Modern Trends in Pseudo-Differential Operators (Operator Theory: Advances and Applications)

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 aj (r) − r (0.1) A = r−µ ∂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 aα (r, y)D(r,y) , (0.3) A= |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 ajα (0, y) − r (rη)α (0.8) σ∧ (A)(y, η) := r−µ ∂r j+|α|≤µ

the so-called homogeneous principal edge symbol. It represents a family of continuous operators (0.9) σ∧ (A)(y, η) : Ks,γ (X ∧ ) → Ks−µ,γ−µ (X ∧ ), (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)

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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

W = Y (1.4) 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 L allows us to define stretched wedge neighborhoods mentioned bijection V → ∼ =

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 (Rt × X) such that defined as the set of all u(t, .) ∈ Hloc

(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,  ∞ M u(z) = rz−1 u(r)dr. 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 ∂ = M −1 zM , and Hs,γ (X ∧ ) for s ∈ N is equal to the subspace line. We have −r ∂r 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 (X ∧ ) that is defined as follows. for a space Hcone 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 (X ∧ ) is defined to be the subspace of all u ∈ Hloc (R × X)|R+ ×X such Then Hcone −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 (X ∧ ) for large r it is enough to look at above characterization of the space Hcone 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 u ˆ(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 {˜ κλ }λ∈R+ , (i) Let E and E  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  (U × (R \ {0}); E, E), j ∈ N, such that S 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 ) (U × Rq ; E, E g(y, η) ∈ Scl 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 (U × Rq ; Ks,β (X ∧ ), SO (X ∧ )) for all s, β ∈ R. g(y, η), g ∗ (y, η) ∈ Scl 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 η2s κ−1 u  (η) dη , E

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

s and Wloc (Ω, E)

s s of vector-valued analogues of the spaces Hcomp (Ω) and Hloc (Ω), respectively. 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) := 
ei(y−y )η a(y, y  , η)u(y  )dy  d¯η for d¯η = (2π)−n dη, the operator s−µ s  (Ω, E) → Wloc (Ω, E) Op(a) : Wcomp

(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 Wcomp(y) (X ∧ × Ω)

s,γ Wloc(y) (X ∧ × Ω),

(1.22)

and, similarly, spaces of the kind and Ω ⊆ Rq 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 Rn+1 regarded as a space with ‘conical exit’ to infinity x ˜ |˜ x| → ∞. Definition 1.11. The space S µ;ν (Rn+1 × Rn+1 ), x ˜ 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 ξ |Dx˜ Dξ˜ a(˜ x, ξ)| x, ξ) × Rn+1 < ∞ sup ˜ x ˜ ∈ 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,
× Rn+1 × Rn+1 ) S µ;ν,ν (Rn+1 x ˜ x ˜
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 (Rn+1 × Rξn+1 ) := Scl (Rn+1 )⊗ (Rn+1 ) x ˜ ˜ x ˜ ˜ 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 x−g H s (Rn+1 ). H s;g (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 ) . Lµ;ν ) := Op(a) : a(˜ x, ξ) (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(˜ ˜ ∈ An operator A ∈ Lµ;ν (Rn+1 ) is called elliptic if the associated symbol a(˜ x, ξ) 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 Analogously, the principal exit symbol σe (a)(˜ x, ξ) ˜ of order ν has the property ˜ = δ ν σe (a)(˜ ˜ for all (˜ ˜ ∈ (Rn+1 \ {0}) × Rn+1 , δ ∈ R+ . σe (a)(δ˜ x, ξ) x, ξ) x, ξ) ˜ in ξ˜ of The third component is the homogeneous principal part of σe (a)(˜ x, ξ) 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)

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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: x ∈ Rn+1 : |˜ x| ≥ R} = {λ˜ x:x ˜ ∈ Γι,R , λ ≥ 1} for Γι,R := Γι ∩ {|˜ x| = R}; (i) Γι ∩ {˜ (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 x) = λτ˜ι,ι (˜ x) τ˜ι,ι (λ˜ 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 ˜ λ ). κλ = κ  be Hilbert spaces with group actions {κλ }λ∈R+ and {˜ Let E and E κλ }λ∈R+ , respectively. Then
 (1.27) S µ;ν,ν (Rq × Rq × Rq ; E, E)   ∞ 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 (2.1) Lµcl (X; R1+q

,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 ˜ |η| )uL2 (R+ ×X) uHcone (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 (2.6) p0 (r, y, , η) := p˜(0, y, r, rη), h0 (r, y, z, η) := ˜h(0, y, z, rη) 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 ˜ (2.9) ω[η] + χ[η] opr (p)(y, η)χ ˜[η] σ a(y, η) := r−µ σ ω[η] opM 2 (h)(y, η)˜ 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 ω|η| + χ|η| opr (p0 )(y, η)χ ˜|η| σ∧ (a)(y, η) := r−µ ω|η| opM 2 (h0 )(y, 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 g(η) := r−µ σ(r)χ[η] (r)(1 − ψ[η] (r, r ))opr (p)(η)χ ˜[η] (r )˜ σ (r ). 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 

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

, η˜) then gives us the homogeneous components The above process applied to rl p˜l (˜ 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 ˜|η| (r ). g(µ) (η) := r−µ χ|η| (r)opr (p)χ 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 (2.24) ψ(r, r )ϕ(r) ∈ S(R × R) 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 ˜[η] (2.25) f (η) := r−µ χ[η] opr (p)(η)χ is a Green symbol, since f (η), f ∗ (η) have the property (2.14) (with f instead of ˜ , we g) and f (η) = g(µ) (η) for large |η|. Applying Remark 1.10 to ϕ := |η|, ϕ˜ := σ 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 )). p˜(0, y, ˜, η˜) ∈ C ∞ (Ω, Lµ−1 )), h(0, 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 ω[η] r−µ ω[η] opM 2 (f )(y)˜

for an f ∈ C ∞ (Ω, M−∞ (X; (when m(y, z) is given as 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 ˜[η] }˜ σ (2.32) r−µ σ{χ[η] opr (p − p0 )(y, η)χ 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; R1+q ρ , 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 (3.1) σψ (ab) = σψ (a)σψ (b), σ∧ (a)(η)σ∧ (b)(η) = σ∧ (ab)(η). 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    b0 ω  + (1 − ω)a1 (1 − ω  )b1 (1 − ω ) P := ωa0 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 − ω  )a0 (ω − ω  )b0 ω  )b0 ω , +ω  a0 (ω − 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  ).  P = ωa0 b0 ω  + (1 − ω)c1 (1 − ω  ) + G7 + G8

for G7 := ωa0 ( ω − 1)b0 ω ,

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

and c1 (η) = σr−(µ+ν) opr (p)(η) σ

(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    G1 := ω  a0 ( ω − ω)b1 (1 − ω  ), G1 := (ω − ω  )a0 ( ω − ω)b1 (1 − ω  ). We have

  a0 ( ω − ω), D := ϕb1 (1 − ω ) G1 = CD for C := ω ∞ for any ϕ ∈ C0 (R+ ) that is equal to 1 on supp ( ω − ω). The function ϕ is interpreted (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, ω − ω), we can write L = DC for taking some ϕ ∈ C0∞ (R+ ), ϕ ≡ 1 on supp (  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
akα (x, y)(xDx )k Dyα (1.3) A = x−m 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   (2.1) x−2 (xDx )2 + a(x, y)(xDx ) + ∆Y (x) , 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
Pˆ (σ) = akα (0, y)σ k Dyα . 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 (2.5) G = {D ∈ D : ind AD = 0}

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   (3.5) ∆∧ = x−2 (xDx )2 + ∆Y , 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,  ◦ bg-res A∧ = Λα . 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. G∧,α . We fix a component Λα of bg-res A∧ and let G∧ = ◦ 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 ˆK∧,λ ,D L (E∧,max ,· ) ≤ C for λ ∈ ΛR , (4.4) 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 κ−1 π κ 1/m = πK |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 . L (x−m/2 L2 ,D∧,max ) |λ| 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

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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 − vk v → ∞ 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   (1 − ω0 )B(λ)ω1 ∈ S Λ, L (x−m/2 L2b , Dmax ) (5.3) 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)ϕ(x) = ep (t, x, y)ϕ(y)dvy , ϕ ∈ Γ ∧p T ∗ (M ) 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 fp = + kj (0 ≤ p ≤ n) n− n−j

(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 CI (x) = ( √ )n ( )m sign(π) sign(σ) 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)ϕ(x) = ep (t, x, y)ϕ(y)dvy , ϕ ∈ Γ ∧p T ∗(0,1) (M ) 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 ∧ dv I . CID (x) = det Ω 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

dv I = ω ¯ j1 ∧ ω j1 ∧ ω ¯ j2 ∧ ω j2 · · · ∧ ω ¯ jn− ∧ ω jn− , 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)

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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 

Then a∗k ak

=

So we have on ∧j (V )

 Mp =

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

1, on vI if k ∈ I; 0, on vI if k ∈ I.

0, if j ≤ p − 1; j  , if j ≥ p. p

(2.8)

Symbolic Calculus and Curvature By (2.8) we have (2.6). Owing to (2.6) it holds   n
j+p j Ψp = (−1) Mj , (0 ≤ p ≤ n). p j=p On the other hand by the definition of Mp we have  j 
n−q Mj = 2−j Γq . n−j q=0

57

(2.9)

(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 cpq = (−1)p+q pq , dpq = 2−p n−p . C −1 and D−1 are matrices whose elements   n−q  c˜pq , d˜pq (0 ≤ p, q, ≤ n) are given by c˜pq = qp , d˜pq = 2q (−1)p+q n−p . So, the following equation holds;   
q n−j (M−1 )pq = (−1)p+j 2j . j n−p 0≤j≤p,q

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 (−1) 2 (2.12) xj = (−1)q 2−q xq (1 − )n−q . n−j 2 j=q Put x = 1 − u in (2.12). Then we have    n
n
n−q p p+j −j j (−1) 2 u = 2−n (u − 1)q (1 + u)n−q . 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  Mpq (−1) = 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  Mpq (−1) = q −n q n −  (−1) , if q ≥ . 2  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  Mpq (−1) = 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  Mpq (−1) = 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 kj = (−1) 2 2 (−1) + αp .  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 = (−1)n−+p n− ; (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 (−1) 2 kj j , ( ≤ p ≤ n). j=+1 2  + So, f defined by (2.18) satisfies (2.17) if and only if kj satisfies      p
p p 2j−n αp = (−1)p 2−n + kj .  j

(2.20)

j=+1

The equation (2.20) means that     p−1
p j p − kj . 2 kp = (−1) 2 αp − 2 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
R= Rqkij a∗i aj a∗k aq . 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)
sign(π) sign(σ) = 2n−r−m



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 if r is odd 0(t− 2 + 2 + 2 ),




(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

e(¯ ω j )∇Z¯j , ϑ” = − ı(Z¯j )∇Zj , d” = 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
Rγ¯βi¯j Zγ . R(Zi , Z¯j )Zβ = 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¯j ∇Zj ) − ∇(∇Zj Z¯j +∇Z¯ Zj ) − R(Zj , Z¯j )}. 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 det g. u ˜0 (t, x, x) = (2πt)−n det exp(tM0 ) − Id Note that e(t, x, x)dv = u˜0 (t, x, x)dx(1 + 0(t)). We shall calculate

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

 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 det = tr (βI Ar )tr + 0(tr+1 ). 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 dv = in ω 1 ∧ ω ¯ 1 ∧ ω2 ∧ ω ¯ 2 · · · ∧ ωn ∧ ω ¯n = (−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

· · · iπ(r) , iσ(r) )¯ ω iπ(1) ∧ ω iσ(1) ∧ ω ¯ iπ(2) ∧ ω iσ(2) · · · ∧ ω ¯ iπ(r) ∧ ω iσ(r) ∧ dv I . 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. ˜0 (t, x, x))dx (4.9) tr (βI u  n   c 1 Ω = (−1)r ) 2n−r t−n+r det( ∧ dv I + 0(t−n+r+1 ). 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, π(g)ϕ)Xπ (π(g)ϕ, y)Xπ dµ(g), (x, y)Xπ = 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= (f, dπ πjk )L2 (G) dπ πjk = απ,j,k πjk , 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 !  tr f (h)π(h−1 )π(g) dµ(h) 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 = λπ = λπ I, [νjk (e)]dj,k=1

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

 I= u(gh )χπ (g ) dµ(g) dµ(h) 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 G(g, h) = χπ (gh−1 ), g, h ∈ G. 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 kt (g, g) dµ(g) − 1 ts ζG (s) = 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 e e , kt (θ) = 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 a

−b

⎠ by b a 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, as the special unitary group. We sometimes denote the matrix ⎝

−1 ∗ = Ua,b = Ua,−b . Ua,b

The group action of SU(2) on C2 is given by ⎛ ⎞⎛ ⎞ ⎛ ⎞ a −b z az − bw ⎝ ⎠⎝ ⎠=⎝ ⎠ b bz + aw a 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, Q)L2 (S 3 ) = P (σ)Q(σ) dν(σ), P, Q ∈ Pm . 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 (πm (a, b)P, Q)L2 (S 3 ) = P (Ua,b ω)Q(ω) dν(ω) 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
1 ∂2 ∂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 Pk (σ (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 P (σ · τ ) 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  ˜ (FE f )(ξ) = fE (ξ) := f (x) e−ix·ξ dx, Rn

˜ = (2π)−n dx. Then FE is a bijection and where dx  f (x) = fE (ξ) eix·ξ dξ, 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 m n n where ξ2 := 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
fT (ξ) eix·ξ . f (x) = 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 !ξ σ(ξ) = (−1) σ(ξ + β). 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) (x)y j + RN (x, y), f (x + y) = N 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

η 1 (α) σ(ξ + η) = (I + !ξ )η σ(ξ) = !α σ(ξ) = . !α 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 ) /N !. Then !N η /N ! = 1, and !α 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

and Ik−b := − . (2.2) Ikb := 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
rα (ξ, θ), rM (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 !ξ p(ξ) rα (ξ, θ) − 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 Lemma 3.2. Let a ∈ Sρ,δ (Tn × Tn × Zn ), and define  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 
˜ (Op(aα )f )(x) = f (y) aα (x, y, ξ) ei(x−y)·ξ dy 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 ⎠ ˜ dy a(x, y, ξ) (!α 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
u(x + 2πξ). pu(x) = 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 Proof. Notice that a ˜ ∈ Sρ,δ (Tn × Zn ). Let f ∈ L1 (Rn ). Then we have
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 Proposition 4.2. Let a ∈ Sρ,δ (Rn × Rn ) satisfy a(x, ξ) = 0 for all x ∈ Rn \ [−π, π]n . 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 , so that Here |σ1 A T (η, ξ)| ≤ C η  ˜ Af 2L2 (Tn ) = |Af (x)|2 dx 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

 2

|σ1 A T (ω − ξ, ξ)|

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 a|Tn ×Zn = b|Tn ×Zn . Then a − b is smoothing, a − b ∈ S (R × Rn ).

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 Thus a ∈ Sρ,δ (Rn × Rn ).

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
 ˜ T u(x) := ei(φ(x,ξ)−y·ξ) a(x, y, ξ) u(y) dy, 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
 ˜ T P u(x) = ei(φ(x,ξ)−z·ξ) c(x, z, ξ) u(z) dz, 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
 ˜ T u(x) := ei(φ(x,ξ)−y·ξ) a(x, y, ξ) u(y) dy, 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
 ˜ p(x, D)T u(x) = ei(φ(x,ξ)−z·ξ) c(x, z, ξ) u(z) dz, 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, ξ) u E (ξ, t) eix·ξ dξ a(x, D)u(x, t) = 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  ∂ w(x, t) = (a1 (x, D) + p2 a0 (X, D))w(x, t), i ∂t (8.4) w(x, 0) = pf (x). Calculus constructed in previous sections provides the solution in the form
eiφ(t,x,k) c(t, x, k)fE (k). w(x, t) = 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
+ 2iaj yj + − 2iaj xj . La = − 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 ∂ ∂ − aj z j and Zj = + aj z j , Zj = ∂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 G(z, ζ) = e− j=1 aj [|zj −ζj | coth(aj s)+zj ζj −zj ζj ] ds 2π n 0 j=1 sinh(aj s) for all z and ζ in Cn , which means that   G(·, ζ)f (ζ) dζ = f La 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 e− j=1 aj [|zj −ζj | coth(2aj t)+zj ζj −zj ζj ] (1.1) 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 u(z, t) = (cos (tL1/2 sin (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 Pt (z, ζ) = √ e−t /(4u) u−1/2 Hu (z, ζ) du 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 etu/2 Ht/(2u) (z, ζ)u−3/2 du, Wt (z, ζ) = 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 √ J1/2 (z) = (3.1) 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 eα(u−(z /u))/2 u−ν−1 du (3.2) 2πiJν (αz) = z ν 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 sin (tL ) = etu/2 e−(t/(2u))La u−3/2 du, L−1/2 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 e(t /(4w))−γ(w,z) ν(w) √ , wt (z) = (1/2)+n 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 et v/4 h1/v (z)v −3/2 dv. wt (z) = √ 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 et /(4w) hw (z) √ , t > 0, z ∈ Cn , wt (z) = √ 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, ∞), t2 (u − iv)
sinh (4aj u) − i sin (4aj v) − 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
1 4aj λj = . 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 ⎨ ⎬
2aj vλ λj vλ ∈ [0, ∞) : =1 . ⎩ ⎭ 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 (Λ) = a ∈ C ∞ (Rd × Rd → C) : Dξα Dxβ a  Λm−ρ|α|+δ|β| , Sρ,δ with −ρ  δ < ρ  σ. It is clear that 1 Λ ∈ Sσ,−σ (Λ).

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

Au(x) = (2π)−d/2

u(ξ) dξ, eix·ξ a(x, ξ)ˆ

(2.3)

which is continuous on the Schwartz’ class S and extends to a continuous operator ˆ is the Fourier (we still denote by A) on temperate distributions S  . In (2.3) u 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 to Sρ,δ (Λ) and satisfying the above estimates. 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 for all u, v ∈ S . A+ u, v¯ = u, Av, 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 (Λ) has a paraTheorem 2.3. A Λ-hypoelliptic operator A with symbol in HSρ,δ −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 (Λ). Then the maximal and the minimal closed extensions of A symbol in HSρ,δ 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 HSρ,δ (Λ). 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 k pα,k (x)xkr−2|α| , Dα e− x = e− x

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 has symbol a ∈ Sρ,δ (Λ) for all  ∈ R and 0 < r  1 + s.




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(uv) = A(γ) uDγ v, (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 u(x) = e x A e− x u(x) = e x A(γ) u(x)Dγ 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) η γ! 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 (Λ) because m0 > 0.
a − λ ∈ HSρ,δ 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 + (2π) eiy·ξ 1 − χ(|y|) q(x, ξ)u(x − y) dydξ

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    ξ−2N dξ dy < ∞. |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 (−∆ξ )N Dxβ q(x, ξ) 1 − χ(|y|) Dxα−β u(x − y)dydξ, = eiy·ξ |y| 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   (−∆ξ )N Dxβ q(x, ξ)  1 − χ(|y|) Dxα−β u(x − y) dydξ |y| β β
α    −2N −2N −2N ρ −2N ρµ0  |y| Λ(x, ξ) dydξ  ξ dξ |y| dy < ∞.
|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 HSρ,δ (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 ∈ 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 pseudo-differential operator with symbol in HSσ,−σ (Λ). 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 (Λ) extends Proposition 5.4. A pseudo-differential operator A with symbol in Sρ,δ 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 ,

with symbol in satisfy super-exponential estimates of type s > 0 if they 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 Proof. Since Λ = Λ2l , Dξα |p| = 0 for |α| > 2lm and Dxβ |x| = 0 for |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 with 0  δ < ω  m , 0 < m0  m = deg p, 0 < n0  n, and R  0. 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 S. belong to e− x 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 + x2n p(ξ) + q(x) = ξ12m1 + ξ22m2 + 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
ΛP (x, ξ) = x2α ξ 2β (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) + atp ξ 2k − btq ξ k u ˆ(ξ, t) = fˆ(ξ, t), (2.2)

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  B(ξ, t, t ) = − as ξ − bsq ξ k ds 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 uˆ(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 rp+1 2k rq+1 k 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 ξ , rM = −rm = ab ξ −k p−q and In Figure 1 we have set r1 = −r0 = a(q+1) p−2q−1

(2.7) gM = −gm = A|ξ|k p−q , q+1 p+1     b b p−q a b p−q − p+1 is a positive constant. Now, since t ≤ t and where A = q+1 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 |ˆ u(ξ, t)| ≤ D eC|ξ| e−|ξ| s dt = 2DT eC|ξ| e−|ξ| s ; −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 ˆ(ξ, t) + (atp ξ 2k − btq ξ k )ˆ u(ξ, t) = 0, ∂t u

(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 t = tξ = ξ − p−q , 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 s < 2 (3.7) 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, a di Bruno formula f, g ∈ C ∞ (R), we have that for every ν ∈ Z+ the following Fa` holds: ν  
   dν  f (h) g(y) Bν,h {g (j) (y)} , (3.11) f (g(y)) = 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     Bν,h {yj } ≤ Bν,h {wj } ; (3.14) (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 1 ν! ν! ≤ ν = ν Bµ,h {C} C = ν C h ehz , = ν Bν,h {C} 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)

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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   −1 β C (1 + µ0 )λ
2kh ˜ (C ) em(λρ) λ B { C} ≤ 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

 ν !s β ∞

  (C −1 )β   (2C −1 )1/s −β 2kh ˜ ˜ Bβ,h {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 C =2 λ =2 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 α 1/s α !s  −1 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 where d1 = s C −1 (1 + µ0 ) . We finally obtain from (3.16), (3.17) and (3.18) 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  in the analysis of the norm of I1 we can limit our attention   that t and so we can find ˜1 and ˜2 with 0 < ˜1 < ˜2 < 1 such that, to t ∈ supp g1 tλρ

˜ defining K = (t, x) ∈ R2 : ttλ ∈ [1 − ˜ 2 , 1 − ˜1 ] ∪ [1 + ˜1 , 1 + ˜2 ], |x| ≤ 1 , we have   ˜ for every ρ ∈ [1, 1 + µ0 ]; then, from Lemma 3.1, supp ∂t gλρ (t, x) ⊂ K 1 I1 s,Kλ , η− = I1 s,K, ˜ 1 η−  1+µ0

p−2q−1

 t 



k k

eixλρ+g(λρ,t) ˜ 1 e−A(λρ) p−q

ϕ(x)

≤ Cλ p−q dρ.

g1 s,K, η ˜ 1 t s,K, λρ 1 η (3.23)

In order to apply Lemma 3.7 to eixλρ+g(λρ,t) s,K, ˜ 1 we observe that η

˜ 0 )(λρ)k m(λρ) = sup g(λρ, t) ≤ (A − L

p−2q−1 p−q

˜ (t,x)∈K

˜ 0 , as we can deduce by (3.6). The conclusion then follows for a positive constant L from (3.23) and Lemmas 3.5 and 3.7.
We want now to estimate the norm of I2 and I3 , cf. (3.21) and (3.22) respectively. We observe at first that we can limit ourselves to consider I2 and I3 for x ∈ supp ϕ , so, defining K  = {(t, x) ∈ R2 : (1 − 2 )tλ ≤ t ≤ (1 + 2 )tλ , x ∈ [−1, −˜ ] ∪ [˜ , 1]} for suitable ˜ we have 1 1 , Ij s,Kλ , η− = Ij s,K
, η−

(3.24)

for j = 2, 3. In order to estimate the norm of I2 and I3 we need the following lemma, that we present here without the proof, since it is a trivial modification of [13, Lemmas 4.5 and 4.6]. Lemma 3.10. (i) For every R ∈ R, s > 1 and C > 0 there exist positive constants d and c satisfying tR s,Kλ ,C ≤ dλ−R p−q ecλ k

k (p−q)(s−1)

for all λ > 0; (ii) we can find a positive constant D such that

1



≤ DM+1



(ix)M s,K
,C for every positive integer M .

,

Gevrey Solvability for Degenerate Parabolic Operators

147

Remark 3.11. In Lemma 3.10, (i), we had convenience in making explicit the dependence of the estimate on R, in order to obtain a suitable result for our purposes k below. The term λ−R p−q may become in fact very important when choosing R depending on the compact set Kλ : in the particular case when R = λ, for example, we get tλ s,Kλ ,C ≤ bλ−λ p−q ecλ k

k (p−q)(s−1)

,

which is very nice when λ → +∞, because the term λ−λ p−q is stronger than the k

k cλ (p−q)(s−1)

exponential e and let the expression tend to zero as λ tends to +∞. In the following we do not use Lemma 3.10, (i) exactly for R = λ, but we use it in (3.32) below, where the corresponding exponent is connected with M that in k its turn shall be linked with λ. The term λ−R p−q is then going to have a crucial importance in the sequel. We can now estimate the norms of (3.21) and (3.22). ˜ c, d1 , d2 and G0 such that for every Lemma 3.12. There exists positive constants C, integer M > 0 we have 1 Ij s,Kλ , η−
≤ C˜ M+1 M !s λ2k−M(1−k

p−2q−1 p−q )

1

ed1 λ s +d2 λ

2k s

k

k

+G0 λ (s−s
)(p−q) +cλ (s−1)(p−q)

, (3.25)

for j = 2, 3. Proof. We limit ourselves to prove (3.25) for j = 2, since the case j = 3 can be treated in the same way. We observe at first that for all positive integers M we have 1 ∂ M  ixλρ  eixλρ = λ−M e ; (3.26) (ix)M ∂ρM moreover, using (3.6) we have that in the compact Kλ we can write g(λρ, t) − A(λρ)k

p−2q−1 p−q

= −e0 t2 (λρ)k

p−2q+1 p−q

+ 2e0 ttλρ (λρ)k

p−2q+1 p−q

− e0 t2λρ (λρ)k

p−2q+1 p−q

(3.27)

plus lower order terms, that for simplicity we do not write in the following, since they can be treated in the same way as the ones appearing in (3.27). We can now put (3.26) and (3.27) in the expression if I2 ; integrating by parts and applying the Leibnitz formula we then get
2k M  K1  K2  K3  K4  (−1)k+M i2k−j atp λ2k−j λ−M I2 = j M1 M2 M3 M4 M5  1+µ0 p−2q+1    k  2k−j  M2  −e0 t2 (λρ)k p−2q+1 1 p−q p−q ixλρ M1 M3 2e0 ttλρ (λρ) e ∂ e ρ ∂ × e ∂ ρ ρ ρ (ix)M 1 p−2q+1     t  k 2 p−q × ∂ρM4 e−e0 tλρ (λρ) ψ (M5 ) (ρ)∂ρM6 g1 ϕ(j) (x) dρ tλρ

148

A. Oliaro and P. Popianov

where the sum is performed over j = 1, . . . , 2k, M5 + M6 = K4 , K4 + M4 = K3 , K3 + M3 = K2 , K2 + M2 = K1 , K1 + M1 = M and M1 ≤ 2k − j. We can now write the derivatives appearing in the previous expression via the formula (3.11); applying (3.27) and the property B2 at page 143 we then obtain
2k M  K1  K2  K3  K4  C(2k − j) · · · (2k − j − M1 + 1) I2 = j M1 M2 M3 M4 M5  1+µ0 p−2q+1 k 1 eixλρ+g(λρ,t) × tp+2h+m+d λ2k−j λ−M λk p−q (h+m+r)− p−q (m+2r−d) (ix)M 1  t  p−2q−1 k p−q (d) ϕ(j) (x) × e−A(λρ) ρ2k−j−M1 ψ (M5 ) (ρ)g1 tλρ   p−2q+1     p−2q+1 k   k p−q − p−q i2 B ∂ ρ × BM2 ,h ∂ρi1 ρk p−q M3 ,m ρ i1 =1,...,M2 i2 =1,...,M3   p−2q+1    k   k  × BM4 ,r ∂ρi3 ρk p−q −2 p−q i =1,...,M BM6 ,d ∂ρi4 ρ p−q i =1,...,M dρ 3

4

4

6

(3.28)   m+2r−d where C = (−1)k+M+h+r i2k−j a eh+r+m 2m ab p−q and the sum is performed 0 over j = 1, . . . , 2k, M5 + M6 = K4 , K4 + M4 = K3 , K3 + M3 = K2 , K2 + M2 = K1 , K1 + M1 = M , M1 ≤ 2k − j, h = 1, . . . , M2 , m = 1, . . . , M3 , r = 1, . . . , M4 and 6 d = 1, . . . , M6 ; observe that j=1 Mj = M . We now observe that, due to the α α inequality β ≤ 2 for all integers α, β with α ≥ β, we have   (2k − j) · · · (2k − j − M1 + 1) ≤ (M1 + 4k + 1)!   M1 + 4k + 1 = M1 ! (4k + 1)! (3.29) 4k + 1 ≤ C M1 +1 M1 ! for a suitable positive constant C (depending on k); similarly we get, using (3.13) and taking into account that ρ ∈ [1, 1 + µ0 ],    k       BM6 ,d ∂ρi4 ρ p−q i4 =1,...,M6  ≤ BM6 ,d C˜ i4 +1 i4 !}  ∞   i4 +1  z m6 M6 !
˜ i4 !} ≤ M6 Bm6 ,d C z m6 ! m6 =d  ∞ d M6 ! 1
˜ i4 +1 z i4 C i4 ! ; = M6 z d! i =1 i4 ! 4

so choosing z =

1 ˜ 2C

we obtain    k     BM6 ,d ∂ρi4 ρ p−q i4 =1,...,M6  ≤ C1M6 +1 M6 !

(3.30)

and the same kind of estimate holds for all the Bell polynomials appearing in (3.28). By these last considerations, using (3.24), Lemma 3.1, (3.29) and the well-known

Gevrey Solvability for Degenerate Parabolic Operators estimate

α β

149

≤ 2α for 0 ≤ β ≤ α integers we then have

1 I2 s,Kλ , η−
p−2q+1 k ≤ C2M+1 λ2k λ−M M1 ! M2 ! M3 ! M4 ! M6 ! λk p−q (h+m+r)− p−q (m+2r−d)

 1+µ0

1

p+2h+m+d ixλρ+g(λρ,t)



1 × t s,K
, η1 s,K
, 1+2

(ix)M
1 e 1 s,K , η+

  p−2q−1 k  

p−q

(d) t

ϕ(j) (x)

× e−A(λρ) ρ2k ψ (M5 ) (ρ) g1 dρ. (3.31) 1 tλρ s,K
, η+2

Now from Lemma 3.10 we have λk

p−2q+1 k p−q (h+m+r)− p−q (m+2r−d)

≤ dλk

p−2q−1 k p−q (h+m+r)−p p−q

ecλ

tp+2h+m+d s,K
, η1

k (p−q)(s−1)

≤ dλMk

(3.32)

p−2q−1 p−q

ecλ

k (p−q)(s−1)

;

the conclusion then follows from (3.31), (3.32), Lemmas 3.5, 3.7, 3.10 and the fact
that ψ ∈ Gs0 (R).
1 ). There exist positive constants Proposition 3.13 (Upper bound for tP uλ s,Kλ , η− ˜ 0 , L0 such that for every h satisfying C, d1 , d2 , G

h>

s

(3.33)

1 − k p−2q−1 p−q

we have 1

k

1 ≤ Cλ2k+ p−q ed1 λ s +d2 λ tP uλ s,Kλ , η−

2k s

k

˜ 0 λ (p−q)(s−s
) +G



k

e−L0 λ

p−2q−1 p−q


1 + e−s λ h .

Proof. It follows immediately from (3.19), Lemma 3.9 and Lemma 3.12 and the fact that s > 1 that k

1 tP uλ 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


+ 2C˜ M+1 M !s λ2k−M(1−k

p−2q−1 ) p−q

1

ed1 λ s +d2 λ

2k s

2k s

(3.34) k

+(G0 +c)λ (p−q)(s−s
)

;

now choosing λ = M , with h satisfying (3.33) (observe that λ → +∞ as M → +∞, due to the hypothesis in Theorem 1.2, (ii)), we have by the Stirling formula that p−2q−1
(3.35) C˜ M+1 M !s λ−M(1−k p−q )
  p−2q−1

s ≤ C˜2 e−s M . = C˜ M+1 M !s M −hM(1−k p−q ) ≤ C˜1 M ! M −M h

The conclusion follows then from (3.34) and (3.35), since M = λ1/h .




Now we can pass to the proof of the nonsolvability part of Theorem 1.2. Proof of Theorem 1.2, (ii)–(ii) . We suppose ab absurdo that P is Gs -locally solvk(p−q) p−q (resp., s > 2 (1−k)p−(1−2k)q+k ). It follows then able at the origin for s > 2 p−2q−1

150

A. Oliaro and P. Popianov

from Theorem 3.2 and Propositions 3.4, 3.8 and 3.13 that there exist constants ˜ L0 satisfying C  , d˜1 , d˜2 , G, k p−2q−1  k p−2q+1 k
1 ˜ 1 ˜ 2k ˜ (p−q)(s−s
) p−q e−L0 λ + e−s λ h λ−k( p−q ) ≤ C  λ2k+ p−q ed1 λ s +d2 λ s +Gλ (3.36) for every λ sufficiently large. Let us observe now that in the hypotheses (ii)–(ii) we have 1 2k  p − 2q − 1 1  k max , , < min k , (3.37)  s s (p − q)(s − s ) p−q h for a suitable choice of s and h satisfying (3.7) and (3.33) respectively. In fact, 2k k 2k 1 choosing s < 2p−2q−1 p−2q−1 we have that s > (p−q)(s−s
) ; then, since s > s , the following is equivalent to (3.37): p − 2q − 1 1  2k < min k , . (3.38) s p−q h Now it is enough to prove that p − 2q − 1 p − 2q − 1  2k < min k ,1 − k , s p−q p−q

(3.39)

since if this last inequality holds then for fixed s we can choose s > 1 suffi1 ciently close to 1 and, consequently, h sufficiently close to 1−k p−2q−1 , cf. (3.33) p−q

in such a way that (3.38) is satisfied. Now we observe that in the case (ii), p−2q−1 , so (3.39) is true by (1.5), while in the = k p−2q−1 min k p−2q−1 p−q , 1 − k p−q p−q

p−2q−1  = 1 − k p−2q−1 case (ii) , min k p−q , 1 − k p−2q−1 p−q p−q , and so (3.39) is a consequence of (1.6). In both cases (3.37) is true, and so (3.36) cannot be satisfied for
λ → +∞, for any constant C  . The proof is complete.

References [1] A. Bove and D. Tartakoff, Propagation of Gevrey regularity for a class of hypoelliptic equations, Trans. Amer. Math. Soc. 348 (1996), 2533–2575. [2] A. Bove and D. Tartakoff, Optimal non-isotropic Gevrey exponents for sum of squares of vector fields, Comm. Partial Differential Equations 22 (1997), 1263–1282. [3] D. Calvo and P. Popivanov, Solvability in Gevrey classes for second powers of the Mizohata operator, C. R. Acad. Bulg. Sci. 57 (2004), 11–18. [4] F. Colombini, L Pernazza, and F Treves, Solvability and nonsolvability of secondorder evolution equations, in Hyperbolic Problems and Related Topics, Grad. Ser. Anal., Int. Press, Somerville, MA, 2003, 111–120. [5] A. Corli, On local solvability in Gevrey classes of linear partial differential operators with multiple characteristics, Comm. Partial Differential Equations 14 (1989), 1–25. [6] T. Gramchev, P. Popivanov and M. Yoshino, Critical Gevrey index for hypoellipticity of parabolic equations and Newton polygons, Ann. Mat. Pura Appl. 170 (1996), 103– 131.

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[7] B. Helffer, Sur l’hypoellipticit´e d’une classe d’op´erateurs paraboliques d´eg´en´er´es, in Sur quelques ´equations aux d´ eriv´ees partielles singuli`eres, Ast´erisque, 19, Soc. Math. France, Paris, 1974, 79–105. [8] Y. Kannai, An unsolvable hypoelliptic differential operator, Israel J. Math. 9 (1971), 306–315. [9] M. Mascarello and L. Rodino, Partial differential equations with multiple characteristics, Wiley-Akademie Verlag, Berlin, 1997. [10] T. Matsusawa, On some degenerate parabolic equations I, Nagoya Math. J. 51 (1973), 57–77. [11] T. Matsusawa, On some degenerate parabolic equations II, Nagoya Math. J. 52 (1973), 61–84. [12] O.A. Ole˘ınik and E.V. Radkeviˇc, The method of introducing a parameter for the investigation of evolution equations, Uspekhi Mat. Nauk 33, 5(203) (1978), 7–76, 237. [13] A. Oliaro, On a Gevrey non-solvable partial differential operator, in Recent Advances in Operator Theory and its Applications, Editors: M.A. Kaashoek, C Van der Mee and S. Seatzu, Birkh¨ auser, 337–356. [14] P. Popivanov, Local properties of linear pseudodifferential operators with multiple characteristics, C. R. Acad. Bulg. Sci. 29, (1976), 461–464. [15] P. Popivanov, Local solvability of several classes of Partial Differential Equations, C. R. Acad. Bulg. Sci. 48, (1995), 15–18. [16] P. Popivanov, On a nonsolvable partial differential operator, Ann. Univ. Ferrara VII, Sc. Mat. 49 (2003), 197–208. [17] L. Rodino, Gevrey hypoellipticity for a class of operators with multiple characteristics, in Analytic Solutions of Partial Differential Equations (Trento), Ast´erisque, 89-90, Soc. Math. France, Paris, 1981, 249–262. [18] R. Rubinstein, Examples of nonsolvable partial differential equations, Trans. Amer. Math. Soc. 199 (1974), 123–129. [19] F. Tr`eves, On the existence and regularity of solutions of linear partial differential equations, in Proceedings of Symposia in Pure Mathematics, Vol. XXIII, 33–60. Alessandro Oliaro Department of Mathematics University of Torino Via Carlo Alberto, 10 I-10123 Torino, Italy e-mail: [email protected] Petar Popivanov Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bontchev Str., bl. 8 1113 Sofia , Bulgaria e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 153–159 c 2006 Birkh¨
auser Verlag Basel/Switzerland

A New Aspect of the Lp-extension Problem for Inhomogeneous Differential Equations Mitsuru Sugimoto Abstract. For the differential operator P of order m and the inhomogeneous data f ∈ S  on Rn , we say that the Lp -extension of the solution holds if u ∈ Lp , m ≤ n(1 − 1/p), and P u = f on Rn \ 0 imply P u = f on Rn . In this article, we discuss which kind of inhomogeneous terms f ∈ S  admit the Lp -extension of the solution. In previous works, this problem was studied by using classical Bochner’s method ([1]) or a new method developed by the author and Uchida ([5], [4]). We consider inhomogeneous terms which are not covered by these results. Mathematics Subject Classification (2000). Primary 35B60; Secondary 35G05. Keywords. Extension of weak solution, removable singularity.

1. The Lp -extension problem First of all, we explain the Lp -extension problem of weak solutions. Let P : differential operator of order m with C ∞ coefficient on Rn . For a given distribution f ∈ D (Rn ), assume that u ∈ Lp (Rn ) (1 < p < ∞) is a weak solution of the equation on a punctured space P u = f on Rn \ 0. Our problem is when u is a weak solution of the same equation on the whole space: P u = f on Rn . Without loss of generality, we may assume that all the derivatives of coefficients of P are bounded and the inhomogeneous term f is a tempered distribution, that is, f ∈ S  (Rn ). The following answer to this problem with the homogeneous case f = 0 was given by Bochner [1]: Theorem 1.1. If u ∈ Lp (Rn ), m ≤ n(1 − 1/p), and P u = 0 on Rn \ 0, then P u = 0 on Rn .

154

M. Sugimoto

The inequality m ≤ n(1 − 1/p) in Theorem 1.1 cannot be removed if we take account of the fundamental solution of an elliptic differential operator P . In fact, if P has analytic coefficients, then u(x) = |x|m−n {A(x) + B(x) log |x|} solves P u = δ on Rn and P u = 0 on Rn \ 0 with some A(x), B(x) bounded in a neighborhood of 0 by John [3]. We remark that u belongs to Lp locally in the neighborhood for m > n(1 − 1/p). On account of this fact, we say that the Lp -extension of the solution holds for the given inhomogeneous term f ∈ S  , if u ∈ Lp , m ≤ n(1 − 1/p), and P u = f on Rn \ 0 imply P u = f on Rn . In this article, we do not care about the existence of such u and P which −n(1−1/p) solve P u = f on Rn \ 0 for each given f ∈ S  . But once they exist, f ∈ Hp p p is necessary for the L -extension, since P maps u ∈ L to f = P u ∈ Hp−m ⊂ −n(1−1/p)

Hp . On the other hand, it is known that the Lp -extension of the solution −n(1−1/p) always holds for f ∈ Hp ([4]). The proof of this fact will be also given in Section 2 of this article. Hence we have the following answer to the problem with the inhomogeneous case: Theorem 1.2. Let f ∈ S  (Rn ). Assume u ∈ Lp (Rn ), m ≤ n(1 − 1/p), and P u = f −n(1−1/p) on Rn \ 0. Then P u = f on Rn if and only if f ∈ Hp (Rn ). Besides this criterion for the inhomogeneous terms, we know more useful ones which can be easily checked. Actually, in a non-linear problem with f replaced by non-linear terms, the following theorem plays an important role ([5]). Theorem 1.3. Let f ∈ L1 (Rn ) (microlocally). If u ∈ Lp (Rn ), m ≤ n(1 − 1/p), and P u = f on Rn \ 0, then P u = f on Rn . Theorem 1.3 with f ∈ L1 can be proved by the argument in [1], and its microlocal version by [5]. The typical example of such f is f = h(x) + (x1 ± i0)−1 ⊗ g(x ) ∈ L1 



microlocally,

where x = (x1 , x ), x = (x2 , . . . , xn ), h(x) ∈ L (R ), and g(x ) ∈ S  (Rn−1 ). We −n(1−1/p) remark that f ∈ L1 (microlocally) dose not always imply f ∈ Hp . But Theorem 1.3 (together with Theorem 1.2) says that we can automatically conclude −n(1−1/p) once we know the existence of u ∈ Lp and P with m ≤ n(1 − 1/p) f ∈ Hp satisfying P u = f on Rn \ 0. 1

n

2. Two methods In this article, we will discuss a generalization of Theorem 1.3. In author’s previous paper [4], two classes B0 and C0 of inhomogeneous terms were introduced, which allow the Lp -extension of the solution. We briefly review these arguments. First we explain a fundamental idea by Bochner [1] to prove Theorem 1.1:

Lp -extension Problem

155

Let a(x) be a cutoff function of the origin, that is, a(x) ∈ C0∞ and a(x) = 1 for x near 0. We use the notation aε (x) = a(x/ε) for ε > 0. Then, for test functions ϕ, we have aε u, t P (aε ϕ), P u, ϕ = aε P u, ϕ = ˜ ˜(x) is another cutoff function which is equal where t P is the transpose of P and a to 1 on the support of a(x). We remark that ˜ε u → 0 in Lp . lim a

ε0

Hence, if {t P (aε ϕ)}ε>0 is bounded in Lq with 1/p + 1/q = 1, we get P u = 0, and this can be realized if m ≤ n(1 − 1/p). By the method above, we have easily the following result for inhomogeneous equations: Definition 2.1. Let f ∈ S  (Rn ). We say that f ∈ B0 if there exist a strictly decreasing sequence {εν }∞ ν=1 of positive numbers and a cutoff function a(x) of the origin such that lim aεν f = 0 in S  (Rn ). ν→∞

Theorem 2.2. Let f ∈ B0 . If u ∈ Lp (Rn ), m ≤ n(1 − 1/p), and P u = f on Rn \ 0, then P u = f on Rn . Theorem 2.2 is a generalization of Theorem 1.3 since L1 ⊂ B0 by Lebesgue’s convergence theorem. On the other hand, there is another method to prove Theorem 1.1, found by the author and Uchida [5]: Assume P u = 0 on Rn \ 0. By the structure theorem of distributions with a point support, we can write P u = Qδ on Rn where Q is a differential operator with constant coefficients. Then, by the microlocal ellipticity of Q (when Q = 0) and the mapping property of pseudo-differential operators on the Sobolev space Hps , we have δ = Q−1 P u ∈ Hp−m (at least in a microlocal sense). It is known that δ ∈ Hps implies s < −n(1 − 1/p). Then this relation yields m > n(1 − 1/p). Hence m ≤ n(1 − 1/p) yields Q = 0, hence P u = 0 on Rn . Thus we have obtained Theorem 1.1 again. We remark that the argument above works for homogeneous equations P u = f with f ∈ H −n(1−1/p) , which proves the if part of Theorem 1.2.

156

M. Sugimoto

By just following the same argument, we also have a result of another type. Denote by 5 W F (f ) = W Fz (f ) z∈Rn

the set of f , which is a subset of the cotangent bundle T ∗ Rn = 6 wave∗ front n z∈Rn Tz R . Definition 2.3. Let f ∈ S  (Rn ). We say that f ∈ C0 if f is microlocally smooth in some direction at the origin, that is, the fiber of the wave front set at 0 has the proper inclusion W F0 (f )  T0∗ Rn . Theorem 2.4. Let f ∈ C0 . If u ∈ Lp (Rn ), m ≤ n(1 − 1/p), and P u = f on Rn \ 0, then P u = f on Rn . In [5], it was shown that C0 in Theorem 2.4 can be replaced by a larger class L1 + C0 , which is the microlocal statement of Theorem 1.3. In [4], larger classes than L1 + C0 were also discussed.

3. Homogeneous distributions In Section 2, we have explained two different methods to prove the Lp -extension of the solution. We remark here that the following examples are not integrable, but covered by these method: 1 ⊗ g(x ) ∈ B0 , fβ (x) = p. v. x1 1 fγ± (x) = ⊗ g(x ) ∈ C0 , x1 ± i0 where x = (x1 , x ), x = (x2 , . . . , xn ), and g(x ) ∈ S  (Rn−1 ). If fact, the proof of fβ ∈ B0 can be reduced to show 8 7 1 aε (x1 ) p. v. , ϕ(x1 ) → 0 as ε ' 0 x1 for all test function ϕ of dimension 1. We take a cutoff function a(x1 ) of dimension 1 such that a(−x1 ) = a(x1 ). Then a cancellation property works, and we have 8 7  (aε ϕ)(x1 ) 1 dx1 aε (x1 ) p. v. , ϕ(x1 ) = lim δ0 |x1 |≥δ x1 x1   aε (x1 ) = lim dx1 · ϕ(0) + (aε H)(x1 ) dx1 δ0 |x |≥δ x1 1 with H bounded. The first term vanishes since aε is an even function, and the second term tends to 0 as ε ' 0. On the other hand, we also have fγ± ∈ C0 since W F0 (fγ± ) ⊂ {0} × {(ξ1 , ξ2 , . . . , ξn ); ξ1  0}  T0∗ Rn . We remark that we have fβ ∈ / C0 if we take non-smooth g(x ).

Lp -extension Problem

157

On account of these examples, it is natural to ask the question as to whether or not the Lp -extension of the solution holds for the inhomogeneous term   a+ a− + fµ (x) = ⊗ g(x ), x1 + i0 x1 − i0 where x = (x1 , x ), x = (x2 , . . . , xn ), a± ∈ C, and g(x ) ∈ S  (Rn−1 ). We remark that 1 1 ⊗ g(x ), δ(x1 ) ⊗ g(x ) ⊗ g(x ), p. v. x1 x1 ± i0 are typical examples, since     1 1 1 1 1 1 1 + − p. v. = , δ(x1 ) = . x1 2 x1 + i0 x1 − i0 2πi x1 − i0 x1 + i0 It is not clear when fµ ∈ B0 , except for the case a+ = a− . Hence Theorem 2.2 cannot answer this question. Theorem 2.4 neither, except for the case a+ = 0 or a− = 0. In the next section, we will discuss how to deal with this example.

4. A new aspect By modifying the idea to prove Theorem 2.4, we can provide a new aspect to explain when the Lp -extension of the solution holds for the inhomogeneous term   a+ a− + fµ (x) = ⊗ g(x ), x1 + i0 x1 − i0 as in Section 3. Assume that |f1 µ (ξ)| → 0

(|ξ| → ∞)

uniformly in a direction. (See Definition 4.1 below for the precise meaning.) It is equivalent to the same property (of dimension n − 1) for g(x ) since the Fourier −1 transforms of (x1 ± i0) are just a constant in each direction. Noticing that, in the direction, f1 µ (ξ) can be regarded as a perturbation to any non-zero polynomial then, let us repeat the argument of the proof of Theorem 2.4. If P u = fµ on Rn \ 0, then we have P u = fµ + Q(D)δ ∈ Hp−m with a polynomial Q by the structure theorem and the mapping property of P . Furthermore, since fµ = fˆµ (D)δ, we have  Q(D)δ ∈ Hp−m ,

 Q(ξ) = Q(ξ) + fˆµ (ξ).

If the polynomial Q = 0, then Q(D) is microlocally elliptic in a direction, and if  Then we have δ ∈ Hp−m fˆµ (ξ) is just a perturbation, the same is true for Q(D). (microlocally), which implies m > n(1 − 1/p). Hence m ≤ n(1 − 1/p) yields Q = 0 and we can conclude P u = fµ on Rn . From this argument, we obtain the following another type of criterion.

158

M. Sugimoto

Definition 4.1. Let f ∈ S  (Rn ). We say that f ∈ M if fˆ is a function which satisfies |fˆ(ξ)| → 0 (|ξ| → ∞) uniformly, that is, sup |fˆ(ξ)| → 0 (R → ∞). |ξ|>R

We remark that M is closed under the multiplication of cutoff functions a(x) of the origin. In fact, since  1 af (ξ) = a ˆ(ξ − η)fˆ(η) dη, we have    1  sup af(ξ) ≤ sup |ξ|>R

|ξ|>R

+

|η|≤R/2

−1

≤ R/2 → 0

 |η|>R/2



    a(ξ − η)fˆ(η) dη ˆ

          a(η)| dη + sup fˆ(ξ) · |ˆ a(η)| dη sup fˆ(ξ) · η|ˆ

ξ∈Rn

|ξ|>R/2

(R → ∞).

M is also closed under the operation of χ(D), where χ(ξ) is a bounded function. On account of these arguments, we have the following main result of this article: Theorem 4.2. Let f ∈ M + C0 . If u ∈ Lp (Rn ), m ≤ n(1 − 1/p), and P u = f on Rn \ 0, then P u = f on Rn . We remark that, by the Riemann-Lebesgue theorem, we have L1 ⊂ M. (To justify it, show S ⊂ M and use the fact that S is dense in L1 .) Hence Theorem 4.2 is also a generalization of Theorem 1.3. Here is another interesting example:   d fν = F −1 ei|ξ| |ξ|−nd/2 ψ(|ξ|) ∈ M, where smooth function ψ(t) is a smooth function of t ≥ 0, which is equal to 0 for −n(1−1/p) 0 ≤ t ≤ 1 and 1 for t ≥ 2. Since it is not straightforward to show fν ∈ Hp except for the case p = 2, Theorem 1.2 does not say anything to this example. Theorem 1.3 neither because fν ∈ / L1 for 0 < d < 1. In fact, fν is of the form fν (x) = K(|x|)|x|−n + O(|x|ω ) as |x| → 0, where |K(|x|)| is a non-zero constant and ω > −n (Ishii [2]). The author showed the weak Lp -extension property for it by a different approach ([4]). As was seen in the above, fµ is a typical example which belongs to M if g(x ) belongs to M of dimension n − 1. More generally, we have f˜µ (x) = g1 (x1 ) ⊗ g2 (x2 ) · · · ⊗ gn (xn ) ∈ M, where x = (x1 , x2 , . . . , xn ), if at least one of gj (t) (j = 1, 2, . . . , n) belongs to M of dimension 1, and all other gl (t) is a linear combination of (t ± i0)−1 . Furthermore,

Lp -extension Problem

159

if such gj (t) ∈ L1 (R) admits a nice regularity and up to (k − 1)th derivatives of it are integrable again, then we have a stronger decaying property   (R → ∞) sup tk−1 gj (t) → 0 |t|>R

In this case, linear combinations of more general homogeneous distributions (t ± i0)−1 , (t ± i0)−2 , . . . , (t ± i0)−k are allowed for all other gl (t) since their Fourier transforms are polynomial of order up to k − 1 in each direction, that is,    ck |τ |k−1 for ± τ > 0 −k (τ ) = F (t ± i0) 0 for ± τ < 0 with complex numbers ck ∈ C (for k = 1, 2, . . .).

References [1] S. Bochner, Weak solutions of linear partial differential equations, J. Math. Pures Appl. 35 (1956), 193–202. [2] H. Ishii, On some Fourier multipliers and partial differential equations, Math. Japon. 19 (1974), 139–163. [3] F. John, The fundamental solution of linear elliptic differential equations with analytic coefficients, Comm. Pure. Appl. Math. 3 (1950), 273–304. [4] M. Sugimoto, A weak extension theorem for inhomogeneous differential equations, Forum Math. 13 (2001), 323–334. [5] M. Sugimoto and M. Uchida, A generalization of Bochner’s extension theorem and its application, Ark. Mat. 38 (2000), 399–409. Mitsuru Sugimoto Department of Mathematics Graduate School of Science Osaka University Machikaneyama-cho 1-16 Toyonaka, Osaka 560-0043, Japan e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 161–172 c 2006 Birkh¨
auser Verlag Basel/Switzerland

Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols Gianluca Garello and Alessandro Morando Abstract. In this paper a result of continuity for pseudo-differential operators with non-regular symbols on spaces of quasi-homogeneous type is given. More precisely, the symbols a(x, ξ) take their values in a quasi-homogeneous Besov space with respect to the x variable; moreover a finite number of derivatives with respect to the second variable satisfies, in Besov norm, decay estimates of quasi-homogeneous type. Mathematics Subject Classification (2000). 35S05; 35A17. Keywords. Pseudo-differential operators, Besov spaces, Sobolev spaces.

1. Introduction The present paper has to be considered in a series of authors articles about the continuity of pseudo-differential operators with non-regular symbols in the Lp framework, see, e.g., [4], [5], [6]. We consider here the pseudo-differential operators in the classical form:  u(ξ)dξ, u ∈ S(Rn ), (1.1) a(x, D)u(x) = (2π)−n eix·ξ a(x, ξ) where the symbol a(x, ξ) is a tempered distribution in the Schwartz class S  (R2n ). With standard notations x·ξ denotes the inner product of x, ξ ∈ Rn , while u  = Fu is the Fourier transform of u. It is well known that the pseudo-differential operators with symbols in the 0 , 0 ≤ δ < ρ ≤ 1, defined for any smooth classical zero order H¨ormander classes: Sρ,δ function a(x, ξ) by the estimates:  α β  ∂ξ ∂x a(x, ξ) ≤ Cα,β (1 + |ξ|)−ρ|α|+δ|β| , cα,β > 0, x, ξ ∈ Rn , (1.2) are not in general Lp continuous for p = 2 unless ρ = 1, see, e.g., Fefferman [2]. The authors are supported by F.I.R.B. grant of Italian Government.

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In order to obtain Lp continuity when 0 ≤ δ < ρ < 1 one needs further assumptions. For instance Taylor [[11], Ch. XI] introduces the symbol classes m , where Marcinkiewicz conditions on the derivatives with respect to Mρm ⊂ Sρ,0 ξ are required. For M = (m1 , . . . , mn ) ∈ Zn+ , such that min mj ≥ 1 let us now define the 1≤j≤n

quasi-homogeneous weight function in Rn : ⎞ 12 ⎛ n
2m |ξ|M := ⎝ ξj j ⎠ ,

ξ ∈ Rn .

(1.3)

j=1 m , m ∈ R, introduced by Lascar The quasi-homogeneous smooth symbol classes SM [7], 1977, are defined by the following:   α β ∂ξ ∂x a(x, ξ) ≤ cα,β (1 + |ξ|M )m− M1 ·α , cα,β > 0, x, ξ ∈ Rn , (1.4)  n 1 · α = j=1 m1j αj . where M min mj 0 0 It is not difficult to see that SM ⊂ Sρ,0 with ρ = < 1, unless M = max mj (k, . . . , k) for some integer k ≥ 1; then the related pseudo-differential operators are not expected to be Lp continuous, p = 2. In the present paper we deal with a class of pseudo-differential operators whose symbol a(x, ξ) takes its values in a quasi-homogeneous Besov space with respect to x and it satisfies estimates of type (1.4), for a suitable finite number of derivatives with respect to ξ. Since the weights |ξ|M display a quasi-homogeneous structure, that is for any t > 0:      1/M  ξ  = t|ξ|M , where t1/M ξ = t1/m1 ξ1 , . . . , t1/mn ξn , t M

n we ∞can consider, see the next Proposition 2.3, a dyadic partition of unity on R , = 1, with ϕh (ξ) supported, for h ≥ 0, in the quasi-annulus of the h=−1 ϕh (ξ)   1 form K ≤ 2−h/M ξ M ≤ K, K > 1. The non-regular symbols a(x, ξ) above considered can then reduce to an expansion of elementary symbols, following the techniques developed by CoifmanMeyer [1], Marschall [9], Taylor [12]. The authors themselves [4] improve such techniques in the more general setup of multi-quasi-elliptic pseudo-differential calculus, where any reference to homogeneity is lost. The zero order pseudo-differential operators considered in this paper satisfy in natural way the Marcinkiewicz-Lizorkin Lemma for Fourier multipliers, thanks to the quasi-homogeneous structure of the symbols; thus their Lp continuity may be proved. A number of applications of the pseudo-differential operators with non-regular symbols to the non linear partial differential equations are given in the literature of the last twenty years; see, e.g., Taylor [12], [13] and the references given there.

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163

The plan of the paper is the following. In §2, after reviewing the main properties of the quasi-homogeneous weight |ξ|M , we define in the details the symbol classes. We also introduce in this section the quasi-homogeneous dyadic partition of unity and the expansion of a(x, ξ) in elementary symbols. A particular stress is given in the proof of the next estimates (2.7), because of their importance in the applications of the Marcinckiewicz Lemma. In §3 the quasi-homogeneous Besov and Sobolev spaces are introduced and s,p r,M , HM , r, s ∈ R, 1 < p < ∞. The Sobolev spaces respectively denoted by B∞,∞ are then characterized by means of the above introduced partition of unity and suitable useful properties are derived. s,p , 1 < p < ∞, of the At last in §4 we prove the continuity on the spaces HM r,M , pseudo-differential operators whose symbols belong to the Besov spaces B∞,∞ with respect to x, for suitable choice of the exponents r, s. Related to our result let us quote a work of Yamazaki [15], where the Lp continuity is proved for a class of pseudo-differential operators with similar structure, but under different conditions.

2. Symbol classes In the whole paper M = (m1 , . . . , mn ) ∈ Zn+ shall be a vector with positive integer components, such that min mj ≥ 1. We will write 1/M := (1/m1 , . . . , 1/mn ), 1≤j≤n  12  2 and ξM = 1 + |ξ|M . Proposition 2.1. For any M we have (i) |ξ + η|M ≤ C(|ξ|M + |η|M ),

C > 0;

(ii) (quasi-homogeneity) for any t > 0, |t1/M ξ|M = t|ξ|M ; 1 1− M ·α

(iii) ξ γ ∂ α+γ |ξ|M ≤ CξM

, for any α, γ ∈ Zn+ and ξ = 0.

The proofs of (i), (ii) are quite trivial and (iii) follows with some small changes from the proof of Lemma 2.1 in [3] Definition 2.2. [symbol classes] For X Banach space and any r ∈ R, N ∈ Z+ we say that the complex-valued measurable function a(x, ξ) on Rnx × Rnξ belongs to r (N ) if for some positive constant C: the symbol class XSM   r− 1 ·α  α  (2.1) ∂ξ a(x, ξ) ≤ CξM M , for any x, ξ ∈ Rn , |α| ≤ N



1 r− ·α

α

(2.2)

∂ξ a(·, ξ) ≤ CξM M , for any ξ ∈ Rn , |α| ≤ N. X

For some K > 1 let us consider φ(t) ∈ C0∞ ([0, +∞]) such that 0 ≤ φ(t) ≤ 1, 1 φ(t) = 1 for 0 ≤ t ≤ 2K ; φ(t) = 0, when t > K. For h ≥ 0 we can then introduce

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the sequence: ϕ−1 (ξ) = φ (|ξ|M ) ,

  ϕh (ξ) = ϕ0 2−h/M ξ ,

   where ϕ0 (ξ) = φ 2−1/M ξ M − φ (|ξ|M ). dyadic partition of unity. In fact we have

(2.3)

∞

{ϕh }h=−1 is a quasi-homogeneous

Proposition 2.3. For any ξ ∈ Rn , α, γ ∈ Zn+ and some positive constant Cα,γ,K : K,M := {ξ ∈ Rn , |ξ|M ≤ K} supp ϕ−1 ⊂ C−1

supp ϕh ⊂ ChK,M := ξ ∈ Rn ; ∞
ϕh (ξ) = 1; h=−1 ∞


ϕh (D)u = u,

1 h−1 K2

≤ |ξ|M ≤ K2h+1 , h ≥ 0;

(2.4)

(2.5)

with convergence in S  (Rn );

(2.6)

h=−1

 γ α+γ  1 ξ ∂ ϕh (ξ) ≤ Cα,γ,K 2−( M ·α)h ,

h = −1, 0, . . . .

(2.7)

Moreover for any fixed ξ ∈ Rn the sum in (2.5) reduces to a finite number of terms, independent of the choice of ξ itself. n Proof. It is trivialto prove (2.4).  For every fixed ξ ∈ R we have for any suitably −h/M   ξ M = 1; then (2.5), (2.6) follow. large integer h: φ 2 For every integer h ≥ 0 we compute

γ      γ α+γ  1 1  ξ ∂ ∂ α+γ ϕ0 2−h/M ξ  2−h( M ·α) ≤ Cα,γ,K 2−h( M ·α) , ϕh (ξ) =  2−h/M ξ where Cα,γ,K = max |η γ ∂ α+γ ϕ0 (η)| is independent of h.




η

Proposition 2.4. [Decomposition in elementary symbols.] For N ≥ n + 1 let us con0 sider a(x, ξ) ∈ XSM (N ). We then obtain with absolute convergence in L∞ (R2n ): a(x, ξ) =




ck ak (x, ξ),

(2.8)

k∈Zn

where {ck }k∈Zn is a real positive-valued sequence such that

 k∈Zn

ck < ∞.

Moreover for some

∞constants L, C

∞> 0, K > 1 we can find two suitable function sequences dkh h=−1 and ψhk h=−1 which satisfy, for any k ∈ Zn+ , the

Continuity in Quasi-homogeneous Spaces

165

following properties: ∞


ak (x, ξ) =

dkh (x)ψhk (ξ);

(2.9)

h=−1

dkh X ≤ L

dkh L∞ ≤ L;

,

C0∞ (Rn )

∈ , ⊂  γ α+γ k  1 − ·α h ( ) ξ ∂ ψh (ξ) ≤ C2 M , ψhk

supp ψhk

(2.10) ChK,M

;

(2.11)

when |α + γ| ≤ N − n − 1.

(2.12)

The symbols defined in (2.9) are called elementary symbols. Proof. Let us write a(x, ξ) =

∞ 

ϕh (ξ)a(x, ξ) =

h=−1

∞ 

ah (x, ξ). Then, for any

h=−1

x ∈ Rn and h = −1, 0, . . . , supp ah (x, ·) ⊂ ChK,M . Moreover when |α| ≤ N the Leibnitz rule gives:  α  ∂ξ ah (x, ξ) ≤ Cα 2−( M1 ·α)h . (2.13) In the same way we obtain

α

∂ξ ah (·, ξ) ≤ Cα 2−( M1 ·α)h , X

ξ ∈ Rn ,

|α| ≤ N.

(2.14)

  By setting now bh (x, ξ) = ah x, 2h/M ξ when h ≥ 0 and b−1 (x, ξ) = a−1 (x, ξ), it follows for some positive constant cα independent of h and all x ∈ Rn : K,M supp bh (x, ·) ⊂ C0K,M , h ≥ 0 , supp b−1 (x, ·) ⊂ C−1 ;  α 



∂ξ bh (x, ξ) ≤ cα , ∂ξα bh (·, ξ) ≤ cα , ξ ∈ Rn , h = −1, 0, . . . . X

(2.15) (2.16)

The functions bh (x, ·) are supported in [−T, T ]n, uniformly with respect to x, when 1

max1≤j≤n (2K) mj ≤ T . We can then construct the (2T )-periodic extension of bh : 1  Bh (x, ξ) = λ∈Zn bh (x, ξ − 2T λ). For some K  > K such that max (2K  ) mj ≤ T 1≤j≤n

let us now consider two positive smooth functions θ(ξ), θ−1 (ξ) supported respec
K
,M , both equal to one in C0K,M or tively in the quasi-annulus C0K ,M and in C−1 K,M respectively in C−1 . Then we plainly obtain bh (x, ξ) = θ(ξ)Bh (x, ξ), h ≥ 0,

b−1 (x, ξ) = θ−1 (ξ)B−1 (x, ξ).

(2.17)

We are now in the position to apply the classical construction of elementary symbols, introduced by Coifman-Meyer [1]; see also [[5] Proposition 2.9, Lemma 2.11]. By means of the Fourier expansion of Bh (x, ξ) we obtain: ah (x, ξ) =


k∈Zn



1 dk (x)ψhk (ξ), π n+1 h 1 + |k| T

(2.18)

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G. Garello and A. Morando

where |k| = |k1 | + · · · + |kn | and  π k −n e−i T k·η M(Dη )bh (x, η) dη, h = −1, 0, . . . ; dh (x) = (2T )

(2.19)

[−T,T ]n

  ψhk (ξ) = ψ k 2−h/M ξ ,

for h ≥ 0, with 1

k

ψ (ξ) =  N −n−1 e 1 + Tπ |k| 1 k i π k·ξ ψ−1 (ξ) =  N −n−1 e T θ−1 (ξ). π 1 + T |k|

(2.20) π iT k·ξ

θ(ξ); (2.21)

M(D) in (2.19) is a differential operator  of integer order N , such that for any φ ∈ C0N (Rn ) and m ∈ Zn : (1 + |m|)N e−im·x φ(x) dx = e−im·x M(D)φ(x) dx. For the existence of M(D) see [[5]; Lemma 2.11]. The functions dkh and ψhk in (2.19), (2.20) satisfy the conditions (2.10)–(2.12). (2.11) we  can assure Since |dkh (x)|, |ψhk (ξ)| are uniformly bounded, using also  that the expansion in (2.18) is absolutely convergent in L∞ Rnx × Rnξ . Finally, +∞ ∞   plugging (2.18) into a(x, ξ) = ah (x, ξ) and setting ak (x, ξ) = dkh (x)ψhk (ξ) for each k ∈ Zn , we obtain:

h=−1

a(x, ξ) =


k∈Zn

h=−1



1 a (x, ξ). π n+1 k 1 + |k| T

(2.22)


3. Quasi-homogeneous Besov and Sobolev spaces Let us state the Hilbert space version of the classical Marcinkiewicz-Lizorkin Fourier multipliers theorem. For the proof we refer to [[10], §IV.6]. Lemma 3.1. Let mk,l (ξ)  (k, l = 1,-2, . . . ) be n times continuously differentiable n 0 ξj = 0 and assume that there exists a positive confunctions in A := Rn \ j=1

stant B such that  ⎛ ⎞ 12 .  ∞  n γj 
 ξj  ⎝ |∂ γ mk,l (ξ)|2 ⎠ ≤ B,  j=1  k,l=1

ξ ∈ A, γ ∈ {0, 1}n.

Then for every p ∈]1, +∞[ and some positive constant c = cB,p,n ,



  12

 12







∞ ∞




∞

−1 2 2





|F m F f | ≤ cB |f | k,l l l





p

k=1

l=1 l=1 L

(3.1) Lp

Continuity in Quasi-homogeneous Spaces

167

n holds true for all sequences {fl }∞ l=1 ⊂ S(R ) such that fl ≡ 0 for all but a finite number of indices l = 1, 2, . . . .

Remark 3.2. By a density argument (3.1) extends to the elements of Lp (Rn ; 2 ), i.e., the space of all the Lp functions taking values in the Banach space 2 . s,M to be the For any s ∈ R we define the quasi-homogeneous Besov space B∞,∞  n set of all u ∈ S (R ), such that for some quasi homogeneous dyadic partition of unity ϕ := {ϕh }∞ h=−1 :

:= uϕ B s,M ∞,∞

sup h=−1,0,...

2sh ϕh (D)uL∞ < ∞.

(3.2)

Exploiting the properties of the quasi-homogeneous dyadic decompositions set in Proposition 2.3 it is not difficult to show that different choices of the partition of s,M unity in (3.2) give equivalent norms, noted by  · B∞,∞ s,M ; moreover B∞,∞ realizes to be a Banach space. s,M For M = (1, . . . , 1) and s > 0 B∞,∞ is the classical H¨older-Zygmund space of order s, deeply studied for example in [14]. We refer also to [[14]; §10.1] for the s,M study of the quasi-homogeneous spaces Bp,q , with arbitrary 0 < p, q ≤ +∞.

Definition 3.3. For arbitrary s ∈ R and p ∈]1, +∞[, the quasi-homogeneous Sobolev s,p is defined as the class of all tempered distributions u ∈ S  (Rn ) such space HM s that DM u ∈ Lp (Rn ). s,p For every s ∈ R and p ∈]1, +∞[, HM is a Banach space with respect to the s s,p := D p u and it admits S(Rn ) as dense subspace. The spaces norm uHM L M s,p HM can be regarded as a particular case of a more general family of weighted Sobolev spaces HΛs,p , defined in terms of an appropriate weight function Λ and deeply studied in [3], [4].

Proposition 3.4. Let us consider s ∈ R and p ∈]1, +∞[; then, for every quasihomogeneous dyadic partition of unity ϕ := {ϕh }∞ h=−1 , there exists a constant C > 1 such that the inequalities



  12

 12







∞ ∞



1

sh 2 sh 2





s,p ≤ C 4 |ϕh (D)u| ≤ uHM 4 |ϕh (D)u| (3.3)





C



p h=−1 h=−1 p L

L

are verified for any u ∈ S  (Rn ). Proposition 3.4 is a straightforward extension of [[14]; Theorem 2.5.6], which provides the characterization of classical Sobolev spaces, H s,p , 1 < p < ∞, by means of dyadic partitions of unity. Essentially we can achieve the proof by using Lemma 3.1 instead of the Michlin-H¨ ormander Lemma (see [14], Proposition 2.5.6). In the same way the following proposition is obtained from [[9] Propositions 1.1, 1.2], where Marschall states the results for classical Sobolev spaces.

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G. Garello and A. Morando

Proposition 3.5. For every s ∈ R and p ∈]1, +∞[ there exists a positive constant C  n such that for all sequences {uh }∞ h=−1 of distributions uh ∈ S (R ) whose Fourier K,M transform is respectively supported in Ch , h = −1, 0, . . . , we have



∞

 12

∞











sh 2

. uh

≤C

4 |uh | (3.4)





s,p



h=−1

h=−1

HM

Lp

Proposition 3.6. When s > 0 and p ∈]1, +∞[ there exists a positive constant C such that estimate (3.4) is true for all the sequences {uh }∞ h=−1 of distributions K,M  n n h+1 uh ∈ S (R ) with supp u ˆh ⊆ Bh := {ξ ∈ R : |ξ|M ≤ K2 }, h = −1, 0, . . . . Remark 3.7. With the assumptions on the supports of u ˆh respectively made in ∞  uh is convergent in S  (Rn ) as long as the Propositions 3.5, 3.6, the series h=−1

right-hand side of (3.4) is finite.

4. Action on quasi-homogeneous Sobolev spaces r,M In this section we apply the results of §2 in the particular case X = B∞,∞ with positive order r. The main result is: r,M m SM (N ), with r > 0, Theorem 4.1. Let a(x, ξ) belong to the symbol class B∞,∞ m ∈ R and N ≥ 2n + 1. Then for every s ∈] − r, r[ and p ∈]1, +∞[ the range of s,p the operator a(x, D) : S(Rn ) → S  (Rn ) is contained in HM and a(x, D) extends to a continuous operator s+m,p s,p a(x, D) : HM → HM .

(4.1)

The proof needs some preliminary remarks. r,M m r,M 0 SM (N ) yields a(x, ξ)ξ−m ∈ B∞,∞ SM (N ). Let us observe that a(x, ξ) ∈ B∞,∞ M Hence, without loss of generality, we are reduced to the case m = 0. r,M 0 Agreeing with Proposition 2.4 the symbols in B∞,∞ SM (N ) reduce to an r,M 0 expansion of ak (x, ξ) ∈ B∞,∞ SM (N − n − 1). Thus we prove at first the result for elementary symbols; from (2.9) it follows

a(x, D)u(x) =

∞


dh (x)ψh (D)u(x),

u ∈ S(Rn ),

(4.2)

h=−1

where the expansion is absolutely convergent in L∞ (Rn ). By means of a given partition of unity {φl }∞ l=−1 we write (4.2) in the form a(x, D)u(x) =

∞ ∞



dh,l (x)uh (x) (4.3)

h=−1 l=−1

=

(N ) T1 0 u(x)

+

(N ) T2 0 u(x)

+

(N ) T3 0 u(x),

Continuity in Quasi-homogeneous Spaces

169

(N0 )

where dh,l := φl (D)dh , uh := ψh (D)u and the operators Tj defined by: (N0 )

T1

u :=

(N0 )

T3

h−N 0

h=N0 −1 l=−1

(N0 )

T2

∞ 

u :=

∞ 

h+N 0 −1

h=−1

l=lh

∞ 

u :=

l−N 0

l=N0 −1 h=−1

, j = 1, 2, 3, are

dh,l uh , dh,l uh ,

(4.4) lh = max{−1; h − N0 + 1},

dh,l uh .

(4.5) (4.6)

N0 is a fixed positive integer to be precised later. (N0 )

Proposition 4.2. For every suitably large positive integer N0 , T1 linear continuous operator (N0 )

T1

extends to a

s,p s,p : HM → HM ,

for all s ∈ R and p ∈]1, +∞[. (N0 )

Proof. Set Uh h−N 0

(x) :=

h−N 0 l=−1

 (N ) dh,l (x)uh (x) (h ≥ N0 − 1). Since (2π)n Uh 0 =

1h , we can find a constant K  > 1 such that, for all N0 sufficiently d9 h,l ∗ u

l=−1
 (N ) large, supp Uh 0 ⊆ ChK ,M for h ≥ N0 − 1. Proposition 3.5 assures that for any s ∈ R, p ∈]1, ∞[ there exists a positive constant C such that

 1

∞  2 2






  (N0 ) (N )

. s,p ≤ C

T1 uHM 4sh Uh 0 



p

h=N0 −1

L

r,M with r > 0: Since dh ∈ B∞,∞ h−N  
0  (N0 )  r,M r,M |uh (x)|, 2−rl ≤ Cdh B∞,∞ Uh (x) ≤ |uh (x)|dh B∞,∞

x ∈ Rn ,

l=−1

with a positive constant C = Cr . Since N − n − 1 ≥ n the previous estimates yield (N0 )

T1

s,p ≤ C uHM

sup h=−1,0,...

dh B∞,∞ r,M uH s,p , M

(4.7)


with C = Cr,s,p . This completes the proof (N0 )

Proposition 4.3. For every suitably large positive integer N0 , T2 linear continuous operator (N0 )

T2 for all s > −r and p ∈]1, +∞[.

s,p s+r,p : HM → HM ,

extends to a

170

G. Garello and A. Morando (N0 )

Proof. We set Uh

:=

h+N 0 −1

dh,l uh for h = −1, 0, . . . . Arguing as in the proof of

l=lh

 (N ) Proposition 4.2, we find the existence of a constant K  > 1 such that supp Uh 0 ⊆
BhK ,M , for all h = −1, 0, . . . and any N0 sufficiently large. For s > −r, Proposition 3.6 gives

 1



∞


(s+r)h  (N0 ) 2 2

(N0 )

,

T2 uH s+r,p ≤ C

4 Uh 

M

p

h=−1 L

with positive constant C depending only on s, r and p. Moreover, for all h = −1, 0, . . . we get for every x ∈ Rn and some positive constant C = CN0 ,r : h+N 0 −1  
 (N0 )  −rh |dh,l (x)| ≤ Cdh B∞,∞ r,M 2 |uh (x)|. Uh (x) ≤ |uh (x)|

(4.8)

l=lh

By the help of Proposition 3.4 we conclude (N0)

T2

uH s+r,p ≤ C M

sup h=−1,0,...

r,M uH s,p . dh B∞,∞ M


(N0 )

Proposition 4.4. For every positive integer N0 sufficiently large, T3 a continuous linear operator (N0 )

T3

extends to

s−r+δ,p s,p : HM → HM ,

for any real s and positive δ such that s + δ < r. (N0 )

Proof. Let us set Vl

:=

l−N 0

dh,l uh , l ≥ N0 − 1. Then, arguing as in the proof

h=−1


 (N ) of Proposition 4.2, we see that for every N0 sufficiently large supp Vl 0 ⊆ ClK ,M , with a suitable K  > 1 independent of N0 . Then by Proposition 3.5, for any real s we get

 1

∞  2 2






 (N )  (N )

, s,p ≤ C

4sl Vl 0  (4.9) T3 0 uHM



l=N0 −1

p

L

with positive constant C depending only on s and p. On the other hand, if s+δ < r, by means of the Cauchy-Schwarz inequality, we compute:   l−N 0 0  (N0 )  l−N −rl (x) ≤ |dh,l (x)||uh (x)| ≤ dh B∞,∞ r,M 2 |uh (x)| Vl h=−1 h=−1   12   12 l−N l−N 0 −(s−r+δ)h  0 (s−r+δ)h −rl 2 r,M 4 4 |uh (x)| ≤ 2 dh B∞,∞ (4.10) h=−1 h=−1   12 ∞  −(s+δ)(l−N0 ) (s−r+δ)h 2 ≤ C2 dh B∞,∞ r,M 4 |uh (x)| , x ∈ Rn . h=−1

Continuity in Quasi-homogeneous Spaces

171

The constant C depends only upon r, s, δ, N0 and p. Then estimates (4.9), (4.10) and Proposition 3.4 give

  12



∞ ∞





(N0) −δl (s−r+δ)h 2



s,p r,M 4 4 |uh | T3 uHM ≤ C sup dh B∞,∞



h=−1,0,...

l=N0 −1 h=−1

p L

≤ C sup dh B∞,∞ r,M u s−r+δ,p H M

h=−1,0,...




where C = C(r, s, δ, N0 , p). This ends the proof.

Proof of Theorem 4.1. For any operator carrying the structure in (4.2) let us take the decomposition (4.3) for some N0 so large that Propositions 4.2, 4.3, 4.4 work; gathering such results we immediately obtain the mapping property (4.1). More precisely, we are able to prove that for every r > 0, s ∈] − r, r[ and p ∈]1, +∞[, there exists a positive constant C such that any pseudo-differential operator of type (4.2) satisfies: s,p ≤ C a(·, D)uHM

sup h=−1,0,...

dh B∞,∞ r,M uH s,p , M

u ∈ S(Rn ).

(4.11)

r,M 0 For a general symbol a(x, ξ) ∈ B∞,∞ SM (N ), (2.8) holds for any ak (x, ξ) ∈ r,M 0 B∞,∞ SM (N − n − 1) of type (2.9) with N − n − 1 ≥ n. Then we get  s,p ≤ s,p ck ak (., D)uHM a(., D)uHM

≤C

k∈Zn

sup k,h=−1,0,...

dkh B∞,∞ r,M uH s,p , M

u ∈ S(Rn ),

with positive constant C = Cr,s,p . Since the sequences {dkh }∞ h=−1 are bounded in r,M B∞,∞ uniformly with respect to k (cf. Proposition 2.4; (2.10)), the result follows s,p from the density of S(Rn ) into HM .
Corollary 4.5. For every r > 0, s ∈] − r, r[ and p ∈]1, +∞[ there exists a positive constant C such that s,p ≤ Cu s,p v r,M uvHM (4.12) HM B∞,∞ r,M . is true for any u ∈ S(Rn ) and v ∈ B∞,∞

Proof. Let ψ = {ψh }∞ h=−1 be an arbitrary quasi-homogeneous dyadic partition of unity. For any u ∈ S(Rn ) we can write: uv =

∞


vψh (D)u,

h=−1

so that in the Schwartz class S(Rn ) the multiplication by v can be regarded as a particular pseudo-differential operator of type (4.2), where dh = v for all h ≥ −1. In this case (4.11) just reduces to (4.12).


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References [1] R.R. Coifman and Y. Meyer, Au del` a des op´erateurs pseudo-differentiels, Ast´erisque 57, Soc. Math. France, 1978. [2] C. Fefferman, Lp bounds for pseudo-differential operators, Israel J. Math. 14 (1973), 413–417. [3] G. Garello and A. Morando, Lp -bounded pseudodifferential operators and regularity for multi-quasi-elliptic equations, Integral Equations Operator Theory 51 (2005), 501– 517. [4] G. Garello and A. Morando, Lp boundedness for pseudodifferential operators with nonsmooth symbols and applications, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8 (2005), 461–503. [5] G. Garello and A. Morando, Continuity in weighted Sobolev spaces of Lp type for pseudo-differential operators with completely nonsmooth symbols, in Advances in Pseudo-differential Operators, Editors: R. Ashino, P. Boggiatto and M.W. Wong, Birkh¨ auser, 2004, 91–106. [6] G. Garello and A. Morando, Continuity in weighted Besov spaces for pseudodifferential operators with non-regular symbols, in Recent Advances in Operator Theory and its Applications, Editors: M.A. Kaashoek, S. Seatzu and C. van der Mee, Birkh¨ auser, 2005, 195–216. [7] R. Lascar, Propagation des singularit´es des solutions d’´equations pseudodiff´erentielles quasi-homog`enes, Ann. Inst. Fourier Grenoble 27 (1977), 79–123. [8] P.I. Lizorkin, (Lp , Lq )-multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR 152 (1963), 808–811. [9] J. Marschall, Pseudo-differential operators with coefficients in Sobolev spaces, Trans. Amer. Math. Soc. 307(1) (1988), 335–361. [10] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. [11] M.E. Taylor, Pseudodifferential Operators, Princeton University Press, 1981. [12] M.E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkh¨ auser, BaselBoston-Berlin, 1991. [13] M.E. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators and Layer Potentials, American Mathematical Society, 2000. [14] H. Triebel, Theory of Function Spaces, Birkh¨ auser Verlag, Basel, Boston, Stuttgart, 1983. [15] M. Yamazaki, The Lp -boundedness of pseudodifferential operators with estimates of parabolic type and product type, J. Math. Soc. Japan 38(2) (1986), 199–225. Gianluca Garello Dipartimento di Matematica, Universit` a di Torino Via Carlo Alberto 10, I-10123 Torino, Italy e-mail: [email protected] Alessandro Morando Dipartimento di Matematica, Facolt` a di Ingegneria – Universit` a di Brescia Via Valotti 9, I-25133 Brescia, Italy e-mail: [email protected], [email protected]

Operator Theory: Advances and Applications, Vol. 172, 173–206 c 2006 Birkh¨
auser Verlag Basel/Switzerland

Continuity and Schatten Properties for Pseudo-differential Operators on Modulation Spaces Joachim Toft p,q Abstract. Let M(ω) be the modulation space with parameters p, q and weight function ω. We prove that if t ∈ R, p, pj , q, qj ∈ [1, ∞], ω1 , ω2 and ω are p,q , then the pseudo-differential operator at (x, D) appropriate, and a ∈ M(ω) p1 ,q1 p2 ,q2 to M(ω . If in addition pj = qj = 2, then we is continuous from M(ω ) 1 2) establish necessary and sufficient conditions on p and q in order to at (x, D) should be a Schatten-von Neumann operator of certain order.

Mathematics Subject Classification (2000). Primary 35S05, 47B10, 47B37; Secondary 42B35, 47B35. Keywords. Pseudo-differential operators, Schatten classes, modulation spaces, non-smooth symbols.

0. Introduction In [25] and [26], Gr¨ ochenig and Heil present a method, based on time-frequency analysis when investigating pseudo-differential operators with non-smooth symbols belonging to non-weighted modulation spaces. Here they make suitable Gabor expansions of the symbols, which in some extent reduce the problems in such way that the symbols are translations and modulations of a fix and well-known function. In that end, they are able to make a somewhat detailed study of compactness, and prove embedding properties between Schatten-von Neumann classes of pseudodifferential operators acting on L2 , and modulation spaces. Furthermore, they prove that any pseudo-differential operator with symbol ostrand) is continuous in the modulation space M ∞,1 (denoted by Sw in [41] by Sj¨ on any non-weighted modulation space M p,q . Since L2 = M 2,2 , it follows in particular that such operators are continuous on L2 , a property which was proved by Sj¨ ostrand in [40], where modulation spaces were used as symbol classes for the first

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time. Furthermore, since S00 , the set of functions which are bounded together with all their derivatives, is contained in M ∞,1 , it follows from these investigations that any pseudo-differential operator with symbol in S00 is continuous on M p,q . The latter result was remarked in the L2 -case in [40], and for general p and q, the result is a special case of Theorem 2.1 in [42] by Tachizawa. Some further improvements and extensions of the results above have been done since [25, 26, 40–42]. In [7], Boulkhemair extend the L2 continuity to Fourier integral operators with symbols in M ∞,1 and phase functions of rather general types. In the independent papers [27] and [48], continuity for pseudo-differential operators with symbol class M p,q acting on modulation spaces, are considered. In [49] these results were further extended in the case of Weyl operators where the symbols belong to weighted modulation spaces. Some further properties concerning embeddings between Schatten-von Neumann classes in the pseudo-differential calculus and modulation spaces can also be found in [48]. Important parts in this context concern modulation spaces, and their properties. These spaces were introduced by Feichtinger in [13] and [15] during the period 1980–1983 as an appropriate family of function and distribution spaces to have in background when discussing certain problems within time-frequency analysis. The basic theory of such spaces were thereafter established and extended by Feichtinger and Gr¨ ochenig (see, e.g., [14, 18, 19, 24], and the references therein). Roughly speaking, for an appropriate weight function ω, the modulation space p,q is obtained by imposing a mixed Lp,q M(ω) (ω) -norm on the short-time Fourier transform of a tempered distribution. The non-weighted modulation space M p,q is then obtained by choosing ω = 1. In terms of modulation spaces it is sometimes easy to obtain information concerning growth and decay properties, as well as certain localization and regularity properties for distributions. In this paper we continue the discussions in [48,49] concerning continuity for pseudo-differential operators in background of modulation space theory. (Cf. [51].) More precisely, we consider pseudo-differential operators where the corresponding symbols belong to appropriate modulation spaces, and discuss continuity for such operators when acting on modulation spaces. Especially we are concerned with a somewhat detailed study of continuity and compactness for pseudo-differential operators acting between modulation spaces of Hilbert type in terms of Schattenvon Neumann classes. In particular we investigate trace-class and Hilbert-Schmidt properties. Except for the Hilbert-Schmidt case it is in general a hard task to find complete characterizations of Schatten-von Neumann classes. One is therefore forced to find embeddings between such classes and other spaces which are more convenient. In Section 4 we discuss embeddings between such classes and modulation spaces, and generalize certain results in [26, 45, 48]. In contrast to the latter papers, the situation in Section 4 is more complicated depending on the fact that we consider operators acting on modulation spaces which involve weight functions of general types, instead of operators acting on L2 . In particular, by choosing the

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involved weight functions in appropriate ways, we may use our results to discuss Schatten-von Neumann properties for pseudo-differential operators acting between weighted Lebesgue spaces and/or Sobolev spaces of Hilbert type. The general types of modulation spaces which are involved in the continuity investigations cause new problems comparing to [26, 48, 49]. These problems are overcome by using a related Gabor technique as in [26], leading to a convenient way to expand the symbols, and discretization of certain parts of the problems. The requested results are thereafter obtained by using techniques in modulation space theory, which are well known within time-frequency analysis, in combination with certain duality properties for Schatten-von Neumann classes in pseudo-differential calculus, presented in Section 3, and harmonic analysis. In order to describe our results in more details we recall the definition of modulation spaces. Assume that χ ∈ S (Rm ) \ 0, p, q ∈ [1, ∞] and that ω is an appropriate function on R2m , and let τx χ(y) = χ(y − x) when x, y ∈ Rm . (We use the same notation for the usual functions and distribution spaces as in, e.g., [28].) p,q Then the modulation space M(ω) (Rm ) consists of all f ∈ S  (Rm ) such that p,q = f  p,q,χ f M(ω) M(ω)   q/p 1/q ≡ dξ < ∞, |F (f τx χ)(ξ)ω(x, ξ)|p dx

(0.1)

with the obvious modifications when p = ∞ and/or q = ∞. Here F denotes the Fourier transform on S  (Rm ), which takes the form  F f (ξ) = f(ξ) = (2π)−m/2 f (x)e−i x,ξ dx when f ∈ S (Rm ). Moreover, the function (x, ξ) → F (f τx χ)(ξ) is called the shorttime Fourier transform of f with window function, or just window, χ for f in the literature. (In the literature, the terms coherent state transform and coherent state also occur.) Next assume that t ∈ R is fixed and that a ∈ S (R2m ). Then the pseudodifferential operator at (x, D) is the continuous operator on S (Rm ) which is defined by the formula (at (x, D)f )(x) = (Opt (a)f )(x)  = (2π)−m a((1 − t)x + ty, ξ)f (y)ei x−y,ξ dydξ.

(0.2)

The definition of at (x, D) extends to any a ∈ S  (R2m ), and then at (x, D) is continuous from S (Rm ) to S  (Rm ). (See, e.g., [28], or Section 1.) If t = 1/2, then at (x, D) is equal to the Weyl operator aw (x, D) for a. If instead t = 0, then the standard (Kohn-Nirenberg) representation a(x, D) is obtained. In Section 4 we discuss continuity for pseudo-differential operators acting on modulation spaces when the operator symbols belong to modulation spaces. Some of these questions were discussed for Weyl operators (i.e., when t = 1/2 in (0.2))

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already in Section 5 in [49]. In the first part of Section 4 we extend the results in [49] to arbitrary t ∈ R. In particular we find appropriate conditions on ω, ω1 , ω2 and p, q, pj , qj for j = 1, 2, in order for p1 ,q1 p2 ,q2 → M(ω at (x, D) : M(ω 1) 2) p,q to be continuous when a ∈ M(ω) . p ,q The second part of Section 4 is devoted to the case when the M(ωjj )j above are Hilbert spaces, i.e., pj = qj = 2 for j = 1, 2. In this case a more detailed compactness study in terms of Schatten-von Neumann properties is performed. These considerations are dependent on some preparations which are made in Section 2 2,2 2,2 and in Section 3. Recall that an operator T from M(ω to M(ω belongs to Ip , 1) 2) the set of Schatten-von Neumann operators of order p ∈ [1, ∞], if and only if 
1/p p 2 sup | < ∞, |(T fj , gj )M(ω ) 2

2,2 where the supremum is taken over all orthonormal sequences (fj ) in M(ω and 1)

2,2 . In particular, this implies that I∞ is the set of linear and continuous (gj ) in M(ω 2) operators, and that T is compact when T ∈ Ip and p < ∞. (Cf. [37, 39].) We are then concerned with classification and embedding properties for the set st,p (ω1 , ω2 ) which consists of all a ∈ S  (R2m ) such that at (x, D) ∈ Ip . In Section 4 we prove that p,q1 p,q2 M(ω) ⊆ st,p (ω1 , ω2 ) ⊆ M(ω) , (0.3)

for appropriate choices of q1 , q2 and ω. In particular, our investigations concern Schatten-von Neumann properties for pseudo-differential operators which map the Sobolev space Hs21 or the weighted Lebesgue space L2s1 to Hs22 or L2s2 , since each 2 one of these spaces agrees with M(ω) if ω is chosen in an appropriate way. Here p m Hs (R ) is the Sobolev space of distributions with s derivatives in Lp , i.e., the set of all f ∈ S  such that (1 − ∆)s/2 f ∈ Lp (Rm ). In the end of Section 4 we also use (0.3) and embedding results in Section 4 in [50] between modulation spaces and Besov spaces, to obtain embedding properties between the st,p -spaces and Besov spaces. Finally we remark that the results here are applied in [52] in the present volume and in a forthcoming joint paper with Paolo Boggiatto, in order to obtain continuity properties for Toeplitz operators. In particular some improvements of certain results in [9, 48] are presented.

1. Preliminaries In this section we discuss basic properties for modulation spaces. The proofs are in many cases omitted since they can be found in [5, 13–16, 18–20, 25, 44–47, 49]. We start by recalling some properties of the weight functions which are inm volved. We say that the function ω ∈ L∞ loc (R ) is v-moderate for some appropriate

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m positive function v ∈ L∞ loc (R ), if there is a constant C > 0 such that

ω(x1 + x2 ) ≤ Cω(x1 )v(x2 ),

x1 , x2 ∈ Rm .

(1.1)

The function v is then said to moderate ω, and if ω is v-moderate for some polynomial v, then ω is called polynomial moderate. If in addition (1.1) holds for ω = v, then v is said to be a moderate or submultiplicative function. As in [49] we let P(Rm ) denote the cone which consists of all polynomial moderate functions on Rm . Note that if ω ∈ P(Rm ), then ω(x) + ω(x)−1 ≤ P (x), x ∈ Rm for some polynomial P on Rm . Let ω1 and ω2 be positive functions. If ω2 /ω1 is bounded, then we write ω2 ≺ ω1 . They are called equivalent when ω1 ≺ ω2 ≺ ω1 , and then and we write ω1 ∼ ω2 . If H is a Hilbert space, then its scalar product is denoted by ( · , · )H , or ( · , · ) when there are no confusions about the Hilbert space structure. The duality between a topological vector space and its dual is denoted by  · , · . For admissible a and b in S  (Rm ), we set (a, b) = a, b, and it is obvious that ( · , · ) on L2 is the usual scalar product. Next assume that B1 and B2 are topological spaces. Then B1 → B2 means that B1 is continuously embedded in B2 . In the case that B1 and B2 are Banach spaces, B1 → B2 is equivalent to B1 ⊆ B2 and xB2 ≤ CxB1 , for some constant C > 0 which is independent of x ∈ B1 . Next let V1 and V2 be vector spaces such that V1 ⊕ V2 = Rm and V2 = V1⊥ , and assume that v0 ∈ S  (V1 ) and that v(x1 , x2 ) = (v0 ⊗ 1)(x1 , x2 ), where xj ∈ Vj for j = 1, 2. Then v(x1 , x2 ) is identified with v0 (x1 ), and we set v(x1 , x2 ) = v(x1 ). Assume that ω ∈ P(R2m ), p, q ∈ [1, ∞], and that χ ∈ S (Rm ) \ 0. Then p,q (Rm ) is the set of all f ∈ S  (Rm ) such that recall that the modulation space M(ω) p,q (0.1) holds. We note that the definition of M(ω) (Rm ) is independent of the choice of window χ, and that different choices of χ give rise to equivalent norms. (See Proposition 1.2 below.) p,q . Moreover we set If ω = 1, then the notation M p,q is used instead of M(ω) p p,p p p,p M(ω) = M(ω) and M = M . Remark 1.1. We are also concerned with the following family of function and distribution spaces which are related to the Wiener amalgam spaces. Assume that p,q (Rm ) consists of all p, q ∈ [1, ∞] and that ω ∈ P(R2m ). Then the space W(ω)  m a ∈ S (R ) such that   q/p 1/q p,q aW(ω) = dx |F (a τx χ)(ξ)ω(x, ξ)|p dξ is finite. (Cf. Definition 4 in [20].)

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p,q p,q = F M(ω when ω0 (x, ξ) = ω(−ξ, x) ∈ P(R2m ). In fact, We recall that W(ω) 0) let χ(x) ˇ = χ(−x) as usual. Then Parseval’s formula and a change of the order of integration shows that

|F −1 ( a τξ χ )(x)| = |F (a τx χ)(ξ)|, ˇ

(1.2)

and the assertion follows. We refer to [15,20] for more facts about the

p,q W(ω) -spaces.

The convention of indicating weight functions with parenthesis is used also in other situations. For example, if ω ∈ P(Rm ), then Lp(ω) (Rm ) is the set of all measurable functions f on Rm such that f ω ∈ Lp (Rm ), i.e., such that f Lp(ω) ≡ f ωLp is finite. Next we consider the Fourier transform of functions and distributions defined on R2m . By interpreting R2m as the phase space with dual variables (y, η), we let :, be defined by the formula the phase space Fourier transform, F  −m :  f (x, ξ)e−i( x,η+ y,ξ) dxdξ, (1.3) (F f )(y, η) = f (y, η) ≡ (2π) :f )(x, ξ) = (F f )(ξ, x). Then F : is a homeomorphism when f ∈ L1 (R2m ), i.e., (F on S (R2m ) which extends to a homeomorphism on S  (R2m ) and to a unitary map on L2 (R2m ), since similar facts hold for F . :p,q , M :p , M :p,q and M :p instead of M p,q , M p , M p,q We use the notation M (ω) (ω) (ω) (ω) : is used instead of F , in the definition of modulation and M p respectively, when F spaces. The following proposition is a consequence of well-known facts in [15,25]. Here and in what follows, we let p denote the conjugate exponent of p, i.e., 1/p+ 1/p = 1. Proposition 1.2. Assume that p, q, pj , qj ∈ [1, ∞] for j = 1, 2, and that ω, ω1 , ω2 , v ∈ P(R2m ) are such that ω is v-moderate. Then the following are true: p,q 1 (1) if χ ∈ M(v) (Rm ) \ 0, then f ∈ M(ω) (Rm ) if and only if (0.1) holds, i.e., p,q p,q m M(ω) (R ) is independent of the choice of χ. Moreover, M(ω) is a Banach space under the norm in (0.1), and different choices of χ give rise to equivalent norms; (2) if p1 ≤ p2 , q1 ≤ q2 and ω2 ≺ ω1 , then p1 ,q1 p2 ,q2 S (Rm ) → M(ω (Rm ) → M(ω (Rm ) → S  (Rm ); 1) 2) p,q (Rm ) × (3) the L2 product ( · , · ) from S extends to a continuous map from M(ω)





p ,q M(1/ω) (Rm ) to C. On the other hand, if a = sup |(a, b)|, where the supre





p ,q (Rm ) such that bM p
,q
≤ 1, then  ·  mum is taken over all b ∈ M(1/ω) p,q are equivalent norms; and  · M(ω)

(1/ω)

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179

p,q p,q (Rm ). The dual space of M(ω) (Rm ) (4) if p, q < ∞, then S (Rm ) is dense in M(ω)





p ,q (Rm ), through the form ( · , · )L2 . Moreover, can be identified with M(1/ω) m ∞ S (R ) is weakly dense in M(ω) (Rm ).

Proposition 1.2 (1) permits us be rather vague about to the choice of χ ∈ 1 M(v) \ 0 in (0.1). For example, if C > 0 is a constant and Ω is a subset of S  , then p,q ≤ C for every a ∈ Ω, means that the inequality holds for some choice of aM(ω)

1 1 \ 0 and every a ∈ Ω. Evidently, for any other choice of χ ∈ M(v) \ 0, a χ ∈ M(v) similar inequality is true although C may have to be replaced by a larger constant, if necessary. Next we discuss weight functions which are common in the applications. For any s, t ∈ R, set σt (x) = xt , σs,t (x, ξ) = ξs xt , (1.4) when x, ξ ∈ Rm . Then it follows that σt ∈ P0 (Rm ) and σs,t ∈ P0 (R2m ) for every s, t ∈ R, and σt is σ|t| -moderate and σs,t is σ|s|,|t| -moderate. Obviously, σs (x, ξ) = (1 + |x|2 + |ξ|2 )s/2 , and σs,t = σt ⊗ σs . Moreover, if ω ∈ P(Rm ), then ω is σt -moderate provided t is chosen large enough. p,q For convenience, we use the notations Lps , Msp,q and Ms,t instead of Lp(σs ) , p,q p,q M(σs ) and M(σs,t ) respectively. m m It is also convenient to let Mp,q (ω) (R ) be the completion of S (R ) under p,q p,q p,q . Then M the norm  · M(ω) (ω) ⊆ M(ω) with equality if and only if p < ∞ and p,q (Rm ), also q < ∞. It follows that most of the properties which are valid for M(ω) p,q hold for M(ω) (Rm ).

Remark 1.3. Assume that p, q, q1 , q2 ∈ [1, ∞]. Then the following properties for modulation spaces hold: (1) if q1 ≤ min(p, p ) and q2 ≥ max(p, p ), then M p,q1 ⊆ Lp ⊆ M p,q2 . In particular, M 2 = L2 ; ∞,1 ; (2) S00 = ∩s∈R Ms,0 p,q (Rm ) → C(Rm ) if (3) if ω ∈ P(R2m ) is such that ω(x, ξ) = ω(x), then M(ω) and only if q = 1; (4) M 1,∞ is a convolution algebra which contains all measures on Rm with bounded mass; (5) if Ω is a subset of P(R2m ) such that for any polynomial P on R2m , there is an element ω ∈ Ω such that P/ω is bounded, then p,q (Rm ), S (Rm ) = ∩ω∈Ω M(ω)

p,q S  (Rm ) = ∪ω∈Ω M(1/ω) (Rm );

(6) if s, t ∈ R are such that t ≥ 0, then 2 = Hs2 , Ms,0

2 M0,s = L2s ,

(See, e.g., [13–15, 17–20, 25, 47–49].)

and Mt2 = L2t ∩ Ht2 .

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Next we recall some facts from Chapter 12 and Chapter 13 in [25] concerning extension of the Gabor theory to modulation spaces. Let (xj )j∈I and (ξk )k∈I be lattices in Rm , and assume that χ ∈ S (Rm ) is fixed and satisfies
|χ(x − xj )|2 ≤ C, x ∈ Rm , (1.5) C −1 ≤ j∈I

for some constant C. If (ξk ) is sufficiently dense, then it follows from [25] and Section 7.3 in [28] that there exists a function ψ ∈ S (Rm ) such that
cj,k (f )ei x,ξk  χ(x − xj ) (1.6) f (x) = j,k∈I

=




dj,k (f )ei x,ξk  ψ(x − xj ),

(1.6)

j,k∈I 

for every f ∈ S (R ), where cj,k (f ) and dj,k (f ) are the “Fourier coefficients” for f , given by the formulas m

cj,k (f ) = cj,k = F (f τxj ψ)(ξk ) and

(1.7)

dj,k (f ) = dj,k = F (f τxj χ)(ξk ).

(1.7)

Here the sums converge in S  (Rm ). In order to present some further properties in the case of modulation spaces, it is convenient to consider the following sequence spaces. Assume that λ = (λj,k )j,k∈I is a (fix) sequence of non-negative numbers, and that p, q ∈ [1, ∞]. p,q Then let l(λ) be the Banach space of sequences (cj,k )j,k∈I of complex numbers such that q/p 1/q 

p ≡ |c λ | (cj,k )j,k∈I lp,q j,k j,k (λ) k

j

p,q (λ)

is finite. Also let be the completion of all finite sequences (i.e., all sequences (cj,k )j,k∈I such that only a finite numbers of cj,k are non-zero) under the norm p,q p p,p  · lp,q . Furthermore we set lp,q = l(λ) when λj = 1 for every j, l(λ) = l(λ) , and (λ) similarly for the p,q -spaces. (λ) The following proposition shows that essential parts of the Gabor theory, in the context of L2 -spaces, carry over to modulation spaces. This shows in particular that there is a convenient way to discretize modulation spaces. The proof is omitted, since the result follows from Chapter 12 and 13 in [25]. Proposition 1.4. Let (xj )j∈I ⊆ Rm be a lattice, ω ∈ P(R2m ), and assume that χ ∈ S (Rm ) satisfies (1.5). Then there is a lattice (ξk0 )k∈I0 ⊆ Rm and a function ψ ∈ S (Rm ) such that if (ξk )k∈I ⊆ Rm is a lattice which contains (ξk0 )k∈I0 , the following are true. (1) if f ∈ S  (Rm ), and cj,k (f ) and dj,k (f ) are given by (1.7) and (1.7) respectively, then (1.6) and (1.6) hold;

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(2) if λ = (ω(xj , ξk ))j,k and f ∈ S  (Rm ), then the following conditions are equivalent: p,q (Rm ); (i) f ∈ M(ω) p,q ; (ii) (cj,k (f ))j,k ∈ l(λ) p,q (iii) (dj,k (f ))j,k ∈ l(λ) .

Moreover, if the conditions (i)–(iii) are fulfilled and in addition p, q < p,q ∞, then the sums in (1.6) and (1.6) converge to f in M(ω) . Furthermore, p,q p,q p,q are the norms f → (cj,k (f ))j,k l(λ) , f → (dj,k (f ))j,k l(λ) and  · M(ω) equivalent. Remark 1.5. We note that the equivalence (i)–(iii) in Proposition 1.4 holds, after p,q p,q p,q and l(λ) have been replaced by Mp,q M(ω) (ω) and (λ) respectively. Next we discuss (complex) interpolation properties for modulation spaces. Such properties were carefully investigated in [15] for non-weighted modulation spaces, and thereafter extended in several directions in [19], were interpolation properties for coorbit spaces were established. As a consequence of [19] we have the following proposition. In order to be more self-contained, we give a short proof based on Theorem 1.4 and Remark 1.5. Proposition 1.6. Assume that 0 < θ < 1 and that p, q, pj , qj ∈ [1, ∞] satisfy 1−θ 1 θ = + p p1 p2

and

1−θ 1 θ = + . q q1 q2

Also assume that ω1 , ω2 ∈ P(R2m ) and let ω = ω11−θ ω2θ . Then m 1 2 (Rm ), Mp(ω2 ,q (Rm ))[θ] = Mp,q (Mp(ω1 ,q (ω) (R ). 1) 2)

Proof. In view of Remark 1.5 it suffices to prove that (p(λ1 1,q)1 , p(λ2 2,q)2 )[θ] = p,q (λ) ,

(1.8)

where λi = (λij,k )j,k≥0 for i = 1, 2 and λ = (λj,k )j,k≥0 are sequences of positive numbers such that λj,k = (λ1j,k )1−θ (λ2j,k )θ . Furthermore, by using the operator T (z) for 0 ≤ Re z ≤ 1, defined by the formula   T (z) (cj,k )j,k≥0 = (cj,k (λ1j,k )1−z (λ2j,k )z )j,k≥0 , it follows that we may assume that each λij,k is equal to 1. This means that we shall prove (p1 ,q1 , p2 ,q2 )[θ] = p,q . The result is now a consequence of Theorem 5.1.1 and Theorem 5.1.2 in [1]. The proof is complete.
Next we recall some facts in Chapter XVIII in [28] concerning pseudo-differential operators. Assume that a ∈ S (R2m ), and that t ∈ R is fixed. Then the pseudo-differential operator at (x, D) in (0.2) is a linear and continuous operator on

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S (Rm ), as remarked in the introduction. For general a ∈ S  (R2m ), the pseudodifferential operator at (x, D) is defined as the continuous operator from S (Rm ) to S  (Rm ) with distribution kernel Kt,a (x, y) = (2π)−m/2 (F2−1 a)((1 − t)x + ty, y − x),

(1.9)

Here F2 F is the partial Fourier transform of F (x, y) ∈ S (R2m ) with respect to the y-variable. This definition makes sense, since the mappings F2 and F (x, y) → F ((1 − t)x + ty, y − x) are homeomorphisms on S  (R2m ). We also note that this definition of at (x, D) agrees with the operator in (0.2) when a ∈ S (R2m ). Furthermore, any linear and continuous operator T from S (Rm ) to S  (Rm ) has a distribution kernel K in S  (R2m ) in view of kernel theorem of Schwartz. By Fourier’s inversion formula we may then find a unique a ∈ S  such that (1.9) is fulfilled with K = Kt,a . Consequently, for every fixed t ∈ R, there is a one to one correspondence between linear and continuous operators from S (Rm ) to S  (Rm ), and Opt (S  (R2m )), the set of all at (x, D) such that a ∈ S  (R2m ). In particular, if a ∈ S  (R2m ) and s, t ∈ R, then there is a unique b ∈  S (R2m ) such that as (x, D) = bt (x, D). By straightforward applications of Fourier’s inversion formula, it follows that as (x, D) = bt (x, D)

⇔

b(x, ξ) = ei(t−s) Dx ,Dξ  a(x, ξ).

(1.10)

(Cf. Section 18.5 in [28].) We may also express the relations between a and b here above in terms of convolution operators. In fact, by Fourier’s inversion formula it follows that if t = 0 and Φ(x, ξ) = x, ξ/t, then there is a constant c such that eit Dx ,Dξ  a = c eiΦ ∗a. If instead t = 0, then eit Dx ,Dξ  a = a = δ ∗ a. These relations motivate us to consider continuity properties for operators of the form f → SΦ f ≡ (eiΦ ⊗ δV2 ) ∗ f,

(1.11)

where δV2 is the delta function on the vector space V2 ⊆ Rm and Φ is a real-valued and non-degenerate quadratic form on V1 = V2⊥ . The operator SΦ is essentially a composition of a partial Fourier transform with respect to the variables in V1 and a non-degenerate matrix AΦ /2. In particular, Fourier’s inversion formula gives that SΦ is a homeomorphism on S (Rm ) which extends uniquely to a homeomorphism on S  (Rm ). In a similar way as in [45–49], we are especially concerned with continuity properties of SΦ when acting on modulation spaces. It was proved in [45] that such operators are continuous on any M p,q . In [49] the latter result was extended p,q , where ω(x, ξ) = ω(ξ). In the following to modulation spaces of the form M(ω) proposition we show that the arguments in [49] can be used in a broader context. Proposition 1.7. Assume that χ ∈ S (Rm ), ω ∈ P(R2m ), p, q ∈ [1, ∞], and that V1 , V2 ⊆ Rm are vector spaces such that V2 = V1⊥ . Also assume that Φ is a realvalued and non-degenerate quadratic form on V1 , and let AΦ /2 be the corresponding

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matrix. If ξ = (ξ1 , ξ2 ) where ξj ∈ Vj for j = 1, 2, then p,q,χ = f  SΦ f M(ω M p,q,ψ , ) (ω)

Φ

where

ωΦ (x, ξ) = ω(x − A−1 Φ ξ1 , ξ)

f ∈ S  (Rm ), (1.12)

and

ψ = SΦ χ.

In particular, the following are true: p,q (1) the map (1.11) on S  (Rm ) restricts to a homeomorphism from M(ω) (Rm ) p,q m to M(ωΦ ) (R ); (2) if t ∈ R, ω0 ∈ P(R2m ⊕ R2m ), and ωt (x, ξ, y, η) = ω0 (x − ty, ξ − tη, y, η), then the map eit Dx ,Dξ  on S  (R2m ) restricts to a homeomorphism from :p,q (R2m ) to M :p,q (R2m ). M (ω0 ) (ωt ) Proof. We only prove the first part of the proposition in the case V1 = Rm , leaving the modifications to the general case for the reader. Assume that f ∈ S  (Rm ) and χ ∈ S (Rm ). It follows easily that the Hilbert adjoint for SΦ is equal to S−Φ . This gives |F (SΦ f τx χ)(ξ)| = |(SΦ f, τx χei ξ,· )| = |(f, S−Φ (τx χei ξ,· ))|.

(1.13)

i ξ,·

We have to analyze S−Φ (τx χe ). By straightforward computations it follows that  i ξ,· )(y) = e−iΦ(y−z) χ(z − x)ei z,ξ dz S−Φ (τx χe  = ei x,ξ e−iΦ(y−x−z) χ(z)ei z,ξ dz  −1 i y,ξ iΨ(x,ξ) =e e e−iΦ(y−(x−AΦ ξ)−z) χ(z) dz = ei y,ξ eiΨ(x,ξ) (τx−A−1 ξ (SΦ χ))(y), Φ

for some real quadratic form Ψ on R2m . By inserting this into (1.13) we get |F (SΦ f τx χ)(ξ)| = |(f, ei · ,ξ (τx−A−1 ξ (SΦ χ)))| = |F (f τx−A−1 ξ ψ)(ξ)|. Φ

Φ

The assertions (1) and (1.12) now follow by applying the Lp,q (ωΦ ) -norm on the latter equalities, and using Proposition 1.2 (1). The assertion (2) follows now by replacing Rm with R2m , and letting V1 = R2m and Φ(x, ξ) = x, ξ/t when t = 0, and V2 = R2m when t = 0. The proof is complete.
We finish the section by giving some remarks on Wigner distributions and Weyl operators of rank one. The Wigner distribution for f ∈ S (Rm ) and g ∈ S (Rm ) is defined by the formula  Wf,g (x, ξ) = (2π)−m/2 f (x − y/2)g(x + y/2)ei y,ξ dy. (1.14)

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For future references we note that the map (f, g) → Wf,g is continuous from S (Rm ) × S (Rm ) to S (R2m ) which extends uniquely to continuous mappings from S  (Rm ) × S  (Rm ) to S  (R2m ), and from L2 (Rm ) × L2 (Rm ) to L2 (R2m ). (See [22, 44, 46].) Next assume that a ∈ S (R2m ) and that f, g ∈ S (Rm ). Then it follows from (0.2) with t = 1/2 and straightforward computations that (aw (x, D)f, g) = (2π)−m/2 (a, Wg,f ),

(1.15)

In particular, (1.15) and Fourier’s inversion formula imply that if f1 , f2 ∈ S (Rm ), then aw (x, D)f (x) = (2π)−m/2 (f, f2 )f1 (x)

⇔

a = Wf1 ,f2 .

(1.16)

Consequently, a Weyl operator is a rank one operator if and only if its symbol is a Wigner distribution.

2. Schatten-von Neumann classes for operators acting on Hilbert spaces In this section we discuss Schatten-von Neumann classes of linear operators from a Hilbert space H1 to another Hilbert space H2 . Such operator classes were introduced by R. Schatten in [35] in the case H1 = H2 . (See also [23, 39].) The general situation when H1 is not necessarily equal to H2 have thereafter been discussed in [2, 37]. Here we give a short introduction, based on an argument which essentially reduces the situation to the case H1 = H2 . For any Hilbert space H , we let ON(H ) be the set of orthonormal sequences in H . Assume that T : H1 → H2 is linear, and that p ∈ [1, ∞]. Then set 1/p 
T Ip = T Ip(H1 ,H2 ) ≡ sup (2.1) |(T fj , gj )H2 |p (with obvious modifications when p = ∞). Here the supremum is taken over all (fj ) ∈ ON(H1 ) and (gj ) ∈ ON(H2 ). Now recall that Ip = Ip (H1 , H2 ), the Schatten-von Neumann class of order p, consists of all linear operators T from H1 to H2 such that T Ip(H1 ,H2 ) is finite. Obviously, I∞ (H1 , H2 ) consists of all continuous operators from H1 to H2 . If H1 = H2 , then the shorter notation Ip (H1 ) is used instead of Ip (H1 , H2 ). We also let I (H1 , H2 ) be the set of all linear and compact operators from H1 to H2 , and equip this space with the norm  · I∞ as usual. The spaces I1 (H1 , H2 ) and I2 (H1 , H2 ) are called the sets of trace-class operators and Hilbert-Schmidt operators respectively. These definitions agree with the old ones when H1 = H2 , and in this case the norms  · I1 and  · I2 agree with the trace-class norm and Hilbert-Schmidt norm respectively. Another description of Schatten-von Neumann classes can be obtained in terms of singular numbers. Assume first that T above is compact. Then by the

Pseudo-differential Operators on Modulation Spaces spectral theorem it follows that ∞
Tf = λj (f, gj )H1 fj ,

f ∈ H1 ,

185

(2.2)

j=1 ∞ for some sequences (fj )∞ j=1 ∈ ON(H2 ) and (gj )j=1 ∈ ON(H1 ), and some sequence λ1 ≥ λ2 ≥ · · · ≥ 0. Here the numbers λj are called the singular numbers for T , and we use the notation σj (T ) for these numbers, i.e., σj (T ) = λj . There is a canonical way to extend the definition of singular values to bounded operators, which are not necessarily compact. More precisely, for any closed subspace V of H1 , set sup T f H2 , µV (T ) = f ∈V, f H1 ≤1

Then let σj (T ) be defined by the formula σj (T ) = σj,H1 ,H2 (T ) ≡

inf

dim V ⊥ =j−1

µV (T ).

(2.3)

It is straightforward to verify that σj (T ) agrees with the earlier definition when T is compact. Moreover, T ∈ Ip if and only if (σj (T )) ∈ lp , and T Ip = (σj (T ))lp .

(2.4)

From now on we assume that the involved Hilbert spaces are separable. Then without loss of generality we may in many situations concerning Ip (H1 , H2 ) reduce ourself to the case H1 = H2 . In fact, assume that (fj0 ) and (gj0 ) are fixed orthonormal bases for H1 and H2 respectively, and let T0 be the linear map, defined by the formula 

αj fj0 = αj gj0 . T0 Here (αj ) ∈ l2 is arbitrary. Then T0 is an isometric bijection from H1 to H2 , T0∗ ◦ T0 = IdH1 and T0 ◦ T0∗ = IdH2 . Consequently, (fj ) → (T0 fj ) is a bijection from ON(H1 ) to ON(H2 ). This in turn implies that T → T0 ◦ T is an isometric homeomorphism from Ip (H1 ) to Ip (H1 , H2 ), i.e., there is a canonical identification between Ip (H1 ) and Ip (H1 , H2 ). In Proposition 2.1 till Proposition 2.7 below we list some properties for spaces of the type Ip (H1 , H2 ), which are well known in the case H1 = H2 . (See [39].) The general case, when H1 is not necessarily equal to H2 is now a consequence of the identification here above and the corresponding results in [39]. In the general case, the results can also be found in [2, 23, 37]. Proposition 2.1. Assume that p, pj ∈ [1, ∞] for 1 ≤ j ≤ 2 such that p1 < p2 < ∞. Also assume that H1 and H2 are separable Hilbert spaces. Then Ip is a Banach space, Ip1 (H1 , H2 ) ⊆ Ip2 (H1 , H2 ) ⊆ I (H1 , H2 ) ⊆ I∞ (H1 , H2 ), and T I∞ ≤ T Ip2 ≤ T Ip1 ,

T ∈ I∞ (H1 , H2 ).

(2.5)

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Moreover, equalities in (2.5) occur if and only if T is a rank one operator, i.e., T f = (f, g1 )H1 g2 for some g1 ∈ H1 and g2 ∈ H2 , and then T Ip = g1 H1 g2 H2 for every p ∈ [1, ∞]. The next result concerns algebraic properties for Schatten-von Neumann classes. Proposition 2.2. Assume that p, q, r ∈ [1, ∞] such that 1/p + 1/q = 1/r. Also assume that Hj for 1 ≤ j ≤ 3 are separable Hilbert spaces. If T1 ∈ Ip (H1 , H2 ) and T2 ∈ Iq (H2 , H3 ), then T = T2 ◦ T1 ∈ Ir (H1 , H3 ), and T2 ◦ T1 Ir ≤ T1 Ip T2 Iq .

(2.6)

On the other hand, for any T ∈ Ir (H1 , H3 ), there are operators T1 ∈ Ip (H1 , H2 ) and T2 ∈ Iq (H2 , H3 ) such that T = T2 ◦ T1 and equality is attained in (2.6). Remark 2.3. Assume that p ∈ [1, ∞] and that Hj for j = 1, . . . , 4 are Hilbert spaces such that H1 = H2 with equivalent norms, and H3 → H4 . Then it follows from Proposition 2.2 that these embeddings induce the embedding Ip (H1 , H3 ) → Ip (H2 , H4 ). This is also a consequence of (2.4), since it follows from the assumptions that there exists a constant C such that if T is linear from H1 = H2 to H3 and j ≥ 1, then σj,H2 ,H4 (T ) ≤ Cσj,H1 ,H3 (T ). In particular, if in addition H3 = H4 with equivalent norms, then Ip (H1 , H3 ) = Ip (H2 , H4 ) with equivalent norms. We note that T ∈ Ip (H1 , H2 ) if and only if T ∗ ∈ Ip (H2 , H1 ), in view of (2.1). The next proposition deals with duality properties. Here recall that p ∈ [1, ∞] denotes the conjugate exponent for p ∈ [1, ∞], i.e., 1/p + 1/p = 1. Proposition 2.4. Assume that p ∈ [1, ∞], and that H1 and H2 are separable Hilbert spaces. Then the form (T1 , T2 ) = (T1 , T2 )H1 ,H2 ≡ trH1 (T2∗ ◦ T1 ) on I1 (H1 , H2 ) extends uniquely to a continuous and sesqui-linear form on Ip (H1 , H2 ) × Ip
(H1 , H2 ), and for every T1 ∈ Ip and T2 ∈ Ip
, then (T1 , T2 )H1 ,H2 = (T2 , T1 )H2 ,H1 |(T1 , T2 )H1 ,H2 | ≤ T1 Ip T2 Ip
,

T1 Ip = sup |(T1 , S)H1 ,H2 |,

where the supremum is taken over all S ∈ Ip
such that SIp
≤ 1. If in addition p < ∞, then the dual space for Ip (H1 , H2 ) can be identified with Ip
(H1 , H2 ) through this form. In view of Proposition 2.4 we note that I2 (H1 , H2 ) is a Hilbert space with scalar product (·, ·)H1 ,H2 , and that the corresponding norm agrees with the Hilbert-Schmidt norm  · I2 . The first part of the next proposition follows immediately from Proposition 2.1, since Ip ⊆ I when p < ∞, and I1 (H1 , H2 ) contains all operators of finite rank.

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Proposition 2.5. Assume that p ∈ [1, ∞), and that H1 and H2 are separable Hilbert spaces. Then I1 (H1 , H2 ) is dense in I (H1 , H2 ) and in Ip (H1 , H2 ). It is dense in I∞ (H1 , H2 ) with respect to the weak∗ topology. The next proposition deals with spectral properties. Here recall that ∞ is the set of all bounded sequences (λj )j≥1 such that λj → ∞ as j tends to infinity. Proposition 2.6. Assume that T ∈ I (H1 , H2 ). Then (2.2) holds for some choice ∞ ∞ ∞ of sequences (fj )∞ j=1 ∈ ON(H1 ), (gj )j=1 ∈ ON(H2 ) and λ = (λj )j=1 ∈  , where the sum on the right-hand side in (2.2) convergences with respect to the operator norm. Moreover, if 1 ≤ p < ∞ then T ∈ Ip (H1 , H2 ), if and only if λ ∈ lp , and then T Ip = λlp and the sum on the right-hand side of (2.2) converges with respect to the norm  · Ip . The next result concerns interpolation properties. Proposition 2.7. Assume that p, p1 , p2 ∈ [1, ∞] and 0 ≤ θ ≤ 1 such that 1/p = (1 − θ)/p1 + θ/p2 . Assume also that H1 and H2 are separable Hilbert spaces. Then the (complex) interpolation space (Ip1 , Ip2 )[θ] is equal to Ip with equality in norms.

3. Schatten-von Neumann classes for operators acting on modulation spaces In this section we present some properties of Schatten-von Neumann classes of 2 2 linear operators acting from M(ω (Rm ) to M(ω (Rm ), where ω1 , ω2 ∈ P(R2m ). 1) 2) We are especially concerned with finding appropriate identifications of the dual of st,p (ω1 , ω2 ) when p < ∞ (see the introduction or below). In the first part we prove that there is a canonical way to identify the dual of st,p (ω1 , ω2 ) with st,p
(ω1 , ω2 ). In the second part we use this property to prove that the dual of st,p (ω1 , ω2 ) can also be identified with st,p
(1/ω1 , 1/ω2 ) through a unique extension of the L2 product from S . We start by considering Schatten-von Neumann classes in the context of pseudo-differential calculus. Let t ∈ R, p ∈ [1, ∞] and ω1 , ω2 ∈ P(R2m ) be fixed. From the introduction we recall that st,p (ω1 , ω2 ) consists of all a ∈ S  (R2m ) 2 2 such that at (x, D) ∈ Ip (M(ω , M(ω ). Also let st, (ω1 , ω2 ) be the set of all 1) 2)  2m 2 2 , M(ω ), We let st,p (ω1 , ω2 ) and a ∈ S (R ) such that at (x, D) ∈ I (M(ω 1) 2) st, (ω1 , ω2 ) be equipped by the norms 2 ast,p = ast,p (ω1 ,ω2 ) ≡ at (x, D)Ip (M(ω

1)

2 ,M(ω

2)

)

and  · st,∞ respectively. Since the Weyl quantization is important in our invesw tigations we also use the notations sw p and s instead of st,p and st, respectively when t = 1/2 and p ∈ [1, ∞]. In the case ω1 = ω2 = ω, then we use the notation w st,p (ω) and sw p (ω) instead of st,p (ω1 , ω2 ) and sp (ω1 , ω2 ) respectively.

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Remark 3.1. Let ωj ∈ P(R2m ) for j = 1, 2, t ∈ R and p ∈ [1, ∞]. Then recall that 2 (Rm ) is independent of the choice of windows in S \ 0, and that different M(ω j) windows give rise to different norms. Hence Remark 2.3 shows that st,p (ω1 , ω2 ) is invariant under the choices of windows and that different windows give rise to equivalent norms. In order to avoid ambiguity of the st,p (ω1 , ω2 ) norm, we assume from now on 2 2 that the windows for M(ω and M(ω are fixed. 1) 2) From the fact that each linear and continuous operator from S (Rm ) to S  (Rm ) is equal to at (x, D), for a unique a ∈ S  (R2m ), it follows that the map 2 2 , M(ω ) a → at (x, D) is an isometric homeomorphism from st,p (ω1 , ω2 ) to Ip (M(ω 1) 2) 2 2 when p ∈ [1, ∞], and from st, (ω1 , ω2 ) to I (M(ω1 ) , M(ω2 ) ). Consequently, most of the properties which are listed in Section 2 carry over to the st,p -spaces. Hence Proposition 2.1 shows that st,p (ω1 , ω2 ) is a scale of Banach spaces which increase with the parameter p ∈ [1, ∞]. Moreover, Proposition 2.4 shows that the norm  · st,2 (ω1 ,ω2 ) in st,2 (ω1 , ω2 ) induces a scalar product (·, ·)st,2 = (·, ·)st,2 (ω1 ,ω2 ) and that if p < ∞, then the dual for st,p (ω1 , ω2 ) can be identified with st,p
(ω1 , ω2 ) through a uniquely extension of the st,2 (ω1 , ω2 ) product from st,1 (ω1 , ω2 ). A problem in this context is the somewhat complicated structure of the form (·, ·)st,2 (ω1 ,ω2 ) , compared to, e.g., the scalar product on L2 , which in general fits pseudo-differential calculus well. In the remaining part of the section we therefore focus on a possible replacement of the form (·, ·)st,2 with L2 when discussing duality. From our investigations it turns out that indeed the dual space of st,p (ω1 , ω2 ) may be identified with st,p
(1/ω1 , 1/ω2 ), by a unique and continuous extension of the L2 product from S . We start by making some preparations. Let ω ∈ P(R2m ) be fixed. By Theo2 2 rem 1.2 it follows that the dual for M(ω) (Rm ) can be identified with M(1/ω) (Rm ) 2 is a Hilbert through the scalar product on L2 (Rm ). On the other hand, since M(ω) 2 space, its dual can also be identified with M(ω) through the scalar product on 2 . Consequently, there exist unique homeomorphisms M(ω)  2 ∗ 2 2 2 Tω : M(ω) → M(ω) and Rω : M(ω) → M(1/ω)  2 ∗ 2 such that if  ∈ M(ω) and h = Tω  ∈ M(ω) , then 2 = (f, Rω h)L2 , (f ) = (f, h)M(ω)

and

2 f ∈ M(ω)

2 2 2 ≤  = hM(ω) ≤ CRω hM(1/ω) C −1 Rω hM(1/ω)

(3.1) (3.2)

for some constant C which is independent of h. We observe that (3.1) and (3.2) 2 imply that if (fj )∞ j=1 ∈ ON(M(ω) ) and gk = Rω fk , then (fj , gk )L2 = δj,k

2 and C −1 ≤ gk M(1/ω) ≤ C.

(3.3)

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189

2 ), and we let ON∗ω be the set of all seFor convenience we set ONω = ON(M(ω) ∞ 2 2 quences (gj )j=1 in M(1/ω) such that gj = Rω fj for some (fj )∞ j=1 ∈ ON(M(ω) ). The following characterization of st,p (ω1 , ω2 ) follows immediately from (1.15), Proposition 2.6 and the homoeomorphism property of Rω .

Proposition 3.2. Assume that t ∈ R and ω1 , ω2 ∈ P(R2m ), Also assume that a ∈ ∗ ∞ ∞ st, (ω1 , ω2 ). Then for some (gj )∞ j=1 ∈ ONω1 , (hj )j=1 ∈ ONω2 and λ = (λj )j=1 ∈ ∞  it holds ∞
λj Whj ,gj (3.4) a= j=1

(with convergence with respect to the norm  · st,∞ ). Moreover, if 1 ≤ p < ∞ then a ∈ st,p (ω1 , ω2 ), if and only if λ ∈ lp , and then ast,p (ω1 ,ω2 ) = λlp for some constant C which is independent of a. ∗ ∞ On the other hand, if a is given by (3.4) for some (gj )∞ j=1 ∈ ONω1 , (hj )j=1 ∈ ∞ p ONω2 and λ = (λj )j=1 ∈ l , then a ∈ st,p (ω1 , ω2 ). Next we prove that the dual space of sw 1 (ω1 , ω2 ) may be identified with through a continuous extension of the scalar product on L2 (R2m ). ∗ Assume that  ∈ sw 1 (ω1 , ω2 ) . It follows from (1.15) that an arbitrary rank one 2 2 operator from M(ω1 ) to M(ω2 ) can be written as sw ∞ (1/ω1 , 1/ω2 )

T f = (f, g)L2 h = bw (x, D)f, 2 2 and h ∈ M(ω . Moreover where b = Wh,g , g ∈ M(1/ω 1) 2) 2 2 C −1 gM(1/ω hM(ω ) 1

2)

≤ bsw 1 (ω1 ,ω2 ) 2 2 = T I1 ≤ CgM(1/ω hM(ω , ) ) 1

(3.5)

2

for some constant C which only depends on ω1 and ω2 . Hence 2 2 hM(ω . |(Wh,g )| ≤ CgM(1/ω ) ) 1

2

In particular it follows that the mappings f1 → (Wf20 ,f1 ) and f2 → (Wf2 ,f10 ) are continuous and linear mappings from S (Rm ) to C, for every fixed f10 and f20 in S (Rm ). Hence the Schwartz kernel theorem and its proof (see Theorem 5.2.1 in [28]) show that there is a unique distribution K ∈ S (Rm ⊕ Rm ) such that (Wf2 ,f1 ) = (K, f2 ⊗ f 1 ) for every f1 , f2 ∈ S (R ). By letting a be the Weyl symbol for the operator with kernel K, it follows that m

(Wf2 ,f1 ) = (aw (x, D)f1 , f2 ) = (a, Wf2 ,f1 )L2 .

(3.6)

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J. Toft

Next we prove that a ∈ sw ∞ (1/ω1 , 1/ω2 ). By the positive linearity in the definition of the trace norm we have  =

sup |(b)| = sup |(b)|

bsw ≤1 1

where the latter supremum is taken over all b ∈ sw ≤1 1 (ω1 , ω2 ) such that bsw 1 w and b (x, D) is an operator of rank one. Hence (3.5) and (3.6) give 2 2 f2 M(ω , |(aw (x, D)f1 , f2 )| ≤ Cf1 M(1/ω ) ) 1

2

(3.7)

2 and it follows from Proposition 1.2 (4) that aw (x, D) is continuous from M(1/ω 1) 2 ∞ to M(1/ω2 ) , i.e., a ∈ sw (1/ω1 , 1/ω2 ).  Assume next that b ∈ sw λj Whj ,gj for some 1 (ω1 , ω2 ) is arbitrary. Then b = (gj ) ∈ ON∗ω1 , (hj ) ∈ ONω2 , and (λj ) ∈ l1 . We let (a, b)L2 be defined as
(a, b)L2 = λj (a, Whj ,gj )L2 .

This definition makes sense, since (3.2), (3.5) and (3.6) give

|λj ||(a, Whj ,gj )| ≤ C |λj | = Cbsw 1 for some constant C. Hence |(a, b)L2 | ≤ Cbsw . By combining the latter estimate 1 with (3.5) it follows that (b) = (a, b)L2 , and that b → (a, b)L2 is continuous from w sw 1 (ω1 , ω2 ) to C. Consequently, the dual space of s1 (ω1 , ω2 ) can be identified with w s∞ (1/ω1 , 1/ω2 ) through the form (·, ·)L2 . We also note that the extension of (·, ·)L2 w to a duality between sw 1 (ω1 , ω2 ) and s∞ (1/ω1 , 1/ω2 ) is unique since S is dense in 2 2 2 M(ωj ) for j = 1, 2, and that finite rank operators are dense in I1 (M(ω , M(ω ). 1) 2) In particular we have proved the following result in the case p = 1 and t = 1/2. Theorem 3.3. Assume that t ∈ R, p ∈ [1, ∞) and that ω1 , ω2 ∈ P(R2m ). Then the scalar product on L2 (R2m ) extends uniquely to a duality between st,p (ω1 , ω2 ) and st,p
(1/ω1 , 1/ω2), and the dual space for st,p (ω1 , ω2 ) can be identified with st,p
(1/ω1 , 1/ω2 ) through this form. Moreover, if  ∈ st,p (ω1 , ω2 )∗ and a ∈ st,p
(1/ω1 , 1/ω2 ) such that (b) = (a, b)L2 when b ∈ st,p (ω1 , ω2 ), then C −1 ast,p
(1/ω1 ,1/ω2 ) ≤  ≤ Cast,p
(1/ω1 ,1/ω2 ) for some constant C which only depends on ω1 and ω2 . Proof. By (1.10) and the fact that ei Dx ,Dξ  is unitary on L2 , it suffices to prove the result in the case t = 1/2. Since we have already proved the result in the case p = 1, we may assume that p > 1. First we prove that (b) = (a, b)L2 defines a w continuous linear form on sw p (ω1 , ω2 ) for every element sp
(1/ω1 , 1/ω2 ). Since I1 is dense in Ip and in Ip
, Proposition 3.2 shows that it suffices to prove |(a, b)L2 | ≤ C(λj )lp
(µj )lp , for some constant C, where
a= λj Whj ,gj

and b =




µj Wh0j ,gj0

(3.8)

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191

are finite sums. Here gj = R1/ω1 fj and gj0 = Rω1 fj0 for some (fj ) ∈ ON1/ω1 , (fj0 ) ∈ ONω1 , (hj ) ∈ ON1/ω2 and (h0j ) ∈ ONω2 . It is also no restriction to assume that (λj )lp
= (µj )lp = 1, By straightforward computations we get 

  λj µk (Whj ,gj , Wh0k ,gk0 )L2  ≤ |λj µk αj,k βj,k |, |(a, b)L2 | =  j,k

j,k

where αj,k = (gj , gk0 )L2 , and βj,k = (hj , h0k )L2 . Hence the inequality between arithmetic and geometric mean-values gives   
 1 1 p
p 1 2 2 |(a, b)L2 | ≤ . |µ |α |λ | + | | + |β | j k j,k j,k p p 2

(3.9)

j,k

By letting κj = R1/ω2 hj and κ0j = Rω2 h0j it follows from (3.3) that 2 αj,k = (gj , gk0 )L2 = (fj , gk0 )M(1/ω

1)

2 βj,k = (hj , h0k )L2 = (hj , κ0k )M(1/ω

2)

2 = (gj , fk0 )M(ω

1)

2 = (κj , h0k )M(ω

2)

= (fj , fk0 )L2 ,

and

= (κj , κ0k )L2 .

Hence the orthogonality assumptions imply

2 2 |αj,k |2 ≤ gk0 M(1/ω , |αj,k |2 ≤ gj M(ω , ) ) 1

j

1

k

and similarly when gj , gk0 and αj,k are replaced by hj , h0k and βj,k respectively. By combining these estimates with (3.9), it follows that (3.8) holds with 2 2 2 2 , gk0 M(1/ω , κj M(ω , κ0k M(1/ω ) < ∞. C = sup(gj M(ω ) ) ) )

j,k

1

1

2

2

This proves the assertion. ∗ Assume next that  ∈ sw p (ω1 , ω2 ) . By Proposition 2.4 there is a unique 2 2 2 2 , M(ω ) such that if H1 = M(ω and H2 = M(ω , then operator T ∈ Ip
(M(ω 1) 2) 1) 2) (b) = (bw (x, D), T )H1 ,H2 holds for any b ∈ sw p (ω1 , ω2 ). Hence Proposition 2.6 gives

2 2 hj , and bw (x, D) = µj (f, fj0 )M(ω h0 Tf = λj (f, fj )M(ω ) ) j 1

1




for some (fj ), (fj0 ) ∈ ONω1 and (hj ), (h0j ) ∈ ONω2 , (λj ) ∈ lp and (µj ) ∈ lp . This implies that
2 2 λj µk (h0k , hj )M(ω (fj , fk0 )M(ω . (b) = ) ) j,k

2



1

λj Wκj ,fj . Then it follows by straightNow let gj0 = Rω1 fj0 , κj = Rω2 h j and a = forward computations that b = µj Wh0j ,gj0 , and that

λj µk (Wκj ,fj , Wh0k ,gk0 )L2 = λj µk (κj , h0k )L2 (gk0 , fj )L2 = (b). (a, b)L2 = j,k

j,k

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J. Toft

From the first part of the proof it also follows that aw (x, D)

2 Ip
(M(1/ω

1)

2 ,M(1/ω

2)

)

≤ C(λj )lp
< ∞,


for some constant C. The proof is complete.

Remark 3.4. Theorem 3.3 appeared after fruitful discussions with Paolo Boggiatto.

4. Continuity and Schatten-von Neumann properties for pseudo-differential operators In this section, we discuss continuity and Schatten-von Neumann properties for pseudo-differential operators, when the operator symbols belong to appropriate classes of modulation spaces. In particular we generalize some of the continuity results in Section 5 in [49]. An important ingredient in these investigations concern continuity properties for the Wigner distributions in context of modulation spaces, as well as the central role for Wigner distributions in the Weyl calculus of pseudodifferential operators. First assume that Bj for j = 1, 2, 3 are Frech´et spaces such that S (R2m ) → B1 → S  (R2m ),

S (Rm ) → B2 , B3 → S  (Rm ),

and that (a, f, g) → (a, Wg,f ) is well defined and sequently continuous from B1 × B2 × B3 to C. Then (1.15) is taken as the definition of aw (x, D)f as an element in B3 when f ∈ B2 , and it follows that aw (x, D) is a continuous operator from B2 to B3 . Next we discuss continuity properties for pseudo-differential operator, and prove in a moment that if t ∈ R, 1/p1 − 1/p2 = 1/q1 − 1/q2 = 1 − 1/p − 1/q,

q ≤ p2 , q2 ≤ p,

(4.1)

p,q M(ω) ,

then at (x, D) is ω1 , ω2 and ω are appropriate weight functions and a ∈ p1 ,q1 p2 ,q2 continuous from M(ω to M . As a first step we consider continuity properties (ω2 ) 1) for Wigner distributions in background of modulation space theory. Proposition 4.1. Assume that pj , qj , p, q ∈ [1, ∞] are such that p ≤ pj , qj ≤ q, for j = 1, 2, and that 1/p1 + 1/p2 = 1/q1 + 1/q2 = 1/p + 1/q. holds. Also assume that ω1 , ω2 ∈ P(R

2m

), and that ω ∈ P(R

(4.2) 2m

⊕R

ω(x, ξ, y, η) ≤ Cω1 (x − y/2, ξ + η/2)ω2 (x + y/2, ξ − η/2).

2m

) satisfy (4.3)

Then the map (f1 , f2 ) → Wf1 ,f2 from S  (Rm ) × S  (Rm ) to S  (R2m ) restricts p1 ,q1 p2 ,q2 :p,q (R2m ), and for to a continuous mapping from M(ω (Rm ) × M(ω (Rm ) to M (ω) 1) 2) some constant C, Wf1 ,f2 M : p,q ≤ Cf1 M p1 ,q1 f2 M p2 ,q2 (ω)

when f1 , f2 ∈ S  (Rm ).

(ω1 )

(ω2 )

(4.4)

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193

The result is essentially a restatement of Theorem 5.1 in [49], where a short proof is presented. A detailed proof in the non-weighted case is available in [48]. In order to being more self-contained, we present here a more detailed proof in the general case. Proof. We only prove the result in the case p, q < ∞. The straightforward modifications to the cases when p = ∞ or q = ∞ are left for the reader. Let χ1 , χ2 ∈ S (Rm ) \ 0, and let Φ = Wχ2 ,χ1 . Then it follows from Fourier’s inversion formula that if X = (x, ξ) ∈ R2m and Y = (y, η) ∈ R2m , then :(Wf ,f τX Φ)(Y )| = F1 (x − y/2, ξ + η/2)F2 (x + y/2, ξ − η/2) |F 1 2 where F1 (X) = |F (f1 τx χ1 )(ξ)| By applying the

and F2 (X) = |F (f2 τx χ2 )(ξ)|.

Lp,q (ω) -norm

Wf1 ,f2 M : p,q (ω)

on the latter equality, and using (4.3) we get  1/q  1/p ≤ C G1 ∗ G2 Lr =C H(η) dη

for some constant C, where Gj = (Fj ωj )p , r = q/p ≥ 1 and    r H(η) = G1 (y − x, η − ξ)G2 (x, ξ) dxdξ dy. By Minkowski’s inequality we get r 1/r r    dξ . H(η) ≤ G1 (y − x, η − ξ)G2 (x, ξ) dx dy Assume now that rj , sj ∈ [1, ∞] for j = 1, 2 are chosen such that 1/r1 + 1/r2 = 1/s1 + 1/s2 = 1 + 1/r. Then Young’s inequality gives r  H(η) ≤ G1 ( · , η − ξ)Lr1 G2 ( · , ξ)Lr2 dξ Hence another application of Young’s inequality gives  1/q 1/p  Wf1 ,f2 M ≤ C ≤ C G1 Lr1 ,s1 G2 Lr2 ,s2 H(η) dη p,q : (ω)

By letting pj = prj and qj = qsj , and using the fact that Gj (x, ξ) = |F (fj τx χj )(ξ)ωj (x, ξ)|p , the last inequality gives (4.4). The proof is complete.




We shall next apply Proposition 4.1 to pseudo-differential calculus. The following result extends Theorem 14.5.2 in [25], Theorem 7.1 in [27] and Theorem 4.3 in [49].

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J. Toft

Theorem 4.2. Assume that t ∈ R and p, q, pj , qj ∈ [1, ∞] for j = 1, 2, satisfy (4.1). Also assume that ω ∈ P(R2m ⊕ R2m ) and ω1 , ω2 ∈ P(R2m ) satisfy ω2 (x − ty, ξ + (1 − t)η) ≤ Cω(x, ξ, y, η) ω1 (x + (1 − t)y, ξ − tη)

(4.5)

:p,q (R2m ), then at (x, D) from S (Rm ) to S  (Rm ) for some constant C. If a ∈ M (ω) p1 ,q1 p2 ,q2 extends uniquely to a continuous mapping from M(ω (Rm ) to M(ω (Rm ). 1) 2) :p,q , then at (x, D) : M p1 ,q1 → M p2 ,q2 is Moreover, if in addition a ∈ M (ω1 ) (ω2 ) (ω) compact. Proof. It suffices to prove the theorem in the case t = 1/2 in view of Proposition 1.7. The conditions on pj and qj implies that p ≤ p1 , q1 , p2 , q2 ≤ q  ,

1/p1 + 1/p2 = 1/q1 + 1/q2 = 1/p + 1/q  .

Hence Proposition 4.1, and (4.5) show that for some constant C > 0, p ,q Wg,f M : p
,q
≤ Cf M 1 1 g (ω1 )

(1/ω)

p
,q


2 2 M(1/ω ) 2

p
,q


p1 ,q1 2 2 when f ∈ M(ω (Rm ) and g ∈ M(1/ω (Rm ). 1) 2) :p,q (R2m ). Then for some constants C > 0 and C  > 0 Now assume that a ∈ M (ω) we get |(a, Wg,f )| ≤ CaM : p,q Wg,f M : p
,q
(ω) (1/ω) (4.6)  p ,q ≤ C aM : p,q f M 1 1 g p
2 ,q2
. (ω)

(ω1 )

M(1/ω

2)

The continuity assertion now follows by combining (4.6) and Proposition 1.2 (3). :p,q , then aw (x, D) : M p1 ,q1 → M p2 ,q2 It remains to prove that if a ∈ M (ω1 ) (ω2 ) (ω) is compact. By straightforward computations it follows that aw (x, D) is compact :p,q , it follows now from the first part of the proof when a ∈ S . For general a ∈ M (ω) that aw (x, D) may be approximated in operator norm by compact operators. Hence
aw (x, D) is compact and the proof is complete. In Corollary 4.3 below, we prove that Theorem 4.2 may be used to obtain continuity results for pseudo-differential operators, where the corresponding symbols belong to S(ω) (R2m ) for some ω ∈ P(R2m ), which consists of all a ∈ C ∞ (R2m ) such that ∂ α a/ω is bounded for every multi-index α. Corollary 4.3. Assume that t ∈ R, p, q ∈ [1, ∞], ω1 , ω2 ∈ P(R2m ), ω0 (x, ξ) = ω2 (x, ξ)/ω1 (x, ξ)

and

ω(x, ξ, y, η) = ω0 (x, ξ)σN (y, η).

where N ≥ 0 is an integer such that ω1 and ω2 are σN/2 -moderate.

Pseudo-differential Operators on Modulation Spaces

195

Then the following are true: :∞,1 (R2m ), then at (x, D) is continuous from M p,q (Rm ) (1) if a ∈ M (ω) (ω1 ) p,q m to M(ω (R ); 2) p,q (Rm ) (2) if a ∈ S(ω0 ) (R2m ), then at (x, D) is continuous from M(ω 1) p,q to M(ω (Rm ). 2)

Proof. The assertion (1) is an immediate consequence of Theorem 4.2, and the ∞,1 assertion (2) follows from (1) and the fact that S(ω0 ) ⊆ M(ω) (see [49, 50]). The proof is complete.
We note that (2) in Corollary 4.3 is deeply related to Theorem 2.1 in [42]. The difference between these two results is that less restrictions are imposed on ω in Corollary 4.3, while less regularity assumptions are imposed in Theorem 2.1 in [42]. In a forthcoming paper we present an alternative proof of (2) in Corollary 4.3, which also holds when at (x, D) acts Banach function spaces, a family of functions and distribution spaces which contains the modulation spaces. (See, e.g., [18].) Remark 4.4. If p = p1 = p2 = ∞ and q = q1 = q2 = 1, or p = q1 = q2 = ∞ and q = p1 = p2 = 1 in Theorem 4.2, then S is not dense in any of the involving spaces. However, in spite of these facts, there are no ambiguity in order to define at (x, D)f p1 ,q1 when f ∈ M(ω . In fact, by (1.10) and Proposition 1.7 it is no restriction to 1) assume that t = 1/2. Then (4.6) shows that aw (x, D)f makes sense as an element p1 ,q1 in S  when f ∈ M(ω . By using (4.6) again in combination with Proposition 1) p2 ,q2 1.2 (3), it follows that indeed then give that aw (x, D)f ∈ M(ω , and the assertion 2) follows. Corollary 4.5. Assume that p, q, pj , qj ∈ [1, ∞] for j = 1, 2, satisfy q ≤ p2 , q2 ≤ p and (4.2) . Assume also that t ∈ R \ {0, 1}, sj , tj ∈ R for j = 0, 1, 2 such that :p,q (R2m ), for some ω ∈ P(R2m ⊕ R2m ). Then s1 , s2 , t1 , t2 ≥ 0, and that a ∈ M (ω) the following are true: (1) if |s0 | ≤ s1 + s2 , and ω(x, ξ, y, η) = σs1 (x, ξ)σs2 (y, η), then at (x, D) is continuous from Msp01 ,q1 (Rm ) to Msp02 ,q2 (Rm ); (2) if |s0 | ≤ s1 + s2 , |t0 | ≤ t1 + t2 , and ω(x, ξ, y, η) = σs1 ,t1 (x, ξ)σs2 ,t2 (y, η), then at (x, D) is continuous from Msp01,t,q01 (Rm ) to Msp02,t,q02 (Rm ). Proof. The assertion (1) follows by letting ω1 = ω2 = σs0 in Theorem 4.2, and using the fact that (x − ty, ξ + (1 − t)η/2)−1 (x + (1 − t)y, ξ − tη/2) ≤ Ct min(x, y) · min(ξ, η)

196

J. Toft

for some constant C. The assertion (2) is proved in a similar way, and is left for the reader. The proof is complete.
In the case p = q = ∞ in Theorem 4.2, the converse is also true, i.e., we have the following result. Theorem 4.6. Assume that t ∈ R, a ∈ S  (R2m ), ω ∈ P(R2m ⊕ R2m ), and ω1 , ω2 ∈ P(R2m ) such that (4.5) holds. Then the operator at (x, D) from S (Rm ) 1 ∞ (Rm ) to M(ω (Rm ), if to S  (Rm ) extends to a continuous mapping from M(ω 1) 2) :∞ (R2m ). and only if a ∈ M (ω)

For the proof we need the following two propositions of independent interest, where the first proposition is a slight generalization of Feichtinger-Gr¨ ochenig’s kernel theorem, named as Schwartz-Gr¨ ochenig’s kernel theorem in Theorem 4.1 in [49]. (See also Theorem 14.4.1 in [25].) Proposition 4.7. Assume that m = m1 + m2 , ωj ∈ P(R2mj ) for j = 1, 2 and ω ∈ P(Rm ⊕ Rm ) such that ω(x, y, ξ, η) = ω2 (x, ξ)/ω1 (y, −η).

(4.7)

Also assume that T is a linear and continuous map from S (Rm1 ) to S  (Rm2 ). 1 ∞ Then T extends to a continuous mapping from M(ω (Rm1 ) to M(ω (Rm2 ), if and 1) 2) ∞ m only if it exists an element K ∈ M(ω) (R ) such that (T f )(x) = K(x, ·), f .

(4.8)

Here the right-hand side in (4.8) should be interpreted as the distribution u, given by the formula u, g = K, g ⊗ f , or alternatively, by the formula (u, g) = (K, g ⊗ f ), when f ∈ S (Rm1 ) and g ∈ S (Rm2 ). 1 ∞ Proof. Assume that T extends to continuous map from M(ω to M(ω . It follows 1) 2) from the kernel theorem of Schwartz that (4.8) holds for some K ∈ S  (Rm ). We ∞ . shall prove that K belongs to M(ω) From the assumptions and Proposition 1.2 (3) it follows that 1 1 |(K, g ⊗ f )L2 | ≤ Cf M(ω gM(1/ω ) 1

(4.9)

2)

holds for some constant C which is independent of f ∈ S (Rm1 ) and g ∈ S (Rm2 ). By letting χ = g ⊗ f be fixed, and replacing f and g with fy,η = e−i ·,η f (· − y) and gx,ξ = ei ·,ξ f (· − x), it follows that (4.9) takes the form 1 1 |F (Kτ(x,y) χ)(ξ, η)| ≤ Cfy,η M(ω gx,ξ M(1/ω . ) ) 1

2

(4.9)

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197

We have to analyze the right-hand side of (4.9) . If v is chosen such that ω1 is v-moderate, and χ1 ∈ S (Rm1 ) \ 0 is a fixed function, then we obtain  1 fy,η M(ω = |F (f τz−y χ1 )(ζ + η)ω1 (z, ζ)| dzdζ 1)  = |F (f τz χ1 )(ζ)ω1 (z + y, ζ − η)| dzdζ 1 ≤ Cω1 (y, −η)f M(v) = C  ω1 (y, −η).

In the same way we get 1 gx,ξ M(1/ω

2)

≤ Cω2 (x, ξ)−1 .

If these estimates are inserted into (4.9) , we obtain |F (Kτ(x,y) χ)(ξ, η)ω(x, y, ξ, η)| ≤ C, for some constant C which is independent of x, y, ξ and η. By taking the supremum ∞ ∞ < ∞. Hence K ∈ M(ω) , and the of the left-hand side it follows that KM(ω) necessity follows. The sufficiency follows by straightforward computations. The details are left for the reader. (See also the proof of Theorem 4.1 in [49].) The proof is complete.
Proposition 4.8. Assume that a ∈ S  (R2m ), and that K ∈ S  (R2m ) is the distribution kernel for the Weyl operator aw (x, D). Also assume that p ∈ [1, ∞], and that ω, ω0 ∈ P(R2m ⊕ R2m ) are such that ω(x, ξ, y, η) = ω0 (x − y/2, x + y/2, ξ + η/2, −ξ + η/2). :p (R2m ) if and only if K ∈ M p (R2m ). Moreover, if χ ∈ S (R2m ) Then a ∈ M (ω) (ω0 ) and  ψ(x, y) = χ((x + y)/2, ξ)ei y−x,ξ dξ, then aM : p,χ = KM p,ψ . (ω)

(ω0 )

Proof. By Fourier’s inversion formula, it follows by straightforward computations that :(a τ(x,ξ) χ)(y, η)|. |F (K τ(x−y/2,x+y/2) ψ)(ξ + η/2, −ξ + η/2)| = |F The result now follows by applying the Lp(ω) -norm on these expressions. The proof is complete.
Proof of Theorem 4.6. In the case t = 1/2, the result follows immediately by combining Theorem 4.2, Proposition 4.7 and Proposition 4.8. For general t the result is now a consequence of (1.10) and Proposition 1.7. The proof is complete.
Remark 4.9. Proposition 4.7 was proved in the case that ω1 and 1/ω2 are moderate functions in [25]. It was proved in the non-weighted case (i.e., ω1 = ω2 = 1) already in [12].

198

J. Toft

1 and Remark 4.10. Let ω1 , ω2 and ω be the same as in Proposition 4.7, f0 ∈ M(ω 2) ∞  1 ∞ let g0 ∈ (M(ω1 ) ) \ M(1/ω1 ) , where the dual space of M(ω1 ) is identified through an 1 is a strict extension of the sesqui-linear form on S . Such g0 exists, since M(1/ω 1) ∞  subset of (M(ω1 ) ) (see, e.g., [25]). Then TK f (x) ≡ (f, g0 )f0 (x) has the distribution kernel K = f0 ⊗ g0 , and defines an element in L(ω1 , ω2 ), the set of all linear and continuous operators from ∞ 1 1 1 M(ω to M(ω . Since K ∈ / M(ω) , it follows that the map K → TK from M(ω) to 1) 2) L(ω1 , ω2 ) is injective but not surjective. (See also [21].) Consequently, the converse of Theorem 4.2 when p = q = 1 (the “dual case” of Theorem 4.6) does not hold. On the other hand, let the operator norm define the topology for L(ω1 , ω2 ), and let L0 (ω1 , ω2 ) be the completion of the set of all operators with kernels in 1 S , under this norm. Then Theorem 7.4.1 in [21] and its proof show that M(ω) is homeomorphic with L0 (ω1 , ω2 ).

Next we discuss embedding properties between Schatten-von Neumann classes in pseudo-differential calculus and modulation spaces. As a first step we characterize Hilbert-Schmidt operators acting on modulation spaces of Hilbert type. Proposition 4.11. Assume that ω1 , ω2 ∈ P(R2m ) and ω ∈ P(R2m ⊕ R2m ) are such that (4.7) holds, and that T is a linear and continuous operator from S (Rm ) 2 2 to S  (Rm ) with distribution kernel K ∈ S  (R2m ). Then T ∈ I2 (M(ω , M(ω ), 1) 2) 2 2m if and only if K ∈ M(ω) (R ), and then 2 . T I2 = KM(ω)

(4.10)

2 (Rm ) Proof. Let (fj ) ∈ ONω1 and (hk ) ∈ ONω2 be orthonormal basis for M(ω 1) 2 and M(ω (Rm ) respectively. Then 2)

2 2 2 T 2I2 = |(T fj , hk )M(ω )|2 = |(K, hk ⊗ fj )M(ω (4.11) ⊗L2 | ) ) 2

j,k

2

j,k

2 Next we consider the operator Tω 0 = IM(ω

which acts from (4.11) gives

2 2 M(ω ⊗ M(1/ω 2) 0)

T 2I2 =




to

⊗ R1/ω0 , where ω0 (x, ξ) = ω1 (x, −ξ), 2) 2 2 M(ω2 ) ⊗ M(ω (Hilbert tensor products). Then 0)

2 |(Tω 0 K, hk ⊗ fj )M(ω

j,k

= Tω 0 K2M 2

(ω2 )

2 ⊗M(ω

0)

2)

2 ⊗M(ω

= K2M 2

Hence (4.10) holds, and the proof is complete.

1)

|2

(ω2 )

2 ⊗M(1/ω

0)

= K2M 2

(ω)




The following result is now an immediate consequence of Proposition 4.11 and Proposition 4.8 for p = 2. Proposition 4.12. Assume that a ∈ S  (R2m ), ω1 , ω2 ∈ P(R2m ) and that ω ∈ P(R2m ⊕ R2m ) are such that equality is attained in (4.5) for t = 1/2 and some

Pseudo-differential Operators on Modulation Spaces

199

2 2 :2 (R2m ). , M(ω ), if and only if a ∈ M constant C. Then aw (x, D) ∈ I2 (M(ω (ω) 1) 2) Moreover, for some constant C > 0 it holds w C −1 aM : 2 ≤ a (x, D)I2 ≤ CaM :2 . (ω)

(ω)

For general Schatten-von Neumann classes, we have the following generalization of Proposition 4.12. Theorem 4.13. Assume that t ∈ R and p, q, pj , qj ∈ [1, ∞] for j = 1, 2, satisfy p1 ≤ p ≤ p2 ,

q1 ≤ min(p, p )

q2 ≥ max(p, p ).

and

(4.12)

Also assume that ω ∈ P(R2m ⊕R2m ) and ω1 , ω2 ∈ P(R2m ) are such that equality is attained in (4.5), for some constant C. Then :p1 ,q1 (R2m ) → st,p (ω1 , ω2 ) → M :p2 ,q2 (R2m ) M (ω) (ω)

(4.13)

For the proof we need the following lemma. Lemma 4.14. Assume that (xj1 )j1 ∈I1 ,

(ξj2 )j2 ∈I2 ,

(yk1 )k1 ∈I1

and

(ηk2 )k2 ∈I2

2

are lattices in Rm . Also assume that ϕ(x) = e−|x| /2 where x ∈ Rm , ω1 , ω2 ∈ P(R2m ), (fl ) ∈ ONω1 , (hl ) ∈ ONω2 , and set κl = Rω2 hl and θ1 (j, k, l) = F (fl τxj1 +yk1 /2 ϕ)(ξj2 − ηk2 /2)ω1 (xj1 + yk1 /2, ξj2 − ηk2 /2), θ2 (j, k, l) = F (κl τxj1 −yk1 /2 ϕ)(ξj2 + ηk2 /2)/ω2 (xj1 − yk1 /2, ξj2 + ηk2 /2), where j = (j1 , j2 ) ∈ I1 × I2 ≡ I, k = (k1 , k2 ) ∈ I. Then for some constant C and integer N ≥ 0 it holds
|θi (j, k, l)|2 ≤ Cϕ2M 2 < ∞, i = 1, 2, (4.14) N

l


k∈I


k∈I

|θ1 (j, k, l)|2 ≤ Cfl 2M 2

≤ C < ∞,

(4.15)

|θ2 (j, k, l)|2 ≤ Chl 2M 2

≤ C < ∞.

(4.16)

(ω1 )

(ω2 )

Proof. Let N ≥ 0 be chosen such that ω1 and ω2 are σN -moderate. Then (4.14) in the case i = 1 follows if we prove that
|F (fl τz ϕ)(ζ)|2 ≤ Cϕ2M 2 /ω1 (z, ζ)2 l

N

for some constant C which is independent of (z, ζ). Since 2 F (fl τz ϕ)(ζ) = (fl , ei · ,ζ τz ϕ)L2 = (fl , R1/ω1 (ei · ,ζ τz ϕ))M(ω

1)

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J. Toft

and (fl ) ∈ ONω1 , we obtain for some χ ∈ S \ 0 that

2 |F (fl τz ϕ)(ζ)|2 = |(fl , R1/ω1 (ei · ,ζ τz ϕ))M(ω |2 ) l

1

l

≤ R1/ω1 (ei · ,ζ τz ϕ)2M 2 ≤ Cei · ,ζ τz ϕ2M 2 (ω1 ) (1/ω1 )  i · ,ζ 2 =C |F (e τz ϕτx χ)(ξ)/ω1 (x, ξ)| dxdξ  =C |F (ϕτx−z χ)(ξ − ζ)/ω1 (x, ξ)|2 dxdξ  =C |F (ϕτx χ)(ξ)/ω1 (x + z, ξ + ζ)|2 dxdξ  ≤ C |F (ϕτx χ)(ξ)σN (x, ξ)|2 dxdξ/ω1 (z, ζ)2 = C  ϕ2M 2 /ω1 (z, ζ)2 . N

This proves the assertion. The case i = 2 in (4.14) follows from similar arguments together with the fact that 2 F (κl τz ϕ)(ζ) = (κl , ei · ,ζ τz ϕ)L2 = (hl , ei · ,ζ τz ϕ)M(ω . ) 2

Next we prove (4.15). For some lattices (zk1 )k1 ∈I1 and (ζk2 )k2 ∈I2 we have

|θ1 (j, k, l)|2 ≤ |F (fl τzk1 ϕ)(ζk2 )ω1 (zk1 , ζk2 )|2 ≤ Cfl 2M 2 k

(ω1 )

k1 ,k2

for some constant C, where the last inequality follows from Proposition 1.4. This proves (4.15). By replacing the lattices (zk1 ) and (ζk2 ) with other ones, if necessary, we obtain

|θ2 (j, k, l)|2 ≤ |F (κl τzk1 ϕ)(ζk2 )ω2 (zk1 , ζk2 )|2 k

k1 ,k2

≤ Cκl 2M 2

(1/ω2 )



≤ C  hl 2M 2

(ω2 )

for some constants C and C , and (4.16) follows. The proof is complete.




Proof of Theorem 4.13. We use the same notations as in Lemma 4.14. In view of Proposition 1.7 it is no restriction to assume that t = 1/2, and that equalities are attained in (4.12). Then the result is an immediate consequence of Proposition 4.12 in the case p = q = 2. Next we consider the case q = 1. Let Xj = (xj1 , ξj2 ) and Yk = (yk1 , ηk2 ). Then it follows that (Xj )j∈I and (Yk )k∈I are lattices in R2m . :p,1 . By Proposition 1.4 it follows that if the lattices here above Assume that a ∈ M (ω) are chosen sufficiently dense, then
cj,k ei X,Yk  (τXj Wϕ,ϕ )(X), a(X) = j,k∈I

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201

for some sequence (cj,k )j,k∈I which satisfies C −1



j∈I

λpj,k

1/p

≤ CaM : p,1 ≤ C (ω)

k∈I



j∈I

λpj,k

1/p ,

k∈I

(4.17)

λj,k = |cj,k ω(xj1 , ξj2 yk1 , ηk2 )|. Here X = (x, ξ) ∈ R2m and X, Yk  = x, ηk2  + yk1 , ξ. Hence if (fl ) ∈ ONω1 and (κl ) ∈ ON∗ω2 , then |(aw (x, D)fl , κl )| ≤




|cj,k (ei ·,Yk  τXj Wϕ,ϕ , Wκl ,fl )|.

j,k

By straightforward computations it follows that ei X,Yk  (τXj Wϕ,ϕ )(X) = eiΦ(Yk ,Xj ) Wϕ1j,k ,ϕ2j,k , where Φ is a real-valued quadratic form on R2m ⊕ R2m , and ϕ1j,k (x) = ei x,ξj2 +ηk2 /2 ϕ(x − xj1 + yk1 /2), ϕ2j,k (x) = ei x,ξj2 −ηk2 /2 ϕ(x − xj1 − yk1 /2). This gives |(ei ·,Yk  τXj Wϕ,ϕ , Wκl ,fl )| = |(Wϕ1j,k ,ϕ2j,k , Wκl ,fl )| = |(ϕ1j,k , κl )(ϕ2j,k , fl )| ω2 (xj1 − yk1 /2, ξj2 + ηj /2) ω1 (xj1 + yk1 /2, ξj2 − ηk2 /2) ≤ C|θ1 (j, k, l)θ2 (j, k, l)|ω(xj1 , ξj2 , yk1 , ηk2 ),

= |θ1 (j, k, l)θ2 (j, k, l)|

for some constant C. Hence |cj,k (ei ·,Yk  τXj Wϕ,ϕ , Wκl ,fl )| ≤ Cλj,k |θ1 (j, k, l)θ2 (j, k, l)|. From these inequalities we obtain 


|(aw (x, D)fl , κl )|p

1/p

≤C

l




λj,k |θ1 (j, k, l)θ2 (j, k, l)| ≤ C(J1 + J2 )/2,

j,k,l

where Ji =



l

j,k

λj,k |(θi (j, k, l)|2

p 1/p ,

j = 1, 2.

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J. Toft

older’s inequalities we get We have to estimate J1 and J2 . By Minkowski’s and H¨ p 1/p


λj,k |θ1 (j, k, l)|2 J1 ≤ C j

=C

l

j

≤C

k




l

k




j




(λj,k |θ1 (j, k, l)|2/p )|θ1 (j, k, l)|2/p

l

λpj,k |θ1 (j, k, l)|2




k

p 1/p

|θ1 (j, k, l)|2

p/p
1/p .

k

Now (4.14)–(4.17) give  
p/p
1/p

p J1 ≤ C1 λj,k |θ1 (j, k, l)|2 sup |θ1 (j, k, l)|2 

j

l

k,l








2/p

≤ C2 sup fl M 2

(ω1 )

l

2/p

≤ C3 ϕM 2

N



j∈I

k∈I

j

λpj,k

k

1/p

λpj,k

k




|θ1 (j, k, l)|2

1/p

l

≤ C4 aM : p,1 , (ω)

for some constants C1 , . . . , C4 . In the same way we get J2 ≤ CaM : p,1 for some (ω)

constant C. Hence it follows from these estimates that 1/p 
|(aw (x, D)fl , κl )|p ≤ CaM p,1 . (ω)

l

For some constant which is independent of the choice of sequences (fl ) and (κl ). The result now follows by taking the supremum of the left-hand side with respect to all sequences (fl ) and (κl ). The first embedding in (4.13) now follows by interpolation of the case q = 1 and the case p = q = 2, using Proposition 1.6 and Proposition 2.7. The second embedding in (4.13) now follows from the first embedding and duality, using Theorem 3.3. The proof is complete.
Next we present some consequences of Theorem 4.13 when the weight functions ω1 and ω2 are the same and satisfy certain properties which are common in the applications. It is for example common that the moderate function v ∈ P(R2m ) satisfies the symmetry condition v(x, ξ) = v(−x, ξ) = v(x, −ξ) = v(−x, −ξ), for every (x, ξ) ∈ R

2m

(4.18)

.

Corollary 4.15. Assume that t ∈ R, p, q, pj , qj ∈ [1, ∞] for j = 1, 2 are the same as in Theorem 4.13, and that ω0 ∈ P(R2m ) is v-moderate for some v ∈ P(R2m ) which satisfies (4.18). Also let v0 (x, ξ, y, η) = v(y, η). Then :p1 ,q1 ⊆ st,p (ω0 ) ⊆ M :p2 ,q2 . M (v0 ) (1/v0 )

Pseudo-differential Operators on Modulation Spaces

203

Proof. It follows from the assumptions that ω0 (x − ty, ξ + (1 − t)η) ≤ Cv(y, η). C −1 v(y, η)−1 ≤ ω0 (x + (1 − t)y, ξ − tη) The result is therefore a consequence of Proposition 1.2 (2) and Theorem 4.13. The proof is complete.
Example 4.16. Assume that t ∈ R and that p, q, pj , qj ∈ [1, ∞] for j = 1, 2 are the same as in Theorem 4.13. Then it follows from the Corollary 4.15 that the following are true: (1) if s ∈ R, then :p2 ,q2 ; :p1 ,q1 ⊆ st,p (σs ) ⊆ M M |s|,0 −|s|,0 (2) if s1 , s2 ∈ R and v(x, ξ, y, η) = σ|s1 |,|s2 | (y, η), then :p1 ,q1 ⊆ st,p (σs1 ,s2 ) ⊆ M :p2 ,q2 . M (v) (1/v) Remark 4.17. By using embedding properties in [50] between modulation spaces and Besov spaces, the embeddings in Theorem 4.13, Corollary 4.15 and Example 4.16 give rise to embeddings between the st,p spaces and Besov spaces. More precisely, assume that t ∈ R, p, q, qj ∈ [1, ∞] for j = 1, 2, ω0 and v are the same as in Corollary 4.15, and let ϑ(p) = |1 − 2/p| and s, s1 , s2 ∈ R. Then Theorem 4.4 and Remark 4.6 in [50] and Corollary 4.15 and Example 4.16 give: p,q1 (R2m ) ⊆ st,p (ω0 ) B(σ 2mϑ(p) v)

p,q2 ⊆B(σ (R2m ), −2mϑ(p) /v)

p,q1 (R2m ) ⊆ st,p (σs ) B2mϑ(p)+|s|

p,q2 ⊆B−2mϑ(p)−|s| (R2m ),

p,q1 p,q2 (R2m ) ⊆ st,p (σs1 ,s2 )⊆B−mϑ(p)−|s (R2m ). Bmϑ(p)+|s 2 |,mϑ(p)+|s1 | 2 |,−mϑ(p)−|s1 |

(See [6, 8] and Section 4 in [50] for strict definitions of the involved Besov spaces.) Remark 4.18. Assume that ω1 = ω2 = 1 in st,p = st,p (ω1 , ω2 ). Then it is proved in Corollary 3.5 in [48] that p,min(p,p
)







p,max(p,p
)

B2m|1−2/p| ⊆ M p,min(p,p ) ⊆ st,p ⊆ M p,max(p,p ) ⊆ B−2m|1−2/p| .

(4.19)

These relations can also be obtained by combining Theorem 4.4 and Remark 4.6 in [50] or Theorem 3.1 in [48], with Example 4.16 in the case s = 0, or s1 = s2 = 0. Note here that the embeddings p,p ⊆ sw B2m|1−2/p| p,

p,p 1 ≤ p ≤ 2 and sw p ⊆ B−2m|1−2/p| ,

2 ≤ p ≤ ∞,

which are consequences of (4.19), can not be improved. On the other hand, in the remaining cases, it was proved in [46] that the embeddings


p,p sw p ⊆ B−m|1−2/p| ,

1≤p≤2




p,p and Bm|1−2/p| ⊆ sw p,

2≤p≤∞

are sharp, which lead to strict improvements of (4.19) when considering embeddings between the sw p -spaces and Besov spaces. Note here that if p = ∞, then the latter embedding was proved already in [8]. Furthermore, it was proved in [6] that ∞,1 s0,∞ ⊆ Bm .

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Acknowledgements I am grateful to A. Holst for a careful reading and valuable comments, leading to improvement of the language and the content. I also thank P. Boggiatto, N. Kruglyak and I. Asekritova for fruitful discussions.

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[16] H.G. Feichtinger, Atomic characterizations of modulation spaces through Gabor-type representations, Rocky Mountain J. Math. 19 (1989), 113–126. [17] H.G. Feichtinger and P. Gr¨ obner, Banach spaces of distributions defined by decomposition methods, I, Math. Nachr. 123 (1985), 97–120. [18] H.G. Feichtinger and K.H. Gr¨ ochenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal. 86 (1989), 307– 340. [19] H.G. Feichtinger and K.H. Gr¨ ochenig, Banach spaces related to integrable group representations and their atomic decompositions, II, Monatsh. Math. 108 (1989), 129–148. [20] H.G. Feichtinger and K. Gr¨ ochenig, Gabor frames and time-frequency analysis of distributions, J. Funct. Anal. 146 (1997), 464–495. [21] H.G. Feichtinger and W. Kozek, Operator quantization on lca groups, in Gabor Analysis and Algorithms, Theory and Applications, Editors: H.G. Feichtinger and T. Strohmer, Birkh¨ auser, Boston, Basel, Berlin, 1998. [22] G.B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, 1989. [23] I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Non-Selfadjoint Operators in Hilbert Space (Russian), Izdat. Nauka, Moscow, 1965. [24] K. Gr¨ ochenig, Describing functions: atomic decompositions versus frames, Monatsh. Math. 112 (1991), 1–42. [25] K. Gr¨ ochenig, Foundations of Time-Frequency Analysis, Birkh¨ auser, Boston, 2001. [26] K. Gr¨ ochenig and C. Heil, Modulation spaces and pseudo-differential operators, Integral Equations Operator Theory 34 (1999), 439–457. [27] K. Gr¨ ochenig and C. Heil, Modulation spaces as symbol classes for pseudodifferential operators, in Wavelets and their Applications, Editors: M. Krishna, R. Radha and S. Thangavelu, Allied Publishers, 2003, 151–170. [28] L. H¨ ormander, The Analysis of Linear Partial Differential Operators I, III, SpringerVerlag, Berlin, Heidelberg, New York, Tokyo, 1983, 1985. [29] C. Heil, J. Ramanathan and P. Topiwala, Singular values of compact pseudodifferential operators, J. Funct. Anal. 150 (1997), 426–452. [30] D. Labate, Pseudodifferential operators on modulation spaces, J. Math. Anal. Appl. 262 (2001), 242–255. [31] S. Pilipovi´c and N. Teofanov, Wilson bases and ultramodulation spaces, Math. Nachr. 242 (2002), 179–196. [32] S. Pilipovi´c and N. Teofanov, On a symbol class of elliptic pseudodifferential operators, Bull. Acad. Serb. Sci. Arts 27 (2002), 57–68. [33] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, London, New York, 1979. [34] R. Rochberg and K. Tachizawa, Pseudo-differential operators, Gabor frames and local trigonometric bases, in Gabor Analysis and Algorithms, Editors: H.G. Feichtinger and T. Strohmer, Birkh¨ auser, Boston, 1998, 171–192. [35] R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, Berlin, 1960.

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[36] B.W. Schulze, Boundary Value Problems and Singular Pseudo-differential Operators, Wiley, Chichester, 1998. [37] B.W. Schulze and N.N. Tarkhanov, Pseudodifferential operators with operatorvalued symbols, in Israel Math. Conf. Proc. 16, 2003. [38] M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987. [39] B. Simon, Trace Ideals and their Applications, Cambridge University Press, Cambridge, London, New York, Melbourne, 1979. [40] J. Sj¨ ostrand, An algebra of pseudodifferential operators, Math. Res. Letters 1 (1994), 185–192. [41] J. Sj¨ ostrand, Wiener type algebras of pseudodifferential operators, in S´eminaire Equations aux D´eriv´ees Partielles, Ecole Polytechnique, 1994/1995, Expos´ e n◦ IV. [42] K. Tachizawa, The boundedness of pseudo-differential operators on modulation spaces, Math. Nachr. 168 (1994), 263–277. [43] N. Teofanov, Ultramodulation Spaces and Pseudodifferential Operators, Endowment Andrejevi´c, Beograd, 2003. [44] J. Toft, Continuity and Positivity Problems in Pseudo-differential Calculus, Thesis, Department of Mathematics, University of Lund, Lund, 1996. [45] J. Toft, Subalgebras to a Wiener type algebra of pseudo-differential operators, Ann. Inst. Fourier (Grenoble) 51 (2001), 1347–1383. [46] J. Toft, Continuity properties for non-commutative convolution algebras with applications in pseudo-differential calculus, Bull. Sci. Math. 126 (2002), 115–142. [47] J. Toft, Modulation spaces and pseudo-differential operators, Research Report, Blekinge Institute of Technology, Karlskrona, 2002. [48] J. Toft, Continuity properties for modulation spaces with applications to pseudodifferential calculus, I, J. Funct. Anal. 207 (2004), 399–429. [49] J. Toft, Continuity properties for modulation spaces with applications to pseudodifferential calculus, II, Ann. Global Anal. Geom. 26 (2004), 73–106. [50] J. Toft, Convolution and embeddings for weighted modulation spaces, in Advances in Pseudo-differential Operators, Editors: R. Ashino, P. Boggiatto and M.W. Wong, Birkh¨ auser, Basel 2004, 165–186. [51] J. Toft, Continuity and Schatten-von Neumann properties for pseudo-differential operators on modulation spaces, Research Report, V¨ axj¨ o University, V¨ axj¨ o, 2005. [52] J. Toft, Continuity and Schatten properties for Toeplitz operators on modulation spaces, in Modern Trends in Pseudo-differential Operators, Editors: J. Toft, M.W. Wong and H. Zhu, Birkh¨ auser, Basel, this Volume, 313–328. [53] H. Triebel, Modulation spaces on the Euclidean n-space, Z. Anal. Anwendungen 2 (1983), 443–457. [54] M.W. Wong, Weyl Transforms, Springer-Verlag, 1998. Joachim Toft Department of Mathematics and Systems Engineering V¨ axj¨ o University, Sweden e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 207–233 c 2006 Birkh¨
auser Verlag Basel/Switzerland

Algebras of Pseudo-differential Operators with Discontinuous Symbols Yu.I. Karlovich Abstract. Using the boundedness of the maximal singular integral operator related to the Carleson-Hunt theorem we prove the boundedness and study the compactness of pseudo-differential operators a(x, D) with bounded measurable V (R)-valued symbols a(x, ·) on the Lebesgue spaces Lp (R) with 1 < p < ∞, where V (R) is the Banach algebra of all functions of bounded total variation on R. Replacement of absolutely continuous functions of bounded total variation by arbitrary functions of bounded total variation allows us to study pseudo-differential operators with symbols admitting discontinuities of the first kind with respect to the spatial and dual variables. Appearance of discontinuous symbols leads to non-commutative algebras of Fredholm symbols. Three different Banach algebras of pseudo-differential operators with discontinuous symbols acting on the spaces Lp (R) are studied. We construct Fredholm symbol calculi for these algebras and establish Fredholm criteria for the operators in these algebras in terms of their Fredholm symbols. For the operators in the first algebra we also obtain an index formula. An application to the Haseman boundary value problem is given. Mathematics Subject Classification (2000). Primary 47G30, 47L15; Secondary 47A53, 47G10, 30E25. Keywords. Maximal singular integral operator, function of bounded total variation, slow oscillation, pseudo-differential operator, Lebesgue space, boundedness, compactness, algebra, Fredholm symbols, Fredholmness, index, Haseman problem.

1. Introduction The study of pseudo-differential operators with non-regular symbols, motivated by an extension of the sphere of their applications, is one of the natural directions in the development of the theory of pseudo-differential operators (see, for example, Partially supported by CONACYT (M´exico) Project No. 49992.

208

Yu.I. Karlovich

[10], [11], [40], [41]). There are several calculi for pseudo-differential operators with non-smooth symbols, which were elaborated by J. Marschall [33]–[35], W. Hoh [21], [22], and by N. Jacob and A.G. Tokarev [24]. Recently a new calculus was worked out in [27] for the Banach algebra of pseudo-differential operators with slowly oscillating V0 (R)-valued symbols on the Lebesgue spaces Lp (R) with 1 < p < ∞, where V0 (R) is the Banach algebra of absolutely continuous functions of bounded total variation on R (also see [28] for an application to singular integral operators). The present paper continues the investigations of [27]. Applying the integral analogue [30] of the famous Carleson-Hunt theorem (see [9], [23] and [25]) we study the boundedness and compactness of pseudo-differential operators a(x, D) with bounded measurable V (R)-valued symbols a(x, ·) on the Lebesgue spaces Lp (R) with 1 < p < ∞, where V (R) is the Banach algebra of all functions of bounded total variation on R. Replacement of absolutely continuous functions of bounded total variation by arbitrary functions of bounded total variation essentially extends the class of pseudo-differential operators in comparison with [27]. In particular, this allows one to study pseudo-differential operators with symbols admitting discontinuities of the first kind with respect to the spatial and dual variables. Appearance of discontinuous symbols leads to non-commutative algebras of Fredholm symbols, which completely changes the strategy of the Fredholm study for corresponding pseudo-differential operators because we cannot approximate initial symbols by infinitely differentiable ones. As usual, Fredholm symbols depend only on the behavior of initial symbols a(x, λ) in the neighborhood of infinity but the presence of discontinuities results in an appearance of an additional variable µ in the Fredholm symbols A(x, λ, µ) associated with the initial symbols a(x, λ), where µ runs through a set Lp depending on p ∈ (1, ∞). Given p ∈ (1, ∞), let B := B(Lp (R)) denote the Banach algebra of all bounded linear operators acting on the Banach space Lp (R) and let K := K(Lp (R)) be the closed two-sided ideal of all compact operators in B. As is well known (see, for example, [7], [18], [5]), an operator A ∈ B is said to be Fredholm, if its image is closed and the spaces ker A and ker A∗ are finite-dimensional. In that case Ind A = dim ker A − dim ker A∗ is referred to as the index of A.  and A)  of pseudo-differential operators Three different Banach algebras (A, A with discontinuous symbols acting on the space Lp (R) are studied in the present paper. We construct Fredholm symbol calculi for these algebras and establish Fredholm criteria for the operators in these algebras in terms of their Fredholm symbols. For the operators A ∈ A we also obtain an index formula. The results  are applied to studying the Haseman boundary value obtained for the algebra A problem on C \ R+ (see [1], [26], [32], and the references therein). The paper is organized as follows. In Section 2 we introduce necessary classes of functions and symbols of pseudo-differential operators. In Sections 3 and 4 we study, respectively, the boundedness and compactness of pseudo-differential operators with symbols in L∞ (R, V (R)) on the Lebesgue spaces Lp (R) where 1 < p < ∞. In Section 5, following [14, Section 7], [13], [37, Section 15] and [29, Section 6], we

Algebras of Pseudo-differential Operators

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expose the Fredholm theory for the Banach subalgebra D of B(Lp (R)) generated by convolution type operators cWb := cF −1 bF where F is the Fourier transform, c and b are piecewise continuous functions on R, and b is also a Fourier multiplier on Lp (R). In Section 6 we study the Banach algebra A of pseudo-differential operators with piecewise continuous V (R)-valued symbols on the spaces Lp (R) by  generated reduction to the algebra D. In Section 7 we study the Banach algebra A by pseudo-differential operators with slowly oscillating (in the sense of [12]) V0 (R)valued symbols and piecewise continuous V0 (R)-valued symbols. In Section 8 we  generated by pseudo-differential operators with slowly study the Banach algebra A oscillating (in the sense of [36]) V0 (R)-valued symbols and piecewise continuous V (R)-valued symbols. In Section 9 we study a singular integral operator N with a shift which is associated with the Haseman boundary value problem on C \ R+ . We reduce the operator N to a Mellin pseudo-differential operator related to an  Finally, applying results of Section 7, we establish a operator in the algebra A. Fredholm criterion for the operator N provided that the shift has a slowly oscillating derivative in contrast to preceding papers on the Haseman problem.

2. Classes of symbols As usual, let C0∞ (R) be the set of all functions in C ∞ (R) of compact support. Functions of bounded total variation. Let a : R → C be a function of bounded total variation V (a) on R where  n
  a(xk ) − a(xk−1 ) : −∞ < x0 < x1 < ···< xn < +∞, n ∈ N . V (a) := sup k=1

˙ := R ∪ {∞} there Hence (see, for example, [20, Chapter 9]), at every point x ∈ R exist finite one-sided limits a(x ± 0) = lim a(t), where a(±∞) = a(∞ ∓ 0), the t→x±

set of discontinuities is at most countable, and thus a is piecewise continuous on R := [−∞, +∞]. Without loss of generality (see the proof of Theorem 3.1 below) we will assume that functions of bounded total variation are continuous from the ˙ The set V (R) of all continuous from the left at every discontinuity point x ∈ R. left functions of bounded total variation on R is a unital non-separable Banach algebra with the norm aV := aL∞ (R) + V (a). Absolutely continuous functions of bounded total variation. Let V0 (R) be the unital Banach subalgebra of V (R) consisting of all absolutely continuous functions of bounded total variation on R. Hence (see [20, Chapter 9]), every function a ∈ V0 (R) is continuous on R = [−∞, +∞], and  x  a ∈ L1 (R), a(x) = a (y)dy + a(−∞) for all x ∈ R, V (a) = |a (y)|dy. −∞

R

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multipliers on Fourier multipliers. Let Mp denote the Banach algebra of Fourier

Lp (R) for 1 < p < ∞. From Stechkin’s inequality bMp ≤ SR B(Lp (R)) bV (see, for example, [13, Theorem 2.11]) it follows that V (R) ⊂ Mp for all p ∈ (1, ∞). Let P Cp (R) and Cp (R) be the closures in Mp of the sets of all piecewise constant functions on R with at most finite sets of jumps and, respectively, of all continuous functions of bounded total variation on R. Clearly, P Cp (R) and Cp (R) are Banach subalgebras of Mp . Moreover, P Cp (R) is also the closure of V (R) in Mp (see [13, Remark 2.12]), and, by [39, Lemma 1.1], Cp (R) is the closure of V0 (R) in Mp . ˙ := P Cp (R) ∩ C(R) ˙ is the closure in Mp of the set V0 (R) ∩ C(R). ˙ Hence, Cp (R) Slowly oscillating functions. Let Cb (R) := C(R) ∩ L∞ (R) be the C ∗ -algebra of all bounded continuous functions a : R → C. For a continuous function a : R → C, let    (2.1) cmx (a) := max a(x + h) − a(x) : h ∈ R, |h| ≤ 1 be the local oscillation of a at a point x ∈ R. According to [12, p. 122], a function a ∈ Cb (R) is called slowly oscillating at ∞ if lim cmx (a) = 0. Clearly, the set |x|→∞

SO of all functions in Cb (R) which are slowly oscillating at ∞ is a C ∗ -subalgebra of Cb (R). Following [36] we also consider the C ∗ -subalgebra SO♦ of Cb (R) which consists of functions that slowly oscillate at ∞ in another sense. Namely,    a(t) − a(τ ) = 0 . sup SO♦ := a ∈ Cb (R) : lim x→+∞ t,τ ∈[−2x,−x]∪[x,2x]

Obviously, SO♦ ⊂ SO. Let M (A) stand for the maximal ideal space of a unital C ∗ -algebra A. Identifying the points t ∈ R with the evaluation functionals t(f ) = f (t) for f ∈ C(R), where f (±∞) = lim f (x), we obtain M (C(R)) = R. Consider the fibers x→±∞

M±∞ (SO) := ξ ∈ M (SO) : ξ|C(R) = ±∞ of the maximal ideal space M (SO) over the points t = ±∞. Analogously, the fiber M∞ (SO♦ ) of the maximal ideal space M (SO♦ ) at the point t = ∞ is defined by

M∞ (SO♦ ) := ξ ∈ M (SO♦ ) : ξ|C(R) ˙ = ∞ . According to [27, Proposition 2.4], [4, Proposition 5] and [6, Proposition 4.1], M−∞ (SO) ∪ M+∞ (SO) = (closSO∗ R) \ R,

M∞ (SO♦ ) = (clos(SO♦ )∗ R) \ R,

where closSO∗ R is the weak-star closure of R in SO∗ , the dual space of SO, and clos(SO♦ )∗ R is the weak-star closure of R in (SO♦ )∗ , the dual space of SO♦ . Bounded symbols. In what follows we denote by L∞ (R, V (R)) the set of all functions a : R×R → C such that  a : x → a(x, ·) is a bounded measurable V (R)-valued function on R. Since the Banach space V (R) is non-separable, the measurability of  a means that the map  a : R → V (R) possesses the Luzin property: for any compact K ⊂ R and any δ > 0 there is a compact Kδ ⊂ K such that the Lebesgue

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a : Kδ → V (R) is conmeasure of the set K \ Kδ is less than δ and the function  tinuous (see, for example, [38, Chapter IV, § 4]). This implies that the functions



˙ and the function x → a(x, ·) are measurable on R x → a(x, λ ± 0) for all λ ∈ R V ˙ and for almost all x ∈ R, as well. Note that the limits a(x, λ ± 0) exist for all λ ∈ R and a(x, λ) = a(x, λ − 0) for all λ ∈ R and a(x, ±∞) = lim a(x, λ). Therefore, λ→±∞



˙ and the function x → a(x, ·) , where the functions a(·, λ ± 0) for every λ ∈ R V





 

a(x, ·) := a(x, ·) ∞ + V a(x, ·) , (2.2) V L (R) belong to L∞ (R). Clearly, L∞ (R, V (R)) is a unital Banach algebra with the norm





a ∞ := ess sup a(x, ·) . L

(R,V (R))

x∈R

V

Piecewise continuous symbols. Let P C(R, V (R)) stand for the set of all functions a : R × R → C such that  a : x → a(x, ·) is a piecewise continuous V (R)-valued function on R. If a ∈ P C(R, V (R)), then the functions x → a(x, λ ± 0) for all ˙ and the function x → a(x, ·) given by (2.2) belong to P C(R). Clearly, λ∈R V P C(R, V (R)) ⊂ P C(R × R) is a unital Banach subalgebra of L∞ (R, V (R)). Slowly oscillating symbols. Let Cb (R, V0 (R)) denote the Banach algebra of all bounded continuous V0 (R)-valued functions on R. Following [27], for a function a ∈ Cb (R, V0 (R)) by analogy with (2.1), we put



cmVx (a) := max a(x + h, ·) − a(x, ·) V : h ∈ R, |h| ≤ 1 . Let SO(R, V0 (R)) be the Banach algebra of all bounded continuous V0 (R)-valued functions on R that slowly oscillate at ∞, that is, satisfy the condition lim cmVx (a) = 0.

|x|→∞

Clearly, every V0 (R)-valued function x → a(x, ·) in SO(R, V0 (R)) is uniformly continuous on R. We also introduce the Banach subalgebra SO♦ (R, V0 (R)) of Cb (R, V0 (R)) consisting of all bounded continuous V0 (R)-valued functions on R that slowly oscillate at ∞ in the sense of [36]. Thus, SO♦ (R, V0 (R)) is the set 



a(t, ·) − a(τ, ·) = 0 . sup f ∈ Cb (R, V0 (R)) : lim V x→+∞ t,τ ∈[−2x,−x]∪[x,2x]

Let E denote the subset of all functions a : R × R → C in SO(R, V0 (R)) that satisfy the conditions     +∞ M a(x, ·) = lim sup VM a(x, ·) = 0 (2.3) lim sup V−∞ M→−∞ x∈R

and

M→+∞ x∈R



lim sup a(x, ·) − ah (x, ·) V = 0

|h|→0 x∈R

(2.4)

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where Vbc (f ) is the variation of f on an interval (b, c), and ah (x, λ) := a(x, λ + h) for all (x, λ) ∈ R×R. Analogously, let E be the subset of all functions a : R×R → C in SO♦ (R, V0 (R)) that satisfy the conditions (2.3) and (2.4). Obviously, E and E  are Banach subalgebras of the Banach algebra Cb (R, V0 (R)), and E ⊂ E. By [27, Lemmas 2.7 and 2.9], for every a(x, λ) ∈ E and every ξ ∈ M±∞ (SO), there exist a sequence xn → ±∞ and a function a(ξ, ·) ∈ V0 (R) such that



lim a(xn , ·) − a(ξ, ·) V = 0; (2.5) n→∞

and, conversely, every sequence yn → ±∞ contains a subsequence xn → ±∞ such that (2.5) holds for some ξ ∈ M±∞ (SO). Obviously, the functions a(x, λ) ∈ E also possess such the property for every ξ ∈ M∞ (SO♦ ) (with xn → ∞ and yn → ∞)  because E ⊂ E. Discontinuous slowly oscillating symbols. Finally, we consider discontinuous slowly  := P C(R, V0 (R)) ∪ E and Ω  := oscillating symbols that belong to the subsets Ω P C(R, V (R)) ∪ E of L∞ (R, V (R)).

3. Boundedness of pseudo-differential operators on Lebesgue spaces For −∞ < a < b < +∞, we consider the partial sum operators S(a,b) given by  b   1 S(a,b) f (x) = (3.1) f(λ)eixλ dλ, for x ∈ R, 2π a where f(λ) := (F f )(λ) :=



f (y)e−iλy dy,

R

for λ ∈ R,

is the Fourier transform of f . Since S(a,b) = F −1 χ(a,b) F where χ(a,b) is the characteristic function of the interval (a, b), we infer from Stechkin’s inequality that the operators S(a,b) are bounded on all the spaces Lp (R) with 1 < p < ∞. According to the famous Carleson-Hunt theorem on almost everywhere convergence (more precisely, by its integral analog for Lp (R), see [30]), the maximal singular integral operator S∗ given by   (S(a,b) f )(x), for x ∈ R, sup (S∗ f )(x) = −∞
is also bounded on every space Lp (R) with 1 < p < ∞ (also see [15, Chapter 2, Section 2.2]). In particular, for every f ∈ Lp (R) and for almost every x ∈ R,   (3.2) sup (S(0,λ) f )(x) ≤ (S∗ f )(x) < ∞. λ∈R

Modifying the proofs of [15, Chapter 3, Section 3.3] and [27, Theorem 3.1], we obtain the following result.

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Theorem 3.1. If a ∈ L∞ (R, V (R)), then the pseudo-differential operator a(x, D) defined for functions u ∈ C0∞ (R) by the iterated integral     1 a(x, D)u (x) = dλ a(x, λ)ei(x−y)λ u(y)dy, f or x ∈ R, (3.3) 2π R R extends to a bounded linear operator on every Lebesgue space Lp (R) with p ∈ (1, ∞), and







a(x, D) p ≤ 2 a L∞ (R,V (R)) S∗ B(Lp (R)) . (3.4) B(L (R)) Proof. Fix u ∈ C0∞ (R). Due to (3.3),    1 a(x, λ) u(λ)eiλx dλ, a(x, D)u (x) = 2π R

for x ∈ R,

(3.5)

where the integral on the right of (3.5) converges a.e. on R because the function λ → a(x, λ) is bounded on R for almost all x ∈ R and u  is a rapidly decreasing function. Since for every x ∈ R the function λ → S(0,λ) u (x) given by (3.1) is continuous on R and for almost every x ∈ R the function λ → a(x, λ) is of bounded total variation on R, we conclude that the Riemann-Stieltjes integral  M   S(0,λ) u (x) da(x, λ) −M

with the parameter x ∈ R also converges a.e. on R for every M > 0. Therefore, it follows from the formula of integration by parts for the Riemann-Stieltjes integrals that  M  M   1 S(0,λ) u (x) da(x, λ) a(x, λ) u(λ)eiλx dλ = − 2π −M −M       + a(x, M ) S(0,M) u (x) + a(x, −M ) S(−M,0) u (x) for almost all x ∈ R. Therefore, taking into account (3.2) and (2.2), for almost every x ∈ R and every M > 0 such that the function λ → a(x, λ) is continuous at the points λ = ±M , we obtain   M    M     1   iλx ≤   S a(x, λ) u (λ)e dλ u (x) da(x, λ) (0,λ)     2π −M −M         + a(x, M ) S(0,M) u (x) + a(x, −M ) S(−M,0) u (x) 



     

 ≤ S∗ u (x) V a(x, ·) + 2 a(x, ·) L∞ (R) ≤ 2 S∗ u (x) a(x, ·) V . The latter estimate and (3.5) imply that

    

 a(x, D)u (x) ≤ 2 S∗ u (x) a(x, ·)

V

a.e. on R.

Consequently, in view of the boundedness of the maximal singular operator S∗ on the spaces Lp (R) for p ∈ (1, ∞), we obtain









a(x, D)u p ≤ 2 a L∞ (R,V (R)) S∗ B(Lp (R)) u Lp (R) , L (R) which implies (3.4).




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4. Compactness of pseudo-differential operators on Lebesgue spaces Here, by analogy with [27, Theorem 4.1], we get sufficient conditions for compactness of pseudo-differential operators σ(x, D) with symbols σ(x, λ) ∈ L∞ (R, V (R)). Theorem 4.1. The pseudo-differential operator σ(x, D) with a discontinuous symbol σ(x, λ) ∈ L∞ (R, V (R)) is compact on every Lebesgue space Lp (R) (1 < p < ∞) if   σ(x, ±∞) = 0 f or almost all x ∈ R, lim V σ(x, ·) = 0, (4.1) |x|→∞      −M +∞ σ(x, ·) + VM σ(x, ·) = 0 f or every N > 0. (4.2) lim ess sup V−∞ M→+∞ |x|≤N

Proof. Since σ(x, λ) ∈ L∞ (R, V (R)), the operator σ(x, D) is bounded on all the spaces Lp (R) with 1 < p < ∞ by virtue of Theorem 3.1. Following the proof of [27, Theorem 4.1] we consider the sequence of pseudo-differential operators {σk,n (x, D)}k,n∈N with the symbols σk,n (x, λ) = ψk (x)σ(x, λ)ψn (λ)

(4.3)

where ψn (x) = 1 for |x| < n, ψn (x) = ψ(|x| − n) for |x| ∈ [n, n + 1], ψn (x) = 0 for |x| > n + 1, and ψ(x) = (x − 1)2 (2x + 1). Obviously, ψn ∈ C01 (R) because ψ : [0, 1] → [1, 0], ψ  (0) = ψ  (1) = 0, and ψ  (x) < 0 for x ∈ (0, 1).

(4.4)

Since σ(x, ·) ∈ V (R) for almost all x ∈ R, we conclude from the first equality in (4.1) that for almost all x ∈ R and for all M > 0,     −M +∞ σ(x, ·) if λ ≤ −M, |σ(x, λ)| ≤ VM σ(x, ·) if λ ≥ M. |σ(x, λ)| ≤ V−∞ Therefore, taking into account the facts that according to (4.4),  1



   

ψn ∞ = 1, V n+1 ψn = V −n ψn = |ψ  (x)|dx = |ψ(1)−ψ(0)| = 1, (4.5) n −n−1 L (R) 0

we obtain     V σk,n (x, ·) = |ψk (x)| V σ(x, ·)ψn (·)    

n+1 n+1 σ(x, ·)ψn (·) ≤ ψn L∞ [−n−1,n+1] V−n−1 σ(x, ·) ≤ V−n−1



 

  −n ψn + σ(x, ·) L∞ [n,n+1] Vnn+1 ψn + σ(x, ·) L∞ [−n−1,−n] V−n−1         −n σ(x, ·) + Vn+∞ σ(x, ·) ≤ 2V σ(x, ·) , (4.6) ≤ V σ(x, ·) + V−∞







 

σk,n (x, ·) ≤ σk,n (x, ·) ∞ + V σk,n (x, ·) ≤ 2 σ(x, ·) < ∞. (4.7) V L (R) V Clearly, (4.7) means that σk,n (x, λ) ∈ L∞ (R, V (R)). Hence, by Theorem 3.1, the operators σk,n (x, D) are bounded on all the spaces Lp (R) (1 < p < ∞).

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Analogously to (4.6), making use of the definition of ψn and (4.5), we obtain          −n σ(x, ·) 1 − ψn (·) + Vn+∞ σ(x, ·) 1 − ψn (·) V σ(x, ·) 1 − ψn (·) = V−∞





    −n ≤ σ(x, ·) L∞ (−∞,−n] + σ(x, ·) L∞ [n,+∞) + V−∞ σ(x, ·) + Vn+∞ σ(x, ·)      −n ≤ 2 V−∞ σ(x, ·) + Vn+∞ σ(x, ·) . (4.8) Since σ(x, ±∞) − σk,n (x, ±∞) = 0, applying (4.8) we get



  ess sup σ(x, ·) − σk,n (x, ·) ≤ 2 ess sup V σ(x, ·) − σk,n (x, ·) V

x∈R

x∈R

        ≤ 2 ess sup 1 − ψk (x)V σ(x, ·) + 2 ess sup ψk (x)V σ(x, ·) 1 − ψn (·) x∈R x∈R        −n (4.9) ≤ 2 ess sup V σ(x, ·) + 4 ess sup V−∞ σ(x, ·) + Vn+∞ σ(x, ·) . |x|≥k

|x|≤k+1

Fix ε > 0. By (4.1), there exists a k0 > 0 such that   ess sup V σ(x, ·) < ε for all k > k0 .

(4.10)

According to (4.2), for every k > k0 there is an nk ∈ N such that      −nk σ(x, ·) + Vn+∞ σ(x, ·) < k −1 . ess sup V−∞ k

(4.11)

|x|≥k

|x|≤k+1



From (4.9), (4.10) and (4.11) it follows that lim σ − σk,nk L∞ (R,V (R)) = 0. This k→∞

implies according to (3.4) that



lim σ(x, D) − σk,nk (x, D) B(Lp (R)) = 0. k→∞

(4.12)

Obviously, by (4.3), all the operators σk,n (x, D) are compact on the space L2 (R). By the Krasnoselskii theorem on interpolation of compactness (see [31, Theorem 3.10]), every operator σk,n (x, D), being bounded on all the spaces Lp (R) for 1 < p < ∞ and compact on the space L2 (R), is compact on all Lp (R) (1 < p < ∞). Finally, from (4.12) it follows that the operator σ(x, D), being the uniform limit of compact operators σk,nk (x, D), is itself compact.
Corollary 4.2. If σ(x, λ) ∈ P C(R, V (R)) and (4.1) holds, then the pseudo-differential operator σ(x, D) is compact on every Lebesgue space Lp (R) with 1 < p < ∞. Proof. Since σ(x, λ) ∈ P C(R, V (R)) and for every M ∈ R,  +∞         +∞ +∞ V σ(x, ·) − VM σ(y, ·)  ≤ VM σ(x, ·) − σ(y, ·) ≤ V σ(x, ·) − σ(y, ·) , M     +∞ M the function x → VM σ(x, ·) and, analogously, the function x → V−∞ σ(x, ·) are piecewise continuous on R. Furthermore, for every x ∈ R, lim σ(x ± 0, λ) = σ(x ± 0, −∞),

λ→−∞

whence lim

M→+∞

lim σ(x ± 0, λ) = σ(x ± 0, +∞),

λ→+∞

     −M +∞ V−∞ σ(x ± 0, ·) + VM σ(x ± 0, ·) = 0 for all x ∈ R.

(4.13)

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 −M   +∞ + VM As the function x → V−∞ σ(x, ·) is piecewise continuous on every segment [−N, N ] ⊂ R, choosing an appropriate finite covering of this segment and taking into account (4.13) it is easily seen that (4.2) is fulfilled automatically. It remains to apply Theorem 4.1.


5. Algebra D of convolution type operators Recall the symbol calculus and the Fredholm criterion for the Banach algebra

D := alg cWb : c ∈ P C(R), b ∈ P Cp (R) ⊂ B(Lp (R)) where cWb = cF −1 bF (see [14, Section 7], [13], and also [37, Section 15], [8], [3]). Let K be the closed two-sided ideal of all compact operators in the Banach algebra B = B(Lp (R)). By analogy with [17] one can show that K ⊂ D. As is known, the Fredholmness of operators A ∈ D is equivalent to the invertibility of the cosets Aπ := A + K in the Calkin algebra B π := B/K. Given 1 < p < ∞, we consider the set       (5.1) M := R × {∞} × Lp ∪ {∞} × R × Lp ∪ {∞} × {∞} × {0, 1} equipped with the discrete topology, where    Lp := µ = 2−1 1 + coth(πx + πi/p) : x ∈ R is a circular arc with the endpoints 0 and 1 if p = 2, and the line segment joining these points if p = 2. Following [3, Subsection 2.4], for A = cWb (with c ∈ P C(R) and b ∈ P Cp (R)) and for (x, λ, µ) ∈ M , we define the matrix function A(x, λ, µ) = ;     < c(x + 0) b(λ − 0)µ + b(λ + 0)(1 − µ) c(x + 0) b(λ − 0) − b(λ + 0) ν(µ)     , c(x − 0) b(λ − 0) − b(λ + 0) ν(µ) c(x − 0) b(λ − 0)(1 − µ) + b(λ + 0)µ (5.2) a(∓∞), b(∞ ± 0)= b(∓∞), and ν(µ) is the where, by convention, a(∞ ± 0) =   continuous branch of the function µ(1 − µ) in C \ (−∞, 0) ∪ (1, +∞) which satisfies the condition ν(1/2) = 1/2. Let BC(M, C2×2 ) stand for the set of bounded continuous functions of M into 2×2 C . From [14, Section 7], [13], [37, Section 15] and [8] we obtain the following. Theorem 5.1. Let 1 < p < ∞ and let M be given by (5.1). (i) The map Ψ associating with the operator A = cWb (c ∈ P C(R), b ∈ P Cp (R)) the matrix function (5.2) extends to a Banach algebra homomorphism Ψ : D → BC(M, C2×2 ) whose kernel contains K. (ii) The quotient algebra Dπ = D/K is inverse closed in the Calkin algebra B π . (iii) An operator A ∈ D is Fredholm on the space Lp (R) if and only if [det Ψ(A)](x, λ, µ) = 0

f or all (x, λ, µ) ∈ M.

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Now we equip the set M given by (5.1) with the following topology. Let the neighborhood base for (x, λ, µ) ∈ M consist of the sets U (x, λ, µ) of the form ⎧ ∞)} × Vµ if x ∈ R, µ ∈ Lp \ {0, 1}; ⎪ ⎨ {(x,     if x ∈ R, µ = 0; {(x, ∞)} × V0 ∪ (x − ε, x) × {∞} × Lp U (x, ∞, µ) = ⎪     ⎩ if x ∈ R, µ = 1; {(x, ∞)} × V1 ∪ (x, x + ε) × {∞} × Lp ⎧ λ)} × Vµ if λ ∈ R, µ ∈ Lp \ {0, 1}; ⎪ ⎨ {(∞,     if λ ∈ R, µ = 0; {(∞, λ)} × V0 ∪ {∞} × (λ, λ + ε) × Lp U (∞, λ, µ) = ⎪    ⎩  if λ ∈ R, µ = 1; {(∞, λ)} × V1 ∪ {∞} × (λ − ε, λ) × Lp     U (∞, ∞, 0) = {(∞, ∞, 0)} × (c, +∞) × {∞} × Lp ∪ {∞} × (−∞, d) × Lp ,     U (∞, ∞, 1) = {(∞, ∞, 1)} × (−∞, d) × {∞} × Lp ∪ {∞} × (c, +∞) × Lp , where ε > 0, c, d ∈ R, and Vµ := Lp ∩ {ν ∈ C : |ν − µ| < ε}. Then M becomes a compact Hausdorff space. Let A(x, λ, µ) := [Ψ(A)](x, λ, µ) for A ∈ A. With the Fredholm symbol 2  A(x, λ, µ) = aij (x, λ, µ) i,j=1 of any Fredholm operator A ∈ D we associate the function det A(x, λ, µ) a22 (x, ∞, 0) , for (x, λ, µ) ∈ M. (5.3) fA (x, λ, µ) := a11 (x, ∞, 0) a22 (x, λ, 0) a22 (x, λ, 1) By analogy with [29, Section 6] one can prove that the function (5.3) is continuous on M and separated from zero. Let us show that fA (M ) is the graph of a bounded, closed and continuous ˙ define A1 (x) := fA (x, ∞, 0). It follows curve A# in the complex plane. For x ∈ R, ˙ and from (5.2) that the limits A1 (x ± 0) exist at each point x ∈ R, A1 (x − 0) = fA (x, ∞, 0),

A1 (x + 0) = fA (x, ∞, 1).

Thus, these limits are finite, non-zero, and A1 (x − 0) = A1 (x). Let J1 denote the set of points x ∈ R at which the functions fA (x, ∞, ·) are not constant on Lp . It is clear that the set J1 is at most countable. For every x ∈ J1 we join the one-sided limits A1 (x ± 0) by the continuous curve A# x,∞ := fA (x, ∞, µ) : µ ∈ Lp oriented from A1 (x − 0) = fA (x, ∞, 0) to A1 (x + 0) = fA (x, ∞, 1). Therefore, we obtain the open continuous oriented curve  

A# A# x,∞ ∪ fA (∞, ∞, 0) 1 := fA (∞, ∞, 1) ∪ A1 (R) ∪ x∈J1

with the starting point A1 (−∞) = fA (∞, ∞, 1) and the terminating point A1 (+∞) = fA (∞, ∞, 0). Moreover, 0 ∈ / A# 1 . Analogously, for λ ∈ R, we define the function A2 (λ) := fA (∞, λ, 1). It follows ˙ and from (5.2) that the limits A2 (λ ± 0) also exist at each point λ ∈ R, A2 (λ − 0) = fA (∞, λ, 1),

A2 (λ + 0) = fA (∞, λ, 0).

These limits also are finite, non-zero, and A2 (λ − 0) = A2 (λ). Let J2 denote the at most countable set of points λ ∈ R at which the functions fA (∞, λ, ·) are

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not constant on Lp . For each λ ∈ J2 we join the one-sided limits A2 (λ ± 0) of the

function A2 at the point λ by the continuous curve A# ∞,λ := fA (∞, λ, µ) : µ ∈ Lp oriented from A2 (λ − 0) = fA (∞, λ, 1) to A2 (λ + 0) = fA (∞, λ, 0). Therefore, we obtain the open continuous oriented curve  

# := f (∞, ∞, 0) ∪ A (R) ∪ A A# A 2 2 ∞,λ ∪ fA (∞, ∞, 1) λ∈J2

with the starting point A2 (−∞) = fA (∞, ∞, 0) and the terminating point A2 (+∞) = fA (∞, ∞, 1), where again 0 ∈ / A# 2 . # As A2 (+∞) = A1 (−∞) and A2 (−∞) = A1 (+∞), the curve A# := A# 1 ∪ A2 is closed. Thus, for every Fredholm operator A ∈ D, we get the bounded, closed, continuous and oriented curve A# ⊂ C. Obviously, A# = fA (M ) and 0 ∈ / A# . Finally, from [14, Section 7], [13] and [29, Section 6] we deduce the following. Theorem 5.2. If an operator A ∈ D is Fredholm on the space Lp (R) (1 < p < ∞), then (5.4) Ind A = −wind A# where wind A# denotes the winding number of the curve A# about the origin.

6. Algebra A of pseudo-differential operators Let A = Ap denote the minimal Banach subalgebra of B(Lp (R)) that contains all pseudo-differential operators a(x, D) with symbols a ∈ P C(R, V (R)). Repeating literally the proof of [27, Lemma 10.1] we obtain the following. Lemma 6.1. For every p ∈ (1, ∞), the Banach algebra A contains all compact operators acting on the space Lp (R). Let us show that pseudo-differential operators A ∈ A are related to convolution type operators in the Banach algebra D. To this end with every function a(x, λ) ∈ P C(R, V (R)) we associate the function  a(x, λ) = a(−∞, λ)ζ− (x) + a(+∞, λ)ζ+ (x) + c− (x)ζ− (λ) + c+ (x)ζ+ (λ)

(6.1)

where x, λ ∈ [−∞, +∞] and c± (x) ζ± (λ)

= =

a(x, ±∞) − a(−∞, ±∞)ζ− (x) − a(+∞, ±∞)ζ+ (x), (1 ± tanh λ)/2.

(6.2) (6.3)

Since c± (−∞) = c± (+∞) = 0, it is easily seen that  a(x, ±∞) = a(x, ±∞),

 a(±∞, λ) = a(±∞, λ) for all x, λ ∈ [−∞, +∞]. (6.4)

By (6.1)–(6.3), the function  a(x, λ) also is in the Banach algebra P C(R, V (R)). Hence, by Theorem 3.1, the pseudo-differential operator  a(x, D) is bounded on each Lebesgue space Lp (R) with 1 < p < ∞. On the other hand, the operator  a(x, D) is represented in the form  a(x, D) = ζ− Wa(−∞,·) + ζ+ Wa(+∞,·) + c− Wζ− + c+ Wζ+ ,

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that is,  a(x, D) is a convolution type operator that belongs to the Banach subalgebra D of B(Lp (R)) generated by all operators of the form cWb = cF −1 bF with c ∈ P C(R) and b ∈ P Cp (R). Lemma 6.2. If a ∈ P C(R, V (R)), then the pseudo-differential operator a(x, D) −  a(x, D) is compact on every Lebesgue space Lp (R) with 1 < p < ∞. Proof. Since a(x, λ) −  a(x, λ) = 0 for all (x, λ) ∈ ∂(R × R), we infer that the function σ(x, λ) = a(x, λ) −  a(x, λ) belongs to P C(R, V (R)) and satisfies (4.1). Hence, by Corollary 4.2, the operator σ(x, D) = a(x, D) −  a(x, D) is compact on
every Lebesgue space Lp (R) with 1 < p < ∞. Hence the quotient Banach algebras Aπ = A/K and Dπ = D/K coincide, and therefore the Banach algebra Aπ is not commutative. By (6.4) and (5.2), to every pseudo-differential operator A = a(x, D) with symbol a(x, λ) ∈ P C(R, V (R)) we assign the Fredholm symbol A(x,λ,µ) = < ;   a(x + 0,λ − 0)µ + a(x + 0,λ + 0)(1 − µ) a(x + 0,λ − 0) − a(x + 0,λ + 0) ν(µ)   a(x − 0,λ − 0) − a(x − 0,λ + 0) ν(µ) a(x − 0,λ − 0)(1 − µ) + a(x − 0,λ + 0)µ (6.5) where (x, λ, µ) ∈ M . Then Theorems 5.1 and 5.2 immediately give the following. Theorem 6.3. Let 1 < p < ∞ and let M be given by (5.1). (i) The map Ψ : A → A(x, λ, µ) associating matrix functions of the form (6.5) to pseudo-differential operators A = a(x,D) with symbols a(x,λ) ∈ P C(R,V (R)) extends to a Banach algebra homomorphism Ψ : A → BC(M, C2×2 ) whose kernel contains K. (ii) The quotient algebra Aπ is inverse closed in the Calkin algebra B π . (iii) An operator A ∈ A is Fredholm on the space Lp (R) if and only if det A(x, λ, µ) = 0

f or all (t, x, µ) ∈ M,

where A(x, λ, µ) = [Ψ(A)](x, λ, µ) is the Fredholm symbol of the operator A. (iv) If A ∈ A is Fredholm on the space Lp (R), then its index is calculated by (5.4). Thus, the algebra Ψ(A) of Fredholm symbols for A ∈ A is non-commutative.

 of pseudo-differential operators 7. Algebra A  where Ω  = P C(R, V0 (R)) ∪ E is a subset of L∞ (R, V0 (R)). Let now a(x, λ) ∈ Ω Combining results of [27] and [14] we obtain the following.  then the commutator ˙ and a(x, λ) ∈ Ω, Lemma 7.1. If c ∈ SO, b ∈ Cp (R), cWb a(x, D)−a(x, D) cWb is compact on every Lebesgue space Lp (R) (1 < p < ∞).

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˙ and p ∈ (1, ∞). Since Cp (R) ˙ is the closure of Proof. Fix c ∈ SO, b ∈ Cp (R), ˙ V0 (R) ∩ C(R) in Mp , we conclude that every operator cWb with c ∈ SO and b ∈ ˙ can be approximated in B(Lp (R)) by pseudo-differential operators with symCp (R)  Then, by [27, Corollary 8.4], the commutator cWb a(x, D) − a(x, D) cWb bols in E.  If a ∈ P C(R, V0 (R)), then is compact on the space Lp (R) for every a(x, λ) ∈ E.  Therefore, by (6.1) and by [27, Corolthe functions a(±∞, ·) and ζ± belong to E. lary 8.4], the commutator c  a(x, D)− a(x, D) cI is compact on the space Lp (R). By [14, Lemma 7.3] and by (6.1) and (6.2), the commutator Wb  a(x, D)− a(x, D) Wb is a(x, D) −  a(x, D) cWb is also compact on Lp (R). As a result, the commutator cWb  compact on Lp (R) too. Finally, applying Lemma 6.2, we deduce the compactness of the commutator cWb a(x, D) − a(x, D) cWb for a ∈ P C(R, V0 (R)).
=A  p of B(Lp (R)) generated By [27, Lemma 10.1], the Banach subalgebra A  contains all by all pseudo-differential operators a(x, D) with symbols a(x, λ) ∈ Ω  π := A/K.  compact operators in B(Lp (R)). Consider the quotient Banach algebra A p  Let Z be the Banach subalgebra of B(L (R)) generated by all operators cWb , ˙ and by all compact operators on Lp (R). In virtue where c ∈ SO and b ∈ Cp (R),  is a central subalgebra of of Lemma 7.1, the quotient Banach algebra Zπ := Z/K π  A . From [27, Sections 10–12] it follows that the maximal ideal space of Zπ can be identified with the set ˙ ∪ (R × {∞}) ∪ (M+∞ (SO) × R), ˙ (7.1) M (Zπ ) := (M−∞ (SO) × R)  be the Banach algebra where M±∞ (SO) are the fibers of M (SO) at ±∞. Let Λ p generated by all operators in B(L (R)) that commute with every operator cWb ∈ Z  are called operators of local type to within compact operators (the elements of Λ π π   π := Λ/K.   Let Jξ,η be with respect to Z). Then A is a Banach subalgebra of Λ π π  generated by the maximal ideal of Z identified the closed two-sided ideal of Λ with the point (ξ, η) ∈ M (Zπ ). Applying the Allan-Douglas local principle (see, for example, [7, Theorem 1.34]), we obtain the following criterion.  is Fredholm on the space Lp (R) with 1 < p < ∞ Theorem 7.2. An operator A ∈ A π := Aπ +Jπ is invertible if and only if for every point (ξ, η) ∈ M (Zπ ) the coset A ξ,η ξ,η  π /Jπ .  π := Λ in the Banach algebra Λ ξ,η ξ,η + Given y ∈ R, let χ− y and χy denote the characteristic functions of (−∞, y) and (y, +∞), respectively. Put χ± := χ± 0 . One can easily obtain the following.

 Then Lemma 7.3. Let a(x, λ) ∈ Ω.  − a(x, D) − a(ξ − 0, −∞)χξ Wχ− − a(ξ + 0, −∞)χ+ Wχ π ξ π − − + −a(ξ − 0, +∞)χξ Wχ+ − a(ξ + 0, +∞)χξ Wχ+ ∈ Jξ,η if (ξ, η) ∈ R × {∞},  π π if ξ ∈ M±∞ (SO), η ∈ R, a(x, D) − a(ξ, η)I ∈ Jξ,η π  π if ξ ∈ M±∞ (SO), η = ∞. a(x, D)−a(ξ, −∞)ζ± Wζ− −a(ξ, +∞)ζ± Wζ+ ∈ Jξ,η

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Lemma 7.3 immediately implies the following. π , defined on Lemma 7.4. For every (ξ, η) ∈ M (Zπ ), the mapping δξ,η : A → A ξ,η  by the rule the generators a(x, D) of the Banach algebra A δξ,η [a(x, D)] := ⎧  + a(ξ − 0, −∞)χ− ⎪ ξ Wχ− + a(ξ + 0, −∞)χξ Wχ− ⎪ ⎪  ⎪ ⎨ + a(ξ − 0, +∞)χ− Wχ + a(ξ + 0, +∞)χ+ Wχ π , (ξ, η) ∈ R × {∞}, + + ξ,η ξ ξ  π ⎪ a(ξ, η)I ξ,η , (ξ, η) ∈ M±∞ (SO) × R, ⎪ ⎪ ⎪ ⎩ a(ξ, −∞)ζ W + a(ξ, +∞)ζ W π , (ξ, η) ∈ M±∞ (SO) × {∞}, ± ζ− ± ζ+ ξ,η →Λ  π , and extends to a Banach algebra homomorphism δξ,η : A ξ,η



π 

 sup δξ,η (A) Λ π ≤ A  := inf A + K f or all A ∈ A. π ) (ξ,η)∈M(Z

ξ,η

K∈K

 ⊂ {A π = Aπ + Jπ : A ∈ D} ⊂ Λ  π for every (ξ, η) ∈ M (Zπ ). Thus, δξ,η (A) ξ,η ξ,η ξ,η π in the local algebras Λ  π in the To study the invertibility of the cosets A ξ,η ξ,η case (ξ, η) ∈ R × {∞} we need the following special version of the two projections theorem (see [16], [19] and [5]). Theorem 7.5. Let A be a Banach algebra with identity e, let r and q be idempotents in A, and let R = alg (e, r, q) denote the smallest closed subalgebra of A which contains e, r, q. Let x = e − (r − q)2 and suppose that 0 and 1 are not isolated points of the spectrum spA x of x in A and that spA x = spR x. Then R is inverse closed in A and (i) for every µ ∈ spA x, the map Υµ : {e, r, q} → C2×2 given by ! ! ! 1 0 1 0 µ ν(µ) Υµ (e) = , Υµ (r) = , Υµ (q) = 0 1 0 0 ν(µ) 1 − µ extends to a Banach algebra homomorphism Υµ : R → C2×2 , where ν(µ) :=  µ(1 − µ) means an arbitrary value of the square root; (ii) an element a ∈ R is invertible in the algebra A (equivalently, in R) if and only if det Υµ (a) = 0 for all µ ∈ spA x. By analogy with (5.1) and (7.1) we introduce the sets    MA := (M−∞ (SO) × R) ∪ R × {∞} × Lp ∪ M+∞ (SO) × R),  : M

:=

(M−∞ (SO) × R) ∪ (R × {±∞}) ∪ (M+∞ (SO) × R).

Applying Lemma 7.4 and Theorem 7.5 we infer the following (cf. Theorem 5.1). Lemma 7.6. For every (x, λ, µ) ∈ R × {∞} × Lp , the map A → A(x, λ, µ) given  by (6.5) extends for pseudo-differential operators a(x, D) with symbols a(x, λ) ∈ Ω  into C2×2 . For every (ξ, λ) ∈ M :, the to a Banach algebra homomorphism of A map A → A(ξ, λ) given for pseudo-differential operators a(x, D) with symbols

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 by A(ξ, λ) := a(ξ, λ) extends to a Banach algebra homomorphism of a(x, λ) ∈ Ω  A onto C.  is Fredholm on the space Lp (R) Theorem 7.7. A pseudo-differential operator A ∈ A (1 < p < ∞) if and only if min | det A(x, ∞, µ)| > 0

µ∈Lp

lim inf min |A(x ± 0, λ)| > 0, x→−∞ λ∈R

f or every x ∈ R,

lim inf min |A(x ± 0, λ)| > 0. x→+∞ λ∈R

(7.2) (7.3)

˙ the Proof. From Lemmas 7.4 and 7.6 it follows that for (ξ, η) ∈ M±∞ (SO) × R π π   cosets Aξ,η are invertible in the quotient algebras Λξ,η if and only if A(ξ, η) = 0 in case η ∈ R and A(ξ, ±∞) =  0 in π case η = ∞. On the π other hand, if ξ ∈ R and η = ∞, then setting r := χ+ I and q := W χ+ ξ,∞ in Theorem 7.5, we ξ ξ,∞ conclude from [8, Subsection 4.6] (also see [3, Corollary 5.3]) that the spectra of  π coincide the element e − (r − q)2 in the Banach algebras R = alg (e, r, q) and Λ ξ,∞ with Lp . Therefore, Theorem 7.5 together with Lemmas 7.4 and 7.6 immediately π are invertible in the quotient imply that for (ξ, η) ∈ R × {∞} the cosets A ξ,η  π if and only if det A(ξ, ∞, µ) = 0 for every (ξ, µ) ∈ R × Lp . Applying algebras Λ ξ,η  in the form Theorem 7.2 we obtain the Fredholm criterion for the operators A ∈ A det A(x, ∞, µ) = 0

for every

(x, µ) ∈ R × Lp ,

A(ξ, λ) = 0

for every

(ξ, λ) ∈ M±∞ (SO) × R,

or, equivalently, in the form inf min | det A(x, ∞, µ)| > 0,

x∈R µ∈Lp

inf

min |A(ξ, λ)| > 0.

(7.4)

ξ∈M±∞ (SO) λ∈R

Since the property (2.5) holds for A(ξ, λ) too, we infer from [27, Lemma 11.9] that the inequalities (7.4) are equivalent to (7.2)–(7.3).
  Thus, the algebra of pairs (x, µ) → A(x, ∞, µ), (ξ, λ) → A(ξ, λ) , where (x, µ) ∈ R × Lp and (ξ, λ) ∈ M±∞ (SO) × R, is a non-commutative algebra of  Fredholm symbols for the operators A ∈ A.

 of pseudo-differential operators 8. Algebra A  where Ω  = P C(R, V (R))∪E is a subset of L∞ (R, V (R)). Combining Let a(x, λ) ∈ Ω results of [27], [2] and [14] we obtain the following.  then the commutator ˙ and a(x, λ) ∈ Ω, Lemma 8.1. If c ∈ SO♦ , b ∈ Cp (R), cWb a(x, D)−a(x, D) cWb is compact on every Lebesgue space Lp (R) (1 < p < ∞). ˙ and p ∈ (1, ∞). Since SO♦ ⊂ SO and since Cp (R) ˙ Proof. Fix c ∈ SO♦ , b ∈ Cp (R), ˙ in Mp , we conclude that every operator cWb with is the closure of V0 (R) ∩ C(R) ˙ can be approximated in B(Lp (R)) by pseudo-differential c ∈ SO♦ and b ∈ Cp (R)

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 Therefore, taking into account the inclusion E ⊂ E operators with symbols in E. we infer from [27, Corollary 8.4] that the commutator cWb a(x, D) − a(x, D) cWb  If a ∈ P C(R, V (R)), then is compact on the space Lp (R) for every a(x, λ) ∈ E. the function  a given by (6.1) also belongs to P C(R, V (R)). Therefore, by (6.1) and by [2, Theorem 4.2], the commutator c  a(x, D) −  a(x, D) cI is compact on the space Lp (R). By [14, Lemma 7.3] and by (6.1) and (6.2), the commutator Wb  a(x, D)− a(x, D) Wb is also compact on the space Lp (R). Thus, the commutator cWb  a(x, D) −  a(x, D) cWb is compact on the space Lp (R). Applying Lemma 6.2, we obtain the compactness of the commutator cWb a(x, D) − a(x, D) cWb for a ∈ P C(R, V (R)), which completes the proof.
=A  p of B(Lp (R)) generated By [27, Lemma 10.1], the Banach subalgebra A  contains all by all pseudo-differential operators a(x, D) with symbols a(x, λ) ∈ Ω p  π := A/K.  compact operators in B(L (R)). Consider the quotient Banach algebra A p Let Z be the Banach subalgebra of B(L (R)) generated by all operators cWb , where ˙ and by all compact operators in B(Lp (R)). In virtue of c ∈ SO♦ and b ∈ Cp (R), π.  Lemma 8.1, the quotient Banach algebra Zπ := Z/K is a central subalgebra of A π From [3, Section 6] it follows that the maximal ideal space of Z can be identified with the set ˙ ∪ (R × {∞}), M (Zπ ) := (M∞ (SO♦ ) × R)  be the Banach algebra where M∞ (SO♦ ) is the fiber of M (SO♦ ) at ∞. Let Λ p generated by all operators in B(L (R)) that commute with every operator cWb ∈ Z  are called operators of local type to within compact operators (the elements of Λ π π  is a Banach subalgebra of Λ  π := Λ/K.   Then A Let Jξ,η be with respect to Z). π π   the closed two-sided ideal of Λ generated by the maximal ideal (ξ, η) ∈ M (Z ). Applying again the Allan-Douglas local principle, we obtain the following criterion.  is Fredholm on the space Lp (R) with 1 < p < ∞ Theorem 8.2. An operator A ∈ A  the coset A π := Aπ + Jπ is invertible in if and only if for every (ξ, η) ∈ M (Z) ξ,η ξ,η  π /Jπ .  π := Λ the Banach algebra Λ ξ,η ξ,η ˙ we set For (ξ, η) ∈ M∞ (SO♦ ) × R,  a(∓∞, η ± 0) if a(x, λ) ∈ P C(R, V (R)), a(ξ ± 0, η ± 0) :=  a(ξ, η ± 0) if a(x, λ) ∈ E. By analogy with Lemma 7.3 one can easily obtain the following.  Then Lemma 8.3. Let a(x, λ) ∈ Ω.  − − a(x, D) − a(ξ − 0, −∞)χξ Wχ− − a(ξ + 0, −∞)χ+ ξ Wχ− − a(ξ − 0, +∞)χξ Wχ+  π π −a(ξ + 0, +∞)χ+ ∈ Jξ,η if (ξ, η) ∈ R × {∞}, ξ Wχ+  a(x, D) − a(ξ − 0, η + 0)χ+ Wχ+ − a(ξ + 0, η + 0)χ− Wχ+ − a(ξ − 0, η − 0)χ+ Wχ− η η η π π ♦ −a(ξ + 0, η − 0)χ− W − ∈ J if (ξ, η) ∈ M∞ (SO ) × R, χη

ξ,η

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a(x, D) − a(ξ − 0, −∞)χ+ Wχ− − a(ξ + 0, −∞)χ− Wχ− − a(ξ − 0, +∞)χ+ Wχ+ π π −a(ξ + 0, +∞)χ− Wχ+ ∈ Jξ,η if (ξ, η) ∈ M∞ (SO♦ ) × {∞}. ∓ ± For ξ ∈ M∞ (SO♦ ) and η = ∞, we put χ± ξ = χη = χ0 . Then we immediately infer from Lemma 8.3 the following assertion.  ˙ ∪ (R × {∞}), the Lemma 8.4. For every (ξ, η) ∈ M (Zπ ) = M∞ (SO♦ ) × R) π  defined on the generators a(x, D) of the Banach algebra mapping δξ,η : A → A ξ,η  A by the rule  δξ,η [a(x, D)] := a(ξ − 0, η + 0)χ− + a(ξ + 0, η + 0)χ+ ξ Wχ+ ξ Wχ+ η η π − + + a(ξ − 0, η − 0)χξ Wχ− + a(ξ + 0, η − 0)χξ Wχ− , η η ξ,η

→Λ  π , and extends to a Banach algebra homomorphism δξ,η : A ξ,η



π



 sup δξ,η (A) Λ π ≤ A  := inf A + K f or all A ∈ A. π ) (ξ,η)∈M(Z

ξ,η

K∈K

Theorem 8.2, Lemma 8.4 and Theorem 7.5 immediately imply the following. 1 where the set Theorem 8.5. Let 1 < p < ∞ and let (ξ, λ, µ) ∈ M       1 := R × {∞} × Lp ∪ M∞ (SO♦ ) × R × Lp ∪ M∞ (SO♦ ) × {∞} × {0, 1} M  : A → A(ξ,  λ, µ) given for is equipped with the discrete topology. The map Ψ   by pseudo-differential operators A = a(x, D) ∈ A with symbols a(x, λ) ∈ Ω  A(ξ,λ,µ) = ;   < a(ξ + 0,λ − 0)µ + a(ξ + 0,λ + 0)(1 − µ) a(ξ + 0,λ − 0) − a(ξ + 0,λ + 0) ν(µ) ,   a(ξ − 0,λ − 0) − a(ξ − 0,λ + 0) ν(µ) a(ξ − 0,λ − 0)(1 − µ) + a(ξ − 0,λ + 0)µ  → BC(M  :A 1, C2×2 ) whose kernel extends to a Banach algebra homomorphism Ψ  is Fredholm on the space Lp (R) if and only if contains K. An operator A ∈ A  [det Ψ(A)](ξ, λ, µ) = 0

1. f or all (ξ, λ, µ) ∈ M

 A)  of Fredholm symThus, we again obtain the non-commutative algebra Ψ(  bols A possessing the property: the Fredholmness of any pseudo-differential ope on the space Lp (R) is equivalent to the invertibility of A on M 1. rator A ∈ A

9. An application to the Haseman boundary value problem Let Lp (R+ , w) be the weighted Lebesgue space over R+ := (0, ∞) with the norm f Lp(R+ ,w) = wf Lp (R+ ) where 1 < p < ∞ and w ∈ Ap (R+ ), that is, w is a nonnegative weight on R+ that satisfies the Muckenhoupt condition 1/p   1/q  1 p −q w(τ ) dτ w(τ ) dτ <∞ (9.1) sup sup ε>0 t∈R+ ε R+ (t,ε) R+ (t,ε)

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where R+ (t, ε) := R+ ∩ (t − ε, t + ε) and 1/p + 1/q = 1. Then (see, for example, [5]) the Cauchy singular integral operator SR+ given by  1 ϕ(τ ) dτ, for t ∈ R+ , (SR+ ϕ)(t) = lim ε→0 πi R \(t−ε,t+ε) τ − t + is bounded on the space Lp (R+ , w). Consider an orientation-preserving diffeomorphism α of R+ onto itself with ln α ∈ Cb (R+ ). Assume that (w ◦ α)/w, w/(w ◦ α) ∈ L∞ (R+ ).

(9.2)

Then the shift operator Bα : ϕ → ϕ◦α is also bounded on the space B(Lp (R+ , w)). Obviously, Bα is invertible and (Bα )−1 = Bα−1 ∈ B(Lp (R+ , w)). Let us apply the results of Section 7 to studying the Haseman boundary value problem (see, for example, [1], [32]): Find a function Φ analytic in C \ R+ , represented by the Cauchy type integral over R+ with a density ϕ ∈ Lp (R+ , w), where 1 < p < ∞ and w ∈ Ap (R+ ), and satisfying the boundary condition Φ+ (α(t)) = G(t)Φ− (t) + g(t),

for t ∈ R+ ,

(9.3)

where Φ± (t) are angular boundary values of Φ on R+ , G ∈ L∞ (R+ ), g ∈ Lp (R+ ,w). By the Sokhotski-Plemelj formulas Φ± = ±P± ϕ, with the Haseman problem (9.3) we can associate the equivalent singular integral operator with a shift N = Bα P+ + GP−

(9.4)

where P± = ± SR+ ), I is the identity operator and SR+ is the Cauchy singular integral operator. Clearly, N ∈ B(Lp (R+ , w)). Let us study the Fredholmness of the operator (9.4) on the space Lp (R+ , w). First, following [6], we introduce the slowly oscillating data of the operator (9.4). A function a ∈ Cb (R+ ) is called slowly oscillating at the points 0 and ∞ if the function x → a(ex ) is in SO, that is, if

lim max |a(x) − a(y)| : x, y ∈ [r, 2r] = 0, for s ∈ {0, ∞}. 1 2 (I

r→s

Let SO(R+ ) stand for the C ∗ -algebra of all functions in Cb (R+ ) which are slowly oscillating at 0 and ∞. We call a function w : R+ → R+ a slowly oscillating weight (at 0 and ∞) if w(r) = ev(r) ,

for r ∈ R+ ,

(9.5)

where v is a real-valued function in C 3 (R+ ) and the function r → rv  (r) is in SO(R+ ). One can show (see, for example, [5, Theorem 2.36] and [29, Section 5]) that the weight (9.5) satisfies the Muckenhoupt condition (9.1) if and only if −1/p < lim inf rv  (r) ≤ lim sup rv  (r) < 1/q, r→0

r→0

r→∞

r→∞

−1/p < lim inf rv  (r) ≤ lim sup rv  (r) < 1/q.

(9.6)

Let w ∈ ASO p (R+ ) denote the set of all slowly oscillating weights on R+ that satisfy (9.6).

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We call an orientation-preserving diffeomorphism α : R+ → R+ a slowly oscillating shift (at 0 and ∞) if for r ∈ R+ ,

α(r) = reω(r) ,

(9.7)

where ω is a real-valued function in C 3 (R+ ) and the functions ω and r → rω  (r) are in SO(R+ ). Let α be a slowly oscillating shift. Hence α > 0 on R+ , and since α (r) = (1 + rω  (r))eω(r) for all r ∈ R+ , we conclude that ln α ∈ Cb (R+ ), α ∈ SO(R+ ), and (9.8) inf (1 + rω  (r)) > 0. r∈R+

Moreover, because rω  (r) is slowly oscillating at 0 and ∞, the slow oscillation of ω at 0 and ∞ is equivalent to the property: lim rω  (r) = 0,

r→0

lim rω  (r) = 0.

(9.9)

r→∞

Finally, let G belong to the C ∗ -algebra P SO(R+ ) ⊂ L∞ (R+ ) generated by all functions in SO(R+ ) and by all piecewise continuous functions on R+ := [0, +∞]. Under these conditions singular integral operators in the Banach subalgebra S := alg {cI, SR+ : c ∈ P SO(R+ )} of B(Lp (R+ , w)) can be reduced to Mellin pseudo-differential operators OP (b) = E a(x, D)E −1 acting on the Lebesgue space Lp (R+ , dµ) with measure dµ(r) = dr/r, where a(x, D) are pseudo-differential ope studied in Section 7, b(r, λ) = a(ln r, λ) for (r, λ) ∈ R+ × R, rators in the algebra A and E is the following isometric isomorphism: E : Lp (R) → Lp (R+ , dµ),

(Ef )(x) = f (ln x), for x ∈ R+ .

(9.10)

Hence, Theorem 3.1 implies the following. Theorem 9.1. If b ∈ L∞ (R+ , V (R)), then the Mellin pseudo-differential operator OP (b) defined for functions u ∈ C0∞ (R+ ) by the iterated integral  iλ     r d 1 , f or r ∈ R+ , dλ b(r, λ) f () OP (b)f (r) = 2π R   R+ extends to a bounded linear operator on every space Lp (R+ , dµ) with p ∈ (1, ∞). Consider the operator R given by  f (τ ) 1 dτ, (Rf )(t) = πi R+ τ + t

for t ∈ R+ .

By [29, Corollary 4.2], the operator R with fixed singularities at 0 and ∞ is bounded on every space Lp (R+ , w) with 1 < p < ∞ and w ∈ Ap (R+ ). Put ma (r, ) := (a(r) − a())/(ln r − ln ) for a ∈ C(R+ ) and consider the map Γ : B(Lp (R+ , w)) → B(Lp (R+ , dµ)), A → Y AY −1 , p

(9.11)

p

where Y is the isometric isomorphism of L (R+ , w) onto L (R+ , dµ) given by (Y f )(r) = ev(r) r1/p f (r),

for r ∈ R+ .

(9.12)

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Theorem 9.2. If 1 < p < ∞, w ∈ ASO p (R+ ), α is a slowly oscillating shift on R+ , (9.2) is fulfilled and     0 < inf 1/p + mv (r, ) ≤ sup 1/p + mv (r, ) < 1, r, ∈R+

0 < inf

r, ∈R+

r, ∈R+

1/p + mv (r, ) 1/p + mv (r, ) ≤ sup < 1, 1 + mω (r, ) r, ∈R+ 1 + mω (r, )

then for the operators SR+ , Bα SR+ Bα−1 , R, Bα R ∈ B(Lp (R+ , w)) we have Γ(SR+ ) = OP (σ) + K0 ,  0, Γ(R) = OP (ν0 ) + K

Γ(Bα SR+ Bα−1 ) = OP (σα ) + Kα ,

(9.13)

α, Γ(Bα R) = OP (να ) + K

(9.14)

 0, K  α are compact operators on the space Lp (R+ , dµ) and, for where K0 , Kα , K (r, λ) ∈ R+ × R,   (9.15) σ(r, λ) := coth πλ + πi(1/p + rv  (r)) ,    λ + i(1/p + rv (r)) σα (r, λ) := coth π , (9.16) 1 + rω  (r)   (9.17) ν0 (r, λ) := 1/ sinh πλ + πi(1/p + rv  (r)) ,   exp iλω(r) − (1/p + rv  (r))ω(r)   . (9.18) να (r, λ) := sinh πλ + πi(1/p + rv  (r)) Proof. Relations (9.13) follow from [28, Theorems 8.3 and 6.3]. A straightforward computation on the basis of (9.7), (9.11) and (9.12) shows that for r > 0,  d (r/)1/p+mv (r, ) 1 f () [Γ(Bα R) f ](r) = =: I0 . (9.19) πi R+ 1 + eω(r) (r/)  Further, taking into account the Mellin transform identity  1 xη 1 exp(i(λ + iη) ln κ) iλ = x dλ (x > 0, 0 < η < 1, κ > 0) πi 1 + κx 2π R sinh(π(λ + iη)) and setting x = r/, η = 1/p + mv (r, ) and κ = eω(r) , we get    iλ   exp iλω(r) − (1/p + mv (r, ))ω(r) r 1 d   . I0 = dλ f () 2π R   sinh πλ + πi(1/p + mv (r, )) R+

(9.20)

Replacing mv (r, ) by rv  (r) in (9.20), we obtain [OP (να )f ](r), with να given by (9.18). Finally, using (9.19), (9.20) and [28, Theorem 6.3], we infer by analogy with [28, Theorem 8.3] that the operator Γ(Bα R) − OP (να ) is compact on the space Lp (R+ , dµ). This implies for Bα = I and ω = 0 that the operator Γ(R) − OP (ν0 )
with ν0 given by (9.17) is also compact. Thus, relations (9.14) hold too. Consider the isomeric shift operator Uα := cα Bα ∈ B(Lp (R+ , w)) where   ω(r) cα (r) := (α )1/p (w ◦ α)/w (r) = (1 + rω  (r))1/p eω(r)/p ev(re )−v(r) . (9.21) Obviously, cα is invertible in Cb (R+ ) in view of (9.8), and (Uα )−1 = Uα−1 .

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Lemma 9.3. If all the conditions of Theorem 9.2 are fulfilled, then the operator Uα SR+ Uα−1 − SR+ is compact on the space Lp (R+ , w). Proof. As the functions r → rv  (r) and ω are in Cb (R+ ) and   ω(r)    ω(r)    re    v re  − v(r) =  v (t) dt ≤ sup |tv  (t)| ω(r), r



t∈R+



− v(r) also is in Cb (R+ ). Moreover, we deduce that the function η(r) := v re η is slowly oscillating at 0 and ∞ because the function r → rv  (r) is in SO(R+ ) and therefore, due to (9.9),     lim rη  (r) = lim (1 + rω  (r))reω(r) v  reω(r) − rv  (r) = 0 for s ∈ {0, ∞}. r→s

ω(r)

r→s

Hence (w ◦ α)/w ∈ SO(R+ ). Since the function α : r → (1 + rω  (r))eω(r) is in SO(R+ ) too, we conclude from (9.21) that cα ∈ SO(R+ ). It follows from (9.15), (9.16) and (9.9) that σα −σ ∈ C(R+ , V0 (R)) and σα −σ = 0 on ∂(R+ × R). Hence, passing to pseudo-differential operators on R and using Corollary 4.2, we infer that the operators OP (σα −σ) and Γ(Bα SR+ Bα−1 −SR+ ) are compact on the space Lp (R+ , dµ). Analogously, [27, Corollary 8.2] implies that the operator Γ(cα SR+ − SR+ cα I) is also compact on Lp (R+ , dµ). Hence the operator
Γ(Uα SR+ Uα−1 − SR+ ) is compact, which implies the assertion of the lemma. Theorem 9.4. Under the conditions of Theorem 9.2, the operator Tα := Uα P+ +P− is Fredholm on the space Lp (R+ , w) and Ind Tα = 0. Proof. Since −4P+ P− = SR2 + − I = R2 and, by Lemma 9.3, (Uα P+ + P− )(Uα−1 P+ + P− ) = I + (Uα + Uα−1 − 2I)P+ P− + K1 , (Uα−1 P+ + P− )(Uα P+ + P− ) = I + (Uα + Uα−1 − 2I)P+ P− + K2 ,

(9.22)

where K1 , K2 are compact operators, we infer from Theorem 9.2 that Γ(I + (Uα + Uα−1 − 2I)P+ P− ) = OP (βα ) + K

(9.23)

where K is a compact operator in B(L (R+ , dµ)) and p

βα := 1 − 4−1 (cα να ν0 + cα−1 να−1 ν0 − 2ν02 ).

(9.24)

Taking into account the relations


(1 + rω  (r))1/p ev(re )−v(r)−rv (r)ω(r) − 1   , e−iλω(r) sinh πλ + πi(1/p + rv  (r))  reω(r)        tv (t) − rv  (r) dt/t = 0, lim v reω(r) − v(r) − rv  (r)ω(r) = lim ω(r)

cα (r)να (r, λ) − eiλω(r) ν0 (r, λ) =

r→s

r→s

r

where s ∈ {0, ∞}, we deduce that cα (r)να (r, λ) − eiλω(r) ν0 (r, λ) = 0 for all (r, λ) ∈ ∂(R+ × R). Analogously, cα−1 (r)να−1 (r, λ) − e−iλω(r) ν0 (r.λ) = 0

for all (r, λ) ∈ ∂(R+ × R).

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Hence, taking into account (9.24) and setting γα (r, λ) = 1 −

exp(iλω(r)) + exp(−iλω(r)) − 2   , 4 sinh2 πλ + πi(1/p + rv  (r))

for (r, λ) ∈ R+ × R,

(9.25)

we conclude that βα −γα ∈ C(R+ , V0 (R)) and βα −γα = 0 on ∂(R+ ×R). Therefore, Corollary 4.2 implies that the operator OP (βα − γα ) is compact. Thus, by (9.23), Γ(I + (Uα + Uα−1 − 2I)P+ P− ) = OP (γα ) + K  

(9.26)

p

where K is a compact operator on L (R+ , dµ). One can prove that the function (x, λ) → γα (ex , λ) belongs to the Banach algebra E ⊂ SO(R, V0 (R)) defined in Section 2. Then passing to pseudo-differential operators on R we infer from [27, Theorems 12.2 and 12.5] that the operator OP (γα ) is Fredholm on the space Lp (R+ , dµ) if and only if inf |γα (r, ±∞)| > 0,

r∈R+

lim inf min |γα (r, λ)| > 0 for s ∈ {0, ∞}, r→s

(9.27)

λ∈R

and in the case of Fredholmness

 1 arg γα (r, λ) (9.28) m→+∞ 2π (r,λ)∈∂Πm

where Πm = [1/m, m] × R and arg γα (r, λ) (r,λ)∈∂Πm denotes the increment of arg γα (r, λ) when the point (r, λ) traces the boundary ∂Πm of Πm counterclockwise. Clearly, conditions (9.27) are fulfilled because, according to (9.25),   exp − 2πλ − 2πi(1/p + rv  (r))  f+ (r, λ) f− (r, λ)  γα (r, λ) = 4 sinh2 πλ + πi(1/p + rv  (r)) Ind OP (γα ) =

lim

and γα (r, ±∞) = 1 for all r ∈ R+ , where the functions   f± (r, λ) := exp(±iλω(r)) − exp 2πλ + 2πi(1/p + rv  (r)) are separated from 0 for all (r, λ) ∈ R+ × R. Furthermore, since the function γα (r, λ, θ) = 1 − 

exp(iλθω(r)) + exp(−iλθω(r)) − 2   4 sinh2 πλ + πi(1/p + rv  (r))

is continuous and separated from 0 for all (r, λ, θ) ∈ R+ × R × [0, 1], we conclude

that arg  γα (r, λ, θ) (r,λ)∈∂Πm does not depend on θ ∈ [0, 1]. Consequently,   1 1 = = 0, arg γα (r, λ) arg γ α (r, λ, 0) 2π 2π (r,λ)∈∂Πm (r,λ)∈∂Πm

and therefore, by (9.28), Ind OP (γα ) = 0. Hence, applying (9.26) and (9.22), we deduce that the operators Tα = Uα P+ +P− and Tα−1 = Uα−1 P+ +P− are Fredholm on the space Lp (R+ , w) and Ind Tα + Ind Tα−1 = 0.

(9.29)

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It remains to observe that Uα is invertible and, with Cf = f , C(Uα P+ + P− )C = Uα P− + P+ = Uα (Uα−1 P+ + P− ). Hence Ind Tα = Ind Tα−1 , which implies due to (9.29) that Ind Tα = 0.




Theorem 9.5. Let G ∈ P SO(R+ ) and all conditions of Theorem 9.2 hold. Then the operator N = Bα P+ + GP− is Fredholm on the space Lp (R+ , w) if and only if   min G(r + 0)(1 − µ) + G(r − 0)µ > 0 f or every r ∈ R+ , (9.30) µ∈Lp

 
lim inf min eiω(r)(λ+i(1/p+rv (r))) P+ (r, λ) + G(r)P− (r, λ) > 0, r→s

(9.31)

λ∈R

   where s ∈ {0, ∞} and P± (r, λ) := 2−1 1 ± coth πλ + πi(1/p + rv  (r)) . Proof. By analogy with (9.22), we obtain (Bα P+ + GP− )(Uα−1 P+ + P− ) = c−1 α Nα + K,

(9.32)

p

where K is a compact operator on L (R+ , w), Nα := P+ + Gα P− + 4−1 (Uα + Gα Uα−1 − (1 + Gα )I)P+ P− ,

(9.33)

and Gα := cα G. Hence, by (9.32) and Theorem 9.4, the operators N and Nα are Fredholm on the space Lp (R+ , w) only simultaneously, and in the case of Fredholmness Ind N = Ind Nα . It is easily seen that Γ(Nα ) = OP (ζα ) + K  , 

(9.34)

p

where K is a compact operator on the space L (R+ , dµ) and according to (9.33), ζα := (P+ + Gα P− ) − 4−1 (cα να ν0 + Gα cα−1 να−1 ν0 − (1 + Gα )ν02 ). By analogy with (9.25), we define the function    α (r)P− (r, λ) φα (r, λ) := P+ (r, λ) + G  α (r)) α (r) exp(−iλω(r)) − (1 + G exp(iλω(r)) + G   − 2 4 sinh πλ + πi(1/p + rv  (r))  α (r) := e(1/p+rv
(r))ω(r) G(r) and Gα (r) − G α (r) → 0 for (r, λ) ∈ R+ × R, where G as r → s ∈ {0, ∞}. Then ζα − φα δα ∈ P C(R+ , V0 (R)) and ζα − φα δα = 0 on  α )P− ∈ C(R+ , V0 (R)). Therefore, from ∂(R+ × R), where δα := P+ + (Gα /G Corollary 4.2 it follows that the operator OP (ζα − φα δα ) is compact. Lemmas 6.2 and 7.1 imply that the operator OP (φα δα ) − OP (φα )OP (δα ) also is compact. As δα = 0 on ∂(R+ × R), the operator OP (δα ) is Fredholm on the space Lp (R+ , dµ) by Theorem 5.1. Hence, taking into account (9.34) we infer that the operator N is Fredholm on the space Lp (R+ , w) if and only if the Mellin pseudo-differential operator OP (φα ) is Fredholm on the space Lp (R+ , dµ), and in the case of Fredholmness Ind N = Ind OP (φα ) + Ind OP (δα ). (9.35) One can show that the function a(x, λ) := φα (ex , λ) is the symbol of a pseudo (see Section 7). Therefore, passing differential operator belonging to the algebra A

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 where E is given to pseudo-differential operators a(x, D) = E −1 OP (φα )E ∈ A, by (9.10), and applying Theorem 7.7 we infer that the Mellin pseudo-differential operator OP (φα ) is Fredholm on the space Lp (R+ , dµ) if and only if min | det Φα (r, ∞, µ)| > 0

µ∈Lp

lim inf min |φα (r ± 0, λ)| > 0, r→0

λ∈R

for every r ∈ R+ ,

lim inf min |φα (r ± 0, λ)| > 0, r→∞ λ∈R

(9.36) (9.37)

where for r ∈ R+ and λ ∈ Lp , ; − <  −  φα (r + 0) − φ+ φα (r + 0)µ + φ+ α (r + 0)(1 − µ) α (r + 0) ν(µ) Φα (r, ∞, µ) :=  ,  + − + φ− α (r − 0) − φα (r − 0) ν(µ) φα (r − 0)(1 − µ) + φα (r − 0)µ +  with φ− α (r) := φα (r, +∞) = 1 and φα (r) := φα (r, −∞) = Gα (r). Since

det Φα (r, ∞, µ)

α (r + 0)(1 − µ) + G α (r − 0)µ = G  
= e(1/p+rv (r))ω(r) G(r + 0)(1 − µ) + G(r − 0)µ ,

we infer that condition (9.36) is equivalent to (9.30). α (r) := e(1/p+rv
(r))ω(r) G(r), it follows from the equality Finally, since G     α (r)P− (r, λ) e−iλω(r) P+ (r, λ) + P− (r, λ) φα (r, λ) = eiλω(r) P+ (r, λ) + G that inequalities (9.37) are equivalent to the relations (9.31).




If G has a finite set of discontinuous, then combining (9.35), Theorem 6.3 and [27, Theorem 12.5] one can also calculate Ind N .

References [1] A.V. A˘ızenshtat, Yu.I. Karlovich and G.S. Litvinchuk, The method of conformal gluing for the Haseman boundary value problem on an open contour, Complex Variables 28 (1996), 313–346. [2] M.A. Bastos, A. Bravo and Yu.I. Karlovich, Convolution type operators with symbols generated by slowly oscillating and piecewise continuous matrix functions, in Operator Theoretical Methods and Applications to Mathematical Physics, The Erhard Meister Memorial Volume, Editors: I. Gohberg at al., Birkh¨ auser, 2004, 151–174. [3] M.A. Bastos, A. Bravo and Yu.I. Karlovich, Symbol calculus and Fredholmness for a Banach algebra of convolution type operators with slowly oscillating and piecewise continuous data, Math. Nachr. 269–270 (2004), 11–38. [4] M.A. Bastos, Yu.I. Karlovich and B. Silbermann, Toeplitz operators with symbols generated by slowly oscillating and semi-almost periodic matrix functions, Proc. London Math. Soc. 89 (2004), 697–737. [5] A. B¨ ottcher and Yu.I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, Birkh¨ auser, Basel, 1997. [6] A. B¨ ottcher, Yu.I. Karlovich and V.S. Rabinovich, The method of limit operators for one-dimensional singular integrals with slowly oscillating data, J. Operator Theory 43 (2000), 171–198.

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