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 Science...

Author: Joachim Toft | Man Wah Wong | Hongmei Zhu (Editors)

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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)

6

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

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

χreg : V \ Y → L \ Y.

Incidentally we will write

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

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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)

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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)

Rays of Minimal Growth

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 .

40

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 .

42

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

44

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

as k → ∞. Note that for k large we have m k λ0 ∈ res A∧,D , so λ0 ∈ res A∧,κ−1 k D and therefore, Dk ∈ V . Pick v ∈ D0 ∩ K with v = 1. Let πDk denote the orthogonal projection on Dk . Since Dk → D0 as k → ∞, we have πDk → πD0 , so vk = πDk v → πD0 v = v as k → ∞. Since Dk ∈ V , vk − v = 0 and πK,Dk vk = 0. Hence v − 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

Rays of Minimal Growth

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

48

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.

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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

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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}.

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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).

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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)

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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 .

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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 ).

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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.

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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 .

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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 .

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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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M. Ruzhansky and V. Turunen

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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M. Ruzhansky and V. Turunen

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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M. Ruzhansky and V. Turunen

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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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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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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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

eεx Au ∈ V =⇒ eεx 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.

118

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

ηβ

−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¯ = ex Be−x u, v¯ = u, e−x Bex 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 βα α = eiy·ξ ξ−2N Dxβ q(x, ξ) (1 − ∆y )N χ(|y|)Dxα−β u(x − y) dydξ, β βα

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

3

Actually Q is continuous on Cb∞ .

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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 βα −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 βα |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)

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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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A. Oliaro and P. Popianov

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

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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

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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.

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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.

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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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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

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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

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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

Pseudo-diﬀerential Operators on Modulation Spaces

177

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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183

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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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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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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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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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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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 p2 ,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.

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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)

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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].

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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

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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

Pseudo-diﬀerential Operators on Modulation Spaces

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.

202

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

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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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[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.

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[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

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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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[7] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz Operators, Akademie-Verlag, Berlin, 1989 and Springer, Berlin, 1990. [8] A. B¨ ottcher and I.M. Spitkovsky, Pseudodiﬀerential operators with heavy spectrum, Integral Equations Operator Theory 19 (1994), 251–269. [9] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157. [10] R.R. Coifman and Y. Meyer, Au del` a des op´erateurs pseudodiﬀ´ erentiels, Ast´erisque 57 (1978), 1–184. [11] H.O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodiﬀerential operators, J. Funct. Anal. 18 (1975), 115–131. [12] H.O. Cordes, Elliptic Pseudo-diﬀerential Operators – An Abstract Theory, Springer, Berlin, 1979. [13] R.V. Duduchava, Integral equations of convolution type with discontinuous coeﬃcients, Soobshch. Akad. Nauk Gruz. SSR 92 (1978), 281–284 [Russian]. [14] R.V. Duduchava, Integral Equations with Fixed Singularities, Teubner, Leipzig, 1979. [15] E.M. Dynkin, Methods of the theory of singular integrals: Hilbert transform and Calder´ on-Zygmund theory, in Commutative Harmonic Analysis I: General Surveys, Classical Results, Editors: V. P. Khavin and N.K. Nikol’skiy, Encyclopedia of Mathematical Sciences 15, Springer, Berlin, 1991; Russian original, VINITI, Moscow, 1987. [16] T. Finck, S. Roch and B. Silbermann, Two projection theorems and symbol calculus for operators with massive local spectra, Math. Nachr. 162 (1993), 167–185. [17] I. Gohberg and N. Krupnik, Singular integral operators with piecewise continuous coeﬃcients and their symbols, Math. USSR - Izv. 5 (1971), 955–979 [Russian]. [18] I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations, Vols. 1 and 2, Birkh¨ auser, Basel, 1992; Russian original, Shtiintsa, Kishinev, 1973. [19] I. Gohberg and N. Krupnik, Extension theorems for invertibility symbols in Banach algebras, Integral Equations Operator Theory 15 (1992), 991–1010. [20] N.B. Haaser and J.A. Sullivan, Real Analysis, Dover Publications, New York, 1991. [21] W. Hoh, A symbolic calculus for pseudo-diﬀerential operators generating Feller semigroups, Osaka J. Math. 35 (1998), 789–820. [22] W. Hoh, Perturbations of pseudodiﬀerential operators with negative deﬁnite symbol, Appl. Math. Optimization 45 (2002), 269–281. [23] R.A. Hunt, On the convergence of Fourier series, in Orthogonal Expansions and Their Continuous Analogues, Editor: D. Haimo, Southern Illinois Univ. Press, Carbondale, 1968, 235–255. [24] N. Jacob and A.G. Tokarev, A parameter-dependent symbolic calculus for pseudodiﬀerential operators with negative-deﬁnite symbols, J. London Math. Soc. 70 (2004), 780–796. [25] O.G. Jørsboe and L. Melbro, The Carleson–Hunt Theorem on Fourier Series, Springer, Berlin, 1982. [26] Yu.I. Karlovich, On the Haseman problem, Demonstratio Mathematica 26 (1993), No. 3–4, 581–595. [27] Yu.I. Karlovich, An algebra of pseudodiﬀerential operators with slowly oscillating symbols, Proc. London Math. Soc. 92 (2006), 713–761.

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[28] Yu.I. Karlovich, Pseudodiﬀerential operators with compound slowly oscillating symbols, in: Oper. Theory: Adv. Appl., Vol. 171, 2006, 189–224, to appear. [29] Yu.I. Karlovich and E. Ram´ırez de Arellano, Singular integral operators with ﬁxed singularities on weighted Lebesgue spaces, Integral Equations Operator Theory 48 (2004), 331–363. [30] C.E. Kenig and P.A. Tomas, Maximal operators deﬁned by Fourier multipliers, Studia Math. 68 (1980), 79–83. [31] M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustylnik and P.E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Noordhoﬀ I.P., Leyden, 1976; Russian original, Nauka, Moscow, 1966. [32] G.S. Litvinchuk, Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift, Kluwer, Dordrecht, 2000. m , [33] J. Marschall, Pseudo-diﬀerential operators with nonregular symbols of the class Sρδ Comm. Partial Diﬀerential Equations 12 (1987), 921–965. [34] J. Marschall, On the boundedness and compactness of nonregular pseudo-diﬀerential operators, Math. Nachr. 175 (1995), 231–262. [35] J. Marschall, Nonregular pseudo-diﬀerential operators, Z. Anal. Anwendungen 15 (1996), 109–148. [36] S.C. Power, Fredholm Toeplitz operators and slow oscillation, Canad. J. Math. 32 (1980), 1058–1071. [37] S. Roch and B. Silbermann, Algebras of Convolution Operators and Their Image in the Calkin Algebra, Report R-Math-05/90, Karl-Weierstrass-Inst. f. Math., Berlin, 1990. [38] L. Schwartz, Analyse Math´ematique, Vol. 1, Hermann, 1967. [39] I.B. Simonenko and Chin Ngok Min, Local Method in the Theory of One-Dimensional Singular Integral Equations with Piecewise Continuous Coeﬃcients, Noetherity, University Press, Rostov on Don, 1986 [Russian]. [40] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993. [41] M.E. Taylor, Tools for PDE. Pseudodiﬀerential Operators, Paradiﬀerential Operators, and Layer Potentials, American Mathematical Society, Providence, RI, 2000. Yu.I. Karlovich Facultad de Ciencias Universidad Aut´ onoma del Estado de Morelos Av. Universidad 1001, Col. Chamilpa, C.P. 62209 Cuernavaca, Morelos, M´exico e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 235–249 c 2006 Birkh¨ auser Verlag Basel/Switzerland

A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform Paolo Boggiatto, Giuseppe De Donno and Alessandro Oliaro Abstract. Motivated by problems in signal analysis, we deﬁne a class of timefrequency representations which is based on the short-time Fourier transform and depends on two ﬁxed windows. We show that this class can be viewed as a link between the classical Rihaczek representation and the spectrogram. Correspondingly we formulate for this class a suitable general form of the uncertainty principle which have, as limit case, the uncertainty principles for the Rihaczek representation and for the spectrogram. We ﬁnally consider the questions of marginal distributions. We compute them in terms of convolutions with the windows and prove simple conditions for which average and standard deviation of the distributions in our class coincide with that of their marginals. Mathematics Subject Classiﬁcation (2000). Primary 47G30; Secondary 42B10, 47G10. Keywords. Time-frequency representations, short-time Fourier transform, uncertainty principles, Rihaczek distribution.

1. Time-frequency representations The total energy of a signal f (x), (x ∈ Rd ), is commonly interpreted in physics and in signal analysis as its L2 norm f L2 . Accordingly, the positive function |f (x)|2 represents the density of the energy of the signal with respect to the time (although x is the “time” variable we shall assume here x ∈ Rd for generality and also in view of possible other interpretations). As the Fourier transform fˆ(ω) gives the frequencies contained in the signal, then |fˆ(ω)|2 represents the energy density with respect to the frequency variable ω and the equality f L2 = fˆL2 , stated by the Plancherel theorem, is interpreted, in this context, as the natural assertion that the total energy of a signal does not change if we switch from the time representation

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f (x) to the frequency representation fˆ(ω). Actually the sentence “the signal f (x)”, which is of common use and we shall also adopt, is somehow misleading because f (x) and fˆ(ω) should be viewed as two diﬀerent, and in some sense complementary, representations of the same physical phenomenon constituted by the signal. In the L2 frame, which corresponds to the space of ﬁnite energy signals, the fact that the Fourier transform is a bijection means that each of the two representations contain in reality all information about the signal. However in each representation some information are contained in an explicit and others in a more implicit form and this can make the latter diﬃcult to be recognized. For example it is clear that from the time representation f (x) of a signal the support supp f immediately indicates when the signal takes place in time, however the frequencies contained in the signal are not to be detected directly but given only through the “form” of the graph of the function f (x). In a quite symmetrical way the function fˆ(ω) contains explicit information about the frequencies of the signal but the information about when these frequencies are present in the signal is hidden in the “form” of the function fˆ(ω). For example the Fourier transform e2πiaω fˆ(ω) of the delayed signal f (x − a) diﬀers from the transform fˆ(ω) of the original signal f (x) only in the “shape” factor e2πiaω . A clear exposition on these subjects can be found for example in [13]. One fundamental problem of time-frequency analysis is to ﬁnd suitable representations for the energy density of a signal simultaneously with respect to time and frequency. Of course this question is equivalent to the deﬁnition of a 2ddimensional distribution density Q(f )(x, ω) depending on the variables x ∈ Rd and ω ∈ Rd . However, diﬀerently to what happens with usual distributions in statistics and probability theory, here the uncertainty principle put intrinsic limitations to the natural properties one would expect from a 2d-dimensional distribution. The basic problem is that “instantaneous frequency” has no physical meaning, and therefore, no matter which time-frequency representation Q(f ) is used, its point values do not have any univocal physical interpretation. More precisely, let Q(f )(x, ω) be a time-frequency distribution of the energy of a signal f . Then some desirable features are expressed by the following conditions: ◦ Positivity: Q(f )(x, ω) should be real and positive. ◦ No spreading eﬀect: If supp f ⊂ I for an interval I ⊂ Rd then Πx supp Q(f ) ⊂ I (where Πx is the orthogonal projection from Rdx × Rdω to Rdx ); (1) analogously, if supp fˆ ⊂ J then Πω supp Q(f ) ⊂ J. ◦ Marginal distributions: 2 2 ˆ Rd Q(f )(x, ω)dx = |f (ω)| ; Rd Q(f )(x, ω)dω = |f (x)| . It turns out that these conditions are actually incompatible with the uncertainty principle and they can therefore be satisﬁed only with a certain degree of approximation. Many diﬀerent representations have been deﬁned in the literature with the aim of approaching as near as possible an ideal representation (see [5], [8], [9],

A Class of Quadratic Time-frequency Representations

237

[12], [13]). A general frame for the study of these representations is given by the so-called Cohen class, see for reference [3], [4]. Three of the most known representations are the spectrogram, the Rihaczek and the Wigner representations. For what concerns the spectrogram, the idea, originally due to D. Gabor, is to focus the Fourier transform on small intervals in time and to analyze the frequencies present in these intervals. This is performed by multiplying the signal f (t) by a cut-oﬀ, or window, function φ(t) that can be translated by a parameter x along the time axis, before taking the Fourier transform. In this way we are lead to the deﬁnition of the Gabor transform or short-time Fourier transform (brieﬂy STFT) of a signal f (t): e−2πitω φ(t − x)f (t)dt. (2) Vφ f (x, ω) = Rd

It indicates the frequencies ω contained in the signal in a neighborhood of the time x. The conjugation on the window appears just for mathematical convenience so that Vφ f (x, ω) = (f, φx,ω )L2 , where φx,ω (t) = e2πiωt φ(t − x). The quadratic representation corresponding to the STFT is called spectrogram (see [4], [10]) and is deﬁned as Spφ (f )(x, ω) = |Vφ f (x, ω)|2 .

(3)

It is of course a positive distribution but it does not satisﬁes the marginals and has a spreading eﬀect depending on the support of the window φ. The Rihaczek quadratic representation is essentially deﬁned as the product of the signal f (x) with its Fourier transform fˆ(ω), more precisely it is the distribution R(f )(x, ω) = e−2πixω f (x)fˆ(ω).

(4)

Though its elementary deﬁnition it has reasonable physical motivations and was widely used in the time-frequency analysis of signals (see [4], [11]). As one can immediately verify, it satisﬁes the marginals and has no spreading eﬀect, however it is evidently not positive. The third very popular representation is the Wigner distribution (see [18]) deﬁned as e−2πixω f (x + t/2)f (x − t/2)dt. (5) W ig(f )(x, ω) = Rd

As the Rihaczek representation, it is not positive but it has no spreading eﬀect and it satisﬁes the marginals (see [4], [10], [12]). The Wigner distribution and the STFT are the basic tools for the theory of Weyl and localization operators respectively, about these wide ﬁelds there exists a huge literature, see for example [2], [16], [17], [19], [15], [20]. We shall be addressed in this paper essentially to the ﬁrst two above-mentioned representations; related questions about the Wigner representation will be treated in a further paper. We shall establish a link between the spectrogram and the Rihaczek representation by deﬁning in the next section a class of time-

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frequency representations Q(f )(x, ω) = Qφ,ψ (f )(x, ω) which depend on two window functions or distributions φ, ψ. In a suitable general functional setting for windows and signals, we shall show that a sort of path between the spectrogram and the Rihaczek representation is allowed within this class, permitting to pass from one representation to the other by a continuous variation of the windows. Of course the degree of approximation by which the conditions (1) are satisﬁed will also change accordingly. As we explain in Section 2, the motivation of our deﬁnition is not only a mathematical one but it has been suggested by the attempt to reduce the spreading eﬀect of the spectrogram, result which can be easily conﬁrmed by computer simulations. In Section 3 we treat the question of the uncertainty principle. We formulate a simple uncertainty principle for the representation Qφ,ψ in the Lp frame and show that it links diﬀerent form of uncertainty principles for the Rihaczek and spectrogram representations. Finally in Section 4 we consider the problem of the marginal distributions. We compute the marginals of the representation Qφ,ψ in terms of convolution with the product of the windows and deduce from this that the case where our representation reduces to the Rihaczek is the only case where both marginal conditions in (1) are satisﬁed. Further we give in the general case simple conditions on the windows in order that average and standard deviation of the marginals of Qφ,ψ coincide with that of the “expected” marginal distributions |f (x)|2 and |fˆ(ω)|2 .

2. The representation Qφ,ψ Let q : E × E → F be a sesquilinear form from the cartesian product of some functional space E to another space F . Then, as mentioned in [10], there are at least two natural ways of obtaining a quadratic representation Q on E. One can consider Q(f ) = q(f, f ) or, for ﬁxed φ, one can set Q(f ) = |q(f, φ)|2 . The ﬁrst of these methods is followed in order to pass from the “cross” Rihaczek sesquilinear form R(f, g)(x, ω) = e−2πixω f (x)ˆ g (ω) to the (quadratic) Rihaczek representation (4), as well as from the “cross” Wigner distri bution W ig(f, g)(x, ω) = Rd e−2πixω f (x + t/2)g(x − t/2)dt to the quadratic Wigner representation (5). The second is used to pass from the short-time Fourier transform Vφ f (x, ω), which is sesquilinear in the couple (f, φ), to the spectrogram Spφ (f )(x, ω) = |Vφ f (x, ω)|2 . We propose here a third way of deﬁning a quadratic representation. Let q be a given sesquilinear form, then, for ﬁxed φ and ψ, we set Qφ,ψ (f )(x, ω) = q(f, φ)q(f, ψ).

(6)

We observe that, in the case φ = ψ our deﬁnition coincides obviously with the second method mentioned above. On the other hand it can also be viewed as an application of the ﬁrst method to the sesquilinear form q˜(f, g) = q(f, φ)q(g, ψ).

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In order to justify our deﬁnition we shall now focus on the case where q(f, φ) = Vφ f and introduce some considerations about the spreading eﬀect of the spectrogram. As we mentioned in the previous section, the presence of the window φ is responsible for this eﬀect. More precisely, as a consequence of the uncertainty principle, if the support of the window φ is well concentrated around the origin then we have a little spreading eﬀect in time, i.e., a good “localization” in time, but a considerable spreading eﬀect in the frequency, i.e., a bad “localization” in frequency. Vice versa, if the support of φ is “widespread” then the localization is bad in time but good in frequency. This suggests that instead of the spectrogram Spφ (f ) = |Vφ f |2 = Vφ f Vφ f , we can ﬁx a second window ψ and consider the form Qφ,ψ (f ) = Vφ f Vψ f .

(7)

The advantage is that, if φ is very concentrated around the origin and ψ very spread on the plane (i.e., ψˆ is very concentrated), then Vφ f will show a good localization in time and Vψ f a good localization in frequency. Due to the reciprocal cut-oﬀ eﬀect in the product, Vφ f Vψ f will have both good localizations. The drawback is of course the lost of positivity of the distribution. In the limit case where φ tends to δ and ψ to 1 then we shall have no spreading eﬀect at all, i.e., the third condition in (1) will be exactly satisﬁed. We shall prove that this case will coincide with that of the Rihaczek representation. The better behavior with regard to the support property of the representation Qφ,ψ compared with the spectrogram can be made precise in many diﬀerent ways. For example we can remark that Vφ f = (e−2πiω(·) f ) ∗ g and therefore, from the support property of the convolution, Πx (supp Vφ f ) ⊂ supp φ + supp f . From the fundamental equality of time-frequency analysis Vφ f (x, ω) = e−2πixω Vφˆfˆ(ω, −x) we also have Πω (supp Vφ f ) ⊂ supp φˆ + supp fˆ, so we have proved the following property. δ

Proposition 2.1. Let B0j indicate balls in Rd of radius δj > 0, j = 1, 2. i) If supp φ ⊂ B0δ1 then Πx (supp Spφ (f )) ⊂ supp f + B0δ1 ; ii) If supp φˆ ⊂ B0δ2 then Πω (supp Spφ (f )) ⊂ supp fˆ + B0δ2 . Of course hypothesis i) and ii) of this proposition cannot be both satisﬁed, which is one possible form of expressing that for the spectrogram good localization in time and in frequency are complementary. An immediate consequence of this same proposition is that, on the contrary, for the representation Qφ,ψ = Vφ f Vψ f we can have arbitrary good localization both in time and frequency, more precisely: δ

Proposition 2.2. Let B0j indicate balls in Rd of radius δj > 0, j = 1, 2. Suppose that supp φ ⊂ B0δ1 and supp ψˆ ⊂ B0δ2 , then i) Πx (supp Qφ,ψ (f )) ⊂ supp f + B0δ1 ; ii) Πω (supp Qφ,ψ (f )) ⊂ supp fˆ + B0δ2 .

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We present next two possible functional settings for the representation Qφ,ψ . We start by recalling some properties of the STFT. Let f ⊗ φ be the (tensor) prodτa F (x, t) = uct f ⊗ φ (x, t) = f (x)φ(t), τa the asymmetric coordinate transform F (t, t − x), and F2 the partial Fourier transform F2 F (x, ω) = Rd e−2πitω F (x, t) dt of a function F on R2d . Then an easy computation gives (see [7]): Lemma 2.3. If f, φ ∈ L2 (Rd ), then Vφ f (x, ω) = F2 τa (f ⊗ φ). Lemma 2.3 is used to extend the STFT to a map from S (Rd ) × S (Rd ) to S (Rd ). Namely, if f, φ ∈ S (Rd ), then f ⊗ φ ∈ S (R2d ), with (f ⊗ φ, χ) = (fx , (φω , χ)), χ ∈ S(R2d ), as we consider distributions as anti-linear functionals. Further both operators τa and F2 are isomorphisms on S (R2d ). Thus Vφ f is a well-deﬁned tempered distribution, whenever f, φ ∈ S (Rd ). Following these general lines Qφ,ψ (f, g) = Vφ f Vψ g would result as a product of two tempered distributions. To overcome problems of deﬁnition we introduce then some restrictions. Deﬁniton 2.4. Let B ∞ (Rd ) be the space of smooth bounded functions together with all its derivatives, i.e., the space of C ∞ functions h such that for every multi-index α = (α1 , . . . , αd ) ∈ Nd there exists a constant Mα with |∂xα h(x)| ≤ Mα on Rd . By the diﬀerentiation under the integral and the Riemann-Lebesgue theorem one immediately obtain the following result. Lemma 2.5. If ψ ∈ B ∞ (Rd ) and g ∈ S(Rd ), then Vψ g ∈ B ∞ (R2d ). Of course the role of g and ψ in Lemma 2.5 can be exchanged. We observe now that the form qφ,ψ (f, g) = Vφ f Vψ g makes sense whenever Vφ f is a tempered distribution and Vψ g ∈ B ∞ (Rd ). From Lemma 2.5 this is the case for example of the next proposition. Proposition 2.6. Let f, φ ∈ S (Rd ) and g ∈ S(Rd ), ψ ∈ B ∞ (Rd ) (or vice versa g ∈ B ∞ (Rd ), ψ ∈ S(Rd )). Then qφ,ψ (f, g) is a well-deﬁned tempered distribution in S (R2d ). Better regularity for the time-frequency representation qφ,ψ can easily be obtained for example in the following situation. Proposition 2.7. Let f, g ∈ S(Rd ) and φ, ψ ∈ S (Rd ) (or vice versa f, g ∈ S (Rd ), and φ, ψ ∈ S(Rd )), then qφ,ψ (f, g) ∈ C ∞ (R2d ). As a second functional setting for the representation (7) we consider Lp spaces. We shall need the following boundedness result about the STFT, see [14], [1]. Proposition 2.8. Let us ﬁx q and p satisfying q ≥ 2 and q ≤ p ≤ q, where q is the conjugate of q, i.e., 1q + q1 = 1. Then V : Lp (Rd ) × Lp (Rd ) → Lq (R2d ) is continuous, in particular Vφ f Lq ≤ f Lp φLp .

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We can now expresses the behavior of (7) in the context of Lp spaces by the following proposition. Theorem 2.9. Let q ∈ [1, ∞], qj ∈ [2, ∞], pj ∈ [1, ∞], j = 1, 2 satisfy q11 + and qj ≤ pj ≤ qj , where qj is the conjugate of qj , i.e., q1j + q1 = 1. Then

1 q2

= 1q ,

j

p1

p2

q : (f, φ, g, ψ) ∈ L (R ) × L (R ) × L (R ) × L (R ) → qφ,ψ (f, g) ∈ Lq (R2d ) (9) is continuous, in particular d

p1

d

d

p2

d

qφ,ψ (f, g)q ≤ f Lp1 φLp1 gLp2 ψLp2 .

(10)

q2 Proof. We begin by remarking that q = qq11+q ≥ 1 for qj ≥ 2, j = 1, 2, which 2 means that (9) makes sense under our hypothesis. From Proposition 2.8 we have that Vϕ h : Lp (Rd ) × Lp(Rd ) → Lq (R2d ), with q ≥ 2 and q ≤ p ≤ q, is continuous, and in particular Vϕ hLq ≤ hLp ϕLp holds. So we obtain that

q = Vφ f Vψ g : Lp1 (Rd ) × Lp1 (Rd ) × Lp2 (Rd ) × Lp2 (Rd )

→

Lq1 (R2d ) · Lq2 (R2d )

with Vφ f Lq1 ≤ f Lp1 φLp1 , Vψ gLq2 ≤ gLp2 ψLp2 . where Lq1 (R2d ) · Lq2 (R2d ) indicates the subset of S (Rd ) of all the products f g of a function f ∈ Lq1 (R2d ) with a function g ∈ Lq2 (R2d ). The generalized H¨older’s formula: 1 1 1 Vφ f Vψ gq ≤ Vφ f Lq1 Vψ gLq2 , for (11) + = q1 q2 q proves then (10). Moreover (11) means: Lq1 (R2d ) · Lq2 (R2d ) → Lq (R2d ) ,

which proves (9).

We end this section by showing that the form (7) represents a link between the spectrogram Spφ (f ) in (3) and the Rihaczek’s distribution R(f ) in (4). Theorem 2.10. Let f ∈ S(Rd ), and φ ∈ S (Rd ), then i) Qφ,φ (f ) = Spφ (f ); ii) Qδ,1 (f ) = R(f ). Proof. The choice φ = ψ in (7) proves immediately the ﬁrst part of the theorem. Consider now the limit cases φ = δ, the point measure (δ, ϕ) = ϕ(0), and ψ = 1. For χ ∈ S(R2d ) we have: (Vδ f, χ) = (F2 τa (f ⊗ δ), χ) = (f ⊗ δ, τa −1 F2 −1 χ) = (f, (δ, τa −1 F2 −1 χ)) = (f, =

R2d

Rd

e2πixω χ(x − t, ω) dω |t=0 ) = (f,

Rd

e2πixω χ(x, ω) dω)

e−2πixω f (x)χ(x, ω) dxdω = (e−2πixω f (x), χ(x, ω)) .

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Therefore Vδ f (x, ω) = e−2πixω f (x).

(12)

Now by relation (12) we obtain V1 g(x, ω) = e−2πixω Vˆ1 gˆ(ω, −x) = e−2πixω Vδ gˆ(ω, −x) = gˆ(ω). g (ω). We conclude that Vδ f V1 g(x, ω) = e−2πixω f (x)ˆ

We observe that it is of course possible to deﬁne explicitly a “path” between the spectrogram and the Rihaczek representation. For example, using Gaussian 2 functions, we can consider the L1 -normalized window φλ (x) = λd/2 e−πλx and 1 2 set ψλ (x) = φˆλ (x) = e−π λ x with λ ∈ [1, ∞], adopting the convention φ∞ = δ, ψ∞ = 1.

3. Uncertainty principles In this section we examine various forms of the uncertainty principle. We study in particular which forms of this principle apply to the quadratic representation Qφ,ψ and show that suitable Lp formulations have, as limit case, uncertainty principles for the spectrogram and for the Rihaczek representation. We begin with recalling the classical uncertainty principle. Proposition 3.1. For f ∈ L2 (R), a, b ∈ R we have 1/2 1/2 1 2 2 2 ˆ 2 f 2L2 . (x − a) |f (x)| dx (ω − b) |f (ω)| dω ≥ 4π R R As pointed out in [11] this principle is essentially an inequality for the Rihaczek representation and can actually be reformulated as: 1/2 1 2 2 2 f 2L2 . (x − a) (ω − b) |Rf (x, ω)| dxdω ≥ 4π R2 Another expression of the uncertainty principle for the Rihaczek representation is the well-known uncertainty principle of Donoho-Stark for which we refer to [6]. We prove next a form of the uncertainty principle for the Rihaczek representation in terms of the integral U |R(f )(z)| dz instead of U |R(f )(z)|2 dz (as usual, z = (x, ω) ∈ R2d ). Proposition 3.2. Let f ∈ L1 (Rd ) ∩ L∞ (Rd ) and ≥ 0. If U ⊂ R2d is a measurable set and |Rf (z)|dz, (1 − )f L1 f L∞ ≤ U

then µ (U ) ≥ 1 − , where µ (U ) is the Lebesgue measure of U .

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Proof. As |Rf (z)| = |f (x)fˆ(ω)| we have immediately: |Rf (z)|dz (1 − )f L∞ f L1 ≤

U

≤ f L∞ fˆL∞

dz U

≤ f L∞ f L1 µ (U ).

For p ∈ (1, ∞) we have the following result.

Proposition 3.3. Let f ∈ Lp (Rd ) ∩ Lp (Rd ), U be measurable with ﬁnite measure and ≥ 0. If (1 − )f Lp f Lp ≤ |Rf (z)|dz, U

then µ (U ) ≥

Cpdp0 0 (1

3

− ) , where p0 = min(p, p ) and Cp0 = p0

(p0 )1/p0 (p0 )1/p0

.

Proof. In the proof we make use of the continuous inclusion Lp2 (Uj ) ⊂ Lp1 (Uj ) for p1 ≤ p2 , where f Lp2 ≤ µ (Uj )(1/p1 −1/p2 ) f Lp1 and, of the boundedness of the d q 1/q /q 1/q f Lq , Fourier transform F : Lq (Rd ) → Lq (Rd ) with fˆLq ≤ which holds for q ∈ [1, 2]. We consider at ﬁrst the case p ≤ 2, we have the estimates (1 − )f Lp (Rd ) f Lp(Rd ) ≤ U |f (x)fˆ(ω)| dxdω ≤ f ⊗ fˆLp (U) µ(U )α ≤ f Lp (Rd ) fˆLp (Rd ) µ (U )α d ≤ p1/p /p1/p f Lp (Rd ) f Lp(Rd ) µ (U )α (13) where α = 1 − 1/p = 1/p, which proves the thesis. Analogously for p ≥ 2 we have (1 − )f Lp (Rd ) f Lp(Rd )

≤ U |f (x)fˆ(ω)| dxdω ≤ f ⊗ fˆLp (U) µ(U )β ≤ f Lp(Rd ) fˆLp (Rd ) µ (U )β d ≤ p1/p /p1/p f Lp(Rd ) f Lp (Rd ) µ (U )β

where β = 1 − 1/p = 1/p . The thesis follows then from (13) and (14).

(14)

Remark 3.4. We note that we can regard Proposition 3.2 as a particular case of Proposition 3.3 if we allow p ∈ [1, ∞], adopting, in the case p0 = 1, the convention ∞1/∞ = 01/0 = 1.

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Our next goal is the investigation of possible relations between these results and corresponding uncertainty principles for our general class of representations Qφ,ψ . For this class we prove now the following simple uncertainty principle.

Proposition 3.5. Consider non zero functions φ ∈ Lp (Rd ), ψ ∈ Lp (Rd ) and f ∈ Lp (Rd ) ∩ Lp (Rd ), where 1 ≤ p ≤ ∞. Let U be a measurable set in R2d and ≥ 0. If |Qψ,φ (f )(z)| dz, (15) (1 − )f Lp f Lp φLp ψLp ≤ U

then µ (U ) ≥ 1 − . Proof. Writing, as usual, z = (x, ω), φz (t) = e2πitω φ(t − x), ψz (t) = e2πitω ψ(t − x), we have |Qψ,φ (f )(z)| dz = |(f, φz )(ψz , f )| dz (1 − )f Lp f Lp φLp ψLp ≤ U U ≤ f Lp f Lp φLp ψLp dz = f Lp f Lp φLp ψLp µ (U ). (16) U

Though its simplicity, this result is enlightening for what concerns the connections with the uncertainty principles for the spectrogram and the Rihaczek representation. Actually, for ψ = φ ∈ L2 (Rd ), it contains the well-known weakuncertainty principle for the spectrogram (see [14]), namely: Proposition 3.6. For f, φ ∈ L2 (Rd ), U measurable set in R2d , ≥ 0, the condition 2 2 |Vφ f (z)|2 dz (17) (1 − )f L2 φL2 ≤ U

implies µ (U ) ≥ 1 − . On the other hand, for p = 1, keeping φL1 = ψL∞ = 1, we can let φ tend to δ and ψ tend to 1 in S (Rd ) and we see therefore from Theorem 2.10 that the previous proposition yields, in this limit case, exactly the uncertainty principle of Proposition 3.2 for the Rihaczek representation. Proposition 3.5 also permits to understand the substantial reason why, in the cases 1 < p < ∞, we do not recapture the uncertainty principle of the Rihaczek representation as a limit case of that of the generalized spectrogram. Namely, if ψ ∈ Lp (Rd ) tends to 1 in S (Rd ), then ψLp cannot remain bounded, otherwise there would exist a subsequence weakly convergent to an element ψ0 ∈ Lp (Rd ), d contradicting ψ → 1 in S (R ). In order that the product φLp ψLp , which appears in (15), remains bounded, it is then necessary that φLp → 0. This implies φ → 0 in S (Rd ), and is therefore incompatible with the condition φ → δ. This justiﬁes the more involved result for the Rihaczek representation which we have obtained in Proposition 3.3.

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4. Marginal distributions In this section we consider the quadratic form Qφ,ψ (f, f ) = Vφ f Vψ f , and we want to study its marginal distributions, i.e., (1) Qφ,ψ (x) = Vφ f Vψ f dω, and (2) Qφ,ψ (ω)

(18)

=

Vφ f Vψ f dx.

(19)

The marginals (18) and (19) are well deﬁned for instance if Vφ f Vψ f ∈ L1 (R2d ), q2 that is satisﬁed when q1 , q2 in Theorem 2.9 verify qq11+q = 1, that means q1 = q2 . 2 Since qj ≥ 2, j = 1, 2 we have to choose q1 = q2 = 2, and consequently p1 = p2 = 2. Then from now on we shall ﬁx f, φ, ψ ∈ L2 (Rd ). We start now by giving the explicit expression of (18) and (19) and then we study their average E and their standard deviation σ 2 . Proposition 4.1. For every f ∈ L2 (Rd ) and φ, ψ ∈ L2 (Rd ) we have: Vφ f Vψ f dω = |f |2 ∗ ψ˜ φ˜ (x),

(20)

˜ ˜ where as usual φ(x) = φ(−x) and ψ(x) = ψ(−x). Proof. We start by considering f, φ, ψ ∈ S(Rd ), recovering then (20) in the generical case by density arguments. By a simple change of variables we get: Vφ f Vψ f dω = e−2πiqω f (q + s) f (s) ψ(s − x) φ(q + s − x) ds dq dω; now, by integrating at ﬁrst in the ω-variable, since e−2πiqω dω is the δ distribution in the q-variable, we obtain that Vφ f Vψ f dω = f (s) f (s) ψ(s − x) φ(s − x) ds = |f |2 ∗ ψ˜ φ˜ (x); that is what we want to prove.

Let us analyze now (19). From the previous proposition and the formula Vg f (x, ω) = e−2πiωx Vfˆgˆ(ω, −x) we get immediately an analogous result. Proposition 4.2. For every f ∈ L2 (Rd ) and φ, ψ ∈ L2 (Rd ) we have: Vφ f Vψ f dx = |fˆ|2 ∗ ψˆ˜ φˆ˜ (ω), with the same notations as in the previous proposition.

(21)

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In formula (20) we obtain the marginal with respect the x-variable, |f (x)|2 , if ψ˜ φ˜ (x) = δ, (δ, ϕ) = ϕ(0), which is the case when φ = δ and ψ(0) = 1 or vice versa ψ = δ and φ(0) = 1 . In formula (21) we obtain the marginal with respect the ω-variable, |fˆ(ω)|2 , if ˆ ˆ ψ˜ φ˜ (ω) = δ, which happens when ˆ ˆ φˆ = δ and ψ(0) = 1 or vice versa ψˆ = δ and φ(0) =1. We have therefore: Corollary 4.3. For the generalized spectrogram Qφ,ψ (f ) both marginal distributions conditions are satisﬁed in the cases φ = δ, ψ = 1 and φ = 1, ψ = δ, which correspond to the Rihaczek’s distribution R(f ) = e−2πixω f (x)fˆ(ω) and its conjugate R(f ) respectively. More generally, for every multi-index α, the marginal distributions conditions are satisﬁed by generalized spectrograms with windows φ = ∂ α δ and ψ = xα /α!, or vice versa φ = xα /α! and ψ = ∂ α δ. We analyze now some properties of the marginal distributions (18) and (19). Proposition 4.4. We have: (1) E Qφ,ψ = E |f |2 ψ, φ L2 − f 2L2 xψ, φ L2 and

(1) σ 2 Qφ,ψ =

(22)

(1) 2 y − E Qφ,ψ |f (y)|2 dy ψ, φ L2 (1) y − E Qφ,ψ |f (y)|2 dy xψ, φ L2 −2 + f 2L2 x2 ψ, φ L2 .

(23)

Proof. By (20) and a simple exchange of integration order we have: (1) ˜ dx E Qφ,ψ = x |f |2 ∗ (ψ˜ φ)(x) ˜ − y) φ(x ˜ − y) + (x − y)|f (y)|2 ψ(x ˜ − y) φ(x ˜ − y) dy dx = y|f (y)|2 ψ(x = ψ, φ L2 y|f (y)|2 dy − xψ, φ L2 |f (y)|2 dy, and so (22) is proved. Regarding (23) we observe that (1) 2 2 (1) 2 ˜ x − E Qφ,ψ dx |f | ∗ (ψ˜ φ)(x) σ Qφ,ψ = 2 (1) ˜ − y) φ(x ˜ − y) dy dx y − E Qφ,ψ + x − y |f (y)|2 ψ(x =

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247

(1) 2 ˜ − y) φ(x ˜ − y) dy dx = y − E Qφ,ψ |f (y)|2 ψ(x (1) ˜ − y) φ(x ˜ − y) dy dx + 2 (x − y) y − E Qφ,ψ |f (y)|2 ψ(x ˜ − y) φ(x ˜ − y) dy dx + (x − y)2 |f (y)|2 ψ(x (1) 2 y − E Qφ,ψ |f (y)|2 dy ψ, φ L2 = (1) y − E Qφ,ψ |f (y)|2 dy xψ, φ L2 −2 + |f (y)|2 dy x2 ψ, φ L2 ,

that is what we wanted to prove.

With analogous computations results as in Proposition 4.4 can be proved also (2) for Qφ,ψ (ω), namely: Proposition 4.5. We have: (2) ˆ φˆ 2 E Qφ,ψ = E |fˆ|2 ψ, φ L2 − f 2L2 ω ψ, L and

(2) σ 2 Qφ,ψ =

(2) 2 η − E Qφ,ψ |fˆ(η)|2 dη ψ, φ L2 (2) ˆ φˆ 2 η − E Qφ,ψ |fˆ(η)|2 dη ω ψ, −2 L 2 2 ˆ φˆ 2 . + f L2 ω ψ, L

Remark 4.6. Let us remark that in the particular case when xψ, φ L2 = 0, ψ, φ L2 = 1,

(24)

(25)

(26)

˜ ˜ φ(x) is a Gaussian, we have from (22) that that is veriﬁed for example when ψ(x) (1) E Qφ,ψ = E |f |2 ; (27) analogously, when we obtain from (24) Moreover, if (26) and

ψ, φ L2 = 1,

ˆ φˆ 2 = 0 ω ψ, L

(28)

(2) E Qφ,ψ = E |fˆ|2 .

(29)

2 x ψ, φ L2 = 0

(30)

are satisﬁed, we also have from (23) and (27) that (1) σ 2 Qφ,ψ = σ 2 |f |2 ,

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and when the windows φ and ψ satisfy (28) and 2 ˆ φˆ 2 = 0 ω ψ, L

(31)

then we get from (25) and (29) (2) σ 2 Qφ,ψ = σ 2 |fˆ|2 . ˜ φ(x) ˜ A simple example in the one-dimensional case of a function ψ(x) = h(x) satisfying the conditions (26) and (30) is given by h(x) = k(|x|), where ⎧ 3 x ∈ ( 12 , 1) ⎨ 4x2 3 k(x) = − 2 x ∈ (1, 32 ) ⎩ 4x 0 elsewhere . Acknowledgements The authors are thankful to the referee for his valuable remarks and suggestions.

References [1] P. Boggiatto, G. De Donno and A. Oliaro, Weyl Quantization of Lebesgue spaces, to appear. [2] P. Boggiatto, E. Buzano and L. Rodino, Hypoellipticity and spectral theory, Akademie Verlag, Berlin, 1996. [3] L. Cohen, Time-frequency distributions – a review, Proc. IEEE, 77 (7) (1989), 941– 981. [4] L. Cohen, Time-Frequency Analysis, Prentice Hall, New Jersey, 1995. [5] L. Cohen, The uncertainty principle for the short-time Fourier transform, Proc. Int. Soc. Opt. Eng. 22563 (1995), 80–90. [6] D.L. Donoho and P.B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math., 49 (3) (1989), 906–931. [7] G.B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, 1989. [8] G.B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3 (3) (1989), 207–238. [9] L. Galleani and L. Cohen, The Wigner distribution for classical systems, Physics Letters A, 302 (2002), 149–155. [10] K. Gr¨ ochenig, Foundations of Time-Frequency Analysis, Birkh¨ auser, Boston, 2001. [11] K. Gr¨ ochenig, Uncertainty principles for time-frequency representations. in Advances in Gabor analysis, Editors: H.G. Feichtinger and T. Strohmer, Birkh¨auser, Boston, Boston, 2003, 11–30. [12] A.J.A. Janssen, Proof of a conjecture on the supports of Wigner distributions, J. Fourier Anal. Appl., 4 (6) (1998), 723–726. [13] G. Kaiser, A Friendly Guide to Wavelets, Birkh¨ auser, Boston, 1994. [14] E.H. Lieb, Integral bounds for radar ambiguity functions and Wigner distributions, J. Math. Phys., 31(3) (1990), 594–599.

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[15] M.A. Shubin, Pseudodiﬀerential Operators and Spectral Theory, Second Edition, Springer-Verlag, Berlin, 2001. [16] J. Toft, Continuity properties for modulation spaces with applications to pseudodiﬀerential calculus, I, J. Func. Anal. 207 (2) (2004), 399–429. [17] J. Toft, Continuity properties for modulation spaces with applications to pseudodiﬀerential calculus, II, Ann. Global Anal. Geom. 26 (2004), 73–106. [18] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932), 749–759. [19] M.W. Wong, Weyl Transforms, Springer-Verlag, 1998. [20] M.W. Wong, Wavelet Transforms and Localization Operators, Birkh¨ auser, Basel, 2002. Paolo Boggiatto, Giuseppe De Donno and Alessandro Oliaro Department of Mathematics University of Torino Via Carlo Alberto 10 I-10123 Torino, Italy e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 251–257 c 2006 Birkh¨ auser Verlag Basel/Switzerland

A Characterization of Stockwell Spectra M.W. Wong and Hongmei Zhu Abstract. Signals in real applications are typically ﬁnite in duration, dynamic and non-stationary processes with frequency characteristics varying over time. This often requires techniques capable of locally analyzing and processing signals. An integral transform known as the Stockwell transform is a combination of the classic Gabor transform and the current and versatile wavelet transform. It allows more accurate detection of subtle changes and easy interpretation in the time-frequency domain. In this paper, we study the mathematical underpinnings of the Stockwell transform. We look at the Stockwell transform as a stack of simple pseudo-diﬀerential operators parameterized by frequencies and give a complete description of the Stockwell spectra. Mathematics Subject Classiﬁcation (2000). Primary 47G10, 47G30, 65R10; Secondary 92C55, 94A12. Keywords. Stockwell transforms, pseudo-diﬀerential operators, signals, Fourier spectra, Stockwell spectra.

1. Introduction Rapid development of information technology demands innovative mathematical tools to represent a signal in such a way that the representation can provide an accurate and easily understood description of the behavior of the signal. An optimal representation of a signal allows better designs of ﬁlters to improve the quality of signal processing and aids extraction of important information from the signal. The classical Fourier transform forms the foundation of signal processing and modern medical imaging modalities such as computed tomography and magnetic resonance imaging. The Fourier transform F f , also denoted by fˆ, of a function f in L2 (R) is deﬁned by ∞ (F f )(k) = e−2πitk f (t) dt, k ∈ R, −∞

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

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∞ R where the integral −∞ is understood as limR→∞ −R in the mean. Here, f and fˆ describe, respectively, the temporal and frequency behavior of the signal. In general, it is diﬃcult to observe the temporal occurrence of a particular frequency component from fˆ, and it is not easy to understand the frequency information from f . Therefore f and fˆ are two diﬀerent but equivalent representations of the same signal. The Fourier transform is ideal when the frequency characteristics of a signal do not change over time. In reality, however, most signals are non-stationary, and hence a mixed and simultaneous representation of f and fˆ is desirable in signal processing. Two well-developed integral transforms addressing this issue are the Gabor transform and the wavelet transform. The Gabor transform Vg f of a signal f in L2 (R), also known as the short-time Fourier transform, provides a time-frequency representation of f by applying the Fourier transform to the signal conditioned by a window g that translates over time. More precisely, ∞ e−2πitk g(t − τ )f (t) dt, τ, k ∈ R. (Vg f )(τ, k) = −∞

Analysis based on the Gabor transform is intuitive but is limited by the ﬁxed time and frequency resolutions. Chapter 3 of the book [6] by Gr¨ ochenig contains detailed discussions on the Gabor transform. The wavelet transform projects a signal f in L2 (R) on a family of wavelets obtained by scaling and shifting a mother wavelet g. Mathematically, the wavelet transform Wg f of f with respect to a mother wavelet g is deﬁned by ∞ t−τ 1 f (t) g (Wg f )(τ, s) = dt, τ ∈ R, s ∈ R \ {0}. s |s| −∞ The multi-scale analysis provided by the wavelet transform gives a more accurate assessment of the local characteristics of a signal and thereby has extensive applications in many ﬁelds. Nevertheless, the time-scale representation may be diﬃcult to interpret. Moreover, the right choice of a mother wavelet for a given problem may be a diﬃcult task in signal processing. See, for instance, Chapter 2 of the book [3] by Daubechies, Chapter 10 of the book by [6] by Gr¨ ochenig and Chapter 18 of the book [12] by Wong for the mathematical aspects of the wavelet transform. The search for optimal data representations leads to new development in time-frequency analysis and results in generating new integral transforms combining the merits of the Gabor transform and the wavelet transform. Among the remarkable ones is the Stockwell transform. The Stockwell transform, ﬁrst proposed by geophysicists in the paper [9] by Stockwell, Mansinha and Lowe in 1996, can be considered as a multi-scale local Fourier transform that provides a multi-resolution time-frequency data representation. It is a natural extension of the Gabor transform and gives a more accurate description of the local behavior of a signal through multi-scale analysis. The Stockwell transform has gained popularity in the signal processing community because of its applications in geophysics, oceanology, engineering and biomedicine.

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See the papers [1], [2], [4], [5] and [13] for diverse applications of the Stockwell transform. In spite of its manifold applications, the mathematics of the Stockwell transform has not been well investigated. The aim of this paper is to give, among other mathematical properties, a complete description of the range {Sf : f ∈ L2 (R)} of the Stockwell transform. Since each element Sf in the range is simply the Stockwell spectrum of the signal f , the main result in this paper can be seen as a characterization of the Stockwell spectra. In Section 2, we look at the Stockwell transform as a pseudo-diﬀerential operator. Pseudo-diﬀerential operators and their variants, ﬁrst introduced by Kohn and Nirenberg in the seminal paper [8] in 1965 and modiﬁed by H¨ ormander [7] and others for problems in partial diﬀerential equations, have roots in quantization due to Weyl [10] and fertilizations with modern developments in mathematical sciences and engineering. The new way of looking at Stockwell transforms as pseudo-diﬀerential operators is promising and research in this direction is underway by the authors. A compact and self-contained account of pseudo-diﬀerential operators can be found in the book [11] by Wong. In Section 3, we make precise some of the results in the original paper [9]. In particular, we establish the inversion formula and the Plancherel theorem for the Stockwell transform. New results on the Stockwell transform are given in Section 4. We ﬁrst introduce a new function space M , which is an interesting space in its own right. Then, in terms of the function space M , we give a characterization of the Stockwell spectra.

2. Stockwell transforms and pseudo-diﬀerential operators Let f ∈ L2 (R). Then the Stockwell transform Sf with Gaussian window of f is the function on R × R deﬁned by ∞ 2 2 2 −2πikτ e2πiτ v e−2π (v−k) /k fˆ(v) dv, τ ∈ R, k ∈ R\{0}. (2.1) (Sf )(τ, k) = e −∞

Thus, for each nonzero frequency k, the Stockwell transform can be thought of as 2 2 2 the Fourier multiplier F −1 e−2π (·−k) /k F followed by a modulation. We have deﬁned the Stockwell transform in the framework of Fourier multipliers or pseudo-diﬀerential operators. The following proposition gives a formula for the Stockwell transform in terms of an integral operator. Proposition 2.1. Let f ∈ L2 (R). Then ∞ 2 2 |k| (Sf )(τ, k) = √ e−(τ −t) k /2 e−2πitk f (t) dt, 2π −∞

τ, k ∈ R.

The proof of Proposition 2.1 is a simple exercise in the Fourier transform and is omitted.

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By (2.1), we note that the computation of the Stockwell transform is usually implemented using the fast Fourier transform. Figure 1 illustrates the timefrequency representation computed from the Stockwell transform for a signal. The signal consists of two cosine functions with frequencies 0.1 and 0.2 Hz that are long in duration, and two cosine functions with the same frequency 0.4 Hz, which are short in duration. Due to the Heisenberg uncertainty principle, the Stockwell spectrum exhibits good frequency resolution for low frequency components and good time resolution for high frequency components. For users in the non-mathematical community, the close connections between the Fourier and Stockwell spectra make these features transparent. (a) A Signal Amplitude

2 1 0 −1

(b) Amplitude of its Fourier Spectrum

(c) Amplitude of its Stockwell Spectrum 0.5

Frequency (Hz)

0.4 0.3 0.2 0.1 0 31

0

0

20

40

60 80 Time (s)

100

120

Figure 1. Time-frequency representation of the Stockwell transform: (a) A signal consisting of multiple frequency components (b) The amplitude of the corresponding Fourier spectrum, i.e., |(F f )(k)| (c) The contour plotting the amplitude of the corresponding Stockwell spectrum, i.e., |(Sf )(τ, k)|

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255

3. An inversion formula and Plancherel’s theorem For a measurable function F : R × R → C, we deﬁne the time average AF of F to be the function on R by ∞ (AF )(k) = F (τ, k) dτ, k ∈ R. (3.1) −∞

Theorem 3.1. ASf = fˆ,

f ∈ L2 (R).

Proof. Let f be a Schwartz function. Then, by Fubini’s theorem, we get for all k in R \ {0}, ∞ ∞ ∞ 2 2 2 (ASf )(k) = (Sf )(τ,k)dτ = e−2πikτ e2πiτ v e−2π (v−k) /k fˆ(v)dv dτ −∞ −∞ −∞ ∞ ∞ 2 2 2 = e2πiτ (v−k) dτ e−2π (v−k) /k fˆ(v)dv. −∞

−∞

Since for all k in R \ {0} and v in R, ∞ e2πiτ (v−k) dτ = δ(v − k), −∞

where δ is the Dirac delta function, it follows that ∞ 2 2 2 (ASf )(k) = δ(v − k)e−2π (v−k) /k fˆ(v) dv −∞ ∞ 2 2 2 δ(v)e−2π v /k fˆ(v + k) dv = fˆ(k) = −∞

for all k in R \ {0}. The proof is then complete by a density argument.

Remark 3.2. The physical meaning of Theorem 3.1 can be understood visually by means of Figure 1. We can look at Theorem 3.1 as the Stockwell inversion formula in the sense that for all f in L2 (R), f = F −1 ASf, which can be written out explicitly as ∞ ∞ f (t) = e2πitk (Sf )(τ, k) dτ dk, −∞

−∞

t ∈ R.

Another useful consequence of Theorem 3.1 is the Plancherel theorem for the Stockwell transform to the eﬀect that for all f in L2 (R), 2 ∞ ∞ dk = f 2 2 . (Sf )(τ, k) dτ L (R) −∞

−∞

Remark 3.3. The results in this section can be found in the paper [9] by Stockwell, Mansinha and Lowe.

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4. Stockwell spectra The main results in this paper are built on the function space M deﬁned by 2 ∞ ∞ M = F :R×R→C: F (τ, k) dτ dk < ∞ . −∞

−∞

Roughly speaking, M is the space of all functions F such that the timeaverage of F has ﬁnite energy with respect to the frequency k. In fact, we have the following theorem. Theorem 4.1. M is an indeﬁnite Hilbert space in which the indeﬁnite inner product ( , )M is given by (F, G)M = (AF, AG)L2 (R) ,

F, G ∈ M,

where AF is the time-average of F deﬁned by (3.1) and ( , )L2 (R) is the inner product in L2 (R). We note that ( , )M is indeﬁnite in the sense that there exist nonzero functions F in M for which (F, F )M = 0. Proof of Theorem 4.1. We only need to prove that M is complete with respect to the norm M . Let {Fj }∞ j=1 be a Cauchy sequence of functions in M . Then 2 2 {AFj }∞ j=1 is Cauchy in L (R). Since L (R) is complete, AFj → f for some f in 2 L (R) as j → ∞. Therefore ∞ Fj (τ, ·) dτ → f −∞

in L2 (R) as j → ∞. By Theorem 3.1 and the Fourier inversion formula, f = AS fˇ. So, AFj → AS fˇ 2 in L (R) as j → ∞. Hence Fj → S fˇ in M as j → ∞, and the proof is complete. Theorem 4.2. The range R(S) of the Stockwell transform S : L2 (R) → M is a Hilbert space with respect to the inner product ( , )M . In fact, S is an isometry of L2 (R) onto R(S). Proof. That R(S) is complete follows from the proof of Theorem 4.1. To see that ( , )M is a genuine inner product in R(S), we only need to prove that (Sf, Sf )M = 0 ⇒ Sf = 0,

f ∈ L2 (R).

But this follows from the Plancherel theorem for the Stockwell transform. That S : L2 (R) → R(S) is a surjective isometry is also a consequence of the Plancherel theorem for the Stockwell transform. We end this paper with a characterization of R(S). Theorem 4.3. R(S) = M/Z, where Z = {F : R × R → C : AF = 0}.

A Characterization of Stockwell Spectra

257

Proof. Let G ∈ M be such that G ⊥ R(S). Then for all f in L2 (R), (Sf, G)M = 0. This means that ∞ (ASf )(k)(AG)(k) dk = 0, f ∈ L2 (R), −∞

or, by Theorem 3.1,

∞

fˆ(k)(AG)(k) dk = 0,

−∞

f ∈ L2 (R).

Thus, AG = 0 and the proof is complete.

References [1] R.A. Brown, H. Zhu and J.R. Mitchell, Distributed vector processing of a new local multi-scale Fourier transform for medical imaging applications, IEEE Trans. Medical Imaging 24 (2005), 689–691. [2] P.K. Dash, B.K. Panigrahi and G. Panda, Power quality analysis using S-transform, IEEE Trans. Power Delivery 18 (2003), 406–412. [3] I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. [4] M.G. Eramian, R.A. Schincariol, L. Mansinha, and R.G. Stockwell, Generation of aquifer heterogeneity maps using two-dimensional spectral texture segmentation techniques, Math. Geology 31 (1999), 327–348. [5] B.G. Goodyear, H. Zhu, R.A. Brown and J.R. Mitchell, Removal of phase artifacts from fMRI data using a Stockwell transform ﬁlter improves brain activity detection, Magn. Reson. Med. 51 (2004), 16–21. [6] K. Gr¨ ochenig, Foundations of Time-Frequency Analysis, Birkh¨ auser, 2001. [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] R.G. Stockwell, L. Mansinha and R.P. Lowe, Localization of the complex spectrum: the S transform, IEEE Trans. Signal Processing 44 (1996), 998–1001. [10] H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1950. [11] M.W. Wong, An Introduction to Pseudo-diﬀerential Operators, Second Edition, World Scientiﬁc, 1999. [12] M.W. Wong, Wavelet Transforms and Localization Operators, Birkh¨ auser, 2002. [13] H. Zhu, B.G. Goodyear, M.L. Lauzon, R.A. Brown, G.S. Mayer, L. Mansinha, A.G. Law and J.R. Mitchell, A new local multiscale Fourier analysis for MRI, Med. Phys. 30 (2003), 1134–1141. M.W. Wong and Hongmei Zhu Department of Mathematics and Statistics, York University 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada e-mail: [email protected] e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 259–277 c 2006 Birkh¨ auser Verlag Basel/Switzerland

Exact and Numerical Inversion of Pseudo-diﬀerential Operators and Applications to Signal Processing Vladimir S. Rabinovich and Steﬀen Roch Abstract. A large class of time-varying ﬁlters can be described via pseudo0 diﬀerential operators belonging to the H¨ ormander class OP S0, 0 . The questions whether and how an input signal can be reconstructed from a known output lead to the problems of invertibility of pseudo-diﬀerential operators in that class and of (at least, numerical) solution of pseudo-diﬀerential equations. We are going to derive eﬀective conditions for the invertibility for pseudodiﬀerential operators with globally slowly varying symbols as well as for causal pseudo-diﬀerential operators, and we study the stability of the ﬁnite sections method with respect to time and frequency for these operators. Mathematics Subject Classiﬁcation (2000). Primary 35S05; Secondary 65N30, 94A12. Keywords. Pseudo-diﬀerential operators, ﬁnite sections, reconstruction of signals.

1. Introduction We start with recalling some basic deﬁnitions and facts from signal processing theory. Standard references to this ﬁeld are [8, 9, 4, 22], for instance. An analog complex signal (ACS for short) is a function u : R → C. We will only consider ACS with ﬁnite energy, that is, we suppose that u belongs to the Hilbert space L2 (R) with norm 1/2 2 u2 := |u(t)| dt < ∞. R

The authors are grateful for the support by the German Research Foundation (DFG), grant 444 MEX-112/2/05.

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An analog linear ﬁlter is a linear mapping A which transforms every ACS u (which is then called an input signal) into an ACS v := Au, the so-called output signal. We will exclusively consider linear ﬁlters for which A acts as a bounded linear operator A : L2 (R) → L2 (R). An interesting but still easy to analyze class of linear ﬁlters is constituted by the time-invariant ﬁlters. They are characterized by their invariance with respect to shifts. More precisely, an analog ﬁlters A is said to be invariant with respect to shifts if Vh A = AVh for every h ∈ R where the shift operators Vh on L2 (R) acts via (Vh u)(x) := u(x − h). It is well known that every bounded linear operator A : L2 (R) → L2 (R) which is invariant with respect to shifts is in fact a convolution operator kA (t − τ ) u(τ ) dτ, t ∈ R, (1) (Au)(t) = R

where the kernel kA has a bounded Fourier transform kˆA in the distributional sense. Thus, kˆA belongs to the space L∞ (R) of all essentially bounded measurable functions with norm u∞ := esssupt∈R |u(t)| < ∞. For a time-invariant ﬁlter, every input signal can be reconstructed from its output by means of the Fourier transform. Indeed, if A is the ﬁlter (1) and if essinfω∈R |kˆA (ω)| > 0 then the input signal u is obtained from the output v = Au via vˆ(ω) itω 1 e dt. u(t) = (A−1 v)(t) = 2π R kˆA (ω)

(2)

A natural and important from the point of applications generalization of timeinvariant ﬁlters are time-varying ﬁlters. They are described by linear operators A of the form (Au)(t) = R

kA (t, t − τ ) u(τ ) dτ,

t ∈ R.

(3)

Evidently, for time-varying ﬁlters, the problem of reconstruction of the input signal from a given output is much more involved than for time-invariant ﬁlters. In particular, there is not explicit formula for the dependence of the input from the output signal. Thus, it is both necessary and natural to consider numerical methods to determine the input approximately. The present paper is devoted to this circle of problems. The crucial observation in our approach to the problem of reconstruction of input signals is that the operator (3) can be written as the pseudo-diﬀerential operator 1 σA (t, ω) u(τ )ei(t−τ )ω dτ dω (4) (Au)(t) = 2π R R

Inversion of Pseudo-diﬀerential Operators with symbol

σA (t, ω) :=

R

261

kA (t, τ ) e−iτ ω dτ.

The Fredholm properties of pseudo-diﬀerential operators in the H¨ ormander class m OP S0,0 have been studied in [13] (see also [15]) without imposing any additional conditions on the symbol. In particular, we derived necessary and suﬃcient condim thought of as acting from tions for the Fredholmness of an operator A ∈ OP S0,0 s N s−m N (R ), and these conditions are formulated the Sobolev space H (R ) into H in terms of the limit operators of A. On the other hand, for the purpose of reconstruction of the input signals, the Fredholmness is of less use. What one really needs is the (exact) invertibility of the pseudo-diﬀerential operators which model the time-varying ﬁlters rather than their Fredholmness. Thus, the main goals of the present paper are: (a) to consider classes of pseudo-diﬀerential operators which are important for signal processing, and to derive conditions for their invertibility as well as eﬀective formulas for their inversion (which generalize formula (2) for timevarying ﬁlters), and (b) to consider the problem of approximation in L2 (R) of the input signal by a sequence uN of signals with ﬁnite support or ﬁnite spectrum. To make the latter point more explicit, let PN be the operator of multiplication by the characteristic function of the interval [−N, N ], i.e., PN is the projection on L2 (R) acting as u(t) if t ∈ [−N, N ] (PN u)(t) := (5) 0 if t ∈ / [−N, N ]. Together with the equation Au = v where A is the operator (4) we consider the sequence of the equations PN APN uN = vN = PN v,

N ∈ N,

(6)

where uN and vN are functions with support in [−N, N ]. Moreover, let PˆN be the operator which is dual to PN with respect to the Fourier transform, i.e., N 1 (PˆN u)(t) = u ˆ(ω) eiωt dω. (7) 2π −N Then we also consider the sequence of the dual equations PˆN APˆN uN = vN := PˆN v.

(8)

Below we shall discuss under which conditions the solutions uN of (6) and uN of (8) converge in the norm of L2 (R) to the input signal u. From the point of view of signal processing, the more important question is that of the convergence of the uN to u. This is simply because the functions uN have ﬁnite spectra [−N, N ]. Thus, by the well-known Shannon-Nyquist Sampling Theorem (see, for instance, [4]), the signals uN can be reconstructed from their samples uN (nh) where n ∈ Z and h ∈ (0, π/N ].

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The obtained results concerning the invertibility of pseudo-diﬀerential operators arising from problems in signal theory and concerning the applicability of the ﬁnite sections method for them with respect to the sequences (PN ) and (PˆN ) are new both from the point of view of signal theory and of the theory of pseudo-diﬀerential operators. The paper is organized as follows. In Section 2 we recall some auxiliary material on pseudo-diﬀerential operators. Standard references to the theory of pseudodiﬀerential operators are [6, 20, 21], to mention only a few. All cited facts can be also found in [17]. In Section 3 we study the invertibility of slowly varying pseudo-diﬀerential operators, which simulate slowly time-varying ﬁlters. In Section 4 we consider causal pseudo-diﬀerential operators simulating so-called causal time-varying ﬁlters. Finally, Section 5 is devoted to the stability of the ﬁnite sections methods, that is, to the problems of numerical approximation of the input signal by the sequences (uN ) and (uN ) of signals with ﬁnite supports or with ﬁnite spectra.

2. The calculus of pseudo-diﬀerential operators 2.1. Function spaces Let N and N0 stand for the sets of the positive and non-negative integers, respectively, and let N ∈ N. For x ∈ RN , write |x| for the Euclidean norm of x and set x := (1 + |x|2 )1/2 . The usual operators of ﬁrst order partial diﬀerentiation on ∂ , and for every multi-index α = (α1 , . . . , αN ) ∈ N0 RN are denoted by ∂xj := ∂x j α α1 αN we write ∂ := ∂x1 . . . ∂xN . We will have to work in the following standard function spaces where Ω is an open subset of RN : • C ∞ (Ω) is the linear topological space of all real-valued inﬁnitely diﬀerentiable functions on Ω. A sequence (ϕm ) in C ∞ (Ω) converges to zero if, for each compact subset K of Ω and for every multi-index j ∈ NN 0 , lim sup |(∂ j ϕm )(t)| = 0.

m→∞ t∈K

• C0∞ (Ω) is the subspace of C ∞ (Ω) which consists of all functions with compact support. A sequence (ϕm ) in C0∞ (Ω) converges to zero if there is a compact subset K of Ω such that supp ϕm ⊂ K and if, for every multi-index j ∈ NN 0 , lim sup |(∂ j ϕm )(t)| = 0.

m→∞ t∈K

• S(RN ) is the subspace of C ∞ (RN ) of all functions ϕ with convergence deﬁned by the family of semi-norms |ϕ|m,j := sup tm |(∂ j ϕ)(t)| < ∞ t∈RN

where m ∈ N0 and j ∈ NN 0 .

Inversion of Pseudo-diﬀerential Operators

263

• Cb∞ (RN ) is the subspace of C ∞ (RN ) of all functions which are bounded together with all of their derivatives. Convergence in Cb∞ (RN ) is deﬁned by the family of the semi-norms |ϕ|j := sup |(∂ j ϕ)(t)| < ∞ t∈RN

where j ∈ NN 0 . We will use the Fourier transform F for functions on R in the form ϕ(ω) ˆ = (F ϕ)(ω) := ϕ(t) e−itω dt. R

The operator F : S(R) → S(R) acts as an isomorphism. It can be continuously extended to a bounded linear operator on L2 (R) such that Parseval identity ϕ ˆ 22 = 2πϕ22 holds for all functions ϕ ∈ L2 (R). 2.2. Oscillatory integrals In what follows we will have to deal with functions on R and on R×R only. A point in R × R will be usually denoted by (t, ω), and we then write ∂t and ∂ω for the operators of partial diﬀerentiation with respect to the ﬁrst and second variable, respectively. Further, Dt 2 and Dω 2 stand for the operators I − ∂t2 and I − ∂ω2 , respectively. ormander Deﬁnition 1. Let m ∈ N0 . A function a ∈ C ∞ (R × R) belongs to the H¨ m if, for each pair k, l ∈ N0 , class S0,0 sup |∂ωα ∂tβ a(t, ω)| ω−m < ∞. |a|k,l,m := α≤k, β≤l (t, ω)∈R×R

A function χ ∈ S(R × R) is called a cut-oﬀ function if it is identically 1 in a neighborhood of the origin of R×R. For each R > 0, set χR (x, ω) := χ(x/R, ω/R). Deﬁnition 2. Let a be a measurable function on R × R, and let χ be a cut-oﬀ function on R × R. If the limit 1 χR (t, ω) a(t, ω) e−itω dt dω lim R→∞ 2π R R exists and if it is independent of the choice of χ, then this limit is called the oscillatory integral of a, and we denote it by 1 os a(t, ω) e−itω dt dω. (9) 2π R R

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m . Then the oscillatory integral (9) exists, and the folProposition 3. Let a ∈ S0,0 lowing equality holds for all integers k1 , k2 with 2k1 > 1 and 2k2 > 1 + m: 1 os a(t, ω) e−itω dt dω 2π R R 1 e−itω ω−2k2 Dt 2k2 t−2k1 Dω 2k1 a(t, ω) dt dω. = 2π R R

Moreover, for all 2k1 > 1 and 2k2 > 1 + m, the estimate 1 −itω os ≤ C|a|2k1 , 2k2 , m a(t, ω)e dt dω 2π R

(10)

R

holds with a constant C independent of a (but depending on k1 and k2 ). Proposition 4. Let a ∈ C ∞ (R), and suppose there are an m ∈ N0 and constants Cj such that |a(j) (t)| ≤ Cj tm for all j ∈ N0 . Then, for all t ∈ R, 1 a(t + τ ) e−iτ ω dτ dω = a(t). os 2π R R 2.3. Pseudo-diﬀerential operators m Let a ∈ S0,0 . The operator Op(a) deﬁned on S(R) by 1 a(t, ω) uˆ(ω) eitω dω, (Op(a)u)(t) := 2π R

t ∈ R,

(11)

(12)

is called the pseudo-diﬀerential operator with symbol a. The class of all pseudom m is denoted by OP S0,0 . diﬀerential operators with symbol in S0,0 m . Then Op(a) is bounded on S(R). Proposition 5. Let a ∈ S0,0 m and u ∈ S(R). It follows from (12) that Let a ∈ S0,0 1 a(t, ω) u(t + τ ) e−iτ ω dτ dω. (Op(a)u)(t) = 2π R R

One easily checks that this integral can be considered as the oscillatory integral 1 (Op(a)u)(t) = os a(t, ω) u(t + τ ) e−iτ ω dτ dω. (13) 2π R R This observation oﬀers another way to deﬁne the pseudo-diﬀerential operator Op(a) as the oscillatory integral (13) depending on the parameter t. Note that (13) makes sense also for functions u in Cb∞ (R). Thus, this alternative deﬁnition m allows one to consider the pseudo-diﬀerential operator Op(a) ∈ OP S0,0 as being ∞ deﬁned on Cb (R). m . Then the pseudo-diﬀerential operator Op(a) deﬁned Proposition 6. Let a ∈ S0,0 by (13) acts boundedly on Cb∞ (R).

Inversion of Pseudo-diﬀerential Operators

265

Let A be a linear operator acting from Cb∞ (R) into Cb∞ (R). The formal symbol of A is deﬁned by σA (t, ω) = e−itω A(eitω ), Proposition 7. Let A = Op(a) ∈ cides with a.

m OP S0,0 .

(t, ω) ∈ R × R.

(14)

Then the formal symbol σA of A coin-

m Basically, this proposition states that each time-varying ﬁlter A ∈ OP S0,0 can itω be reconstructed from its action on the system of the harmonic signals {e }ω∈R . 0 Proposition 8 (Calder´ on-Vaillancourt). Let a ∈ S0,0 . Then Op(a) is a bounded 2 operator on L (R), and

Op(a)B(L2 (R)) ≤ C|a|2,2,0 . Here and in what follows, B(X) stands for the Banach algebra of all bounded linear operators on X. 0 Proposition 9. Let a ∈ S0,0 . Then Op(a) is a compact operator on L2 (R) if and only if lim a(t, ω) = 0. (t, ω)→∞

The next result describes the symbol of the composition of pseudo-diﬀerential operators. m1 m2 Proposition 10. Let A = Op(a) ∈ OP S0,0 and B = Op(b) ∈ OP S0,0 . Then m1 +m2 and AB ∈ OP S0,0 1 a(t, ω + η) b(t + τ, ω) e−iτ η dτ dη. (15) σAB (t, ω) = os 2π R R Moreover,

|σAB |l1 , l2 , m1 +m2 ≤ C(l1 , l2 )|a|2k1 +l1 , l2 , m1 |b|l1 , 2k2 +l2 , m2

(16)

for all integers k1 , k2 with 2k1 > m1 + 1 and 2k2 > m2 + 1 and for all non-negative integers l1 , l2 . 2.4. Fredholmness of pseudo-diﬀerential operators For α, β ∈ Z, consider the unitary operators Vα and Eβ on L2 (R) deﬁned by (Vα u)(t) := u(t − α)

and (Eβ u)(t) := eiβt u(t),

t ∈ R,

and set U(α, β) := Vα Eβ . Clearly, the U(α, β) are also unitary on L2 (R). Let A be a bounded operator on L2 (R), and let h : N → Z × Z, m → h(m) = (α(m), β(m)) be a sequence which tends to inﬁnity in the sense that |h(m)| → ∞ as m → ∞. The operator Ah ∈ B(L2 (R)) is called the limit operator of A with respect to the sequence h if ∗ AUh(m) , Ah = s-limm→∞ Uh(m)

∗ A∗h = s-limm→∞ Uh(m) A∗ Uh(m) ,

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V.S. Rabinovich and S. Roch

where s-limm→∞ Am denotes the strong limit of the sequence (Am ) of operators in B(L2 (R)). We denote the set of all limit operators of A by σop (A). 0 Let A = Op(a) ∈ OP S0,0 and let h be as above. Then ∗ AUh(m) = Op(ah(m) ) with ah(m) (t, ω) := a(t + α(m), ω + β(m)). Uh(m)

The sequence h has a subsequence g such that the sequence (ag(m) )m∈N converges in the topology of C ∞ (R × R) to a certain function ag . One can show that ag 0 belongs to S0,0 and that Ag := Op(ag ) is just the limit operator of A deﬁned by the sequence g. Recall that a bounded linear operator acting on a Banach space X is called a Fredholm operator if ker A := {x ∈ X : Ax = 0} and coker A := X/(AX) are ﬁnite-dimensional spaces. This means that the equations Ax = 0 and A∗ y = 0 possess only a ﬁnite number of linear independent solutions in the spaces X and X ∗ , respectively, and the equation Ax = f is solvable if and only if yj (f ) = 0 for j = 1, . . . , m where the functionals y1 , . . . , ym form a basis of ker A∗ . The integer ind (A) := dim ker A − dim ker A∗ is called the Fredholm index of the Fredholm operator A. The following theorem is the main result of [13] (see also Theorems 4.3.9 and 4.3.15 in [17]). It states necessary and suﬃcient conditions for operator A = 0 to be Fredholm on the space L2 (R). Op(a) ∈ OP S0,0 0 Theorem 11. The operator A = Op(a) ∈ OP S0,0 is Fredholm on L2 (R) if and only 2 if all limit operators of A are invertible on L (R).

Our next goal is to single out a special class of pseudo-diﬀerential operators for which the Fredholm criterion from Theorem 11 can be made more explicit and 0 is slowly oscillating at inﬁnity with respect eﬀective. We say that a symbol a ∈ S0,0 to t if (17) lim sup |∂t a(t, ω)| = 0, t→∞ ω∈R

and we call a ∈

0 S0,0

slowly oscillating at inﬁnity with respect to ω if lim sup |∂ω a(t, ω)| = 0.

ω→∞ t∈R

(18)

0 Finally, a ∈ S0,0 is said to be slowly oscillating at inﬁnity with respect to both variables (t, ω) if conditions (17) and (18) hold simultaneously. 0 be slowly oscillating at inﬁnity with respect to t (resp. Proposition 12. Let a ∈ S0,0 ω) and let Op(ah ) be the limit operator of Op(a) corresponding to the sequence h = (α, β) : N → Z × Z with α(m) → ∞ (resp. β(m) → ∞). Then the symbol ah of Op(ah ) is independent of t (resp. ω):

ah (ω) =

lim

a(α(m), β(m) + ω)

(α(m), β(m))→∞

resp. ah (t) =

lim (α(m), β(m))→∞

a(α(m) + t, β(m)).

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267

Theorem 11 and Proposition 12 imply the following result. Theorem 13. Let the symbol a be slowly oscillating at inﬁnity with respect to both variables (t, ω). Then Op(a) : L2 (R) → L2 (R) is a Fredholm operator if and only if lim

inf

R→∞ |t|+|ω|>R

|a(t, ω)| > 0.

3. Invertibility of pseudo-diﬀerential operators with globally slowly varying symbols 0 For a ∈ S0,0 and ε > 0, deﬁne aε by aε (t, ω) := a(εt, ω). Clearly, if ε tends to zero, then the variation of aε with respect to t becomes slower and slower. 0 Theorem 14. Let a ∈ S0,0 satisfy

inf

(t, ω)∈R2

|a(t, ω)| > 0,

and set aε (t, ω) := a(εt, ω) for ε > 0. Then there exists ε0 > 0 such that the pseudo-diﬀerential operator Op(aε ) : L2 (R) → L2 (R) is invertible for all ε ∈ (0, ε0 ). Proof. Set bε (t, ω) := a−1 (εt, ω). Via (15) we obtain that Op(bε )Op(aε ) = Op(cε ) where 1 cε (t, ω) = os a(εt, ω + η) a−1 (ε(t + τ ), ω) e−iτ η dτ dη 2π R2 1 = os a(εt, ω + η) e−iτ η dτ dη a−1 (εt, ω) 2π 2 R 1 a(εt, ω + η)(a−1 (ε(t + τ ), ω) − a−1 (εt, ω)) e−iτ η dτ dη + os 2π R2 =: 1 + lε (t, ω) with lε (t, ω) being equal to 1 εi os ∂η a(εt, ω + η)∂τ a−1 (ε(t + θτ ), ω) e−iτ η dθ dτ dη. 2π R2 0 Applying estimate (16) and Proposition 8 one obtains Op(lε )B(L2 (R)) ≤ C(a)ε where the constant C(a) depends on a only. Hence, there exists an ε0 > 0 such that Op(lε ) < 1 for all ε ∈ (0, ε0 ). For ε ∈ (0, ε0 ), the operator (I + Op(lε ))−1 Op(bε ) =

∞

Op(lε )j Op(bε )

j=0

is a left inverse for Op(aε ). In the same way one gets the existence of a right inverse operator for Op(aε ) whenever ε > 0 is suﬃciently small.

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0 satisfy Theorem 15. Let a ∈ S0,0

inf

(t, ω)∈R2

|a(t, ω)| > 0,

and set aε (t, ω) := a(t, εω) for ε > 0. Then there exists ε0 > 0 such that the pseudo-diﬀerential operator Op(aε ) : L2 (R) → L2 (R) is invertible for all ε ∈ (0, ε0 ). The proof runs similarly to the proof of Theorem 14.

4. Invertibility of causal pseudo-diﬀerential operators 4.1. Invertibility on weighted spaces Causal pseudo-diﬀerential operators are realizations of casual ﬁlter, that is, of ﬁlters A owning the property that if u ∈ L2 (R) and u(t) = 0 for t < 0, then (Au)(t) = 0 for t < 0. Casual time-varying ﬁlters admit the following representation t

kA (t, t − τ ) u(τ ) dτ,

(Au)(t) =

t > 0.

0

Hence, the support of the function τ → kA (t, τ ) is located on the positive semiaxis. Consequently, its Fourier transform ∞ kˆA (t, z) = kA (t, τ )e−iτ z dτ 0

(understood, if necessary, in the sense of the distributions) has an analytic extension into the lower half-plane Π− := {z ∈ C : Im z < 0}. Thus, the consideration of causal time-varying ﬁlters leads in a natural way to the study of so-called causal pseudo-diﬀerential operators, i.e., of pseudo-diﬀerential operators with symbols (t, ω) → a(t, ω) admitting an analytic extension with respect to ω into the lower half-plane Π− . Let R+ stand for the set of the positive real numbers. Deﬁnition 16. Let S 0 (Π− ) denote the set of all symbols a ∈ C ∞ (R+ × R) for which the function (t, ω) → a(t, ω) possesses an analytic extension with respect to ω into the half-plane Π− and for which j k (19) |a|jk := sup ∂t ∂z a(t, z) < ∞ t∈R+ , z∈Π−

for all j, k ≥ 0 and lim

sup |∂z a(t, z)| = 0.

Π− z→∞ t∈R+

(20)

Let further OP S 0 (Π− ) denote the set of all pseudo-diﬀerential operators with symbols in S 0 (Π− ). R+

Let L2 (R+ , e−ht ) stand for the Banach space of all measurable functions on with norm 2 |u(t)|2 e−2ht dt < ∞. uL2 (R+ , e−ht ) := R+

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Proposition 17. Let a ∈ S 0 (Π− ). Then Op(a) is bounded on L2 (R+ , e−ht ) for every h ≥ 0, and |a|jk . Op(a)B(L2 (R+ , e−ht )) ≤ C j≤2, k≤2

Proof. Let kA denote the kernel of the operator A := Op(a) understood in the sense of distributions. The analyticity of the function z → a(t, z) in the lower complex half-plane and the estimate (19) imply that the support of the kernel kA (t, τ ) with respect to τ is located on the real interval [0, ∞). Hence, if u is a distribution in D (R) with support in [0, ∞) and if the convolution of τ → kA (t, τ ) with u (with respect to τ ) is well deﬁned, then this convolution is supported in [0, ∞) again. Further, Op(a)B(L2 (R+ , e−ht ))

≤

e−ht Op(a)eht B(L2 (R))

=

Op(a(t, ω − ih))B(L2 (R))

≤

|∂tj ∂ωk a(t, w − ih)| C |a|jk

≤

j≤2, k≤2

by the Calder´ on-Vaillancourt theorem (Proposition 8) and by (19). This estimate yields the boundedness of A as well as the norm estimate. Theorem 18. Let a ∈ S 0 (Π− ) and assume that inf

t∈R+ , z∈Π−

|a(t, z)| > 0.

(21)

Then there exists an h0 > 0 such that the operator Op(a) : L2 (R+ , e−ht ) → L2 (R+ , e−ht ) is invertible for all h ≥ h0 . Proof. Let b := 1/a. Condition (21) implies that b ∈ S 0 (Π− ). Set L := Op(b). Via the composition formula (15) we get LOp(a) = Op(c) where 1 c(t, ω) = os b(t, ω + η) a(t + τ, ω)e−iτ η dτ dη. (22) 2π R R Applying the Lagrange formula

1

b(t, ω + η) = b(t, ω) + η

bω (t, ω + θη) dθ

0

we further obtain from (22) that c = 1 + r where 1 i bω (t, ω + θη) at (t + τ, ω) e−iτ η dθ dτ dη. r(t, ω) = os 2π R R 0 It follows from the deﬁnition of oscillatory integrals that r ∈ S 0 (Π− ). Moreover, condition (20) yields that lim

sup

h→+∞ t∈R+ , ω∈R

|∂tk ∂ωj r(t, ω − ih)| = 0

(23)

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for all non-negative integers k, j. Hence, Proposition 17 together with (23) imply that Op(r)B(L2 (R+ , e−ht ))

=

e−ht Op(r)eht B(L2 (R+ ))

=

Op(r(t, ω − ih))B(L2 (R+ )) → 0.

if h → +∞. Thus, there exists an h0 > 0 such that, for all h ≥ h0 , Op(r)B(L2 (R+ , e−ht )) < 1. Hence, I + Op(r) is invertible, and the operator L := (I + Op(r))−1 Op(b) is a left inverse of Op(a) in the space L2 (R+ , e−ht ) for every h ≥ h0 if only h0 > 0 is large enough. In the same way, one gets the existence a right inverse of Op(a). Thus, L = Op(a)−1 . 4.2. Invertibility on L2 (R+ ) Our next goal is eﬀective conditions for the invertibility of causal pseudo-diﬀerential operators on the (unweighted) Hilbert space L2 (R+ ). Theorem 19. Let a ∈ S 0 (Π− ) satisfy the conditions sup |∂t a(t, z)| = 0

lim

t→+∞ z∈Π−

(24)

and inf

t∈R+ , z∈Π−

|a(t, z)| > 0.

(25)

Then Op(a) : L2 (R+ ) → L2 (R+ ) is invertible. Proof. Let L be as in the proof of Theorem 18. Then LOp(a) = I + Op(r) where r ∈ S 0 (Π− ). Condition (24) implies lim t+|ω|→∞

r(t, ω − ih) = 0

for every h > 0. Hence, and by Proposition 9, Op(r) acts as a compact operator on L2 (R+ , e−th ) for every h ≥ 0. Consequently, L is a left regularizator of Op(a) on L2 (R+ , e−th ) for every h ≥ 0. In the same way one checks that L is also a right regularizator of Op(a) on L2 (R+ , e−th ) for every h ≥ 0. Thus, the operator Op(a) considered as acting from L2 (R+ , e−th ) into L2 (R+ , e−th ) is Fredholm. But then the operator Op(a) h := e−th Op(a)eth : L2 (R+ ) → L2 (R+ ) is a Fredholm operator, too. It follows from the deﬁnition of the class S 0 (Π− ) and from Proposition 8 that the operators Op(a) h depend continuously on the parameter h ≥ 0 in the norm topology of B(L2 (R+ )). Hence, being a continuous function on the set of all Fredholm operators, the index of the Fredholm operators Op(a) h is independent of h. Since the operator Op(a) h is invertible for h large enough, this implies that the index of Op(a) h is zero for all h ≥ 0.

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We prove that the kernel of Op(a) is trivial. Indeed, suppose that u ∈ L2 (R+ ) ∩ ker Op(a). Then e−th u ∈ L2 (R+ ) for every h ≥ 0. By Theorem 18, there exists an h > 0 for which Op(a) h is an invertible operator. The equality Op(a) h e−th u = (e−th Op(a)eth )e−th u = e−th Op(a)u = 0 and the invertibility of Op(a) h imply that e−th u = 0. Hence, u = 0, and since the index of Op(a) : L2 (R+ ) → L2 (R+ ) is zero, this operator is invertible.

5. Finite sections of pseudo-diﬀerential operators Let A ∈ B(L2 (R)) be an invertible operator. Hence, the equation Au = v

(26)

has a unique solution u for every right-hand side v ∈ L2 (R). Let PN ∈ B(L2 (R)) be the operator of multiplication by the characteristic function of the interval [−N, N ]. Together with equation (26), we consider the sequence of its ﬁnite sections (27) PN APN uN = PN v, N ∈ N. The crucial question is whether the equations (27) possess unique solutions for suﬃciently large N and whether the sequence (uN ) of these solutions converges in the norm of L2 (R) to the solution u of equation (26). It is well known (see for instance [5, 3, 17]) that the convergence u − uN 2 → 0 is guaranteed if the operator A : L2 (R) → L2 (R) is invertible and if the sequence (PN APN ) is stable, i.e., if the operators PN APN : im PN → im PN are invertible for suﬃciently large N , say for N ≥ N0 , and if the norms of their inverses are uniform bounded, sup (PN APN )−1 < ∞.

N ≥N0

In this case, we also say that the ﬁnite sections method with respect to the projections PN is stable. In the same way, one deﬁnes the stability of ﬁnite sections method with respect to the sequence of the projections PˆN deﬁned by (7). A general approach to the ﬁnite sections method via limit operators is proposed in [15, 16], see also Chapter 6 in [17]. This approach usually yields results of the following form: The ﬁnite sections method (PN APN ) is stable if and only if the operator A is invertible and if a certain associated family of operators (related with the limit operators of A) is uniformly invertible. In any concrete situation, the discussion of the invertibility of the operators in this auxiliary family requires an additional (in general, large) amount of work. Only in some instances, one can state the additional conditions in a more explicit way (see, e.g., Theorem 6.2.8 in [17] for the ﬁnite sections of convolution type operators). Below we shall discuss this problem for pseudo-diﬀerential operators. The obtained general Theorems 22 and 23, holding for (almost) arbitrary pseudodiﬀerential operators, are certainly of their own interest. But for some special classes of symbols (viz. the multiplicatively slowly oscillating symbols), we will

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be able to formulate the extra conditions in a completely explicit manner, which yields a complete and ﬁnal solution of the problem of the stability of the ﬁnite section method for these operators. One should also mention that the obtained results are well known for some special classes of pseudo-diﬀerential operators with symbols which behave suﬃciently well at inﬁnity (see, for example, [18, 19] where the ﬁnite sections method is considered for operators which are constituted by operators of multiplication and of convolution by piecewise continuous functions). The formulation of our results for the stability of the ﬁnite sections method with respect to the sequences (PN ) and (PˆN ) requires a modiﬁcation of the notion of a limit operator. To each operator A ∈ B(L2 (R)), we now associate two families of limit operators. These families are deﬁned with respect to the families V := {Vα }α∈R and E := {Eβ }β∈R of unitary operators where Vα and Eβ are as in Section 2.5. Deﬁnition 20. Let A ∈ B(L2 (R)), and let the sequence h : N → Z tend to +∞ or −∞. (a) An operator Ah is called the limit operator of A with respect to the family V and to the sequence h if ∗ ∗ AVh(k) − Ah )PN = lim (Vh(k) A∗ Vh(k) − A∗h )PN = 0 lim (Vh(k)

k→∞

k→∞

(28)

for every N ∈ N. We denote the set of all limit operators of A with respect to the family V which correspond to sequences h tending to +∞ (resp. −∞) V V (A) (resp. σop,− (A)). by σop,+ (b) An operator Ah is called the limit operator of A with respect to the family E and to the sequence h if ∗ ∗ AEh(k) − Ah )PˆN = lim (Eh(k) A∗ Eh(k) − A∗h )PˆN = 0 lim (Eh(k)

k→∞

k→∞

(29)

for every N ∈ N. We denote the set of all limit operators of A with respect to the family E which correspond to sequences h tending to +∞ (resp. −∞) E E (A) (resp. σop,− (A)). by σop,+ 0 of the set of all functions We denote by S˜ the closure in the topology of S0,0 of the form N an (t)bn (ω) with an , bn ∈ Cb∞ (R). a(t, ω) = n=1

The class of all pseudo-diﬀerential operators with symbols in S˜ will be denoted by ˜ OP S. ˜ Then every sequence h tending to ∞ possesses a subseProposition 21. Let a ∈ S. quence g for which the limit operator of Op(a) with respect to the family V exists. The same statement holds with respect to the family E.

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273

We denote by P+ ∈ B(L2 (R)) the operator of multiplication by the characteristic function of the real interval (0, ∞) and by Pˆ+ its dual with respect to the Fourier transform, that is, Pˆ+ := F P+ F −1 . Further, let P− := I − P+ 2 2 and Pˆ− := I − Pˆ+ , and write H+ := Pˆ+ (L2 (R)) and H− := Pˆ− (L2 (R)) for the corresponding Hardy spaces. The following theorems state necessary and suﬃcient conditions for the stability of the ﬁnite sections method with respect to the sequences (PN ) and (PˆN ), respectively. Theorem 22. Let A := Op(a) ∈ OP S˜ be an invertible operator on L2 (R). Then the ﬁnite sections method for A with respect to the family (PN )N ∈N is stable if and only if (a) the Toeplitz operators P+ Ah P+ : L2 (R+ ) → L2 (R+ ) are invertible for every V limit operator Ah ∈ σop,− (A) and if (b) the Toeplitz operators P− Ah P− : L2 (R− ) → L2 (R− ) are invertible for every V (A). limit operator Ah ∈ σop,+ Theorem 23. Let A := Op(a) ∈ OP S˜ be an invertible operator on L2 (R). Then the ﬁnite sections method for A with respect to the family (PˆN )N ∈N is stable if and only if 2 2 (a) the Toeplitz operators Pˆ+ Ah Pˆ+ : H+ → H+ are invertible for every limit E operator Ah ∈ σop,− (A) and if 2 2 → H− are invertible for every limit (b) the Toeplitz operators Pˆ− Ah Pˆ− : H− E operator Ah ∈ σop,+ (A).

The proofs of these results follow (with evident modiﬁcations) as in [14]. Theorems 22 and 23 reduce the problem of stability of the ﬁnite sections method to the problem of invertibility of pseudo-diﬀerential operators in OP S˜ and of invertibility of the Toeplitz operators P+ Ah P+ , P− Ah P− and Pˆ+ Ah Pˆ+ , Pˆ− Ah Pˆ− . We are not aware of any eﬀective invertibility criteria for general operators of this form. However, below we will consider certain subclasses of operators in OP S˜ which are both important in signal processing and for which we will be able to derive eﬀective conditions of their invertibility and, thus, for the stability of the related ﬁnite sections methods. Consider Theorem 22 for operators with symbols (t, ω) → a(t, ω) which are slowly oscillating with respect to t → ∞. In this case, the limit operators of A := Op(a) have the form Ah = Op(ah ) where the symbol ah is independent of t and belongs to Cb∞ (R). Hence, P+ Op(ah )P+ and P− Op(ah )P− are Toeplitz operators on the positive and negative half-line, respectively, with symbols ah in Cb∞ (R). Necessary and suﬃcient conditions for the invertibility of the Toeplitz operator P+ Op(ah )P+ on L2 (R+ ) are well known in case the symbol ah possesses one-sided limits a± h := lim ah (ω) ω→±∞

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V.S. Rabinovich and S. Roch

at inﬁnity (consult [2, 5], for example). We are going to consider the more general class of symbols which are multiplicatively slowly oscillating at inﬁnity in the following sense. Deﬁnition 24. A function a ∈ Cb (R) is multiplicatively slowly oscillating at inﬁnity if lim sup |a(λω) − a(ω)| = 0 λ→+∞ ω∈[a, b]

for every interval [a, b] in (0, ∞) ∪ (−∞, 0). The class of all multiplicatively slowly oscillating functions is (with respect to pointwise deﬁned operations and the supremum norm) a C ∗ -subalgebra of Cb (R). We denote it by Cbsl (R). Note that this algebra indeed contains all functions in Cb (R) which possess one-sided limits at ±∞. 2 2 Let a ∈ Cbsl (R). Clearly, the Toeplitz operators Pˆ+ aPˆ+ : H+ → H+ and 2 2 ˆ ˆ P− aP− : H− → H− are Fredholm operators if and only if the operators A± := aPˆ± + Pˆ∓ : L2 (R) → L2 (R) are Fredholm operators (this holds, of course, for every a ∈ L∞ ). Let +∞ u(η) 1 dη (SR u)(ω) = πi −∞ η − ω stand for the Cauchy singular integral operator on R. Then Pˆ± = (I ± SR )/2 and A± =

a−1 a+1 I± SR . 2 2

Hence, the operators A± can be considered as singular integral operators on L2 (R) with multiplicatively slowly oscillating coeﬃcients. Singular integral operators with multiplicatively slowly oscillating coeﬃcients acting on Lp (Γ, w), with p ∈ (1, ∞), Γ is a Carleson curve and w being a Muckenhoupt weight, have been intensively studied in [1, 10, 12, 11]. Following these papers, we associate to the singular integral operator A := aI + bSR : L2 (R) → L2 (R) with a, b ∈ Cbsl (R) the pair σA (ω, λ) := (a(ω) + b(ω)sgn(λ), qA (ω, λ)), where qA (ω, λ) is the 2 × 2 matrix a+ (ω) + b+ (ω)s(λ) qA (ω, λ) = b− (ω)n(λ)

(ω, λ) ∈ R+ × R,

−b+ (ω)n(λ) a− (ω) − b− (ω)s(λ)

with s(λ) := coth π(λ + i/2),

n(λ) := (sinh π(λ + i/2))−1

and a± (ω) := a(±ω),

b± (ω) := b(±ω).

Inversion of Pseudo-diﬀerential Operators

275

Theorem 25. The singular integral operator A = aI + bSR : L2 (R) → L2 (R) is Fredholm if and only if (30) inf |a(ω) ± b(ω)| > 0 ω∈R

and lim inf inf |qA (ω, λ)| > 0. ω→∞ λ∈R

(31)

If conditions (30) and (31) are satisﬁed, then the index of A is equal to !R a(ω) + b(ω) 1 ∞ − lim arg + [arg det qA (R, λ)]λ=−∞ . R→+∞ 2π a(ω) − b(ω) ω=−R In particular, the matrices qA± (ω, λ) for the operators A± are equal to 1 ∓(a+ (ω) − 1)n(λ) (a+ (ω) + 1) ± (a+ (ω) − 1)s(λ) , ±(a− (ω) − 1)n(λ) (a− (ω) + 1) ∓ (a− (ω) − 1)s(λ) 2 and their determinants are ! 1 i (a+ (ω) + a− (ω)) ± (a+ (ω) − a− (ω)) coth π(λ + ) =: M± (a, ω, λ). 2 2 Hence, in our context, Theorem 25 specializes as follows. 2 2 Theorem 26. Let a ∈ Cbsl (R). Then the Toeplitz operator Pˆ± aPˆ± : H± → H± is Fredholm if and only if (32) inf |a(ω)| > 0 ω∈R

and lim inf inf |M± (a, ω, λ)| > 0. ω→+∞ λ∈R

(33)

If conditions (32) and (33) are satisﬁed, then the index of Pˆ± aPˆ± is equal to 1 +∞ [arg a(ω)]R − lim (34) ω=−R − [arg M± (a, R, λ)]λ=−∞ . R→∞ 2π In case the limits a± := limω→±∞ a(ω) exist, Theorem 26 reduces to a wellknown result in the Toeplitz operator theory. We formulate it for the operator Pˆ+ aPˆ+ . Let a ˜(R) denote the closed and naturally oriented curve which results from the essential range of a : R → C by ﬁlling in the line segment [a− , a+ ] between the /a ˜(R), let wind(˜ a, 0) refer to the winding number one-sided limits a− and a+ . If 0 ∈ of the closed curve a ˜(R) with respect to 0. The following result was discovered by many people, including Calder´ on, Spitzer, Widom, Devinatz, Gohberg, Krupnik, and Simonenko. For detailed references, see [2]. Theorem 27. Let a ∈ Cb (R) be a function for which the one-sided limits a± := 2 2 → H+ is a Fredholm operator limω→±∞ a(ω) at inﬁnity exist. Then Pˆ+ aPˆ+ : H+ if and only if 0 ∈ /a ˜(R). In this case, ind (Pˆ+ aPˆ+ ) = wind (˜ a, 0).

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2 2 → H+ is an invertible operator Corollary 28. Let a ∈ Cbsl (R). Then Pˆ+ aPˆ+ : H+ if and only if conditions (32) and (33) hold and if the index of Pˆ+ aPˆ+ which is given by (34) is equal to zero.

Proof. A well-known lemma by Coburn and Simonenko (see, for instance, Theorem 2.38 in [2]) states that every Fredholm Toeplitz operator has a trivial kernel or a trivial cokernel. Hence, the conditions (32) and (33) together with the vanishing of the index already imply the invertibility of Pˆ+ aPˆ+ . The operator P± Op(a)P± on L2 (R± ) is unitarily equivalent to the Toeplitz 2 . Hence, Theorem 26 has the following corollary. operator Pˆ± aPˆ± acting on H± Corollary 29. Let a ∈ Cbsl (R). Then P± Op(a)P± : L2 (R± ) → L2 (R± ) is a Fredholm operator if and only if conditions (32) and (33) are satisﬁed. In this case, the index of P± Op(a)P± is given by (34). Moreover, this operator is invertible if and only if it is a Fredholm operator with index zero.

References [1] A. B¨ ottcher, Yu.I. Karlovich and V.S. Rabinovich, Mellin pseudodiﬀerential operators with slowly varying symbols and singular integrals on Carleson curves with Muckenhoupt weights, Manuscripta Math. 95 (1998), 363–376. [2] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz Operators, Akademie-Verlag, Berlin, 1989 and Springer-Verlag, Berlin, 1990. [3] A. B¨ ottcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Springer-Verlag, New York, 1999. [4] P. Br´emaud, Mathematical Principles of Signal Processing: Fourier and Wavelet Analysis, Springer-Verlag, New York, Berlin, Heidelberg, 2002. [5] I. Gohberg and I. Feldman, Convolution Equations and Projection Methods for Their Solution, Nauka, Moskva, 1971 (Russian, Engl. transl.: Amer. Math. Soc. Transl. of Math. Monographs 41, Providence, R.I., 1974. [6] L. H¨ ormander, The Analysis of Linear Partial Diﬀerential Operators, Vol. I-IV, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983/1985. [7] M. Lindner, V.S. Rabinovich and S. Roch, Finite sections of band operators with slowly oscillating coeﬃcients, Linear Algebra Appl. 390 (2004), 19–26. [8] A.V. Oppenheim and R.W. Schafer, Digital Signal Processing, Prentice-Hall, Inc., 1975. [9] A. Papoulis, Signal Analysis, McGraw-Hill, New York, 1984. [10] V.S. Rabinovich, Singular integral operators on a composed contour with oscillating tangent and pseudodiﬀerential Mellin operators, Soviet Math. Dokl. 44 (1992), 791– 796. [11] V.S. Rabinovich, Algebras of singular integral operators on complicated contours with nodes that are of logarithmic whirl points, Izv. Ross. Akad. Nauk, Ser. Mat. 60 (1996), 6, 169–200 (Russian, Engl. transl.: Izv. Math. 60 (1996), 1261–1292).

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[12] V.S. Rabinovich, Mellin pseudodiﬀerential operators techniques in the theory of singular integral operators on some Carleson curves, in Operator Theory: Advances and Applications, Vol. 102, Birkh¨ auser, Basel, 1998, 201–218. [13] V.S. Rabinovich and S. Roch, Wiener algebras of operators, and applications to pseudodiﬀerential operators, J. Anal. Appl. 23 (2004), 437–482. [14] V.S. Rabinovich, S. Roch and B. Silbermann, Fredholm theory and ﬁnite section method for band-dominated operators, Integral Equations Operator Theory 30 (1998), 452–495. [15] V.S. Rabinovich, S. Roch and B. Silbermann, Band-dominated operators with operator-valued coeﬃcients, their Fredholm properties and ﬁnite sections, Integral Equations Operator Theory 40 (2001), 342–381. [16] V.S. Rabinovich, S. Roch and B. Silbermann, Algebras of approximation sequences: Finite sections of band-dominated operators, Acta Applicande Mathematicae 65 (2001), 315–332. [17] V.S. Rabinovich, S. Roch and B. Silbermann, Limit Operators and Their Applications in Operator Theory, Birkh¨ auser, Basel, 2004. [18] S. Roch, Finite sections of operators generated by convolutions, in Seminar Analysis, Operator Equations and Numerical Analysis 1987/88, Berlin 1988, 118–138. [19] S. Roch, P.A. Santos and B. Silbermann, Finite section method in some algebras of multiplication and convolution operators and a ﬂip, J. Anal. Appl. 16 (1997), 575–606. [20] M.A. Shubin, Pseudodiﬀerential Operators and Spectral Theory, Second Edition, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 2001. [21] M.E. Taylor, Pseudodiﬀerential Operators, Princeton University Press, Princeton, New Jersey, 1981. [22] M. Vetterli and J. Kovaˇceviˇc, Wavelets and Sub-Band Coding, Prentice-Hall, Englewood Cliﬀs, N.J., 1995. Vladimir S. Rabinovich Instituto Politecnico Nacional ESIME Zacatenco Avenida IPN Mexico, D. F. 07738, Mexico e-mail: [email protected] Steﬀen Roch Department of Mathematics Technical University of Darmstadt Schlossgartenstrasse 7 D-64289 Darmstadt, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 279–295 c 2006 Birkh¨ auser Verlag Basel/Switzerland

On the Product of Localization Operators Elena Cordero and Karlheinz Gr¨ochenig Abstract. We provide examples of the product of two localization operators. As a special case, we study the composition of Gabor multipliers. The results highlight the instability of this product and underline the necessity of expressing it in terms of asymptotic expansions. Mathematics Subject Classiﬁcation (2000). Primary 35S05; Secondary 47G30. Keywords. Localization operators, modulation spaces, time-frequency analysis, Weyl calculus, Wigner distribution, short-time Fourier transform.

We study the problem of calculating or estimating the product of two localization operators. The motivation comes either from signal analysis or pseudo-diﬀerential operator theory. On the one hand, in signal analysis the problem of ﬁnding a ﬁlter that has the same eﬀect as two ﬁlters arranged in series amounts to the computation of the product of two localization operators, see [8, 9] and references therein. On the other side, composition of pseudodiﬀerential operators by means of symbolic calculus gives rise to asymptotic expansions, mainly employed in PDEs (see, e.g., [14, 16]). Outcomes are regularity properties of partial diﬀerential operators and the construction of an approximate inverse (so-called parametrix). Since localization operators are a sub-class of pseudodiﬀerential operators, looking for asymptotic expansions of the localization operator product appears to be natural as well. Applications can be found in the framework of PDEs and Quantum Mechanics [1, 6, 7, 15]. In this paper, we survey the known approaches to this problem and provide concrete examples of the composition of localization operators. Indeed, very few cases allow the product to be written as a localization operator as well, consequently the class of localization operators is not closed under composition. Thus the product is unstable with respect to composition. This instability highlights the importance of a symbolic calculus for localization operators [6]. K. Gr¨ ochenig was supported by the Marie-Curie Excellence Grant MEXT-CT 2004-517154.

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We present localization operators using language and tools from time-frequency analysis. First, the deﬁnition of the short-time Fourier transform is required. Given a function f on Rd and a point (x, ω) of the phase space R2d , the operators of translation and modulation are deﬁned to be Tx f (t) = f (t − x)

and Mω f (t) = e2πiωt f (t) .

(1)

We often combine translations and modulations into time-frequency shift (phasespace shifts in physical terminology). Set z = (x, ω) ∈ R2d , then the general time-frequency shift is deﬁned by π(z) = Mω Tx .

(2)

Associated to time-frequency shifts is an important time-frequency representation, the short-time Fourier transform (STFT), also well known as coherent state transform, Gabor transform and windowed Fourier transform. The STFT of a function or distribution f with respect to a ﬁxed non-zero window function g is given by f (t) g(t − x) e−2πiωt dt = f, Mω Tx g = f, π(z)g , (3) Vg f (x, ω) = Rd

whenever the integral or the brackets ·, · (expressing a sesquilinear form) are well deﬁned. The short-time Fourier transform can be deﬁned on many pairs of distribution spaces and test functions. For instance, Vg f maps L2 (Rd ) × L2 (Rd ) into L2 (R2d ) and S(Rd ) × S(Rd ) into S(R2d ). Furthermore, Vg f can be extended to a map from S (Rd ) × S (Rd ) into S (R2d ) [12, p. 41]. The short-time Fourier transform is the appropriate tool for deﬁning localization operators, as we shall see presently. Let a be a symbol on the time-frequency plane R2d and choose two windows 1 ,ϕ2 ϕ1 , ϕ2 on Rd , then the localization operator Aϕ is deﬁned as a 1 ,ϕ2 f (t) = a(x, ω)Vϕ1 f (x, ω)Mω Tx ϕ2 (t) dxdω . (4) Aϕ a R2d

1 ,ϕ2 Taking the inner product with a test function g, the deﬁnition of Aϕ can be a written in a weak sense, namely, 1 ,ϕ2 Aϕ f, g = aVϕ1 f, Vϕ2 g = a, Vϕ1 f Vϕ2 g. a

(5)

If a ∈ S (R ) and ϕ1 , ϕ2 ∈ S(R ), then (5) is a well-deﬁned continuous 2 1 ,ϕ2 operator from S(Rd ) to S (Rd ). If ϕ1 (t) = ϕ2 (t) = 2d/4 e−πt , then Aa = Aϕ a ϕ1 ,ϕ2 is well known as (anti-)Wick operator and the mapping a → Aa is interpreted as a quantization rule [2, 8, 15, 16, 19]. Both the exact and the asymptotic product of localization operators rely upon 1 ,ϕ2 can the connection with the Weyl calculus. Namely, a localization operator Aϕ a be represented as a Weyl transform. Here we need to refer to another time-frequency representation, the cross-Wigner distribution W (g, f ) of the functions g, f deﬁned below (18). The Weyl transform Lσ of σ ∈ S (R2d ) is then deﬁned by 2d

d

Lσ f, g = σ, W (g, f ),

f, g ∈ S(Rd ).

(6)

On the Product of Localization Operators

281

Every continuous operator from S(Rd ) to S (Rd ) can be represented as a Weyl transform, and a calculation in [3, 11, 16] reveals that 1 ,ϕ2 = La∗W (ϕ2 ,ϕ1 ) , Aϕ a

(7)

1 ,ϕ2 is given by so, the (Weyl) symbol of Aϕ a

σ = a ∗ W (ϕ2 , ϕ1 ) .

(8)

The composition of the Weyl operators Lσ and Lτ , with symbols σ and τ belonging to suitable function spaces, can be expressed in the Weyl form [11, Chap. 2.3] (9) Lσ Lτ = Lστ , where στ , the twisted multiplication of the symbols σ and τ , is given by 2d στ (ζ) = 2 σ(z)τ (w)e4πi[ζ−w,ζ−z] dzdw

(10)

R2d

and the brackets [·, ·] express the symplectic form on R2d [(z1 , z2 ), (ζ1 , ζ2 )] = z1 ζ2 − z2 ζ1 ,

z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ R2d .

Thus, the composition of Weyl operators, whenever possible, deﬁnes a bilinear form (the twisted multiplication) on the corresponding symbols. The product of localization operators will be studied along the following steps in Sections 3–5: (i) Rewrite the two localization operators in terms of Weyl transforms (7); (ii) Use the product formula for Weyl symbols (9), (10) to compute the Weyl symbol of their product; (iii) Express, whenever possible, the resulting operator as a localization operator. In view of (8) this amounts to a deconvolution problem. Notation We deﬁne t2 = t · t, for t ∈ Rd , and xy = x · y is the scalar product on Rd . The Schwartz class is denoted by S(Rd ), the space of tempered distributions by S (Rd ). We use the brackets f, g to denote the extension to S (Rd ) × S(Rd ) of the inner product f, g = f (t)g(t)dt on L2 (Rd ). The Fourier transform is normalized to be fˆ(ω) = F f (ω) = f (t)e−2πitω dt. Given a continuous positive function (so-called weight function) m and 1 ≤ p ≤ ∞, we deﬁne as Lpm (Rd ) the space of all (Lebesgue) measurable functions on Rd such that the norm f Lpm := 1/p p p is ﬁnite (obvious changes for p = ∞). For 1 ≤ p, q ≤ ∞, Rd |f (x)| m(x) dx p,q the mixed-norm space L (R2d ) is the Banach space of all (Lebesgue) measurable functions on R2d satisfying 1/q 1/p

F Lp,q :=

Rd

Rd

|F (x, ω)|p dx

dω

again with obvious modiﬁcations whether p = ∞ or q = ∞.

,

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Throughout the paper, we shall use the notation A < ∼ B to indicate A ≤ cB for a suitable constant c > 0, whereas A + B if A ≤ cB and B ≤ kA, for suitable c, k > 0.

1. Product formulae In this section we brieﬂy review two approaches to handle the product of localization operators. 1.1. Exact product We reformulate the result of [8, 9] according to the notation of [11, 12]. 2 We consider the window functions ϕ1 (t) = ϕ2 (t) = ϕ(t) = 2d/4 e−πt , t ∈ Rd . In this case, the Wigner distribution of the Gaussian ϕ is a Gaussian as well. Precisely, we have 2

W (ϕ, ϕ)(z) = 2d e−2πz ,

z ∈ R2d .

(11)

If we compute the Weyl symbol σ of the operator we obtain σ(ζ) = 2 (a ∗ 2 e−2πz )(ζ), z, ζ ∈ R2d . In order to express the product in the form of a localization operator, we rewrite the factors in a Weyl form, as detailed in the end of the previous section. Secondly, we come back to localization operators by means of relation (8). Given the Weyl symbols σ, τ ∈ S(R2d ), we are interested in the Fourier transform of the twisted multiplication στ , that is Aϕ,ϕ a

d

F (στ )(ζ) = σ ˆ ˆ τ (ζ), where the twisted convolution is given by σ ˆ ˆ τ (ζ) = σ ˆ (z)ˆ τ (ζ − z)eπi[z,ζ] dz.

(12)

For any f, g ∈ S(R2d ), we deﬁne the product by 2 f g(ζ) = f (z)g(z − ζ)eπ(zζ+i[z,ζ]) e−πz dz,

(13)

R2d

R2d

then the product of localization operators is given by the following formula. Theorem 1.1. Let a, b ∈ S(R2d ). If there exists a symbol c ∈ S (R2d ) so that cˆ = 2−2d a ˆˆb,

(14)

then we have ϕ,ϕ Aϕ,ϕ = Aϕ,ϕ a Ab c .

The proof is a straightforward consequence of relations (8) and (12). Indeed, ϕ,ϕ one rewrites Aϕ,ϕ in Weyl form and uses relation (12) for the product. The a Ab

On the Product of Localization Operators

283

result is the Weyl operator Lµ , where the Fourier transform of µ is given by µ ˆ(ζ)

= = =

[F (a ∗ W (ϕ, ϕ))F (b ∗ W (ϕ, ϕ))](ζ) 2 2 −2d 2 a ˆ(z)ˆb(z − ζ)e−(π/2)z e−(π/2)(ζ−z) eπi[z,ζ] dz R2d 2 −2d −(π/2)ζ 2 2 e a ˆ(z)ˆb(z − ζ)eπ(zζ+i[z,ζ]) e−πz dz. R2d

Hence, we have 2

µ ˆ(ζ) = cˆ(ζ)(e−(π/2)ζ ) = F (c ∗ W (ϕ, ϕ))(ζ), where cˆ is given by relation (14). The stability of the product when localization symbols live in the linear space spanned by the Gaussian functions is proven in [9, Thm. 2.1]. Here we reformulate the result using our terminology and we shall prove it using the Weyl connection and the twisted multiplication, instead of the one (Section 5). 2

Theorem 1.2. Let ϕi (t) = ϕ(t) = 2d/4 e−πt , i = 1, . . . , 4, t ∈ Rd . Consider the symbols a(z) =

m

2

Ck e−2πdk z ,

b(z) =

l

2

Cj e−2πdj z ,

z ∈ R2d ,

(15)

j=1

k=1

where C1 , . . . , Cm ; C1 , . . . , Cl are complex numbers while d1 , . . . , dm ; d1 , . . . , dm are ϕ,ϕ positive real numbers. Then Aϕ,ϕ = Aϕ,ϕ a Ab c , with c(z) =

l m

2

Ck Cj e−2πrk,j z ,

z ∈ R2d ,

k=1 j=1

with rk,j = dk + dj + 2dk dj . 1.2. Asymptotic product Asymptotic expansions realize the composition of two localization operators as a sum of localization operators plus a controllable remainder. Versions of such a symbolic calculus are developed in [1, 7, 15, 6]. Most of them, as we observed in the introduction, were mainly motivated by PDEs and energy estimates, and therefore used smooth symbols that are deﬁned by diﬀerentiability properties, such as H¨ ormander or Shubin classes. For applications in quantum mechanics and signal analysis, alternative notions of smoothness – “smoothness in phase-space” or quantitative measures of “time-frequency concentration” – have turned out to be useful. This point of view is pursued in [6], and we shall present the corresponding results. The starting point is the following composition formula for two localization operators derived in [7]: 3 ,ϕ4 1 ,ϕ2 Aϕ = Aϕ a b

N −1 |α|=0

(−1)|α| Φα ,ϕ2 Aa∂ α b + EN . α!

(16)

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The essence of this formula is that the product of two localization operators can be written as a sum of localization operators, with new windows Φα suitably deﬁned, and a remainder term EN , which is “small”. In the spirit of the classical symbolic calculus, this formula was derived in [7, Thm. 1.1] for smooth symbols belonging to some Shubin class S m (R2d ) and for windows in the Schwartz class S(Rd ). In [6] the validity of (16) is established on the modulation spaces. The innovative features of this extension are highlighted below (we do not give here detailed statements and proofs). (i) Rough symbols. While in (16) the symbol b must be N -times diﬀerentiable, the symbol a only needs to be locally bounded. The classical results in symbolic calculus require both symbols to be smooth. (ii) Growth conditions on symbols. The symbolic calculus in (16) can handle symbols with ultra-rapid growth (as long as it is compensated by a decay of b or β vice versa). For instance, a may grow subexponentially as a(z) ∼ eα|z| for α > 0 and 0 < β < 1. This goes far beyond the usual polynomial growth and decay conditions. (iii) General window classes. A precise description of the admissible windows ϕj 2 in (16) is provided. Usually only the Gaussian e−πt or Schwartz functions are considered as windows. (iv) Size of the remainder term. Norm estimates for the size of the remainder term EN are derived. They depend explicitly on the symbols a, b and the windows ϕj . (v) The Fredholm property of localization operators. By choosing N = 1, ϕ1 = ϕ2 = ϕ with ϕ = 1, a(z) = 0, and b = 1/a, the composition formula (16) yields the following important special case: ϕ,ϕ ϕ,ϕ Aϕ,ϕ +R = I+R. a A1/a = A1

(17)

2d If the symbol a belongs to L∞ m (R ) and |a| + 1/m, and the ﬁrst partial derivative satisﬁes (∂j a)m ∈ L∞ and vanishes at inﬁnity for j = 1, . . . , 2d, then R is shown to be a compact operator. Besides, Aϕ,ϕ is proven to be a Fredholm operator bea tween suitable modulation spaces. This result works even for ultra-rapidly growing β symbols such as a(z) = eα|z| for α > 0 and 0 < β < 1.

2. Concepts of time-frequency analysis We ﬁrst present the tools and properties from time-frequency analysis that we shall use in the following sections. 2.1. The (cross-)Wigner distribution and the Weyl calculus The cross-Wigner distribution W (f, g) of f, g ∈ L2 (Rd ) is deﬁned to be t t W (f, g)(x, ω) = f (x + )g(x − )e−2πiωt dt. 2 2

(18)

On the Product of Localization Operators

285

The quadratic expression W f = W (f, f ) is usually called the Wigner distribution of f . The Wigner distribution W (f, g) is deﬁned on many pairs of Banach or topological vector spaces. For instance, they both map L2 (Rd ) × L2 (Rd ) into L2 (R2d ) and S(Rd )× S(Rd ) into S(R2d ). Furthermore, they can be extended to a map from S (Rd ) × S (Rd ) into S (R2d ). We ﬁrst report a crucial property of the (cross-)Wigner distribution (for proofs, see [12, Ch. 4] and [11]). Lemma 2.1. For λ = (u, η), µ = (v, γ) ∈ R2d and z = (x, ω) ∈ R2d we have W (π(λ)f, π(µ)g)(x, ω)

=

eπi(u+v)(η−γ) e2πix(η−γ) e−2πiω(u−v) λ+µ ). ×W (f, g)(z − 2

(19)

In particular, if λ = µ, relation (19) becomes W (π(λ)f, π(λ)g)(z) = W (f, g)(z − λ).

(20)

Since the (cross-)Wigner distribution of Schwartz functions f, g ∈ S(Rd ) is a Schwartz function on R2d , its partial derivatives are well deﬁned and may be expressed explicitly as linear combinations of (cross-)Wigner distributions of the functions tγ1 ∂ δ1 f and tγ2 ∂ δ2 g. Lemma 2.2. Let (x, ω) ∈ R2d , f, g ∈ S(Rd ), α = (α1 , α2 ), β = (β1 , β2 ) ∈ Zd+ × Zd+ , then α α1 α2 |α2 | ∂x ∂ω W (f, g)(x, ω)=(−2πi) (−1)|β2 | W (tα2 −β2 ∂ α1 −β1 f, tβ2 ∂ β1 g)(x, ω). β β≤α

(21) Proof. Using the product formula for derivatives, the ﬁrst partial derivative with respect to the time variable xj , with j = 1, . . . , d, is given by ∂xj W (f, g)(x, ω) = W (∂tj f, g)(x, ω) + W (f, ∂tj g)(x, ω), and, by induction or by Leibniz’ formula, we obtain α1 α ∂x W (f, g)(x, ω) = W (∂ α1 −β1 f, ∂ β1 g)(x, ω). β1

(22)

β1 ≤α1

The ﬁrst partial derivative with respect to the frequency variable ωj , with j = 1, . . . , d, is t t (−2πitj )e−2πitω f x + g x− dt. (23) ∂ωj W (f, g)(x, ω) = 2 2 Rd

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E. Cordero and K. Gr¨ ochenig

The cross-Wigner distributions of the functions tj f, g and f, tj g, respectively, are given by 1 t t −2πitω tj e f x+ g x− W (tj f, g)(x, ω) = xj W (f, g)(x, ω) + dt , 2 Rd 2 2 1 t t tj e−2πitω f x + g x− W (f, tj g)(x, ω) = xj W (f, g)(x, ω) − dt. 2 Rd 2 2 Subtracting the latter from the former and using (23) we obtain ∂ωj W (f, g)(x, ω) = (−2πi)[W (tj f, g)(x, ω) − W (f, tj g)(x, ω)]. As for the time case, the induction process yields α2 ∂ωα2 W (f, g)(x, ω) = (−2πi)|α2 | (−1)β2 W (tα2 −β2 f, tβ2 g)(x, ω). β2

(24)

β2 ≤α2

By combining (22) and (24) we obtain (21).

2.2. Modulation spaces The modulation space norm measures the joint time-frequency distribution of a f ∈ S . For their basic properties we refer, for instance, to [12, Ch. 11–13] and the original literature quoted there. The introduction of the general theory of modulation spaces is beyond our scope, therefore we shall limit ourselves to draw on their unweighted version. Given a non-zero window g ∈ S(Rd ), and 1 ≤ p, q ≤ ∞, the modulation space M p,q (Rd ) consists of all tempered distributions f ∈ S (Rd ) such that Vg f ∈ Lp,q (R2d ) (mixed-norm spaces). The norm on M p,q is q/p 1/q p f M p,q = Vg f Lp,q = |Vg f (x, ω)| dx dω . Rd

Rd

If p = q, we write M p instead of M p,p . Modulation spaces M p,q are Banach spaces whose deﬁnition is independent of the choice of the window g. Moreover, if g ∈ M 1 \ {0}, then Vg f Lp,q is an equivalent norm for M p,q (Rd ) (see [12, Thm. 11.3.7]). We recall that M 2 (Rd ) = L2 (Rd ) and, among the weighted modulation spaces one can encounter Sobolev spaces and Shubin-Sobolev spaces. Furthermore, the space of tempered distribution S is recovered as unions of suitable weighted modulation spaces. 2.3. Convolution relations and Wigner estimate In view of the relation between the multiplier a and the Weyl symbol (8), we shall use convolution relations between modulation spaces and some properties of the Wigner distribution. Convolution relations for modulation spaces, studied in [5, 18], yield the following unweighted version.

On the Product of Localization Operators Proposition 2.3. Let 1 ≤ p, q, r, s, t ≤ ∞. If 1 1 1 + −1= , and p q r then

287

1 1 + = 1, t t

M p,st (Rd ) ∗ M q,st (Rd ) → M r,s (Rd ) with norm inequality f ∗ hM r,s < ∼ f M p,st hM q,st .

(25)

The modulation space norm of a cross-Wigner distribution may be controlled by the window norms, as expressed below (see [5]). Proposition 2.4. If ϕ1 , ϕ2 ∈ M 1 (Rd ) we have W (ϕ2 , ϕ1 ) ∈ M 1 (R2d ), with W (ϕ2 , ϕ1 )M 1 < ∼ ϕ1 M 1 ϕ2 M 1 .

(26)

The modulation space M ∞,1 is the so-called Sj¨ostrand class and deserves quite an attention when studying Weyl operators. In particular, Sj¨ ostrand in [17] proved that, if the Weyl symbol σ belongs to M ∞,1 , the corresponding Weyl operator Lσ is bounded on L2 (Rd ). Besides, if σ, τ ∈ M ∞,1 , and Lµ = Lσ Lτ , then µ ∈ M ∞,1 ; thus M ∞,1 is a Banach algebra of pseudo-diﬀerential operators. In [13] the previous result is recaptured by using time-frequency analysis techniques. In particular, the Banach algebra property follows from the continuity of the twisted multiplication (see [13, Thm. 4.2]): Theorem 2.5. The modulation space M ∞,1 is a Banach ∗-algebra with respect to twisted multiplication and the involution σ → σ ¯ . In particular, the M ∞,1 -norm with respect to a window Wigner distribution W (ϕ, ϕ), with ϕ ∈ S, is given by στ M ∞,1 ≤ Cϕ σM ∞,1 τ M ∞,1 ,

∀ σ, τ ∈ M ∞,1 .

(27)

The continuity of the twisted multiplication on M ∞,1 × M ∞,1 will be employed when composing Gabor multipliers (Section 4).

3. Examples of well-localized products We shall provide few examples of products of localization operators. To this aim, we ﬁrst need some results on Weyl operators. Every rank one linear operator acting on L2 (Rd ) can be interpreted as Weyl operator. The characterization of its Weyl symbol is given in [12, Lemma 14.6.3]: Lemma 3.1. Given h, k ∈ L2 (Rd ) and set σ = W (h, k). Then Lσ is the rank one operator (28) Lσ f = f, kh, f ∈ L2 (Rd ). If we express the product of two rank one operators as a Weyl transform, we obtain immediately a formula for the twisted multiplication of cross-Wigner distributions.

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Lemma 3.2. Given ϕ1 , ϕ2 , ψ1 , ψ2 ∈ L2 (Rd ) we have W (ϕ1 , ϕ2 )W (ψ1 , ψ2 ) = ψ1 , ϕ2 W (ϕ1 , ψ2 )

(29)

Proof. Instead of the explicit formula (10), we use Lemma 3.1 and compute the product of the rank one operators LW (ϕ1 ,ϕ2 ) LW (ψ1 ,ψ2 ) . Let f be in L2 (Rd ), then LW (ϕ1 ,ϕ2 ) LW (ψ1 ,ψ2 ) f

=

LW (ψ1 ,ψ2 ) f, ϕ2 ϕ1

= =

f, ψ2 ψ1 , ϕ2 ϕ1 ψ1 , ϕ2 f, ψ2 ϕ1

=

L ψ1 ,ϕ2 W (ϕ1 ,ψ2 ) f.

Hence, the Weyl operator LW (ϕ1 ,ϕ2 )W (ψ1 ,ψ2 ) = LW (ϕ1 ,ϕ2 ) LW (ψ1 ,ψ2 ) possesses the Weyl symbol claimed in (29). With these tools, we can now compute the product of two localization operators whose symbols have minimal support. Proposition 3.3. Let 1 ≤ j ≤ d, and consider the distributions with support at the origin a = ∂xj δ, b = δ. For every ϕk ∈ S(Rd ), k = 1, . . . , 4, we have ϕ ,∂ ϕ

ϕ3 ,ϕ4 2 1 ,ϕ2 3 ,ϕ2 Aϕ = A ϕ34 ,ϕj 1 δ + Aϕ ∂x δ Aδ

ϕ4 ,∂j ϕ1 δ .

(30)

j

Proof. Rewriting the composition of two localization operators as a Weyl transform (7), we reduce ourselves to compute the twisted multiplication of the corresponding Weyl symbols. Using Lemma 3.1 and (29), the desired result follows. Namely, [(∂xj δ) ∗ W (ϕ2 , ϕ1 )] [δ ∗ W (ϕ4 , ϕ3 )] = [δ ∗ ∂xj W (ϕ2 , ϕ1 )] W (ϕ4 , ϕ3 ) = [W (∂j ϕ2 , ϕ1 ) + W (ϕ2 , ∂j ϕ1 )] W (ϕ4 , ϕ3 ) = W (∂j ϕ2 , ϕ1 ) W (ϕ4 , ϕ3 ) + W (ϕ2 , ∂j ϕ1 ) W (ϕ4 , ϕ3 ) = ϕ4 , ϕ1 W (∂j ϕ2 , ϕ3 ) + ϕ4 , ∂j ϕ1 W (ϕ2 , ϕ3 ), = δ ∗ [ϕ4 , ϕ1 W (∂j ϕ2 , ϕ3 )] + δ ∗ [ϕ4 , ∂j ϕ1 W (ϕ2 , ϕ3 )].

The product above is no longer a single localization operator, in this sense the composition is unstable. However, the product in (30) is still a sum of two localization operators, and both have symbols localized at the origin If we choose the window with the optimal time-frequency localization, i.e., the normalized Gaussian 2 ϕ(t) = 2d/4 e−πt and set ϕ1 = ϕ4 = ϕ, then formula (30) reduces to ϕ ,∂j ϕ2

ϕ3 ,ϕ 2 = Aδ 3 Aϕ,ϕ ∂x δ Aδ j

,

∀ ϕ2 , ϕ3 ∈ S

(31)

because ϕ4 , ∂j ϕ1 = 0 and ϕ, ϕ = 1. Next, choosing the partial derivatives ∂ωj δ, j = 1, . . . , d, as symbols, we obtain a similar formula.

On the Product of Localization Operators

289

Proposition 3.4. Let 1 ≤ j ≤ d, and consider the distributions with support at the origin a = ∂ωj δ, b = δ. For every ϕk ∈ S(Rd ), k = 1, . . . , 4, we have ϕ ,t ϕ

ϕ3 ,ϕ4 j 2 1 ,ϕ2 3 ,ϕ2 Aϕ = A ϕ34 ,ϕ + Aϕ ∂ω δ Aδ

ϕ4 ,tj ϕ1 δ . 1 δ

(32)

j

The proof is similar to the one of Proposition 3.3. Again, if ϕ1 = ϕ4 = ϕ = 2 2d/4 e−πt , then product is is stable, and ϕ ,tj ϕ2

ϕ3 ,ϕ 2 Aϕ,ϕ = Aδ 3 ∂ω δ Aδ

,

j

∀ ϕ2 , ϕ3 ∈ S.

If we increase the order of the derivative of the symbol, the product of two localization operators is never a single localization operator, and the stability of the product is deﬁnitely lost, as is shown the the following observation. Nevertheless, the supports of the symbols are all localized at the origin. Proposition 3.5. Let α = (α1 , α2 ), β = (β1 , β2 ) ∈ Zd+ × Zd+ , ϕi ∈ S(Rd ), i = 1, . . . , 4. Then the product of localization operators whose symbols are derivatives of the delta distribution is given by (tν2 ∂ ν1 ϕ ),(tα2 −ν2 ∂ α1 −ν1 ϕ ) 2 1 ,ϕ2 4 Aϕ3β,ϕ = (−2πi)|α2 +β2 | Acα,β,γ,ν δ 3 , (33) Aϕ α α 1 β2 (∂ 1 ∂ 2 δ) x

ω

where cα,β,γ,ν

(∂x ∂ω δ)

γ≤α ν≤β

α b = (−1)|γ2 +ν2 | tβ2 −ν2 ∂ β1 −ν1 ϕ4 , tγ2 ∂ γ1 ϕ1 . γ ν

Again, the proof relies on the same tools as for Proposition 3.3 and therefore we shall omit it. If the derivative order is greater than one, we highlight that neither the Gaussian choice for the windows could help us to have a single localization operator in the right-hand side. In fact, the brackets tβ2 −ν2 ∂ β1 −ν1 ϕ, tγ2 ∂ γ1 ϕ do not vanish if |β − ν + γ| ∈ 2N.

4. Product of Gabor multipliers In this section we shall study the product of Gabor multipliers, for a survey on the topic we refer to [10]. Consider a time-frequency lattice Λ in R2d , for instance, Λ = αZd × βZd , α, β ∈ R and set a = (aλ )λ∈Λ ; moreover, choose two non-zero window functions ϕ1 , ϕ2 ∈ L2 (Rd ). Then, a Gabor multiplier Ga associated to the triple (ϕ1 , ϕ2 , Λ) and with symbol a is given by Ga f = aλ f, π(λ)ϕ1 π(λ)ϕ2 , f ∈ L2 (Rd ). (34) λ∈Λ

The domain of a Gabor multiplier can even be a subspace of tempered distribution rather than simply a space of functions. The distribution/function space of f in (34) depends on the decay and smoothness properties of the pair (ϕ1 , ϕ2 ) of dual windows and on the decay of the symbol a. Gabor multipliers are special cases of localization operators, as we shall see presently.

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ϕ2 ∈ M 1 (Rd ). ConLemma 4.1. Let Λ be a lattice in R2d , (aλ )λ∈Λ ∈ ∞ (Λ), ϕ1 , ∞ ∞ 2d sider the mapping i : (Λ) → M (R ) deﬁned by i(a) = λ∈Λ aλ δλ , then we obtain 1 ,ϕ2 (35) Ga = Aϕ i(a) . Proof. An easy computation shows that i(a) ∈ M ∞ (R2d ). More precisely, choose d support such that, for some constants A, B > 0 a window g ∈ S(R ) with compact 2 we have A ≤ |ˆ g (ξ − µ)| ≤ B < ∞ for all ξ ∈ Rd (as usual Λ⊥ deµ∈Λ⊥ notes the dual lattice of Λ). A result in approximation theory then implies that supx∈Rd | λ∈Λ aλ g(x − λ)| + a∞ . Using this observation, we obtain sup | aλ δλ , Mξ Tx g| i(a)M ∞ = x,ξ∈Rd

=

sup |

λ∈Λ

aλ e−2πiλξ g(x − λ)|

x,ξ∈Rd λ∈Λ

+ Consequently,

sup |aλ | = a∞ .

λ∈Λ

aλ δδ M ∞ + a∞

∀a ∈ ∞ (Λ) .

(36)

λ∈Λ

By assumption ϕ1 , ϕ2 ∈ M 1 (Rd ), we then appeal to [5, Theorem 3.2] to 1 ,ϕ2 deduce the boundedness of the localization operator Aϕ on L2 (Rd ). Using the i(a) d weak deﬁnition (5), we observe that, for every f, g ∈ S(R ), 1 ,ϕ2 Aϕ i(a) f, g = i(a), Vϕ1 f Vϕ2 g = aλ δλ , Vϕ1 f Vϕ2 g

λ∈Λ

=

aλ Vϕ1 f (λ)Vϕ2 g(λ)

λ∈Λ

=

aλ f, π(λ)ϕ1 π(λ)ϕ2 , g.

λ∈Λ 1 ,ϕ2 That is, the localization operator Aϕ coincides with the Gabor multiplier i(a) λ∈Λ aλ ·, π(λ)ϕ1 π(λ)ϕ2 . Finally, we note that for f, g ∈ L2 (Rd ) we have (f, π(λ)ϕ1 ) ∈ 2 (Λ) with 2 a norm estimate f, π(λ)ϕ1 λ∈Λ 2 < ∼ f 2 , and likewise (π(λ)ϕ2 , g) ∈ (Λ), thus (f, π(λ)ϕ1 π(λ)ϕ2 , g) ∈ 1 (Λ) and the 1 -norm is bounded by f 2 g2 . 1 ,ϕ2 Consequently, the series deﬁning Aϕ i(a) f, g converges absolutely. This implies that the partial sums SN = λ∈Λ:|λ|≤N aλ ·, π(λ)ϕ1 π(λ)ϕ2 converge in the strong 1 ,ϕ2 operator topology to Aϕ i(a) .

Since Gabor multipliers are a special case of localization operators, we may compute their product via Weyl calculus. We shall see that the result is a Gabor

On the Product of Localization Operators

291

multiplier plus a remainder term. The remainder operator is no more expressible in terms of Gabor multipliers but it can be recaptured as series of suitable localization operators. Proposition 4.2. Let Λ be a time-frequency lattice in R2d . Consider ϕi ∈ M 1 , i = 1, . . . , 4, two sequences a, b ∈ ∞ (Λ) and the Gabor multipliers Ga and Gb associated to the triple (ϕ1 , ϕ2 , Λ) and (ϕ3 , ϕ4 , Λ). Set c = (aλ bλ )λ , and let Gc be the Gabor multiplier associated to the triple (ϕ3 , ϕ2 , Λ). Then, π(µ)ϕ ,π(λ)ϕ Ga Gb = ϕ4 , ϕ1 Gc + A π(µ)ϕ34 ,π(λ)ϕ21 a(λ)b(µ)δ . (37) λ,µ∈Λ λ=µ

Proof. In virtue of (35), we rewrite the Gabor multipliers Ga , Gb as localization operators and then use the Weyl connection as in the preceding proofs. Precisely, if we name σ and τ the Weyl symbols of Ga and Gb , respectively, their expression is given by aλ δλ ∗ W (ϕ2 , ϕ1 ), τ = bλ δλ ∗ W (ϕ4 , ϕ3 ). σ= λ∈Λ

λ∈Λ

Next, we need to compute σ τ . In this framework, Proposition 2.4 guarantees W (ϕ2 , ϕ1 ), W (ϕ4 , ϕ3 ) ∈ M 1 (R2d ), whereas Proposition 2.3 provides the continuity of the convolution acting from M ∞ × M 1 into M ∞,1 . Getting the preceding results all together, we observe that the Weyl symbols σ and τ belong to M ∞,1 (R2d ). Thus, the corresponding operators Lσ and Lτ are bounded operators on L2 (Rd ) and the same for their product Lσ τ , with σ τ ∈ M ∞,1 . In the following, we shall calculate σ τ explicitly, using the boundedness properties of the convolution and twisted multiplication listed above. Relation (20) will be repeatedly used in the sequel, ⎤ ⎞ < ⎡⎛ ; aλ δλ ∗ W (ϕ2 , ϕ1 ) ⎣⎝ bµ δµ ⎠ ∗ W (ϕ4 , ϕ3 )⎦ στ = λ∈Λ

; = ; =

<

(aλ δλ ∗ W (ϕ2 , ϕ1 )) ⎣

λ∈Λ

<

µ∈Λ

;

⎤

(bµ δµ ∗ W (ϕ4 , ϕ3 ))⎦

µ∈Λ

aλ W (π(λ)ϕ2 , π(λ)ϕ1 )

λ∈Λ

=

⎡

< bµ W (π(µ)ϕ4 , π(µ)ϕ3 )

λ∈Λ

aλ bµ [W (π(λ)ϕ2 , π(λ)ϕ1 ) W (π(µ)ϕ4 , π(µ)ϕ3 )]

λ,µ∈Λ

= ϕ4 , ϕ1

aλ bλ W (π(λ)ϕ2 , π(λ)ϕ3 )+

λ∈Λ

+

λ,µ∈Λ, λ=µ

aλ bµ π(µ)ϕ4 , π(λ)ϕ1 W (π(λ)ϕ2 , π(µ)ϕ3 ).

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E. Cordero and K. Gr¨ ochenig = ϕ4 , ϕ1

(aλ bλ ) δλ ∗ W (ϕ2 , ϕ3 )+

λ∈Λ

+

aλ bµ π(µ)ϕ4 , π(λ)ϕ1 W (π(λ)ϕ2 , π(µ)ϕ3 ).

λ,µ∈Λ, λ=µ

We notice that the convergence of the series in the line above is assumed to be in the M ∞,1 -norm. Passing to the corresponding Weyl operator we obtain the desired result. In the above argument we have assumed that all series converge in norm, in particular that λ aλ δλ converges in the M ∞ -norm. According to (36) this is the case if and only if a ∈ c0 (Λ), i.e., if lim|λ|→∞ |aλ | = 0. For arbitrary bounded sequences we now give an alternative argument. Second proof of Proposition 4.2. Assume ﬁrst that a and b have ﬁnite support, so that there are no issues about the convergence of the series involved. Then we calculate directly that aλ bµ f, π(µ)ϕ3 π(µ)ϕ4 , π(λ)ϕ1 π(λ)ϕ2 Ga Gb f = λ∈Λ µ∈Λ

=

µ=λ

=

+

...

µ=λ

aλ bλ ϕ4 , ϕ1 f, π(λ)ϕ3 π(λ)ϕ2 +

λ∈Λ

+

aλ bµ f, π(µ)ϕ3 π(µ)ϕ4 , π(λ)ϕ1 π(λ)ϕ2

λ,µ∈Λ,µ=λ

=

ϕ4 , ϕ1 Gc +

π(µ)ϕ ,π(λ)ϕ

A π(µ)ϕ34 ,π(λ)ϕ21 a(λ)b(µ)δ .

λ,µ∈Λ,λ=µ ∞

If a, b ∈ (λ), we consider the partial sums GaM f = aλ f, π(λ)ϕ1 π(λ)ϕ2 λ∈Λ,|λ|≤M

and GbN f =

bµ f, π(µ)ϕ1 π(µ)ϕ2 .

µ∈Λ,|≤N

Since GaM and GbN f converge strongly to Ga and Gb , the above argument carries over to general bounded sequences a and b.

5. Gaussian functions as symbols In this section we prove Theorem 1.2. Instead of using the product formula (12), as done in [9], we use the techniques of the preceding two sections. The result is a mere consequence of the Gaussian nice behavior under convolution and twisted products.

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293

Lemma 5.1. Let a, b > 0, then (i) Gaussian convolution: 2

2

ab

2

a+b

2

[e−πat ∗ e−πbt ](x) = (a + b)−d/2 e−π a+b x ,

t, x ∈ Rd .

(38)

z, ζ ∈ R2d .

(39)

(ii) Gaussian twisted multiplication: 2

2

[e−2πaz e−2πbz ](ζ) = (1 + ab)−d e−2π 1+ab ζ ,

Proof. (i) The semigroup property of Gaussians is well known, see for instance [11], and follows by an easy calculation (ii) We use the deﬁnition of in (10) and make a direct computation using Gaussian integrals. All integrals converge absolutely and exchanging the order of integration is justiﬁed by Fubini’s Theorem. 2 2 2 2 e−2πaz e−2πbw e4πi[ζ−w,ζ−z] dzdw [e−2πaz e−2πbz ](ζ) = 22d 2d R 2 2 = 22d e−2πaz e−4πi[ζ−w,z] dz e−2πbw e−4πi[w,ζ] dw. R2d

R2d

To begin with, we compute the interior integral with respect to the variable z. Namely, if z = (z1 , z2 ), w = (w1 , w2 ), ζ = (ζ1 , ζ2 ) ∈ R2d , −2πaz 2 −4πi[ζ−w,z] −2πaz12 −2πi(−2(ζ2 −w2 )z1 ) e e dz = e e dz1 R2d Rd −2πaz22 −2πi(2(ζ1 −w1 )z2 ) · e e dz2 Rd

2 −d − 2π a (ζ−w)

= (2a)

e

.

Then, we have [e

−2πaz 2

e

−2πbz 2

](ζ)

= = =

d 2 2 2π 2 e− a (ζ−w) e−2πbw e−4πi[w,ζ] dw a R2d d 1+ab 2 4πi 2 − 2π ζ2 a e e−2π a w e a ζw e−4πi[w,ζ] dw a 2d R !2 d 1/2 ζ −2π ( 1+ab w− 2 1 ) 2 a 1/2 − 2π 1− ζ (a(1+ab)) e a ( 1+ab ) e a R2d ·

e−4πi[w,ζ] dw.

Splitting up the variables again, the twisted multiplication reduces to d (1+ab) 2 2 2 2 2π 1 2 [e−2πaz e−2πbz ](ζ) = e− a (1− 1+ab )ζ T ζ1 e−2π a w1 a Rd (a(1+ab))1/2 (1+ab) 2 · e−2πi(2ζ2 )w1 dw1 T ζ2 e−2π a w2e−2πi(−2ζ1 )w2dw2 Rd

(a(1+ab))1/2

294

E. Cordero and K. Gr¨ ochenig d (1+ab) 2 1 2 − 2π 1− 1+ab ζ2 ( ) a e e−2π a w1 e−2πi(2ζ2 )w1 dw1 = a d R (1+ab) −2π a w22 −2πi(−2ζ1 )w2 · e e dw2 Rd

2

= (1 + ab)−d e−2π 1+ab ζ , a+b

as desired. 2

Corollary 5.2. Let ϕ(t) = 2d/4 e−πt anc c > 0. Then, 2

2

(e−2πcz ∗ W (ϕ, ϕ))(ζ) = (c + 1)−d e−2π c+1 ζ , c

z, ζ ∈ R2d .

Proof. It is a straightforward consequence of the relations (11) and (38).

(40)

Now we have all we need to prove the main result of this section. Proof of Theorem 1.2. We use relation (7) and the bilinearity of both the convolution and twisted multiplication. This yields [a ∗ W (ϕ, ϕ)] [b ∗ W (ϕ, ϕ)](ζ) =

m l

Ck Cj pk,j (ζ),

k=1 j=1

2

2

with pk,j (ζ) := [e−2πdk z ∗W (ϕ, ϕ)] [e−2πdj z ∗W (ϕ, ϕ)](ζ). The convolution products are achieved by (40) and the outcomes are pk,j (ζ) = (dk + 1)−d (dj + 1)−d (e

−2π d

dk z2 k +1

d

e

j −2π d +1 z2 j

)(ζ).

Finally, we compute the twisted multiplication using (39) and we get pk,j (ζ)

−d

= (dk + 1)

(dj

−d

+ 1)

(e

−2π d

= (dk + dj + 2dk dj + 1)−d e

dk z2 k +1

d

e

j −2π d +1 z2 j

dk +dj +2dk dj −2π d +d +2d k dj +1 k j

)(ζ)

ζ2

2

= [e−2π(dk +dj +2dk dj )z ∗ W (ϕ, ϕ)](ζ), where in the last equality we used relation (40) backwards.

Acknowledgment This work was started while both authors were visiting the Erwin Schr¨ odinger Institute in Vienna. Its hospitality and great working conditions are gratefully acknowledged.

References [1] H. Ando and Y. Morimoto, Wick calculus and the Cauchy problem for some dispersive equations. Osaka J. Math. 39 (1) (2002), 123–147. [2] F.A. Berezin, Wick and anti-Wick symbols of operators, Mat. Sb. (N.S.) 86 (128) (1971), 578–610.

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[3] P. Boggiatto, E. Cordero, and K. Gr¨ ochenig, Generalized Anti-Wick operators with symbols in distributional Sobolev spaces. Integral Equations Operator Theory 48 (2004), 427–442. [4] E. Cordero, Gelfand-Shilov windows for weighted modulation spaces, Integral Transforms Spec. Funct., to appear. [5] E. Cordero and K. Gr¨ ochenig, Time-frequency analysis of localization operators. J. Funct. Anal. 205 (1) (2003), 107–131. [6] E. Cordero and K. Gr¨ ochenig. Symbolic calculus and Fredholm property for localization operators, J. Fourier Anal. Appl. 12 (4) (2006), 371–392. [7] E. Cordero and L. Rodino, Wick calculus: a time-frequency approach, Osaka J. Math. 42 (1) (2005), 43–63. [8] J. Du and M.W. Wong, A product formula for localization operators, Bull. Korean Math. Soc. 37 (1) (2000), 77–84. [9] J. Du and M.W. Wong, Gaussian functions and Daubechies operators, Integral Equations Operator Theory 38 (1) (2000), 1–8. [10] H.G. Feichtinger and K. Nowak, A first survey of Gabor multipliers, in Advances in Gabor Analysis, Editors: H.G. Feichtinger and T. Strohmer, Birkh¨ auser, Boston, 2002, 99–128. [11] G.B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ, 1989. [12] K. Gr¨ ochenig, Foundations of Time-Frequency Analysis, Birkh¨ auser, Boston, 2001. [13] K. Gr¨ ochenig, Composition and spectral invariance of pseudodiﬀerential operators on modulation spaces, J. Anal. Math., J. Anal. Math. 98 (2006), 65–82. [14] L. H¨ ormander, The Analysis of Linear Partial Diﬀerential Operators III, SpringerVerlag, Berlin, 1994. [15] N. Lerner, The Wick calculus of pseudo-diﬀerential operators and energy estimates, in New Trends in Microlocal Analysis, Springer, 1997, 23–37. [16] M.A. Shubin, Pseudodiﬀerential Operators and Spectral Theory, Second Edition, Springer-Verlag, Berlin, 2001. [17] J. Sj¨ ostrand, An algebra of pseudodiﬀerential operators, Math. Res. Lett. 1 (2) (1994), 185–192. [18] J. Toft, Continuity properties for modulation spaces with applications to pseudodiﬀerential calculus, I, J. Funct. Anal. 207 (2) (2004), 399–429. [19] M.W. Wong, Wavelet Transforms and Localization Operators, Birkh¨ auser, Basel, 2002. Elena Cordero Department of Mathematics University of Torino, Italy e-mail: [email protected] Karlheinz Gr¨ ochenig Faculty of Mathematics University of Vienna, Austria e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 297–312 c 2006 Birkh¨ auser Verlag Basel/Switzerland

Gelfand–Shilov Spaces, Pseudo-diﬀerential Operators and Localization Operators Marco Cappiello, Todor Gramchev and Luigi Rodino Abstract. We present new results concerning pseudo-diﬀerential operators in the function spaces Sνµ (Rn ) of Gelfand and Shilov. In particular we discuss Sνµ (Rn )-regularity of solutions to SG-elliptic pseudo-diﬀerential equations, allowing lower order semilinear perturbations. The results apply to SG-elliptic partial diﬀerential equations with polynomial coeﬃcients. We also study the action of Weyl operators and localization operators on Sµµ (Rn ). Mathematics Subject Classiﬁcation (2000). Primary 35S05; Secondary 47G30. Keywords. Ultradiﬀerentiable functions, Gelfand–Shilov spaces, pseudo-differential operators, time-frequency analysis.

1. Introduction In [14] I.M. Gelfand and G.E. Shilov deﬁned general classes of ultradiﬀerentiable test functions, including as relevant case Sνµ (Rn ), µ ≥ 0, ν ≥ 0, space of all functions f ∈ C ∞ (Rn ) satisfying for suitable positive constants C, ε the estimates in Rn 1/ν |∂xα f (x)| ≤ C |α|+1 (α!)µ e−ε|x| . (1.1) Hence Sνµ (Rn ) is a subspace of the Schwartz space S(Rn ). In fact, motivation in [14] was to introduce general ultradistributions spaces (Sνµ ) (Rn ), containing the Schwartz dual S (Rn ); in the corresponding functional frame, existence and uniqueness was proved for the (non-Kowalewskian) Cauchy problem for heat equation and other parabolic equations with constant coeﬃcients. It is easy to verify that f ∈ Sνµ (Rn ) if and only if there exists a positive constant C such that sup

sup C −|α|−|β| (α!)−µ (β!)−ν |xβ ∂xα f (x)| < +∞.

(1.2)

n α,β∈Zn + x∈R

We have Sνµ (Rn ) = {0} iﬀ µ + ν > 1 or µ + ν = 1 and µν > 0, Sνµ (Rn ) → Sνµ (Rn ) if µ ≤ µ , ν ≤ ν and the Fourier transform acts as an isomorphism interchanging

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M. Cappiello, T. Gramchev and L. Rodino

the indices µ and ν F : Sνµ (Rn ) −→ Sµν (Rn ).

(1.3)

for µ ≥ 1/2, and In particular, we obtain that F is an automorphism of Sµµ (Rn ), 1/2 ≤ µ < 1, is a subset of the restrictions of the entire functions on Cn of exponential type 1/µ (cf. [14], [20] and the references therein). There are equivalent ways of deﬁning Sνµ (Rn ) as inductive limit of scales of Banach spaces, cf. [9] and Section 3 below. Gelfand–Shilov spaces, namely the above classes Sνµ (Rn ), played an important role in the development of the general theory of partial diﬀerential equations, after [14]. Recently, they have been also considered as functional framework for pseudo-diﬀerential operators and localization operators. We want to present some new results in this line, improving in part preceding contributions of the authors and their collaborators ([3], [5], [6], [7], [8], [10], [15]). We ﬁrst recall that in [5] Fourier integral operators in Rn were treated, of the form (2π)−n eiϕ(x,ξ) p(x, ξ)ˆ u(ξ)dξ, Sµµ (Rn )

where uˆ denotes the Fourier transform of u. Suitable assumptions on the phase ϕ(x, ξ) and the symbol p(x, ξ) grant a calculus on Sνµ classes. In the pseudodiﬀerential case ϕ(x, ξ) = x, ξ, these results were applied in [6] to the study of the class of the SG-elliptic equations, having as a model P = (1 + |x|2k )(−∆ + 1) + L1 (x, D),

(1.4)

where L1 is a ﬁrst order operator with polynomial coeﬃcients of degree 2k − 1, k ≥ 1. The main concern in [6], deriving from problems in Quantum Mechanics (see [1]) was the exponential decay of the solutions, expressed by the index ν in the estimates (1.1). In the next Section 2 we present a (light) generalization of the calculus in [6]. To be precise, we ﬁx attention on the linear problem P u = f,

(1.5)

where f belongs to the Gelfand–Shilov classes and P is an SG-elliptic diﬀerential operator with polynomial coeﬃcients or, more generally, an SG elliptic pseudodiﬀerential operator. Namely, we assume that the symbol p(x, ξ) of P satisﬁes the following estimates α β Dξ Dx p(x, ξ) ≤ C |α|+|β|+1 (α!)ν (β!)µ ξm1 −|α| xm2 −|β| (1.6) for every (x, ξ) ∈ R2n and α, β ∈ Zn+ , and a global ellipticity condition of the form |p(x, ξ)| ≥ Co ξm1 xm2

for

|x| + |ξ| ≥ R > 0.

(1.7)

For example (1.6) and (1.7) are satisﬁed by P in (1.4) with m1 = 2, m2 = 2k. The result of [6] was that for f ∈ Sθθ (Rn ) in (1.5), with θ ≥ µ+ν −1, µ ≥ 1, ν ≥ 1, under the further assumption θ > 1, all the solutions u ∈ (Sθθ ) (Rn ) are in fact in Sθθ (Rn ). In Section 2 we shall allow f ∈ Sνµ (Rn ), with µ > 1, ν > 1, min{µ , ν } ≥ µ+ν −1, and for u ∈ (Sνµ ) (Rn ) conclude that u ∈ Sνµ (Rn ), see Corollary 2.8 below. This

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result is more general than [6], but it does not eliminate the drawback of the assumption θ > 1. So for example, for SG-elliptic operators satisfying (1.6) with µ = ν = 1, in particular for P in (1.4), the solutions of the homogeneous equation P u = 0 are proved by Corollary 2.8 to belong to Sνµ (Rn ) for any µ > 1, ν > 1, hence in particular 1/ν |u(x)| ≤ Ce−ε|x| (1.8) 1 n with 1/ν < 1. However the expected optimal result is u ∈ S1 (R ), that implies ν = 1 in (1.8), according to the known results for the one-dimensional case. The fact is that in [6] and Section 2 below the technique does not give satisfactory results for analytic symbols since it involves Gevrey cut-oﬀ functions of order µ and ν, which must be assumed both strictly greater than 1. On the other hand, in a recent paper [7], the authors proved some kinds of S11 estimates for SG-elliptic diﬀerential operators with polynomial coeﬃcients. Here we want to improve and generalize these results to pseudo-diﬀerential operators, see the next Theorem 3.1. To do this, we use an iterative scheme developed in [3], [15] for studying the S11 regularity of travelling waves, cf. [4]. This method requires some technical assumptions we shall give in Section 3. In Section 4, we show that for a self-adjoint diﬀerential operator with polynomial coeﬃcients of the form P (x, D) = cαβ xβ Dxα (1.9) |α|≤m1

|β|≤m2

satisfying (1.7) these conditions are fulﬁlled. On the other hand, the iterative scheme mentioned above allows to prove analogous results also for semilinear perturbations of (1.5) of the form P u = f + F [u] where the nonlinear term F is of the form F [u] = Fγjk xj uk (Dγ u)hγ ,

(1.10)

(1.11)

with j, k ∈ Z+ , 0 ≤ k ≤ M for some M, |j| ≤ m2 − 1, 0 < |γ| ≤ mo for some mo < m1 , hγ < m1 /|γ|. Namely, if P in (1.10) is of the form (1.9), under the only assumption of SG-ellipticity we get solutions in Sνµ (Rn ) if f ∈ Sνµ (Rn ), for any µ ≥ 1, ν ≥ 1, hence u ∈ S11 (Rn ) if f = 0, see Theorem 4.5 below. This applies in particular to P in (1.4). The method has been tested already in [7] on some examples of SG-elliptic operators and in [8] on globally elliptic operators of Shubin type (cf. [24]) for semilinear autonomous equations. In Section 5 ﬁnally, in a diﬀerent direction, we consider localization operators and Weyl pseudo-diﬀerential operators on Gelfand–Shilov spaces. We do not state new results here, but just observe that localization operators with windows in Sµµ (Rn ), as well as generic Weyl operators, deﬁne continuous maps from Sµµ (Rn ) to (Sµµ ) (Rn ), provided the symbol is in (Sµµ ) (R2n ), see Propositions 5.1 and 5.2 below. This is an easy consequence of the results of [17], [25] concerning the Sνµ (Rn )-regularity of the short-time Fourier transform and Wigner transform.

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We obtain in this way a very general setting for localization operators, which was already adopted in [10] in the case µ = 1, to characterize ultradistribution symbols giving L2 -bounded operators.

2. Pseudo-diﬀerential operators and Gelfand–Shilov spaces In this section, we brieﬂy describe the action of SG pseudo-diﬀerential operators with symbols deﬁned by (1.6) on the spaces Sνµ (Rn ), µ ≥ 1, ν ≥ 1 and give some results about composition of operators and parametrices. The statements in the sequel provide a generalization of the calculus developed in [6] for µ = ν > 1 following essentially the same approach. We shall omit the proofs which are just a repetition of [6], and only detail the new part. Let m = (m1 , m2 ) ∈ R2 and let µ, ν be real numbers such that µ ≥ 1, ν ≥ 1. m 2n Deﬁnition 2.1. We shall denote by Γm ν,µ = Γν,µ (R ) the space of all functions ∞ 2n p(x, ξ) ∈ C (R ) satisfying the estimates (1.6). We shall use a separate notation m for the class of analytic symbols Γm 1,1 denoting it by Γa .

Given p ∈ Γm ν,µ , we can consider the pseudo-diﬀerential operator deﬁned as standard by P u(x) = p(x, D)u(x) = (2π)−n ei x,ξ p(x, ξ)ˆ u(ξ)dξ, u ∈ S(Rn ), (2.1) Rn

where u ˆ denotes the Fourier transform of u. We shall denote by OP Γm ν,µ the space . of all operators of the form (2.1), with symbol in Γm ν,µ Theorem 2.2. Given p ∈ Γm ν,µ , the operator P deﬁned by (2.1) is linear and con

tinuous from Sνµ into itself for any µ , ν with µ ≥ µ, ν ≥ ν. Furthermore, P can be extended to a linear and continuous map from (Sνµ ) into itself. Proof. For any α, β ∈ Zn+ and for any positive integer N , we can write: β α β −n α x Dx P u(x) = (2π) x ei x,ξ ξ β1 Dxβ2 p(x, ξ)ˆ u(ξ)dξ β1 n R β1 +β2 =β β ei x,ξ (1 − ∆ξ )N [ξ β1 Dxβ2 p(x, ξ)ˆ u(ξ)]dξ. = (2π)−n xα x−2N β1 n R β1 +β2 =β 2 + 1, by (1.3) and by standard factorial inequalities, we Choosing N = |α|+m 2 obtain 1/µ x|α|−2N (1 − ∆ξ )N [ξ β1 Dxβ2 p(x, ξ)ˆ u(ξ)] ≤ C |α|+|β|+1 (α!)ν (β1 !)µ (β2 !)µ e−a ξ for some positive constants C, a. Then, by the conditions µ ≥ µ, ν ≥ ν, it follows that P is continuous from Sνµ into itself. In order to extend P on the dual space

Pseudo-diﬀerential Operators and Localization Operators

301

(Sνµ ) , we observe that, for u, v ∈ Sνµ , P u(x)v(x)dx = Rn

Rn

where −n

pv (ξ) = (2π)

u ˆ(ξ)pv (ξ)dξ

ei x,ξ p(x, ξ)v(x)dx Rn

Furthermore, by the same argument of the ﬁrst part of the proof, it follows that the map v → pv is linear and continuous from Sνµ to Sµν . Then, we can deﬁne, for

u ∈ (Sνµ ) , P u(v) = u ˆ(pv ),

v ∈ Sνµ .

This is a linear continuous map from (Sνµ ) into itself and it extends P . For t ≥ 0, denote by Qt the set Qt = {(x, ξ) ∈ R2n : ξ < t and x < t} and Qet = R2n \ Qt . m Deﬁnition 2.3. We denote by F Sν,µ the space of all formal sums

pj such that

j≥0

pj ∈ C ∞ (R2n ) for all j ≥ 0 and there exist positive constants B, C such that ∀j ≥ 0 : α β Dξ Dx pj (x, ξ) ≤ C |α|+|β|+2j+1 (α!)ν (β!)µ (j!)µ+ν−1 · ·ξm1 −|α|−j xm2 −|β|−j

(2.2)

for all α, β ∈ Zn+ and for all (x, ξ) ∈ QeBj µ+ν−1 . m Deﬁnition 2.4. We say that two sums pj , qj ∈ F Sν,µ are equivalent if there j≥0

j≥0

exist positive constants B, C such that for every N = 1, 2, . . . α β D D (pj − qj ) ≤ C |α|+|β|+2N +1 (α!)ν (β!)µ (N !)µ+ν−1 · ξ x j

pj ∼

j≥0

j≥0

Proposition 2.5. Let p ∈ Γ0ν,µ such that p ∼ 0 and let µ , ν be real numbers with min{µ , ν } ≥ µ + ν − 1. Then the operator P is Sνµ -regularizing, i.e., it extends to a continuous linear map from (Sνµ ) into Sνµ . In particular, if p ∈ Γ0a and p ∼ 0 in F Sa0 , then P is S11 -regularizing.

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Proof. Denote θ = min{µ , ν }. Since Sθθ (Rn ) ⊂ Sνµ (Rn ), by duality it is suﬃcient to show that P is Sθθ -regularizing to prove the assertion. Arguing as in [6], it is easy to prove that if p ∼ 0, then p ∈ Sθθ (R2n ). Then, the assertion follows invoking Proposition 2.11 in [5]. m As standard we can identify Γm ν,µ with a subspace of F Sν,µ by associating to m m each symbol p ∈ Γν,µ a sum pj ∈ F Sν,µ , with p0 = p and pj = 0 ∀j ≥ 1. In the j≥0 m pj ∈ F Sν,µ , opposite direction, by Theorem 2.14 in [5], starting from a sum j≥0 m we can ﬁnd a symbol p ∈ Γm pj in F Sν,µ . However, the ν,µ such that p ∼ j≥0

arguments used to prove this statement involve Gevrey cut-oﬀ functions of order µ and ν. This leads to assume the additional restriction µ > 1, ν > 1. From now on in this section, we shall assume µ > 1, ν > 1. The next results hold also for m analytic symbols of Γm a only considering it as a subclass of Γν,µ for any choice of µ > 1, ν > 1. We conclude this section recalling and stating more precisely the composition theorem for SG operators in Gelfand–Shilov spaces. We address to [6] for the proof. Construction of parametrices and regularity theorem follow then as standard.

m 2 Theorem 2.6. Let p ∈ Γm ν,µ , q ∈ Γν,µ for some m, m ∈ R . Then, there exists a

symbol s ∈ Γm+m such that p(x, D)q(x, D) = s(x, D)+R for some Sνµ -regularizing ν,µ operator R, where min{µ , ν } ≥ µ + ν − 1. Theorem 2.7. Let p ∈ Γm ν,µ be SG-elliptic, i.e., let (1.7) be satisﬁed. Then there such that p(x, D)q(x, D) = I + R1 , q(x, D)p(x, D) = I + R2 , for exists q ∈ Γ−m ν,µ

some Sνµ -regularizing operators R1 , R2 , with min{µ , ν } ≥ µ + ν − 1. Corollary 2.8. Let p ∈ Γm ν,µ , µ ≥ 1, ν ≥ 1 be SG-elliptic. Let µ > 1, ν > 1 satisfy

min{µ , ν } ≥ µ + ν − 1. Then, every solution u ∈ (Sνµ ) (Rn ) of the equation P u = f ∈ Sνµ (Rn ) belongs to Sνµ (Rn ).

3. Semilinear equations In this section, we prove the Sνµ estimates for the solutions of the equation (1.10). To do this, we ﬁrst need to introduce some scales of Sobolev norms deﬁning Sνµ (Rn ). Consider s ∈ R. Let us ﬁx µ ≥ 1, ν ≥ 1 and deﬁne, for δ, ε > 0 : u ε,δ = [u(α,β) ]µ,ν (3.1) ε,δ s , α,β∈Zn +

where [u(α,β) ]µ,ν ε,δ (x) =

ε|α| δ |β| β α x ∂x u(x). (α!)µ (β!)ν

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Equivalence with the deﬁnition from (1.2) is easily proved by standard factorial inequalities and Sobolev embedding estimates. Namely, for any ﬁxed s ∈ R, u ∈ Sνµ (Rn ) iﬀ u ε,δ < ∞ for some positive δ, ε. We also deﬁne the partial sums ε,δ [u] = [u(α,β) ]µ,ν N ∈ Z+ (3.2) SN ε,δ s , α,β∈Zn +

|α|+|β|≤N

and ε [u] = EN

α∈Zn +

ε|α| ∂ α u . (α!)µ x s

|α|≤N

Let P ∈ OP Γm ν,µ be SG-elliptic, i.e., let (1.7) be satisﬁed. In the subsequent proof we need the existence of λ ∈ C \ spec (P ). This hypothesis is always satisﬁed if, for example, P is selfadjoint, what we shall assume in the following for simplicity. Furthermore, in view of the results in [11], [12], [19], [23] on SG pseudo-diﬀerential calculus, if m1 > 0, m2 > 0 the operator P − λ is also SG-elliptic and (P − λ)−1 ◦ xq ∂xp : H s (Rn ) → H s (Rn )

(3.3)

is continuous for all p, q ∈ Zn+ , |p| ≤ m1 , |q| ≤ m2 and for every s ∈ R. We will assume the following condition on the operator (P − λ)−1 :

(P − λ)−1 [P, xβ ∂ α ]v ≤ (α!)µ (β!)ν x s

γ≤α,σ≤β

|α−γ|+|β−σ|

C1 xσ ∂xγ vs (γ!)µ (σ!)ν

(3.4)

|γ+σ|<|α+β|

Zn+

and for a positive constant C1 independent of α, β. for every s ∈ R, α, β ∈ We shall prove in Section 4 that SG-ellipticity actually implies the technical condition (3.4) for partial diﬀerential operators P with polynomial coeﬃcients (1.9). We are not able to prove the same for general operators P , but address to [3], [15] for examples of pseudo-diﬀerential operators satisfying (3.4). Theorem 3.1. Let P ∈ OP Γm ν,µ , m = (m1 , m2 ), m1 > 0, m2 > 0, µ ≥ 1, ν ≥ 1 be SG-elliptic, selfadjoint and satisfy the condition (3.4). Let f ∈ Sνµ (Rn ). Assume F to be of the form (1.11) and let u ∈ H s+mo (Rn ) for some s > n/2 be a solution of (1.10). Then u ∈ Sνµ (Rn ). To prove Theorem 3.1 we need a preliminary result. / spec (P ). Let Proposition 3.2. Let P ∈ OP Γm ν,µ be as in Theorem 3.1. Fix λ ∈ s > n/2. Assume moreover that F = xj uk (∂xγ u)r for some k, r ∈ Z+ and j, γ ∈ Zn+ with r|γ| ≤ m1 − 1, 0 ≤ j ≤ m2 − 1. Then, there exists a positive constant C > 0 such that for every N ∈ Z+ , ε, δ ∈ (0, 1) α,β∈Zn +

|α|+|β|≤N

ε|α| δ |β|

(P − λ)−1 ◦ xβ ∂xα F ≤ Cuk+r s+|γ| µ ν s (α!) (β!)

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M. Cappiello, T. Gramchev and L. Rodino ε,δ ε,δ ε k+r k−1 r +C (ε + δ)SN [u](E [u]) + δS [u] · u · u N −1 s s+|γ| . −1 N −1

(3.5)

Proof. First ﬁx α ∈ Zn+ with |α| ≤ r|γ|. By (3.3), the operator (P − λ)−1 ◦ ∂xα ◦ xj is bounded on H s (Rn ) for every s ∈ R, then applying Schauder’s lemma we have ε|α|

(P − λ)−1 ◦ ∂xα xj uk (∂xγ u)r ≤ C k+r+1 uks · ur (3.6) 1 s+|γ| µ s (α!) |α|≤N

|α|≤r|γ|

for some positive constant C1 depending only on m1 , m2 . On the other hand, if |α| > r|γ|, then we can write α = (α − θ) + θ with |θ| = r|γ| + 1 ≤ m1 and then ∂xα xj uk (∂xγ u)r = ∂xθ ∂xα−θ xj uk (∂xγ u)r ⎛ ⎜ j (α − θ)! = ∂xθ ⎜ ⎝ σ ! . . . σ !ρ ! . . . ρ ! σ 1 k 1 r 0 σ +σ +···+σ +ρ +···+ρ =α−θ 0

1

k

1

σ0 ≤j

r

× xj−σ0 ∂xσ1 u · · · · · ∂xσk u · ∂xρ1 +γ u · · · · · ∂xρr +γ u . By (3.3), the operators (P − λ)−1 ◦ ∂xθ ◦ xj−σ0 are bounded in H s (Rn ). Then, applying Schauder’s lemma and observing that C (α − θ)! 1 ≤ · , (α)!µ σ1 ! . . . σk !ρ1 ! . . . ρr ! (σ1 !)µ . . . (σk !)µ (ρ1 + γ)!µ . . . (ρr + γ)!µ for some constant C depending only on r, γ, we obtain

ε|α|

(P − λ)−1 ◦ ∂ α xj uk (∂ γ u)r

x x µ s (α!) ε|σ1 | ε|σk | ≤ Cε ∂xσ1 us · · · · · ∂xσk us µ µ (σ !) (σ !) 1 k σ +σ +···+σ +ρ +···+ρ =α−θ 0

×

1

k

1

σ0 ≤j

r

ε|ρ1 +γ| ε|ρr +γ| ∂xρ1 +γ us · · · · · ∂ ρr +γ us . µ (ρ1 + γ)! (ρr + γ)!µ x

then r|γ|<|α|≤N

ε|α|

(P − λ)−1 ◦ ∂ α xj uk (∂ γ u)r ≤ C k+r+1 ε E ε [u] k+r . x x N −1 2 µ s (α!) (3.7)

for some constant C2 depending only on m1 , m2 . From (3.6) and (3.7), we deduce that ε|α|

(P − λ)−1 ◦ ∂ α xj uk (∂ γ u)r

x x µ s (α!) |α|≤N k+r ε (3.8) ≤ C3k+r+1 ε EN + uks · urs+|γ| . −1 [u]

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Now, let α, β ∈ Zn+ , with βi = 0 for some i ∈ {1, . . . , n} and |α| ≥ r|γ| + 1. We can decompose xβ ∂xα xj uk (∂xγ u)r = xi xβ−ei ∂xθ ∂xα−θ xj uk (∂xγ u)r = xi ∂xθ xj xβ−ei ∂xα−θ uk (∂xγ u)r + xi xβ−ei ∂xα−θ , ∂xθ xj uk (∂xγ u)r = xi ∂xθ xj xβ−ei ∂xα−θ uk (∂xγ u)r j j−ρ+β α−ρ k γ r α! u (∂x u) ∂x + x (α − ρ)! ρ 0=ρ≤α ρ≤j

+

0=σ≤θ

θ j−σ+β α−σ k γ r (j + β − ei )! u (∂x u) . ∂x x (j + β − ei − σ)! σ

σ≤j+β−ei

Now, the operators (P − λ)−1 ◦ xi ∂xθ ◦ xj , (P − λ)−1 ◦ xj−ρ and (P − λ)−1 ◦ xj−σ are bounded in H s (Rn ), then, applying Leibniz rule and arguing as for (3.7), we obtain

ε|α| δ |β|

(P − λ)−1 ◦ xβ ∂xα xj uk (∂xγ u)r

s (α!)µ (β!)ν |α|+|β|≤N

r|γ|<|α|,β=0 ε,δ ε k+r−1 ≤ C4k+r+1 (ε + δ)SN . −1 [u](EN −1 [u])

(3.9)

Finally, arguing as before, we derive the estimate

ε|α| δ |β|

(P − λ)−1 ◦ xβ ∂xα xj uk (∂xγ u)r

µ ν s (α!) (β!) |α|+|β|≤N

|α|≤r|γ|,β=0 ε,δ k−1 ≤ C4k+r+1 (ε + δ)SN urs+|γ|. −1 [u]us

(3.10)

The estimates (3.8), (3.9), (3.10) directly yield (3.5). Proof of Theorem 3.1. Now, if u is a solution of (1.10), then P u − λu = −λu + f + F. Hence, for every α, β ∈

Zn+ ,

we have

(P − λ)xβ ∂xα u = −λxβ ∂xα u + xβ ∂xα f + P, xβ ∂xα u + xβ ∂xα F.

from which we get ε,δ SN [u] =

(3.11)

−1 (P − λ)−1 ([f (α,β) ]µ,ν ([u(α,β) ]µ,ν ε,δ ) − λ(P − λ) ε,δ )

α,β∈Zn +

|α|+|β|≤N

+

! ε|α| δ |β| −1 β α −1 (α,β) µ,ν P, x u + (P − λ) (P − λ) ∂ ([F ] ) . (3.12) x ε,δ (α!)µ (β!)ν

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ε,δ [u] converges for some To conclude the proof, it is suﬃcient to show that SN ε, δ > 0. The assumption (3.4) implies that

α,β∈Zn +

ε|α| δ |β|

(P − λ)−1 P, xβ ∂xα u ≤ C(ε + δ)S ε,δ [u] N −1 µ ν s (α!) (β!)

(3.13)

|α|+|β|≤N

for every ε, δ ∈ (0, 1) and for some positive constant C. Moreover, (3.3) yields that for some C1 > 0 δ λ(P − λ)−1 ([u(α,β) ]µ,ν [u(α,β−e ) ]µ,ν (3.14) ε,δ )s ≤ C1 ε,δ s (β )ν if β ≥ 1 for some ∈ {1, . . . , n} and λ(P − λ)−1 ([u(α,0) ]µ,ν ε,δ )s ≤ C1

ε [u(α−ej ,0) ]µ,ν ε,δ s (αj )µ

(3.15)

if αj ≥ 1 for some j ∈ {1, . . . , n}. Hence, from (3.13), (3.14), (3.15) and Proposition 3.2, we deduce that ε,δ ε,δ SN [u] ≤ C uk+h s+mo + f ε,δ + (δ + ε)SN −1 [u] k+h ε,δ ε,δ k−1 h + δSN −1 [u]us us+mo . (3.16) +(δ + ε) SN −1 [u] for every ε, δ ∈ (0, 1) and for some positive constant C and positive integers h, k, mo . Then, iterating (3.16) and choosing δ and ε suﬃciently small, we obtain ε,δ that SN [u] is bounded and hence it converges. Remark 3.3. We proved Theorem 3.1 under the assumption m2 > 0. The Sνµ estimates for u solution of (1.10) in the case m2 = 0 have been already proved in [15]. We recall that this case presents some additional diﬃculties and requires stronger assumptions on the a priori H s regularity of u. Namely, one must assume xτ u ∈ H s+mo for some s > n/2, τ > 0. Moreover, in this case, P has not compact resolvent in general. In [15], the operator P is assumed to be invertible.

4. Commutator identities for diﬀerential operators In this section, we introduce some commutator identities. These allow to show easily that the assumption (3.4) is fulﬁlled in the case of SG-elliptic diﬀerential operators with polynomial coeﬃcients, then Theorem 3.1 applies to this class of operators. Lemma 4.1. Let α, β, ρ, σ ∈ Zn+ and u ∈ S(Rn ). Then the following identity holds: σ σ−γ β ρ α−γ α! xβ ∂xα (xσ ∂xρ u) = x ∂x (∂x u). (4.1) x (α − γ)! γ γ≤α γ≤σ

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Proof. We have, by the Leibniz rule, α ∂xγ (xσ )∂xα−γ+ρ u xβ ∂xα (xσ ∂xρ u) = xβ γ γ≤α γ≤σ

=

γ≤α

σ σ−γ β α−γ+ρ α! x ∂x u. x (α − γ)! γ

γ≤σ

(4.2)

Next, we need another commutator identity. Lemma 4.2. Let β, ρ ∈ Zn+ and w ∈ S(Rn ). Then the following identity holds ρ β! (4.3) xβ ∂xρ w = (−1)|j| ∂xρ−j (xβ−j w). (β − j)! j j≤β j≤ρ

Proof. Denote by F

−1

the inverse Fourier transform −1 Fξ→x w ˆ = (2π)−n exp(ixξ)w(ξ) ˆ dξ

(4.4)

Rn

ˆ We have, by (4.4) and the and recall that Fx→ξ xβ ∂xρ w (ξ) = i|β|+|ρ| ∂ξβ (ξ ρ w(ξ)). Leibnitz rule −1 ∂ξβ (ξ ρ w(ξ)) ˆ xβ ∂xρ w(x) = i|β|+|ρ| Fξ→x ρ β! −1 |β|+|ρ| (ξ ρ−j ∂ξβ−j w(ξ)) ˆ Fξ→x =i (β − j)! j j≤β j≤ρ

=

j≤β

ρ β! (−1)|j| ∂xρ−j (xβ−j w(x)). (β − j)! j

j≤ρ

(4.5)

As an immediate consequence of Lemma 4.1 and Lemma 4.2, we are able to show the following commutator identity. Lemma 4.3. Let α, β, ρ, σ ∈ Zn+ . Then the following commutator decomposition is true (4.6) [xβ ∂xα , xσ ∂xρ ]u = xβ ∂xα (xσ ∂xρ u) − xσ ∂xρ (xβ ∂xα u) σ ρ α! β! = (−1)|j| xσ−γ ∂xρ−j (xβ−j ∂xα−γ u) (α − γ)! γ (β − j)! j γ≤α j≤β γ≤σ

where

j≤ρ

means that |γ| + |j| > 0.

Proposition 4.4. Let P be a selfadjoint SG elliptic operator of the form (1.9) with m1 > 0, m2 > 0 and let λ ∈ C \ spec (P ). Then, for every α, β ∈ Zn+ and for every s ∈ R the condition (3.4) holds with µ = ν = 1.

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Proof. By Lemma 4.3, we can write: −1

(P − λ)

[P, xβ ∂xα ]u

=

c˜ρσ

|ρ|≤m

γ≤α

|σ|≤m2

γ≤σ

α! β! σ ρ (−1)|j| (α − γ)! γ j≤β (β − j)! j

j≤ρ

−1 σ−γ

× (P − λ)

x

∂xρ−j (xβ−j ∂xα−γ u)

(4.7)

with c˜ρσ = (−i)|ρ| cρσ . Using the smoothing property (3.3), we deduce that the operator (P − λ)−1 ◦ xσ−γ ∂xρ−j is bounded on H s (Rn ) for every s ∈ R. Then

(P − λ)−1 [P, xβ ∂xα ]u

s σ ρ α! β! ≤C xβ−j ∂xα−γ us (α − γ)! γ (β − j)! j γ≤α j≤β |ρ|≤m

|σ|≤m2 γ≤σ

≤ C1

γ≤α

j≤ρ

α! β! xβ−j ∂xα−γ us (α − γ)! (β − j)!

(4.8)

j≤β

which gives (3.4), since |γ| + |j| > 0 in (4.8). To be explicit, we state again the result we have obtained. Theorem 4.5. Consider the semilinear partial diﬀerential equation cαβ xβ Dxα u = f + F [u].

(4.9)

|α|≤m1

|β|≤m2

We assume that the linear part is self-adjoint and SG-elliptic, that is the constants cαβ ∈ R are ﬁxed in such a way that β α cαβ x ξ ≥ Co xm2 ξm1 for |x| + |ξ| ≥ R > 0. |α|≤m1

|β|≤m2

The nonlinearity F is of the form (1.11) and f ∈ Sνµ (Rn ), µ ≥ 1, ν ≥ 1. Let u ∈ H s+m1 −1 (Rn ), for some s > n/2, be a solution of (4.9), then u ∈ Sνµ (Rn ). In particular, if f = 0, then u ∈ S11 (Rn ).

5. Localization operators, Weyl operators and Gelfand–Shilov spaces: Some remarks Arguing ﬁrst at a formal level, we recall the deﬁnition of localization operator in terms of time-frequency representation. Building blocks are the translation and modulation operators, with t, x, ω ∈ Rn : Tx f (t) = f (t − x),

Mω f (t) = e2πiωt f (t).

(5.1)

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For a ﬁxed non-zero function ϕ, called “window”, the short time Fourier transform (STFT) of f with respect to ϕ is given by (5.2) Vϕ f (x, ω) = f, Mω Tx ϕ = f (t)ϕ(t − x)e−2πiωt dt. (here and in the following we use the normalization of Fourier integrals with a factor 2π in the phase, which is standard in time-frequency analysis). For ϕ1 , ϕ2 with ϕ1 , ϕ2 = 1, the following inversion formula is valid: f (t) = Vϕ1 f (x, ω)Mω Tx ϕ2 (t)dxdω. (5.3) 1 ,ϕ2 , with Given a function a(x, ω), we then deﬁne the localization operator Aϕ a symbol a and windows ϕ1 , ϕ2 : 1 ,ϕ2 f (t) = a(x, ω)Vϕ1 f (x, ω)Mω Tx ϕ2 (t)dxdω (5.4) Aϕ a

or, in the weak sense, 1 ,ϕ2 f, g = a, Vϕ1 f Vϕ2 g. (5.5) Aϕ a A precise functional setting for the above formal computations is obtained by taking f, g, ϕ and ϕ1 , ϕ2 in the Schwartz space S(Rn ). Since Vϕ is a continuous map from S(Rn ) to S(R2n ), we may give sense to (5.5) for any a ∈ S (Rn ). Note that in (5.5) we read ·, · as the extension to S(Rn ), S (Rn ) of the L2 -inner 1 ,ϕ2 product, i.e., f, g = (f, g). So we get Aϕ : S(Rn ) → S (Rn ). Similar settings a are obtained by ﬁxing less regular windows ϕ1 , ϕ2 and more general symbols in such a way that an extension of L2 -inner product in (5.5) makes sense, see [16], [26]. In the opposite direction, we may take more regular windows, belonging to Gelfand–Shilov spaces, ϕ1 , ϕ2 ∈ Sµµ (Rn ), µ ≥ 1/2. This allows to consider even more general symbols than a ∈ S (R2n ).

Proposition 5.1. Let ϕ1 , ϕ2 ∈ Sµµ (Rn ), µ ≥ 1/2, and let a be an element of the dual space (Sµµ ) (Rn ). Then (5.5) deﬁnes a linear map 1 ,ϕ2 Aϕ : Sµµ (Rn ) → (Sµµ ) (Rn ). a

(5.6)

Proof. In fact, it follows from the results in [17] that for windows ϕ ∈ Sµµ (Rn ), µ ≥ 1/2, the STFT Vϕ maps continuously Sµµ (Rn ) into Sµµ (R2n ). Since Sµµ (Rn ) is an algebra, then Vϕ1 f Vϕ2 g ∈ Sµµ (R2n ). By L2 -duality, the right-hand side of (5.5) is well deﬁned for a ∈ (Sµµ ) (R2n ) and we get (5.6). For more details concerning the setting given by Proposition 5.1, we refer to [25] in the case µ > 1 and to [10] in the case µ = 1. In particular, in [10] it was proved that for windows ϕ1 , ϕ2 ∈ S11 (Rn ), every ultradistribution with compact support a ∈ Eµ (R2n ), µ > 1, cf. [22], considered as a symbol, deﬁnes a trace class, hence L2 -bounded, localization operator. The properties of the operators from

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Proposition 5.1 in the case 1/2 ≤ µ < 1 are rather unexplored, but for the case ϕ1 (t) = ϕ2 (t) = 2n/4 e−π|t|

2

(5.7) ick AW a

1 ,ϕ2 Aϕ a

coming back to [2]. With the choice of ϕ1 , ϕ2 in (5.7), the operator = comes in [2] as a result of a quantization procedure and it is used in the PDE 2 1/2 1/2 context, see [24]. Since e−π|t| ∈ S1/2 (Rn ), we have for a ∈ (S1/2 ) (R2n ): ick : S1/2 (Rn ) → (S1/2 ) (Rn ). AW a 1/2

1/2

(5.8)

Let us pass now to consider Weyl pseudo-diﬀerential operators Lσ with symbol σ, deﬁned by x+y Lσ f (x) = σ , ω f (y)e2πi(x−y)ω dydω. 2 Connected with Weyl operators is the Wigner transform of a pair of functions f, g, deﬁned by t −2πitω t g x− dt. (5.9) W (f, g)(x, ω) = f x + e 2 2 In fact, an easy formal computation shows that Lσ f, g = σ, W (f, g).

(5.10)

Taking (5.10) as deﬁnition of Lσ , we have the following analogous to Proposition 5.1. Proposition 5.2. If σ ∈ (Sµµ ) (Rn ), µ ≥ 1/2, then Lσ : Sµµ (Rn ) → (Sµµ ) (Rn ).

(5.11)

Proof. Basic fact is that the Wigner transform maps Sµµ (Rn ) ⊗ Sµµ (Rn ) continuously into Sµµ (R2n ). See for the proof Theorem 3.8 in [25], or else observe that W (f, g)(x, ω) can be seen as STFT, namely: W (f, g)(x, ω) = 2n e4πixω Vg˜ f (2x, 2ω),

(5.12)

where g˜(t) = g(−t) (see [16] Lemma 4.3.1). In (5.12) we may then appeal to the results of [17] used to prove Proposition 5.1. The right-hand side of (5.10) is then well deﬁned for σ ∈ (Sµµ ) (R2n ) and we get (5.11). There is a simple connection between Weyl symbols and localization symbols. 1 ,ϕ2 Namely, Aϕ = Lσ for a (5.13) σ = a ∗ W (ϕ2 , ϕ1 ), see for example [13]. In the Gelfand–Shilov spaces, taking a ∈ (Sµµ ) (R2n ), µ ≥ 1/2, we assume ϕ1 , ϕ2 ∈ Sµµ (Rn ) to apply Proposition 5.1. Hence, W (ϕ1 , ϕ2 ) ∈ Sµµ (R2n ) and from (5.13) and the convolution properties in [14], we get σ ∈ (Sµµ ) (R2n ). So we may apply Proposition 5.2 as well, to obtain 1 ,ϕ2 = Lσ : Sµµ (Rn ) → (Sµµ ) (Rn ). Aϕ a

(5.14)

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2

In particular, if we assume ϕ1 = ϕ2 = 2n/4 e−π|t| , by observing that in this case 2 2 W (x, ω) = 2n e−2π(|x| +|ω| ) , we have σ = a ∗ 2n e−2π(|x| cf. [24]. Summing up, starting from a ∈ 1/2 S1/2 (Rn )

→

1/2 (S1/2 ) (Rn )

2

+|ω|2 )

,

1/2 (S1/2 ) (R2n ),

(5.15) ick we may consider AW : a

and write ick AW = Lσ a

where the Weyl symbol σ ∈ (S1/2 ) (R2n ) is given by (5.15). Note however that the 1/2

convolution (5.15) has a strong smoothing eﬀect, in particular a ∈ (S1/2 ) (R2n ) im1/2

plies the analyticity of σ, cf. [14]. So, given a Weyl symbol σ ∈ (S1/2 ) (R2n ) or even 1/2

ick . σ ∈ S (R2n ), we cannot ﬁnd in general a ∈ (S1/2 ) (R2n ) such that Lσ = AW a The study of the convolution equation (5.15) in appropriate spaces of distributions is in fact a challenging problem, see for example [18] and the references there. 1/2

References [1] S. Agmon, Lectures on Exponential Decay of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-body Schr¨ odinger Operators, Princeton University Press, Princeton, 1982. [2] F.A. Berezin, Wick and anti-Wick symbols of operators, Mat. Sb. (N.S.) 86 (1971), 578–610. [3] H.A. Biagioni and T. Gramchev, Fractional derivative estimates in Gevrey spaces, global regularity and decay for solutions to semilinear equations in Rn , J. Diﬀ. Equations, 194 (2003), 140–165. [4] J. Bona and Y. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl. 76 (1997), 377–430. [5] M. Cappiello, Fourier integral operators of inﬁnite order and applications to SGhyperbolic equations, Tsukuba J. Math. 28 (2004), 311–361. [6] M. Cappiello and L. Rodino, SG-pseudo-diﬀerential operators and Gelfand–Shilov spaces, Rocky Mountain J. Math. 36 (2006), no. 4, 1117–1148. [7] M. Cappiello, T. Gramchev and L. Rodino, Exponential decay and regularity for SGelliptic operators with polynomial coeﬃcients, in Hyperbolic Problems and Regularity Questions, Editors: M. Padula and L. Zanghirati, Birkh¨auser, Basel, to appear. [8] M. Cappiello, T. Gramchev and L. Rodino, Superexponential decay and holomorphic extensions for semilinear equations with polynomial coeﬃcients, J. Funct. Anal. 237 (2006), 634–654. [9] J. Chung, S.Y. Chung and D. Kim, Characterization of the Gelfand–Shilov spaces via Fourier transforms, Proc. Amer. Math. Soc. 124 (1996), 2101–2108. [10] E. Cordero, S. Pilipovic, L. Rodino and N. Teofanov, Localization operators and exponential weights for modulation spaces, Mediterranean. J. Math. 2 (2005), 381–394. [11] H.O. Cordes, The Technique of Pseudodiﬀerential Operators, Cambridge Univ. Press, 1995.

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[12] Y.V. Egorov and B.-W. Schulze, Pseudo-diﬀerential Operators, Singularities, Applications, Birkh¨ auser, Basel, 1997. [13] G.B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, 1989. [14] I.M. Gel’fand and G.E. Shilov, Generalized Functions II, Academic Press, New York, 1968. [15] T. Gramchev, Perturbative Methods in Scales of Banach Spaces: Applications for Gevrey Regularity of Solutions to Semilinear Partial Diﬀerential Equations, Rend. Sem. Mat. Univ. Pol. Torino 61 (2003), 101–134. [16] K. Gr¨ ochenig, Foundations of Time-Frequency Analysis, Birkh¨ auser, Boston 2001. [17] K. Gr¨ ochenig and G. Zimmermann, Spaces of test functions via the STFT, J. Function Spaces Appl. 2 (2004), 25–53. [18] B.C. Hall, Harmonic analysis with respect to heat kernel measure, Bull. Amer. Math. Soc. 38 (2000), 43–78. [19] C. Parenti, Operatori pseudodiﬀerenziali in Rn e applicazioni, Ann. Mat. Pura Appl. 93 (1972), 359–389. [20] S. Pilipovic, Tempered ultradistributions, Boll. Unione Mat. Ital., VII. Ser., B, 2 (1988), 235–251. [21] V.S. Rabinovich, Exponential estimates for eigenfunctions of Schr¨odinger operators with rapidly increasing and discontinuous potentials, Contemporary Math. 364 (2004), 225–236. [22] L. Rodino, Linear Partial Diﬀerential Operators in Gevrey Spaces, World Scientiﬁc Publishing Co., Singapore, 1993. [23] E. Schrohe, Spaces of weighted symbols and weighted Sobolev spaces on manifolds, in Pseudodiﬀerential Operators, Editors: H.O. Cordes, B. Gramsch and H. Widom, Springer, 1987, 360–377. [24] M. Shubin, Pseudodiﬀerential Operators and Spectral theory, Second Edition, Springer-Verlag, Berlin, 2001. [25] N. Teofanov, Ultradistributions and time-frequency analysis, in Pseudo-diﬀerential Operators and Related Topics, Editors: P. Boggiatto, L. Rodino, J. Toft, M.W. Wong, Birkh¨ auser, Basel, 2006, 173–191. [26] M.W. Wong, Wavelet Transforms and Localization Operators, Birk¨ auser, Basel, 2002. Marco Cappiello Dipartimento di Matematica, Universit` a di Ferrara Via Machiavelli 35, I-44100 Ferrara, Italy e-mail: [email protected] Todor Gramchev Dipartimento di Matematica e Informatica, Universit` a di Cagliari Via Ospedale 72, I-09124 Cagliari, Italy e-mail: [email protected] Luigi Rodino 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, 313–328 c 2006 Birkh¨ auser Verlag Basel/Switzerland

Continuity and Schatten Properties for Toeplitz 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 p, q, p1 , p2 , q1 , q2 ∈ [1, ∞], ω1 , ω2 , ω and h1 , h2 are p,q p1 ,q1 , then the Toeplitz operator Tph1 ,h2 (a) : M(ω → appropriate, and a ∈ M(ω) 1) p2 ,q2 M(ω2 ) is continuous. If in addition p1 = p2 = q1 = q2 = 2, then we present suﬃcient conditions on p, q, h1 and h2 in order for Tph1 ,h2 (a) should be a Schatten-von Neumann operator of certain degree.

Mathematics Subject Classiﬁcation (2000). Primary 47B10, 47B35, 47B37: Secondary 35S05, 42B35. Keywords. Toeplitz operators, localization operators, anti-Wick operators, modulation spaces, Schatten classes.

0. Introduction In this paper we discuss continuity properties for Toeplitz operators (in the literature, the terms localization operators or anti-Wick operators also occur) in background of modulation space theory. More precisely, we consider Toeplitz operators where the corresponding window functions (or coherent state) and symbols belong to appropriate modulation spaces, and discuss continuity for such operators when acting on modulation spaces. At the same time we clarify some details and correct some mistakes in [26], and show that stronger results can be obtained by a slight improvement of the techniques there. Furthermore, if such operators act on modulation spaces of Hilbert type, then a detailed study of continuity and compactness is performed in terms of Schattenvon Neumann classes. In particular we investigate trace-class and Hilbert-Schmidt properties. In Section 1 we discuss general properties for such classes, and recall from [28, 29] embedding relations between such operators in Weyl calculus and modulation spaces. Thereafter we apply, in Section 2, a standard technique to analyze Toeplitz operators in background of Weyl calculus, using the fact that the

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Weyl symbol of a Toeplitz operator is equal to a convolution between the Toeplitz symbol and a Wigner distribution (see [2–4,17,21,24–29]). In that end, we are able to extend certain continuity and Schatten-von Neumann results in [2, 4, 25, 26]. Toeplitz operators have been applied in quantum physics (see, e.g., [18] and the references therein). They were introduced in time-frequency analysis in [6] by Daubechies. In this ﬁeld, Toeplitz operators can be used as a tool for ﬁltering signals. A convenient way to measure the time-frequency content of these signals then can be done by means of modulation spaces. It might therefore not be surprising that Toeplitz operators have been considered in background of modulation space theory before (see, e.g., [2–5, 8, 25, 26]). As a consequence of the investigations in the present paper we are able to present a general result which essentially contains the most of the results in the latter papers. The basic theory of modulation spaces were established by Feichtinger and Gr¨ ochenig. (See [9, 10] and and the references therein.) Roughly speaking, for an p,q is obtained by imposing appropriate weight function ω, the modulation space M(ω) p,q a mixed L(ω) -norm on the short-time Fourier transform of a tempered distribution. The non-weighted (or classical) modulation space M p,q is then obtained by choosing ω = 1. By using modulation spaces it might be easy to obtain information concerning growth and decay properties, as well as certain localization and regularity properties for distributions. In order to describe our results more in detail we recall the deﬁnition of modulation spaces, and start to consider appropriate conditions on the involved m weight functions. Assume that 0 < ω, v ∈ L∞ loc (R ). Then recall that ω is called v-moderate if there is a constant C such that ω(x1 + x2 ) ≤ Cω(x1 )v(x2 ) holds for every x1 , x2 ∈ Rm , and if ω is v-moderate for some polynomial v, then ω is called polynomial moderate. (See [13].) As in [25] we let P(Rm ) be the set of all polynomial moderate functions on Rm . Next assume that χ ∈ S (Rm ) \ 0, p, q ∈ [1, ∞] and that ω ∈ P(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., [16].) Then the modulation space p,q (Rm ) consists of all f ∈ S (Rm ) such that M(ω) 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 ), and the function (x, ξ) → F (f τx χ)(ξ) in (0.1) is called the short-time Fourier transform of f with respect to the window function χ.

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For functions and distributions on the phase space R2m , we sometimes use the symplectic Fourier transform (see (1.1) below) instead of the Fourier transform F above, in the deﬁnition of modulation space norm. Then we use the notation :p,q (R2m ) and · : p,q instead of M p,q (R2m ) and · M p,q respectively. M (ω) (ω) M (ω) (ω)

Another “advanced Fourier transform”, similar to the short-time Fourier transform, is the Wigner distribution which we shall discuss now. Assume that f1 , f2 ∈ S (Rm ). Then Wf1 ,f2 (x, ξ) = (2π)−m/2 f1 (x − y/2)f2(x + y/2)ei y,ξ dy (0.2) is the Wigner distribution of f1 and f2 . The map (f1 , f2 ) → Wf1 ,f2 is continuous from S (Rm ) × S (Rm ) to S (R2m ), which extends to a continuous map from S (Rm ) × S (Rm ) to S (R2m ). (See [11].) Furthermore, some extensions to continuity in terms of modulation spaces can be found in [25, 26]. Next assume that a ∈ S (R2n ), and that h1 , h2 ∈ S (Rn ). Then the Toeplitz operator, with symbol a and window functions h1 and h2 , is deﬁned by the formula (Tph1 ,h2 (a)f1 , f2 ) = (a(2 · )Wf1 ,h1 , Wf2 ,h2 )

(0.3)

when f1 , f2 ∈ S (R ). The deﬁnition extends in several ways (see, e.g., [2–4, 24– 29]). In Section 2 we discuss continuity for Toeplitz operators acting on modulation spaces when the window functions and operator symbols belong to modulation spaces. In particular we ﬁnd appropriate conditions on the weight functions ω, ωj , νj and the exponents p, q, pj , qj , rj , sj for j = 1, 2, in order for the map m

p1 ,q1 p2 ,q2 Tph1 ,h2 (a) : M(ω (Rm ) → M(ω (Rm ) 1) 2)

(0.4)

:p,q (R2m ) and hj ∈ M j j (Rm ) for j = 1, 2. should be continuous when a ∈ M (ω) (νj ) As it is seen above, the general situation involves at least ten diﬀerent exponents and ﬁve weight functions attached to the modulation spaces. The interplay between them is rather complicated and is regulated with certain equalities and inequalities. Here we present some examples of possible conditions, and consequences of our results. At the same time we get a link about the generality of our investigations. First assume ﬁrst that rj = sj = 1 and q = ∞ here above. Then it turns out that appropriate conditions for the remaining exponents are 1 1 1 1 1 (0.5) − = − = . p2 p1 q2 q1 p If instead rj = sj = 2, then appropriate conditions are 1 1 1 1 1 1 − = − = + − 1, q ≤ p2 , q2 ≤ p. (0.5) p2 p1 q2 q1 p q In order to establish appropriate conditions for the involved weight functions, it is convenient to include weight functions κ, κj and vj for j = 1, 2, such that ω1 is v1 -moderate and ω2 is vˇ2 -moderate. Here and in what follows we let fˇ(x) = f (−x) r ,s

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for every function or distribution f . Then we sometimes assume that the involved weight functions satisfy ω2 (X) , for j = 1, 2, νj = κj vj , ω(X, Y ) = ω1 (X)κ(2Y ) (0.6) and κ(X + Y ) ≤ Cκ1 (−X)κ2 (Y ), for some constant C. As a consequence of our general result we have the following result. Proposition 0.1. Assume that p, pj , qj ∈ [1, ∞] for j = 1, 2 satisfy (0.5). Also assume that (0.6) is fulﬁlled, for some κ, κj , vj , νj , ωj ∈ P(R2m ) such that ω1 1,1 is v1 -moderate and ω2 is vˇ2 -moderate. If hj ∈ M(ν (Rm ) for j = 1, 2 and a ∈ j) :p,∞ (R2m ), then the map (0.4) is continuous. M (ω)

We note that if we choose p = ∞, ω1 = ω2 = 1 and κ(X) = κj (X) = Xs for some s ≥ 0, then Proposition 0.1 agrees with Theorem 3.2 in [4]. Here and in what follows we let x = (1 + |x|2 )1/2 . If instead ω1 = ω2 and κ(X) = κj (X) = 1, then Proposition 0.1 agrees with Corollary 4.2 in [4]. Furthermore, from embedding properties between Lebesgue spaces and modulation spaces (cf., e.g., [27]), it follows that Proposition 3.3 in [2] in the case q0 ≥ p0 is an immediate consequence of Proposition 0.1 in the case κ = κj = 1. (See also Corollary 2.7 and Remark 2.11 below.) 2,2 , then our results give the following. If, more generally, hj ∈ M(ν j) Proposition 0.1 . Assume that p, q, pj , qj ∈ [1, ∞] for j = 1, 2 satisfy (0.5) . Also assume that (0.6) is fulﬁlled, for some κ, κj , vj , νj , ωj ∈ P(R2m ) such that ω1 2,2 is v1 -moderate and ω2 is vˇ2 -moderate. If hj ∈ M(ν (Rm ) for j = 1, 2 and a ∈ j) :p,q (R2m ), then the map (0.4) is continuous. M (ω)

If we let p1 = p2 = q1 = q2 = 2, then the operator Tph1 ,h2 (a) in Proposition 2,2 0.1 and Proposition 0.1 acts between the Hilbert spaces H1 = M(ω and H2 = 1)

2,2 . In this case, a more detailed compactness investigation in terms of SchattenM(ω 2) von Neumann classes is performed in Section 3 (see Section 1 for strict deﬁnitions). These investigations are based on certain results in [29] in the present volume, where similar questions are considered for pseudo-diﬀerential operators. (See also [28].) As a consequence of these investigations, we are able to prove a general result for Toeplitz operators, which in particular implies that the following two propositions hold. Here and in what follows we let p denote the conjugate exponent of p ∈ [1, ∞], i.e., 1/p + 1/p = 1.

Proposition 0.2. Assume that p ∈ [1, ∞]. Also assume that (0.6) is fulﬁlled, for some κ, κj , vj , νj , ωj ∈ P(R2m ) such that ω1 is v1 -moderate and ω2 is vˇ2 -moderate. 1,1 :p,∞ (R2m ), then (Rm ) for j = 1, 2 and a ∈ M If hj ∈ M(ν (ω) j) 2,2 2,2 Tph1 ,h2 (a) ∈ Ip (M(ω , M(ω ). 1) 2)

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Proposition 0.2 . Assume that p, q ∈ [1, ∞] satisfy q ≤ min(p, p ). Also assume that (0.6) is fulﬁlled for some κ, κj , vj , νj , ωj ∈ P(R2m ) such that ω1 is v1 -moderate 2,2 :p,q (R2m ), then and ω2 is vˇ2 -moderate. If hj ∈ M(ν (Rm ) for j = 1, 2 and a ∈ M (ω) j)

2,2 2,2 Tph1 ,h2 (a) ∈ Ip (M(ω , M(ω ). 1) 2)

1. Preliminaries In the ﬁrst part of the section we discuss basic properties for modulation spaces, Wigner distributions and Weyl operators in pseudo-diﬀerential calculus. The proofs are in the most cases omitted since they can be found in [7, 9, 10, 13, 24–26]. In the second part we discuss Schatten-von Neumann properties and recall some facts which can be found in [1, 22, 25–29]. We start to present some notations which are used. 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. Furthermore, the restriction of ( · , · ) to S × S is the (standard) sesqui-linear form on S . 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 . Assume that ω ∈ P(R2m ), p, q ∈ [1, ∞], and that χ ∈ S (Rm ) \ 0. Then p,q recall that the modulation space M(ω) (Rm ) is the set of all f ∈ S (Rm ) such that (0.1) holds. Obviously, this is equivalent to that the short-time Fourier transform 2m ), the set of all F ∈ L1loc (R2m ) such that F (f τx χ)(ξ) belongs to Lp,q (ω) (R F

Lp,q (ω)

≡

q/p 1/q dξ |F (x, ξ)ω(x, ξ)|p dx

is ﬁnite. 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 . The convention of indicating weight functions with parenthesis is also used 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 consisting of all pairs (x, ξ), where x ∈ Rm and ξ ∈ Rm are dual variables, it is sometimes convenient to use

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the symplectic Fourier transform Fσ which is given by −m (Fσ f )(y, η) = f (y, η) ≡ π f (x, ξ)e2i( x,η− y,ξ) dxdξ,

(1.1)

when f ∈ L1 (R2m ). Then it follows that Fσ is a homeomorphism on f ∈ S (R2m ) which extends to a homeomorphism on f ∈ 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) and M p respectively, when Fσ is used instead of F , in the deﬁnition of modulation :p,q here does not agree with M :p,q spaces of distributions on R2m . Note that M (ω) (ω) in [29], since Fσ does not agree with the phase space Fourier transform which is :p,q in [29]. used in the deﬁnition of M (ω) The following proposition is a consequence of well-known facts in [7, 13]. Proposition 1.1. 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 (Rm ) \ 0, then f ∈ M(ω) (Rm ) if and only if (0.1) holds, i.e., 1. if χ ∈ M(v) p,q p,q M(ω) (Rm ) 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 3. the L2 product ( · , · ) from S extends to a continuous map from M(ω) (Rm ) ×

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

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

p,q are equivalent norms; and · M(ω)

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

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 1.1 (1) permits us be rather vague about to the choice of χ ∈ 1 \ 0 in (0.1). For example, if C > 0 is a constant and Ω is a subset of S , then M(v) 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. m :p,q 2m )) be the completion of S (Rm ) It is convenient to let Mp,q (ω) (R ) (M(ω) (R p,q p,q p,q ( · (S (R2m )) under the norm · M(ω) : p,q ). Then M(ω) ⊆ M(ω) with equality M (ω)

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319

if and only if p < ∞ and q < ∞. It follows that most of the properties which are m valid for the usual modulation spaces also hold for Mp,q (ω) (R ). Remark 1.2. Assume that p, q, q1 , q2 ∈ [1, ∞]. If q1 ≤ min(p, p ) and q2 ≥ max(p, p ), then M p,q1 ⊆ Lp ⊆ M p,q2 . In particular, M 2 = L2 . (See, e.g., [25, 26].) Remark 1.3. Assume that pj , qj ∈ [1, ∞] and ωj ∈ P(R2m ) for 0 ≤ j ≤ 2, satisfy 1 1 1 1 1 1 + =1+ , + = , p1 p2 p0 q1 q2 q0 and ω0 (x1 + x2 , ξ) ≤ Cω1 (x1 , ξ)ω2 (x2 , ξ), for some constant C independent on x1 , x2 , ξ ∈ Rm . Then Theorem 5.5 in [27] shows that there is a canonical way to extend the the usual convolution ∗ on S p ,q p1 ,q1 p2 ,q2 p0 ,q0 to a continuous multiplication from M(ω × M(ω to M(ω , and if fj ∈ M(ωjj )j 1) 2) 0) for j = 1, . . . , 2, then f1 ∗ f2 M p0 ,q0 ≤ Cf1 M p1 ,q1 f2 M p2 ,q2 , (ω0 )

(ω1 )

(ω2 )

(1.2)

for some constant C, independent of f1 and f2 . (See also [4].) Next we recall the deﬁnition of Weyl operators. Assume that a ∈ S (R2m ). Then the Weyl operator aw (x, D) with respect to the symbol a is the continuous operator on S (Rm ), deﬁned by the formula w −m a((x + y)/2, ξ)f (y)ei x−y,ξ dydξ. (a (x, D)f )(x) = (2π) The deﬁnition of aw (x, D) extends to any a ∈ S (R2m ), and then aw (x, D) is continuous from S (Rm ) to S (Rm ). This is a consequence of the relation (aw (x, D)f, g) = (2π)−m/2 (a, Wg,f ),

(1.3)

when a ∈ S (R2m ) and f, g ∈ S (Rm ), and the fact that the map (f, g) → Wf,g is continuous from S (Rm ) × S (Rm ) to S (R2m ). (See [11, 16].) Next 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 [19] in the case H1 = H2 . (See also [12, 22].) The general situation when H1 is not necessarily equal to H2 have thereafter been discussed in [1, 20]. Here we give a short introduction, based on [28, 29]. 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 |(T fj , gj )H2 |p (1.4) (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, is the Banach space which consists of all linear operators T from H1 to H2 such that T Ip(H1 ,H2 ) is ﬁnite. Obviously,

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I∞ (H1 , H2 ) consists of all linear and 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. We refer to [1, 12, 20, 22, 28, 29] for more properties concerning Schatten-von Neumann classes on general Hilbert spaces. Next we consider Schatten-von Neumann classes in background of Weyl calculus. Let p ∈ [1, ∞] and ω1 , ω2 ∈ P(R2m ) be ﬁxed. Then let sw p (ω1 , ω2 ) be the 2 2 , M(ω ). Also let set of all a ∈ S (R2m ) such that aw (x, D) belongs to Ip (M(ω 1) 2) w 2m w 2 2 , M(ω ). s (ω1 , ω2 ) be the set of all a ∈ S (R ) such that a (x, D) ∈ I (M(ω 1) 2) w w We let sp (ω1 , ω2 ) and s (ω1 , ω2 ) be equipped by the norms 2 asw = asw ≡ aw (x, D)Ip (M(ω p p (ω1 ,ω2 )

1)

2 ,M(ω

2)

)

and · sw respectively. ∞ Since any linear and continuous operator from S (Rm ) to S (Rm ) is equal to aw (x, D), for a unique a ∈ S (R2m ) (cf. the introduction or [16]), it follows that the map a → aw (x, D) is an isometric homeomorphism from sw p (ω1 , ω2 ) to 2 2 w 2 2 , M ) when p ∈ [1, ∞], and from s (ω , ω ) to I (M , M Ip (M(ω 1 2 (ω2 ) (ω1 ) (ω2 ) ). Con1) sequently, most of the properties for the Ip spaces which are listed earlier in this w section carry over to the sw p -spaces. For example, sp (ω1 , ω2 ) is a Banach space which increases with the parameter p ∈ [1, ∞]. Next we discuss continuity for Wigner distributions in terms of modulation spaces. The following result is essentially a restatement of Theorem 5.1 in [26] and Proposition 4.1 in [29]. Proposition 1.4. Assume that pj , qj , p, q ∈ [1, ∞] such that p ≤ pj , qj ≤ q, for j = 1, 2, and that 1/p1 + 1/p2 = 1/q1 + 1/q2 = 1/p + 1/q. Also assume that ω1 , ω2 ∈ P(R

2m

), and that ω ∈ P(R

2m

⊕R

(1.5) 2m

) satisfy

ω(X, Y ) ≤ Cω1 (X − Y )ω2 (X + Y ). 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 (Rm ) × M(ω (Rm ) to M to a continuous mapping from M(ω (ω) 1) 2) some constant C it holds Wf1 ,f2 M : p,q ≤ Cf1 M p1 ,q1 f2 M p2 ,q2 (ω)

when f1 , f2 ∈ S (Rm ).

(ω1 )

(ω2 )

(1.6)

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321

We ﬁnish this section by recalling continuity and Schatten-von Neumann properties for Weyl operators in [25–29]. The following result is a restatement of Theorem 4.2 in [29] in the case of Weyl operators. A similar result, containing a mistake is also available in [26]. Theorem 1.5. Assume that p, q, pj , qj ∈ [1, ∞] for j = 1, 2, satisfy (0.5) . Also assume that ω ∈ P(R2m ⊕ R2m ) and ω1 , ω2 ∈ P(R2m ) satisfy ω2 (X − Y ) = ω(X, Y ). ω1 (X + Y )

(1.7)

:p,q (R2m ), then aw (x, D) from S (Rm ) to S (Rm ) extends uniquely to a If a ∈ M (ω) p1 ,q1 p2 ,q2 (Rm ) to M(ω (Rm ). continuous map from M(ω 1) 2) :p,q , then Moreover, if in addition a ∈ M (ω)

p1 ,q1 p2 ,q2 a (x, D) : M(ω → M(ω 1) 2) w

is compact. The next result concerns Schatten-von Neumann properties for Weyl operators with symbols in appropriate modulation spaces. For the proof we refer to [28, 29] in the general case, and to [25] in the unweighted case (see also [14]). Theorem 1.6. Assume that p, q, pj , qj ∈ [1, ∞] for j = 1, 2, satisfy p1 ≤ p ≤ p2 , Also assume that ω ∈ P(R

q1 ≤ min(p, p ) 2m

⊕R

2m

and

q2 ≥ max(p, p ).

) and ω1 , ω2 ∈ P(R

2m

(1.8)

) satisfy (1.7). Then

:p1 ,q1 (R2m ) → sw (ω1 , ω2 ) → M :p2 ,q2 (R2m ). M p (ω) (ω)

(1.9)

2. Continuity for Toeplitz operators In this section, we extend certain continuity properties in Section 5 in [26] concerning Toeplitz operators, when the operator symbols belong to appropriate classes of modulation spaces. We start by recalling the deﬁnition of Toeplitz operators. Assume that h1 , h2 ∈ S (Rm ) \ 0 are ﬁxed. For any a ∈ S (R2m ), the Toeplitz operator Tph1 ,h2 (a) on S (Rm ), with respect to h1 , h2 and symbol a is deﬁned by (0.3). Obviously, Tph1 ,h2 (a) extends to a continuous map from S (Rm ) to S (Rm ). As in [2–4, 21, 24–30] we study Toeplitz operators in background of Weyl calculus, using the fact that if a ∈ S (R2m ) and h1 , h2 ∈ S (Rm ), then Tph1 ,h2 (a) = (a ∗ uh1 ,h2 )w (x, D), where

uh1 ,h2 (X) = (2π)−m/2 Wh2 ,h1 (−X).

(2.1)

If, more generally, a ∈ S (R2m ) and h1 , h2 ∈ S (Rm ) are distributions such that a ∗ uh1 ,h2 makes sense as an element in S (R2m ), then (2.1) is taken as the

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deﬁnition of the Toeplitz operator Tph1 ,h2 (a). We refer to [2–4, 21, 24, 26, 30–32] for more facts about Toeplitz operators. Next we discuss Tph1 ,h2 (a) when h1 , h2 and a belong to certain types of modulation spaces. In the introduction it was stated that Proposition 0.1 and Proposition 0.1 are special cases of more general results. The ﬁrst result is Proposition 2.1 below, which in some sense is the most general one. Here the exponentials for the involved modulation spaces are given by p, q, p0 , q0 , rj , sj ∈ [1, ∞] for j = 1, 2, and they should satisfy 1 1 1 1 1 1 1 1 + = + =1− − + + r1 r2 s1 s2 p q p0 q0 1 1 1 1 0≤ − ≤ min , (2.2) p p0 rj sj 1 1 1 1 , j = 1, 2. 0≤ − ≤ min , q0 q rj sj w It is also convenient to let Lp,q (ω) be the set of all operators of the type p,q 2m w : (R ), where the topology for Lp,q a (x, D) such that a ∈ M (ω) is inherited (ω) p,q 2m : (R ). from M w

(ω)

Proposition 2.1. Assume that rj , sj , p, p0 , q, q0 ∈ [1, ∞] for j = 1, 2 satisfy (2.2). r ,s Also assume that νj ∈ P(R2m ), hj ∈ M(νjj ) j (Rm ), j = 1, 2, and that ω, ω0 ∈ P(R2m ⊕ R2m ) satisfy ω0 (X1 + X2 , Y ) ≤ Cω(X1 , Y ) νˇ1 (Y + X2 )ν2 (Y − X2 )

X1 , X2 , Y ∈ R2m ,

for some constant C > 0. Then the map a → Tph1 ,h2 (a) from S (R2m ) to the set of linear and continuous operators from S (Rm ) to S (Rm ) extends to a continuous :p,q (R2m ) to L w (ω0 ). map from M p0 ,q0 (ω) Proof. By (2.2) it is possible to ﬁnd p1 , q1 ∈ [1, ∞] such that p1 ≤ rj , sj ≤ q1 for j = 1, 2 and 1/p − 1/p0 = 1 − 1/p1 ,

1/q0 − 1/q = 1/q1 ,

1/r1 + 1/r2 = 1/p1 + 1/q1 .

:p1 ,q1 , M (ω1 )

where ω1 (X, Y ) = νˇ1 (Y + Hence Proposition 1.4 implies that uh1 ,h2 ∈ p,q 2m : X)ν2 (Y − X). If a ∈ M(ω) (R ), then Remark 1.3 gives :p,q ∗ M :p1 ,q1 ⊆ M :p0 ,q0 , a ∗ uh1 ,h2 ∈ M (ω1 ) (ω0 ) (ω)

and the result follows from (2.1). The proof is complete.

Next we combine Proposition 2.1 with Theorem 1.5 in order to obtain more explicit continuity results for Tph1 ,h2 (a) when acting on modulation spaces. More precisely we establish continuity properties of the form Tph1 ,h2 (a)M p1 ,q1 →M p2 ,q2 ≤ CaM : p,q h1 M r1 ,s1 h2 M r2 ,s2 . (ω1 )

(ω2 )

(ω)

(ν1 )

(ν2 )

(2.3)

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323

It turns out that an appropriate condition for the involving weight functions is ω2 (X − 2Y − Z) ≤ Cω(X, Y ). (2.4) sup ω1 (X − Z)ν1 (Z)ν2 (2Y + Z) Z and appropriate conditions for the exponents p, q, pj , qj , rj , sj is that they should satisfy (2.2) for some p0 , q0 ∈ [1, ∞], and 1 1 1 1 1 1 − = − = + − 1, p2 p1 q2 q1 p0 q0

(0.5)

q0 ≤ p2 , q2 ≤ p0 .

Theorem 2.2. Assume that p, q, pj , qj , rj , sj ∈ [1, ∞] for j = 0, 1, 2 satisfy (0.5) and (2.2). Also assume that νj , ωj ∈ P(R2m ) for j = 1, 2 and ω ∈ P(R2m ⊕ R2m ) satisfy (2.4) for some constant C > 0 independent of X, Y ∈ R2m . If hj ∈ r ,s :p,q (R2m ), then the map (0.4) is continuous, M(νjj ) j (Rm ) for j = 1, 2 and a ∈ M (ω) :p,q and hj ∈ and (2.3) holds for some constant C which is independent of a ∈ M (ω)

r ,s

M(νjj ) j for j = 1, 2. Proof. Let ω0 (X, Y ) = ω2 (X − Y )/ω1 (X + Y ) . Then it follows from (2.4) that ω0 (X1 + X2 , Y ) ≤ Cω(X1 , Y ) νˇ1 (X2 + Y )ν2 (Y − X2 ), for some constant C which is independent of X1 , X2 , Y ∈ R2m . Hence Tph1 ,h2 (a) ∈ Lpw0 ,q0 (ω0 ) by Proposition 2.1. It follows now from the deﬁnition of ω0 , Theorem p1 ,q1 p2 ,q2 1.5, (2.2) and (0.5) that Tph1 ,h2 (a) is continuous from M(ω to M(ω . The proof 1) 2) is complete. As a consequence of Theorem 2.2 we have the following result. Proposition 0.1 . Assume that p, q, pj , qj , rj , sj ∈ [1, ∞] for j = 0, 1, 2 satisfy (0.5) and (2.2). Also assume that (0.6) is fulﬁlled, for some κ, κj , vj , νj , ωj ∈ P(R2m ) r ,s such that ω1 is v1 -moderate and ω2 is vˇ2 -moderate. If hj ∈ M(νjj ) j (Rm ) for j = 1, 2 :p,q (R2m ), then the map (0.4) is continuous. and a ∈ M (ω)

Proof. The assertion is an immediate consequence of Theorem 2.2, since the condition that ω1 is ν1 -moderate and ω2 is νˇ2 -moderate immediately gives (2.4). The proof is complete. Theorem 2.2 may be used to give other results, parallel to Proposition 0.1 . For example, if ω(X, Y ) = (v1 (X)v2 (X)κ(2Y ))−1 , (0.6) and κ(X + Y ) ≤ Cκ1 (X)κ2 (Y ), are imposed on the involved weight functions, then we get.

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Proposition 2.3. Assume that p, q, pj , qj , rj , sj ∈ [1, ∞] for j = 0, 1, 2 satisfy (0.5) and (2.2). Also assume that (0.6) is fulﬁlled, for some κ, κj , vj , ωj ∈ P(R2m ) r ,s such that ωj is vj -moderate for j = 1, 2. If hj ∈ M(ωj j κjj ) (Rm ) for j = 1, 2 and :p,q (R2m ), then the map a∈M (ω)

p1 ,q1 p2 ,q2 Tph1 ,h2 (a) : M(v → M(1/v 1) 2)

is continuous. Proof. The result follows if we let ω1 = v1 , ω2 = 1/v2 and νj = ωj κj in (2.4). The proof is complete. Remark 2.4. If κ = κ1 = κ2 = 1, then Proposition 2.3 is equivalent to Proposition 0.1 . In fact, it follows from (0.2) that Wf,g = Wg,f for every f, g ∈ S . This in turn implies that if a ∈ S (R2m ) and fj , hj ∈ S (Rm ), then (Tph1 ,h2 (a)f1 , f2 ) is well deﬁned if and only if (Tpf2 ,f1 (a)h2 , h1 ) is well deﬁned, and then (Tph1 ,h2 (a)f1 , f2 ) = (Tpf2 ,f1 (a)h2 , h1 ). It is now straightforward to control that the asserted equivalence follows from the latter identiﬁcation formula and Proposition 1.1. Remark 2.5. Theorem 2.2 can be used to obtain other results, similar to Proposition 0.1 and Proposition 2.3. For example, the proof of Proposition 0.1 gives the following more general result. Assume that θ, θ1 , θ2 ∈ P(R2m ) satisﬁes θ(X + Y ) ≤ C

θ2 (−Y ) , θ1 (X)

X, Y ∈ R2m ,

for some constant C, and that the hypothesis in Proposition 0.1 holds, except that the conditions on ω in (0.6) is replaced by ω(X, Y ) =

ω2 (X)θ(2Y ) . ω1 (X)κ(2Y )

Then p1 ,q1 p2 ,q2 (Rm ) → M(ω (Rm ) Tph1 ,h2 (a) : M(ω 1 θ1 ) 2 θ2 )

is continuous. Next we present a consequence of Proposition 0.1 . Here and in what follows we let P0 (Rm ) be the set of all ω ∈ P(Rm ) ∩ C ∞ (Rm ) such that ∂ α ω/ω ∈ L∞ for every multi-index α. Corollary 2.6. Assume that p, q, pj , qj , rj , sj ∈ [1, ∞] for j = 0, 1, 2 satisfy (0.5) and (2.2). Also assume that vj , ωj ∈ P0 (R2m ) such that ω1 is v1 -moderate and ω2 is vˇ2 -moderate such that ω2 (X) ω(X, Y ) = (2.5) ω1 (X) r ,s :p,q (R2m ), then the is fulﬁlled. If hj ∈ M(vjj ) j (Rm ) for j = 1, 2 and ω(X, D)a ∈ M map (0.4) is continuous.

Toeplitz Operators on Modulation Spaces

325

:p,q , if and only if Proof. The result follows immediately from the fact that a ∈ M (ω) :p,q (see, e.g., Theorem 3.2 in [27]), and Proposition 0.1 . ω(X, D)a ∈ M Corollary 2.7. Assume that p, pj , qj ∈ [1, ∞] for j = 0, 1, 2 satisfy (0.5) and (2.2). Also assume that vj , ωj ∈ P0 (R2m ) such that ω1 is v1 -moderate and ω2 is vˇ2 1,1 (Rm ) for j = 1, 2 and ω(X, D)a ∈ moderate such that (2.5) is fulﬁlled. If hj ∈ M(v j) p 2m L (R ), then the map (0.4) is continuous. Proof. The result follows immediately by letting rj = sj = 1 in Corollary 2.6 and using the embedding Lp ⊆ M p,q when q = max(p, p ) (see Remark 1.2). Remark 2.8. By straightforward computations it follows that if it in addition is :p,q (R2m ) and hj ∈ Mrj ,sj (Rm ) for j = 1, 2 in Theorem 2.2, assumed that a ∈ M (ω) (νj ) Proposition 0.1 or Proposition 2.6, then (0.4) is compact. Remark 2.9. In [4], Cordero and Gr¨ ochenig present a converse of Corollary 0.1 in the case p = q = ∞ and rj = sj = 1 for j = 1, 2, and the weight functions satisfy some extra conditions. (Cf. Theorem 4.3 in [4].) Remark 2.10. For each ω ∈ P(Rm ), it exists an element ω0 ∈ P0 (Rm ) which is equivalent to ω, i.e., there exists a constant C such that C −1 ω ≤ ω0 ≤ Cω holds. In fact, it suﬃces to let ω0 = ω ∗ ϕ, for some 0 ≤ ϕ ∈ S (Rm ) \ 0. (Cf. Lemma 1.2 in [26].) It follows from this equivalence that the conditions ωj , vj ∈ P0 (R2m ) in Corollary 2.7 can be replaced by ωj , vj ∈ P(R2m ). Remark 2.11. From the equivalence property in Remark 2.10 it also follows that Proposition 3.3 in [2] can be formulated as: Assume that p, q, pj , qj ∈ [1, ∞] for j = 1, 2 are such that 1 1 1 − = , p2 p1 p

1 1 1 − = . q2 q1 q

Also assume that ωj , vj ∈ P0 (R2m ) for j = 1, 2 are such that ω1 is v1 -moderate 1,1 and ω2 is vˇ2 -moderate, aω2 /ω1 ∈ Lp,q (R2m ) and that hj ∈ M(v for j = 1, 2. j) Then the map (0.4) is continuous. We note that Proposition 1.1 (2) implies that this result is a consequence of Corollary 2.7 when p ≤ q. On the other hand, if instead q < p, then Proposition 1.1 (2) shows that Corollary 2.7 is a consequence of Proposition 3.3 in [2].

3. Schatten-von Neumann properties for Toeplitz operators In this section, we give examples on how the results in the previous section can be combined with Theorem 1.6 in order to establish Schatten-von Neumann properties for Toeplitz operators acting on modulation spaces of Hilbert types. This means

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that p1 = p2 = q1 = q2 = 2 in Proposition 2.1, Theorem 2.2 and Proposition 0.1 , and that (0.4) takes the form (0.4)

2,2 2,2 Tph1 ,h2 (a) : M(ω → M(ω . 1) 2)

By combining Theorem 1.6 with Proposition 2.1 we obtain. Proposition 3.1. Assume that rj , sj , p, p0 , q, q0 ∈ [1, ∞] for j = 1, 2 satisfy (2.2) r ,s and q0 ≤ min(p0 , p0 ). Also assume that νj ∈ P(R2m ), hj ∈ M(νjj ) j (Rm ), j = 1, 2, and that ω, ω0 ∈ P(R2m ⊕ R2m ) satisfy ω0 (X1 + X2 , Y ) ≤ Cω(X1 , Y ) νˇ1 (Y + X2 )ν2 (Y − X2 )

X1 , X2 , Y ∈ R2m ,

for some constant C > 0. Then a → Tph1 ,h2 (a) from S (R2m ) to the set of linear and continuous operators from S (Rm ) to S (Rm ) extends to a continuous map :p,q (R2m ) to Ip0 (M 2,2 , M 2,2 ). from M (ω) (ω1 ) (ω2 ) If we instead combine Theorem 1.6 with Theorem 2.2, then we get the following. Theorem 3.2. Assume that p, q, q0 , rj , sj ∈ [1, ∞] for j = 1, 2 satisfy (2.2) and q0 ≤ min(p, p ). Also assume that νj , ωj ∈ P(R2m ) for j = 1, 2 and ω ∈ P(R2m ⊕R2m ) r ,s satisfy (2.4) for some constant C > 0 independent of X, Y ∈ R2m . If hj ∈ M(νjj ) j :p,q (R2m ), then Tp for j = 1, 2 and a ∈ M (a) ∈ Ip (M 2,2 , M 2,2 ). h1 ,h2

(ω)

(ω1 )

(ω2 )

The next result is a consequence of Proposition 0.1 and Proposition 3.1. Proposition 0.2 . Assume that p, q, , q0 , rj , sj ∈ [1, ∞] for j = 1, 2 satisfy (2.2) and q0 ≤ min(p, p ). Also assume (0.6) is fulﬁlled for some κ, κj , vj , νj , ωj ∈ P(R2m ) r ,s such that ω1 is v1 -moderate and ω2 is vˇ2 -moderate. If hj ∈ M(νjj ) j (Rm ) for j = 1, 2 :p,q (R2m ), then Tph ,h (a) ∈ Ip (M 2,2 , M 2,2 ). and a ∈ M (ω)

1

(ω1 )

2

(ω2 )

The next result is a consequence of Proposition 2.3 and Proposition 3.1. Proposition 3.3. Assume that p, q, q0 , rj , sj ∈ [1, ∞] for j = 0, 1, 2 satisfy (2.2) and q0 ≤ min(p, p ). Also assume (0.6) is fulﬁlled, for some κ, κj , vj , ωj ∈ P(R2m ) r ,s such that ωj is vj -moderate for j = 1, 2. If hj ∈ M(ωj j κjj ) for j = 1, 2 and a ∈ :p,q (R2m ), then Tph ,h (a) ∈ Ip (M 2,2 , M 2,2 ). M (ω)

1

2

(v1 )

(1/v2 )

Finally we remark that more Schatten-von Neumann results can be obtained by combining Proposition 3.1 with other results in Section 2. We leave the details for the reader.

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References [1] M.S. Birman and M.Z. Solomyak, Estimates for the singular numbers of integral operators (Russian), Uspehi Mat. Nauk. 32, (1977), 17–84. [2] P. Boggiatto, Localization operators with Lp symbols on modulation spaces, in Advances in Pseudo-diﬀerential Operators, Editors: R. Ashino, P. Boggiatto and M.W. Wong, Birkh¨ auser, Basel 2004, 149–163. [3] P. Boggiatto, E. Cordero and K. Gr¨ ochenig, Generalized anti-Wick operators with symbols in distributional Sobolev spaces, Integral Equations Operator Theory, 48 (2004), 427–442. [4] E. Cordero and K. Gr¨ ochenig, Time-frequency analysis of localization operators, J. Funct. Anal. 205 (2003), 107–131. [5] E. Cordero, S. Pilipovi´c, L. Rodino and N. Teofanov, Localization operators and exponential weights for modulation spaces, Mediterranean J. Math. 2 (2005), 381– 394. [6] I. Daubechies, Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inform. Theory 34 (1988), 605–612. [7] H. G. Feichtinger, Modulation spaces on locally compact abelian groups, in Wavelets and their Applications, Editors: M. Krishna, R. Radha and S. Thangavelu, Allied Publishers, 2003, 99–140. [8] C. Fern´ andez and A. Galbis, Compactness of time-frequency localization operators on L2 (Rd ), J. Funct. Anal. 233 (2006), 307–340. [9] 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), 335–350. [10] 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. [11] G.B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, 1989. [12] I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Non-Selfadjoint Operators in Hilbert Space (Russian), Izdat. Nauka, Moscow, 1965. [13] K. Gr¨ ochenig, Foundations of Time-Frequency Analysis, Birkh¨ auser, Boston, 2001. [14] K. Gr¨ ochenig and C. Heil Modulation spaces and pseudo-diﬀerential operators, Integral Equations Operator Theory 34 (1999), 439–457. [15] K. Gr¨ ochenig, 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. [16] L. H¨ ormander, The Analysis of Linear Partial Diﬀerential Operators I, III, SpringerVerlag, Berlin, Heidelberg, New York, Tokyo, 1983, 1985. [17] Z. He and M.W. Wong, Localization operators associated to square integrable group representations, Panamer. Math. J. 6 (1996), 93–104. [18] E.H. Lieb, Integral bounds for radar ambiguity functions and Wigner distributions, J. Math. Phys. 31 (1990), 594–599. [19] R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, Berlin, 1960.

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[20] B.W. Schulze and N.N. Tarkhanov, Pseudodiﬀerential operators with operatorvalued symbols, in Israel Math. Conf. Proc. 16, 2003. [21] M.A. Shubin, Pseudodiﬀerential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987. [22] B. Simon, Trace Ideals and their Applications, Cambridge University Press, Cambridge, London, New York, Melbourne, 1979. [23] N. Teofanov, Ultramodulation spaces and pseudodiﬀerential operators, Endowment Andrejevi´c, Beograd, 2003. [24] J. Toft, Continuity and Positivity Problems in Pseudo-diﬀerential Calculus, Thesis, Department of Mathematics, University of Lund, Lund, 1996. [25] J. Toft, Continuity properties for modulation spaces with applications to pseudodiﬀerential calculus, I, J. Funct. Anal., 207 (2004), 399–429. [26] J. Toft, Continuity properties for modulation spaces with applications to pseudodiﬀerential calculus, II, Ann. Global Anal. Geom. 26 (2004), 73–106. [27] 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. [28] 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. [29] J. Toft, Continuity and Schatten properties for pseudo-diﬀerential 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, 173–206. [30] M.W. Wong, Weyl Transforms, Springer-Verlag, 1998. [31] M.W. Wong, Localization operators on the Weyl-Heisenberg group, in Geometry, Analysis and applications, Editor: R.S. Pathak,, World Scientiﬁc, 2001, 303–314. [32] M.W. Wong Wavelet transform and localization operators, Birkh¨ auser, Basel, 2002. Joachim Toft Department of Mathematics and Systems Engineering V¨ axj¨ o University, Sweden e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 172, 329–342 c 2006 Birkh¨ auser Verlag Basel/Switzerland

Microlocalization within Some Classes of Fourier Hyperfunctions Richard D. Carmichael, Atsuhiko Eida and Stevan Pilipovi´c Abstract. New presheaves of hyperfunction spaces with the growth estimates with respect to |x| → ∞ and y → 0 in a cone Γ are introduced. Then it is shown that the Laplace transform is a bijective mapping of the space of tempered ultradistributions on Rn of non-quasianalytic class onto the corresponding hyperfunction space of sections over Dn , the compactiﬁcation of Rn . Microlocalization of tempered ultradistributions at (x0 ∞, ξ0 ) is introduced as well as a new microlocalization within some classes of hyperfunctions. Mathematics Subject Classiﬁcation (2000). Primary 32A40, 32A45, 46F15. Keywords. Fourier hyperfunctions, tempered ultradistributions, analytic wavefront.

1. Introduction Two of the authors recently studied the microlocal decomposition of several classes of Sato’s hyperfunctions. They showed in [4] that the sheaves of certain classes of microfunctions, e.g., C ∗ , C d,∗ , are supple; whereas the sheaf of microfunctions itself is ﬂabby (cf. [19], [1]). One of the authors, S. Pilipovi´c, has introduced in [16] and [13] the spaces of tempered ultradiﬀerentiable functions and tempered ultradistributions which are globally expressed for the usage with Fourier transforms on Rn . These spaces S ∗ (Rn ), S ∗ (Rn ) are topologically isomorphic to themselves through the Fourier transform. On the other hand, T. Kawai [10] introduced the spaces of Fourier hyperfunctions which are homologically constructed on Dn , the compactiﬁcation of Rn with inﬁnity. These spaces are also topologically isomorphic to themselves through the Fourier transform. The diﬀerence between the tempered ultradistributions and the Fourier hyperfunctions is that the spaces of Fourier hyperfunctions can be locally expressed from the viewpoint of sheaf. Here the authors will apply the ideas of Fourier hyperfunctions to the new presheaf of spaces of tempered ultradistributions of Roumieu

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type S {t} (Ω), Ω open in Dn , and will proceed to the microlocalization as Kaneko did ([8], [7]). The authors will do this by introducing a presheaf Sbs , s ∈ (0, 1), which is used for the deﬁnition of the wave front at (x0 ∞, ξ) ∈ Dn × Rn \{0}. The authors will also introduce a sheaf DSC s , a subsheaf of Rs , and prove the softness of Dbs /P s , DRC s and DSC s . In order to point out the main idea of the paper recall that Sato had constructed hyperfunctions using analytic functions without a growth estimate and Kawai [10], Zharinov [21], Kaneko [8], [9], Saburi [18] and others had introduced various Fourier hyperfunctions by the means of appropriate growth rate at inﬁnity with respect to real variable. In our analysis of introduced hyperfunction spaces bounds appear depending also on the imaginary variable as it approaches to the boundary.

2. Tempered ultradistributions We refer to [11] and [12] for the deﬁnitions of spaces of ultradiﬀerentiable functions and ultradistributions of Beurling and Roumieu type. Here we recall the deﬁnition of E {t} (Ω), where Ω is open in Rn and t > 1 : E {t} (Ω) = indlim indlim E t,h (K), K⊂⊂Ω h→∞

where E t,h (K) = {ϕ ∈ C ∞ (K); ||ϕ||K,h < ∞, h|α| |ϕ(α) (x)| ; x ∈ K, α ∈ Nn0 }. α!t (K ⊂⊂ Ω means that compact sets K run over a set of compact sets exhausting Ω.) n−1 Let Dn = Rn , S∞ be the directional compactiﬁcation of Rn with the n−1 natural topology. A fundamental system of neighborhoods of a∞ ∈ S∞ is given by sets of the form ||ϕ||K,h = sup{

(Γ ∩ {|x| > R}) , (Γ ∩ S n−1 )∞ , where Γ runs over open cones containing a and R & 0. Let t > 1 and h > 0. Let Ω be an open set of Dn . A space of smooth functions ϕ on Rn ∩ Ω with compact supports contained in Ω such that ⎛ ⎞1/2 |α+β| 2 h 2 |β|/2 (α) σh,2 (ϕ) := ⎝ ϕ (x) dx⎠ < ∞, α!t β!t (1 + |x| ) n α,β∈Nn 0

R ∩Ω

equipped with the topology induced by the norm σh,2 , is denoted by S˜2h (Ω). Denote by S2h (Ω) its completion. The strong dual of S {t} (Ω) = indlim S2h (Ω), h→0

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S {t} (Ω), is called the space of tempered ultradistributions of Roumieu type. In case Ω = Dn , we have S {t} (Dn ) = S {t} (Rn ), the space of tempered ultradistributions. For the properties of S {t} (Rn ) as well as of Beurling type spaces, we refer to [13] and [17]. It is clear that a function ϕ ∈ S {t} (Ω) can be extended out of Ω. More precisely, there is a Φ ∈ S {t} (Ωε ), where Ωε = {x ∈ Dn ; d(x, Ω) < ε}, ε > 0, such that supp ϕ ⊂ Ωε and Φ|Ω = ϕ. We refer to [2] for the extension of ultradiﬀerentiable functions of appropriate classes forming Whitney’s jets on a closed set. Using the part of the proof of Theorem 2 in [17] we have that an element f ∈ D{t} (Rn ) is in S {t} (Rn ) if and only if it is of the form F (t)P (∂)ϕ(t)dt, ϕ ∈ S {t} (Rn ), (1) f, ϕ = Rn

where P is an ultradiﬀerential operator of {t} – class ([11]) and a function F s satisﬁes F (x)e−k|x| ∈ L2 (Rn ), s = 1/t. Let Ω be an arbitrary open set of Dn . Then by the same proof as for Ω = Dn , we have the representation F (t)P (∂)φ(t)dt, φ ∈ S {t} (Ω), (2) f, φ = Rn ∩Ω

where P (∂) is an ultradiﬀerential operator of {t} – class and F is a function such s that F (x)e−k|x| ∈ L2 (Rn ∩ Ω) for every k > 0. The representations in (1) and (2) are not unique. We will show how the representation (2) can be used for the deﬁnition of the extension of an f ∈ S {t} (Ω) to an element fext ∈ S {t} (Rn ). With the notation as in (2), put Fext = F on Rn ∩ Ω, Fext = 0 on Rn \ (Rn ∩ Ω) and deﬁne fext ∈ S {t} (Rn ) by Fext (t)P (∂)ϕ(t)dt, ϕ ∈ S {t} (Rn ). fext , ϕ =

(3)

Rn

We have fext |Ω = f, where the left side is deﬁned by Fext (t)P (∂)ϕ(t)dt, ϕ ∈ S {t} (Ω). fext , ϕ = Rn ∩Ω

Let Ω1 , Ω2 be arbitrary open sets of Dn and Ω1 ⊂ Ω2 . We deﬁne the restriction mapping rΩ2 ,Ω1 : S {t} (Ω2 ) → S {t} (Ω1 ) as follows. Let f ∈ S {t} (Ω2 ) be of the form (2) (with Ω1 instead of Ω). Then rΩ2 ,Ω1 (f ) is deﬁned by F (t)P (∂)ϕ(t)dt, ϕ ∈ S {t} (Ω1 ). rΩ2 ,Ω1 (f ), ϕ = Rn ∩Ω1

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This deﬁnition does not depend on the representatives F and P (∂) in (2) since compactly supported elements of S {t} (Ω1 ) make a dense set in S {t} (Ω1 ) and they are contained also in S {t} (Ω2 ). The given explanation proves the next proposition: Proposition 1. Let Ω1 ⊂ Ω2 ⊂ Ω3 be open sets of Dn and let f ∈ S {t} (Ω3 ). Then rΩ3 ,Ω1 (f ) = rΩ2 ,Ω1 (rΩ3 ,Ω2 f ). Thus we have constructed a presheaf Ω → S {t} (Ω), Ω runs through all open sets in Dn . It is not a sheaf. In fact, the ﬁrst sheaf property related to the support is satisﬁed but the second sheaf property (gluing of sections) is not satisﬁed. The corresponding associated sheaf is denoted by Ω → D{t} (Ω), Ω ∈ Dn . If Ω ⊂ Rn , then it is the Roumieu space of ultradistributions. This is a consequence of the fact that any f in the associated sheaf has the representation of an ultradistribution of Roumieu type over any open bounded set G such that G ⊂⊂ Ω ⊂ Rn . We denote the associated sheaf by D{t} . Recall that the Fourier transform is an isomorphism of S {t} (Rn ) onto itself.

3. Sheaves of Fourier hyperfunctions of type [s, 0] ([8]) Let O be the sheaf of holomorphic functions on Cn . Kaneko has introduced in [8] ˜ s,δ , s > 0, δ ∈ R. We need such sheaves with s ∈ (0, 1) and δ = 0. the sheaves O √ Recall, for open U ⊂ Dn + −1Rn , ˜ s,0 (U ) = O ˜ s (U ) O |F (z)|e−ε|z| < ∞}, √ where K ⊂⊂ U means that the closure of K in Dn + −1Rn is a compact set contained in U . Let Ω be an open subset of Dn and Γ be an open cone in Rn . Then √ ˜ s (U ), ˜ s (Ω + −1Γ0) := indlim O O = {F (z) ∈ O(U ∩ Cn ); ∀K ⊂⊂ U, ∀ε > 0,

sup

s

z∈K∩Cn

U

U running over inﬁnitesimal wedges with edge Ω and opening Γ (for the deﬁnition of an inﬁnitesimal wedge we refer to [7], Deﬁnition 2.2.9). ˜ s |Dn and the sheaf of Fourier hyperfuncThe sheaf P s on Dn , is deﬁned by O n ˜s tions on Dn is deﬁned by Qs = HD n (O ) ([8]). n ˜ s ). This cohomological For every open set Ω ⊂ D , set Qs (Ω) = HΩn (Cn , O s expression can be translated in the following way: Q (Ω) = X/Y , where √ ˜ s (Ω + −1Γ0), Γ varying over all open convex cones, O X= Γ

and Y is a C− linear subspace of X generated by all the elements of the form √ ˜ s (Ω + −1Γj 0), j = 1, 2, 3, F1 (z) + F2 (z) − F3 (z), Fj (z) ∈ O

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such that Γ3 ⊂ Γ1 ∩ Γ2 and F1 (z) + F2 (z) = F3 (z) on a common domain of deﬁnition. √ N We use the same notation j=1 Fj (x + −1Γj 0) for an element in X and the corresponding class in X/Y . Note that N

Fj (x +

√ −1Γj 0) ∼ 0

if and only if there exists

j=1

˜ s (Ω + Hjk (z) ∈ O

√ −1(Γj + Γk )0) with Hjk (z) = −Hkj (z), j, k, = 1, . . . , N, and

Fj (z) =

N

Hjk (z), j = 1, . . . , N, on a common domain.

k=1

The singular spectrum of an f ∈ Qs (Ω) is deﬁned as follows: f is microanalytic at (a, ξ) ∈ Ω × S n−1 if there exists a local boundary value expression r √ Fj (x + −1Γj 0) on a neighborhood of a such that Γj ∩ {y, ξ < 0} = ∅ f = j=1

for all j. The complement of the set of points where f is microlocally analytic is called the singular spectrum of f and denoted by SS s f . The corresponding (analytic) singular support is deﬁned by singsuppsa f = π(SS s f ) (Refer to [7] and [8] for details.) Next, the sheaf Rs of Fourier microfunctions on Dn × S n−1 is deﬁned as the one associated to the presheaf Dn × S n−1 ⊃ Ω × ∆ → Qs (Ω)/{f ∈ Qs (Ω); SS s (f ) ∩ Ω × ∆ = ∅}. Let π : Dn × S n−1 → Dn be a natural (ﬁrst) projection. Kaneko has proved a generalized version of Sato’s fundamental exact sequence: 0 → P s → Qs → π∗ Rs → 0. The sheaves Qs , Qs /P s , and Rs are ﬂabby and H k (Ω, P s ) = 0, k ≥ 1, for real open Ω ⊂ Dn . So we have a global exact sequence 0 → P s (Ω) → Qs (Ω) → π∗ Rs (Ω × S n−1 ) → 0. If s = 1, we omit the superindex. Is this case Q and R are well-known spaces for Fourier hyperfunctions and microfunctions. We have the topological isomorphisms through the Fourier transform: ∼

∼

F : Q(Dn ) → Q(Dn ), F : R(Dn × S n−1 ) → R(Dn × S n−1 ).

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4. Tempered hyperfunctions In the sequel we will use notation s ∈ (0, 1) and t = 1/s. n n Deﬁnition √ 1. Let Ω be an open set in D and Γ be an open convex cone in R . s ˜ Oτ (Ω + −1Γ0) is the inductive limit of spaces of functions F analytic in an appropriate inﬁnitesimal wedge with the basis Ω and the opening Γ such that

(∀K ⊂⊂ Ω) (∀∆ ⊂⊂ Γ)(∀a > 0)(∀b > 0)(∃c > 0) √ |F (x + −1y)| < c exp(a|x|s + b|y|s/(1−s) ), (4) √ √ n x + −1y ∈ (R ∩ K) + −1∆, |y| < 1/c. Sbs (Ω) is a C-vector space deﬁned as X /Y , where X and Y are deﬁned (as were X and Y in Section 3) by ˜ s (Ω + iΓ0), Γ varying over all open convex cones; O X = τ Γ

and Y is a C-linear subspace of X generated by all elements of the form √ ˜ τs (Ω + −1Γj 0), j = 1, 2, 3, F1 (z) + F2 (z) − F3 (z), Fj (z) ∈ O such that Γ3 ⊂ Γ1 ∩ Γ2 and that F1 (z) + F2 (z) = F3 (z) on a common domain of deﬁnition. X consists of sums of the form N √ ˜ s (Ω + −1Γj 0). Fj (z), Fj ∈ O τ

(5)

j=1

√ N The corresponding equivalence class to (5) is denoted by j=1 Fj (x + −1Γj 0) and, as for Fourier hyperfunctions, we will use the same notation for the elements of X and the corresponding classes.

5. Laplace transform For a cone γ ⊂ Rn , γ 0 , int γ and γ¯ denote the polar cone, the interior of γ and the closure of γ, respectively. Recall that a cone is said to be acute if it does not contain straight lines. In this case int(γ 0 ) = ∅. We use the deﬁnition of the Laplace transform with the weight function √ √ exp −1(x + −1y)t which involves ty = ti yi ≥ 0 for a t belonging to a cone and y belonging to a polar cone. Thus, for an f ∈ S {t} (Rn ), such that supp f ⊂ γ, its Laplace transformation Lf is deﬁned by √ √ √ √ f (t)(exp −1(x + −1y)t)dt, x + −1y ∈ Rn + −1 int(γ 0 ), Lf (z) = γ

and in this domain it is a holomorphic function.

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Isomorphism j : S {t} (Rn ) → Sbs (Dn ). With the notation Γ = int(γ√0 ) and f ∈ S {t} (Rn ), supp f ⊂ γ, we have proved in [3] that F = Lf ∈ O(Rn + −1Γ) satisﬁes (∀a > 0)(∀b > 0)(∀ε > 0)(∃M > 0) √ √ √ |F (x + −1y)| ≤ M exp(a|x|s + b|y|s/(s−1) + ε|y|), x + −1y ∈ Rn + −1Γ (6) and √ (7) F (x + −1Γi 0) = bvF = (2π)n F −1 (f )(x) ∈ S {t} (Rn ) i = 1, . . . , p. Let K ⊂ Rn be a compact set such that 0 ∈ int K; and let γi , i = 1, . . . , p, be open convex cones such that ∪pi=1 γi = Rn \{0}. Assume that cones γ i , i = 1, . . . , p, are acute. Let η0 , η1 , . . . , ηp ∈ C ∞ be a partition of unity such that supp η0 ⊂ K, and supp ηi ⊂ γi , i = 1, . . . , p. Let f ∈ S {t} (Rn ). Clearly, f˜ = (2π)−n F f ∈ S {t} (Rn ). We make the decomposition f˜ = f˜0 +

p

f˜i ,

i=1

f˜0 = f˜η0 ∈ E {t} (Rn ), f˜i = f˜ηi ∈ S {t} (Rn ), supp f˜i ⊂ γi , i = 1, . . . , p. We deﬁne the Laplace transform by p Lf := F0 + Fi ,

(8)

i=1

where F0 (z) = Lf˜0 (z), z ∈ Cn ; Fi (z) = Lf˜i (z), Im z ∈ Γi = int(γi0 ), i = 1, . . . , p. By the Paley-Wiener theorem √ we know that F0 is an entire function. More˜ s (Ω + −1Γ0), for any open cone Γ. By (6) and (7) we over, it is an element of √ O τ have that Fi ∈ O(Rn + −1Γi ) satisﬁes (∀a > 0)(∀b > 0)(∀ε > 0)(∃M > 0) √ √ √ |Fi (x + −1y)| ≤ M exp(a|x|s + b|y|s/(s−1) + ε|y|), x + −1y ∈ Rn + −1Γi and √ Fi (x + −1Γi 0) = bvFi = (2π)n F −1 (f˜ηi )(x), i = 1, . . . , p. √ p −1Γi 0), So by (8), we let f correspond to the formal sum i=0 Fi (x + where (4) holds in corresponding cones. We refer to [3] for the proof of (6); for tempered distributions supported by a cone the corresponding proof is given in [20]. In the ﬁrst part of the next theorem we prove that L : S {t} (Rn ) → Sbs (Dn ) does not depend on the given decomposition. So, for this purpose, let T, δj , j = 1, . . . , r be another covering of Rn and νj , j = 0, . . . , r, be a corresponding partition r of unity with the same properties as above. We have f˜ = g˜0 + 1 g˜j , where g˜0 = f˜ν0 ∈ E {t} (Rn ), g˜j = f˜νj ∈ S {t} (Rn ), supp g˜j ⊂ δj . Let Gj = L(˜ gj ) j = 0, . . . , r.

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Theorem 1. (i) With the above notation, p

r Fi − ( Gj ) ∈ Y .

i=0

j=0

(ii) Let j(f ) := (2π)−n LF(f ), f ∈ S {t} (Rn ). Then j is an algebraic isomorphism of S {t} (Rn ) onto Sbs (Dn ) and bv(j(f )) :=

p i=0

lim

yi →0,yi ∈Γi

Fi (x +

√ −1yi )

equals f in the sense of convergence in S {t} (Rn ). Proof. (i) For every i = 0, 1, . . . , p, r Fi (z) = L( f˜i νj )(z), Im z ∈ Γi = int(γi0 ). j=0

This implies Fi − p i=0

Fi −

r j=0

p r

L(f˜i νj ) ∈ Y and

L(νj ηi f˜) =

i=0 j=0

=

p i=0 p

p r

Fi −

L(ηi g˜j )

j=0 i=0 r

Fi −

i=0

L(˜ gj ) =

j=0

p

Fi −

i=0

r

Gj ∈ Y .

j=0

This implies assertion (i). (ii) Let f ∈ S {t} (Rn ). Again, we put f˜ = (2π)−n F (f ) and use the decomp position f˜ = f˜η0 + f˜ηi , using the partition of unity ηi , i = 1, . . . , p. With i=1 √ ˜τs (Rn + −1Γi 0) and by (7), we have Γi = int(γi0 ), i = 1, . . . , p, Fi = L(f˜ηi ) ∈ O √ Fi (x + −1Γi 0) = (bvFi )(x) = (2π)n F −1 (f˜ηi )(x) ∈ S {t} (Rn ), i = 1, . . . , p, p

Fi (x +

p √ −1Γi 0) = (bvFi )(x) = f (x).

i=0

So j(f ) is the formal sum

i=0 p

Fi (x +

√ −1Γi 0) such that (6) holds in corre-

i=0

sponding cones. Let f, g ∈ S {t} (Rn ) so that j(f ) − j(g) ∈ Y .

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By deﬁnition, any element in Y has the boundary value equaling 0, in S {t} (Rn ) (the sum of limits when the imaginary part approaches to 0 in corresponding cones). This implies that bv(j(f )) − bv(j(g)) = f − g = 0 and this implies the injectivity of j. Let us show that the mapping j : S {t} (Rn ) → Sbs√(Dn ) is surjective. Let f ∈ Sbs (Dn ) be determined by a single element F (x + −1Γ0) which satisﬁes (4). We will show that the boundary value of F as y → 0, y ∈ Γ, in the sense of convergence in S {t} (Rn ), is an element f˜ ∈ S {t} (Rn ). Then, by the proof of injectivity it follows that this f is the image of f˜, j(f˜) = f . So let us ﬁnd the boundary value of F in S {t} (Rn ). The idea of proof is similar to the proof of the corresponding assertions in D (Rn ) ([7], Theorem 3.1.15). It is based on the almost analytic extension in which appropriate “cut oﬀ” functions appear. As in [13], let κ be a non-negative function which belongs to S {t} (R) such that supp κ ⊂ [−2, 2], κ |[−1,1] = 1 and let κp (y) = κ(4ypt−1 /h), y > 0, p ∈ N0 (remember, t = 1/s > 1). Let ϕ be a smooth function such that sh (ϕ) < ∞. The almost analytic extension of ϕ is deﬁned by Φ(z) =

ϕ(p) (x) √ √ ( −1y)p κp (y), z = x + −1y ∈ Cn , p! n

p∈N0

where

√ √ √ p! = p1 ! . . . pn !, ( −1y)p = ( −1y1 )p1 . . . ( −1yn )pn , κp (y) = κp1 (y1 ) . . . κpn (yn ).

This is a smooth function in R2n . Let 0 = Y ∈ Γ, | Y |< γ, Y = (Y1 , . . . , Yn ), Yi = 0, i = 1, . . . , n. (If some Yi = 0, the proof is the same.) Using 1 1 h/pt−1 ≤ Yi y ≤ h/pt−1 , on the support of κp (yY ), p ∈ N0 , 4 2 max{|κ(y)|, |κ (y)|} ≤ C, y ∈ R, it follows that there are Ch > 0 and Hh > 0 such that √ 1 ∂ sup {exp(h | x |s + )| Φ(x + −1yY ) |} ≤ Ch σh,2 (ϕ), s/(1−s) ∂ z ¯ H |y| i=1,...,n i h s

eh|x| | Φ(x +

√ −1yY ) |≤ Ch σh,2 (ϕ), x ∈ Rn , y ∈ [0, 1).

(9)

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√ Put ZY = {z = x + −1yY ; x ∈ Rn , 0 ≤ y ≤ 1}. Let ψ be an analytic function and θ, ∂∂z¯i θ, i = 1, . . . , n, be continuous functions in a neighborhood of ZY . Assume that ψθ is integrable in this neighborhood and that √ √ | (ψθ)(. . . , ai1 + −1tYi1 , . . . , aim + −1tYim , . . . ) |→ 0 as a → ∞, uniformly in y ∈ [0, 1], and xj ∈ R, j = i1 , . . . , im , where | aij |= a, a > 0, i1 , . . . , im ∈ {1, . . . , n}, 1 ≤ m ≤ n. Then, θ(x1 , . . . , xn )ψ(x1 , . . . , xn )dx1 , . . . , dxn = Rn

√ +2 −1

n

1

Rn

0

i=1

Rn

Yi

θ(x +

√ √ −1Y )ψ(x + −1Y )dx

√ √ ∂θ (x + −1yY )ψ(x + −1yY )dydx. ∂ z¯i

The proof of this assertion follows from Stokes’ formula on √ ZYa = {z = x + −1yY ; −a ≤ xi ≤ a, i = 1, . . . , n, 0 ≤ y ≤ 1} letting a → ∞. There exists a constant C > 0 such that y ≤ C | u + yY |, u ∈ Γ, y > 0 ([7], p. 66). Let u ∈ Γ, ϕ ∈ S {t} (Rn ). Then functions √ ψ(z) = F (z + −1u), θ(z) = Φ(z), z ∈ ZY , satisfy conditions given above. This follows from the assumptions and (9). Letting u → 0, u ∈ Γ, we have √ √ √ F (x + −1Y )Φ(x + −1Y )dx F (x + −10), ϕ(x) = √ +2 −1

n i=1

Yi 0

1

Rn

Rn

√ √ ∂ Φ(x + −1yY )F (x + −1Y y)dydx. ∂ z¯i

Moreover, for some C > 0 there holds √ | F (x + −10), ϕ(x) |≤ Cσh,2 (ϕ), ϕ ∈ S {t} . This completes the proof of Theorem 1. Isomorphism j : S {t} (Ω) → Sbs (Ω), Ω ⊂ Dn . Let f ∈ S {t} (Ω). Then we deﬁne fext as in Section 2 through (2) and (3). By Theorem 1 we have a unique element j(fext ) ∈ Sbs (Dn ). The arguments of Theorem 1 imply that bvj(fext )|Ω = fext |Ω = f. Thus we have the next theorem: Theorem 2. The mapping j is an isomorphism of S {t} (Ω) onto Sbs (Ω).

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339

6. Tempered hyperfunctions (Continuation) Since the mapping j is an isomorphism of S {t} (Ω) onto Sbs (Ω), Ω ⊂ Dn and Ω → S {t} (Ω), is a presheaf, we obtain the presheaf Ω → Sbs (Ω). We denote the associated sheaf as Dbs . Note that the canonical mapping Sbs (Ω) → Dbs (Ω) = Γ(Ω, Dbs ) is injective. If f ∈ S {t} (Ω), we denote by jf the corresponding element of Dbs (Ω). The singular spectrum of an f ∈ Sbs (Ω), denoted by SSτs f , is deﬁned as the complement of Sbsτ -microanalytic points, where we deﬁne: f is Sbsτ -microanalytic at (a, ξ) ∈ Ω × S n−1 if there exists a local boundary value expression of the form r √ √ ˜ s (Ω + −1Γj 0)) on a neighborhood of a such Fj (x + −1Γj 0) (Fj ∈ O f = τ j=1

that Γj ∩ {y, ξ < 0} = ∅ for all j. We deﬁne singsuppsτ f = π(SSτs f ), f ∈ Sbs (Ω). The canonical mapping k : Dbs → Qs deﬁned by: N

Fj (x +

N √ √ −1Γj 0) → Fj (x + −1Γj 0)

j=1

j=1

is injective. So we have the injective mapping k : S {t} (Ω) → Γ(Ω, Q)) The next assertions are consequences of deﬁnitions. ˜ s := k(Dbs ) is a soft subsheaf of Qs . Theorem 3. 1) Db 2) Let f ∈ Sbs (Ω). Then SSτs f ⊃ SS s k(f ). In particular, if we deﬁne singsuppsa f := π(SS s k(f )), then singsuppsτ f ⊃ singsuppsa f .

7. Microlocalization ˜ s = k(Dbs ), we deﬁne quotient sheaves Dbs /P s and Qs /Db ˜ s by With the notation Db 0 → P s → Dbs → Dbs /P s → 0, ˜ s → Qs → Qs /Db ˜ s → 0. 0 → Db Theorem 4. For every open Ω ⊂ Dn , Dbs /P s (Ω) = Dbs (Ω)/P s (Ω), ˜ s (Ω) = Qs (Ω)/Db ˜ s (Ω). Qs /Db Proof. The ﬁrst assertion follows from H k (Ω, P s ) = 0, k ≥ 1, Ω ⊂ Dn . The soft˜ s implies H k (Ω, Db ˜ s ) = 0, k ≥ 1, Ω ⊂ Dn and this implies the second ness of Db assertion. Denote by ρ and ρ˜ the natural mappings ˜ s (Ω) → Db ˜ s (Ω)/P s (Ω) ρ : Dbs (Ω) → Dbs (Ω)/P s (Ω), ρ˜ : Db

340

R.D. Carmichael, A. Eida and S. Pilipovi´c

and deﬁne the s-analytic singular support of an f ∈ S {t} (Rn ) by singsuppsa f := supp ρ(j(f )). The proof of the next theorem follows from the deﬁnitions. Proposition 2. Let f ∈ S {t} (Rn ). Then singsuppsa f = supp ρ˜(k(f )), Rn ∩ singsuppsa f = singsuppa f, where singsuppa f denotes the analytic singular support of f . The isomorphism j enables us to deﬁne the analytic singular support and the analytic wave front set of an f ∈ S {t} (Rn ) at x0 ∞ and (x0 ∞, ξ0 ), respectively, n−1 where (x0 ∞, ξ0 ) denotes a point on the cosphere bundle at x0 ∈ S∞ with the direction at component ξ0 ∈ Rn \ {0}. Let f ∈ S {t} (Rn ). Then we deﬁne: / SSτs f iﬀ (x0 ∞, ξ) ∈ / SSτs j(f ). (x0 ∞, ξ) ∈ This leads to: x0 ∞ ∈ / singsuppsτ f if x0 ∞ ∈ / singsuppsτ j(f ). Proposition 3. Let f ∈ S {t} (Rn ). Then / singsuppsτ f ⇐⇒ ∀ξ ∈ Rn \ {0}, (x0 ∞, ξ) ∈ / SSτs j(f ). x0 ∞ ∈ We denote by Sps the canonical surjective spectrum mapping Sps : π −1 Qs → Rs which gives the sheaf homomorphisms ˜ s → Rs . π −1 Db This leads us to the deﬁnitions of new microfunction spaces and new microlocalizations of hyperfunctions in Qs . Deﬁnition 2. ˜ s ). (a) DRC s := Sps (π −1 Db s (b) DSC is deﬁned by the sequence 0 → DRC s → Rs → DSC s → 0. (c) Sp1,s : π −1 Qs → DSC s is deﬁned as the composition of the mapping Sps and the canonical embedding Rs → DSC s . (d) Let u ∈ Qs Then SS 1,s (u) := supp Sp1,s u. One can easily prove the exactness of the next sequence. Theorem 5.

˜ s → π∗ DRC s → 0. 0 → P s → Db

Theorem 6. Sheaves Dbs /P s , DRC s and DSC s are soft sheaves.

Microlocalization within Some Classes of Fourier Hyperfunctions

341

Proof. We will only show the last part of the theorem. By Dbs /P s (Ω) = Dbs (Ω)/P s (Ω) and Qs /P s (Ω) = Qs (Ω)/P s (Ω), Ω ⊂ Dn , n−1 it follows that for every open V ⊂ Dn × S∞ Γ(V, Rs ) → Γ(V, DSC s ) → 0 is exact; and the softness of DSC s follows from the diagram Γ(V, Rs ) → Γ(V, DSC s ) → 0 ↓ ↓ Γ(W, Rs ) → Γ(W, DSC s ) → 0 ↓ 0 exact,

exact exact

n−1 and W ⊂ V . where V, W are open in Dn × S∞

References [1] G. Bengel and P. Schapira, D´ecomposition mircrolocal analytique des distributions, Ann. Inst. Fourier, Grenoble, 29 (1979), 101–124. [2] A. Bonet, R.W. Braun, R. Meise and B.A. Taylor, Whitney’s extension theorem for nonquasianalytic classes of ultradiﬀerentiable functions, Studia Mathematica, 99 (1991), 155–184. [3] R. Carmichael and S. Pilipovi´c, On the convolution and Laplace transformation in the space of Beurling–Gevrey tempered ultradistributions, Math. Nachr., 158 (1992), 119–132. [4] A. Eida and S. Pilipovi´c, On the microlocal decomposition of some classes of ultradistributions, Math. Proc. Cambridge Phil. Soc. 125 (1999), 455–461. [5] A. Eida and S. Pilipovi´c, Suppleness of some quotient spaces of microfunctions, Rocky Mountain J. Math. 30 (2000), 165–177. [6] L. H¨ ormander, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coeﬃcients, Comm. Pure Appl. Math. 24 (1971), 617–704. [7] A. Kaneko, Introduction to Hyperfunctions, Kluwer Acad. Press, Dordrecht, Boston, London, 1988. [8] A. Kaneko, On the global existence of real analytic solutions of linear partial diﬀerential equations on unbounded domains, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), 319–372. [9] A. Kaneko, On the ﬂabbiness of the sheaf of Fourier microfunctions, Sci. Papers of the College of Arts and Sci. 36 (1986), 1–14. [10] T. Kawai, On the theory of Fourier hyperfunctions and its applications to partial diﬀerential equations with constant coeﬃcients, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 17 (1970), 467–517. [11] H. Komatsu, Microlocal analysis in Gevrey classes and in convex domains, in Lecture Notes in Mathematics 1495 Springer-Verlag, 1991, 161–236.

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[12] H. Komatsu, Ultradistributions, I: Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo. Sect. IA Math. 20 (1973), 25–105. [13] D. Kova˘cevi´c and S. Pilipovi´c, Structural properties of the space of tempered ultradistributions, in Complex Analysis and Generalized Functions, Varna, 1991, 169–184. [14] P. Laubin, Front d’onde analytique et d´ecomposition microlocale des distributions, Ann. Inst. Fourier, Grenoble, 33 (1983), 179–199. [15] A. Melin and J. Sj¨ ostrand, Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem, Comm. Partial Diﬀerential Equations 1 (1976), 283–311. [16] S. Pilipovi´c, Microlocal analysis of ultradistributions, Proc. Amer. Math. Soc. 126 (1998), 105–113. [17] S. Pilipovi´c, Characterization of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), 1191–1206. [18] Y. Saburi, Fundamental properties of modiﬁed Fourier hyperfunctions, Tokyo J. Math. 8 (1985), 231– 273. [19] M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-diﬀerential equations in Lecture Notes in Mathematics 287 Springer-Verlag, 1973, 265-529. [20] V. S. Vladimirov, Generalized Functions in Mathematical Physics, Mir, Moscow, 1979. [21] V.V. Zharinov, The Laplace transform of Fourier hyperfunctions and other similar cases of analytic functionals, Part I in Teoret. Mat. Fiz. 33 (1977), 291–309, Part II in Teoret. Mat. Fiz. 37 (1978), 12–29. Richard D. Carmichael Department of Mathematics Wake Forest University Winston-Salem NC 27109-7388, USA e-mail: [email protected] Atsuhiko Eida Department of Engineering Science Tokyo Engineering University 1401-1, Katakura Hachioji, Tokyo 192, Japan e-mail: [email protected] Stevan Pilipovi´c Institute of Mathematics University of Novi Sad Trg D. Obradovica 4 21000 Novi Sad, Yugoslavia e-mail: [email protected]

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