 0: X˜ k (t)X > n .
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Then for any δ > 0, by (i) and (C2) there exists an n large enough such that $ #
sup P˜ τ˜nk < T = sup P˜ sup X˜ k (s)X > n
k∈N
k∈N
= sup P
#
k∈N
s∈[0,T )
$ sup X k (s)X > n
s∈[0,T )
= sup P τnk < T δ. k∈N
Hence, for any δ > 0, by BDG’s inequality (2.17) and (H2) we have
P˜ J3k (t)X δ P˜ J3k (t) δ ; τ˜nk T + P˜ τ˜nk < T ˜
EP J3k (t) · 1{tτ˜nk } 2X
+δ δ 2 ˜ t∧τ˜ k k · Cn EP ( 0 n κ1,n (t, s) ds) +δ δ 2 k · Cn,t + δ. δ 2
Thus, we get
lim P˜ J3k (t)X δ = 0.
k→∞
Let Ji (t), i = 1, 2, be the corresponding terms in Eq. (4.7). In order to prove that X h solves Eq. (4.7), it is now enough to show that for any t ∈ [0, T ] and y ∈ X∗ , lim
k→∞
X
Jik (t) − Ji (t), y
X∗
= 0,
i = 1, 2, P˜ -a.s.
Observe that k X J (t) − J2 (t), y ∗ y X∗ · 2 X
t
B t, s, X˜ k (s) − B t, s, X h (s) h˙˜ k (s) ds X
0
t ˙ k h ˙ + X B t, s, X (s) h˜ (s) − h(s) , y X∗ ds 0 k k =: y X∗ · J21 (t) + J22 (t).
By the weak convergence of h˜ k to h in DN , we have k (t) = 0. lim J22
k→∞
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
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Noting that by (ii), for almost all ω˜ ∈ Ω˜ and some K(ω) ˜ ∈ N, ˜ X ∨ sup n(ω) ˜ := sup X h (s, ω) s∈[0,T ]
sup X˜ k (s, ω) ˜ X < +∞,
kK(ω) ˜ s∈[0,T ]
we have, by Hölder’s inequality and (H3)
k J21 (t, ω) ˜
h˜ k (ω) ˜ 2 ·
t
2 B t, s, X˜ k (s, ω) ˜ − B t, s, X h (s, ω) ˜
L2 (l 2 ;X)
T
1/2 ds
0
t N·
k 2 ˜ κ2,n(ω) ˜ − X h (s, ω) ˜ X ds ˜ (t, s) · X (s, ω)
1/2
0 (ii)
→ 0 as k → ∞, ˜ ∈ DN . where we have used h˜ k (ω) Similarly, we have lim J1k (t) − J1 (t)X = 0,
k→∞
P˜ -a.s.
Combining the above estimates, we find that X h solves Eq. (4.7).
2
Let I (f ) be defined by I (f ) :=
1 inf
h 22 , 2 {h∈2T : f =Xh } T
f ∈ CT (X),
(4.8)
where X h is defined by Eq. (4.7). In order to identify I (f ), we assume that: (C3) For any N ∈ N, sup
sup X h (t)X < +∞.
h∈DN t∈[0,T ]
Similar to the proof of Lemma 4.4, we can prove that: Lemma 4.5. Under (C3), (LD)2 in Section 2.3 holds. Thus, by Theorem 2.12 we have proven: Theorem 4.6. Assume that (H1) –(H2) , (H2) –(H4) and (C1)–(C3) hold. Then, {X , ∈ (0, 1)} satisfies the large deviation principle in CT (X) with the rate function I (f ) given by (4.8).
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Remark 4.7. Conditions (C2) and (C3) are satisfied if (H1) , (H2) and (H4) hold, and κ1 in (H2) belongs to K>1 . In fact, we can prove as in the proof of Theorem 3.8 sup sup E n∈N ∈(0,1)
p " X (t) CT ,p,κ , 1 X
! sup
t∈[0,T ∧τn ]
which then implies (C2). Condition (C3) is more direct in this case. 5. Semilinear stochastic evolutionary integral equations In this section, we consider the following semilinear stochastic evolutionary integral equation: t X(t) = x0 −
t a(t − s)LX(s) ds +
0
Φ s, X(s) ds +
0
t
Ψ s, X(s) dW (s),
(5.1)
0
where a : R+ → R+ is a measurable function, and Φ : R+ × Ω × X → X ∈ M × B(X)/B(X) and Ψ : R+ × Ω × X → L2 l 2 ; X ∈ M × B(X)/B L2 l 2 ; X . Here and below, M stands for the progressively measurable σ -algebra over R+ × Ω. Consider first the following deterministic integral equation: t x(t) = x0 −
a(t − s)Lx(s) ds.
(5.2)
0
The solution of this equation is called the resolvent of (a, L), and denoted by St x0 = x(t). Note that in general St+s = St ◦ Ss . We make the following assumptions: (S1) The resolvent {St : t 0} is of analyticity type (ω0 , θ0 ) in the sense of [53, Definition 2.1], where ω0 ∈ R and θ0 ∈ (0, π/2]. (S2) For any R > 0, there exist CR > 0 and β ∈ [0, 1) such that for all s > 0, ω ∈ Ω and x, y ∈ X with x X , y X R, Φ(s, ω, x) + Ψ (s, ω, x)2 L X
2 (l
and
2 ;X)
CR , (s ∧ 1)β
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1401
Φ(s, ω, x) − Φ(s, ω, y) X
CR
x − y X , (s ∧ 1)β Ψ (s, ω, x) − Ψ (s, ω, y)2 2 CR x − y 2 . X L2 (l ;X) (s ∧ 1)β (S3) For all s > 0, ω ∈ Ω and x ∈ X, it holds that Φ(s, ω, x) X
C 1 + x X , β (s ∧ 1) 2 C Ψ (s, ω, x) 2 1 + x 2X . L2 (l ;X) (s ∧ 1)β The following property of analytic resolvent {St : t > 0} is crucial for the proof of Theorem 5.2 below (cf. [53, Corollary 2.1]). Proposition 5.1. Let St be an analytic resolvent of type (ω0 , θ0 ). Then for any T > 0, sup St L(X;X) CT
(5.3)
˙ t L(X;X) CT t −1 ,
S
(5.4)
t∈[0,T ]
and for any t ∈ (0, T ],
where the dot denotes the operator derivative and · L(X;X) denotes the norm of bounded linear operators. By a solution of Eq. (5.1) we mean that X(t) satisfies the following stochastic Volterra equation: t X(t) = St x0 +
St−s Φ s, X(s) ds +
0
t
St−s Ψ s, X(s) dW (s).
(5.5)
0
Let us define A(t, s, ω, x) := St−s Φ(s, ω, x),
B(t, s, ω, x) := St−s Ψ (s, ω, x).
We have: Theorem 5.2. Under (S1) and (S2), there exists a unique maximal solution (X, τ ) for Eq. (5.5) in the sense of Definition 3.5. Moreover, if (S3) holds, then τ = +∞, a.s. Proof. First of all, it is easy to see by (5.3) that (H2) and (H3) hold with κ1,R (t, s) = κ2,R (t, s) =
CR ∈ K>1 . (s ∧ 1)β
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For 0 s < t < t , ω ∈ Ω and x ∈ X with x X R, we have A t , s, ω, x − A(t, s, ω, x) X
= (St −s − St−s )Φ(s, ω, x)X CR (s ∧ 1)β
t −s (5.4) ˙ r L(X;X) dr
S t−s
CR
St −s − St−s L(X;X) (s ∧ 1)β CR (s ∧ 1)β
t −s CR log = (s ∧ 1)β t −s
t −s
1 dr r
t−s
and B t , s, ω, x − B(t, s, ω, x)2
L2
(l 2 ;X)
CR 2 t −s log . (s ∧ 1)β t −s
Note that the following elementary inequality holds for any γ ∈ (0, 1), log(1 + s) Cs γ ,
∀s > 0.
Therefore, for 0 s < t < t , ω ∈ Ω and x ∈ X with x X R, A t , s, ω, x − A(t, s, ω, x) + B t , s, ω, x − B(t, s, ω, x)2 2 L2 (l ;X) X γ γ (t − t) CR (t − t) 1+ =: λR t , t, s . β γ γ (s ∧ 1) (t − s) (t − s) Thus, we find that (H4) holds if γ ∈ (0, (1 − β)/2). C Lastly, if (S3) is satisfied, it is clear that (H2) holds with κ1 (t, s) = (s∧1) β ∈ K>1 , and (H4) also holds from the above calculations. The non-explosion then follows from Theorem 3.8. 2 We now turn to the small perturbation of Eq. (5.5) and assume that Φ and Ψ are non-random. Consider t X (t) = St x0 +
√ St−s Φ s, X (s) ds +
0
t
St−s Ψ s, X (s) dW (s).
0
In order to use Theorem 4.6 to get the LDP for {X , ∈ (0, 1)}, we also assume: (S4) Let {St : t 0} be an analytic resolvent of type (ω0 , θ0 ). Assume that for some ω1 > ω0 , 0 < θ1 < θ0 , C > 0 and α1 > 0, −1 a(λ) ˆ C |λ − ω1 |α1 + 1 ,
∀λ ∈ C with arg(λ − ω) < θ1 ,
where aˆ denotes the Laplace transform of a. Moreover, we also assume that
(5.6)
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
r
t
a(r + s) − a(s) ds CT |r|δ ,
a(s) ds + 0
1403
(5.7)
0
where r, t ∈ [0, T ] and T , δ > 0. We have: Theorem 5.3. Under (S1)–(S4) and (C1), for any x0 ∈ D(L), {X , ∈ (0, 1)} satisfies the large deviation principle in CT (X) with the rate function I (f ) given by (4.8). Proof. From the proof of Theorem 5.2, it is enough to check (H1) and (H2) . By (5.6) and [53, p. 57, Theorem 2.2 (ii)], we have
LSt L(X;X) Ceω1 t 1 + t −α1 ,
∀t > 0,
which together with (v) of Proposition 4.1 yields that for any α ∈ (0, 1) and T > 0, α L St
L(X;X)
CT 1 + t −α1 ·α ,
∀t ∈ (0, T ].
Thus, (H2) holds by choosing α < 1−β α1 , where β is from (S3). For (H1) , since x0 ∈ D(L) = X1 , by (5.3) we have
LSt x0 X = St Lx0 X C Lx0 X . On the other hand, by the resolvent equation (5.2) and (5.7) we have, for any 0 t < t T , t
S x0 − St x0 X t
a t − s − a(t − s) · LSs x0 X ds
0
t +
a t − s · LSs x0 X ds
t
δ CT Lx0 X · t − t . The proof is thus complete by Theorem 4.6 and Remark 4.7.
2
Example 5.4. Let a be a completely monotonic kernel function, i.e., ∞ a(t) = 0
e−st dρ(s),
t > 0,
(5.8)
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∞ where s → ρ(s) is non-decreasing and such that 1 dρ(s)/s < ∞. Then the resolvent {St : t 0} associated with a is of analyticity type (0, θ ) for some θ ∈ (0, π/2) (cf. [53, p. 55, Example 2.2]), i.e., (S1) holds. For (S4), besides (5.8) and (5.7), we also assume that for some C, α1 > 0, −α1
C(1 + λ)
∞
e−λt · a(t) dt < +∞,
∀λ > 0,
(5.9)
0
which implies by [53, p. 221, Lemma 8.1(v)] that (5.6) holds. In particular, aα (t) =
t α−1 , (α)
α ∈ (0, 1],
is completely monotonic and satisfies (5.7) and (5.9), where denotes the usual Gamma function. Moreover, for the kernel function aα , if 1<α<2−
2φ < 2, π
where φ comes from (4.1), then St is analytic (cf. [53, p. 55, Example 2.1]). Notice that in [53], −L is considered. In this case, (5.6) and (5.7) clearly hold since aˆ α (λ) = λ−α , Re λ > 0. 6. Semilinear stochastic partial differential equations When a = 1 in Eq. (5.1), one sees that Eq. (5.1) contains a class of semilinear SPDEs. However, it cannot deal with the equation like stochastic Navier–Stokes equation. In this section, we shall discuss strong solutions of a large class of semilinear SPDEs by using the properties of analytic semigroups. Consider the following semilinear stochastic partial differential equation: dX(t) = −LX(t) + Φ t, X(t) dt + Ψ t, X(t) dW (t),
X(0) = x0 .
We introduce the following assumptions on the coefficients: (M1) For some α ∈ (0, 1), Φ : R+ × Ω × Xα → X ∈ M × B(Xα )/B(X) and Ψ : R+ × Ω × Xα → L2 l 2 ; X α2 ∈ M × B(Xα )/B L2 l 2 ; X α2 . (M2) For any R > 0, there exist CR > 0 and β ∈ [0, 1) with α+β <1
(6.1)
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1405
such that for all s > 0, ω ∈ Ω and x, y ∈ Xα with x Xα , y Xα R, Φ(s, ω, x) + Ψ (s, ω, x)2 L X
2 (l
2 ;X α 2
)
CR (s ∧ 1)β
and Φ(s, ω, x) − Φ(s, ω, y) X Ψ (s, ω, x) − Ψ (s, ω, y)2
L2 (l 2 ;X α )
2
CR
x − y Xα , (s ∧ 1)β CR
x − y 2Xα . (s ∧ 1)β
(M3) For all s > 0, ω ∈ Ω and x ∈ Xα , it holds that Φ(s, ω, x) X
C 1 + x Xα , β (s ∧ 1) 2 C Ψ (s, ω, x) 2 1 + x 2Xα . L2 (l ;X α ) β (s ∧ 1) 2 By a mild solution of equation (6.1), we mean that X(t) solves the following stochastic Volterra integral equation: t X(t) = Tt x0 +
Tt−s Φ s, X(s) ds +
0
t
Tt−s Ψ s, X(s) dW (s).
(6.2)
0
Theorem 6.1. Under (M1) and (M2), for any x0 ∈ Xα (α is from (M1)), there exists a unique maximal solution (X, τ ) for Eq. (6.2) so that: (i) t → X(t) ∈ Xα is continuous on [0, τ ) almost surely; (ii) limt↑τ X(t) Xα = +∞ on {ω: τ (ω) < +∞}; (iii) it holds that, P -a.s., on [0, τ ), t X(t) = Tt x0 +
Tt−s Φ s, X(s) ds +
0
t
Tt−s Ψ s, X(s) dW (s).
0
Moreover, if (M3) holds, then τ = +∞, a.s. Proof. We first consider the following stochastic Volterra integral equation t Y (t) = L Tt x0 + α
L Tt−s Φ s, L−α Y (s) ds +
t
α
0
0
Lα Tt−s Ψ s, L−α Y (s) dW (s).
(6.3)
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Define g(t) := Lα Tt x0 ,
A(t, s, ω, y) := Lα Tt−s Φ s, ω, L−α y , B(t, s, ω, y) := Lα Tt−s Ψ s, ω, L−α y . Let us verify (H1) –(H4) . Clearly, (H1) holds since x0 ∈ Xα . By (iii) of Proposition 4.1 and (M2), for all t > s > 0, ω ∈ Ω and x, y ∈ X with x X , y X R we have A(t, s, ω, x) + B(t, s, ω, x)2 L X
2 (l
2 ;X)
1 Φ s, ω, L−α x + Ψ s, ω, L−α x 2 2 α) L X (l ;X α 2 (t − s) 2 CR , (t − s)α (s ∧ 1)β
and A(t, s, ω, x) − A(t, s, ω, y) X 1 −α Φ s, ω, L x − Φ s, ω, L−α y X (t − s)α −α CR CR L x − L−α y =
x − y X , X α β α (t − s) (s ∧ 1) (t − s)α (s ∧ 1)β as well as B(t, s, ω, x) − B(t, s, ω, y)2
L2 (l 2 ;X)
1 Ψ s, ω, L−α x − Ψ s, ω, L−α y 2 2 L2 (l ;X α ) α (t − s) 2 CR
x − y 2X . (t − s)α (s ∧ 1)β Hence, if we take κ1,R (t, s) = κ2,R (t, s) :=
CR ∈ K>1 , (t − s)α (s ∧ 1)β
then (H2) and (H3) hold. Let 0 < γ < 1 − (α + β). By (iv) of Proposition 4.1 and (M2) we have A t , s, ω, x − A(t, s, ω, x) X −α α = (Tt −t − 1)L Tt−s Φ s, ω, L x X γ t − t Lα+γ Tt−s Φ s, ω, L−α x X
CR (t − t)γ (t − s)α+γ (s ∧ 1)β
(6.4)
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1407
and B t , s, ω, x − B(t, s, ω, x)2
L2 (l 2 ;X)
2 (Tt −t − 1)Lα Tt−s Ψ s, ω, L−α x L (l 2 ;X) 2 γ α+γ /2 2 t − t L Tt−s Ψ s, L−α x L (l 2 ;X) 2
(t − t)γ CR (t − t)γ L α2 Ψ s, L−α x 2 2 . L2 (l ;X) (t − s)α+γ (t − s)α+γ (s ∧ 1)β So, if we take λR t , t, s :=
CR (t − t)γ , (t − s)α+γ (s ∧ 1)β
then (H4) holds. Hence, by Theorem 3.7 there is a unique maximal solution (Y, τ ) for Eq. (6.3) in the sense of Definition 3.5. Set X(t) = L−α Y (t). It is easy to see that (X, τ ) is a unique maximal solution for Eq. (6.2), which satisfies (i), (ii) and (iii) in the theorem. Lastly, if (M3) is satisfied, then as estimating (6.4), for the above A and B, (H2) holds with some κ1 ∈ K>1 , and also (H4) holds. So, by Theorem 3.8 we have τ = ∞ a.s. 2 Remark 6.2. The solution (X, τ ) in Theorem 6.1 is clearly a local solution of Eq. (6.2) in X. However, it may be not a maximal solution in X because it may happen that
lim X(t, ω)X < +∞ on ω: τ (ω) < +∞ .
t↑τ (ω)
Next, we study the large deviation estimate for Eq. (6.1), and assume that Φ and Ψ are nonrandom. Consider the following small perturbation of Eq. (6.1): √ dX (t) = −LX (t) + Φ t, X (t) dt + Ψ t, X (t) dW (t),
X (0) = x0 .
(6.5)
In order to apply Theorem 4.6 to this situation, we need the non-explosion assumptions as (C2) and (C3). For a family of processes {h , ∈ (0, 1)} in ATN (see (2.20) for the definition of ATN ), consider t X (t) = Tt x0 +
0
+
√
t 0
Tt−s Φ s, X (s) ds +
t 0
Tt−s Ψ s, X (s) dW (s),
Tt−s Ψ s, X (s) h˙ (s) ds
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
and for h ∈ 2T (see (2.19)) t X (t) = Tt x0 + h
t
Tt−s Φ s, X h (s) ds +
0
˙ ds. Tt−s Ψ s, X h (s) h(s)
0
Below, for n ∈ N we define
τn := inf t > 0: X (t)X > n . α
Our large deviation principle can be stated as follows: Theorem 6.3. Assume (M1) and (M2). Let x0 ∈ Xδ for some 1 δ > α, where α is from (M1). We also assume that D(L) = X1 ⊂ X is compact, and lim
sup P ω: τn (ω) < T = 0
(6.6)
sup X h (t)X < +∞.
(6.7)
n→∞ ∈(0,1)
and for any N > 0, sup
h∈DN t∈[0,T ]
α
Then {X , ∈ (0, 1)} satisfies the large deviation principle in CT (Xα ) with the rate function I (f ) given by I (f ) :=
1 inf
h 22 , 2 {h∈2T : f =Xh } T
f ∈ CT (Xα ).
(6.8)
Proof. By Theorem 4.6, it only needs to check (H1) and (H2) for Eq. (6.3). Since x0 ∈ Xδ with δ > α, by (iv) of Proposition 4.1, (H1) holds with δ = δ − α and α ∈ (0, δ − α). As the calculations given in (6.4), one finds that (H2) holds with α ∈ (0, 1 − α − β). 2 Remark 6.4. If (M3) is satisfied, one can see that (6.6) and (6.7) hold by Remark 4.7. Below we study the existence of strong solutions for Eq. (6.1). For this aim, in addition to (M1) and (M2) with β = 0, we also assume: (M4) For any R, T > 0, there exist δ > 0 and α > 1 such that for all s, s ∈ [0, T ], ω ∈ Ω and x ∈ Xα with x Xα R, Φ s , ω, x − Φ(s, ω, x) CT ,R |s − s|δ , X 2 Ψ (s, ω, x) 2 CT ,R . L2 (l ;X α )
(6.9) (6.10)
2
Let us recall the following result (cf. [28, Theorem 3.2.2] or [48, p. 114, Theorem 3.5]).
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1409
Lemma 6.5. Let [0, T ] s → f (s) ∈ X be a Hölder continuous function. Then t t →
Tt−s f (s) ds ∈ C [0, T ]; X1 .
0
Using this lemma, we can prove the following result. Theorem 6.6. Assume that (M1), (M2) and (M4) hold. For any x0 ∈ X1 , let (X, τ ) be the unique maximal solution of Eq. (6.2) in Theorem 6.1. Then: (i) t → X(t) ∈ X1 is continuous on [0, τ ) a.s.; (ii) it holds that in X, t X(t) = x0 −
t LX(s) ds +
0
Φ s, X(s) ds +
0
t
Ψ s, X(s) dW (s)
0
for all t ∈ [0, τ ), P -a.s. We shall call (X, τ ) the unique maximal strong solution of Eq. (6.1). Proof. For n ∈ N, set
τn := inf t > 0: X(t)X > n α
and G(t, s) := Tt−s Ψ s, X(s) . Then by (iii) and (iv) of Proposition 4.1 we have G(t, s)2
L2 (l 2 ;X1 )
2 1 2 , Ψ s, X(s) L2 (l ;Xα /2 ) 2−α (t − s)
and in view of α > 1, G t , s − G(t, s)2 L
2
(l 2 ;X
1
)
2 (t − t)(α −1)/2 2 . )/2 Ψ s, X(s) L2 (l ;Xα /2 ) (3−α (t − s)
Hence, by Lemma 3.4 and (6.10), t t →
Tt−s Ψ s, X(s) dW (s) ∈ X1
0
admits a continuous modification on [0, τn ).
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
Moreover, starting from (6.3), as in the proof of Theorem 3.3, there exists an a > 0 such that for p sufficiently large E
X(t ) − X(t) Xα |t − t|ap t=t ∈[0,T ∧τn ] p
sup
Cn,T ,p .
Thus, by (M2) and (M4) we know that s → Φ s, X(s) ∈ X is Hölder continuous on [0, T ∧ τn ] P -a.s. Therefore, by Lemma 6.5 we have t t →
Tt−s Φ s, X(s) ds ∈ C [0, T ∧ τn ], X1 ,
P -a.s.
0
Noting that x0 ∈ X1 and t 1{tτn } · X(t) = 1{tτn } · Tt x0 + 1{tτn } ·
Tt−s Φ s, X(s) ds
0
t + 1{tτn } ·
Tt−s Ψ s, X(s) dW (s),
∀t 0, P -a.s.,
0
by τn τ , we therefore have that t → X(t) ∈ X1 is continuous on [0, τ ) P -a.s. Lastly, by stochastic Fubini’s theorem (cf. [45, Section 6]) we have t
t LX(s) ds =
0
t s LTs x0 ds +
0
LTs−r Φ r, X(r) dr ds
0 0
t
s
+
LTs−r Ψ r, X(r) dW (r) ds
0 0
t t = x0 − Tt x0 +
LTs−r Φ r, X(r) ds dr
0 r
t
t
+
LTs−r Ψ r, X(r) ds dW (r)
0 r
t = x0 − Tt x0 + 0
Φ r, X(r) − Tt−r Φ r, X(r) dr
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
t +
1411
Ψ r, X(r) − Tt−r Ψ r, X(r) dW (r)
0
t = x0 − X(t) +
t
Φ s, X(s) ds +
0
Ψ s, X(s) dW (s)
0
on {t τn }. The proof is thus complete by letting n → ∞.
2
7. Application to stochastic Navier–Stokes equations 7.1. Unique maximal strong solution for SNSEs Let O be a bounded smooth domain in Rd (d 2), or the whole space Rd , or d-dimensional torus Td . Let d Wm,p (O) := W m,p (O) ,
m,p
W0
m,p d (O) := W0 (O)
and ∞ d
C∞ 0,σ (O) := u ∈ C0 (O) : div(u) = 0 . m,p
m,p
Notice that Wm,p (Rd ) = W0 (Rd ) and Wm,p (Td ) = W0 (Td ). p p p d Let Lσ (O) be the closure of C∞ 0,σ (O) with respect to the norm in L (O) := (L (O)) . Let P2 be the orthonormal projection from L2 (O) to L2σ (O). It is well known that P2 can be exp tended to a bounded linear operator from Lp (O) to Lσ (O) (cf. [23]) so that for every u ∈ Lp (O), u = Pp u + ∇π,
p d π ∈ Lloc (O) .
The stokes operator is defined by Ap u := −Pp u,
p
D(Ap ) := H2 ∩ Lpσ (O),
(7.1)
where p
1,p
H2 := W2,p (O) ∩ W0 (O) = D(I − p ) and p is the Laplace operator on Lp (O). p It is well known that (Ap , D(Ap )) is a sectorial operator on Lσ (O) (cf. [24]). It should be d d noticed that when O = R or T , since the projection Pp can commute with ∇ (cf. [37, p. 84]), we have Ap u = −Pp u = −u,
u ∈ D(Ap ). p
That is, the stokes operator is just the restriction of −p on W2,p (O) ∩ Lσ (O), where O = Rd or Td .
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Below, we write Lp := I + Ap and α/2 Hpα := D Lp . Giga [25] proved that for α ∈ [0, 1], Hpα = Lpσ (O), D(Ap ) α = Hpα ∩ Lpσ (O), p
(7.2)
p
where Hα = [Lp (O), H2 ]α and [·,·]α stands for the complex interpolation space between two Banach spaces. In particular, the following embedding results hold (cf. [48]): for p > 1 and 0 α < 12 < α 1,
u Hp u 1,p u Hp , 2α
2α
and for q p, k −
d q
u ∈ Hpα ,
(7.3)
< 2α − pd , H2α → Wk,q (O),
p
(7.4)
Hpα → Cb (O).
(7.5)
and for α > pd ,
In what follows, we fix p > d,
1 < α < 1, 2
(7.6)
and consider the following stochastic Navier–Stokes equation with Dirichlet boundary (only for bounded smooth domain): ⎧ ⎪ ⎨ du(t) = u(t) + u(t) · ∇ u(t) + ∇π(t) dt + F t, u(t) dt + Ψ t, u(t) dW (t), u(t, ·)|∂ O = 0, div u(t) = 0, (7.7) ⎪ ⎩ u(0, x) = u0 (x), where u and π are unknown functions, and p
p
F : R+ × H2α → H0 are two measurable functions.
p
and Ψ : R+ × H2α → Hpα
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1413
Assume that: (N1) For each T , R > 0, there exist δ > 0 and CT ,R,δ > 0 such that for all t, s ∈ [0, T ] and p u, v ∈ H2α with u Hp , v Hp R, 2α
2α
F (t, u) − F (s, v)
p
H0
CT ,R,δ |t − s|δ + u − v Hp . 2α
(N2) For each T , R > 0, there exist α > 1 and CT ,R > 0 such that for all t ∈ [0, T ] and p u, v ∈ H2α with u Hp , v Hp R, 2α
2α
Ψ (t, u) − Ψ (t, v)
p
L2 (l 2 ;Hα )
CT ,R u − v Hp
2α
and Ψ (t, u)
p
L2 (l 2 ;Hα )
CT ,R .
(7.8)
Set Φ(t, u) := u + Pp (u · ∇)u + F (t, u).
(7.9)
Then Eq. (7.7) can be written as the following abstract form: du(t) = −Lp u(t) + Φ(t, u) dt + Ψ (t, u) dW (s),
u(0) = u0 . p
(7.10)
Theorem 7.1. Let p > d and 12 < α < 1. Under (N1) and (N2), for any u0 ∈ H2 , there exists a unique maximal strong solution (u, τ ) for Eq. (7.10) so that: p
(i) t → u(t) ∈ H2 is continuous on [0, τ ) a.s.; (ii) limt↑τ u(t) Hp = ∞ on {τ < +∞}; 2α
p
p
(iii) it holds that in Lσ (O) = H0 , t u(t) = u0 +
−Lp u(s) + Φ s, u(s) ds +
0
t = u0 +
t
Ψ s, u(s) dW (s)
0
Ap u(s) + Pp u(s) · ∇ u(s) ds
0
t + 0
for all t ∈ [0, τ ), P -a.s.
F s, u(s) ds +
t 0
Ψ s, u(s) dW (s),
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425 p
Proof. In view of (7.6), (7.3) and (7.5), for any u, v ∈ H2α we have Pp (u · ∇)u − (v · ∇)v
p
Lσ
(u · ∇)u − (v · ∇)vLp
u − v L∞ · ∇u Lp + v L∞ · ∇(u − v)Lp u − v Hp · u Hp + v Hp · u − v Hp . 2α
2α
2α
2α
Thus, by (N1) and (N2), it is easy to see that (M2) and (M4) hold for the above Φ and Ψ . The result now follows by Theorem 6.6. 2 7.2. Non-explosion and large deviation for 2D SNSEs In this subsection, we study the non-explosion and large deviation for SNSE in the case of two dimensions. For this aim, in addition to (N1) and (N2), we also suppose that: p
(N3) For any T > 0, there exists CT > 0 such that for all t ∈ [0, T ] and u ∈ H2 , F (t, u) F (t, u)
H20 p
H0
CT u H2 + 1 , 1 CT u Hp + 1 2α
and for i = 0, 1, Ψ (s, u)
CT 1 + u H2 , i Ψ (s, u) 2 p CT 1 + u p , H L (l ;H ) L2 (l 2 ;H2i ) 2
α
2α
where p and α satisfy (7.6). We have the following result, the proof will be given in Lemma 7.7 below. Theorem 7.2. Let p > d and 12 < α < 1. Assume that (N1)–(N3) hold. Let (u, τ ) be the unique maximal solution of Eq. (7.11) in Theorem 7.1. Then τ = +∞ a.s. We now consider the small perturbation for 2D stochastic Navier–Stokes equation: √ du (t) = −Lp u (t) + Φ t, u (t) dt + Ψ t, u (t) dW (t),
u (0) = u0
as well as the control equation: √ du (t) = −Lp u (t) + Φ t, u (t) + Ψ t, u (t) h˙ (t) dt + Ψ t, u (t) dW (t), u (0) = u0 , where h ∈ ATN (see (2.20) for the definition of ATN ), and T > 0 is fixed below.
(7.11)
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1415
Let (u , τ ) be the unique maximal strong solution of Eq. (7.11) with the properties:
lim u (t)Hp = +∞ on τ < ∞ ,
t↑τ
2α
p
and t → u (t) ∈ H2 is continuous on [0, τ ). Before proving the non-explosion result (Lemma 7.7), we first prepare a series of lemmas. Lemma 7.3. There exists a constant CT > 0 such that for any t ∈ [0, T ] and u ∈ H22 , 1 u, −L2 u + Φ(s, u) H2 − u 2H2 + CT u 2H2 + 1 , 0 2 1 0 2 4 L2 u, −L2 u + Φ(s, u) H2 C u H2 u H2 + CT 1 + u 2H2 0
0
1
(7.12) (7.13)
1
and Φ(t, u)
p
H0
CT 1 + u H2 · 1 + u Hp . 2α
1
(7.14)
Proof. Let u ∈ H22 . Noting that 1 u, P2 (u · ∇)u H2 = u, (u · ∇)u L2 = 0 2
2 u(x) · ∇ u(x) dx = 0,
O
by (N3) and Young’s inequality we have
1 u, −L2 u + Φ(s, u) H2 = − u 2H2 + u, u + F (t, u) H2 − u 2H2 + CT u 2H2 + 1 . 0 0 2 1 1 0
Thus, (7.12) is proved. For (7.13), noting that by Gagliado–Nirenberge’s inequality (cf. [22, p. 24 Theorem 9.3]) and (7.2)
u 2L∞ u H2 · u H2 u H2 · u H2 , 2
0
2
0
by Young’s inequality we have 2 1 L2 u, P2 (u · ∇)u H2 u 2H2 + P2 (u · ∇)u H2 0 0 4 2 1 2 u 2H2 + C (u · ∇)uL2 4 2 1 u 2H2 + C u 2L∞ · ∇u 2L2 4 2 1 u 2H2 + C u H2 · u H2 · u 2H2 0 2 4 2 1 1 u 2H2 + C u 2H2 · u 4H2 , 2 2 0 1
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
and by (N3)
1 L2 u, F (s, u) H2 u 2H2 + CT 1 + u 2H2 . 0 2 2 1
Thus, (7.13) holds. Let d
p
1+
d p
− 2α
q∗ =
,
qp . q −p
By Hölder’s inequality we have Pp (u · ∇)u
(7.4)
p
H0
u · ∇u Lp u Lq ∗ · ∇u Lq u H2 · u Hp . 2α
1
2
Estimate (7.14) now follows by (N3). Below, set for n ∈ N,
τn := inf t 0: u (t)Hp > n . 2α
Lemma 7.4. There exists a constant CT > 0 such that for all ∈ (0, 1) and n ∈ N,
E
! sup
2 " u (s) 2 + E H0
s∈[0,T ∧τn ]
T∧τn 2 u (s) 2 ds CT . H1
0
Proof. By Itô’s formula we have 2 u (t) 2 = u0 2 2 + 2 H H 0
t
u (s), −L2 u (s) + Φ s, u (s) H2 ds 0
0
0
t +2
u (s), Ψ s, u (s) h˙ (s) H2 ds 0
0
√ +2
t
u (s), Ψk s, u (s) H2 dW k (s) 0
k 0
Ψk s, u (s) 2 2 ds t
+
k 0
H0
=: u0 2H2 + J1 (t) + J2 (t) + J3 (t) + J4 (t). 1
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1417
Set f (t) := E
! sup
2 " u (s) 2 . H0
s∈[0,t∧τn ]
First of all, noting that by (7.12) t J1 (t) −
2 u (s) 2 + CT
t
H1
0
2 u (s) 2 + 1 ds, H0
0
we have
E
! sup
s∈[0,t∧τn ]
t∧τ n t " 2 f (s) + 1 ds. J1 (s) + E u (s) H2 ds CT 1
0
0
By (N3) and Young’s inequality we have
E
! sup
s∈[0,t∧τn ]
t∧τ n " u (s) 2 · Ψ s, u (s) J2 (s) 2E H L
2 (l
0
2 ;H2 ) 0
· h˙ (s) 2 ds l
0 t∧τ n 2 2 u (s) 2 · Ψ s, u (s) 2N E
L2 (l 2 ;H20 )
H0
0
1 f (t) + CN E 4
t∧τ n 2 1 + u (s)H2 ds 0
0
1 f (t) + CN 4
t
1 + f (s) ds.
0
Similarly, we also have t " 1 E sup J3 (s) f (t) + C 1 + f (s) ds 4 s∈[0,t∧τn ] !
0
and E
! sup
s∈[0,t∧τn ]
t " 1 + f (s) ds. J4 (s) C 0
1/2 ds
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
Combining the above calculations we get t∧τ n
2 u (s) 2 ds 2 u0 2 2 + CN + CN H H
f (t) + 2E
1
t
0
0
1 + f (s) ds.
0
2
The desired estimate follows by Gronwall’s inequality. Set for n ∈ N, t∧τ n
2 2 u (s) 2 · u (s) 2 ds + t
ηn (t) :=
H1
H0
0
t =
2 2 u (s) 2 · u (s) 2 · 1[0,τ ] (s) ds + t H1
H0
n
0
and
θn (t) := inf s 0: ηn (s) t . Clearly, t → ηn (t) is a continuous and strictly increasing function, and the inverse function of t → θn (t) is just given by ηn . Moreover, since ηn (t) > t, we have θn (t) < t. Lemma 7.5. For any K > 0, there exists a constant CK,N > 0 such that for all ∈ (0, 1) and n ∈ N, E
!
2 " u (s) 2 CK,N .
sup
s∈[0,θn (K)∧τn ]
H1
Proof. Consider the following evolution triple H22 ⊂ H21 ⊂ H20 . By Itô’s formula (cf. [58]), we have 2 u (t) 2 = u0 2 2 + 2 H H 1
t
L2 u (s), −L2 u (s) + Φ s, u (s) H2 ds 0
1
0
t +2
L2 u (s), Ψ s, u (s) h˙ (s) H2 ds 0
0
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
√ +2 k
t
1419
u (s), Ψk s, u (s) H2 dW k (s) 1
0
Ψk s, u (s) 2 2 ds + t
H1
k
0
=: u0 2H2 + J1 (t) + J2 (t) + J3 (t) + J4 (t). 1
Set f (t) := E
! 2 " sup u θn (s) ∧ τn H2 = E
!
1
s∈[0,t]
sup
2 " u (s) 2 .
s∈[0,θn (t)∧τn ]
H1
For J1 (t), by (7.13) we have, for t ∈ [0, K], J1 θn (t) ∧ τn
θn (t)∧τn
2 2 4 C u (s)H2 · u (s)H2 + CK 1 + u (s)H2 ds 0
1
1
0 θn (t)
u s ∧ τ 2 2 dη (s) + CK
C
n
n
H1
0
t =C
u θ (s) ∧ τ 2 2 ds + CK , n
n
H1
0
where the last step is due to the substitution of variable formula. So,
E
!
t " sup J1 θn (s) ∧ τn C f (s) ds + CK .
s∈[0,t]
0
Using the same trick as used in Lemma 7.4 and by (N3), we also have t " 1 E sup Ji θn (s) ∧ τn f (t) + CN,K f (s) + 1 ds, 2 s∈[0,t] !
0
Thus, we get t f (t) 2 u0 2H2 1
+ CN,K
f (s) + 1 ds,
0
which yields the desired estimate by Gronwall’s inequality.
2
i = 2, 3, 4.
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X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
Set for M > 0,
ζn (M) := inf t 0: u t ∧ τn H2 M . 1
Lemma 7.6. For any M > 0 and q 2, there exists a constant CT ,M,N > 0 such that for all ∈ (0, 1) and n ∈ N, E
q & u (t) p CT ,M,N . H
% sup
t∈[0,T ∧τn ∧ζn (M)]
2α
Proof. Set for t ∈ [0, T ], ξn (t) := t ∧ τn ∧ ζn (M) and for q 2, f (t) := E
% sup
q & u (t) p . H
t ∈[0,ξn (t)]
2α
Note that t u (t) = Tt u0 +
Tt−s Φ s, u (s) ds +
0
+
√
t
t
Tt−s Ψ s, u (s) h˙ (s) ds
0
Tt−s Ψ s, u (s) dW (s).
0
By (iii) of Proposition 4.1, Hölder’s inequality and Lemma 7.14, we have, for q >
1 1−α ,
t q E sup Tt −s Φ s, u (s) ds p t ∈[0,ξn (t)]
H2α
0
E
t sup
t ∈[0,ξn (t)]
0
1 Φ s, u (s) p ds H0 (t − s)α
q
ξn (t) q Φ s, u (s) p ds E H0
0
ξn (t) t q q 1 + u (s)H2 · 1 + u (s)Hp ds CM f (s) + 1 ds. E
(7.14)
1
2α
0
0
On the other hand, set G(t, s) := Tt−s Ψ s, u (s) .
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1421
Then by (iii) and (iv) of Proposition 4.1, we have G(t, s)2 p H
C Ψ s, u (s) 2 2 p L2 (l ;Hα ) α (t − s)
2α
and for γ ∈ (0, (1 − α)/2), G t , s − G(t, s)2 p H 2α
|t − t|γ Ψ s, u (s) 2 2 p . L2 (l ;Hα ) (t − s)α+2γ
Therefore, using Lemma 3.4 for q large enough, we get t q E sup G t , s dW (s) p t ∈[0,T ∧ξn (t)]
H2α
0
T ∧ξ n (t) q Ψ s, u (s) CT E
p
L2 (l 2 ;Hα )
ds
0
t
(N3)
CT
f (s) + 1 ds.
0
Similarly, we have t q E sup Tt−s Ψ s, u (s) h˙ (s) ds p t ∈[0,T ∧ξn (t)]
H2α
0
t CT ,N
f (s) + 1 ds.
0
Combining the above calculations, we obtain t f (t) CT ,M,N
f (s) ds + CT ,M,N , 0
which yields the desired estimate by Gronwall’s inequality.
2
Lemma 7.7. It holds that lim
sup P ω: τn (ω) T = 0.
n→∞ ∈(0,1)
(7.15)
1422
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
Proof. First of all, for any M, K > 0 we have
P ζn (M) < T P ζn (M) < T ; θn (K) T + P θn (K) < T # $ = P sup u t ∧ τn H2 > M; θn (K) T 1
t∈[0,T )
+P P E
sup ηn (s) > K
$
s∈[0,T )
# !
#
$
> M + P ηn (T ) > K
sup
u (t)
sup
2 " 2 u (t) 2 /M + E η (T ) /K.
t∈[0,θn (K)∧τn ]
t∈[0,θn (K)∧τn )
H21
n
H1
Hence, by Lemmas 7.4 and 7.5 we have
lim sup P ζn (M) < T = 0.
M→∞ n,
Secondly, we also have
P τn < T P τn < T ; ζn (M) T + P ζn (M) < T .
(7.16)
For the first term, by Lemma 7.6 we have # $
P τn < T ; ζn (M) T = P sup u (t)Hp > n; ζn (M) T P P E
# #
t∈[0,T )
2α
sup
u (t)
t∈[0,T ∧τn ]
sup
!
p H2α
n; ζn (M) T
u (t)
s∈[0,T ∧ζn (M)∧τn ]
p
H2α
n
$
$
q " q CT ,M,N u (t) p /n , H2α nq s∈[0,T ∧ζn (M)∧τn ] sup
where CT ,M,N is independent of and n. The desired limit now follows by taking limits for (7.16), first n → ∞, then M → ∞. 2 Thus, using Theorem 6.3 we get: p
Theorem 7.8. Let O = T2 or a bounded smooth domain in R2 . Under (N1)–(N3), for u0 ∈ H2 , p {u , ∈ (0, 1)} satisfies the large deviation principle in CT (H2α ) with the rate function I (f ) given by I (f ) :=
1 inf
h 22 , 2 {h∈2T : f =uh } T
p f ∈ CT H2α ,
X. Zhang / Journal of Functional Analysis 258 (2010) 1361–1425
1423
where uh solves the following equation: t uh (t) = u0 + 0
t + 0
t uh (s) ds +
Pp uh (s) · ∇ uh (s) ds
0
F s, uh (s) ds +
t
˙ ds. Ψ s, uh (s) h(s)
0
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