Modern Trends in Pseudo-Differential Operators

Operator Theory: Advances and Applications Vol. 172 Editor: I. Gohberg Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Böttcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. E. Curto (Iowa City) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) B. Gramsch (Mainz) J. A. Helton (La Jolla) M. A. Kaashoek (Amsterdam) H. G. Kaper (Argonne) Subseries: Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze Universität Potsdam Germany Sergio Albeverio Universität Bonn Germany

S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. Olshevsky (Storrs) M. Putinar (Santa Barbara) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) P. R. Halmos (Santa Clara) T. Kailath (Stanford) H. Langer (Vienna) P. D. Lax (New York) M. S. Livsic (Beer Sheva) H. Widom (Santa Cruz) Michael Demuth Technische Universität Clausthal Germany

Modern Trends in Pseudo-Differential Operators

Joachim Toft M.W. Wong Hongmei Zhu Editors

Advances in Partial Differential Equations

Birkhäuser Basel . Boston . Berlin

Editors: Joachim Toft School of Mathematics and Systems Engineering Växjö University SE-351 95 Växjö Sweden e-mail: [email protected]

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Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

C.-I. Martin and B.-W. Schulze The Quantization of Edge Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

J.B. Gil, T. Krainer and G.A. Mendoza On Rays of Minimal Growth for Elliptic Cone Operators . . . . . . . . . . . . .

33

C. Iwasaki Symbolic Calculus of Pseudo-differential Operators and Curvature of Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

M.W. Wong Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups . . . . . . . . . . . . . . . .

67

M. Ruzhansky and V. Turunen On the Fourier Analysis of Operators on the Torus . . . . . . . . . . . . . . . . . . .

87

J. Tie and M.W. Wong Wave Kernels of the Twisted Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 E. Buzano Super-exponential Decay of Solutions to Differential Equations in Rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 A. Oliaro and P. Popivanov Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 M. Sugimoto A New Aspect of the Lp -extension Problem for Inhomogeneous Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 G. Garello and A. Morando Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols . . . . . . . . . . . . . . . . 161 J. Toft Continuity and Schatten Properties for Pseudo-differential Operators on Modulation Spaces . . . . . . . . . . . . . . 173

vi

Contents

Yu.I. Karlovich Algebras of Pseudo-differential Operators with Discontinuous Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 P. Boggiatto, G. De Donno and A. Oliaro A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 235 M.W. Wong and H. Zhu A Characterization of Stockwell Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251

V.S. Rabinovich and S. Roch Exact and Numerical Inversion of Pseudo-differential Operators and Applications to Signal Processing . . . . . . . . . . . . . . . . . . . . . 259 E. Cordero and K. Gr¨ ochenig On the Product of Localization Operators . . . . . . . . . . . . . . . . . . . . . . . . . . .

279

M. Cappiello, T. Gramchev and L. Rodino Gelfand–Shilov Spaces, Pseudo-differential Operators and Localization Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297

J. Toft Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313

R.D. Carmichael, A. Eida and S. Pilipovi´c Microlocalization within Some Classes of Fourier Hyperfunctions . . . . . 329

Preface The ISAAC Group in Pseudo-differential Operators (IGPDO) was formed at the Fourth ISAAC Congress held at York University in Toronto in 2003 and the first volume entitled Advances in Pseudo-differential Operators and devoted to papers focussing on pseudo-differential operators and its diverse applications was then initiated and published in Professor Israel Gohberg’s series Operator Theory: Advances and Applications in 2004. As a satellite conference to the Fourth Congress of European Mathematics held at Stockholm University in 2004, the International Conference on Pseudo-differential Operators and Related Topics was held at V¨axj¨ o University in Sweden. Prompted by the enthusiasm of the participants, the second volume with similar scope and entitled Pseudo-differential Operators and Related Topics was published in the same series in 2006. Members of IGPDO met again at the Fifth ISAAC Congress held at Universit`a di Catania in Italy in July 2005. Core members of the group encouraged the publication of a sequel to the Toronto Volume and the V¨ axj¨ o Volume. The vision is to seek new directions for the broad subject on pseudo-differential operators and the strategy is to devote the Catania Volume not only to papers based on lectures given at the special session on pseudo-differential operators, but also invited papers that bear on the themes of IGPDO. In order to reflect the goal and vision of IGPDO, the Catania Volume is entitled Modern Trends in Pseudo-differential Operators. This Catania Volume consists of nineteen peer-reviewed papers representing modern trends in pseudo-differential operators. Topics include partial differential equations, global analysis, geometry, quantization, Wigner transforms, Weyl transforms on Lie groups, mathematical physics and time-frequency analysis. The nineteen articles in this volume should be of great interest to graduate students and researchers in analysis, mathematical physics and mathematical sciences. The sustained growth of the subject on pseudo-differential operators can be testified by the much wider scope and the significantly greater number of contributors in this Catania Volume.

Operator Theory: Advances and Applications, Vol. 172, 1–31 c 2006 Birkh¨
auser Verlag Basel/Switzerland

The Quantization of Edge Symbols C.-I. Martin and B.-W. Schulze Abstract. We investigate operators on manifolds with edges from the point of view of the symbolic calculus induced by the singularities. We discuss new aspects of the quantization of edge-degenerate symbols which lead to continuous operators in weighted edge spaces. Mathematics Subject Classification (2000). 35S15, 35J70, 35S05, 58J40. Keywords. Operators on manifolds with edges, quantization of edge-degenerate symbols, composition in the edge algebra.

Introduction This paper is aimed at studying a number of new properties of edge amplitude functions belonging to the calculus of operators on a manifold with edges. (The notation ‘manifold’ is used here for convenience; in general, our spaces are manifolds only outside their subset of singularities). The investigation is motivated by the program of establishing operator algebras on configurations with higher singularities, e.g., when the cones of local wedges have cross sections with singularities, based on an iterative definition of such ‘corner manifolds’, cf. [2], [3]. The hope is that the ideas of the cone and edge pseudo-differential calculus with smooth model cones, cf. [13], [15], are more or less iterative. However, the symbolic structures in the higher floors of the calculus are by no means straightforward. Recall that already for manifolds with smooth edges there are different ways of constructing quantizations of corresponding edge-degenerate symbols; they reflect different aspects of the calculus. Compared with [14] an alternative quantization is given in [5]; remainders belong to the class of flat Green edge symbols. The result of [5] is applied in [11] and [6] to the construction of holomorphic representatives of corner Mellin symbols, given in terms of parameter-dependent families of edge operators of first generation (i.e., for smooth edges).

2

C.-I. Martin and B.-W. Schulze

Here we study edge symbols in another way, namely, with an additional localisation near the diagonal with respect to the cone axis variable at infinity, depending on the edge covariable. In higher corner algebras localisations of a similar kind seem to be necessary, cf. [1], [3]. The (operator-valued) symbols of the edge algebra from [12] or [15] are specific families of pseudo-differential operators on infinite cones X ∧ := R+ × X
(r, x) with r → 0 representing the tip and r → ∞ a conical exit to infinity, where (in the simplest case) the cross section X is a closed compact C ∞ manifold. These symbols may be interpreted as a specific parameter-dependent quantization of edge-degenerate scalar symbols (or, alternatively, of an edge-degenerate family of classical pseudo-differential operators on X). Quantizations are usually not canonical, even in simpler situations. The choice of the edge quantization in the above mentioned expositions (which we briefly recall below) gives rise to an algebra of continuous operators in weighted edge spaces, with all the expected features, such as ellipticity with respect to a principal symbolic hierarchy and existence of a parametrix within the calculus. However, some aspects of the theory remained complicated, for instance, the rigorous proof of the composition behavior and other necessary elements. For that reason the authors of [5] proposed another edge quantization which entails the composition result in a very simple way. It would be desirable to apply a similar strategy also for manifolds with singularities of higher order. Unfortunately, the attempt to generalise [5], for instance, for singularities of order 2 (i.e., when the base X of the model cone itself has singularities of conical or edge type) leads to an enormous blow-up of technicalities, and the construction of corresponding alternative quantizations along the lines of [5] compared with the ‘usual’ ones seems to be more complicated than a direct approach. Moreover, the analogue of the usual edge quantization for the edge calculus of second generation as studied in [1] gives rise to difficulties; so there was to be invented a modification. Since this is new in an analogous form also in the edge calculus of first generation, we discuss here such an edge quantization and characterise the remainders compared with the former one. This is one of the points of the present paper, and we prove that the remainders are again flat Green symbols. Moreover, we give a new relatively elementary proof for the composition theorem of edge symbols of first generation. Our considerations are also motivated by the expectation that analogous conclusions simplify and unify the structure of operator algebras on manifolds with higher corners, according to the program of successive conifications and edgifications of the operator algebras, opened in [16] and continued in [17], [2], [3]. By a manifold with corners we understand a topological space M that contains a subspace M  of singularities such that M \ M  is a C ∞ manifold, and near every m ∈ M  the space M is modelled on a cone X ∆ := (R+ × X)/({0} × X) or a wedge X ∆ × Ω, for an open set Ω ⊆ Rq , where X is again a manifold with corners (of ‘lower order’ than M ). In addition we require specific properties of the transition maps belonging to different such local representations, cf. [2].

The Quantization of Edge Symbols

3

Examples are manifolds with conical singularities or edges; in this case we assume the base spaces X of the model cones to be closed, compact C ∞ manifolds. Manifolds with higher singularities can be obtained by iteratively forming cones or wedges and pasting them together to corresponding global spaces. In order to illustrate the nature of operator-valued amplitude functions on a singular manifold we first consider the case of differential operators on a manifold B with conical singularities. Assume, for convenience, that B has one conical point v, and let B denote the associated stretched manifold which is a C ∞ manifold with compact boundary ∂B ∼ = X and int B ∼ = B \ {v}. Then B is equal to the quotient space B/∂B (with ∂B collapsed to the point v), and B is locally near ∂B identified with R+ × X. A differential operator A with smooth coefficients on intB is said to be of Fuchs type, if locally near ∂B in the splitting of variables (r, x) ∈ R+ × X the operator has the form µ 
∂ j A = r−µ aj (r) − r (0.1) ∂r j=0 with coefficients aj ∈ C ∞ (R+ , Diff µ−j (X)) (here Diff ν (·) denotes the space of all differential operators of order ν with smooth coefficients on the C ∞ manifold in the parentheses). In this case the principal symbolic structure consists of a pair σ(A) = (σψ (A), σc (A)), where σψ (A) is the standard homogeneous principal symbol of order µ and σc (A)(z) :=

µ


aj (0)z j

(0.2)

j=0

the principal conormal symbol. The function (0.2) is operator-valued and acts between the standard Sobolev spaces on X as a family of continuous operators σc (A)(z) : H s (X) → H s−µ (X) depending on the variable z ∈ C. Another example is the case of a C ∞ manifold with boundary, locally near the boundary modelled on R+ × Ω
(r, y), Ω ⊆ Rq open. Then, if A is a differential operator of order µ with smooth coefficients up to the boundary, the principal symbolic hierarchy of A is a pair σ(A) = (σψ (A), σ∂ (A)). The first component is again the standard homogeneous principal symbol of order µ. Moreover, writing A near the boundary as
α A= aα (r, y)D(r,y) , (0.3) |α|≤µ

4

C.-I. Martin and B.-W. Schulze

we set σ∂ (A)(y, η) :=




aα (0, y)Drk η γ ,

(0.4)

|α|=µ α=(k,γ)

(y, η) ∈ T ∗ Ω \ 0, called the homogeneous principal boundary symbol of A. The symbol (0.4) is operator-valued and acts between the standard Sobolev spaces on R+ as a family of continuous operators σ∂ (A)(y, η) : H s (R+ ) → H s−µ (R+ ), (0.5)  depending on (y, η) ∈ T ∗ Ω \ 0; here H s (R+ ) := H s (R)R+ . A third category of examples are operators on manifolds with edges (cf. also Section 1 below). Manifolds with (smooth) edges can be regarded as a generalisation of manifolds with (smooth) boundary. In this case the local model near the edge is equal to X ∆ × Ω for a closed compact C ∞ manifold X. A manifold with boundary just corresponds to the case dim X = 0. We often employ the stretched manifolds rather than the manifolds with singularities themselves. In the case of a wedge X ∆ × Ω the associated stretched space is equal to R+ × X × Ω. In the corresponding splitting of variables (r, x, y) the typical differential operators A are assumed to be of the form 
∂ j A = r−µ ajα (r, y) − r (rDy )α , (0.6) ∂r j+|α|≤µ

∞

with coefficients ajα ∈ C (R+ × Ω, Diff µ−(j+|α|) (X)). Such operators will also be called edge-degenerate. Their principal symbolic hierarchy is a pair σ(A) = (σψ (A), σ∧ (A))

(0.7)

with σψ (A) being the homogeneous principal symbol of A of order µ in the usual sense, and 
∂ j σ∧ (A)(y, η) := r−µ ajα (0, y) − r (rη)α (0.8) ∂r j+|α|≤µ

the so-called homogeneous principal edge symbol. It represents a family of continuous operators σ∧ (A)(y, η) : Ks,γ (X ∧ ) → Ks−µ,γ−µ (X ∧ ), (0.9) (y, η) ∈ T ∗ Ω \ 0, acting between weighted Sobolev spaces on the (open infinite stretched) cone X ∧ := R+ × X, cf. Section 1.3 below. Comparing (0.8) with (0.1) we see that there is a family of subordinate conormal symbols σc σ∧ (A)(y, z) =

µ


aj0 (0, y)z j

(0.10)

j=0

from the interpretation of (0.9) as an operator of Fuchs type for every fixed (y, η) ∈ T ∗ Ω \ 0. In the case of ellipticity the adequate weights γ ∈ R for (0.9) are to be fixed in connection with the non-bijectivity points of (0.10) in the complex plane.

The Quantization of Edge Symbols

5

The difference between the notation ‘boundary’ and ‘edge’ symbol is motivated by the fact that the choice of spaces is different. If we write the operator (0.3) in edge-degenerate form (0.6) (with coefficients ajα (r, y) ∈ C ∞ (R+ × Ω); in this case we have dim X = 0) we also obtain an edge symbol (0.9). In the present paper we concentrate on edge-degenerate operators (0.6) and their pseudo-differential versions. The analogy with boundary value problems will not play a major role, but it is worth to recall that the edge calculus for dim X = 0 corresponds to the pseudo-differential calculus of boundary value problems without the transmission property at the boundary, cf. [18].

1. Operators on manifolds with edges 1.1. Manifolds with conical singularities and edges Let X be a topological space. As in the introduction we set X ∧ := (R+ × X)/({0} × X) and X ∧ := R+ × X. In the following discussion, for convenience, topological spaces are assumed to be countable unions of compact sets; C ∞ manifolds are assumed to be oriented and equipped with a Riemannian metric. The definition of a manifold W with edge Y is based on a certain kind of (locally trivial) X ∆ -bundles L over Y ; here Y and X are C ∞ manifold. In order to describe the specific structure we first consider a (locally trivial) R+ × X-bundle L over Y , the stretched space associated with L. The transition maps  l : R+ × X × Ω → R+ × X × Ω  ⊆ Rq open, q = dim Y , are assumed to be between trivialisations of L, Ω, Ω  belonging to an R × Xrestrictions of transition maps R × X × Ω → R × X × Ω bundle 2L over Y to R+ × X × Ω. Then L contains a subspace Lsing which is an X-bundle over Y , represented by the trivialisations {0} × X × Ω. Let us set Lreg := L \ Lsing ; this is an X ∧ -bundle over Y with the trivialisations X ∧ × Ω. From the construction of L and Lsing it follows that we obtain an X ∆ -bundle L by passing to the quotient space R+ × X → X ∆ = (R+ × X)/({0} × X) in every fibre of L. The bundle L contains Y as a subspace, interpreted as an edge, and L \ Y is a C ∞ manifold which can be identified with Lreg in a natural way. More precisely, we have a projection π : L → L, (1.1) fibrewise defined by R+ × X → X ∆ , and π restricts to the bundle projection πsing : Lsing → Y

(1.2)

and to an isomorphism of X ∧ -bundles πreg : Lreg → L \ Y.

(1.3)

6

C.-I. Martin and B.-W. Schulze

Definition 1.1. Let W be a topological space and Y ⊂ W a subspace. Then W is called a manifold with edge Y , if (i) W \ Y and Y are C ∞ manifolds; (ii) there exists a neighborhood V of Y in W and a homeomorphism χ:V →L to an X ∆ -bundle L over Y for a C ∞ manifold X, such that χ restricts to diffeomorphisms χ0 : V ∩ Y → Y,

χreg : V \ Y → L \ Y.

Incidentally we will write

(1.4) W = Y From W we can pass to a so-called stretched manifold W when we first replace V by the stretched set V that is defined to be a C ∞ manifold with boundary, diffeomorphic to L, such that V \ ∂V is identified with V \ Y and ∂V isomorphic to Lsing as an X-bundle over Y . In other words, W is a C ∞ manifold with boundary ∂W =: Wsing which is an X-bundle over Y . Let us set W\∂W =: Wreg . We then have a canonical continuous map π:W→W which restricts to the bundle projection πsing : Wsing → Y and to a diffeomorphism πreg : Wreg → W \ Y. With W we can associate the double 2W which is a C ∞ manifold (without boundary) by gluing together two copies W± of W along the common boundary ∂W. Example. For W = X ∆ × Ω we have W = R+ × X × Ω and 2W = R × X × Ω. Remark 1.2. In the case dim Y = 0 we speak about manifolds with conical singularities. Remark 1.3. Let W be a manifold with edge Y . Then for every C ∞ manifold M the Cartesian product W × M is a manifold with edge Y × M . Definition 1.4. Let Wi be manifolds with edges Yi , i = 1, 2, and let Xi be the base of the model cone for Wi , i = 1, 2. A continuous map T : W1 → W2 is called an M1 -morphism, if there is a differentiable map T : W1 → W2 between the respective stretched manifolds as manifolds with C ∞ boundary, such that  T∂W : ∂W1 → ∂W2 1

The Quantization of Edge Symbols

7

 is a homomorphism between the corresponding Xi -bundles (in particular, T Y1 : Y1 → Y2 is then a differentiable map). T : W1 → W2 is called an M1 -isomorphism if there is an M1 -morphism T −1 : W2 → W1 which is a two-sided inverse to T . In this case we also write W1 ∼ =M1 W2 . In this way, the manifolds with edges form a category M1 with the subcategory of manifolds with conical singularities. Let W be a manifold with edge Y of dimension q > 0. Then the above mentioned bijection V → L allows us to define stretched wedge neighborhoods ∼ =

U ⊂ V that correspond to a trivialisation of L, i.e., U∼ =M1 R+ × X × Ω

(1.5)

for open sets Ω ⊆ Rq . Set Ureg := U \ ∂W. Let W be a manifold with edge Y and W its stretched manifold. By Diff µdeg (W ) we denote the space of all differential operators A ∈ Diff µ (W \ Y ) such that for every (stretched) wedge neighborhood Ureg in the splitting of variables (r, x, y) ∈ R+ × X × Ω the operator A has the form 
∂ j A = r−µ ajα (r, y) − r (rDy )α (1.6) ∂r j+|α|≤µ

with coefficients ajα ∈ C ∞ (R+ × Ω, Diff µ−(j+|α|) (X)). In local coordinates (r, x, y) ∈ R+ ×Σ×Ω, Σ ⊆ Rn , Ω ⊆ Rq open (n = dim X, q = dim Y ) the homogeneous principal symbol σψ (A) of A of order µ has the form σψ (A)(r, x, y, , ξ, η) = r−µ σ ˜ψ (A)(r, x, y, r, ξ, rη) for a function σ ˜ψ (A)(r, x, y, ˜, ξ, η˜) which is smooth up to r = 0 (and homogeneous of order µ in (˜ , ξ, η˜)). In addition, as noted in the introduction, we have the homogeneous principal edge symbol σ∧ (A)(y, η), (y, η) ∈ T ∗ Y \ 0, the second component of the principal symbolic hierarchy (0.7). Note that for a(y, η) := r−µ


j+|α|≤µ

and (κλ u)(r, x) := λ

n+1 2

 ∂ j ajα (r, y) −r (rη)α , ∂r

u(λr, x), λ ∈ R+ , we have

σ∧ (A)(y, η) = lim λ−µ κλ a(y, λη)κ−1 λ , λ→∞

and σ∧ (A)(y, λη) = λµ κλ σ∧ (A)(y, η)κ−1 λ , for all λ ∈ R+ .

(1.7)

8

C.-I. Martin and B.-W. Schulze

1.2. Weighted Sobolev spaces We now formulate Sobolev spaces on stretched cones X ∧ = R+ × X
(r, x) and wedges X ∧ × Rq
(r, x, y), first for the case of a smooth compact manifold X. To this end we first consider the cylindrical Sobolev space H s (R × X) s defined as the set of all u(t, .) ∈ Hloc (Rt × X) such that

(ϕu) ◦ (1 × χ)−1 ∈ H s (Rt × Rn ) for every chart χ : U → Rn on X and every ϕ ∈ C0∞ (U ); here (1 × χ)(r, ·) := (r, χ(·)). Let 1 (Sβ u)(t) := e−( 2 −β)t u(e−t ), t ∈ R for any β ∈ R. Then we set

−1 s  H (R × X) Hs,γ (X ∧ ) := Sγ− n2

for n = dim X. Let us interpret these spaces in connection with the Mellin transform on R+
r,  ∞ rz−1 u(r)dr. M u(z) = 0  For u ∈ C0∞ (R+ ) the function, M u(z) is an entire function, and we have M uΓ ∈ β S(Γβ ) for (1.8) Γβ := {z ∈ C : Rez = β} for every β ∈ R, uniformly in compact intervals. Here and in the sequel the ‘weight line’ Γβ is treated as a real axis in connection with the spaces, e.g., the Schwartz space, Sobolev spaces, etc., or amplitude functions, given with respect to the corresponding covariable. The Mellin transform will also be applied to vector- or operator-valued functions depending on r ∈ R; then the covariable z will often vary on a certain weight ∂ line. We have −r ∂r = M −1 zM , and Hs,γ (X ∧ ) for s ∈ N is equal to the subspace n −n +γ of all u(r, x) ∈ r 2 L2 (X ∧ ) such that (−r∂r )k Dxα u(r, x) ∈ r− 2 +γ L2 (X ∧ ) for all |α| k + |α| ≤ s. Here Dxα runs over all elements of Diff (X), and L2 (X ∧ ) refers to the measure drdx. From this definition we can recover Hs,γ (X ∧ ) for arbitrary s ∈ R by duality and interpolation. It will be adequate to modify the spaces Hs,γ (X ∧ ) at infinity by setting s Ks,γ (X ∧ ) := {ωu + (1 − ω)v : u ∈ Hs,γ (X ∧ ) v ∈ Hcone (X ∧ )} s for a space Hcone (X ∧ ) that is defined as follows. Set B := {x ∈ Rn : |x| < 1} and B ∨ := {(r, rx) ∈ R1+n : (r, x) ∈ R+ × B} which is a conical set in R1+n . Let χ : U → B be a chart on X, and consider 1 × χ : R+ × U → R+ × B. Together with

β : (r, x) → (r, rx),

R+ × B → B ∨ ,

The Quantization of Edge Symbols

9

we have the composition β ◦ (1 × χ) : R+ × U → B ∨ . s s Then Hcone (X ∧ ) is defined to be the subspace of all u ∈ Hloc (R × X)|R+ ×X such −1 −1 s 1+n ∈ H (R ) for every chart χ : U → B and that (1 − ω)ϕu ◦ (1 × χ) ◦ β s arbitrary ϕ ∈ C0∞ (U ). Another equivalent characterization of Hcone (X ∧ ) is given in Remark 2.1 below. Let us introduce some convenient terminology in connection with the variety of spaces that will occur in our calculus. If E0 , E1 are Fr´echet spaces, embedded in a Hausdorff topological vector space H, we set E0 + E1 = {e0 + e1 : e0 ∈ E0 , e1 ∈ E1 }. (1.9) ∼ There is then an algebraic isomorphism E0 + E1 = E0 ⊕ E1 /∆ for ∆ := {(e, −e) : e ∈ E0 ∩ E1 }, and we endow (1.9) with the Fr´echet topology of the quotient space. We then call (1.9) the non-direct sum of E0 and E1 . Moreover, let E be a Fr´echet space that is a left module over an algebra A. Then [a]E for a ∈ A will denote the closure of the space {ae : e ∈ E} in E. In a similar sense we employ the notation E[b] or [a]E[b] for a, b ∈ E if E is a right or two-sided A-module. s Example. The spaces Hcone (X ∧ ) are two-sided modules over the algebra of all ∞ ϕ(r, x) ∈ C (R+ × X) that do not depend on r for r > R for some R > 0. In particular, for every cut-off function ω(r) we can form the spaces [ω]Hs,γ (X ∧ ) and s [1 − ω]Hcone (X ∧ ), and we have s Ks,γ (X ∧ ) = [ω]Hs,γ (X ∧ ) + [1 − ω]Hcone (X ∧ ).

(1.10)

Clearly the space (1.10) is independent of the specific choice of ω. Remark 1.5. If E0 and E1 are Hilbert spaces, also E0 + E1 becomes a Hilbert space by the identification with the orthogonal complement of ∆ in E0 ⊕ E1 . Thus, if we fix the cut-off function ω, we get a Hilbert space structure in Ks,γ (X ∧ ) via (1.10). For s = γ = 0 we take the scalar product from the identification n K0,0 (X ∧ ) = r− 2 L2 (R+ × X), where L2 (R+ ×X) refers to drdx with dx being connected with a fixed Riemannian metric on X. For purposes below we tacitly identify a coordinate neighborhood U on X with B ⊂ Rn , with the coordinates x via a chart χ : U → B. Then, for the s above characterization of the space Hcone (X ∧ ) for large r it is enough to look at s ∧ ∞ (1 − ω)ϕHcone (B ) for any ϕ ∈ C0 (B) and a cut-off function ω. We then have x ˜ s u ∈ (1 − ω)ϕHcone (X ∧ ) ⇐⇒ (β ∗ u)(r, ) ∈ H s (R1+n (1.11) r,˜ x ) r for the map β : B ∧ → B ∨ ⊂ R1+n .

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C.-I. Martin and B.-W. Schulze

Remark 1.6. Let A ∈ Diff µdeg (R+ × X × Ω) be given in the form (1.6), and set 
∂ j a(y, η) := r−µ ajα (r, y) − r (rη)α . (1.12) ∂r j+|α|≤µ

Assume that there is an R > 0 such that the coefficients ajα are independent of r for r > R. Then Dyα Dηβ a(y, η) : Ks,γ (X ∧ ) → Ks−µ+|β|,γ−µ+|β| (X ∧ ), (y, η) ∈ Ω×Rq , is a family of continuous operators for every s, γ ∈ R and α, β ∈ Nq . In fact, writing a(y, η) = ω ˜ a(y, η) + (1 − ω ˜ )a(y, η), for some cut-off function ω ˜ (r) we first have ω ˜ a(y, η) : Hs,γ (X ∧ ) → ω ˜ Hs−µ,γ−µ (X ∧ ) which is fairly obvious. On the other hand, to obtain s s (X ∧ ) → (1 − ω ˜ )Hcone (X ∧ ) (1 − ω ˜ )a(y, η) : Hcone

(1.13)

we express a(y, η) in local coordinates on X via a chart χ : U → B. If we write
ajα;γ (r, x, y)Dxγ ajα (r, y) = |γ|≤µ−(j+|α|)

with coefficients ajα;γ ∈ C ∞ (R+ × U × Ω) for a coordinate neighborhood U ⊂ X, it suffices to consider the summands  ∂ j r−µ ajα;γ (r, x, y)Dxγ − r (rη)α (1.14) ∂r separately. For simplicity, let us interpret x as coordinates in the unit ball B ⊂ Rn . Then (1.14) is a family of operators in R+ × B
(r, x). Applying the substitution x ˜ = rx and the characterization (1.11), for (1.13) it is enough to observe that  x  x ˜  ∂ j ˜  −µ r ajα r, , y r|γ| Dxγ˜ −r (1 − ω ˜ )ϕ (rη)α r r ∂r for ϕ(x) ∈ C0∞ (U ) respects the standard Sobolev spaces up to r = ∞. 1.3. Abstract edge spaces and symbols with twisted homogeneity A Hilbert space E is said to be equipped with a group action if there is given a strongly continuous group of isomorphisms κλ : E → E, λ ∈ R+ , such that κλ κλ
= κλλ
for all λ, λ ∈ R+ . An example is the space E = Ks,γ (X ∧ ) with (κλ u)(r, x) = λ

n+1 2

u(λr, x), λ ∈ R+

for n = dim X. More generally, if E is a Fr´echet space, written as a projective limit limj∈N E j of Hilbert spaces E j with continuous embeddings E j+1 → E j for ←− all j, and if {κλ }λ∈R+ is a group action on E 0 which restricts to group actions on E j for every j we say that E is equipped with a group action.

The Quantization of Edge Symbols

11

An example is the space Sεγ (X ∧ ) := lim E j ←−

(1.15)

j∈N

−1

for E j := r−j Kj,γ+ε−(1+j) (X ∧ ), ε > 0. Let us define SO (X ∧ ) := lim Sεγ (X ∧ ) ←−

(1.16)

ε>0

(the notation indicates that the left-hand side is independent of γ). Definition 1.7. Let E be a Hilbert space with group action {κλ }λ∈R+ . Then W s (Rq , E) (the ‘abstract’ edge Sobolev space on Rq of smoothness s ∈ R) is defined to be the completion of S(Rq , E) with respect to the norm 

2  12

dη ; u ˆ (η) η2s κ−1

η E here uˆ(η) = Fy→η u(η) is the Fourier transform in Rq , and κ−1

η acts on the values of u ˆ(η) for every η. Definition 1.8.  be Hilbert spaces with group actions {κλ }λ∈R+ and {˜ (i) Let E and E κλ }λ∈R+ ,  respectively. Then the space of (operator-valued) symbols S µ (U × Rq ; E, E) for an open set U ⊆ Rp , µ ∈ R, is defined to be the set all a(y, η) ∈ C ∞ (U ×  such that Rq , L(E, E)) 

−1 α β

q ˜ η Dy Dη a(y, η) κ η L(E,E) sup η−µ+|β| κ  : (y, η) ∈ K × R is finite for every K ⊂ U and all multi-indices α ∈ Np , β ∈ Nq .  denotes the set of all functions f(µ) ∈ C ∞ (U × (ii) S (µ) (U × (Rq \ {0}); E, E) q  such that (R \ {0}), L(E, E)) f(µ) (y, λη) = λµ κ ˜ λ f(µ) (y, η)κ−1 λ for all (y, η) ∈ U × (Rq \ {0}), λ ∈ R+ . µ  of classical symbols is defined as the set of (U × Rq ; E, E) (iii) The space Scl  such that there are elements a(µ−j) (y, η) ∈ all a(y, η) ∈ S µ (U × Rq ; E, E) (µ−j) q  S (U × (R \ {0}); E, E), j ∈ N, such that a(y, η) −

N


 χ(η)a(µ−j) (y, η) ∈ S µ−(N +1) (U × Rq ; E, E)

j=0

for every N ∈ N. Here χ(η) is any excision function, i.e., χ ∈ C ∞ (Rq ), χ(η) = 0 for |η| < c0 , χ(η) = 1 for |η| > c1 for certain 0 < c0 < c1 . In the case E = C we always set κλ = idC for every λ ∈ R+ . Then for  = C the Definition 1.8 reproduces the standard spaces of scalar symbols. E=E If a notation or a relation is valid both in the classical and the general case we

12

C.-I. Martin and B.-W. Schulze

µ  denote the corresponding spaces (Rq ; E, E) also write ‘(cl)’ as subscript. Let S(cl) of symbols a(η) that are independent of y, i.e., with constant coefficients.  that are projective limits of Definition 1.8 extends to Fr´echet spaces E, E, Hilbert spaces with group action, cf. [13].

Remark 1.9. The symbol spaces of Definition 1.8 depend on the choice of the group  respectively. If κ = { κλ }λ∈R+ in the spaces E and E, actions κ = {κλ }λ∈R+ and  necessary we use the notation µ  κ,κ S(cl) (U × Rq ; E, E)

(1.17)

µ  (U × Rq ; E, E). instead of S(cl)

Example. For the operator function (1.7) we have S µ (Ω × Rq ; Ks,γ (X ∧ ), Ks−µ,γ−µ (X ∧ )), provided that the coefficients ajα satisfy a suitable condition for r → ∞, e.g., to be independent of r for large r. Example. Let us set ε := Sεγ−µ (X ∧ ) ⊕ Cj+ E := Ks,γ (X ∧ ) ⊕ Cj− , E and F := Ks,−γ+µ (X ∧ ) ⊕ Cj+ , Fε := Sε−γ (X ∧ ) ⊕ Cj− for γ, µ ∈ R and ε > 0. A family of operators g(y, η) : Ks,γ (X ∧ ) ⊕ Cj− → K∞,γ (X ∧ ) ⊕ Cj+ (continuous for every s ∈ R ) is called a Green symbol, if there is an ε = ε(g) > 0 such that µ ε ), g ∗ (y, η) ∈ S µ (U × Rq ; F, Fε ) g(y, η) ∈ Scl (U × Rq ; E, E cl

(1.18)

for all s ∈ R. A Green symbol is flat (of infinite order) if the conditions (1.18) hold for all ε > 0. Especially, if g(y, η) is an upper left corner, that means µ g(y, η), g ∗ (y, η) ∈ Scl (U × Rq ; Ks,β (X ∧ ), SO (X ∧ )) for all s, β ∈ R. Note that Green operators (for fixed y, η) admit beautiful kernel characterizations, cf. [22]. Those imply corresponding descriptions of the symbols themselves, cf. [20]. Remark 1.10. The operator of multiplication by a function ϕ ∈ C0∞ (R+ ) belongs to S 0 (Rq ; Ks,γ (X ∧ ), Ks,γ (X ∧ )) for every s, γ ∈ R. If g(y, η) is a Green symbol in the sense of Example 1.3, also diag(ϕ, 1)g(y, η)diag(ϕ, ˜ 1) is a Green symbol for every ϕ, ϕ˜ ∈ C0∞ (R+ )

The Quantization of Edge Symbols

13

Parallel to the spaces of symbols of Definition 1.8 we have vector-valued analogues of Sobolev spaces, based on a Hilbert space E with group action {κλ }λ∈R+ . Recall the corresponding definition from [12]. By W s (Rq , E) we denote the completion of S(Rq , E) with respect to the norm 1/2  2 u  (η) dη , η2s κ−1 E

η u (η) := (Fy→η u)(η). For an open set Ω ⊆ Rq we also have the spaces s (Ω, E) Wcomp

s and Wloc (Ω, E)

s s (Ω) and Hloc (Ω), respectively. of vector-valued analogues of the spaces Hcomp Concerning useful functional analytic properties of the abstract edge spaces, see [13] or [8]. There is an immediate generalisation of these spaces to the case of Fr´echet spaces E that are projective limits of Hilbert spaces with group action. Let us recall the following general continuity result.  be a symbol in the sense of Definition Let a(y, y  , η) ∈ S µ (Ω × Ω × Rq ; E, E) 1.8, here with U := Ω × Ω for an open set Ω ⊆ Rq . Then, setting Op(a)u(y) :=  i(y−y
)η a(y, y  , η)u(y  )dy  d¯η for d¯η = (2π)−n dη, the operator e s−µ s  Op(a) : Wcomp (Ω, E) → Wloc (Ω, E)

(1.19)

is continuous for all s ∈ R. A proof may be found in [13] under some natural  which are satisfied, for instance, for the spaces E := Ks,γ (X ∧ ), conditions on E, E,  := Ks−µ,γ−µ (X ∧ ), occurring in the edge calculus. The continuity (1.19) for E  without any extra condition is proved in Seiler Ω = Rq , µ = 0 and s = 0 for E, E [21], as a generalisation of the Calder´ on-Vaillancourt theorem in the version of [9]. This entails other continuity results, for instance, for arbitrary µ, s ∈ R, when we specify the symbol a by imposing a certain exit property with respect to y, cf. the following section in the scalar case; vector-valued variants are studied in [10]. Similarly as (1.17) the spaces W s (Rq , E) depend on the choice of κ = {κλ }λ∈R+ . If necessary we write W s (Rq , E)κ (1.20) instead of W s (Rq , E). It is reasonable, cf. Tarkhanov [23], or Schulze and Tarkha the spaces nov [19], also to take for E and E K s,γ (X ∧ ) := rs−γ Ks,γ (X ∧ ) and K s−µ,γ−µ (X ∧ ) := rs−γ Ks−µ,γ−µ (X ∧ ), (1.21) s−γ+ n+1 2 u(λr, x), λ ∈ R . In respectively, with the group action (κs−γ u)(r, x) := λ + λ that case we can form W s,γ (X ∧ × Rq ) := W s (Rq , K s,γ (X ∧ ))κs−γ , s,γ (X ∧ × Ω) Wcomp(y)

s,γ Wloc(y) (X ∧ × Ω),

(1.22)

and Ω ⊆ Rq and, similarly, spaces of the kind open, where ‘comp(y)’ and ‘loc(y)’ means ‘comp’ and ‘loc’ with respect to the variables y ∈ Ω.

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C.-I. Martin and B.-W. Schulze

1.4. Conical exits to infinity In this section we want to deepen the information of Remark 1.6 in the sense of a more systematic discussion between edge symbols, standard operators in polar coordinates, and symbols within the exit pseudo-differential calculus. Let us first consider Rxn+1 regarded as a space with ‘conical exit’ to infinity ˜ |˜ x| → ∞. Definition 1.11. The space S µ;ν (Rxn+1 × Rn+1 ), ˜ ξ˜

(1.23)

˜ ∈ C ∞ (Rn+1 × Rn+1 ) such that µ, ν ∈ R, is defined to be the set of all a(˜ x, ξ) x ˜ ξ˜ −ν+|α| −µ+|β| α β

n+1 ˜ ˜ : (˜ ˜ ∈R sup ˜ x ξ |Dx˜ Dξ˜ a(˜ x, ξ)| x, ξ) × Rn+1 < ∞ ˜ ∈ S µ (Rn+1 × Rn+1 ) has for every α, β ∈ Nn+1 . We also say that a symbol a(˜ x, ξ) the exit property, if it belongs to the space (1.23); then µ is called the pseudodifferential order, ν the exit order of a. More generally,
S µ;ν,ν (Rxn+1 × Rxn+1 × Rn+1 ) ˜ ˜
ξ˜ ˜ ∈ C ∞ (R
) such that denotes the set of all a(˜ x, x ˜ , ξ) x ˜,˜ x ,ξ 3(n+1)

x −ν sup{˜ x−ν+|α| ˜




+|α
|




˜ −µ+|β| |Dα Dα
a(˜ ˜ ∈ R3(n+1) } < ∞ ξ x, x ˜ , x ˜)| : (˜ x.˜ x , ξ) x ˜ x ˜

for every α, α , β ∈ Nn+1 . Moreover, we set µ;ν µ ν ˆ π Scl Scl (Rxn+1 × Rn+1 ) := Scl (Rxn+1 )⊗ (Rn+1 ) ˜ ˜ ˜ ξ˜ ξ˜ ξ,˜ x

which is the space of classical symbols in ξ˜ and x ˜. Example. Let ω(t) ∈ C0∞ (R+ ) be a cut-off function such that ω(t) = 1 for t < 12 , ω(t) = 0 for t > 23 , and set   (r − r )2  . ψ(r, r ) := ω 1 + (r − r )2 Then ψ represents an element of S 0;0,0 (Rr × Rr
× R ) (which is independent of the covariable ). Let us set H s;g (Rn+1 ) := ˜ x−g H s (Rn+1 ). ˜ ∈ S µ;ν (Rn+1 × Rn+1 ) the associated pseudoTheorem 1.12. For every a(˜ x, ξ) differential operator Op(a) induces continuous operators Op(a) : H s;g (Rn+1 ) → H s−µ;g−ν (Rn+1 ) for all s, g ∈ R.

The Quantization of Edge Symbols

15

Remark 1.13. By virtue of S(Rn+1 ) = lim H N ;N (Rn+1 ) ←− N ∈N

it follows that Op(a) also induces a continuous operator Op(a) : S(Rn+1 ) → S(Rn+1 ). Let us set

n+1 ˜ ∈ S µ;ν (Rn+1 × Rn+1 ) . ) := Op(a) : a(˜ x, ξ) Lµ;ν (cl) (R (cl ˜ ) ξ;x ˜

As is known, Op(·) induces an isomorphism µ;ν n+1 Op : S(cl (Rn+1 × Rn+1 ) → Lµ;ν ) (cl) (R ˜ ) ξ;x ˜

(1.24)

for µ, ν ∈ R, including µ = −∞ or ν = −∞. Note that L−∞;−∞ (Rn+1 ) =  every µ;ν (Rn+1 ) is equal to the space of all integral operators with kernel in µ,ν L n+1 S(R × Rn+1 ). ˜ ∈ S µ;ν (Rn+1 × Rn+1 ) is called elliptic, if there Definition 1.14. An element a(˜ x, ξ) ˜ ∈ S −µ;−ν (Rn+1 × Rn+1 ) such that exists a p(˜ x, ξ) ˜ x, ξ) ˜ = 1 mod S µ−1;ν−1 (Rn+1 × Rn+1 ). a(˜ x, ξ)p(˜ ˜ ∈ x, ξ) An operator A ∈ Lµ;ν (Rn+1 ) is called elliptic if the associated symbol a(˜ µ;ν n+1 n+1 S (R ×R ) is elliptic (cf. the bijection (1.24)). ˜ ∈ S µ;ν (Rn+1 × Rn+1 ) we have a triple of prinFor classical symbols a(˜ x, ξ) clξ; ˜x ˜ cipal homogeneous components, σ(a) := (σψ (a), σe (a), σψe (a)) given on Rn+1 × (Rn+1 \ {0}), (Rn+1 \ {0})× Rn+1, and (Rn+1 \ {0})× (Rn+1 \ {0}), ˜ in ξ˜ order µ is as usual, x, ξ) respectively. The homogeneous principal symbol σψ (a)(˜ namely, ˜ = λµ σψ (a)(˜ ˜ for all (˜ ˜ ∈ Rn+1 × (Rn+1 \ {0}), λ ∈ R+ . σψ (a)(˜ x, λξ) x, ξ) x, ξ) ˜ in x x, ξ) ˜ of order ν has the property Analogously, the principal exit symbol σe (a)(˜ ˜ = δ ν σe (a)(˜ ˜ for all (˜ ˜ ∈ (Rn+1 \ {0}) × Rn+1 , δ ∈ R+ . x, ξ) x, ξ) x, ξ) σe (a)(δ˜ ˜ in ξ˜ of x, ξ) The third component is the homogeneous principal part of σe (a)(˜ order µ and has the homogeneity ˜ = δ ν λµ σψe (a)(˜ ˜ σψe (a)(δ˜ x, λξ) x, ξ) ˜ ∈ (Rn+1 \ {0}) × (Rn+1 \ {0}), λ, δ ∈ R+ . for all (˜ x, ξ)

16

C.-I. Martin and B.-W. Schulze

˜ ∈ S µ;ν (Rn+1 × Rn+1 ) is elliptic in the sense of Remark 1.15. A symbol a(˜ x, ξ) clξ; ˜x ˜ Definition 1.14 if and only if ˜ = 0 for all (˜ ˜ ∈ Rn+1 × (Rn+1 \ {0}), σψ (a)(˜ x, ξ) x, ξ) ˜ = 0 for all (˜ ˜ ∈ (Rn+1 \ {0}) × Rn+1 , σe (a)(˜ x, ξ) x, ξ) ˜ = 0 for all (˜ ˜ ∈ (Rn+1 \ {0}) × (Rn+1 \ {0}). σψe (a)(˜ x, ξ) x, ξ) These three conditions are independent. The following results are well known. Theorem 1.16. Let A ∈ Lµ;ν (Rn+1 ), µ, ν ∈ R. Then the following conditions are equivalent: (i) the operator A is elliptic; (ii) A induces a Fredholm operator A : H s;g (Rn+1 ) → H s−µ;g−ν (Rn+1 )

(1.25)

for some fixed s = s0 , g = g0 ∈ R. Theorem 1.17. n+1 ) be elliptic. Then A has a parametrix P ∈ L−µ;−ν (Rn+1 ) (i) Let A ∈ Lµ;ν (cl) (R (cl) in the sense P A − 1, AP − 1 ∈ L−∞;−∞ (Rn+1 ).

(ii) The Fredholm property of the operator (1.25) for some s = s0 , g = g0 ∈ R entails the Fredholm property of (1.25) for all s, g ∈ R. (iii) Let A ∈ Lµ;ν (Rn+1 ) be elliptic. Then ker A and cokerA of the Fredholm operator (1.25) are independent of s, g, and there are subspaces of finite dimension V, W ⊂ S(Rn+1 ) such that ker A = V and im A + W = H s−µ;g−ν (Rn+1 ), W ∩ im A = {0}. We now interpret the stretched cone X ∧
(r, x) as a space with conical exit r → ∞. In order to avoid clumsy precautions for r → 0 we first pass to the cylinder R × X and later on localise the operators on the plus side. Definition 1.18. We say that the cylinder R × X is equipped with the structure of a manifold with conical exits r → ±∞ if there is given a locally finite atlas of charts χι : R × Uι → Γι , ι ∈ I, for coordinate neighborhoods Uι on X and open sets Γι ⊂ Rn+1 such that for a constant R > 0 independent of ι for every ι, ˜ι ∈ I we have: (i) Γι ∩ {˜ x ∈ Rn+1 : |˜ x| ≥ R} = {λ˜ x:x ˜ ∈ Γι,R , λ ≥ 1} for Γι,R := Γι ∩ {|˜ x| = R}; (ii) χι (λr, x) = λχι (r, x) for every λ ≥ 1, |r| ≥ R;

The Quantization of Edge Symbols

17

(iii) the transition maps : χι (R × (Uι ∩ U˜ι )) → χ˜ι (R × (Uι ∩ U˜ι )) τ˜ι,ι := χ˜ι χ−1 ι have the property τ˜ι,ι (λ˜ x) = λτ˜ι,ι (˜ x) for every |˜ x| ≥ R, λ ≥ 1. If R × X is equipped with a structure in that sense we also write X instead of R × X. Let us fix a locally finite partition of unity {ϕι }ι∈I on X subordinate to {Uι }ι∈I , moreover, let {ψι }ι∈I be a system of functions ψι ∈ C0∞ (Uι ) such that ψι ≡ 1 on supp ϕι . Moreover, set ϕι (r, x) := ϕι (x), ψι (r, x) := ψι (x) for all (r, x) ∈ R × X. Let us endow X with a Riemannian metric that is equal to a cone metric gX := dr2 + r2 gX for |r| > R for a Riemannian metric gX on X. If dx denotes a corresponding measure on X the associated measure on X for |r| > R is of the form |r|n drdx (1.26) for n = dim X. µ;ν n+1 Now let Lµ;ν ) to Γι (in the sense of (cl) (Γι ) denote the restriction of L(cl) (R ∞ ∞ operators A : C0 (Γι ) → C (Γι )), and let Lµ;ν (cl) (X )

 denote the space of all operators ι∈I ϕι (χ−1 ι )∗ Aι ψι + C for arbitrary Aι ∈ µ;ν L(cl) (Γι ) and C having a kernel in S((R × X) × (R × X)) := S(R × R, C ∞ (X × X)). The concept of operators on manifolds with conical exits to infinity will be necessary also in the set-up of operator-valued symbols, cf. Definition 1.1. Let us recall some basic technicalities (the formulations will be slightly more general than  = C and trivial in the scalar case; so we also give additional information for E = E κλ = κ ˜ λ ).  be Hilbert spaces with group actions {κλ }λ∈R+ and {˜ Let E and E κλ }λ∈R+ , respectively. Then
 S µ;ν,ν (Rq × Rq × Rq ; E, E) (1.27)   ∞ q q q  for µ, ν, ν ∈ R denotes the set of all a(y, y , η) ∈ C (R × R × R , L(E, E)) such that







−1 α α
β

κ η Dy Dy
Dη a(y, y  , η) κ η

≤ cηµ−|β| yν−|α| y  ν −|α | (1.28)

˜  L(E,E)



for all (y, y , η) ∈ R and α, α , β ∈ Nq , with constants c = c(α, α , β) > 0. Similarly we have spaces of the kind 3q



 S µ;ν (Rq × Rq ; E, E),

(1.29)

18

C.-I. Martin and B.-W. Schulze

where the elements a(y, η) satisfy analogues of the estimates (1.28) (obtained by simply omitting y  ). The elements of (1.27) are regarded as double symbols of corresponding pseudo-differential operators Op(a). There are then unique left and right symbols aL (y, η) and aR (y  , η)
 such that Op(a) = Op(aL ) = Op(aR ). For inbelonging to S µ;ν+ν (Rq × Rq ; E, E)  −izζ stance, aL (y, η) can be calculated as an oscillatory integral aL (y, η) = e a(y, y + z, η + ζ)dzd¯ζ; a similar expression holds for aR (y  , η). Setting

 := {Op(a) : S µ;ν (Rq × Rq ; E, E)}  Lµ;ν (Rq ; E, E) we have an isomorphism  → Lµ;ν (Rq ; E, E).  Op : S µ;ν (Rq × Rq ; E, E)

2. Edge quantization 2.1. Pseudo-differential edge symbols We now pass to the algebra of pseudo-differential edge symbols. These will be parameter-dependent families of operators on the infinite (stretched) cone X ∧ with a specific dependence on edge variables and covariables (y, η) ∈ Ω × Rq . In this section X is assumed to be a closed compact C ∞ manifold. In order to model families of edge-degenerate pseudo-differential operators we start from the space Lµcl (X; R1+q (2.1)

,η ) of classical parameter-dependent pseudo-differential operators on X which is a Fr´echet space in a natural way. With p(r, y, , η) ∈ C ∞ (R+ × Ω, Lµcl (X; R1+q

,η ) we can associate a family of pseudo-differential operators opr (p)(y, η) ∈ C ∞ (Ω, Lµcl (X ∧ ; Rq )). Remark 2.1. Let p˜(˜ , η˜) ∈ Lµcl (X; R1+q

,˜ ˜ η ) be parameter-dependent elliptic of order µ and set b(r, , η) := r−µ p˜(r, rη). Then for every η = 0 there is an R(η) > 0 such that b(r, , η) : H s (X) → H s−µ (X) is a family of isomorphisms for all r ∈ R, r ≥ R and  ∈ R. In addition for a suitable choice of cut-off functions ω(r) and ω ˜ (r) the expression s uHcone ˜ |η| )uL2 (R+ ×X) (X ∧ ) := ω|η| uH s (R+ ×X) + (1 − ω|η| )opr (b)(η)(1 − ω

s is an equivalent norm in the space Hcone (X ∧ ).

The Quantization of Edge Symbols

19

Incidentally the parameter  plays the role of Imz for a complex variable z = β + i with fixed β ∈ R. In this case instead of (2.1) we also write Lµcl (X; Γβ × Rq ),

(2.2) ∞

(R+ × Ω, Lµcl (X; Γ 21 −γ × Rq ))

cf. also the notation (1.8). Elements f (r, y, z, η) ∈ C will be regarded as amplitude functions of Mellin pseudo-differential operators   r −( 12 −γ+i ) 1 dr f (r, y, − γ + i, η)u(r )  d¯, (2.3) opγM (f )(y, η)u(r) :=  r 2 r d¯ = (2π)−1 d. Also in this case we have opγM (f )(y, η) ∈ C ∞ (Ω, Lµcl (X ∧ ; Rq )). Definition 2.2. By Lµcl (X; Cz × Rqη ) we denote the space of all operator families f (z, η) ∈ A(C, Lµcl (X; Rqη )) such that f (β + i, η) ∈ Lµcl (X; R1+q

,η ) for every β ∈ R, uniformly in compact β-intervals. Moreover, let M−∞ (X; Γδ )ε for any β ∈ R, ε > 0, denote the set of all f (z) ∈ A({δ − ε < Rez < δ + ε}, L−∞ (X)) such that

f (β + i) ∈ L−∞ (X; R ) for every β ∈ (δ − ε, δ + ε), uniformly in compact β-intervals. Finally, we set  M−∞ (X; Γδ ) := M−∞ (X; Γδ )ε . ε>0

The spaces and M−∞ (X; Γδ )ε are Fr´echet with natural seminorm systems that immediately follow from the definition. In the sequel, if ϕ(r) ∈ C ∞ (R+ ) is any function and β ∈ R+ , we set ϕβ (r) := ϕ(βr) (for instance, ω[η] (r) = ω(r[η]) when ω is a cut-off function). Lµcl (X; C × Rq )

Remark 2.3. (i) If ω(r), ω ˜ (r) are arbitrary cut-off functions and f (y, z) ∈ C ∞ (Ω, −∞ M (X; Γ n+1 −γ )), Ω ⊆ Rq open, we have 2

m(y, η) := ∈

γ− n 2

r−ν ω[η] opM ν Scl (Ω

× R ;K q

(f )(y)˜ ω[η] s,γ

(X ∧ ), K∞,γ−ν (X ∧ ))

(2.4)

for every s ∈ R. For the principal part of order ν we have γ− n 2

σ∧ (m)(y, η) = r−ν ω|η| opM

(f )(y)˜ ω|η| .

(2.5)

× (ii) Let h(r, y, z, η) :=  h(r, y, z, rη) for  h(r, y, z, η) ∈ C (R+ × q ∞ Rη )), and let ϕ0 (r), ϕ1 (r) ∈ C0 (R+ ), and ϕ1 ≡ 0 on supp ϕ0 (e.g., ϕ1 may be a cut-off function, ϕ0 ∈ C0∞ (R+ )). Then ∞

γ− n 2

g(y, η) := ϕ1 (r[η])r−µ opM is a flat Green symbol of order µ.

(h)(y, η)ϕ0 (r[η])

Ω, Lµcl (X; Cz

20

C.-I. Martin and B.-W. Schulze The following result may be interpreted as a Mellin quantization.

Theorem 2.4. For every p(r, y, , η) of the form

 p(r, y, , η) = p˜(r, y, ˜, η˜) =r ,˜ ˜ η =rη

for a p˜(r, y, ˜, η˜) ∈ C ∞ (R+ × Ω, Lµcl (X; R1+q

,˜ ˜ η )) there exists an h(r, y, z, η) of the form ˜ y, z, η˜)|η˜=rη h(r, y, z, η) = h(r, µ ∞ ˜ y, z, η˜) ∈ C (R+ × Ω, L (X; C × Rq )) such that for an h(r, cl

opr (p)(y, η) = opγM (h)(y, η) mod C ∞ (Ω, L−∞ (X ∧ ; Rq )), ˜ y, z η˜) is unique mod C ∞ (R+ × Ω, L−∞ (X; C × Rq )). for every γ ∈ R, and h(r, Moreover, setting p0 (r, y, , η) := p˜(0, y, r, rη), h0 (r, y, z, η) := ˜h(0, y, z, rη) (2.6) we also have opr (p0 )(y, η) = opγM (h0 )(y, η) mod C ∞ (R+ × Ω, L−∞ (X ∧ ; Rq )) for every γ ∈ R. A proof may be found in [13], see also [15], or [5]. Let us fix cut-off functions ω(r), ω ˜ (r) such that ω ˜ ≡ 1 on supp ω; in that case we write ω ≺ ω  . Set ˜ χ(η) = 1 − ω(r), χ(η) ˜ =1−ω ˜ (r)

(2.7) ˜ ˜ for another cut-off function ω ˜ (r) such that ω ˜ ≺ ω. Moreover, choose cut-off functions σ(r), σ ˜ (r). Let p(r, y, , η), h(r, y, z, η) (2.8) be operator families related to each other as in Theorem 2.4, and set

γ− n a(y, η) := r−µ σ ω[η] opM 2 (h)(y, η)˜ ˜ (2.9) ω[η] + χ[η] opr (p)(y, η)χ ˜[η] σ n = dim X; recall that ωc (r) := ω(rc), χc (r) := χ(rc), etc., for any c > 0. Remark 2.5. Let a(y, η) be an operator function of the form (2.9). Then we have a(y, η) ∈ C ∞ (Ω, Lµcl (X ∧ ; Rq )). The parameter-dependent homogeneous principal symbol of a(y, η) σψ (a)(r, x, y, , ξ, η) ∈ C ∞ (T ∗ (R+ × X × Ω × R1+n+q ) \ 0) has the form σψ (a)(r, x, y, , ξ, η) = σ(r)˜ σ (r)r−µ p˜(µ) (r, x, y, r, ξ, rη), where p˜(µ) (r, x, y, ˜, ξ, η˜) denotes the parameter-dependent homogeneous principal symbol of p˜(r, y, ˜, η˜) ∈ C ∞ (R+ × Ω, Lµcl(X; R1+q

,˜ ˜ η )) of order µ.

The Quantization of Edge Symbols

21

Theorem 2.6. We have a(y, η) ∈ S µ (Ω × Rq ; Ks,γ (X ∧ ), Ks−µ,γ−µ (X ∧ )) for every s ∈ R. Moreover, for every ε > 0 we have a(y, η) ∈ S µ (Ω × Rq ; Sεγ (X ∧ ), Sεγ−µ (X ∧ )). A proof of Theorem 2.6 may be found in [4, Section 2.1.3]. Theorem 2.6 can be regarded as a quantization for edge-degenerate families of pseudo-differential operators as in Theorem 2.4. In fact, Theorem 2.6 gives rise to continuous operators s−µ s Opy (a) : Wcomp (Ω, Ks,γ (X ∧ )) → Wloc (Ω, Ks−µ,γ−µ (X ∧ ))

(2.10)

for all s, γ ∈ R. In other words, the correspondence p → a → Op(a) represents an operator convention that associates with p continuous operators (2.10). Let us set

γ− n σ∧ (a)(y, η) := r−µ ω|η| opM 2 (h0 )(y, η)˜ ω|η| + χ|η| opr (p0 )(y, η)χ ˜|η| for the families h0 and p0 as in Theorem 2.4 and (y, η) ∈ T ∗ Ω \ 0. Then we have σ∧ (a)(y, λη) = λµ κλ σ∧ (a)(y, η)κ−1 λ

(2.11)

for all λ ∈ R+ . Remark 2.7. An inspection of the proof of Theorem 2.6 shows that we also have a(y, η) ∈ S µ (Ω × Rq ; K s,γ (X ∧ ), K s−µ,γ−µ (X ∧ ))κs−γ ,κs−γ with the spaces (1.21). Similarly as (2.10) we thus obtain the continuity s,γ s−µ,γ−µ Opy (a) : Wcomp(y) (X ∧ × Ω) → Wloc(y) (X ∧ × Ω)

(2.12)

for all s ∈ R, cf. the notation (1.22). What we see is that there is a quite different quantization of edge-degenerate families compared with (2.10). It can be proved (cf. [7, Chapter 7]) that (2.10) as well as (2.12) belong to a continuously parametrised family of different (non-equivalent) quantizations. All of them are possible choices for a consistent edge calculus in the (stretched) wedge X ∧ × Ω. 2.2. A new edge quantization The new quantization of edge degenerate symbols consists of taking an operator family of the form  γ− 1 a(y, η) := r−µ σ ω[η] opM 2 (h)(y, η)  (2.13) ω[η] + χ[η] ψ[η] opr (p)(y, η) χ[η] σ instead of (2.9), where ω, ω , ω  , σ, σ  and p, h are as before, while ψ[η] (r, r ) := ψ(r[η], r [η]), with the function ψ(r, r ) of Example 1.4. In order to compare (2.9) and (2.13) we analyse operator families associated with p(r, , η) := p˜(r, r, rη)

22

C.-I. Martin and B.-W. Schulze

for an element p˜(r, ˜, η˜) ∈ C ∞ (R+ , Lµcl (X; R1+q

,˜ ˜ η )) (it is enough to assume the yindependent case). We assume that p˜(r, ˜, η˜) is independent of r for r ≥ R some constant R > 0. Theorem 2.8. Let σ, σ ˜ , χ, χ ˜ be as in (2.9) and form ˜[η] (r )˜ σ (r ). g(η) := r−µ σ(r)χ[η] (r)(1 − ψ[η] (r, r ))opr (p)(η)χ Then we have µ g(η), g ∗ (η) ∈ Scl (Rq ; Ks,γ (X ∧ ), SO (X ∧ ))

(2.14)

for every s, γ ∈ R, i.e., g(η) is a flat Green symbol of order µ, (cf. Example 1.3 and the notation (1.16)). Proof. We first show the property g(η), g ∗ (η) ∈ C ∞ (Rq , L(Ks,γ (X ∧ ), SO (X ∧ )). Let us consider g(η); the corresponding relation for g ∗ (η) can be obtained in an analogous manner. It is enough to show g(η) ∈ C ∞ (Rq , L(Ks,γ (X ∧ )U , SO (X ∧ )U ))

(2.15)

for every coordinate neighborhood U on X where Ks,γ (X ∧ )U := {u(r, x) ∈ Ks,γ (X ∧ ) : suppu ⊆ R+ × U }, ∧

∧

SO (X )U := {u(r, x) ∈ SO (X ) : suppu ⊆ R+ × U }.

(2.16) (2.17)

Here we use the fact that there is an atlas {U1 , . . . , UN } on X in such a way that for every two indices 1 ≤ j, k ≤ N also Uj ∪ Uk is a coordinate neighborhood on X. Without loss of generality we may assume that the coordinate neighborhood  such that U in (2.16), (2.17) are contained in other coordinate neighborhoods U U are compact subsets. Now, if we pass to local coordinates on X we can write 

−µ ei(r−r ) +i(x−x )ξ g(η)u(r, x) = r σ(r)χ[η] (r) p˜(r, x, x , r, ξ, rη)(1 − ψ[η] (r, r ))χ ˜[η] (r )˜ σ (r )

(2.18)

u(r , x )dr dx d¯d¯ξ, where supp u is contained in R+ ×K for a compact set K ⊂ Rn . Here p˜(r, x, x , ˜, ξ, η˜) is a classical symbol in (˜ , ξ, η˜) ∈ R1+n+q of order µ. After multiplying p(r, ˜ x, x , ˜, ξ, η˜) from the left and the right by localising functions ϕ0 (x) and ψ0 (x ), respectively, belonging C0∞ (K), we can assume that p˜(r, x, x , ˜, ξ, η˜) has compact support with respect to (x, x ). We want to compute the distributional kernel of (2.18). To this end we choose

an N and write ei(r−r ) = |r − r |−2N D 2N ei(r−r ) . Inserting this in (2.18) and

The Quantization of Edge Symbols

23

integrating by parts under the integral gives us  |r[η] − r [η]|−2N (1 − ψ[η] (r, r )) g(η)u(r, x) =r−µ σ(x)χ[η] (r)





ei(r−r ) +i(x−x )ξ (r[η])2N (D 2N ˜)(r, x, x , r, ξ, rη) ˜ p χ ˜[η] (r )˜ σ (r )u(r , x )dr dx d¯d¯ξ. The kernel of this operator can be written as  K(g)(r, r , x, x ; η) = r−µ σ(r)χ[η] (r) |r[η] − r [η]|2N (1 − ψ[η] (r, r ))





ei(r−r ) +i(x−x )ξ (r[η])2N (D 2N ˜) ˜ p

(2.19)

(r, x, x , r, ξ, rη)χ ˜[η] (r )˜ σ (r )d¯d¯ξ. By virtue of the symbolic estimates |D 2N ˜(r, x, x , r, ξ, rη)| ≤ cr, ξ, rηµ−2N ˜ p

(2.20)



for all x, x ∈ K and r ∈ supp σχ[η] we see that the integral (2.19) converges, together with all derivatives in r, r , x, x up to some order M when N = N (M ) is chosen sufficiently large. This shows that for every fixed η ∈ Rq the kernel of (2.19) belongs to C0∞ (R+ × Rn × R+ × Rn ), and this remains true if η varies in a compact set ⊂ Rq . It is also clear that the kernel smoothly depends on η ∈ Rq . Thus we have verified the relation (2.15). For the formal adjoint we can argue in a similar manner. For the proof of (2.14) we first generate the homogeneous principal symbol of order µ σ∧ (g)(η) = lim λ−µ κ−1 λ g(λη)κλ for η = 0. λ→∞

In local coordinates on X it takes the form 

ei(r−r ) +i(x−x )ξ (1 − ψ|η| (r, r )) σ∧ (g)(η)u(r, x) = r−µ χ|η| (r) ˜|η| (r )u(r , x )dr dx d¯d¯ξ. p˜(0, x, x ; r, ξ, rη)χ Integration by parts gives us again  |r|η| − r |η||−2N (1 − ψ|η| (r, r )) σ∧ (g)(η)u(r, x) = r−µ χ|η| (r)





ei(r−r ) +i(x−x )ξ (r|η|)2N (D 2N ˜)(0, x, x , r, ξ, rη)χ ˜|η| (r ) ˜ p u(r , x )dr dx d¯d¯ξ for every N ∈ N. Using the estimate |r − r |−2N (1 − ψ(r, r )) ≤ r−N r −N

(2.21)

and the symbolic estimate (2.20) we can easily verify that σ∧ (g)(η) has a kernel in ˆ π SO (X ∧ )U . Assuming for the moment that p˜ = p˜(˜ SO (X ∧ )U ⊗ , η˜) is independent of r we have g(η) − χ(η)σ∧ (g)(η) ∈ S −∞ (Rq ; Ks,γ (X ∧ ), SO (X ∧ ))

24

C.-I. Martin and B.-W. Schulze

for any excision function χ(η) ∈ C ∞ (Rq ). Otherwise, we can successively compute the lower order terms by a Taylor expansion p˜(r, ˜, η˜) =

L


rl p˜l (˜ , η˜) + rL+1 p˜(L+1) (r, ˜, η˜).

l=0

The above process applied to rl p˜l (˜ , η˜) then gives us the homogeneous components of g(η) of order µ−l. Another elementary calculation with rL+1 p˜(L+1) (r, ˜, η˜) yields a remainder in S µ−(L+1) (Rq ; Ks,γ (X ∧ ), SO (X ∧ )) which shows that our symbol is classical. For the formal adjoints we can do the same.
Proposition 2.9. Let p˜(r, ˜, η˜) ∈ C ∞ (R+ , L−∞ (X; R1+q

,˜ ˜ η )) be independent of r for r ≥ R for some R > 0. Then g(η) := r−µ σ(r)χ[η] (r)opr (p)(η)χ ˜[η] (r )˜ σ (r )

(2.22)

is a flat Green symbol of order µ; in particular, it satisfies the relations (2.14). The same is true of g1 (η) := r−µ σ(r)χ[η] (r)ψ[η] (r, r )opr (p)(η)χ[η] (r )˜ σ (r ). Proof. It is evident that g(η) has the property (2.14). It remains to show that g(η) is a classical symbol. Let us first consider the case that p˜ = p˜(˜ , η˜) is independent of r. Then for η = 0 we can set g(µ) (η) := r−µ χ|η| (r)opr (p)χ ˜|η| (r ). Write g(µ) (η)

r−µ χ|η| (r)ψ|η| (r, r )opr (p)(η)χ ˜|η| (r )

=

˜|η| (r ) +r−µ χ|η| (r)(1 − ψ|η| (r, r ))opr (p)(η)χ 



(2.23)



with the function ψ(r, r ) of Example 1.14, and ψ|η| (r, r ) := ψ(r|η|, r |η|). Observe that we have ψ(r, r )ϕ(r) ∈ S(R × R) (2.24) for every ϕ ∈ S(R). The first summand on the right of (2.23) is a Schwartz function in r, r with values in L−∞ (X). In fact, χ|η| (r)˜ p(˜ , rη) belongs to S(Rr , L−∞ (X; R ˜)) for every fixed η = 0; then applying a relation of the kind (2.24) gives us this property. For the second summand on the right of (2.23) we can proceed in a similar manner as in the proof of Theorem 2.8.  π SO (X ∧ ) We obtain altogether that the kernel of g(η) (η) belongs to SO (X ∧ )⊗ for every η = 0. Thus f (η) := r−µ χ[η] opr (p)(η)χ ˜[η] (2.25) is a Green symbol, since f (η), f ∗ (η) have the property (2.14) (with f instead of g) and f (η) = g(µ) (η) for large |η|. Applying Remark 1.10 to ϕ := |η|, ϕ˜ := σ ˜ , we obtain that (2.22) itself has the desired property. In the case of an r-dependence of p˜(r, ˜, η˜) we assume that p˜(r, ˜, η˜) is independent of r for large r; otherwise we subtract p˜(∞, ˜, η˜) which is constant in r

The Quantization of Edge Symbols

25

and can be treated as before and then consider the difference p˜(r, ˜, η˜)− p˜(∞, ˜, η˜). Then we can write ∞
λj ϕj (r)˜ pj (˜ , η˜) p˜(r, ˜, η˜) = j=0



with λj ∈ C, |λj | < ∞, ϕj ∈ C0∞ (R+ ), p˜j (˜ , η˜) ∈ L−∞ (X; R1+q

,˜ ˜ η ) tending to 0 in the respective spaces as j → ∞. This easily reduces the general case to r-independent p˜ when we apply other standard arguments, in particular, that r−µ χ[η] opr (pj )(η)χ ˜[η] tends to zero for j → ∞ in the space of Green operators of the kind discussed before. The assertion on g1 (η) is included in the proof, cf. the discussion around the formula (2.23).
2.3. Edge symbols as parameter-dependent cone operator families By an edge amplitude function of order µ, referring to the weight data g = (γ, γ − µ), we understand an operator family of the form γ− n 2

a(y, η) = r−µ σ{ω[η] opM

(h)(y, η)˜ ω[η] + χ[η] opr (p)(y, η)χ ˜[η] }˜ σ + m(y, η) + g(y, η),

where the first expression on the right-hand side is as in (2.9), moreover, m(y, η) is a smoothing Mellin edge symbol and g(y, η) a Green symbol as in Example 1.3 (for j− = j+ = 0). Let Rµ (Ω × Rq ; g) denote the space of all those a(y, η). Let us set σ(a) := (σψ (a), σ∧ (a)),

(2.26)

where σψ (a) is defined as in Remark 2.5; we take into account that σψ (m + g) = 0. Moreover, we set γ− n 2

σ∧ (a)(y, η) := r−µ {ω|η| opM

(h0 )(y, η)˜ ω|η|

+ χ|η| opr (p0 )(y, η)χ ˜|η| } + σ∧ (m + g)(y, η) with the notation (2.6) and σ∧ (m + g)(y, η) as the homogeneous principal symbol of (m + g)(y, η) as a classical symbol, cf. also (2.5) for µ = ν. Theorem 2.10. Let a(y, η) ∈ Rµ (Ω × Rq , g) and assume that σ(a) = 0. Then we have a(y, η) ∈ S µ−1 (Ω × Rq ; Ks,γ (X ∧ ), Ks−µ,γ−µ (X ∧ )), for every s ∈ R, and a(y, η) takes values in the space of compact operators Ks,γ (X ∧ ) → Ks−µ,γ−µ (X ∧ ), s ∈ R. Proof. The relation σψ (a) = 0 implies that 1+q σ(r)˜ σ (r)p(r, y, ˜, η˜) ∈ C ∞ (R+ × Ω, Lµ−1 ˜ η )) cl (X; R ,˜

and ˜ y, z, η˜) ∈ C ∞ (R+ × Ω, Lµ−1 (X; C × Rq )). σ(r)˜ σ (r)h(r, η ˜ cl

26

C.-I. Martin and B.-W. Schulze

This yields 1+q ˜ y, z, η˜) ∈ C ∞ (Ω, Lµ−1 (X; C × Rq )). )), h(0, p˜(0, y, ˜, η˜) ∈ C ∞ (Ω, Lµ−1 cl (X; R cl

Now σ∧ (a)(y, η) = 0 implies that γ− n 2

r−µ {ω|η| opM

(h0 )(y, η)˜ ω|η| + χ|η| opr (p0 )(y, η)χ ˜|η| } = −σ∧ (m + g)(y, η). (2.27)

Both sides of the latter relation are operators in the cone algebra on X ∧ . That means their subordinate symbols (interior, conormal, exit) symbols coincide. In particular, we have γ− n 2

σc (r−µ ω|η| opM

˜ 0 (0, y, z, 0) = −σc σ∧ (m(y, η))(z). (h0 )(y, η)˜ ω|η| )(z) = h

In other words, setting h00 (y, z) := f˜(0, y, z, 0), we have h00 (y, z) = −f (y, z)

(2.28)

γ− n r−µ ω[η] opM 2 (f )(y)˜ ω[η]

(when m(y, z) is given as for an f ∈ C ∞ (Ω, M−∞ (X; Γ n+1 −γ )ε ) which we assume without loss of generality). In addition we may assume 2 ˜ [η] . It follows that that σ ≡ 1, σ ˜ ≡ 1 on supp ω[η] and supp ω γ− n 2

δ(η){r−µ {ω[η] opM

(h0 − h00 )(y, η)˜ ω[η] + χ[η] opr (p0 )(y, η)χ ˜[η] } + g(y, η)} (2.29)

is a Green symbol of order µ − 1 for every excision function δ(η) in Rq . In fact, the relation (2.27) shows that γ− n 2

r−µ {ω|η| opM

(h0 − h00 )˜ ω|η| + χ|η| opr (p0 )(y, η)χ ˜|η| } = −σ∧ (g)(y, η).

This entails the identity (2.29) for all η with |η| ≥ const for a constant > 0. On the other hand, using the technique of proving Theorem 2.10 we see that the left-hand side of (2.29) is an operator-valued symbol, even classical in this situation, with −σ∧ (g)(y, η) as the homogeneous principal symbol. Then the same is true of g0 (y, η)

γ− n 2

:= r−µ σδ(η){ω[η] opM +

(h0 − h00 )(y, η)˜ ω[η] + χ[η] opr (p0 )(y, η)χ ˜[η] }˜ σ

δ(η)g(y, η),

cf. Remark 1.10. Now the symbol a(y, η) can be written in the form γ− n 2

r−µ σ{ω[η] opM

(h − h0 )(y, η)˜ ω[η] + χ[η] opr (p − p0 )(y, η)χ ˜[η] }˜ σ γ− n 2

+ δ(η){r−µ σ{ω[η] opM

+ g(y, η) + (1 − δ(η)){r

(h0 − h00 )(y, η)˜ ω[η] + χ[η] opr (p0 )(y, η)χ ˜[η] }˜ σ}

−µ

γ− n 2

σ{ω[η] opM

(h0 − h00 )(y, η)˜ ω[η]

+ χ[η] opr (p0 )(y, η)χ ˜[η] }˜ σ + g(y, η)} =

γ− n r−µ σ{ω[η] opM 2 (h

(2.30)

− h0 )(y, η)˜ ω[η] + χ[η] opr (p − p0 )(y, η)χ ˜[η] }˜ σ

+ g0 (y, η) + (1 − δ(η)){r−µ σ{ω[η] opM

γ−n/2

+ χ[η] opr (p0 )(y, η)χ ˜[η] }˜ σ}

(h0 − h00 )(y, η) ω[η]

The Quantization of Edge Symbols

27

The summand g0 (y, η) is a Green symbol of order µ − 1 and takes values in compact operators. Also γ− n 2

r−µ σ{ω[η] opM

(h − h0 )(y, η)˜ ω[η] }˜ σ

(2.31)

˜ (1) (r, y, z, rη) for an is compact for every (y, η), since we can write h − h0 = rh µ−1 η ˜ (1) (r, y, z, η˜) ∈ C ∞ (R+ × Ω, L h cl (X; C × Rξ˜)) which yields an improvement of the weight at r = 0, together with the improvement of the order. For a similar reason also r−µ σ{χ[η] opr (p − p0 )(y, η)χ ˜[η] }˜ σ (2.32) is compact for every (y, η). Finally, (2.31) and (2.32) are operator-valued symbols of order µ − 1, by similar arguments as for the proof of Theorem 2.6, and the last summand on the right of (2.30) takes values in compact operators and is of order −∞ in η.


3. Composition properties 3.1. Composition of edge symbols In this section we analyse the composition properties of edge amplitude functions of the form (2.9). It will be more convenient here to employ the cut-off functions  ω(r), ω  (r), ω  (r) rather than the excision functions (2.7). Moreover, since the presence of the variables y ∈ Ω only causes minor modifications of the arguments, we content ourselves with the y-independent case. Let :=

r−µ σ{aM (η) + aψ (η)} σ,

b(η) :=

σ, r−ν σ{bM (η) + bψ (η)}

a(η)

where σ(r) and σ (r) are arbitrary cut-off functions, and γ−ν− n 2

aM (η) := ω[η] opM

γ− n 2

bM (η) := ω[η] opM

(h1 )(η) ω[η] ,

(h2 )(η) ω[η] ,

 aψ (η) := (1 − ω[η] )opr (p1 )(η)(1 − ω  [η] ),  bψ (η) := (1 − ω[η] )opr (p2 )(η)(1 − ω  [η] ).

Here pj (r, ρ, η) = pi (r, rρ, rη) for families pi (r, ρ, η) ∈ C ∞ (R+ , Lµcli (X; Rρ1+q , η ))   with µ1 = µ, µ2 = ν and hi (r, z, η) = hi (r, z, rη) for elements hi (r, z, η) ∈ C ∞ (R+ , Lµcli (X; C × Rq )), i = 1, 2, where we assume that hi is the Mellin quantization of pi in the sense of Theorem 2.4. Theorem 3.1. We have (with respect to the pointwise composition of operator functions) σ + g(η) (ab)(η) = σr−(µ+ν) {cM (η) + cψ (η)} where γ− n 2

cM (η) = ω[η] opM

 (h)(η) ω[η] + (1 − ω[η] )opr (p)(η)(1 − ω  [η] )

28

C.-I. Martin and B.-W. Schulze

for operator functions p(r, ρ, η) and h(r, z, η) which are of the same nature as those in Theorem 2.4, now of order µ + ν, and g(η) is a flat Green symbol of order µ + ν. We have σψ (ab) = σψ (a)σψ (b), σ∧ (a)(η)σ∧ (b)(η) = σ∧ (ab)(η). (3.1) Proof. For the proof we employ the abbreviations a = a(η) for  a = ωa0 ω  + (1 − ω)a1 (1 − ω  ),  where the cut-off functions ω, ω , ω  have the meaning ω = ω[η] , etc., and γ−ν− n 2

a0 := σr−µ opM

(h1 ) σ , a1 := σr−µ opr (p1 ) σ,

and, similarly,  b = ωb0 ω  + (1 − ω)b1 (1 − ω ) for

γ− n 2

b0 := σr−ν opM

(h2 ) σ , b1 := σr−µ opr (p2 ) σ.

≺ω≺ω In the following computations we systematically employ the properties ω   6   which implies ω ω = ω, ω ω = ω  . We then obtain ab = P + k=1 Gk after elementary rearrangements of summands in the composition ab for   P := ωa0 ω  b0 ω  + (1 − ω)a1 (1 − ω  )b1 (1 − ω ) and G1

:=

G2

:=

G3

:=

G4

:=

G5

:=

G6

:=

We now write

   ω  a0 ( ω − ω)b1 (1 − ω  ) + (ω − ω  )a0 ( ω − ω)b1 (1 − ω  ),    , (1 − ω  )a1 (ω − ω  )b0 ω  + ( ω − ω)a1 (ω − ω  )b0 ω    +ω  a0 (ω − ω (ω − ω  )a0 (ω − ω  )b0 ω  )b0 ω ,  − ω)b1 (1 − ω   (1 − ω  )a1 (ω   ) + ( ω − ω)a1 (ω  − ω)b1 (1 − ω  ),   ( ω − ω)a0 (ω  − ω)(b1 − b0 )( ω−ω  ),   (ω − ω  )(a1 − a0 )( ω − ω)b0 ( ω−ω  ).   + (1 − ω)c1 (1 − ω  ) + G7 + G8 P = ωa0 b0 ω

for G7 := ωa0 ( ω − 1)b0 ω ,

  G8 := (1 − ω){a1 (1 − ω  )b1 − c1 }(1 − ω  ),

and σ c1 (η) = σr−(µ+ν) opr (p)(η)

(3.2)

for an operator function p(r, ρ, η) = p(r, rρ, rη) which is defined by an element 1+q p(r, ρ, η) ∈ C ∞ (R+ , Lµ+ν cl (X; Rρ , η )),

obtained by computing the Leibniz product #r with respect to r, (r)σ(r)r−ν p2 (r, rρ, rη). r−(µ+ν) p(r, rρ, rη) = r−µ p1 (r, rρ, rη)r σ

(3.3)

The Quantization of Edge Symbols

29

In particular, we may assume p(r, ρ, η) to be smooth up to r = 0. Moreover, we have a0 b 0

= =

γ−ν− n 2

σr−µ opM σr

−(µ+ν)

γ− n 2

(h1 )(η) σ σr−ν opM

(h2 )(η) σ n γ− n γ− ν opM 2 (T h1 )(η)opM 2 (σ σ h2 )(η) σ.

We therefore obtain a Mellin symbol h(r, z, η) =  h(r, z, rη) for an  h(r, w, η) ∈ µ+ν ∞ q C (R+ , Lcl (X; C × R )) such that γ− n 2

a0 b0 = σr−(µ+ν) opM

(h)(η) σ.

(3.4)

Summing up it follows that  ab = ωc0 ω  + (1 − ω)c1 (1 − ω  ) + g, where c0 (η) and c1 (η) are given by (3.1) and (3.2), respectively, and g=

8


Gj .

(3.5)

j=1

It turns out that (3.5) is a flat Green symbol. This will be verified in the following section. The relation (3.1) follows from the fact that we can apply the above computations for the compositions to corresponding operator functions, where [η] is replaced by |η|, η = 0, and all the first r-variables are frozen at zero.
3.2. Characterization of remainders In order to characterise (3.5) as a flat Green symbol we consider the summands separately. Let us write G1 := G1 + G1 for     a0 ( ω − ω)b1 (1 − ω  ), G1 := (ω − ω  )a0 ( ω − ω)b1 (1 − ω  ). G1 := ω We have

 G1 = CD for C := ω  a0 ( ω − ω), D := ϕb1 (1 − ω ) ∞ ω − ω). The function ϕ is interfor any ϕ ∈ C0 (R+ ) that is equal to 1 on supp ( preted (similarly as the cut-off functions) as an η-dependent factor, i.e., ϕ = ϕη for ϕη (r) := ϕ(r[η]). The family of operators C is a flat Green symbol, see Example 1.3 and Remark 2.3. It is then easy to verify that the composition with D gives again a Green symbol. For G1 we choose another cut-off function ω0 that is equal to 1 on supp ω and such that ω  is equal to 1 on supp ω0 (this is always possible). Then we can write G1 = H + L for H

:=

L := For any ϕ ∈

C0∞ (R+ )

 ω − ω)b1 (1 − ω  ), (ω − ω  )a0 (1 − ω0 )(  (ω − ω  )a0 ω0 ( ω − ω)b1 (1 − ω  ).

such that ϕ ≡ 1 on supp ( ω − ω) we have H = CD for

 ω − ω)b1 (1 − ω  ), C := (ω − ω  )a0 (1 − ω0 )ϕ, D := (

30

C.-I. Martin and B.-W. Schulze

(in this proof, C and D occur in different meaning which will be clear from the context). Since (1 − ω) vanishes on supp (ω − ω  ), the factor C is smoothing and can be treated in a similar manner as the corresponding factor occurring in G1 . As above it is again easy to verify that then also H is a flat Green symbol. Moreover, taking some ϕ ∈ C0∞ (R+ ), ϕ ≡ 1 on supp ( ω − ω), we can write L = DC for  D := (ω − ω  )a0 ( ω − ω), C := ω0 ϕb1 (1 − ω  ). Since 1− ω  vanishes on supp ω0 , the family C is smoothing and a flat Green symbol. This implies the same for L. The arguments for Gk , 1 < k ≤ 8, are similar and left to the reader.

References [1] D. Calvo and B.-W. Schulze, Operators on corner manifolds with exits to infinity, Journal of Differential Equ. 19, 2 (2006), 147–192. [2] D. Calvo, C.-I. Martin and B.-W. Schulze, Symbolic structures on corner manifolds, in RIMS Conference on Microlocal Analysis and Asymptotic Analysis, Keio University, Tokyo, 2005, 22–35. [3] D. Calvo and B.-W. Schulze, Edge calculus of second generation, Preprint, Institut f¨ ur Mathematik, Universit¨ at Potsdam, 2005. [4] N. Dines and B.-W. Schulze, Mellin-edge-representations of elliptic operators, Math. Meth. Appl. Sci. 28 (2005), 2133–2172. [5] J.B. Gil, B.-W. Schulze and J. Seiler, Cone pseudodifferential operators in the edge symbolic calculus, Osaka J. Math. 37 (2000), 219–258. [6] G. Harutjunjan and B.-W. Schulze, The relative index for corner singularities, Integral Equations Operator Theory 54, 3 (2006), 385–426. [7] G. Harutjunjan and B.-W. Schulze, Elliptic Mixed, Transmission and Singular Crack Problems, European Mathematical Soc., Z¨ urich, 2007, to appear. [8] T. Hirschmann, Functional analysis in cone and edge Sobolev spaces, Ann. Global Anal. Geom. 8 (1990), 167–192. [9] I.L. Hwang, The L2 -boundness of pseudodifferential operators, Trans. Amer. Math. Soc. 302 (1987), 55–76. [10] D. Kapanadze and B.-W. Schulze, Crack Theory and Edge Singularities, Kluwer Academic Publ., Dordrecht, 2003. [11] L. Maniccia and B.-W. Schulze, An algebra of meromorphic corner symbols, Bull. des Sciences Math. 127 (2003), 55–99. [12] B.-W. Schulze, Pseudo-differential operators on manifolds with edges, in Partial Differential Equations, Teubner, Teubner-Texte zur Mathematik, Leipzig, 1989, 259– 287. [13] B.-W. Schulze, Pseudo-differential Operators on Manifolds with Singularities, NorthHolland, Amsterdam, 1991. [14] B.-W. Schulze, Corner Mellin operators and conormal asymptotics, in Sem. Equ. aux D´eriv. Part. 1990–1991, Exp. XII, Ecole Polytechnique, Mars 1991.

The Quantization of Edge Symbols

31

[15] B.-W. Schulze, Boundary Value Problems and Singular Pseudo-differential Operators. J. Wiley, Chichester, 1998. [16] B.-W. Schulze, Operator algebras with symbol hierarchies on manifolds with singularities, in Advances in Partial Differential Equations (Approaches to Singular Analysis), Editors: J. Gil, D. Grieser and M. Lesch, Birkh¨auser, Basel, 2001, 167– 207. [17] B.-W. Schulze, Operators with symbol hierarchies and iterated asymptotics, Publ. RIMS, Kyoto University 38 (2002), 735–802. [18] B.-W. Schulze and J. Seiler, The edge algebra structure of boundary value problems, Ann. Global Anal. and Geom. 22 (2002), 197–265. [19] B.-W. Schulze and N. Tarkhanov, New algebras of boundary value problems for elliptic pseudodifferential operators. Preprint, Institut f¨ ur Mathematik, Universit¨ at Potsdam, 2005. [20] B.-W. Schulze and A. Volpato, Green operators in the edge Quantisation of elliptic operators. Preprint, Institut f¨ ur Mathematik, Universit¨ at Potsdam, 2004. [21] J. Seiler, Continuity of edge and corner pseudodifferential operators. Math. Nachr., 205, (1999), 163–182. [22] J. Seiler, The cone algebra and a kernel characterization of Green operators, in Advances in Partial Differential Equations (Approaches to Singular Analysis), Editors: J. Gil, D. Grieser and M. Lesch, Birkh¨ auser, Basel, 2001, 1–29. [23] N. Tarkhanov, Harmonic integrals on domains with edges, Preprint, Institut f¨ ur Mathematik, Universit¨ at Potsdam, 2004. C.-I. Martin and B.-W. Schulze Institut f¨ ur Mathematik Universit¨ at Potsdam Am Neuen Palais 10 D-14469 Potsdam, Deutschland e-mail: [email protected] e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 33–50 c 2006 Birkh¨
auser Verlag Basel/Switzerland

On Rays of Minimal Growth for Elliptic Cone Operators Juan B. Gil, Thomas Krainer and Gerardo A. Mendoza Abstract. We present an overview of some of our recent results on the existence of rays of minimal growth for elliptic cone operators and two new results concerning the necessity of certain conditions for the existence of such rays. 2000 Mathematics Subject Classification. Primary 58J50; Secondary 35J70, 47A10. Keywords. Resolvents, conical singularities, spectral theory.

1. Introduction The aim of this article is twofold. On the one hand, we present an overview of some of the results contained in [6, 7] on the subject in the title, and of the geometric perspective we developed in the course of the investigations leading to the aforementioned papers. We illustrate the main ideas of our approach by means of examples concerning Laplacians on a compact 2-manifold. Already this simple situation exhibits the structural richness and complexity of the general theory. On the other hand, we offer some improvements, cf. Theorems 4.3 and 5.5, regarding necessary and sufficient conditions for a closed sector Λ ⊂ C to be a sector of minimal growth for a certain class of elliptic cone operators A and for the associated model operator A∧ . Recall that a closed sector of the form Λ = {λ ∈ C : λ = reiθ for r ≥ 0, θ ∈ R, |θ − θ0 | ≤ a}

(1.1)

is called a sector of minimal growth (or of maximal decay) for a closed operator A : D ⊂ H → H, where H is a Hilbert space and D is dense in H, if there is a constant R > 0 such that A − λ is invertible for every λ ∈ ΛR = {λ ∈ Λ : |λ| ≥ R}, and the resolvent

34

J.B. Gil, T. Krainer and G.A. Mendoza

(A − λ)−1 satisfies either of the equivalent



(A − λ)−1

≤ C/|λ|, L (H)

estimates



(A − λ)−1

L (H,D)

≤C

(1.2)

for some C > 0 and all λ ∈ ΛR . We are interested in cone operators on smooth manifolds with boundary. Specifically, let M be a smooth n-manifold with boundary Y = ∂M and let E → M be a Hermitian vector bundle over M . Fix a defining function x for Y . A differential cone operator of order m acting on sections of E → M is an operator of the form A = x−m P with P in the class Diff m b (M ; E) of totally characteristic differential operators of order m,◦cf. Melrose [14]. We write A ∈ x−m Diff m b (M ; E). More explicitly, in the interior M of M , A is a differential operator with smooth coefficients, and near the boundary, in local coordinates (x, y) ∈ (0, ε) × Y , it is of the form
A = x−m akα (x, y)(xDx )k Dyα (1.3) k+|α|≤m

with coefficients akα smooth up to x = 0; here Dx = −i∂/∂x and likewise Dyj . We will say that A (or P ) has coefficients independent of x near Y , if the coefficients akα in (1.3) do not depend on x (this notion depends on the choice of tubular neighborhood map, defining function x, and connection on E. For a precise definition see [6]). This paper consists of 5 sections. In Section 2 we review some basic properties of cone operators while in Section 3 we discuss the associated model operators. The new results on rays of minimal growth can be found in Sections 4 and 5. Apart from the works explicitly cited in the text, our list of references contains additional items referring to related works on resolvents and rays of minimal growth for elliptic operators. The investigation of spectral properties for elliptic cone operators was initiated by Cheeger [4] in the seventies and has since undergone major developments. Concerning the large-parameter behavior of resolvents, the interested reader is referred to [3, 5, 12, 13, 15], among others. A more detailed account on the subject is provided in the introduction of the papers [6, 7].

2. Preliminaries on cone operators Let A be a differential cone operator. As introduced in [6], the principal symbol of A, c σ (A), is ◦defined on the c-cotangent c T ∗ M of M rather than on the cotangent itself. Over M it is essentially the usual principal symbol, and equal to
akα (x, y)ξ k η α k+|α|=m

near the boundary Y , see (1.3). Example. Let M be a compact 2-manifold with boundary Y = S 1 . Let gY (x) be a smooth family of Riemannian metrics on S 1 such that gY (0) is the standard metric, dy 2 . We equip M with a “cone metric” g that near Y takes the form

Rays of Minimal Growth

35

g = dx2 + x2 gY (x) (g is a regular Riemannian metric in the interior of M ). Then, near Y , the Laplace-Beltrami operator ∆ has the form   x−2 (xDx )2 + a(x, y)(xDx ) + ∆Y (x) , (2.1) where a(x, y) is a smooth function with a(0, y) = 0 and ∆Y (x) is the nonnegative Laplacian on S 1 associated with gY (x). In this case, near the boundary, we have c

σ (∆) = ξ 2 + σ (∆Y (x)).

Ellipticity and boundary spectrum c c ∗ An operator A ∈ x−m Diff m b (M ; E) is c-elliptic if σ (A) is invertible on T M \0. Moreover, the family A − λ is said to be c-elliptic with parameter λ ∈ Λ ⊂ C if c σ (A) − λ is invertible on (c T ∗ M × Λ)\0. Associated with A = x−m P there is an operator-valued polynomial

C
σ → Pˆ (σ) ∈ Diff m (Y ; E|Y ) called the conormal symbol of P (and of A). If we write A as in (1.3), then
akα (0, y)σ k Dyα . Pˆ (σ) = k+|α|≤m

If A is c-elliptic, then Pˆ (σ) is invertible for all σ ∈ C except a discrete set specb (A), the boundary spectrum of A, cf. [14]; Pˆ (σ) is a holomorphic family of elliptic operators on Y and σ → Pˆ (σ)−1 is a meromorphic operator-valued function on C. Example. The Laplacian (2.1) is clearly c-elliptic. If y is the angular variable on S 1 , then Pˆ (σ) = σ 2 + ∆Y (0) = σ 2 + Dy2 , and the boundary spectrum of ∆ is given by specb (∆) = {±ik : k ∈ N0 }. Closed extensions Let m be a positive b-density on M , that is, xm is a smooth everywhere positive the density on M . Let L2b (M ; E) be the L2 space of sections of E with respect to ◦ Hermitian form on E and the density m. Consider A initially defined on C0∞ (M ; E) and look at it as an unbounded operator on the Hilbert space x−m/2 L2b (M ; E) = L2 (M ; E; xm m). The particular weight x−m/2 is just a convenient normalization and represents no loss. If we are interested in A on xµ L2b (M ; E) for µ ∈ R, we can base all our analysis on the space x−m/2 L2b (M ; E) by considering the operator x−µ−m/2 A xµ+m/2 . Typically, A has a large class of closed extensions AD : D ⊂ x−m/2 L2b (M ; E) → x−m/2 L2b (M ; E).

(2.2)

36

J.B. Gil, T. Krainer and G.A. Mendoza

There are two canonical closed extensions, namely the ones with domains ◦

Dmin (A) = closure of C0∞ (M ; E) with respect to  · A , Dmax (A) = {u ∈ x−m/2 L2b (M ; E) : Au ∈ x−m/2 L2b (M ; E)}, where uA = u + Au is the graph norm in Dmax (A). Both domains are dense in x−m/2 L2b (M ; E), and for any closed extension (2.2), Dmin (A) ⊆ D ⊆ Dmax (A). Let D(A) = {D ⊂ Dmax (A) : D is a vector space and Dmin (A) ⊂ D}. The elements of D(A) are in one-to-one correspondence with the subspaces of Dmax (A)/Dmin (A). If the operator A is fixed and there is no possible ambiguity, we will omit A from the notation and will write simply Dmin , Dmax , and D. Theorem 2.1 (Lesch [12]). If A ∈ x−m Diff m b (M ; E) is c-elliptic, then dim Dmax /Dmin < ∞ and all closed extensions of A are Fredholm. Moreover, ind AD = ind ADmin + dim D/Dmin .

(2.3)

Modulo Dmin , the elements of Dmax are determined by their asymptotic behavior near the boundary of M . The structure of these asymptotics depends on the conormal symbols of A and on the part of specb (A) in the strip {|σ| < m/2}. More details will be discussed in the next section. Corollary 2.2. If A is c-elliptic and symmetric (formally selfadjoint), then 1 ind ADmax = − ind ADmin and ind ADmin = − dim Dmax /Dmin . 2 Example. Consider the cone Laplacian ∆, cf. (2.1). Then (3.2) and (3.6) imply dim Dmax (∆)/Dmin (∆) = 2 and thus, by the previous corollary, ind ∆min = −1 and

ind ∆max = 1.

(2.4)

−m/2 2 If A ∈ x−m Diff m Lb (M ; E) b (M ; E) is c-elliptic, the embedding Dmax → x is compact. Therefore, for every D ∈ D and λ ∈ C, the operator AD − λ is also Fredholm with ind(AD − λ) = ind AD . Consequently, if spec(AD ) = C, then we necessarily have ind AD = 0. For this reason, we will primarily be interested in the set of domains G = {D ∈ D : ind AD = 0} (2.5)

which is empty unless ind ADmin ≤ 0 and ind ADmax ≥ 0. Let d = − ind ADmin . Using that the map D
D → D/Dmin is a bijection, we identify G with the complex Grassmannian of d -dimensional subspaces of Dmax /Dmin .

Rays of Minimal Growth Example. For ∆ we have

37

G(∆) ∼ = CP1 = S 2 .

Note that by (2.3) and (2.4), ind ∆D = 0 if and only if dim D/Dmin = 1. We finish this section with the following proposition that gives a first glimpse of the complexity of the spectrum of elliptic cone operators. Proposition 2.3. If A is c-elliptic and dim G > 0, then for any λ ∈ C there is a domain D ∈ G such that λ ∈ spec(AD ). If, in addition, A is symmetric on Dmin , then for any λ ∈ R there is a D ∈ G such that AD is selfadjoint and λ ∈ spec(AD ). A proof is given in [6, Propositions 5.7 and 6.7]. A surprising consequence of the second statement is that for any arbitrary negative number λ there is always a selfadjoint extension of A having λ as eigenvalue, even if A is positive on Dmin .

3. The model operator Let A ∈ x−m Diff m b (M ; E) be c-elliptic. The model operator A∧ associated with A is an operator on N+ Y , the closed inward normal bundle of Y , that in local coordinates takes the form
A∧ = x−m akα (0, y)(xDx )k Dyα , k+|α|≤m

if A is written as in (1.3). A Taylor expansion in x (at x = 0) of the coefficients of the operator A induces a decomposition xm A =

N −1


Pk xk + xN P˜N

for every N ∈ N,

(3.1)

k=0

where each Pk has coefficients independent of x near Y . Thus the model operator can be written, near Y , as A∧ = x−m P0 . In other words, A∧ can be thought of as the “most singular” part of A. ∧ We trivialize N+ Y as Y ∧ = [0, ∞)×Y . The operator A∧ ∈ x−m Diff m b (Y ; E) ◦ ∞ ∧ acts on C0 (Y ; E) and can be extended as a densely defined closed operator in x−m/2 L2b (Y ∧ ; E). The space L2b (Y ∧ ; E) is the L2 space with respect to a density ∗ ∗ of the form dx x ⊗ π mY and the canonically induced Hermitian form on π (E|Y ), ∧ where π : Y → Y is the projection on the factor Y . The density mY is related to m and, by abuse of notation, we denote π ∗ (E|Y ) by E, cf. [6]. Again, there are two canonical domains D∧,min and D∧,max and we denote by D∧ the set of subspaces of D∧,max that contain D∧,min . There is a natural (and useful) linear isomorphism θ : Dmax /Dmin → D∧,max /D∧,min , cf. Section 5. As a consequence we have dim D∧,max /D∧,min = dim Dmax /Dmin

(3.2)

38

J.B. Gil, T. Krainer and G.A. Mendoza

which by Theorem 2.1 is finite. It is known (cf. Lesch [12]) that D∧,max /D∧,min is isomorphic to a finite-dimensional space E∧,max consisting of functions of the form m  σ

k ϕ= cσ,k (y) log x xiσ (3.3) σ∈specb (A) | σ|<m/2

k=0

where cσ,k ∈ C ∞ (Y ; E). More precisely, for every u ∈ D∧,max there is a function ϕ ∈ E∧,max such that u(x, y) − ω(x)ϕ(x, y) ∈ D∧,min for some (hence any) cut-off function ω ∈ C0∞ ([0, 1)), ω = 1 near 0. The function ϕ is uniquely determined by the equivalence class u + D∧,min. We identify E∧,max = D∧,max /D∧,min and let π∧,max : D∧,max → E∧,max be the canonical projection. Contrary to the situation in Theorem 2.1, the closed extensions of A∧ do not need to be Fredholm. However, if A − λ is c-elliptic with parameter, then the canonical extensions A∧,min − λ and A∧,max − λ are both Fredholm for λ = 0, cf. [7, Remark 5.26]. Moreover, we have ind(A∧,min − λ) = ind ADmin ,

(3.4)

cf. Corollary 5.35 in [7]. Example. On Y ∧ = [0, ∞) × S 1 with the cone metric dx2 + x2 dy 2 , the LaplaceBeltrami operator is given by   ∆∧ = x−2 (xDx )2 + ∆Y , (3.5) where ∆Y is the nonnegative Laplacian on S 1 . ∆∧ is precisely the model operator associated with the cone Laplacian ∆ discussed in the previous section, cf. (2.1). It is easy to check that for any cut-off function ω ∈ C0∞ ([0, 1)), the functions ω(x) · 1, ∆∧ (ω(x) · 1), ω(x) log x, and ∆∧ (ω(x) log x) are all in the space x−1 L2b (Y ∧ ). Thus ω(x)·1 and ω(x) log x are elements of D∧,max . In fact, E∧,max = span{1, log x}. (3.6) Observe that ∆ − λ is c-elliptic with parameter λ ∈ C\R+ and therefore the closed extensions of ∆∧ − λ are Fredholm for every λ ∈ C\R+ . The model operator has a dilation/scaling property that can be exploited to analyze its closed extensions and their resolvents from a geometric point of view. In order to describe this property we first introduce the one-parameter group of isometries R+
 → κ : x−m/2 L2b (Y ∧ ; E) → x−m/2 L2b (Y ∧ ; E)

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39

which on functions is defined by (κ f )(x, y) = m/2 f (x, y).

(3.7)

It is easily verified that the operator A∧ satisfies the relation κ A∧ = −m A∧ κ . This implies

A∧ − λ = m κ (A∧ − λ/m )κ−1 (3.8)

for every  > 0 and λ ∈ C. This homogeneity property, called κ-homogeneity, will be used systematically to describe the closed extensions of A∧ with nonempty resolvent sets. It is convenient to introduce the set bg-res A∧ = {λ ∈ C : A∧,min − λ is injective and A∧,max − λ is surjective}, the background resolvent set of A∧ , cf. [6]. Lemma 3.1 (Lemma 7.3 in [6]). If λ ∈ bg-res A∧ and D ∈ D∧ , then A∧,D − λ is Fredholm. The set bg-res A∧ is a disjoint union of open sectors,  ◦ Λα . bg-res A∧ = α∈I⊂N

This lemma follows immediately from (3.8). For λ ∈ bg-res A∧ and D ∈ D∧ we have ind(A∧,D − λ) = ind(A∧,min − λ) + dim D/D∧,min . Moreover, the map

(3.9)

◦

Λα
λ → ind(A∧,D − λ) is constant since the embedding D → x−m/2 L2b (Y ∧ ; E) is continuous. Now, in analogy with (2.5) we define ◦

G∧,α = {D ∈ D∧ : ind(A∧,D − λ) = 0 for λ ∈ Λα } ◦

and let dα = − ind(A∧,min − λ) for λ ∈ Λα . We identify G∧,α with the complex Grassmannian of dα -dimensional subspaces of E∧,max . The canonical domains D∧,min and D∧,max are both κ-invariant. Thus the group action κ induces an action on E∧,max . In general, κ does not preserve the elements of D∧ . In fact, the set of κ-invariant domains in D∧ is an analytic variety because it consists of the stationary points of a holomorphic flow, cf. Section 7 in [6]. To ◦better analyze the resolvents of the closed extensions of A∧ over the open sector Λα , we will consider the manifold G∧,α together with the flow generated by the induced action of κ given by κ (D/D∧,min ) = κ (D)/D∧,min . Example. The background resolvent set of ∆∧ is the open sector C\R+ ; this is easily seen after noting that ∆∧ is the standard Laplacian in R2 written in polar coordinates. Moreover, since ind(∆∧,min − λ) = −1 for every λ ∈ C\R+ , we have that D ∈ D∧ belongs to G∧ if and only if dim D/D∧,min = 1. Thus G∧ ∼ = CP1 = S 2 .

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J.B. Gil, T. Krainer and G.A. Mendoza

We identify E∧,max with D∧,max /D∧,min and use (3.6) to write E∧,max = span {1, log x} . For D ∈ G∧ we then have π∧,max D = span {ζ0 · 1 + ζ1 log x} for some ζ0 , ζ1 ∈ C, (ζ0 , ζ1 ) = 0.

(3.10)

Hence, with κ as defined in (3.7), we get π∧,max κ −1 D = span{(ζ0 − ζ1 log ) · 1 + ζ1 log x}.

(3.11)

Clearly, the only κ-invariant domain in G∧ is the domain DF such that π∧,max DF = span{1}; DF is precisely the domain of the Friedrichs extension of ∆∧ , cf. [8]. Every domain D ∈ G∧ with ζ1 = 0 in (3.10) generates a nontrivial orbit as given by (3.11). In order to describe the flow of κ on these nonstationary points, rewrite (3.10) as  π∧,max D = span ζζ01 · 1 + log x . Then the projection to E∧,max of the dilation κ−1

D is given by    π∧,max κ −1 D = span ζζ01 − log  · 1 + log x .

(3.12)

If [ζ0 : ζ1 ] ∈ CP1 is the point corresponding to D, then κ−1

D is represented by [ζ0 − ζ1 log  : ζ1 ]. In other words, in the situation at hand, the flow generated by κ on G∧ ∼ = CP1 consists of curves that in projective coordinates are lines parallel to the real axis, see Figure 1.

C r

r ζ0 /ζ1

Figure 1. Orbit in G∧ (∆∧ ) generated by D ↔ ζ0 /ζ1 ∈ C. Observe that the Friedrichs extension corresponds to the point [1 : 0] ∈ CP1 . Using  π∧,max κ −1 D = span 1 + ζ0 −ζζ11log log x if  = eζ0 /ζ1

Rays of Minimal Growth

41

we see that π∧,max κ −1 D → span{1} = π∧,max DF as  → ∞ or  → 0.

(3.13)

Let D0 ∈ G∧ be such that π∧,max D0 = span{log x}. This domain gives a selfadjoint extension of ∆∧ which on the sphere corresponds to the point [0 : 1]. The circle consisting of the orbit of D0 together with DF is the set of domains of selfadjoint extensions of ∆∧ .

4. Ray conditions on the model cone In this section we will discuss the existence of sectors of minimal growth for the ∧ model operator A∧ ∈ ◦x−m Diff m b (Y ; E) associated with a c-elliptic cone operator. We fix a component Λα of bg-res A∧ and let G∧ = G∧,α . ◦ Let Λ be a closed sector such that Λ\0 ⊂ Λα , cf. (1.1), let res A∧,D be the resolvent set of A∧,D . The κ-invariant domains are the simplest domains to analyze. Proposition ◦4.1 (Proposition 8.4 in [6]). Suppose D ∈ G◦∧ is κ-invariant. If there exists λ0 ∈ Λα such that A∧,D − λ0 is invertible, then Λα ⊂ res A∧,D and Λ is a sector of minimal growth for A∧,D . If D ∈ G∧ is not κ-invariant, the situation is more complicated. Nonetheless, in [6] we found a condition necessary and sufficient for a sector Λ to be a sector of minimal growth for A∧,D . This condition is expressed in terms of finite-dimensional spaces and projections that we proceed to discuss briefly. For λ ∈ bg-res A∧ we let K∧,λ = ker(A∧,max − λ). Then res A∧,D = bg-res A∧ ∩ {λ : K∧,λ ∩ D = 0}, and for λ ∈ res A∧,D we have D∧,max = K∧,λ ⊕ D.

(4.1)

Projecting on E∧,max , this direct sum induces the decomposition E∧,max = π∧,max K∧,λ ⊕ π∧,max D,

(4.2)

and the projection on π∧,max K∧,λ according to (4.2) is given by the map π ˆK∧,λ ,D : E∧,max → E∧,max (u + D∧,min ) → πK∧,λ ,D u + D∧,min ,

(4.3)

where πK∧,λ ,D is the projection on K∧,λ according to (4.1). The following theorem gives a condition on the operator norm of (4.3) for a sector Λ to be a sector of minimal growth for A∧,D . Define u2λ = u2 + |λ|−2 A∧ u2 for λ = 0 and u ∈ D∧,max .

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J.B. Gil, T. Krainer and G.A. Mendoza ◦

Theorem 4.2. Let D ∈ G∧ , let Λ be a closed sector with Λ\0 ⊂ Λα . Then Λ is a sector of minimal growth for A∧,D if and only if there are C, R > 0 such that ΛR ⊂ res A∧,D , and



π (4.4) ˆK∧,λ ,D L (E∧,max ,· ) ≤ C for λ ∈ ΛR , λ

where π ˆK∧,λ ,D is the projection (4.3). This theorem is a rephrasing of [6, Theorem 8.7]. There the condition (4.4) appears in the equivalent form



π ˆK ˆ ,κ−1 D L (E∧,max ) ≤ C for λ ∈ ΛR , (4.4 ) ∧,λ

ˆ = λ/|λ|, and π where λ ˆK

|λ|1/m

−1 D ˆ ,κ ∧,λ |λ|1/m

is the projection on K∧,λˆ induced (following

the steps (4.1)–(4.3)) by the direct sum D∧,max = K∧, −m λ ⊕ κ−1

D

(4.5)

for λ ∈ res A∧,D and  > 0. This decomposition is a consequence of (4.1) and the κ-invariance of D∧,max , as follows. First, the κ-homogeneity of A∧ − λ, cf. (3.8), implies κ−1

(K∧,λ ) = K∧, −m λ for  > 0. Furthermore, if D ∈ G∧ and λ ∈ bg-res A∧ , then ⇐⇒ λ ∈ res A∧,D . −m λ ∈ res A∧,κ−1  D In particular, K∧, −m λ ∩ κ−1

D = {0} ⇐⇒ K∧,λ ∩ D = {0}, as claimed. The equivalence of (4.4) and (4.4 ) follows immediately from the identity π κ 1/m = πK κ−1 |λ|1/m K∧,λ ,D |λ|

−1 D ˆ ,κ ∧,λ |λ|1/m

using the relation (3.8) and the fact that κ is an isometry on x−m/2 L2b . The virtue of (4.4 ) is that the norm is fixed, while the advantage of (4.4) lies in that it gives a more explicit dependence on λ and deals with a projection on a subspace of E∧,max with fixed complement π∧,max D. In [6, Corollary 8.22] it is proved that Λ is a sector of minimal growth for A∧,D if and only if there are constants C, R > 0 such that ΛR ⊂ res A∧,D and

−1

κ 1/m (A∧,D − λ)−1

≤ C/|λ|, λ ∈ ΛR . |λ| L (x−m/2 L2 ,D∧,max ) b

It can be shown that this estimate is equivalent to (4.4) and (4.4 ). Example. We consider again the model Laplacian ∆∧ from the previous section. Recall that bg-res ∆∧ = C\R+ . For λ ∈ C\R+ , we have π∧,max K∧,λ = span {−k0 log(−λ) + k1 log x} for some k0 , k1 > 0,

Rays of Minimal Growth

43

where log means the principal branch of the logarithm. Moreover, by (3.12),    π∧,max κ −1 D = span ζζ01 − log  · 1 + log x . The projection in (4.4 ) can be computed explicitly. Namely, if u = α0 + α1 log x ∈ E∧,max = span{1, log x} and λ = m λ0 , then π ˆK∧,λ

,κ−1  D 0

u=

−α0 + α1 ( ζζ01 − log ) k0 log(−λ0 ) + k1 ( ζζ01 − log )

(−k0 log(−λ0 ) + k1 log x) .

(4.6)

Let Λ be a closed sector in C\R+ containing the half-plane {λ < 0}. Since the family of projections (4.6) is bounded as  → ∞, uniformly for |λ0 | = 1 in Λ, regardless of the specific choice of α0 , α1 , Theorem 4.2 implies that every closed extension ∆∧,D , D ∈ G∧ , of the model Laplacian admits Λ as a sector of minimal growth. Equivalent geometric condition We identify G∧ with the Grassmannian Grd

(E∧,max ) where d = − ind(A∧,min −λ) ◦ for λ ∈ Λα ⊂ bg-res A∧ . Let d = dim K∧,λ . The condition that in the Grassmannian Grd

(E∧,max ), the curve [R, ∞)
 → π∧,max κ−1

D does not approach the set VK∧,λ = {D ∈ Grd

(E∧,max ) : D ∩ π∧,max K∧,λ = 0} as  → ∞, is sufficient for the validity of (4.4 ). This is [6, Theorem 8.28]. The following theorem states that the condition is also necessary. For D ∈ Grd

(E∧,max ) let Ω− (D) = D ∈ Grd

(E∧,max ) : ∃ {k }∞ k=1 ⊂ R+ such that

 k → ∞ and κ−1

k D → D as k → ∞ . ◦

Theorem 4.3. Let λ0 ∈ Λα . The ray through λ0 is a ray of minimal growth for A∧,D if and only if Ω− (π∧,max D) ∩ VK∧,λ0 = ∅. ◦

Proof. Let λ0 ∈ Λα and D ∈ G∧ . For simplicity, we use the notation D = π∧,max D,

V = VK∧,λ0 ,

K = π∧,max K∧,λ0 and πK,D = π ˆK∧,λ0 ,D .

Suppose Ω− (D) ∩ V = ∅. Since Ω− (D) and V are closed sets, there are a neighborhood U of V and a constant that if  > R then κ−1

D ∈ U. Then

R > 0 such

is uniformly bounded as  → ∞, and Lemma 5.24 in [6] gives that πK,κ−1  D therefore, by Theorem 4.2 the ray through λ0 is a ray of minimal growth for A∧,D . Assume now that there are C, R > 0 such that ΛR ⊂ res A∧,D and the condition (4.4 ) is satisfied. Suppose Ω− (D) ∩ V = ∅ and let D0 ∈ Ω− (D) ∩ V . Since D0 ∈ V , we have D0 ∩ K = {0}. On the other hand, D0 ∈ Ω− (D) implies −1 that there is a sequence {k }∞ k=1 ⊂ R+ such that k → ∞ and Dk = κ k D → D0

44

J.B. Gil, T. Krainer and G.A. Mendoza

as k → ∞. Note that for k large we have m k λ0 ∈ res A∧,D , so λ0 ∈ res A∧,κ−1 k D and therefore, Dk ∈ V . Pick v ∈ D0 ∩ K with v = 1. Let πDk denote the orthogonal projection on Dk . Since Dk → D0 as k → ∞, we have πDk → πD0 , so vk = πDk v → πD0 v = v as k → ∞. Since Dk ∈ V , vk − v = 0 and πK,Dk vk = 0. Hence   v v − vk = → ∞ as k → ∞, πK,Dk v − vk  v − vk  since v = 1 and vk → v as k → ∞. But this implies that πK,Dk  → ∞ contradicting the boundedness of the norm in (4.4 ). Thus Ω− (D) ∩ V = ∅.
Example. Let ∆∧ be the model Laplacian and let D ∈ G∧ . In this case, the limiting set Ω− (π∧,max D) consists of the one element of CP1 corresponding to the Friedrichs extension of ∆∧ , cf. (3.13). From this new perspective, it is evident that every closed extension ∆∧,D of ∆∧ with D ∈ G∧ must admit a sector of minimal growth.

5. Rays of minimal growth We continue to assume that A ∈ x−m Diff m b (M ; E) is c-elliptic. Unlike the case of a differential operator with smooth coefficients on a closed manifold, that a ray Γ is a ray of minimal growth for the principal symbol c σ (A) of A is not expected to imply that Γ is a ray of minimal growth for A. In this context, it is useful to think of A∧ as a symbol (the wedge symbol) associated with A, cf. Schulze [16], so that it is natural to impose ray conditions on A∧ . For this to work, however, we need a procedure to transfer the information on the given domain D of A on M to equivalent information for A∧ on Y ∧ , and vice versa. Theorem 5.1 (Theorem 4.12 in [6]). There is a natural isomorphism θ−1 : D∧,max /D∧,min → Dmax /Dmin given by a finite iterative procedure that involves the boundary spectrum of A and the decomposition (3.1). In particular, if A has coefficients independent of x near Y , then θ is the identity map. A constructive proof of this theorem can be found in [6]. For illustration purposes, we will give here only a simplified description of θ−1 and will make some convenient identifications that can be justified via the Mellin transform, cf. [6, 7]. According to the decompositions (1.3) and (3.1) we define
1 (∂ k ajα )(0, y)(σ + ik)j Dyα . Pˆk (σ) = k! x j+|α|≤m

Note that Pˆ0 (σ) is the conormal symbol of A. Again, we identify D∧,max /D∧,min with the space E∧,max of singular functions of the form (3.3). As in loc. cit., we let

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45

E∧,σ0 be the space of all functions in E∧,max that are of the form ψ=

σ0 m


 cσ0 ,k (y) logk x xiσ0 .

(5.1)

k=0

These spaces give a splitting E∧,max =



E∧,σ0 ,

σ0 ∈Σ

where Σ = specb (A) ∩ {σ ∈ C : −m/2 < (σ) < m/2}. Every function ψ ∈ E∧,σ0 of the form (5.1) corresponds (via the Mellin transform) to a meromorphic function of the form  ψ(σ) =

mσ 0


(−i)k−1 k! cσ ,k (y) 0 . (σ − σ0 )k+1 k=0

We use this correspondence ψ → ψ to define linear operators ◦

eσ0 ,k : E∧,σ0 → C ∞ (Y ∧ ; E) through the following inductive algorithm. For ψ ∈ E∧,σ0 we let eσ0 ,0 (ψ) = ψ, and given eσ0 ,0 , . . . , eσ0 ,ϑ−1 for some ϑ ∈ N, we define eσ0 ,ϑ (ψ) to be the unique singular function of the form σ0 −iϑ m
 cσ0 −iϑ,k (y) logk x xi(σ0 −iϑ) k=0

such that ∧ eσ0 ,ϑ (ψ) (σ) + Pˆ0 (σ)−1

ϑ 


 ∧ Pˆk (σ)eσ0 ,ϑ−k (ψ) (σ + ik)

k=1

is holomorphic at σ = σ0 − iϑ. Finally, we define θ−1 on E∧,σ0 as the sum θ−1 |E∧,σ0 =




N (σ0 )

eσ0 ,k ,

(5.2)

k=0

where N (σ0 ) ∈ N0 is the largest integer such that σ0 − N◦ (σ0 ) ≥ −m/2. Analogous to Section 3, the range Emax of θ−1 in C ∞ (Y ∧ ; E) can be identified with the quotient Dmax /Dmin via the mapping Emax
ψ → ωψ + Dmin , where ω ∈

C0∞ ([0, 1))

is an arbitrary cut-off function near zero.

Example. Let M be a compact 2-manifold with boundary Y = S 1 . Let A be a differential operator in x−2 Diff 2b (M ) that over the interior of M coincides with some Laplacian, and near Y , is of the form   A = x−2 (xDx )2 + q(x)∆Y ,

46

J.B. Gil, T. Krainer and G.A. Mendoza

where ∆Y is the standard nonnegative Laplacian on S 1 and q is a smooth function. We assume q to have the form q(x) = α2 + βx + x2 γ(x), where α, β are constants with 12 < α < 1, β = 0, and γ is smooth up to x = 0. The associated model operator is then given by   A∧ = x−2 (xDx )2 + α2 ∆Y , and specb (A) = {±iαk : k ∈ N0 }. Since 12 < α < 1, only the set {−iα, 0, iα} is relevant for the spaces Emax and E∧,max , cf. (3.3). If y denotes the angular variable on S 1 , then E∧,−iα = span{e−iy xα , eiy xα }, E∧,0 = span{1, log x}, E∧,iα = span{e−iy x−α , eiy x−α }, and E∧,max = E∧,−iα ⊕ E∧,0 ⊕ E∧,iα . Moreover, we have Pˆ0 (σ) = σ 2 + α2 ∆Y and Pˆ1 (σ) = β∆Y . ˆ = 0. Thus e0,1 (ψ) = 0 Since every ψ ∈ E∧,0 is independent of y, we have Pˆ1 (σ)ψ(σ) and so θ−1 |E∧,0 = e0,0 is the identity. On the other hand, N (−iα) = 0 and therefore θ−1 |E∧,−iα = e−iα,0 is also the identity. Finally, for ψ = e±iy x−α we have   β β xψ, so θ−1 ψ = 1 − x ψ. e−iα,1 (ψ) = − 2α − 1 2α − 1 In other words,   β Emax = span 1, log x, e±iy xα , e±iy x−α 1 − 2α−1 x . The map θ induces an isomorphism Θ : D → D∧ that we use to define D∧ = ΘD for any given D ∈ D. The operator A∧,D∧ is the closed extension of A∧ in x−m/2 L2b (Y ∧ ; E) uniquely associated with AD . As in [7, Section 6], and motivated by the importance of κ in studying the model operator A∧ , we introduce on Dmax (A)/Dmin (A) the one-parameter group κ ˜ = θ−1 κ θ for  > 0. Similar to the situation on the model cone, the spectrum and resolvent of the closed extensions of A can be geometrically analyzed by considering the manifold G, cf. (2.5), together with the flow generated by κ ˜ . An interesting consequence of Theorem 5.1 is the following. Proposition 5.2. If A − λ is c-elliptic with parameter λ = 0, then ind(A∧,D∧ − λ) = ind AD .

Rays of Minimal Growth

47

Proof. The existence of θ implies dim D∧ /D∧,min = dim D/Dmin . Now, the proposition follows by combining this identity with the relative index formulas (2.3) and (3.9), together with the equation (3.4).
The following theorem describes the pseudo-differential structure of the resolvent of a cone operator A and gives tangible conditions over a given sector Λ on the symbols c σ (A) and A∧ for A to have Λ as a sector of minimal growth. Theorem 5.3 (Theorem 6.9 in [7]). Let A ∈ x−m Diff m b (M ; E) be such that A − λ is c-elliptic with parameter λ ∈ Λ. If Λ is a sector of minimal growth for A∧,D∧ , then it is a sector of minimal growth for AD . Moreover, (AD − λ)−1 = B(λ) + GD (λ), where B(λ) is a parametrix of ADmin −λ with B(λ)(ADmin −λ) = 1 for λ sufficiently large, and GD (λ) is a pseudo-differential regularizing operator of finite rank. The following lemma gives further information about the behavior at large of the resolvent along a sector of minimal growth. Given two cut-off functions ω0 and ω1 , the notation ω1 ≺ ω0 will indicate that ω0 = 1 in a neighborhood of the support of ω1 . Lemma 5.4. Let A ∈ x−m Diff m b (M ; E) be c-elliptic and let Λ be a sector of minimal growth for AD . For every pair of cut-off functions ω1 ≺ ω0 , supported near the boundary, we have   (1 − ω0 )(AD − λ)−1 ω1 ∈ S Λ, L (x−m/2 L2b , Dmax ) , where S stands for Schwartz (rapidly decreasing as |λ| → ∞). Proof. Since Λ is a sector of minimal growth for AD , the family A − λ must be c-elliptic with parameter λ ∈ Λ, and A∧,min − λ must be injective for every λ ∈ Λ, λ = 0. A proof of this can be found in [7, Theorem 4.1]. As a consequence (cf. [7, Section 5]), there is a parametrix B(λ) such that B(λ)(ADmin − λ) = 1 for large λ ∈ Λ, and   (5.3) (1 − ω0 )B(λ)ω1 ∈ S Λ, L (x−m/2 L2b , Dmax ) for all cut-off functions ω1 ≺ ω0 supported near the boundary. We now make use of the identity (AD − λ)−1 = B(λ) + (1 − B(λ)(A − λ))(AD − λ)−1 . Multiplying by (1 − ω0 ) from the left and by ω1 from the right, (5.3) proves the assertion for the first term involving B(λ). On the other hand, since 1−B(λ)(A−λ) vanishes on Dmin for large λ, we have for such λ, (1 − ω0 )(1 − B(λ)(A − λ)) = (1 − ω0 )(1 − B(λ)(A − λ))ω2 = −(1 − ω0 )B(λ)(A − λ)ω2 = −(1 − ω0 )B(λ)ω1 (A − λ)ω2

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J.B. Gil, T. Krainer and G.A. Mendoza

whenever ω2 ≺ ω1 . Thus, by (5.3), (1 − ω0 )(1 − B(λ)(A − λ)) : Dmax → Dmax is rapidly decreasing as |λ| → ∞. Finally, the assertion of the lemma can be completed using the fact that (AD − λ)−1 ω1 : x−m/2 L2b → Dmax is uniformly bounded.
Necessity of the conditions The converse of Theorem 5.3 involves proving that the minimal growth of the resolvent (AD − λ)−1 over a sector Λ implies a corresponding behavior for the inverse of c σ (A) − λ and for the resolvent (A∧,D∧ − λ)−1 . While in [7, Theorem 4.1] we established the necessity of the condition on c σ (A), we did not address the question whether Λ must necessarily be a sector of minimal growth for A∧,D∧ . In the next theorem we prove that this is indeed the case when A has coefficients independent of x near Y = ∂M . Theorem 5.5. Let A ∈ x−m Diff m b (M ; E) be c-elliptic with coefficients independent of x near Y . If Λ is a a sector of minimal growth for AD , then A − λ is c-elliptic with parameter λ ∈ Λ, and Λ is a sector of minimal growth for A∧,D∧ . Proof. As stated in the proof of Lemma 5.4, the assumption on the resolvent of AD implies that A − λ is c-elliptic with parameter λ ∈ Λ and that A∧,min − λ is injective for every λ = 0. Thus we only need to prove the statement about A∧,D∧ . By Proposition 5.2, and since ind AD = 0, we have ind(A∧,D∧ − λ) = 0 for λ = 0. For this reason, in order to show that Λ is a sector of minimal growth for A∧,D∧ , it suffices to find (for large λ ∈  Λ) a right-inverse of A∧,D∧ − λ that is uniformly bounded in L x−m/2 L2b , D∧ as |λ| → ∞. Since A is assumed to have coefficients independent of x near the boundary, there is a cut-off function ω0 such that Aω0 = A∧ ω0

and ω0 D = ω0 D∧ .

Let ω1 , ω2 be cut-off functions with ω2 ≺ ω1 ≺ ω0 . Then the operator B(λ) = ω1 (AD − λ)−1 ω2 can be regarded as an operator on M with values in D or as an operator on Y ∧ with values in D∧ . Depending on the context we will write B(λ) as BD (λ) : x−m/2 L2b (M ; E) → D

or BD∧ (λ) : x−m/2 L2b (Y ∧ ; E) → D∧ .

On M we consider (AD − λ)BD (λ) = ω0 (AD − λ)ω1 (AD − λ)−1 ω2 = ω2 − ω0 (AD − λ)(1 − ω1 )(AD − λ)−1 ω2 = ω2 + R(λ) with R(λ) = −ω0 (AD − λ)(1 − ω1 )(AD − λ)−1 ω2 . By Lemma 5.4, R(λ) is rapidly decreasing in the norm as |λ| → ∞.

Rays of Minimal Growth

49

Because of the presence and nature of the cut-off functions  ω0 and ω2 , R(λ)  can also be regarded as an operator on Y ∧ , say R∧ (λ) ∈ S Λ, L (x−m/2 L2b ) . Now, using that (AD − λ)ω1 = (A∧,D∧ − λ)ω1 , we get on Y ∧ the identity (A∧,D∧ − λ)BD∧ (λ) = ω2 + R∧ (λ).

(5.4)

Furthermore, we have BD∧ (λ)L (x−m/2 L2b ,D∧,max ) = O(1) as |λ| → ∞, since, by assumption, BD (λ)L (x−m/2 L2b ,Dmax ) has the same asymptotic behavior. On the other hand, as A− λ is c-elliptic with parameter, by [7, Theorem 5.24] there is a family of pseudo-differential operators B2,∧ (λ) : x−m/2 L2b → D∧,min (uniformly bounded in λ) such that (A∧ − λ)B and for  2,∧ (λ) − 1 is regularizing,  ω3 ≺ ω2 , the families ω3 B2,∧ (λ)(1 − ω2 ) and (A∧ − λ)B2,∧ (λ) − 1 (1 − ω2 ) are rapidly decreasing in the norm as |λ| → ∞. Thus, as A∧,D∧ (1 − ω3) = A∧ (1 − ω3), (A∧,D∧ − λ)(1 − ω3 )B2,∧ (λ)(1 − ω2 ) = (1 − ω2 ) + S∧ (λ)   with S∧ (λ) ∈ S Λ, L (x−m/2 L2b ) . Finally, the operator family

(5.5)

Q∧ (λ) = BD∧ (λ) + (1 − ω3 )B2,∧ (λ)(1 − ω2 ) : x−m/2 L2b → D∧,max is bounded in the norm as |λ| → ∞ and by (5.4) and (5.5) we have   (A∧,D∧ − λ)Q∧ (λ) − 1 ∈ S Λ, L (x−m/2 L2b ) . By a Neumann series argument, it follows that A∧,D∧ − λ : D∧ → x−m/2 L2b has a uniformly bounded right-inverse for large λ ∈ Λ.
Acknowledgment The new results contained herein reflect part of work carried out by the three authors at the Mathematisches Forschungsinstitut Oberwolfach under their “Research in Pairs” program. They gratefully acknowledge the Institute’s support and hospitality.

References [1] M. Agranovich and M. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Russ. Math. Surveys 19 (1963), 53–159. [2] S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math. 15 (1962), 119–147. [3] J. Br¨ uning and R. Seeley, The expansion of the resolvent near a singular stratum of conical type, J. Funct. Anal. 95 (1991), 255–290. [4] J. Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), 2103–2106. [5] J. Gil, Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators, Math. Nachr. 250 (2003), 25–57. [6] J. Gil, T. Krainer, and G. Mendoza, Geometry and spectra of closed extensions of elliptic cone operators, Canad. J. Math., to appear.

50

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[7] J. Gil, T. Krainer, and G. Mendoza, Resolvents of elliptic cone operators, J. Funct. Anal. 241 (2006), no. 1, 1–55. [8] J. Gil and G. Mendoza, Adjoints of elliptic cone operators, Amer. J. Math. 125 (2003), no. 2, 357–408. [9] G. Grubb, Functional calculus of pseudodifferential boundary problems, Second Edition, Birkh¨ auser, Basel, 1996. [10] V. Kondrat’ev, Boundary problems for elliptic equations in domains with conical or angular points, Trans. Mosc. Math. Soc. 16 (1967), 227–313. [11] T. Krainer, Resolvents of elliptic boundary problems on conic manifolds, to appear in Communications in PDE. [12] M. Lesch, Operators of Fuchs type, conical singularities, and asymptotic methods, B.G. Teubner, Stuttgart, Leipzig, 1997. [13] P. Loya, On the resolvent of differential operators on conic manifolds, Comm. Anal. Geom. 10 (2002), no. 5, 877–934. [14] R. Melrose, Transformation of boundary value problems, Acta Math. 147 (1981), 149–236. [15] E. Schrohe and J. Seiler, The resolvent of closed extensions of cone differential operators, Canad. J. Math. 57 (2005), 771–811. [16] B.-W. Schulze, Pseudo-differential operators on manifolds with edges, in Proc. Symp. Partial Differential Equations, Holzhau 1988 (Leipzig), Teubner-Texte zur Math. Vol. 112, Teubner, 1989, 259–288. [17] R. Seeley, Complex powers of an elliptic operator, in Singular Integrals, Amer. Math. Soc., Providence, 1967, 288–307. Juan B. Gil Department of Mathematics and Statistics Penn State Altoona 3000 Ivyside Park Altoona, PA 16601, USA e-mail: [email protected] Thomas Krainer Institut f¨ ur Mathematik Universit¨ at Potsdam D-14415 Potsdam, Germany e-mail: [email protected] Gerardo A. Mendoza Department of Mathematics Temple University Philadelphia, PA 19122, USA e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 51–66 c 2006 Birkh¨
auser Verlag Basel/Switzerland

Symbolic Calculus of Pseudo-differential Operators and Curvature of Manifolds Chisato Iwasaki Abstract. The method of construction of the fundamental solution for heat equations using pseudo-differential operators with parameter time variable is discussed, which is applicable to calculate traces of operators. This gives extensions of both the Gauss-Bonnet-Chern Theorem and the Riemann-Roch Theorem. Mathematics Subject Classification (2000). Primary 58J35; Secondary 58J40, 58J20. Keywords. Symbolic calculus, fundamental solution, curvature, local index.

1. Introduction In this paper we give, by means of symbolic calculus of pseudo-differential operators, both an extension theorem of a local version of the Gauss-Bonnet-Chern theorem given in [8] and that of a local version of the Riemann-Roch theorem given in [9]. Let M be a Riemannian manifold of dimension n without boundary. The Gauss-Bonnet-Chern theorem is stated as follows:  n
p (−1) dim Hp (M ) = Cn (x, M )dv, M

p=0

where Hp is the set of harmonic p-forms, Cn (x, M )dv is the Euler form if n is even and Cn (x, M )dv = 0 if n is odd. Its analytical proof is based on the formula 
n n
p (−1) dim Hp (M ) = (−1)p trep (t, x, x)dv, M p=0

p=0

where ep (t, x, y) denotes the kernel of the fundamental solutionEp (t) of the  Cauchy problem for the heat equation of ∆p on differential p-forms Γ ∧p T ∗ (M ) ;    ep (t, x, y)ϕ(y)dvy , ϕ ∈ Γ ∧p T ∗ (M ) Ep (t)ϕ(x) = M

52

C. Iwasaki

satisfies



 d + ∆p Ep (t) dt Ep (0)

= 0

in (0, T ) × M ,

= I

in M .

So, we may say that a local version of the Gauss-Bonnet-Chern theorem holds, if n
√ (−1)p trep (t, x, x) = Cn (x, M ) + 0( t) (1.1) p=0

as t tends to 0. The author proved (1.1) in [8], using an algebraic theorem on linear spaces stated in [2], and the method of construction of the fundamental solution by the technique of pseudo-differential operators with new weights on symbols. In this paper, a generalization of a local version of the Gauss-Bonnet-Chern theorem is obtained. Before stating our theorems, we introduce the notation. We denote by I the index set I = {I = (i1 , i2 , . . . , ir ) : 0 ≤ r ≤ n, 1 ≤ i1 < · · · < i ≤ n},   a =0 b

and

if a < b, or b < 0,

  0 = 1. 0

In the rest of this paper, fix an integer  such that 0 ≤  ≤ n . Define the constants {fp }p=0,1,...,n     n
n−p n−p (0 ≤ p ≤ n) fp = + kj n−j n−

(1.2)

j=max{p,+1}

with arbitrary constants {kj }j=+1,...,n . Theorem 1.1 (Main Theorem I). Let M be a Riemannian manifold of dimension n without boundary and let Ep (t) be the fundamental solution on Γ(∧p T ∗ (M )). Suppose the fp are of the form (1.2). Then we have n


1

(−1)p fp trep (t, x, x) = C (x)t− 2 + 2 + 0(t− 2 + 2 + 2 ) n



n



as t → 0,

p=0

where C (x) is given as follows: (1) If  is odd, C (x) = 0.  (2) If  is even ( = 2m), C (x) = I∈I,(I)= CI (x), for I = (i1 , i2 , . . . , i ) ∈ I
1 1 1 sign(π) sign(σ) CI (x) = ( √ )n ( )m 2 π m! 2 π,σ∈S

× Riπ(1) iπ(2) iσ(1) iσ(2) · · · Riπ( −1) iπ( ) iσ( −1) iσ( ) for I = (i1 , i2 , . . . , i ) ∈ I.

Symbolic Calculus and Curvature

53

Remark 1.2. Assume  = n. Then fp = 1 of (1.2) for all p. Theorem 1.1 is a local version of the Gauss-Bonnet-Chern theorem.   Remark 1.3. Assume kj = 0 for all j. Then fp = n−p n− (0 ≤ p ≤ ), fp = 0 ( + 1 ≤ p ≤ n). So Theorem 1.1 coincides with the result in [3]. Now consider the similar problem for Dolbeault complex on a Kaehler manifold M , that is, a local version of the Riemann-Roch theorem. Let ep (t, x, y) denote the kernel of the fundamental solution Ep (t) of the Cauchy  problem for the heat p ∗(0,1) (M ) ; equation of
p on differential (0, p)-forms Γ ∧ T    ep (t, x, y)ϕ(y)dvy , ϕ ∈ Γ ∧p T ∗(0,1) (M ) Ep (t)ϕ(x) = M

satisfies



 d +
p Ep (t) = dt Ep (0) =

0 I

in (0, T ) × M , in M .

In [9] the author has given a proof of a local version of the Riemann-Roch theorem, constructing the fundamental solution according to the method of symbolic calculus for a degenerate parabolic operator in [7]. There are several papers about a local version of the Riemann-Roch theorem. In [11] this formula is proved for manifolds of dimension 1. Then in [13] it is proved for Kaehler manifolds of any dimension. In [6] it is also shown, using invariant theory. In [5] this problem is treated by a different approach. We obtain an extension of this problem as follows. Theorem 1.4 (Main Theorem II). Let M be a compact Kaehler manifold with complex dimension n, and let Ep (t) be the fundamental solution on A0,p (M ) = Γ(∧p T ∗(0.1) (M )). Suppose fp are of the form (1.2). Then we have n  1 n
(−1)p fp trep (t, x, x)dv = CD (x)t−n+ + 0(t−n++1 ) as t → 0, 2πi p=0 where DD (x) are defined as follows:  CD (x) = I∈I,(I)= CID (x), where for I = (i1 , i2 , . . . , i ) ∈ I   Ω  c CID (x) = det Ω ∧ dv I . e − Id 2 Here Ω is a matrix whose (j, k) element is a 2-form defined as n
  a Ω jk = Rkja ¯b ¯ ¯ bω ∧ ω a,b=1

and c

¯ j1 ∧ ω j1 ∧ ω ¯ j2 ∧ ω j2 · · · ∧ ω ¯ jn− ∧ ω jn− , dv I = ω where I c = (j1 , j2 , . . . , jn− ) ∈ I such that I ∪ I c = {1, 2, . . . , n}.

54

C. Iwasaki

Remark 1.5. Assume  = n. Then fp = 1 of (1.2) for all p. In this case Theorem 1.4 is a local version of the Riemann-Roch theorem. Our point is that one can prove the above theorems by only calculating the main term of the symbol of the fundamental solution, introducing a new weight of symbols of pseudo-differential operators. The plan of this paper is following. In Section 2 an algebraic theorem, which is the key of the proof, is proved. The sketch of proof is given in Section 3 and Section 4.

2. Algebraic properties for the calculation of the trace Let V be a vector space of dimension n withan inner product and let ∧p (V ) be n p its anti-symmetric p tensors. Set ∧∗ (V ) = p=0 ∧ (V ). Let {v1 , . . . , vn } be an ∗ orthonormal basis for V . Let ai be a linear transformation on ∧∗ (V ) defined by a∗i v = vi ∧ v and let ai be the adjoint operator of a∗i on ∧∗ (V ). We note that {a∗i , aj }1≤i,j≤n satisfy the following relations. ai aj + aj ai = 0, a∗i a∗j + a∗j a∗i = 0, ai a∗j

(2.1)

a∗j ai

+ = δij .

Definition 2.1. Set A = (µ1 , . . . , µk ) : 1 ≤ k ≤ 2n, 1 ≤ µ1 < · · · < µk ≤ 2n , γ2k−1 = ak + a∗k , γ2k = i−1 (ak − a∗k ) for k ∈ {1, 2, . . . , n}, γA = i for A = (µ1 , . . . , µk ) ∈ A and γφ = 1.

k(k−1) 2

γµ1 · · · γµk

By (2.1) we have γµ γν + γν γµ = 2δµν , 1 ≤ µ, ν ≤ 2n and 2 = 1 for any A ∈ A. γA

The following propositions are shown in [2] under the above assumptions. Proposition 2.2. We have the following equality for transformation on ∧∗ (V ).  0, if A = φ; tr(γA ) = n 2 , if A = φ. Corollary 2.3. For any A, B ∈ A tr(γA γB ) =



0, 2n ,

if A = B; if A = B.

Definition 2.4. Set βφ = 1, βj = iγ2j−1 γ2j for 1 ≤ j ≤ n and βI = βi1 · · · βik for I = (i1 , . . . , ik ) ∈ I.

Symbolic Calculus and Curvature

55

It holds that for I = (i1 , i2 , . . . , ik ) ∈ I βI = γI˜, where I˜ = (2i1 − 1, 2i1, 2i2 − 1, 2i2 , . . . , 2ik − 1, 2ik ) ∈ A. It is clear that 1 a∗k ak = (1 + βk ), βj βk = βk βj , βj2 = 1 2 by the properties of γj . Proposition 2.5. We have for any I = (i1 , . . . , ik ) ∈ I the following assertions: (1) If p < k tr[βI aj1 aj2 · · · ajp a∗h1 a∗h2 · · · a∗hp ] = 0. (2) Suppose p = k and {j1 , j2 , . . . , jk } = {i1 , i2 , . . . , ik } or {h1 , h2 , . . . , hk } = {i1 , i2 , . . . , ik }. Then tr[βI aj1 aj2 · · · ajp a∗h1 a∗h2 · · · a∗hp ] = 0. (3) Let π, σ be elements of the permutation group of degree k. Then we have tr[βI a∗iπ(1) aiσ(1) a∗iπ(2) aiσ(2) · · · a∗iπ(k) aiσ(k) ] = 2n−k sign(π) sign(σ). Proof. (1) It holds that 1 1 (2.2) (γ2k−1 + iγ2k ), a∗k = (γ2k−1 − iγ2k ). 2 2 So aj1 aj2 · · · ajp a∗h1 a∗h2 · · · a∗hp is represented by a linear combination of γA with ˜ = 2k. We get the A ∈ I, (A) = 2p. On the other hand we have βI = γI˜ with (I) result by Corollary 2.3. ˜ More precisely (2) In this case k = p but A ∈ A does not coincide with I. ∗ ∗ ∗ aj1 aj2 · · · ajp ah1 ah2 · · · ahp are represented by a linear combination of the form γ21 −1 γ22 −1 · · · γ2k −1 γ2˜1 γ2˜2 · · · γ2˜k with ak =

{j1 , j2 , . . . , jk , h1 , h2 , . . . , hk } = {1 , 2 , . . . , k , ˜1 , ˜2 , . . . , ˜k }. By the assumption this set does not coincide with {i1 , i2 , . . . , ik }. So, by the above corollary we get (2). (3) We have for π, σ ∈ Sk a∗iπ(1) aiσ(1) a∗iπ(2) aiσ(2) · · · a∗iπ(k) aiσ(k) sign(π) sign(σ)a∗ik a∗ik−1

=

· · · a∗i1 ai1

· · · aik

+ lower order terms, because a∗iπ(1) aiσ(1) a∗iπ(2) aiσ(2) · · · a∗iπ(k) aiσ(k) = (−1)

k(k−1) 2

a∗iπ(1) a∗iπ(2) · · · a∗iπ(k) aiσ(1) aiσ(2) · · · aiσ(k)

+ lower order terms,

(2.3)

56

C. Iwasaki

a∗iπ(1) a∗iπ(2) · · · a∗iπ(k) = sign(π)a∗i1 a∗i2 · · · a∗ik = (−1)

k(k−1) 2

sign(π)a∗ik a∗ik−1 · · · a∗i1 .

By (2.3) and the statement of (1) we have   tr βI a∗iπ(1) aiσ(1) a∗iπ(2) aiσ(2) · · · a∗iπ(k) aiσ(k)

  = 2−k sign(π) sign(σ)tr βI (1 + βi1 ) · · · (1 + βik ) ,

where we use a∗ik a∗ik−1 · · · a∗i1 ai1 · · · aik = a∗i1 ai1 a∗i2 ai2 · · · a∗ik aik = 2−k (1+βi1 ) · · · (1+βik ). (2.4) By the above corollary we get (3), noting βJ = γJ˜ for any J ∈ I. ∗




Let Ψp be the projection of ∧ (V ) on ∧ (V ). The following proposition is the key algebraic argument of the proof in this paper. p

Proposition 2.6. For any p (0 ≤ p ≤ n) we have the equation n
Ψp = Mpq Γq , q=0

where




Mpq = and Γ0 = 1, Γk =

p+j −j

(−1)

2

p,q≤j≤n



I∈I,(I)=k

   j n−q p n−j

βI .

Proof. Set M0 = 1 and for 1 ≤ p ≤ n
Mp = 2−p

(1 + βi1 )(1 + βi2 ) · · · (1 + βip ).

(2.5)

(I)=p,I=(i1 ,...,ip )∈I

Then it holds that for 0 ≤ p ≤ n

n  
j Mp = Ψj . p j=p

In fact, we have




Mp =

a∗i1 ai1 a∗i2 ai2 · · · a∗ip aip ,

(2.6)

(2.7)

(I)=p,I=(i1 ,...,ip )∈I

noting (1 + βj ) = 2a∗j aj . We denote vI = vi1 ∧ · · · ∧ vip

for I = (i1 , . . . , ip ).



Then a∗k ak

1, on vI if k ∈ I; 0, on vI if k ∈ I.

=

So we have on ∧j (V )

 Mp =

if j ≤ p − 1; , if j ≥ p.

0, j  p

(2.8)

Symbolic Calculus and Curvature

57

By (2.8) we have (2.6). Owing to (2.6) it holds   n
j+p j Mj , (0 ≤ p ≤ n). (−1) Ψp = p j=p

(2.9)

On the other hand by the definition of Mp we have  j 
n−q Γq . Mj = 2−j n−j q=0

(2.10)


By (2.9) and (2.10) we get the assertion.

Note that a (n + 1) × (n + 1) matrix M = (Mpq )0≤p,q≤n is nonsingular because, with the argument of Proposition 2.6, M = CD where C and D are (n + 1) × (n +1) matrices whose cpq , dpq (0 ≤ p, q, ≤ n) are given by n−q elements . C −1 and D−1 are matrices whose elements cpq = (−1)p+q pq , dpq = 2−p n−p     c˜pq , d˜pq (0 ≤ p, q, ≤ n) are given by c˜pq = q , d˜pq = 2q (−1)p+q n−q . So, the p

following equation holds; (M−1 )pq =




(−1)p+j 2j

0≤j≤p,q

n−p

   q n−j . j n−p

Then we have Corollary 2.7. Let fj , hj be constants. Then n
q=0

fq Ψq =

n


h p Γp

p=0

holds if and only if

f =t M−1 h with h = (h0 , h1 , . . . , hn ) and f = (f0 , f1 , . . . , fn ).

We are interested in obtaining the exact formula of f for h of the following special form. Proposition 2.8. Assume hj = 0 (0 ≤ j ≤  − 1), h = 2−n (−1) . Then we obtain the following typical solutions f =t (f0 , f1 , . . . , fn ) of t

Mf = h, h =t (h0 , h1 , . . . , hn ).

(2.11)

(1) If f satisfies fp = 0 ( + 1 ≤ p ≤ n), the unique solution of (2.11) is   n−p fp = (−1)p (0 ≤ p ≤ ). n−   In this case hp = (−1)p 2−n p ( + 1 ≤ p ≤ n). (2) If f satisfies fp = 0 (0 ≤ p ≤ n −  − 1), the unique solution of (2.11) is   p n−+p fp = (−1) (n −  ≤ p ≤ n). n−   In this case hp = (−1) 2−n p ( + 1 ≤ p ≤ n).

58

C. Iwasaki

Proof. Fix an integer q (0 ≤ q ≤ n). The following equation holds for arbitrary x.   n
1 j −j n − q xj = (−1)q 2−q xq (1 − )n−q . (−1) 2 (2.12) 2 n−j j=q Put x = 1 − u in (2.12). Then we have    n n

n−q p p+j −j j u = 2−n (u − 1)q (1 + u)n−q . (−1) 2 p n − j p=0 j≥p,q

The above equation means for any q n
Mpq up = 2−n (u − 1)q (1 + u)n−q .

(2.13)

p=0

Put u =

1 v−1

in (2.13). Then we have n


Mpq (v − 1)n−p = 2−n v n−q (2 − v)q .

p=0

Comparing the coefficients of (−1)r v n−r for any r (0 ≤ r ≤ n) of both sides of the above equation, we obtain for any q    n
n−p 0, if r > q; p  (−1) = Mpq q r−n q n − r , if r ≤ q. 2 (−1) r p=0 This means that for any integer  (0 ≤  ≤ n)    n
0, if q ≤  − 1; n−p p  (−1) = Mpq q −n q n −  , if q ≥ . 2 (−1)  p=0 The above equations mean that t

MF = H

(2.14)

if we set a pair of (n + 1) vectors F =t (f,0 , f,1 , . . . , f,n ) and H =t (h,0 , h,1 , . . . , h,n ) 

as follows: f,p =

 h,p =

(−1)p 0,

n−p n−

, (0 ≤ p ≤ ); ( + 1 ≤ p ≤ n) ,

0, (0 ≤ p ≤  − 1);   p −n p (−1) 2  , ( ≤ p ≤ n).

The assertion (1) is proved. For the proof of (2), put u = w − 1 in (2.12). Then we have for any q n
p=0

Mpq (w − 1)p = 2−n wn−q (w − 2)q .

(2.15)

Symbolic Calculus and Curvature

59

Comparing the coefficients of (−w)n−r for any r (0 ≤ r ≤ n) of both sides of the above equation, we obtain for any q    n
0, if r > q; p p q  (−1) = Mpq n r−n n−r (−1) 2 r , if r ≤ q. p=0 This means that for any integer  (0 ≤  ≤ n)    n
0, if q ≤  − 1; p p q  (−1) = Mpq n −n n− (−1) 2  , if q ≥ . . p=0 The above equations mean that t

Mf = h



if we set

0, (0 ≤ p ≤ n −  − 1);   n−+p p (−1) n− , (n −  ≤ p ≤ n),  0, (0 ≤ p ≤  − 1);   hp = (−1) 2−n p , ( ≤ p ≤ n).

fp =

(2.16)




The assertion (2) is proved. Theorem 2.9. Let αp ( + 1 ≤ p ≤ n) be constants. The equation n


fq Ψq = (−1) 2−n Γ +

q=0

n


αp Γp

(2.17)

p=+1

has a solution as follows: n − p p + fp = (−1) n−

n
j=max(+1,p)

  n−p  for any p kj n−j

with constants kj ( + 1 ≤ j ≤ n) defined by      j
j j n−j −n  j + αp . 2 (−1) kj = (−1) 2  p

(2.18)

(2.19)

p=+1

Especially: (1) If  = n, then

n


fq Ψq = (−1)n Γn

q=0

holds if and only if fp = (−1)p for any p. (2) If αp = (−1)p 2  ( + 1 ≤ p ≤ n), we have fp of the following form:    (−1)p n−p n− , (0 ≤ p ≤ ); fp = 0, ( + 1 ≤ p ≤ n).   −n p

60

C. Iwasaki

  (3) If αp = (−1) 2−n p ( + 1 ≤ p ≤ n), we have fp of the following form:  0, (0 ≤ p ≤ n −  − 1);  p  fp = ; (n −  ≤ p ≤ n). (−1)n−+p n− Proof. It is shown that by (2.14) for any j (0 ≤ j ≤ n), t MFj = Hj hold. {Fj }nj= are linearly independent because fj,j = (−1)j and fj,k = 0 for (j < k). Put f =t (f0 , f1 , . . . , fn ) = F +

n


kj Fj .

j=+1

Then we have by (2.15) t Mf = h with h =t (h0 , h1 , . . . , hn ) as follows:    n n−p , (0 ≤ p ≤ ); (−1)p n−p j=+1 kj n−j n− + fp = n−p n p (−1) ( + 1 ≤ p ≤ n), j=p kj n−j ,  0, (0 ≤ p ≤  − 1);   p p hp = p −n p j−n kj j , ( ≤ p ≤ n). (−1) 2 j=+1 2  + So, f defined by (2.18) satisfies (2.17) if and only if kj satisfies      p
p p + kj . 2j−n αp = (−1)p 2−n  j

(2.20)

j=+1

The equation (2.20) means that     p−1
p j p − kj . 2 2 kp = (−1) 2 αp − 2  j p

p n



j=+1

We get (2.19) solving the above equation inductively. The other statements are clear by (2.14) and (2.15).




3. The proof of Main Theorem I (Riemannian manifold) We give a rough review of the method of construction of the fundamental solution in [8]. Let M be a smooth Riemannian manifold of dimension n with a Riemannian metric g. Let X1 , X2 , . . . , Xn be a local orthonormal frame of T (M ) in a local path U . And let ω 1 , ω 2 , . . . , ω n be its dual. The differential d and its dual ϑ acting on Γ(∧p T ∗ (M )) are written as follows, using the L´evi-Civit`a connection ∇ (see Appendix A of [12]): d=

n


e(ω j )∇Xj ,

j=1

where we use the following notation.

ϑ=−

n
j=1

ı(Xj )∇Xj ,

Symbolic Calculus and Curvature

61

Notation. e(ω j )ω = ω j ∧ ω, ı(Xj )ω(Y1 , . . . , Yp−1 ) = ω(Xj , Y1 , . . . , Yp−1 ). Let R(X, Y ) be the curvature transformation, that is   R(X, Y ) = ∇X , ∇Y − ∇[X,Y ] . Set R(Xi , Xj )Xk =

n


Rkij X 1 ≤ i, j, k, ≤ n.

n=1 p ∗ p=0 Γ(∧ T (M ))

The Laplacian ∆ = dϑ + ϑd on has the following Weitzenb¨ ock formula; n n n


∇Xj ∇Xj − ∇(∇Xj Xj ) + e(ω))ı(Xj )R(Xi , Xj )}. ∆ = −{ j=1

j=1

i,j=1

We use the following notation in the rest of this section. a∗j = e(ω∗),

ak = ι(Xk ).

The fundamental solution E(t) has an expansion, due to [8],
E(t) ∼ uj (t, x, D), j=0

where uj (t, x, D) are pseudo-differential operators with parameter t. The following statement is obtained in p. 255 of [8]. The kernel of a pseudodifferential operator with symbol u0 (t, x, ξ) is obtained as  −n u0 (t, x, ξ)dξ u ˜0 (t, x, x) = (2π) Rn (3.1)   √  1 = ( √ )n detge−tR 1 + 0( t) , 2 πt where n
Rqkij a∗i aj a∗k aq . R= i,j,k,q=1

We shall calculate

√ 1 tr (βI u ˜0 (t, x, x))dx = ( √ )n tr (βI e−tR )dv(1 + 0( t)), 2 πt

(3.2)

for I ∈ I, (I) = r. Using e−tR =

∞
(−1)k k k R t }, { k! k=0

and by Proposition 2.5 we have  m m m m+1 ), tr(βI (−1) −tR m! R )t + 0(t )= tr (βI e r+1 0(t 2 ),

if r = 2m; if r is odd.

(3.3)

62

C. Iwasaki We have the following proposition.

Proposition 3.1. For I = (i1 , i2 , . . . , ir ) ∈ I (r = 2m)
tr(βI (−1)m Rm ) = 2n−r−m sign(π) sign(σ)

(3.4)

π,σ∈Sr

×Riπ(1) iπ(2) iσ(1) iσ(2) · · · Riπ(r−1) iπ(r) iσ(r−1) iσ(r) . Proof. We can write −R =

n


Rijkq a∗i aj a∗k aq

i,j,k,q=1

=

1 { 2

n
i,j,k,q=1

=2

Rjkjq a∗k aq }.

j,k,q=1

By Proposition 2.5 we have   1 m m tr(βI (−1) R ) = tr βI 2

n


Rikjq a∗i aj a∗k aq

m 

i,j,k,q=1




−m

n


Rikjq a∗i aj a∗k aq −

 tr βI Riπ(1) iπ(2) iσ(1) iσ(2) · · · Riπ(r−1) iπ(r) iσ(r−1) iσ(r)

π,σ∈Sr

a∗iπ(1) aiσ(1) a∗iπ(2) aiσ(2) · · · a∗iπ(r) aiσ(r)
= 2n−r−m sign(π) sign(σ)



π,σ∈Sr

Riπ(1) iπ(2) iσ(1) iσ(2) · · · Riπ(r−1) iπ(r) iσ(r−1) iσ(r) . By (3.2),(3.3) and Proposition 3.1 we have  n r n r 2n−r t− 2 + 2 CI (x)dv + 0(t− 2 + 2 +1 ), ifr = 2m; ˜0 (t, x, x))dx = tr (βI u n r 1 0(t− 2 + 2 + 2 ), if r is odd




(3.5)

with CI (x) defined in Definition 1.1. Similarly we have j

tr (βI u ˜j (t, x, x))dx = 0(t− 2 + 2 + 2 ). n

By Theorem 2.9 we obtain  n 
tr fp ep (t, x, x)




= (−1) 2−n

p=0

r

tr (βI e(t, x, x))

I∈I,(I)=

+

n


αp tr (Γp e(t, x, x)).

p=+1

By (3.5) and (3.6) we have 1

tr (Γp e(t, x, x)) = 0(t− 2 + 2 + 2 ). n



(3.6)

Symbolic Calculus and Curvature

63

Applying (3.5) (3.6), we have    n n n 1
CI (x)t− 2 + 2 + 0(t− 2 + 2 + 2 ), if  is even; fp ep (t, x, x) = tr n 1 0(t− 2 + 2 + 2 ), if  is odd. p=0

4. The proof of Main Theorem II (Kaehler manifolds) We give a rough review of the method of construction of the fundamental solution in [9]. Let M be a compact Kaehler manifold whose complex dimension is n with a Hermitian metric g. Let Z1 , Z2 , . . . , Zn be a local orthonormal frame of T 1,0 (M ) in a local patch of a chart U . And let ω 1 , ω 2 , . . . , ω n be its dual. The differential d” and its dual ϑ” acting on A0,p (M ) are given as follows, using the L´evi-Civit`a connection ∇. n n

d” = e(¯ ω j )∇Z¯j , ϑ” = − ı(Z¯j )∇Zj , j=1

j=1

where we use the following notation. Notation. Z¯j = Z¯j ,

ω¯j = ω ¯j

(j = 1, . . . , n),

e(ω )ω = ω ∧ ω, α

α

ı(Zα )ω(Y1 , . . . , Yp−1 ) = ω(Zα , Y1 , . . . , Yp−1 ), ¯ ...,n (α ∈ Λ = {1, . . . , n, 1, ¯ }).

Let R(Zα , Z¯β ) be the curvature transformation,   R(Zα , Zβ ) = ∇Zα , ∇Zβ − ∇[Zα ,Zβ ] . The curvature transformations satisfy R(Zi , Zj ) = 0,

R(Z¯i , Z¯j ) = 0,

because M is a Kaehler manifold. Set
R(Zi , Z¯j )Zβ = Rγ¯βi¯j Zγ . γ∈Λ

n The Laplacian
= d ϑ + ϑ d on A (M ) = p=0 A0,p (M ) has the following Bochner-Kodaira formula; n n n

1
∇(∇Zj Z¯j +∇Z¯ Zj ) − R(Zj , Z¯j )}.
= − { (∇Zj ∇Z¯j + ∇Z¯j ∇Zj ) − j 2 j=1 j=1 j=1 ” ”

” ”

0,∗

We use the following notation in the rest of this section. e(¯ ω j ) = a∗j , ι(Z¯k ) = ak . The fundamental solution has an expansion, due to [9]
E(t) ∼ uj (t, x, D), j=0

64

C. Iwasaki

where uj (t, x, D) are pseudo-differential operators with parameter t and the main part of their symbols u0 (t, x, ξ) is represented by the precise form. The kernel u ˜0 (t, x, x) of a pseudo-differential operator with symbol u0 (t, x, ξ) is obtained as in p. 90 [9]    tM0 u ˜0 (t, x, x) = (2πt)−n det det g. exp(tM0 ) − Id Note that e(t, x, x)dv = u˜0 (t, x, x)dx(1 + 0(t)). We shall calculate

 tr (βI u ˜0 (t, x, x))dx = (2πt)−n tr βI det(

 tM0 ) dv, exp(tM0 ) − Id

(4.1) (4.2)

for I ∈ I, (I) = r, where

n n (M0 )jk = R(Zj , Z¯k ) = − p,q=1 Rp¯q jk¯ a∗q ap = p,q=1 Rj k¯ q¯p a∗q ap .

Set

 det

tM0 exp(tM0 ) − Id

 =

n


Aj tj .

j=0

Then we have by Proposition 2.5    tM0 = tr (βI Ar )tr + 0(tr+1 ). tr βI det exp(tM0 ) − Id

(4.3)

Set Ω a matrix whose (j, k) element is a 2-form defined by n n

q p p (Ω)jk = − Rj k¯ q¯p ω ¯ ∧ω = Rkjp¯ ¯q. ¯ qω ∧ ω p,q=1

p,q=1

Then we have Proposition 4.1. tr (βI Ar )dv = (−1)r Proof. If Ar is of the form
Ar =

 n   c 1 Ω ) 2n−r det( ∧ dv I i exp Ω − Id 2r

(4.4)

B(j1 , k1 , j2 , k2 , . . . , jr , kr )a∗j1 ak1 a∗j2 ak2 · · · a∗jr akr ,

1≤j1 ≤n,1≤j2 ≤n,...,1≤jr ≤n 1≤j1 ≤n,1≤j2 ≤n,...,1≤jr ≤n

then we have for I = (i1 , i2 , . . . , ir ) by Proposition 2.5 
tr (βI Ar ) = tr βI B(iπ(1) , iσ(1) , iπ(2) , iσ(2) , . . . , iπ(r) , iσ(r) ) π,σ∈Sr

= 2n−r


π,σ∈Sr

×a∗iπ(1) aiσ(1) a∗iπ(2) aiσ(2) · · · a∗iπ(r) aiσ(r)



sign(π) sign(σ)B(iπ(1) , iσ(1) , iπ(2) , iσ(2) , . . . , iπ(r) , iσ(r) ).

(4.5)

Symbolic Calculus and Curvature

65

On the other hand we have the equations ¯ 1 ∧ ω2 ∧ ω ¯ 2 · · · ∧ ωn ∧ ω ¯n dv = in ω 1 ∧ ω = (−i)n ω ¯ 1 ∧ ω1 ∧ ω ¯ 2 ∧ ω2 · · · ∧ ω ¯ n ∧ ωn c

= (−i)n ω ¯ i1 ∧ ω i1 ∧ ω ¯ i2 ∧ ω i2 · · · ∧ ω ¯ ir ∧ ω ir ∧ dv I ,

(4.6)

sign(π) sign(σ)¯ ω i1 ∧ ω i1 ∧ ω ¯ i2 ∧ ω i2 · · · ∧ ω ¯ ir ∧ ω ir =ω ¯

iπ(1)

∧ω

iσ(1)

∧ω ¯

iπ(2)

∧ω

iσ(2)

···∧ ω ¯

iπ(r)

∧ω

(4.7) iσ(r)

.

We obtain by (4.5), (4.6) and (4.7)  n
1 tr (βI Ar )dv = 2n−r B(iπ(1) , iσ(1) , iπ(2) , iσ(2) , . . . , i

(4.8)

π,σ∈Sr

c

ω iπ(1) ∧ ω iσ(1) ∧ ω ¯ iπ(2) ∧ ω iσ(2) · · · ∧ ω ¯ iπ(r) ∧ ω iσ(r) ∧ dv I . · · · iπ(r) , iσ(r) )¯ Then by (4.8) we have

 n   c 1 Ω tr (βI Ar )dv = (−1) ) 2n−r det( ∧ dv I . i exp Ω − Id 2r r




From (4.2), (4.3) and Proposition 4.1 the following equation holds. tr (βI u ˜0 (t, x, x))dx (4.9) n    c 1 Ω ) 2n−r t−n+r det( ∧ dv I + 0(t−n+r+1 ). = (−1)r 2πi exp Ω − Id 2r Now by Theorem 2.9 we obtain  n 
tr fp ep (t, x, x) = (−1) 2−n p=0




tr (βI e(t, x, x))

I∈I,(I)=

+

n


αp tr (Γp e(t, x, x))

p=+1

with some constants aj ( + 1 ≤ j ≤ n). Applying (4.9) we have n


αp tr (Γp e(t, x, x))dv = 0(t−n++1 ).

p=+1

By (4.1) and (4.9) we obtain  n  n 
1 tr fp ep (t, x, x) dv = t−n+ 2πi p=0 + 0(t

−n++1




 det(

I∈I,(I)=

).

 c Ω ) ∧ dv I exp Ω − Id 2

66

C. Iwasaki

References [1] N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators, SpringerVerlag, 1992. [2] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators, Springer, 1987. [3] P. G¨ unther and R. Schimming, Curvature and spectrum of compact Riemannian manifolds, J. Diff. Geom. 12 (1977), 599–618. [4] E. Getzler, The local Atiyah-Singer index theorem, in Critical Phenomena, Random Systems, Gauge Theories, Editors: K. Sterwalder and R. Stora, Les Houches, NorthHolland, 1984, 967–974. [5] E. Getzler, A short proof of the local Atiyah-Singer index theorem, Topology 25 (1986), 111–117. [6] P.B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Publish or Perish, Inc., 1984. [7] C. Iwasaki and N. Iwasaki, Parametrix for a degenerate parabolic equation and its application to the asymptotic behavior of spectral functions for stationary problems, Publ. Res. Inst. Math. Sci. 17 (1981), 557–655. [8] C. Iwasaki, A proof of the Gauss-Bonnet-Chern theorem by the symbol calculus of pseudo-differential operators, Japanese J. Math.21 (1995), 235–285. [9] C. Iwasaki, Symbolic calculus for construction of the fundamental solution for a degenerate equation and a local version of Riemann-Roch theorem, in Geometry, Analysis and Applications, Editor: R.S. Pathak, World Scientific, 83–92. [10] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, I, II, John Wiley & Sons, 1963. [11] T. Kotake, An analytic proof of the classical Riemann-Roch theorem, in Global Analysis, Proc. Symp. Pure Math. XVI Providence, 1970. [12] S. Murakami, Manifolds, Kyoritsusshuppan, 1969 (in Japanese). [13] V.K. Patodi, An analytic proof of Riemann-Roch-Hirzebruch theorem for Kaehler manifold, J. Diff. Geom. 5 (1971), 251–283. Chisato Iwasaki Department of Mathematics University of Hyogo 2167 Shosha Himeji Hyogo 671-2201, Japan e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 67–85 c 2006 Birkh¨
auser Verlag Basel/Switzerland

Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups M.W. Wong Abstract. The Plancherel formula and the inversion formula for Weyl transforms on compact and Hausdorff groups are given. A formula expressing the relationships of the wavelet constant, the degree of the irreducible and unitary representation and the volume of an arbitrary compact and Hausdorff group is derived. The role of the Weyl transforms in the derivation of the formulas for the heat kernels of Laplacians on compact Lie groups is explicated. The Green functions and the Riemann zeta functions of Laplacians on compact Lie groups are constructed using the corresponding heat kernels. Mathematics Subject Classification (2000). Primary 22A10, 35K05, 43A77, 47G30; Secondary 22C05, 35J05. Keywords. Weyl transforms, pseudo-differential operators, compact groups, Lie groups, Laplacians, irreducible and unitary representation, wavelet constants, resolution of the identity formula, Plancherel formula, Weyl inversion formula, eigenvalues and eigenvectors, heat kernels, Green functions, Riemann Zeta functions.

1. Introduction Pseudo-differential operators and their invariants, first systematically developed by Kohn and Nirenberg [8] in 1965 and modified almost immediately by H¨ ormander [7] and others for problems in partial differential equations, have roots in quantization due to Hermann Weyl [13] more than thirty years earlier. As such, they are known as Weyl transforms. The product formula for these classical Weyl transforms entails a twisted convolution, which is essentially a convolution on the Heisenberg group and can be found in the paper [17] by Wong. This suggests a notion of Weyl This research has been partially supported by the Natural Sciences and Engineering Research Council of Canada.

68

M.W. Wong

transforms on locally compact and Hausdorff topological groups, which we present in the context of compact groups in this paper. These Weyl transforms play a very fundamental role in the formulas for the heat kernels of the Laplacians on compact Lie groups. We first recall in Section 2 the construction of the Laplacian on a compact Lie group. In Sections 3–6, we have the freedom of working in a more general setting and we study harmonic analysis on compact and Hausdorff topological groups. To be specific, we give in Section 3 a streamlined presentation of irreducible and unitary representations, admissible wavelets and the resolution of the identity formula for compact and Hausdorff groups. An interesting formula relating the wavelet constant, the degree of the representation and the volume of the compact group is derived. The notion of a Weyl transform on a compact and Hausdorff group is introduced. We give a version of the Plancherel formula for Weyl transforms on compact groups in Section 4, the Weyl inversion formula in Section 5 and a formulation of the Weyl inversion formula in terms of characters in Section 6. The technique that underpins the results in Sections 4–6 is the Peter-Weyl theorem and the role played by the Weyl transforms is emphasized. Sections 7–9 are devoted to compact Lie groups in the context of partial differential equations. The eigenvalues and eigenfunctions of Laplacians are computed in Section 7. The heat kernels are constructed Section 8. The ubiquity of the heat kernels is demonstrated in Sections 9 and 10 by constructing, respectively, the Green functions and the Riemann zeta functions for the Laplacians on compact Lie groups. The Riemann zeta function of the Laplacian on a compact Lie group is the Mellin transform of the regularized trace of the heat kernel, and we express the Riemann zeta function in terms of the eigenvalues of the Laplacian. Issues on the regions of convergence of the series defining the Riemann zeta functions are beyond the scope of this paper and hence omitted. The book [3] by Fegan contains discussions on the impact of the trace of the heat kernel and hence the Riemann zeta function on the geometry of compact Lie groups. The unit circle S 1 with center at the origin of R2 is given as an example of an abelian compact Lie group in Section 11. SU(2) is given as an example of a non-commutative compact Lie group in Section 12. Since SU(2) can be identified as the unit sphere S 3 with center at the origin in R4 and the Laplacian on S 3 can best be studied using spherical harmonics, it is more convenient and natural to construct the heat kernel and Green function for the Laplacian ∆S n−1 on the unit sphere S n−1 with center at the origin in Rn using spherical harmonics in Rn . This is done in Section 13. It is worth pointing out that with the exception of S 1 and S 3 , S n−1 is a compact symmetric space instead of a compact Lie group.

2. Laplacians Let G be a compact Lie group. This means that G is a compact and Hausdorff topological group, which is also an n-dimensional C ∞ manifold. A vector field X

Compact Lie Groups

69

on G is a linear mapping X : C ∞ (G) → C ∞ (G) such that ϕ, ψ ∈ C ∞ (G).

X(ϕψ) = X(ϕ)ψ + ϕX(ψ),

For all vector fields X and Y on G, we can define the bracket [X, Y ] of X and Y by [X, Y ] = XY − Y X. The bracket [ , ] is bilinear, anti-symmetric and satisfies the Jacobi identity [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 for all vector fields X, Y and Z on G. The vector space of all vector fields on G is infinite-dimensional and hence too big to be useful. Therefore we look at instead the vector space of all left-invariant vector fields on G. A vector field X on G is said to be left-invariant if L(h)X = XL(h),

h ∈ G,

where L(h) is the left translation by h defined by (L(h)ϕ)(g) = ϕ(h−1 g),

g ∈ G,

∞

for all ϕ in C (G). It is obvious that the set of all left-invariant vector fields on G is closed with respect to the bracket operation. Let X be a vector field on G. If we denote the identity element in G by e, then we define the element Xe in the tangent space Te (G) of G at e by Xe (ϕ) = X(ϕ)(e),

ϕ ∈ C ∞ (G).

If we let LIV(G) be the vector space of all left-invariant vector fields on G, then we have the following proposition. Proposition 2.1. The mapping LIV(G)
X → Xe ∈ Te (G) is a bijective isomorphism. The isomorphism alluded to in the preceding proposition induces a natural bracket operation on Te (G). Equipped with this bracket operation, Te (G) is the Lie algebra g of G. The nice thing about g is that it is n-dimensional. As such, it has a basis consisting of n elements. We choose a basis for g, say, {v1 , v2 , . . . , vn }. Then we can get a basis {X1 , X2 , . . . , Xn } for LIV(G) as follows. For j = 1, 2, . . . , n, let γj : R → G be a homomorphic curve in G such that γj (0) = e and γj (0) = vj . Then for j = 1, 2, . . . , n, we define the left-invariant vector field Xj on G by  d  (Xj f )(g) = f (g · γj (s)), g ∈ G, ds s=0 for all f ∈ C ∞ (G).

70

M.W. Wong Here comes the Laplacian ∆G on G, which we define by n
∆G = − Xj2 . j=1

It can be proved that ∆G is positive and bi-invariant.

3. Weyl transforms Let G be a compact and Hausdorff group on which the left (and right) Haar measure is denoted by µ. Let π be an irreducible and unitary representation of G on a complex and separable Hilbert space Xπ . We denote the inner product and the norm in Xπ by ( , )Xπ and  Xπ respectively. Since G is compact, it is well known that X is finite-dimensional. A good reference is Theorem 5.2 in the book [4] by Folland. We let dπ be the dimension of Xπ . The number dπ is also known as the degree of the representation π of G on Xπ . For every element ϕ in X, the compactness of G implies that  |(ϕ, π(g)ϕ)|2 dµ(g) < ∞. G

Using the terminology in the general theory of wavelet transforms in the book [15] by Wong, we see that every element ϕ in the unit sphere of X is an admissible wavelet and the wavelet constant cϕ,π for the admissible wavelet ϕ is defined by  cϕ,π = |(ϕ, π(g)ϕ)Xπ |2 dµ(g). G

The starting point is the following theorem. Theorem 3.1. Let ϕ ∈ Xπ be such that ϕXπ = 1. Then  1 (x, y)Xπ = (x, π(g)ϕ)Xπ (π(g)ϕ, y)Xπ dµ(g), cϕ,π G

x, y ∈ Xπ .

(3.1)

See, for instance, Theorem 3.1 in the paper [5] by Grossmann, Morlet and Paul and Theorem 6.1 in the book [15] by Wong for more general results. The essence of the formula is that the identity operator (matrix) on X can be resolved into an infinite sum of rank-one operators on Xπ . Thus, it is reasonable to call the formula (3.1) a resolution of the identity formula. As an immediate consequence of Theorem 3.1 and the unimodularity of G, we have the following interesting result on wavelet constants. Theorem 3.2. Let ϕ and ψ be in Xπ such that ϕXπ = ψXπ = 1. Then cϕ,π = cψ,π . Proof. By (3.1),

 cϕ,π = G

(ψ, π(g)ϕ)Xπ (π(g)ϕ, ψ)Xπ dµ(g)

(3.2)

Compact Lie Groups and

71

 cψ,π = G

(ϕ, π(g)ψ)Xπ (π(g)ψ, ϕ)Xπ dµ(g).

By (3.3) and the unimodularity of G, we get  cψ,π = (ψ, π(g −1 )ϕ)Xπ (π(g −1 )ϕ, ψ)Xπ dµ(g) G (ψ, π(g)ϕ)Xπ (π(g)ϕ, ψ)Xπ dµ(g). =

(3.3)

(3.4)

G




So, by (3.2) and (3.4), the proof is complete.

Another interesting consequence of Theorem 3.1 is the following formula relating the wavelet constant, the degree of the representation and the volume of the compact group. Theorem 3.3. Let ϕ ∈ X be such that ϕXπ = 1. Then cϕ,π =

µ(G) . dπ

Proof. Let {ψ1 , ψ2 , . . . , ψdπ } be an orthonormal basis for Xπ . Then, by Theorem 3.1 and Parseval’s identity, dπ

=

dπ


(ψj , ψj )Xπ

j=1

=

 dπ
1 |(ψj , π(g)ϕ)Xπ |2 dµ(g) c ϕ,π G j=1

= =


dπ

1 cϕ,π 1 cϕ,π

G j=1

|(ψj , π(g)ϕ)Xπ |2 dµ(g)



G

π(g)ϕ2Xπ dµ(g).

Since π is a unitary representation of G on Xπ and ϕXπ = 1, we get dπ =

µ(G) , cϕ,π


which is the same as asserted.

From now on, we assume that µ(G) = 1. Let F ∈ L1 (G). Then we define the linear operator (matrix) WF,π : Xπ → Xπ by

 (WF,π x, y)Xπ =

G

F (g)(x, π(g)y)Xπ dµ(g),

x, y ∈ Xπ .

(3.5)

72

M.W. Wong

We call WF,π : Xπ → Xπ the Weyl transform associated to the symbol F and the representation π. A more transparent formula for WF,π than (3.5) is given by  WF,π = F (g)π(g −1 )dµ(g). G

The terminology is due to the fact that the classical Weyl transform in, say, the book [14] by Wong is in fact given by (3.5) when the compact group and the representation π are replaced, respectively, by the non-compact Weyl-Heisenberg group and the Schr¨ odinger representation. A seasoned harmonic analyst can recognize immediately that this is nothing but the Fourier transform of F on the group G evaluated at π. Such a viewpoint has been exploited in the paper [16] by Wong. Moreover, motivated by the formula of the Weyl transform provided by Theorem 4.3 in [14], we call G
g → (π(g)x, y)Xπ ∈ C the Wigner transform of x and y. Using the realization of Weyl transforms as Fourier transforms on the group G, and the fact that the Fourier transform of the convolution of two functions F ˆ and H in L1 (G) at π is the product Fˆ (π)H(π) of the bounded linear operators Fˆ (π) ˆ and H(π), we have the following result on the product of two Weyl transforms. Theorem 3.4. Let F and H be functions in L1 (G). Then the product of the Weyl transforms WF,π : Xπ → Xπ and WH,π : Xπ → Xπ is the Weyl transform WF ∗H,π : Xπ → Xπ , where F ∗ H is the convolution of F and H defined by  (F ∗ H)(g) = F (gh−1 )H(h) dµ(h), g ∈ G. G

4. The Plancherel formula ˆ be the set of all irreducible and unitary representations of G. Let ν ∈ G. ˆ Let G Then ν is finite-dimensional. We denote its degree by dν and its representation space by Xν . Let {ψ1 , ψ2 , . . . , ψdν } be an orthonormal basis for Xν . Then for all j, k = 1, 2, . . . , we define νjk by νjk (g) = (ν(g)ψk , ψj )Xν , g ∈ G. √ ˆ j, k = 1, 2, . . . } forms an orthonormal basis for The fact that { dν νjk : ν ∈ G, 2 L (G) is the core of the Peter-Weyl theorem. Theorem 4.1. (The Plancherel formula) Let F ∈ L2 (G). Then
dπ WF,π 2S2 (Xπ ) = F 2L2 (G) , ˆ π∈G

where WF,π S2 (Xπ ) is the Hilbert-Schmidt norm of WF,π : Xπ → Xπ .

Compact Lie Groups

73

Proof. By the core of the Peter-Weyl theorem, we can write F as

 cν,j,k dν νjk . Then WF,π = cν,j,k W√dν νjk ,π . F = ν,j,k

ν,j,k

By Parseval’s identity,




F 2L2 (G) =

|cν,j,k |2 .

ν,j,k

Let {ψ1 , ψ2 , . . . , ψdπ } be an orthonormal basis for Xπ . Then for all dπ


x=

xl ψl

and

y=

(WF,π x, y)Xπ =




cν,j,k (W√dν νjk ,π x, y)Xπ

ν,j,k

=




cν,j,k

ν,j,k

=







  xl ym dν νjk (g)πlm (g) dµ(g) G

l,m

xl ym

l,m



1 dν cν,j,k (νjk , πlm )L2 (G) = xl ym cπ,l,m . dπ

ν,j,k

l,m

Now, for j = 1, 2, . . . , dπ , (WF,π ψj , y)Xπ =




ym cπ,j,m

m

where zm =

ym ψm

m=1

l=1

in Xπ , we get

dπ


√1 cπ,j,m , dπ

1 = (z, y)Xπ , dπ

m = 1, 2, . . . , dπ . Thus,

WF,π ψj =

1 (cπ,j,1 , cπ,j,2 , . . . , cπ,j,n ). dπ

Therefore WF,π ψj 2Xπ

dπ 1
= |cπ,j,k |2 dπ k=1

and consequently, dπ


WF,π ψj 2Xπ =

j=1

dπ 1
|cπ,j,k |2 . dπ j,k=1

Therefore dπ WF,π 2S2 (Xπ ) =

dπ


|cπ,j,k |2

j,k=1

and we get




dπ WF,π 2S2 (Xπ ) = F 2L2 (G) ,

ˆ π∈G

as asserted.




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M.W. Wong

5. The Weyl inversion formula The Peter-Weyl theorem states that for every f in L2 (G),

  (f, dπ πjk )L2 (G) dπ πjk = απ,j,k πjk , f= π,j,k

π,j,k

where

 απ,j,k = dπ

f (g)πjk (g) dµ(g). G

ˆ the entry (Wf,π )kj of the matrix Wf,π is given by For every π in G,  1 f (g)πjk (g) dµ(g) = απ,j,k . (Wf,π )kj = d π G Therefore for j, k = 1, 2, . . . , dπ , and g ∈ G,

απ,j,k πjk (g) = dπ (Wf,π )kj πjk (g) = dπ tr[Wf,π π(g)]. j,k

j,k

So, we have the Weyl inversion formula to the effect that for every f ∈ L2 (G),
dπ tr[Wf,π π(·)]. f= ˆ π∈G

6. Characters For every finite-dimensional unitary representation π of G, the character χπ of π is defined by χπ (g) = tr[π(g)], g ∈ G. Then for all g in G,

=

! f (h)π(h−1 ) dµ(h) π(g) G !  f (h)π(h−1 )π(g) dµ(h) tr G !  f (h)π(h−1 g) dµ(h) tr  G f (h)tr[π(h−1 g)] dµ(h)

=

(χπ ∗ f )(g).



tr[Wf,π π(g)] = = =

tr

G

Thus, for every f in L2 (G), f=


ˆ π∈G

dπ (χπ ∗ f ).

Compact Lie Groups

75

7. Eigenvalues and eigenfunctions of Laplacians We prove in this section that for every Laplacian ∆G on a compact Lie group G, there exists an orthonormal basis for L2 (G) consisting of eigenfunctions of ∆G . The precise result is the following theorem. ˆ there exists a nonnegative number λπ such that Theorem 7.1. For all π in G, ∆G πjk = λπ πjk ,

j, k = 1, 2, . . . , dπ .

ˆ Then for j, k = 1, 2, . . . , dπ , we let Proof. Let π ∈ G. νjk = ∆πjk . Since for all j, k = 1, 2, . . . , dπ , and all g and h in G, dπ
l=1

πkl (g)πlj (h) =

dπ
(π(g)ψl , ψk )Xπ (ψj , π(h−1 )ψl )Xπ l=1

dπ
= (π(h)ψj , ψl )Xπ (ψl , π(g −1 )ψk )Xπ l=1

= (π(h)ψj , π(g −1 )ψk )Xπ = (π(gh)ψj , ψk )Xπ = πkj (hg), it follows from the left-invariance of ∆G that νkj (gh) =

dπ


πkl (g)νlj (h).

l=1

Letting h = e, we get for j, k = 1, 2, . . . , dπ and all g in G, νkj (g) =

dπ


πkl (g)νlj (e).

l=1

Similarly, if we use the right-invariance of ∆G , we get νkj (g) =

dπ


νkl (e)πlj (g).

l=1 π Thus, the matrix [νjk (e)]dj,k=1 commutes with the irreducible and unitary representation π. So, by Schur’s lemma, there exists a constant λπ such that π [νjk (e)]dj,k=1 = λπ = λπ I,

where I is the identity matrix. Therefore for j, k = 1, 2, . . . , dπ and all g in G, (∆G πjk )(g) = νjk (g) =

dπ


πjl (g)νlk (e) = λπ πjk (g).

l=1

So, we are done if we can show that λπ ≥ 0. But this follows from the fact that (∆G f, f )L2 (G) ≥ 0,

f ∈ L2 (G).




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M.W. Wong

8. Heat kernels We give a formula for the heat kernels of Laplacians on compact Lie groups. The formula involves Weyl transforms in a very fundamental way. To this end, let u and v be functions in L2 (G). Then for t > 0, Theorem 7.1 gives (e−t∆G u, v)L2 (G) =




e−tλπ dπ (u, πjk )L2 (G) (πjk , v)L2 (G) .

π,j,k

We know from Section 5 that (u, πjk )L2 (G) = (Wu,π )jk and a simple calculation gives (πjk , v)L2 (G) = (Wv∗ ,π )jk , where v ∗ (g) = v(g −1 ),

g ∈ G.

So, if we let {ψ1 , ψ2 , . . . , ψdπ } be an orthonormal basis for Xπ , then (e−t∆G u, v)L2 (G)




=

e−tλπ dπ

ˆ π∈G




=

dπ


(Wu,π )jk (Wv∗ ,π )jk

j,k=1

e−tλπ dπ tr(Wu,π Wv∗ ,π )

ˆ π∈G




=

e−tλπ dπ tr(Wu∗v∗ ,π )

ˆ π∈G




=

e

−tλπ

dπ

ˆ π∈G




=

k=1

e

−tλπ

 dπ




ˆ π∈G




=

G

(u ∗ v ∗ )(g)(π(g −1 )ψk , ψk )Xπ dµ(g)

(u ∗ v ∗ )(g)tr(π(g −1 )) dµ(g)

G

ˆ π∈G

=

dπ 


e−tλπ dπ



(u ∗ v ∗ )(g)χπ (g −1 ) dµ(g)

G

e−tλπ dπ I,

ˆ π∈G

where

 

u(gh−1 )v ∗ (h)χπ (g −1 ) dµ(g) dµ(h).

I= G

G

Compact Lie Groups So,



77

 u(gh )χπ (g ) dµ(g) dµ(h) I= G G   −1 −1 −1 v(h ) u(k)χπ (h k ) dµ(k) dµ(h) = G G  = v(h−1 )(χπ ∗ u)(h−1 ) dµ(h) = v(h)(χπ ∗ u)(h) dµ(h) = (χπ ∗ u, v)L2 (G) . 

−1

v(h−1 )

−1

G

G

Thus, (e−t∆G u, v)L2 (G) =




e−tλπ dπ (χπ ∗ u, v)L2 (G) ,

ˆ π∈G

and we get e−t∆G u =




e−tλπ dπ χπ ∗ u.

ˆ π∈G

The conclusion then is that the heat kernel kt , t > 0, of the Laplacian ∆G on the compact Lie group G is given by
kt (g, h) = e−tλπ dπ χπ (gh−1 ), g, h ∈ G. ˆ π∈G

The preceding formula for the heat kernel can be found on page 45 of the book [9] by Stein.

9. Green functions Green functions are the kernels of the integral operators for the inverse. We derive in this section the Green function of the Laplacian ∆G on a compact Lie group G. By the very definition of a Laplacian, the constant functions on G are eigenfunctions of ∆G corresponding to the eigenvalue 0. As such, it is desirable to quotient out these constant functions from L2 (G). To do this, we write L2 (G) = L2C (G)⊕ C, where L2C (G) is the orthogonal complement of C in L2 (G). Then ∆G is reduced in the sense that L2C (G) and C are both invariant subspaces with respect to ∆G . 2 2 It is then clear that ∆−1 G : LC (G) → LC (G) exists and as in the derivation of the heat kernel, we have
dπ (∆−1 χπ ∗ u G u)(g) = λπ λ=0

for all u ∈

L2C (G).

So, the Green function G of ∆G is given by
dπ χπ (gh−1 ), g, h ∈ G. G(g, h) = λπ λπ =0

An alternative way of getting the Green function is to integrate the heat kernel with respect to t from 0 to ∞. We get the same formula for the Green function.

78

M.W. Wong

10. Riemann zeta functions We can now define the Riemann zeta function ζG of the Laplacian ∆G by   ∞  1 dt ζG (s) = kt (g, g) dµ(g) − 1 ts Γ(s) 0 t G for all s ∈ C such that the integral makes sense. It is the Mellin transform of G kt (g, g) dµ(g) − 1, which is the regularized trace of the heat kernel at time t. Theorem 10.1. Let s ∈ C be such that ζ(s) is defined. Then
d2 π . ζ(s) = λsπ λπ =0

Proof. By the definition of the Riemann zeta function, we get ⎞ ⎛  ∞ 
dt 1 ⎝ e−tλπ d2π dµ(g) − 1⎠ ts ζG (s) = Γ(s) 0 t G ˆ π∈G ⎞ ⎛  ∞
dt 1 ⎝ = e−tλπ d2π − 1⎠ ts Γ(s) 0 t ˆ π∈G  ∞
1 dt = e−tλπ d2π ts Γ(s) 0 t λπ =0  1
∞ −tλπ 2 s dt e dπ t = Γ(s) t λπ =0 0  1
2 ∞ −tλπ s dt dπ e t = Γ(s) t 0 λπ =0  1
d2π ∞ −δ s dδ e δ = Γ(s) λsπ 0 δ λπ =0

=


d2 π , λsπ

λ=0




as asserted.

11. The unit circle S 1 in R2 The unit circle S 1 with center at the origin in R2 is given by S 1 = {eiθ : θ ∈ R}. It is obviously a compact Lie group in which the group law is given by the multiplication of complex numbers. So, the group S 1 can be identified with the group R/2πZ in which the group law is given by addition modulo 2πZ. The striking

Compact Lie Groups

79

feature to note here is that S 1 is abelian. As such, all its irreducible and unitary representations are one-dimensional. Let us find all of them. For all n ∈ Z, let χn : S 1 → C be the mapping defined by θ ∈ R.

χn (θ) = ei n θ ,

Then χn is an irreducible and unitary representation of S 1 . In fact, deg(χn )=1 for all n in Z. To see that every irreducible and unitary representation χ of degree 1 is equal to χn for some integer n, we let χ(1) = eiα . Then, using the homogeneity and the continuity of χ, we get θ ∈ R.

χ(θ) = χ(1)θ = eiαθ , Thus, for every integer k, we get

1 = χ(0) = χ(2kπ) = eiα2kπ ⇒ α2kπ = 2nπ for some integer n. If we let k = 1, then α has to be an integer. Thus, we have proved the following theorem. Theorem 11.1. S 1 = {χn : n ∈ Z}. Consequently, we can identify S 1 with Z. It follows from the Peter-Weyl theorem that {χn : n ∈ Z} is an orthonormal basis for L2 (S 1 ). Let F ∈ L1 (S 1 ). Then the Weyl transform WF,χn : C → C associated to the symbol F and the representation χn is given by  2π 1 WF,χn = F (θ)e−inθ dθ, 2π 0 which is nothing but the nth Fourier coefficient of the function F . Using polar coordinates, the Laplacian ∆ on R2 is given by ∆=

1 ∂2 1 ∂ ∂2 + 2 2. + 2 ∂r r ∂r r ∂θ

Hence ∆S 1 = −

∂2 ∂θ2

and we have ∆S 1 χn = n2 χn ,

n ∈ Z.

The heat kernel kt is then given by 1
−n2 t inθ kt (θ) = e e , 2π

t > 0, θ ∈ R.

n∈Z

The Green function G of ∆S 1 is given by 1
1 inθ G(θ) = e , 2π n2 n=0

θ ∈ R.

80

M.W. Wong

Remark 11.2. There is an explicit formula for G(θ). Indeed, G(θ) =

∞ ∞ 1 1 2 1
1 π 1
1 inθ −inθ θ (e + e ) = cos nθ = − θ + 2 2 2π n=1 n π n=1 n 6 2 4π

for 0 ≤ θ ≤ 2π. See, for instance, the formula (3.8.5) in the book [6] by Hardy and Rogosinski. By Theorem 10.1, the Riemann zeta function ζS 1 for the Laplacian ∆S 1 is given by ∞
1 ζS 1 (s) = 2 = 2ζ(2s) n2s n=1 for all s in C such that ζ(2s) is defined, where ζ is the classical Riemann zeta function, which can be found in the book [12] by Titchmarsh among others.

12. The unit sphere S 3 in R4 In order to see that the unit sphere S 3 with center at the origin in R4 is a compact Lie group, we use the identification of R4 with C2 and the identification R4
(x, y, ξ, η) → (z, w) ∈ C2 , where z = x + iy and w = ξ + iη. We let SU(2) be the set defined by ⎧⎛ ⎫ ⎞ ⎨ a −b ⎬ ⎠ : a, b ∈ C, |a|2 + |b|2 = 1 . SU(2) = ⎝ ⎩ b ⎭ a It is obviously a group with respect to the multiplication of matrices ⎛ and is known ⎞ as the special unitary group. We sometimes denote the matrix ⎝

a

−b

⎠ by a b Ua,b . Matrices in SU(2) are unitary matrices with determinant 1, and hence we get for all a and b in C with |a|2 + |b|2 = 1, −1 ∗ Ua,b = Ua,b = Ua,−b .

The group action of SU(2) on C2 is given by ⎛ ⎞⎛ ⎞ ⎞ ⎛ a −b z az − bw ⎝ ⎠⎝ ⎠ ⎠=⎝ a b bz + aw w or  for all

a −b b a



Ua,b (z, w) = (az − bw, bz + aw) in SU(2) and ( wz ) in C2 . Therefore the identification SU(2)
Ua,b ↔ (a, b) = Ua,b (1, 0)

Compact Lie Groups

81

allows us to look at SU(2) as the unit sphere S 3 with center at the origin in R4 , and the identity element in SU(2) is identified with the north pole (1, 0) in C2 . Therefore SU(2) is isomorphic to S 3 , which is a group with respect to the multiplication ◦ given by σ◦τ =ω for all σ and τ in S , where ω ∈ S is such that 3

3

Uω = Uσ Uτ . It is now clear that S 3 is a compact group. Our first task is to identify all the irreducible and unitary representations of S 3 . Since S 3 is compact, they have to be finite-dimensional. To see what they are, we let P be the set of all polynomials in two complex variables. To wit, a polynomial P in P is given by
P (z, w) = cjk z j wk , cjk , z, w ∈ C. 1≤j,k≤N

We can look at P as a subspace of L2 (S 3 ), where the surface measure ν on S 3 is chosen to be such that ν(S 3 ) = 1. For every nonnegative integer m, we let Pm be the subset of P consisting of all homogeneous polynomials of degree m. Thus, ⎧ ⎫ m ⎨
⎬ cj z j wm−j : c1 , c2 , . . . , cm , z, w ∈ C . Pm = ⎩ ⎭ j=0

Since Pm is a finite-dimensional vector space with dimension dm , it follows that Pm is a Hilbert space with respect to the product ( , )L2 (S 3 ) in L2 (S 3 ) given by  P (σ)Q(σ) dν(σ), P, Q ∈ Pm . (P, Q)L2 (S 3 ) = S3

The exact value of dm can be found on Pages 138-139 of the book [10] by Stein and Weiss. The following result is Proposition 5.34 in the book [4] by Folland. Proposition 12.1. The family {Pm : m = 0, 1, 2, . . . } of Hilbert spaces is mutually orthogonal in L2 (S 3 ) and for m = 0, 1, 2, . . . , , (m + 1)! j m−j z w :0≤j≤m j!(m − j)! is an orthonormal basis for Pm . The group action of SU(2) on C2 induces a representation π of S 3 on P given by −1 (π(a, b)P )(z, w) = P (Ua,b (z, w)) = P (az + bw, −bz + aw)

for all (a, b) in S 3 and (z, w) in C2 .

82

M.W. Wong

Proposition 12.2. For m = 0, 1, 2, . . . , Pm is an invariant subspace of π. Proof. Let P ∈ Pm . Then for all (a, b) in S 3 , (π(a, b)P )(z, w) = P (az + bw, −bz + aw),

z, w ∈ C.

It is then clear that π(a, b)P is in Pm .




For m = 0, 1, 2, . . . , let πm be the sub-representation of π on Pm . In other words, πm is the representation of S 3 on Pm given by πm (U (a, b)P = π(a, b)P for all (a, b) in S and P in Pm . 3

Proposition 12.3. For m = 0, 1, 2, . . . , πm is a unitary representation of S 3 on Pm . Proof. It suffices to show that πm (a, b) is unitary for all (a, b) in S 3 . But, for all P and Q in Pm , using the fact that the surface measure ν on S 3 is rotation invariant,  −1 P (Ua,b ω)Q(ω) dν(ω) (πm (a, b)P, Q)L2 (S 3 ) = 3 S  = P (ω)Q(Ua,b ω) dω S3

−1 = (P, πm (Ua,b )Q)L2 (S 3 ) .

Therefore πm (a, b)∗ = πm (a, b)−1 and we are done.




The following result is Theorem 5.37 in the book [4] by Folland. Theorem 12.4. For m = 0, 1, 2, . . . , πm is an irreducible representation of S 3 on Pm . In fact, it is proved in [4] that up to unitary equivalence, the family {πm : m = 0, 1, 2, . . . } of representations of S 3 on Pm gives all the irreducible and unitary representations of S 3 on Pm . Realizing SU(2) as S 3 , we see that for m = 0, 1, 2, . . . , πm can be expressed as (πm (τ )P )(σ) = P (τ −1 ◦ σ), σ, τ ∈ S 3 , for all P in Pm . Now, let F ∈ L2 (S 3 ). Then the Weyl transform WF,πm : Pm → Pm is given by  F (τ )(πm (τ )P, Q)L2 (S 3 ) dν(τ ), P, Q ∈ Pm . (WF,πm P, Q)L2 (S 3 ) = S3

Thus, for all P and Q in Pm , (WF,πm P, Q)L2 (S 3 )

=





F (τ ) S3  

dν(τ )  F (τ )P (τ −1 ◦ σ) dν(τ ) Q(σ) dν(σ).

= S3

S3

S3

P (τ −1 ◦ σ)Q(σ)



Thus, for all P in Pm , WF,πm P is the same as the convolution F ∗ P of F and P .

Compact Lie Groups

83

Now, a basis for the Lie algebra su(2) of SU(2) is given by the 2 × 2 skewsymmetric matrices with zero trace ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ 0 1 0 i 0 1⎝ i ⎠ , A3 = ⎝ ⎠. ⎠ , A2 = ⎝ A1 = 2 −1 0 i 0 0 −i For j = 1, 2, 3, let γj : R → SU(2) be a homomorphic curve such that γj (0) = I

and

γj (0) = Aj ,

where I is the identity matrix in SU(2). Then the Laplacian ∆SU(2) on SU(2) can be identified with the Laplacian ∆S 3 on S 3 . This identification allows us to use spherical harmonics to construct the heat kernel and Green function for ∆SU(2) . Using Theorem 10.1, the Riemann zeta function ζS 3 of the Laplacian ∆S 3 is given by ∞ ∞
d2n 1
(n + 1)2 ζS 3 (s) = = λs 4 n=1 ns (n + 2)2−s n=1 n for all s in C such that the series is defined.

13. Spherical harmonics In this section, we use spherical harmonics in, e.g., the work [1] by Chavel, the book [10] by Stein and Weiss, and the book [11] by Terras, to construct the heat kernel and Green function for the Laplacian ∆S n−1 on the unit sphere S n−1 in n ∂ 2 Rn , where n > 2. The Laplacian ∆S n−1 is obtained from the Laplacian j=1 ∂x 2 j using polar coordinates and is given by n
∂2 1 ∂2 n−1 ∂ − 2 ∆S n−1 . = + 2 2 ∂xj ∂r r ∂r r j=1

A spherical harmonic Y k of degree k is the restriction to S n−1 of a homogeneous and harmonic polynomial of degree k. If we denote by Hk the set of all spherical harmonics of degree k, then Hk is a finite-dimensional Hilbert space with respect to the inner product in L2 (S n−1 ) and its dimension dk is given by     n+k−1 n+k−3 − . dk = k k−2 If we let {Y1k , . . . , Ydkk } be an orthonormal basis for Hk , then it can be proved that k k 2 n−1 ). It can also be proved ∪∞ k=0 {Y1 , . . . , Ydk } is an orthonormal basis for L (S k that for all Y ∈ Hk , ∆S n−1 Y k = k(k + n − 2)Y k .

84

M.W. Wong

So, for each nonnegative integer k, Hk is an eigenspace of ∆S n−1 corresponding to the eigenvalue k(k + n − 2). Therefore the heat kernel kt for ∆S n−1 is given by kt (σ, τ ) =

∞


e

dk


−k(k+n−2)t

t > 0; σ, τ ∈ S n−1 .

Yjk (σ)Yjk (τ ),

j=1

k=0

For k = 0, 1, 2, . . . , there exists a constant ck,n such that dk


(n−2)/2

Yjk (σ)Yjk (τ ) = ck,n Pk

(σ · τ ),

σ, τ ∈ S n−1 ,

j=1

where Pkλ is the Gegenbauer polynomial given by the generating function (1 − 2rt + r2 )−λ =

∞


|r| < 1, |t| ≤ 1, λ > 0.

Pkλ (t)rk ,

k=0

For more information about Gegenbauer polynomials, see, for instance, the work [2] by Erd´elyi, Magnus, Oberhettinger and Tricomi. In order to compute ck,n , we note that  dk

dk


=

Yjk (σ)Yjk (σ) dν(σ)

S n−1 j=1



= ck,n S n−1

 = ck,n

S n−1 n−1

= ck,n |S

(n−2)/2

(σ · σ) dν(σ)

(n−2)/2

(1) dν(σ)

Pk Pk

(n−2)/2

|Pk

(1).

Therefore ck,n =

dk (n−2)/2 |S n−1 |Pk (1)

.

So, kt (σ, τ ) =

1

∞


|S n−1 |

k=0

e−k(k+n−2)t

dk (n−2)/2 (σ Pk (n−2)/2 Pk (1)

· τ)

for all t > 0 and all σ and τ in S n−1 . The Green function G for ∆S n−1 is given by G(σ, τ ) =

1

∞


|S n−1 |

k=1

for all σ and τ in S n−1 .

1 dk (n−2)/2 (σ · τ ) P k(k + n − 2) P (n−2)/2 (1) k k

Compact Lie Groups

85

References [1] I. Chavel, The Laplacian on Riemannian manifolds, in Spectral Theory and Geometry, Editors: E.B. Davies and Y. Safarov, Cambridge University Press, 1999, 30–75. [2] A. Erd´elyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. 2, Krieger, 1981. [3] H.D. Fegan, An Introduction to Compact Lie Groups, World Scientific, 1991. [4] G.B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995. [5] A. Grossmann, J. Morlet and T. Paul, Transforms associated to square integrable group representations I: General results, J. Math. Phys. 26 (1985), 2473–2479. [6] G.H. Hardy and W.W. Rogosinski, Fourier Series, Dover, 1999. [7] L. H¨ ormander, The Analysis of Linear Partial Differential Operators III, SpringerVerlag, 1985. [8] J.J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 269–305. [9] E.M. Stein, Topics in Harmonic Analysis, Princeton University Press, 1971. [10] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. [11] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications I, SpringerVerlag, 1985. [12] E.C. Titchmarsh, The Theory of the Riemann Zeta-function, Second Edition, Oxford University Press, 1986. [13] H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1950. [14] M.W. Wong, Weyl Transforms, Springer-Verlag, 1998. [15] M.W. Wong, Wavelet Transforms and Localization Operators, Birkh¨ auser, 2002. [16] M.W. Wong, A new look at pseudo-differential operators, in Wavelets and their Applications, Editors: M. Krishna, R. Radha and S. Thangavelu, Allied Publishers, 2003, 283–290. [17] M.W. Wong, Weyl transforms and convolution operators on the Heisenberg group, in Pseudo-differential Operators and Related Topics, Editors: P. Boggiatto, L. Rodino, J. Toft and M.W. Wong, Birkh¨ auser, 2006, 115–121. M.W. Wong Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3, Canada e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 87–105 c 2006 Birkh¨
auser Verlag Basel/Switzerland

On the Fourier Analysis of Operators on the Torus Michael Ruzhansky and Ville Turunen Abstract. Basic properties of Fourier integral operators on the torus Tn = (R/2πZ)n are studied by using the global representations by Fourier series instead of local representations. The results can be applied in studying hyperbolic partial differential equations. Mathematics Subject Classification (2000). Primary 35L40; Secondary 58J40. Keywords. Fourier integral operators, torus, pseudo-differential operators.

1. Introduction In this paper we will discuss the version of the Fourier analysis and pseudodifferential operators on the torus. Using the toroidal Fourier transform we will show several simplifications of the standard theory. We will also discuss the corresponding toroidal version of Fourier integral operators. To distinguish them from those defined using the Euclidean Fourier transform, we will call them Fourier series operators. The use of discrete Fourier transform will allow to use global representation of these operators, thus eliminating a number of topological obstructions known in the standard theory. We will prepare the machinery and describe how it can be further used in the calculus of Fourier series operators and applications to hyperbolic partial differential equations. In fact, the form of the required discrete calculus is not a priori clear, for example, the form of the discrete Taylor’s theorem best adopted to the calculus. We will develop the corresponding version of the periodic analysis similar in formulations to the standard Euclidean theory. It was realised already in the 1970s that on the torus, one can study pseudodifferential operators globally using Fourier series expansions, in analogy to Euclidean pseudo-differential calculus. These periodic pseudo-differential operators were treated, e.g., by Agranovich [1, 2]. Contributions have been made by many authors, and the following is a non-comprehensive list of the research on the torus: This first author would like to thank the UK Royal Society for its support. The second author thanks the Academy of Finland and Magnus Ehrnrooth Foundation for their support.

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Agranovich, crediting the idea to Volevich, proposed the Fourier series representation of pseudo-differential operators. Later, he proved the equivalence of the Fourier series representation and H¨ormander’s definition for (1, 0)-symbol classes; the case of classical pseudo-differential operators on the circle had been treated by Saranen and Wendland [15]; McLean [9] proved the equivalence of these approaches for H¨ ormander’s general (ρ, δ)-classes, by using charts; in [17], the equivalence for the case of (1, 0)-classes is proven by studying iterated commutators of pseudo-differential operators and smooth vector fields. Elschner [5] and Amosov [3] constructed asymptotic expansions for classical pseudo-differential operators; these results were generalized for (ρ, δ)-classes in [18]. There are plenty of papers considering applications and numerical computation of pseudo-differential equations on torus, e.g., spline approximations by Pr¨ ossdorf and Schneider [10], physical applications by, e.g., Vainikko and Lifanov [19, 20], and many others. On the other hand, the use of operators which are discrete in the frequency variable allows one to weaken regularity assumptions on symbols with respect to ξ. Symbols with low regularity in x have been under intensive study for many years, e.g., Kumano-go and Nagase [8], Sugimoto [16], Boulkhemair [4], Garello and Morando [6], and many others. However, in these papers one assumes symbols to be smooth or sufficiently regular in ξ. The discrete approach in this paper will allow us to reduce regularity assumptions with respect to ξ. For example, no regularity with respect to ξ is assumed for L2 estimates, and for elements of the calculus. Moreover, one can consider scalar hyperbolic equations with C 1 symbols with respect to ξ. For example, this allows to construct parametrices to certain hyperbolic systems with variable multiplicities. Details of such constructions will appear in our forthcoming paper [14]. Let us now fix the notation. Let S(Rn ) be the Schwartz test function space with its usual topology, and let S  (Rn ) be its dual, the space of tempered distributions. Let FE : S(Rn ) → S  (Rn ) be the Euclidean Fourier transform (hence the subscript E ) defined by  ˜ f (x) e−ix·ξ dx, (FE f )(ξ) = fE (ξ) := Rn

˜ = (2π)−n dx. Then FE is a bijection and where dx  fE (ξ) eix·ξ dξ, f (x) = Rn

and this Fourier transform can be uniquely extended to FE : S  (Rn ) → S  (Rn ). The main symbol class in the sequel consists of H¨ormander’s (ρ, δ)-symbols of order m: Let  m ∈ R and 0 ≤ δ < ρ ≤ 1. For ξ ∈ Rn define ξ := (1 + ξ2 )1/2 , n 2 2 m n n where ξ := j=1 |ξj | . Then Sρ,δ (R × R ) consists of those functions σ ∈ C ∞ (Rn × Rn ) for which   α β ∂ ∂ σ(x, ξ) ≤ Cσαβm ξm−ρ|α|+δ|β| (1.1) ξ x for every x ∈ Rn and for every α, β ∈ Nn .

On the Fourier Analysis of Operators on the Torus

89

Let Tn = (R/2πZ)n denote the n-dimensional torus. We may identify Tn with the hypercube [0, 2π[n ⊂ Rn (or [−π, π[n ). Functions on Tn may be thought as those functions on Rn that are 2π-periodic in each of the coordinate directions. Let D(Tn ) be the vector space C ∞ (Tn ) endowed with the usual test function topology, and let D (Tn ) be its dual, the space of distributions on Tn . Inclusion D(Tn ) ⊂ D (Tn ) is interpreted by  φ(ψ) := φ(x) ψ(x) dx, Tn

where we identify the measure on torus with the corresponding restriction of the Euclidean measure on the hypercube. Let S(Zn ) denote the space of rapidly decaying functions Zn → C. Let FT : D(Tn ) → S(Zn ) be the toroidal Fourier transform (hence the subscript T ) defined by  ˜ (FT f )(ξ) = fT (ξ) := f (x) e−ix·ξ dx, Tn

˜ = (2π)−n dx. Then FT is a bijection and where dx
f (x) = fT (ξ) eix·ξ . ξ∈Zn

This Fourier transform is extended uniquely to FT : D (Tn ) → S  (Zn ). Notice that S  (Zn ) consists of those functions Zn → C growing at infinity at most polynomially. Any continuous linear operator A : D(Tn ) → D(Tn ) can be presented by a formula
σA (x, ξ) f(ξ) eix·ξ , (Af )(x) = ξ∈Zn ∞

where the unique function σA ∈ C (Tn × Zn ) is called the symbol of A: σA (x, ξ) = e−ix·ξ Aeξ (x), where eξ (x) := eix·ξ . Notice that when s A (x)T (ξ) = σA (x, ξ), the Schwartz kernel KA of A satisfies KA (x, y) = sA (x)(x − y) in the sense of distributions. Next, in analogy to the classical differential calculus, we discuss difference calculus, which is needed when dealing with Fourier series operators.

2. Difference calculus Let σ : Zn → C. Let vj ∈ Nn , (vj )j = 1 and (vj )i = 0 if i = j. Let us define the partial difference operator !ξj by !ξj σ(ξ) := σ(ξ + vj ) − σ(ξ), and define α1 αn !α ξ := !ξ1 · · · !ξn

for

α = (αj )nj=1 ∈ Nn .

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Lemma 2.1. By the binomial theorem,  
α |α−β| α σ(ξ + β). (−1) !ξ σ(ξ) = β β≤α

By induction, one can show: Lemma 2.2 (Leibnitz formula for differences). Let φ, ψ : Zn → C. Then 
α  β ! !α (φψ)(ξ) = φ(ξ) !α−β ψ(ξ + β). ξ ξ ξ β β≤α

“Integration by parts” has the discrete analogy “summation by parts”

φ(ξ) (!α ((!ξ α )t φ)(ξ) ψ(ξ), ξ ψ)(ξ) = − ξ∈Zn

ξ∈Zn

where ((!ξj )t φ)(ξ) = φ(ξ)−φ(ξ −vj ), provided that the series converge absolutely. For ξ ∈ Zn and γ ∈ Zn , let us define ξ (γ) =

n .

(γj )

ξj

,

j=1

where (γj )

ξj

⎧ 0 γj −1 ⎪ γj > 0, ⎨ i=0 (ξj − i), 1, γj = 0, := ⎪ −1 ⎩ 00 , γj < 0. i=γj +1 (ξj − i)

Then (γ) !α = γ (α) ξ (γ−α) , ξξ

in analogy to ∂ξα ξ γ = γ (α) ξ γ−α . Let us now discuss the discrete version of the Taylor’s theorem. For simplicity, let us consider the one-dimensional case first. Theorem 2.3 (Discrete Taylor’s theorem). For a function σ : Z → C, σ(ξ + η) =

N −1
α=0

where |!α ξ RN (ξ, η)|

≤

≤

1 (α) (!α + RN (ξ, η), ξ σ)(ξ) η α!

⎧ ⎪ ⎪ ⎨

1 (N ) N! η

0,

(ξ, η ∈ Z, N ∈ N),

    +α max0≤κ<η !N σ(ξ + κ) , ξ

η ≥ N, 0 ≤ η < N,

   (N )  η  maxη≤κ<0 !N +α σ(ξ + κ) , η < 0, ξ   1  (N )    +α σ(ξ + κ) η  max !N . ξ N! κ∈{0,...,η}

⎪ ⎪ ⎩

1 N!

On the Fourier Analysis of Operators on the Torus

91

Notice that the estimate above resembles closely the Lagrange form of the error term in the traditional Taylor theorem:   −1 1 (j) f (x + y) = N (x)y j + RN (x, y), j=0 j! f 1 (N ) RN (x, y) = N ! f (x + θ)y N , θ ∈ [min{0, y}, max{0, y}]. Proof. First assume that η ≥ 0. Then, by the binomial formula, η   η

1 η (α) σ(ξ + η) = (I + !ξ )η σ(ξ) = !α !α σ(ξ) = . ξ ξ σ(ξ) η α! α α=0 α=0

(2.1)

Thus RN (ξ, η) = 0 for 0 ≤ η < N . Therefore !α η RN (ξ, η)|η=0 = 0, (α) when 0 ≤ α < N . Now let η be an arbitrary integer. We notice that !N = η η (N ) (α−N ) N α η = 0 for 0 ≤ α < N , so that when we apply !η , we get N !N η σ(ξ + η) = !η RN (ξ + η).

We have hence the Cauchy problem  N !η RN (ξ, η)= !N η σ(ξ + η),  !α η RN (ξ, η) η=0 = 0,

0 ≤ α ≤ N − 1.

It is enough to prove the estimate for |RN (ξ, η)| (i.e., α = 0). Let us define σ(η) := (N ) (N −N ) η /N ! = 1, and !α η (N ) /N !. Then !N η σ(η) = N ξ σ(ξ)|ξ=0 = 0 when 0 ≤ α < N , so by the uniqueness of the solution of the Cauchy problem it has to be  −1 κ −1 κ2 −1 N 1 (N ) , η ≥ N, κN =0 κN −1 =1 · · · κ1 =1 1 = N ! η    −1 −1 −1 1  (N )  , η < 0, κN =m κN −1 =κN · · · κ1 =κ2 1 = N ! η completing the proof. Let us now deal with discrete Taylor polynomial-like expansions for a function f : Zn → C. For b ∈ N, let us denote

Ikb := and Ik−b := − . (2.2) 0≤k
−b≤k<0

It is useful to think of · · · as of a discrete version of the one-dimensional integral θ · · · dξ. In this discrete context, the difference !ξ takes the role of the differential 0 operator d/dξ. In the sequel, we adopt the notational conventions ⎧ ⎪ if α = 0, ⎨1, k Ikθ1 Ikk21 · · · Ikαα−1 1 = Ikθ 1, if α = 1, ⎪ ⎩ θ k1 Ik1 Ik2 1, if α = 2, Iξθ

and so on.

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Lemma 2.4. If θ ∈ Z and α ∈ N then k

Ikθ1 Ikk21 · · · Ikαα−1 1 =

1 (α) θ . α!

(2.3)

Proof. The result follows step-by-step from the observations k (0) ≡ 1, !k k (i) = i k (i−1) and Ikb !k k (i) = b(i) . Remark 2.5. This is like applying a discrete trivial version of the fundamental θ theorem of calculus: 0 f  (ξ) dξ = f (θ) − f (0) for smooth enough f : R → C corresponds to Iξθ !ξ f (ξ) = f (θ) − f (0) for f : Z → C. Corollary 2.6. If θ ∈ Zn and α ∈ Nn then n .

k(j,α −1)

k(j,1)

θ Ik(j,1) Ik(j,2) · · · Ik(j,αjj)

j=1

where

0n

j=1 Ij

θ

1=

1 (α) θ , α! k(j,1)

(2.4) k(j,α −1)

j means I1 I2 · · · In , where Ij := Ik(j,1) Ik(j,2) · · · Ik(j,αjj)

.

We now have Theorem 2.7. Let p : Zn → C and




rM (ξ, θ) := p(ξ + θ) −

|α|<M

Then

 ω  ! rM (ξ, θ) ≤ cM ξ

max

|α|=M, ν∈Q(θ)

1 (α) α θ !ξ p(ξ). α!    (α) α+ω  !ξ p(ξ + ν) , θ

(2.5)

where Q(θ) := {ν ∈ Zn : min(0, θj ) ≤ νj ≤ max(0, θj )}. Proof. For 0 = α ∈ Nn , let us denote mα := min{j : αj = 0}. For θ ∈ Zn and i ∈ {1, . . . , n}, let us define ν(θ, i, k) ∈ Zn by ν(θ, i, k) := (θ1 , . . . , θi−1 , k, 0, . . . , 0), i.e.,

⎧ ⎪ ⎨θj , if 1 ≤ j < i, ν(θ, i, k)j = k, if j = i, ⎪ ⎩ 0, if i < j ≤ n.

We claim that the remainder can be written in the form
rM (ξ, θ) = rα (ξ, θ),

(2.6)

|α|=M

where for each α, rα (ξ, θ) =

n . j=1

θ

k(j,1)

k(j,α −1)

j Ik(j,1) Ik(j,2) · · · Ik(j,αjj)

!α ξ p(ξ + ν(θ, mα , k(mα , αmα )));

(2.7)

On the Fourier Analysis of Operators on the Torus

93

recall (2.2) and (2.4). The proof of (2.7) is by induction: The first remainder term r1 is of the claimed form, since r1 (ξ, θ) = p(ξ + θ) − p(ξ) =

n


rvi (ξ, θ),

i=1

where rvi (ξ, θ) = Ikθi !vξ i p(ξ + ν(θ, i, k)); here rvi is of the form (2.7) for α = vi , m(α) = i and αmα = 1. So suppose that the claim (2.7) is true up to order |α| = M . Then rM+1 (ξ, θ)

=




rM (ξ, θ) −

|α|=M

1 (α) α θ !ξ p(ξ) α!


 1 (α) α θ rα (ξ, θ) − !ξ p(ξ) α!

=

|α|=M

n
.

=

θ

k(j,α −1)

k(j,1)

j Ik(j,1) Ik(j,2) · · · Ik(j,αjj)

|α|=M j=1 !α ξ [p(ξ +

ν(θ, mα , k(mα , αmα ))) − p(ξ)] ,

where we used (2.7) and (2.4) to obtain the last equality. Combining this to the observation p(ξ + ν(θ, mα , k)) − p(ξ) =

mα


ν(θ,mα ,k)i

I

!vξ i p(ξ + ν(θ, i, )),

i=1

we get rM+1 (ξ, θ)

=

n
. |α|=M j=1

=

θ

k(j,1)

k(j,α −1)

j Ik(j,1) Ik(j,2) · · · Ik(j,αjj)

ν(θ,mα ,k(mα ,αmα ))i

I(i)

i=1

i !α+v p(ξ + ξ n
.

|β|=M+1 j=1

mα


ν(θ, i, (i))) θ

k(j,1)

k(j,β −1)

j Ik(j,1) Ik(j,2) · · · Ik(j,βjj)

!βξ p(ξ + ν(θ, mβ , k(mβ , βmβ ))); the last step here is just simple tedious book-keeping. Thus the induction proof of (2.7) is complete. Finally, let us prove estimate (2.5). By (2.7), we obtain       ω  
ω !ξ rM (ξ, θ) =  !ξ rα (ξ, θ)  |α|=M 

94

M. Ruzhansky and V. Turunen   n 
. θj k(j,α −1) k(j,1) =  Ik(j,1) Ik(j,2) · · · Ik(j,αjj) |α|=M j=1   !α+ω p(ξ + ν(θ, m , k(m , α )))  α α mα ξ    
1     p(ξ + ν) ≤ θ(α)  max !α+ω , ξ α! ν∈Q(θ) |α|=M

where in the last step we used (2.4). The proof is complete. Remark 2.8. If n ≥ 2, there are many alternative forms for remainders rα (ξ, θ). This is due to the fact that there may be many different shortest discrete step-bystep paths in the space Zn from ξ to ξ + θ. In the proof above, we chose just one such path, travelling via the points ξ,

ξ + θ1 v1 , . . . ,

ξ+

j


θi vi , . . . ,

ξ + θ.

i=1

But if n = 1, there is just one shortest discrete path from ξ ∈ Z to θ ∈ Z, and in that case kM −1 M rM (ξ, θ) = Ikθ1 Ikk21 · · · IkM !ξ p(ξ + kM ). Notice also that the discrete Taylor theorem presented above implies the following smooth Taylor result: Corollary 2.9. Let p ∈ C ∞ (Rn ) and rM (ξ, θ) := p(ξ + θ) −


|α|<M

Then

 ω  ∂ξ rM (ξ, θ) ≤ cM

max

1 α θ α!

|α|=M, ν∈QRn (θ)



∂ ∂ξ

α p(ξ).

   α α+ω  θ ∂ξ p(ξ + ν) ,

(2.8)

where QRn (θ) := {ν ∈ Rn : min(0, θj ) ≤ νj ≤ max(0, θj )}. Remark 2.10. We see that in the remainder estimates above, the cubes Q(θ) ⊂ Zn and QRn (θ) ⊂ Rn could be replaced by (discrete, resp. continuous) paths from 0 to θ; especially, QRn (θ) could be replaced by the straight line from 0 to θ.

3. Pseudo-differential operators on the torus m Let m ∈ R and 0 ≤ δ < ρ ≤ 1. Then Sρ,δ (Tn × Zn ) consists of those functions σ ∈ C ∞ (Tn × Zn ) for which  α β  !ξ ∂x σ(x, ξ) ≤ Cσαβm ξm−ρ|α|+δ|β| (3.1)

On the Fourier Analysis of Operators on the Torus

95

m (Tn × Zn ), we denote A ∈ for every x ∈ Tn , for every α, β ∈ Nn . If σA ∈ Sρ,δ m n n m n n n OpSρ,δ (T × Z ). The class Sρ,δ (T × T × Z ) consists of the functions a ∈ C ∞ (Tn × Tn × Zn ) such that  α β γ  !ξ ∂x ∂y a(x, y, ξ) ≤ Caαβγm ξm−ρ|α|+δ|β+γ| (3.2)

for every x, y ∈ Tn , for every α, β, γ ∈ Nn ; such a function a is called an amplitude of order m ∈ R of type (ρ, δ). Formally we may define 
˜ (Op(a)f )(x) := f (y) a(x, y, ξ) ei(x−y)·ξ dy Tn

ξ∈Zn

for f ∈ D(Tn ). Remark 3.1. On Tn , H¨ ormander’s usual (ρ, δ)-symbol class of order m ∈ R coinm cides with the class OpSρ,δ (Tn × Zn ) [9]. m (Tn × Tn × Zn ), and define Lemma 3.2. Let a ∈ Sρ,δ  α aα (x, y, ξ) := ei(y−x) − 1 a(x, y, ξ), m−ρ|α|

where α ∈ Nn . Then !α ξ a ∈ Sρ,δ

(Tn × Tn × Zn ) and Op(aα ) = Op(!α ξ a).

m−ρ|α|

(Tn × Tn × Zn ). Now Proof. Clearly !α ξ a ∈ Sρ,δ 
˜ f (y) aα (x, y, ξ) ei(x−y)·ξ dy (Op(aα )f )(x) = Tn

 = Tn

 =

ξ∈Zn

⎛

⎞ α
 ˜ ei(y−x) − 1 a(x, y, ξ) ei(x−y)·ξ ⎠ dy f (y) ⎝ ⎛ f (y) ⎝

Tn

 =

⎛ f (y) ⎝

Tn

ξ∈Zn




⎞ t i(x−y)·ξ ⎠ ˜ a(x, y, ξ) (!α dy ξ) e

ξ∈Zn




⎞ ˜ ⎠ dy. ei(x−y)·ξ !α ξ a(x, y, ξ)

ξ∈Zn

Thus Op(aα ) = Op(!α ξ a). m (Tn × Tn × Zn ) there exists a unique Theorem 3.3. For every amplitude a ∈ Sρ,δ m n n symbol σ ∈ Sρ,δ (T × Z ) satisfying Op(a) = Op(σ), where

σ(x, ξ) ∼


1 !α ∂ (α) a(x, y, ξ)|y=x . α! ξ y

α≥0

Proof. Essentially the same as in [18].

(3.3)

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M. Ruzhansky and V. Turunen

4. Periodisation Let us define the periodisation operator p : S(Rn ) → D(Tn ) by pu(x) := FT−1 ((FE u)|Zn )(x). Let us describe the extension of this periodisation for some nice-enough classes of distributions. From the proof of the Poisson summation formula it follows that
pu(x) = u(x + 2πξ). ξ∈Zn

This formula makes sense almost everywhere for u ∈ L1 (Rn ). Indeed, formally    

eix·k e−iy·k u(y) dy = u(y) ei(x−y)·k dy (pu)(x) = Rn

k∈Zn



= Rn

Rn

u(y) δZn (2π(x − y)) dy =




k∈Zn

u(x + 2πξ).

ξ∈Zn

This calculation can be justified in the standard way. Now pu ∈ L1 (Tn ) with puL1(Tn ) ≤ uL1 (Rn ) . Moreover, if ξ ∈ Zn then p1 uT (ξ) = u E (ξ). Clearly p : L (R ) → L (T ) is a surjection. We will also use that

(pu)(x) = eix·ξ p1 uT (ξ) = eix·ξ u E (ξ). 1

n

1

n

ξ∈Zn

ξ∈Zn

Let us establish basic properties of pseudo-differential operators with respect to periodisation. Let us call symbol a(x, ξ) periodic if the function x → a(x, ξ) is 2π-periodic. We will use tildes to denote corresponding restricted operators. m Proposition 4.1. Let a ∈ Sρ,δ (Rn × Rn ) be a periodic symbol. Let a ˜ = a|Tn ×Zn . Then p ◦ a(X, D) = a ˜(X, D) ◦ p.

Notice that a does not have to be a symbol, as the same property holds when we define a(X, D) in the usual sense, by even quite irregular amplitude a(x, ξ). m (Tn × Zn ). Let f ∈ L1 (Rn ). Then we have Proof. Notice that a ˜ ∈ Sρ,δ
a(x + 2πk, D)f (x + 2πk) p(a(X, D)f )(x) = k∈Zn

=




k∈Zn

 =

Rn

 =

Rn



Rn

ei(x+2πk)·ξ a(x + 2πk, ξ) fE (ξ) dξ




 e

i2πk·ξ

eix·ξ a(x, ξ) fE (ξ) dξ

k∈Zn

eix·ξ a(x, ξ) fE (ξ) δZn (ξ) dξ

On the Fourier Analysis of Operators on the Torus =




97

eix·ξ a(x, ξ) fE (ξ)

ξ∈Zn

=




1 (ξ) eix·ξ a(x, ξ) pf T

ξ∈Zn

=a ˜(X, D)(pf )(x); these calculations are justified in the sense of distributions. The proof is complete. Since we will not always work with periodic symbols it may be convenient to periodize them. If a(X, D) is a pseudo-differential operator with symbol a(x, ξ), by (pa)(X, D) we will denote a pseudo-differential operator with symbol (pa)(x, ξ) =  k∈Zn a(x + 2πk, ξ). This makes sense if, for example, a in integrable in x. m (Rn × Rn ) satisfy a(x, ξ) = 0 for all x ∈ Rn \ [−π, π]n . Proposition 4.2. Let a ∈ Sρ,δ Then we have a(X, D)f = (pa)(X, D)f + Rf, for all f supported in [−π, π]n . Here R : S  (Rn ) → S(Rn ) is a smoothing pseudodifferential operator.

Proof. By our definition we can write
 (pa)(X, D)f (x) = k∈Zn

Rn

eix·ξ a(x + 2πk, ξ) fE (ξ) dξ,

and let Rf = a(X, D)f −(pa)(X, D)f . The assumption on the support of a implies that for every x there is only one k ∈ Zn for which a(x + 2πk, ξ) = 0, so the sum consists of only one term. It follows that Rf (x) = 0 for x ∈ [−π, π]n . Let now x ∈ Rn \ [−π, π]n . Since
  ei(x−y)·ξ a(x + 2πk, ξ) f (y) dy dξ Rf (x) = − k∈Zn ,k=0

Rn

Rn

is just a single term and |x − y| > 0, we can integrate by parts with respect to ξ any number of times. This implies that R ∈ Ψ−∞ and that Rf decays at infinity faster than any power. The proof is complete since the same argument can be applied to the derivatives of Rf . Remark 4.3. Note that if f is compactly supported, but not necessarily in the cube [−π, π]n , sums in the proof may consist of finite number of terms. This means that on E  (Rn ), modulo a smoothing operator, we can write a(X, D) as a finite sum of operators with periodic symbols. Moreover, the same argument applies if a(x, ξ) is compactly supported in x, but not necessarily in [−π, π]n . This proposition allows us to extend formula of Proposition 4.1 to perturbations of periodic symbols. We will use it when a(x, D) is a sum of a constant coefficient operator and an operator with symbol having compact x-support. Corollary 4.4. Let a(X, D) be an operator with symbol a(x, ξ) = a1 (x, ξ) + a0 (x, ξ),

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m m (Rn × Rn ) is periodic in x and a0 ∈ Sρ,δ (Rn × Rn ) has compact where a1 ∈ Sρ,δ m x-support. Then there is a symbol ˜b ∈ Sρ,δ (Tn × Zn ) such that

p(a(X, D)f ) = ˜b(X, D)(pf ) + p(Rf ), f ∈ E  (Rn ), where R : S  (Rn ) → S(Rn ). In particular, if supp(a0 (·, ξ)), supp(f ) ⊂ [−π, π]n , a0 (X, D). we can take ˜b(X, D) = a1 (X, D) + p2 We assumed that symbols are smooth, but the requirement of the smoothness of a1 (x, ξ) is not necessary similar to Proposition 4.1. Proof. By Proposition 4.2, a(X, D) = a1 (X, D) + (pa0 )(X, D) + R. Since operator b(X, D) = a1 (X, D)+(pa0 )(X, D) has periodic symbol, by Proposition 4.1 we have a0 (X, D)◦p. Since R : S  (Rn ) → S(Rn ), p◦b(X, D) = ˜b(X, D)◦p = a1 (X, D)◦p+ p2 we also have p ◦ R : S  (Rn ) → D(Tn ). The proof is complete.

5. Conditions for L2 -boundedness Next we study conditions on a toroidal symbol σA that guarantee L2 -boundedness for the corresponding operator A : D(Tn ) → D(Tn ). Notice that x → σA (x, ξ) ∈ C ∞ (Tn ) for every ξ ∈ Zn . Proposition 5.1. If

  β ∂x σA (x, ξ) ≤ C

when |β| ≤ n/2 + 1 then A ∈ L(L2 (Tn )). Proof. Now Af (x) =




σA (x, ξ) fT (ξ) eix·ξ

ξ∈Zn

=




ix·(ξ+η)  σ1 AT (η, ξ) fT (ξ) e

ξ,η∈Zn

=




ω∈Zn

eix·ω




 σ1 AT (ω − ξ, ξ) fT (ξ).

ξ∈Zn

−k Here |σ1 , so that A T (η, ξ)| ≤ C η  ˜ |Af (x)|2 dx Af 2L2 (Tn ) = Tn
1 (ω)|2 |Af = T ω∈Zn

2  

    = σ1 A T (ω − ξ, ξ) fT (ξ)    n n ω∈Z ξ∈Z

On the Fourier Analysis of Operators on the Torus ⎛ ≤ ⎝ sup  sup ≤C

2⎠ |σ1 AT (ω − ξ, ξ)|

ξ∈Zn




ξ∈Zn ω∈Zn 

⎞




ω∈Zn

99



|σ1 A T (ω − ξ, ξ)|

2

2
  fT (ξ) ξ∈Zn

f 2L2(Tn ) .

Note that no difference conditions for the ξ-variable were needed. In fact, this is related to the following more general result. Theorem 5.2. Let T u(x) =




eiφ(x,k) a(x, k) uˆ(k).

k∈Zn

Assume that for every α for which |α| ≤ 2n + 1, |∂xα a(x, k)| ≤ C,

|∂xα ∆k φ(x, k)| ≤ C.

(5.1)

Assume also that for every x ∈ Tn and for every k, l ∈ Zn , |∇x φ(x, k) − ∇x φ(x, l)| ≥ C|k − l|.

(5.2)

Then T ∈ L(L2 (Tn )). Note that condition (5.2) is a discrete version of the usual local graph condition for Fourier integral operators, necessary for the local L2 -boundedness. We also note that these conditions roughly correspond to C 1 properties of the phase in ξ. Finally, if φ and a are not 2π–periodic in x, operator T is bounded from L2 (Tn ) to L2loc (Rn ). Theorem 5.2 is the discrete version of the global boundedness theorem in [12].

6. Extending symbols It is often useful to extend toroidal symbols from Tn × Zn to Tn × Rn . This can be done with a suitable convolution that respects the symbol inequalities. The idea of the following lemma goes probably back to Y. Meyer. Lemma 6.1. There exist θ, φα ∈ S(Rn ) such that θE |Zn (ξ) = δ0,ξ and ∂ξα θE (ξ) = t (!α ξ ) φα (ξ) for every multi-index α. Proof. Let us first consider the case n = 1. Let θ = θ1 ∈ C ∞ (R1 ) such that supp(θ1 ) ⊂] − 2π, 2π[,

θ1 (−x) = θ1 (x),

θ1 (π − y) + θ1 (π + y) = 1

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for x ∈ R and for 0 ≤ y ≤ π. These assumptions for θ are enough for us, and of course the choice is not unique. In any case, θ1E ∈ S(R1 ). If ξ ∈ Zn then  ˜  θ1 (x) e−ix·ξ dx θ1 E (ξ) = R1 2π

 =

˜ (θ1 (x − 2π) + θ1 (x)) e−ix·ξ dx

0



2π

=

˜ e−ix·ξ dx

0

=

δ0,ξ .

If desired φα ∈ S(Rn ) exists, it must satisfy   ix·ξ (α) t e φ (ξ) dξ = eix·ξ (!α ξ ) φα (ξ) dξ R1

R1

because FE : S(R1 ) → S(R1 ) is bijective. Integration by parts yields (−ix)α θ(x) = (1 − eix )α (FE−1 φα )(x). Thus (FE−1 φα )(x) =



−ix 1 − eix

α θ(x).

The general n-dimensional case is reduced to the 1-dimensional case, since θ = (x → θ1 (x1 )θ1 (x2 ) · · · θ1 (xn )) ∈ S(Rn ) has desired properties. The following two results can be easily obtained from the discrete Taylor’s theorem. m Lemma 6.2. Let a : Tn × Rn → C belong to a ∈ Sρ,δ (Rn × Rn ). Then the restriction m n n σ = a|Tn ×Zn ∈ Sρ,δ (T × Z ). m (Rn × Rn ) and Proposition 6.3. Let a, b : Tn × Rn → C such that a, b ∈ Sρ,δ −∞ n (R × Rn ). a|Tn ×Zn = b|Tn ×Zn . Then a − b is smoothing, a − b ∈ S

The main theorem of this paragraph is that we can extend toroidal symbols in a unique smooth way. m m Theorem 6.4. Let σ ∈ Sρ,δ (Tn × Zn ). Then there exists a ∈ Sρ,δ (Rn × Rn ) such that σ = a|Tn ×Zn ; this extended symbol is unique up to smoothing.

Proof. Uniqueness up to smoothing follows from Proposition 6.3, so the existence is the main issue here. Let θ ∈ S(Rn ) be as in Lemma 6.1. Define a : Rn × Rn → C by
a(x, ξ) := θE (ξ − η) σ(x, η). η∈Zn

On the Fourier Analysis of Operators on the Torus

101

It is easy to see that σ = a|Tn ×Zn . Furthermore,   
    α β  α β  ∂ ∂ a(x, ξ) =  ∂ (ξ − η) ∂ σ(x, η) θ ξ x ξ E x   η∈Zn    
   t β  =  (!α ) φ (ξ − η) ∂ σ(x, η) α ξ x  η∈Zn      
 β |α|  =  φα (ξ − η) !α ∂ σ(x, η) (−1) η x  η∈Zn 
≤ |φα (ξ − η)| Cαβm ηm−ρ|α|+δ|β| η∈Zn




 ≤ Cαβm ξm−ρ|α|+δ|β|

|φα (η)| η|m−ρ|α|+δ|β||

η∈Zn  ≤ Cαβm ξm−ρ|α|+δ|β| . m (Rn × Rn ). Thus a ∈ Sρ,δ

7. Fourier series operator calculus In this section we will describe composition formulae of Fourier series operators with pseudo-differential operators. They are similar to the global composition formulae in [11] and [13] in Rn . However, the situation on the torus is technically much simpler since it does not require the global in space analysis of the corresponding remainders. Theorem 7.1 (composition T P ). Let T : D(Tn ) → D (Tn ) be defined by
 ˜ ei(φ(x,ξ)−y·ξ) a(x, y, ξ) u(y) dy, T u(x) := ξ∈Zn

Tn

where the amplitude a ∈ C ∞ (Tn × Tn × Zn ) satisfies   α β ∂ ∂ a(x, y, ξ) ≤ Cαβm ξm x y for every x, y ∈ Tn , ξ ∈ Zn and α, β ∈ Nn ; no restrictions for φ here. Let p ∈ S t (Tn × Zn ). Then
 ˜ ei(φ(x,ξ)−z·ξ) c(x, z, ξ) u(z) dz, T P u(x) = Tn

ξ∈Zn

where c(x, z, ξ) =


 η∈Zn

Tn

˜ ei(y−z)·(η−ξ) a(x, y, ξ) p(y, η) dy

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satisfying

  α β ∂x ∂z c(x, z, ξ) ≤ Cαβmt ξm+t

for every x, z ∈ Tn , ξ ∈ Zn and α, β ∈ Nn . Moreover, c(x, z, ξ) ∼


1   ∂y(α) a(x, y, ξ) !α ξ p(y, ξ) |y=z . α!

α∈≥0

Composition in the other direction is given by the following theorem. Theorem 7.2 (composition P T ). Let T : D(Tn ) → D (Tn ) such that
 ˜ ei(φ(x,ξ)−y·ξ) a(x, y, ξ) u(y) dy, T u(x) := ξ∈Zn

Tn

where a ∈ C ∞ (Tn × Tn × Zn ) satisfying  α β  ∂x ∂y a(x, y, ξ) ≤ Cαβm ξm for every x, y ∈ Tn , ξ ∈ Zn and α, β ∈ Nn ; we assume that φ ∈ C ∞ (Tn × Zn ) satisfies C −1 ξ ≤ ∇x φ(x, ξ) ≤ C ξ for some C for every x ∈ Tn , ξ ∈ Zn , and that     |∂xα φ(x, ξ)| ≤ Cα ξ, ∂xα !βξ φ(x, ξ) ≤ Cαβ for every x ∈ Tn , ξ ∈ Zn and α, β ∈ Nn \ {0}. Let p ∈ S t (Tn × Zn ). Then
 ˜ ei(φ(x,ξ)−z·ξ) c(x, z, ξ) u(z) dz, p(x, D)T u(x) = ξ∈Zn

where

Rn

 α β  ∂x ∂z c(x, z, ξ) ≤ Cαβ ξm+t

for every x, z ∈ Tn , ξ ∈ Zn and α, β ∈ Nn . Moreover, c(x, z, ξ) ∼

 
i−|α| ∂ηα p(x, η)|η=∇x φ(x,ξ) ∂yα eiΨ(x,y,ξ)a(y, z, ξ) |y=x , α!

α≥0

(here we use a smooth extension for the symbol p(x, η)) where Ψ(x, y, ξ) := φ(y, ξ) − φ(x, ξ) + (x − y) · ∇x φ(x, ξ), when x ≈ y.

(7.1)

On the Fourier Analysis of Operators on the Torus

103

8. Applications to hyperbolic equations Let a(x, D) ∈ Ψm (Rn ) (with some properties to be specified). If u depends on x and t, we write  a(x, D)u(x, t) = a(x, ξ) u E (ξ, t) eix·ξ dξ Rn   ˜ dξ. ei(x−y)·ξ a(x, ξ) u(y, t) dy = Rn

Rn

Let u(·, t) ∈ L (R ) (0 < t < t0 ) be a solution to the hyperbolic problem  ∂ i ∂t u(x, t) = a(x, D)u(x, t), u(x, 0) = f (x), 1

n

(8.1)

where f ∈ L1 (Rn ) is compactly supported. Assume now that a(X, D) = a1 (X, D) + a0 (X, D) where a1 (x, ξ) is periodic and a0 (x, ξ) is compactly supported in x (assume even that supp(a0 (·, ξ)) ⊂ [−π, π]n ). Typically, we will want to have a1 (x, ξ) = a1 (ξ) a constant coefficient operator, not necessarily smooth in ξ. Let us also assume that supp(f ) ⊂ [−π, π]n . We will now describe a way to periodise problem (8.1). According to Proposition 4.2, we can replace (8.1) by  ∂ i ∂t u(x, t) = (a1 (x, D) + (pa0 )(X, D))u(x, t) + Ru(x, t), (8.2) u(x, 0) = f (x), where the symbol a1 + pa0 is periodic and R is a smoothing operator. To study singularities of (8.1), it is sufficient to analyse the Cauchy problem  ∂ i ∂t v(x, t) = (a1 (x, D) + (pa0 )(X, D))v(x, t), (8.3) v(x, 0) = f (x) since by Duhamel’s formula WF(u − v) = ∅. This problem can be transferred to the torus. Let w(x, t) = pv(·, t)(x). In view of Proposition 4.1 it will solve the Cauchy problem  ∂ i ∂t w(x, t) = (a1 (x, D) + p2 a0 (X, D))w(x, t), (8.4) w(x, 0) = pf (x). Calculus constructed in previous sections provides the solution in the form
w(x, t) = eiφ(t,x,k) c(t, x, k)fE (k). k∈Zn

1 (k) = fE (k). Also, if the symbol a1 (x, ξ) = a1 (ξ) has conHere we note that pf T stant coefficients and a0 is of order zero, we have φ(t, x, k) = x · k + ta1 (k). In particular, ∇x φ(x, k) = k, so composition formulas for b(x, D)w in previous sections can be applied. Details of this analysis and investigation of the corresponding properties will appear in [14].

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References [1] M.S. Agranovich, Spectral properties of elliptic pseudodifferential operators on a closed curve, (Russian) Funktsional. Anal. i Prilozhen. 13 (1979), 54–56. [2] M.S. Agranovich, Elliptic pseudodifferential operators on a closed curve, (Russian) Trudy Moskov. Mat. Obshch. 47 (1984), 22–67, 246. [3] B.A. Amosov, On the theory of pseudodifferential operators on the circle, (Russian) Uspekhi Mat. Nauk. 43 (1988), 169–170; Translation in Russian Math. Surveys 43 (1988), 197–198. [4] A. Boulkhemair, L2 estimates for pseudodifferential operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), 155–183. [5] J. Elschner, Singular ordinary differential operators and pseudodifferential equations, Springer-Verlag, Berlin, 1985. [6] G. Garello and A. Morando, Lp -boundedness for pseudodifferential operators with non-smooth symbols and applications, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 8 (2005), 461–503. [7] L. H¨ ormander, The Analysis of Linear Partial Differential Operators IV, SpringerVerlag, 1985. [8] H. Kumano-go and M. Nagase, Pseudo-differential operators with non-regular symbols and applications, Funkcial. Ekvac. 21 (1978), 151–192. [9] W. McLean, Local and global description of periodic pseudodifferential operators, Math. Nachr. 150 (1991), 151–161. [10] S. Pr¨ ossdorf and R. Schneider, Spline approximation methods for multidimensional periodic pseudodifferential equations, Integral Equations Operator Theory 15 (1992), 626–672. [11] M. Ruzhansky and M. Sugimoto, Global calculus of Fourier integral operators, weighted estimates, and applications to global analysis of hyperbolic equations, in Pseudo-differential Operators and Related Topics, Editors: P. Boggiatto, L. Rodino, J. Toft and M.W. Wong, Birkh¨ auser, 2006, 65–78. [12] M. Ruzhansky and M. Sugimoto, Global L2 boundedness theorems for a class of Fourier integral operators, Comm. Partial Differential Equations, to appear. [13] M. Ruzhansky and M. Sugimoto, Weighted L2 estimates for a class of Fourier integral operators, Preprint. [14] M. Ruzhansky and V. Turunen, Fourier integral operators on the torus, in preparation. [15] J. Saranen and W.L. Wendland, The Fourier series representation of pseudodifferential operators on closed curves, Complex Variables 8 (1987), 55–64. [16] M. Sugimoto, Pseudo-differential operators on Besov spaces, Tsukuba J. Math. 12 (1988), 43–63. [17] V. Turunen, Commutator characterization of periodic pseudodifferential operators, Z. Anal. Anw. 19 (2000), 95–108. [18] V. Turunen and G. Vainikko, On symbol analysis of periodic pseudodifferential operators, Z. Anal. Anw. 17 (1998), 9–22.

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[19] G.M. Vainikko and I.K. Lifanov, Generalization and use of the theory of pseudodifferential operators in the modeling of some problems in mechanics, (Russian) Dokl. Akad. Nauk. 373 (2000), 157–160. [20] G.M. Vainikko and I.K. Lifanov, The modeling of problems in aerodynamics and wave diffraction and the extension of Cauchy-type integral operators on closed and open curves, (Russian) Differ. Uravn. 36 (2000), 1184–1195, 1293; Translation in Diff. Equ. 36 (2000), 1310–1322. Michael Ruzhansky Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2AZ, United Kingdom e-mail: [email protected] Ville Turunen Institute of Mathematics Helsinski University of Technology P.O. Box 1100 FIN-02015, Finland e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 107–115 c 2006 Birkh¨
auser Verlag Basel/Switzerland

Wave Kernels of the Twisted Laplacian Jingzhi Tie and M.W. Wong Abstract. We construct the wave kernels of the non-isotropic twisted Laplacian by means of its heat kernel. We then express the wave kernels at the origin in terms of complex Fourier integrals and we exploit the connections of the phase functions with the complex integrals. Mathematics Subject Classification (2000). Primary 35A22, 35L05, 35S30; Secondary 47G30. Keywords. Twisted Laplacian, wave kernels, finite propagation speed.

1. Introduction Let a = (a1 , a2 , . . . , an ), where aj > 0, j = 1, 2, . . . , n. Then the non-isotropic twisted Laplacian La associated to a on Cn = R2n is given by  2  2 n ∂ 1
∂ . La = − + 2iaj yj + − 2iaj xj 2 j=1 ∂xj ∂yj If aj = 12 , j = 1, 2, . . . , n, then La is the isotropic twisted Laplacian. For j = 1, 2, . . . , n, we let ∂ ∂ Zj = − aj z j and Zj = + aj z j , ∂zj ∂zj where ∂ 1 = ∂zj 2



∂ ∂ −i ∂xj ∂yj

 and

∂ 1 = ∂zj 2



∂ ∂ +i ∂xj ∂yj

 .

Then a straightforward calculation gives n
La = − (Zj Zj + Zj Zj ). j=1

This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

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J. Tie and M.W. Wong

When n = 1, the fundamental solution, also known as the Green function, and the heat kernel of the isotropic twisted Laplacian are derived from scratch in [9, 10, 11]. The underpinnings of these formulas are Weyl transforms, which can be found in [8]. Similar formulas for the non-isotropic twisted Laplacian for n > 1 using the heat kernel of the sub-Laplacian on the Heisenberg group can be found in [7]. We recall these formulas in the following theorems. Theorem 1.1. The Green function or the fundamental solution G of the nonisotropic twisted Laplacian La is given by    ∞. n  n 2 aj 1 e− j=1 aj [|zj −ζj | coth(aj s)+zj ζj −zj ζj ] ds G(z, ζ) = 2π n 0 j=1 sinh(aj s) for all z and ζ in Cn , which means that   La G(·, ζ)f (ζ) dζ = f Cn n

for all Schwartz functions on C .

Theorem 1.2. The heat kernel Ht of the non-isotropic twisted Laplacian La is given by ⎧ ⎫ n ⎬ n 2 1 ⎨. aj Ht (z, ζ) = n (1.1) e− j=1 aj [|zj −ζj | coth(2aj t)+zj ζj −zj ζj ] π ⎩ sinh(2aj t) ⎭ j=1

for all z and ζ in C , which means that Ht is the kernel of the strongly continuous one-parameter semigroup e−tLa , t > 0, generated by La . n

Remark 1.3. The construction of the heat kernel of the sub-Laplacian on the Heisenberg group can be traced back to [2, 4] and expositions can be found in [1, 6]. The heat kernel of the isotropic twisted Laplacian can also be found on page 85 of [6]. Now, we look at the initial value problem for the wave equation governed by the non-isotropic twisted Laplacian, i.e., ⎧ ∂2u ⎪ ⎪ z ∈ Cn , t > 0, ⎪ ∂t2 (z, t) = (La u)(z, t), ⎨ (1.2) u(z, 0) = f (z), z ∈ Cn , ⎪ ⎪ ⎪ ⎩ ∂u (z, 0) = g(z), z ∈ Cn . ∂t The unique solution u of the preceding initial value problem is given by −1/2 sin (tL1/2 u(z, t) = (cos (tL1/2 a )f )(z) + (La a )g)(z),

z ∈ Cn , t > 0.

The aim of this paper is to give explicit formulas for the wave kernels of the −1/2 1/2 initial value problem (1.2), i.e., the kernel Wt of the linear operator La sin (tLa ) 1/2 and the kernel Ωt of the linear operator cos (tLa ) for positive time t. Since ∂ {L−1/2 sin (tL1/2 cos (tL1/2 t > 0, a )= a )}, ∂t a

Wave Kernels of the Twisted Laplacian

109 −1/2

1/2

it is enough to look at the wave kernel Wt of the operator La sin (tLa ) for positive time t. As a motivation for the explicit formulas for the wave kernel Wt of the nonisotropic twisted Laplacian La in this paper, we first give in Section 2 a recall of the finite propagation speed of the wave kernel Wt in [7]. The main result tells us that for all z and ζ in Cn , Wt (z, ζ) = 0 whenever t < |z − ζ|. This phenomenon poses the interesting problem on finding explicit formulas for Wt for all positive numbers t. The formula for the wave kernel Wt , t > 0, in terms of the heat kernel is given in Section 3. In Section 4, we express the wave kernel Wt , t > 0, at the origin as a complex integral. The contour of the complex integral in Section 4 can be deformed as in [3] to another contour and this is carried out in Section 5.

2. Finite propagation speed The starting point is the identity e

−α

1 =√ π



∞

2

e−1/u u−3/2 e−α

/4

du,

0

which is valid for all real numbers α. Differentiating with respect to α, we get  ∞ 2 1 −1 −α e−1/u u−1/2 e−uα /4 du α e = √ 2 π 0 for every real number α. Thus, we can obtain an integral representation of the 1/2 −1/2 linear operator La e−tLa given by  ∞ 2 1 −1/2 −tL1/2 a =√ e−t /(4u) u−1/2 e−uLa du, t > 0. La e π 0 1/2

So, the kernel Pt of the linear operator L−1/2 e−tL is given by  ∞ 2 1 e−t /(4u) u−1/2 Hu (z, ζ) du Pt (z, ζ) = √ π 0 for all positive numbers t and all z and ζ in Cn . Since sin u =

eiu − e−iu , 2i

it follows that a plausible way to get Wt is to complexify the positive time t to imaginary times it and −it. Now, for z and ζ in Cn , ⎧ ⎫ n ⎨ ⎬ n . 2 aj 1 |Hu (z, ζ)| = n e− j=1 aj |zj −ζj | . π ⎩j=1 sinh (2aj u) ⎭

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Since sinh u behaves like eu /2 and coth u is bounded for large and positive values of u, it follows that ⎧ ⎫  ∞ n ⎨. ⎬ n 2 2 a j e−τ /(4u) u−1/2 e− j=1 aj |zj −ζj | coth (2aj u) du ⎩ ⎭ sinh (2a u) j 1 j=1 is absolutely convergent and is an entire function of τ . For small and positive values of u, sinh u behaves like u and coth u behaves like u1 . Therefore there exists a positive constant C such that  ⎧  ⎫   1 n ⎬ n  −τ 2 /4u −1/2 ⎨ .  2 a j − j=1 aj |zj −ζj | coth (2aj u)  e u e   du ⎩ ⎭ sinh (2a u) j 0   j=1  1  n 2 1  −τ 2 /(4u)  −1/2 −n − 2u j=1 |zj −ζj | du ≤C u e e u 0 ∞ 2 2 =C un−(3/2) e−Re (τ +2|z−ζ| )u/4 du, 1

which is convergent provided that Re (τ 2 + 2|z − ζ|2 ) > 0. Letting τ = ±it, t > 0, we see that the integral  ∞ 2 e−(±it) /(4u) u−1/2 Hu (z, ζ) du 0

exists provided that t<

√ 2|z − ζ|.

Thus, the wave kernel Wt is explicitly given by Wt (z, ζ) =

Pit (z, ζ) − P−it (z, ζ) =0 2i

for all z and ζ in Cn provided that t<

√ 2|z − ζ|.

So, we have the following theorem on the finite propagation speed of the wave kernel. Theorem 2.1. For all z and ζ in Cn and all positive numbers t, Wt (z, ζ) = 0 whenever t < |z − ζ|.

Wave Kernels of the Twisted Laplacian

111

3. The wave kernel Wt −1/2

Theorem 3.1. The wave kernel Wt of the linear operator La 3 √  c+i∞ π t Wt (z, ζ) = etu/2 Ht/(2u) (z, ζ)u−3/2 du, 2 2πi c−i∞

1/2

sin(tLa ) is given by

z, ζ ∈ Cn , t > 0,

where c is a positive number, Ht/(2u) is obtained by complexifying the time in √ the heat kernel given by (1.1), and for every nonzero complex√ number z, z is understood throughout this paper to be the principal branch of z. Proof. We begin with the well-known formula for the Bessel function of the first kind and order 12 to the effect that 3 2 sin z √ (3.1) J1/2 (z) = π z for all nonzero complex numbers z. We also need Schl¨afli’s integral for the Bessel function of the first kind and order ν telling us that  c+i∞ 2 2πiJν (αz) = z ν eα(u−(z /u))/2 u−ν−1 du (3.2) c−i∞

for all z in C such that the formula is defined, where c, α and ν are positive numbers and the principal branch of z ν is adopted. This can be obtained by a change of variable and a deformation of the contour in the integral (9.12) on page 58 in [5]. Using (3.1) and (3.2), we get for all positive numbers c and α, 3 √  c+i∞ 2 sin (αz) π α = eα(u−(z /u))/2 u−3/2 du z 2 2πi c−i∞ 1/2

for all nonzero complex numbers z. Thus, letting z = La , we get 3 √  c+i∞ π t 1/2 L−1/2 sin (tL ) = etu/2 e−(t/(2u))La u−3/2 du, a a 2 2πi c−i∞




which gives the asserted formula.

4. A complex Fourier integral Let ν be the function on (0, ∞) defined by ν(t) =

n . j=1

aj , sinh (2aj t)

t ∈ (0, ∞).

Let γ be the function on (0, ∞) × Cn defined by γ(t, z) =

n
j=1

t > 0,

aj |zj |2 coth (2aj t),

(t, z) ∈ (0, ∞) × Cn .

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J. Tie and M.W. Wong

Then for t > 0, the heat kernel ht at the origin is given by 1 ht (z) = Ht (z, 0) = n ν(t)e−γ(t,z) , z ∈ Cn . π For t > 0, we let wt be the wave kernel at the origin defined by wt (z) = Wt (z, 0),

z ∈ Cn .

Theorem 4.1. Let c be a positive number. Then for t > 0,  2 1 dw wt (z) = (1/2)+n e(t /(4w))−γ(w,z) ν(w) √ , w 2π i Γt

z ∈ Cn ,

where Γt is the circle with center at (t/(4c), 0) and radius t/(4c). Proof. Using Theorem 3.1 and the heat kernel at the origin, we get for all positive numbers t and all z in Cn , 3 √  c+i∞ π t etu/2 ht/(2u) (z)u−3/2 du. wt (z) = 2 2πi c−i∞ Letting v = 2u/t, we get for all t in (0, ∞) and z in Cn ,  (2c/t)+i∞ 2 1 wt (z) = √ et v/4 h1/v (z)v −3/2 dv. 2 πi (2c/t)−i∞ Letting w = 1/v, the contour Re v = 2c/t is transformed into the circle Γt with center at (t/(4c), 0) and radius t/(4c), and  2 1 dw wt (z) = √ et /(4w) hw (z) √ , t > 0, z ∈ Cn , 2 πi Γt w


and the proof is complete.

5. The phase function The phase function Φ of the wave kernel at the origin given by Theorem 4.1 is the function on C\{0} × Cn × (0, ∞) defined by
t2 t2 Φ(w, z, t) = − γ(w, z) = − aj |zj |2 coth (2aj w) 4w 4w j=1 n

for all (w, z, t) in C\{0} × Cn × (0, ∞). In this section, we use the method in [3] to deform the contour Γ in Theorem 3.1 to a contour C, which is symmetric with respect to the real axis in the w-plane and is contained in the region {(w, z, t) ∈ C\{0} × Cn × (0, ∞) : Re (Φ(w, z, t)) ≤ 0}. To that end, we begin with the task of finding the set of points (w, z, t) in C\{0} × Cn × (0, ∞) for which Re (Φ(w, z, t)) = 0.

Wave Kernels of the Twisted Laplacian

113

If we let w = u + iv and use the formula sinh (2u) − i sin (2v) coth (u + iv) = 2(sinh2 u + sin2 v) for all w = u + iv such that the denominator does not vanish, then for all (w, z, t) in C\{0} × Cn × (0, ∞), sinh (4aj u) − i sin (4aj v) t2 (u − iv)
− aj |zj |2 4(u2 + v 2 ) j=1 2[sinh2 (2aj u) + sin2 (2aj v)] n

Φ(w, z, t) = and hence


sinh (4aj u) t2 u − . aj |zj |2 2 2 2 4(u + v ) j=1 2[sinh (2aj u) + sin2 (2aj v)] n

Re (Φ(w, z, t)) =

For a fixed point (z, t) ∈ Cn × (0, ∞), let λ = (λ1 , λ2 , . . . , λn ), where 2aj |zj |2 , j = 1, 2, . . . , n, t2 and let Θλ be set of all points w in C\{0} such that λj =

Re (Φ(w, z, t)) = 0. Then Θλ =

⎧ ⎨ ⎩

w = u + iv ∈ C\{0} :

u = u2 + v 2

n
j=1

⎫ ⎬

λj sinh (4aj u) . sinh2 (2aj u) + sin2 (2aj v) ⎭

It is obvious that Θλ is symmetric with respect to the u-axis and the v-axis, and it contains the v-axis. To find the intersection points of Θλ with the u-axis, let v = 0 in Θλ . Then n 1
λj sinh (4aj u) = . u j=1 sinh2 (2aj u) Thus,

n
j=1

2λj u =1 tanh (2aj u)

and the set of all its nonnegative solutions is denoted by ⎧ ⎫ n ⎨ ⎬
2λj uλ uλ ∈ [0, ∞) : =1 . ⎩ ⎭ tanh (2aj uλ ) j=1 To find the intersection points of Θλ with the v-axis, we see that there is no information by simply letting u = 0 in Θλ . Instead, we write the equation in Θλ as
sinh (4aj u) λj 1 = 2 2 2 u +v u sinh (2aj u) + sin2 (2aj v) j=1 n

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J. Tie and M.W. Wong

and take the limit as u → 0. This gives n
4aj λj 1 . = v2 sin2 (2aj v) j=1 Hence

n
λj j=1

aj

2aj v sin (2aj v)

!2 =1

and the set of all its nonnegative solutions is denoted by ⎫ ⎧  n ⎬ ⎨
λj 2aj vλ =1 . vλ ∈ [0, ∞) : ⎭ ⎩ a sin (2aj vλ ) j=1 j It is clear that the poles of the integrand in wt are given by the zeros of sinh (2aj w) = 0,

j = 1, 2, . . . , n,

i.e., im π, m ∈ Z. 2aj So, we can enumerate the poles on the positive v-axis as 0 < w1 < w2 < · · · . To w=

v v λ w2

C

w1

Γ ε

u λ

u

− w1

− w2

Figure 1. The contours Γ and C in the w-plane, where w = u + iv get a contour C, we begin with the point uλ closest to the origin, and join it with a path having a vertical tangent at (uλ ) to the point vλ , which is closest to the

Wave Kernels of the Twisted Laplacian

115

origin. Then we go down along the v-axis to the origin in such a way that we make a small detour into the right half-plane when we come across a pole. The path is then symmetrized with respect to the u-axis. As an illustration by Figure 1, we assume that two poles w1 and w2 are met as we move down to the origin along the v-axis from above.

References [1] C. Berenstein, D.-C. Chang and J. Tie, Laguerre Calculus and its Applications on the Heisenberg Group, AMS/International Press, 2001. [2] B. Gaveau, Principe de moindre action, propagation de la chaleur et estim´ees sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), 95–153. [3] P.C. Greiner, D. Holcman and Y. Kannai, Wave kernels related to second order operators, Duke Math. J. 114 (2002), 329–386. [4] A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group, Studia Math. 56 (1976), 165–173. [5] F.W.J. Olver, Introduction to Asymptotics and Special Functions, Academic Press, 1974. [6] S. Thangavelu, An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups, Birkh¨ auser, 2004. [7] J. Tie, The non-isotropic twisted Laplacian on Cn and the sub-Laplacian on Hn , Comm. Partial Differential Equations 31 (2006), 1047–1069. [8] M.W. Wong, Weyl Transforms, Springer-Verlag, 1998. [9] M.W. Wong, The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. J. 34 (2005), 394–404. [10] M.W. Wong, Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator, Ann. Global Anal. Geom. 28 (2005), 271–283. [11] M.W. Wong, Weyl transforms and a degenerate elliptic partial differential equation, Proc. Royal Soc. London A 461 (2005), 3863–3870. Jingzhi Tie Department of Mathematics University of Georgia Athens, GA 30602-7403, USA e-mail: [email protected] M.W. Wong Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3, Canada e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 117–133 c 2006 Birkh¨
auser Verlag Basel/Switzerland

Super-exponential Decay of Solutions to Differential Equations in Rd Ernesto Buzano Abstract. We study the exponential decay of solutions to differential equations of hypoelliptic type (see Definition 2.1). In particular we find sufficient conditions on the differential operator A in order for estimates of the kind r

r

eε
x Au ∈ V =⇒ eε
x u ∈ V to hold, for several types of functions and distribution spaces V . For example, V may be S or S  or Cb∞ . Mathematics Subject Classification (2000). Primary 35B05, 35B40; Secondary 35H10, 35S05, 47F05, 47G30. Keywords. Super-exponential decay, solutions and eigenfunctions of differential operators, pseudo-differential calculus, Λ-hypoellipticity.

1. Introduction Exponential and super-exponential decay of solutions to differential equations in Rd have been studied by several people. See for example Agmon [1], Hislop and Sigal [6], Rabinovich [11], Rabier [8, 9] and Rabier and Stuart [10]. We also draw attention on the recent paper of Cappiello, Gramchev and Rodino [3] on superexponential estimates of solutions to semilinear differential equations with polynomial coefficients. Here we deal with Sobolev and pointwise super-exponential estimates of solutions to differential operators generalizing the harmonic oscillator. Our results are related to the recent work of Cappiello, Gramchev, Rodino [4, 5] on the exponential decay of solutions to differential equations with elliptic symbol in SG classes and Shubin’s classes.

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E. Buzano

2. Symbol classes A weight-function is a C ∞ -function Λ : Rd × Rd → R for which • there exist 0 < µ0  µ such that 1 x, ξµ0  Λ(x, ξ)  x, ξµ ,

for all x, ξ ∈ Rd ;

• there exists 0 < σ  1 such that for all α, β ∈ N we have  α β  D D Λ(x, ξ)  Λ(x, ξ)1−σ|α+β| , for all x, ξ ∈ Rd . ξ x

(2.1)

d

(2.2)

In (2.1) and (2.2) we employ the following notations: 1/2  2 2 , x, ξ = 1 + |x| + |ξ| Dξj = −i

∂ , ∂ξj

Dxj = −i

∂ ; ∂xj

moreover f (x)  g(x), for all x ∈ X, means that there exists a constant C > 0 such that f (x)  Cg(x),

for all x ∈ X.

Given a weight-function Λ, we define the following symbol-classes  

m Sρ,δ (Λ) = a ∈ C ∞ (Rd × Rd → C) : Dξα Dxβ a  Λm−ρ|α|+δ|β| , with −ρ  δ < ρ  σ. It is clear that 1 (Λ). Λ ∈ Sσ,−σ

When Λ(x, ξ) = x, ξ and δ = −ρ, reduces to Shubin’s symbol class Γm ρ (see [12]). m (Λ) defines a pseudo-differential Under these hypotheses a symbol a ∈ Sρ,δ operator  m (Λ) Sρ,δ

Au(x) = (2π)−d/2

eix·ξ a(x, ξ)ˆ u(ξ) dξ,

(2.3)

which is continuous on the Schwartz’ class S and extends to a continuous operator (we still denote by A) on temperate distributions S  . In (2.3) u ˆ is the Fourier Transform  u ˆ(ξ) = (2π)−d/2 e−ix·ξ u(x) dx. 1 Actually one can prove that the second assumption in (2.1) is a consequence of assumption (2.2).

Super-exponential Decay

119

m (Λ) is Λ-hypoelliptic if there exists a constant Definition 2.1. A symbol a ∈ Sρ,δ R  0 such that: • for all α, β ∈ Nd we have  α β 

Dξ Dx a(x, ξ)  |a(x, ξ)| Λ(x, ξ)−ρ|α|+δ|β| , for max |x| , |ξ|  R;

• there exists m0  m such that |a(x, ξ)|  Λ(x, ξ)m0 ,



for max |x| , |ξ|  R.

When m0 = m, the symbol is called Λ-elliptic. m,m0 We denote by HSρ,δ (Λ) the class of all Λ-hypoelliptic symbols belonging m (Λ) and satisfying the above estimates. to Sρ,δ An operator with Λ-hypoelliptic symbol is called Λ-hypoelliptic. In Theorems 2.2 to 2.5 we state some properties of pseudo-differential operators. The proofs are a straightforward modification of the corresponding statements in [2] and [12], and therefore are omitted. Recall that the formal adjoint A+ of a pseudo-differential operator A is defined by A+ u, v¯ = u, Av, for all u, v ∈ S . Theorem 2.2. The formal adjoint and the composition of Λ-hypoelliptic operators are Λ-hypoelliptic. More precisely, if A and B are operators with symbols a ∈ m,m0 n,n0 HSρ,δ (Λ) and b ∈ HSρ,δ (Λ), we have that A+ and BA have symbols a+ ∈ m,m0 n+m,n0 +m0 HSρ,δ (Λ) and b#a ∈ HSρ,δ (Λ). m,m0 Theorem 2.3. A Λ-hypoelliptic operator A with symbol in HSρ,δ (Λ) has a para−m0 ,−m metrix, in the sense that there exist operators Q with symbol in HSρ,δ (Λ) and K1 and K2 with symbols in S such that  QA = I + K1 , AQ = I + K2 .

In particular we have the following regularity property Au ∈ S =⇒ u ∈ S ,

for all u ∈ S  .

Theorem 2.4. Denote by A¯ the closure in L2 of a Λ-hypoelliptic operator A with m,m0 symbol in HSρ,δ (Λ). Then the maximal and the minimal closed extensions of A 2 to L coincide, that is

¯ = u ∈ S  : Au ∈ L2 . D(A) Moreover the adjoint A∗ is the closure of the formal adjoint A+ and has domain

D(A∗ ) = u ∈ S  : A+ u ∈ L2 . In particular A¯ is self-adjoint iff A is formally self-adjoint. Finally, the operator A¯ is Fredholm when m0  0.

120

E. Buzano

m,m0 Theorem 2.5. Assume that A is a Λ-hypoelliptic operator with symbol a ∈ HSρ,δ  with m0  0. Then A is a Fredholm operator both on S and on S , in the sense that ker A = {u ∈ S : Au = 0} = {u ∈ S  : Au = 0}

has finite dimension, and A(S ) = {u ∈ S : u, v¯ = 0, for all v ∈ ker A} , A(S  ) = {u ∈ S  : u, v¯ = 0, for all v ∈ ker A} . In particular A is an isomorphism on S and S  iff ker A = 0. Denote by Cc∞ the space of C ∞ -functions with compact support. Later on we need the following Theorem 2.6. Assume that A is a Λ-hypoelliptic formally self-adjoint operator, m,m0 with symbol a ∈ HSρ,δ (Λ) and m0  0. Then there exists a real-valued function ∞ φ ∈ Cc such that A+φ is a pseudo-differential, formally self-adjoint isomorphism m,m0 on S and S  with symbol a + φ ∈ HSρ,δ (Λ), where δ+ = max {δ, 0}. + m,m0 0 Proof. Since φ ∈ S1,0 (Λ), we have a + φ ∈ HSρ,δ (Λ). + Now we prove the existence of φ. It suffices to find φ such that A + φ is an isomorphism on S . By Theorem 2.5 it suffices to find φ real-valued and such that A + φ is one-to-one. Thanks to Theorem 2.4, we have that A¯ is Fredholm, because m0  0, thus from Lemma 8 of [10] we have that there exists a real-valued function φ ∈ Cc∞ such that A¯ + φ is one-to-one. This in particular implies that also A + φ : S → S is one-to-one and the proof is complete.


3. Basic estimate Now we state our main result on super-exponential estimates. Theorem 3.1. Consider a Λ-hypoelliptic differential operator A with symbol a ∈ m,0 (Λ). HSρ,δ Let s  ρµ0 > 0 be such that xs  Λ(x, ξ)ρ , where, as usual,

for all x, ξ ∈ Rd ,

(3.1)

1/2  2 x = 1 + |x| .

Then there exists 0 > 0 such that for all 0 <   0 and 0 < r  1 + s we have e x Au ∈ S  =⇒ e x u ∈ S  , r

r

for all u ∈ S  .

Proof. In order to prove this theorem we need two lemmas.

(3.2)

Super-exponential Decay Lemma 3.2. Given , r ∈ R and α ∈ Nd \ 0, we have
r r Dα e− x = e− x k pα,k (x)xkr−2|α| ,

121

(3.3)

0
where pα,k is a polynomial of degree less than or equal to 2 |α| − k. Proof. By induction on |α|. We have r

r

Dj e− x = e− x irxr−2 xj , which takes care of the case |α| = 1. Now we differentiate both sides of (3.3): Dβ e− x = Dj Dα e− x
r = e− x irxr−2 xj k pα,k (x)xkr−2|α| r

r

+ e− x

r




0
  k Dj pα,k (x)xkr−2|α| − pα,k (x)i(kr − 2 |α|)xkr−2|α|−2 xj

0
=e

− xr




k+1 ixj pα,k (x)x(k+1)r−2(|α|+1)

0
+e

− xr




  k x2 Dj pα,k (x) − i(kr − 2 |α|)xj pα,k (x) xkr−2(|α|+1)

0
=e




− xr

1
+e

− xr




k ixj pα,k−1 (x)xkr−2(|β|)   k x2 Dj pα,k (x) − i(kr − 2 |α|)xj pα,k (x) xkr−2(|β|)

0
= e− x

r




k pβ,k (x)xkr−2(|β|) ,

0
with pβ,1 = x2 Dj pα,1 (x) − i(r − 2 |α|)xj pα,1 (x), pβ,k = ixj pα,k−1 (x) + x2 Dj pα,k (x) − i(kr − 2 |α|)xj pα,k (x), for 1 < k < |β|, and pβ,|β| = ixj pα,|α| (x). Eventually, one verifies that deg pβ,k  2 |β| − k. Lemma 3.3. Under the hypotheses of Theorem 2.3, the differential operator   r r A u(x) = e x A e− x u(x) m (Λ) for all  ∈ R and 0 < r  1 + s. has symbol a ∈ Sρ,δ




122

E. Buzano Moreover, we have




a (x, ξ) = a(x, ξ) +

k bk (x, ξ),

(3.4)

0
where n is the degree of the polynomial ξ → a(x, ξ), and for all α, β ∈ Nd  α β  Dξ Dx bk (x, ξ)  |a(x, ξ)| Λ(x, ξ)−ρ|α|+δ|β| , for all x, ξ ∈ Rd .

(3.5)

In particular there exists 0 > 0 such that a is Λ-hypoelliptic and belongs to m,m0 HSρ,δ (Λ) for all ||  0 and 0 < r  1 + s. Proof. Recall Leibnitz’ rule for differential operators: ([7], formula (1.1.10)):
(−1)|γ| A(γ) uDγ v, A(uv) = (3.6) γ! |γ|
n

where A(γ) is the differential operator with symbol Dξγ a(x, ξ). By using (3.6) one computes   |γ| r r r
(−1) r A(γ) u(x)Dγ e− x , A u(x) = e x A e− x u(x) = e x γ! |γ|
n

and therefore, thanks to Lemma 3.2, we obtain
(−1)|γ|
a (x, ξ) = a(x, ξ) + k pγ,k (x)xkr−2|γ| Dξγ a(x, ξ) γ! 0<|γ|
n 0
with bk (x, ξ) =


k
|γ|
n

(−1)|γ| pγ,k (x)xkr−2|γ| Dξγ a(x, ξ). γ!

In order to prove (3.5), we observe that, for k  |γ| and 0 < r  1 + s, thanks to (3.1) for all β ∈ Nd we have     η  Dx pγ,k (x)xkr−2|γ|   xk(r−1)−|η|  xs|γ|−|η|  Λ(x, ξ)ρ|γ|+δ|η| , for all x, ξ ∈ Rd . It follows that  α β  D D bk (x, ξ)  ξ x    
(−1)|γ|
β     γ+α   Dxη pγ,k (x)xkr−2|γ|  Dξ Dxβ−η a(x, ξ) γ! η k
|γ|
n

 |a(x, ξ)| Λ

η
β

−ρ|α|+δ|β|

.

Eventually, from (3.4) and (3.5) we obtain |a (x, ξ)| − |a(x, ξ)|  − |a(x, ξ)| ,

Super-exponential Decay

123

for 0 <   1 and all x, ξ, which implies that a is Λ-hypoelliptic when we choose  small enough.
Now we prove the theorem. Since A+ A is Λ-hypoelliptic with symbol a+ #a ∈ and formally self-adjoint, then from Theorem 2.6 there exists φ ∈ Cc∞ such that B = A+ A + φ is an isomorphism on S and S  . Moreover one easily verifies that also 2m,2m0 (Λ), Hρ,δ

B = e x Be− x = (A+ A) + φ r

r

is one-to-one with formal adjoint given by

2

B− = e− x Be x r

r

which again is Λ-hypoelliptic and one-to-one. Then Theorem 2.5 implies that B is an isomorphism on S  . From Theorem 2.3 we have that B−1 has symbol in −2m0 ,−2m (Λ). HSρ,δ r From e x Au(x) ∈ S  we obtain e x A+ Au(x) = (A+ ) e x Au(x) ∈ S  , r

r

  r r v = B−1 e x A+ Au + e x φu ∈ S  .

and therefore

On the other hand we have e x Be− x v = B v = e x A+ Au + e x φu = e x Bu, r

r

r

which implies

r

r

u = e− x v, r

that is

e x u = v ∈ S  . r




4. Pointwise estimates Proposition 4.1. Under the hypotheses of Theorem 2.6 we have that there exists 0 > 0 such that for all 0 <   0 and 0 < r  1 + s we have r

r

for all u ∈ S  .

e x Au ∈ S =⇒ e x u ∈ S ,

(4.1)

Proof. From Theorem 3.1 we have that e x u ∈ S  . Since   r r A e x u = e x Au ∈ S r

r

and A is Λ-hypoelliptic, by Theorem 2.3 it follows that e x u ∈ S . 2

As a matter of facts we have r

r

B u, v¯ = e
x Be−
x u, v¯ = u, e−
x Be
x v = u, B− v, for all u, v ∈ Cc∞ .

r

r




124

E. Buzano

m,m0 (Λ) Corollary 4.2. Under the hypotheses of Theorem 2.6, assume that a ∈ HSρ,δ with m0 > 0. Then there exists  > 0 such that

e x

1+s

u ∈ S,

for all generalized eigenfunctions u of A : S  → S  . Proof. Assume that u ∈ ker(A − λI)n , with n ∈ Z+ . Then the result follows from previous proposition applied to (A − λI)n , which is Λ-hypoelliptic, since m,m0 a − λ ∈ HSρ,δ (Λ) because m0 > 0.
The hypothesis that |a|  Λm0 with m0 > 0 is essential to estimate the eigenfunctions. Consider for example the ordinary differential operator  −1 2 A = 1 + x2 D +1 with symbol a(x, ξ) =

ξ2 + 1. (1 + x2 )

We have that 2,0 a ∈ HS1,0 (Λ),

where Λ(x, ξ) = x, ξ. In fact, for all |α| + |β| > 0 we have  α β        Dξ Dx a(x, ξ) = Dξα ξ 2 Dxβ 1 + x2 −1  1 + x2 + ξ 2 1−α/2 1 + x2 −1−β/2  1−α/2 1 + x2 + ξ 2  = a(x, ξ)Λ(x ξ)−α , for all x, ξ ∈ Rd . 1 + x2 Then the solutions to Au = f satisfy super-exponential estimate (4.1) with 0 < r  2, but the eigenfunctions corresponding to the eigenvalue λ = 1 are polynomials of degree 1 and therefore do not decay exponentially. Denote by Cb∞ the space of functions u ∈ C ∞ , such that Dα u ∈ L∞ for all α ∈ Nd . Proposition 4.3. Consider a Λ-hypoelliptic differential operator A with symbol a ∈ m,0 HSρ,0 (Λ). Let s  ρµ0 > 0 be such that xs  Λ(x, ξ)ρ ,

for all x, ξ ∈ Rd .

Then there exists 0 > 0 such that for all 0 <   0 and 0 < r  1 + s we have e x Au ∈ Cb∞ =⇒ e x u ∈ Cb∞ , r

r

Proof. First one proves the following simple

for all u ∈ S  .

(4.2)

Super-exponential Decay

125

0 (Λ), we Lemma 4.4. Given a pseudo-differential operator Q with symbol q ∈ Sρ,0 ∞ ∞ 3 have that Q(Cb ) ⊂ Cb .

Proof. Let χ : R → R+ be a smooth function such that  1, if |t|  1, χ(t) = 0, if |t|  2 and write −n



Qu(x) = (2π)

= (2π)−n



eiy·ξ q(x, ξ)u(x − y) dydξ

eiy·ξ χ(|y|)q(x, ξ)u(x − y) dydξ    −n eiy·ξ 1 − χ(|y|) q(x, ξ)u(x − y) dydξ + (2π)

From  Dxα eiy·ξ χ(|y|)q(x, ξ)u(x − y) dydξ
α  eiy·ξ χ(|y|)Dxβ q(x, ξ)Dxα−β u(x − y) dydξ = β β
α
α    = eiy·ξ ξ−2N Dxβ q(x, ξ) (1 − ∆y )N χ(|y|)Dxα−β u(x − y) dydξ, β β
α

we obtain that     α  Dx eiy·ξ χ(|y|)q(x, ξ)u(x − y) dydξ       
α     N  ξ−2N Dxβ q(x, ξ) (1 − ∆y ) χ(|y|)Dxα−β u(x − y)  dydξ  β β
α   dy < ∞.  ξ−2N dξ |y|
2

While, from    Dxα eiy·ξ 1 − χ(|y|) q(x, ξ)u(x − y) dydξ
α    2 eiy·ξ 1 − χ(|y|) Dxβ q(x, ξ)Dxα−β u(x − y) dydξ = β β
α 
α    −2N = eiy·ξ |y| (−∆ξ )N Dxβ q(x, ξ) 1 − χ(|y|) Dxα−β u(x − y)dydξ, β β
α

3

Actually Q is continuous on Cb∞ .

126

E. Buzano

we obtain that      α   Dx eiy·ξ 1 − χ(|y|) q(x, ξ)u(x − y) dydξ    
α      −2N   |y| (−∆ξ )N Dxβ q(x, ξ)  1 − χ(|y|) Dxα−β u(x − y) dydξ β β
α    −2N −2N −2N ρ −2N ρµ0 Λ(x, ξ) dydξ  ξ dξ |y| dy < ∞.
 |y| |y|2

Now we prove the proposition. From Lemma 3.3 and Theorem 2.2 we have r r that for 0 <   0 the operator A = e x Ae− x has a parametrix Q with 0 symbol in Sρ,0 (Λ). This in particular means that K = I − Q A is regularizing. r Assume now that e x Au(x) ∈ Cb∞ . Since Cb∞ ⊂ S  , from the basic estimate we r have that e x u(x) ∈ S  . Thus from the above lemma we obtain     r r r e x u(x) = Q A e x u(x) + K e x u(x)     r r = Q e x Au(x) + K e x u(x) ∈ Cb∞ .
In order to handle the case δ > 0, we introduce the space Ct∞ of C ∞ -functions u such that for all α ∈ Nd there exists κα ∈ R such that Dα u(x)  xκα ,

for all x ∈ Rd .

Then we have the following Proposition 4.5. Consider a Λ-hypoelliptic differential operator A with symbol a ∈ m,0 (Λ). HSρ,δ Let s  ρµ0 > 0 be such that xs  Λ(x, ξ)ρ ,

for all x, ξ ∈ Rd .

Then there exists 0 > 0 such that for all 0 <   0 and 0 < r  1 + s we have

e x Au ∈ Ct∞ =⇒ e x u ∈ Ct∞ , r

r

for all u ∈ S  .

(4.3)

Proof. The proof is the same as that one of Proposition 4.3, with the following lemma instead of Lemma 4.4: 0 (Λ), we Lemma 4.6. Given a pseudo-differential operator Q with symbol q ∈ Sρ,δ ∞ ∞
have that Q(Ct ) ⊂ Ct .

5. Estimates in Sobolev spaces Now we introduce a scale of Sobolev-like spaces, in order to give further estimates. For all s ∈ R, we define the operator:  Ws u(x) = Λ(y, η)s (Φy,η , u)L2 Φy,η (x) dydη,

Super-exponential Decay where 1

127

2

Φy,η (x) = π −n/4 eix·η e− 2 |y−x| . Ws is the anti-Wick quantization of Λs . The following proposition is a variant of Theorem 8.2 of [2] and therefore its proof is omitted (see [2] and [12] for details on anti-Wick operators). Proposition 5.1. Ws is a formally self-adjoint Λ-hypoelliptic pseudo-differential s,s operator with symbol in HSσ,−σ (Λ). Ws is an isomorphism on S and S  ; its inverse Ws−1 is a Λ-hypoelliptic −s,−s (Λ). pseudo-differential operator with symbol in HSσ,−σ For each s ∈ R we introduce the following Sobolev-like spaces:

HΛs = u ∈ S  : Ws u ∈ L2 , uH s = Ws uL2 . Λ

Observe that W0 is the identity and HΛ0 = L2 . Moreover, when Λ(x, ξ) = x, ξ, HΛs reduce to Shubin’s space Qs (see [12]). The proofs of Propositions 5.2 to 5.5 are standard and therefore are omitted (see [2] and [12]). Proposition 5.2. For all s ∈ R, HΛs is a Hilbert space. The bilinear form ⎧ ⎨S × S → C,  ⎩(u, v) → u, v =

uv dx

extends to a continuous bilinear mapping HΛs × HΛ−s → C, which we shall continue to denote by ·, ·. The spaces HΛs and HΛ−s are dual to each other with respect to this pairing. Proposition 5.3. We have the compact inclusion HΛs ⊂ HΛt whenever s < t. Moreover  4 HΛs = S , HΛs = S  , s∈R

s∈R

where the topologies of S and S  coincide with the initial and the final one, respectively. m Proposition 5.4. A pseudo-differential operator A with symbol in Sρ,δ (Λ) extends s−m s to a continuous operator from HΛ to HΛ . m,m0 Proposition 5.5. Given an operator A with Λ-hypoelliptic symbol a ∈ HSρ,δ (Λ), we have the following regularity property:

Au ∈ HΛs =⇒ u ∈ HΛs+m0 ,

for all u ∈ S  .

128

E. Buzano

Proposition 5.6. Consider a Λ-hypoelliptic differential operator A with symbol a ∈ m,m0 (Λ) and m0  0. HSρ,δ Let s  ρµ0 > 0 be such that xs  Λ(x, ξ)ρ ,

for all x, ξ ∈ Rd .

Then there exists 0 > 0 such that for all t ∈ R, 0 <   0 and 0 < r  1 + s we have r

r

e x Au ∈ HΛt =⇒ e x u ∈ HΛt+m0 ,

for all u ∈ S  .

Proof. It follows from Theorem 3.1 and Proposition 5.5.

(5.1)


6. Examples Before considering some examples, we give the following Definition 6.1. We say that the solutions to a Λ-hypoelliptic differential equation Au = f, m,m0 , HSρ,δ

satisfy super-exponential estimates of type s > 0 if they with symbol in satisfy estimates (3.2), (4.1), (4.3), and (5.1), for δ < ρ and (4.2) when δ  0. As a first example we consider operators of Schr¨ odinger type, that is operators satisfying the assumptions of Proposition 6.3. Proposition 6.2. Let p(ξ) be a hypoelliptic polynomial such that  α  Dξ p(ξ)  |p(ξ)|1−ω|α| , for |ξ|  R, ξ

m0

 |p(ξ)|  ξ , m

(6.1)

for |ξ|  R,

with 0 < ω  1/m, 0 < m0  m = deg p and R  0. Then for all l, k ∈ Z+ 1  2l 2k 2l Λ(x, ξ) = 1 + |p(ξ)| + |x|

(6.2)

is a weight function such that for all α, β ∈ Nd we have   α β Dξ Dx Λ(x, ξ)  Λ(x, ξ)1−σ|α+β| , for all x, ξ ∈ Rd , and x, ξµ0  Λ(x, ξ)  x, ξµ , for all x, ξ ∈ Rd , where

  l . σ = min ω, k     k k µ0 = min m0 , and µ = max m, . l l

(6.3)

Super-exponential Decay

129

 1/(2l) 2l 2k , Dξα |p| = 0 for |α| > 2lm and Dxβ |x| = 0 for Proof. Since Λ = Λ2l |β| > 2k, it suffices to show that   α β Dξ Dx Λ(x, ξ)2l   Λ(x, ξ)2l−ω|α+β| , for all x, ξ ∈ Rd , for 0 < |α|  2lm and |β| = 0, and for |α| = 0 and 0 < |β|  2k. We have    α    
α  β    α−β   D Λ(x, ξ)2l  = Dα p(ξ)l p¯(ξ)l  = p¯(ξ)l  Dξ p(ξ)l  Dξ ξ ξ β β
α  |p(ξ)|2l−ω|α|  Λ(x, ξ)2l−ω|α| , for |ξ|  R,  1  Λ(x, ξ)2l−ω|α| , for |ξ| < R, and    1− |β|  β  2k D Λ(x, ξ)2l  = Dβ |x|2k  1 + |x|2k x x  Λ(x, ξ)2l− k |β| , l

for all x, ξ ∈ Rd ,

because 2l − ω |α|  0,

for |α|  2lm,

and 2l −

l |β|  0, k

for |β|  2k.




Proposition 6.3. Consider a differential operator A = p(D) + q(x), where p(ξ) is a hypoelliptic polynomial satisfying the estimates of Proposition 6.2 and q(x) is a C ∞ function such that  β  Dx q(x)  |q(x)| xδ|β| , for |x|  R, (6.4) xn0  |q(x)|  xn ,

for |x|  R,

1 , 0 < m0  m = deg p, 0 < n0  n, and R  0. with 0  δ < ω  m Assume that |p(ξ)| + |q(x)|  |p(ξ) + q(x)| ,

for max |x| , |ξ|  R, then, if δ < n0 ω,

we have that ln/k,1

p + q ∈ HSσ,δ

(Λ)

where Λ and σ are defined in (6.2) and (6.3) and l, k ∈ Z+ satisfy k  n0 . l

(6.5)

(6.6)

130

E. Buzano

Remark 6.4. It follows from this proposition that the solutions to the equation p(D)u + qu = f satisfy super-exponential estimates of type s = kσ/l = min {kω/l, 1}  min {n0 ω, 1} . In particular there exists  > 0 such that the eigenfunctions of p(D) + q 1+kσ/l belong to e− x S. Proof. Since Dξα p = 0 for |α| > m, thanks to (6.1), (6.2), (6.4), (6.5) and (6.6) it suffices to observe that 1. Λ(x, ξ)  |p(ξ)| + xk/l  |p(ξ) + q(x)|  |p(ξ)| + xn  Λ(x, ξ)ln/k , 2. for 0 < |α|  m we have  1−ω|α|  2l 1−ω|α| 2l 2k α Dξ p(ξ)  |p(ξ)|  1 + |p(ξ)| + |x|    |p(ξ)| + |q(x)| Λ(x, ξ)−ω|α| , for |ξ|  R,  1−ω|α|  2l 2l 2k Dξα p(ξ)  1  1 + |p(ξ)| + |x|    |p(ξ)| + |q(x)| Λ(x, ξ)−ω|α| , for |ξ|  R and |x|  R, 3. for |β| > 0 we have  β    D q(x)  |q(x)| xδ|β|  |p(ξ)| + |q(x)| Λ(x, ξ)δ|β| , x and  β    Dx q(x)  1  ξm0  |p(ξ)| + |q(x)| Λ(x, ξ)δ|β| ,

for |x|  R

for |x|  R and |ξ|  R.

For example, the differential operator 1 2 1 2 + Dx2m + x2n + x2n Dx2m 1 2 , 1 2

with m1 , m2 , n1 , n2 ∈ Z+ , has symbol 1 2 p(ξ) + q(x) = ξ12m1 + ξ22m2 + x2n + x2n 1 2 ,

where for all α ∈ N2 we have |Dα p(ξ)|  |p(ξ)|

1−ω|α|

,

for |ξ|  1,

with ω=

1 , 2m

m = max {m1 , m2 } ,

and for all β ∈ N2 we have  β  Dx q(x)  |q(x)| ,

for |x|  1.

It follows that n/n0 ,1

p + q ∈ Hσ,0

(Λ),




Super-exponential Decay

131

with  2  2n0 1/2  , Λ(x, ξ) = 1 + ξ12m1 + ξ22m2 + x21 + x22 n = max {n1 , n2 } , n0 = min {n1 , n2 } ,   1 1 , σ = min . 2m 2n0 This means that the solutions to the differential equation 1 2 1 2 Dx2m u1 + Dx2m u2 + x2n + x2n = f, 1 2 1 2

satisfy super-exponential estimates with s = 2n0 σ = min

n

0

 ,1 .

m In particular there exists  > 0 such that all the eigenfunctions of p(D) + q(x) 1+s belong to e− x S . As a second example we describe the special class of globally multi-quasielliptic differential operators. We begin with some geometric preliminaries, see [2] for details and proofs. A convex polyhedron P ⊂ Rk is the convex hull of a finite set of points in Rk . One can show that P can be obtained as the convex hull of a finite subset V (P) ⊂ Rk of convex-linearly independent points, called the vertices of P and univocally determined by P. Moreover there exists a finite set N (P) = N0 (P)∪N1 (P) ⊂ Rk such that |ν| = 1, for all ν ∈ N0 (P), and





P = z ∈ Rk : ν · z  0, ∀ ν ∈ N0 (P) ∩ z ∈ Rk : ν · z  1, ∀ ν ∈ N1 (P) .

N0 (P) and N1 (P) are univocally determined by P, if P has non-empty interior. Definition 6.5. A complete polyhedron is a convex polyhedron with non-empty interior and such that 1. (0, . . . , 0) ∈ V (P) ⊂ Nk ;

2. N0 (P) = (0, . . . , 1 , . . . , 0) ∈ Rk : 1  j  k ; j -entry

k

3. N1 (P) ⊂ (R+ ) . Proposition 6.6. Given a complete polyhedron P ⊂ Rd × Rd , we have that 1/2 
x2α ξ 2β ΛP (x, ξ) = (α,β)∈V (P)

is a weight function such that x, ξµ0  ΛP (x, ξ)  x, ξµ ,

132

E. Buzano

and  α β  Dξ Dx ΛP (x, ξ)  ΛP (x, ξ)1−σ|α+β| , for all x, ξ ∈ Rd , with σ = inf {νj : j = 1, . . . , d and ν ∈ N1 (P)} , and µ0 = µ=

min (α,β)∈V (P)\0

max (α,β)∈V (P)\0

|α + β| ,

|α + β| .

Proof. See [2], Chapter I, Proposition 1.1.




A differential operator A with symbol
a(x, ξ) = aα (x)ξ α α∈A

is globally multi-quasi-elliptic if there exists a complete polyhedron P such that m,m a ∈ HSσ,δ (ΛP ),

with −σ  δ < σ. If the set N1 (P) has only one element ν, the operator is called globally quasielliptic. If furthermore ν1 = ν2 = · · · = ν2d , the operator is called globally elliptic. Proposition 6.7. The solutions to a globally multi-quasi-elliptic differential equam,m tion Au = f with symbol a ∈ HSσ,δ (ΛP ), satisfy super-exponential estimates of type s = σ min {|β| : (0, β) ∈ V (P) \ 0} . Proof. It follows from Proposition 6.6 and Theorem 3.1.




References [1] S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N -Body Schr¨ odinger Operators, Princeton University Press, Princeton, NJ, 1982. [2] P. Boggiatto, E. Buzano and L. Rodino, Global Hypoellipticity and Spectral Theory, Akademie Verlag, Berlin, 1996. [3] M. Cappiello, T. Gramchev and L. Rodino, Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients, J. Funct. Anal. 237 (2006), no. 2, 634–654. [4] M. Cappiello, T. Gramchev and L. Rodino Exponential decay and regularity for SG-elliptic operators with polynomial coefficients, to appear. [5] M. Cappiello, T. Gramchev and L. Rodino, Gelfand-Shilov spaces, pseudo-differential operators and localization operators, to appear.

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133

[6] P.D. Hislop and I.M. Sigal, Introduction to Spectral Theory with Applications to Schr¨ odinger Operators, Springer-Verlag, New York, 1996. [7] L. H¨ ormander, The Analysis of Linear Partial Differential Operators I, SpringerVerlag, Berlin, 1983. [8] P.J. Rabier, Fredholm operators, semigroups and the asymptotic and boundary behavior of solutions of PDEs, J. Diff. Equations 193 (2003), 460–480. [9] P.J. Rabier, Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on RN , Trans. Amer. Math. Soc. 356 (2004), 1889–1907. [10] P.J. Rabier and C.A. Stuart, Fredholm properties of Schr¨odinger operators in Lp (RN ), Differential Integral Equations 13 (2000), 1429–1444. [11] V.S. Rabinovich, Exponential estimates for eigenfunctions of Schr¨odinger operators with rapidly increasing and discontinuous potentials, Complex analysis and dynamical systems, Contemp. Math. 364, Amer. Math. Soc., Providence, RI, 2004, 225–236. [12] M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987. Ernesto Buzano Dipartimento di Matematica Universit` a di Torino Via Carlo Alberto 10 I-10123 Torino, Italy e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 135–151 c 2006 Birkh¨
auser Verlag Basel/Switzerland

Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order Alessandro Oliaro and Petar Popivanov Abstract. In this paper we study the local solvability in Gevrey classes for degenerate parabolic operators of order ≥ 2. We assume that the lower order term vanishes at a suitably smaller rate with respect to the principal part; we then analyze its influence on the behavior of the operator, proving local solvability in Gevrey spaces Gs for small s, and local nonsolvability in Gs for large s. Mathematics Subject Classification (2000). Primary 35K65. Keywords. Degenerate parabolic operators, Gevrey classes, local solvability, non local solvability.

1. Introduction In this paper we study the local solvability of the following degenerate parabolic partial differential operator: P (t, ∂t , Dx ) = ∂t + atp Dx2k − btq Dxk ,

(1.1)

where a and b are positive constants, k ∈ Z+ and p, q are integers, p, q ≥ 0. Degenerate parabolic operators have been studied by several authors. The first observation is that they may be locally nonsolvable in C ∞ , in fact in 1971 Kannai [8] considered the second order equation ∂t u + t

n


∂x2j u = f,

(1.2)

j=1

proving that for some datum f ∈ C ∞ it does not admit (locally near the origin) any solution u ∈ D . Colombini, Pernazza and Treves [4] studied a class that generalizes (1.2), finding, for the C ∞ local solvability, a necessary and sufficient condition of the kind of the (Ψ) of Nirenberg and Treves for operators of principal type. Results on local nonsolvability in D for degenerate parabolic equations can be found in

136

A. Oliaro and P. Popianov

Popivanov [15], where also the degenerate elliptic and hyperbolic cases are treated. Considering in particular the operator (1.1), it was studied in the simple case k = 1 by Rubinstein [18], who proved non local solvability in C ∞ under the condition p > 2q + 1; this last requirement is crucial, in fact for p ≤ 2q + 1, p even, (1.1) is C ∞ -locally solvable, as we can deduce by a general result in Helffer [7], who proved C ∞ hypoellipticity for a wide class of equations. In this frame let us mention also Matsuzawa [10], [11] where, for degenerate parabolic second order operators, C ∞ hypoellipticity and correctness of the Cauchy problem is treated; similar equations are studied in Ole˘ınik and Radkeviˇc [12], from the point of view of initial-boundary value problems and asymptotic behavior of the solutions. Interesting results on operators of the kind of (1.1) can be found in Popivanov [14], where the case k = 2 is analyzed, proving hypoellipticity, positive and negative solvability results in the frame of D ; in the same paper other equations are also treated, in particular necessary and sufficient conditions for C ∞ -local solvability are given for a class of two space variable degenerate parabolic operators, for which a subelliptic estimate is also proved. The results that we have recalled till now, in particular the negative ones, give rise to the questions if we can prove Gevrey Gs -local solvability for (1.1) in the cases when C ∞ -local solvability fails, and for which s the operator (1.1) is Gs locally solvable and for which it is not. For degenerate parabolic equations there are several open problems in Gevrey spaces: in Bove and Tartakoff [1], [2] Gevrey hypoellipticity is studied, but only for degenerate elliptic equations, and the same problem is treated in Rodino [17], for degenerate quasi-elliptic. The parabolic case has been studied for example in Gramchev, Popivanov and Yoshino [6], where positive and negative results about Gs hypoellipticity are showed; in particular it follows from the results in this paper that, in the case k = 1, if p > 2q+1 then (1.1) is not Gs hypoelliptic for any s ≥ 1: this leaves open the question if (1.1) is or not Gs -locally solvable, and this is the frame where we work in the present paper. We consider the operator (1.1), for a generic integer k, in the case p > 2q + 1, proving its Gs -local solvability for small s and its Gs -local nonsolvability for large s. In order to precisely state our results let us recall some definitions, starting from Gevrey spaces. Given s > 1, an open set Ω ⊂ Rn , a relatively compact set K ⊂⊂ Ω and a positive constant C, the space Gs0 (Ω, K, C) is the set of all the functions f ∈ C0∞ (Ω) with support contained in K and such that the following norm is finite:   f s,K,C := sup C −|α| (α!)−s ∂ α f L∞ (K) . (1.3) α∈Zn +

Similarly we write Gs (Ω, K, C), s ≥ 1, for the space of all the functions f ∈ C ∞ (Ω) for which the norm (1.3) of the restriction of f to K is finite. We then define Gs0 (Ω) =

ind lim

KΩ C+∞

Gs0 (Ω, K, C),

Gs (Ω) = proj lim ind lim Gs (Ω, K, C); KΩ

C+∞

Gevrey Solvability for Degenerate Parabolic Operators

137

the space Gs0 (Ω), s > 1, turns out to be the set of all the functions f ∈ Gs (Ω) with compact support contained in Ω. We may then define Ds (Ω) and Es (Ω) as topological duals of Gs0 (Ω), Gs (Ω) respectively. Definition 1.1. We say that a partial differential operator P with coefficients belonging to Gs (Ω) is Gs -locally solvable at a point x0 ∈ Ω if there exists a neighborhood U of x0 such that for every function f ∈ Gs0 (U ) there is a solution u ∈ Ds (U ) of the equation P u = f . Since for every t < s we have Gt ⊂ Gs and Ds ⊂ Dt , we deduce immediately that if P is Gs -locally solvable then it is also Gt -locally solvable for every t < s. Therefore, when an operator is not locally solvable in C ∞ the natural question is to investigate its local solvability in Gevrey classes, looking for the largest s up to which it remains Gs -locally solvable. The main result of this paper is the following. Theorem 1.2. Let us consider the operator (1.1), where we suppose that p > 2q +1. (i) If p is even and for

p−2q−1 p−q

< k1 , then P (t, ∂t , Dx ) is Gs -locally solvable at (0, 0) s<

< (ii) if p−2q−1 p−q for

1 2k

p−q ; k(p − 2q − 1)

we have that P (t, ∂t , Dx ) is not Gs -locally solvable at (0, 0) s>2

(ii) if

1 2k

≤

p−2q−1 p−q

(1.4)

p−q . p − 2q − 1

(1.5)

< k1 , then P (t, ∂t , Dx ) is not Gs -locally solvable at (0, 0) for s>2

k(p − q) . (1 − k)p − (1 − 2k)q + k

(1.6)

Regarding the techniques, we shall prove the point (i) by constructing explicitly a solution, following the idea of Tr`eves [19], cf. also Calvo and Popivanov [3]; the points (ii) and (ii) are proved by violating the well-known necessary condition of Corli, following a construction that was used in Popivanov [16] in the C ∞ frame and then in Oliaro [13] in Gevrey.

2. Local solvability in Gevrey classes In this Section we want to prove the local solvability result of Theorem 1.2. We shall proceed as in Tr´eves [19], by constructing an explicit solution u(t, x) of the equation P (t, ∂t , Dx )u = f (2.1) for an arbitrary datum f (t, x) ∈ Gs0 (Ω), with s satisfying (1.4). At first, a partial Fourier transform of (2.1) gives us   ∂t u ˆ(ξ, t) = fˆ(ξ, t), (2.2) ˆ(ξ, t) + atp ξ 2k − btq ξ k u

138

A. Oliaro and P. Popianov

    where u ˆ(ξ, t) = Fx→ξ u(x, t) , and similarly fˆ(ξ, t) = Fx→ξ f (x, t) . Now writing  t  p 2k  as ξ − bsq ξ k ds B(ξ, t, t ) = − t


we have that the following function is a solution of (2.2):  t
u ˆ(ξ, t) = eB(ξ,t,t ) fˆ(ξ, t ) dt ,

(2.3)

−T

where T > 0 is fixed. We can then write (formally) a solution of the equation (2.1) in the following way:  +∞ 1 u(x, t) = eixξ u ˆ(ξ, t) dξ. (2.4) 2π −∞ We shall show that in our hypotheses the function (2.4) is well defined; in particular, we prove now that u ˆ(ξ, t) is exponentially decreasing at infinity in the ξ-variable. Lemma 2.1. For every t ∈ [−T, t] we have that B(ξ, t, t ) ≤ C|ξ|k

p−2q−1 p−q

,

(2.5)

C being a positive constant independent of t, t and ξ. Proof. We observe at first that B(ξ, t, t ) = g(ξ, t) − g(ξ, t ),

(2.6)

where rq+1 k rp+1 2k ξ +b ξ . p+1 q+1 In order to estimate (2.6) we need to analyze the behavior of g(ξ, r), that depends on q and k. g(ξ, r) = −a

(i) if q is even and k is odd, the function g(ξ, r) has the following form:  1   1  b(p+1) −k p−q In Figure 1 we have set r1 = −r0 = a(q+1) ξ , rM = −rm = ab ξ −k p−q and p−2q−1

gM = −gm = A|ξ|k p−q , (2.7) q+1 p+1     b b p−q a b p−q where A = q+1 − p+1 is a positive constant. Now, since t ≤ t and a a taking into account the particular form of B(ξ, t, t ), cf. (2.6), we can prove the following estimates (we assume that T ≥ r1 , that is the less restrictive case): – in the case ξ < 0, since the function r → g(ξ, r) is decreasing we have that for every t ∈ [−T, T ] and t ∈ [−T, t], g(ξ, t) ≤ g(ξ, t ), and so by (2.6) we obtain B(ξ, t, t ) ≤ 0; the same arguments show that this last inequality holds when ξ > 0, t ∈ [−T, r0 ], t ∈ [−T, 0] ∪ [r1 , T ];

Gevrey Solvability for Degenerate Parabolic Operators g(ξ, r)

139

g(ξ, r)

....... .......... .. .. ... .... .. .. ... ... .. .. ... .. ... .. ... ... .. ... .. ... ... .. .................................... M .. . ... .. .. ... ... .. ... ... .. ... .. ... ... ... ... .. .. .. ... .. ... . .. .. .. .. ... .. .. .. ... .. .. ... .. .... . .. 1 .. . . . . m . . . .............................................• ....................................................................................................................• ................................................................ . . . .. . .. .. . . . . . .. M . . . 0 ... .. . . .. . . . . . ... .. ... ... ... .. .... .. . .. .. ..... .. . . .. .. .. .. . . .. .. ... .. .. ..... .. .. . .......................... ... .. m ... .... .. ... ... .. ... .. ... .. ... .. ... .... .. .. ... ... . ... ... .. .

....... ......... .. ... ... .... ... .. ... ... .. .. ... .. ... .. .. ... ... ... ... ... .. ... .. ... ... ... ... ... ... ... ... ... .... ... ..... ... ........ . . ................................................................................................................................................................................................................... ...... .. .... ... .... ... ... .. ... ... ... ... ... .. .. ... .. ... .. ... ... .. ... .. ... .. .. ... .. ... ... ... .... .. .. ... ... .. ... ... ..

Fig. 1: g(ξ, r) for fixed ξ > 0

Fig. 2: g(ξ, r) for fixed ξ < 0

g

r

r

r

r

g

r

r

0

– if ξ > 0 and t, t satisfy t ∈ [−T, r0 ], t ∈ [0, r1 ], or t ∈ [r0 , 0], t ∈ [r1 , T ], or t, t ∈ [0, T ], t ≤ t, we have that B(ξ, t, t ) ≤ gM , and then by (2.7) the estimate (2.5) is satisfied with C = A; – in the remaining case t ∈ [r0 , 0], t ∈ [r0 , r1 ], t ≤ t we have B(ξ, t, t ) ≤ gM − gm = 2gM , which means as in the previous point that (2.5) is satisfied with C = 2A; for q even and k odd the proof of Lemma 2.1 is then complete, with C = 2A in (2.5). Now we want to analyze the other cases: (ii) if q and k are even, then the function g(ξ, r) behaves as in Fig. 1 for both ξ > 0 and ξ < 0, and then the same proof as in (i) gives (2.5) with C = 2A; (iii) let us suppose now that q and k are odd. Then one can easily see that in the case ξ > 0 we have: – if t ≤ 0, since t ≤ t by hypothesis and r → g(ξ, r) is decreasing for r ≤ 0 we immediately have by (2.6) that B(ξ, t, t ) ≤ 0; – in the case t > 0, t ∈ [−T, t] we obtain B(ξ, t, t ) ≤ gM , and then by (2.7) we get (2.5) with C = A. If ξ < 0 we can proceed in the same way, since the behavior of g(ξ, r) is similar as in the case ξ > 0. Summarizing, for both q and k odd we have proved (2.5) with C = A; (iv) the last case that we have to consider is when q is odd and k is even: since in this case the function g(ξ, r) is as in Fig. 3 for both ξ > 0 and ξ < 0 we can use the same arguments as in (iii). The proof is then complete.
We can now prove the solvability part of Theorem 1.2.

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Proof of Theorem 1.2, (i). We have already written (at least formally) a solution of (2.1) in the form (2.4); now we are interested in giving sense to the formal solution (2.4). We observe at first that, since f ∈ Gs0 (Ω) we have that there exist positive constants D and  satisfying 1

sup |fˆ(ξ, r)| ≤ De−|ξ| s ,

(2.8)

r

for every ξ ∈ R. We then have from (2.3), (2.8) and Lemma 2.1 that  T p−2q−1 p−2q−1 1 1 k k p−q p−q eC|ξ| e−|ξ| s dt = 2DT eC|ξ| e−|ξ| s ; |ˆ u(ξ, t)| ≤ D −T

since by hypothesis satisfying

k p−2q−1 p−q

<

1 s

we then have that there exist constants C  , µ > 0 1

|ˆ u(ξ, t)| ≤ C  e−µ|ξ| s

for all t, and this last estimate shows that (2.4) is a well-defined solution of the linear equation (2.1).
Remark 2.2. The same construction as above gives us C ∞ local solvability in the case p ≤ 2q + 1, p even; recall, however, that this result was already known, cf. Helffer [7], as we have pointed out in the Introduction.

3. Non local solvability in Gevrey classes The aim of this section is to prove Theorem 1.2, (ii)–(ii) . We recall at first the following known property of the Gevrey seminorms (1.3) (a proof can be found in Corli [5]). Lemma 3.1. Let us fix a compact set K and two constants η >  > 0. There exists C > 0 satisfying 1 f gs,K, η− ≤ Cf s,K, η1 gs,K, η1 for all f, g ∈ Gs (Ω, K, η1 ). We shall use the following necessary condition for the local solvability in Gevrey classes, proved by Corli in [5, Proposition 2.2]. Theorem 3.2. Let us fix s > 1 and let P be a linear partial differential operator with coefficients belonging to Gs (Ω); we suppose that the linear equation P u = f admits a solution u ∈ Ds (Ω) for all f ∈ Gs0 (Ω). Then for every compact set K ⊂ Ω, for every η >  > 0, there exists a positive constant C such that  2 t 1  P u 1 uL∞ (K) ≤ Cus,K, η− (3.1) s,K, η− for all u ∈ Gs0 (Ω, K, η1 ), where tP is the transposed of P .

Gevrey Solvability for Degenerate Parabolic Operators

141

We are going now to construct a suitable function uλ (t, x) that shall violate the condition (3.1). We follow here an idea developed in Popivanov [16] for the C ∞ case and then applied in Oliaro [13] to the Gevrey frame. Let us consider now the homogeneous equation P (t, ∂t , Dx )u = 0; by making a partial Fourier transform with respect to the x-variable we obtain ∂t u ˆ(ξ, t) + (atp ξ 2k − btq ξ k )ˆ u(ξ, t) = 0,

(3.2)

with notations as in (2.2). The equation (3.2) has the following solution: u ˆ(ξ, t) = eg(ξ,t) , where

tp+1 2k tq+1 k ξ +b ξ ; p+1 q+1 recall that g(ξ, t) has (for fixed ξ > 0) a positive maximum for 1  b  p−q k ξ − p−q , t = tξ = a and there exists a constant A = A(p, q, a, b) > 0 such that g(ξ, t) = −a

g(ξ, tξ ) = Aξ k

p−2q−1 p−q

p−2q−1 p−q

− e0 (t − tξ )2 ξ k

(3.4)

.

We can then find a constant e0 > 0 satisfying g(ξ, t) = Aξ k

(3.3)

p−2q+1 p−q

(3.5) 

 1 + o(1)

(3.6)

with |o(1)| < 12 , for |t − tξ | ≤ 0 |tξ |, 0 > 0 being a sufficiently small constant. Let us fix now p−q (3.7) s < 2 p − 2q − 1


and three cut-off functions ϕ(x), ψ(ρ) and g1 (t) belonging to Gs0 (R) in the following way: – ϕ ≡ 1 for |x| % 1, ϕ ≡ 0 for |x| > 1 and 0 ≤ ϕ(x) ≤ 1 for every x ∈ R; – supp ψ = [1, 1 + µ0 ] for a positive fixed µ0 , ψ(ρ) > 0 for every ρ ∈ (1, 1 + µ0 )  +∞ and −∞ ψ(ρ) dρ = 1; – there exists a (sufficiently small) 1 > 0 such that g1 ≡ 1 for |t − 1| ≤ 1 , g1 ≡ 0 for |t − 1| > 21 and 0 ≤ g1 (t) ≤ 1 for every t ∈ R. We set now, for λ > 0,

 t  , (3.8) tλρ is given by (3.4) with λρ instead of ξ. Let us consider now the function  +∞ p−2q−1 k p−q uλ (t, x) = eixλρ eg(λρ,t)−A(λρ) ψ(ρ)gλρ (t, x) dρ, (3.9) gλρ (t, x) = ϕ(x)g1

where tλρ

−∞

g(λρ, t) being given by (3.3) with ξ = λρ; we shall prove that uλ (t, x) does not satisfy the condition (3.1) for λ → +∞.

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Remark 3.3. The function uλ (t, x) is compactly supported, and supp uλ ⊂ Kλ := {(1 − 2 )tλ ≤ t ≤ (1 + 2 )tλ } × {|x| ≤ 1} for a suitable 0 < 2 < 1. In the estimate of the quantities appearing in (3.1) for the function uλ (t, x) we can then limit our attention to K = Kλ . Proposition 3.4 (Lower bound for uλ L∞ (Kλ ) ). There exists a positive constant E0 such that p−2q+1

uλ L∞ (Kλ ) ≥ E0 λ−k 2(p−q) ,

(3.10)

for λ sufficiently large. Proof. We observe at first that, since supp g1 λ & 0 and every ρ ∈ [1, 1 + µ0 ], we have



t tλρ



⊂ K1 := {t ∈ R : |t| ≤ 12 } for

uλ L∞ (Kλ ) ≥ uλ (t, 0)L∞ (K1 ) ≥ uλ (t, 0)L1 (K1 ) . Now by applying (3.6) in the expression of uλ (t, 0) and by making the change of p−2q+1 variables (t − tλρ )λk 2(p−q) = y in the integral we obtain that for λ sufficiently ˜0 such that large there exists a positive constant E     1+µ0 uλ (t, 0)L1 (K1 ) y −e0 y 2 = e ψ(ρ)g1 + 1 dρ dy → E˜0 −k p−2q+1 k p−2q−1 2(p−q) 2(p−q) R 1 λ λ tρ for λ → +∞, as we can deduce from the Lebesgue Dominated Convergence The˜ orem. Then the estimate (3.10) is satisfied for λ & 0 with E0 = E20 .
1 ; to this aim we need some Now we pass to the estimate of uλ s,Kλ , η− technical results. The following lemma can be proved in the same way as in Oliaro [13, Lemma 4.4]; the proof is omitted.

Lemma 3.5. Let us consider the cut-off function (3.8). We can find positive constants D and G0 such that for every α, β ∈ Z+ , λ > 0, ρ ∈ [1, 1 + µ0 ] and K ⊂ R2 compact we have k



 t 

α

α+β+1 s
G0 λ (p−q)(s−s
) Dβ ϕ(x)

≤ D (α!β!) e ,

(D g1 ) tλρ s,K, η1 for arbitrary s > s , where s is the Gevrey order of the functions ϕ(·) and g1 (·). Let us recall now some results concerning derivatives of composite functions and Bell Polynomials (here in the simplest case of dimension 1, see for example Mascarello and Rodino [9, Section 5.5] for the n-dimensional case). Given f, g : R → R, f, g ∈ C ∞ (R), we have that for every ν ∈ Z+ the following Fa` a di Bruno formula holds: ν 
    dν  f (g(y)) = f (h) g(y) Bν,h {g (j) (y)} , (3.11) ν dy h=1

Gevrey Solvability for Degenerate Parabolic Operators

143

where,  given  an arbitrary sequence of real numbers {yj }j∈Z+ , the ‘Bell Polynomial’ Bν,h {yj } is given by 






Bν,h {yj } = ν!

∞  j=1

∞ 

j=1

hj =h

∞ . 1  1 hj , yj h ! j! j=1 j

(3.12)

jhj =ν

  for every ν, h ∈ Z+ , where hj are positive integers. The polynomials Bν,h {yj } satisfy the following properties: (B1 ) For every z ∈ R we have ∞ h
∞   zν 1
zj ; (3.13) yj = Bν,h {yj } h! j=1 j! ν! ν=h   (B2 ) the Bell polynomials Bν,h {yj } are homogeneous of degree h, with integer non-negative coefficients; (B3 ) for {yj }j∈Z+ and {wj }j∈Z+ with the property 0 ≤ yj ≤ wj for all j we get     (3.14) Bν,h {yj } ≤ Bν,h {wj } ; (B4 ) we have

  Bν,0 {yj } =



0 1

if ν > 0 if ν = 0

Remark 3.6. Let us shortly analyze the case when {yj } in the Bell polynomial (3.12) is a positive constant sequence, namely, yj = C for every j ∈ Z+ and for a constant C > 0. From the property  (B2 ) above we have that Bµ,h {C} ≥ 0 for every µ and h. Consider now Bν,h {C} , for ν ≥ h; for every z ∈ R, z > 0, from (3.13) we have:   Bν,h {C} ∞ h ∞   zν   zµ ν!
1 ν!
z j ν! 1 ν! ≤ ν = ν = ν Bν,h {C} Bµ,h {C} C = ν C h ehz , z ν! z µ! z h! j=1 j! z h! µ=h

which implies that, by keeping z fixed, we can find positive constants C1 and C2 satisfying   ν! Bν,h {C} ≤ C1ν C2h . h! Lemma 3.7. Let K ⊂ R2 be a compact set and fix C > 0. There exist positive constants C1 , d1 , d2 such that 1

eixλρ+g(λρ,t) s,K,C ≤ C1 em(λρ)+d1 λ s +d2 λ

2k s

,

where the function g(·, t) is defined in (3.3) and m(λρ) := sup g(λρ, t). (t,x)∈K

(3.15)

144

A. Oliaro and P. Popianov

Proof. We observe at first that there exists a constant C ≥ 1 such that |t| ≤ C for every (t, x) ∈ K; then we have that for every j ∈ Z+ and λ ≥ 1 |∂tj g(λρ, t)| ≤ ap!C

p+1

(1 + µ0 )2k λ2k + bq!C

q+1

˜ 2k , (1 + µ0 )k λk ≤ Cλ

where C˜ is a constant independent of λ. We then obtain from (3.11), (3.14) and the property (B2 ) at page 143 that for every α, β ∈ Z+ β
 α β  ixλρ+g(λρ,t)    ∂ ∂ e  ≤ (λρ)α eg(λρ,t) Bβ,h {|∂ j g(λρ, t)|} x

t

t

h=1

≤ (λρ)α

β


  ˜ 2k } eg(λρ,t) Bβ,h {Cλ

h=1

≤ (1 + µ0 ) λ e

α α m(λρ)

β


  ˜ . λ2kh Bβ,h {C}

h=1

We then have

   C −α−β (α!β!)−s sup ∂xα ∂tβ eixλρ+g(λρ,t)  (t,x)∈K



α β   −1 β C −1 (1 + µ0 )λ
2kh ˜ (C ) em(λρ) { C} λ B ≤ β,h α!s β!s

(3.16)

h=1

˜ Now we observe that  the property (B2) at page 143 guarantees that for C ≥ 1 we ˜ ˜ can only have Bβ,h {C} = 0 or Bβ,h {C} ≥ 1, which implies that, since we can take C˜ ≥ 1 without loss of generality,     s ˜ ≤ Bβ,h {C} ˜ ; Bβ,h {C} we then obtain from (3.13) that β
h=1

2kh

λ

Bβ,h

 ν !s β ∞

 (C −1 )β   (2C −1 )1/s −β 2kh ˜ ˜ {C} ≤2 λ Bν,h {C} β!s ν! h=1 ν=h  2k h !s β β  s

C˜1 λ s −β 2kh 1 ˜ h −β =2 λ =2 C h! 1 h!



h=1

≤ 2−β

h=1

β 


˜

eC1 λ

2k s s

≤ C1 ed2 λ

2k s

,

(3.17)

h=1 β ˜ (2C −1 )1/s , d2 = sC˜1 and C1 = sup 2−β  1 is a constant indepenwhere C˜1 = Ce β∈Z+

h=1

dent of β. In an analogous way we obtain  −1  −1 α 1/s α !s 1 C (1 + µ0 )λ C (1 + µ0 )λ = ≤ ed 1 λ s , s α! α!

(3.18)

Gevrey Solvability for Degenerate Parabolic Operators

145

 1/s . We finally obtain from (3.16), (3.17) and (3.18) where d1 = s C −1 (1 + µ0 ) that 1 2k    C −α−β (α!β!)−s sup ∂xα ∂tβ eixλρ+g(λρ,t)  ≤ em(λρ)+d1 λ s +d2 λ s , (t,x)∈K




which implies (3.15).

1 ). Let uλ (t, x) be the function (3.9). Proposition 3.8 (Upper bound for uλ s,Kλ , η− There exist positive constants C, d1 , d2 and G0 satisfying 1

1 uλ s,Kλ , η− ≤ Ced1 λ s +d2 λ

2k s

k

+G0 λ (p−q)(s−s
)

,

where s is the Gevrey order of the functions ϕ, ψ and g1 , cf. page 141. Proof. At first we have by Lemma 3.1 that 1 ≤ uλ s,Kλ , η−  1+µ0 p−2q−1 k

ixλρ+g(λρ,t)

p−q

e

ψ(ρ)e−A(λρ) ≤ s,K

1

1 λ, η

gλρ (t, x)s,Kλ , η1 dρ;

the conclusion then follows from Lemma 3.5 and Lemma 3.7, since now m(λρ) =

sup

g(λρ, t) = A(λρ)k

p−2q−1 p−q

,

(t,x)∈Kλ




cf. (3.5).

t 1 . Since P (t, ∂t , Dx ) = Let us pass now to the analysis of tP uλ s,Kλ , η− P (t, ∂t , Dx ) we have by a simple computation that   t P (t, ∂t , Dx ) eixλρ+g(λρ,t) = 0;

we then obtain P (t, ∂t , Dx )uλ (t, x) = I1 + I2 − I3 ,

t

where



1+µ0

I1 = I2 = (−i)j atp

1+µ0

× 1

I3 = (−i)h btq  ×

k

p−2q−1 p−q

ψ(ρ)∂t gλρ (t, x) dρ,

(3.20)

1 2k 
j=1



eixλρ eg(λρ,t)−A(λρ)

(3.19)

 2k 2k−j λ j

(3.21)

ρ2k−j eixλρ eg(λρ,t)−A(λρ)

k

p−2q−1 p−q

ψ(ρ)∂xj gλρ (t, x) dρ,

k  
k k−h λ h

h=1 1+µ0

ρ

k−h ixλρ g(λρ,t)−A(λρ)

e

e

(3.22) k

p−2q−1 p−q

ψ(ρ)∂xh gλρ (t, x) dρ.

1

Let us start by analyzing I1 ; we have the following result.

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A. Oliaro and P. Popianov

Lemma 3.9. There exist positive constants C1 , L0 , G0 , d1 and d2 satisfying k

1 I1 s,Kλ , η− ≤ C1 λ p−q e−L0 λ k

p−2q−1 p−q

k

1

+G0 λ (p−q)(s−s
) +d1 λ s +d2 λ

2k s

,

s being as usual the Gevrey order of the cut-offs. Proof. Observe   that  in the analysis of the norm of I1 we can limit our attention t to t ∈ supp g1 tλρ and so we can find ˜1 and ˜2 with 0 < ˜1 < ˜2 < 1 such that,

˜ 2 , 1 − ˜1 ] ∪ [1 + ˜1 , 1 + ˜2 ], |x| ≤ 1 , we have defining K = (t, x) ∈ R2 : ttλ ∈ [1 − ˜   ˜ for every ρ ∈ [1, 1 + µ0 ]; then, from Lemma 3.