Time-Frequency Analysis of Operators 9783110530353, 9783110532456, 9783110530605, 2020942033

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Elena Cordero, Luigi Rodino Time-Frequency Analysis of Operators

De Gruyter Studies in Mathematics

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Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China Karl-Hermann Neeb, Erlangen, Germany

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Volume 75

Elena Cordero, Luigi Rodino

Time-Frequency Analysis of Operators

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Mathematics Subject Classification 2010 42B10, 81S30, 42B37, 35S05

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Authors Prof. Dr. Elena Cordero Dipartimento di Matematica Università degli Studi di Torino Via Carlo Alberto 10 10123 Torino Italy [email protected]

Prof. Dr. Luigi Rodino Dipartimento di Matematica Università degli Studi di Torino Via Carlo Alberto 10 10123 Torino Italy [email protected]

ISBN 978-3-11-053035-3 e-ISBN (PDF) 978-3-11-053245-6 e-ISBN (EPUB) 978-3-11-053060-5 ISSN 0179-0986 Library of Congress Control Number: 2020942033 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

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To Elisa and Francesco – E.C. To Mia – L.R.

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Preface The main topics of this book are the applications of the time–frequency analysis methods to the study of the partial differential equations, with the emphasis on the theory of the pseudodifferential and Fourier integral operators reconsidered in terms of Gabor transform and Gabor frames. The material comes largely from a series of papers of the authors in collaboration with Fabio Nicola. Although our friend Fabio does not appear as a coauthor of the present volume, much depends here on his results. In the last ten years several other people contributed to the field, and we believe a book reporting, at least in part, on their studies is useful, despite the fact that research is going on and a large part of the work has still to be done. Our ambition is also to attract people working in partial differential equations from the theoretical and numerical point of view, convincing them that the time–frequency approach can be useful. Besides, the material is organized starting from scratch so that also nonexperts and young students can enter the subject. We wish to thank the many friends and colleagues who have contributed in this monograph in various ways. In particular, two groups of scientists working on time– frequency analysis, one at the University and Politecnico of Torino and the other in Vienna (NuHAG), had an essential role; their research results, reported in the bibliography, give essential background to the book. We wish also to express our gratitude to Stevan Pilipović and Nenad Teofanov (Novi Sad University), to Hans G. Feichtinger and Maurice de Gosson (University of Vienna), to Joachim Toft and Patrik Wahlberg (Linnaeus University of Växjö), to Federico Bastianoni and S. Ivan Trapasso (Politecnico of Torino), and to Jörg Seiler (University of Torino) for suggesting several improvements of the text. Many classical and recent volumes have influenced the writing, the choice of topics and the proofs. Those that have had the most profound influence on this book are as follows: K. Gröchenig, Foundations of time–frequency analysis [160], C. Heil, A basis theory primer [183]. We greatly appreciate these texts and encourage the reader to consult them. Additional papers are listed in the references. We wish to thank Elsevier, holding the copyright, for granting permission to include 13 figures in Chapter 6. Finally, we thank De Gruyter for all the support and patient waiting of the completion of the monograph. Torino, April 2020

https://doi.org/10.1515/9783110532456-201

Elena Cordero, Luigi Rodino

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Introduction Two different names, “micro-local analysis” and “time–frequency analysis”, are both used in Fourier analysis to identify the simultaneous study of a function in the space (or time) variable and in the dual variable, looking for possible localization of the frequencies. The two terms are not exactly synonymous though. In fact, aim of the microlocal analysis is the study of the partial differential equations, the main concern being the investigation of local singularities of the solutions and related issues of the general theory. In this perspective, we may refer to the monumental work of Lars Hörmander in the years 1970–2000, providing in particular the setting of the pseudodifferential and Fourier integral operator theory. Because of the problems under consideration, attention is limited to high frequencies, and final information is usually expressed in terms of asymptotic expansions rather than numerical values. The functional framework is given by the classical Lebesgue–Sobolev spaces. The history of time–frequency analysis is more recent, despite the fact that its roots are in Fourier analysis used widely in the applied sciences. The main present motivation is the signal theory, with the aim of analyzing the local frequencies of the function (called signal in this context) by means of the short-time Fourier transform defined in the spirit of Dennis Gabor 1946. It is then evident that numerical aspects are dominant, with attention to low or, say, bounded frequencies. In fact, the short-time Fourier transform is often replaced by its discrete version, the Gabor frames. Pseudodifferential operators still appear, but with the new role of filters of the signal, with limited attention to differential equations. The new functional setting is given by modulation spaces and Wiener amalgam spaces. The present book is devoted to exploring the interface between micro-local and time–frequency analysis, with intentions as before. Namely, we want to apply the time–frequency methods to study the partial differential equations. The usual microlocal methods are not abandoned, but the standard Fourier transform is systematically replaced by time–frequency representations as the short-time Fourier transform, the Wigner distribution, and Gabor frames. Correspondingly, Sobolev spaces are substituted by modulation and Wiener amalgam spaces. This new point of view presents advantages and disadvantages. Precisely, the applications to partial differential equations in this book, reflecting the present state of the literature, are essentially limited to constant coefficient equations, with wave equation as a basic example, and Schrödinger equations. In fact, we cannot reproduce the classical results of Hörmander concerning propagation of singularities for variable-coefficient hyperbolic equations simply because short-time Fourier transform and Gabor frames ignore the exact location of the singularities generated by high frequencies. An advantage of our approach is the possibility of numerical applications by using Gabor frames. Other advantages are given by the use of modulation spaces which eliminate in some sense the poisoned part of the Lebesgue spaces Lp . Estimates in their framework are easy for some problems where Lp estimates fail. https://doi.org/10.1515/9783110532456-202

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X | Introduction Let us describe in short the contents of the six chapters of the book. For details we address to the respective introductions, providing exhaustive summary, historical notes, and motivations. The first three chapters are addressed to the nonexperts, presenting the basics of the time–frequency analysis. They have a preliminary character with respect to the sequel and can be read independently as a short textbook, showing the state-of-theart in the field. In particular, Chapter 1 treats the short-time Fourier transform and the Wigner distribution, as well as other related time–frequency representations. The symplectic and metaplectic groups are reviewed, having applications in the following chapters of the book. Chapter 2 is devoted to modulation and Wiener amalgam spaces. Chapter 3 addresses Gabor frames and kernel theorems in modulation spaces. The subsequent three chapters contain the aforesaid applications to partial differential equations. Precisely, Chapter 4 is devoted to pseudodifferential operators, playing the role of filters for signal theory and resolvent operators for equations. The reader may be surprised that the standard classes of the micro-local analysis, and related symbolic calculus, do not appear here. In fact, the approach of the time–frequency analysis prevails, and we mainly address the boundedness properties in modulation and Wiener amalgam spaces. Chapter 4 contains also a rather detailed presentation of the Born–Jordan quantization and related pseudodifferential calculus; the material there represents a novelty for textbooks. Chapter 5 concerns applications to operators with constant coefficients. In particular, Strichartz estimates are revisited, together with the use of Wiener amalgam spaces allowing an improvement of the classical results obtained in the framework of Lebesgue spaces. The second part of Chapter 5 considers hyperbolic and parabolic-type equations with constant coefficients. The use of Gabor matrix representations of the corresponding propagators provides optimally sparse decompositions. Chapter 6 is devoted to the study of general Schrödinger equations. To this end, the calculus of the Fourier integral operators in ℝd is reconsidered in terms of the time–frequency analysis. Beside solving the problem of the global representation of the classical theory, this again allows for numerical applications. In Appendix A we collect some basic results from functional analysis, that we use throughout the volume. Concerning the bibliography, we have tried to be complete as far as possible, but some relevant omissions are certainly possible, and we apologize for them.

Contents Preface | VII

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Introduction | IX 0 0.1 0.2 0.2.1 0.2.2

Preliminaries | 1 Notation and background | 1 Function spaces and Fourier transform | 3 Fundamental operators | 6 Gaussian functions | 7

1 1.1 1.1.1 1.1.2 1.1.3 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.3.6 1.3.7

Basics of time–frequency representations and related properties | 9 Hints on symplectic and metaplectic groups | 12 The symplectic group | 12 The metaplectic group | 13 The symplectic Fourier transform | 17 Time–frequency representations | 18 The short-time Fourier transform | 18 Orthogonality relations | 24 Short-time Fourier transform and symplectic matrices | 25 Reproducing formula for the short-time Fourier transform | 26 Short-time Fourier transform of tempered distributions | 28 Decay and further properties | 36 Ambiguity function, Wigner distribution, and related topics | 38 The ambiguity function | 38 The Wigner distribution | 39 Marginal densities | 44 The Cohen class | 46 τ-Wigner distributions | 53 Time–frequency representations of Gaussians | 60 Short-time Fourier transform of time–frequency representations | 64

2 2.1 2.2 2.3 2.3.1 2.3.2

Function spaces | 69 Weight functions | 72 Mixed norm spaces | 76 Modulation spaces | 79 Properties of modulation spaces | 81 Alternative definition of modulation spaces using frequency-uniform localization techniques | 95 Wiener amalgam spaces | 100

2.4

XII | Contents 2.4.1 2.5 2.6 2.7 2.7.1 2.7.2 3 3.1 3.1.1 3.2 3.2.1 3.2.2 3.3 3.3.1

Gabor frames and linear operators | 143 Frames in Hilbert spaces | 145 Frame expansions and frame operator | 153 Gabor frames | 162 Existence and density of Gabor frames | 166 Gabor frames for modulation spaces | 176 Kernel theorems for modulation spaces | 181 Operators acting on Mp,q | 187

4 4.1 4.1.1 4.2 4.3 4.4 4.4.1

Pseudodifferential operators | 191 Boundedness on Lp and Wiener amalgam spaces | 197 Boundedness on Wiener amalgam spaces: necessary conditions | 201 Symbols in Wiener amalgam spaces | 208 Pseudodifferential operators on modulation spaces | 211 Continuity results for the (cross-)Wigner distribution | 214 Continuity results for the short-time Fourier transform and the ambiguity function | 223 Pseudodifferential operators on modulation spaces, conclusions | 226 Localization operators on modulation spaces | 227 Opτ (a) operators with symbols in Wiener amalgam spaces | 230 Boundedness properties of τ-Wigner distributions | 232 Continuity properties of Opτ (a) operators | 239 Born–Jordan quantization and related pseudodifferential calculus | 242 Division of distributions | 245 Changes of coordinates for temperate distributions | 250 Born–Jordan pseudodifferential operators | 251 Invertibility of the Born–Jordan quantization | 253 Born–Jordan pseudodifferential operators on modulation spaces | 261 Analysis of the Cohen kernel Θ | 263 Proof of Theorem 4.7.1 | 268

4.4.2

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Inclusion, convolution, and multiplication relations | 107 Dilation properties for Wiener amalgam spaces W (Lp , Lq ) | 113 Dilation properties for modulation spaces | 118 Sharpness of convolution, inclusion, and multiplication relations for modulation spaces | 135 Time-frequency tools | 138 Further extensions of modulation spaces (quasi-Banach setting) | 140

4.4.3 4.5 4.5.1 4.5.2 4.6 4.6.1 4.6.2 4.6.3 4.6.4 4.7 4.7.1 4.7.2

Contents | XIII

5 5.1 5.1.1 5.1.2 5.1.3 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6

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6

Time–frequency analysis of constant-coefficient partial differential equations | 277 Strichartz estimates for the Schrödinger equation in Wiener amalgam spaces | 281 Fixed-time estimates | 283 Strichartz estimates | 287 Sharpness of fixed-time and Strichartz estimates | 295 Gabor representations of evolution operators | 301 Gelfand–Shilov spaces | 301 Almost diagonalization for pseudodifferential operators | 308 Sparsity of the Gabor matrix | 313 Boundedness of pseudodifferential operators on modulation and Gelfand–Shilov spaces | 314 Evolution equations | 315 Some numerics for the heat equation | 321

6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.1.5 6.1.6 6.2 6.2.1 6.2.2 6.2.3 6.3 6.3.1 6.4 6.5 6.6 6.6.1 6.6.2 6.6.3 6.6.4

Fourier integral operators and applications to Schrödinger equations | 325 Fourier integral operators | 326 Phase functions and canonical transformations | 328 Almost diagonalization of FIOs | 329 p Continuity of FIOs on Mμ (ℝd ) | 333 p,q Continuity of FIOs on M | 336 Modulation spaces as symbol classes | 338 The case of quadratic phases: metaplectic operators | 342 Sparsity of Gabor representation of Schrödinger propagators | 349 p Continuity of FIOs on Mμ (ℝd ), 0 < p < 1 | 351 Sparsity of the Gabor matrix and nonlinear approximation | 352 Application to the solution of the Schrödinger equation | 356 Wiener algebras of Fourier integral operators | 363 Wiener algebra properties | 367 FIOs of type I | 371 Generalized metaplectic operators | 380 Propagation of Gabor wave front set for Schrödinger equations | 383 Preliminaries and Shubin classes | 386 Unperturbed Schrödinger equations | 393 Schrödinger equations with bounded perturbations | 397 Propagation of singularities | 400

A A.1 A.1.1

Appendix | 409 Basis theory and series expansions | 412 Bases and orthonormal sequences | 412

XIV | Contents A.1.2 A.2

Convergence of series | 414 Some results concerning real interpolation theory | 421

Bibliography | 425 Index | 439

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Index of Notation | 441

0 Preliminaries In this part we present the notation used in this book and collect some basic results which will be useful in the sequel.

0.1 Notation and background We use the standard set-theoretic notation, with one peculiarity: set-theoretic inclusion A ⊂ B does not exclude equality. Thus proper inclusion has to be stated explicitly: A ⊂ B and A ≠ B. We set ℕ = {0, 1, 2, . . . }, ℕ+ = {1, 2, 3, . . . }, whereas ℤ, ℝ, ℂ stand for the set of all integer, real, and complex numbers, respectively. We define ℝ+ = {t ∈ ℝ : t > 0}, ℝ− = {t ∈ ℝ : t < 0}. For R > 0, x0 ∈ ℝd , we denote by BR (x0 ) the ball centered at x0 with radius R: BR (x0 ) = {y ∈ ℝd : |y − x0 | < R}. If X is a nonempty set and f , g : X → [0, +∞), we set f (t) ≲ g(t),

t ∈ X,

if there exists C > 0 such that f (t) ≤ Cg(t), for all t ∈ X. Moreover, if f and g depend on a further variable z ∈ Z, the statement that, for all z ∈ Z, f (t, z) ≲ g(t, z),

t ∈ X,

means that for every z ∈ Z there exists a real number Cz > 0 such that f (t, z) ≤ Cz g(t, z) for every t ∈ X. It may happen that supz∈Z Cz = ∞. We set

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f (t) ≍ g(t),

t ∈ X,

if f (t) ≲ g(t) and g(t) ≲ f (t), t ∈ X, and similarly as above if the functions depend on a further parameter. The dimension of the space is denoted by d. We will employ the multiindex notation. Given α, β ∈ ℕd and t ∈ ℝd , we set |α| = α1 + ⋅ ⋅ ⋅ + αd ,

α ≤ β ⇐⇒ αj ≤ βj , α < β ⇐⇒ α ≠ β

α! = α1 ! ⋅ ⋅ ⋅ αd !,

for j = 1, . . . , d,

and α ≤ β,

d α α α! ( ) = ∏( j) = , β β!(α − β)! β j j=1 α

α

t α = t1 1 ⋅ ⋅ ⋅ td d . https://doi.org/10.1515/9783110532456-001

β ≤ α,

2 | 0 Preliminaries Observe that α ∑ ( ) = 2|α| , β β≤α

(0.1)

|α|! ≤ d|α| α!.

(0.2)

and

Functions are always understood to be complex-valued, if not stated otherwise. Partial derivatives are denoted by 𝜕j = 𝜕tj =

𝜕 , 𝜕tj

j = 1, . . . , d.

More generally, we set α

α

α

α

𝜕α = 𝜕1 1 ⋅ ⋅ ⋅ 𝜕d d = 𝜕tα = 𝜕t11 ⋅ ⋅ ⋅ 𝜕tdd . If t, ξ ∈ ℝd , we set d

tξ = t ⋅ ξ = ∑ tj ξj ,

|t| =

j=1

d

t 2 = |t|2 = ∑ tj2 , j=1

1/2 2 (∑ tj ) , j=1 d

1/2

⟨t⟩ = (1 + t 2 ) .

Observe that ⟨t⟩ is a smooth function satisfying 𝜕tj ⟨t⟩ = tj /⟨t⟩. In general, one can verify that

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󵄨󵄨 α 󵄨󵄨 |α|+1 |α|!⟨t⟩1−|α| , 󵄨󵄨𝜕t ⟨t⟩󵄨󵄨 ≤ 2

(0.3)

for all t ∈ ℝd and all α ∈ ℕd . The following inequality, also called Peetre’s inequality, will be used throughout the book: ⟨x + y⟩s ≤ 2s ⟨x⟩s ⟨y⟩|s| ,

x, y ∈ ℝd , s ∈ ℝ.

(0.4)

The power of multiindex notation is well explained by the following formulas. For an open subset X of ℝd and m ∈ ℕ+ , consider the spaces C m (X) of functions having continuous partial derivatives of order ≤ m, and C ∞ (X) = ⋂m∈ℕ+ C m (X). Given f , g ∈ C m (X), we have Leibniz’ formula: α 𝜕α (fg) = ∑ ( )𝜕β f 𝜕α−β g, β β≤α

|α| ≤ m.

(0.5)

0.2 Function spaces and Fourier transform

| 3

If we assume furthermore that X is convex, we have the Taylor’s formula: 1 α 𝜕 f (x)(y − x)α α! |α| d/2.

(0.12)

Finally, we recall that the space H s (ℝd ), for s > d/2, is an algebra with respect to pointwise multiplication, and the following Schauder estimates hold: ‖fg‖H s (ℝd ) ≤ Cs ‖f ‖H s (ℝd ) ‖g‖H s (ℝd ) ,

f , g ∈ H s (ℝd ).

(0.13)

0.2.1 Fundamental operators We introduce the fundamental operators in time–frequency analysis: the translation operator Tx of a function or distribution f on ℝd Tx f (t) = f (t − x),

t, x ∈ ℝd

and the modulation operator Mξ f (t) = e2πiξ ⋅t f (t),

ξ , t ∈ ℝd .

Translation and modulation operators satisfy the so-called commutation relations Tx Mξ = e−2πix⋅ξ Mξ Tx .

(0.14)

In fact, for any function/distribution f on ℝd , we can write Tx Mξ f (t) = e2πiξ ⋅(t−x) f (t − x) = e−2πiξ ⋅x e2πiξ ⋅t f (t − x) = e−2πiξ ⋅x Mξ Tx f (t).

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We denote a point in the time–frequency space ℝ2d as z = (x, ξ ) ∈ ℝ2d , and the corresponding phase-space shift (time–frequency shift) acting on a function or distribution as π(z)f (t) = Mξ Tx f (t) = e2πiξ ⋅t f (t − x).

(0.15)

Using the commutation relations it is straightforward to compute the adjoint π ∗ of the time–frequency shift π as π ∗ (z) = e−2πiz1 ⋅z2 π(−z),

z = (z1 , z2 ) ∈ ℝ2d .

(0.16)

Similarly, the composition of two time–frequency shifts is given by π(z)π(w) = e−2πiz1 ⋅w2 π(z + w),

z = (z1 , z2 ), w = (w1 , w2 ).

The reflection operator is given by ℐ f (t) = f (−t).

(0.17)

0.2 Function spaces and Fourier transform

| 7

We define the involution operator as ∀t ∈ ℝd .

f ∗ (t) = f (−t),

(0.18)

For λ > 0, the dilation operator is given by Dλ f (t) = λd/2 f (λt),

t ∈ ℝd .

(0.19)

Observe that ∀f ∈ L2 (ℝd ).

‖Dλ f ‖2 = ‖f ‖2 ,

The previous operators enjoy the following properties: – The operators Tx and Mξ are isometries on Lp (ℝd ), 1 ≤ p ≤ ∞. In particular, f ∈ Lp (ℝd ).

‖Mξ Tx f ‖p = ‖Tx Mξ f ‖p = ‖f ‖p , –

For f ∈ L1 (ℝd ), we have ∀ξ , h ∈ ℝd ,

̂f (ξ ) = M f ̂(ξ ), T h −h f̂∗ (ξ ) = f ̂(ξ ), ̂f (ξ ) = λ D λ

−d/2

(0.20)

d

̂f (ξ ) = T f ̂(ξ ), M η η

∀ξ , η ∈ ℝ ,

∀ξ ∈ ℝd , f ̂(ξ /λ),

(0.21) (0.22) d

∀ξ ∈ ℝ , λ > 0.

(0.23)

0.2.2 Gaussian functions In the following we shall use profusely rescaled Gaussian functions and their behavior 2 under convolution and Fourier transform. From now on, we set φ(t) = e−πt , t ∈ ℝd (recall t 2 = t ⋅ t). For a > 0, let us denote 2

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φ√a (t) = φ(√at) = e−πat ,

t ∈ ℝd .

(0.24)

Then, −d/2 ̂ φ φ1/√a (ξ ). √a (ξ ) = a

(0.25)

2 2 ? −πt 2 )(ξ ) = e−πξ , that is, the Gaussian e−πt is an eigenfunction In particular, for a = 1, (e of the Fourier transform. We recall the semigroup property of Gaussians: for a, b > 0,

φ1/√a ∗ φ1/√b (t) = (

d/2

ab ) a+b

φ1/√a+b (t),

t ∈ ℝ2d .

(0.26)

We recall from [145, page 257] the following well-known formula for Gaussian integrals.

8 | 0 Preliminaries Lemma 0.2.2. Let A be a d × d complex matrix such that A = A∗ and Re A is positive definite. Then for every z ∈ ℂn , ∫ e−πx⋅Ax−2πiz⋅x dx = (det A)−1/2 e−πz⋅A

−1

z

(0.27)

,

where the branch of the square root is determined requiring that (det A)−1/2 > 0 when A is real and positive definite. In particular, we have the following identity. 2

Lemma 0.2.3. For Re c ≥ 0, c ≠ 0. Let ϕ(c) (t) = e−πct , then ξ ∈ ℝd ,

̂ (c) (ξ ) = c−d/2 ϕ(1/c) (ξ ), ϕ

(0.28)

where the square root is chosen to have a positive real part. 2

Lemma 0.2.4. For c ∈ ℂ, c ≠ 0, consider the complex Gaussian function ϕ(c) (t) = e−πct , t ∈ ℝd . For every c1 , c2 ∈ ℂ, with Re c1 ≥ 0, Re c2 > 0, we have c c

( c 1+c2 )

ϕ(c1 ) ∗ ϕ(c2 ) = (c1 + c2 )−d/2 ϕ

1

2

(0.29)

.

Proof. Using equality (0.27), 2

2

(ϕ(c1 ) ∗ ϕ(c2 ) )(t) = ∫ e−πc1 (t−y) −πc2 y dt 2

= e−πc1 t ∫ e−π(c1 +c2 )y

2

2

−2πc1 t⋅y

= (c1 + c2 )−d/2 e−πc1 t e = (c1 + c2 )

−d/2

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as desired.

ϕ

πc12 t 2 1 +c2

−c

c c

( c 1+c2 ) 1

2

(t),

,

dy

1 Basics of time–frequency representations and related properties The first chapter of the book is devoted to the presentation of the basic tools of the time–frequency analysis, with emphasis on the short-time Fourier transform. To introduce the nonexpert readers to the subject, we shall start with a brief discussion on the so-called problem of the instantaneous frequencies. Namely, when considering the Fourier transform of a function or distribution f ,

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f ̂(ξ ) = ℱ f (ξ ) = ∫ f (t)e−2πit⋅ξ dt, and evaluating |f ̂(ξ )|, one obtains a precise description of the relevance of each frequency ξ in the Fourier decomposition. However, |f ̂(ξ )| does not give information on which instants t contributed to such frequency. Roughly speaking, we may have different oscillations at different points t, and their presence is revealed by |f ̂(ξ )|, but we cannot detect any more their origin in the t-space. Searching then for instantaneous frequencies, one is led to cut off f (t) at the point t = x of interest, and then take the Fourier transform of this restriction. The first problem of technical nature to this end is the definition of suitable cut-off functions φ(t). A natural choice of φ(t) is the characteristic function χQ (t) of a neighborhood Q of x, so that we have simply to perform the Fourier transform of the segment χQ (t)f (t) of f (t) near x. However, one realizes that the jump of χQ (t) at the boundary of Q causes the presence of unexpected frequencies in the Fourier transform χ̂ Q f (ξ ). The solution of the problem is to consider a more regular cut-off. As a curiosity, observe the use of a trapezoidal cut-off in the book of E. C. Titchmarsh [285], providing a Lipschitz φ(t). Nowadays we may, of course, refer to φ(t) in 𝒟(ℝd ), the space of smooth functions with compact support on ℝd , and the additional frequencies contributed by such cutoff functions are negligible. So the search for instantaneous frequencies seems successful by taking φ ∈ 𝒟(ℝd ) with very small support, to isolate the behavior of f (t) at x. At this moment we have to face a second problem: the Heisenberg uncertainty principle. There are several mathematical formulations of it. In the present context, let us limit ourselves to saying that by shrinking the support of φ(t) we loose precision in determining frequencies at x. Actually, one cannot beat the Heisenberg principle, but it is possible to compromise, by contenting oneself with a description of the asymptotic behavior of the frequency spectrum in suitable unbounded domains of ℝdξ . A milestone in this connection is given by the definition of wave front set (WF) by L. Hörmander [191]. Namely, we consider a distribution f in ℝd (or in a open subset of ℝd ), the more interesting case being now d > 1, with t representing a space variable rather than time. We say that (x, ξ0 ) ∉ WFf , ξ0 ≠ 0, if there exists φ ∈ 𝒟(ℝd ), φ(t) = 1 https://doi.org/10.1515/9783110532456-002

10 | 1 Basics of time–frequency representations and related properties for t in a neighborhood of x (it will be actually sufficient to assume φ(x) ≠ 0), and there exists a conic neighborhood Γξ0 of ξ0 in ℝdξ such that 󵄨󵄨 ̂ 󵄨󵄨 −N 󵄨󵄨φf (ξ )󵄨󵄨 ≤ CN ⟨ξ ⟩

for ξ ∈ Γξ0 ,

for every N > 0, and a suitable constant CN > 0. This definition had an enormous impact on the general theory of the partial differential equations, in particular, concerning propagation of singularities of solutions. In fact, the growth of the spectrum is responsible for a singularity of f at x, and we can use WFf to describe which high frequency directions are exactly causing the singularity. However, WFf does not give quantitative information on the spectrum of f and may turn out to be of limited interest in some applications. For example, in psychoacoustics, high frequencies are not audible! In time–frequency analysis, in particular, in the applications to signal processing, one prefers to bargain with Heisenberg principle in a different way, by considering the Gabor transform [152] 2

Vf (x, ξ ) = ∫ f (t)e−π|t−x| e−2πit⋅ξ dt,

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ℝd

and accepting Vf (x, ξ ) as a substitute for the instantaneous frequency spectrum at x. 2 One can raise immediately the objection that the Gaussian φ(t) = e−π|t−x| cannot be accepted as cut-off function since its support is not compact. On the other hand, the Gaussian decreases so rapidly at infinity that its numerical approximation vanishes outside a neighborhood of x. So the contributions to the spectrum from other points are not excluded in principle, but their value is extremely reduced. In favor of Gabor transform, with respect to the cut-off functions in 𝒟(ℝd ), we may say that the regularity properties of the Gaussian are optimal as it is an entire function, and symbolic computations are easier. Compared to the wave front set of Hörmander, the localization proceeding is not so precise, but relevant frequencies are satisfactorily identified, with possibility of practical applications. The short-time Fourier transform (STFT), which is a basic tool of the time– frequency analysis, is defined by replacing the Gaussian with a general window g(t), namely Vg f (x, ξ ) = ∫ f (t)g(t − x)e−2πit⋅ξ dt,

(x, ξ ) ∈ ℝ2d .

ℝd

We shall choose g ∈ 𝒮 (ℝd ), or g ∈ L2 (ℝd ), or even g belonging to more general function spaces. Our applications to partial differential equations will be essentially based on the STFT. We shall also produce, in the last chapter of the book, the counterpart of the WF of Hörmander, namely the so-called Gabor wave front set WFG , defined in terms of the STFT.

1 Basics of time–frequency representations and related properties | 11

Besides the STFT, in the present chapter we shall also consider other time– frequency representations, in particular, the Wigner distribution [314] Wf (x, ξ ) = ∫ f (x + t/2)f (x − t/2)e−2πit⋅ξ dt. ℝd

Because of the manifold applications, Wf (x, ξ ) is maybe the most popular time– frequency representation. It can be related easily to the STFT: Wf (x, ξ ) is roughly the STFT of f (t) with window given by f itself. This has the advantage of giving a selfcontained definition of the spectrum, without the help of the window g as mediator. More generally, one considers the Cohen representation [53] of f Qσ f = Wf ∗ σ, for some function or distribution σ. By convolving with suitable σ, one may obtain more flexible representations. Wigner distributions Wf will play an important role as well in the sequel of the book. Let us describe more precisely the contents of the next sections. In Section 1.1 we report on basic facts concerning symplectic matrices, metaplectic operators, and symplectic Fourier transform. These topics can be seen as natural complements of the definition of Fourier transform, and metaplectic operators will play an important role in the applications to partial differential equations. Section 1.2 is the core of the chapter. We introduce here the definition of the STFT Vg f and prove its properties. Attention is mainly fixed on the L2 (ℝd ) setting, with proof of the orthogonality relations and the analysis of the conjugation by metaplectic operators. The inversion formula is also first proved in L2 (ℝd ). By taking windows g ∈ 𝒮 (ℝd ), the properties of Vg f are then extended to the case of a distribution f ∈ 𝒮 󸀠 (ℝd ). Finally, a formula is proved connecting Vg1 f and Vg2 f for different windows g1 , g2 ; this will be frequently used in the sequel of the book. Section 1.3 addresses the analysis of the ambiguity function

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A(f , g)(x, ξ ) = eπix⋅ξ Vg f (x, ξ ). The classical properties of the Wigner distribution are then reviewed, in particular, Moyal’s identity ‖Wf ‖2 = ‖f ‖22 ,

f ∈ L2 (ℝd ),

the connection with metaplectic operators and the relations with STFT and ambiguity function. Cohen representations are also discussed in this section, extending the properties of the Wigner distribution. Finally, attention is given to the τ-Wigner distribution, τ ∈ [0, 1], defined by Wτ f (x, ξ ) = ∫ f (x + τt)f (x − (1 − τ)t)e−2πit⋅ξ dt. ℝd

12 | 1 Basics of time–frequency representations and related properties The τ-Wigner distributions can be seen as Cohen representations, for a suitable choice of the kernel σ. The final part of the section is devoted to the explicit computations of the STFT in the case when f or g is a Gaussian function. In conclusion, we observe that it turned out to be difficult for us to determine the priority of results, concerning the basics of time–frequency representations. For this reason, in this chapter we decided to omit precise references. We refer the reader to the textbooks [53, 105, 160, 183, 305] for a more complete bibliography.

1.1 Hints on symplectic and metaplectic groups 1.1.1 The symplectic group Let us first introduce the standard symplectic matrix J, defined by 0 J=( d −Id

Id ), 0d

(1.1)

where 0d is the d × d zero matrix and Id is the d × d identity matrix. Observe that J 2 = −I2d ,

J T = J −1 = −J

(the superscript T denotes the transposition). Definition 1.1.1. A bilinear form on ℝ2d is called a symplectic form if it is antisymmetric and nondegenerate. The antisymmetric bilinear form σ on ℝ2d defined by σ(z1 , z2 ) = ξ1 ⋅ x2 − ξ2 ⋅ x1 ,

z1 = (x1 , ξ1 ),

z2 = (x2 , ξ2 ) ∈ ℝ2d ,

(1.2)

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is symplectic; it is called the standard symplectic form on ℝ2d . To be precise, let us recall that antisymmetry means σ(z1 , z2 ) = −σ(z2 , z1 ), for every z1 , z2 ∈ ℝ2d , and nondegeneracy means σ(z0 , z) = 0

∀z ∈ ℝ2d ⇐⇒ z0 = 0.

The definition in (1.2) can be rewritten using the standard symplectic matrix σ(z1 , z2 ) = Jz1 ⋅ z2 = z2T Jz1 . Definition 1.1.2. The symplectic group Sp(d, ℝ) is the subgroup of 2d × 2d invertible matrices GL(2d, ℝ), defined by Sp(d, ℝ) = {𝒜 ∈ GL(2d, ℝ) : 𝒜T J 𝒜 = J}.

(1.3)

1.1 Hints on symplectic and metaplectic groups |

13

Observe that if 𝒜 satisfies the identity in (1.3), then also the transpose 𝒜T and the inverse 𝒜−1 fulfill the same identity, and so are symplectic matrices as well. Writing 𝒜 ∈ Sp(d, ℝ) in the following d × d block decomposition: A C

𝒜=(

B ), D

(1.4)

from (1.3) we have that the four blocks must satisfy the following conditions: DT A − BT C = Id ,

(1.5)

T

T

A C − C A = 0d ,

(1.6)

T

T

D B − B D = 0d .

(1.7)

Moreover, since also 𝒜

−1

DT −C T

−BT ), AT

=(

is a symplectic matrix, relations (1.6) and (1.7) for 𝒜−1 give CA−1 − A−T C T = 0d ,

(1.8)

−AB + BA = 0d .

(1.9)

T

T

Algebraic arguments give then the following two propositions. Proposition 1.1.3. The subsets L−1 D = {( 0d

0d ) : L ∈ GL(d, ℝ)} , LT

I N = {( d C

0d ) : C ∈ M(d, ℝ), C = C T } Id

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of GL(2d, ℝ) are subgroups of Sp(d, ℝ). Recall that a group G is generated by some elements g1 , . . . , gn ∈ G if every element of the group can be expressed as a combination (under the group operation) of finitely many elements gj , j = 1, . . . , n, and their inverses. Proposition 1.1.4. With the notation of the previous proposition, the symplectic group Sp(d, ℝ) is generated by D ∪ N ∪ {J}. Definition 1.1.5. The symplectic algebra sp(d, ℝ) is the set of all 2d × 2d real matrices 𝒜 such that et𝒜 ∈ Sp(d, ℝ) for all t ∈ ℝ. 1.1.2 The metaplectic group There are many constructions of the metaplectic group Mp(d, ℝ), doubly covering the symplectic group Sp(d, ℝ), and the metaplectic representation of Mp(d, ℝ) into

14 | 1 Basics of time–frequency representations and related properties 𝒰 (L2 (ℝd )), the group of unitary operators acting on L2 (ℝd ). In [145] G. B. Folland in-

troduced the metaplectic representation μ of Mp(d, ℝ) as an intertwining operator. In short, in terms of the time–frequency shift operator π(z) in (0.15), z = (x, ξ ) ∈ ℝ2d , for 𝒜 ∈ Sp(d, ℝ), one defines μ(𝒜) ∈ 𝒰 (L2 (ℝd )) by π(𝒜z) = c𝒜 μ(𝒜)π(z)μ(𝒜)−1

(1.10)

where c𝒜 , |c𝒜 | = 1, is a phase factor. To be precise, recall first that the Heisenberg group ℍd is the group obtained by defining on ℝ2d+1 the product law 1 (z, t) ⋅ (z 󸀠 , t 󸀠 ) = (z + z 󸀠 , t + t 󸀠 + σ(z, z 󸀠 )), 2

z, z 󸀠 ∈ ℝ2d , t, t 󸀠 ∈ ℝ,

where σ is the standard symplectic form defined in (1.2). The Schrödinger representation of the group ℍd on L2 (ℝd ) is then defined by ρ(q, p, t)f (x) = e2πit e−πip⋅q e2πip⋅x f (x − q),

x, q, p ∈ ℝd , t ∈ ℝ.

Observe that, for t = 0, we have ρ(q, p, 0)f (x) = e−πip⋅q e2πip⋅x f (x − q) = e−πip⋅q Mp Tq f (x) = e−πip⋅q π(z)f (x),

(1.11)

with z = (q, p) ∈ ℝ2d . The symplectic group acts on ℍd via automorphisms that leave the center {(0, t) : t ∈ ℝ} ⊂ ℍd of ℍd pointwise fixed A ⋅ (z, t) = (Az, t). Therefore for any fixed 𝒜 ∈ Sp(d, ℝ), there is a representation

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ρ𝒜 : ℍd → 𝒰 (L2 (ℝd )),

(z, t) �→ ρ(𝒜 ⋅ (z, t))

whose restriction to the center is a multiple of the identity. By the Stone–von Neumann theorem (cf., e. g., Theorem 9.3.1 in [160]), ρ𝒜 is equivalent to ρ. So there exists an intertwining unitary operator μ(𝒜) ∈ 𝒰 (L2 (ℝd )) such that ρ𝒜 (z, t) = μ(𝒜) ∘ ρ(z, t) ∘ μ(𝒜)−1

(z, t) ∈ ℍd .

(1.12)

By Schur’s lemma (cf., e. g., Lemma 9.3.2 in [160]), μ is determined up to a phase factor eis , s ∈ ℝ. Actually, the phase ambiguity is only a sign, so that μ lifts to a representation of the (double cover of the) symplectic group. This gives a precise meaning to (1.10). Another construction of the metaplectic group Mp(d, ℝ) was presented by M. de Gosson in [105]. Since the metaplectic representation turns out to be faithful (one-toone), the metaplectic group elements can be thought as unitary operators themselves.

1.1 Hints on symplectic and metaplectic groups | 15

Hence Mp(n, ℝ) is defined as a subgroup of the unitary group 𝒰 (L2 (ℝd )) and the corresponding representation is just the inclusion. A difficult point is to prove the existence of a projection π̃ : Mp(d, ℝ) → Sp(d, ℝ) which makes Mp(d, ℝ) the double covering of Sp(d, ℝ). We recall here the main points of the construction, and we refer to [105] for details. It can be proved that the symplectic group Sp(d, ℝ) is generated by the so-called free symplectic matrices A C

S=(

B ) ∈ Sp(d, ℝ), D

det B ≠ 0.

To each such matrix we associate the generating function 1 1 W(x, ξ ) = DB−1 x ⋅ x − B−1 x ⋅ ξ + B−1 Aξ ⋅ ξ . 2 2 Conversely, to every polynomial of the type 1 1 W(x, ξ ) = Px ⋅ x − Lx ⋅ ξ + Qξ ⋅ ξ 2 2 with P = P T , Q = QT and det L ≠ 0, we can associate a free symplectic matrix, namely

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L−1 Q SW = ( −1 PL Q − LT

L−1 ). PL−1

Now, given SW as above and m ∈ ℤ such that mπ ≡ arg det L mod 2π, we define the operator ŜW,m by setting, for ψ ∈ 𝒮 (ℝd ), 1 ŜW,m ψ(x) = d/2 Δ(W) ∫ e2πiW(x,ξ ) ψ(ξ ) dξ i ℝd

(to be clear, id/2 = eiπd/4 ) where Δ(W) = im √| det L|.

(1.13)

16 | 1 Basics of time–frequency representations and related properties The operator ŜW,m is called a quadratic Fourier transform associated to the free symplectic matrix SW . The class modulo 4 of the integer m is called Maslov index of ŜW,m . Observe that if m is one choice of Maslov index, then m + 2 is another equally good choice. Hence to each function W, we associate two operators, namely ŜW,m and ŜW,m+2 = −ŜW,m . The quadratic Fourier transform corresponding to the choices SW = J and m = 0 is denoted by ̂J. The generating function of J being simply W(x, ξ ) = −x ⋅ ξ , it follows that ̂Jψ(x) = 1 ∫ e−2πix⋅ξ ψ(ξ ) dξ = 1 ℱ ψ(x) id/2 n id/2

(1.14)

ℝ

for ψ ∈ 𝒮 (ℝn ), where ℱ is the usual Fourier transform. The quadratic Fourier transforms ŜW,m form a subset of the group 𝒰 (L2 (ℝd )) of unitary operators acting on L2 (ℝd ), which is closed under the operation of inversion, and they generate a subgroup of 𝒰 (L2 (ℝd )) which is, by definition, the metaplectic group Mp(d, ℝ). Every Ŝ ∈ Mp(d, ℝ) is thus, by definition, a product ŜW1 ,m1 ⋅ ⋅ ⋅ ŜWk ,mk of metaplectic operators associated to free symplectic matrices. In fact, it can be shown that every Ŝ ∈ Mp(d, ℝ) can be written as a product of only two quadratic Fourier transforms Ŝ = ŜW,m ŜW 󸀠 ,m󸀠 . Furthermore (cf. Theorem 114 in [105]), the map ŜW,m ��→ SW extends to a surjective group homomorphism

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π̃ : Mp(d, ℝ) → Sp(d, ℝ), with kernel ker π̃ = {−I, +I}. Hence π̃ is a twofold covering of the symplectic group. We also observe that each metaplectic operator is, by construction, a unitary operator in L2 (ℝd ), but also an automorphism of 𝒮 (ℝd ) and of 𝒮 󸀠 (ℝd ). Alternatively, Mp(d, ℝ) is generated by the following three type of operators (see Proposition 1.1.3 and [193]): ̂J in (1.14) and the operators L−1

μ ((

0d

0d

LT

)) f (x) = √| det L|f (Lx),

(1.15)

1.1 Hints on symplectic and metaplectic groups | 17

I μ (( d C

0d )) f (x) = eπiCx⋅x f (x). Id

(1.16)

Another definition of metaplectic operators is also presented in [284]. This involves a time–frequency representation, the so-called Wigner distribution Wf of a function f ∈ L2 (ℝd ) defined in the following Section 1.3.2. 1.1.3 The symplectic Fourier transform We recall here the notion and main properties of the symplectic Fourier transform for function/distributions defined on the phase space ℝ2d . Definition 1.1.6. The symplectic Fourier transform ℱσ is defined, for a ∈ 𝒮 (ℝ2d ), by ℱσ a(z) = ∫ e

−2πiσ(z,z 󸀠 )

a(z 󸀠 ) dz 󸀠 .

(1.17)

ℝ2d

We will often use the shorthand notation aσ = ℱσ a. We list here the main properties of the operator ℱσ . Proposition 1.1.7. We have (i) The Fourier transform and symplectic Fourier transform are related by ℱσ a(z) = ℱ a(Jz) = ℱ (a ∘ J)(z),

(1.18)

for every a ∈ 𝒮 (ℝ2d ). (ii) The symplectic Fourier transform ℱσ is a topological isomorphism from 𝒮 (ℝ2d ) onto itself, which extends by duality to a topological isomorphism on 𝒮 󸀠 (ℝ2d ). (iii) ℱσ is involutive, that is, ℱσ2 = I. (iv) For every a ∈ L2 (ℝ2d ), ℱσ is a unitary operator on L2 (ℝ2d )

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‖ℱσ a‖L2 (ℝ2d ) = ‖a‖L2 (ℝ2d ) .

(1.19)

(v) We have ℱσ (a ∗ b) = ℱσ a ⋅ ℱσ b,

a, b ∈ 𝒮 (ℝ2d ).

(1.20)

Proof. (i) Using the definition of the standard symplectic form σ(z, z 󸀠 ) = Jz ⋅ z 󸀠 , we can write, for every a ∈ 𝒮 (ℝ2d ), ℱσ a(z) = ∫ e

−2πiJz⋅z 󸀠

a(z 󸀠 ) dz 󸀠 = ℱ a(Jz).

ℝ2d

The second equality in (1.18) follows by observing that Jz ⋅ z 󸀠 = z ⋅ J −1 z 󸀠 and performing the change of variables J −1 z 󸀠 = w (dz 󸀠 = dw) in the above integral.

18 | 1 Basics of time–frequency representations and related properties (ii) It immediately follows by the previous claim since ℱ : 𝒮 (ℝ2d ) → 𝒮 (ℝ2d ) is a topological isomorphism which extends by duality on 𝒮 󸀠 (ℝ2d ). (iii) Using (i), ℱσ2 a(z) = ℱ (ℱ a)(−z) = a(z), for every z ∈ ℝ2d and a ∈ 𝒮 (ℝ2d ). (iv) It follows by (i) and the fact that ℱ is a unitary operator on L2 (ℝ2d ). (v) It is a consequence of (i). From the previous property (iii), we have that the symplectic Fourier transform satisfies ℱσ−1 = ℱσ , hence it coincides with its own inverse. Let us study the behavior of ℱσ under the action of the symplectic group. Proposition 1.1.8. For a ∈ 𝒮 󸀠 (ℝ2d ), 𝒜 ∈ Sp(d, ℝ), we have ℱσ a(𝒜z) = ℱσ (a ∘ 𝒜)(z),

z ∈ ℝ2d .

(1.21)

Proof. It is enough to consider a ∈ 𝒮 (ℝ2d ) and then extend the result by duality. Using the fact that symplectic matrices leave invariant the standard symplectic form, we can write σ(𝒜z, z 󸀠 ) = σ(𝒜𝒜−1 z, 𝒜−1 z 󸀠 ) = σ(z, 𝒜−1 z 󸀠 ), then making the change of variables 𝒜−1 z 󸀠 = w in the integral expression of ℱσ a(𝒜z) (observe d𝒜z 󸀠 = dw), we immediately obtain (1.21).

1.2 Time–frequency representations Many results presented in this section are borrowed from the textbook [160].

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1.2.1 The short-time Fourier transform In what follows we often will call a function f ∈ L2 (ℝd ) a signal, according to the language of signal analysis. In order to obtain information about the local frequency spectrum of a signal f , we restrict f to an interval centered at the instant x which is the object of interest and then take the Fourier transform of this restriction. Such a localization in time is made by multiplying f with a smooth cut-off function g, called the window function. We shall work mainly with g ∈ 𝒮 (ℝd ). This is the idea under the construction of the short-time Fourier transform. Definition 1.2.1. Fix a window function g ∈ L2 (ℝd ) \ {0}. The short-time Fourier transform (STFT) of a function f ∈ L2 (ℝd ) with respect to g is defined by Vg f (x, ξ ) = ⟨f , Mξ Tx g⟩ = ∫ f (t)g(t − x)e−2πit⋅ξ dt, ℝd

x, ξ ∈ ℝd .

(1.22)

1.2 Time–frequency representations | 19

Remark 1.2.2. Assume that the window g is a compactly supported function centered at the origin; then Vg f (x, ⋅) involves the Fourier transform of a segment of f centered around x. As x varies, the window g slides along the x-axis to different positions, and Vg f (x, ⋅) is a substitute for the instantaneous frequency spectrum at x (impossible to be determined, by the uncertainty principle!) We can rewrite expression (1.22) in terms of the time–frequency shift π(z) (cf. (0.15)), z = (x, ξ ) ∈ ℝ2d , as follows: z ∈ ℝ2d .

Vg f (z) = ⟨f , π(z)g⟩,

(1.23)

Consider f ∈ Lp (ℝd ), g ∈ Lp (ℝd ), then by Hölder’s inequality, the integral in (1.22) is absolutely convergent and the STFT is well defined, see the subsequent Proposition 1.2.10. Easy computations yield the following equivalent formulae for the STFT: 󸀠

̄ ) Vg f (x, ξ ) = (f? ⋅ Tx g)(ξ

(1.24)

= ⟨f ̂, Tξ M−x g⟩̂

(1.25)

? ̄̂ = e−2πix⋅ξ (f ̂ ⋅ Tξ g)(−x)

(1.26)

=e

(1.27)

−2πix⋅ξ

Vĝ f ̂(ξ , −x)

= e−2πix⋅ξ (f ∗ Mξ g ∗ )(x)

(1.28)

= (f ̂ ∗ M−x ĝ ∗ )(ξ )

(1.29)

x x ̄ − )e−2πit⋅ξ dt. = e−πix⋅ξ ∫ f (t + )g(t 2 2

(1.30)

ℝd

In particular, formula (1.27), Vg f (x, ξ ) = e−2πix⋅ξ Vĝ f ̂(ξ , −x),

(1.31)

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is called the fundamental identity of time–frequency analysis. In fact, it combines both f and f ̂ into a joint time–frequency representation. Since ξ 0 ( )=( d −x −Id

Id x x )( ) = J ( ), 0d ξ ξ

formula (1.31) can be rewritten using the standard symplectic matrix as Vg f (z) = e−2πix⋅ξ Vĝ f ̂(Jz),

z = (x, ξ ) ∈ ℝ2d .

In this identity the switch to Fourier transforms corresponds to a rotation of the time– frequency plane by an angle of π/2. Lemma 1.2.3 (Switching f and g). For f , g ∈ L2 (ℝd ), we have Vf g(x, ξ ) = e−2πix⋅ξ Vg f (−x, −ξ ).

(1.32)

20 | 1 Basics of time–frequency representations and related properties Proof. It is a straightforward computation. In details Vf g(x, ξ ) = ⟨f , Mξ Tx g⟩ = ⟨Mξ Tx g, f ⟩ = ⟨g, T−x M−ξ f ⟩ = e−2πix⋅ξ ⟨g, M−ξ T−x f ⟩. Another easy computation gives the relation below. Lemma 1.2.4 (Fourier transform of the STFT). For f , g ∈ L2 (ℝd ), we have ̄̂ ̂f (y, η) = e2πiy⋅η f (−η)g(y). V g

(1.33)

Remark 1.2.5. The STFT is linear in f and conjugate-linear in g. Keeping the window g fixed, we have that Vg : f �→ Vg f maps functions f on ℝd to functions Vg f on ℝ2d . We shall now exhibit the basic properties of the STFT. To reach this goal, we first establish some features of the translation operator Th . Theorem 1.2.6. If 1 ≤ p < ∞, h ∈ ℝd , then lim ‖Th f − f ‖p = 0,

h→0

∀f ∈ Lp (ℝd ).

(1.34)

Proof. (First step) We prove (1.34) when f ∈ 𝒞c (ℝd ). We can limit ourselves to the case h ∈ ℝd , |h| ≤ 1. The set F = {supp f + h, |h| ≤ 1} is bounded since supp f is compact, hence there exists a compact set K ⊂ ℝd such that F ⊂ K. Of course, supp Th f = supp f + h, hence 󵄨 󵄨p ‖Th f − f ‖pp = ∫󵄨󵄨󵄨Th f (t) − f (t)󵄨󵄨󵄨 dt Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

K

p

󵄨 󵄨 ≤ (sup󵄨󵄨󵄨f (t − h) − f (t)󵄨󵄨󵄨) meas(K). t∈K

The function f is continuous on the compact set K, hence uniformly continuous on K by the Heine–Cantor’s theorem, that is, 󵄨 󵄨 lim sup󵄨󵄨󵄨f (t − h) − f (t)󵄨󵄨󵄨 = 0,

h→0 t∈K

and it follows limh→0 ‖Th f − f ‖pp = 0.

(Second step) Consider f ∈ Lp (ℝd ). We shall prove ∀ϵ > 0 ∃δ > 0,

∀h ∈ ℝd , |h| < δ �⇒ ‖Th f − f ‖p < ϵ.

1.2 Time–frequency representations | 21

Given ϵ > 0, by the density of 𝒞c (ℝd ) in Lp (ℝd ), there exists g ∈ 𝒞c (ℝd ) such that ‖f −g‖p < ϵ/3. By the first step, there exists a δ > 0 such that ∀h, |h| < δ, ‖Th g−g‖p < ϵ/3. Then ‖Th f − f ‖p = ‖Th f − Th g + Th g − g + g − f ‖p

≤ ‖Th f − Th g‖p + ‖Th g − g‖p + ‖g − f ‖p = 2‖f − g‖p + ‖Th g − g‖p ϵ ϵ < 2 + = ϵ, 3 3

for every h ∈ ℝd with |h| < δ. Remark 1.2.7. The function h ∈ ℝd ,

ωp (h) = ‖Th f − f ‖p ,

is often called the Lp -modulus of continuity of f . If f ∈ L∞ (ℝd ), then f satisfies (1.34) with p = ∞ if and only if f is uniformly continuous, up to a null set. In particular, lim ‖Th f − f ‖∞ = 0,

h→0

∀f ∈ 𝒞0 (ℝd ).

(1.35)

Corollary 1.2.8. Consider h ∈ ℝd . Then lim ‖Mh f − f ‖2 = 0,

h→0

∀f ∈ L2 (ℝd ).

Proof. By Plancherel identity, ‖Mh f − f ‖2 = ‖Th f ̂ − f ̂‖2 , and the claim is a consequence of Theorem 1.2.6 for p = 2. Remark 1.2.9. If we consider the linear mapping

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τ : ℝd → ℒ(Lp (ℝd )),

τ : h �→ Th

(1.36)

for 1 ≤ p < ∞, we have lim ‖Th f − Ts f ‖p = 0,

h→s

∀f ∈ Lp (ℝd ),

that is, τ(h) → τ(s) in the strong operator topology when h → s. This follows by (1.34), with the change of variables h → h − s (hence h → 0 ⇐⇒ h → s), and by 󵄩 󵄩 ‖Th−s f − f ‖p = 󵄩󵄩󵄩Ts (Th−s f − f )󵄩󵄩󵄩p = ‖Th f − Ts f ‖p . We may therefore say that {Th }h∈ℝd is a strongly continuous one-parameter family of operators on Lp (ℝd ), 1 ≤ p < ∞. Analogously, {Th }h∈ℝd is a strongly continuous oneparameter family of operators on 𝒞0 (ℝd ).

22 | 1 Basics of time–frequency representations and related properties The information given by Corollary 1.2.8 extends easily to Lp (ℝd ), namely the same arguments as in Theorem 1.2.6 yield lim ‖Mh f − f ‖p = 0,

h→0

∀f ∈ Lp (ℝd ), 1 ≤ p < ∞

(1.37)

∀f ∈ 𝒞0 (ℝd ).

(1.38)

and lim ‖Mh f − f ‖∞ = 0,

h→0

Hence {Mh }h∈ℝd is a strongly continuous one-parameter family of operators on Lp (ℝd ), 1 ≤ p < ∞, and on 𝒞0 (ℝd ). Proposition 1.2.10. Consider 1 < p < ∞. If f ∈ Lp (ℝd ), g ∈ Lp (ℝd ), with 1/p + 1/p󸀠 = 1, then Vg f is uniformly continuous on ℝ2d . The same claim is valid for f ∈ L1 (ℝd ), g ∈ 󸀠

L∞ (ℝd ) and uniformly continuous on ℝd , or vice versa. Moreover, we have ‖Vg f ‖∞ ≤ ‖f ‖p ‖g‖p󸀠 .

(1.39)

Proof. Consider f ∈ Lp (ℝd ), g ∈ Lp (ℝd ), 1 ≤ p ≤ ∞. By Hölder’s inequality, 󸀠

󵄨󵄨 󵄨 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 ≤ ‖f ‖p ‖Mξ Tx g‖p󸀠 = ‖f ‖p ‖g‖p󸀠 so that Vg f is defined everywhere on ℝ2d and we obtain (1.39). Let us check that Vg f

is uniformly continuous for 1 < p < ∞. We assume f ∈ Lp (ℝd ) \ {0}, g ∈ Lp (ℝd ) \ {0}, otherwise Vg f (x, ξ ) = 0 and we are done. For h = (h1 , h2 ) ∈ ℝ2d ,

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󸀠

󵄨󵄨 󵄨 󵄨󵄨Vg f (x − h1 , ξ − h2 ) − Vg f (x, ξ )󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨Vg f (x − h1 , ξ − h2 ) − Vg f (x − h1 , ξ ) + Vg f (x − h1 , ξ ) − Vg f (x, ξ )󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨⟨f , Mξ −h2 Tx−h1 g⟩ − ⟨f , Mξ Tx−h1 g⟩ + ⟨f , Mξ Tx−h1 g⟩ − ⟨f , Mξ Tx g⟩󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨⟨Mh2 f − f , Mξ Tx−h1 g⟩ + ⟨M−ξ f , (Tx−h1 g − Tx g)⟩󵄨󵄨󵄨 󵄩 󵄩 ≤ ‖Mh2 f − f ‖p ‖Mξ Tx−h1 g‖p󸀠 + ‖M−ξ f ‖p 󵄩󵄩󵄩Tx (T−h1 g − g)󵄩󵄩󵄩p󸀠 = ‖Mh2 f − f ‖p ‖g‖p󸀠 + ‖f ‖p ‖T−h1 g − g‖p󸀠 . Now, take ϵ > 0. By Theorem 1.2.6, there exists δ1 > 0 such that, if |h1 | < δ, ‖T−h1 g − g‖p󸀠 < ϵ/(2‖f ‖p ), and analogously, by (1.37), there exists δ2 > 0 such that if |h2 | < δ2 , then ‖Mh2 f − f ‖p < ϵ/(2‖g‖p󸀠 ). Taking δ = min{δ1 , δ2 }, if |h| < δ, we have the claim. The proof in the case p = 1 and p󸀠 = ∞, with g uniformly continuous (and vice versa) is analogous. As we shall see in the next sections, the domain of the STFT can be extended to other function spaces. Indeed, the formula Vg f (x, ξ ) = ⟨f , Mξ Tx g⟩

(1.40)

1.2 Time–frequency representations | 23

makes Vg f well defined whenever f and g belong to any pair of dual spaces, extending the duality of L2 (ℝd ), namely linear in the first component and (conjugate)-linear in the second. We need, of course, the continuity of the time–frequency shifts in such spaces. For example, we can consider f ∈ 𝒮 󸀠 (ℝd ) and a window function g ∈ 𝒮 (ℝd ). We will see below that the time–frequency shifts are continuous on 𝒮 (ℝd ) and on 𝒮 󸀠 (ℝd ). Moreover, the STFT can be defined for both f , g ∈ 𝒮 󸀠 (ℝd ). This is a consequence of interpreting the STFT as a sesquilinear form (f , g) �→ Vg f . Define f ⊗ g to be the tensor product x, y ∈ ℝd .

f ⊗ g(x, y) = f (x)g(y),

Let F(x, y) be a function on ℝ2d . Consider the linear operator Ta (change of coordinates), defined by x, y ∈ ℝd .

Ta F(x, y) = F(y, y − x)

(1.41)

Let ℱ2 be the partial Fourier transform with respect to the second variable y: ℱ2 F(x, ξ ) = ∫ e

−2πiy⋅ξ

F(x, y) dy.

(1.42)

ℝd

Using these operators in the definition (1.22), we obtain another expression for the STFT; for f , g ∈ L2 (ℝd ), ̄ Vg f = ℱ2 Ta (f ⊗ g).

(1.43)

The claim is obtained by writing down explicitly formula (1.24). Indeed, ̄ − x) dy Vg f (x, ξ ) = ∫ e−2πiy⋅ξ f (y)g(y ℝd

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̄ = ∫ e−2πiy⋅ξ Ta (f ⊗ g)(x, y) dy ℝd

̄ = ℱ2 Ta (f ⊗ g)(x, ξ ). Thanks to (1.43), we can extend the domain of STFT to the space of tempered distributions 𝒮 󸀠 (ℝd ). Let us first recall a few definitions. The complex conjugate f ∈ 𝒮 󸀠 (ℝd ) of a distribution f ∈ 𝒮 󸀠 (ℝd ) is defined by ⟨f , φ⟩ = ⟨f , φ⟩,

φ ∈ 𝒮 (ℝd ).

The tensor product of functions f , g ∈ 𝒮 󸀠 (ℝd ) is a distribution in 𝒮 󸀠 (ℝ2d ) defined by the rule ⟨f ⊗ g, Φ⟩ = ⟨fx , ⟨g, Φ(x, ⋅)⟩⟩,

Φ ∈ 𝒮 (ℝ2d ),

24 | 1 Basics of time–frequency representations and related properties where fx indicates that f acts only on variable x. In particular, f ⊗ g is the unique tempered distribution such that ⟨f ⊗ g, φ1 ⊗ φ2 ⟩ = ⟨f , φ1 ⟩⟨g, φ2 ⟩, for every φ1 , φ2 ∈ 𝒮 (ℝd ). Now, it is easy to check that both operators Ta and ℱ2 are topological isomorphisms on the Schwartz class 𝒮 (ℝ2d ) and on its dual space 𝒮 󸀠 (ℝ2d ). Therefore if f , g ∈ 𝒮 󸀠 (ℝd ), the tensor product f ⊗ g is a well-defined distribution in 𝒮 󸀠 (ℝ2d ), and consequently Vg f = ℱ2 Ta (f ⊗ g) ∈ 𝒮 󸀠 (ℝ2d ). We note that when f and g are tempered distributions, the STFT Vg f is no longer a function but a tempered distribution in 𝒮 󸀠 (ℝ2d ).

1.2.2 Orthogonality relations The orthogonality relations for the STFT correspond to Parseval’s formula for the Fourier transform, which is indeed the key ingredient to obtain such a result. Theorem 1.2.11 (Orthogonality relations for the STFT). Consider fi , gi ∈ L2 (ℝd ), i = 1, 2. Then Vgi fi ∈ L2 (ℝ2d ) and ⟨Vg1 f1 , Vg2 f2 ⟩L2 (ℝ2d ) = ⟨f1 , f2 ⟩L2 (ℝd ) ⟨g1 , g2 ⟩L2 (ℝd ) .

(1.44)

Proof. Using that the operators ℱ2 and Ta are unitary on L2 (ℝd ), we can write ⟨Vg1 f1 , Vg2 f2 ⟩ = ⟨ℱ2 Ta (f1 ⊗ g1̄ ), ℱ2 Ta (f2 ⊗ g2̄ )⟩ = ⟨f1 ⊗ g1̄ , f2 ⊗ g2̄ ⟩

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= ⟨f1 , f2 ⟩⟨g1̄ , g2̄ ⟩, yielding the claim. Corollary 1.2.12. For f , g ∈ L2 (ℝd ), we have

‖Vg f ‖2 = ‖f ‖2 ‖g‖2 .

(1.45)

In particular, if ‖g‖2 = 1, the STFT becomes an isometry from L2 (ℝd ) into L2 (ℝ2d ) ‖Vg f ‖2 = ‖f ‖2 ,

∀f ∈ L2 (ℝd ).

(1.46)

Observe that Vg is not onto. Indeed, recall from Proposition 1.2.10 that Vg f is a continuous function, whereas in L2 (ℝ2d ) there are functions which are not continuous.

1.2 Time–frequency representations | 25

1.2.3 Short-time Fourier transform and symplectic matrices We use the orthogonality relations in (1.44) to prove the following properties of the STFT. Proposition 1.2.13. For f , fi , g, gi ∈ L2 (ℝd ), i = 1, 2, we have (i) If w ∈ ℝ2d , Vπ(w)g (π(w)f )(z) = e−2πiσ(z,w) Vg f (z),

z ∈ ℝ2d

(1.47)

where σ is the standard symplectic form and π(w) is the time–frequency shift of w ∈ ℝ2d . (ii) Fourier transform of a product of STFTs: ℱ (Vg1 f1 Vg2 f2 )(z) = (Vf2 f1 Vg2 g1 )(−Jz),

z ∈ ℝ2d ,

(1.48)

where J is the standard symplectic matrix. Proof. (i) It is a straightforward computation. Indeed, Vπ(w)g (π(w)f )(z) = ⟨π(w)f , π(z)π(w)g⟩ = ⟨f , π ∗ (w)π(z)π(w)g⟩ = e2πiw1 ⋅w2 ⟨f , π(−w)π(z)π(w)g⟩

= e2πiw1 ⋅w2 e2πiz1 ⋅w2 ⟨f , π(−w)π(z + w)g⟩ = e−2πiσ(z,w) ⟨f , π(z)g⟩,

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which gives (1.47). (ii) Observe that σ(w, −Jz) = −z ⋅ w, so that using (1.47) and the orthogonality relations (1.44) gives ℱ (Vg1 f1 Vg2 f2 )(z) = ∫ Vg1 f1 (w)Vg2 f2 (w)e

2πiσ(w,−Jz)

dw

ℝ2d

= ⟨Vg1 f1 , Vπ(−Jz)g2 π(−Jz)f2 ⟩

= ⟨f1 , π(−Jz)f2 ⟩⟨g1 , Vπ(−Jz)g2 ⟩,

as desired. Proposition 1.2.14 (Symplectic covariance formula for the STFT). Consider f , g ∈ ̂ with projection 𝒜 ∈ Sp(d, ℝ). For z = (x, ξ ) ∈ ℝ2d , L2 (ℝd ), the metaplectic operator 𝒜 we set 𝒜−1 z = z 󸀠 = (x󸀠 , ξ 󸀠 ). Then πi(x ⋅ξ ̂ V𝒜g ̂ (𝒜f )(z) = e 󸀠

󸀠

−x⋅ξ )

Vg f (𝒜−1 z).

(1.49)

26 | 1 Basics of time–frequency representations and related properties Proof. Using the intertwining property of the metaplectic representation with respect to the Schrödinger one (1.12) and the relation between time–frequency shifts and Schrödinger representation (1.11), we calculate ̂ ̂ ̂ ̂−1 ̂ V𝒜g ̂ (𝒜f )(z) = ⟨𝒜f , π(z)𝒜g⟩ = ⟨f , 𝒜 π(z)𝒜g⟩ ̂ −1 ρ(z, 0)𝒜 ̂g⟩ = e−πix⋅ξ ⟨f , ρ(𝒜−1 z, 0)g⟩ = e−πix⋅ξ ⟨f , 𝒜 = eπi(x ⋅ξ 󸀠

󸀠

−x⋅ξ )

⟨f , π(𝒜−1 z)g⟩ = eπi(x ⋅ξ 󸀠

󸀠

−x⋅ξ )

Vg f (𝒜−1 z).

This concludes the proof. Let us see the action of the STFT on a time–frequency shift. Proposition 1.2.15. Whenever Vg f is defined, we have Vg (Mv Tu f )(x, ξ ) = e−2πiu⋅(ξ −v) Vg f (x − u, ξ − v),

u, v, x, ξ ∈ ℝd .

(1.50)

In particular, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Vg (π(z)f )(w)󵄨󵄨󵄨 = 󵄨󵄨󵄨Tz Vg f (w)󵄨󵄨󵄨,

z, w ∈ ℝ2d .

(1.51)

Proof. Using that Mv and Tu are unitary operators, we can write Vg (Mv Tu f )(x, ξ ) = ⟨Mv Tu f , Mξ Tx g⟩ = ⟨f , T−u M−v Mξ Tx g⟩ = ⟨f , T−u Mξ −v Tx g⟩ = e−2πiu⋅(ξ −v) ⟨f , Mξ −v Tx−u g⟩, proving the claim. The formula in (1.50) is also known as the covariance property of the STFT.

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1.2.4 Reproducing formula for the short-time Fourier transform In what follows we are going to show that every function in L2 (ℝd ) can be represented by means of the STFT. Namely, once we have computed the STFT of a signal f , we can go back to the original signal f . Let us first give the definition of vector-valued integrals, here understood in a weak sense. If g : ℝd → B, where B is a Banach space, then f = ∫ g(x) dx ℝd

means that ℓ(h) = ⟨f , h⟩ = ∫ ⟨g(x), h⟩ dx, ℝd

∀h ∈ B∗ .

1.2 Time–frequency representations | 27

If the mapping ℓ : B∗ → ℂ is a bounded (conjugate)-linear functional on B∗ , then ℓ defines a unique element f ∈ B∗∗ , the bidual of B. We shall work with reflexive Banach spaces, so that B∗∗ = B. Relevant choice will be B = L2 (ℝd ). The vector-valued integrals object of our study are given by superpositions of time–frequency shifts f = ∫ F(x, ξ )Mξ Tx g dx dξ .

(1.52)

ℝ2d

Assuming g ∈ L2 (ℝd ), we have also Mξ Tx g ∈ L2 (ℝd ), for every (x, ξ ) ∈ ℝ2d . If we choose F ∈ L2 (ℝ2d ), the (conjugate)-linear functional ℓ(h) = ∫ F(x, ξ )⟨h, Mξ Tx g⟩ dx dξ

(1.53)

ℝ2d

is bounded on L2 (ℝd ). Indeed, by Cauchy–Schwarz inequality and the orthogonality relations in (1.46), we estimate 󵄨󵄨 󵄨 󵄨󵄨ℓ(h)󵄨󵄨󵄨 ≤ ‖F‖2 ‖Vg h‖2 = ‖F‖2 ‖h‖2 ‖g‖2 ,

∀h ∈ L2 (ℝd ).

(1.54)

Hence ℓ defines a unique function f = ∫ F(x, ξ )Mξ Tx g dx dξ ∈ L2 (ℝd ), ℝ2d

with norm ‖f ‖2 =

sup

h∈L2 ,‖h‖

2 =1

󵄨󵄨 󵄨 󵄨󵄨ℓ(h)󵄨󵄨󵄨 ≤ ‖F‖2 ‖g‖2 ,

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satisfying ℓ(h) = ⟨f , h⟩. Theorem 1.2.16 (Inversion formula for the STFT). Assume g, γ ∈ L2 (ℝd ) with ⟨g, γ⟩ ≠ 0. Then for all f ∈ L2 (ℝd ), f =

1 ∫ Vg f (x, ξ )Mξ Tx γ dx dξ . ⟨γ, g⟩

(1.55)

ℝ2d

Proof. By Theorem 1.2.11, the STFT Vg f is in L2 (ℝ2d ). This implies that the vectorvalued integral f̃ =

1 ∫ Vg f (x, ξ )Mξ Tx γ dx dξ ⟨γ, g⟩ ℝ2d

28 | 1 Basics of time–frequency representations and related properties is a well-defined function f ̃ ∈ L2 (ℝd ). It remains to prove that f ̃ = f . Using the orthogonality relations (1.44), for every h ∈ L2 (ℝd ), ⟨f ̃, h⟩ =

1 ∫ Vg f (x, ξ )⟨Mξ Tx γ, h⟩ dx dξ ⟨γ, g⟩ ℝ2d

=

1 ∫ Vg f (x, ξ )Vγ h(x, ξ ) dx dξ ⟨γ, g⟩ ℝ2d

= ⟨f , h⟩ hence f = f ̃, as desired. The adjoint of the short-time Fourier transform. Fix g ∈ L2 (ℝd ) and consider the linear operator Ag defined by Ag F = ∫ F(x, ξ )Mξ Tx g dx dξ . ℝ2d

By (1.54), the operator Ag is a bounded operator from L2 (ℝ2d ) onto L2 (ℝd ). Moreover, Ag = Vg∗ , where Vg : L2 (ℝd ) → L2 (ℝ2d ). Indeed, ⟨Ag F, h⟩ = ∫ F(x, ξ )⟨Mξ Tx g, h⟩ dx dξ ℝ2d

= ⟨F, Vg h⟩ = ⟨Vg∗ F, h⟩ for every h ∈ L2 (ℝd ), F ∈ L2 (ℝ2d ). The inversion formula (1.55) can be then rephrased as

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f =

1 V ∗V f , ⟨γ, g⟩ γ g

f ∈ L2 (ℝd ).

(1.56)

1.2.5 Short-time Fourier transform of tempered distributions We come now to the study of the STFT on the Schwartz class and on the space of tempered distributions. In what follows we will make heavily use of the Schwartz seminorms, for which there are equivalent definitions. Namely, for m, n ∈ ℕ, the Schwartz seminorms can be defined, for f ∈ 𝒮 (ℝd ), by ‖f ‖m,n =

∑

󵄨 󵄨 sup󵄨󵄨󵄨t α 𝜕β f (t)󵄨󵄨󵄨 < ∞.

d |α|≤m,|β|≤n t∈ℝ

(1.57)

1.2 Time–frequency representations | 29

The linear space 𝒮 (ℝd ), endowed with the family of seminorms {‖ ⋅ ‖m,n }, m, n ∈ ℕ, is a Fréchet space. Other equivalent families of seminorms are provided by 󵄩 󵄨 󵄩 󵄨 pα,β (f ) = sup󵄨󵄨󵄨t α 𝜕β f (t)󵄨󵄨󵄨 = 󵄩󵄩󵄩t α 𝜕β f 󵄩󵄩󵄩∞ , t∈ℝd

α, β ∈ ℕd ,

or 󵄩 󵄩 ‖f ‖k = ∑ 󵄩󵄩󵄩(1 + |t|k )𝜕α f 󵄩󵄩󵄩∞ ,

k ∈ ℕ.

(1.58)

|α|≤k

The following result shows how to exchange differential and power operators with time–frequency shifts. Lemma 1.2.17. For g ∈ 𝒮 (ℝd ), for every α, β ∈ ℕd , we have t α 𝜕β (Mξ Tx g) =

α β ∑ ( )( )(2πiξ )γ x δ Mξ Tx (t α−δ 𝜕β−γ g). δ γ δ≤α,γ≤β

(1.59)

Proof. Using α t α = (t − x + x)α = ∑ ( )x δ (t − x)α−δ δ δ≤α and Leibniz’ formula (0.5) β β 𝜕t (e2πiξ ⋅t g(t − x)) = ∑ ( )(2πiξ )γ e2πiξ ⋅t (𝜕β−γ g)(t − x), γ γ≤β we immediately obtain (1.59). Corollary 1.2.18. The operator-valued map

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(x, ξ ) �→ Mξ Tx is strongly continuous on 𝒮 (ℝd ) and 𝒮 󸀠 (ℝd ) (with the weak∗ -topology). Proof. For φ ∈ 𝒮 (ℝd ), we have to show that 󵄩 󵄩 lim 󵄩󵄩󵄩t α 𝜕β (Mξ Tx φ − φ)󵄩󵄩󵄩∞ = 0,

|x|,|ξ |→0

∀α, β ∈ ℕ.

Using (1.59), 󵄩󵄩 α β 󵄩 󵄩 α β α β 󵄩 󵄩󵄩t 𝜕 (Mξ Tx φ − φ)󵄩󵄩󵄩∞ ≤ 󵄩󵄩󵄩Mξ Tx (t 𝜕 φ) − t 𝜕 φ󵄩󵄩󵄩∞ α β 󵄨 󵄨󵄩 󵄩 + ∑ ( )( )󵄨󵄨󵄨(2πiξ )γ x δ 󵄨󵄨󵄨󵄩󵄩󵄩Mξ Tx (t α−δ 𝜕β−γ g)󵄩󵄩󵄩∞ . δ γ δ≤α,γ≤β, (δ,γ)=(0,0) ̸

(1.60)

30 | 1 Basics of time–frequency representations and related properties Observe that t α 𝜕β φ ∈ 𝒮 (ℝd ) ⊂ 𝒞0 (ℝd ), hence 󵄩 󵄩 lim 󵄩󵄩󵄩Mξ Tx (t α 𝜕β φ) − t α 𝜕β φ󵄩󵄩󵄩∞ = 0,

|x|,|ξ |→0

∀α, β ∈ ℕ,

in view of (1.35) and (1.38). Obviously, 󵄩 󵄨 󵄨󵄩 lim 󵄨󵄨󵄨(2πiξ )γ xδ 󵄨󵄨󵄨󵄩󵄩󵄩t α−δ 𝜕β−γ g 󵄩󵄩󵄩∞ = 0,

whenever (δ, γ) ≠ (0, 0).

|x|,|ξ |→0

This proves (1.60). For f ∈ 𝒮 󸀠 (ℝd ), for every φ ∈ 𝒮 (ℝd ), lim ⟨Mξ Tx f , φ⟩ =

|x|,|ξ |→0

lim ⟨f , T−x M−ξ φ⟩ =

|x|,|ξ |→0

lim e2πix⋅ξ ⟨f , M−ξ T−x φ⟩

|x|,|ξ |→0

= ⟨f , φ⟩, by the first part of the proof. For f ∈ 𝒮 󸀠 (ℝd ), g ∈ 𝒮 (ℝd ) \ {0}, let us define pointwise (x, ξ ) ∈ ℝ2d .

Vg f (x, ξ ) = ⟨f , Mξ Tx g⟩,

An easy consequence of Corollary 1.2.18 is the continuity of the STFT. Corollary 1.2.19. For g ∈ 𝒮 (ℝd ) \ {0}, f ∈ 𝒮 󸀠 (ℝd ), the STFT Vg f is a continuous function on ℝ2d . Proof. Set h = (h1 , h2 ) ∈ ℝ2d . We have, for every (x, ξ ) ∈ ℝ2d , lim

|h1 |,|h2 |→0

Vg f (x + h1 , ξ + h2 ) = = =

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

=

lim

⟨f , Mξ +h2 Tx+h1 g⟩

lim

⟨M−ξ f , Mh2 Tx+h1 g⟩

lim

e−2πih2 ⋅x ⟨T−x M−ξ f , Mh2 Th1 g⟩

lim

⟨T−x M−ξ f , Mh2 Th1 g⟩

|h1 |,|h2 |→0 |h1 |,|h2 |→0 |h1 |,|h2 |→0 |h1 |,|h2 |→0

= ⟨T−x M−ξ f , g⟩ = Vg f (x, ξ ), where the result of the last but one equality follows by Corollary 1.2.18. We now investigate the behavior at infinity of the STFT. We recall the notation s

⟨(x, ξ )⟩ = (1 + |x|2 + |ξ |2 )

s/2

,

for s ∈ ℝ.

Proposition 1.2.20. For f ∈ 𝒮 󸀠 (ℝd ), g ∈ 𝒮 (ℝd ) \ {0}, there exist a constant C > 0 and an integer N ∈ ℕ such that N 󵄨󵄨 󵄨 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 ≤ C⟨(x, ξ )⟩ ,

(x, ξ ) ∈ ℝ2d .

(1.61)

1.2 Time–frequency representations | 31

Proof. For f ∈ 𝒮 󸀠 (ℝd ), g ∈ 𝒮 (ℝd ) \ {0}, Mξ Tx g ∈ 𝒮 (ℝd ) \ {0}, hence there exist C > 0, m, n ∈ ℕ, such that 󵄨 󵄨󵄨 󵄨󵄨⟨f , Mξ Tx g⟩󵄨󵄨󵄨 ≤ C

󵄩 󵄩󵄩 α β 󵄩󵄩t 𝜕 (Mξ Tx g)󵄩󵄩󵄩∞

∑ |α|≤m,|β|≤n

≤C ≤C

α β 󵄨 󵄩 󵄨󵄩 ∑ ( )( )󵄨󵄨󵄨(2πiξ )γ x δ 󵄨󵄨󵄨󵄩󵄩󵄩Mξ Tx (t α−δ 𝜕β−γ g)󵄩󵄩󵄩∞ δ γ |α|≤m,|β|≤n δ≤α,γ≤β ∑

∑ Cα,β,γ,δ |x||δ| |ξ ||γ|

∑

|α|≤m,|β|≤n δ≤α,γ≤β N

≤ K⟨(x, ξ )⟩

for N = m + n (in the second inequality we applied (1.59)). Let us now show a pair of examples where it is easy to compute the STFT of a tempered distribution. Example 1.2.21. (i) In dimension d = 1, consider the delta distribution δ ∈ 𝒮 󸀠 (ℝ). For any g ∈ 𝒮 (ℝ) \ {0}, Vg δ(x, ξ ) = ⟨δ, Mξ Tx g⟩ = (Mξ Tx g)(0) = g(−x). (ii) Consider now δ󸀠 ∈ 𝒮 󸀠 (ℝ). Then for every (x, ξ ) ∈ ℝ2d Vg δ󸀠 (x, ξ ) = ⟨δ󸀠 , Mξ Tx g⟩ = −⟨δ, (Mξ Tx g)󸀠 ⟩ = −(⟨δ, (2πiξ )Mξ Tx g⟩ + ⟨δ, Mξ Tx g 󸀠 ⟩) = (2πiξ )g(−x) − g 󸀠 (−x).

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In what follows we study a vector-valued integral, obtained by multiplying the time–frequency shifts with a function F of rapid decay on the phase space. Theorem 1.2.22. Fix g ∈ 𝒮 (ℝd ) \ {0} and consider a measurable function F on ℝ2d with rapid decay ∀n ≥ 0 ∃Cn > 0,

−n 󵄨󵄨 󵄨 2d 󵄨󵄨F(x, ξ )󵄨󵄨󵄨 ≤ Cn ⟨(x, ξ )⟩ , ∀(x, ξ ) ∈ ℝ .

(1.62)

Then the integral f (t) = ∫ F(x, ξ )Mξ Tx g(t) dx dξ ℝ2d

defines a function f ∈ 𝒮 (ℝd ).

(1.63)

32 | 1 Basics of time–frequency representations and related properties Proof. For every fixed t ∈ ℝd , 󵄨 󵄨󵄨 󵄨 ∫ 󵄨󵄨󵄨F(x, ξ )󵄨󵄨󵄨󵄨󵄨󵄨Mξ Tx g(t)󵄨󵄨󵄨 dx dξ ≤ Cn ‖g‖∞ ∫ ℝ2d

ℝ2d

1 dx dξ . ⟨(x, ξ )⟩n

Taking n > 2d, we see that the integral above converges to a number f (t). Moreover, for 1 ≤ j ≤ d, 𝜕tj f (t) = ∫ F(x, ξ )𝜕tj (Mξ Tx g)(t) dx dξ . ℝ2d

Differentiation is legitimate since the obtained integral is uniformly convergent. In fact, writing A(x, ξ , t) = 𝜕tj (Mξ Tx g)(t) = (2πiξj )Mξ Tx g(t) + Mξ Tx 𝜕tj g(t), and taking n > 2d + 1 in (1.62), we have 󵄨 󵄨 󵄨 󵄨 ∫ 󵄨󵄨󵄨A(x, ξ , t)F(x, ξ )󵄨󵄨󵄨 dx dξ ≤ C ∫ (1 + |ξ |)󵄨󵄨󵄨F(x, ξ )󵄨󵄨󵄨 dx dξ < ∞. ℝ2d

ℝ2d

Generalizing the argument, we derive that f ∈ 𝒞 ∞ (ℝd ) and, using (1.59), obtain t α 𝜕β f (t) = ∫ F(x, ξ )t α 𝜕β (Mξ Tx g)(t) dx dξ ℝ2d

α β = ∑ ∑ ( )( ) ∫ F(x, ξ )(2πiξ )γ x δ Mξ Tx (t α−δ 𝜕β−γ g) dx dξ . δ γ δ≤α γ≤β ℝ2d

Hence α β 󵄩󵄩 α β 󵄩󵄩 󵄨 󵄨󵄨 γ δ 󵄨󵄩 α−δ β−γ 󵄩 󵄩󵄩t 𝜕 f 󵄩󵄩∞ ≤ ∑ ∑ ( )( ) ∫ 󵄨󵄨󵄨F(x, ξ )󵄨󵄨󵄨󵄨󵄨󵄨(2πiξ ) x 󵄨󵄨󵄨󵄩󵄩󵄩t 𝜕 g 󵄩󵄩󵄩∞ dx dξ . δ γ δ≤α γ≤β ℝ2d

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If we set 󵄩 󵄩 cα,β := max{󵄩󵄩󵄩t α−δ 𝜕β−γ g 󵄩󵄩󵄩∞ , δ ≤ α, γ ≤ β}, we infer 󵄩󵄩 α β 󵄩󵄩 󵄨 󵄨 󵄩󵄩t 𝜕 f 󵄩󵄩∞ ≤ cα,β ∫ 󵄨󵄨󵄨F(x, ξ )󵄨󵄨󵄨p(x, ξ ) dx dξ ,

(1.64)

ℝ2d

where d α β 󵄨 α β 󵄨 p(x, ξ ) = ∑ ∑ ( )( )󵄨󵄨󵄨(2πξ )γ xδ 󵄨󵄨󵄨 = ∏(1 + |xj |) j (1 + 2π|ξj |) j . δ γ j=1 δ≤α γ≤β

Taking n > 2d + |α| + |β| in (1.62), we conclude the convergence of the integral. This gives f ∈ 𝒮 (ℝd ).

1.2 Time–frequency representations | 33

Thanks to the previous results, we can now provide a characterization of the Schwartz class 𝒮 (ℝd ) in terms of the STFT as follows. Theorem 1.2.23. Fix g ∈ 𝒮 (ℝd ) \ {0}. For f ∈ 𝒮 󸀠 (ℝd ), the following conditions are equivalent: (i) f ∈ 𝒮 (ℝd ); (ii) Vg f ∈ 𝒮 (ℝ2d ); (iii) For every n ≥ 0, there exists Cn > 0 such that −n 󵄨󵄨 󵄨 2d (1.65) 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 ≤ Cn ⟨(x, ξ )⟩ , (x, ξ ) ∈ ℝ . ̄ If Proof. (i) �⇒ (ii). We use the representation of the STFT in (1.43) Vg f = ℱ2 Ta (f ⊗ g). f , g ∈ 𝒮 (ℝd ), then f ⊗ ḡ ∈ 𝒮 (ℝ2d ) and Ta and ℱ2 are topological isomorphisms on 𝒮 . This gives Vg f ∈ 𝒮 (ℝ2d ). (ii) �⇒ (iii). This is trivial since members of the Schwartz class are rapidly decreasing functions. (iii) �⇒ (i). Set f ̃(t) =

1 ∫ Vg f (x, ξ )Mξ Tx g(t) dx dξ . ‖g‖22 ℝ2d

Theorem 1.2.22 gives f ̃ ∈ 𝒮 (ℝd ) ⊂ L2 (ℝd ). The inversion formula for the STFT in L2 (ℝd ) (1.56) reads 1 V ∗ Vg f = f , ‖g‖22 g

so that f = f ̃ in L2 (ℝd ) and hence f ∈ 𝒮 (ℝd ). Corollary 1.2.24. The STFT can be interpreted as a continuous mapping 𝒮 (ℝd ) × 𝒮 (ℝd ) → 𝒮 (ℝ2d ), namely (f , g) �→ Vg f , Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

in the respective topology. Proof. The proof follows from the argument (i) �⇒ (ii) of Theorem 1.2.23. The proof of Theorem 1.2.23 shows also that the inversion formula 1 V ∗ Vg = f ‖g‖22 g

is valid in 𝒮 (ℝd ). Corollary 1.2.25. Fix g ∈ 𝒮 (ℝd ) \ {0}. Then the collection 󵄨 󵄨 ‖Vg f ‖L∞ = sup ⟨z⟩s 󵄨󵄨󵄨Vg f (z)󵄨󵄨󵄨, v s

z∈ℝ2d

is an equivalent family of seminorms for 𝒮 (ℝd ).

s ≥ 0,

(1.66)

34 | 1 Basics of time–frequency representations and related properties Proof. Define d

d

𝒮̃(ℝ ) = {f ∈ 𝒮 (ℝ ) : ‖Vg f ‖L∞ < ∞}. v 󸀠

s

Theorem 1.2.23 gives d

d

𝒮̃(ℝ ) = 𝒮 (ℝ )

as an equality between sets. Let us show that they have the same topology. Consider α, β ∈ ℕd . Using the inversion formula 1 ∫ Vg f (z)π(z)g dz ‖g‖2

f =

ℝ2d

and proceeding exactly as in the proof of Theorem 1.2.22 (cf. estimate (1.64)), we can write, for a sufficiently large n, 󵄩󵄩 α β 󵄩󵄩 󵄩󵄩t 𝜕 f 󵄩󵄩∞ ≤ Cα,β,g ∫ |Vg f |p(x, ξ ) dx dξ ℝ2d

󵄨 󵄨 ≤ C 󸀠 ∫ 󵄨󵄨󵄨Vg f (z)󵄨󵄨󵄨⟨z⟩n dz ℝ2d

≤ C 󸀠 ‖Vg f ‖L∞ v

n+2d+ϵ

≤ C ‖Vg f ‖L∞ v

∫ ℝ2d

󸀠󸀠

n+2d+ϵ

1 dz ⟨z⟩2d+ϵ

,

for every ϵ > 0. Hence for every α, β ∈ ℕd , there exists an s > 0 such that 󵄩󵄩 α β 󵄩󵄩 . 󵄩󵄩t 𝜕 f 󵄩󵄩∞ ≲ ‖Vg f ‖L∞ vs

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This implies that the identity operator I : 𝒮̃(ℝd ) → 𝒮 (ℝd ) is continuous. By the open mapping theorem (valid also for Fréchet spaces), the identity I is a topological isomorphism. The claim is proved. Theorem 1.2.26. Assume g, γ ∈ 𝒮 (ℝd ) \ {0}. (i) If there exist C, N > 0 such that N 󵄨󵄨 󵄨 󵄨󵄨F(x, ξ )󵄨󵄨󵄨 ≤ C⟨(x, ξ )⟩ ,

then f = ∫ F(x, ξ )Mξ Tx g dx dξ ∈ 𝒮 󸀠 (ℝd ), ℝ2d

that is, for every φ ∈ 𝒮 (ℝd ), ⟨f , φ⟩ = ∫ F(x, ξ )⟨Mξ Tx g, φ⟩ dx dξ . ℝ2d

(1.67)

1.2 Time–frequency representations | 35

(ii) In particular, if F = Vg f for some f ∈ 𝒮 󸀠 (ℝd ), then f =

1 ∫ Vg f (x, ξ )Mξ Tx γ dx dξ ⟨γ, g⟩

(1.68)

ℝ2d

for ⟨γ, g⟩ ≠ 0. Extending the notation (1.56), we can write f =

1 V ∗V f . ⟨γ, g⟩ γ g

(1.69)

(Observe that this implies the injectivity of the STFT in 𝒮 󸀠 (ℝd ).) Proof. (i) The integral ∫ F(x, ξ )⟨Mξ Tx g, φ⟩ dx dξ ℝ2d

converges absolutely because |F(x, ξ )| = O((⟨(x, ξ )⟩)N ) for |x|, |ξ | → +∞ and ⟨Mξ Tx g,

φ⟩ = Vφ g(x, ξ ) ∈ 𝒮 (ℝ2d ). Indeed,

N 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨⟨f , φ⟩󵄨󵄨󵄨 ≤ C ∫ 󵄨󵄨󵄨Vg φ(x, ξ )󵄨󵄨󵄨⟨(x, ξ )⟩ dx dξ ℝ2d

󵄩 󵄩 ≤ C 󵄩󵄩󵄩Vg φ⟨⋅⟩N+2d+1 󵄩󵄩󵄩∞ ∫ =C

ℝ2d N+2d+1 󵄩 󵄩󵄩Vg φ⟨⋅⟩ 󵄩󵄩󵄩∞ .

1 dx dξ ⟨(x, ξ )⟩2d+1

󸀠󵄩 󵄩

̄ Finally, using Vg φ(x, ξ ) = ℱ2 Ta (φ ⊗ g)(x, ξ ) and the continuity of the operators ℱ2 and Ta on 𝒮 (ℝ2d ), we obtain that there exists an m ∈ ℕ such that

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N+2d+1 󵄩 󵄩󵄩 󵄩󵄩 ≤ C 󵄩󵄩Vg φ(x, ξ )⟨(x, ξ )⟩ 󵄩∞

m󵄩 󵄩 β ̄ ξ )⟩ 󵄩󵄩󵄩∞ ∑ 󵄩󵄩󵄩𝜕xα 𝜕ξ (φ ⊗ g)⟨(x,

|α|,|β|≤m

≤C

∑

󵄩󵄩 α m󵄩 󵄩 β ̄ )⟨ξ ⟩m 󵄩󵄩󵄩󵄩∞ 󵄩󵄩𝜕 φ(x)⟨x⟩ 󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝜕 g(ξ

|α|≤m,|β|≤m

󵄩 󵄩 = C ∑ 󵄩󵄩󵄩𝜕α φ(x)⟨x⟩m 󵄩󵄩󵄩∞ , 󸀠

|α|≤m

hence f ∈ 𝒮 󸀠 (ℝd ). (ii) By the previous part of the proof and Proposition 1.2.20, the integral (1.68) defines an element f ̃ ∈ 𝒮 󸀠 (ℝd ). Now combining the inversion formula (1.55) with (1.63), we obtain the identity in 𝒮 (ℝd ) φ=

1 ∫ Vγ φ(x, ξ )Mξ Tx g dx dξ . ⟨g, γ⟩ ℝ2d

36 | 1 Basics of time–frequency representations and related properties Inserting the previous equality in (1.68), we infer ⟨f , φ⟩ =

1

⟨g, γ⟩

∫ Vγ φ(x, ξ )⟨f , Mξ Tx g⟩ dx dξ ℝ2d

1 = ∫ Vg f (x, ξ )Vγ φ(x, ξ ) dx dξ ⟨γ, g⟩ ℝ2d

= ⟨f ̃, φ⟩,

∀φ ∈ 𝒮 (ℝd ),

and so f = f ̃. For g ∈ 𝒮 (ℝd ), the notation of the adjoint of the STFT Vg : Vg∗ : L2 (ℝ2d ) → L2 (ℝd ), defined by

Vg∗ F = ∫ F(x, ξ )Mξ Tx g dx dξ ,

F ∈ L2 (ℝ2d ),

ℝ2d

can be extended in this setting; it maps functions F on ℝ2d satisfying (1.67) to distributions f = Vg∗ F ∈ 𝒮 󸀠 (ℝd ). With this notation we can therefore write the reproducing formula (1.69). Remark 1.2.27. (i) The STFT is not surjective on 𝒮 󸀠 (ℝ2d ). Indeed, we recall that Vg f has polynomial growth and it is continuous, whereas there are tempered distributions in 𝒮 󸀠 (ℝ2d ) which are neither continuous nor of polynomial growth. Take, for instance, in dimension d = 2 the tempered distribution F(x, ξ ) = ex cos(ex )δξ = 𝜕x 𝜕ξ G(x, ξ ),

(x, ξ ) ∈ ℝ2

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where G(x, ξ ) = sin(ex )H(ξ ) and H(ξ ) is the Heaviside function, defined by H(ξ ) = 0 for ξ ≤ 0, H(ξ ) = 1, for ξ > 0. (ii) Observe that if F satisfies (1.67) and g ∈ 𝒮 (ℝd ), with the notation of Theorem 1.2.26, for every ψ ∈ 𝒮 (ℝd ), ⟨Vg∗ F, ψ⟩ = ⟨F, Vg ψ⟩, where the brackets are meant as duality between 𝒮 󸀠 and 𝒮 .

1.2.6 Decay and further properties The properties of the STFT on the Schwartz class are the key ingredient to show the following result. Proposition 1.2.28. For 1 < p < ∞, f ∈ Lp (ℝd ), g ∈ Lp (ℝd ), the STFT Vg f is in 𝒞0 (ℝ2d ). The same is true whenever f ∈ L1 (ℝd ) and g ∈ 𝒞0 (ℝd ) (and vice versa). 󸀠

1.2 Time–frequency representations | 37

Proof. Assume first f ∈ Lp (ℝd ), g ∈ Lp (ℝd ), with 1 < p < ∞. Since the Schwartz class is dense in both spaces, there exist sequences {fn }, {gn } ⊂ 𝒮 (ℝd ) such that fn → f 󸀠 in Lp (ℝd ) and gn → g in Lp (ℝd ). Observe that Vgn fn ∈ 𝒮 (ℝ2d ) ⊂ 𝒞0 (ℝ2d ) by Theorem 1.2.23, hence 󸀠

‖Vgn fn − Vg f ‖∞ = ‖Vgn fn − Vg fn + Vg fn − Vg f ‖∞ 󵄩 󵄩 = 󵄩󵄩󵄩V(gn −g) fn + Vg (fn − f )󵄩󵄩󵄩∞ ≤ ‖fn ‖p ‖gn − g‖p󸀠 + ‖g‖p ‖fn − f ‖p󸀠 , where the last inequality is due to Proposition 1.2.10. Since the sequence {‖fn ‖p } is bounded, we infer lim ‖Vgn fn − Vg f ‖∞ = 0.

n→∞

This implies Vg f ∈ 𝒞0 (ℝ2d ), since 𝒞0 (ℝ2d ) is a Banach space with respect to the supnorm. The same argument applies to the latter case. We end up with the short-time Fourier transform by showing a very important inequality, which will be exploit heavily in the following chapters. Lemma 1.2.29. Consider g, h, γ ∈ 𝒮 (ℝd ) \ {0} such that ⟨h, γ⟩ ≠ 0 and f ∈ 𝒮 󸀠 (ℝd ). Then 1 󵄨󵄨 󵄨 (|V f | ∗ |Vg γ|)(x, ξ ), 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 ≤ |⟨h, γ⟩| h

∀(x, ξ ) ∈ ℝ2d .

(1.70)

Proof. Using the inversion formula (1.68) with the windows h, γ, f =

1 V ∗V f , ⟨h, γ⟩ γ h

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we obtain Vg f =

1 V V ∗V f . ⟨h, γ⟩ g γ h

Now, the covariance of the STFT (1.50) gives Vg Vγ∗ Vh f (x, ξ ) = ⟨Vγ∗ Vh f , π(x, ξ )g⟩ = ⟨Vh f , Vγ (π(x, ξ )g)⟩ = ∫ Vh f (x, ξ )e2πix⋅(η−ξ ) Vγ g(y − x, η − ξ ) dy dη ℝ2d

= ∫ Vh f (x, ξ )e2πiy⋅(η−ξ ) Vg γ(x − y, ξ − η) dy dη ℝ2d

(1.71)

38 | 1 Basics of time–frequency representations and related properties where in the last equality we used (1.32). Taking the modulus in (1.71) gives 1 󵄨 󵄨 󵄨󵄨 󵄨 ∗ ∫ |Vh f |󵄨󵄨󵄨Vg γ(x − y, ξ − η)󵄨󵄨󵄨 dy dη 󵄨󵄨Vg Vγ Vh f (x, ξ )󵄨󵄨󵄨 ≤ |⟨h, γ⟩| ℝ2d

1 = (|V f | ∗ |Vg γ|)(x, ξ ), |⟨h, γ⟩| h

∀(x, ξ ) ∈ ℝ2d ,

as desired.

1.3 Ambiguity function, Wigner distribution, and related topics 1.3.1 The ambiguity function We are now going to introduce another time–frequency representation: the ambiguity function, which is used in radar identifications and other applications. For background and motivation, we refer to [320]. Definition 1.3.1. Given f ∈ L2 (ℝd ), the ambiguity function Af is defined by x x Af (x, ξ ) = A(f , f )(x, ξ ) = ∫ f (t + )f (t − )e−2πit⋅ξ dt. 2 2

(1.72)

ℝd

Its polarized version is called a cross-ambiguity function x x A(f , g)(x, ξ ) = ∫ f (t + )g(t − )e−2πit⋅ξ dt, 2 2

f , g ∈ L2 (ℝd ).

(1.73)

ℝd

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Remark 1.3.2. (i) By the change of variables t �→ t −

x 2

in (1.72) and (1.73), we obtain

Af (x, ξ ) = eπix⋅ξ Vf f (x, ξ ),

A(f , g)(x, ξ ) = eπix⋅ξ Vg f (x, ξ )

(1.74)

so that “up to a chirp” the cross-ambiguity function is just the STFT. (ii) We observe that the (cross)-ambiguity function is defined only up to phase factors, i. e., c ∈ ℂ, with |c| = 1. In fact, a straightforward computation gives A(cf , cg)(x, ξ ) = A(f , g)(x, ξ ), for every c ∈ ℂ, with |c| = 1. Relations (1.74) imply that the cross-ambiguity function enjoys most of the properties of the STFT. In what follows we limit ourselves to listing the main issues, the proofs are easy consequences of the previous remark.

1.3 Ambiguity function, Wigner distribution, and related topics | 39

Proposition 1.3.3. The cross-ambiguity function has the following properties: (i) (Moyal’s identity) For fi , gi ∈ L2 (ℝd ), i = 1, 2, ⟨A(f1 , g1 ), A(f2 , g2 )⟩ = ⟨f1 , f2 ⟩⟨g1 , g2 ⟩.

(1.75)

In particular, ‖Af ‖2 = ‖f ‖22 ,

f ∈ L2 (ℝd ).

(ii) For 1 < p < ∞, f ∈ Lp (ℝd ), g ∈ Lp (ℝd ), the cross-ambiguity function is in 𝒞0 (ℝ2d ), with 󸀠

󵄩󵄩 󵄩 󵄩󵄩A(f , g)󵄩󵄩󵄩∞ ≤ ‖f ‖p ‖g‖p󸀠 . The same holds true if f ∈ L1 (ℝd ) and g ∈ 𝒞0 (ℝd ) (and vice versa). (iii) For f , g ∈ L2 (ℝd ), A(f , g)(x, ξ ) = A(ℐ f , ℐ g)(x, ξ ),

(x, ξ ) ∈ ℝ2d ,

(1.76)

(we use the notation ℐ h(t) = h(−t)). (iv) For f , g ∈ L2 (ℝd ), the involution is given by (A(f , g)) (x, ξ ) = A(f , g)(−x, −ξ ) = A(f , g)(x, ξ ), ∗

(x, ξ ) ∈ ℝ2d .

(1.77)

(v) It is a continuous mapping from 𝒮 (ℝd ) × 𝒮 (ℝd ) into 𝒮 (ℝ2d ).

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1.3.2 The Wigner distribution The Wigner distribution (or function) is a time–frequency representation closely related to the ambiguity function. We will see in the sequel that these representations are symplectic Fourier transforms of each other. It was introduced by E. Wigner in 1932 [314] in the field of quantum mechanics and then employed in signal analysis by J. Ville [300]. Since then the Wigner distribution has been applied in several fields of mathematics, engineering, and physics, and it is for sure the most popular time–frequency representation. Definition 1.3.4. For f ∈ L2 (ℝd ), the Wigner distribution Wf is defined by t t Wf (x, ξ ) = W(f , f )(x, ξ ) = ∫ f (x + )f (x − )e−2πit⋅ξ dt. 2 2

(1.78)

ℝd

Similarly, the cross-Wigner distribution is t t W(f , g)(x, ξ ) = ∫ f (x + )g(x − )e−2πit⋅ξ dt, 2 2 ℝd

f , g ∈ L2 (ℝd ).

(1.79)

40 | 1 Basics of time–frequency representations and related properties The relation between the cross-Wigner distribution and the STFT is provided in what follows. Lemma 1.3.5. For f , g ∈ L2 (ℝd ), we have W(f , g)(x, ξ ) = 2d e4πix⋅ξ Vℐg f (2x, 2ξ ), Proof. We make the change of variables u = x +

t 2

(x, ξ ) ∈ ℝ2d .

(1.80)

in (1.79) so that dt = 2d du and

W(f , g)(x, ξ ) = 2d ∫ f (u)g(2x − u)e−2πi2(u−x)⋅ξ du. ℝd

= 2d e4πix⋅ξ ∫ f (u)(ℐ g)(u − 2x)e−2πiu⋅(2ξ ) du ℝd

= 2d e4πix⋅ξ Vℐg f (2x, 2ξ ), as desired. Observe that, using the connection between Schrödinger representation and time–frequency shifts in (1.11), formula (1.80) can be rephrased as W(f , g)(x, ξ ) = 2d ⟨f , ρ(2x, 2ξ , 0)ℐ g⟩.

(1.81)

Using this result, many properties of the STFT can be carried over to the cross-Wigner distribution as well. In what follows we list the most relevant ones. Proposition 1.3.6. The (cross-)Wigner distributions has the following properties: (i) (Moyal’s identity) For fi , gi ∈ L2 (ℝd ), i = 1, 2, ⟨W(f1 , g1 ), W(f2 , g2 )⟩ = ⟨f1 , f2 ⟩⟨g1 , g2 ⟩.

(1.82)

In particular,

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‖Wf ‖2 = ‖f ‖22 ,

f ∈ L2 (ℝd ).

(ii) For 1 < p < ∞, f ∈ Lp (ℝd ), g ∈ Lp (ℝd ), the Wigner function is in 𝒞0 (ℝ2d ), with 󸀠

󵄩󵄩 󵄩 d 󵄩󵄩W(f , g)󵄩󵄩󵄩∞ ≤ 2 ‖f ‖p ‖g‖p󸀠 . The same holds true if f ∈ L1 (ℝd ) and g ∈ 𝒞0 (ℝd ) (and vice versa). (iii) For f , g such that W(f , g) is well defined, W(f , g) = W(g, f ). In particular, Wf is always a real-valued function.

1.3 Ambiguity function, Wigner distribution, and related topics | 41

(iv) Consider f , g such that W(f , g) is well defined. Recall that, for z = (x, ξ ), π(z) = Mξ Tx is the time–frequency shift operator. Then for z0 , z1 ∈ ℝ2d , we have W(π(z0 )f , π(z1 )g)(z) = e−2πi[σ(z,z0 −z1 )+σ(z0 ,z1 )] W(f , g)(z −

z0 + z1 ). 2

In particular, for z0 = z1 , W(π(z0 )f , π(z0 )g)(z) = W(f , g)(z − z0 ) = Tz0 W(f , g)(z).

(1.83)

(v) For f , g ∈ L2 (ℝd ), ̂ W(f ̂, g)(x, ξ ) = W(f , g)(−ξ , x),

(x, ξ ) ∈ ℝ2d .

(1.84)

(vi) It is a continuous mapping from 𝒮 (ℝd ) × 𝒮 (ℝd ) into 𝒮 (ℝ2d ). Proof. The proofs are straightforward. Let us only detail claim (v). Observe that −2πit⋅ξ ̂ ̂ ). (̂ ℐ g)(ξ ) = ∫ e g(−t) dt = ∫ e−2πi(−t)⋅ξ g(t) dt = g(−ξ ) = ℐ g(ξ ℝd

ℝd

Hence, by Parseval’s formula (for the inverse Fourier transform) and the commutation relations (0.14), ̂ W(f ̂, g)(x, ξ ) = 2d e4πix⋅ξ ⟨f ̂, M2ξ T2x ℐ g⟩̂

= 2d e4πix⋅ξ ⟨f , T−2ξ M2x ℐ g⟩

= 2d e4πix⋅(−ξ ) ⟨f , M2x T−2ξ ℐ g⟩

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= W(f , g)(−ξ , x).

Relation (1.84) means that the Fourier transform produces a rotation by π/2 of the Wigner distribution in the time–frequency plane. Setting z = (x, ξ ), relation (1.84) can be rewritten in terms of the standard symplectic matrix as follows: W(̂Jf , ̂Jg)(z) = W(f , g)(J −1 z),

f , g ∈ L2 (ℝd ),

with J, ̂J as in (1.1), (1.14). This can be seen as a particular case of the symplectic covariance property for the Wigner distribution. ̂ is a metaProposition 1.3.7 (Covariance property for the Wigner distribution). If 𝒜 plectic operator with projection the symplectic matrix 𝒜 ∈ Sp(d, ℝ), then ̂f , 𝒜 ̂g)(z) = W(f , g)(𝒜−1 z), W(𝒜

f , g ∈ L2 (ℝd ).

(1.85)

42 | 1 Basics of time–frequency representations and related properties Proof. Observe that the reflection operator ℐ g(t) = g(−t) is a metaplectic operator with ? projection the symplectic matrix −I2d ∈ Sp(d, ℝ), hence we can write (−I 2d )g = ℐ g. Let us show first the following relation: for every 𝒜 ∈ Sp(d, ℝ), ∀g ∈ L2 (ℝd ).

̂g) = 𝒜 ̂(ℐ g), ℐ (𝒜

(1.86)

This follows from the previous observation and from the fact that −I2d commutes with any matrix 𝒜 ∈ Sp(d, ℝ), so that ? ? ? ̂g) = (−I ̂ ℐ (𝒜 2d )(𝒜g) = (−I2d 𝒜)g = (𝒜(−I2d ))g ? ̂(−I ̂ =𝒜 2d )g = 𝒜(ℐ g). Using (1.81) and (1.86), we now can compute ̂g)⟩ ̂f , 𝒜 ̂g)(x, ξ ) = 2d ⟨𝒜 ̂f , ρ(2x, 2ξ , 0)ℐ (𝒜 W(𝒜 ̂−1 ρ(2x, 2ξ , 0)𝒜 ̂(ℐ g)⟩ = 2d ⟨f , 𝒜 ̂ −1 ρ(2x, 2ξ , 0)𝒜 ̂(ℐ g)⟩ = 2d ⟨f , 𝒜 = 2d ⟨f , ρ(𝒜−1 (2x, 2ξ ), 0)ℐ g⟩ = W(f , g)(𝒜−1 (x, ξ )). This concludes the proof. The Wigner distribution possesses a factorization similar to the STFT in Lemma 1.43. Definition 1.3.8. We define Ts to be the symmetric coordinate change y y Ts F(x, y) = F(x + , x − ), 2 2

x, y ∈ ℝd .

(1.87)

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Observe that the inverse change T−1 s is given by T−1 s F(x, y) = F(

x+y , x − y). 2

(1.88)

Lemma 1.3.9. For f , g ∈ L2 (ℝd ), ̄ W(f , g) = ℱ2 Ts (f ⊗ g),

(1.89)

where ℱ2 is the partial Fourier transform with respect to the second variable. Proof. It follows by the definition of the cross-Wigner distribution in (1.79). Remark 1.3.10. Moyal’s formula (1.82) can be proved also by using Lemma 1.3.9 and following the pattern of Theorem 1.2.11 (second proof).

1.3 Ambiguity function, Wigner distribution, and related topics | 43

As for the STFT, using Lemma 1.3.9, we can enlarge the domain of the Wigner distribution to 𝒮 󸀠 (ℝd ). Indeed, the operators ℱ2 and Ts are isometries on 𝒮 (ℝ2d ) and extend to isometries on 𝒮 󸀠 (ℝ2d ). Hence ℱ2 Ts (f ⊗ g)̄ is well defined as a tempered distrī bution on ℝ2d whenever f , g ∈ 𝒮 󸀠 (ℝd ), and we may define W(f , g) = ℱ2 Ts (f ⊗ g). Recall that we consider a distribution (conjugate)-linear instead of linear, so that the duality 𝒮 󸀠 ⟨⋅, ⋅⟩𝒮 extends the inner product in L2 (ℝd ). In particular, the Dirac’s delta δ is defined by ⟨δ, f ⟩ = f (0). Let us compute the Wigner distribution of the Dirac’s delta δ defined above. Let φ ∈ 𝒮 (ℝ2d ), then ̄ φ⟩ = ⟨δ ⊗ δ,̄ T−1 ℱ −1 φ⟩ ⟨Wδ, φ⟩ = ⟨ℱ2 Ts (δ ⊗ δ), s 2 −1 = T−1 s ℱ2 φ(0, 0) = ∫ φ( ℝd

x+ξ , t)e2πit⋅(x−ξ ) dt|x=ξ =0 . 2

= ∫ φ(0, t) dt = ⟨1, φ(0, ⋅)⟩ ℝd

= ⟨δ ⊗ 1, φ⟩. Therefore Wδ = δ ⊗ 1 ∈ 𝒮 󸀠 (ℝ2d ). We now show the link between the Wigner distribution and the ambiguity function. Lemma 1.3.11. For any f , g ∈ 𝒮 󸀠 (ℝ2d ),

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A(f , g) = ℱσ W(f , g).

(1.90)

Proof. We prove the result for f , g ∈ 𝒮 (ℝd ). The general case follows by duality arguments. Define by ℱ1 F(x, ξ ) = ∫ℝd F(t, ξ )e−2πit⋅x dt the partial Fourier transform in the first variable, so that the Fourier transform can be factorized as ℱ = ℱ2 ℱ1 ,

ℱ

−1

= ℱ1−1 ℱ2−1 .

Moreover, from (1.18), ℱσ W(f , g)(z) = ℱ W(f , g)(Jz) = ℱ W(f , g)(−Jz). −1

For z = (x, ξ ), we have −Jz = (−ξ , x) and using definition (1.89), we can write ̄ ℱσ W(f , g)(x, ξ ) = ℱ1 ℱ2 ℱ2 Ts (f ⊗ g)(−ξ , x) −1

−1

̄ = ∫ Ts (f ⊗ g)(t, x)e2πit⋅x dt ℝd

44 | 1 Basics of time–frequency representations and related properties x x ̄ − )e2πit⋅x dt = ∫ f (t + )g(t 2 2 ℝd

= A(f , g)(x, ξ ), as desired. Lemma 1.3.12. Consider the Gaussian function φ1/√a (x) = e−πx

2

/a

, x ∈ ℝd . Then

Wφ1/√a (x, ξ ) = (2a)d/2 φ√ 2 (x)φ√2a (ξ ).

(1.91)

a

2

In particular, for φ(x) = e−πx we have Wφ(x, ξ ) = 2d/2 e−2π(x

2

+ξ 2 )

.

Proof. It is an easy computation, cf. (0.25), π

2

2

Wφ1/√a (x, ξ ) = ∫ e− a [(x+t/2) +(x−t/2) ] e−2πit⋅ξ dt ℝd π

= ∫ e− a (2x

2

+t 2 /2) −2πit⋅ξ

e

dt

ℝd x2

π 2

= e−2π a ∫ e−2πit⋅ξ e− 2a t dt ℝd d

2 ̂ = φ a (x)φ 2a (ξ ) = (2a) φ√ 2 (x)φ√2a (ξ ), 2

a

as required.

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1.3.3 Marginal densities The fundamental problem of time–frequency analysis and quantum mechanics is to discover a good mathematical device able to simultaneously represent a given signal in terms of its intensity in time and frequency. Indeed, introducing Wf in quantum mechanics, Wigner wanted to find a joint probability density function of the position and momentum variables. The quantity Wf (x, ξ )ΔxΔξ is interpreted as the probability that the particle in the quantum state f ∈ L2 (ℝd ), ‖f ‖2 = 1, is located in [x, x + Δx] with momentum in [ξ , ξ + Δξ ]. The marginal densities implied by a joint probability density are also implied by the Wigner distribution. Proposition 1.3.13. Consider f , f ̂ ∈ L1 (ℝd ) ∩ L2 (ℝd ), then 󵄨 󵄨2 ∫ Wf (x, ξ ) dξ = 󵄨󵄨󵄨f (x)󵄨󵄨󵄨

x ∈ ℝd ,

(1.92)

ξ ∈ ℝd .

(1.93)

ℝd

󵄨 󵄨2 ∫ Wf (x, ξ ) dx = 󵄨󵄨󵄨f ̂(ξ )󵄨󵄨󵄨 ℝd

1.3 Ambiguity function, Wigner distribution, and related topics | 45

In particular, ∫ ∫ Wf (x, ξ ) dx dξ = ‖f ‖22 .

(1.94)

ℝd ℝd

Proof. First, observe that by the identity (1.80) and the representation of the STFT in (1.29), we can write ̄ 󵄨 󵄨 󵄨󵄨 d d d󵄨 󵄨󵄨Wf (x, ξ )󵄨󵄨󵄨 = 2 󵄨󵄨󵄨Vℐf f (2x, 2ξ )󵄨󵄨󵄨 = 2 |f ̂ ∗ M−2x f ̂|(2ξ ) ≤ 2 |f ̂| ∗ |f ̂|(2ξ ). Since f ̂ ∈ L1 (ℝd ), by Young’s inequality, we have |f ̂| ∗ |f ̂| ∈ L1 (ℝd ), which is invariant under dilations. Hence, for every fixed x ∈ ℝd , Wf (x, ⋅) is in L1 (ℝd ), and we can apply the inverse Fourier formula to obtain ∫ Wf (x, ξ ) dξ = ℱ2−1 [ℱ2 Ts (f ⊗ f ̄)](x, 0) ℝd

󵄨 󵄨2 = Ts (f ⊗ f ̄)(x, 0) = 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 .

By Proposition 1.3.6 (v), W f ̂(ξ , −x) = Wf (x, ξ ), so we can apply the previous computations to obtain 󵄨 󵄨2 ∫ Wf (x, ξ ) dx = ∫ W f ̂(ξ , −x) dx = ∫ W f ̂(ξ , x) dx = 󵄨󵄨󵄨f ̂(ξ )󵄨󵄨󵄨 ℝd

ℝd

ℝd

for every ξ ∈ ℝ2d , and so (1.93) is attained. The last formula is trivial. Observe that for the marginal property (1.92) we only require that the signal f is in L (ℝd ) with f ̂ ∈ L1 (ℝd ), whereas formula (1.93) needs f ∈ L1 (ℝd ) ∩ L2 (ℝd ). Both formulas (1.92) and (1.93) (with Plancherel’s equality) give (1.94), but be careful that in both cases the integrals in (1.94) are only understood as iterated integrals. In general, the assumption f , f ̂ ∈ L1 (ℝd ) does not imply Wf ∈ L1 (ℝ2d ). From the point of view of quantum mechanics, ∫ℝd Wf (x, ξ ) dξ gives the probability density |f (x)|2 of the position variable and ∫ℝd Wf (x, ξ ) dx yields the probability density |f ̂(ξ )|2 of the momentum variable.

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2

So far we have seen the nice properties of the Wigner distribution that would make it a nice candidate for being a joint probability density. Observe that, if the Wigner distribution were a joint probability density, then it would be nonnegative. Unfortunately, this is not the case for the Wigner distribution, in general. For instance, take f ∈ L2 (ℝd ) odd, then 󵄨󵄨 t 󵄨󵄨2 t t 󵄨 󵄨 Wf (0, 0) = ∫ f ( )f (− ) dt = − ∫ 󵄨󵄨󵄨f ( )󵄨󵄨󵄨 dt = −2d ‖f ‖22 < 0. 󵄨󵄨 2 󵄨󵄨 2 2 ℝd

ℝd

Moreover, the following theorem by Hudson [198] shows that the Wigner distribution is positive only for f being a generalized Gaussian.

46 | 1 Basics of time–frequency representations and related properties Theorem 1.3.14. Assume f ∈ L2 (ℝd ). Then Wf (x, ξ ) > 0 for all (x, ξ ) ∈ ℝ2d if and only if f is a generalized Gaussian of the form f (x) = e−πx⋅Ax+2πb⋅x+c ,

(1.95)

where A ∈ GL(d, ℂ) (the space of invertible d×d complex matrices) is a symmetric matrix with positive definite real part (A = AT , ℜA > 0), b ∈ ℂd , c ∈ ℂ. We refer to [198] or [160] for the proof of Hudson’s theorem, cf. Lemma 1.3.12 for the Wigner of the rescaled Gaussian function φ. For recent applications of the Wigner distribution to PDEs, we refer the reader to [27, 39]. For links between the Wigner distribution and Paley–Wiener theorems, we point to [29]. 1.3.4 The Cohen class Since Wf fails to be positive in general, a remedy could be the averaging of Wf by convolving it with suitable smooth functions σ, obtaining a more useful time–frequency representation. This is the idea under the definition of the Cohen class, introduced by L. Cohen in [53]: a time–frequency representation belongs to the Cohen class if it is of the form 𝒬σ f = Wf ∗ σ,

∀f ∈ 𝒮 (ℝd ),

(1.96)

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for some function or distribution σ ∈ 𝒮 󸀠 (ℝ2d ), called kernel of the Cohen distribution. Observe that if f ∈ 𝒮 (ℝd ) then Wf ∈ 𝒮 (ℝ2d ) by Proposition 1.3.6 (vi), and the convolution Wf ∗ σ is a well-defined distribution in 𝒮 󸀠 (ℝ2d ). The Wigner distribution trivially belongs to the Cohen class since we may take as kernel the Dirac distribution σ = δ ∈ 𝒮 󸀠 (ℝ2d ). Proposition 1.3.15. For f ∈ L2 (ℝd ) and σ ∈ L1 (ℝd ), the element of the Cohen class 𝒬σ f is well defined in L2 (ℝ2d ). Proof. The condition f ∈ L2 (ℝd ) implies that Wf ∈ L2 (ℝ2d ) by Moyal’s formula, and consequently 𝒬σ f = Wf ∗ σ ∈ L2 (ℝd ) by Young’s inequality. It is hard task to determine those kernels σ for which Wf ∗ σ is always positive. A simple case is provided by Gaussian kernels. The distribution 𝒬σ in (1.96) with ker2 nel σ(z) = e−2π|z| is called Husimi distribution since it was introduced by K. Husimi in his paper [199]. Theorem 1.3.16. Define x2

ξ2

σa,b (x, ξ ) = e−2π( a + b ) = φ√ 2 (x)φ√ 2 (ξ ), a

b

a, b > 0, (x, ξ ) ∈ ℝ2d .

(1.97)

1.3 Ambiguity function, Wigner distribution, and related topics | 47

Then (i) If ab = 1, then Wf ∗ σa,b ≥ 0 for all f ∈ L2 (ℝd ). (ii) If ab > 1, then Wf ∗ σa,b > 0 for all f ∈ L2 (ℝd ) \ {0}. (iii) If 0 < ab < 1, then Wf ∗ σa,b may be negative. In the case a = b = 1, σ1,1 is the Husimi’s kernel. The proof of the above theorem needs the following lemma, which is an easy consequence of the semigroup property of Gaussian functions (0.26). Lemma 1.3.17. Choose real numbers a, b, c, d such that 0 < c < a, 0 < d < b and cd = 1. Then it holds d

σa,b

2 4ab ) σc,d ∗ σa−c,b−d . =( (a − c)(b − d)

(1.98)

Proof. Using the semigroup property (0.26), d

2 4ab ) σc,d ∗ σa−c,b−d (x, ξ ) ( (a − c)(b − d) d

2 4ab =( ) (φ√ 2 ∗ φ√ 2 )(x)(φ√ 2 ∗ φ√ 2 )(ξ ) (a − c)(b − d) a−c b−d c d d

d

d

2 c(a − c) 2 d(b − d) 2 4ab ) ( ) ( ) φ√ 2 (x)φ√ 2 (ξ ) =( (a − c)(b − d) 2a 2b b a

= σa,b (x, ξ ),

that is, (1.98) holds. We can now prove the theorem above. Proof of Theorem 1.3.16. First, we observe the following action of the reflection operator on the Wigner distribution:

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ℐ (Wf ) = W(ℐ f ),

(1.99)

which follows from this straightforward computation: t 2

t 2

ℐ (Wf )(x, ξ ) = Wf (−x, −ξ ) = ∫ f (−x + )f (−x − )e ℝd

2πiξ ⋅t

dt

t t = ∫ f (−x − )f (−x + )e−2πiξ ⋅t dt = W(ℐ f )(x, ξ ). 2 2 ℝd

(i) If ab = 1, that is, b = 1/a, we can write the kernel σa,b as Wigner distribution, using the Lemma 1.3.12, d

σa,b = φ√ 2 (x)φ√2a (ξ ) = (2a)− 2 Wφ 1 (x, ξ ). a

√a

48 | 1 Basics of time–frequency representations and related properties By means of formula (1.99), the covariance of Wf , cf. Proposition 1.2.15, and Moyal’s formula, cf. Proposition 1.3.6, we can write (Wf ∗ σa, 1 )(x, ξ ) = (σa, 1 ∗ Wf )(x, ξ ) = ∫ σa, 1 (t, ν)Wf (x − t, ξ − ν) dt dν a

a

ℝ2d

a

d

= ∫ Wf (x − t, ξ − ν)(2a)− 2 Wφ 1 (t, ν) dt dν ℝ2d

√a

d

= (2a)− 2 ∫ T(x,ξ ) (ℐ Wf )(t, ν)Wφ 1 (t, ν) dt dν √a

ℝ2d d

= (2a)− 2 ∫ W(Mξ Tx ℐ f )(t, ν)Wφ 1 (t, ν) dt dν d

√a

ℝ2d

= (2a)− 2 ⟨W(Mξ Tx ℐ f ), Wφ 1 ⟩ √a

d 󵄨 󵄨2 = (2a)− 2 󵄨󵄨󵄨⟨Mξ Tx (ℐ f ), φ 1 ⟩󵄨󵄨󵄨 ≥ 0, √a

and so we obtain the claim. (ii) Assume ab > 1. Then we can choose c, d ∈ ℝ such that 0 < c < a and 0 < b < d with cd = 1. By Lemma 1.3.17, we have d

Wf ∗ σa,b

2 4ab =( ) (Wf ∗ σc,d ) ∗ σa−c,b−d . (a − c)(b − d)

The right-hand side is strictly positive since it is the convolution of the function Wf ∗ σc,d , nonnegative by the previous part of the proof (observe that Wf ∗ σc,d ≥ 0 is not the zero function, since f is not null) and a strictly positive one, σa−c,b−d > 0. (iii) For the case 0 < ab < 1, we exhibit a counterexample. If we choose f (x) = −πx 2 xe on ℝ, we can verify that Wf ∗ σa,b (0, 0) < 0. Indeed, by direct calculation, we 2 2 have that Wf (x, ξ ) = √2e−2πx e−2πξ (x2 + ξ 2 − 1/4π) and hence

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(Wf ∗ σa,b )(0, 0) =

√2 ab ab − 1 √ ( ) < 0. 8π (a + 1)(b + 1) (a + 1)(b + 1)

Coming back to general Cohen kernels, let us first observe that we can use the cross-Wigner distribution to provide an equivalent definition of the Cohen class. Definition 1.3.18. A sesquilinear form 𝒬σ : 𝒮 (ℝd ) × 𝒮 (ℝd ) → 𝒮 (ℝ2d ) belongs to the Cohen class if there exists a kernel σ ∈ 𝒮 󸀠 (ℝ2d ) such that 𝒬σ (f , g) = W(f , g) ∗ σ,

∀f , g ∈ 𝒮 (ℝd ).

(1.100)

For f = g from (1.100), we obtain (1.96). On the other hand, starting from (1.96), we may define 𝒬σ (f , g) uniquely. In fact, the polarization identity for the cross-Wigner distribution W(f , g) =

1 (W(f + g) − W(f − g) + iW(f + ig) − iW(f − ig)) 4

1.3 Ambiguity function, Wigner distribution, and related topics | 49

gives 1 (W(f + g) − W(f − g) + iW(f + ig) − iW(f − ig)) ∗ σ 4 1 = (𝒬σ (f + g) − 𝒬σ (f − g) + i𝒬σ (f + ig) − i𝒬σ (f + g)) 4 = 𝒬σ (f , g),

W(f , g) ∗ σ =

that is, (1.100) is true. We list the main properties for members of the Cohen class. We say that 𝒬σ is real-valued if 𝒬σ f is real-valued for every f ∈ 𝒮 (ℝd ). Proposition 1.3.19. The members of the Cohen class satisfy the following properties: (i) 𝒬σ is covariant, namely, for every z ∈ ℝ2d , 𝒬σ (π(z)f ) = Tz 𝒬σ f ,

∀f ∈ 𝒮 (ℝd ).

(1.101)

(ii) 𝒬σ is real-valued if and only if σ = σ. (iii) (Marginal properties). The conditions 󵄨 󵄨2 ∫ 𝒬σ f (x, ξ )dξ = 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 ,

󵄨 󵄨2 ∫ 𝒬σ f (x, ξ )dx = 󵄨󵄨󵄨f ̂(ξ )󵄨󵄨󵄨 ,

ℝd

ℝd

hold if and only if ∫ σ(t, ν) dν = δ(t),

∫ σ(t, ν) dt = δ(ν),

ℝd

ℝd

(1.102)

respectively. Moreover, ∫ 𝒬σ f (x, ξ ) dx dξ = ‖f ‖22 ℝ2d

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holds if and only if ∫ σ(t, ν) dx dν = 1.

(1.103)

ℝ2d

(iv) (Moyal’s formula) Assume σ̂ ∈ L∞ (ℝ2d ). Then for every f , g ∈ L2 (ℝd ), 󵄨 󵄨2 ⟨𝒬σ f , 𝒬σ g⟩L2 (ℝ2d ) = 󵄨󵄨󵄨⟨f , g⟩󵄨󵄨󵄨 ,

∀f , g ∈ L2 (ℝd )

if and only if 󵄨󵄨 ̂ 󵄨 2d 󵄨󵄨σ(x, ξ )󵄨󵄨󵄨 = 1 a. e. (x, ξ ) ∈ ℝ .

(1.104)

Remark 1.3.20. The integrals in (1.102) and (1.103) have no meaning in general for σ ∈ 𝒮 󸀠 (ℝ2d ). Concerning ∫ℝ2d σ(t, ν) dt dν, it makes sense, of course, if σ ∈ L1 (ℝ2d ).

50 | 1 Basics of time–frequency representations and related properties Otherwise, for σ ∈ 𝒮 󸀠 (ℝ2d ), we may read (1.103) as ⟨σ, 1⟩ = 1, provided ⟨σ, 1⟩ is well defined for σ in suitable subclasses of 𝒮 󸀠 (ℝ2d ), for example, for σ ∈ ℰ 󸀠 (ℝ2d ), the space of the distributions with compact support. Similarly, we may argue on the meaning of (1.102). Note that for σ ∈ 𝒮 󸀠 (ℝ2d ) equivalent formulations to (1.102) are obtained by imposing on the Fourier transform σ̂ ∈ 𝒮 󸀠 (ℝ2d ) the requirement ̂ 0) = 1, σ(x,

̂ ξ) = 1 σ(0,

(1.105)

̂ 0) and σ(0, ̂ ξ ) make sense. provided again σ(x, The computations in the subsequent proof of (iii) will be valid in all the cases, the integrals to be understood in the weak sense whenever σ is a distribution. Proof of Proposition 1.3.19. (i) Using the covariant property of Wigner distribution (1.50) and the property, namely Tz (F ∗ G) = (Tz F) ∗ G = F ∗ (Tz G), we have 𝒬σ (π(z)f )(u) = (W(π(z)f ) ∗ σ)(u) = (Tz Wf ) ∗ σ(u)

= Tz (Wf ∗ σ)(u) = Tz 𝒬σ f (u).

(ii) Using the fact that Wf is a real function, 𝒬σ f = Wf ∗ σ

= Wf ∗ σ = Wf ∗ σ. Hence 𝒬σ is real-valued if and only Wf ∗ σ = Wf ∗ σ,

∀f ∈ 𝒮 (ℝd ).

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Taking the symplectic Fourier transform of both sides, this is equivalent to saying Af ⋅ ℱσ (σ) = Af ⋅ ℱσ (σ)̄

(1.106)

for every f ∈ 𝒮 (ℝd ), cf. Lemma 1.3.11 and Proposition 1.18 (v). Consider the Gaussian 2 2 f (t) = φ(t) = e−πt . The Wigner distribution is Wφ(z) = 2d/2 e−2πz by Lemma 1.3.12 and 2 the ambiguity function is given by Aφ(z) = 2−d/2 e−πz /2 . Writing (1.106) for f = φ and ̄ which gives σ = σ,̄ as desired. dividing by Aφ, we obtain ℱσ (σ) = ℱσ (σ), (iii) We argue according to Remark 1.3.20: ∫ 𝒬σ f (x, ξ ) dξ = ∫ (Wf ∗ σ)(x, ξ ) dξ ℝd

ℝd

= ∫ ( ∫ Wf (x − t, ξ − ν)σ(t, ν) dt dν) dξ ℝd ℝ2d

1.3 Ambiguity function, Wigner distribution, and related topics | 51

= ∫ σ(t, ν)( ∫ Wf (x − t, ξ − ν) dξ ) dt dν ℝ2d

ℝd

= ∫ σ(t, ν)( ∫ Wf (x − t, ξ ) dξ ) dt dν ℝd

ℝ2d

󵄨2 󵄨 = ∫ σ(t, ν)󵄨󵄨󵄨f (x − t)󵄨󵄨󵄨 dt dν ℝ2d

󵄨 󵄨2 = ∫ 󵄨󵄨󵄨f (x − t)󵄨󵄨󵄨 ( ∫ σ(t, ν) dν) dt, ℝd

ℝd

where we have made the change of coordinate ξ �→ ξ − ν and used the marginal property of the Wigner distribution (1.92). We deduce that 󵄨 󵄨2 ∫ 𝒬σ f (x, ξ ) dξ = 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 ⇐⇒ ∫ σ(t, ν) dν = δ(t). ℝd

ℝd

The second marginal property is proved in the same way as the previous one, employing the second marginal property of the Wigner distribution. By combining the preceding arguments, ∫ 𝒬σ f (x, ξ ) dx dξ = ∫ (Wf ∗ σ)(x, ξ ) dx dξ ℝ2d

ℝ2d

= ( ∫ Wf (t, ν) dt dν)( ∫ σ(t 󸀠 , ν󸀠 ) dt 󸀠 dν󸀠 ) ℝ2d

ℝ2d

= ‖f ‖22 ∫ σ(t 󸀠 , ν󸀠 ) dt 󸀠 dν󸀠 , ℝ2d

and the equality ∫ℝ2d 𝒬σ f (x, ξ ) dx dξ = ‖f ‖22 holds for every f ∈ 𝒮 (ℝd ) if and only if

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∫ σ(t 󸀠 , ν󸀠 ) dt 󸀠 dν󸀠 = 1. ℝ2d

̂ ⋅ σ)̂ is a well(iv) If σ̂ ∈ L∞ (ℝ2d ) and f ∈ L2 (ℝd ), the convolution Wf ∗ σ = ℱ −1 (Wf 2 2d 2 2d defined function in L (ℝ ), since Wf ∈ L (ℝ ) and L2 (ℝ2d ) ⋅ L∞ (ℝ2d ) �→ L2 (ℝ2d ). Using Parseval’s formula, we deduce that ⟨𝒬σ f , 𝒬σ g⟩L2 (ℝ2d ) = ⟨Wf ∗ σ, Wg ∗ σ⟩L2 (ℝ2d )

̂ ⋅ σ), ̂ ⋅ σ)⟩ ̂ ℱ −1 (Wg ̂ L2 (ℝ2d ) = ⟨ℱ −1 (Wf ̂ ⋅ σ,̂ Wg ̂ ⋅ σ⟩̂ 2 2d = ⟨Wf L (ℝ )

̂ , Wg| ̂ σ|̂ 2 ⟩ 2 2d . = ⟨Wf L (ℝ )

52 | 1 Basics of time–frequency representations and related properties ̂ ξ )| = 1 a. e., then we obtain the claim by Parseval’s formula and Moyal’s idenIf |σ(x, tity (1.82): ̂ , Wg| ̂ σ|̂ 2 ⟩ 2 2d = ⟨Wf , Wg⟩ 2 2d = 󵄨󵄨󵄨⟨f , g⟩󵄨󵄨󵄨2 . ⟨Qσ f , Qσ g⟩L2 (ℝ2d ) = ⟨Wf L (ℝ ) 󵄨 󵄨 L (ℝ ) On the other hand, if we impose ⟨Qσ f , Qσ g⟩L2 (ℝ2d ) = |⟨f , g⟩|2 , we obtain ̂ , Wg| ̂ σ|̂ 2 ⟩ 2 2d = ⟨Qσ f , Qσ g⟩ 2 2d = 󵄨󵄨󵄨⟨f , g⟩󵄨󵄨󵄨2 ⟨Wf L (ℝ ) 󵄨 󵄨 L (ℝ ) ̂ ̂ = ⟨Wf , Wg⟩ 2 2d = ⟨Wf , Wg⟩

L2 (ℝ2d ) ,

L (ℝ )

̂ , Wg| ̂ , Wg⟩ ̂ σ|̂ 2 ⟩ 2 2d = ⟨Wf ̂ 2 2d , for every f , g ∈ L2 (ℝd ). This easily gives and thus ⟨Wf L (ℝ ) L (ℝ ) 2

̂ ξ )| = 1 a. e. (take f (t) = g(t) = φ(t) = e−t , t ∈ ℝd ), which ends the the condition |σ(x, proof.

Hence not every element of Cohen class satisfies the marginal properties. For example, the member of Cohen class Wf ∗ σa,b , where σa,b = exp(−2π(x 2 /a + ξ 2 /b)), does not satisfy conditions (1.102) and thus it does not verify the marginal properties, whereas (1.103) is obviously satisfied after normalization. The next proposition gives a sufficient condition for being a member of the Cohen class. Theorem 1.3.21. Suppose that a quadratic time-frequency representation Q, i. e., a sesquilinear form Q : 𝒮 (ℝd ) × 𝒮 (ℝd ) → 𝒮 (ℝ2d ), is covariant (1.101) and weakly continuous, that is, 󵄨󵄨 󵄨 2 2 󵄨󵄨Q(f , g)(0, 0)󵄨󵄨󵄨 ≤ ‖f ‖2 ‖g‖2

(1.107)

for all f , g ∈ L2 (ℝd ), or f , g ∈ 𝒮 (ℝd ). Then there exists a tempered distribution σ ∈ 𝒮 󸀠 (ℝ2d ) such that Qf = Q(f , f ) = Wf ∗ σ

(1.108)

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for all f ∈ 𝒮 (ℝd ). Proof. The assumption (1.107) means that the map (f , g) ��→ Q(f , g)(0, 0) is a bounded sesquilinear form. Hence, there exists a bounded operator A on L2 (ℝd ) such that Q(f , g)(0, 0) = ⟨Af , g⟩. (cf. Theorem A.0.10 in Appendix A). Using the covariance property, we obtain that Qf can be expressed as Qf (z) = T(−z) Qf (0, 0) = Q(π(−z)f )(0, 0) = ⟨A(π(−z)f ), π(−z)f ⟩.

1.3 Ambiguity function, Wigner distribution, and related topics | 53

It follows, in particular, that A maps 𝒮 (ℝd ) into 𝒮 󸀠 (ℝd ), we can then apply Schwartz kernel theorem and thus there exists a distribution k ∈ 𝒮 󸀠 (ℝ2d ) such that ⟨Af , g⟩ = ⟨k, g ⊗ f ⟩,

∀f , g ∈ 𝒮 (ℝd ).

Recalling that the unitary operators ℱ2 and Ts are continuous bijections on 𝒮 (ℝ2d ) and 𝒮 󸀠 (ℝ2d ), and the Wigner distribution is real-valued, we deduce by Lemma 1.3.9 that Qf (0, 0) = ⟨Af , f ⟩ = ⟨k, f ⊗ f ⟩ = ⟨ℱ2 Ts k, ℱ2 Ts (f ⊗ f )⟩ = ⟨ℱ2 Ts k, Wf ⟩ = ⟨ℱ2 Ts k, Wf ⟩. Similarly, Qf (z) = Q(π(−z)f )(0, 0) = ⟨A(π(−z)f ), π(−z)f ⟩ = ⟨ℱ2 Ts k, W(π(−z)f )⟩ = ⟨ℱ2 Ts k, T(−z) Wf ⟩ = Wf ∗ ℐ (ℱ2 Ts k). The desired kernel is ℐ (ℱ2 Ts k) ∈ 𝒮 󸀠 (ℝ2d ), and this concludes the proof. 1.3.5 τ-Wigner distributions We discuss here a generalization of the Wigner distribution, depending on a parameter τ ∈ [0, 1]. For more general perturbations of the Wigner distribution, we refer to [8]. Definition 1.3.22. For τ ∈ [0, 1], f , g ∈ L2 (ℝd ), we define the (cross-)τ-Wigner distribution by Wτ (f , g)(x, ξ ) = ∫ e−2πit⋅ξ f (x + τt)g(x − (1 − τ)t) dt.

(1.109)

ℝd

For f = g, Wτ f := Wτ (f , f ) is called the τ-Wigner distribution of f .

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1/2.

So Wf is a particular case of τ-Wigner distribution corresponding to the value τ =

Proposition 1.3.23. For f , g ∈ L2 (ℝd ), the cross-τ-Wigner distribution has the following properties: (i) W1−τ (f , g) = Wτ (g, f ) for every τ ∈ [0, 1]. (ii) Wτ f (x, ξ ) = W1−τ f ̂(ξ , −x) for every τ ∈ [0, 1]. Equivalently, Wτ f ̂(z) = W1−τ f (Jz). Proof. (i) It is an immediate consequence of definition of τ-Wigner distribution. Indeed, W1−τ (f , g)(x, ξ ) = ∫ e−2πit⋅ξ f (x + (1 − τ)t)g(x − τt) dt ℝd

= ∫ e2πit⋅ξ g(x − τt)f (x + (1 − τ)t) dt ℝd

54 | 1 Basics of time–frequency representations and related properties

= ∫ e−2πit⋅ξ g(x + τt)f (x − (1 − τ)t) dt ℝd

= Wτ (g, f )(x, ξ ). (ii) We detail the computations for f ∈ 𝒮 (ℝd ), so that all the integrals below are absolutely convergent. The general case follows by standard density arguments. We have W1−τ f ̂(ξ , −x) = ∫ e2πit⋅x f ̂(ξ + (1 − τ)t)f ̂(ξ − τt) dt ℝd

= ∫ e−2πiξ ⋅(y−z) e2πit⋅(x−y+τ(y−z)) f (y)f (z) dt dy dz ℝ3

= ∫ e−2πiξ ⋅z e2πit⋅(x+τz) e−2πit⋅y f (y)f (y − z)̃ dt dy dz̃ ̃

̃

ℝ3

= ∫ e−2πiξ ⋅z e2πit⋅(x+τz) [ ∫ e−2πit⋅y f (y)Tz̃ f (y) dy] dt dz̃ ̃

̃

ℝ2d

ℝd

̃ ̃ ? = ∫ e−2πiξ ⋅z [ ∫ e2πit⋅(x+τz) f ⋅ Tz̃ f (t) dt] dz̃ ℝd

ℝd

̃ ? = ∫ e−2πiξ ⋅z ℱ −1 (f ⋅ Tz̃ f )(x + τz)̃ dz̃ ℝd

̃ (x − (1 − τ)z)̃ dz̃ = Wτ f (x, ξ ), = ∫ e−2πiξ ⋅z f (x + τz)f ̃

ℝd

as desired. The last formula immediately follows by observing that

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W1−τ (ℱ 2 f )(z) = W1−τ (ℐ f )(z) = W1−τ f (−z). From Proposition 1.3.23, we infer Wτ f + W1−τ f = 2ℜ(Wτ f ),

Wτ f − W1−τ f = 2ℑ(Wτ f ).

The τ-Wigner distribution Wτ f possesses similar properties to those of the classical Wigner distribution as stated in the next proposition. Proposition 1.3.24. The τ-Wigner distribution satisfies the marginal properties for every τ ∈ [0, 1], namely, for every f ∈ 𝒮 (ℝd ), 󵄨 󵄨2 ∫ Wτ f (x, ξ ) dx = 󵄨󵄨󵄨f ̂(ξ )󵄨󵄨󵄨 ,

󵄨 󵄨2 ∫ Wτ f (x, ξ ) dξ = 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 .

ℝd

ℝd

1.3 Ambiguity function, Wigner distribution, and related topics | 55

Proof. Since f ∈ 𝒮 (ℝd ), all the integrals below are absolutely convergent, and we can make the changes of coordinate y �→ x + τt and subsequently s �→ y − t as follows: ∫ Wτ f (x, ξ ) dx = ∫ e−2πit⋅ξ f (x + τt)f (x − (1 − τ)t) dt dx ℝd

ℝ2d

= ∫ e−2πit⋅ξ f (y)f (y − t) dy dt ℝ2d

= ∫ f (y)[ ∫ e−2πit⋅ξ f (y − t) dt] dy ℝd

ℝd

󵄨 󵄨2 = ∫ e−2πiy⋅ξ f (y) dy ∫ e−2πis⋅ξ f (s) ds = 󵄨󵄨󵄨f ̂(ξ )󵄨󵄨󵄨 . ℝd

ℝd

Using Proposition 1.3.23 (ii), we obtain the second equality as well: ∫ Wτ f (x, ξ ) dξ = ∫ W1−τ f ̂(ξ , −x) dξ ℝd

ℝd

󵄨 ̂ 󵄨2 󵄨 󵄨2 󵄨 󵄨2 = 󵄨󵄨󵄨f ̂(−x)󵄨󵄨󵄨 = 󵄨󵄨󵄨ℱ −1 (f ̂)(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 .

Moreover, Wτ (f , g) has a factorization similar to that of Wigner distribution.

Proposition 1.3.25. For f , g ∈ L2 (ℝd ),

Wτ (f , g)(x, ξ ) = ℱt→ξ [T [x] 1

( 2 −τ)t

Ts (f ⊗ g)](x, ξ ),

(1.110)

where Ts is the symmetric coordinate change (1.87) and T [x] denotes the translation operator acting on the x-variable. Proof. By definition of Wτ (f , g), we have

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Wτ (f , g)(x, ξ ) = ∫ e−2πit⋅ξ f (x + τt)g(x − (1 − τ)t) dt ℝd

= ∫ e−2πit⋅ξ T [x] 1

( 2 −τ)t

ℝd

= ℱt→ξ [T [x] 1

( 2 −τ)t

Ts (f ⊗ g)(x, t) dt

Ts (f ⊗ g)](x, ξ ),

as desired. Observe that, since the linear operators appearing in (1.110) are isomorphisms on

𝒮 (ℝ2d ) and L2 (ℝ2d ), we immediately obtain

Corollary 1.3.26. For τ ∈ [0, 1], we have (i) f , g ∈ 𝒮 (ℝd ) �⇒ Wτ (f , g) ∈ 𝒮 (ℝ2d ); (ii) f , g ∈ L2 (ℝd ) �⇒ Wτ (f , g) ∈ L2 (ℝ2d ).

56 | 1 Basics of time–frequency representations and related properties Proposition 1.3.27. The τ-Wigner distribution belongs to the Cohen class for every τ ∈ [0, 1]. In particular, Wτ (f , g) = W(f , g) ∗ στ ,

∀f , g ∈ 𝒮 (ℝd )

(1.111)

where the kernel στ ∈ 𝒮 󸀠 (ℝ2d ) is given by 2d

{ de στ (x, ξ ) = { |2τ−1| {δ,

2 x⋅ξ 2πi 2τ−1

,

τ ≠ 21 ,

(1.112)

τ = 21 .

Proof. If τ = 21 , Wτ (f , g) coincides with the classical cross-Wigner distribution and so the claim becomes trivial. For τ ≠ 21 , applying the inverse Fourier transform to (1.111), the claim turns into ℱ (Wτ (f , g))(ν, t) = (ℱ W(f , g)) ⋅ (ℱ στ )(ν, t), −1

−1

−1

(1.113)

for all f , g ∈ 𝒮 (ℝd ). Let us to compute ℱ −1 στ (ν, t). For short, we use integrals in the weak sense, application to test functions in 𝒮 (ℝd ) being understood. Since τ ≠ 21 , by (1.112), ℱ στ (ν, t) = −1

2 2d ∫ e2πiν⋅x e2πit⋅ξ e2πi 2τ−1 x⋅ξ dx dξ d |2τ − 1|

ℝ2d

d

=

2 2τ−1 2 ∫ e2πi( 2τ−1 x+t)⋅(ξ + 2 ν) e−πi(2τ−1)t⋅ν dx dξ . d |2τ − 1|

ℝ2d

By making the change of variables

2 x 2τ−1

ℱ στ (ν, t) = e −1

+ t = y and ξ +

−πi(2τ−1)t⋅ν

2τ−1 ν 2

= ρ, we get

∫ e2πiy⋅ρ dy dρ,

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ℝ2d

consequently ℱ στ (ν, t) = e −1

−πi(2τ−1)t⋅ν

(1.114)

,

since ∫ℝ2d e2πiy⋅ρ dy dρ = 1. Let us now compute the left-hand side of (1.113). To this end we use the factorization (1.110), the factorization of Wigner distribution (1.89), and the decomposition of the inverse Fourier transform ℱ −1 = ℱ1−1 ℱ2−1 as follows: ℱ (Wτ (f , g))(ν, t) = ℱx→ν ℱξ →t ℱt→ξ [T −1

−1

−1

−1 = ℱx→ν [T [x] 1

( 2 −τ)t

[x] T (f ( 21 −τ)t s

Ts (f ⊗ g)](ν, t)

⊗ g)](ν, t)

1.3 Ambiguity function, Wigner distribution, and related topics | 57

= M [ν] 1

ℱx→ν (Ts (f ⊗ g))(ν, t) −1

( 2 −τ)t

−1 −1 ℱξ →t (ℱt→ξ (Ts (f ⊗ g)))(ν, t) = e−πi(2τ−1)t⋅ν ℱx→ν

= e−πi(2τ−1)t⋅ν ℱ −1 (W(f , g))(ν, t). In view of (1.114), this completes the proof. Thanks to this proposition, we can give an alternative proof of the marginal properties of Wτ . Since the τ-Wigner distribution is a member of Cohen class, applying Proposition 1.3.19 and Remark 1.3.20, we can easily see that the kernel στ in (1.112) satisfies conditions (1.105). Indeed, for τ ≠ 21 , ℱ στ (x, 0) =

2 2d ∫ e−2πix⋅t e2πi 2τ−1 t⋅ξ dt dξ d |2τ − 1|

ℝ2d

d

=

2 2 ∫ e2πit⋅( 2τ−1 ξ −x) dt dξ d |2τ − 1|

ℝ2d

= ∫ e2πit⋅ω dt dω = 1, ℝ2d

where again integrals are in the weak sense. Similarly, one can check that ℱ σ(0, ξ ) = 1. Corollary 1.3.28 (Moyal’s formula for τ-Wigner distributions). For τ (cross-)τ-Wigner distribution satisfies Moyal’s formula ⟨Wτ (f1 , g1 ), Wτ (f2 , g2 )⟩ = ⟨f1 , f2 ⟩⟨g1 , g2 ⟩,

∈

[0, 1], the

f1 , f2 , g1 , g2 ∈ L2 (ℝd ).

(1.115)

Proof. It is enough to prove (1.115) for f1 = f2 , g1 = g2 and then apply the polarization identity. The Cohen kernel στ of Wτ f is given by (1.112). Hence, using (1.114), we compute ℱ (στ )(x, ξ ) = ℱ (στ )(−x, −ξ ) = ℱ (στ )(x, ξ ) = e Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

−1

−1

−πi(2τ−1)x⋅ξ

so that 󵄨󵄨 ̂ 󵄨 󵄨󵄨στ (x, ξ )󵄨󵄨󵄨 = 1,

∀(x, ξ ) ∈ ℝ2d ,

and condition (1.104) is satisfied. This completes the proof. We now want to find a relation between the τ-Wigner distribution and the STFT which generalizes Lemma 1.3.5 for the Wigner distribution. To this aim, we need to introduce a new operator depending on parameter τ ∈ [0, 1], which plays the same role of the reflection operator in (1.80). Definition 1.3.29. For τ ∈ (0, 1), we define the operator Aτ as follows: Aτ : f (t) ��→ ℐ f (

1−τ t). τ

(1.116)

58 | 1 Basics of time–frequency representations and related properties For τ ∈ (0, 1) and f ∈ Lp (ℝd ), 1 ≤ p ≤ ∞, we obtain d

‖Aτ f ‖Lp (ℝd ) =

|τ| p d

|τ − 1| p

‖f ‖Lp (ℝd ) ,

with the convention d/∞ = 0. Proposition 1.3.30. For τ ∈ (0, 1), we have Wτ (f , g)(x, ξ ) =

1 1 1 2πi τ1 x⋅ξ e VAτ g f ( x, ξ ), 1−τ τ τd

f , g ∈ L2 (ℝd ).

(1.117)

Proof. Using definition of τ-Wigner distribution and performing the change of variables y = x + τt, we can write Wτ (f , g)(x, ξ ) = ∫ e−2πit⋅ξ f (x + τt)g(x − (1 − τ)t) dt ℝd

=

ξ 1 2πi τ1 ξ ⋅x 1−τ x −2πiq⋅ τ e e f (y)ℐ g( y − ) dy ∫ τ τ τd

ℝd

ξ 1 1 1 = d e2πi τ ξ ⋅x ∫ e−2πiy⋅ τ f (y)Aτ g(y − x) dy τ−1 τ

ℝd

1 1 1 1 = d e2πi τ ξ ⋅x VAτ g f ( x, ξ ), 1−τ τ τ

as required. It remains to consider the cases τ = 0 and τ = 1. For τ = 0, W0 (f , g) is the wellknown (cross-)Rihaczek distribution. In detail, ̂ ), W0 (f , g)(x, ξ ) = R(f , g)(x, ξ ) = e−2πix⋅ξ f (x)g(ξ

f , g ∈ L2 (ℝd ).

(1.118)

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For τ = 1, W1 (f , g) is the conjugate-(cross-)Rihaczek distribution R∗ (f , g) given by W1 (f , g)(x, ξ ) = R∗ (f , g)(x, ξ ) = R(g, f ) = e2πix⋅ξ g(x)f ̂(ξ ),

f , g ∈ L2 (ℝd ).

(1.119)

Moreover, we have R(f , g) = W(1, δ) ∗ W(f , g) = e−4πix⋅ξ ∗ W(f , g)(x, ξ ),

R∗ (f , g) = W(δ, 1) ∗ W(f , g) = e4πix⋅ξ ∗ W(f , g)(x, ξ ),

hence recapturing the Cohen kernels σ0 , σ1 in (1.112). Using (1.33), we obtain the following relationship between the Fourier transform of the STFT and the cross-Rihaczek distribution: ̂f (x, ξ ) = W (f , g)(−ξ , x) = W (f , g)(−J(x, ξ )), V g 0 0

(x, ξ ) ∈ ℝ2d .

(1.120)

1.3 Ambiguity function, Wigner distribution, and related topics | 59

In terms of the symplectic Fourier transform (cf. (1.18)), ℱσ Vg f (z) = W0 (f , g)(z),

z ∈ ℝ2d ,

f , g ∈ L2 (ℝd ),

(1.121)

or, in terms of the conjugate-(cross-)-Rihaczek distribution, ℱσ Vg f (z) = W1 (g, f )(z),

z ∈ ℝ2d ,

f , g ∈ L2 (ℝd ).

(1.122)

It is then natural to determine the symplectic Fourier transform of a τ-Wigner distribution. In what follows we write fλ (t) = f (λt), λ ∈ ℝ, for short. Lemma 1.3.31. For τ ∈ (0, 1), f , g ∈ L2 (ℝd ), we have d

d

ℱσ Wτ (f , g)(y, η) = τ (1 − τ) W1−τ (fτ , (ℐ g)1−τ )(y, τ(1 − τ)η),

(y, η) ∈ ℝ2d .

(1.123)

Proof. Using the inversion formula of the Fourier transform in 𝒮 󸀠 (ℝd ), the relation ℱσ Wτ (f , g)(y, η) = ℱ Wτ (f , g)(η, −y) = ℱ1 ℱ2 Wτ (f , g)(η, −y), and the factorization (1.110), we can write 2

ℱσ Wτ (f , g)(y, η) = ℱ Wτ (f , g)(η, −y) = ℱ1 ℱ2 [T

[t] T (f ( 21 −τ)(−y) s

⊗ g)](η, −y)

= ∫ e−2πiη⋅t f (t + τy)g(t − (1 − τ)y) dt ℝd

1 1 = ∫ e−2πiη⋅t f (τ(y + t))g(−(1 − τ)(y − t)) dt τ 1−τ ℝd

= τd (1 − τ)d ∫ e−2πiη⋅τ(1−τ)t f (τ(y + (1 − τ)t 󸀠 ))g(−(1 − τ)(y − τt 󸀠 )) dt 󸀠 󸀠

ℝd

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where we made the change of variables t = τ(1 − τ)t 󸀠 . Corollary 1.3.32. For f , g ∈ L2 (ℝd ), we have the following formulas: (i) ℱσ W(f , g)(y, η) = 2

−2d

η W(f 1 , (ℐ g) 1 )(y, ), 2 2 4

(y, η) ∈ ℝ2d ;

(1.124)

(ii) y η ), 2 2

ℱσ W(f , g)(y, η) = 2 W(f , ℐ g)( , −d

Proof. (i) It follows by choosing τ = 1/2 in formula (1.123).

(y, η) ∈ ℝ2d .

(1.125)

60 | 1 Basics of time–frequency representations and related properties (ii) Using the inversion formula of the Fourier transform, ℱσ W(f , g)(y, η) = ∫ e

−2πiη⋅t

ℝd

y y f (t + )g(t − ) dt 2 2

t󸀠 y t󸀠 y t󸀠 = 2−d ∫ e−2πiη⋅ 2 f ( + )g(−( − )) dt 󸀠 2 2 2 2

ℝd

η 󸀠 y t󸀠 y t󸀠 = 2−d ∫ e−2πi 2 ⋅t f ( + )(ℐ g)( − ) dt 󸀠 2 2 2 2

ℝd

y η = W(f , ℐ g)( , ), 2 2 as desired. For more general perturbations of the Wigner distribution and a characterization of the Cohen class, we refer to [90]. 1.3.6 Time–frequency representations of Gaussians In what follows we shall compute STFT and Wigner distributions related to Gaussian functions. Such formulas will be very useful in the next chapters to prove the sharpness of continuity properties. To shorten notation, in what follows we write x ⋅ ξ = xξ and x2 = x ⋅ x. Proposition 1.3.33. Given a, b, c > 0, consider the generalized Gaussian function 2

2

f (x, ξ ) = fa,b,c (x, ξ ) = e−πax e−πbξ e2πicxξ ,

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

For Φ(x, ξ ) = e−π(x

2

VΦ f (z, ζ ) =

+ξ 2 )

(x, ξ ) ∈ ℝ2d .

(1.126)

, z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ ℝ2d , we obtain 1

[(a + 1)(b + 1) + c2 ] ×e

d 2

e

2πi − a+1 [z1 ζ1 +(cz1 −(a+1)ζ2 )

−π

[a(b+1)+c2 ]z12 +[(a+1)b+c2 ]z22 +(b+1)ζ12 +(a+1)ζ22 −2c(z1 ζ2 +z2 ζ1 ) (a+1)(b+1)+c2

cζ1 +(a+1)z2 (a+1)(b+1)+c2

]

(1.127)

.

Proof. We write VΦ f (z, ζ ) = ∫ e−πax

2

−πbξ 2 +2πicxξ −2πi(ζ1 x+ζ2 ξ ) −π[(x−z1 )2 +(ξ −z2 )2 ]

e

e

ℝ2d 2

2

= e−π(z1 +z2 ) ∫ ( ∫ e−π[(a+1)x ℝd

× e−π[(b+1)ξ

2

2

ℝd

−2ξz2 ] −2πiζ2 ξ

e

e

−2xz1 ] −2πix(ζ1 −cξ )

dξ

dx)

dx dξ

1.3 Ambiguity function, Wigner distribution, and related topics | 61

1

2

2

= e−π(1− a+1 )z1 −πz2 ∫ ( ∫ e × e−π[(b+1)ξ

ℝd

2

e

dx)

ℝd

−2ξz2 ] −2πiζ2 ξ

e

z1 √a+1

With the change of variables √a + 1x − VΦ f (z, ζ ) =

z

1 ]2 −π[√a+1x− √a+1 −2πix(ζ1 −cξ )

dξ .

= t, dx =

dt , (a+1)d/2

we obtain

z1 t a 2 2 2 1 −2πi( √a+1 )(ζ1 −cξ ) + a+1 dt) e−π a+1 z1 −πz2 ∫ ( ∫ e−πt e d/2 (a + 1)

× e−π[(b+1)ξ

2

ℝd ℝd

−2ξz2 ] −2πiζ2 ξ

e

dξ

(ζ −cξ )2 cz1 z1 ζ1 a 2 1 −π a+1 z1 −πz22 −2πi a+1 −π 1a+1 = e e e2πi a+1 ξ ∫ d/2 (a + 1)

×e

−π[(b+1)ξ 2 −2ξz2 ] −2πiζ2 ξ

e

ℝd

dξ

ζ1 z1 ζ1 a 2 2 1 e−π a+1 z1 −πz2 −2πi a+1 −π a+1 = d/2 (a + 1) 2

× ∫ e−π[(b+1)ξ

2

2

cζ

cz

c + a+1 ξ 2 −2 a+11 ξ −2ξz2 ] 2πi( a+11 −ζ2 )ξ

e

dξ .

ℝd

The last integral can be computed as follows: I := ∫ e−π[

(a+1)(b+1)+c2 a+1

ξ 2 −2

cζ1 +z2 (a+1) ξ] a+1

cz1

e2πi( a+1 −ζ2 )ξ dξ

ℝd π

= ∫ e− a+1 {[(a+1)(b+1)+c

2

cz

]ξ 2 −2[cζ1 +z2 (a+1)]ξ } 2πi( a+11 −ζ2 )ξ

e

dξ

ℝd

= ∫e

π √ [ (a+1)(b+1)+c2 ξ − − a+1

cζ1 +z2 (a+1) ]2 √(a+1)(b+1)+c2

π

e a+1

[cζ1 +z2 (a+1)]2 (a+1)(b+1)+c2

cz1

e2πi( a+1 −ζ2 )ξ dξ

ℝd π

= e a+1

[cζ1 +z2 (a+1)]2 (a+1)(b+1)+c2

∫e

cζ1 +z2 (a+1) ]2 √(a+1)(b+1)+c2

π √ − a+1 [ (a+1)(b+1)+c2 ξ −

cz1

e2πi( a+1 −ζ2 )ξ dξ .

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

ℝd

Making the change of variables t = dξ = [

√(a+1)(b+1)+c2 ξ − √a+1

cζ1 +z2 (a+1) √(a+1)(b+1)+c2

d/2

a+1 ] (a + 1)(b + 1) + c2

, so that

dt,

we can write d

I=

(a + 1) 2

π

[(a + 1)(b + 1) + c2 ] 2

× ∫ e−πt e ℝd

cz

d 2

2πi( a+11 −ζ2 )[

e a+1

[cζ1 +z2 (a+1)]2 (a+1)(b+1)+c2

√a+1

√(a+1)(b+1)+c2

t+

cζ1 +(a+1)z2 (a+1)(b+1)+c2

]

dt

(1.128)

62 | 1 Basics of time–frequency representations and related properties d

=

(a + 1) 2

π

[(a + 1)(b + 1) + c2 ] ×e

−π[

a+1 (a+1)(b+1)+c2

e a+1

d 2

[cζ1 +z2 (a+1)]2 (a+1)(b+1)+c2

e

cz

−2πi( a+11 −ζ2 )

cζ1 +(a+1)z2 (a+1)(b+1)+c2

cz

( a+11 −ζ2 )2 ]

.

The result then follows by substituting the value of the integral I in (1.128). 2

2

Lemma 1.3.34. Consider φ(t) = e−πt and its rescaled version φ√λ (t) = e−πλt . Then W(φ, φ√λ )(x, ξ ) =

2d

4πλ 2

(λ + 1)

d 2

4π

λ−1

2

e− λ+1 x e− λ+1 ξ e−4πi λ+1 xξ .

(1.129)

Proof. The proof is a straightforward computation. Indeed, t 2

t 2

W(φ, φ√λ )(x, ξ ) = ∫ e−π(x+ 2 ) −πλ(x− 2 ) e−2πitξ dt ℝd π

2

= e−π(1+λ)x ∫ e− 4 [(1+λ)t

2

e

+4(1−λ)xt] −2πitξ

dt

ℝd 2

= e−π(1+λ)x ∫ e

2

(1−λ) 2 − π4 (√1+λt+2 √ x) π (1−λ) x2 −2πitξ 1+λ 1+λ

e

e

dt

ℝd

= e−π[(1+λ)−

(1−λ)2 1+λ

]x 2

∫e

(1−λ) 2 − π4 (√1+λt+2 √ x) −2πitξ 1+λ

e

dt.

ℝd

Making the change of variables 2u = √1 + λt + 2 4λ

(1 − λ) x, 1+λ

2

2

W(φ, φ√λ )(x, ξ ) = e−π 1+λ x ∫ e−πu e

2u

t=

√1 + λ

− 2(1−λ) x)ξ −2πi( √2u 1+λ 1+λ

ℝd

−2

(1 − λ) x, √1 + λ

dt =

2d du (1 + λ)d/2

2d du, (1 + λ)d/2

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

d

=

2ξ 4λ 2 1−λ 2 2 −2πiu( √1+λ ) du e−π 1+λ x e4πi 1+λ xξ ∫ e−πu e d/2 (1 + λ)

ℝd

d

=

2 e−π (1 + λ)d/2

4λ 2 x 1+λ

e−π

4 2 ξ 1+λ

1−λ

e4πi 1+λ xξ

as desired. We now get Corollary 1.3.35. Consider φ and φ√λ as in the assumptions of Lemma 1.3.34. Then ℱ W(φ, φ√λ )(ζ1 , ζ2 ) =

1 (λ + 1)

π

d 2

2

πλ

2

λ−1

e− λ+1 ζ1 e− λ+1 ζ2 e−πi λ+1 ζ1 ζ2 .

(1.130)

1.3 Ambiguity function, Wigner distribution, and related topics | 63

Proof. Formula (1.130) is easily obtained from (1.129) using well-known Gaussian integral formulas; it can also be painlessly obtained from (1.129) and (1.125), we leave the details to the interested reader. Observe that ℱσ F(ζ ) = ℱ F(Jζ ). Hence the symplectic Fourier transform of the Wigner distribution (1.129) is given by 1

ℱσ W(φ, φ√λ )(ζ1 , ζ2 ) =

πλ

(λ + 1)

d 2

2

π

λ−1

2

e− λ+1 ζ1 e− λ+1 ζ2 eπi λ+1 ζ1 ζ2 .

(1.131)

We now study the STFT of a rescaled Gaussian by a matrix A ∈ GL(d, ℝ). Given a function f on ℝd and A ∈ GL(d, ℝ), we set fA (t) = f (At). We also consider the unitary operator 𝒰A on L2 (ℝd ) defined by 1/2

𝒰A f (t) = | det A|

f (At) = | det A|1/2 fA (t).

(1.132)

2

Lemma 1.3.36. Let A ∈ GL(d, ℝ), φ(t) = e−πt , then T

Vφ φA (x, ξ ) = cA e−π(I−(A

A+I)−1 )x2

M−((AT A+I)−1 )x e−π(A

T

A+I)−1 ξ 2

,

where cA = (det(AT A + I))−1/2 . Proof. By definition of the STFT, 2

Vφ φA (x, ξ ) = ∫ e−πAy⋅Ay e−2πiξ ⋅y e−π(y−x) dy ℝd 2

T

= e−πx ∫ e−π(A

A+I)y⋅y+2πx⋅y −2πiξ ⋅y

e

dy.

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ℝd

Now, we rewrite the generalized Gaussian above using the translation and dilation operators, that is, T

e−π(A

A+I)y2 +2πxy

T

= cA1/2 eπ(A

A+I)−1 x2

(T(AT A+I)−1 x 𝒰(AT A+I)1/2 )φ(y),

and use the properties ℱ𝒰B = 𝒰(BT )−1 ℱ , for every B ∈ GL(d, ℝ), and ℱ Tx = M−x ℱ . Thereby, T

Vφ φA (x, ξ ) = cA1/2 e−π(I−(A T

= cA e−π(I−(A as desired.

A+I)−1 )x2

A+I) )x −1

2

ℱ (T(AT A+I)−1 x 𝒰(AT A+I)1/2 )φ(ξ )

M−(AT A+I)−1 x e−π(A

T

A+I)−1 ξ 2

,

64 | 1 Basics of time–frequency representations and related properties 1.3.7 Short-time Fourier transform of time–frequency representations To investigate the local properties of the STFT and the Wigner distribution, we will need to compute their STFTs. We remark that the STFT of a function on ℝ2d is a function on ℝ4d , so one has to be careful when dealing with the STFT Vg f (x, ξ ), (x, ξ ) ∈ ℝ2d , of f ∈ 𝒮 󸀠 (ℝd ), and the STFT VΦ F(z, ζ ), (z, ζ ) ∈ ℝ4d , of F ∈ 𝒮 󸀠 (ℝ2d ). We write z = (z1 , z2 ) ∈ ℝ2d and ζ = (ζ1 , ζ2 ) ∈ ℝ2d , when necessary. Lemma 1.3.37. Fix a nonzero φ ∈ 𝒮 (ℝd ) and let f , g ∈ 𝒮 (ℝd ). (i) Set Φ = Vφ φ ∈ 𝒮 (ℝ2d ). Then the STFT of Vg f with respect to the window Φ is given by VΦ (Vg f )(z, ζ ) = e−2πiz2 ⋅ζ2 Vφ g(−z1 − ζ2 , ζ1 )Vφ f (−ζ2 , z2 + ζ1 ),

(1.133)

where z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ ℝ2d . (ii) Let Φ = W(φ, φ) ∈ 𝒮 (ℝ2d ). Then the STFT of W(g, f ) with respect to the window Φ is given by VΦ (W(g, f ))(z, ζ ) = e−2πiz2 ⋅ζ2 Vφ g(z1 −

ζ2 ζ ζ ζ , z2 + 1 )Vφ f (z1 + 2 , z2 − 1 ), 2 2 2 2

(1.134)

where z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ ℝ2d . Proof. Before we calculate VΦ (Vg f )(z, ζ ) = ⟨Vg f , Mζ Tz Φ⟩ = (Vg fTz Φ)̂(ζ ), we rewrite the time–frequency shift Mζ Tz Φ. We use Proposition 1.2.15 to evaluate Tz Φ and find that Tz Φ(x, ξ ) = Vφ φ(x − z1 , ξ − z2 ) = e2πi(ξ −z2 )⋅z1 Vφ (Mz2 Tz1 φ)(x, ξ ).

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Now we substitute this expression into the formula for VΦ (Vg f ), and after rearranging some terms, we apply Proposition 1.2.13 (ii) to get VΦ (Vg f )(z, ζ ) = ⟨Vg f , Mζ Tz Φ⟩ = ∬ Vg f (x, ξ )Vφ (Mz2 Tz1 φ)(x, ξ )e−2πi[(x⋅ζ1 +ξ ⋅ζ2 )+(ξ −z2 )⋅z1 ] dx dξ ℝ2d

= e2πiz1 ⋅z2 ∬ Vg f (x, ξ )Vφ (Mz2 Tz1 φ)(x, ξ )e−2πi[ζ1 ⋅x+(z1 +ζ2 )⋅ξ ] dx dξ =e

2πiz1 ⋅z2

ℝ2d

(Vg f ⋅ Vφ (Mz2 Tz1 φ))̂(ζ1 , z1 + ζ2 )

= e2πiz1 ⋅z2 (V(Mz

T φ) 2 z1

=e This proves (i).

−2πiz2 ⋅ζ2

f ⋅ Vφ g)(−z1 − ζ2 , ζ1 )

(Vφ f )(−ζ2 , z2 + ζ1 )Vφ g(−z1 − ζ2 , ζ1 ).

1.3 Ambiguity function, Wigner distribution, and related topics | 65

We now work out the case (ii). Using Lemma 1.3.5 and formulas (1.83) and (1.47), we compute VΦ (W(g, f ))(z, ζ ) = ⟨W(g, f ), Mζ Tz Φ⟩ = ⟨W(g, f ), Mζ Tz W(φ, φ)⟩ = ∫ W(g, f )(x, ξ )W(φ, φ)(x − z1 , ξ − z2 )e−2πi(x⋅ζ1 +ξ ⋅ζ2 ) dx dξ ℝ2d

= 22d ∫ Vℐf g(2x, 2ξ )VℐTz ℝ2d −2πi(x⋅ζ1 +ξ ⋅ζ2 )

×e

1

Mz2 φ (Tz1 Mz2 φ)(2x, 2ξ )

dx dξ ζ2

ζ1

= ∫ Vℐf g(x, ξ )e2πi(−x⋅ 2 −ξ ⋅ 2 ) VℐTz

1

ℝ2d

= ∫ VT ζ

2 2

ℝ2d

× VℐTz

1

= ⟨VT ζ

2 2

M

M

ζ − 21

ℐf (T ζ2 2

Mz2 φ (Tz1 Mz2 φ)(x, ξ ) dx dξ

M− ζ1 g)(x, ξ ) 2

Mz2 φ (Tz1 Mz2 φ)(x, ξ ) dx dξ

ζ − 21

ℐf (T ζ2 2

M− ζ1 g), VℐTz

1

2

Mz2 φ (Tz1 Mz2 φ)⟩.

We can apply the orthogonality relations to obtain VΦ (W(g, f ))(z, ζ ) = ⟨T ζ2 M− ζ1 g, Tz1 Mz2 φ⟩⟨T ζ2 M− ζ1 ℐ f , ℐ Tz1 Mz2 φ⟩. 2

2

2

2

Applying the commutation relations (0.14) to the first factor on the right-hand side, we have ζ2

⟨T ζ2 M− ζ1 g, Tz1 Mz2 φ⟩ = e2πiz2 ⋅(z1 − 2 ) Vφ g(z1 − 2

2

ζ2 ζ , z + 1 ). 2 2 2

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Whereas, observing that the operator ℐ is unitary, the second factor becomes ⟨T ζ2 M− ζ1 ℐ f , ℐ Tz1 Mz2 φ⟩ = ⟨f , M− ζ1 Tz + ζ2 Mz2 φ⟩ 2

2

2

=e

ζ 2πiz2 ⋅(z1 + 22

1

)

2

Vφ f (z1 +

ζ2 ζ , z − 1 ), 2 2 2

thereby proving our claim. We now generalize formula (1.134) to τ-Wigner distributions defined in (1.109). An easy computation shows the following commutation relation between the operators Aτ in (1.116) and the time–frequency shifts π(y, η), (y, η) ∈ ℝ2d : π(y, η)Aτ = Aτ π(−

1−τ τ y, − η), τ 1−τ

τ ∈ (0, 1).

(1.135)

66 | 1 Basics of time–frequency representations and related properties Lemma 1.3.38. Consider τ ∈ (0, 1). Let φ1 , φ2 ∈ 𝒮 (ℝd ) f , g ∈ 𝒮 (ℝd ) and set Φτ = Wτ (φ1 , φ2 ). Then VΦτ Wτ (g, f )(z, ζ ) = e−2πiz2 ⋅ζ2 Vφ1 g(z1 − τζ2 , z2 + (1 − τ)ζ1 )

(1.136)

× Vφ2 f (z1 + (1 − τ)ζ2 , z2 − τζ1 ) where z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ ℝ2d . Proof. The argument follows the pattern of the τ = 1/2-case, already shown in Lemma 1.3.37 (ii). Using the covariant property (1.101) and the representation of the τ-Wigner distribution as an STFT in (1.117), VΦτ Wτ (g, f )(z, ζ ) = ⟨Wτ (g, f ), Mζ Tz Wτ (φ1 , φ2 )⟩ = ⟨Wτ (g, f ), Mζ Wτ (π(z)φ1 , π(z)φ2 )⟩ =

1 1 1 1 1 x, ξ )e−2πi(x,ξ )⋅(ζ1 ,ζ2 ) VAτ π(z)φ2 π(z)φ1 ( x, ξ ) dx dξ ∫ VAτ f g( 1−τ τ 1−τ τ τ2d ℝ2d

=

(1 − τ)d ∫ VAτ f g(x, ξ )e−2πi(ζ1 ,ζ2 )⋅((1−τ)x,τξ ) VAτ π(z)φ2 π(z)φ1 (x, ξ ) dx dξ . τd ℝ2d

To shorten notation, we write cτ =

(1 − τ)d . τd

Using formula (1.47), the orthogonality relations (1.44), and the commutation relations between π and Aτ in (1.135), we can write

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VΦτ Wτ (g, f )(z, ζ ) = cτ ∫ Vπ(τζ2 ,−(1−τ)ζ1 )Aτ f π(τζ2 , −(1 − τ)ζ1 )gVAτ π(z)φ2 π(z)φ1 (x, ξ ) dx dξ ℝ2d

= cτ ⟨π(τζ2 , −(1 − τ)ζ1 )g, π(z1 , z2 )φ1 ⟩⟨π(τζ2 , −(1 − τ)ζ1 )Aτ f , Aτ π(z1 , z2 )φ2 ⟩ = cτ e−2πiτz2 ⋅ζ2 ⟨g, π(z1 − τζ2 , z2 + (1 − τ)ζ1 )φ1 ⟩ × ⟨Aτ f , π(−τζ2 , (1 − τ)ζ1 )Aτ π(z1 , z2 )φ2 ⟩ = cτ e−2πiτz2 ⋅ζ2 ⟨g, π(z1 − τζ2 , z2 + (1 − τ)ζ1 )φ1 ⟩ × ⟨Aτ f , Aτ π((1 − τ)ζ2 , −τζ1 )π(z1 , z2 )φ2 ⟩ = e−2πiτz2 ⋅ζ2 e−2πi(1−τ)z2 ⋅ζ2 ⟨g, π(z1 − τζ2 , z2 + (1 − τ)ζ1 )φ1 ⟩

1.3 Ambiguity function, Wigner distribution, and related topics | 67

× ⟨f , π(z1 + (1 − τ)ζ2 , z2 − τζ1 )φ2 ⟩ = e−2πiz2 ⋅ζ2 Vφ1 g(z1 − τζ2 , z2 + (1 − τ)ζ1 )Vφ2 f (z1 + (1 − τ)ζ2 , z2 − τζ1 ). The claim is proved. We now consider the case τ = 0 and compute the STFT of the (cross-)Rihaczek distribution defined in (1.118). Lemma 1.3.39 (STFT of the Rihaczek distribution). Let φ1 , φ2 ∈ 𝒮 (ℝd ), f , g ∈ 𝒮 (ℝd ) and set Φ0 = W0 (φ1 , φ2 ). Then VΦ0 W0 (g, f )(z, ζ ) = e−2πiz2 ⋅ζ2 Vφ1 g(z1 , z2 + ζ1 )Vφ2 f (z1 + ζ2 , z2 )

(1.137)

where z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ ℝ2d . Proof. We use the definition in (1.118) and formula (1.47) in the following computations: VΦ0 W0 (g, f )(z, ζ ) = ⟨W0 (g, f ), Mζ Tz W0 (φ1 , φ2 )⟩ ̂2 (ξ − z2 ) dx dξ = ∫ e−2πix⋅ξ g(x)f ̂(ξ )e−2πi(x⋅ζ1 +ξ ⋅ζ2 ) e2πi(x−z1 )⋅(ξ −z2 ) φ1 (x − z1 )φ ℝ2d

̂2 (ξ − z2 ) dξ = e2πiz1 ⋅z2 ∫ e−2πix⋅(z2 +ζ1 ) g(x)φ1 (x − z1 ) dx ∫ f ̂(ξ )e−2πiξ ⋅(z1 +ζ2 ) φ ℝd

ℝd

= e2πiz1 ⋅z2 Vφ1 g(z1 , z2 + ζ1 )Vφ̂2 f ̂(z2 , −(z1 + ζ2 ))

= e2πiz1 ⋅z2 Vφ1 g(z1 , z2 + ζ1 )Vφ2 f (z1 + ζ2 , z2 )e−2πiz2 ⋅(z1 +ζ2 )

= e−2πiz2 ⋅ζ2 Vφ1 g(z1 , z2 + ζ1 )Vφ2 f (z1 + ζ2 , z2 ),

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as desired. Corollary 1.3.40 (STFT of the conjugate-Rihaczek distribution). Let φ1 , φ2 ∈ 𝒮 (ℝd ), f , g ∈ 𝒮 (ℝd ) and set Φ1 = W1 (φ1 , φ2 ). Then VΦ1 W1 (g, f )(z, ζ ) = e−2πiz2 ⋅ζ2 Vφ1 g(z1 − ζ2 , z2 )Vφ2 f (z1 , z2 − ζ1 ),

(1.138)

where z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ ℝ2d . Proof. Using the connection between the Rihaczek and the conjugate-Rihaczek distribution in (1.119) and the result of Lemma 1.3.39, we can write VΦ1 W1 (g, f )(z, ζ ) = ⟨W1 (g, f ), Mζ Tz W1 (φ1 , φ2 )⟩ = ⟨W0 (f , g), Mζ Tz W0 (φ2 , φ1 ) = ⟨W0 (f , g), M−ζ Tz W0 (φ2 , φ1 )⟩

68 | 1 Basics of time–frequency representations and related properties = VW0 (φ2 ,φ1 ) W0 (f , g)(z, −ζ ) = e2πiz2 ⋅ζ2 Vφ2 f (z1 , z2 − ζ1 )Vφ1 g(z1 − ζ2 , z2 ) = e−2πiz2 ⋅ζ2 Vφ1 g(z1 − ζ2 , z2 )Vφ2 f (z1 , z2 − ζ1 ). The proof is completed.

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Remark 1.3.41. (i) Heuristically, formulas (1.137) and (1.138) can be inferred by putting τ = 0 and τ = 1, respectively, into expression (1.136). (ii) The STFT of a multilinear version of the Rihaczek distribution was computed in [13, Lemma 3.3], cf. formula (3.3). However, there is a flaw in the phase factor of that formula. Indeed, the exponential e2πiu0 ⋅(u1 +⋅⋅⋅+um ) should be replaced by m e2πi ∑i=1 ui ⋅vi , as the linear case m = 1 in (1.137) shows.

2 Function spaces The modern functional analysis requires a precise setting of the involved function spaces as a preliminary step in the study of mathematical problems. The Lebesgue spaces Lp , 1 ≤ p ≤ ∞, play a relevant role in this connection. Once the short-time Fourier transform (STFT) Vg f (x, ξ ) is introduced as a basic tool of the time–frequency analysis, it is then natural to define Lebesgue spaces involving simultaneously x and ξ variables. Namely, for a function or distribution f , one defines for a couple of indices p, q, with 1 ≤ p, q ≤ ∞, and a weight function m(x, ξ ) in ℝ2d , the mixed Lebesgue norm q

1

p q 󵄨 󵄨p ‖f ‖Mmp,q (ℝd ) = ( ∫ ( ∫ 󵄨󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 m(x, ξ )p dx) dξ ) .

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ℝd ℝd

p,q According to H. G. Feichtinger, in the early 1980s, the corresponding Mm (ℝd ) were p d called modulation spaces. If p = q, the notation is shortened to Mm (ℝ ). If m ≡ 1, the spaces are simply denoted by M p,q (ℝd ), M p (ℝd ). It is important to acknowledge that H. G. Feichtinger was the scientist introducing the word modulation into the mathematical standard terminology. Unlike engineers who modulate a given frequency by changing the amplitude, he was thinking of imposing a particular frequency on a given function and looking at modulation invariant Segal algebras, see, e. g., [118] or, in connection with the factorization problem of Segal algebras, the paper [129], where this property played an important role. The STFT of f with window g can be rewritten as a convolution of f with a modulated version of g, see formula (1.28) in the previous chapter. So modulation spaces got their name because they capture the behavior/effect of modulation of the window function g in the filtering process (convolution). They were used for signal theory in the past years, and still they are subject of intensive study in this context. In the present book we adopt them for the analysis of the partial differential operators, with a detailed investigation of the properties necessary in this new perspective. p,q Let us shortly compare Mm (ℝd ) with other function spaces in the literature. The Sobolev spaces Wsp (ℝd ), based on Lebesgue Lp spaces, have nowadays a dominant role in the theory of the partial differential equations. In the case p = q = 2, the mod2 ulation space Mm (ℝd ) with a suitable weight m may reproduce the standard Sobolev s d spaces H (ℝ ) = Ws2 (ℝd ), see Proposition 2.3.6 in the sequel. The situation is drastically different in the case of potential Sobolev spaces Wsp (ℝd ) with p ≠ 2, for which we cannot establish exact identities, but only inclusions, see Proposition 2.3.21 (for more general inclusion properties, we refer to the paper by Sugimoto and Tomita [274]). In fact, as it concerns local regularity, we may say that f ∈ M p,q (ℝd ) is of class ℱ Lq (ℝd ). More precisely, in Proposition 2.3.26 we shall see that if f ∈ 𝒮 󸀠 (ℝd ) is compactly supported, then we have f ∈ M p,q (ℝd ) if and only if its Fourier transform f ̂ belongs to Lq (ℝd ). From this point of view, ancestors of modulation spaces are the spaces https://doi.org/10.1515/9783110532456-003

70 | 2 Function spaces Bq,k (ℝd ), used by Hörmander (see [196, Section 10.1]) and others in the study of the fundamental solutions of the equations with constant coefficients. Namely, f ∈ Bq,k (ℝd ) means that f ̂k ∈ Lq (ℝd ) (definition of weight function m(x, ξ ) = k(ξ )), so modulation spaces and Bq,k spaces locally coincide, apart from notation and definition of weight functions. The properties of functions or distributions in ℱ Lq (ℝd ) represent one of the main problems of harmonic analysis, but once the mysterious ℱ Lq regularity is assumed as true definition, several computations simplify. In fact, the key idea for the modulation spaces is to join the local ℱ Lq regularity with an Lp behavior at infinity; see again Proposition 2.3.26, where it is also proved that if f ̂ is compactly supported then f ∈ M p,q (ℝd ) if and only if f ∈ Lp (ℝd ). Such an interplay between local and asymptotic properties provides a nice theory for M p,q spaces, a basic example being the well-known Feichtinger algebra M 1 (ℝd ), the smallest Banach algebra invariant with respect to pointwise and convolution products, see Propositions 2.4.19 and 2.4.23. p,q Let us also emphasize the action of the Fourier transform on Mm (ℝd ), namely for p = q we have (cf. Proposition 2.3.28) p

d

p

d

ℱ M (ℝ ) = M (ℝ ).

In this chapter we shall also consider the so-called Wiener amalgam spaces. Though not directly related to Fourier transform, they enter the picture in a natural way. Namely, one defines the Wiener amalgam space W(B, C), with B and C being Banach spaces satisfying certain assumptions, as the set of all distributions f ∈ 𝒮 󸀠 (ℝd ) with local regularity expressed by B and such that for a window g ∈ 𝒟(ℝd ) the function Fg (x) = ‖fTx g‖B belongs to the global space C, see Definition 2.4.7 for details. Wiener amalgam spaces (by a different name then) in full generality have been introduced in [122]. The role of the function g as a window is strictly related to that of g in the definition of the STFT Vg f , and the basic connection is given by

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ℱM

p,q

= W(ℱ Lp , Lq ).

So that, for p = q, we obtain M p = W(ℱ Lp , Lp ), and in particular M 1 = W(ℱ L1 , L1 ), see the examples following Definition 2.4.7. By using properties of the Wiener amalgams, we can infer results for inclusion, convolution and multiplication relations for p,q Mm . The next chapters of the book will show that the use of modulation spaces offers relevant advantages in the study of partial differential operators, with respect to the

2 Function spaces | 71

p,q standard frame of Sobolev spaces. Natural applications of Mm will concern, in pard ticular, equations globally defined on ℝ and requiring a precise Fourier analysis of the solutions. On the other hand, drawbacks may obviously originate from the more p,q p,q involved definition of Mm . In particular, a complication in the Mm setting is represented by the behavior of the norm under dilations. The issue is essential for the applications to linear and nonlinear partial differential equations, and we shall devote to it a large part of the chapter, see Theorem 2.6.6. Let us list more precisely the contents of the next sections. In Section 2.1 we give the definition of the weight functions under our consideration and study their properties. In harmonic analysis the term weight function is usually reserved to a submultiplicative function

v(z1 + z2 ) ≤ v(z1 )v(z2 ),

∀z1 , z2 ∈ ℝ2d ,

basic example being v(z) = (1 + |z|)s , s ≥ 0. In the definition of modulation spaces, we shall refer, more generally, to v-moderate weights m(z), z ∈ ℝ2d , satisfying for some C > 0 the inequality

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m(z1 + z2 ) ≤ Cv(z1 )m(z2 ),

∀z1 , z2 ∈ ℝ2d .

Note that in the theory of partial differential equations this is also named temperance property, see Hörmander [196, Section 10.1] and Nicola-Rodino [236, Section 1.1]. Submultiplicative weights (also called Beurling weights) are quite important. However, the pointwise inverse of such a weight is not submultiplicative anymore. But any positive or negative power of a submultiplicative function is, of course, a moderate weight. It is not wrong to think of moderate functions as mostly being of that type. In Section 2.2 we consider mixed-norm Lebesgue spaces. In fact, the definition of p,q Mm involves a repeated integration of this type, with different exponents in the x and ξ variables if p ≠ q. So, as a preparation for the study of ‖f ‖Mmp,q , we collect here the properties of the mixed-norms used in the sequel. p,q Section 2.3 is the core of the chapter: the definition of Mm is given here and basic properties are proved, in particular, it is shown that the definition does not depend on the choice of the window g in the expression of the STFT Vg f . Banach properties, relations with Sobolev spaces, density of the Schwartz class 𝒮 , duality, and interpolation p,q are discussed. An equivalent definition of Mm (ℝd ) is also given in terms of a smooth d p,q partition of unity in the dual space ℝξ . This provides alternative norms for Mm (ℝd ), which turn out to be useful in the proof of the local ℱ Lq regularity and the global Lp behavior. Finally, the action of the Fourier transform on modulation spaces is considered. As observed before, this introduces in a natural way to the Wiener amalgam spaces, studied in the subsequent section. The first part of Section 2.4 is a short survey on the general theory of the Wiener amalgams. Relevant examples are W(Lp , Lqm ), 1 ≤ p, q ≤ ∞, that for p = q reduce to the

72 | 2 Function spaces standard Lebesgue spaces, W(Lp , Lpm ) = Lpm . Coming back to modulation spaces, besides the fundamental identities ℱ M p,q = W(ℱ Lp , Lq ), M p = W(ℱ Lp , Lp ), it is observed p,q 2d that if f ∈ Mm (ℝd ), then the STFT Vg f is in the Wiener amalgam W(ℱ L1 , Lp,q m )(ℝ ). The rather long analysis of norms of dilations begins in Section 2.5, concerning Wiener amalgams, and it is concluded in Section 2.6, on modulation spaces. The problem is to study the dependence on λ > 0 of ‖fλ ‖M p,q , where fλ (t) = f (λt). More generally, one may consider ‖fA ‖M p,q , A ∈ GL(d, ℝ), with fA (t) = f (At). The main result of the section gives sharp estimates of the type ‖fλ ‖M p,q ≲ λr ‖f ‖M p,q , where r depends on d, p, q and takes different values for λ → 0+ , λ → ∞. Section 2.7 proves the sharpness of the results of Section 2.3 for convolution, inp,q clusion, and multiplication in modulation spaces. Here the definition of Mm is also extended to the case 0 < p, q < 1 in the quasi-Banach setting.

2.1 Weight functions A weight function m on ℝ2d is a positive, locally bounded, and measurable function on ℝ2d , that is, m(z) > 0, for every z ∈ ℝ2d , and mχK ∈ L∞ (ℝ2d ), for every compact set K ⊂ ℝ2d , χK being the characteristic function of the set K. Weights on the time–frequency space ℝ2d may quantify the growth and smoothness of functions on ℝd , as we shall see in the sequel. We say that two weights m1 , m2 on ℝ2d are equivalent if there exist positive constants C1 , C2 such that C1 m1 (z) ≤ m2 (z) ≤ C2 m1 (z),

∀z ∈ ℝ2d .

(2.1)

Among many possible choices of weights, we shall focus on the following types.

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Definition 2.1.1. A weight function v on ℝ2d is named submultiplicative if v(z1 + z2 ) ≤ v(z1 )v(z2 ),

∀z1 , z2 ∈ ℝ2d .

(2.2)

For a fixed submultiplicative weight v, a weight function m on ℝ2d is called v-moderate if there exists a positive constant C such that m(z1 + z2 ) ≤ Cv(z1 )m(z2 ),

∀z1 , z2 ∈ ℝ2d .

(2.3)

The set of all v-moderate weights on ℝ2d is denoted by ℳv (ℝ2d ). Important examples of submultiplicative weights are the functions v(z) = (1+|z|)s , for s ≥ 0. In fact, using the triangle inequality |z1 + z2 | ≤ |z1 | + |z2 |, z1 , z2 ∈ ℝ2d , we infer 1 + |z1 + z2 | ≤ 1 + |z1 | + |z2 | ≤ (1 + |z1 |)(1 + |z2 |)

2.1 Weight functions |

73

so that s

s

s

(1 + |z1 + z2 |) ≤ (1 + |z1 |) (1 + |z2 |) ,

∀z1 , z2 ∈ ℝ2d .

Other examples are the exponential weights v(z) = ea|z| , for any a ≥ 0, as it follows b immediately from |z1 + z2 | ≤ |z1 | + |z2 |, and the subexponential weights v(z) = ea|z| , a ≥ 0, 0 ≤ b < 1. The last case can be proved in this way. First, we write |z1 + z2 |b ≤ (|z1 | + |z2 |)b and then we notice that b

(|z1 | + |z2 |) ≤ |z1 |b + |z2 |b . The claim follows by observing that the function f (t) = (t + 1)b − t b − 1 for t ≥ 0 is a decreasing function whenever b < 1. Further examples of submultiplicative weights are the logarithmic functions t

v(z) = log(e + |z|) ,

t ≥ 0, z ∈ ℝ2d .

If v1 and v2 are submultiplicative weights, then it is straightforward to check that the product v1 v2 enjoys the same property. Therefore the functions s

b

t

va,b,s,t (z) = (1 + |z|) ea|z| log(e + |z|) ,

a, s, t ≥ 0, 0 ≤ b ≤ 1

are submultiplicative weights. It can be checked easily that the weights s

b

t

ma,b,s,t (z) = (1 + |z|) ea|z| log(e + |z|) ,

a, s, t ∈ ℝ, 0 ≤ b ≤ 1

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are va,b,s,t -moderate. In the sequel we shall mainly work with the weight function ⟨z⟩s = (1 + z 2 )s/2 , which is equivalent to (1 + |z|)s , but ⟨z⟩s fails to be submultiplicative for s ≥ 0. In fact, Peetre’s inequality (0.4) (which is sharp!) gives ⟨z1 + z2 ⟩s ≤ 2s ⟨z1 ⟩s ⟨z2 ⟩s ,

s ≥ 0.

Lemma 2.1.2. If a weight function v on ℝ2d satisfies v(z1 + z2 ) ≤ Cv(z1 )v(z2 ),

∀z1 , z2 ∈ ℝ2d

̃ = Cv(z) is a submultiplicative weight. for a suitable constant C > 0, then v(z) Proof. We have, by definition of ṽ and using (2.4), ̃ 1 + z2 ) = Cv(z1 + z2 ) ≤ C 2 v(z1 )v(z2 ) = v(z ̃ 1 )v(z ̃ 2 ), v(z as desired.

(2.4)

74 | 2 Function spaces Hence the function v(z) = 2s ⟨z⟩s , s ≥ 0, is a submultiplicative weight. Since the constant 2s will be irrelevant when considering weighted function spaces, we shall mainly use the weights vs (z) = ⟨z⟩s

(2.5)

and, by abuse of notation, we shall denote by ℳvs (ℝ2d ) the class of vs -moderate weights, which indeed coincides with the class ℳ2s vs (ℝ2d ). For s ≤ 0, the weights vs are v|s| -moderate by Peetre’s inequality ⟨z1 + z2 ⟩s ≤ 2s ⟨z1 ⟩|s| ⟨z2 ⟩s , i. e., vs ∈ ℳv|s| (ℝ2d ). Equivalent weights originate the same modulation spaces. This suggests considering only functions v which are continuous, since any submultiplicative weight is equivalent to a continuous function. Lemma 2.1.3. Consider a submultiplicative weight v and a function ψ such that ψ ∈

𝒞c (ℝ2d ) \ {0}, ψ ≥ 0, then

̃ = v ∗ ψ(z) v(z) is a weight function equivalent to v. Moreover, ṽ is v-moderate. Proof. Observe that ṽ > 0 is well defined and continuous by Young’s inequality and the smoothing property of the convolution. Moreover, ṽ is equivalent to v. Let K be the support of ψ, then ̃ = ∫ v(z − y)ψ(y) dy ≤ v(z) ∫ v(−y)ψ(y) dy ≤ v(z)‖v‖L∞ (−K) ‖ψ‖L1 (K) = Cv(z), v(z) K

K

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with C = ‖v‖L∞ (−K) ‖ψ‖L1 (K) . Vice versa, consider a compact set K̃ ⊂ K where ψ(z) > 0 for every z ∈ K,̃ then ∫ v(z) dy = ∫ v(z − y + y) K̃

ψ(y) v(y) dy ≤ ∫ v(z − y)ψ(y) dy ψ(y) ψ(y) K̃

K̃

≤ ‖v/ψ‖L∞ (K)̃ ∫ v(z − y)ψ(y) dy ≤ ‖v/ψ‖L∞ (K)̃ ∫ v(z − y)ψ(y) dy K̃

K

̃ = ‖v/ψ‖L∞ (K)̃ v(z) ̃ ̃ hence v(z) ≤ C v(z), with C = (1/meas(K))‖v/ψ‖ L∞ (K)̃ . Finally, observe that ṽ is v-moderate: ̃ 1 + z2 ) = ∫ v(z1 + z2 − y)ψ(y) dy ≤ v(z1 )v(z ̃ 2 ), v(z K

z1 , z2 ∈ ℝ2d .

2.1 Weight functions | 75

Observe that if v is continuous and satisfies (2.2), then v(0) = v(0 + 0) ≤ v(0)2 and, dividing by v(0) > 0, it follows that v(0) ≥ 1. Another desirable property is the following symmetry: for z = (x, ξ ) ∈ ℝ2d , m(x, ξ ) = m(−x, ξ ) = m(x, −ξ ) = m(−x, −ξ ) enjoyed by the weights ma,b,s,t . Other weights can be constructed by restricting weight functions on ℝ2d to ℝm , m < 2d. For instance, we can consider only weights in the time (resp. frequency) variables m(x, ξ ) = ⟨x⟩s ,

m(x, ξ ) = ⟨ξ ⟩s , x, ξ ∈ ℝd , s ∈ ℝ.

Further examples we shall use are m(x, ξ ) = ⟨x⟩s ⟨ξ ⟩t ,

s, t ∈ ℝ.

We end this section by observing that submultiplicative weights grow at most exponentially. Lemma 2.1.4. Let v be a submultiplicative weight on ℝ2d . Then v grows at most exponentially. That is, there exist constants a ≥ 0, C > 0 such that v(z) ≤ Cea|z| ,

z ∈ ℝ2d .

Proof. Since any submultiplicative function is equivalent to a continuous one by Lemma 2.1.3, we assume without loss of generality that v is continuous. Define a by ea = sup|z|≤1 v(z). Because v is continuous, it satisfies v(0) ≥ 1 so that a ≥ 0. Take any z ∈ ℝ2d and select n ∈ ℕ+ such that n − 1 < |z| ≤ n. Then |z|/n ≤ 1 and using the submultiplicativity property n

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z z v(z) = v(n ) ≤ (v( )) ≤ ean < ea(|z|+1) = Cea|z| , n n with C = ea , as desired. In the sequel we shall only work with submultiplicative weights v which are even, that is, v(−z) = v(z), for z ∈ ℝ2d . Under such an assumption, v-moderate weights satisfy the following inequalities. Lemma 2.1.5. If m ∈ ℳv (ℝ2d ) then there exists a constant C > 0 such that 1 m(z) ≤ m(z − t) ≤ Cv(t)m(z), C v(t)

t, z ∈ ℝ2d .

In particular, 1 1 ≤ m(z) ≤ Cv(z), C v(z)

z ∈ ℝ2d ,

(2.6)

76 | 2 Function spaces and for every z ∈ l + [0, 1]2d , l ∈ ℤ2d ,

1 m(l) ≤ m(z) ≤ C 󸀠 m(l). C󸀠

(2.7)

Proof. Using (2.3), we write m(z − t + t) ≤ Cv(t)m(z − t) and obtain the left-hand side inequality in (2.6). Similarly, m(z −t) ≤ Cv(−t)m(z) = Cv(t)m(z), since v is even, and the right-hand side inequality in (2.6) is shown. The latter inequalities follow from (2.6), exchanging the roles of the variables t and z and setting t = 0. Finally, writing z = l+z 󸀠 , with z 󸀠 ∈ [0, 1]2d , we have m(z 󸀠 + l) ≤ Cm(l)v(z 󸀠 ) and m(l) ≤ Cm(l + z 󸀠 )v(−z 󸀠 ). So we can choose C 󸀠 = C maxz 󸀠 ∈[0,1]2d v(z 󸀠 ) and obtain (2.7).

2.2 Mixed norm spaces The Lebesgue mixed norm spaces were studied in detail by Benedek and Panzone in their work [10]. Such spaces can be considered as an extension of the classical Lp -spaces, enjoying similar properties, which we summarize below. For the proofs we refer mainly to the original paper [10]. We limit ourselves to presenting their definition in the Euclidean space ℝd , d ∈ ℕ+ , endowed with the Lebesgue measure dx1 ⋅ ⋅ ⋅ dxd . This can be extended to more general measure spaces, in particular, we recall the discrete case with the counting measure. Given pi ∈ (0, ∞], i = 1, . . . , d, and a weight function m on ℝd , we say that a p ,...,p measurable function f : ℝd → ℂ belongs to Lm1 d (ℝd ) if the following norm is finite: p2

p ,p ,...,p Lm1 2 d

‖f ‖

ℝ

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1

p1 pd 󵄨 󵄨p = (∫ ⋅ ⋅ ⋅ (∫󵄨󵄨󵄨f (x1 , . . . , xd )󵄨󵄨󵄨 1 m(x1 , . . . , xd )p1 dx1 ) ⋅ ⋅ ⋅ dxd )

(2.8)

ℝ

with standard modification when pi = ∞ for some i. That is, we assume that the number obtained by taking successively the p1 -norm in x1 , the p2 -norm in x2 , . . . , the pd -norm in xd , in that order for the function fm, is finite. p ,p ,...,p Observe that if pi = p for every i = 1, . . . , d, m(x) ≡ 1, then Lm1 2 d (ℝd ) = Lp (ℝd ), p ,p ,...,p the standard Lebesgue space. If pi = p for every i = 1, . . . , d, then Lm1 2 d (ℝd ) = p d p Lm (ℝ ) is the weighted L -space of measurable functions f such that ‖f ‖Lpm = ‖fm‖Lp < ∞. If we replace ℝ with a countable set Xi , i = 1, . . . , d, endowed with the counting meap ,...,p sure, then the weighted mixed norm spaces are denoted by ℓm1 d (X), X = ∏di=1 Xi , m = (mλ ) : X → ℂ, and are spaces of sequences a = (aλ ), λ = (λ1 , . . . , λd ) ∈ X such that p2

‖a‖ℓp1 ,...,pd m

1

pd 󵄨 󵄨p p1 = ( ∑ ⋅ ⋅ ⋅ ( ∑ 󵄨󵄨󵄨(am)(λ1 , . . . , λd )󵄨󵄨󵄨 1 ) . . . ) < ∞.

λ1 ∈X1

λd ∈Xd

We next recall a particular instance of such discrete mixed norm spaces that we shall often use in the sequel.

2.2 Mixed norm spaces | 77

p,q Definition 2.2.1. For 1 ≤ p, q ≤ ∞, m ∈ ℳv (ℝ2d ), the space ℓm (ℤ2d ) consists of all sequences a = (ak,n )k,n∈ℤd such that the norm

p,q ℓm

‖a‖

p

p

q p

= ( ∑ ( ∑ |ak,n | m(k, n) ) )

1 q

n∈ℤd k∈ℤd

is finite. Remark 2.2.2. Writing λ = (k, n) ∈ ℤ2d , by (2.7) one can easily show the normequivalence 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ aλ Tλ χ[0,1]2d 󵄩󵄩󵄩 p,q ≍ ‖a‖ℓmp,q . 󵄩󵄩 2d 󵄩󵄩Lm λ∈ℤ

(2.9)

Next, let us list the main properties of the mixed norm spaces. We exhibit the Euclidean case, the discrete one is analogous. To simplify notations, we use the multiindex convention P = (p1 , . . . , pd ),

P 󸀠 = (p󸀠1 , . . . , p󸀠d ),

1 1 = 1, i = 1, . . . , d. + pi p󸀠i

We write 1 ≤ P ≤ ∞ if 1 ≤ pi ≤ ∞ for every i = 1, . . . , d (and similarly if the inequalities p ,...,p are strict). Hence the mixed norm space Lm1 d (ℝd ) is shortened to LPm (ℝd ). Theorem 2.2.3. Consider a weight function m on ℝd . Then (i) For 1 ≤ P ≤ ∞, LPm (ℝd ) is a Banach space, and every sequence converging in LPm (ℝd ) contains a subsequence convergent almost everywhere to the limit function. (ii) If f is a measurable function over ℝd , then ‖f ‖LPm =

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ f (x)g(x) dx 󵄨󵄨󵄨, 󵄨 󵄨󵄨 󸀠 g∈LP , ‖g‖=1󵄨 d sup

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1 m

(2.10)

ℝ

where dx = dx1 ⋅ ⋅ ⋅ dxd . 󸀠 (iii) For 1 < P < ∞, LPm (ℝd ) is reflexive and for 1 ≤ P < ∞, (LPm (ℝd ))󸀠 = LP1 (ℝd ). m

(iv) (Hölder’s inequality) If 1 ≤ P ≤ ∞, f ∈ LPm (ℝd ), g ∈ LP1/m (ℝd ), then fg ∈ L1 (ℝd ), with 󸀠

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ f (x)g(x) dx󵄨󵄨󵄨 ≤ ‖f ‖LPm ‖g‖LP󸀠 . 󵄨󵄨 󵄨󵄨 1 m n

(2.11)

ℝ

(v) (Young’s inequality) Let m be a v-moderate weight. Consider 1 ≤ P, Q, R ≤ ∞ such that 1 1 1 + =1+ , P Q R

78 | 2 Function spaces that is, for P = (p1 , . . . , pd ), Q = (q1 , . . . , qd ), R = (r1 , . . . , rd ), 1 1 1 + =1+ , pi qi ri

i = 1, . . . d.

If f ∈ LPv (ℝd ), g ∈ LQm (ℝd ) then f ∗ g ∈ LRm (ℝd ), with ‖f ∗ g‖LRm ≤ C‖f ‖LPv ‖g‖LQ . m

(2.12)

Proof. The proofs when m ≡ 1 are carried out in [10] by using similar arguments as for standard Lp -spaces. The weight m does not change the proofs of (i)–(iv), one just has to replace the function f with fm and follow the standard pattern. Let us spend a few words for Young’s inequality. It is enough to prove it for nonnegative functions f and g. Here m is a v-moderate weight, so that in view of (2.3), (f ∗ g)(x)m(x) = ∫ m(x − y + y)f (x − y)g(y) dy

(2.13)

ℝd

≤ C ∫ v(x − y)m(y)f (x − y)g(y) dy ℝd

= C[(fv) ∗ (mg)](x).

(2.14) (2.15)

Hence using the unweighted Young’s inequality, 󵄩 󵄩 ‖f ∗ g‖LRm ≤ C 󵄩󵄩󵄩(fv) ∗ (mg)󵄩󵄩󵄩LR ≤ C‖fv‖LP ‖mg‖LQ = C‖f ‖LPv ‖g‖LQ , m as desired. If we choose P = Q = R = 1, m = v submultiplicative, then Young’s inequality gives

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L1v (ℝd ) ∗ L1v (ℝd ) �→ L1v (ℝd ), that is, L1v (ℝd ) is a Banach algebra under the convolution product. The following result will be useful in the sequel. Lemma 2.2.4. Consider 1 ≤ P ≤ ∞ and m ∈ ℳv . For z ∈ ℝd , the translation operator Tz is bounded on LPm (ℝd ) with ‖Tz f ‖LPm ≤ Cv(z)‖f ‖LPm ,

∀f ∈ LPm (ℝd ).

Proof. Consider f ∈ LPm (ℝd ), then using m(x) ≤ Cv(z)m(x − z), 󵄩 󵄩 󵄩 󵄩 ‖Tz f ‖LPm = 󵄩󵄩󵄩f (⋅ − z)m(⋅)󵄩󵄩󵄩LP ≤ Cv(z)󵄩󵄩󵄩f (⋅ − z)m(⋅ − z)󵄩󵄩󵄩LP = Cv(z)‖fm‖LP = Cv(z)‖f ‖LPm .

This concludes the proof.

(2.16)

2.3 Modulation spaces | 79

2.3 Modulation spaces Modulation spaces were introduced and studied by H. G. Feichtinger in the early 1980s. The original setting was that of locally compact Abelian (LCA) groups [117]. The main point was to define smoothness over LCA groups, where one cannot use dilation or differentiation. The corresponding spaces in Euclidean spaces were introduced somewhat later [126], to model the family of the Besov spaces from H. Triebel [297] and J. Peetre [240]. The conditions formulated in the original paper [117, 126] already include spaces of ultra-distributions introduced by N. Teofanov in [282], as noted in [127]. The subject is so vast that an entire book should be devoted to its study. Here we limit ourselves to considering the case of weights with at most polynomial growth, so that we remain in the field of tempered distributions. Notice that a large part of the results we shall present stay valid also for weights with at most exponential growth, replacing the Schwartz class and the space of tempered distributions with suitable Gelfand–Shilov classes and their duals. From now on, we assume that there exist s0 , C > 0 such that the weight m satisfies m(z) ≤ C⟨z⟩s0 = Cvs0 (z),

z ∈ ℝ2d .

(2.17)

The STFT Vg f plays the central role in the definition of modulation spaces: the measure of its decay on the phase space provides information on smoothness and decay of the function f . Definition 2.3.1. Fix g ∈ 𝒮 (ℝd ) \ {0}, a weight function m ∈ ℳv . Consider the indices P = (p1 , . . . , pd ),

Q = (q1 , . . . , qd ),

pi , qi ∈ [1, ∞], i = 1, . . . , d.

(2.18)

Then P,Q Mm (ℝd ) = {f ∈ 𝒮 󸀠 (ℝd ) : ‖f ‖M P,Q (ℝd ) < ∞},

(2.19)

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m

where ‖f ‖M P,Q (ℝd ) = ‖Vg f ‖Lp1 ,...,pd ,q1 ,...,qd . m

m

p,q P,Q If p = p1 = ⋅ ⋅ ⋅ = pd , q = q1 = ⋅ ⋅ ⋅ = qd , we write Mm in place of Mm . In this case we have q

‖f ‖Mmp,q (ℝd )

1

p q 󵄨 󵄨p = ( ∫ ( ∫ 󵄨󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 m(x, ξ )p dx) dξ )

ℝd

(2.20)

ℝd

(with obvious modifications when p = ∞ or q = ∞). These are the original modulation spaces introduced by Feichtinger in [126]. Roughly speaking, a weight in ξ regulates

80 | 2 Function spaces p,q the smoothness of f ∈ Mm (ℝd ), whereas a weight in x regulates the decay of f ∈ p,q d p,p p Mm (ℝ ). If p = q, the notation Mm is shortened to Mm . In this case, ‖f ‖Mmp (ℝd ) = ‖(Vg f )m‖Lp (ℝ2d ) . P,Q If m ≡ 1, the space Mm is simply denoted by M P,Q . The extension of the original modulation spaces in Definition 2.3.1 was widely studied and applied to the investigation of boundedness and sampling properties for pseudodifferential operators in [228, 229, 241]. P,Q Proposition 2.3.2. The definition of Mm (ℝd ) is independent of the window g ∈ 𝒮 (ℝd ). P,Q Different windows yield equivalent norms on Mm (ℝd ).

Proof. Set P = (p1 , . . . , pd ), Q = (q1 , . . . , qd ). Let g, g0 be two different window functions in 𝒮 (ℝd ). Take first g0 as fixed window function to measure the modulation space norm P,Q of f ∈ Mm (ℝd ). Applying Lemma 1.2.29 with h = γ = g0 , we obtain the pointwise estimate 1 󵄨󵄨 󵄨 (|Vg0 f | ∗ |Vg g0 |)(x, ξ ), 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 ≤ ‖g0 ‖22

(x, ξ ) ∈ ℝ2d .

Young’s inequality (2.12) with R = (p1 , . . . , pd , q1 , . . . , qd ), P = 1 and hence Q = (p1 , . . . , pd , q1 , . . . , qd ), gives ‖Vg f ‖Lp1 ,...,pd ,q1 ,...,qd ≲ ‖Vg0 f ‖Lp1 ,...,pd ,q1 ,...,qd ‖Vg g0 ‖L1v m

m

since, by Theorem 1.2.23, Vg g0 ∈ 𝒮 (ℝ2d ) �→ L1v (ℝ2d ). Switching the roles of g and g0 yields the reverse inequality ‖Vg0 f ‖Lp1 ,...,pd ,q1 ,...,qd ≲ ‖Vg f ‖Lp1 ,...,pd ,q1 ,...,qd ‖Vg0 g‖L1v m

m

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so that ‖Vg0 f ‖Lp1 ,...,pd ,q1 ,...,qd and ‖Vg f ‖Lp1 ,...,pd ,q1 ,...,qd are equivalent norms. The proof is comm m pleted. Let us present a further extension of Definition 2.3.1, introduced by S. Bishop in [19], in her study of Schatten p-class properties for integral operators. This is a natural generalization of the previous definition. Assume that c is a permutation of the set {1, . . . , 2d}. We identify c with the linear bijection c̃ : ℝ2d → ℝ2d ,

̃ 1 , . . . , x2d ) = (xc(1) , . . . , xc(2d) ). c(x

(2.21)

Definition 2.3.3 (Mixed modulation spaces). Consider g ∈ 𝒮 (ℝd ) \ {0}, indices P, Q as in (2.18), and let c be a permutation corresponding to the map c̃ as above. Then d 󸀠 d M(c)P,Q m (ℝ ) is the mixed modulation space of tempered distributions f ∈ 𝒮 (ℝ ) for which ‖f ‖M(c)P,Q = ‖Vg f ∘ c‖̃ LP,Q < ∞. m

m

(2.22)

2.3 Modulation spaces |

81

If p = p1 = p2 = ⋅ ⋅ ⋅ = pd , q = q1 = q2 = ⋅ ⋅ ⋅ = qd , we simply write M(c)p,q m ; if p = p1 = p2 = ⋅ ⋅ ⋅ = pd = q1 = q2 = ⋅ ⋅ ⋅ = qd , we shorten to M(c)pm . Remark 2.3.4. (i) If c is the identity permutation and P = (p1 , . . . , pd ), Q = (q1 , . . . , qd ), we come back to the modulation spaces in Definition 2.3.1. (ii) If the indices satisfy p = p1 = p2 = ⋅ ⋅ ⋅ = pd = q1 = q2 = ⋅ ⋅ ⋅ = qd , and m ≡ 1, given any permutation c, a simple change of variables yields M(c)p (ℝd ) = M p (ℝd ). (iii) In general, the mixed modulation spaces differ from the classical ones of Definition 2.3.1. A simple example is provided by the permutation c0 (1, 2, . . . , d, d + 1, . . . , 2d) = (d + 1, d + 2, . . . , 2d, 1, 2, . . . , d).

(2.23)

If we choose p = p1 = p2 = ⋅ ⋅ ⋅ = pd ,

q = q1 = q2 = ⋅ ⋅ ⋅ = qd ,

then M p,q (c0 ) is the space of tempered distributions such that q

‖f ‖M p,q (c0 )

ℝd

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1

p q 󵄨 󵄨p = ( ∫ ( ∫ 󵄨󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 dξ ) dx) ≍ ‖f ̂‖M p,q = ‖f ‖ℱ M p,q .

(2.24)

ℝd

The second to last equivalence follows by using the fundamental identity of time– frequency analysis: Vg f (x, ξ ) = e−2πix⋅ξ Vĝ f ̂(ξ , −x), hence |Vg f (x, ξ )| = |Vĝ f ̂(ξ , −x)|. Moreover, when computing the M p,q -norm of f ̂ we used the window ĝ ∈ 𝒮 (ℝd ) in place of g. This does not matter in view of Proposition 2.3.2. (iv) The definition of mixed modulation spaces does not depend on the choice of the window g ∈ 𝒮 (ℝd ). The proof follows a similar pattern as for the standard modulation spaces. 2.3.1 Properties of modulation spaces In this section we shall recall the main properties of modulation spaces. We shall prove that they are Banach spaces. Moreover, we shall exhibit inclusion relations, duality and density properties. For sake of simplicity, we limit our exposition to the original modulation spaces p,q Mm (ℝ2d ) in [126]. The general case of Definition 2.3.1 enjoys analogous properties. 2d Let us first recall the following convolution property for L∞ vs (ℝ ).

82 | 2 Function spaces 2d Lemma 2.3.5. Consider the weight function vs in (2.5). If s > 2d and f , g ∈ L∞ vs (ℝ ), then there exists Cs > 0 such that

‖f ∗ g‖L∞ ≤ Cs ‖f ‖L∞ ‖g‖L∞ . v v v s

s

(2.25)

s

2d Proof. If f , g ∈ L∞ vs (ℝ ), then

z ∈ ℝ2d ,

󵄨 󵄨󵄨 v (z), 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ ‖f ‖L∞ vs −s

and the same estimate holds for g. Since the weight vs is equivalent to (1 + |z|)s , we have 󵄨󵄨 󵄨 ‖g‖L∞ ∫ v−s (w)v−s (z − w) dw 󵄨󵄨f ∗ g(z)󵄨󵄨󵄨 ≤ ‖f ‖L∞ vs vs ℝ2d

≤ Cs ‖f ‖L∞ ‖g‖L∞ ∫ (1 + |w|) (1 + |z − w|) dw v v −s

s

≤

−s

s

ℝ2d 󸀠 Cs ‖f ‖L∞ ‖g‖L∞ (1 + vs vs

−s

|z|) ,

and the last bound can be proved as follows. For a fixed z ∈ ℝ2d , we can define the following partition of ℝ2d : Nz = {w ∈ ℝ2d ||w − z| ≤

Nzc = {w ∈ ℝ2d ||w − z| >

|z| }, 2

|z| }. 2

When w ∈ Nz , we have |w| ≥ |z|/2 and −s

(1 + |w|)

−s

≤ (1 +

|z| ) 2

≤ 2s (1 + |z|) , −s

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therefore ∫ (1 + |w|) (1 + |z − w|) dw ≤ 2s (1 + |z|)

∫ (1 + |z − w|) dw.

Nz

Nz

−s

−s

−s

−s

(2.26)

Analogously, when w ∈ Nzc , since (1 + |z − w|)

−s

−s

≤ (1 +

|z| ) 2

≤ 2s (1 + |z|) , −s

one has ∫ (1 + |w|) (1 + |z − w|) dw ≤ 2s (1 + |z|)

∫ (1 + |w|) dw.

Nzc

Nzc

−s

−s

−s

−s

(2.27)

2.3 Modulation spaces | 83

Now, since s > 2d, the integrals in the right-hand sides of (2.26) and (2.27) converge, and we can write ∫ (1 + |w|) (1 + |z − w|) dw ≤ Cs (1 + |z|) −s

−s

−s

ℝ2d

where Cs = 2s+1 ∫ (1 + |w|) dw. −s

ℝ2d

The proof is completed. Observe that the estimate (2.25) in particular proves that, for s > 2d, (

1 1 1 ∗ )(z) ≤ Cs (z), vs vs vs

z ∈ ℝ2d .

2.3.1.1 Relations between modulation and other function spaces We now present the first collection of useful results that specify the relations of some modulation spaces with other function spaces. We borrow many results from the textbook [160].

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Proposition 2.3.6. Fix a nonzero window g ∈ 𝒮 (ℝd ). (i) If |f (x)| ≲ ⟨x⟩−s and s > d, then |Vg f (x, ξ )| ≲ ⟨x⟩−s ; analogously, if |f ̂(ξ )| ≲ ⟨ξ ⟩−s and s > d, then |Vg f (x, ξ )| ≲ ⟨ξ ⟩−s . 2 (ii) If m ∈ ℳv and m(x, ξ ) = m(x) = (m⊗1)(x, ξ ), then Mm⊗1 (ℝd ) = L2m (ℝd ); analogously, 2 d 2 if m(x, ξ ) = m(ξ ) = (1 ⊗ m)(x, ξ ), then M1⊗m (ℝ ) = ℱ Lm (ℝd ). 2 (iii) If vs (ξ ) = ⟨ξ ⟩s for some s ∈ ℝ, then M1⊗v (ℝd ) = H s (ℝd ) (the potential Sobolev s space). (iv) If vs (z) = ⟨z⟩s , 1 ≤ p, q ≤ ∞, then d

p,q

d

𝒮 (ℝ ) = ⋂ Mvs (ℝ ), s≥0

p,q

d

d

𝒮 (ℝ ) = ⋃ M1/v (ℝ ). 󸀠

s≥0

s

(2.28)

Proof. (i) We write |Vg f (x, ξ )| ≤ ∫ℝd |f (t)||g(t −x)| dt = |f |∗|ℐ g|(x), and apply the convolution properties (2.25) in dimension d to the functions f and g, which, by assumption, d are in L∞ vs (ℝ ) 󵄩󵄩 󵄩 ‖g‖L∞ . 󵄩󵄩|f | ∗ |ℐ g|󵄩󵄩󵄩L∞ ≤ Cs ‖f ‖L∞ vs vs vs This implies 󵄨󵄨 󵄨 ‖g‖L∞ ⟨x⟩−s , 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 ≤ Cs ‖f ‖L∞ vs vs

∀ξ ∈ ℝd .

84 | 2 Function spaces The other estimate can be retrieved by the same argument applied to 󵄨 󵄨󵄨 ̂ ), 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 ≤ |f ̂| ∗ |ℐ g|(ξ

∀x ∈ ℝd ,

cf. (1.29). (ii) Since 󵄩2 󵄩 = 󵄩󵄩󵄩Vg f (x, ξ ) ⋅ m(x)󵄩󵄩󵄩L2 (ℝ2d ) < ∞,

‖f ‖2M 2

m⊗1

we have Vg f (x, ξ ) = ℱ [f ⋅ Tx g](ξ ) ∈ L2 (ℝdξ ),

for a. e. x ∈ ℝd .

By Plancherel’s theorem, 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨ℱ [f ⋅ Tx g](ξ )󵄨󵄨󵄨 dξ = ∫ 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 󵄨󵄨󵄨g(t − x)󵄨󵄨󵄨 dt,

ℝd

ℝd

and thus ‖f ‖2M 2

m⊗1

󵄨 󵄨2 󵄨 󵄨2 = ∫ [ ∫ 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 󵄨󵄨󵄨g(t − x)󵄨󵄨󵄨 m(x)2 dt] dx ℝd ℝd

󵄨 󵄨2 󵄨 󵄨2 = ∫ 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 󵄨󵄨󵄨g(u)󵄨󵄨󵄨 m(t − u)2 dt du. ℝ2d

Now, using m ∈ ℳv and the weights’ inequalities (2.6), we obtain C −2 ‖f ‖2L2 ‖g‖2L2 ≤ ‖f ‖2M 2

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m

1/v

m⊗1

≤ C 2 ‖f ‖2L2 ‖g‖2L2 ; m

v

(2.29)

2 then Mm⊗1 (ℝd ) = L2m (ℝd ) with equivalent norms. Observe that for this argument it is enough to assume g ∈ L2v (ℝd ). If m(x, ξ ) = m(ξ ), the fundamental identity of time– frequency analysis (1.31) yields

󵄨2 󵄨 󵄨2 󵄨 ∫ 󵄨󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 m(ξ )2 dx dξ = ∫ 󵄨󵄨󵄨Vĝ f ̂(ξ , −x)󵄨󵄨󵄨 m(ξ )2 dx dξ .

ℝ2d

ℝ2d

Then we apply the previous argument. (iii) It follows from the preceding claim since H s (ℝd ) = ℱ L2vs (ℝd ). (iv) The left-hand side equality follows from Theorem 1.2.23, whereas the other from Proposition 1.2.20. Let us study the properties of Vg∗ on modulation spaces. Theorem 2.3.7. Consider m ∈ ℳv and γ ∈ 𝒮 (ℝd ). Then for 1 ≤ p, q ≤ ∞,

2.3 Modulation spaces | 85

2d p,q d (i) Vγ∗ : Lp,q m (ℝ ) → Mm (ℝ ) and the following estimate holds:

󵄩󵄩 ∗ 󵄩󵄩 p,q 󵄩󵄩 ∗ 󵄩 ≲ ‖Vg γ‖L1v ‖F‖Lp,q . 󵄩󵄩Vγ F 󵄩󵄩Mm = 󵄩󵄩Vg (Vγ F)󵄩󵄩󵄩Lp,q m m

(2.30)

p,q (ii) If F = Vg f and ⟨γ, g⟩ ≠ 0, we have the inversion formula in Mm (ℝd )

f =

1 ∫ Vg f (x, ξ )Mξ Tx γ dx dξ . ⟨γ, g⟩

(2.31)

ℝ2d

In short, IdMmp,q = ⟨γ, g⟩−1 Vγ∗ Vg . 2d ∗ 󸀠 d d Proof. (i) Consider F ∈ Lp,q m (ℝ ). We first prove that Vγ F ∈ 𝒮 (ℝ ). Taking ϕ ∈ 𝒮 (ℝ ) and using Hölder’s inequality for mixed norm spaces in (2.11), we majorize

󵄨󵄨 ∗ 󵄨 󵄨 󵄨 ‖Vγ ϕ‖ 󵄨󵄨⟨Vγ F, ϕ⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨F, Vγ ϕ⟩󵄨󵄨󵄨 ≤ ‖F‖Lp,q m

Lp1/m,q

󸀠 󸀠

󵄩󵄩 n 󵄩 󵄩 −n 󵄩 ≤ ‖F‖Lp,q 󵄩⟨⋅⟩ Vγ ϕ󵄩󵄩󵄩∞ 󵄩󵄩󵄩⟨⋅⟩ 󵄩󵄩󵄩Lp󸀠 ,q󸀠 . m 󵄩 1/m

For sufficiently large n, this expression is finite and, from Corollary 1.2.25, we obtain the claim. Therefore Vγ∗ F has a continuous STFT in view of Corollary 1.2.19, given by Vg Vγ∗ F(u, η) = ⟨Vγ∗ F, Mη Tu g⟩ = ∫ F(x, ξ )Vγ (Mη Tu g)(x, ξ ) dx dξ ℝ2d

= ∫ F(x, ξ )Vg γ(u − x, η − ξ )e−2πix⋅(η−ξ ) dx dξ . ℝ2d

Taking the absolute values of the previous equality, we obtain the pointwise estimate

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󵄨󵄨 󵄨 ∗ 󵄨󵄨Vg Vγ F(u, η)󵄨󵄨󵄨 ≤ (|F| ∗ |Vg γ|)(u, η), and Young’s inequality (2.12) gives 󵄩󵄩 ∗ 󵄩 ≲ ‖F‖Lp,q ‖Vg γ‖L1v . 󵄩󵄩Vg Vγ F 󵄩󵄩󵄩Lp,q m m Since g, γ ∈ 𝒮 (ℝd ), by Theorem 1.2.23, we have Vg γ ∈ 𝒮 (ℝ2d ) �→ L1v (ℝ2d ) and the rightp,q hand side of the inequality above is finite. Hence the Mm -norm of Vγ∗ F, related to the window g, is controlled by 󵄩󵄩 ∗ 󵄩󵄩 p,q 󵄩󵄩 ∗ 󵄩 ≲ ‖F‖Lp,q ‖Vg γ‖L1v < ∞, 󵄩󵄩Vγ F 󵄩󵄩Mm = 󵄩󵄩Vg (Vγ F)󵄩󵄩󵄩Lp,q m m p,q that is, Vγ∗ F ∈ Mm (ℝd ).

2d −1 ∗ p,q d ̃ (ii) If Vg f ∈ Lp,q m (ℝ ), then f = ⟨γ, g⟩ Vγ Vg f ∈ Mm (ℝ ) by the preceding step. The equality f ̃ = f follows from the inversion formula in 𝒮 󸀠 (ℝd ), cf. (1.68).

86 | 2 Function spaces Proposition 2.3.8 (Density). If the weight m satisfies (2.17) and 1 ≤ p, q < ∞, then p,q 𝒮 (ℝd ) is a dense subspace of Mm (ℝd ). Proof. Since by assumption there exist C, s0 > 0 such that m(z) ≤ Cvs0 (z) for every p,q z ∈ ℝ2d , we can majorize the Mm -norm of f ∈ 𝒮 (ℝd ) as follows: 󵄩 󵄩 . ‖f ‖Mmp,q = ‖Vg f ‖Lp,q ≤ C‖Vg fvs ‖∞ 󵄩󵄩󵄩vs−1 󵄩󵄩󵄩Lp,q m m If f ∈ 𝒮 (ℝd ) and s ≥ s0 is large enough, both terms on the right-hand side are finite p,q because Vg f ∈ 𝒮 (ℝ2d ) by Theorem 1.2.23, and thus 𝒮 (ℝd ) ⊆ Mm (ℝd ). Now choose, for example, the exhausting sequence of compact sets Kn = {z | |z| ≤ n} (closed balls of p,q radius n) and a window g ∈ 𝒮 (ℝd ) with ‖g‖2 = 1. For f ∈ Mm (ℝd ), set Fn = (Vg f )χKn and fn = Vg∗ Fn = ∫ Fn (x, ξ )Mξ Tx g dx dξ = ∫ Vg f (x, ξ )χKn (x, ξ )Mξ Tx g dx dξ . ℝ2d

ℝ2d

Since Fn is of rapid decay, we see that fn ∈ 𝒮 (ℝd ) by Theorem 1.2.22. Using the in2d p,q d version formula and the continuity of the mapping Vg∗ : Lp,q m (ℝ ) → Mm (ℝ ) (cf. Theorem 2.3.7), we can write 󵄩 󵄩 ‖f − fn ‖Mmp,q = 󵄩󵄩󵄩Vg∗ (Vg f − Fn )󵄩󵄩󵄩Mmp,q ≲ ‖Vg f − Fn ‖Lp,q . m → 0 as n → ∞ and then ‖f − fn ‖Mmp,q → 0, hence If p, q < ∞, we have ‖Vg f − Fn ‖Lp,q m p,q 𝒮 (ℝd ) �→ Mm (ℝ2d ), as desired.

Theorem 2.3.9. Consider m ∈ ℳv , then p,q (i) If p, q ∈ [1, ∞], the space Mm (ℝd ) is a Banach space. p,q d (ii) The space Mm (ℝ ) is invariant under time–frequency shifts, and there exists a constant C > 0 such that

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󵄩󵄩 󵄩 󵄩󵄩π(z)f 󵄩󵄩󵄩Mmp,q ≤ Cv(z)‖f ‖Mmp,q ,

z ∈ ℝ2d .

(2.32)

(iii) If p = q and there exists C > 0 such that m(Jz) ≤ Cm(z), for every z ∈ ℝ2d , then p Mm (ℝd ) is invariant under Fourier transform. p,q Proof. (i) Observe that Mm (ℝd ) is a normed space. Let us show that it is complete. For a fixed window function g ∈ 𝒮 (ℝd ) \ {0}, set 2d 󸀠 d V(g) = {F ∈ Lp,q m (ℝ ) | F = Vg f , f ∈ 𝒮 (ℝ )}. 2d By definition V(g) is a linear subspace of Lp,q m (ℝ ) isometrically isomorphic to p,q p,q Mm (ℝd ). Let us now show that Mm (ℝd ) is complete. p,q Consider a Cauchy sequence {fn } ⊂ Mm (ℝd ), then {Vg fn } is a Cauchy sequence p,q 2d 2d in Lm (ℝ ) which is complete, and thus there exists F ∈ Lp,q m (ℝ ) such that ‖Vg fn −

2.3 Modulation spaces | 87

p,q F‖Lp,q → 0, as n → ∞. Now define f = ‖g‖−2 V ∗ F. Theorem 2.3.7 gives f ∈ Mm (ℝd ) and L2 g m −2 ∗ fn = ‖g‖L2 Vg Vg fn . By the same result, we have the estimate

󵄩󵄩 ∗ 󵄩󵄩 →0 ‖f − fn ‖Mmp,q = ‖g‖−2 󵄩Vg (F − Vg fn )󵄩󵄩Mmp,q ≲ ‖F − Vg fn ‖Lp,q L2 󵄩 m as n → ∞, which gives the claim. (ii) We use the covariance formula (1.51), |Vg (π(z)f )(w)| = |Tz Vg f (w)|, so that 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 = 󵄩󵄩󵄩Vg f (⋅ − z)m󵄩󵄩󵄩Lp,q = 󵄩󵄩󵄩Vg f (⋅)m(⋅ + z)󵄩󵄩󵄩Lp,q 󵄩󵄩π(z)f 󵄩󵄩󵄩Mmp,q = 󵄩󵄩󵄩Vg (π(z)f )󵄩󵄩󵄩Lp,q m 󵄩 󵄩 ≤ Cv(z)󵄩󵄩󵄩Vg (f )m󵄩󵄩󵄩Lp,q = Cv(z)‖f ‖Mmp,q , as desired. (iii) It follows by an explicit computation involving the fundamental identity (1.31) and the norm equivalence in Proposition 2.3.2: 󵄨 󵄨p ‖f ̂‖pM p = ‖Vg f ̂‖pLp ≲ ‖Vĝ f ̂‖pLp = ∫ 󵄨󵄨󵄨Vĝ f ̂(z)󵄨󵄨󵄨 m(z)p dz m m m ℝ2d

󵄨 󵄨p 󵄨 󵄨p = ∫ 󵄨󵄨󵄨Vg f (−Jz)󵄨󵄨󵄨 m(z)p dz = ∫ 󵄨󵄨󵄨Vg f (z)󵄨󵄨󵄨 m(Jz)p dz ℝ2d

≲

ℝ2d

‖f ‖pM p , m

as desired, since m(Jz) ≲ m(z) for z ∈ ℝ2d . p ,q p,q Theorem 2.3.10 (Duality). If 1 ≤ p, q < ∞ then (Mm (ℝd ))∗ ≃ M1/m (ℝd ) and the duality 󸀠

󸀠

p ,q p,q between f ∈ Mm and h ∈ M1/m is concretely given by the continuous sesquilinear form 󸀠

󸀠

(which extends the inner product of L2 (ℝd ), see Theorem 1.2.11) ⟨f , h⟩ = ∫ Vg f (z)Vg h(z) dz,

(2.33)

ℝ2d

where g ∈ 𝒮 (ℝd ) \ {0} is a fixed window function. p ,q Proof. Suppose that h ∈ M1/m (ℝd ). Then ℓh (f ) := ∫ Vg f (z)Vg h(z) dz defines a bounded

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󸀠

󸀠

p,q linear functional over Mm (ℝd ). From Hölder’s inequality (2.11), we have

󵄨󵄨 󵄨 ‖Vg h‖ p󸀠 ,q󸀠 = ‖f ‖Mmp,q ‖h‖ p󸀠 ,q󸀠 . 󵄨󵄨ℓh (f )󵄨󵄨󵄨 ≤ ‖Vg f ‖Lp,q m L1/m M1/m p,q p,q Conversely, if we assume that ℓ ∈ (Mm (ℝd ))∗ , since Mm (ℝd ) is isometrically isomorp,q 2d 2d phic to the close subspace V(g) = {F ∈ Lm (ℝ ) | F = Vg f , f ∈ 𝒮 󸀠 (ℝd )} of Lp,q m (ℝ ), ̃ g f ). Hahn–Banach theℓ provides a linear functional ℓ̃ on V(g) by setting ℓ(f ) = ℓ(V p,q 2d ̃ orem allows extending ℓ to a bounded functional on Lm (ℝ ) and Theorem 2.2.3 (iii)

,q shows that there is a function H ∈ Lp1/m (ℝ2d ) such that 󸀠

󸀠

̃ g f ) = ∫ Vg f (z)H(z) dz. ℓ(V ℝ2d

88 | 2 Function spaces p ,q Now set h = Vg∗ H, therefore h ∈ M1/m (ℝd ) by Theorem 2.3.7 and 󸀠

󸀠

⟨f , h⟩ = ∫ Vg f (z)H(z) dz = ∫ Vg f (z)Vg h(z) dz = ℓ(f ), ℝ2d

ℝ2d

since Vg∗ (H − Vg h) = 0 if ‖g‖2 = 1 and ∫ Vg f (z)H(z) dz = ∫ f (x)Vg∗ H(x) dx, ℝ2d

ℝd

cf. Remark 1.2.27 (ii), and argue by density. Hence every linear functional has the prescribed form and this gives the claim. By the previous result, we can compute the M p,q -norm by duality. In fact, we have p,q Corollary 2.3.11. Assume 1 < p, q ≤ ∞ and f ∈ Mm (ℝd ). Then

‖f ‖Mmp,q =

󵄨 󵄨 sup 󵄨󵄨󵄨⟨f , g⟩󵄨󵄨󵄨.

‖g‖

p󸀠 ,q󸀠 M 1/m

≤1

(2.34)

Notice that (2.34) still holds true whenever p = 1 or q = 1 and f ∈ 𝒮 (ℝd ), simply by extending Theorem 2.3.10 to the duality 𝒮 󸀠 ⟨⋅, ⋅⟩𝒮 . We now widen the space of admissible windows from 𝒮 (ℝd ) to Mv1 (ℝd ).

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Theorem 2.3.12. Assume that m ∈ ℳv (ℝ2d ) and let g, γ ∈ Mv1 (ℝd )\{0}. Then 2d p,q d (i) The mapping Vγ∗ : Lp,q m (ℝ ) → Mm (ℝ ) is bounded and the estimate (2.30) still holds. p,q 2d (ii) For f ∈ Mm (ℝd ), we have Vg f ∈ Lp,q m (ℝ ), and, if ⟨g, γ⟩ ≠ 0, the inversion formula (2.31) remains valid. p,q (iii) ‖Vg f ‖Lp,q is an equivalent norm on Mm (ℝd ). m 2d ∗ Proof. (i) Given F ∈ Lp,q m (ℝ ), consider the mapping γ �→ Vγ F. Then by Theorem 2.3.7,

p,q it maps 𝒮 (ℝd ) into Mm (ℝd ) and

󵄩󵄩 ∗ 󵄩󵄩 p,q ‖Vg γ‖L1v = ‖F‖Lp,q ‖γ‖Mv1 . 󵄩󵄩Vγ F 󵄩󵄩Mm ≲ ‖F‖Lp,q m m Moreover, since 𝒮 (ℝd ) is dense in Mv1 (ℝd ), γ �→ Vγ∗ F extends to a bounded map from p,q Mv1 (ℝd ) to Mm (ℝd ) by the density principle (cf. Theorem A.0.9). p,q (ii) (First step) Let us show that for f ∈ Mm (ℝd ) and g ∈ Mv1 (ℝd ) the linear map1 d p,q 2d ping g �→ Vg f is bounded from Mv (ℝ ) to Lm (ℝ ). Using Lemma 1.2.29 for g, h = γ = g0 ∈ 𝒮 (ℝd ), we have the pointwise estimate

󵄨󵄨 󵄨 −2 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 ≤ ‖g0 ‖2 (|Vg0 f | ∗ |Vg g0 |)(x, ξ ).

2.3 Modulation spaces | 89

Young’s inequality yields −2 p,q p,q ‖Vg f ‖Lp,q ≲ ‖g0 ‖−2 2 ‖Vg0 f ‖Lm ‖Vg g0 ‖L1v ≲ ‖g0 ‖2 ‖f ‖Mm ‖g‖Mv1 . m

(2.35)

p,q By density of 𝒮 (ℝd ) in Mv1 , this estimate implies that for fixed f ∈ Mm the map g �→ Vg f 1 p,q is bounded from Mv into Lm . (Second step) Consider g, γ ∈ Mv1 (ℝd )\{0} as in the assumptions. By the density of 𝒮 (ℝd ) in Mv1 (ℝd ), we can find two sequences {gn }, {γn } ⊂ 𝒮 (ℝd ) such that ‖gn − g‖Mv1 → 0, ‖γn − γ‖Mv1 → 0 and ⟨γn , gn ⟩ ≠ 0. Using the estimates in (2.35) and (i), p,q 2d we obtain, for f ∈ Mm (ℝd ) and F ∈ Lp,q m (ℝ ),

≲ ‖gn − g‖Mv1 → 0, ‖Vgn f − Vg f ‖Lp,q m

󵄩󵄩 ∗ ∗ 󵄩 󵄩󵄩Vγn F − Vγ F 󵄩󵄩󵄩Mmp,q ≲ ‖γn − γ‖Mv1 → 0.

Finally, writing the inversion formula (2.31) for γn , gn and taking the limit n → +∞, ⟨γ, g⟩−1 Vγ∗ Vg f = lim ⟨γn , gn ⟩−1 Vγ∗n Vgn f = f , n→∞

we obtain the result. (iii) The proof of the norm equivalence is similar to the proof of Proposition 2.3.2. We now study the action of the multiplication operator f �→ ⟨⋅⟩s f and of the Fourier multiplier ⟨D⟩s f = ℱ −1 ⟨⋅⟩s ℱ f

(2.36)

on modulation spaces, cf. [287]. We need the following preliminary result. Lemma 2.3.13. For s ∈ ℝ, take m = s + d + 1, and define Φ(x, t) =

vs (t) , vs (x)vm (t − x)

x, t ∈ ℝd

(2.37)

x, ξ ∈ ℝd ,

(2.38)

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and Φ2 (x, ξ ) = [ℱ2 Φ(x, ⋅)](ξ ),

the partial Fourier transform of Φ with respect to the t-variable. Then for any r ≥ 0, we have 󵄨 󵄨 ∫ ⟨ξ ⟩r sup 󵄨󵄨󵄨Φ2 (x, ξ )󵄨󵄨󵄨 dξ < ∞. 2d

ℝd

x∈ℝ

Proof. Observe that Φ ∈ 𝒞 ∞ (ℝ2d ), moreover, 0 ≤ Φ(x, t) =

vs (t − x + x) ≤ Cv−d−1 (t − x), vs (x)vm (t − x)

(2.39)

90 | 2 Function spaces so that ∫ Φ(x, t) dt ≤ C ∫ v−d−1 (t − x) dt = C ∫ v−d−1 (y) dy < ∞ ℝd

ℝd

ℝd

and RΦ := sup ∫ Φ(x, t) dt < ∞. x∈ℝd

ℝd

Using (0.3), one easily verifies that ∀α ∈ ℕd , 󵄨󵄨 α 󵄨 󵄨󵄨𝜕t Φ(x, t)󵄨󵄨󵄨 ≤ Cα Φ(x, t). By applying the partial Fourier transform with respect to the t-variable and using the Riemann–Lebesgue lemma, 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 sup󵄨󵄨󵄨ξ α Φ2 (x, ξ )󵄨󵄨󵄨 ≤ C sup󵄨󵄨󵄨ℱ2 [𝜕tα Φ(x, t)](ξ )󵄨󵄨󵄨 ≤ C sup ∫ 󵄨󵄨󵄨𝜕tα Φ(x, t)󵄨󵄨󵄨 dt ≤ C 󸀠 RΦ , d x,ξ

x,ξ

x∈ℝ

ℝd

for every α ∈ ℕd . The previous estimate easily implies (2.39). Theorem 2.3.14. Consider 1 ≤ p, q ≤ ∞, m ∈ ℳv (ℝ2d ), s ∈ ℝ. (i) For (vs ⊗ 1)(x, ξ ) = vs (x), the mapping f �→ vs f is a topological isomorphism from p,q p,q Mm(v (ℝd ) to Mm (ℝd ). s ⊗1) (ii) For (1 ⊗ vs )(x, ξ ) = vs (ξ ), the Fourier multiplier ⟨D⟩s defined in (2.36) is a topological p,q p,q isomorphism from Mm(1⊗v (ℝd ) to Mm (ℝd ). ) s

Proof. Consider a real-valued window function g0 ∈ 𝒮 (ℝd ) \ {0} and set m = s + d + 1, p,q g = v−m g0 ∈ 𝒮 (ℝd ) \ {0}. Then for f ∈ Mm(v (ℝd ), ⊗1)

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s

Vg (vs f )(x, ξ ) = ℱ (vs fTx g)(ξ ) = ℱ (vs fv−m (⋅ − x)Tx g0 )(ξ ) = vs (x)ℱ (fTx g0 Φ(x, ⋅))(ξ ) = vs (x)ℱ (fTx g0 ) ∗ ℱ [Φ(x, ⋅)](ξ ) = vs (x)ℱ (fTx g0 ) ∗ Φ2 (x, ⋅)(ξ ) with Φ, Φ2 defined in (2.37) and (2.38), respectively, and ℱ = ℱ2 . Consider the case p, q < ∞, then 󵄩 󵄩 ‖vs f ‖Mmp,q ≍ 󵄩󵄩󵄩Vg (vs f )󵄩󵄩󵄩Lp,q m

q

1

p q 󵄨 󵄨p = ( ∫ ( ∫ 󵄨󵄨󵄨ℱ (fTx g0 ) ∗ Φ2 (x, ⋅)(ξ )󵄨󵄨󵄨 mp (x, ξ )vsp (x) dx) dξ )

ℝd ℝd

q

1

󵄨󵄨 󵄨󵄨p p q 󵄨 󵄨 = ( ∫ ( ∫ 󵄨󵄨󵄨 ∫ ℱ (fTx g0 )(ξ − η)Φ2 (x, η) dη󵄨󵄨󵄨 mp (x, ξ )vsp (x) dx) dξ ) . 󵄨󵄨 󵄨󵄨 d d d ℝ

ℝ ℝ

2.3 Modulation spaces | 91

Define h(ξ ) = supx∈ℝd |Φ2 (x, ξ )| and observe that the estimate (2.39) implies h ∈ L1vr (ℝd ), for every r ≥ 0. Using Minkowski’s integral inequality (cf. Appendix A, Theorem A.0.1), 1

󵄨󵄨p 󵄨󵄨 p 󵄨 󵄨 ( ∫ 󵄨󵄨󵄨 ∫ ℱ (fTx g0 )(ξ − η)Φ2 (x, η) dη󵄨󵄨󵄨 mp (x, ξ )vsp (x) dx) 󵄨󵄨 󵄨󵄨 d d ℝ ℝ

1

p 󵄨p 󵄨p 󵄨 󵄨 ≤ ∫ ( ∫ 󵄨󵄨󵄨ℱ (fTx g0 )(ξ − η)󵄨󵄨󵄨 󵄨󵄨󵄨Φ2 (x, η)󵄨󵄨󵄨 mp (x, ξ )vsp (x) dx) dη

ℝd ℝd

1

p 󵄨 󵄨p ≤ ∫ ( ∫ 󵄨󵄨󵄨ℱ (fTx g0 )(ξ − η)󵄨󵄨󵄨 mp (x, ξ )vsp (x) dx) h(η) dη

ℝd ℝd

1

p 󵄨 󵄨p ≤ C ∫ ( ∫ 󵄨󵄨󵄨ℱ (fTx g0 )(ξ − η)󵄨󵄨󵄨 mp (x, ξ − η)vp (0, η)vsp (x) dx) h(η) dη

ℝd ℝd

1

p 󵄨 󵄨p ≤ C 󸀠 ∫ ( ∫ 󵄨󵄨󵄨ℱ (fTx g0 )(ξ − η)󵄨󵄨󵄨 mp (x, ξ − η)vsp0 (η)vsp (x) dx) h(η) dη

ℝd ℝd

1

p 󵄨 󵄨p = C ∫ ( ∫ 󵄨󵄨󵄨ℱ (fTx g0 )(ξ − η)󵄨󵄨󵄨 mp (x, ξ − η)vsp (x) dx) vs0 (η)h(η) dη

󸀠

ℝd ℝd

where we used m ∈ ℳv and the polynomial growth in (2.17) (observe that vs0 (0, η) = vs0 (η)). Coming back to the computation of ‖vs f ‖Mmp,q , we apply Minkowski’s integral inequality again, so that ‖vs f ‖Mmp,q

ℝ

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1

󵄩 󵄩󵄩 p 󵄨 󵄨p 󵄩 ≤ C 󵄩󵄩 ∫ ( ∫ 󵄨󵄨󵄨ℱ (fTx g0 )(⋅ − η)󵄨󵄨󵄨 mp (x, ⋅ − η)vsp (x) dx) vs0 (η)h(η) dη󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩q d d 󵄩󵄩 󸀠󵄩

ℝ

1

󵄩󵄩 p󵄩 󵄩󵄩 󵄩 󵄨 󵄨p ≤ C 󸀠 ∫ 󵄩󵄩󵄩( ∫ 󵄨󵄨󵄨ℱ (fTx g0 )(⋅ − η)󵄨󵄨󵄨 mp (x, ⋅ − η)vsp (x) dx) 󵄩󵄩󵄩 vs0 (η)h(η) dη 󵄩󵄩q 󵄩󵄩 d d ℝ

ℝ

1

󵄩󵄩 p󵄩 󵄩󵄩 󵄩 󵄨 󵄨p = C 󸀠 ∫ 󵄩󵄩󵄩( ∫ 󵄨󵄨󵄨ℱ (fTx g0 )(⋅)󵄨󵄨󵄨 mp (x, ⋅)vsp (x) dx) 󵄩󵄩󵄩 vs0 (η)h(η) dη 󵄩󵄩q 󵄩󵄩 d d ℝ

ℝ

= C ‖f ‖M p,q 󸀠

m(vs ⊗1)

‖hvs0 ‖1 < ∞,

since ‖hvs0 ‖1 < ∞ because h ∈ L1vr (ℝd ), for every r ≥ 0. The cases p = ∞ or q = ∞ follow by using the same arguments above. Finally, using vs v−s = 1 and the continuity of the mapping f �→ vs f from p,q p,q Mm(v (ℝd ) to Mm shown above, we have ⊗1) s

‖f ‖M p,q

m(vs ⊗1)

󵄩 󵄩 = 󵄩󵄩󵄩v−s (vs f )󵄩󵄩󵄩M p,q

m(vs ⊗1)

≤ C‖vs f ‖Mmp,q .

92 | 2 Function spaces p,q p,q to Mm . This Hence the mapping f �→ vs f is a linear continuous bijection from Mm(v s ⊗1) gives the claim. ̂ m , with m = s + d + 1 as in the (ii) Consider g ∈ 𝒮 (ℝd ) \ {0} and define g0 = gv previous part. Using the fundamental identity of time–frequency analysis (1.31), we can write

󵄨 󵄨 󵄨 󵄨 󵄨󵄨 s ̄̂ 󵄨󵄨󵄨 󵄨󵄨Vg (⟨D⟩ f )(x, ξ )󵄨󵄨󵄨 = 󵄨󵄨󵄨Vĝ (vs f ̂)(ξ , −x)󵄨󵄨󵄨 = 󵄨󵄨󵄨ℱ (vs f ̂Tξ g)(−x) 󵄨 󵄨 󵄨 = vs (ξ )󵄨󵄨󵄨ℱ (f ̂Tξ g0 Φ(ξ , ⋅))(−x)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = vs (ξ )󵄨󵄨󵄨 ∫ ℱ (f ̂Tξ g0 )(−x − y)Φ2 (ξ , y) dy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ℝd

󵄨 󵄨 ≤ vs (ξ ) ∫ 󵄨󵄨󵄨ℱ (f ̂Tξ g0 )(−x − y)Φ2 (ξ , y)󵄨󵄨󵄨 dy ℝd

where, as in the previous step, Φ and Φ2 are defined in (2.37) and (2.38), respectively. Setting χ0 = ℱ −1 g0 and using (1.26), 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨ℱ (f ̂Tξ g0 )(−x − y)󵄨󵄨󵄨 = 󵄨󵄨󵄨Vχ0 f (x + y, ξ )󵄨󵄨󵄨, so that, using Minkowski’s integral inequality and proceeding as in the previous part, q

p

1

p q 󵄩󵄩 s 󵄩󵄩 p,q 󵄨 󵄨 p p 󵄩󵄩⟨D⟩ f 󵄩󵄩Mm ≤ ( ∫ ( ∫ ( ∫ 󵄨󵄨󵄨Vχ0 f (x + y, ξ )Φ2 (ξ , y)󵄨󵄨󵄨 dy) m (x, ξ )vs (ξ ) dx) dξ )

ℝd ℝd ℝd

q

1

p q 󵄨 󵄨p ≤ C( ∫ ( ∫ 󵄨󵄨󵄨Vχ0 f (x, ξ )󵄨󵄨󵄨 mp (x, ξ )vsp (ξ ) dx) dξ ) ‖vs0 h‖1

ℝd ℝd

= C ‖f ‖M p,q 󸀠󸀠

m(1⊗vs )

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where h(y) = supξ |Φ2 (ξ , y)| ∈ L1vs (ℝd ) by (2.39), with s0 coming from (2.17). So we 0 obtain the result. In the sequel we shall often use the closure of the Schwartz class with respect to the modulation norm. d Definition 2.3.15. For m ∈ ℳv (ℝ2d ), 1 ≤ p, q ≤ ∞, we define by ℳp,q m (ℝ ) the closure p,q d p,q d of 𝒮 (ℝd ) in the Mm -norm. Observe that ℳp,q m (ℝ ) = Mm (ℝ ), whenever the indices p and q are finite. p ,q ∗ Notice that these spaces enjoy the duality property (ℳp,q m ) = ℳ1/m , with 1 ≤ 󸀠 󸀠 p, q ≤ ∞, and p , q being the conjugate exponents. Finally, we recall the behavior of modulation spaces with respect to complex interpolation. For the proof, we refer to the original literature [131, 132] and [126, Corollary 2.3]. 󸀠

󸀠

2.3 Modulation spaces | 93

Proposition 2.3.16. Let 0 < θ < 1, pj , qj ∈ [1, ∞] and mj ∈ ℳv (ℝ2d ), j = 1, 2. Set θ 1 1−θ = + , p p1 p2

1 1−θ θ = + , q q1 q2

θ m = m1−θ 1 m2 ,

then p1 ,q1 d (ℳm (ℝd ), ℳpm22,q2 (ℝd ))[θ] = ℳp,q m (ℝ ). 1

Proposition 2.3.17. Let 1 ≤ p1 , p2 , q1 , q2 ≤ ∞, with q2 < ∞. If T is a linear operator such that, for i = 1, 2, ∀f ∈ M pi ,qi ,

‖Tf ‖M pi ,qi ≤ Ai ‖f ‖M pi ,qi then θ ‖Tf ‖M p,q ≤ CA1−θ 1 A2 ‖f ‖M p,q

∀f ∈ M p,q ,

where 1/p = (1 − θ)/p1 + θ/p2 , 1/q = (1 − θ)/q1 + θ/q2 , 0 < θ < 1, and C is independent of T. We shall also use the following interpolation result. Proposition 2.3.18. Let 0 < θ < 1, pj , qj ∈ [1, ∞] and mj ∈ ℳv (ℝ2d ), j = 1, 2. Then ∞,1 ∞,1 ∞,1 (Mm (ℝd ), Mm (ℝd ))[θ] = Mm⊗1 (ℝd ), 1 ⊗1 2 ⊗1 θ with m = m1−θ 1 m2 . 1 ∞ Proof. We limit to an outline. By [132], this amounts to computing [ℓ1 (L∞ m1 ), ℓ (Lm2 )][θ] = ∞ ℓ1 ([L∞ m1 , Lm2 ][θ] ), where the equality is due to [17, Theorem 5.1.2]. Using the connection between complex and real interpolation given by [17, Theorem 4.7.2], we have

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

∞ ∞ ∞ [L∞ m1 , Lm2 ][θ] = (Lm1 , Lm2 )θ,∞ .

Finally, the interpolation theorem by Stein–Weiss [17, Theorem 5.4.1] gives ∞ ∞ (L∞ m1 , Lm2 )θ,∞ = Lm ,

θ m = m1−θ 1 m2 ,

as desired. 2.3.1.2 Potential Sobolev spaces As a consequence of the previous fact, we can establish relations between modulation spaces and potential Sobolev spaces. We have already seen in Proposition 2.3.6 (iii) that the Sobolev spaces H s (ℝd ), s ∈ ℝ, are basic examples of modulation spaces. We now study the more general case Wsp (ℝd ), 1 ≤ p ≤ ∞.

94 | 2 Function spaces Definition 2.3.19. Given s ∈ ℝ, the Fourier multiplier ⟨D⟩s in (2.36) is also called the Bessel potential operator. The potential Sobolev space Wsp (ℝd ) is defined by 󵄩 󵄩 Wsp (ℝd ) = {f ∈ 𝒮 󸀠 (ℝd ), 󵄩󵄩󵄩⟨D⟩s f 󵄩󵄩󵄩p < ∞},

1 ≤ p ≤ ∞.

(2.40)

For s ∈ ℝ the Bessel kernel is Gs = ℱ −1 {(1 + | ⋅ |2 )

−s/2

(2.41)

} = ℱ −1 (⟨⋅⟩−s ),

so that the potential Sobolev space can be equivalently defined by Wsp (ℝd ) = Gs ∗ Lp (ℝd ) = {f ∈ 𝒮 󸀠 (ℝd ), f = Gs ∗ g, g ∈ Lp (ℝd )}.

(2.42)

Of course, if s = 0, W0p (ℝd ) = Lp (ℝd ). Moreover, Ws2 (ℝd ) = H s (ℝd ). Let us first report the following isomorphism property for Bessel potentials. Theorem 2.3.20. For 1 ≤ p, q ≤ ∞, s1 , s2 ∈ ℝ given, the Bessel potential f ∈ 𝒮 (ℝd )

Tf = Gs ∗ f ,

with kernel Gs defined in (2.41) and s = s2 − s1 , gives rise to an isomorphism between p,q p,q M1⊗v (ℝd ) and M1⊗v (ℝd ). More precisely, s1

s2

p,q p,q M1⊗v (ℝd ) = Gs ∗ M1⊗v (ℝd ). s2

(2.43)

s1

Proof. It follows immediately from Theorem 2.3.14 (ii) since Tf = Gs ∗ f = ⟨D⟩−s f . According to the previous isomorphism, one can also define the weighted modup,q lation spaces M1⊗v (ℝd ) as images of the unweighted ones via Bessel potentials s

p,q M1⊗v (ℝd ) = Gs ∗ M p,q (ℝd ), s

1 ≤ p, q ≤ ∞, s ∈ ℝ.

(2.44)

Proposition 2.3.21. For 1 ≤ p ≤ ∞ and s ∈ ℝ p,∞ Wsp (ℝd ) �→ M1⊗v (ℝd ). Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

s

Proof. Consider g ∈ 𝒮 (ℝd ). Assume s = 0 first. Using formula (1.28) to express the STFT and Young’s inequality, we find that 1/p

󵄨 󵄨p ‖f ‖M p,∞ ≍ ‖Vg f ‖Lp,∞ = sup ( ∫ 󵄨󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 dx) ξ ∈ℝd

ℝd

󵄩 󵄩 󵄩 󵄩 = sup 󵄩󵄩󵄩(f ∗ Mξ g ∗ )󵄩󵄩󵄩p ≤ sup ‖f ‖p 󵄩󵄩󵄩Mξ g ∗ 󵄩󵄩󵄩1 ≲ ‖f ‖p . d d ξ ∈ℝ

ξ ∈ℝ

Consequently, Lp (ℝd ) ⊆ M p,∞ (ℝd ). For arbitrary s ∈ ℝ, we observe that Wsp = Gs ∗ Lp p,∞ by definition, and M1⊗v (ℝd ) = Gs ∗ M p,∞ (ℝd ) by (2.44). So the embedding follows for s all s ∈ ℝ.

2.3 Modulation spaces | 95

Concerning weighted modulation spaces Lpm (ℝd ), with m ∈ ℳv (ℝd ), they are inp,∞ cluded in the modulation spaces Mm⊗1 (ℝd ) as follows. Proposition 2.3.22. For 1 ≤ p ≤ ∞ and m ∈ ℳv (ℝd ), p,∞ Lpm (ℝd ) �→ Mm⊗1 (ℝd ).

Proof. Consider g ∈ 𝒮 (ℝd ). Using formula (1.28) to express the STFT, we can write 1/p

󵄨 󵄨p ‖f ‖M p,∞ ≍ ‖Vg f ‖Lp,∞ = sup ( ∫ 󵄨󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 m(x)p dx) m⊗1

m⊗1

ξ ∈ℝd

ℝd

󵄩 󵄩 󵄩 󵄩 = sup 󵄩󵄩󵄩(f ∗ Mξ g ∗ )󵄩󵄩󵄩Lpm ≤ sup ‖f ‖Lpm 󵄩󵄩󵄩Mξ g ∗ 󵄩󵄩󵄩L1 ≲ ‖f ‖Lpm , v d d ξ ∈ℝ

ξ ∈ℝ

where in the last row we used the weighted Young’s inequality in (2.12) in the form Lpm ∗ L1v �→ Lpm . Also, ‖Mξ g ∗ ‖L1v = ‖g ∗ ‖L1v < ∞, since g ∈ 𝒮 (ℝd ) by assumption. 2.3.2 Alternative definition of modulation spaces using frequency-uniform localization techniques Frequency-uniform localization techniques can be applied to discretize modulation spaces, see [298, 303, 176]. For k ∈ ℤd , we denote by 𝒬k the unit closed cube centered at k. The family {Qk }k∈ℤd is a covering of ℝd . We define |ξ |∞ := maxi=1,...,d |ξi |. Consider now a smooth function ρ : ℝd → [0, 1] satisfying ρ(ξ ) = 1 for |ξ |∞ ≤ 1/2 and ρ(ξ ) = 0 for |ξ |∞ ≥ 3/4. Define ρk (ξ ) = Tk ρ(ξ ) = ρ(ξ − k),

k ∈ ℤd ,

(2.45)

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that is, ρk is the translation of ρ at k. By the assumption on ρ, we infer that ρk (ξ ) = 1 for ξ ∈ 𝒬k and ∑ ρk (ξ ) ≥ 1,

k∈ℤd

∀ξ ∈ ℝd .

Denote by σk (ξ ) =

ρk (ξ ) , ∑l∈ℤd ρl (ξ )

ξ ∈ ℝd , k ∈ ℤd .

(2.46)

Observe that σk (ξ ) = σ0 (ξ −k) ∈ 𝒟(ℝd ) and the sequence {σk }k∈ℤd is a smooth partition of unity ∑ σk (ξ ) = 1,

k∈ℤd

∀ξ ∈ ℝd .

96 | 2 Function spaces Definition 2.3.23. For k ∈ ℤd , we define the frequency-uniform decomposition operator by ◻k := ℱ −1 σk ℱ .

(2.47)

The previous operators allow introducing an alternative norm on the weighted p,q modulation spaces Mu⊗w (ℝd ) as follows. Definition 2.3.24. For 1 ≤ p, q ≤ ∞, u, w ∈ ℳv (ℝd ) (that is, v-moderate weights on ℝd ), we define the norm 1 q

‖f ‖M̃ p,q

d)

u⊗w (ℝ

= ( ∑ ‖◻k f ‖qLp w(k)q ) , u

k∈ℤd

f ∈ 𝒮 󸀠 (ℝd ),

(2.48)

with an obvious modification for q = ∞. Let us show that ‖f ‖M̃ p,q

d)

u⊗w (ℝ

p,q is an equivalent norm for Mu⊗w (ℝd ).

Proposition 2.3.25. For 1 ≤ p, q ≤ ∞, u, w ∈ ℳv (ℝd ), the norm ‖ ⋅ ‖M̃ p,q is an equivalent u⊗w

p,q norm on Mu⊗w (ℝd ).

Proof. We prove the case q < ∞; the case q = ∞ can be obtained similarly. Consider ̂ ) = 1 in a window function ϕ ∈ 𝒮 (ℝd ) such that supp ϕ̂ ⊂ B10√d (0), satisfying ϕ(ξ 󸀠 d B5√d (0). For any f ∈ 𝒮 (ℝ ), we reckon ̄ ◻k f = ℱ −1 σk ℱ f = ℱ −1 σk Tξ ϕ̂ ℱ f ,

for ξ ∈ 𝒬k ,

̄ since Tξ ϕ̂ = 1 in supp σk for ξ ∈ 𝒬k . Using Young’s inequality, for ξ ∈ 𝒬k , we obtain ̄ 󵄩 ̄ 󵄩 󵄩 󵄩 󵄩 󵄩 ‖◻k f ‖Lpu ≲ 󵄩󵄩󵄩ℱ −1 σk 󵄩󵄩󵄩L1 󵄩󵄩󵄩ℱ −1 Tξ ϕ̂ ℱ f 󵄩󵄩󵄩Lpu ≲ 󵄩󵄩󵄩ℱ −1 Tξ ϕ̂ ℱ f 󵄩󵄩󵄩Lpu . v

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Observing that w(ξ ) ≍ w(k) for ξ ∈ 𝒬k , cf (2.7) in Lemma 2.1.5, we infer ̄ 󵄩q ̄ 󵄩q 󵄩 󵄩 ‖◻k f ‖qLp w(k)q ≲ 󵄩󵄩󵄩ℱ −1 Tξ ϕ̂ ℱ f 󵄩󵄩󵄩Lpu w(ξ )q ≲ ∫ 󵄩󵄩󵄩ℱ −1 Tξ ϕ̂ ℱ f 󵄩󵄩󵄩Lpu w(ξ )q dξ . u 𝒬k

Hence using the equivalent formula (1.26) for the STFT, we obtain ̄ 󵄩q 󵄩 ∑ ‖◻k f ‖qLp w(k)q ≲ ∑ ∫ 󵄩󵄩󵄩ℱ −1 Tξ ϕ̂ ℱ f 󵄩󵄩󵄩Lpu w(ξ )q dξ u

k∈ℤd

k∈ℤd 𝒬

k

= ∫ ‖Vϕ f ‖qLp w(ξ )q dξ , ℝd

p,q . which gives ‖f ‖M̃ p,q ≲ ‖f ‖Mu⊗w u⊗w

u

2.3 Modulation spaces | 97

Vice versa, for ξ ∈ 𝒬k , we can write 󵄨󵄨 󵄨 ̄ ̄ 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 −1 󵄨󵄨 −1 󵄨󵄨Vϕ f (x, ξ )󵄨󵄨󵄨 = 󵄨󵄨󵄨ℱ Tξ ϕ̂ ℱ f (x)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∑ ℱ σl Tξ ϕ̂ ℱ f (x)󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨|l−k|≤C Thus, for ξ ∈ 𝒬k , using Young’s inequality again, 󵄩󵄩 ̄ 󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 −1 󵄩󵄩Vϕ f (⋅, ξ )󵄩󵄩󵄩Lpu = 󵄩󵄩󵄩 ∑ ℱ σl Tξ ϕ̂ ℱ f 󵄩󵄩󵄩 p 󵄩󵄩Lu 󵄩󵄩 |l−k|≤C ̄ 󵄩 󵄩 ≲ ∑ 󵄩󵄩󵄩ℱ −1 [Tξ ϕ]̂ 󵄩󵄩󵄩Lpv ‖◻l f ‖Lpu |l−k|≤C

̄ 󵄩 󵄩 = ∑ 󵄩󵄩󵄩Mξ ℱ −1 [ϕ]̂ 󵄩󵄩󵄩Lpv ‖◻l f ‖Lpu |l−k|≤C

≲ ∑ ‖◻l f ‖Lpu . |l−k|≤C

Taking the q-power integration over ξ ∈ 𝒬k with weight w yields 󵄩 󵄩 ∫ 󵄩󵄩󵄩Vϕ f (⋅, ξ )󵄩󵄩󵄩Lpu w(ξ )q dξ ≲ ∑ ‖◻l f ‖qLp w(k)q . |l−k|≤C

𝒬k

u

Summing over k, p,q ≲ ‖f ‖ ̃ p,q . ‖f ‖Mu⊗w M u⊗w

This concludes the proof.

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The previous proposition is an ingredient for the following characterization. For tempered distributions compactly supported either in time or in frequency, the M p,q -norm is equivalent to the ℱ Lq -norm or Lp -norm, respectively, cf. [244, 255]. Proposition 2.3.26. Let 1 ≤ p, q ≤ ∞. (i) For every f ∈ 𝒮 󸀠 (ℝd ), supported in a compact set K ⊂ ℝd , we have f ∈ M p,q ⇐⇒ f ∈ ℱ Lq , and CK−1 ‖f ‖M p,q ≤ ‖f ‖ℱ Lq ≤ CK ‖f ‖M p,q ,

(2.49)

where CK > 0 depends only on K. (ii) For every f ∈ 𝒮 󸀠 (ℝd ), whose Fourier transform is supported in a compact set K ⊂ ℝd , we have f ∈ M p,q ⇐⇒ f ∈ Lp , and CK−1 ‖f ‖M p,q ≤ ‖f ‖Lp ≤ CK ‖f ‖M p,q , where CK > 0 depends only on K.

(2.50)

98 | 2 Function spaces Proof. (i) Consider f ∈ 𝒮 󸀠 (ℝd ) supported in a compact set K ⊂ ℝd and R > 0 such that K ⊂ BR (0). Assume first f ∈ ℱ Lq (ℝd ). Let g ∈ 𝒟(ℝd ) with supp g ⊂ BR (0). Then for each ω ∈ ℝd , Vg f (⋅, ω) is supported in B2R (0). Thus, using (1.26), 󵄨󵄨 󵄨 󵄨 −1 󵄨󵄨 ̄̂ 󵄨󵄨, 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 = 󵄨󵄨󵄨ℱ (f ̂Tξ g)(x) and setting meas(B2R (0))1/p = CR,p , we have the following estimates: 1/p 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 −1 󵄩 󵄩󵄩Vg f (⋅, ξ )󵄩󵄩󵄩Lp ≤ meas(B2R (0)) 󵄩󵄩󵄩Vg f (⋅, ξ )󵄩󵄩󵄩L∞ = CR,p 󵄩󵄩󵄩ℱ (f ̂Tξ g)̄̂ 󵄩󵄩󵄩L∞ ̌ ). ≤ CR,p ‖f ̂Tξ g‖̄̂ L1 ≤ CR,p |f ̂| ∗ |g|(ξ

Finally, taking the Lq -norm in the above inequalities, 󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩󵄩󵄩Vg f (x, ξ )󵄩󵄩󵄩Lpx 󵄩󵄩󵄩Lq ≤ CR,p 󵄩󵄩󵄩|f ̂| ∗ |g|̌ 󵄩󵄩󵄩Lq ≤ CR,p ‖f ̂‖Lq ‖g‖̌ L1 , ξ i. e., ‖f ‖M p,q ≲ ‖f ‖ℱ Lq . Vice versa, consider f ∈ M p,q (ℝd ), g ∈ 𝒟(ℝd ) with g ≡ 1 on B2R (0). Observe that g(t − x) = 1,

∀t, x ∈ BR (0).

Hence for every ξ ∈ ℝd f ̂(ξ )χBR (0) (x) = Vg f (x, ξ )χBR (0) (x) and, taking the Lp -norm with respect to the x-variable, 1/p 󵄨

meas(BR (0))

󵄨󵄨f ̂(ξ )󵄨󵄨󵄨 = 󵄩󵄩󵄩Vg f (⋅, ξ )χB (0) (⋅)󵄩󵄩󵄩 p ≤ 󵄩󵄩󵄩Vg f (⋅, ξ )󵄩󵄩󵄩 p . 󵄨 󵄨 󵄩 󵄩L 󵄩 󵄩L R

This yields −1/p 󵄩󵄩 󵄩󵄩󵄩󵄩Vg f (x, ξ )󵄩󵄩󵄩 p 󵄩󵄩󵄩 q , ‖f ̂‖Lq ≤ meas(BR (0)) 󵄩󵄩 󵄩Lx 󵄩L Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

ξ

as desired. (ii) Using the equivalent M p,q -norm in (2.48), 1/q

‖f ‖M p,q ≍ ( ∑ ‖◻k f ‖qLp ) k∈ℤd

.

Now, if f ̂ has compact support, the above sum is finite, and one deduces at once the first estimate in (2.50), since the multipliers ◻k are (uniformly) bounded on Lp . To obtain the second estimate in (2.50), we write f = ∑k∈ℤd ◻k f , then apply the triangle inequality and the finiteness of the sum over k again. For another characterization of modulation spaces using symplectic rotations, we refer the reader to the recent contribution [64].

2.3 Modulation spaces | 99

2.3.2.1 Action of the Fourier transform on modulation spaces We now consider the action of the Fourier transform on modulation spaces. Theorem 2.3.27. For 1 ≤ p ≤ ∞, u, w even weights on ℝd , the Fourier transform ℱ is a p p (ℝd ) with inverse ℱ −1 . In particular, (ℝd ) onto Mw⊗u topological isomorphism from Mu⊗w p d if u = w, ℱ is an automorphism on Mu⊗u (ℝ ). Proof. The key idea is to use the fundamental identity of time–frequency analysis (1.31), which gives Vĝ f ̂(x, ξ ) = Vg f (−ξ , x),

(x, ξ ) ∈ ℝ2d .

p In fact, consider p < ∞, let f ∈ Mu⊗w (ℝd ), and a window function g ∈ 𝒮 (ℝd ). Then d ĝ ∈ 𝒮 (ℝ ) and can be chosen as window function for f ̂, so that, using Tonelli theorem,

‖f ̂‖pM p

w⊗u

≍ ‖Vĝ f ̂‖pLp

w⊗u

󵄨 󵄨p = ∫ ( ∫ 󵄨󵄨󵄨Vĝ f ̂(x, ξ )󵄨󵄨󵄨 wp (x) dx)up (ξ ) dξ ℝd ℝd

󵄨 󵄨p = ∫ ( ∫ 󵄨󵄨󵄨Vg f (−ξ , x)󵄨󵄨󵄨 up (ξ ) dξ )wp (x) dx ℝd ℝd

= ‖Vg f ‖pLp

u⊗w

≍ ‖f ‖pM p . u⊗w

The case p = ∞ is analogous. Similarly one can show that the inverse Fourier transform ℱ −1 is bounded from p p Mw⊗u (ℝd ) to Mu⊗w (ℝd ), and the claim follows. Using the same pattern, we obtain the following result. Proposition 2.3.28. Consider 1 ≤ p ≤ ∞, m a weight function on ℝ2d such that m(x, ξ ) = m(ξ , x) for every (x, ξ ) ∈ ℝ2d and even with respect to x and ξ , m(−x, ξ ) = m(x, ξ ) = m(x, −ξ ) = m(−x, −ξ ),

∀(x, ξ ) ∈ ℝ2d .

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

p Then the Fourier transform ℱ is an automorphism on Mm (ℝd ) with inverse ℱ −1 .

Of course, the previous arguments do not apply for the more general case of modulation spaces with two different indices p ≠ q. In this case, we enter the realm of Wiener amalgam spaces, see the next section. Using the fundamental identity of time–frequency analysis (1.31), we write the STFT as follows: 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨Vg f (x, ω)󵄨󵄨󵄨 = 󵄨󵄨󵄨Vĝ f ̂(ω, −x)󵄨󵄨󵄨 = 󵄨󵄨󵄨ℱ [f ̂Tω ĝ ](−x)󵄨󵄨󵄨.

(2.51)

Since the weights are even, for 1 < p, q < ∞, we compute 1

q q 󵄩 ̂ p,q = ( ∫ 󵄩 ̂ ω)󵄩󵄩󵄩󵄩ℱ Lp wq (ω) dω) . ‖f ‖Mu⊗w 󵄩󵄩f Tω g(x, u

ℝd

(2.52)

100 | 2 Function spaces (The cases p = ∞ and q = ∞ are similar.) This allows, via the Fourier isomorphism, p,q (ℝd ) with norm giving a Banach space structure to ℱ Mu⊗w q

p,q ‖f ‖ℱ Mu⊗w

1

p q 󵄨p 󵄨 = ( ∫ ( ∫ 󵄨󵄨󵄨Vg f (x, ω)󵄨󵄨󵄨 up (ω) dω) wq (x) dx) ,

ℝd

(2.53)

ℝd

for 1 ≤ p, q < ∞. If p = ∞ or q = ∞, the integral on ℝd is replaced by the supremum on ℝd . As observed by H. G. Feichtinger in his insightful retrospective survey [127], it would be natural to include such spaces into a more general class of modulation spaces, where a switch in the order of integration is considered as well in Definition 2.3.1. In the sequel, following a standard procedure, we shall rather regard p,q

ℱ Mu⊗w (ℝd ) as the special case of the Wiener amalgam spaces with local components ℱ Lpu and global component Lqw , 1 ≤ p, q ≤ ∞, cf. (2.52). Namely, with the notation of

the next section,

p,q

d

p

q

d

ℱ Mu⊗w (ℝ ) = W(ℱ Lu , Lw )(ℝ ).

(2.54)

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

2.4 Wiener amalgam spaces We now introduce a class of spaces of functions (or distributions) defined by a norm which “amalgamates” (i. e., mixes) a local criterion for membership with a global one. More precisely, the norm provides a global criterion for a local property, in a sense that will be specified in the sequel. The first appearance of these spaces traces back to several works by N. Wiener (cf. [310, 311, 313]), and we refer to the standard papers [38, 124, 148, 181, 189] for a systematic review of the subject. A remark on the name is needed as the current formulation of the theory for these function spaces is due to H. G. Feichtinger [120, 121, 122] who, following a suggestion of J. J. Benedetto, named them “Wiener amalgam spaces.” For the sake of clarity, let us first introduce a particular kind of amalgam space, modeled on the Lp spaces, but much more flexible since it is possible to control the local regularity of a function and its decay at infinity separately. Definition 2.4.1 (Weighted Wiener amalgams W(Lp , Lqm )). Consider p, q d

d

∈

[1, ∞],

a weight function m ∈ ℳv (ℝ ), and a compact set Q ⊂ ℝ with nonempty interior. The Wiener amalgam space W(Lp , Lqm )(ℝd ) consists of the functions f : ℝd → ℂ such that f ∈ Lploc (ℝd ) and for which the control function FQ (x) := ‖f ⋅ Tx χQ ‖Lp ∈ Lqm (ℝd ),

x ∈ ℝd .

(2.55)

2.4 Wiener amalgam spaces | 101

The norm on W(Lp , Lqm ) is given by 󵄩 󵄩 ‖f ‖W(Lp ,Lqm ) := 󵄩󵄩󵄩FQ (x)󵄩󵄩󵄩Lqm 󵄩 󵄩 = 󵄩󵄩󵄩‖f ⋅ Tx χQ ‖Lp 󵄩󵄩󵄩Lqm q

1

p q 󵄨p 󵄨 = ( ∫ ( ∫ 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 χQ (t − x) dt) mq (x) dx) ,

ℝd

(2.56)

ℝd

with suitable adjustments for the cases p, q = ∞.

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Remark 2.4.2. (i) This special definition allows us to grasp the sense of the amalgam; we first view f “locally”, through translations Tx χQ of the sharp cut-off function χQ , and measure those local pieces in the Lp -norm, then we measure the global behavior of those local pieces according to the Lqm -norm. The “window”, through which we view f locally need not be a unit d-dimensional cube. (ii) There is no gain in generality by allowing the local component to be a weighted Lp space. It can be shown that W(Lpv , Lqm ) = W(Lp , Lqvm ); so it suffices to only consider the case where the global component is a weighted space. Generalizing the previous argument, we now want to consider a space of suitable functions with global behavior (decay and/or summability) expressed by the global component C and with local properties given by the local component B, for certain Banach spaces B, C. Even if the construction can be carried out in a more abstract context (cf. [124]) we will restrict ourselves to Euclidean spaces ℝd and to subspaces of temperate distributions, following [125], and we will provide technical conditions on B and C in order to properly define W(B, C). We observe that general Wiener amalgams W(B, C) are not simply “gluing together disjoint local pieces.” In particular, overlaps are essential when moving to more general amalgams determined by continuity, smoothness, or other criteria. In this case the simple cut-off function χQ of Definition 2.4.1 is clearly inadequate. To define general Wiener amalgams W(B, C), an appropriate smooth cutoff function needs to be employed in place of the sharp cut-off function χQ above. For local components, we require Definition 2.4.3 (Local component space). A Banach space (B, ‖ ⋅ ‖B ) is a local component space if (i) 𝒮 (ℝd ) �→ B �→ 𝒮 󸀠 (ℝd ) are continuous embeddings; (ii) For every test function g ∈ 𝒟(ℝd ) and f ∈ B one has gf ∈ B. Moreover, there exists a constant C = C(g) > 0 such that ‖fTx g‖B ≤ C‖f ‖B

∀x ∈ ℝd , ∀f ∈ B.

102 | 2 Function spaces In view of the previous condition, the following space is well defined: Bloc := {f ∈ 𝒮 󸀠 (ℝd ) : gf ∈ B, ∀g ∈ 𝒟(ℝd )}. Following the standard terminology of time–frequency analysis, any given nonzero g ∈ 𝒟(ℝd ) will be called a window function (windows usually can be taken from a much larger space), and we can define the related control function as Fg (x) := ‖fTx g‖B .

(2.57)

The role of the control function is to localize f by means of the window g, measured in the B-norm. Example 2.4.4. Examples of local component Banach spaces are the following: (i) The ordinary Lp (ℝd ) spaces, p ∈ [1, ∞]; (ii) For m ∈ ℳv (ℝd ), the Fourier–Lebesgue spaces ℱ Lpm (ℝd ), 1 ≤ p ≤ ∞. These are local component spaces, since the embeddings 𝒮 �→ ℱ L1m �→ 𝒮 󸀠 and Young’s inequality L1v ∗Lpm ⊆ Lpm imply that the elements of ℱ L1v define bounded multipliers on ℱ Lpm . (iii) The mixed norm spaces Lp,q (ℝ2d ), p, q ∈ [1, ∞]. For the global behavior, we introduce the following conditions. Definition 2.4.5 (Global component space). A Banach space (C, ‖ ⋅ ‖C ) can serve as a global component space if (i) (C, ‖ ⋅ ‖C ) is a solid BF-space, i. e., a Banach space of locally integrable functions, continuously embedded into L1loc (ℝd ), such that for every f ∈ C, g ∈ L1loc (ℝd ), 󵄨󵄨 󵄨 󵄨 󵄨 d 󵄨󵄨g(x)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 for a. e. x ∈ ℝ �⇒ g ∈ C and ‖g‖C ≤ ‖f ‖C .

(2.58)

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

(ii) (C, ‖ ⋅ ‖C ) is with tempered translation, that is, it satisfies the following estimate for some s ≥ 0: ‖Tx f ‖C ≲ vs (x)‖f ‖C ,

vs (x) = ⟨x⟩s .

(2.59)

(iii) (C, ‖ ⋅ ‖C ) is a Banach convolution module with respect to the Beurling algebra L1vs (ℝd ), s ≥ 0, i. e., f ∗g ∈C

for f ∈ C, g ∈ L1vs (ℝd ),

and we have the norm estimate ‖f ∗ g‖C ≤ ‖g‖L1v ‖f ‖C . s

Example 2.4.6. Examples of global component spaces are the following:

(2.60)

2.4 Wiener amalgam spaces | 103

(i) The ordinary Lp spaces, p ∈ [1, ∞]; (ii) Weighted Lpm (ℝd ) spaces, where m ∈ ℳvs (ℝd ), s ≥ 0; 2d 2d (iii) The mixed norm spaces Lp,q m (ℝ ), p, q ∈ [1, ∞], m ∈ ℳvs (ℝ ), s ≥ 0; (iv) Orlicz spaces, as described in a recent paper [293]. Definition 2.4.7 (Wiener amalgam spaces W(B, C)). Let (B, ‖⋅‖B ) be a local component space and (C, ‖ ⋅ ‖C ) a global component space. We define the Wiener amalgam space W(B, C) with local component B and global component C as W(B, C)(ℝd ) = {f ∈ Bloc : Fg ∈ C},

(2.61)

where g ∈ 𝒟(ℝd )\{0} is a fixed window function and Fg is the associated control function. The natural norm on W(B, C) is ‖f ‖W(B,C) := ‖Fg ‖C . Example 2.4.8. Combining Examples 2.4.4 and 2.4.6, we obtain some Wiener amalgam spaces, relevant for the applications in the sequel. (i) W(Lp , Lqm )(ℝd ) are the spaces in Definition 2.4.1. The use of the window χ𝒬 does not produce more generality than g ∈ 𝒟(ℝd ). p,q (ii) W(ℱ Lpu , Lqw )(ℝd ) = ℱ Mu⊗w (ℝd ), according to (2.54). (iii) For p = q and under the assumptions of Theorem 2.3.27, we infer p W(ℱ Lpu , Lpw )(ℝd ) = Mu⊗w (ℝd ).

(2.62)

For conditions on weights u and w which guarantee invariance under Fourier transform, we refer to [124]. Notice that, when p = q = 1, the space

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W(ℱ L1 , L1 )(ℝd ) = M 1 (ℝd )

(2.63)

is the well-known Feichtinger algebra, also denoted by S0 (ℝd ), see [202]. It can be shown that it is the smallest Banach algebra invariant with respect to pointwise and convolution products. The algebra’s property follows from the convolution and multiplication relations for modulation spaces shown in Propositions 2.4.19 and 2.4.23. 2d p,q d (iv) The space W(ℱ L1 , Lp,q m )(ℝ ) will describe the regularity of Vg f for f ∈ Mm (ℝ ) 1 d (g ∈ M (ℝ )); see the sequel. We now state the main properties of W(B, C) in the general setting, for proofs we refer to the original literature [130], see also the master thesis of T. Dobler [112]. First, we recall that a Banach space (Y, ‖ ⋅ ‖Y ) has an absolutely continuous norm if the Banach dual Y 󸀠 coincides with the Köthe-dual Y α := {H ∈ L1loc (ℝd ) : HF ∈ L1 (ℝd ), ∀F ∈ Y}. For example, for 1 ≤ p < ∞, the Lebesgue space Lp (ℝd ) has an absolutely continuous norm, but L∞ (ℝd ) does not.

104 | 2 Function spaces Theorem 2.4.9 (Properties of Wiener amalgam spaces W(B, C)). Let B, Bi local component spaces and C, Ci global component spaces (i = 1, 2, 3). (i) (Banach space property) (W(B, C)(ℝd ), ‖ ⋅ ‖W(B,C) ) is a Banach space continuously embedded into 𝒮 󸀠 (ℝd ); different choices of window function in 𝒟(ℝd ) provide equivalent norms. (ii) (Convolution) If B1 ∗ B2 �→ B3 and C1 ∗ C2 �→ C3 , then W(B1 , C1 ) ∗ W(B2 , C2 ) �→ W(B3 , C3 ). If, moreover, (B1 , B2 , B3 ) and (C1 , C2 , C3 ) are Banach convolution triples, i. e., ‖f ∗ g‖B3 ≲ ‖f ‖B1 ‖g‖B2

‖f ∗ g‖C3 ≲ ‖f ‖C1 ‖g‖C2 ,

then (W(B1 , C1 ), W(B2 , C2 ), W(B3 , C3 )) is a Banach convolution triple ‖f ∗ g‖W(B3 ,C3 ) ≲ ‖f ‖W(B1 ,C1 ) ‖g‖W(B2 ,C2 ) . (iii) (Inclusions) If B1 �→ B2 and C1 �→ C2 , then W(B1 , C1 ) �→ W(B2 , C2 ). (iv) (Complex interpolation) For 0 < θ < 1, we have [W(B1 , C1 ), W(B2 , C2 )][θ] = W([B1 , B2 ][θ] , [C1 , C2 ][θ] )

(2.64)

if C1 or C2 has absolutely continuous norm. The same holds if every Wiener amalgam space is replaced by the closure of the Schwartz space into itself. (v) (Duality) If B󸀠 , C 󸀠 are the topological dual spaces of B, C respectively, and the space of test functions 𝒟 is dense in both B and C, then W(B, C)󸀠 = W(B󸀠 , C 󸀠 ). (vi) (Pointwise products) If B1 ⋅ B2 �→ B3 and C1 ⋅ C2 �→ C3 , we have

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W(B1 , C1 ) ⋅ W(B2 , C2 ) �→ W(B3 , C3 ). One class of Wiener amalgam spaces we are going to work with is given by W(Lp , Lqm )(ℝ2d ), 1 ≤ p, q ≤ ∞. Let us show that they are translation-invariant. Lemma 2.4.10. If m ∈ ℳv (ℝd ), then W(Lp , Lqm )(ℝd ) is translation invariant, with ‖Ty f ‖W(Lp ,Lqm ) ≤ Cv(y)‖f ‖W(Lp ,Lqm ) ,

∀y ∈ ℝd .

In particular, if m ≡ 1, the translation Ty is an isometry on W(Lp , Lqm )(ℝd ). Proof. Fix a window function g in 𝒟(ℝd ) and consider f ∈ W(Lp , Lqm ). Since Lp is translation-invariant, we have Ty f ∈ Lp and ‖Ty fTx g‖Lp = ‖fTx−y g‖Lp = Fg (x − y) = Ty Fg (x).

2.4 Wiener amalgam spaces | 105

Since weighted Lebesgue space Lqm is translation invariant as well, by using (2.3) as standard, we have ‖Ty f ‖W(Lp ,Lqm ) = ‖Ty Fg ‖Lqm ≤ Cv(y)‖f ‖W(Lp ,Lqm ) . If m ≡ 1, we use the fact that translation is an isometry on Lq . In the next result, we show that if p = q then the Wiener amalgam space W(Lp , Lpm ) turns into the Lebesgue space Lpm and in particular W(Lp , Lp ) = Lp . To this end, we need a lemma which introduces an important property of weight functions. Lemma 2.4.11. Let m be a v-moderate weight. Then for any compact set K ⊂ ℝd with positive (Lebesgue) measure, there exist constants C1 (K) > 0 and C2 (K) > 0 such that C1 (K)m(y) ≤ ∫ m(t) dt ≤ C2 (K)m(y),

x ∈ ℝd

(2.65)

K+x

for all y ∈ K + x. Proof. First, we check that given a compact set K ⊂ ℝd with positive measure, there exists a constant C(K) > 0 such that ∀x ∈ ℝd .

sup m(t) ≤ C(K) inf m(t), t∈K+x

t∈K+x

(2.66)

Fix any compact set K ⊂ ℝd and any x ∈ ℝd . Let t = r + x and u = s + x be any points in K + x. Then m(t) = m(r + x) ≤ Am(x)v(r) ≤ A sup v(r)m(x) ≤ A1 (K)m(x) r∈K

and m(x) = m(u − s) ≤ Bm(u)v(−s) ≤ sup v(s)m(u) ≤ B1 (K)m(u). Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

s∈−K

If we set C(K) = A1 (K)B1 (K), we get m(t) ≤ C(K)m(u) for all t, u ∈ K + x and consequently (2.66). Next, given any y ∈ K + x, we deduce m(y) ≤ sup m(t) ≤ C(K) inf m(t) ≤ t∈K+x

t∈K+x

C(K) ∫ m(t) dt, |K| K+x

and ∫ m(t) dt ≤ |K| sup m(t) ≤ |K|C(K) inf m(t) ≤ |K|C(K)m(y). K+x

t∈K+x

t∈K+x

Therefore we obtain (2.65), with C1 (K) = |K|/C(K) and C2 (K) = |K|C(K).

106 | 2 Function spaces Proposition 2.4.12. Let m be a v-moderate weight. We have W(Lp , Lpm )(ℝd ) = Lpm (ℝd ).

(2.67)

Proof. Suppose that 1 ≤ p < ∞, take a window function g ∈ 𝒟(ℝd ), then 󵄨p 󵄨p 󵄨 󵄨 ‖f ‖pW(Lp ,Lp ) = ∫ ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 󵄨󵄨󵄨g(x − y)󵄨󵄨󵄨 mp (x) dx dy m ℝd ℝd

󵄨 󵄨p 󵄨 󵄨p = ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 ( ∫ 󵄨󵄨󵄨g(x − y)󵄨󵄨󵄨 mp (x) dx) dy. ℝd

ℝd

We can take as compact set K = supp g. Consequently, 󵄨 󵄨p 󵄨 󵄨p ∫ 󵄨󵄨󵄨g(x − y)󵄨󵄨󵄨 mp (x) dx = ∫ 󵄨󵄨󵄨g(x − y)󵄨󵄨󵄨 mp (x) dx. K+y

ℝd

Lemma 2.4.11 yields Amp (y) ≤ ∫ mp (x) dx ≤ Bmp (y),

(2.68)

K+y

so that, using the left-hand side inequality above, 󵄨 󵄨p A‖f ‖pLp = A ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 mp (y) dy m

ℝd

󵄨 󵄨p ≤ ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 ( ∫ mp (x) dx) dy K+y

ℝd

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󵄨 󵄨p = ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 ( ∫ χK (x − y)m(x)p dx) dy ℝd

≍

ℝd p ‖f ‖W(Lp ,Lp ) . m

On the other hand, using the right-hand side inequality in (2.68), 󵄨 󵄨p 󵄨 󵄨p ‖f ‖pW(Lp ,Lp ) ≤ sup󵄨󵄨󵄨g(x)󵄨󵄨󵄨 ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 ( ∫ mp (x) dx) dy m

x∈K

ℝd

󵄨 󵄨p ≤ B sup󵄨󵄨󵄨g(x)󵄨󵄨󵄨 ‖f ‖pLp . m

K+y

x∈K

Hence W(Lp , Lpm ) is equal to Lpm with equivalent norm. The remaining case p = ∞ is similar.

2.4 Wiener amalgam spaces | 107

The same arguments show that, for m ∈ ℳv (ℝ2d ), 2d p,q 2d W(Lp,q , Lp,q m )(ℝ ) = Lm (ℝ ).

(2.69)

For further references, we recall the action of the Fourier transform on the Wiener amalgam spaces W(ℱ Lq , Lp ) (cf. [124]). Proposition 2.4.13. For every 1 ≤ p ≤ q ≤ ∞, the Fourier transform ℱ maps W(ℱ Lq , Lp ) in W(ℱ Lp , Lq ) continuously. For p = q, this was already proved in Theorem 2.3.27, cf. (ii) in Example 2.4.8. Note also that in view of Proposition 2.4.13, exchanging the role of p, q and using (2.54), (ii) in Example 2.4.8, we have, for p ≥ q, M p,q = ℱ W(ℱ Lp , Lq ) �→ W(ℱ Lq , Lp ) and, in particular, M ∞,1 �→ W(ℱ L1 , L∞ ).

(2.70)

The above inclusion is strict, this will play a crucial role in Chapter 4. 2.4.1 Inclusion, convolution, and multiplication relations We now mainly focus on modulation spaces and present their main properties in terms of inclusion, convolution, and multiplication relations. For this reason, a finer estimate in terms of Wiener amalgam spaces of the STFT is needed, see Lemma 2.4.15. 2d Lemma 2.4.14. Fix g ∈ Mv1 (ℝ2d ) \ {0}. For f ∈ W(ℱ L1 , Lp,q m )(ℝ ), we can express the norm by means of the STFT as follows:

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

p

q/p

󵄨 󵄨 ‖f ‖W(ℱ L1 ,Lp,q = ( ∫ ( ∫ ( ∫ 󵄨󵄨󵄨Vg f (z1 , z2 , ζ )󵄨󵄨󵄨 dζ ) m(z1 , z2 )p dz1 ) m ) ℝd

ℝd

1/q

dz2 )

(2.71)

.

ℝ2d

Proof. It is a simple computation. In fact, consider first g ∈ 𝒮 (ℝ2d ) \ {0}, f ∈ 𝒮 (ℝ2d ), q/p

‖f ‖W(ℱ L1 ,Lp,q = ( ∫ ( ∫ ‖f ⋅ T(z1 ,z2 ) g‖̄ pℱ L1 m(z1 , z2 )p dz1 ) m ) ℝd ℝd

1/q

dz2 )

p

q/p

󵄨 󵄨 ̄ = ( ∫ ( ∫ ( ∫ 󵄨󵄨󵄨(f ⋅ T(z1 ,z2 ) g)̂(ζ )󵄨󵄨󵄨 dζ ) m(z1 , z2 )p dz1 ) ℝd ℝd ℝ2d

p

q/p

󵄨 󵄨 = ( ∫ ( ∫ ( ∫ 󵄨󵄨󵄨Vg f (z1 , z2 , ζ )󵄨󵄨󵄨 dζ ) m(z1 , z2 )p dz1 ) ℝd ℝd ℝ2d

1/q

dz2 ) 1/q

dz2 )

.

108 | 2 Function spaces The extension to arbitrary f , g is done by approximation (see, e. g., the proof of Theorem 2.3.12). p,q Lemma 2.4.15. Let 1 ≤ p, q ≤ ∞. If f ∈ Mm (ℝd ) and g ∈ Mv1 (ℝd ), then Vg f ∈ 1 p,q 2d W(ℱ L , Lm )(ℝ ) with norm estimate

‖Vg f ‖W(ℱ L1 ,Lp,q ≲ ‖f ‖Mmp,q ‖g‖Mv1 . m )

(2.72)

Proof. Let φ ∈ 𝒮 (ℝd ) \ {0} and set Φ = Vφ φ ∈ 𝒮 (ℝ2d ). We only consider f , g ∈ 𝒮 (ℝd ), the extension to arbitrary f , g can be obtained by approximation. We use (2.71) and then Lemma 1.3.37 (i) to evaluate the W(ℱ L1 , Lp,q m )-norm of Vg f . ‖Vg f ‖W(ℱ L1 ,Lp,q m )

p

q/p

󵄨 󵄨 ≍ ( ∫ ( ∫ ( ∫ 󵄨󵄨󵄨VΦ (Vg f )(z1 , z2 , ζ1 , ζ2 )󵄨󵄨󵄨 dζ1 dζ2 ) m(z1 , z2 )p dz1 ) ℝd ℝd ℝ2d

1/q

dz2 )

p

󵄨 󵄨󵄨 󵄨 = ( ∫ ( ∫ ( ∫ 󵄨󵄨󵄨Vφ g(−z1 − ζ2 , ζ1 )󵄨󵄨󵄨󵄨󵄨󵄨Vφ f (−ζ2 , z2 + ζ1 )󵄨󵄨󵄨 dζ1 dζ2 ) ℝd

ℝd ℝ2d

q/p

× m(z1 , z2 )p dz1 )

1/q

dz2 )

p

󵄨 󵄨󵄨 󵄨 = ( ∫ ( ∫ ( ∫ 󵄨󵄨󵄨Vφ f (w1 , w2 )󵄨󵄨󵄨󵄨󵄨󵄨Vφ g(w1 − z1 , w2 − z2 )󵄨󵄨󵄨 dw1 dw2 ) ℝd

ℝd ℝ2d

q/p

× m(z1 , z2 )p dz1 )

1/q

dz2 ) p

q/p

= ( ∫ ( ∫ (|Vφ f | ∗ |Vφ g|∗ ) (z1 , z2 )m(z1 , z1 )p dz1 )

1/q

dz2 )

ℝd ℝd

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

󵄩 󵄩 = 󵄩󵄩󵄩|Vφ f | ∗ |Vφ g|∗ 󵄩󵄩󵄩Lp,q . m Now Young’s inequality for mixed norm spaces (cf. Theorem 2.2.3 (v)) yields the desired estimate ‖Vg f ‖W(ℱ L1 ,Lp,q ≲ ‖Vφ f ‖Lp,q ‖Vφ g‖L1v ≍ ‖f ‖Mmp,q ‖g‖Mv1 . m ) m p,q Corollary 2.4.16. If g ∈ Mv1 (ℝd ) \ {0} and f ∈ Mm (ℝd ), then 2d Vg f ∈ W(L∞ , Lp,q m )(ℝ )

with ‖Vg f ‖W(𝒞0 ,Lp,q ≤ C‖Vg g‖W(L∞ ,L1v ) ‖f ‖Mmp,q . m )

(2.73)

2.4 Wiener amalgam spaces | 109

Proof. It follows from Lemma 2.4.15 and the inclusion relations for Wiener amalgam spaces since, by Riemann–Lebesgue lemma, ℱ L1 (ℝd ) �→ 𝒞0 (ℝd ) �→ L∞ (ℝd ). We have now all the instruments to prove the inclusion relations for modulation spaces. Theorem 2.4.17 (Inclusion relations). If p1 ≤ p2 , q1 ≤ q2 and m2 ≲ m1 , then p1 ,q1 p2 ,q2 Mm (ℝd ) �→ Mm (ℝd ). 1 2

(2.74)

Proof. The following inclusion relation between ℓp (ℤd ) spaces is well known: ℓp1 (ℤd ) ⊆ ℓp2 (ℤd )

for p1 ≤ p2 .

Such relation extends easily to the discrete spaces ℓp,q (ℤ2d ). In fact, let us show ℓp1 ,q1 (ℤ2d ) ⊆ ℓp2 ,q2 (ℤ2d ) for p1 ≤ p2 , q1 ≤ q2 . Given a ∈ ℓp1 ,q1 (ℤ2d ), define the sequences c = (cn ), d = (dn ) by p1

1 p1

p2

1 p2

dn = ( ∑ |ak,n | ) .

cn = ( ∑ |ak,n | ) ,

k∈ℤd

k∈ℤd

Then dn ≤ cn because p1 ≤ p2 and ‖a‖ℓp2 ,q2 = ‖d‖q2 ≤ ‖d‖q1 ≤ ‖c‖q1 = ‖a‖ℓp1 ,q1 . The inclusion p1 ,q1 p2 ,q2 ℓm (ℤ2d ) ⊆ ℓm (ℤ2d ) 1 2

for p1 ≤ p2 , q1 ≤ q2 , m2 ≤ Cm1 ,

(2.75)

follows then from ‖a‖ℓp2 ,q2 = ‖am2 ‖ℓp2 ,q2 ≤ ‖am2 ‖ℓp1 ,q1 ≤ C‖am1 ‖ℓp1 ,q1 = C‖a‖ℓmp1 ,q1 . Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

m2

1

p,q Fix a window g ∈ 𝒮 (ℝd ) \ {0}. Given f ∈ Mm (ℝd ), we define the sequence a = (ak,n ) by

ak,n =

sup

(x,ξ )∈[0,1]2d

󵄨󵄨 󵄨 󵄨󵄨Vg f (x + k, ξ + n)󵄨󵄨󵄨.

Hence we may write ‖a‖ℓmp,q = ‖Vg f ‖W(L∞ ,Lp,q . Finally, by Proposition 2.4.12 and (2.75), m ) ‖f ‖M p2 ,q2 = ‖Vg f ‖Lp2 ,q2 = ‖Vg f ‖W(Lp2 ,q2 ,Lp2 ,q2 ) m2

m2

m2

≤ ‖Vg f ‖W(L∞ ,Lp2 ,q2 ) = ‖a‖ℓp2 ,q2 ≤ C‖a‖ℓmp1 ,q1 m2

m2

= C‖Vg f ‖W(L∞ ,Lpm1 ,q1 ) ≤ C 󸀠 ‖f ‖Mmp1 ,q1 , 1

where the last estimate follows from Corollary 2.4.16.

1

1

110 | 2 Function spaces Other useful embeddings are as follows, cf. [126]. Proposition 2.4.18. Given 1 ≤ p1 , p2 , q1 , q2 ≤ ∞, the weight vs defined in (2.5), with s = s1 or s = s2 in ℝ, one has p ,q

p ,q

1 1 2 2 M1⊗v (ℝd ) �→ M1⊗v (ℝd )

(2.76)

p1 ≤ p2

(2.77)

s1

s2

if and only if

and q1 ≤ q2 ,

s1 ≥ s2

or

q1 > q2 ,

s s1 1 1 + > 2 + . d q1 d q2

(2.78)

Proof. We use the discrete modulation norm defined in (2.48), so that ‖f ‖M p,q ≍ ( ∑ 1⊗vs

k∈ℤd

‖◻k f ‖qLp ⟨k⟩sq )

1 q

.

The necessity of (2.77) follows from the fact that ℱ Lp1 is locally contained in ℱ Lp2 if and only if p1 ≤ p2 (with strict inclusion if p1 < p2 ), cf. [20, 147]. The set of conditions in (2.78) in turn describes the inclusions between weighted ℓq spaces. Namely, recall that 󵄩 󵄩 ℓsq = {a = (ak )k∈ℤd : 󵄩󵄩󵄩⟨k⟩s ak 󵄩󵄩󵄩ℓq < ∞} and

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

ℓsq11 ⊂ ℓsq22 if and only if the indices’ relations in (2.78) are satisfied; cf., for instance, [177, Lemma 2.10]. It is important to notice that the spaces above are equal if and only if all the parameters are equal. We now present the convolution and multiplication properties for modulation spaces. Proposition 2.4.19. Consider m ∈ ℳv (ℝ2d ). We write m1 (x) = m(x, 0) and m2 (ξ ) = m(0, ξ ) for the restrictions to ℝd × {0} and {0} × ℝd , and likewise for v. Let ν be an arbitrary weight function on ℝd satisfying (2.17) and 1 ≤ p, q, p1 , p2 , q1 , q2 ≤ ∞. If 1 1 1 +1≤ + p p1 p2

(2.79)

2.4 Wiener amalgam spaces | 111

and 1 1 1 ≤ + , q q1 q2

(2.80)

then p ,q

p ,q

p,q Mm11 ⊗ν1 (ℝd ) ∗ Mv 2⊗v2 ν−1 (ℝd ) �→ Mm (ℝd ) 1

(2.81)

2

with norm inequality ‖f ∗ h‖Mmp,q ≲ ‖f ‖M p1 ,q1 ‖h‖M p2 ,q2 m1 ⊗ν

v1 ⊗v2 ν−1

(2.82)

.

Proof. We measure the modulation space norm with respect to the Gaussian windows 2 2 g0 (x) = e−πx and g(x) = 2−d/2 e−πx /2 = (g0 ∗ g0 )(x) ∈ 𝒮 (ℝd ). Recall that different windows yield equivalent norms for the modulation spaces, specifically, . ≍ ‖Vg f ‖Lp,q ‖f ‖Mmp,q ≍ ‖Vg0 f ‖Lp,q m m p ,q

p ,q

Consider f ∈ Mm11 ⊗ν1 (ℝd ), h ∈ Mv 2⊗v2 ν−1 (ℝd ) such that 1

2

1 1 1 +1= + , p p1 p2

1 1 1 = + . q q1 q2

Formula (1.28) for the STFT Vg (f ∗ h)(x, ξ ) and the identity Mξ (g0∗ ∗ g0∗ ) = Mξ g0∗ ∗ Mξ g0∗ , allow expressing the STFT of f ∗ h as follows:

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Vg (f ∗ h)(x, ξ ) = e−2πix⋅ξ ((f ∗ h) ∗ Mξ g ∗ )(x) = e−2πix⋅ξ ((f ∗ Mξ g0∗ ) ∗ (h ∗ Mξ g0∗ ))(x). To estimate the M p,q -norm of f ∗ h, we first majorize m by m(x, ξ ) ≲ m(x, 0)v(0, ξ ) = m1 (x)v2 (ξ ) and then we use Young’s inequality with 1/p1 +1/p2 = 1+1/p in the x-variable and Hölder’s inequality with 1/q1 + 1/q2 = 1/q in the ξ -variable, with weights as in Theorem 2.2.3. We obtain that 󵄩 󵄩 ‖f ∗ h‖Mmp,q ≍ 󵄩󵄩󵄩Vg (f ∗ h)󵄩󵄩󵄩Lp,q m 󵄨 󵄨p ≲ ( ∫ ( ∫ 󵄨󵄨󵄨(f ∗ Mξ g0∗ ) ∗ (h ∗ Mξ g0∗ )(x)󵄨󵄨󵄨 m1 (x)p dx) ℝd ℝd

1/q

󵄩 󵄩q = ( ∫ 󵄩󵄩󵄩(f ∗ Mξ g0∗ ) ∗ (h ∗ Mξ g0∗ )󵄩󵄩󵄩Lp v2 (ξ )q dξ ) m1

ℝd

1/q

󵄩 󵄩q 󵄩 󵄩q ≲ ( ∫ 󵄩󵄩󵄩f ∗ Mξ g0∗ 󵄩󵄩󵄩Lp1 󵄩󵄩󵄩h ∗ Mξ g0∗ 󵄩󵄩󵄩Lp2 v2 (ξ )q dξ ) ℝd

m1

v1

q/p

1/s

v2 (ξ )q dξ )

112 | 2 Function spaces 1

q

1

q1 q2 󵄩 󵄩q 󵄩 󵄩q v (ξ ) 2 ≲ ( ∫ 󵄩󵄩󵄩f ∗ Mξ g0∗ 󵄩󵄩󵄩L1p1 ν(ξ )q1 dξ ) ( ∫ 󵄩󵄩󵄩h ∗ Mξ g0∗ 󵄩󵄩󵄩L2p2 2 q dξ ) m1 v1 ν(ξ ) 2

ℝd

ℝd

≍ ‖f ‖M p1 ,q1

m1 ⊗ν

‖h‖M p2 ,q2

v1 ⊗v2 ν−1

The estimate (2.82) is proved when the indices satisfy (2.79) and (2.80) with the equalities. The case of the inequalities follows by the inclusion relations (2.74). Remark 2.4.20. Despite the large number of indices, the statement of this proposition has some intuitive meaning: a function f ∈ M p,q behaves like f ∈ Lp and f ̂ ∈ Lq ; so the parameters related to the x-variable behave like those in Young’s theorem for convolution, whereas the parameters related to ξ behave like in Hölder’s inequality for pointwise multiplication. For Wiener amalgam spaces, we have the following multiplication relations. Proposition 2.4.21. For every 1 ≤ p, q ≤ ∞, we have ‖fu‖W(ℱ Lp ,Lq ) ≤ ‖f ‖W(ℱ L1 ,L∞ ) ‖u‖W(ℱ Lp ,Lq ) . Proof. From the norm equivalence (2.54), the estimate to prove is equivalent to ‖f ̂ ∗ u‖̂ M p,q ≤ ‖f ̂‖M 1,∞ ‖u‖̂ M p,q , but this a special case of Proposition 2.4.19. Convolution relations for W(ℱ Lp , Lq ) spaces follow immediately from Theorem 2.4.9 (ii). Precisely, Proposition 2.4.22. For every 1 ≤ p, q ≤ ∞, we have ‖f ∗ u‖W(ℱ Lp ,Lq ) ≤ ‖f ‖W(ℱ L∞ ,L1 ) ‖u‖W(ℱ Lp ,Lq ) .

(2.83)

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Finally, returning to modulation spaces, we have the following multiplication properties. Proposition 2.4.23. Consider m ∈ ℳv (ℝ2d ). We write m1 (x) = m(x, 0) and m2 (ξ ) = m(0, ξ ) for the restrictions to ℝd ×{0} and {0}×ℝd , and likewise for v. Let ν be an arbitrary weight function on ℝd satisfying (2.17). Consider 1 ≤ p, q, p1 , p2 , q1 , q2 ≤ ∞ such that

and

p ,q

1 1 1 ≤ + p p1 p2

(2.84)

1 1 1 +1≤ + . q q1 q2

(2.85)

p ,q

p,q 1 If f ∈ Mv11ν⊗m (ℝd ), h ∈ Mν−12 ⊗v2 (ℝd ), the product fh is in Mm (ℝd ), with 2 2

‖fh‖Mmp,q ≲ ‖f ‖M p1 ,q1 ‖h‖M p2 ,q2 . v1 ν⊗m2

ν−1 ⊗v2

(2.86)

2.5 Dilation properties for Wiener amalgam spaces W (Lp , Lq )

| 113

Proof. We measure the modulation space norm with respect to the Gaussian windows 2 g(x) = g02 (x), with g0 (x) = e−πx . Observe that m(x, ξ ) ≲ v(x, 0)m(0, ξ ) = v1 (x)m2 (ξ ) and 󵄩 󵄩 󵄩 󵄩 ≲ 󵄩󵄩󵄩Vg (fh)󵄩󵄩󵄩Lp,q . ‖fh‖Mmp,q ≍ 󵄩󵄩󵄩Vg (fh)󵄩󵄩󵄩Lp,q m v1 ⊗m2 Using (1.24), we write ? ? ? ̄ ) = fT Vg (fh)(x, ξ ) = fhT x g0 ∗ hTx g0 (ξ ). x g(ξ p ,q

p ,q

1 Consider f ∈ Mv11ν⊗m (ℝd ), h ∈ Mν−12 ⊗v2 (ℝd ) such that relations (2.84) and (2.85) are 2 2 satisfied with the equality signs. Using Minkowskii’s integral inequality first, Hölder’s inequality with 1/p1 + 1/p2 = 1/p in the x-variable, and finally Young’s inequality with 1/q1 + 1/q2 = 1 + 1/q in the ξ -variable, we reckon

󵄩 󵄩 󵄩 󵄩 ‖fh‖Mmp,q ≍ 󵄩󵄩󵄩Vg (fh)󵄩󵄩󵄩Lp,q ≲ 󵄩󵄩󵄩Vg (fh)󵄩󵄩󵄩Lp,q m v1 ⊗m2 󵄩󵄩󵄩󵄩 ? ? 󵄩󵄩 p 󵄩 = 󵄩󵄩󵄩󵄩(fTx g0 ∗ hTx g0 )(ξ )󵄩󵄩Lv (ℝd ) 󵄩󵄩󵄩Lqm (ℝd ) x ξ 1 2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ? 󵄩 󵄩 ? ≤ 󵄩󵄩 ∫ 󵄩󵄩fTx g0 (ξ − η)hT g0 (η)󵄩󵄩󵄩Lpv (ℝd ) dη󵄩󵄩󵄩 x x 󵄩󵄩 󵄩󵄩Lqm (ℝd ) 1 ξ 2 d ℝ

󵄩󵄩 󵄩󵄩 󵄩 󵄩? 󵄩 󵄩 ? 󵄩󵄩 p 󵄩 ≤ 󵄩󵄩󵄩 ∫ 󵄩󵄩󵄩fT g0 (ξ − η)󵄩󵄩󵄩Lpv 1ν (ℝd ) 󵄩󵄩󵄩hT g0 (η)󵄩󵄩L 2 (ℝd ) dη󵄩󵄩󵄩 x x x x 󵄩󵄩 󵄩󵄩Lqm (ℝd ) 1 ν−1 ξ 2 d ℝ

󵄩󵄩 ? 󵄩󵄩 p 󵄩󵄩 ? 󵄩󵄩 󵄩󵄩 q = 󵄩󵄩󵄩󵄩󵄩󵄩(fT 2 (ℝd ) )(ξ )󵄩 x g0 󵄩 󵄩Lv11ν (ℝdx ) ∗ 󵄩󵄩hTx g0 󵄩󵄩Lpν−1 󵄩Lm2 (ℝdξ ) x 󵄩󵄩 ? 󵄩󵄩 p 󵄩󵄩 󵄩󵄩󵄩󵄩 ? 󵄩󵄩 󵄩󵄩 q ≤ 󵄩󵄩󵄩󵄩󵄩󵄩fT d 2 (ℝd ) 󵄩 x g0 󵄩 󵄩Lv11ν (ℝdx ) 󵄩󵄩Lqm2 (ℝdξ ) 󵄩󵄩󵄩󵄩hTx g0 󵄩󵄩Lpν−1 x 󵄩Lm2 (ℝξ ) ≍ ‖f ‖M p1 ,q1 ‖h‖M p2 ,q2 . v1 ν⊗m2

ν−1 ⊗v2

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

The inequalities in (2.84) and (2.85) follow by the inclusion relations (2.74). This concludes the proof. In a recent contribution [176], W. Guo, J. Chen, D. Fan, and G. Zhao characterize some properties of weighted modulation and Wiener amalgam spaces by the corresponding properties on weighted Lebesgue spaces. As applications, they obtain sharp conditions for product and convolution inequalities, and embedding on weighted modulation and Wiener amalgam spaces. The weights considered there are of the type m(x, ξ ) = m(x)μ(ξ ). We refer the interested reader to the original paper, see also [292].

2.5 Dilation properties for Wiener amalgam spaces W (Lp , Lq ) In this section we study the dilation properties of Wiener amalgam spaces W(Lp , Lq ), 1 ≤ p, q ≤ ∞.

114 | 2 Function spaces For λ > 0, we set fλ (x) = f (λx). Then Wiener amalgam spaces fulfill the following estimates. Proposition 2.5.1. For 1 ≤ p, q ≤ ∞, ‖fλ ‖W(Lp ,Lq ) ≲ λ

−d max{ p1 , q1 }

‖f ‖W(Lp ,Lq ) ,

∀λ, 0 < λ ≤ 1,

(2.87)

∀λ ≥ 1.

(2.88)

∀λ, 0 < λ ≤ 1,

(2.89)

∀λ ≥ 1.

(2.90)

and ‖fλ ‖W(Lp ,Lq ) ≲ λ

−d min{ p1 , q1 }

‖f ‖W(Lp ,Lq ) ,

Also, we have ‖fλ ‖W(Lp ,Lq ) ≳ λ

−d min{ p1 , q1 }

‖f ‖W(Lp ,Lq ) ,

and ‖fλ ‖W(Lp ,Lq ) ≳ λ

−d max{ p1 , q1 }

‖f ‖W(Lp ,Lq ) ,

We first prove the following weaker estimates. Lemma 2.5.2. For 1 ≤ p, q ≤ ∞, ‖fλ ‖W(Lp ,Lq ) ≲ λ

−d( p1 + q1 )

‖f ‖W(Lp ,Lq ) ,

∀λ, 0 < λ ≤ 1,

(2.91)

∀λ ≥ 1.

(2.92)

and ‖fλ ‖W(Lp ,Lq ) ≲ λ

d(1− p1 − q1 )

‖f ‖W(Lp ,Lq ) ,

Proof. To compute the Wiener norm, we choose the window function g = χ[0,1]d , the characteristic function of the box [0, 1]d . Then

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󵄩󵄩 󵄩 󵄩 ‖fλ ‖W(Lp ,Lq ) ≍ 󵄩󵄩󵄩󵄩󵄩󵄩f (λt)g(t − x)󵄩󵄩󵄩Lp 󵄩󵄩󵄩Lqx t 󵄩󵄩 󵄩 󵄩 − dp 󵄩 󵄩 = λ 󵄩󵄩󵄩󵄩f (t)g1/λ (t − λx)󵄩󵄩󵄩Lp 󵄩󵄩󵄩Lqx t 󵄩󵄩󵄩 󵄩 󵄩 −d( p1 + q1 ) 󵄩 =λ 󵄩󵄩󵄩󵄩f (t)g1/λ (t − x)󵄩󵄩󵄩Lp 󵄩󵄩󵄩Lqx . t

(2.93)

If 0 < λ ≤ 1, the window function g fulfills g1/λ (y) ≤ g(y), and (2.91) follows. Let now λ ≥ 1. To prove (2.92), we observe that g1/λ (t) ≤

∑

j∈ℤd ∩[0,λ]d

g(t − j).

Notice that the above sum contains Nλ = O(λd ) terms. Using this inequality, we dominate the expression in (2.93) by λ

−d( p1 + q1 )

∑

j∈ℤd ∩[0,λ]d

󵄩󵄩󵄩󵄩 󵄩 󵄩 −d( 1 + 1 ) 󵄩󵄩󵄩󵄩f (t)g(t − x − j)󵄩󵄩󵄩Lp 󵄩󵄩󵄩Lqx = λ p q Nλ ‖f ‖W(Lp ,Lq ) , t

where we also performed a change of variable in the integral with respect to x. This concludes the proof.

2.5 Dilation properties for Wiener amalgam spaces W (Lp , Lq )

| 115

The complex interpolation of Wiener amalgam spaces (having Lebesgue spaces as global components) in Theorem 2.4.9, can be rephrased as follows (see also [120]). Corollary 2.5.3. Let B0 , B1 , be local components of Wiener amalgam spaces, as in Definition 2.4.3. Then for 1 ≤ q0 , q1 ≤ ∞ with q0 < ∞ or q1 < ∞, and 0 < θ < 1 we have [W(B0 , Lq0 ), W(B1 , Lq1 )][θ] = W([B0 , B1 ][θ] , Lqθ ), with 1/qθ = (1 − θ)/q0 + θ/q1 . Proof of Proposition 2.5.1. We first prove (2.87) and (2.88) when p = ∞. We see at once that (2.87) coincides with (2.91) when p = ∞. On the other hand, (2.88) for p = ∞ follows by complex interpolation (cf. Corollary 2.5.3) from (2.92) with (p, q) = (∞, 1), i. e., ‖fλ ‖W(L∞ ,L1 ) ≲ ‖f ‖W(L∞ ,L1 ) and the trivial estimate ‖fλ ‖W(L∞ ,L∞ ) ≍ ‖fλ ‖L∞ = ‖f ‖L∞ ≍ ‖f ‖W(L∞ ,L∞ ) . Since the estimates (2.87) and (2.88) also hold for p = q (because W(Lp , Lp ) = Lp with equivalent norms, in view of Proposition 2.4.12), by interpolation with the case p = ∞, 1 ≤ q ≤ ∞, we see that they hold for any pair (p, q), with 1 ≤ q ≤ p ≤ ∞. When 1 < p < q ≤ ∞ they follow by duality arguments as follows, see (v) in Theorem 2.4.9. Suppose first λ ≥ 1. Then relation (2.87), applied to the pair (p󸀠 , q󸀠 ), yields ‖fλ ‖W(Lp ,Lq ) =

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

=

sup

‖g‖

󸀠 󸀠 W(Lp ,Lq )

sup

‖g‖

≤λ

󸀠 󸀠 W(Lp ,Lq )

=1

󵄨 󵄨 λ−d 󵄨󵄨󵄨⟨f , g1/λ ⟩󵄨󵄨󵄨

sup

−d ‖g‖

≲ λ−d λ

󵄨󵄨 󵄨 󵄨󵄨⟨fλ , g⟩󵄨󵄨󵄨

=1

󸀠 󸀠 W(Lp ,Lq )

d max{

=1

1 1 , p󸀠 q󸀠

}

‖f ‖W(Lp ,Lq ) ‖g1/λ ‖W(Lp󸀠 ,Lq󸀠 )

‖f ‖W(Lp ,Lq ) ,

which is (2.88). Similarly, one proves (2.87) in that case. It remains to prove the estimates (2.87) and (2.88) for p = 1, 1 < q ≤ ∞. They follow from interpolation from the case (p, q) = (1, 1) and the case (p, q) = (1, ∞), where (2.87) and (2.88) coincide with (2.91) and (2.92), respectively. Finally, (2.89) and (2.90) follow at once from (2.88) and (2.87), respectively, applied to the function f1/λ . We now show that the result above is sharp.

116 | 2 Function spaces Proposition 2.5.4 (Sharpness of (2.87) and (2.88)). (i) Suppose that, for some α ∈ ℝ, ‖fλ ‖W(Lp ,Lq ) ≲ λα ‖f ‖W(Lp ,Lq ) ,

∀ λ, 0 < λ ≤ 1, ∀f ∈ W(Lp , Lq ).

(2.94)

Then 1 1 α ≤ −d max{ , }. p q

(2.95)

(ii) Suppose that, for some α ∈ ℝ, ‖fλ ‖W(Lp ,Lq ) ≲ λα ‖f ‖W(Lp ,Lq ) ,

∀λ ≥ 1, ∀f ∈ W(Lp , Lq ).

(2.96)

Then 1 1 α ≥ −d min{ , }. p q

(2.97)

This also shows the sharpness of the estimates (2.89) and (2.90), since they are equivalent to (2.88) and (2.87), respectively. Proof. (i) First, consider the case p ≥ q. We have W(Lp , Lq ) �→ W(Lq , Lq ) = Lq . Hence λ

− dq

‖f ‖Lq = ‖fλ ‖Lq ≲ ‖fλ ‖W(Lp ,Lq ) .

Combining this estimate with (2.94) and letting λ → 0+ , we obtain α ≤ −d/q. Assume now p < q. It suffices to verify that, for every ϵ > 0, there exists f ∈ W(Lp , Lq ) such that ‖fλ ‖W(Lp ,Lq ) ≥ Cλ

− dp +ϵ

(2.98)

.

We study the case of dimension d = 1. The general case follows by tensor products of functions of one variable. To this end, we choose

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− p1 +ϵ

for |t| ≤ 1, for |t| > 1.

|t| f (t) = { 0

Observe that f ∈ W(Lp , L1 ) �→ W(Lp , Lq ), for every 1 ≤ q ≤ ∞, and fλ (t) = f (λt) = λ

− p1 +ϵ

− p1 +ϵ

|t|

,

1 . λ

for |t| ≤

Now, take g = χB1 (0) as window function. Of course, 1/q

‖fλ ‖W(Lp ,Lq ) = (∫ ‖fλ Ty g‖qLp dy)

1/q

≥ ( ∫ ‖fλ Ty g‖qLp dy) B1 (0)

.

(2.99)

2.5 Dilation properties for Wiener amalgam spaces W (Lp , Lq )

By using (2.99) and the choice g = χB1 (0) , for λ ≤ 1/2 we have fλ (t) ≥ λ

− p1 +ϵ

last expression is estimated from below by

| 117

f (t) and the

1/q

λ−1/p+ϵ ( ∫ ‖fTy g‖qLp dy)

,

B1 (0)

that is, (2.98) holds. (ii) Again, we first consider the case p ≥ q. Then W(Lp , Lq ) �→ W(Lp , Lp ) = Lp . Hence ‖fλ ‖W(Lp ,Lq ) ≳ ‖fλ ‖Lp = λ

− dp

‖f ‖Lp .

Combining this estimate with (2.96) and letting λ → +∞, we obtain α ≥ − dp .

Suppose now p < q. As before it suffices to prove, in dimension d = 1, that for every ϵ > 0 there exists a function f ∈ W(Lp , Lq ) such that ‖fλ ‖W(Lp ,Lq ) ≥ Cλ

− q1 −ϵ

.

Therefore choose {|t| f (t) = { {0

− q1 −ϵ

for |t| ≥ 1, for |t| < 1.

Then f ∈ W(L∞ , Lq ) �→ W(Lp , Lq ), for every 1 ≤ p ≤ ∞, and f (λt) = λ

− q1 −ϵ

− q1 −ϵ

|t|

,

for |t| ≥

1 . λ

(2.100)

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Again, choose g = χB1 (0) as window function. We have 1/q

‖fλ ‖W(Lp ,Lq ) ≥ ( ∫ ‖fλ Ty g‖qLp dy)

.

B2 (0)

By using (2.100) and the choice g = χB1 (0) , for λ ≥ 1 we have fλ (t) ≥ λ last expression can be estimated from below by

1/q

λ−1/q−ϵ ( ∫ ‖fTy g‖qLp dy) B2 (0)

which concludes the proof of (ii).

,

− q1 −ϵ

f (t) and the

118 | 2 Function spaces

2.6 Dilation properties for modulation spaces These properties were essentially proved in the paper [70], extending the case A = λI, λ > 0, already treated in [274]. Given a function f on ℝd and A ∈ GL(d, ℝ), we set fA (t) = f (At). We also consider the unitary operator 𝒰A on L2 (ℝd ) defined by 1/2

𝒰A f (t) = | det A|

f (At) = | det A|1/2 fA (t).

(2.101)

In this section we study the boundedness of this operator on modulation and Wiener amalgam spaces. We need the following preliminary results. Lemma 1.3.36 helps compute the modulation norm of the rescaled Gaussian φA . 2

Lemma 2.6.1. Let 1 ≤ p, q ≤ ∞, A ∈ GL(d, ℝ) and φ(t) = e−πt . Then ‖φA ‖M p,q = p−d/(2p) q−d/(2q) | det A|−1/p (det(AT A + I))

−(1−1/q−1/p)/2

.

(2.102)

Proof. Since the modulation space norm is independent of the choice of the window function, we choose the Gaussian φ, so that ‖φA ‖M p,q ≍ ‖Vφ φA ‖Lp,q . Since T

∫ e−πp(I−(A

A+I)−1 )x⋅x

dx = (det(I − (AT A + I) ))

ℝd

−1

p

−1/2 −d/2 1/2

= p−d/2 | det A|−1 (det(AT A + I)) T

and, analogously, ∫ℝd e−πq(A follows from Lemma 1.3.36.

A+I)−1 ξ ⋅ξ

dξ = (det(AT A+I))1/2 q−d/2 , the result immediately

As a consequence, if we consider the matrix A = √λI, λ > 0, and the rescaled 2 Gaussian functions φ√λ (t) = φ(√λt) = e−πλt , we obtain the following result.

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Lemma 2.6.2. For 1 ≤ p, q ≤ ∞, we have ‖φ√λ ‖M p,q ≍ λ ‖φ√λ ‖M p,q ≍ λ

−

d 2q󸀠

d − 2p

as λ → +∞,

(2.103)

as λ → 0+ .

(2.104)

Proposition 2.6.3. Let 1 ≤ p, q ≤ ∞ and A ∈ GL(d, ℝ). Then for every f ∈ M p,q (ℝd ) 1/2

‖fA ‖M p,q ≲ | det A|−(1/p−1/q+1) (det(I + AT A)) ‖f ‖M p,q .

(2.105)

Proof. First, by a change of variable, the dilation is transferred from the function f to the window φ: Vφ fA (x, ξ ) = | det A|−1 Vφ

f (Ax, (AT ) ξ ). −1

A−1

2.6 Dilation properties for modulation spaces |

119

Whence performing the change of variables Ax = u, (AT )−1 ξ = v, q/p

−1 󵄨p f (Ax, (AT ) ξ )󵄨󵄨󵄨 dx) A−1

󵄨 ‖fA ‖M p,q = | det A|−1 ( ∫ ( ∫ 󵄨󵄨󵄨Vφ ℝd ℝd

= | det A|−(1/p−1/q+1) ‖Vφ

A−1

1/q

dξ )

f ‖Lp,q .

Note that the same arguments apply to φA , hence −( p1 − q1 +1)

‖φA ‖M p,q = | det A|

‖Vφ

A−1

φ‖Lp,q ,

so that combining with Lemma 2.6.1 we obtain, in particular, 1/2

‖Vφ

A−1

φ‖L1 = (det(I + AT A)) .

(2.106)

2

Now, Lemma 1.2.29, written for h(t) = γ(t) = φ(t) = e−πt , yields the following bound: 󵄨 󵄨󵄨 −2 󵄨󵄨VφA−1 f (x, ξ )󵄨󵄨󵄨 ≤ ‖φ‖L2 (|Vφ f | ∗ |VφA−1 φ|)(x, ξ ). Finally, Young’s inequality and (2.106) provide the desired result 󵄩 ‖fA ‖M p,q ≲ | det A|−(1/p−1/q+1) 󵄩󵄩󵄩|Vφ f | ∗ |Vφ

A−1

≲ | det A|−(1/p−1/q+1) ‖Vφ f ‖Lp,q ‖Vφ

󵄩 φ|󵄩󵄩󵄩Lp,q

A−1

φ‖L1 1/2

≍ | det A|−(1/p−1/q+1) (det(I + AT A)) ‖f ‖M p,q . Corollary 2.6.4. Let 1 ≤ p, q ≤ ∞ and λ ∈ ℝ \ {0}. Then there exists a constant C > 0 such that d/2

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‖fλ ‖M p,q ≤ C|λ|−d(1/p−1/q+1) (1 + λ2 )

‖f ‖M p,q .

Proof. It follows by Proposition 2.6.3 with the matrix A = diag[λ, . . . , λ]. Corollary 2.6.5. Let 1 ≤ p, q ≤ ∞ and A ∈ GL(d, ℝ). Then for every f ∈ W(ℱ Lp , Lq )(ℝd ), 1/2

‖fA ‖W(ℱ Lp ,Lq ) ≲ | det A|(1/p−1/q−1) (det(I + AT A)) ‖f ‖W(ℱ Lp ,Lq ) .

(2.107)

Proof. It follows immediately from the relation between Wiener amalgam spaces ̂) = and modulation spaces given by W(ℱ Lp , Lq ) = ℱ M p,q and by the relation (f A −1 ̂ | det A| (f ) T −1 . (A )

In the sequel we give a more precise result about the behavior of the dilation operator M p,q → M p,q in terms of A, when A is a symmetric matrix.

120 | 2 Function spaces For 1 ≤ p ≤ ∞, let p󸀠 be the conjugate exponent of p, 1/p + 1/p󸀠 = 1. For (1/p, 1/q) ∈ [0, 1] × [0, 1], we define the subsets I1 = max(1/p, 1/p󸀠 ) ≤ 1/q,

I2 = max(1/q, 1/2) ≤ 1/p󸀠 ,

I3 = max(1/q, 1/2) ≤ 1/p,

I1∗ = min(1/p, 1/p󸀠 ) ≥ 1/q,

I2∗ = min(1/q, 1/2) ≥ 1/p󸀠 ,

I3∗ = min(1/q, 1/2) ≥ 1/p,

as shown in Figure 2.1.

Figure 2.1: The index sets.

We introduce the indices −1/p { { { μ1 (p, q) = {1/q − 1 { { {−2/p + 1/q

if (1/p, 1/q) ∈ I1∗ , if (1/p, 1/q) ∈ I2∗ ,

if (1/p, 1/q) ∈

(2.108)

I3∗ ,

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and −1/p { { { μ2 (p, q) = {1/q − 1 { { {−2/p + 1/q

if (1/p, 1/q) ∈ I1 ,

if (1/p, 1/q) ∈ I2 ,

(2.109)

if (1/p, 1/q) ∈ I3 .

We are now able to present the dilation properties for modulation spaces. Let us first recall the basic result for A = λI in [274, Theorem 1.1]. Theorem 2.6.6. Let 1 ≤ p, q ≤ ∞, and A = λI, λ ≠ 0. (i) We have ‖fA ‖M p,q ≲ |λ|dμ1 (p,q) ‖f ‖M p,q ,

|λ| ≥ 1, ∀f ∈ M p,q (ℝd ).

2.6 Dilation properties for modulation spaces |

121

Conversely, if there exists α ∈ ℝ such that |λ| ≥ 1, ∀f ∈ M p,q (ℝd ),

‖fA ‖M p,q ≲ |λ|α ‖f ‖M p,q , then α ≥ dμ1 (p, q). (ii) We have

0 < |λ| ≤ 1, ∀f ∈ M p,q (ℝd ).

‖fA ‖M p,q ≲ |λ|dμ2 (p,q) ‖f ‖M p,q , Conversely, if there exists β ∈ ℝ such that ‖fA ‖M p,q ≲ |λ|β ‖f ‖M p,q ,

0 < |λ| ≤ 1, ∀f ∈ M p,q (ℝd ),

then β ≤ dμ2 (p, q). For simplicity, from now on, we use the abuse of notation fλ = fA , with A = λI, λ ≠ 0. Proof Theorem 2.6.6 (ii), with (1/p, 1/q) ∈ I1 and (i), with (1/p, 1/q) ∈ I1∗ . Case (ii) with (1/p, 1/q) ∈ I1 . Then μ2 (p, q) = −1/p. Let 1 ≤ r ≤ ∞, by Corollary 2.6.4, ‖fλ ‖M r,1 ≤ C|λ|−d/r ‖f ‖M r,1 ,

f ∈ M r,1 , 0 < |λ| ≤ 1.

(2.110)

On the other hand, since M 2,2 = L2 , we can write ‖fλ ‖M 2,2 ≤ C|λ|−d/2 ‖f ‖M 2,2 ,

f ∈ M 2,2 , 0 < |λ| ≤ 1.

(2.111)

For 1 ≤ r ≤ ∞, 0 < θ < 1 such that 1/p = (1 − θ)/r + θ/2 and 1/q = (1 − θ)/1 + θ/2, by complex interpolation (Proposition 2.3.17), we obtain 1−θ

‖fλ ‖M p,q ≤ C(|λ|−d/r )

θ

(|λ|−d/2 ) ‖f ‖M p,q ,

f ∈ M p,q , 0 < |λ| ≤ 1.

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Since (1 − θ)/r = 1/p + 1/q − 1 and θ/2 = 1 − 1/q, it follows that ‖fλ ‖M p,q ≤ C|λ|−d/p ‖f ‖M p,q ,

f ∈ M p,q , 0 < |λ| ≤ 1,

which is the first part of the claim with (1/p, 1/q) ∈ I1 . We next prove the necessary condition (second part of the claim) with (1/p, 1/q) ∈ I1 . Assume that there exists β ∈ ℝ such that ‖fλ ‖M p,q ≲ |λ|β ‖f ‖M p,q ,

0 < |λ| ≤ 1, ∀f ∈ M p,q (ℝd ).

Test the inequality on the Gaussian function φ. By Lemma 2.6.1 with A = λI, ‖φλ ‖M p,q = p−d/(2p) q−d/(2q) |λ|−d/p (1 + λ2 )

−d(1−1/q−1/p)/2

≤ |λ|β ‖φ‖M p,q ,

for every 0 < |λ| ≤ 1. Letting λ → 0, we get β ≤ −d/p, as desired.

122 | 2 Function spaces Case (i) with (1/p, 1/q) ∈ I1∗ . Note that, in this occurrence, μ1 (p, q) = −1/p. Consider 1 ≤ p ≤ ∞, q ≥ 2 such that (1/p, 1/q) ∈ I1∗ . Then (1/p󸀠 , 1/q󸀠 ) ∈ I1 . First, assume p ≠ 1. Since 1 < p, q ≤ ∞, by duality (cf. Eq. (2.34)) 󵄨 󵄨 󵄨 󵄨 ‖fλ ‖M p,q = sup󵄨󵄨󵄨⟨fλ , g⟩󵄨󵄨󵄨 = |λ|−d sup󵄨󵄨󵄨⟨f , g1/λ ⟩󵄨󵄨󵄨 = |λ|−d sup ‖f ‖M p,q ‖g1/λ ‖M p󸀠 ,q󸀠 ≤ |λ|−d ‖f ‖M p,q sup(C(|λ|−1 )

−d/p󸀠

= C|λ|−d/p ‖f ‖M p,q ,

‖g‖M p󸀠 ,q󸀠 )

for all f ∈ M p,q (ℝd ), |λ| ≥ 1, where the supremum is taken over all g ∈ 𝒮 (ℝd ) such that ‖g‖M p󸀠 ,q󸀠 = 1. In the case p = 1, by Corollary 2.6.4, ‖fλ ‖M 1,∞ ≤ C|λ|−d ‖f ‖M 1,∞ ,

f ∈ M 1,∞ , |λ| ≥ 1.

Hence we obtain the first part of case (i), with (1/p, 1/q) ∈ I1∗ . Assume 1 < p < ∞, 2 ≤ q < ∞ such that (1/p, 1/q) ∈ I1∗ , and there exist C > 0 and α > 0 such that ‖fλ ‖M p,q ≤ C|λ|α ‖f ‖M p,q ,

∀ |λ| ≥ 1, ∀f ∈ M p,q (ℝd ).

Then by duality and our assumption 󵄨 󵄨 󵄨 󵄨 ‖fλ ‖M p󸀠 ,q󸀠 = sup󵄨󵄨󵄨⟨fλ , g⟩󵄨󵄨󵄨 = |λ|−d sup󵄨󵄨󵄨⟨f , g1/λ ⟩󵄨󵄨󵄨 = |λ|−d sup ‖f ‖M p󸀠 ,q󸀠 ‖g1/λ ‖M p,q

α

≤ |λ|−d sup ‖f ‖M p󸀠 ,q󸀠 (C(|λ|−1 ) ‖g‖M p,q ) = C|λ|−d−α ‖f ‖M p󸀠 ,q󸀠 ,

for all f ∈ M p ,q (ℝd ), 0 < |λ| ≤ 1, where the supremum is taken over all g ∈ 𝒮 (ℝd ) such that ‖g‖M p,q = 1. Since (1/p󸀠 , 1/q󸀠 ) ∈ I1 , by case (ii) with (1/p󸀠 , 1/q󸀠 ) ∈ I1 , we get −d − α ≤ −d/p󸀠 , that is α ≥ −d/p. The duality argument extends to the case p < ∞, q = ∞. As for the case p = ∞ and q = ∞ we may observe that for f (t) = 1, t ∈ ℝd , we have fλ = f , ‖fλ ‖M ∞ = ‖f ‖M ∞ . The proof is then completed.

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󸀠

󸀠

Next goal is to prove Case (i) with (1/p, 1/q) ∈ I2∗ and Case (ii) with (1/p, 1/q) ∈ I2 . We need some preliminaries. Lemma 2.6.7. Let 1 ≤ p, q ≤ ∞ such that (1/p, 1/q) ∈ I2∗ and 1/p ≥ 1/q. Then there exists a constant C > 0 such that ‖fλ ‖M p,q ≤ C|λ|−d(2/p−1/q) (1 + λ2 ) for all f ∈ M p,q (ℝd ), and |λ| > 0.

d(1/p−1/2)

‖f ‖M p,q ,

2.6 Dilation properties for modulation spaces |

123

Proof. Consider 1 ≤ r ≤ ∞. By Corollary 2.6.4, we have d/2

‖fλ ‖M 1,r ≤ C|λ|d(1/r−2) (1 + λ2 )

‖f ‖M 1,r ,

for every f ∈ M 1,r (ℝd ) and |λ| > 0. Let 1 ≤ p, q ≤ ∞ such that (1/p, 1/q) ∈ I2∗ and 1/p ≥ 1/q. Take 1 ≤ r ≤ ∞ and 0 < θ < 1 such that 1/p = (1 − θ)/1 + θ/2 and 1/q = (1 − θ)/r + θ/2. Since M 2,2 = L2 , we can write f ∈ M 2,2 , |λ| ≥ 1.

‖fλ ‖M 2 2 ≤ C|λ|−d/2 ‖f ‖M 2,2 ,

(2.112)

Then by complex interpolation with (2.111) and (2.112), d/2 1−θ

‖fλ ‖M p,q ≤ C(|λ|d(1/r−2) (1 + λ2 )

)

|λ|−dθ/2 ‖f ‖M p,q ,

for all f ∈ M p,q , and |λ| > 0. Using (1 − θ)/r = 1/p + 1/q − 1, 1 − θ = 2/p − 1, θ/2 = −1/p + 1, we get the desired estimate. Lemma 2.6.8. There exists a constant C > 0 such that ‖fλ ‖M 2,∞ ≤ C|λ|−d ‖f ‖M 2,∞ , for all f ∈ M 2,∞ (ℝd ), and 0 < |λ| ≤ 1.

Proof. We use the characterization of the M 2,∞ -norm in (2.48), that is, ‖f ‖M 2,∞ (ℝd ) ≍ sup ‖◻k f ‖L2 . k∈ℤd

Hence 󵄩 󵄩 ‖fλ ‖M 2,∞ ≲ sup 󵄩󵄩󵄩σ0 (⋅ − k)fλ̂ 󵄩󵄩󵄩L2 d k∈ℤ

󵄩 󵄩 = |λ|−d/2 sup 󵄩󵄩󵄩σ0 (λ ⋅ −k)f ̂󵄩󵄩󵄩L2 k∈ℤd

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󵄩󵄩 󵄩󵄩 󵄩 󵄩 = |λ|−d/2 sup 󵄩󵄩󵄩σ0 (λ ⋅ −k)( ∑ σ0 (⋅ − l))f ̂󵄩󵄩󵄩 . 󵄩 󵄩󵄩L2 d 󵄩 k∈ℤ l∈ℤd Observe that 󵄨 󵄨2 ∑ 󵄨󵄨󵄨σ0 (λt − k)σ0 (t − l)f ̂(t)󵄨󵄨󵄨 ≤ Cd

l∈ℤd

󵄨󵄨 󵄨2 󵄨󵄨σ0 (λt − k)σ0 (t − l)f ̂(t)󵄨󵄨󵄨 ,

∑ |lj −kj /λ|≤2/λ, j=1,...,d

then we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩σ0 (λ ⋅ −k) ∑ σ0 (⋅ − l)f ̂󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩L2 d l∈ℤ ≤ Cd (

∑

1/2

󵄨 󵄨2 ∫ 󵄨󵄨󵄨σ0 (λt − k)σ0 (t − l)f ̂(t)󵄨󵄨󵄨 dt)

|lj −kj /λ|≤2/|λ|, d ℝ j=1,...,d

124 | 2 Function spaces 1/2

≤ Cd ‖σ0 ‖∞ (

󵄩󵄩 󵄩2 −1 󵄩󵄩Ml ℱ σ0 ∗ f 󵄩󵄩󵄩2 )

∑ |lj −kj /λ|≤2/|λ|, j=1,...,d

󵄩 󵄩 ≤ Cd ‖σ0 ‖∞ sup 󵄩󵄩󵄩Mm ℱ −1 σ0 ∗ f 󵄩󵄩󵄩L2 ( m∈ℤd

1/2

∑

1)

|lj −kj /λ|≤2/|λ|, j=1,...,d

≤ Cd󸀠 |λ|−d/2 sup ‖◻m f ‖L2 . m∈ℤd

Hence ‖fλ ‖M 2,∞ ≲ |λ|−d ‖f ‖M 2,∞ , as desired. Lemma 2.6.9. Let 1 ≤ p ≤ ∞, λ ∈ ℝ \ {0}. Then the following statements are true: (i) If p ≤ 2 then there exists a constant C > 0 such that ∀f ∈ M p,1 (ℝd ), |λ| ≥ 1.

‖fλ ‖M p,1 ≤ C‖f ‖M p,1 ,

(2.113)

(ii) If p ≥ 2 then there exists a constant C > 0 such that ‖fλ ‖M p,1 ≤ C|λ|−d(2/p−1) ‖f ‖M p,1 ,

∀f ∈ M p,1 (ℝd ), |λ| ≥ 1.

Proof. (i) Using Lemma 2.6.8, we can write 󵄨 󵄨 󵄨 󵄨 ‖fλ ‖M 2,1 = sup󵄨󵄨󵄨⟨fλ , g⟩󵄨󵄨󵄨 = |λ|−d sup󵄨󵄨󵄨⟨f , g1/λ ⟩󵄨󵄨󵄨

≤ |λ|−d sup ‖f ‖M 2,1 ‖g1/λ ‖M 2,∞ ≤ |λ|−d sup ‖f ‖M 2,1 (C|1/λ|−d ‖g‖M 2,∞ ) = C‖f ‖M 2,1

for all f ∈ 𝒮 (ℝd ) and |λ| ≥ 1, where the supremum is taken over all g ∈ M 2,∞ (ℝd ) such that ‖g‖M 2,∞ = 1. Since 𝒮 (ℝd ) is dense in M 2,1 (ℝd ), we get

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‖fλ ‖M 2,1 ≤ C‖f ‖M 2,1 ,

∀f ∈ M 2,1 (ℝd ), |λ| ≥ 1.

(2.114)

On the other hand, by Corollary 2.6.4, we see ‖fλ ‖M 1,1 ≤ C‖f ‖M 1,1 ,

∀f ∈ M 1,1 (ℝd ), |λ| ≥ 1.

Hence by complex interpolation, we get (2.113). (ii) By Corollary 2.6.4, we can write ‖fλ ‖M ∞,1 ≤ C|λ|d ‖f ‖M ∞,1 ,

∀f ∈ M ∞,1 (ℝd ), |λ| ≥ 1.

(2.115)

Interpolating the latter inequality with (2.114), we obtain 1−θ

‖fλ ‖M p,1 ≤ C(|λ|0 )

θ

(|λ|d ) ‖f ‖M p,1 ,

for all f ∈ M p,1 (ℝd ), |λ| ≥ 1, with 1/p = (1 − θ)/2 + θ/∞, 1 ≤ θ ≤ 1. Since θ = −2/p + 1, the claim is reached.

2.6 Dilation properties for modulation spaces |

125

Lemma 2.6.10. Let 1 ≤ p ≤ ∞, λ ∈ ℝ \ {0}. Then the following statements are true: (i) If p ≤ 2 then there exists a constant C > 0 such that ∀f ∈ M p,∞ (ℝd ), 0 < |λ| ≤ 1.

‖fλ ‖M p,∞ ≤ C|λ|−2d/p ‖f ‖M p,∞ ,

(2.116)

(ii) If p ≥ 2 then there exists a constant C > 0 such that ∀f ∈ M p,∞ (ℝd ), 0 < |λ| ≤ 1.

‖fλ ‖M p,∞ ≤ C|λ|−d ‖f ‖M p,∞ ,

Proof. (i) Let 1 < p ≤ 2. Using Lemma 2.6.9 (ii) and the duality argument, 󵄨 󵄨 󵄨 󵄨 ‖fλ ‖M p,∞ = sup󵄨󵄨󵄨⟨fλ , g⟩󵄨󵄨󵄨 = |λ|−d sup󵄨󵄨󵄨⟨f , g1/λ ⟩󵄨󵄨󵄨 ≤ |λ|−d sup ‖f ‖M p,∞ ‖g1/λ ‖M p󸀠 ,1

≤ |λ|−d sup ‖f ‖M p,∞ (C|1/λ|−d(2/p −1) ‖g‖M p󸀠 ,1 ) 󸀠

= C|λ|−2d/p ‖f ‖M p,∞

for all f ∈ M p,∞ (ℝd ) and 0 < |λ| ≤ 1, where the supremum is taken over all g ∈ 𝒮 (ℝd ) such that ‖g‖M p󸀠 ,1 = 1. For p = 1, by Corollary 2.6.4, ∀f ∈ M 1,∞ (ℝd ), 0 < |λ| ≤ 1.

‖fλ ‖M 1,∞ ≤ C|λ|−2d ‖f ‖M 1,∞ ,

This gives the claim. (ii) This case is obtained by using the same pattern as case (i). We leave the details to the interested reader. Lemma 2.6.11. Let 1 ≤ p, q ≤ ∞, (p, q) ≠ (1, ∞), (∞, 1), and ϵ > 0. Set f =

∑

k∈ℤd \{0}

|k|−d/q−ϵ Mk φ

in 𝒮 󸀠 (ℝd ),

(2.117)

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

with φ being the standard Gaussian function. Then f ∈ M p,q (ℝd ) and there exists a constant C > 0 such that ‖fλ ‖M p,q ≥ C|λ|d(1/q−1)+ϵ ,

0 < |λ| ≤ 1.

Proof. First, we show f ∈ M p,q (ℝd ). Using Leibniz’ formula (0.5), 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2πik⋅t φ(t)φ(t − x)e−2πiξ ⋅t dt 󵄨󵄨󵄨 󵄨󵄨 ∫ e 󵄨󵄨 󵄨󵄨 ℝd 󵄨󵄨 󵄨󵄨󵄨 −2d 󵄨 = 󵄨󵄨󵄨 ∫ φ(t)φ(t − x){⟨2π(ξ − k)⟩ (I − Δt )d e−2πi(ξ −k)⋅t } dt 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ℝd 󵄨 󵄨󵄨 −2d 󵄨󵄨 󵄨 ≲ ⟨2π(ξ − k)⟩ 󵄨󵄨󵄨 ∑ ∑ Cα,β ∫ 𝜕β φ(t)(𝜕α−β φ)(t − x)e−2πi(ξ −k)⋅t dt 󵄨󵄨󵄨 󵄨󵄨 |α|≤2d β≤α 󵄨󵄨 d ≲ ⟨2π(ξ − k)⟩

−2d

ℝ

󵄨 󵄨 ∑ ∑ 󵄨󵄨󵄨𝜕β φ󵄨󵄨󵄨 ∗ |𝜕α−β φ|(x).

|α|≤2d β≤α

(2.118)

126 | 2 Function spaces Hence ‖f ‖M p,q ≍ ‖Vφ f ‖Lp,q

q/p 1/q 󵄨󵄨 󵄨󵄨p 󵄨 󵄨 = ( ∫ ( ∫ 󵄨󵄨󵄨 ∑ |k|−d/q−ϵ ∫ Mk φ(t)φ(t − x)e−2πiξ ⋅t dt 󵄨󵄨󵄨 dx) dξ ) 󵄨󵄨 d 󵄨󵄨 d d k∈ℤ \{0} d ℝ

ℝ

ℝ

≤ C( ∫ ( ∑

k∈ℤd \{0}

ℝd

= C( ∑

|k|−d/q−ϵ ⟨2π(ξ − k)⟩

∫

l∈ℤd l+[−1/2,1/2]d

≤ C( ∑ ( ∑

l∈ℤd k∈ℤd \{0}

( ∑

k∈ℤd \{0}

−2d

q

1/q

) dξ )

|k|−d/q−ϵ ⟨2π(ξ − k)⟩

|k|−d/q−ϵ ⟨2π(l − k)⟩

−2d

−2d

q

1/q

) dξ )

q 1/q

) )

.

Since (|k|−d/q−ϵ )k∈ℤd \{0} ∈ ℓq (ℤd ), by Young’s inequality, we get f ∈ M p,q (ℝd ). We next show (2.118). The Gaussian function φ is in M p ,q (ℝd ), so by duality 󸀠

‖fλ ‖M p,q =

󸀠

󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨⟨fλ , g⟩󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨⟨fλ , φ⟩󵄨󵄨󵄨

‖g‖

󸀠 󸀠 =1 M p ,q

󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨 ∑ |k|−d/q−ϵ ∫ Mλk φ(λt)φ(t) dt 󵄨󵄨󵄨 󵄨󵄨 d 󵄨󵄨 k∈ℤ \{0} d ℝ

= C(1 + λ2 ) ≥C

∑

∑

k∈ℤd \{0}

0 d(1/q − 1). Then we can choose an ϵ > 0 such that d(1/q − 1) + ϵ < β and for this ϵ we consider the function f defined in (2.117). This function f is in M p,q (ℝd ) by Lemma 2.6.11 and there exists a constant C 󸀠 > 0 such that ‖fλ ‖M p,q ≥ C 󸀠 |λ|d(1/q−1)+ϵ , so that

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C 󸀠 |λ|d(1/q−1)+ϵ ≤ ‖fλ ‖M p,q ≤ C|λ|β ‖f ‖M p,q ,

0 < |λ| ≤ 1.

Though, since d(1/q − 1) + ϵ < β, this is a contradiction. Therefore β must satisfy β ≤ d(1/q − 1), that is the claim. Proof Theorem 2.6.6 (i), with (1/p, 1/q) ∈ I2∗ . We note that μ1 (p, q) = 1/q−1 if (1/p, 1/q) ∈ I2∗ . In every case, except for (p, q) ≠ (1, ∞), the result follows from the preceding proofs, namely by duality, Theorem 2.6.6 (ii), with (1/p󸀠 , 1/q󸀠 ) ∈ I2 , and the same argument as in the proof of Theorem 2.6.6 (i), with (1/p, 1/q) ∈ I1∗ . For the case (p, q) = (1, ∞), the result was already proved in Theorem 2.6.6 (i), with (1/p, 1/q) ∈ I1∗ . The last goal is to prove Theorem 2.6.6 (i), with (1/p, 1/q) ∈ I3∗ , and (ii), with (1/p, 1/q) ∈ I3 . We use the fact that there exists γ ∈ 𝒮 (ℝd ) such that supp γ ⊂ [−1/8, 1/8]d and |γ|̂ ≥ 1 on [−2, 2]d (see, for example, the proof of [150, Theorem 2.6]).

128 | 2 Function spaces Lemma 2.6.12. Let 1 ≤ p ≤ ∞, 1 ≤ q < ∞ and ϵ > 0. Suppose that γ, ψ ∈ 𝒮 (ℝd ) satisfy supp γ ⊂ [−1/8, 1/8]d , supp ψ ⊂ [−1/2, 1/2]d , |γ|̂ ≥ 1 on [−2, 2]d and ψ = 1 on [−1/4, 1/4]d . Set f =

∑

k∈ℤd \{0}

|k|−d/q−ϵ Mk Tk ψ

in 𝒮 󸀠 (ℝd ).

(2.120)

Then f ∈ M p,q (ℝd ) and there exists a constant C > 0 such that ‖fλ ‖M p,q ≍ ‖Vγ fλ ‖Lp,q ≥ C|λ|−d(2/p−1/q)+ϵ ,

0 < |λ| ≤ 1.

(2.121)

Proof. In the same way as in the proof of Lemma 2.6.11, we can prove that f is a function in M p,q (ℝd ). Let us consider the second part. Since ‖Vγ fλ ‖Lp,q = |λ|−d(1/p−1/q+1) ‖Vγ1/λ f ‖Lp,q , it is enough to show ‖Vγ1/λ f ‖Lp,q ≥ C|λ|−d/p+ϵ+d for every 0 < |λ| ≤ 1. We note that

supp γ((⋅ − x)/λ) ⊂ l + [−1/4, 1/4]d for all 0 < λ ≤ 1, l ∈ ℤd and x ∈ l + [−1/8, 1, 8]d . Since supp ψ(⋅ − k) ⊂ k + [−1/2, 1/2]d and ψ(t − k) = 1 if t ∈ k + [−1/4, 1/4]d , it follows that 1/p

󵄨 󵄨p ( ∫ 󵄨󵄨󵄨Vγ1/λ f (x, ξ )󵄨󵄨󵄨 dx) ℝd

1/p 󵄨󵄨 󵄨󵄨󵄨p 󵄨 = ( ∫ 󵄨󵄨󵄨 ∑ |k|−d/q−ϵ ∫ e−2πi(ξ −k)⋅t ψ(t − k)γ((t − x)/λ) dt 󵄨󵄨󵄨 dx) 󵄨󵄨 d 󵄨󵄨 d k∈ℤ \{0} d ℝ

≥( ∑

ℝ

∫

1/p 󵄨󵄨p 󵄨󵄨 󵄨󵄨 󵄨󵄨 −d/q−ϵ −2πi(ξ −k)⋅t − x)/λ) dt dx) |k| e ψ(t − k)γ((t ∑ ∫ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 d d k∈ℤ \{0} d

∫

1/p 󵄨󵄨p 󵄨󵄨 󵄨󵄨 −d/q−ϵ 󵄨 ∫ e−2πi(ξ −l)⋅t γ((t − x)/λ) dt 󵄨󵄨󵄨 dx) 󵄨󵄨|l| 󵄨󵄨 󵄨󵄨 d d

l∈ℤd \{0} l+[−1/8,1/8]

=( ∑

ℝ

l∈ℤd \{0} l+[−1/8,1/8]

ℝ

1/p

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󵄨p 󵄨 ̂ − l))󵄨󵄨󵄨 ) = 4−d/p ( ∑ 󵄨󵄨󵄨|l|−d/q−ϵ λd γ(−λ(ξ l∈ℤd \{0}

.

Using |γ|̂ ≥ 1 on [−2, 2]d , q/p

󵄨 󵄨p ̂ ‖Vγ1/λ f ‖Lp,q ≥ 4−d/p |λ|d ( ∫ ( ∑ 󵄨󵄨󵄨|l|−d/q−ϵ γ(−λ(ξ − l))󵄨󵄨󵄨 ) ℝd

l∈ℤd \{0}

q/p

p 󵄨 ̂ + λl)󵄨󵄨󵄨󵄨 ) = 4−d/p |λ|d−d/q ( ∫ ( ∑ 󵄨󵄨󵄨|l|−d/q−ϵ γ(ξ ℝd

l∈ℤd \{0}

1/q

dξ )

q/p

≥ 4−d/p |λ|d−d/q ( ∫ ( [−1,1]d

1/q

dξ )

∑

0 0, βj > 0 such that d

α

β

‖fA ‖M p,q ≤ C ∏(max{1, |λj |}) j (min{1, |λj |}) j ‖f ‖M p,q , j=1

∀f ∈ M p,q (ℝd ),

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with A = diag[λ1 , . . . , λd ], then αj ≥ μ1 (p, q) and βj ≤ μ2 (p, q). Proof. The necessary conditions are an immediate consequence of the one dimensional case, already contained in Theorem 2.6.6. Indeed, it can be seen by taking f as tensor product of functions of one variable and by leaving free to vary just one eigenvalue, the remaining eigenvalues being all equal to one. Let us come to the first part of the theorem. It suffices to prove it in the diagonal case A = D = diag[λ1 , . . . , λd ]. Indeed, since A is symmetric, there exists an orthogonal matrix T such that A = T −1 DT, and D is a diagonal matrix. On the other hand, by Proposition 2.6.3, we have ‖fA ‖M p,q ≲ ‖fT −1 D ‖M p,q = ‖(fT −1 )D ‖M p,q and ‖fT −1 ‖M p,q ≲ ‖f ‖M p,q ; hence the general case in (2.126) follows from the diagonal case A = D, with f replaced by fT −1 . From now onward, A = D = diag[λ1 , . . . , λd ]. If the theorem holds true for a pair (p, q), with (1/p, 1/q) ∈ [0, 1] × [0, 1], then it is also true for their dual pair (p󸀠 , q󸀠 ) (with f ∈ 𝒮 if p󸀠 = 1 or q󸀠 = 1, see (2.34)). Indeed, ‖fD ‖M p󸀠 ,q󸀠 =

󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨⟨fD , g⟩󵄨󵄨󵄨 = | det D|−1 sup 󵄨󵄨󵄨⟨f , gD−1 ⟩󵄨󵄨󵄨

‖g‖M p,q ≤1

≤ | det D| ‖f ‖M p󸀠 ,q󸀠 −1

‖g‖M p,q ≤1

sup ‖gD−1 ‖M p,q

‖g‖M p,q ≤1

132 | 2 Function spaces d

d

j=1

j=1

μ1 (p,q)

≲ ∏ |λj |−1 ∏(max{1, |λj |−1 }) d

μ1 (p󸀠 ,q󸀠 )

= ∏(max{1, |λj |}) j=1

μ2 (p,q)

(min{1, |λj |−1 }) μ2 (p󸀠 ,q󸀠 )

(min{1, |λj |})

‖f ‖M p󸀠 ,q󸀠

‖f ‖M p󸀠 ,q󸀠 ,

for the index functions μ1 and μ2 fulfill μ1 (p󸀠 , q󸀠 ) = −1 − μ2 (p, q),

μ2 (p󸀠 , q󸀠 ) = −1 − μ1 (p, q).

(2.127)

Hence it suffices to prove the estimates (2.126) for the case p ≥ q. Notice that the es󸀠 timate in M 1,q , q󸀠 > 1, are proved for Schwartz functions only, but they extend to all 󸀠 󸀠 functions in M 1,q , q󸀠 < ∞, for 𝒮 (ℝd ) is dense in M 1,q . The uncovered case (1, ∞) will be verified directly at the end of the proof. From Figure 2.1, it is clear that the estimate (2.126) follows by complex interpolation (Proposition 2.3.17) from the diagonal case p = q, and the three cases (p, q) = (∞, 1), (p, q) = (1, ∞), and (p, q) = (2, 1). We shall then limit to consider such values of (p, q) Case p = q. If d = 1 the claim is true by Theorem 2.6.6 in dimension d = 1. We then use the induction method. Namely, we assume that (2.126) is fulfilled in dimension d − 1 and prove that still holds in dimension d. For x, ξ ∈ ℝd , we write x = (x󸀠 , xd ), ξ = (ξ 󸀠 , ξd ), with x 󸀠 , ξ 󸀠 ∈ ℝd−1 , xd , ξd ∈ ℝ, 2 󸀠 2 2 D󸀠 = diag[λ1 , . . . , λd−1 ], and choose the Gaussian φ(x) = e−π|x| = e−π|x | e−π|xd | = φ󸀠 (x󸀠 )φd (xd ) as window function. Observe that Vφ fD admits the two representations Vφ fD (x 󸀠 , xd , ξ 󸀠 , ξd ) = ∫ f (λ1 t1 , . . . , λd td )Mξ 󸀠 Tx󸀠 φ󸀠 (t 󸀠 ) Mξd Txd φd (td ) dt 󸀠 dtd ℝd

= Vφ󸀠 ((Fxd ,ξd ,λd )D󸀠 )

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= Vφd ((Gx󸀠 ,ξ 󸀠 ,D󸀠 )λd )

where Fxd ,ξd ,λd (t 󸀠 ) = Vφd (f (t 󸀠 , λd ⋅))(xd , ξd ),

Gx󸀠 ,ξ 󸀠 ,D󸀠 (td ) = Vφ󸀠 (f (D󸀠 ⋅, td ))(x 󸀠 , ξ 󸀠 ).

By the inductive hypothesis, we have ‖fD ‖M p,p (ℝd ) = ‖Vφ fD ‖Lp (ℝ2d )

1

p 󵄨 󵄨p = (∫ ( ∫ 󵄨󵄨󵄨Vφ󸀠 ((Fxd ,ξd ,λd )D󸀠 )(x󸀠 , ξ 󸀠 )󵄨󵄨󵄨 dx 󸀠 dξ 󸀠 ) dxd dξd )

ℝ2 ℝ2(d−1)

d−1

≲ ∏(max{1, |λj |}) j=1

μ1 (p,p)

(min{1, |λj |})

μ2 (p,p)

2.6 Dilation properties for modulation spaces |

133

1/p

󵄨p 󵄨 × ( ∫ 󵄨󵄨󵄨Vφ󸀠 (Fxd ,ξd ,λd )(x 󸀠 , ξ 󸀠 )󵄨󵄨󵄨 dx dξ ) d−1

ℝ2d

= ∏(max{1, |λj |})

μ1 (p,p)

j=1

(min{1, |λj |})

μ2 (p,p) 1/p

󵄨p 󵄨 × ( ∫ (∫ 󵄨󵄨󵄨Vφd ((Gx󸀠 ,ξ 󸀠 ,I )λd )(xd , ξd )󵄨󵄨󵄨 dxd dξd ) dx󸀠 dξ 󸀠 ) d

ℝ2(d−1) ℝ2

≲ ∏(max{1, |λj |})

μ1 (p,p)

j=1

(min{1, |λj |})

μ2 (p,p)

‖f ‖M p,p (ℝd ) ,

where in the last raw we used Theorem 2.6.6 for d = 1. Case (p, q) = (2, 1). First, we prove the case (p, q) = (2, ∞) and then obtain the claim by duality as above, since 𝒮 is dense in M 2,1 . Namely, we want to show that d

−1

‖fD ‖M 2,∞ ≲ ∏(max{1, |λj |})−1/2 (min{1, |λj |}) ‖f ‖M 2,∞ , j=1

∀f ∈ M 2,∞ .

We use the characterization of the M 2,∞ -norm in (2.48), that is, ‖f ‖M 2,∞ (ℝd ) ≍ sup ‖◻k f ‖L2 . k∈ℤd

(2.128)

Hence 󵄩 󵄩 ‖fD ‖M 2,∞ ≲ | det D|−1/2 sup 󵄩󵄩󵄩σ0 (D ⋅ −k)f ̂󵄩󵄩󵄩L2 k∈ℤd

󵄩󵄩 󵄩󵄩 󵄩 󵄩 = | det D|−1/2 sup 󵄩󵄩󵄩σ0 (D ⋅ −k)( ∑ σ0 (⋅ − l))f ̂󵄩󵄩󵄩 . 󵄩 󵄩󵄩L2 d 󵄩 d k∈ℤ l∈ℤ

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Observe that 󵄨󵄨 󵄨󵄨2 󵄨󵄨 󵄨 󵄨 󵄨2 󵄨󵄨σ0 (Dt − k)( ∑ σ0 (t − l))f ̂(t)󵄨󵄨󵄨 ≤ Cd ∑ 󵄨󵄨󵄨σ0 (Dt − k)σ0 (t − l)f ̂(t)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 d d l∈ℤ l∈ℤ 󵄨 󵄨2 = Cd ∑ 󵄨󵄨󵄨σ0 (Dt − k)σ0 (t − l)f ̂(t)󵄨󵄨󵄨 l∈Λk

where 󵄨󵄨 kj 󵄨󵄨󵄨 1 󵄨 Λk = {l ∈ ℤd : 󵄨󵄨󵄨lj − 󵄨󵄨󵄨 ≤ 1 + } 󵄨󵄨 λj 󵄨󵄨 |λj | and d

#Λk ≤ C ∏ min{1, |λj |} , j=1

−1

∀k ∈ ℤd

(2.129)

134 | 2 Function spaces (C being a constant depending on d only). Since |λj | = max{1, |λj |} min{1, |λj |}, the expression on the right-hand side of (2.129) is dominated by d

C 󸀠 ∏(max{1, |λj |})

−1/2

j=1

(min{1, |λj |})

−1

sup ‖◻k f ‖L2 .

k∈ℤd

Thereby the norm equivalence (2.128) gives the desired estimate. Case (p, q) = (∞, 1). We have to prove that d

‖fD ‖M ∞,1 ≲ ∏ max{1, |λj |}‖f ‖M ∞,1 , j=1

∀f ∈ M ∞,1 .

This estimate immediately follows from (2.105), written for A = D = diag[λ1 , . . . , λd ]: d

d

1/2

‖fD ‖M ∞,1 ≲ ∏(1 + λj2 ) ‖f ‖M ∞,1 ≲ ∏ max{1, |λj |}‖f ‖M ∞,1 . j=1

j=1

Case (p, q) = (1, ∞). We are left to prove that d

−1

∀f ∈ M 1,∞ .

−2

‖fD ‖M 1,∞ ≲ ∏(max{1, |λj |}) (min{1, |λj |}) ‖f ‖M 1,∞ , j=1

This is again the estimate (2.105), written for A = D = diag[λ1 , . . . , λd ], d

d

j=1

j=1

‖fD ‖M 1,∞ ≲ ∏ |λj |−2 ∏ max{1, |λj |}‖f ‖M 1,∞ .

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Corollary 2.6.15. Let 1 ≤ p, q ≤ ∞. There exists a constant C > 0 such that, for every symmetric matrix A ∈ GL(d, ℝ), with eigenvalues λ1 , . . . , λd , we have d

μ1 (p󸀠 ,q󸀠 )

‖fA ‖W(ℱ Lp ,Lq ) ≤ C ∏(max{1, |λj |}) j=1

μ2 (p󸀠 ,q󸀠 )

(min{1, |λj |})

‖f ‖W(ℱ Lp ,Lq ) ,

(2.130)

for every f ∈ W(ℱ Lp , Lq )(ℝd ). Conversely, if there exist αj > 0, βj > 0 such that d

α

β

‖fA ‖W(ℱ Lp ,Lq ) ≤ C ∏(max{1, |λj |}) j (min{1, |λj |}) j ‖f ‖W(ℱ Lp ,Lq ) , j=1

for every f ∈ W(ℱ Lp , Lq )(ℝd ), with A = diag[λ1 , . . . , λd ], then αj ≥ μ1 (p󸀠 , q󸀠 ) and βj ≤ μ2 (p󸀠 , q󸀠 ).

2.7 Sharpness of convolution, inclusion, and multiplication relations | 135

Proof. It is a mere consequence of Theorem 2.6.14 and the index relation (2.127). Namely, ‖fA ‖W(ℱ Lp ,Lq ) = ‖f̂A ‖M p,q = | det A|−1 ‖f −1 ‖M p,q d

d

j=1

j=1

μ1 (p,q)

≤ C ∏ |λj |−1 ∏(max{1, |λj |−1 }) d

μ1 (p󸀠 ,q󸀠 )

= C ∏(max{1, |λj |}) j=1

μ2 (p,q)

(min{1, |λj |−1 })

(min{1, |λj |})

μ2 (p󸀠 ,q󸀠 )

‖f ̂‖M p,q

‖f ‖W(ℱ Lp ,Lq ) .

The necessary conditions use the same argument.

2.7 Sharpness of convolution, inclusion, and multiplication relations for modulation spaces In this section we study the optimality of the convolution, inclusion, and pointwise multiplication relations for modulation spaces. We need a few preliminary results concerning modulation space norms of Gaussians. Lemma 2.7.1. For a, b ∈ ℝ, a > 0, set 2

πt −d/2 − a+ib

𝒢(a+ib) (t) = (a + ib)

e

t ∈ ℝd .

,

Then for every 1 ≤ p, q ≤ ∞, d

‖𝒢(a+ib) ‖W(ℱ Lp ,Lq ) =

d

((a + 1)2 + b2 ) 2

( p1 − 21 )

d

d

p 2p (aq) 2q (a(a + 1) + b2 ) 2

( p1 − q1 )

(2.131)

.

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2

Proof. We use the Gaussian g(y) = e−πy as window function, so that the norm W(ℱ Lp , Lq ) reads 󵄩 󵄩 ‖𝒢(a+ib) ‖W(ℱ Lp ,Lq ) ≍ 󵄩󵄩󵄩‖𝒢(a+ib) Tx g‖ℱ Lp 󵄩󵄩󵄩Lqx . Now, ? 𝒢(a+ib) Tx g(ξ ) = (𝒢? (a+ib) ∗ M−x g)(ξ )

(2.132)

2

2

= ∫ e−π(a+ib)|ξ −y| e−2πix⋅y e−πy dy 2

= e−π(a+ib)ξ ∫ e−π(a+1+ib)y

2

−2πi(x+i(a+ib)ξ )⋅y 2

= (a + 1 + ib)−d/2 e−π(a+ib)ξ e−π

dy

(x+i(a+ib)ξ )⋅(x+i(a+ib)ξ ) a+1+ib

,

136 | 2 Function spaces where we used (0.27). Hence, after a simple computation, π 2 2 2 󵄨 󵄨󵄨 ? 2 2 −d/4 − (a+1)2 +b2 [(a(a+1)+b )ξ +2bx⋅ξ +(a+1)x ] e . 󵄨󵄨𝒢(a+ib) Tx g(ξ )󵄨󵄨󵄨 = ((a + 1) + b )

It follows that ‖𝒢(a+ib) Tx g‖ℱ Lp = ((a + 1)2 + b2 ) × (∫ e

−

pπ (a+1)2 +b2

= ((a + 1)2 + b2 ) × (∫ e

−

e

−d/4 −

[(a2 +b2 +a)ξ 2 +2bx⋅ξ ]

e

−d/4 −

dξ )

1 p

πax 2 a(a+1)+b2

󵄨󵄨 󵄨󵄨2 󵄨󵄨√ 󵄨 b x󵄨󵄨 󵄨󵄨 a(a+1)+b2 ξ + 󵄨󵄨 √a(a+1)+b2 󵄨󵄨󵄨

pπ (a+1)2 +b2

= ((a + 1)2 + b2 )

π(a+1)x2 (a+1)2 +b2

e

−d/4 −

πax 2 a(a+1)+b2

dξ )

1 p

(a(a + 1) + b2 )

d − 2p

d

(a + 1)2 + b2 2p ×( ) p d

=

((a + 1)2 + b2 ) 2 d

( p1 − 21 ) d

p 2q (a(a + 1) + b2 ) 2p

e

−

πax 2 a(a+1)+b2

.

By taking the Lq -norm of this expression, one obtains (2.131). Now, we turn our attention to the sharpness of the convolution properties for modulation spaces. Proposition 2.7.2. Let 1 ≤ p, q, p1 , p2 , q1 , q2 ≤ ∞. Then ‖f ∗ g‖M p,q ≲ ‖f ‖M p1 ,q1 ‖g‖M p2 ,q2

(2.133)

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if and only if the indices’ relations (2.79) and (2.80) hold true. Proof. The sufficiency is shown in Proposition 2.4.19. We prove the necessity. We con2 sider the family of Gaussians φ√λ (x) = e−πλx , for λ > 0. Obviously, φ√λ ∈ 𝒮 (ℝd ) ⊂ −d/2 ̂ M p,q (ℝd ), for every 1 ≤ p, q ≤ ∞. Since ‖f ‖M p,q ≍ ‖f ̂‖W(ℱ Lp ,Lq ) and φ φ√1/λ , √λ = λ Lemma 2.7.1 yields ‖φ√λ ‖M p,q ≍ λ−d/2 ‖φ√1/λ ‖W(ℱ Lp ,Lq ) ≍ ‖𝒢(λ) ‖W(ℱ Lp ,Lq ) ≍

(λ + 1)d(1/p−1/2) . λd/(2q) (λ2 + λ)(1/p−1/q)d/2

(2.134)

d

A straightforward calculation shows that (φ√λ ∗ φ√λ )(x) = (2λ)− 2 φ√λ/2 (x). Hence using (2.134), we obtain ‖φ√λ ∗ φ√λ ‖M p,q ≍ λ−(1+1/p)d/2 ,

for λ → 0+ .

(2.135)

2.7 Sharpness of convolution, inclusion, and multiplication relations | 137

Using (2.134) again, we also obtain ‖φ√λ ‖M pi ,qi ≍ λ

− 2pd

i

,

i = 1, 2,

for λ → 0+ .

(2.136)

Substituting in (2.82), we obtain (2.79). The relation (2.80) can be obtained similarly. In fact, the estimate (2.134) gives, for λ → +∞, ‖φ√λ ∗ φ√λ ‖M p,q ≍ λ

−d(1− 2q1 )

,

‖φ√λ ‖M pi ,qi ≍ λ

− d2 (1− q1 ) i

,

i = 1, 2,

and, using (2.82) again, the relation (2.80) follows. An alternative proof of the necessary conditions (2.79) and (2.80) is provided by Lemma 2.3.26. Precisely, to prove (2.80), consider two compactly supported smooth functions f , g and their scaling fλ (x) = f (λx), gλ (x) = g(λx), with λ ≥ 1. Since λ ≥ 1, fλ and gλ (and therefore fλ ∗ gλ ) are all supported in a compact subset K, independent of λ. By Lemma 2.3.26 (i), the bilinear estimate (2.82) for fλ and gλ becomes ‖fλ ∗ gλ ‖ℱ Lq ≲ ‖fλ ‖ℱ Lq1 ‖gλ ‖ℱ Lq2 . Using fλ ∗ gλ = λ−d (f ∗ g)λ , the dilation property for ℱ Lq spaces −d 󵄩󵄩 󵄩 󵄩󵄩h(λ⋅)󵄩󵄩󵄩ℱ Lq = λ q󸀠 ‖h‖ℱ Lq ,

and letting λ → +∞, we obtain (2.80). In order to prove (2.79), one argues similarly. Here the functions f , g have Fourier transforms f ̂, ĝ compactly supported and the scale λ satisfies 0 < λ ≤ 1. By Lemma 2.3.26 (ii), the estimate (2.81) becomes

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‖fλ ∗ gλ ‖Lp ≲ ‖fλ ‖Lp1 ‖gλ ‖Lp2 . Using fλ ∗ gλ = λ−d (f ∗ g)λ , the dilation property ‖h(λ⋅)‖Lp = λ−d/p ‖h‖Lp , and letting λ → 0+ , we prove (2.79). The family of Gaussians φ√λ also provides a proof for the sharpness of the inclusion relation for modulation spaces. Proposition 2.7.3. Let 1 ≤ p1 , p2 , q1 , q2 ≤ ∞. Then ‖f ‖M p2 ,q2 ≲ ‖f ‖M p1 ,q1

(2.137)

if and only if the following indices’ relation holds: p1 ≤ p2

and

q1 ≤ q2 .

(2.138)

138 | 2 Function spaces Proof. The sufficiency is proved in Theorem 2.4.17. We show the necessity of (2.138). 2 Let φ√λ (x) = e−πλx , λ > 0. From the proof of Proposition 2.7.2, ‖φ√λ ‖M pi ,qi ≍ λ

− 2pd

i

for λ → 0+ ,

,

‖φ√λ ‖M pi ,qi ≍ λ

and

− d2 (1− q1 ) i

for λ → +∞. (2.139)

,

Hence for (2.137) to be satisfied, it must be that λ

− 2pd

2

≲λ

− 2pd

1

,

for λ → 0+ ,

and λ

− d2 (1− q1 ) 2

≲λ

− d2 (1− q1 ) 1

,

for λ → +∞,

which give the indices’ relations in (2.138). In what follows we study the pointwise multiplication operator in modulation spaces (which is equivalent to studying the convolution operator for the Wiener amalgam spaces W(ℱ Lp , Lq )). Proposition 2.7.4. Let 1 ≤ p, q, p1 , p2 , q1 , q2 ≤ ∞. Then (2.140)

‖fg‖M p,q ≲ ‖f ‖M p1 ,q1 ‖g‖M p2 ,q2 if and only if the indices’ relations (2.84) and (2.85) hold true.

Proof. The sufficiency can be found in Proposition 2.4.23. For the necessity of the conditions (2.84) and (2.85), we test the estimate (2.140) on the Gaussians f (x) = g(x) = 2 φ√λ (x) = e−λπx . We observe that φ√λ φ√λ = φ√2λ . Hence by applying (2.139) and substituting in (2.140), relation (2.85) follows by letting λ → 0+ , whereas (2.84) follows by letting λ → +∞.

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2.7.1 Time-frequency tools For further purpose we need to compute the modulation norm of the generalized Gaussian f defined in (1.126), namely for a, b, c > 0, 2

2

f (x, ξ ) = fa,b,c (x, ξ ) = e−πax e−πbξ e2πicx⋅ξ ,

(x, ξ ) ∈ ℝ2d .

Proposition 2.7.5. Consider the generalized Gaussian f defined in (1.126) and the win2 2 dow function Φ(x, ξ ) = e−π(x +ξ ) . Then for every 1 ≤ p, q ≤ ∞, we have ‖f ‖M p,q ≍ ‖VΦ f ‖Lp,q d

≍

[(a + 1)(b + 1) + c2 ] p

+ dq − d2

d

[(c2 + ab + a)(c2 + ab + b)] 2q d

d − 2p d

[b2 (a + 1) + b(c2 + a + 1)] 2q [a2 (b + 1) + a(c2 + b + 1)] 2q

.

(2.141)

The cases p = ∞ or q = ∞ can be obtained by using the rule 1/∞ = 0 in formula (2.141).

2.7 Sharpness of convolution, inclusion, and multiplication relations | 139

Proof. By Proposition 1.3.33, we can write 1

󵄨 󵄨󵄨 󵄨󵄨VΦ f (z, ζ )󵄨󵄨󵄨 =

d

[(a + 1)(b + 1) + c2 ] 2 ×e

−π

[a(b+1)+c2 ]z12 +[(a+1)b+c2 ]z22 +(b+1)ζ12 +(a+1)ζ22 −2c(z1 ⋅ζ2 +z2 ⋅ζ1 ) (a+1)(b+1)+c2

.

It remains to compute the mixed Lp,q -norm of the previous function. We treat the cases 1 ≤ p, q < ∞. The cases if either p = ∞ or q = ∞ are obtained with obvious modifications. For simplicity, we set c2 + a(b + 1) , (a + 1)(b + 1) + c2 (a + 1) δ= , (a + 1)(b + 1) + c2

c2 + (a + 1)b , (a + 1)(b + 1) + c2 c σ= . (a + 1)(b + 1) + c2

α=

β=

(b + 1) , (a + 1)(b + 1) + c2

γ=

(2.142) (2.143)

Hence ‖VΦ f ‖Lp,q [(a + 1)(b + 1) + c2 ] 2

q p

=(∫ I e

d − 2p

−πq(γζ12 +δζ22 )

1 q

dζ1 dζ2 ) =: A,

(2.144)

ℝ2d

2

where I := ∫ℝ2d e−πpαz1 −πpβz2 e2πpσ(z1 ⋅ζ2 +z2 ⋅ζ1 ) dz1 dz2 . Now straightforward computations and change of variables yield 2

2

I = ∫ ( ∫ e−πp(αz1 −2σζ2 ⋅z1 ) dz1 )e−πβz2 +2πpσz2 ⋅ζ1 dz2 ℝd ℝd σ2

2

= eπp α ζ2 e

2

πp σβ ζ12

∫e ℝd

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d

d

d

d

σ −πp(√αz1 − √α ζ2 )2

σ2

2

= p− 2 α− 2 p− 2 β− 2 eπp α ζ2 e

dz1 ∫ e

−πp(√βz2 −

σ √β

ζ1 )2

dz2

ℝd

2

πp σβ ζ12

.

Substituting the value of the integral I in (2.144), we obtain d d − dp − 2p − 2p

A=p

α

β

(∫ e

2

πq σβ ζ12 −πqγζ12

ℝd d d − dp − 2p − 2p

=p

α

β

ℝd

(∫ e

2

−πq(γ− σβ )ζ12

ℝd d d − dp − 2p − 2p − dq

=p

α

β

q

dζ1 ∫ e

2

πq σα ζ22 −πqδζ22

dζ1 ∫ e

d − 2q

σ2 (γ − ) β

1 q

2

−πq(δ− σα )ζ22

ℝd

d − 2q

σ2 (γ − ) α

.

)

dζ2 )

1 q

140 | 2 Function spaces Finally, the goal is attained by substituting in A the values of the parameters α, β, γ, δ, σ in (2.142) and (2.143) and observing that − d2

‖f ‖M p,q ≍ ‖VΦ f ‖Lp,q = A[(a + 1)(b + 1) + c2 ]

.

This concludes the proof.

2.7.2 Further extensions of modulation spaces (quasi-Banach setting) p,q We end up this chapter recalling the extension of modulation spaces Mm (ℝd ) to the indices 0 < p, q ≤ ∞ (quasi-Banach spaces) performed first by Y. V. Galperin and S. Samarah in [153].

Definition 2.7.6. Fix a nonzero window g ∈ 𝒮 (ℝd ), a weight m ∈ ℳv (ℝ2d ), and 0 < p,q p, q ≤ ∞. The modulation space Mm (ℝd ) consists of all tempered distributions f ∈ 󸀠 d 𝒮 (ℝ ) such that the (quasi-)norm q

‖f ‖Mmp,q = ‖Vg f ‖Lp,q m

1

p q 󵄨 󵄨p = ( ∫ ( ∫ 󵄨󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 m(x, ξ )p dx) dξ )

ℝd

(2.145)

ℝd

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(obvious changes with p = ∞ or q = ∞) is finite. They were proved to be quasi-Banach spaces, whose definition does not depend on the window g ∈ 𝒮 (ℝd ). In this framework, it appears that the largest natural class of p,q windows universally admissible for all spaces Mm (ℝd ), 0 < p, q ≤ ∞ (with m having at most polynomial growth) is the Schwartz class 𝒮 (ℝd ). They were further investigated in [211, 212, 248, 305] (see also references therein). Actually, several papers involve modulation spaces with indices 1 ≤ p, q ≤ ∞, whereas very few works deal with the quasi-Banach case 0 < p, q ≤ 1. In fact, many properties related to the latter case are still unexplored. We recall here the convolution relations proved in the recent contribution [6], extending the Banach setting in Proposition 2.4.19. Proposition 2.7.7. Let ν(ξ ) > 0 be an arbitrary weight function on ℝd , 0 < p, q, r, t, u, γ ≤ ∞, with 1 1 1 + = u t γ and 1 1 1 + =1+ , p q r

for 1 ≤ r ≤ ∞,

2.7 Sharpness of convolution, inclusion, and multiplication relations | 141

whereas p = q = r,

for 0 < r < 1.

For m ∈ ℳv (ℝ2d ), m1 (x) = m(x, 0) and m2 (ξ ) = m(0, ξ ) are the restrictions to ℝd × {0} and {0} × ℝd , and likewise for v. Then p,u r,γ Mm (ℝd ) ∗ Mvq,t⊗v ν−1 (ℝd ) �→ Mm (ℝd ) 1 ⊗ν 1

2

with norm inequality ‖f ∗ h‖Mmr,γ ≲ ‖f ‖Mmp,u⊗ν ‖h‖M q,t

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1

v1 ⊗v2 ν−1

.

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3 Gabor frames and linear operators In the applications of time–frequency analysis, in particular signal theory, one refers to the discrete version of the short-time Fourier transform, and this will have also an important role in our book. Namely, one can consider the sampling of Vg f (x, ξ ) = ∫ f (t)g(t − x)e−2πit⋅ξ dt ℝd

at the points x = αm, ξ = βn of the lattice Λ = αℤd × βℤd , for some fixed α > 0, β > 0. Alternatively, one refers to the Gabor coefficients cm,n = ⟨f , Tαm Mβn g⟩ = e2πiαβn⋅m Vg f (αm, βn).

(3.1)

The basic issue is to discuss whether and how we may reconstruct f from the coefficients c = (cm,n ), possibly after normalization. Reminiscent of the classical orthonormal (o. n.) bases, we should try to recapture f by an expression of the type ∑ cm,n Tαm Mβn g.

m,n∈ℤd

However, the Gabor system

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{Tαm Mβn g, (m, n) ∈ ℤ2d } is not an o. n. basis, apart very particular cases. In fact, the so fruitful links with spectral theory, namely identification with eigenfunctions of a self-adjoint operator, are lost. The analysis of Gabor systems represents a challenging and fascinating field of the mathematical analysis. In our book we shall not discuss advanced topics in this context. Rather, aiming to the applications to partial differential equations, we shall limit ourselves to some basic notions and results necessary for proofs and numerical experiments in the sequel. Starting from scratch, we first address nonexpert readers to Appendix A, Section A.1, concerning bases theory and series expansions in Hilbert and Banach spaces. A relevant point there, suggested by the previous expansion in terms of a Gabor system, is the unconditional convergence of a series. With respect to the classical theorems of Dirichlet and Riemann–Dini stating the equivalence in ℝ with the absolute convergence, in Hilbert and Banach spaces absolute convergence implies unconditional convergence, but not vice versa, and a more precise analysis is needed, including in particular the concept of Riesz bases. The first part of the present Chapter 3 has a preliminary character as well, being devoted to the theory of Frames in Hilbert spaces. Frames were proposed by Duffin and Schaeffer in 1952 [113] as a more flexible method of approximation, with respect to o. n. bases, by allowing nonorthogonal and redundant complete systems. https://doi.org/10.1515/9783110532456-004

144 | 3 Gabor frames and linear operators Even in problems where o. n. bases are available, frames are preferred because of their efficiency in computations. They represent also the correct setting for Gabor systems; let us describe in short the theory in this context. A Gabor system is called a Gabor frame if it satisfies the frame condition A‖f ‖22 ≤

󵄨2 󵄨 ∑ 󵄨󵄨󵄨⟨f , Tαm Mβn g⟩󵄨󵄨󵄨 ≤ B‖f ‖22 ,

m,n∈ℤd

∀f ∈ L2 (ℝd ),

where 0 < A ≤ B and the Gabor coefficients cm,n are defined in (3.1). Under such a condition, the Gabor system is complete in L2 (ℝd ) and the series Sf =

∑ cm,n Tαm Mβn g

m,n∈ℤd

is convergent. This is not a reproducing formula, that is, the operator S is not the identity in L2 (ℝd ). Nevertheless, S : L2 (ℝd ) → L2 (ℝd ) is invertible in L2 (ℝd ), and the application of its inverse S−1 provides the expected reconstruction of f . We may then reelaborate the formula by defining the dual window γ = S−1 g, and this provides after some computations f = =

∑ cm,n Tαm Mβn γ

m,n∈ℤd

∑ ⟨f , Tαm Mβn γ⟩Tαm Mβn g.

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m,n∈ℤd

So we have reduced the situation to the classical o. n. setting, apart the appearance of the dual window in the expression of the system or, alternatively, in the computation of the Gabor coefficients. Of course, it remains to check whether the initial frame condition is satisfied. This depends on the window g and the lattice parameters α, β. Intuitively, we may imagine that for sufficiently small α, β the frame condition is satisfied, but precise results are very difficult, and will be not given in this book, addressing the interested reader to the textbooks [50, 160, 183], see also the paper by I. Daubechies [101]. We shall limit to some examples, in particular for the Gaussian 2 window g(t) = e−πt , t ∈ ℝd , the Gabor system is a frame if and only if αβ < 1. The role of Gabor frames in the sequel, apart numerical experiments, will be to provide an alternative definition of modulation spaces. Namely, we can use the sep,q p,q quence spaces ℓm in Definition 2.2.1 to state that f ∈ Mm (ℝd ) if and only if the Gabor p,q coefficients c = (cm,n ) belong to ℓm , with equivalence of norms ‖f ‖Mmp,q ≍ ‖c‖ℓmp,q . The last part of this chapter is devoted to integral expressions of operators, timefrequency counterpart of the classical Schwartz’ kernel theorem for tempered distributions. There the material is preliminary to the next chapters, and the proofs use in

3.1 Frames in Hilbert spaces | 145

an essential way a discretization of the problem by Gabor frames. Precisely, we characterize the continuity of a linear operator on the modulation space M p (ℝd ), 1 ≤ p ≤ ∞, by the membership of its kernel K in (mixed norm) modulation spaces. We recapture the result of H. Feichtinger announced in [126] and proved in [134], stating that linear continuous operators A : M 1 (ℝd ) → M ∞ (ℝd ) are characterized by the condition K ∈ M ∞ (ℝ2d ). An obvious advantage, with respect to the Schwartz’s kernel theorem in the topological vector spaces (𝒮 (ℝd ), 𝒮 󸀠 (ℝd )), is that M p (ℝd ), 1 ≤ p ≤ ∞, are Banach spaces. This reveals a superiority, in some respects, of the modulation space formalism, as emphasized in the Feichtinger’s manifesto 2015 [128] for a post-modern harmonic analysis. The contents of the chapter is organized in the following way. Section 3.1 is devoted to the main properties of frames in Hilbert spaces. Section 3.2 presents first the Gabor frames, and then treats the equivalent definition of modulation spaces. Finally, Section 3.3 concerns the kernel theorems for modulation spaces.

3.1 Frames in Hilbert spaces Frames were originally introduced by Duffin and Schaeffer in 1952 [113], in the framework of nonharmonic Fourier series. In what follows we exhibit a brief survey on the topic, mainly borrowed from [183], to which we refer the interested reader. Definition 3.1.1 (Frame). A sequence {xn } in a Hilbert space H is a frame if there exist constants 0 < A ≤ B such that ∀x ∈ H,

∞

󵄨 󵄨2 A‖x‖2H ≤ ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 ≤ B‖x‖2H . n=1

(3.2)

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The constants A and B are called lower and upper frame bound, respectively. The largest possible lower frame bound is called the optimal lower frame bound and the smallest possible upper frame bound is called the optimal upper frame bound. Thus if {xn } is a frame, ‖(⟨x, xn ⟩)‖ℓ2 is an equivalent norm for H. In particular, if A = B = 1, then actually we have 󵄩 󵄩 ‖x‖H = 󵄩󵄩󵄩(⟨x, xn ⟩)󵄩󵄩󵄩ℓ2 . Although Definition 3.1.1 says nothing explicitly about basis-like properties of {xn }, we shall see that the equivalence (3.2) implies unconditional convergence, basis-like representations of vectors in H. We first exhibit special types of frames. Definition 3.1.2. Let {xn } be a frame for a Hilbert space H. (i) If A = B (optimal frame bounds are equal) then {xn } is called a tight frame. We say in this case that {xn } is an A-tight frame. (ii) If A = B = 1 then {xn } is called a Parseval frame.

146 | 3 Gabor frames and linear operators (iii) A frame {xn } is called an exact frame if it ceases to be a frame whenever any single element is deleted from the sequence. The easiest example of frame is provided by an orthonormal basis (o. n. basis, for short) {en }. In this case A = B = 1, so the frame is a Parseval frame and, moreover, it is exact. Remark 3.1.3. (i) In a frame a repetition of elements and the presence of the zero vector are allowed. For instance, for an o. n. basis {en }, the sequence {0, e1 , e2 , . . . , en , . . . } is a Parseval frame but it is not exact. Another example is {e1 , e1 , e2 , . . . , en , . . . } where we have a frame which is again not exact. (ii) If {xn } is a frame, then ∑n |⟨x, xn ⟩|2 is an absolutely convergent series, so by Lemma A.1.13, it is unconditionally convergent. So for any permutation σ of ℕ, we have that {xσ(n) } is still a frame, and therefore it does not matter what countable set we use to index a frame. (iii) We will see that if a frame is exact then it is a basis for H, cf. Definition A.1.1. An inexact frame is redundant, or overcomplete, in the sense that a proper subset of it is still complete (in fact, still a frame). (iv) Observe that, by (3.2), a frame is a complete sequence. Indeed, if we assume ⟨x, xn ⟩ = 0, for every n, then by (3.2), 󵄨 󵄨2 A‖x‖2H ≤ ∑󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 = 0 �⇒ x = 0. n

(v) Since any space that contains a countable complete set is separable, we conclude that a Hilbert space which has a frame is separable. Conversely, every separable Hilbert space contains an o. n. basis and hence has a frame.

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Example 3.1.4. For H = ℝ2 , consider the vectors x1 = (1, 0),

x2 = (0, 1),

x3 = (

1 1 , ), √2 √2

x4 = (−

1 1 , ). √2 √2

The sets {x1 , x2 } and {x3 , x4 } are orthonormal bases for ℝ2 . So 4

󵄨 󵄨2 2‖x‖2H = ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 , n=1

that is, {x1 , . . . , x4 } is an inexact tight frame with frame bounds A = B = 2. If we take yn = xn /√2, n = 1, . . . , 4, then the set {yn } is a Parseval frame, but observe that its elements are not orthogonal and are not a basis. Example 3.1.5. Let {en } be an orthonormal basis for a Hilbert space H. Then

3.1 Frames in Hilbert spaces | 147

(i) {e1 , e1 , e2 , e2 , . . . , en , en , . . . } is a tight inexact frame with frame bounds A = B = 2. Observe that this frame is not orthogonal and is not a basis. Similarly, for another orthonormal basis {fn }, the set {en } ∪ {fn } is a tight inexact frame. (ii) {e1 , e2 /2, e3 /3, . . . } is a complete orthogonal sequence and it is a basis for H, but it is not a frame, because it does not possess a lower bound. (iii) {xn } = {e1 , e2 /√2, e2 /√2, e3 /√3, e3 /√3, e3 /√3, . . . }, for which we have ‖x‖2H = ∑n |⟨x, xn ⟩|2 , is an inexact Parseval frame. (iv) {√2e1 , e2 , e3 , . . . } is a nontight exact frame with optimal frame bounds A = 1 and B = 2. We now study more interesting examples of frames, coming from trigonometric systems. Example 3.1.6 (Trigonometric systems). We consider the trigonometric system {e2πibnt }n∈ℤ , in L2 (�), where b > 0 is fixed. Recall that L2 (�) = {f : ℝ → ℂ, f measurable : f (t + 1) = f (t) a. e., and ‖f ‖2 < ∞}/ ∼, 1

where ‖f ‖22 = ∫0 |f (t)|2 dt < ∞ and f ∼ g ⇐⇒ f = g a. e.

Since functions in L2 (�) are 1-periodic, we are implicitly considering e2πibnt to be defined on the interval [0, 1) and then extended by 1-periodicity on ℝ. (i) When b = 1, it is well known that {e2πint }n∈ℤ is an o. n. basis for L2 (�). (ii) Suppose b > 1. Then the function e2πibnt is b1 -periodic on ℝ. Since b1 < 1 the interval Ib := [0, 1 − b1 ] is well defined and we have the inclusion Ib ⊂ [0, 1]. Moreover, for every t ∈ Ib , t + b1 ∈ [0, 1] and 1

e2πibn(t+ b ) = e2πibnt , so every function fn (t) := e2πibnt satisfies

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fn (t +

1 ) = fn (t), b

t ∈ Ib .

(3.3)

This implies that span{fn } satisfies (3.3). Moreover, span {fn } (in the L2 (�)-norm) satisfies (3.3) a. e. In fact, f = limn fn in L2 (�) implies pointwise convergence a. e. for a subsequence {fnk }. However, not every function f ∈ L2 (�) fulfills (3.3). For example, the function f (t) = t ∈ [0, 1] is one of these. Hence {e2πibnt }n∈ℤ is incomplete and thus is not a frame. (iii) Consider 0 < b < 1. The function e2πibnt is b1 -periodic but this time b1 > 1, so the previous argument does not apply to this case. For special choices of b, we can see that the system {e2πibnt }n∈ℤ is a frame for L2 (�). For example, for b = 1/2, setting n = 2k for n even and n = 2k + 1 for n odd, we can write 1

{eπint }n∈ℤ = {e2πikt }k∈ℤ ∪ {e2πi(k+ 2 )t }k∈ℤ = {e2πikt }k∈ℤ ∪ {eπit e2πikt }k∈ℤ

148 | 3 Gabor frames and linear operators so the set is the union of two o. n. bases and it is an inexact tight frame with A = B = 2, that is, 󵄨2 󵄨 2‖f ‖22 = ∑ 󵄨󵄨󵄨⟨f , eπint ⟩󵄨󵄨󵄨 . n∈ℤ

Similarly, if b = 1/M with M ∈ ℕ+ , then M−1

{e2πibnt }n∈ℤ = ⋃ {e2πi

Mk+j t M

j=0

}k∈ℤ

and M‖f ‖22 = ∑n∈ℤ |⟨f , e2πibnt ⟩|2 . This means that the system is an inexact M-tight frame (it has redundancy M). We now show that {e2πibnt } is a tight frame in L2 (�) with A = B = 1/b, for all b with 0 < b < 1. Observe that the sequence {b1/2 e2πibnt } is an o. n. basis for L2 ([0, b−1 ]). The space L2 (�) = L2 ([0, 1]) can be viewed as a closed subspace of L2 ([0, b−1 ]) by extending a function f ∈ L2 ([0, 1]) by zero on (1, b−1 ]. Hence every function in L2 (�) can be view as the orthogonal projection of a function on [0, b−1 ], where the projection P : L2 ([0, b−1 ]) → L2 (�) is defined by Pf = fχ[0,1] . Hence for every f ∈ L2 ([0, 1]), 󵄨 󵄨2 ‖f ‖2L2 (�) = ‖fχ[0,1] ‖2L2 ([0,b−1 ]) = b ∑󵄨󵄨󵄨⟨fχ[0,1) , e2πibnt ⟩L2 ([0,b−1 ]) 󵄨󵄨󵄨 n

󵄨 󵄨2 = b ∑󵄨󵄨󵄨⟨f , e2πibnt ⟩L2 (�) 󵄨󵄨󵄨 . n

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So the sequence {e2πibnt } is a tight frame with A = B = 1/b. This frame is not a basis in the sense of Definition A.1.1, and so (as we will see in the sequel) it is not exact. For instance, we can represent the constant function f (t) = 1 in two different ways (with two different choices of coefficients). Indeed, Pg = f for g(t) = 1 on [0, b−1 ], but also g(t) = χ[0,1] (as function on [0, b−1 ]) has the same projection. The former example g = 1 is an element of the sequence {e2πibnt } (n = 0) and so 1 = ∑ δ0,n e2πibnt . n∈ℤ

(3.4)

For the latter one, we have χ[0,1] ∈ L2 ([0, b−1 ]) and χ[0,1] ≠ 1 in [0, b−1 ]. So there exists (cn ) ∈ ℓ2 , (cn ) ≠ (δ0,n ) such that χ[0,1] = b1/2 ∑n cn e2πibnt , hence the representation of 1 ∈ L2 (�) in the series expansion (3.4) is not unique. We now define Bessel sequences in Hilbert spaces. Definition 3.1.7 (Bessel sequence). A sequence {xn } in a Hilbert space H is a Bessel sequence if ∀x ∈ H,

∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 < ∞.

n=1

(3.5)

3.1 Frames in Hilbert spaces | 149

Definition 3.1.8. Given a sequence {xn } in a Hilbert space H, the coefficient operator, also called analysis operator C, is defined to be Cx = (⟨x, xn ⟩),

x ∈ H.

(3.6)

Proposition 3.1.9. If {xn } is a Bessel sequence, the coefficient operator C maps H into ℓ2 and is a linear and continuous operator; that is, there exists a constant B > 0 such that ∀x ∈ H,

∞

󵄨 󵄨2 ‖Cx‖2ℓ2 = ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 ≤ B‖x‖2H . n=1

(3.7)

The constant B > 0 in (3.7) is called Bessel constant. Inequality (3.7) is called Bessel’s inequality. Proof. We are going to present two different demonstrations. First proof. If {xn } is a Bessel sequence, the coefficient operator C maps H into ℓ2 by (3.5). Observe that C is a linear operator. We prove that C is bounded. We use the closed graph theorem (cf. Theorem A.0.2). Assume {yk } ⊂ H is such that yk → y in H and Cyk → c in ℓ2 . We shall show that Cy = c. We write c = (cn ); since Cy = (⟨y, xn ⟩), to reach our goal we need to show that cn = ⟨y, xn ⟩, for every n ∈ ℕ+ . For every m ∈ ℕ+ , we need to prove that lim ⟨yk , xm ⟩ = cm ,

(3.8)

k→∞

or equivalently, 󵄨 󵄨2 lim 󵄨󵄨󵄨⟨yk , xm ⟩ − cm 󵄨󵄨󵄨 = 0.

k→∞

We have ∞

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󵄨󵄨 󵄨2 󵄨 󵄨2 󵄨󵄨⟨yk , xm ⟩ − cm 󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨⟨yk , xm ⟩ − cm 󵄨󵄨󵄨 m=1

= ‖Cyk − c‖2ℓ2 → 0

for k → +∞,

and (3.8) is proved. Finally, by assumption that yk → y in H and by the continuity of the inner product, lim ⟨yk , xm ⟩ = ⟨y, xm ⟩ ∀m

k→+∞

so that, by (3.8), we obtain the claim. Second proof. Consider the sequence of operators {CN }N∈ℕ+ , with CN : x �→ (⟨x, x1 ⟩, ⟨x, x2 ⟩, . . . , ⟨x, xN ⟩, 0, 0, . . . ).

150 | 3 Gabor frames and linear operators The operator CN is trivially linear and maps H into ℓ2 , as ∑Nn=1 |⟨x, xn ⟩|2 < ∞. By using the Cauchy–Schwarz inequality, we infer that CN is bounded with N

N

n=1

n=1

󵄨2 󵄨 ‖CN x‖2ℓ2 = ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 ≤ ‖x‖2H ∑ ‖xn ‖2H = BN ‖x‖2H . Moreover, the sequence (‖CN x‖ℓ2 ) is an increasing sequence of nonnegative numbers indexed by N, so that, for every x ∈ H, ∞

󵄨 󵄨2 sup ‖CN x‖ℓ2 = lim ‖CN x‖ℓ2 = √ ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 = ‖Cx‖ℓ2 < ∞.

N∈ℕ+

N→∞

n=1

We can apply the uniform boundedness principle (cf. Theorem A.0.3) and obtain sup ‖CN ‖B(H,ℓ2 ) =: √B < ∞,

N∈ℕ+

for a suitable B > 0. Finally, the inequality ‖CN x‖ℓ2 ≤ ‖CN ‖B(H,ℓ2 ) ‖x‖H ≤ √B‖x‖H ,

∀x ∈ H

for N → ∞ yields ‖Cx‖ℓ2 ≤ √B‖x‖H ,

∀x ∈ H,

as desired.

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In particular, if {xn } is an o. n. basis of H, then by Plancherel theorem (see Theorem A.1.11 (iv)) we obtain that C : H → ℓ2 is a (topological) isometric isomorphism. Remark 3.1.10. We observe that a more well-known definition of Bessel sequence (which follows in our case from Proposition 3.1.9) is the following: A sequence {xn } in a Hilbert space H is a Bessel sequence if there exists B > 0 such that ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 ≤ B‖x‖2H ,

n=1

∀x ∈ H.

(3.9)

Remark 3.1.11. Let {xn } be a Bessel sequence in a Hilbert space H with Bessel bound B > 0. Then it is straightforward to prove the following inequalities: (i) We have ‖xn ‖2H ≤ B for every n. (ii) If ‖xm ‖2H = B for some m, then xm is orthogonal to {xn }n=m ̸ . Proposition 3.1.12. Let {xn } be a Bessel sequence in a Hilbert space H with Bessel constant B. Then

3.1 Frames in Hilbert spaces | 151

(i) If c = (cn ) ∈ ℓ2 , then the series ∑n cn xn converges unconditionally in H and D(c) = ∑ cn xn n

defines a linear and bounded operator from ℓ2 into H. (ii) D∗ = C and ‖D‖ = ‖C‖ ≤ √B. Consequently, 󵄩󵄩2 󵄩󵄩 󵄩 󵄩󵄩 2 󵄩󵄩∑ cn xn 󵄩󵄩󵄩 ≤ B ∑ |cn | . 󵄩󵄩H 󵄩󵄩 n n

∀(cn ) ∈ ℓ2

(iii) If {xn } is a frame, then C is injective and D is surjective. Proof. (i) We use the equivalent formulation of unconditional convergence of Theorem A.1.21 (iii). Let F ⊂ ℕ+ be a finite set. Then by the Cauchy–Schwarz inequality and using the assumption, we have 󵄩󵄩 󵄩󵄩 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄩󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 󵄩󵄩 ∑ cn xn 󵄩󵄩󵄩 = sup 󵄨󵄨󵄨⟨ ∑ cn xn , y⟩󵄨󵄨󵄨 = sup 󵄨󵄨󵄨 ∑ cn ⟨xn , y⟩󵄨󵄨󵄨 󵄩󵄩 󵄩󵄩H y∈H, ‖y‖H =1󵄨󵄨 󵄨󵄨 y∈H, ‖y‖H =1󵄨󵄨 󵄨󵄨 n∈F n∈F n∈F 1

1

2 󵄨 󵄨2 2 ≤ sup ( ∑ |cn |2 ) ( ∑ 󵄨󵄨󵄨⟨xn , y⟩󵄨󵄨󵄨 )

‖y‖H =1 n∈F

1 2

n∈F

1

󵄨 󵄨2 2 ≤ sup ( ∑ |cn | ) ( ∑ 󵄨󵄨󵄨⟨xn , y⟩󵄨󵄨󵄨 ) 2

n∈ℕ+

‖y‖H =1 n∈F

1 2

1

≤ sup ( ∑ |cn |2 ) (B‖y‖2H ) 2 ‖y‖H =1 n∈F

1 2

= √B( ∑ |cn |2 ) . n∈F

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2

2

Now, (cn ) ∈ ℓ , so ∑n |cn | is unconditionally convergent. For any choice of ϵ > 0, there exists N0 ∈ ℕ+ such that for every finite F ⊂ ℕ+ with min(F) > N0 , ∑n∈F |cn |2 < ϵ2 /B. Thus 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩 2 2 2 󵄩󵄩 ∑ cn xn 󵄩󵄩󵄩 ≤ B ∑ |cn | < Bϵ /B = ϵ , 󵄩󵄩 󵄩󵄩H n∈F n∈F proving our claim. Setting F = {1, . . . , N} and letting N → ∞, we obtain 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩 2 󵄩󵄩∑ cn xn 󵄩󵄩󵄩 ≤ B ∑ |cn | . 󵄩󵄩 n 󵄩󵄩H n So the operator Dc = ∑n cn xn maps ℓ2 into H and ‖D‖ ≤ √B. The operator D is linear. In fact, for every α, β ∈ ℂ and every c = (cn ), d = (dn ) ∈ ℓ2 , D(α(cn ) + β(dn )) = ∑(αcn + βdn )xn = α ∑ cn xn + β ∑ dn xn , n

since the series is unconditionally convergent.

n

n

152 | 3 Gabor frames and linear operators (ii) Observe that by (3.7) the operator C : H → ℓ2 is linear and bounded with ‖C‖ ≤ √B. It remains to prove that C ∗ = D. The operator C ∗ : ℓ2 → H is linear and bounded; it remains to check that C ∗ (c) = ∑n cn xn for every c = (cn ) ∈ ℓ2 . For every x ∈ H, c = (cn ) ∈ ℓ2 , ⟨x, C ∗ (c)⟩H = ⟨Cx, c⟩ℓ2 = ⟨(⟨x, xn ⟩)n , (cn )n ⟩ℓ2 = ∑⟨x, xn ⟩c̄n = ∑⟨x, cn xn ⟩ = ⟨x, ∑ cn xn ⟩ n

n

n

hence C ∗ (c) = ∑n cn xn , that is, C ∗ = D. (iii) If {xn } is also frame, then the inequality A‖x‖2H ≤ ∑n |⟨x, xn ⟩|2 holds for every x ∈ H. The claim follows by applying Theorem A.0.4 to the coefficient operator C. Definition 3.1.13. Consider a Bessel sequence {xn } in a Hilbert space H. (i) The adjoint operator C ∗ = D : ℓ2 → H is called a synthesis, or reconstruction, operator. (ii) The frame operator is defined as S = DC : H → H. (iii) The Gram operator or Gram matrix is G = CD : ℓ2 → ℓ2 . Now consider the Gram operator G : ℓ2 → ℓ2 . It can be represented as multiplication by an infinite matrix. Indeed, it easy to see that the Gram operator can be represented as G = [⟨xn , xm ⟩]m,n∈ℕ .

(3.10)

That is, if we think of c = (cn ) ∈ ℓ2 as an infinite column vector, Gc is the product of the infinite matrix (3.10) with c. So the m-entry of Gc is (Gc)m = ∑ cn ⟨xn , xm ⟩.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

n

Remark 3.1.14. Since the analysis C and synthesis D operators associated to a Bessel sequence {xn } are bounded, so are the frame S and Gram G operators. Moreover, S = DC = C ∗ C = DD∗ and G = CD = CC ∗ = D∗ D, so they are self-adjoint. We shall show that they are both positive. Recall that a self-adjoint operator P is positive if ⟨Px, x⟩ ≥ 0, for every x ∈ H. First, by definition, Sx = ∑⟨x, xn ⟩xn n

∀x ∈ H.

So 󵄨 󵄨2 ⟨Sx, x⟩ = ∑󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 ≥ 0, n

∀x ∈ H.

(3.11)

3.1 Frames in Hilbert spaces | 153

Now, consider the Gram operator G. For every sequence c ∈ ℓ2 , define x = ∑n cn xn , then ⟨Gc, c⟩ = ∑ ∑ cn ⟨xn , xm ⟩c̄m = ∑ ∑⟨cn xn , cm xm ⟩ m n

m n

= ⟨∑ cn xn , ∑ cm xm ⟩ = ‖x‖2H ≥ 0. n

m

3.1.1 Frame expansions and frame operator Frames are Bessel sequences, so all the properties of Bessel sequences transfer to frames. For U, V ∈ B(H), U, V being positive operators, we use the operator notation U ≤ B as follows: U ≤ V ⇐⇒ ⟨Ux, x⟩ ≤ ⟨Vx, x⟩ ∀x ∈ H. Theorem 3.1.15 (Reproducing formulas for a frame). Let {xn } be a frame for the Hilbert space H with frame bounds A, B. Then (i) The frame operator S is a topological isomorphism of H onto itself, self-adjoint and positive, with AI ≤ S ≤ BI, where I is the identity operator. (ii) S−1 is a topological isomorphism, self-adjoint and positive, and B−1 I ≤ S−1 ≤ A−1 I. (iii) {S−1 xn } is a frame for H, with frame bounds 0 < B−1 ≤ A−1 . (iv) For each x ∈ H, we have the reproducing formulae x = ∑⟨x, S−1 xn ⟩xn , Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

n

x = ∑⟨x, xn ⟩S−1 xn , n

(3.12)

and these series converge unconditionally in the norm of H. (v) If the frame is A-tight, then S = AI, S−1 = A−1 I, and ∀x ∈ H,

x=

1 ∑⟨x, xn ⟩xn . A n

Proof. (i) Since {xn } is a Bessel sequence, S is a continuous (self-adjoint) positive operator on H, and, by (3.11), ⟨Sx, x⟩ = ∑n |⟨x, xn ⟩|2 . Since ⟨AIx, x⟩ = A‖x‖2H , the frame definition can be rewritten as ⟨AIx, x⟩ ≤ ⟨Sx, x⟩ ≤ ⟨BIx, x⟩ ∀x ∈ H, which means AI ≤ S ≤ BI.

154 | 3 Gabor frames and linear operators Applying the Cauchy–Schwarz inequality, A‖x‖2H ≤ ⟨Sx, x⟩ ≤ ‖Sx‖H ‖x‖H , so that A‖x‖H ≤ ‖Sx‖H for every x ∈ H. We then apply Theorem A.0.4 which gives that ker S = {0} and S∗ is surjective. But S∗ = S, so S is a topological isomorphism, by the Banach isomorphism theorem. (ii) Since S is a self-adjoint positive topological isomorphism, the same holds for S−1 (left as an exercise). Since AI ≤ S, we have ⟨AIy, y⟩ ≤ ⟨Sy, y⟩ for every y ∈ H. Choosing y = S−1 x, we can write 󵄩 󵄩2 0 ≤ A󵄩󵄩󵄩S−1 x󵄩󵄩󵄩H ≤ ⟨S(S−1 x), S−1 x⟩ = ⟨x, S−1 x⟩ 󵄩 󵄩 ≤ ‖x‖H 󵄩󵄩󵄩S−1 x 󵄩󵄩󵄩H .

Consequently, ‖S−1 x‖H ≤ A−1 ‖x‖H and hence 󵄩 󵄩 ⟨S−1 x, x⟩ ≤ 󵄩󵄩󵄩S−1 x󵄩󵄩󵄩H ‖x‖H ≤ A−1 ‖x‖2H = ⟨A−1 Ix, x⟩. We now prove S−1 ≥ B−1 I. We use the generalized Cauchy–Schwarz inequality (see (A.2) in Appendix A) applied to the semiinner product (x, y) = ⟨S−1 x, y⟩, which gives 󵄨󵄨 −1 󵄨2 −1 −1 󵄨󵄨⟨S u, v⟩󵄨󵄨󵄨 ≤ ⟨S u, u⟩⟨S v, v⟩ u, v ∈ H. Choosing u = Sx, v = x,

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‖x‖4H ≤ ⟨x, Sx⟩⟨S−1 x, x⟩ ≤ B‖x‖2H ⟨S−1 x, x⟩, hence ⟨S−1 x, x⟩ ≥ B−1 ‖x‖2H , that is, S−1 ≥ B−1 I. (iii) It is an easy exercise to show that if S is self-adjoint then S−1 is self-adjoint. Hence ∑⟨x, S−1 xn ⟩S−1 xn = S−1 (∑⟨x, S−1 xn ⟩xn ) n

n

= S−1 (∑⟨S−1 x, xn ⟩xn ) n

= S S(S−1 x) = S−1 x. −1

(3.13)

As a consequence, 󵄨 󵄨2 ∑󵄨󵄨󵄨⟨x, S−1 xn ⟩󵄨󵄨󵄨 = ∑⟨x, S−1 xn ⟩⟨S−1 xn , x⟩ = ⟨∑⟨x, S−1 xn ⟩S−1 xn , x⟩ = ⟨S−1 x, x⟩. n

n

n

3.1 Frames in Hilbert spaces | 155

Using B−1 I ≤ S−1 ≤ A−1 I, we obtain 󵄨2 󵄨 B−1 ‖x‖2H ≤ ∑󵄨󵄨󵄨⟨x, S−1 xn ⟩󵄨󵄨󵄨 ≤ A−1 ‖x‖2H . n

This proves that {S−1 xn } is a frame with bounds B−1 , A−1 . (iv) Since {xn } and {S−1 xn } are, in particular, Bessel sequences, for every c = (cn ) ∈ ℓ2 , the series ∑n cn xn and ∑n cn S−1 xn converge unconditionally. Since the sequences (⟨x, xn ⟩) and (⟨x, S−1 xn ⟩) belong to ℓ2 and S, S−1 are continuous operators on H, we can write x = S(S−1 x) = ∑⟨S−1 x, xn ⟩xn = ∑⟨x, S−1 xn ⟩xn n

n

and x = S−1 (Sx) = ∑⟨Sx, S−1 xn ⟩S−1 xn = ∑⟨x, xn ⟩S−1 xn , n

n

with unconditional convergence in H. (v) This follows from the preceding statements. Let X be a dense subset of H. If A ∈ B(H) satisfies ⟨Ax, x⟩ ≥ 0 for every x ∈ X, then ⟨Ax, x⟩ ≥ 0 for every x ∈ H. Applying this fact to the inequalities ⟨AIx, x⟩ ≤ ⟨Sx, x⟩ ≤ ⟨BIx, x⟩,

x ∈ X,

it is enough to check the frame inequalities (3.2) for x in X. Previous considerations yield the definition of a canonical dual frame. Definition 3.1.16 (Canonical dual frame). Let {xn } be a frame for a Hilbert space H with frame operator S. Then the frame {S−1 xn } is called a canonical dual frame for {xn }. Observe that the frame operator of the canonical dual frame {S−1 xn } is the operator S , see (3.13). The canonical dual of {S−1 xn } is the original frame {xn }. Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

−1

Definition 3.1.17 (Alternative duals). Let {xn } be a frame for a Hilbert space H. A sequence {yn } ⊂ H such that x = ∑⟨x, yn ⟩xn , n

∀x ∈ H

with unconditional convergence is called an alternative dual for {xn }. If {yn } is a frame, then it is called an alternative dual frame. For example, if {en } is an o. n. basis for H, then the frame {e1 , e1 , e2 , e2 , . . . } has infinitely many duals. The canonical dual frame is {e1 /2, e1 /2, e2 /2, e2 /2, . . . } but also {e1 , 0, e2 , 0, . . . } and {0, e1 , 0, e2 . . . } are dual frames. The canonical dual frame is certainly the dual used more often. However, in applications, it may be useful to deal

156 | 3 Gabor frames and linear operators with a noncanonical dual frame. Indeed, sparse representations are the main goals in frame expansions. Just to give an idea of the problem, consider the previous example. If we choose alternative duals, we observe that half of the coefficients in the frame expansion are zero, whereas this is not the case for the canonical dual. The need of well-concentrated frame expansions is crucial in many applications. Let us denote by {x̃n } the canonical dual frame of {xn }, that is, x̃n = S−1 xn , where S is the frame operator for {xn }. We now show that among the choices of scalars (cn ) ∈ ℓ2 such that ∑n cn xn = x, the coefficients ⟨x, x̃n ⟩, n ∈ ℕ+ , provided by the canonical dual frame, are those with minimal ℓ2 -norm. Theorem 3.1.18. Let {xn } be a frame for a Hilbert space H and fix x ∈ H. If c = (cn ) is a sequence of scalars such that x = ∑n cn xn , then 󵄨 󵄨2 󵄨 󵄨2 ∑ |cn |2 = ∑󵄨󵄨󵄨⟨x, x̃n ⟩󵄨󵄨󵄨 + ∑󵄨󵄨󵄨cn − ⟨x, x̃n ⟩󵄨󵄨󵄨 . n

n

n

(3.14)

In particular, the sequence (⟨x, x̃n ⟩) has the minimal ℓ2 -norm among all such sequences. Proof. By Theorem 3.1.15, we have x = ∑⟨x, x̃n ⟩xn , with (⟨x, x̃n ⟩) ∈ ℓ2 . So if (cn ) ∉ ℓ2 , then ∑n |cn |2 = +∞ and equality (3.14) is trivially satisfied. Assume now (cn ) ∈ ℓ2 and write an = ⟨x, x̃n ⟩ for short. We have ⟨x, S−1 x⟩ = ⟨∑ an xn , S−1 x⟩ n

= ∑ an ⟨x̃n , x⟩ n

= ∑ an ā n n

= ⟨(an ), (an )⟩ℓ2

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and ⟨x, S−1 x⟩ = ⟨∑ cn xn , S−1 x⟩ n

= ∑ cn ⟨x̃n , x⟩ n

= ∑ cn ā n n

= ⟨(cn ), (an )⟩ℓ2 . So the sequence (an − cn ) is orthogonal to (an ) in ℓ2 and, by the Pythagorean theorem, we can write 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩(cn )󵄩󵄩󵄩ℓ2 = 󵄩󵄩󵄩(an ) − (an − cn )󵄩󵄩󵄩ℓ2 = 󵄩󵄩󵄩(an )󵄩󵄩󵄩ℓ2 + 󵄩󵄩󵄩(an − cn )󵄩󵄩󵄩ℓ2 , and the claim is proved.

3.1 Frames in Hilbert spaces | 157

Theorem 3.1.19. Let {xn } be a frame for a Hilbert space H. (i) For every m ∈ ℕ, 2 2 󵄨2 1 − |⟨xm , x̃m ⟩| − |1 − ⟨xm , x̃m ⟩| 󵄨 . ∑ 󵄨󵄨󵄨⟨xm , x̃n ⟩󵄨󵄨󵄨 = 2 n=m ̸

(3.15)

(ii) If ⟨xm , x̃m ⟩ = 1, then ⟨xm , x̃n ⟩ = 0 for every n ≠ m. (iii) The removal of an element xm from a frame leaves either a frame or an incomplete set. Specifically, ⟨xm , x̃m ⟩ ≠ 1 �⇒ {xn }n=m ̸ is a frame,

⟨xm , x̃m ⟩ = 1 �⇒ {xn }n=m ̸ is incomplete. Proof. (i) Set an = ⟨xm , x̃n ⟩ for every n, then xm = ∑n an xn . Observe that xm can also be represented by xm = ∑n δm,n xn , so that using Theorem 3.1.18, formula (3.14) here becomes 1 = ∑ |δm,n |2 = ∑ |an |2 + ∑ |δm,n − an |2 n

n

2

n

= |am | + ∑ |an |2 + |1 − am |2 + ∑ |an |2 . n=m ̸

n=m ̸

This yields ∑ |an |2 =

n=m ̸

1 − |am |2 − |1 − am |2 . 2

2 (ii) Suppose ⟨xm , x̃m ⟩ = 1, then by equation (3.15), ∑n=m ̸ |⟨xm , x̃n ⟩| = 0, which implies ⟨xm , x̃n ⟩ = 0, for every n ≠ m. (iii) Assume ⟨xm , x̃m ⟩ = 1, then by part (ii), ⟨xm , x̃n ⟩ = 0, for every n ≠ m. Hence

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0 = ⟨xm , x̃n ⟩ = ⟨xm , S−1 xn ⟩ = ⟨S−1 xm , xn ⟩ = ⟨x̃m , xn ⟩. This means that x̃m is orthogonal to the set {xn }n=m ̸ . However, x̃m ≠ 0 since ⟨xm , x̃m ⟩ = 1. So {xn }⊥ = ̸ {0} and this gives the incompleteness of the set {xn }n=m ̸ . Suppose now n=m ̸ ⟨xm , x̃m ⟩ ≠ 1, that is, in our notation am ≠ 1. Then xm = ∑n an xn can be rewritten as xm =

1 ∑ an xn . 1 − am n=m ̸

For every x ∈ H, by the Cauchy–Schwarz inequality and the previous representation, 󵄨 󵄨󵄨 󵄨󵄨2 󵄨󵄨󵄨2 1 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨2 󵄨󵄨 ̄ a ⟨x, x ⟩ ∑ an xn ⟩󵄨󵄨󵄨 = ∑ 󵄨 󵄨󵄨⟨x, xm ⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨x, n n 󵄨󵄨󵄨 2 󵄨󵄨 󵄨󵄨 󵄨 1 − am n=m |1 − a | 󵄨 󵄨 󵄨 m ̸ n=m ̸ ≤

1 󵄨 󵄨2 󵄨 󵄨2 ( ∑ |an |2 )( ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 ) ≤ C ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 |1 − am |2 n=m ̸ n=m ̸ n=m ̸

158 | 3 Gabor frames and linear operators where C=

1 ∑ |an |2 > 0. |1 − am |2 n=m ̸

Hence 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨2 󵄨 󵄨 ∑󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨x, xm ⟩󵄨󵄨󵄨 + ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 ≤ (1 + C) ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 . n

n=m ̸

n=m ̸

If the constants A, B > 0 are the frame bounds for {xn }, then A 1 󵄨 󵄨2 󵄨 󵄨2 ‖x‖2 ≤ ∑󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 ≤ B‖x‖2H , 1+C H 1+C n 󵄨 n=m ̸ that is, {xn }n=m ̸ is a frame. Let us recall the definition of biorthogonal sequences. Definition 3.1.20. Two sequences {xn }, {yn } in a Hilbert space H are called biorthogonal if ⟨xm , yn ⟩ = δm,n ,

∀m, n.

Corollary 3.1.21. Let {xn } be a frame for a Hilbert space H. Then the following are equivalent: (i) The sequence {xn } is an exact frame. (ii) {xn } and {x̃n } are biorthogonal. (iii) ⟨xn , x̃n ⟩ = 1 for all n.

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Consequently, if {xn } is a A-tight frame, then the following are equivalent: (i) The sequence {xn } is an exact frame. (ii) {xn } is an orthogonal basis for H. (iii) ‖xn ‖2H = A for all n. Proof. (i) �⇒ (iii) If {xn } is exact, then for every m, the sequence {xn }n=m ̸ is no more a frame, hence by Theorem 3.1.19, ⟨xm , x̃m ⟩ = 1. (iii) �⇒ (i) For every fixed m, if ⟨xm , x̃m ⟩ = 1 then, by Theorem 3.1.19, {xn }n=m ̸ is incomplete. This means that {xn } is exact. (ii) �⇒ (iii) It follows immediately from Definition 3.1.20. (iii) �⇒ (ii) It follows immediately from Theorem 3.1.19. If {xn } is a A-tight frame, then S = AI and S−1 = A−1 I, so that x̃n = S−1 xn =

1 x . A n

Then the A-tight frame {xn } is exact iff ⟨xn , A−1 xm ⟩ = δm,n , that is, {xn } is an orthogonal system and it is complete (because it is a frame), so it is a basis.

3.1 Frames in Hilbert spaces | 159

We now present a characterization for Riesz bases, defined in Definition A.1.26, which we shall use in the sequel. For the proof we refer to [183, Theorem 7.13]. Theorem 3.1.22 (Characterization of Riesz bases). Let {xn } be a sequence for a Hilbert space H. Then the following are equivalent: (i) {xn } is a Riesz basis for H. (ii) {xn } is a bounded unconditional basis for H. (iii) {xn } is a basis for H and ∑ cn xn converges ⇐⇒ ∑ |cn |2 < ∞. n

n

(iv) {xn } is complete in H and there are constants A, B > 0 such that ∀c1 , . . . , cN ,

󵄩󵄩 N 󵄩󵄩2 N 󵄩󵄩 󵄩󵄩 󵄩 A ∑ |cn | ≤ 󵄩󵄩 ∑ cn xn 󵄩󵄩󵄩 ≤ B ∑ |cn |2 . 󵄩󵄩n=1 󵄩󵄩 n=1 n=1 󵄩 󵄩H N

2

(v) {xn } is a complete Bessel sequence and possesses a biorthogonal system {yn } that is also a complete Bessel sequence. The system {e1 , e2 /√2, e2 /√2, e3 /√3, e3 /√3, e3 /√3, . . . }

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(cf. Example 3.1.5 (iii)) is an inexact Parseval frame which contains an orthogonal basis {e1 , e2 /√2, e3 /√3, . . . }, not norm-bounded below, so it is not a Riesz basis. Proposition 3.1.23. Frames are preserved by topological isomorphisms. Specifically, if {xn } is a frame for a Hilbert space H and T : H → K is a topological isomorphism between the Hilbert spaces H and K, then {Txn } is a frame for K. Moreover, (i) If A, B are the frame bounds for {xn }, then {Txn } has frame bounds A‖T −1 ‖−2 and B‖T‖2 . (ii) If S is the frame operator for {xn }, then {Txn } has frame operator TST ∗ . (iii) {xn } is exact if and only if {Txn } is exact. Proof. First, observe that T ∗ is a topological isomorphism. Indeed, ker T ∗ = R(T)⊥ = {0}; using Theorem A.0.4, having ker T = {0} and R(T) being closed is equivalent to saying that T ∗ is surjective. Since T being bounded implies T ∗ is bounded (see, e. g., [37]), a corollary of the open mapping theorem gives that T ∗ is a topological isomorphism. Moreover, observe that TST ∗ : K → K; indeed, the following diagram is commutative: H T∗

S

- H T

6 K

TST ∗

? - K

160 | 3 Gabor frames and linear operators (the composition TST ∗ of topological isomorphisms is a topological isomorphism). For every y ∈ K, we have TST ∗ y = T(∑⟨T ∗ y, xn ⟩H xn ) = ∑⟨y, Txn ⟩K Txn n

n

with unconditional convergence because T is bounded. Let us prove that 󵄩 󵄩−2 A󵄩󵄩󵄩T −1 󵄩󵄩󵄩 I ≤ TST ∗ ≤ B‖T‖2 I, so that (i) and (ii) are satisfied. For every y ∈ K, ⟨TST ∗ y, y⟩ = ⟨ST ∗ y, T ∗ y⟩ hence 󵄩 󵄩2 󵄩 󵄩2 A󵄩󵄩󵄩T ∗ y󵄩󵄩󵄩H ≤ ⟨ST ∗ y, T ∗ y⟩ ≤ B󵄩󵄩󵄩T ∗ y󵄩󵄩󵄩H .

(3.16)

Setting y = (T ∗ )−1 T ∗ y, so that ‖y‖K ≤ ‖(T ∗ )−1 ‖‖T ∗ y‖H = ‖T −1 ‖‖T ∗ y‖H , we get ‖y‖K 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩T ∗ y󵄩󵄩󵄩H ≤ 󵄩󵄩󵄩T ∗ 󵄩󵄩󵄩‖y‖K = ‖T‖‖y‖K . ‖T −1 ‖ 󵄩

(3.17)

Combining (3.16) and (3.17) gives A

‖y‖2K ≤ ⟨TST ∗ y, y⟩ ≤ B‖T‖2 ‖y‖2K ‖T −1 ‖2

which yields the claim. Statement (iii) follows from the fact that a topological isomorphism preserves completeness.

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Corollary 3.1.24. A sequence {xn } is an exact frame for a Hilbert space H if and only if it is a Riesz basis. Proof. Assume that sequence {xn } is an exact frame and so a complete Bessel sequence. By Theorem 3.1.15, also {S−1 xn } is a frame for H, so a complete Bessel sequence. Finally, by Corollary 3.1.21, {xn } and {S−1 xn } are biorthogonal systems. The claim follows by the characterization of Riesz bases in Theorem 3.1.22 (v). Conversely, if {xn } is a Riesz basis, then by definition there exist a topological isomorphism T such that Ten = xn with {en } o. n. basis. Since o. n. bases are frames and, by Proposition 3.1.23, frames are preserved by topological isomorphisms, we obtain that {xn } is a frame. We can represent explicitly the isomorphism T giving an alternative to the first part of the proof of Corollary 3.1.24. Since S is a positive operator, which is a topological isomorphism from H onto itself, it has a square root S1/2 , which is a positive topological isomorphism (see Theorem A.0.6). Similarly, the inverse S−1 is a positive operator with square root S−1/2 = (S1/2 )−1 . Since {xn } is exact, then {S−1 xn } and {xn } are biorthogonal, and we have ⟨S−1/2 xm , S−1/2 xn ⟩ = ⟨xm , S−1/2 S−1/2 xn ⟩ = ⟨xm , S−1 xn ⟩ = δm,n .

3.1 Frames in Hilbert spaces | 161

Thus {S−1/2 xn } is an o. n. sequence. Moreover, it is complete since topological isomorphisms preserve completeness. This implies that {S−1/2 xn } is an o. n. basis for H and the topological isomorphism T = S1/2 maps this o. n. basis onto the frame {xn }. We can consider the sequence {S−1/2 xn } for any frame, not just exact. If this is the case, then {S−1/2 xn } will not be an o. n. basis for H, but it will be a Parseval frame. Corollary 3.1.25. Let {xn } be a frame for a Hilbert space H, with frame operator S. Then (i) {S−1/2 xn } is a Parseval frame for H. (ii) ⟨xn , x̃n ⟩ = ‖S−1/2 xn ‖2H and 0 ≤ ⟨xn , x̃n ⟩ ≤ 1 for every n. (iii) {xn } is an exact frame if and only if {S−1/2 xn } is an o. n. basis for H. Proof. (i) The operator S−1/2 is a topological isomorphism because S is such. Since frames are preserved by topological isomorphisms, we have that {S−1/2 xn } is a frame. We prove that it is a Parseval frame. For every x ∈ H, ∑⟨x, S−1/2 xn ⟩S−1/2 xn = S−1/2 (∑⟨S−1/2 x, xn ⟩xn ) = S−1/2 SS−1/2 x = Ix, n

n

which is equivalent to saying that {S−1/2 xn } is a Parseval frame. (ii) Using the self-adjointness of S−1/2 , we can write 󵄩 󵄩2 ⟨xn , x̃n ⟩ = ⟨xn , S−1/2 S−1/2 xn ⟩ = ⟨S−1/2 xn , S−1/2 xn ⟩ = 󵄩󵄩󵄩S−1/2 xn 󵄩󵄩󵄩H . Since {S−1/2 xn } is a 1-tight frame, we have ‖S−1/2 xn ‖H ≤ 1, for every n, by Remark 3.1.11. (iii) The “�⇒” direction is already proved. Conversely, we have xn = S1/2 S−1/2 xn . Then xn is the image of an o. n. basis by the topological isomorphism S1/2 , meaning that {xn } is a Riesz basis and the claim follows by Corollary 3.1.24.

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Let us end this general introduction to frames in Hilbert spaces by a few observations on the convergence of the frame expansions ∑n cn xn , for arbitrary choice of scalars. By Proposition 3.1.12, if c = (cn ) ∈ ℓ2 , then the frame expansion ∑n cn xn converges unconditionally. The converse is not true in general, as shown by the following example. Example 3.1.26. (i) Let {xn } be a frame with infinitely many zero elements. Consider the sequence c = (cn ) defined as follows: cn = {

1, if xn = 0, 0, if xn ≠ 0.

Then ∑n cn xn = 0 unconditionally, whereas ∑n |cn |2 = +∞. (ii) Let {en }n∈ℕ+ be an o. n. basis for H and define xn =

1 e , n n

yn = (1 −

1/2

1 ) en . n2

162 | 3 Gabor frames and linear operators Then {xn } ∪ {yn } is a Parseval frame for H (left as an exercise). Now consider the element x = ∑n n−1 en ∈ H, since the sequence (n−1 )n∈ℕ+ is in ℓ2 . We can write x = ∑n (1xn + 0yn ) and ∑n (12 + 02 ) = +∞, so again the sequence of coefficients does not belong to ℓ2 . Of course, in the previous example we have not considered the coefficient sequence (⟨xn , x̃n ⟩), obtained by the canonical dual frame and which, as we already know, belongs to ℓ2 . (Recall ∑n |⟨xn , x̃n ⟩|2 < ∞.) Let us emphasize that nonuniqueness is one of the major reasons why frames have been introduced as an alternative to bases. So it is important to consider alternative representation of elements with respect to a frame {xn }. For frames norm-bounded below, there is the following characterization, due to C. Heil. Theorem 3.1.27. Let {xn } be a frame for a Hilber space H which is norm-bounded below. Then ∑ |cn |2 < ∞ ⇐⇒ ∑ cn xn converges unconditionally. n

n

Proof. The “�⇒” direction follows by Proposition 3.1.12. On the other hand, assume that ∑n cn xn converges unconditionally. By Orlicz’s theorem (cf. Theorem A.1.23), letting 0 < a = infn ‖xn ‖H , we obtain a2 ∑ |cn |2 ≤ ∑ |cn |2 ‖xn ‖2H = ∑ ‖cn xn ‖2H < ∞, n

n

n

and the claim follows.

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For exact frames the same result is given by Corollary 3.1.24 and Theorem 3.1.22 (iii). If a frame is not exact with infn ‖xn ‖H = 0, the result fails, as shown by the preceding Example 3.1.26. Regarding frames, we refer to an interesting paper by M. Frank, V. I. Paulsenand and T. Tiballi [149].

3.2 Gabor frames The general theory seen so far finds an application for a particular type of frame, having time–frequency shifts as building blocks. Definition 3.2.1 (Gabor frame). Given a nonzero window function g ∈ L2 (ℝd ) and lattice parameters α, β > 0, the set of time–frequency shifts d

𝒢 (g, α, β) = {Tαm Mβn g : m, n ∈ ℤ }

is called a Gabor system.

(3.18)

3.2 Gabor frames | 163

If 𝒢 (g, α, β) is a frame for L2 (ℝd ), it is called a Gabor frame. The associated frame operator, the Gabor frame operator S, has the form Sf = =

∑ ⟨f , Tαm Mβn g⟩Tαm Mβn g

m,n∈ℤd

(3.19)

∑ Vg f (αm, βn)Mβn Tαm g.

m,n∈ℤd α,β

We write Sg,g or Sg,g whenever it is necessary to emphasize the dependence of the frame operator on g, α, β. The series expansions above have coefficients that can be viewed as a sampling of the STFT on the lattice Λ = αℤd ×βℤd . The order of translation and modulation in (3.19) is not important, due to the commutation relation (0.14). The order Mξ Tx is natural in the context of the STFT, whereas the order Tx Mξ is advantageous in connection with representation theory. Gabor frames are named after the electrical engineer and physicist, most notable for inventing holography, Dennis Gabor (1900–1979). In his paper [152] Gabor conjec2 tured that the Gabor system 𝒢 (φ, 1, 1), where φ(t) = e−πt is the 1-dimensional Gaussian, is a basis for L2 (ℝ). In fact, he claimed that every function f ∈ L2 (ℝ) could be represented as f = ∑ cm,n (f )Mn Tm φ, m,n

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for some scalars cm,n (f ). This is the main reason why Gabor systems and Gabor frames are named in his honor. We shall see that his conjecture was false, but the previous expansions may make sense using frame theory. The product αβ of the lattice generators appears in many calculations involving Gabor systems. It is usually the product αβ which is important, despite the single values of α and β. Indeed, by dilating g, we can change the value of α at the expense of a complementary change of β. This is contained in the following lemma. Lemma 3.2.2. Fix g ∈ L2 (ℝd ), α, β, λ > 0 and consider the dilation operator Dλ defined in (0.19). Then the Gabor system 𝒢 (g, α, β) is a Gabor frame for L2 (ℝd ) iff 𝒢 (Dλ g, α/λ, λβ) is. Proof. We compute the action of the dilation operator Dλ on the sequence {Tαm Mβn g}: Dλ (Tαm Mβn g)(x) = λd/2 e2πiβn⋅(λx−αm) g(λx − αm) α

= e2πiλβn⋅(x− λ m) λd/2 g(λ(x −

α m)) = Tαm/λ Mλβn Dλ g(x). λ

So the Gabor system 𝒢 (g, α, β) is mapped by Dλ to the Gabor system 𝒢 (Dλ g, α/λ, λβ). Since Dλ is a bijective isometry of L2 (ℝd ), hence a topological isomorphism, by the

164 | 3 Gabor frames and linear operators general theory of frames (Proposition 3.1.23), we have that 𝒢 (g, α, β) is a frame if and only if 𝒢 (Dλ g, α/λ, λβ) is. The first task concerns the structure of the dual frame of a Gabor frame, first stated in the PhD thesis of O. Christensen [49], and for more general lattices in the paper by H. Feichtinger and W. Kozek [138]. Proposition 3.2.3. If 𝒢 (g, α, β) is a frame for L2 (ℝd ), with frame bounds 0 < A ≤ B, then there exists a window γ ∈ L2 (ℝd ) such that the canonical dual frame of 𝒢 (g, α, β) is 𝒢 (γ, α, β). Consequently, every f ∈ L2 (ℝd ) has the frame expansion f =

∑ ⟨f , Tαm Mβn g⟩Tαm Mβn γ

m,n∈ℤd

(3.20)

∑ ⟨f , Tαm Mβn γ⟩Tαm Mβn g,

=

m,n∈ℤd

with unconditional convergence in L2 (ℝd ). Further, the following norm equivalences hold: A‖f ‖22 ≤ B

−1

‖f ‖22

≤

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , Tαm Mβn g⟩󵄨󵄨󵄨 ≤ B‖f ‖22 ,

m,n∈ℤd

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , Tαm Mβn γ⟩󵄨󵄨󵄨 ≤ A−1 ‖f ‖22 .

m,n∈ℤd

Proof. Let us first show that the Gabor frame operator S in (3.19) commutes with the time–frequency shifts Tαm Mβn . Given f ∈ L2 (ℝd ), r, s ∈ ℤd , (Tαr Mβs )−1 STαr Mβs f = (Tαr Mβs )−1 ∑ ⟨Tαr Mβs f , Tαm Mβn g⟩Tαm Mβn g m,n∈ℤd

=

∑ ⟨Tαr Mβs f , Tαm Mβn g⟩(Tαr Mβs )−1 Tαm Mβn g

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m,n∈ℤd

=

∑ ⟨f , M−βs T−αr Tαm Mβn g⟩M−βs T−αr Tαm Mβn g

m,n∈ℤd

Observe that, by the commutation relations (0.14), M−βs T−αr Tαm = M−βs Tα(m−r) = e−2πiαβs⋅(m−r) Tα(m−r) M−βs . The phase factor e−2πiαβs⋅(m−r) cancels, and we obtain (Tαr Mβs )−1 STαr Mβs f = after renaming the indices.

∑ ⟨f , Tα(m−r) Mβ(n−s) g⟩Tα(m−r) Mβ(n−s) g = Sf ,

m,n∈ℤd

3.2 Gabor frames | 165

As a consequence, also the inverse S−1 commutes with Tαr Mβs (left as an exercise) and the canonical dual frame consists of the functions r, s ∈ ℤd .

S−1 (Tαr Mβs g) = Tαr Mβs S−1 g,

We set γ = S−1 g ∈ L2 (ℝd ) and the claim follows by the general theory of frames, see Theorem 3.1.15. As a consequence of the previous proposition, we obtain the expression of the inverse frame operator. Corollary 3.2.4. Consider 𝒢 (g, α, β) to be a frame for L2 (ℝd ) with frame operator Sg,g . −1 −1 Setting γ = Sg,g g, the inverse frame operator Sg,g is provided by −1 Sg,g f = Sγ,γ f =

∑ ⟨f , Tαm Mβn γ⟩Tαm Mβn γ.

m,n∈ℤd

The window function γ is called a canonical dual window of the window g. Remark 3.2.5. (i) Proposition 3.2.3 provides a discrete time–frequency representation of signals. If 𝒢 (g, α, β) is a frame for L2 (ℝd ), then (3.20) is a discrete version of the inversion formula (1.55) for the STFT. In addition, (3.20) provides a Gabor expansion of f with the canonical set of coefficients given by cm,n = ⟨f , Tαm Mβn γ⟩. The series expansion (3.20) can be rephrased in terms of the STFT as

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f =

∑ Vg f (αm, βn)Mβn Tαm γ,

m,n∈ℤd

and it is a reconstruction of the signal f from the samples of its STFT. (ii) The reconstruction formula (3.20) is based on the additional structure of Gabor −1 frames. Since Sg,g = Sγ,γ , the canonical dual window γ determines the inverse frame operator completely. So it suffices to solve a single linear equation Sγ = g. α,β (iii) The Gabor frame operator Sg,g obviously depends on the window g and the lattice parameters α and β. It is proved in [140] that, when increasing the sampling rate, that is, (αβ)−1 → +∞, the dual window γ becomes similar to the original one, g. Precisely, lim

α,β (αβ)−1 (Sg,g ) g=g −1

(α,β)→(0,0)

in the L2 -norm. (iv) Instead of separable lattices Λ = αℤd × βℤd of the time–frequency plane, we can also consider more general regular lattices of the form Λ = Mℤ2d where M ∈ GL(2d, ℝ), or even nonuniform discrete sets X ⊂ ℝ2d . In this latter case, however, the structure of the dual frame is lost, and we only know that it is a frame, but not a Gabor frame anymore. For this subject, we refer to [133].

166 | 3 Gabor frames and linear operators (v) The work by F. Weisz [308] is to be mentioned, where a signal f is approximated with the Riemannian sums Sk f =

1 k 2d

∑ Vg f (

m,n∈ℤd

m n , )M n T m γ, k k k k

of the inverse short-time Fourier transform. For f ∈ M p,q (ℝd ), 1 ≤ p, q < ∞, it is shown that Sk f → f in the M p,q (ℝd )-norm for k → ∞, provided g, γ ∈ M 1 (ℝd ). 3.2.1 Existence and density of Gabor frames In what follows we shall provide a few examples of windows g and conditions on the lattice parameters so that the Gabor system 𝒢 (g, α, β) becomes a frame. We suggest the reader to look at the classical paper by C. Heil [182] for a detailed survey of this topic and for even more references. We discuss first the following periodization trick. Lemma 3.2.6. If f ∈ L1 (ℝd ), then for all α > 0, ∫ f (x) dx = ∫ ( ∑ f (x + αk)) dx. ℝd

[0,α]d

k∈ℤd

Proof. Observe that the translated cubes αk + [0, α]d are disjoint except for an overlap of measure zero on the boundary, hence form a partition of ℝd . Using the σ-additivity of the Lebesgue integral and performing the change of variables x �→ x + αk, we can write 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx = ∑ ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx = ∑ ∫ 󵄨󵄨󵄨f (x + αk)󵄨󵄨󵄨 dx < ∞, k∈ℤd αk+[0,α]d

ℝd

k∈ℤd [0,α]d

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since f ∈ L1 (ℝd ) by assumption. This gives T−αk f ∈ L1 ([0, α]d × ℤd , dx × δ), where here δ is the discrete measure. An application of Fubini theorem yields ∫ f (x) dx = ∑

∫

k∈ℤd αk+[0,α]d

ℝd

f (x) dx = ∑

∫ f (x + αk) dx

k∈ℤd [0,α]d

= ∫ ( ∑ f (x + αk)) dx. [0,α]d

k∈ℤd

This concludes the proof. Definition 3.2.7. Given g, γ ∈ L2 (ℝd ), α, β > 0, we define the correlation function of the pair (g, γ) as ̄ − Gn (x) = ∑ g(x k∈ℤd

for n ∈ ℤd .

n − αk)γ(x − αk), β

(3.21)

3.2 Gabor frames | 167

Observe that, for any fixed n, the function Gn is a periodization of the function ̄ ∈ L1 (ℝd ) of period αℤd . Setting (Tn/β g)γ d

𝒬α = [0, α] ,

(3.22)

we have Gn ∈ L1 (𝒬α ) by Lemma 3.2.6. In particular, for g = γ, n = 0, we obtain 󵄨2 󵄨 G0 (x) = ∑ 󵄨󵄨󵄨g(x − αk)󵄨󵄨󵄨 . k∈ℤd

(3.23)

Hence G0 (x) ≥ 0 a. e. and, by Lemma 3.2.6, 󵄨 󵄨2 ‖G0 ‖L1 (𝒬α ) = ∫ 󵄨󵄨󵄨g(x)󵄨󵄨󵄨 dx = ‖g‖2L2 (ℝd ) . ℝd

Example 3.2.8. The simplest example of Gabor frame, in dimension d = 1, is provided by 𝒢 (χ[0,1] , 1, 1) = {e

2πinx

χ[m,m+1] },

m, n ∈ ℤ.

For every fixed m, the sequence {e2πinx χ[m,m+1] }n∈ℤ is an o. n. basis for L2 ([m, m + 1]). Hence 𝒢 (χ[0,1] , 1, 1) is simply the union of o. n. bases for L2 ([m, m + 1]) over all m ∈ ℤ and hence is an o. n. basis for L2 (ℝ). (Recall that this implies the frame property.) Unfortunately, the previous example is not very useful in practice because of the lack of smoothness of the window χ[0,1] . Indeed, it is discontinuous and this implies slow convergence of functions in such o. n. basis. Such a basis will capture the decay property of a given f correctly, but it is impossible to distinguish a C ∞ function from a nonsmooth function, because of the artificial discontinuities introduced by the cutoff. On the frequency side, observe that

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ℱ (χ[0,1] )(ξ ) = e

−πiξ

sin(πξ ) , πξ

ξ ≠ 0,

ℱ (χ[0,1] )(0) = 1,

that is, ℱ (χ[0,1] )(ξ ) = e−πiξ sinc(ξ ) and this function does not belong to L1 (ℝ), since its decay at infinity is of order 1/|ξ |. For applications, we need Gabor frames which are generated by functions that are both smooth and well-localized. Exercise 3.2.9. Fix g ∈ L2 (ℝd ) and α, β > 0. Using Parseval formula, show that 𝒢 (g, α, β) is a frame ⇐⇒ 𝒢 (g,̂ β, α) is a frame.

Exercise 3.2.10. Assume 𝒢 (g, α, β) is a frame for L2 (ℝ). Show that 𝒢 (g, α, β) is a Riesz basis if and only if ⟨g, γ⟩ = 1, where γ = S−1 g is the canonical dual window. (Hint: use Corollary 3.1.24 and show that 𝒢 (g, α, β) is an exact frame. To prove this claim, use Corollary 3.1.21 (iii)).

168 | 3 Gabor frames and linear operators If we consider dimension d = 1 and concentrate on functions g compactly supported in an interval of length 1/β, then there exist Gabor frames 𝒢 (g, α, β) for L2 (ℝ) with smooth windows g, provided we choose the lattice parameters α, β properly. This was done first by Daubechies, Grossmann, Meyer in [102], and they were called painless nonorthogonal expansions, since they were easy to construct. Theorem 3.2.11 (Painless nonorthogonal expansions). Fix α, β > 0 and g ∈ L2 (ℝ). (i) If supp g ⊆ [0, β−1 ] then 𝒢 (g, α, β) is a frame for L2 (ℝ) if and only if there exist constants A, B such that 󵄨 󵄨2 Aβ ≤ ∑ 󵄨󵄨󵄨g(x − αk)󵄨󵄨󵄨 ≤ Bβ k∈ℤ

a. e. x ∈ ℝ,

(3.24)

that is, with the notation (3.23), Aβ ≤ G0 (x) ≤ Bβ

a. e. x ∈ ℝ.

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In this case, A and B are frame bounds for 𝒢 (g, α, β). (ii) If 0 < αβ < 1, then there exists a g such that supp g ⊆ [0, β−1 ], satisfying (3.24) and being as smooth as we like, even infinitely differentiable. (iii) If αβ = 1, then any function that is supported in [0, β−1 ] and satisfies (3.24) is discontinuous. (iv) If αβ > 1 and g is supported in [0, β−1 ], then equation (3.24) is not satisfied and 𝒢 (g, α, β) is incomplete in L2 (ℝ). Proof. (i) Assume supp g ⊆ [0, β−1 ] and that (3.24) holds. We check the frame inequalities (3.2) for f in a dense subset of L2 (ℝ). This is enough to assure that 𝒢 (g, α, β) is a frame. We choose Cc (ℝ), the space of continuous functions with compact support, which is dense in L2 (ℝ). Since g ∈ L2 (ℝ) is supported in [0, β−1 ], the translates Tαk g are supported in Ik = [αk, αk + β−1 ] and belong to L2 (Ik ). Since f is bounded, fTαk g ∈ L2 (Ik ) as well. Now, {e2πinx }n∈ℤ is an o. n. basis for L2 ([0, 1]), so, by making a change of variables, we obtain that the sequence {β1/2 e2πiβnx } is an o. n. basis for L2 (Ik ). Applying Plancherel inequality and keeping in mind that g is supported in [0, β−1 ] (so that Tαk g is supported in Ik ), we can write +∞

󵄨 󵄨2 ∫ 󵄨󵄨󵄨f (x)g(x − αk)󵄨󵄨󵄨 dx =

−∞

αk+β−1

󵄨2 󵄨 ∫ 󵄨󵄨󵄨f (x)Tαk g(x)󵄨󵄨󵄨 dx

αk

= ‖fTαk g‖2L2 (Ik )

󵄨 󵄨2 = ∑ 󵄨󵄨󵄨⟨fTαk g, β1/2 e2πiβnx ⟩L2 (I ) 󵄨󵄨󵄨 k n∈ℤ

󵄨󵄨 󵄨󵄨󵄨2 󵄨 = β ∑ 󵄨󵄨󵄨∫ f (x)g(x − αk)e−2πiβnx dx󵄨󵄨󵄨 󵄨 󵄨󵄨 n∈ℤ󵄨 Ik

3.2 Gabor frames | 169

󵄨󵄨2 󵄨󵄨 +∞ 󵄨󵄨 󵄨󵄨 −2πiβnx 󵄨 dx 󵄨󵄨󵄨 = β ∑ 󵄨󵄨 ∫ f (x)g(x − αk)e 󵄨󵄨 󵄨 n∈ℤ󵄨󵄨−∞ 󵄨 2 +∞ 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨 󵄨 = β ∑ 󵄨󵄨󵄨 ∫ f (x)e2πiβnx g(x − αk) dx󵄨󵄨󵄨 󵄨 󵄨󵄨 󵄨 n∈ℤ󵄨−∞ 󵄨 󵄨󵄨2 󵄨󵄨 = β ∑ 󵄨󵄨⟨f , Mβn Tαk g⟩󵄨󵄨 . n∈ℤ

Now, using Tonelli’s theorem to interchange sum and integral, +∞

󵄨 󵄨2 1 󵄨 󵄨2 ∑ ∑ 󵄨󵄨󵄨⟨f , Mβn Tαk g⟩󵄨󵄨󵄨 = ∑ ∫ 󵄨󵄨󵄨f (x)g(x − αk)󵄨󵄨󵄨 dx β k∈ℤ n∈ℤ k∈ℤ −∞

+∞

=

1 󵄨 󵄨2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨g(x − αk)󵄨󵄨󵄨 dx β k∈ℤ −∞

and, by the assumption (3.24), +∞

󵄨 󵄨2 1 󵄨 󵄨2 󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , Mβn Tαk g⟩󵄨󵄨󵄨 = ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨g(x − αk)󵄨󵄨󵄨 dx β k,n∈ℤ k∈ℤ −∞ +∞

󵄨 󵄨2 ≥ A ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx = A‖f ‖2L2 (ℝ) . −∞

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The upper bound ∑k,n∈ℤ |⟨f , Mβn Tαk g⟩|2 ≤ B‖f ‖2L2 (ℝ) is obtained analogously. Hence 𝒢 (g, α, β) is a frame. The converse will be proved in more generality in the subsequent Theorem 3.2.14. (ii) Assume 0 < αβ < 1 and let g be any continuous function such that g(x) = 0 if x ∉ [0, β−1 ] and g(x) > 0 on (0, β−1 ). For example, we can choose the smooth bell function centered at 1/(2β), that is, −

{ { e g(x) = { { 0, {

1 x −x 2 β

,

0 < x < β1 ,

x ≤ 0 ∨ x ≥ β1 .

Now, since α < β−1 , the α-periodic function G0 (x) = ∑k∈ℤ |g(x − αk)|2 is strictly positive at every point. Further, at each point x, the function G0 (x) is continuous (or in 𝒞 ∞ (ℝ) if we choose g properly; take, for instance, the bell function above) since it is a finite sum of continuous functions. So 0 < inf G0 (x) = min G0 (x) ≤ max G0 (x) = sup G0 (x) < ∞, x∈ℝ

x∈[0,α]

x∈[0,α]

x∈ℝ

that is, G0 satisfies (3.24) and, using part (i), 𝒢 (g, α, β) is a frame.

170 | 3 Gabor frames and linear operators (iii) If αβ = 1, then α = β−1 . If supp g ⊆ [0, β−1 ] = [0, α], then Tαk g is supported in [αk, α(k + 1)]. By contradiction, assume that g is continuous, then g(0) = g(α) = 0. Since the intervals [αk, α(k + 1)] overlap at most one point, it follows that G0 is continuous and G0 (αk) = 0 for every k ∈ ℤ, so it does not satisfy (3.24) and, by part (i), 𝒢 (g, α, β) is not a frame. Note that, arguing as in (ii), we may easily produce examples of discontinuous g with supp g ⊂ [0, β−1 ], such that 𝒢 (g, β−1 , β) is a frame, cf. Example 3.2.8. (iv) If αβ > 1, then α > β−1 . Hence the function G0 (x) is equal to zero on the interval −1 (β , α) and cannot satisfy (3.24). By part (i), 𝒢 (g, α, β) is not a frame. We can also check directly that the Gabor system 𝒢 (g, α, β) is incomplete in L2 (ℝ): indeed, consider the characteristic function χ[β−1 ,α] , which is orthogonal to every element of 𝒢 (g, α, β). Remark 3.2.12. Note that it is the product αβ which is important in the previous result. Indeed, by Lemma 3.2.2, one can change the value of α at the expense of a complementary change of β. Also the interval [0, β−1 ] can be replaced by any interval of length β−1 , simply by translating the function g. – – –

Summarizing some important points of the painless nonorthogonal expansions: If 0 < αβ < 1 then we can find a nice function g (even g ∈ 𝒟(ℝ)) such that 𝒢 (g, α, β) is a frame for L2 (ℝ). If αβ = 1 then there exist Gabor frames 𝒢 (g, α, β) for L2 (ℝ), with g discontinuous. If αβ > 1 and supp g ⊆ [0, β−1 ] then the Gabor system 𝒢 (g, α, β) is incomplete and it is not a frame for L2 (ℝ).

Exercise 3.2.13. Assume the hypotheses of Theorem 3.2.11 is satisfied, that is, supp g ⊆ [0, β−1 ] and (3.24) holds. Prove the following statements about the frame 𝒢 (g, α, β): – The frame operator S is pointwise multiplication by β−1 G0 , that is, Sf = β−1 G0 f ,

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

– –

f ∈ L2 (ℝ).

(Hint: use the definition of a frame operator.) The canonical dual window is γ = (βg)/G0 . (Hint: use the definition of a canonical dual window γ = S−1 g.) If αβ = 1, then 𝒢 (g, α, β) is a Riesz basis. (Hint: use Exercise 3.2.10.)

Theorem 3.2.11 gives necessary and sufficient conditions for the existence of Gabor frames 𝒢 (g, α, β) when the window g is supported in [1, 1/β]. This equivalence does not extend to general functions in L2 (ℝ), but the necessity still holds. Theorem 3.2.14. If g ∈ L2 (ℝ) and α, β are such that 𝒢 (g, α, β) is a frame for L2 (ℝ) with frame bounds A, B > 0, then we must have Aβ ≤ G0 (x) ≤ βB

a. e.,

that is, (3.24) holds. In particular, g must be bounded.

3.2 Gabor frames | 171

Proof. We use similar arguments as in the proof of Theorem 3.2.11. In this case we do not know if g is compactly supported and so we start with a function f ∈ L2 (ℝ) which is bounded and such that supp f is in an interval I of length 1/β. So the product fTαk ḡ is in L2 (I), which has the sequence {β1/2 e2πiβnx } as o. n. basis, and again we can write +∞

2 2 󵄨 ̄ 󵄨󵄨󵄨󵄨 dx = β ∑ 󵄨󵄨󵄨󵄨⟨f , Mβn Tαk g⟩󵄨󵄨󵄨󵄨 . ∫ 󵄨󵄨󵄨fTαk g(x) n∈ℤ

−∞

Using Tonelli’s theorem and employing the lower frame bound A, βA‖f ‖2L2 (ℝ)

+∞

󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 ≤ β ∑ 󵄨󵄨󵄨⟨f , Mβn Tαk g⟩󵄨󵄨󵄨 = ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨Tαk g(x)󵄨󵄨󵄨 dx k,n∈ℤ

−∞

k∈ℤ

+∞

󵄨 󵄨2 = ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 G0 (x) dx. −∞

So every bounded f ∈ L2 (I) fulfills +∞

󵄨 󵄨2 ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 (G0 (x) − βA) dx ≥ 0.

(3.25)

−∞

Now, if G0 < βA on some subset E of I of positive measure, then we could take f = χE and obtain a contradiction to (3.25). Therefore, we must have G0 ≥ βA a. e. on I, and a similar computation with the upper bound gives also G0 ≤ βB. Since I is an arbitrary interval of length 1/β and the real line can be covered by countably many translates of I, we conclude βA ≤ G0 ≤ βB a. e. on ℝ. Moreover ℓ2 ⊂ ℓ∞ , so for a. e. x, 1/2

󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨2 󵄨󵄨g(x)󵄨󵄨󵄨 ≤ sup󵄨󵄨󵄨g(x − αk)󵄨󵄨󵄨 ≤ ( ∑ 󵄨󵄨󵄨g(x − αk)󵄨󵄨󵄨 ) Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

k∈ℤ

k∈ℤ

≤ √βB,

so that ‖g‖∞ < ∞. Exercise 3.2.15. Show that, if 𝒢 (g, α, β) is a Gabor frame with frame operator S and g ♯ = S−1/2 g, then 𝒢 (g ♯ , α, β) is a Parseval frame for L2 (ℝ). (Hint: use Corollary 3.1.25 and Theorem A.0.6 in Appendix A.) The following corollary gives several features on Gabor frames on L2 (ℝ). Corollary 3.2.16 (Density and frame bounds). Consider α, β > 0, g ∈ L2 (ℝ) such that 𝒢 (g, α, β) is a Gabor frame for L2 (ℝ) with frame bounds A and B. Then (i) Aαβ ≤ ‖g‖22 ≤ Bαβ. (ii) If 𝒢 (g, α, β) is a Parseval frame, then ‖g‖22 = αβ. (iii) 0 < αβ ≤ 1.

172 | 3 Gabor frames and linear operators (iv) ⟨g, γ⟩ = αβ where γ = S−1 g is the canonical dual window. (v) 𝒢 (g, α, β) is a Riesz basis if and only if αβ = ⟨g, γ⟩ = 1. Proof. (i) and (ii) Integrating (3.24) over the interval [0, α] and using Lemma 3.2.6 (observe that |g|2 ∈ L1 (ℝ)), we have α

α

+∞

0

0 k∈ℤ

−∞

󵄨2 󵄨 󵄨2 󵄨 Aαβ = ∫ Aβ dx ≤ ∫ ∑ 󵄨󵄨󵄨g(x − αk)󵄨󵄨󵄨 dx = ∫ 󵄨󵄨󵄨g(x)󵄨󵄨󵄨 dx = ‖g‖22 . ‖g‖22

Similarly, one can prove ≤ Bαβ. If the frame is a Parseval frame then A = B = 1, 2 thus part (i) implies ‖g‖2 = αβ. (iii) By Exercise 3.2.15, 𝒢 (g ♯ , α, β), where g ♯ = S−1/2 g, is a Parseval frame. Then by part (ii), ‖g ♯ ‖22 = αβ. By Remark 3.1.11, the elements Mβn Tαk g ♯ must satisfy 󵄩 󵄩2 󵄩 󵄩2 αβ = 󵄩󵄩󵄩g ♯ 󵄩󵄩󵄩2 = 󵄩󵄩󵄩Mβn Tαk g ♯ 󵄩󵄩󵄩2 ≤ 1, so that αβ ≤ 1. (iv) Using the previous step and the self-adjointness of S−1/2 , ⟨g, γ⟩ = ⟨g, S−1/2 S−1/2 g⟩ = ⟨S−1/2 g, S−1/2 g⟩ = ‖g♯ ‖22 = αβ. (v) It follows by Exercise 3.2.10. Summarizing, the Gabor systems 𝒢 (g, α, β) for g ∈ L2 (ℝ) can be divided into three parts: – If 0 < αβ < 1 then if 𝒢 (g, α, β) is a Gabor frame then it is a redundant frame. – If αβ = 1 then if 𝒢 (g, α, β) is a Gabor frame then it is a Riesz basis. – If αβ > 1 then the system 𝒢 (g, α, β) is not a frame. The value 1/(αβ) is called the density of the Gabor system 𝒢 (g, α, β).

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Remark 3.2.17. Corollary 3.2.16 and the necessity condition (3.24) is still true for g ∈ L2 (ℝd ) (d ≥ 1). For details on this subject we refer, for instance, to [160]. We end up this section by observing that there are window functions γ ∈ L2 (ℝd ) other than the canonical dual window S−1 g that give rise to Gabor frames and are dual frames to the Gabor frame 𝒢 (g, α, β). Precisely, given a window γ ∈ L2 (ℝd ), we can consider the reconstruction operator Dγ , defined on finite sequences c = (cm,n ) by Dγ c = ∑m,n cm,n Mβn Tαm γ. If Dγ is a bounded operator from ℓ2 (ℤ2d ) into L2 (ℝd ), with a slight abuse of language we can define the frame operator Sg,γ f = Dγ Cg f =

∑ ⟨f , Mβn Tαm g⟩Mβn Tαm γ,

m,n∈ℤ2d

(3.26)

which is a well defined operator Sg,γ : L2 (ℝd ) → L2 (ℝd ), since Cg : L2 (ℝd ) → ℓ2 (ℤ2d ) because the system 𝒢 (g, α, β) is a frame.

3.2 Gabor frames | 173

Definition 3.2.18. A function γ ∈ L2 (ℝd ) is called dual window for a Gabor frame 𝒢 (g, α, β) if the synthesis operator Dγ is a bounded operator from ℓ2 (ℤ2d ) to L2 (ℝd ) and if Sg,γ f = IL2 (ℝd ) , that is, the frame operator is the identity operator on L2 (ℝd ). The following condition characterizes all dual windows. Theorem 3.2.19 (Wexler–Raz biorthogonality relations). Let g, γ be in L2 (ℝd ) such that the synthesis operators Dg , Dγ are bounded. Then the following conditions are equivalent: (i) Sg,γ = Sγ,g = IL2 (ℝd ) . (ii) (αβ)−d ⟨M n T m γ, M n󸀠 T m󸀠 g⟩ = δm,m󸀠 δn,n󸀠 , m, m󸀠 , n, n󸀠 ∈ ℤd . α

β

α

β

Hence the Gabor systems 𝒢 (g, β1 , α1 ) and 𝒢 (γ, β1 , α1 ) are biorthogonal systems in

L2 (ℝd ). The proof can be found in [160, Theorem 7.3.1]. One of the most used window functions for Gabor frames is represented by the Gaussian. A more general result on this topic is contained in [107, Proposition 10]. It extends to the multidimensional case the classical result of Lyubarskii [221] and Seip and Wallsten [262, 263]. We limit ourselves to the statement of such a result in our setting, with a more general lattice than the case αℤd × βℤd , α, β > 0. First, we need a more general definition of Gabor systems. Definition 3.2.20. For a general lattice Λ ⊂ ℝ2d and a nonzero window function g ∈ L2 (ℝd ), we define the Gabor system as 𝒢 (g, Λ) = {π(λ)g : λ ∈ Λ}.

(3.27)

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The Gaussian Gabor frames are then given by the following characterization. Theorem 3.2.21. Consider the lattice Λα,β = αℤd × βℤd , with α = (α1 , . . . , αd ), β = (β1 , . . . , βd ), and αj , βj > 0, for every j = 1, . . . , d. Then the Gabor system 𝒢 (φ, Λα,β ), with 2

the Gaussian function φ(t) = e−πt , t ∈ ℝd , is a frame for L2 (ℝd ) if and only if αj βj < 1, for every j = 1, . . . , d.

Observe that in the critical case αj βj = 1, for every j = 1, . . . , d, the system 𝒢 (φ, 1, 1) is not a frame and, in particular, it is not a basis. Another important tool for analyzing Gabor frames when Λ = αℤd × βℤd , αβ = 1, is the Zak transform. We recommend the article [204] for a survey of the properties of the Zak transform. Definition 3.2.22. For α > 0, the Zak transform of a function f on ℝd is (formally) Zα f (x, ξ ) = ∑ f (x − αk)e2πiαk⋅ξ = ∑ Mαk Tαk f (x), k∈ℤd

k∈ℤd

(x, ξ ) ∈ ℝ2d .

(3.28)

174 | 3 Gabor frames and linear operators 2

For example, the Zak transform Z1 (α = 1) of the Gaussian φ(t) = e−πt is given by 2 Z1 f (x, ξ ) = ∑k∈ℤd e−π(x−k) e2πik⋅ξ . Formally, we have the quasiperiodicity relations Zα f (x + αm, ξ ) = e2πiαm⋅ξ Zα (x, ξ )

and Zα f (x, ξ +

m ) = Zα f (x, ξ ), α

for m ∈ ℤd . Thus Zα f is completely determined by its values on [0, α)d × [0, 1/α)d ⊂ ℝ2d . Recalling the cube 𝒬α = [0, α]d , α > 0, the Plancherel’s theorem for the Zak transform reads as follows [203] (see also [160, Theorem 8.2.3] or [186, Theorem 4.3.2]). Theorem 3.2.23. The Zak transform is a unitary map of L2 (ℝd ) onto L2 (𝒬α × 𝒬1/α ). The unitary nature of the Zak transform allows us to translate conditions on frames for L2 (ℝ2d ) into those in L2 (𝒬α × 𝒬1/α ), where things are frequently easier to deal with. For (u, η) ∈ ℝ2d , we easily compute the Zak transform of the time–frequency shifts Tu Mη f of a function f in L2 (ℝd ). Namely, Zα (Tu Mη f )(x, ξ ) = ∑ e2πiη⋅(x−u−αk) f (x − u − αk)e2πiαk⋅ξ k∈ℤd

= e2πiη⋅(x−u) ∑ f (x − u − αk)e2πiαk⋅(ξ −η) =e

2πiη⋅(x−u)

k∈ℤd

Zα f (x − u, ξ − η).

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Hence Zα (Tu Mη f )(x, ξ ) = e2πiη⋅(x−u) T(u,η) Zα f (x, ξ ), which places great restrictions on the form Zα g can take if 𝒢 (g, α, 1/α) is to generate a frame [100, 203] (see also [160, Theorem 8.2.3] or [186, Section 8.3]). Theorem 3.2.24. Suppose αβ = 1 and g ∈ L2 (ℝd ). (i) The Gabor system 𝒢 (g, α, 1/α) is complete in L2 (ℝd ) if and only if Zα f ≠ 0 a.e. (ii) The following statements are equivalent: (a) 0 < A ≤ |Zα f (x, ξ )|2 ≤ B < ∞ a.e. (b) 𝒢 (g, α, 1/α) is a frame for L2 (ℝd ) with frame bounds A and B. (c) 𝒢 (g, α, 1/α) generates an exact frame for L2 (ℝd ) with frame bounds A, B. (iii) 𝒢 (g, α, 1/α) generates an orthonormal basis for L2 (ℝd ) if and only if 󵄨󵄨 󵄨 󵄨󵄨Zα f (x, ξ )󵄨󵄨󵄨 = 1 a. e. As an example, in dimension d = 1, the Zak transform Zα of the characteristic function χ[0,α) is Zα χ[0,α) (x, ξ ) = ∑ χ[0,α) (x − αk)e2πiαkξ k∈ℤd

3.2 Gabor frames | 175

and χ[0,α) (x − αk) ≠ 0 on [0, α) only if x − αk ∈ [0, α) which forces k = 0 and hence Zα χ[0,α) (x, ξ ) = 1 on the rectangle [0, α) × [0, 1/α). Similarly in higher dimensions the Zak transform of the characteristic function χ𝒬α is Zα χ𝒬α = 1

for (x, ξ ) ∈ 𝒬α × 𝒬1/α .

Using the previous theorem, we obtain that 𝒢 (χ𝒬α , α, 1/α) is an orthonormal basis for L2 (ℝd ), cf. Example 3.2.8. Theorem 3.2.25 ([203]). Let f ∈ L2 (ℝd ) be such that Zα f is continuous on ℝ2d . Then Zα f has a zero. 2

Example 3.2.26 ([102]). The Zak transform Z1 φ of the Gaussian function φ(t) = e−πt , t ∈ ℝ, is continuous and has a single zero in [0, 1) × [0, 1). Therefore Theorem 3.2.24 states that 𝒢 (φ, α, 1/α) is complete but it is not a frame, cf. Theorem 3.2.21. While no function whose Zak transform is continuous can generate a Gabor frame when αβ = 1, this does not exclude smooth window functions from generating Gabor frames. The following theorem, due to R. Balian [4] (and independently to F. Low [220]), shows that, in fact, if 𝒢 (g, α, β) generates a frame when αβ = 1 then either g is not smooth or does not decay very fast. An elegant proof by G. Battle for the orthonormal basis case, based on the Heisenberg uncertainty principle, is in [7]. See also the results by Daubechies and Janssen in [104] and by Benedetto, Heil, and Walnut [11]. We state it in the original version in dimension d = 1.

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Theorem 3.2.27. Consider g ∈ L2 (ℝ) and α, β > 0 with αβ = 1. If 𝒢 (g, α, β) generates a ̂ ) ∉ L2 (ℝ). frame, then either tg(t) ∉ L2 (ℝ) or ξ g(ξ Example 3.2.28. With reference to the preceding Theorem 3.2.11 (iii), every function g generating a frame for αβ = 1 and being compactly supported, hence tg(t) ∈ L2 (ℝ), is ̂ ) ∈ ̸ L2 (ℝ) from the Balian– discontinuous, that is, consistent with the information ξ g(ξ Low theorem. In the opposite direction, consider the Fourier transform of such g, that is, h = g.̂ It generates a frame for αβ = 1, see the previous Exercise 3.2.9. In this case ̂ ) ∈ L2 (ℝ) but th(t) ∉ L2 (ℝ). Note that h is an entire function, we have obviously ξ h(ξ being the Fourier transform of a function with compact support. A simple example is ℱ (χ[0,1] ), namely h(t) = e−πit sinc(t), see the computations before Exercise 3.2.9. We conclude by presenting the so-called amalgam version of the Balian–Low theorem due to C. Heil [179]. Observe that Heil’s formulation does not depend on the dimension d and uses the Wiener amalgam spaces W(L∞ , L1 ) for the windows. Let us define W0 (L∞ , L1 )(ℝd ) := W(L∞ , L1 )(ℝd ) ∩ 𝒞 (ℝd ). Theorem 3.2.29. If 𝒢 (g, α, 1/α), α > 0 is a frame (and hence a Riesz basis) for L2 (ℝd ), then both g ∉ W0 (L∞ , L1 )(ℝd ) and ĝ ∉ W0 (L∞ , L1 )(ℝd ).

176 | 3 Gabor frames and linear operators In summary, Gabor frames with αβ = 1 are bases for L2 (ℝd ) but have unpleasant window functions. As a consequence we obtain a stronger version of the density theorem for nice windows. Corollary 3.2.30. If 𝒢 (g, α, β) is a frame for L2 (ℝd ) and if either g ∈ W0 (L∞ , L1 )(ℝd ) or ĝ ∈ W0 (L∞ , L1 )(ℝd ), then αβ < 1. For the Walnut representation of the Gabor frame operator, we refer the reader to Walnut’s work [302], see also [160, Chapter 5]. Existence results for Gabor frames by pseudodifferential methods were obtained by P. Boggiatto and G. Garello in [25]. We end up this section by recalling the following conjecture known as the HRT conjecture, which is an open problem deeply rooted in time–frequency analysis. It was posed about 20 years ago by C. Heil, J. Ramanathan, and P. Topiwala in [184] as follows Conjecture (The HRT conjecture). If g ∈ L2 (ℝ) and Λ is a finite subset of ℝ2 , then 𝒢 (g, Λ) is linearly independent. Although the HRT conjecture is still unresolved, there are quite a few results that might be regarded as evidence for an affirmative answer. 3.2.2 Gabor frames for modulation spaces

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In this section we shall provide a discrete characterization of the modulation norm p,q (2.20). Precisely, we shall extend Proposition 3.2.3 to the spaces Mm , showing that p,q a function/distribution f is in Mm if and only if the sequence of Gabor coefficients p,q {⟨f , Tαm Mβn g⟩}m,n is in ℓm (ℤ2d ), cf. Definition 2.2.1. Notice that the first connection between modulation spaces and atomic Gabor expansions in the spirit modern coorbit theory was given in [123]. We need some preliminaries. Proposition 3.2.31. Consider a weight function m ∈ ℳv (ℝ2d ), and a lattice Λ = αℤd × βℤd . For every α, β > 0 there is a constant Cα,β > 0 such that the estimate 󵄩󵄩 󵄩 󵄩󵄩{F(αk, βn)}k,n 󵄩󵄩󵄩ℓmp,q (Λ) ≤ Cα,β ‖F‖W(L∞ ,Lp,q m ) 2d is true for any function F ∈ 𝒞 (ℝ2d ) ∩ W(L∞ , Lp,q m )(ℝ ), for any 1 ≤ p, q ≤ ∞. p,q (Λ). Observe that this implies that the sequence {F(αk, βn)}k,n∈ℤd is in ℓm

Proof. The continuity of F guarantees that F(αk, βn) is well defined, for every k, n ∈ ℤd . Since the weight m is v-moderate, using (2.7), there exists a positive constant C satisfying m(αk, βn) ≤ Cm(r, s),

for (αk, βn) ∈ (r, s) + [0, 1]2d .

3.2 Gabor frames | 177

This yields 󵄨 󵄨󵄨 󵄨󵄨F(αk, βn)󵄨󵄨󵄨m(αk, βn) ≤ C‖FTr,s χ‖∞ m(r, s), with χ characteristic function of [0, 1]2d . Since there are at most Cα = ([1/α] + 1)d points αk ∈ r + [0, 1]d , the ℓp -norm with respect to k ∈ ℤd can be controlled by 1

1

p p 󵄨p 󵄨 ( ∑ 󵄨󵄨󵄨F(αk, βn)󵄨󵄨󵄨 m(αk, βn)p ) ≤ (CCα ∑ ‖FTr,s χ‖p∞ m(r, s)p ) ,

k∈ℤd

r∈ℤd

for every βn ∈ s + [0, 1]d , of which there are at most Cβ = ([1/β] + 1)d points. Taking the ℓq -norm of the above inequality and estimating the right-hand side by the norm in W(L∞ , Lp,q m ), cf. Definition 2.4.1, we get the claim. p,q We next show the boundness of the coefficient operator Cg from Mm (ℝd ) into d d for every lattice Λ = αℤ × βℤ .

p,q ℓm (Λ),

Theorem 3.2.32. Consider a weight m ∈ ℳv (ℝ2d ), a nonzero window function g ∈ Mv1 (ℝd ), and a lattice Λ = αℤd × βℤd , for every α, β > 0. Then the coefficient operap,q p,q tor Cg is bounded from Mm (ℝd ) into ℓm (Λ), with ‖Cg ‖B(Mmp,q ,ℓmp,q ) ≤ C(v, α, β)‖g‖Mv1 , independently of p, q and m. p,q Proof. For f ∈ Mm (ℝd ) and the window g ∈ Mv1 (ℝd ), the STFT Vg f is in the space 2d 1 ∞ W(ℱ L1 , Lp,q m )(ℝ ), by Lemma 2.4.15. Since ℱ L ⊂ L , by the inclusion relations for ∞ p,q Wiener amalgam spaces, we infer that Vg f ∈ W(L , Lm )(ℝ2d ). The continuity of the STFT, together with Proposition 3.2.31 and estimate (2.72), give

󵄩 󵄩 ‖Cg f ‖ℓmp,q (Λ) = 󵄩󵄩󵄩Vg f (αk, βn)󵄩󵄩󵄩ℓmp,q (Λ) ≤ Cα,β ‖Vg f ‖W(L∞ ,Lp,q m )

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≤ Cα,β ‖Vg f ‖W(ℱ L1 ,Lp,q ≲ Cα,β ‖f ‖Mmp,q ‖g‖Mv1 . m )

p,q We now present the continuity of the synthesis operator Dg from ℓm (Λ), for Λ = d d p,q d αℤ × βℤ , into Mm (ℝ ).

Theorem 3.2.33. Consider a weight m ∈ ℳv (ℝ2d ), a nonzero window function g ∈ Mv1 (ℝd ), and a lattice Λ = αℤd × βℤd , for every α, β > 0. Then the synthesis operator p,q p,q Dg is bounded from ℓm (Λ) into Mm (ℝd ), with ‖Dg ‖B(ℓmp,q ,Mmp,q ) ≤ C(v, α, β)‖g‖Mv1 , independently of p, q and m. If p, q < ∞, then Dg c = ∑ ck,n Tak Mβn g k,n∈ℤd

p,q ∞ converges unconditionally in Mm (ℝd ). Otherwise, Dg c converges weak* in M1/v (ℝd ).

178 | 3 Gabor frames and linear operators p,q Proof. Consider a sequence c = (ck,n ) in ℓm (Λ) and, for instance, the Gaussian φ(t) = 2

p,q e−πt as window function to measure the Mm -norm. We shall show that Vφ (Dg c) is in p,q 2d Lm (ℝ ). Recalling the action of the STFT on the time–frequency shifts (1.51), we can write Vφ (Tαk Mβn g)(x, ξ ) = T(αk,βn) Vφ g(x, ξ ), so that

󵄨 󵄨󵄨 󵄨󵄨Vφ (Dg c)󵄨󵄨󵄨 ≤ ∑ |ck,n |T(ak,βn) |Vφ g|. k,n∈ℤd

Since φ, g ∈ Mv1 (ℝd ), by Lemma 2.4.15 and the inclusion relations for Wiener amalgam spaces, we infer Vφ g ∈ W(ℱ L1 , L1v )(ℝ2d ) ⊂ W(L∞ , L1v )(ℝ2d ), hence |Vφ g|(x, ξ ) ≤ ∑ ar,s T(αr,βs) χ[0,α]d ×[0,β]d (x, ξ ), r,s∈ℤd

with a = (ar,s ), ar,s ≥ 0 satisfying ‖a‖ℓv1 = ∑ ar,s v(αr, βs) ≲ ‖Vφ g‖W(ℱ L1 ,L1v ) ≲ ‖g‖Mv1 . r,s∈ℤd

This implies, for every (x, ξ ) ∈ ℝ2d , ∑ |ck,n |T(αk,βn) |Vφ g| ≤

k,n∈ℤd

∑ k,n,r,s∈ℤd

|ck,n |ar,s T(α(k+r),β(n+s)) χ[0,α]d ×[0,β]d

= ∑ ( ∑ |ck,n |al−k,m−n )T(αl,βn) χ[0,α]d ×[0,β]d l,m∈ℤd k,n∈ℤd

= ∑ (|c| ∗ a)(l, m)T(αl,βn) χ[0,α]d ×[0,β]d . l,m∈ℤd

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Using now the norm equivalence (2.9), 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ∑ |ck,n |T(αk,βn) |Vφ g|󵄩󵄩󵄩 p,q ≲ 󵄩󵄩󵄩|c| ∗ a󵄩󵄩󵄩ℓmp,q ≲ ‖c‖ℓmp,q ‖a‖ℓv1 . 󵄩󵄩 󵄩󵄩Lm d k,n∈ℤ To sum up, 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ‖Dg c‖Mmp,q = 󵄩󵄩󵄩 ∑ ck,n Tαk,βn g 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩Mmp,q d k,n∈ℤ 󵄩󵄩 󵄩󵄩󵄩 󵄩 ≍ 󵄩󵄩󵄩Vφ ( ∑ ck,n Tαk,βn g)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩Lp,q d m k,n∈ℤ ≲ ‖c‖ℓmp,q . It remains to prove the unconditional convergence for 1 ≤ p, q < ∞. Recall that in p,q this case the finite sequences are dense in ℓm (ℤ2d ). Given ϵ > 0, there is a finite set

3.2 Gabor frames | 179

F0 ∈ ℤ2d such that ‖c − cχF ‖ℓmp,q < ϵ, for every F ⊇ F0 , cf. Theorem A.1.21 in Appendix A. Then 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩Dg c − ∑ ck,n Tαk Mβn g 󵄩󵄩󵄩 p,q = 󵄩󵄩󵄩Dg (c − c ⋅ χF )󵄩󵄩󵄩Mmp,q 󵄩󵄩Mm 󵄩󵄩 (k,n)∈F ≤ C‖c − c ⋅ χF ‖ℓmp,q < Cϵ, yielding the claim. p,q If p = ∞ or q = ∞, taking any c = (ck,n ) ∈ ℓm , we have ⟨Dg c, f ⟩ = ∑ ⟨f , Tαk Mβn g⟩, k,n∈ℤd

and, similarly as above, the series is unconditionally convergent for every f ∈ Mv1 (ℝd ), ∞ that is to say, Dg c converges weak* in (M1/v , Mv1 ). It can be shown that the constant C(v, α, β) can be chosen to be the same for compact ranges of lattice constants inside the (open) first quadrant of ℝ2 . As a consequence of Theorem 3.2.33, we infer the boundedness of the frame operp,q ator on Mm (ℝd ). Corollary 3.2.34. If m ∈ ℳv (ℝ2d ), g, γ ∈ Mv1 (ℝd ), then the Gabor frame operator Sg,γ = p,q Dγ Cg is bounded on Mm (ℝd ), for every 1 ≤ p, q ≤ ∞ and every α, β > 0. The operator norm can be estimated uniformly by ‖Sg,γ ‖B(Mmp,q ) ≤ C(v, α, β)‖g‖Mv1 ‖γ‖Mv1 ,

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with a constant C(v, α, β) depending only on v, α, β, but not on m, p, q. It may be worthwhile to mention that the constants C(v, α, β) appearing in Theorems 3.2.32 and 3.2.33, as well as in Corollary 3.2.34, are in fact uniformly bounded for compact ranges of parameters (α, β) (assuming one has frames for the full range), for good windows (see [137], in contrast to [136], where it is demonstrated that Bessel bounds may vary like crazy for ordinary L2 -functions). If, in addition, the function γ is a dual window for g, that is, Sg,γ = I, the identity operator on L2 (ℝd ), the modulation spaces can be characterized as follows. Theorem 3.2.35. If m ∈ ℳv (ℝ2d ), g, γ ∈ Mv1 (ℝd ) are such that Sg,γ = I on L2 (ℝd ), then f = ∑ ⟨f , Tαk Mβn g⟩Tαk Mβn γ k,n∈ℤd

(3.29)

= ∑ ⟨f , Tαk Mβn γ⟩Tαk Mβn g, k,n∈ℤd

p,q ∞ with unconditional convergence in Mm for p, q < ∞ and weak* convergence in M1/v otherwise. Further, the following norm equivalences hold:

󵄩 󵄩 A‖f ‖Mmp,q ≤ 󵄩󵄩󵄩{⟨f , Tαk Mβn g⟩}k,n 󵄩󵄩󵄩ℓmp,q ≤ B‖f ‖Mmp,q ,

(3.30)

180 | 3 Gabor frames and linear operators 󵄩 󵄩 B−1 ‖f ‖Mmp,q ≤ 󵄩󵄩󵄩{⟨f , Tαk Mβn γ⟩}k,n 󵄩󵄩󵄩ℓmp,q ≤ A−1 ‖f ‖Mmp,q

(3.31)

for suitable 0 < A ≤ B. p,q Proof. Using the boundedness of the frame operator Sg,γ = Dγ Cg on Mm (ℝd ), shown p,q in Corollary 3.2.34, we obtain that the equality Sg,γ = I holds for all f ∈ Mm (ℝd ). The left-hand side of the norm equivalence (3.30) is given by

‖f ‖Mmp,q = ‖Dγ Cg f ‖Mmp,q ≤ ‖Dγ ‖B(ℓmp,q ,Mmp,q ) ‖Cg f ‖ℓmp,q , with A = 1/‖Dγ ‖B(ℓmp,q ,Mmp,q ) . The right-hand side of (3.30) follows by the continuity of the coefficient operator Cg : ‖Cg f ‖ℓmp,q ≤ ‖Cg ‖B(Mmp,q ,ℓmp,q ) ‖f ‖Mmp,q , with B = ‖Cg ‖B(Mmp,q ,ℓmp,q ) . The inequalities in (3.31) are obtained analogously, using the identity Dg Cγ = I. Corollary 3.2.36. If g ∈ 𝒮 (ℝd ), Λ = αℤd × βℤd and 𝒢 (g, Λ) is a Gabor frame, then 󵄨 󵄨 f ∈ 𝒮 (ℝd ) ⇐⇒ sup⟨λ⟩N 󵄨󵄨󵄨⟨f , π(λ)g⟩󵄨󵄨󵄨 < ∞, λ∈Λ

∀N ∈ ℕ,

(3.32)

Proof. The characterization is a straightforward consequence of the norm equivalence in (3.30) (with λ = (αk, βn)) and the representation d

d

𝒮 (ℝ ) = ⋂ Mvs (ℝ ) ∞

s≥0

obtained in (2.28) with p = q = ∞.

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An extension of Theorem 3.2.35 to the quasi-Banach setting is proved in [248, Theorem 8.3]. Theorem 3.2.37. Let μ ∈ ℳv , 𝒢 (g, α, β) be a frame for L2 (ℝd ), with lattice Λ = αℤd × βℤd , and g ∈ 𝒮 (ℝd ). (i) For every 0 < p, q ≤ ∞, Cg : Mμp,q → ℓμp,q and Dg : ℓμp,q → Mμp,q continuously. If f ∈ Mμp,q (ℝd ), then the Gabor expansions (3.29) converge unconditionally in Mμp,q for 0 < p, q < ∞, and weak∗ -Mμ∞ unconditionally if p = ∞ or q = ∞.

(ii) The following (quasi-)norms are equivalent on Mμp,q (ℝd ): ‖f ‖Mμp,q ≍ ‖Cg f ‖ℓμp,q .

(3.33)

We end up this section by recalling an important result about the invertibility of the Gabor frame operator on Mv1 (ℝd ), with v being a submultiplicative weight. For generalizations and the proof of the result below, we refer to [159]; see also [160].

3.3 Kernel theorems for modulation spaces | 181

Theorem 3.2.38. Assume that m ∈ ℳv . Consider a Gabor frame 𝒢 (g, α, β) with lattice Λ = αℤd × βℤd satisfying αβ ∈ ℚ and window g ∈ Mv1 . Then the Gabor frame operator α,β α,β Sg,g is invertible on Mv1 (ℝd ). As a consequence, Sg,g is invertible on all modulation spaces α,β

p,q Mm (ℝd ), 1 ≤ p, q ≤ ∞, and the canonical dual window γ = (Sg,g )−1 g belongs to Mv1 (ℝd ).

It is noteworthy to mention the characterization of modulation spaces via Wilson bases in [36, 54, 103, 135, 315].

3.3 Kernel theorems for modulation spaces Schwartz’ kernel theorem certainly is one of the most important results in modern functional analysis. It states, in the framework of tempered distributions, that every linear continuous operator A : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) can be regarded as an integral operator in a generalized sense, namely ⟨Af , g⟩ = ⟨K, g ⊗ f ⟩,

f , g ∈ 𝒮 (ℝd ),

in the context of distributions, for some kernel K ∈ 𝒮 󸀠 (ℝ2d ), and vice versa [194]. We write, formally, Af (x) = ∫ K(x, y)f (y) dy,

f ∈ 𝒮 (ℝd ).

(3.34)

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ℝd

This result provides a general framework that includes most of the operators of interest in harmonic analysis. However, for many applications in time–frequency analysis and mathematical signal processing, the spaces 𝒮 (ℝd ) and 𝒮 󸀠 (ℝd ) can be fruitfully replaced by the Feichtinger’s algebra M 1 (ℝd ) and its dual M ∞ (ℝd ). One of the obvious advantages is that M 1 (ℝd ) and M ∞ (ℝd ) are Banach spaces; on the other hand, their norm gives a direct and transparent information on the time–frequency concentration of a function or tempered distribution. The same holds, more generally, for the whole scale of intermediate spaces M p,q (ℝd ), 1 ≤ p, q ≤ ∞. A kernel theorem in the framework of modulation spaces was announced by H. Feichtinger in [119] and proved in [134]; see also [160, Theorem 14.4.1]. It states that linear continuous operators A : M 1 (ℝd ) → M ∞ (ℝd ) are characterized by the membership of their distributional kernel K to M ∞ (ℝ2d ). Theorem 3.3.1. (i) Every distribution K ∈ M ∞ (ℝ2d ) defines a bounded linear operator A : M 1 (ℝd ) → M ∞ (ℝd ) according to ⟨Af , g⟩ = ⟨K, g ⊗ f ⟩, with ‖A‖M 1 →M ∞ ≤ ‖K‖M ∞ .

∀f , g ∈ M 1 (ℝd ),

182 | 3 Gabor frames and linear operators (ii) For any bounded operator A : M 1 (ℝd ) → M ∞ (ℝd ) there exists a unique kernel K ∈ M ∞ (ℝ2d ) such that ⟨Af , g⟩ = ⟨K, g ⊗ f ⟩,

∀f , g ∈ M 1 (ℝd ).

In this section we provide a similar characterization of linear continuous operators acting on the following spaces: – M 1 (ℝd ) → M p (ℝd ), for a fixed 1 ≤ p ≤ ∞; – M p (ℝd ) → M ∞ (ℝd ), for a fixed 1 ≤ p ≤ ∞; – M p (ℝd ) → M p (ℝd ), for every 1 ≤ p ≤ ∞. In all cases the characterization is given in terms of the membership of their distributional kernels to certain mixed modulation spaces. In this case we shall use their wider definition contained in (2.3.3). These results are contained in [76]. From now on we assume that c is a permutation of the set {1, . . . , 2d}. We identify c with the linear bijection c̃ defined in (2.21). From now on we shall work with the quadratic lattice Λ = αℤ2d , for suitable α > 0 and use the time–frequency shift π(λ) = Mλ1 Tλ2 ,

λ = (λ1 , λ2 ) ∈ Λ.

If the window function belongs to the Feichtinger’s algebra M 1 (ℝd ), then a Gabor frame for L2 (ℝd ) is a frame also for mixed modulation spaces. The arguments are similar to the classical case contained in Theorem 3.2.35 (cf. [19, Theorem 4.6]). Theorem 3.3.2. Let g be a window function in M 1 (ℝd ) such that 𝒢 (g, Λ) is a frame for L2 (ℝd ) with a dual frame 𝒢 (γ, Λ). Consider p1 , . . . , p2d ∈ [1, ∞]. (i) There exist constants 0 < A ≤ B independent of p1 , . . . , p2d such that

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A‖f ‖M(c)p1 ,...,p2d ≤ ‖Vg f ∘ c|̃ Λ ‖ℓp1 ,...,p2d ≤ B‖f ‖M(c)p1 ,...,p2d ,

∀f ∈ M(c)p1 ,...,p2d .

(ii) We have f = ∑ ⟨f , π(λ)g⟩π(λ)γ = ∑ ⟨f , π(λ)γ⟩π(λ)g, λ∈Λ

λ∈Λ

(3.35)

with unconditional convergence in M(c)p1 ,...,p2d if p1 , . . . , p2d ∈ [1, ∞), and weak* convergence in M ∞ (ℝd ) if p1 , . . . , p2d ∈ [1, ∞] Indeed, this generalizes the basic fact, valid for the identity permutation, that the coefficient operator Cg and the reconstruction operator Dγ , here represented by (Cg f )λ = ⟨f , π(λ)g⟩,

Dγ a = ∑ aλ π(λ)γ, λ∈Λ

3.3 Kernel theorems for modulation spaces | 183

satisfy Cg : M p,q (ℝd ) → ℓp,q (Λ),

Dγ : ℓp,q (Λ) → M p,q (ℝd ),

(3.36)

with Dγ Cg = I, the identity on M p,q (ℝd ), cf. (3.35) (similarly, Dg Cγ = I). Concerning Theorem 3.3.2, we remark that even when pk = ∞ for some or several indices k ∈ {1, . . . , 2d} one still has unconditional weak* convergence. Definition 3.3.3. Let g be a window function in 𝒮 (ℝd ) such that 𝒢 (g, Λ) is a frame for L2 (ℝd ) with a dual frame 𝒢 (γ, Λ), γ ∈ 𝒮 (ℝd ). Given a linear continuous operator A : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ), we define the Gabor matrix related to A by Kλ,μ = ⟨Aπ(μ)γ, π(λ)g⟩,

λ, μ ∈ Λ.

(3.37)

For heuristic purposes at least, one can view Kλ,μ as an infinite matrix, with indices λ, μ ∈ Λ. Properties of the operator A can be inferred from the related Gabor matrix Kλ,μ , as we shall show in the sequel. Notice that this matrix turns out to be the kernel of the operator à := Cg ADγ , acting on sequences on Λ, where Cg and Dγ are the coefficient and synthesis operator, respectively. First, we need some preliminaries coming from the general theory of operators, see, for instance, Tao’s lecture notes [278, Proposition 5.2]. Proposition 3.3.4. Assume that (X, νX ), (Y, μY ) are measure spaces and p ∈ [1, ∞]. Suppose that K : Y × X → ℂ, and ‖K(⋅, x)‖Lp (Y) is uniformly bounded on X. Then the operator A with kernel K maps L1 (X) into Lp (Y) and 󵄩 󵄩 ‖A‖L1 (X)→Lp (Y) = ess supx∈X 󵄩󵄩󵄩K(⋅, x)󵄩󵄩󵄩Lp (Y) .

(3.38)

Proof. For any f ∈ L1 (X), Minkowski’s inequality gives 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄨󵄨 󵄨 󵄩 󵄩󵄨 󵄨󵄨 󵄨󵄩 ≤ ∫󵄩󵄩󵄩󵄨󵄨󵄨K(⋅, x)󵄨󵄨󵄨 󵄨󵄨󵄨f (x)󵄨󵄨󵄨󵄩󵄩󵄩Lp (Y) dνX 󵄩󵄩∫󵄨󵄨K(⋅, x)f (x)󵄨󵄨󵄨 dνX 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩Lp (Y) X

X

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󵄨 󵄨 󵄩 󵄩 ≤ ∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 ess supx∈X 󵄩󵄩󵄩K(⋅, x)󵄩󵄩󵄩Lp (Y) dνX X

󵄩 󵄩 = ‖f ‖L1 (X) ess supx∈X 󵄩󵄩󵄩K(⋅, x)󵄩󵄩󵄩Lp (Y) < ∞, so that we infer ‖A‖L1 (X)→Lp (Y) ≤ ess supx∈X ‖K(⋅, x)‖Lp (Y) . To get the converse inequality, we observe from limiting arguments that we may take the kernel K to be a product of simple functions, i. e., a linear combination of tensor products of indicator functions on X and Y, separately. In such a case, we can perform a partition of X into finitely many sets of positive measure, on each of which K is independent of x up to sets of measure zero. On one of these sets, call it B, the supremum of ‖K(⋅, x)‖Lp (Y) is attained. If we then test A against χB󸀠 , for some B󸀠 ⊂ B of finite measure, we obtain the claim. Similarly, we have a dual statement (see, e. g., [278, Proposition 5.4]).

184 | 3 Gabor frames and linear operators Proposition 3.3.5. Assume that (X, νX ), (Y, μY ) are measure spaces and p ∈ [1, ∞]. Suppose that K : Y ×X → ℂ, and ‖K(⋅, x)‖Lp󸀠 (Y) is uniformly bounded on X. Then the operator A with kernel K maps Lp (X) into L∞ (Y) and 1 1 + 󸀠 = 1. p p

󵄩 󵄩 ‖A‖Lp (X)→L∞ (Y) = ess supx∈X 󵄩󵄩󵄩K(⋅, x)󵄩󵄩󵄩Lp󸀠 (Y) ,

(3.39)

We shall need the following Schur-type test. Proposition 3.3.6. Consider an operator defined on sequences on the lattice Λ = αℤd × βℤd by (Kc)m󸀠 ,n󸀠 = ∑ Km󸀠 ,n󸀠 ,m,n cm,n . m,n

∞ 1 1 ∞ (i) If K ∈ ℓn∞ ℓn1 󸀠 ℓm 󸀠 ℓm , K is continuous on ℓn ℓm .

∞ 1 1 (ii) If K ∈ ℓn∞󸀠 ℓn1 ℓm ℓm󸀠 , K is continuous on ℓn∞ ℓm .

∞ 1 ∞ 1 ∞ 1 ∞ 1 ∞ 1 (iii) If K ∈ ℓn∞ ℓn1 󸀠 ℓm 󸀠 ℓm ∩ℓn󸀠 ℓn ℓm ℓm󸀠 , and moreover K ∈ ℓm󸀠 ,n󸀠 ℓm,n ∩ℓm,n ℓm󸀠 ,n󸀠 , the operator p K is continuous on ℓp,q = ℓnq ℓm for every 1 ≤ p, q ≤ ∞. ̃ , 1 ≤ p, q ≤ ∞, where (iv) Assume the hypotheses in (iii). Then K is continuous on all ℓp,q p,q ℓ̃ is the closure of the space of eventually zero sequences in ℓp,q .

Proof. (i) We have ‖Kc‖ℓ1 󸀠 ℓ∞󸀠 ≤ ∑ sup ∑ |Km󸀠 ,n󸀠 ,m,n ||cm,n | n

m

n󸀠

m󸀠 m,n

≤ ∑(∑ sup ∑ |Km󸀠 ,n󸀠 ,m,n |) sup |cm,n | n

m󸀠

n󸀠

m

m

≤ ‖K‖ℓn∞ ℓ1 󸀠 ℓ∞󸀠 ℓm1 ‖c‖ℓ1 ℓ∞ . n

m

(ii) It turns out that ‖Kc‖ℓ∞󸀠 ℓ1 󸀠 ≤ sup ∑(∑ |Km󸀠 ,n󸀠 ,m,n ||cm,n |) Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

n

m

n󸀠

m󸀠

m,n

≤ sup ∑(sup ∑ |Km󸀠 ,n󸀠 ,m,n | ∑ |cm,n |) n󸀠

n

m

m󸀠

m

≤ ‖K‖ℓ∞󸀠 ℓn1 ℓm∞ ℓ1 󸀠 ‖c‖ℓ∞ ℓ1 . n

m

(iii) Since the statement holds for p = q by the classical Schur’s test, and for (p, q) = (1, ∞) and (p, q) = (∞, 1), by parts (i) and (ii), it follows by complex interpolation (see (3) on page 128 and (15) on page 134 of [296]) that it holds for all (p, q), except possibly in the cases q = ∞, 1 < p < ∞. For these cases we argue by duality as follows. In order to prove the continuity of K on ℓ∞ ℓp , it suffices to verify that for any sep󸀠 p quences c = (cm,n ) ∈ ℓn∞ ℓm and d = (dm󸀠 ,n󸀠 ) ∈ ℓn1 󸀠 lm 󸀠 , with dm󸀠 ,n󸀠 ≥ 0, we have 󵄨 󵄨 ∑ 󵄨󵄨󵄨(Kc)m󸀠 ,n󸀠 󵄨󵄨󵄨dm󸀠 ,n󸀠 ≲ ‖c‖ℓn∞ ℓmp ‖d‖ 1 p󸀠 . ℓn ℓm

m󸀠 ,n󸀠

(3.40)

3.3 Kernel theorems for modulation spaces | 185

Now 󵄨 󵄨 ∑ 󵄨󵄨󵄨(Kc)m󸀠 ,n󸀠 󵄨󵄨󵄨dm󸀠 ,n󸀠 ≤ ∑ ∑ |Km󸀠 ,n󸀠 ,m,n ||cm,n |dm󸀠 ,n󸀠 m󸀠 ,n󸀠 m,n

m󸀠 ,n󸀠

= ∑ ( ∑ |Km󸀠 ,n󸀠 ,m,n |dm󸀠 ,n󸀠 )|cm,n | m,n m󸀠 ,n󸀠

̃ 1 p󸀠 , ≤ ‖c‖ℓ∞ ℓp ‖Kd‖ ℓℓ where K̃ is the operator with matrix kernel K̃ m,n,m󸀠 ,n󸀠 = |Km󸀠 ,n󸀠 ,m,n |. Since it satisfies the

same assumptions as K, it is continuous on ℓ1 ℓp , which gives (3.40). ̃ , it suffices to verify (iv) Since K is continuous on ℓp,q and by the definition of ℓp,q ̃ . This follows from the fact that K that K maps every eventually zero sequence in ℓp,q ̃ , because K is bounded on ℓ1 . maps every eventually zero sequence in ℓ1 �→ ℓp,q 󸀠

Let us introduce the permutation c1 of the set {1, . . . , 4d} with related c̃1 given by, for xi ∈ ℝd , i = 1, . . . , 4,

c̃1 (x1 , x2 , x3 , x4 ) = (

Id

0d

0d

0d

0d

Id

0d

0d

0d

0d

0d 0d

Id

0d

0d Id

x1

x2 ) ( ) = (x1 , x3 , x2 , x4 ). x3

(3.41)

x4

We consider again lattices of the form Λ = αℤ2d . Theorem 3.3.7. Suppose p ∈ [1, ∞]. Let g be a window function in 𝒮 (ℝd ) such that 𝒢 (g, Λ) is a frame for L2 (ℝd ) with a dual frame 𝒢 (γ, Λ). A linear continuous operator A : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) is bounded M 1 (ℝd ) → M p (ℝd ) if and only if its distributional kernel K satisfies

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VG K ∘ c1|̃ Λ×Λ ∈ ℓp,∞ (Λ × Λ),

(3.42)

where G = g ⊗ γ.̄ Moreover, ‖A‖M 1 →M p ≍ ‖VG K ∘ c1̃ ‖ℓp,∞ (Λ×Λ) . Proof. Assume first that (3.42) holds. For λ = (λ1 , λ2 ), μ = (μ1 , μ2 ) ∈ Λ, observe that the Gabor matrix of the operator A in (3.37) is given by Kλ,μ = ⟨Aπ(μ)γ, π(λ)g⟩ ̄ = ⟨K, π(λ1 , λ2 )g ⊗ π(μ1 , −μ2 )γ⟩ = VG K ∘ c̃1 (λ1 , λ2 , μ1 , −μ2 ).

(3.43)

186 | 3 Gabor frames and linear operators The operator à := Cg ADγ with kernel Kλ,μ acts on sequences on Λ. By (3.42) and Proposition 3.3.4, à is a bounded operator ℓ1 (Λ) → ℓp (Λ). Since Cg : M p (ℝd ) → ℓp (Λ) and ̃ g : M 1 (ℝd ) → M p (ℝd ), with Dγ : ℓp (Λ) → M p (ℝd ) are continuous, we have A = Dγ AC ‖A‖M 1 →M p ≤ C sup ‖K⋅,μ ‖ℓp (Λ) . μ∈Λ

Conversely, consider the finite dimensional space EN = {(aλ )λ∈Λ : aλ = 0

for |λ| > N}

and the natural projection PN : ℓ∞ (Λ) → EN . The composition operator PN Cg ADγ |EN : (EN , ‖ ⋅ ‖ℓ1 ) → (EN , ‖ ⋅ ‖ℓp ) is represented by the (finite) matrix {Kλ,μ }λ,μ∈Λ,|λ|≤N,|μ|≤N . Using formula (3.38) in Proposition 3.3.4 (we work on finite sets, so that the assumptions are trivially satisfied), we have sup

μ∈Λ,|μ|≤N

‖Kλ,μ ‖ℓp (Λ,|λ|≤N) = ‖PN Cg ADγ |EN ‖ℓ1 →ℓp ≤ ‖Cg ‖‖Dγ ‖‖A‖M 1 →M p .

Since the right-hand side is independent of N, taking the supremum with respect to N, we obtain sup ‖K⋅,μ ‖ℓp (Λ) ≤ C‖A‖M 1 →M p . μ∈Λ

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This concludes the proof. Observe that, given a window function γ such that the Gabor system 𝒢 (γ, Λ) is frame for L2 (ℝd ), and replacing γ by its complex conjugate γ,̄ the Gabor system 𝒢 (γ,̄ Λ) is still a frame. We choose now a frame 𝒢 (g, Λ) for L2 (ℝd ), with dual frame 𝒢 (γ, Λ). Then 𝒢 (g ⊗ γ,̄ Λ × Λ) is a frame for L2 (ℝ2d ). Using Theorem 3.3.2, we reformulate the previous result in terms of mixed modulation spaces. Corollary 3.3.8. Suppose p ∈ [1, ∞]. A linear continuous operator A : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) is bounded from M 1 (ℝd ) into M p (ℝd ) if and only if its distributional kernel K satisfies K ∈ M(c1 )p,∞ , where c1 is identified with (3.41). Moreover ‖A‖M 1 →M p ≍ ‖K‖M(c1 )p,∞ .

(3.44)

3.3 Kernel theorems for modulation spaces | 187

Remark 3.3.9. When p = ∞, condition (3.44) becomes K ∈ M ∞ (ℝ2d ), and we recapture Feichtinger’s kernel Theorem (Theorem 3.3.1). A similar characterization of bounded operators A : M p (ℝd ) → M ∞ (ℝd ) can be obtained arguing as above, using Proposition 3.3.5, (3.43) and Theorem 3.3.2. Precisely, define the permutation c2 of the set {1, . . . , 4d} with related c̃2 by 0d

0d

Id c̃2 (x1 , x2 , x3 , x4 ) = ( 0d

0d

0d

Id

0d

0d

0d

Id

0d

x1

0d

x2 ) ( ) = (x3 , x1 , x4 , x2 ), x3 Id

0d

(3.45)

x4

0d

for xi ∈ ℝd , i = 1, . . . , 4. Theorem 3.3.10. Suppose p ∈ [1, ∞]. A linear continuous operator A : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) is bounded from M p (ℝd ) into M ∞ (ℝd ) if and only if its distributional kernel K satisfies K ∈ M(c2 )p ,∞ , 󸀠

where 1/p + 1/p󸀠 = 1. Moreover, ‖A‖M p →M ∞ ≍ ‖K‖M(c )p󸀠 ,∞ . 2

As a consequence of Theorem 3.3.7 and Corollary 3.3.8, we have the following result.

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Corollary 3.3.11. Let A : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) be an operator with distributional kernel K ∈ 𝒮 󸀠 (ℝ2d ). The following conditions are equivalent: (i) A : M p (ℝd ) → M p (ℝd ), bounded for every 1 ≤ p ≤ ∞. (ii) A : M 1 (ℝd ) → M 1 (ℝd ) and A : M ∞ (ℝd ) → M ∞ (ℝd ) bounded. (iii) K ∈ M(c1 )1,∞ ∩ M(c2 )1,∞ . Proof. All the equivalences follow trivially from Theorems 3.3.7 and 3.3.10 but (ii) �⇒ (i), which follows by complex interpolation for modulation spaces, see Proposition 2.3.17. 3.3.1 Operators acting on Mp,q Let us now consider the problem of boundedness of an integral operator on M p,q (ℝd ), 1 ≤ p, q ≤ ∞. We no longer expect a characterization as in the case p = q, studied in the previous section (see Remark 3.3.13 below). Nevertheless, sufficient conditions can be inferred.

188 | 3 Gabor frames and linear operators

by

Consider the permutations c3 and c4 of the set {1, . . . , 4d} with related c̃3 , c̃4 defined 0d 0 c̃3 (x1 , x2 , x3 , x4 ) = ( d Id 0d

Id 0d 0d 0d

0d Id 0d 0d

0d x1 0d x2 ) ( ) = (x2 , x3 , x1 , x4 ) 0d x3 Id x4

(3.46)

Id 0 c̃4 (x1 , x2 , x3 , x4 ) = ( d 0d 0d

0d 0d Id 0d

0d 0d 0d Id

x1 0d Id x ) ( 2 ) = (x1 , x4 , x2 , x3 ), 0d x3 0d x4

(3.47)

and

for xi ∈ ℝd , i = 1, . . . , 4. Proposition 3.3.6 can be rephrased in this framework as follows. Proposition 3.3.12. Consider an at most countable index set J and the operator defined on sequences (aλ1 ,λ2 ) on the set Λ = J × J by (Aa)λ1 ,λ2 = ∑ Kλ1 ,λ2 ,μ1 ,μ2 aμ1 ,μ2 . μ1 ,μ2 ∈J

(i) If K ∘ c3̃ ∈ ℓ1,∞,1,∞ (J 4 ), then A is continuous on ℓ∞,1 (Λ). Moreover, ‖A‖ℓ∞,1 →ℓ∞,1 ≤ ‖K ∘ c3̃ ‖ℓ1,∞,1,∞ .

(3.48)

(ii) If K ∘ c4̃ ∈ ℓ1,∞,1,∞ (J 4 ), then A is continuous on ℓ1,∞ (Λ), with

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‖A‖ℓ1,∞ →ℓ1,∞ ≤ ‖K ∘ c4̃ ‖ℓ1,∞,1,∞ .

(3.49)

(iii) If K satisfies assumptions (i) and (ii) and, moreover, K ∘ c0 , K ∈ ℓ1,∞ (Λ × Λ), where the permutation c0 is defined in (2.23), then the operator A is continuous on ℓp,q (Λ), for every 1 ≤ p, q ≤ ∞. Remark 3.3.13. Observe that the reverse inequalities in (3.48), (3.49) do not hold, even if one allows a multiplicative constant. Consider, for example, the estimate (3.48), with J = {0, 1, . . . , N − 1}, N ∈ ℕ, where Kλ1 ,λ2 ,μ1 ,μ2 = Fλ2 ,μ1 =

1 −2πiλ2 μ1 e √N

is the Fourier matrix. We have (K ∘ c3̃ )μ1 ,λ1 ,λ2 ,μ2 = Kλ1 ,λ2 ,μ1 ,μ2

3.3 Kernel theorems for modulation spaces | 189

so that ‖K ∘ c3̃ ‖ℓ1,∞,1,∞ = ‖F‖ℓ1 = N 3/2 but ‖A‖ℓ∞,1 →ℓ∞,1 ≤ N,

(3.50)

which blows up, as N → +∞, at a lower rate. Let us verify (3.50). We have 󵄩󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩󵄩󵄩 ∑ Kλ1 ,λ2 ,μ1 ,μ2 aμ1 ,μ2 󵄩󵄩󵄩 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ Fλ2 ,μ1 ∑ aμ1 ,μ2 󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩󵄩ℓ∞ 󵄩󵄩ℓ1 󵄩󵄩 󵄩󵄩ℓ1 μ1 ,μ2 ∈J μ ∈J μ ∈J λ1 λ2 λ2 1 2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 = 󵄩󵄩 ∑ Fλ2 ,μ1 ã μ1 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩ℓ1 μ1 ∈J λ2 with ã μ1 := ∑μ2 ∈J aμ1 ,μ2 . Now, using the embeddings ‖b‖ℓ1 (J) ≤ √N‖b‖ℓ2 (J) ,

‖b‖ℓ2 (J) ≤ √N‖b‖ℓ∞ (J)

and the fact that F is a unitary transformation of ℓ2 (J), we get (3.50).

by

This Schur-type test yields the following boundedness result. Consider the permutations c5 and c6 of the set {1, . . . , 4d} with related c̃5 , c̃6 defined 0d I c̃5 = c̃1 c̃3 = ( d 0d 0d

Id 0d 0d 0d

0d 0d Id 0d

0d 0d ) 0d Id

(3.51)

Id 0d c̃6 = c̃1 c̃4 = ( 0d 0d

0d Id 0d 0d

0d 0d 0d Id

0d 0d ). Id 0d

(3.52)

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and

Theorem 3.3.14. (i) A linear continuous operator A : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) is bounded on M ∞,1 (ℝd ) if its distributional kernel K satisfies K ∈ M(c5 )1,∞,1,∞ . (ii) A linear continuous operator A : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) is bounded on M 1,∞ (ℝd ) if its distributional kernel K satisfies K ∈ M(c6 )1,∞,1,∞ .

190 | 3 Gabor frames and linear operators Proof. The result is a consequence of (3.43), Proposition 3.3.12 and Theorem 3.3.2, by using the same arguments as in the first part of the proof of Theorem 3.3.7. By interpolation we also obtain the following result. Corollary 3.3.15. Let A : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) be an operator with distributional kernel K ∈ M(c5 )1,∞,1,∞ ∩ M(c6 )1,∞,1,∞ . Then A : M p ,p (ℝd ) → M p ,p (ℝd ) continuously, for every 1 ≤ p ≤ ∞, with 1/p + 1/p󸀠 = 1. 󸀠

󸀠

By combining Corollaries 3.3.11 and 3.3.15, we finally obtain the following result. Corollary 3.3.16. Let A : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) be an operator with distributional kernel K ∈ M(c1 )1,∞ ∩ M(c2 )1,∞ ∩ M(c5 )1,∞,1,∞ ∩ M(c6 )1,∞,1,∞ .

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Then A : M p,q (ℝd ) → M p,q (ℝd ) continuously, for every 1 ≤ p, q ≤ ∞.

4 Pseudodifferential operators Let us first introduce pseudodifferential operators in the form arising as a generalization of the variable-coefficient differential operators ∑ cα (x)𝜕α , |α|≤N

for some N ∈ ℕ+ , where we recall that α = (α1 , . . . , αd ) is a multiindex in ℕd with length |α| = α1 + ⋅ ⋅ ⋅ + αd ≤ N. The coefficients cα (x) belong to suitable function spaces. We start with the Fourier inversion formula for a suitable function f , say, in 𝒮 (ℝd ): f (x) = ∫ e2πix⋅ξ f ̂(ξ ) dξ ℝd

and apply the monomial differential operator 𝜕xα 𝜕xα f (x) = ∫ (2πiξ )α e2πix⋅ξ f ̂(ξ ) dξ , ℝd α

α

where we recall the notation ξ α = ξ1 1 ⋅ ⋅ ⋅ ξd d , for any ξ = (ξ1 , . . . , ξd ) ∈ ℝd . Thus we can write ∑ cα (x)𝜕α f (x) = ∫ a(x, ξ )e2πix⋅ξ f ̂(ξ ) dξ , |α|≤N

ℝd

where the function a(x, ξ ) = ∑ cα (x)(2πiξ )α |α|≤N

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is called the symbol of the previous operator. Inspired by this, we give the following definition. Definition 4.0.1 (Kohn–Nirenberg operator). Given a function or distribution a in ℝ2d , let us first assume a ∈ 𝒮 (ℝ2d ) and define the pseudodifferential operator a(x, D), acting initially on 𝒮 (ℝd ), as follows: a(x, D)f (x) = ∫ e2πix⋅ξ a(x, ξ )f ̂(ξ ) dξ ,

f ∈ 𝒮 (ℝd ).

ℝd

This kind of pseudodifferential operator is called Kohn–Nirenberg operator with Kohn–Nirenberg symbol a. We are going to exhibit the connection between a pseudodifferential operator and its related time–frequency distribution. First, recall the Rihaczek distribution defined in (1.118): ̂ ), W0 (f , g)(x, ξ ) = R(f , g)(x, ξ ) = e−2πix⋅ξ f (x)g(ξ https://doi.org/10.1515/9783110532456-005

f , g ∈ 𝒮 (ℝd ).

192 | 4 Pseudodifferential operators Then it is straightforward to see that ⟨a, R(f , g)⟩ = ⟨a(x, D)f , g⟩,

(4.1)

for a symbol a ∈ 𝒮 (ℝ2d ) and functions f , g ∈ 𝒮 (ℝd ), where the brackets ⟨⋅, ⋅⟩ denote possibly the extension to 𝒮 󸀠 × 𝒮 of the inner product on L2 . The proof is just a matter of changing variables and interchanging the integration order. Recall that if f , g ∈ 𝒮 (ℝd ), the Rihaczek distribution R(f , g) is in 𝒮 (ℝ2d ) (cf. Corollary 1.3.26), this implies that the left-hand side in (4.1) is well defined for any a ∈ 𝒮 󸀠 (ℝ2d ) and therefore a(x, D)f ∈ 𝒮 󸀠 (ℝd ), that is, a(x, D) is a linear operator mapping 𝒮 (ℝd ) into 𝒮 󸀠 (ℝd ). This shows that we can always study the action of a pseudodifferential operator by means of its symbol and a related time–frequency representation. We shall see in the sequel that the time–frequency distributions studied in Chapter 1 give rise to other particular types of pseudodifferential operators. Remark 4.0.2. Formally, if we apply a(x, D) to a plane wave f (x) := e2πix⋅η , we see that a(x, D)f (x) = a(x, η)f (x). Thus we can (in principle) reconstruct the symbol a from the operator a(x, D) by testing it against plane waves. If the symbol depends only on x, that is, a(x, ξ ) = a(x), then a(x, D)f (x) = ∫ e2πix⋅ξ a(x)f ̂(ξ ) dξ = a(x)f (x),

f ∈ 𝒮 (ℝd ),

ℝd

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that is a multiplication operator. In particular, if a(x, ξ ) ≡ C, for a suitable complex constant, then a(x, D) = CI, a multiple of the identity operator. If the symbol depends only on ξ , that is, a(x, ξ ) = a(ξ ), then a(x, D)f (x) = ∫ e2πix⋅ξ a(ξ )f ̂(ξ ) dξ = ℱ −1 (af ̂)(ξ ),

f ∈ 𝒮 (ℝd ),

ℝd

which is a Fourier multiplier. In other words, it can be regarded as a convolution operator, a(x, D)f = h ∗ f , with h = ℱ −1 a. Let us now write a pseudodifferential operator as an integral operator with kernel K. We first introduce the linear change of coordinates for measurable functions F on ℝ2d : Tb F(x, y) = F(x, y − x)

x, y ∈ ℝd .

(4.2)

Let ℱ2 be the partial Fourier transform with respect to the second variable, already defined in (1.42); then for f ∈ 𝒮 (ℝd ) and a ∈ 𝒮 (ℝ2d ),

4 Pseudodifferential operators | 193

a(x, D)f (x) = ∫ e2πix⋅ξ a(x, ξ )f ̂(ξ ) dξ ℝd

= ∫ ( ∫ e−2πi(y−x)⋅ξ a(x, ξ )dξ )f (y) dy ℝd ℝd

= ∫ K(x, y)f (y) dy ℝd

where we used f ̂(ξ ) = ∫ℝd f (y)e−2πiy⋅ξ dy and interchanged the order of integration. Hence the Kohn–Nirenberg operator a(x, D) is an integral operator with kernel K(x, y) = ℱ2 a(x, y − x) = Tb ℱ2 a(x, y).

(4.3)

Notice that Tb and ℱ2 are topological isomorphisms on 𝒮 (ℝ2d ), 𝒮 󸀠 (ℝ2d ), and L (ℝ2d ), as well as on many other function spaces (see in the sequel). The use of the integral expression of a(x, D), as an alternative to (4.1), depends on the problem we are dealing with. Let us give a short historical survey, with particular reference to boundedness problems. Pseudodifferential operators were first introduced (in the so-called Weyl form) in the framework of quantum mechanics. In fact, in 1931 this form was introduced by Hermann Weyl [309] as a quatization rule, that is, roughly speaking, a mapping which to any observable a(x, ξ ) on the phase space ℝ2d assigns an operator Opw (a) on a Hilbert space, see the next Definition 4.0.3. Besides quantum mechanics, they arise in the framework of the partial differential equations (PDEs) and engineering. In the context of PDEs they were introduced independently in [193] and [213]. Since then many symbol classes have been considered, according to several applications to PDEs. In particular, a deep analysis has been m carried on for Hörmander’s classes Sρ,δ , m ∈ ℝ, 0 ≤ δ ≤ ρ ≤ 1, of smooth functions 2

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β

a(x, ξ ) satisfying the estimates |𝜕xα 𝜕ξ a(x, ξ )| ≤ Cα,β (1 + |ξ |)m+δ|α|−ρ|β| . Boundedness results on Lp -based Sobolev spaces for the corresponding pseudodifferential operators are of special interest because they imply regularity results for the solutions of the related PDEs. The basic result is the boundedness on L2 of operators in the above classes, with δ < ρ, m = 0, which can be achieved by means of the symbolic calculus. In fact, L2 -boundedness still holds for 0 ≤ δ = ρ < 1, which is the classical Calderón– Vaillancourt theorem [40]. It is actually sufficient to suppose that the estimates on a are valid for a bounded number of derivatives, in particular, for ρ = δ = 0, the assumption a ∈ 𝒞b2d+1 (ℝ2d ) grants L2 -continuity. 0 Boundedness on Lp , 1 < p < ∞, holds for symbols in S1,δ , 0 ≤ δ < 1, but generally fails for ρ < 1 and a loss of derivatives may then occur. We have no time to enter this topic and refer the interested reader to the works [172, 192, 197] and textbooks [193, 265].

194 | 4 Pseudodifferential operators There are also many results for symbols which are smooth and behave as usual with respect to ξ , but less regular with respect to x, e. g., just belonging to some Hölder class. In this connection see the books [280, 281], where important applications to nonlinear equations are presented as well. Hence, the smoothness of the symbol or the boundedness of all derivatives of the symbol is not necessary for the boundedness of pseudodifferential operators on L2 (ℝd ). Being motivated by this argument, many authors (see, e. g., [55, 93, 207, 231]) have investigated the minimal assumption on the regularity of symbols for the corresponding operators to be bounded on L2 . In particular, Sugimoto [273] showed that symbols in the Besov space B(∞,∞),(1,1) imply L2 -boundedness (see also [272] and the d/2,d/2 references therein for extensions to the Lp framework). In 1994–1995 Sjöstrand intro0 duced a new symbol class, larger than S0,0 , which was then recognized to be the modu∞,1 d lation space M (ℝ ), first introduced in time–frequency analysis by Feichtinger [122, 120, 126]. In [266, 267] Sjöstrand proved that symbols in M ∞,1 give rise to L2 -bounded operators. In view of the inclusion 𝒞b2d+1 ⊂ M ∞,1 , this result represented an important generalization of the classical Calderón–Vaillancourt theorem. Since then several extensions appeared. In particular, in [160, 163], symbols in M ∞,1 were proved to produce bounded operators on all M p,q . A nice survey of results on the boundedness and the Schatten-class property of pseudodifferential operators using a time–frequency approach is for sure the chapter by Gröchenig [161]. Soon after Gröchenig and Strohmer [168] introduced the M ∞,1 class of symbols of Sjöstrand type for pseudodifferential operators on a locally compact Abelian (LCA) group 𝒢 and proved that such operators are continuous on L2 (𝒢 ), extending the ℝd case to LCA groups. Further refinements appeared in [162, 216, 286]. The case of weighted modulation spaces and Schatten–von Neumann properties were treated in [180, 287, 288]. Here we present more general results for boundedness on Lp spaces that were shown in [74]. The pseudodifferential operators considered there are in the Weyl form.

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Definition 4.0.3 (Weyl operators). For a ∈ 𝒮 (ℝ2d ), we define the Weyl operator Opw (a) with Weyl symbol a as Opw (a)f (x) = ∫ e2πi(x−y)⋅ξ a( ℝ2d

x+y , ξ )f (y) dy dξ , 2

f ∈ 𝒮 (ℝd ).

(4.4)

Such an operator is also known as Weyl transform [317]. For f ∈ 𝒮 (ℝd ) and a ∈ 𝒮 (ℝ ) (or in other suitable function spaces), 2d

Opw (a)f (x) = ∫ e2πi(x−y)⋅ξ a( ℝ2d

x+y , ξ )f (y) dy dξ 2

= ∫ ( ∫ e−2πi(y−x)⋅ξ a( ℝd ℝd

= ∫ K(x, y)f (y) dy ℝd

x+y , ξ )dξ )f (y) dy 2

4 Pseudodifferential operators | 195

where −1 K = T−1 s ℱ2 a

(4.5)

2d 2 2d 󸀠 2d and T−1 s is the isomorphism on 𝒮 (ℝ ) (or L (ℝ ), 𝒮 (ℝ ) or other suitable space), defined in (1.88). Hence a Weyl operator is an integral operator with kernel K in (4.5). An easy consequence of a change of variables and an interchange of integration order in the integrals gives

⟨Opw (a)f , g⟩ = ⟨a, W(g, f )⟩,

f , g ∈ 𝒮 (ℝd ).

(4.6)

Consequently, the time–frequency representation related to a Weyl operator is the Wigner distribution (cf. (1.79)). Observe that from (4.3) and (4.5) we infer the equality a(x, D) = Opw (σ), where the relations between σ, a and the kernel K of a(x, D) are given by σ = ℱ2 Ts K,

a = 𝒰 σ,

σ = 𝒰 −1 a,

(4.7)

with ℱ2 being the partial Fourier transform in the second variable, Ts the symmetric coordinate transformation defined in (1.87), and 𝒰 defined by

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πiξ ⋅ξ ̂ 1 , ξ2 ). (̂ 𝒰 σ)(ξ1 , ξ2 ) = e 1 2 σ(ξ

(4.8)

Let us describe the contents of the next sections. We first discuss some generalizations of the results by Sjöstrand. A natural question is which modulation spaces M p,q give rise to symbols of bounded operators on Lp , p ≠ 2. We provide in Section 4.1 a complete answer to this problem in the more general framework of the Wiener amalgam spaces W(Lr , Ls ) (see Corollary 4.1.6 and Proposition 4.1.15 below). Section 4.2 is concerned with the following related question. Examples show that symbols merely on L∞ generally do not produce bounded operators in L2 , but some additional regularity condition should be assumed. The above Sjöstrand’s result is just an instance of this. There is a space larger than M ∞,1 which still consists of bounded functions having locally the same regularity as a function whose Fourier transform is integrable. It is the modulation space W(ℱ L1 , L∞ ), the subspace of tempered dis1 tributions f such that the function ℱ (g(⋅ − x)f )(ξ ) belongs to L∞ x Lξ , see (2.70). A natural question which arises is whether pseudodifferential operators with symbols in W(ℱ L1 , L∞ ) are L2 -bounded. Fourier multipliers with symbols in W(ℱ L1 , L∞ ) are, in fact, bounded on L2 and M p , and the same holds (more generally) for symbols in W(ℱ L1 , L∞ ) of the type a(x, ξ ) = a1 (x)a2 (ξ ) (see Proposition 4.2.2 below). However, contrary to what these special cases could suggest, we shall show in Proposition 4.2.4 that, for more general symbols in that class, boundedness on L2 may fail. Sections 4.3 and 4.4 are the core of the chapter. They concern boundedness of pseudodifferential operators on M p,q spaces. The symbols are also assumed to belong

196 | 4 Pseudodifferential operators to modulation spaces. We want to present a generalization to the fundamental result by Gröchenig and Heil [163] 1999 (and its earlier precursor, the Heil–Ramanathan– 0 Topiwala paper [185]), stating that symbols in M ∞,1 (ℝ2d ), in particular in S0,0 , produce bounded operators on modulation spaces. Such a property is extremely important in the applications to partial differential equations. In fact, the optimal regularity behavior of the solutions arises from the stability (i. e., no loss of derivatives) of the function spaces appearing in the a priori estimates. In turn, this corresponds to stable continuity estimates for the pseudodifferential operators expressing the solutions. As we 0 observed, as soon as the symbol class S0,0 and classes of nonsmooth symbols are involved, stable estimates in the Lebesgue Lp spaces fail. Modulation spaces M p are then excellent substitutes of Lp in such situations. In Section 4.3 we shall first introduce the τ-quantization Opτ (a) of a symbol a, defined formally by ⟨Opτ (a)f , g⟩ = ⟨a, Wτ (g, f )⟩ where Wτ (g, f ) is the (cross)-τ-Wigner distribution defined in (1.109), see Chapter 1. Note that Op1 (a) = a(x, D), Op1/2 (a) = Opw (a). We shall prove that results for modulation symbols do not depend on the choice of the τ-quantization, 0 ≤ τ ≤ 1. Our strategy in Section 4.3 will be then to refer to the Weyl quantization, defined by ⟨Opw (a)f , g⟩ = ⟨a, W(g, f )⟩ in (4.6). Hence, we shall discuss the validity of estimates of the type

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󵄩󵄩 󵄩 󵄩󵄩W(f , g)󵄩󵄩󵄩M p,q ≲ ‖f ‖M p1 ,q1 ‖g‖M p2 ,q2 for suitable indices p, q, p1 , q1 , p2 , q2 . Using these inequalities, it will be easy to deduce Theorem 4.4.15, extending the most part of the results in the literature about boundedness of Opw (a) on modulation spaces, cf. Toft [286, 287]. In Section 4.4 we shall also discuss shortly localization operators; they provide an alternative method of quantization, and are important mathematical tools in signal theory and quantum mechanics. Section 4.5 is devoted to the study of operators with symbols a in the Wiener amalgam spaces W(ℱ Lp , Lq ). Here we continue the analysis of Section 4.2, where boundedness of a(x, D) and Opw (a) was considered on W(Lr , Ls ). Attention is now focused on the τ-quantization Opτ (a) acting on modulation spaces. The strategy will be the same as in Section 4.4, namely taking as definition ⟨Opτ (a)f , g⟩ = ⟨a, Wτ (g, f )⟩, we have reduced the situation to the study of estimates of the type 󵄩󵄩 󵄩 󵄩󵄩Wτ (f , g)󵄩󵄩󵄩W(ℱ Lr ,Lq ) ≤ C(τ)‖f ‖M p1 ,p2 ‖f ‖M q1 ,q2 where the constant C(τ) depends on τ. The corresponding results for Opτ (a) turn out to be surprising, namely boundedness on M p,q holds for 0 < τ < 1, whereas may fail for τ = 0, τ = 1. This is in particular the case for symbols a in W(ℱ L1 , L∞ ). Section 4.6 is devoted to Born–Jordan quantization and related pseudodifferential operators. Born and Jordan [33] 1925 provided the first mathematical basis to the

4.1 Boundedness on Lp and Wiener amalgam spaces | 197

Heisenberg’s matrix mechanics, and recently we may note a strong regain in interest on it. Our rather long presentation is intended to give a detailed and rigorous treatment of Born–Jordan pseudodifferential operators OpBJ (a). Continuity properties on modulation spaces are discussed in Section 4.7. Following our time–frequency approach, we shall use the definition ⟨OpBJ (a)f , g⟩ = ⟨a, Q(g, f )⟩, where Q(g, f ) is the cross-Born–Jordan distribution. We address to Sections 4.6 and 4.7 for a precise setting and historical motivations. We do not report here on other recent contributions to the theory of the pseudodifferential operators; nevertheless, we would like to mention the works of Ruzhansky and collaborators [46, 98, 109, 144, 154, 230, 256, 257, 258] concerning operators on Lie groups and applications. It would be interesting to reconsider this setting from the point of view of the time–frequency analysis.

4.1 Boundedness on Lp and Wiener amalgam spaces We first give a proof of Sjöstrand’s results [266, 267]. In the following remark we anticipate the contents of the discussion in Section 4.3, concerning different types of quantization. Remark 4.1.1. Since a(x, D) = Opw (𝒰 −1 a), with 𝒰 in (4.8), we shall show in the subsequent Proposition 4.3.6 that the modulation spaces M p,q , 1 ≤ p, q ≤ ∞, are invariant under 𝒰 , so that boundedness results for pseudodifferential operators with symbols in modulation spaces can be obtained using either the Weyl or the Kohn–Nirenberg form (for further references, see [286, 287, 288]). In the sequel, we shall adopt the operator form which is more convenient. We also need the following useful remark. Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

W(Lr ,L∞ )

Remark 4.1.2. Denote by 𝒮 the closure of the Schwartz class in W(Lr , L∞ ). If 1 ≤ r < ∞, its dual space is given by (𝒮

W(Lr ,L∞ ) 󸀠

) = W(Lr , L1 ), 󸀠

see, e. g., [130, Theorem 2.8]. Theorem 4.1.3. If σ ∈ M ∞,1 (ℝ2d ), then the Weyl operator Opw (σ) is bounded on W(L2 , Ls )(ℝd ), for every 1 ≤ s ≤ ∞, with the uniform estimate 󵄩󵄩 󵄩 󵄩󵄩Opw (σ)f 󵄩󵄩󵄩W(L2 ,Ls ) ≲ ‖σ‖M ∞,1 ‖f ‖W(L2 ,Ls ) . In view of Proposition 2.4.12, we have W(L2 , L2 ) = L2 , hence the estimate above implies the L2 -boundedness of Opw (σ).

198 | 4 Pseudodifferential operators Proof of Theorem 4.1.3. Let us show the estimate 󵄨 󵄨󵄨 󵄨󵄨⟨Opw (σ)f , g⟩󵄨󵄨󵄨 ≲ ‖σ‖M ∞,1 ‖f ‖W(L2 ,Ls ) ‖g‖W(L2 ,Ls󸀠 ) ,

∀f , g ∈ 𝒮 (ℝd ),

where 1/s + 1/s󸀠 = 1. This will give at once the desired result if s > 1, whereas the case s = 1 follows by Remark 4.1.2. Let φ ∈ 𝒟(ℝd ) and set Φ = W(φ, φ) ∈ 𝒮 (ℝ2d ). By the definition of the Weyl operator via cross-Wigner distribution in (4.6), the orthogonality relations in (1.44) and Hölder’s inequality, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄨󵄨⟨Opw (σ)f , g⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨σ, W(g, f )⟩󵄨󵄨󵄨 ≍ 󵄨󵄨󵄨⟨VΦ σ, VΦ W(g, f )⟩󵄨󵄨󵄨 ≤ ‖VΦ σ‖L∞,1 󵄩󵄩󵄩VΦ W(g, f )󵄩󵄩󵄩L1,∞ 󵄩 󵄩 ≍ ‖σ‖M ∞,1 󵄩󵄩󵄩W(g, f )󵄩󵄩󵄩M 1,∞ . Then the result is proved if we show that ‖W(g, f )‖M 1,∞ ≲ ‖f ‖W(L2 ,Ls ) ‖g‖W(L2 ,Ls󸀠 ) . If ζ = (ζ , ζ ) ∈ ℝ2d , we write ζ ̃ = (ζ , −ζ ). Then Lemma 1.3.37 says that 1

2

2

1

󵄨 ζ ̃ 󵄨󵄨󵄨󵄨󵄨󵄨 ζ ̃ 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨VΦ (W(g, f ))(z, ζ )󵄨󵄨󵄨 = 󵄨󵄨󵄨Vφ f (z + )󵄨󵄨󵄨󵄨󵄨󵄨Vφ g(z − )󵄨󵄨󵄨. 󵄨 󵄨󵄨 󵄨 2 󵄨󵄨 2 󵄨󵄨 Consequently, 󵄨󵄨 󵄩󵄩 󵄩 󵄨 󵄩󵄩W(g, f )󵄩󵄩󵄩M 1,∞ ≍ sup ∫ 󵄨󵄨󵄨Vφ f (z + 󵄨󵄨 2d ζ ∈ℝ 2d ℝ

ζ ̃ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 )󵄨󵄨V g(z − 2 󵄨󵄨󵄨󵄨󵄨󵄨 φ

ζ ̃ 󵄨󵄨󵄨󵄨 )󵄨 dz. 2 󵄨󵄨󵄨

We set π(ζ ̃ ) = Mζ ̃ Tζ ̃ . In what follows, we make the change of variables z �→ z− ζ ̃ /2, and 2 1 use the relations in (1.24) and (1.47), the Cauchy–Schwarz’s and Parseval’s inequalities with respect to the z2 variable, so that 󵄩󵄩 󵄩 󵄨 󵄨󵄨 󵄨 󵄩󵄩W(g, f )󵄩󵄩󵄩M 1,∞ ≍ sup ∫ 󵄨󵄨󵄨Vφ f (z)󵄨󵄨󵄨󵄨󵄨󵄨Vφ g(z − ζ ̃ )󵄨󵄨󵄨 dz 2d ζ ∈ℝ

ℝ2d

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󵄨 󵄨󵄨 󵄨 = sup ∫ 󵄨󵄨󵄨Vφ f (z)󵄨󵄨󵄨󵄨󵄨󵄨Vφ (π(ζ ̃ )g)(z)󵄨󵄨󵄨 dz 2d ζ ∈ℝ

ℝ2d

󵄨 󵄨󵄨 󵄨 = sup ∫ ∫ 󵄨󵄨󵄨ℱ (fTz1 φ)(z2 )󵄨󵄨󵄨󵄨󵄨󵄨ℱ (π(ζ ̃ )gTz1 φ)(z2 )󵄨󵄨󵄨 dz1 dz2 ζ ∈ℝ2d

ℝd ℝd

󵄩 󵄩 ≲ sup ∫ ‖fTz1 φ‖2 󵄩󵄩󵄩π(ζ ̃ )gTz1 φ󵄩󵄩󵄩2 dz1 ζ ∈ℝ2d

ℝd

󵄩 󵄩 ≲ sup ‖f ‖W(L2 ,Ls ) 󵄩󵄩󵄩π(ζ ̃ )g 󵄩󵄩󵄩W(L2 ,Ls󸀠 ) ζ ∈ℝ2d

= ‖f ‖W(L2 ,Ls ) ‖g‖W(L2 ,Ls󸀠 ) , where we have used Hölder’s inequality in the second to last step and the invariance of the W(L2 , Ls ) spaces under time–frequency shifts π(ζ ̃ ) in the last one.

4.1 Boundedness on Lp and Wiener amalgam spaces | 199

If we choose symbols with a stronger decay, namely symbols a ∈ M p,1 ⊂ M ∞,1 , 1 ≤ p ≤ 2, then the corresponding pseudodifferential operators a(x, D) are bounded on every Wiener amalgam spaces W(Lr , Ls ), as shown in the following result. Theorem 4.1.4. If a ∈ M p,1 (ℝ2d ), 1 ≤ p ≤ 2, then the operator a(x, D) is bounded on W(Lr , Ls )(ℝd ), for every 1 ≤ r, s ≤ ∞, with the uniform estimate 󵄩 󵄩󵄩 󵄩󵄩a(x, D)f 󵄩󵄩󵄩W(Lr ,Ls ) ≲ ‖a‖M p,1 ‖f ‖W(Lr ,Ls ) . Proof. By the inclusion relations for modulation spaces, we can just consider the case a ∈ M 2,1 . We shall show that the integral kernel K(x, y) = (ℱ2 a)(x, y − x) (ℱ2 stands for the partial Fourier transform with respect to the second variable) of a(x, D) can be controlled from above by |K(x, y)| ≤ F(x − y), where F is a positive function in L1 (ℝd ) with norm estimated by ‖a‖M 2,1 . If it is so, |a(x, D)f (x)| ≤ (F ∗ |f |)(x) the convolution relations for Wiener amalgam spaces in Lemma 2.4.9 give the desired result. We use the inversion formula (1.2.16) for the symbol a. Namely, for any window g ∈ 𝒮 (ℝ2d ), with ‖g‖2 = 1, we have a(x, ξ ) = ∫ (Vg a)(α, β)(Mβ Tα g)(x, ξ ) dα dβ.

(4.9)

ℝ4d

Hence, if α = (α1 , α2 ), β = (β1 , β2 ), it turns out that K(x, y) = ∫ e2πiα2 ⋅β2 (Vg a)(α1 , α2 , β1 , β2 )M(β1 ,α2 ) T(α1 ,−β2 ) (ℱ2−1 g)(x, x − y) dα1 dα2 dβ1 dβ2 . ℝ4d

Setting H(α1 , t; β1 , β2 ) = ∫ (Vg a)(α, β)e2πi(t−β2 )⋅α2 dα2 , Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

ℝd

and using the Cauchy–Schwarz’s inequality with respect to the α2 variable, we obtain 󵄨󵄨 󵄨 󵄨 󵄨 −1 󵄨󵄨K(x, y)󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨󵄨H(α1 , x − y + β2 ; β1 , β2 )ℱ2 g(x − α1 , x − y + β2 )󵄨󵄨󵄨 dα1 dβ1 dβ2 ℝ3d

󵄩 󵄩󵄩 󵄩 ≤ ∫ 󵄩󵄩󵄩H(⋅, x − y + β2 ; β1 , β2 )󵄩󵄩󵄩2 󵄩󵄩󵄩T(0,−β2 ) ℱ2−1 g(⋅, x − y)󵄩󵄩󵄩2 dβ1 dβ2 . ℝ2d

For simplicity, let us set 󵄩 󵄩󵄩 󵄩 F(t) := ∫ 󵄩󵄩󵄩H(⋅, t + β2 ; β1 , β2 )󵄩󵄩󵄩2 󵄩󵄩󵄩T(0,−β2 ) ℱ2−1 g(⋅, t)󵄩󵄩󵄩2 dβ1 dβ2 , ℝ2d

so that |K(x, y)| ≤ F(x − y).

200 | 4 Pseudodifferential operators We are left to estimate ‖F‖1 . Using the Cauchy–Schwarz’s inequality with respect to the t variable, 󵄩 󵄩󵄩 󵄩 ‖F‖1 = ∫ ∫ 󵄩󵄩󵄩H(⋅, t + β2 ; β1 , β2 )󵄩󵄩󵄩2 󵄩󵄩󵄩T(0,−β2 ) ℱ2−1 g(⋅, t)󵄩󵄩󵄩2 dβ1 dβ2 dt ℝd ℝ2d

󵄩 󵄩󵄩 󵄩 ≤ ∫ 󵄩󵄩󵄩H(⋅, ⋅; β1 , β2 )󵄩󵄩󵄩2 󵄩󵄩󵄩T(0,−β2 ) ℱ2−1 g 󵄩󵄩󵄩2 dβ1 dβ2 ℝ2d

󵄩 󵄩 = ‖g‖2 ∫ 󵄩󵄩󵄩H(⋅, ⋅; β1 , β2 )󵄩󵄩󵄩2 dβ1 dβ2 ℝ2d

= ‖H‖L2,1 = ‖Vg a‖L2,1 ≍ ‖a‖M 2,1 , where in the last row we used Parseval’s formula and the assumption ‖g‖2 = 1. This concludes the proof. Theorem 4.1.5. Let 1 ≤ p, q, r, s ≤ ∞ such that 1 󵄨󵄨󵄨󵄨 1 1 󵄨󵄨󵄨󵄨 1 ≥󵄨 − 󵄨+ , p 󵄨󵄨󵄨 r 2 󵄨󵄨󵄨 q󸀠

q ≤ min{r, r 󸀠 , s, s󸀠 }.

(4.10)

Then every symbol a ∈ M p,q gives rise to a bounded operator a(x, D) on W(Lr , Ls ) with the uniform estimate 󵄩󵄩 󵄩 󵄩󵄩a(x, D)f 󵄩󵄩󵄩W(Lr ,Ls ) ≲ ‖a‖M p,q ‖f ‖W(Lr ,Ls ) . Proof. We first make the complex interpolation between the estimates of Theorems 4.1.3 and 4.1.4 (which deal with the cases in which q = 1). Using the interpolation relations for Wiener amalgam and modulation spaces of Theorem 2.4.9 and Proposition 2.3.16, we obtain, for every 1 ≤ s < ∞,

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

󵄩󵄩 󵄩 󵄩󵄩a(x, D)f 󵄩󵄩󵄩W(Lr ,Ls ) ≲ ‖a‖M p,1 ‖f ‖W(Lr ,Ls ) , where 1 󵄨󵄨󵄨󵄨 1 1 󵄨󵄨󵄨󵄨 ≥ 󵄨 − 󵄨. p 󵄨󵄨󵄨 r 2 󵄨󵄨󵄨 The remaining cases, when s = ∞, p > 2 (and therefore r > 1), follow by duality, 󸀠 for W(Lr , L∞ ) = W(Lr , L1 )󸀠 in view of Theorem 2.4.9 and, considering the Weyl form ̄ Opw (σ) of a(x, D), we have (Opw (σ))∗ = Opw (σ). Finally, by interpolation between what we just proved and the obvious case p = q = r = s = 2 (Weyl pseudodifferential operators with symbols in M 2 (ℝ2d ) = L2 (ℝ2d ) are bounded on W(L2 , L2 )(ℝd ) = L2 (ℝd ) by (4.6) and Moyal’s identity), we obtain the claim.

4.1 Boundedness on Lp and Wiener amalgam spaces | 201

Recalling that, for s = r, we have W(Lr , Lr ) = Lr , the above boundedness result can be rephrased for pseudodifferential operators acting on Lp spaces as follows (see Figure 4.1, where we take into account the necessary conditions in the sequel).

Figure 4.1: The triples (1/r, 1/q, 1/p) inside the convex polyhedron are exactly those for which every symbol in Mp,q produces a bounded operator on Lr .

Corollary 4.1.6. Let 1 ≤ p, q, r ≤ ∞ such that 1 󵄨󵄨󵄨󵄨 1 1 󵄨󵄨󵄨󵄨 1 ≥󵄨 − 󵄨+ , p 󵄨󵄨󵄨 r 2 󵄨󵄨󵄨 q󸀠

q ≤ min{r, r 󸀠 }.

(4.11)

Then every symbol in a ∈ M p,q gives rise to a bounded operator a(x, D) on Lr with the uniform estimate

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󵄩󵄩 󵄩 󵄩󵄩a(x, D)f 󵄩󵄩󵄩r ≲ ‖a‖M p,q ‖f ‖r . 4.1.1 Boundedness on Wiener amalgam spaces: necessary conditions In this section we show the optimality of Theorem 4.1.5 (and Corollary 4.1.6). We need the following auxiliary results. Lemma 4.1.7. Suppose that for some 1 ≤ p, q, r, s ≤ ∞ the following estimate holds: 󵄩󵄩 󵄩 󵄩󵄩Opw (σ)f 󵄩󵄩󵄩W(Lr ,Ls ) ≤ C‖σ‖M p,q ‖f ‖W(Lr ,Ls ) ,

∀σ ∈ 𝒮 (ℝ2d ), ∀f ∈ 𝒮 (ℝd ).

Then the same estimate is satisfied with r, s replaced by the dual indices r 󸀠 , s󸀠 (even if r = ∞ or s = ∞). Proof. Indeed, observe that ⟨Opw (σ)f , g⟩ = ⟨f , Opw (σ)g⟩, ∀f , g ∈ 𝒮 (ℝd ). Hence, by Theorem 2.4.9 and the assumptions rewritten for Opw (σ) (observe that ‖σ‖M p,q = ‖σ‖̄ M p,q ),

202 | 4 Pseudodifferential operators we have 󵄨 󵄨󵄨 󵄨󵄨⟨Opw (σ)f , g⟩󵄨󵄨󵄨 ≤ C‖σ‖M p,q ‖f ‖W(Lr󸀠 ,Ls󸀠 ) ‖g‖W(Lr ,Ls ) ,

∀f ∈ 𝒮 (ℝd ), ∀g ∈ 𝒮 (ℝd ).

(4.12)

Since Opw (σ)f is a Schwartz function, it belongs to W(Lr , Ls ) ⊂ W(Lr , Ls )󸀠 , and 󸀠 󸀠 ‖Opw (σ)f ‖W(Lr ,Ls )󸀠 = ‖Opw (σ)f ‖W(Lr󸀠 ,Ls󸀠 ) , because W(Lr , Ls ) is isometrically embed󸀠

󸀠

ded in W(Lr , Ls )󸀠 . Hence it suffices to prove that the estimate in (4.12) holds for every g ∈ W(Lr , Ls ). This follows by a density argument. Namely, consider, for a given g ∈ W(Lr , Ls ), a sequence gn of Schwartz functions, with gn → g in 𝒮 󸀠 (ℝd ) and ‖gn ‖W(Lr ,Ls ) ≤ ‖g‖W(Lr ,Ls ) (1 ). Letting n → ∞ in the above estimate (written with gn in place of g) gives the desired conclusion.

Lemma 4.1.8. Let h ∈ 𝒟(ℝd ), and consider the family of functions 2

hλ (x) = h(x)e−πiλx ,

λ ≥ 1.

Then for 1 ≤ q ≤ ∞ d

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‖ĥλ ‖q ≍ λ q

− d2

(4.13)

.

Proof. The result is known and outlined, e. g., in [277, Exercise 2.34]. We report on a sketch of the proof for the sake of completeness. Let c > 0 be such that h(x) vanishes for |x| > c. First one shows the estimate |ĥλ (ξ )| ≤ CN ⟨ξ ⟩−N λ−N , for every N > 0 and ξ ∈ ℝd such that |ξ | ≥ 2cλ. To this end, we observe that by rotational symmetry we can assume ξ = (ξ1 , 0, . . . , 0). The claim then follows by applying the nonstationary phase theorem [269, Proposition 1, page 331] with the asymptotic parameter ξ1 and the phase ϕ(x1 ) := −2πx1 − π ξλ x12 (the assump1 tions being satisfied for |x1 | ≤ c, uniformly with respect to the parameter λ/ξ1 ). In the region |ξ | < 2cλ we have the estimate |ĥλ (ξ )| ≤ Cλ−d/2 , as a consequence of the stationary phase theorem (see [269, Section 5.13 (a), page 363]) with the phase given by the quadratic polynomial ϕ(x) := −πx2 . One hence obtains the upper bound ‖ĥλ ‖q ≲ λd(1/q−1/2) . Since ‖h‖22 = ‖ĥλ ‖22 ≤ ‖ĥλ ‖q ‖ĥλ ‖q󸀠 ≲ λd(1/q −1/2) ‖ĥλ ‖q , 󸀠

the lower bound follows as well. Finally, we establish a useful estimate for Wiener amalgam spaces, based on the classical Bernstein’s inequality (see, e. g., [316]), that we are going to recall first. Let BR (x0 ) be the ball of center x0 ∈ ℝd and radius R > 0 in ℝd . 1 For example, take gn (x) = nd φ1 (x/n)(g ∗ φ2 (n⋅))(x), with φ1 , φ2 ∈ 𝒟(ℝd ), φ1 (0) = 1, ‖φ2 ‖1 = 1.

4.1 Boundedness on Lp and Wiener amalgam spaces | 203

Lemma 4.1.9 (Bernstein’s inequality). Let f ∈ 𝒮 󸀠 (ℝd ) be such that f ̂ is supported in BR (x0 ), and let 1 ≤ p ≤ q ≤ ∞. Then there exists a positive constant C independent of f , x0 , R, p, q, such that ‖f ‖q ≤ CR

d( p1 − q1 )

(4.14)

‖f ‖p .

Lemma 4.1.10. Let 1 ≤ p ≤ q ≤ ∞. For every R > 0, there exists a constant CR > 0 such that, for every f ∈ 𝒮 󸀠 (ℝd ) whose Fourier transform is supported in any ball of radius R, it turns out ‖f ‖W(Lq ,Lp ) ≤ CR ‖f ‖p . Proof. Choose a Schwartz function g whose Fourier transform ĝ has compact support in B1 (0), as window function arising in the definition of the norm in W(Lq , Lp ). Then the function (Tx g)f , x ∈ ℝd , has Fourier transform supported in a ball of radius R + 1. Therefore it follows from Bernstein’s inequality (Lemma 4.1.9) that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(Tx g)f 󵄩󵄩󵄩q ≤ CR 󵄩󵄩󵄩(Tx g)f 󵄩󵄩󵄩p . Taking the Lp -norm with respect to x gives the conclusion. We now establish a version of the upper bound in Lemma 4.1.8, for Wiener amalgam spaces. Lemma 4.1.11. With the notation of Lemma 4.1.8 we have, for 1 ≤ p, q ≤ ∞, d

‖ĥλ ‖W(Lp ,Lq ) ≲ λ q

− d2

λ ≥ 1.

,

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Proof. When p ≥ q the desired result follows from Lemmata 4.1.10 and 4.1.8. When p < q the result follows from the inclusion Lq �→ W(Lp , Lq ) and Lemma 4.1.8. Proposition 4.1.12. Let χ ∈ 𝒟(ℝd ), χ ≥ 0, χ(0) = 1. Suppose that, for some 1 ≤ p, q, r, r1 , r2 ≤ ∞, C > 0, the estimate 󵄩󵄩 󵄩 󵄩󵄩χa(x, D)f 󵄩󵄩󵄩r ≤ C‖a‖M p,q ‖f ‖W(Lr1 ,Lr2 ) ,

∀a ∈ 𝒮 (ℝ2d ), f ∈ 𝒮 (ℝd ),

(4.15)

holds. Then q ≤ r2󸀠 and 1 1 1 1 ≥ − + 󸀠. p 2 r q

(4.16)

Proof. First we prove the constraint q ≤ r2󸀠 . Let h ∈ 𝒟(ℝd ), h ≥ 0, h(0) = 1. Let hλ be as in Lemma 4.1.8. We test (4.15) on the family of symbols 2

aλ (x, ξ ) = h(x)hλ (ξ ) = h(x)h(ξ )e−πiλ|ξ | ,

204 | 4 Pseudodifferential operators and functions fλ = ℱ −1 (hλ ) ∈ 𝒮 (ℝd ). An explicit computation shows that χ(x)aλ (x, D)fλ (x) = ∫ e2πix⋅ξ χ(x)h(x)h2 (ξ ) dξ , which is a nonzero Schwartz function independent of λ. On the other hand, by Lemma 4.1.8, we have d

‖aλ ‖M p,q ≍ ‖aλ ‖ℱ Lq ≲ ‖hλ ‖ℱ Lq ≲ λ q

− d2

.

Similarly, by Lemma 4.1.11, d

‖fλ ‖W(Lr1 ,Lr2 ) ≲ λ r2

− d2

.

Taking into account these estimates and letting λ → +∞, (4.15) then gives q ≤ r2󸀠 . Let us now prove (4.16). Let hλ be as above. We now test the estimate (4.15) on the family of symbols 2

a󸀠λ (x, ξ ) = e−πλ|x| ĥλ (ξ ), and functions fλ󸀠 = hλ . The operator a󸀠λ (x, D) has integral kernel Kλ (x, y) = e−πλ

2

|x|2

hλ (x − y),

so that 󵄨 󵄨󵄨 2 2 󵄨 󵄨 󵄨󵄨 󵄨 χ(x)󵄨󵄨󵄨a󸀠λ (x, D)fλ󸀠 (x)󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ e−πλ |x| +2πiλx⋅y h(x − y)χ(x)h(y) dy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≥ Re ∫ e−πλ

2

|x|2 +2πiλx⋅y

h(x − y)χ(x)h(y) dy.

Now, h(y) has compact support, say, in the ball |y| ≤ C. Moreover, if |x| ≤ λ−1 for λ ≥ λ0 2 2 large enough, and |y| ≤ C we have Re(e−πλ |x| +2πiλx⋅y ) ≥ 21 . Hence we deduce 󵄨 󵄨 χ(x)󵄨󵄨󵄨a󸀠λ (x, D)fλ󸀠 (x)󵄨󵄨󵄨 ≳ 1,

for |x| ≤ λ−1 ,

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which implies 󵄩󵄩 󸀠 󸀠󵄩 −d 󵄩󵄩χaλ (x, D)fλ 󵄩󵄩󵄩r ≳ λ r . On the other hand, ‖fλ󸀠 ‖W(Lr1 ,Lr2 ) is clearly independent of λ. Moreover, by Lemma 2.6.1, −d 󵄩󵄩 −πλ2 |⋅|2 󵄩󵄩 󵄩󵄩e 󵄩󵄩M p,q ≲ λ q󸀠

and, by Lemmas 2.3.26 (ii) and 4.1.8, we have d

‖ĥλ ‖M p,q ≲ ‖ĥλ ‖p ≲ λ p

− d2

.

Hence d d − −d 󵄩󵄩 󸀠 󵄩󵄩 󵄩 −πλ2 |⋅|2 󵄩󵄩 󵄩󵄩aλ 󵄩󵄩M p,q = 󵄩󵄩󵄩e 󵄩󵄩M p,q ‖ĥλ ‖M p,q ≲ λ p 2 q󸀠 .

Putting all together and letting λ → ∞ we obtain (4.16).

4.1 Boundedness on Lp and Wiener amalgam spaces | 205

Proposition 4.1.13. Suppose that, for some 1 ≤ p, q, r, s, r1 , r2 ≤ ∞, C > 0, the estimate 󵄩 󵄩󵄩 󵄩󵄩a(x, D)f 󵄩󵄩󵄩W(Lr1 ,Lr2 ) ≤ C‖a‖M p,q ‖f ‖W(Lr ,Ls ) ,

∀a ∈ 𝒮 (ℝ2d ), f ∈ 𝒮 (ℝd ),

(4.17)

holds. Then q ≤ r. ̂ are real-valued, Proof. Let h1 , h2 be two Schwartz functions in ℝd such that h1 and h 2 ̂ with h1 (0) = 1, h2 (0) = 1, and satisfying ̂ ⊂ B (0), supp h 1 1

supp ĥ2 ⊂ B1 (0).

(4.18)

Consider, for every N ≥ 1, the finite lattice ΛN = {n = (n1 , . . . , nd ) ∈ 4ℤd : 0 ≤ nj ≤ 4(N − 1), j = 1, . . . , d}.

(4.19)

Observe that ΛN has cardinality N d , and |n| ≤ d1/2 4(N − 1), ∀n ∈ ΛN

and |n − m| ≥ 4, ∀n, m ∈ ΛN , n ≠ m.

(4.20)

Moreover, let h be a smooth real-valued function, h ≥ 0, h(0) = 1, supported in a ball Bϵ (0), for a small ϵ to be chosen later. We test the estimate (4.17) on the family of functions fN (x) = h(Nx) and symbols aN (x, ξ ) = ∑ bn (x, ξ ), n∈ΛN

where bn (x, ξ ) = (M−n h1 )(x)(Tn h2 )(ξ ).

(4.21)

The integral kernel of the operator aN (x, D) is given by KN (x, y) = (ℱ2−1 aN )(x, x − y) = ∑ ℱ2−1 (bn )(x, x − y) = ∑ e−2πin⋅y h1 (x)ĥ2 (y − x). n∈ΛN

n∈ΛN

We now show that, for a suitable δ > 0,

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󵄨󵄨 󵄨 󵄨󵄨aN (x, D)fN (x)󵄨󵄨󵄨 ≳ 1,

for x ∈ Bδ (0),

(4.22)

which implies 󵄩󵄩 󵄩 󵄩󵄩aN (x, D)fN 󵄩󵄩󵄩W(Lr1 ,Lr2 ) ≳ 1.

(4.23)

In order to prove (4.22), observe that, by the above computation, aN (x, D)fN (x) = ∫ ( ∑ e−2πin⋅y )h1 (x)ĥ2 (y − x)h(Ny) dy. ℝd

n∈ΛN

Now, as a consequence of the first condition in (4.20), we see that on the support of h(Ny), hence where |y| ≤ ϵN −1 , we have 1 Re(e−2πin⋅y ) ≥ , 2

∀n ∈ ΛN ,

206 | 4 Pseudodifferential operators if ϵ ≤ d−1/2 /24. This implies that d 󵄨 N 󵄨󵄨 ∫ h1 (x)ĥ2 (y − x)h(Ny) dy. 󵄨󵄨aN (x, D)fN (x)󵄨󵄨󵄨 ≥ 2 ℝd

Since h1 (0) = ĥ2 (0) = 1, if δ and ϵ are small enough, so as h1 (x) ≥ 1/2 and ĥ2 (y −x) ≥ 1/2 for |y| ≤ ϵ, |x| ≤ δ. It turns out that d ‖h‖1 󵄨󵄨 󵄨 N , ∫ h(Ny) dy = 󵄨󵄨aN (x, D)fN (x)󵄨󵄨󵄨 ≥ 8 8

for |x| ≤ δ,

ℝd

which implies (4.22). We now prove that ‖aN ‖M p,q ≲ N d/q .

(4.24)

̂)(ζ )(M ĥ )(ζ ). b̂n (ζ1 , ζ2 ) = (T−n h 1 1 −n 2 2

(4.25)

To see this, observe that

We choose a window Φ = φ ⊗ φ, where φ̂ is a Schwartz function supported in B1 (0). It follows from (4.25), the second condition in (4.20), and (4.18) that the functions ̂ ∗ )(ω) (with z = (z1 , z2 ), ω = (ω1 , ω2 ), Φ∗ (z) = Φ(−z)), VΦ bn (z1 , z2 , ω1 , ω2 ) = (b̂n ∗ M−z Φ vanish unless ω1 ∈ B2 (−n), ω2 ∈ B2 (0). Hence, when n varies in ΛN , they have pairwise disjoint supports, as well as the functions ‖VΦ bn (⋅, ⋅, ω1 , ω2 )‖p . We deduce that 1/q 󵄩󵄩 󵄩󵄩󵄩q 󵄩 ‖aN ‖M p,q ≍ ( ∫ 󵄩󵄩󵄩 ∑ VΦ bn (⋅, ⋅, ω1 , ω2 )󵄩󵄩󵄩 dω1 dω2 ) 󵄩󵄩 󵄩󵄩p 2d n∈ΛN ℝ

q/p

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󵄩 󵄩p = ( ∫ ( ∑ 󵄩󵄩󵄩VΦ bn (⋅, ⋅, ω1 , ω2 )󵄩󵄩󵄩p ) n∈ΛN

ℝd

1/q

dω1 dω2 ) 1/q

󵄩 󵄩q = ( ∫ ∑ 󵄩󵄩󵄩VΦ bn (⋅, ⋅, ω1 , ω2 )󵄩󵄩󵄩p dω1 dω2 ) ℝ2d

n∈ΛN

1/q

=( ∑

n∈ΛN

∫

󵄩 󵄩q ∫ 󵄩󵄩󵄩VΦ bn (⋅, ⋅, ω1 , ω2 )󵄩󵄩󵄩p dω1 dω2 )

.

(4.26)

B2 (−n) B2 (0)

On the other hand, since VΦ bn (z, ω) = e−2πiz⋅ω (bn ∗ Mω Φ∗ )(z), by Young’s inequality, we have 󵄩󵄩 󵄩 󵄩󵄩VΦ bn (⋅, ⋅, ω1 , ω2 )󵄩󵄩󵄩p ≤ ‖bn ‖p ,

4.1 Boundedness on Lp and Wiener amalgam spaces | 207

and the expression for bn in (4.21) shows that ‖bn ‖p is, in fact, independent of n. Hence the expression in (4.26) is ≲( ∑

n∈ΛN

∫ dz1 dz2 )

∫

1/q

= Cd N d/q ,

B2 (−n) B2 (0)

which gives (4.24). Finally, since the functions fN are supported in a fixed compact subset, we have ‖fN ‖W(Lr ,Ls ) ≍ ‖fN ‖r = ‖h‖r N −d/r .

(4.27)

Combining this estimate with (4.23), (4.24), and (4.17), and letting N → ∞ yields the desired constraint q ≤ r. Theorem 4.1.14. Suppose that, for some 1 ≤ p, q, r, s ≤ ∞, the estimate 󵄩󵄩 󵄩 󵄩󵄩a(x, D)f 󵄩󵄩󵄩W(Lr ,Ls ) ≤ C‖a‖M p,q ‖f ‖W(Lr ,Ls ) ,

∀a ∈ 𝒮 (ℝ2d ), f ∈ 𝒮 (ℝd ),

(4.28)

holds. Then q ≤ min{r, r 󸀠 , s, s󸀠 } and 1 󵄨󵄨󵄨󵄨 1 1 󵄨󵄨󵄨󵄨 1 ≥󵄨 − 󵄨+ . p 󵄨󵄨󵄨 2 r 󵄨󵄨󵄨 q󸀠

(4.29)

Proof. We already know from Proposition 4.1.13 that q ≤ r. The constraints q ≤ r 󸀠 follows by duality arguments, namely by Lemma 4.1.7. Now, we shall prove that q ≤ s󸀠 ,

1 1 1 1 ≥ − + . p 2 r q󸀠

(4.30)

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The remaining constraints will follow by duality as above. Let χ ∈ 𝒟(ℝd ), with 0 ≤ χ ≤ 1, χ(0) = 1. Then (4.28) implies 󵄩󵄩 󵄩 󵄩󵄩χa(x, D)f 󵄩󵄩󵄩W(Lr ,Ls ) ≤ C‖a‖M p,q ‖f ‖W(Lr ,Ls ) ,

∀f ∈ 𝒮 (ℝd ), ∀a ∈ 𝒮 (ℝ2d ).

Since for functions u supported in a fixed compact subset we have ‖u‖W(Lr ,Ls ) ≍ ‖u‖r , we deduce 󵄩󵄩 󵄩 󵄩󵄩χa(x, D)f 󵄩󵄩󵄩r ≤ C‖a‖M p,q ‖f ‖W(Lr ,Ls )

∀f ∈ 𝒮 (ℝd ), ∀a ∈ 𝒮 (ℝ2d ).

Then Proposition 4.1.12 implies (4.30). Proposition 4.1.15. Suppose that, for some 1 ≤ p, q, r, s ≤ ∞, every symbol a ∈ M p,q gives rise to an operator a(x, D) bounded on W(Lr , Ls ). Then the constraints q ≤ min{r, r 󸀠 , s, s󸀠 } and (4.29) must hold.

208 | 4 Pseudodifferential operators Proof. Let 𝒲 (Lr , Ls ) be the closure of 𝒮 (ℝd ) in W(Lr , Ls ). By assumption, the map M p,q ∋ a �→ a(x, D) ∈ B(𝒲 (Lr , Ls ), W(Lr , Ls )) is well defined. By an application of the closed graph theorem and Theorem 4.1.14 we see that the desired conclusion follows if we prove that this map has a closed graph. To this end, let an → a in M p,q , with an (x, D) → A in B(𝒲 (Lr , Ls ), W(Lr , Ls )). We have to prove that A = a(x, D), i. e., ⟨Af , g⟩ = ⟨a(x, D)f , g⟩ ∀f , g ∈ 𝒮 (ℝd ). Now, clearly, ⟨an (x, D)f , g⟩ → ⟨Af , g⟩. On the other hand, ⟨an (x, D)f , g⟩ = ⟨an , G⟩, where G(x, ω) = e−2πix⋅ω f ̂(ω)g(x) is a fixed Schwartz function. Hence ⟨a (x, D)f , g⟩ tends to n

⟨a, G⟩ = ⟨a(x, D)f , g⟩, which concludes the proof.

4.2 Symbols in Wiener amalgam spaces As observed in the introduction to the chapter, a natural question which can be raised is whether the boundedness of pseudodifferential operators on L2 or, more generally, on M p,q , can be attained by widening the symbol Sjöstrand class M ∞,1 to the Winer amalgam space W(ℱ L1 , L∞ ). An example of a function belonging to W(ℱ L1 , L∞ )(ℝd ) \ 2 M ∞,1 (ℝd ) is the chirp φ(x) = eπix , x ∈ ℝd , see Proposition 4.3.3 below. The multiplier 2 operator a(x, D)f (x) = eπix f (x) is a pseudodifferential operator with symbol a(x, ξ ) = 2 (eπi|⋅| ⊗ 1)(x, ξ ) ∈ W(ℱ L1 , L∞ )(ℝ2d ) \ M ∞,1 (ℝ2d ), it is obviously bounded on L2 (ℝd ) and it is bounded on M p,q if and only if p = q, as shown below. 2

Proposition 4.2.1. The multiplication A : f → eπi|⋅| f is unbounded on M p,q , for every 1 ≤ p, q ≤ ∞, with p ≠ q. 2

Proof. For λ > 0, we consider the one-parameter family of functions f (x) = e−πλx ∈ 2 M p,q , so that f ̂(ξ ) = λ−d/2 e−π(1/λ)ξ . For every 1 ≤ p, q ≤ ∞, by (2.6.1), we have

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‖f ‖M p,q

= ‖f ̂ ‖

W(ℱ Lp ,Lq )

≍

d( p1 − 21 )

(λ + 1)

d

d

λ 2q (λ2 + λ) 2

( p1 − q1 )

.

2

Since Af (x) = e−π(λ−i)x , its Fourier transform is given by 1

2

̂ (ξ ) = (λ − i)−d/2 e−π λ−i ξ , Af so that the same formula as above yields d

‖Af ‖M p,q ≍

[(λ + 1)2 + 1] 2 d

( p1 − 21 ) d

λ 2q (λ2 + λ + 1) 2

( p1 − q1 )

.

As λ → 0, we have ‖Af ‖M p,q ≍ λ

d − 2q

,

‖f ‖M p,q ≍ λ

d − 2q − d2 ( p1 − q1 )

so that, if we assume ‖Af ‖M p,q ≤ C‖f ‖M p,q , then 1/p − 1/q ≥ 0, that is, p ≤ q.

4.2 Symbols in Wiener amalgam spaces | 209

2

Moreover, the same argument applies to the adjoint operator A∗ f (x) = e−πix f (x). Now we show that p = q. By contradiction, if A were bounded on M p,q , with p < q, its adjoint A∗ would satisfy 󵄩󵄩 ∗ 󵄩󵄩 󸀠 󸀠 󵄩󵄩A f 󵄩󵄩M p ,q ≤ C‖f ‖M p󸀠 ,q󸀠 ,

∀f ∈ 𝒮 (ℝd ),

with q󸀠 < p󸀠 , which is a contradiction to what has just been proved. The subsequent Proposition 4.2.4 shows that generally symbols in W(ℱ L1 , L∞ ) do not produce bounded operators even on L2 (ℝd ). However, symbols expressed as tensor products a(x, ξ ) = a1 (x)a2 (ξ ) are bounded on M p (ℝd ), 1 ≤ p ≤ ∞ (and hence on L2 (ℝd )), as shown in the next result. Proposition 4.2.2. If a(x, ξ ) = a1 (x)a2 (ξ ), ai ∈ W(ℱ L1 , L∞ ), i = 1, 2, then a(x, D) is bounded on M p (ℝd ), 1 ≤ p ≤ ∞. Proof. We have a(x, D)f (x) = a1 (x)ℱ −1 (a2 f ̂)(x) = a1 (x)[ℱ −1 (a2 ) ∗ f ](x). If a2 ∈ W(ℱ L1 , L∞ )(ℝd ), then ℱ −1 (a2 ) ∈ M 1,∞ (ℝd ) and ℱ −1 (a2 ) ∗ f ∈ M 1,∞ (ℝd ) ∗ M p (ℝd ) �→ M p (ℝd ); see Proposition 2.4.19. It remains to show that the multiplier Aa1 f (x) = a1 (x)f (x) is bounded on M p . This immediately follows by the pointwise products for Wiener amalgam spaces in Theorem 2.4.9. Indeed, ℱ L1 ⋅ ℱ Lp = ℱ (L1 ∗ Lp ) �→ ℱ Lp , L∞ ⋅ Lp �→ Lp , so that, for M p = W(ℱ Lp , Lp ), we have W(ℱ L1 , L∞ ) ⋅ W(ℱ Lp , Lp ) �→ W(ℱ Lp , Lp ), as desired. 2

In particular, we obtain that the multiplier operator a(x, D)f (x) = eπi|x| f (x), which is not bounded on M p,q , p ≠ q, as discussed above, is bounded on M p . We now come to a necessary condition.

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Proposition 4.2.3. Suppose that, for some 1 ≤ p, q, r, s, r1 , r2 ≤ ∞, C > 0, either the estimate 󵄩󵄩 󵄩 󵄩󵄩a(x, D)f 󵄩󵄩󵄩W(Lr1 ,Lr2 ) ≤ C‖a‖W(ℱ Lp ,Lq ) ‖f ‖W(Lr ,Ls ) ,

∀a ∈ 𝒮 (ℝ2d ), f ∈ 𝒮 (ℝd ),

(4.31)

󵄩󵄩 󵄩 󵄩󵄩Opw (σ)f 󵄩󵄩󵄩W(Lr1 ,Lr2 ) ≤ C‖σ‖W(ℱ Lp ,Lq ) ‖f ‖W(Lr ,Ls ) ,

∀a ∈ 𝒮 (ℝ2d ), f ∈ 𝒮 (ℝd ),

(4.32)

or

holds. Then q ≤ r. Proof. We first suppose that (4.31) holds true. We test that estimate on the same families of functions and symbols as in the proof of Proposition 4.1.13, but with (4.18) replaced by supp h1 ⊂ B1 (0), (the other conditions being unchanged).

supp h2 ⊂ B1 (0),

(4.33)

210 | 4 Pseudodifferential operators Then (4.23) and (4.27) still hold, because in their proof we did not use (4.18). On the other hand, we can prove that ‖aN ‖W(ℱ Lp ,Lq ) ≲ N d/q ,

(4.34)

by using the same arguments in the proof of (4.24), with the roles of h1 , h2 replaced by ̂, respectively. Precisely, we now choose a window Φ = φ ⊗ φ with φ ∈ 𝒮 (ℝd ) ĥ2 and h 1 supported in B1 (0). Then the functions VΦ bn (z1 , z2 , ω1 , ω2 ) vanish unless z1 ∈ B2 (−n), z2 ∈ B2 (0), so that they have disjoint supports, as well as ‖VΦ bn (z1 , z2 , ⋅, ⋅)‖p . Hence one deduces that 1/q 󵄩󵄩 󵄩󵄩q 󵄩 󵄩 ‖an ‖W(ℱ Lp ,Lq ) ≍ ( ∫ 󵄩󵄩󵄩 ∑ VΦ bn (z1 , z2 , ⋅, ⋅)󵄩󵄩󵄩 dz1 dz2 ) 󵄩󵄩 󵄩󵄩p 2d n∈ΛN ℝ

=( ∑

n∈ΛN

1/q

∫

󵄩 󵄩q ∫ 󵄩󵄩󵄩VΦ bn (z1 , z2 , ⋅, ⋅)󵄩󵄩󵄩p dz1 dz2 )

.

B2 (−n) B2 (0)

On the other hand, by Young’s inequality, we have 󵄩󵄩 󵄩 󵄩󵄩VΦ bn (z1 , z2 , ⋅, ⋅)󵄩󵄩󵄩p ≤ ‖b̂n ‖p and the right-hand side of the above inequality is independent of n. Hence (4.34) follows. We now suppose that the estimate (4.32) holds. Since the arguments are similar to those just used, we only sketch the main point of the proof. We test the estimate (4.32) on the family of functions fN (x) = h(Nx), where h is a smooth real-valued function, with h ≥ 0, h(0) = 1, supported in a ball Bϵ (0), for a sufficiently small ϵ, and symbols σN (x, ξ ) = ∑ bn (x, ξ ),

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n∈ΛN

where bn (x, ξ ) = (M−2n h1 )(x)(Tn h2 )(ξ ).

The lattice ΛN is defined in (4.19) and h1 , h2 are two Schwartz functions in ℝd such that ̂ are real valued, with h (0) = 1, ĥ (0) = 1, and satisfying (4.33). By using the h1 and h 2 1 2 definition (4.6), we see that Opw (σ) has integral kernel K(x, y) = ℱ2−1 σ(

x+y ̂ x+y , x − y) = ∑ e−4πin⋅y h1 ( )h2 (y − x). 2 2 n∈Λ N

Hence, by arguing as in the proof of (4.22), one obtains, for a suitable δ > 0, 󵄨󵄨 󵄨 󵄨󵄨Opw (σN )fN (x)󵄨󵄨󵄨 ≳ 1,

for x ∈ Bδ (0),

which implies 󵄩󵄩 󵄩 󵄩󵄩Opw (σN )fN 󵄩󵄩󵄩W(Lr1 ,Lr2 ) ≳ 1.

4.3 Pseudodifferential operators on modulation spaces | 211

The arguments in the first part of the present proof, with essentially no changes, show that ‖σN ‖W(ℱ Lp ,Lq ) ≲ N d/q . Combining these estimates with ‖fN ‖W(Lr ,Ls ) ≲ N −d/r and letting N → +∞ gives the desired conclusion. Proposition 4.2.4. Suppose that, for some 1 ≤ p, q, r, s ≤ ∞, every symbol a ∈ W(ℱ Lp , Lq ) gives rise to a bounded operator on W(Lr , Ls ). Then q ≤ r. The same happens if one replaces the Kohn–Nirenberg operator a(x, D) by the Weyl one Opw (a). Proof. The result follows from Proposition 4.2.3 by arguing as in the proof of Proposition 4.1.15. In particular, we obtain that for generic symbols in W(ℱ L1 , L∞ )(ℝ2d ) continuity on L2 (ℝd ) may fail. We also deduce that L2 -boundedness fails in general for a ∈ W(ℱ L2 , Lq )(ℝ2d ) if q > 2. This result is sharp for Weyl operators, since for a ∈ W(ℱ L2 , L2 )(ℝ2d ) = L2 (ℝ2d ) the operator Opw (a) is L2 -continuous, whereas obviously there are symbols a ∈ L2 (ℝ2d ) for which a(x, D) is not bounded. We finally tackle the problem of the invariance of the Wiener amalgam spaces W(ℱ Lp , Lq ) under the action of the operator 𝒰 in (4.8), which expresses the Kohn– Nirenberg symbol of an operator in terms of the Weyl one. The lack of invariance, expressed by the following result, justifies the fact that the necessary conditions in this section were proved for both Kohn–Nirenberg and Weyl quantizations. Proposition 4.2.5. Let 1 ≤ p, q ≤ ∞. If p ≠ q, the operator 𝒰 in (4.8) does not map W(ℱ Lp , Lq ) into itself.

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The proof, limited to the relevant case p = 1, q = ∞, is given by the following remark. For the general proof, we refer the reader to Proposition 6.1.21 in Chapter 6. Remark 4.2.6. A concrete example of a Weyl symbol σ ∈ W(ℱ L1 , L∞ ) such that the corresponding Kohn–Nirenberg symbol a = 𝒰 σ does not belong to W(ℱ L1 , L∞ ) is provided by σ = 𝒰 −1 δ; therefore a = δ. It is clear that δ does not belong to W(ℱ L1 , L∞ ) ̂ (which consists only of continuous functions), whereas from (4.8) we have δ̂ = 𝒰 σ= πiξ1 ⋅ξ2 ̂ −1 −πiξ1 ⋅ξ2 1 ∞ e σ(ξ1 , ξ2 ). Hence σ = ℱ (e ), belonging to W(ℱ L , L ), see (4.40) and Proposition 4.3.3 in the next section for details.

4.3 Pseudodifferential operators on modulation spaces If we consider τ ∈ [0, 1], the Shubin τ-representation of a pseudodifferential operator with symbol a ∈ 𝒮 (ℝ2d ) is defined by Opτ (a)f (x) = ∫ e2πi(x−y)⋅ξ a((1 − τ)x + τy, ξ )f (y) dy dξ . ℝ2d

(4.35)

212 | 4 Pseudodifferential operators For τ = 1/2, we recapture the Weyl operator in (4.4), that is, Op1/2 (a) = Opw (a). The case τ = 0 gives Op0 (a) = a(x, D), that is, the Kohn–Nirenberg operator with symbol a. Indeed, we can generalize the weak definition of a Kohn–Nirenberg operator in (4.1) as follows. Proposition 4.3.1. Consider a ∈ 𝒮 (ℝ2d ), f , g ∈ 𝒮 (ℝd ). Then ⟨Opτ (a)f , g⟩ = ⟨a, Wτ (g, f )⟩,

f , g ∈ 𝒮 (ℝd ),

(4.36)

where Wτ (g, f ) is the (cross-)τ-Wigner distribution defined in (1.109). As already said for Weyl operators, the result is an easy consequence of a change of variables and an interchange of integration order. The case τ = 1/2 gives the equality for Weyl operators in (4.5). Almost diagonalization of τ-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces is shown in [86]. Extensions of τ-operators with τ ∈ ℝ are treated in [45]. More general matrix formulations can be found in [8, 90, 290]. The goal of this section is the study of the continuity properties of pseudodifferential operators on modulation spaces. Their symbol will belong to modulation spaces as well. A direct computation, cf. [193] (see also [286, Remark 1.5]), shows that for every choice τ1 , τ2 ∈ [0, 1], a1 , a2 ∈ 𝒮 󸀠 (ℝ2d ), ̂2 (ξ1 , ξ2 ) = e−2πi(τ2 −τ1 )ξ1 ⋅ξ2 a ̂1 (ξ1 , ξ2 ), Opτ1 (a1 ) = Opτ2 (a2 ) ⇐⇒ a

(4.37)

so for τ1 ≠ τ2 , applying the Fourier transform to both sides of the equality on the righthand side of (4.37), we obtain

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a2 (x, ξ ) =

1 e2πi(τ2 −τ1 )Φ ∗ a1 (x, ξ ), |τ1 − τ2 |d

(4.38)

where Φ(x, ξ ) = x ⋅ ξ , see the subsequent formula (4.40). The first step will be to show that the mapping a �→ TΦ a = e2πiτΦ ∗ a is a homeop,q morphism on M1⊗v (ℝ2d ), 1 ≤ p, q ≤ ∞. s As a consequence, the results for pseudodifferential operators in the Weyl form with symbols in the modulation spaces above are valid for any other τ-pseudodifferential operator. Let us recall the distribution Fourier transform of the generalized chirp below (cf. [145, Appendix A, Theorem 2]): Proposition 4.3.2. Let B a real, invertible, symmetric n×n matrix, and let FB (x) = eπix⋅Bx . Then the distribution Fourier transform of FB is given by ̂B (ξ ) = eπi♯(B)/4 | det B|e−πiξ ⋅B F

−1

ξ

,

(4.39)

where ♯(B) is the number of positive eigenvalues of B minus the number of negative eigenvalues.

4.3 Pseudodifferential operators on modulation spaces | 213

If we choose 0d×d Id×d

Id×d ) 0d×d

B = B−1 = (

(hence n = 2d) then formula (4.39) becomes ℱ (e

2πiζ1 ⋅ζ2

̂B (z1 , z2 ) = e−2πiz1 ⋅z2 . )(z1 , z2 ) = F

(4.40)

We now show that FB belongs to W(ℱ L1 , L∞ )(ℝ2d ). 2

Proposition 4.3.3. The chirp function φ(t) = eπit , t ∈ ℝd , belongs to W(ℱ L1 , L∞ )(ℝd ). The function F(ζ1 , ζ2 ) = e2πiζ1 ⋅ζ2 belongs to W(ℱ L1 , L∞ )(ℝ2d ). Proof. We fix attention on F, the argument for φ is similar. Consider the Gaussian func2 2 tion g(ζ1 , ζ2 ) = e−πζ1 e−πζ2 as window function to compute the W(ℱ L1 , L∞ )-norm. Then we have 󵄩 󵄩 ‖F‖W(ℱ L1 ,L∞ )(ℝ2d ) = sup 󵄩󵄩󵄩ℱ (FTu g)󵄩󵄩󵄩L1 (ℝ2d ) . 2d u∈ℝ

Let us compute ℱ (FTu g)(z). For z = (z1 , z2 ) by using (4.40), ̂ 1 , z2 ) ℱ (FTu g)(z1 , z2 ) = (ℱ (F) ∗ M−u g)(z 2

2

= ∫ e−2πi(z1 −y1 )⋅(z2 −y2 ) e−2πi(u1 ,u2 )⋅(y1 ,y2 ) e−πy1 e−πy2 dy1 dy2 ℝ2d 2

2

= e−2πiz1 ⋅z2 ∫ e−2πiy1 ⋅y2 +2πi(z2 ⋅y1 +z1 ⋅y2 )−2πi(u1 ⋅y1 +u2 ⋅y2 ) e−πy1 e−πy2 dy1 dy2 ℝ2d 2

2

= e−2πiz1 ⋅z2 ∫ e2πi(z1 ⋅y2 −u2 ⋅y2 ) e−πy2 ( ∫ e−2πiy1 ⋅(y2 −z2 +u1 ) e−πy1 dy1 ) dy2 ℝd

ℝd

= e−2πiz1 ⋅z2 ∫ e−2πiy2 ⋅(u2 −z1 ) e

−πy22

2

e−π(y2 −z2 +u1 ) dy2

ℝd Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

2

2

= e−2πiz1 ⋅z2 e−π(u1 −z2 ) ∫ e−2πiy2 ⋅(u2 −z1 ) e−2π(y2 +(u1 −z2 )⋅y2 ) dy2 ℝd 2

π

2

= e−2πiz1 ⋅z2 e−π(u1 −z2 ) + 2 (u1 −z2 ) ∫ e−2πiy2 ⋅(u2 −z1 ) e−2π(y2 + ℝd π

2

2

= e−2πiz1 ⋅z2 e− 2 (u1 −z2 ) ℱ (T− u1 −z2 e−2π|⋅| )(u2 − z1 ) π

2

= 2−d/2 e−2πiz1 ⋅z2 e− 2 (u1 −z2 ) e

2 − π2 (u2 −z1 )2 π

eπi(u1 −z2 )⋅(u2 −z1 ) 2

π

2

= 2−d/2 eπi(u1 ⋅u2 −z1 ⋅z2 −u1 ⋅z1 −u2 ⋅z2 ) e− 2 (u1 −z2 ) e− 2 (u2 −z1 ) . Hence π 2 󵄩󵄩 󵄩 −d/2 󵄩 󵄩󵄩e− 2 |⋅| 󵄩󵄩󵄩 1 = C, 󵄩󵄩ℱ (FTu g)󵄩󵄩󵄩L1 = 2 󵄩 󵄩L

u1 −z2 2 ) 2

dy2

214 | 4 Pseudodifferential operators for a constant C independent of the variable u and, consequently, 󵄩 󵄩 sup 󵄩󵄩󵄩ℱ (FTu g)󵄩󵄩󵄩L1 < ∞

u∈ℝ2d

as desired. Corollary 4.3.4. For ζ = (ζ1 , ζ2 ), consider the function FJ (ζ ) = F(Jζ ) = e−2πiζ1 ⋅ζ2 , where the symplectic matrix J is defined in (1.1). Then FJ ∈ W(ℱ L1 , L∞ )(ℝ2d ). Proof. The result immediately follows by Proposition 4.3.3 and by the dilation properties for modulation spaces (2.107). Corollary 4.3.5. For 0 < |τ| ≤ 1, the function e2πiτΦ , Φ(x, ξ ) = x ⋅ ξ , is in M 1,∞ (ℝ2d ). Proof. By Proposition 4.3.3, the function e2πiΦ is in M 1,∞ (ℝ2d ), being the Fourier transform of the function FJ defined in Corollary 4.3.4. The dilation properties for modulation spaces in Theorem 2.6.6 give the claim. Proposition 4.3.6. For 0 < |τ| ≤ 1, the mapping a �→ TΦ a = e2πiτΦ ∗ a is a homeomorp,q phism on M1⊗v (ℝ2d ), 1 ≤ p, q ≤ ∞. s

Proof. Using the convolution properties for modulation spaces in (2.81) with p1 = p, q1 = q, p2 = 1 and q2 = ∞, we get 󵄩 󵄩 󵄩 󵄩 ‖TΦ a‖M p,q = 󵄩󵄩󵄩e2πiτΦ ∗ a󵄩󵄩󵄩M p,q ≲ 󵄩󵄩󵄩e2πiτΦ 󵄩󵄩󵄩M 1,∞ ‖a‖M p,q , 1⊗vs

1⊗vs

1⊗vs

as desired.

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From now on, we shall focus only on boundedness properties for Weyl operators, but the same results hold for any τ-pseudodifferential operator by the previous observations. The key idea is the study of boundedness properties for the Wigner distribution.

4.4 Continuity results for the (cross-)Wigner distribution An important issue related to such a distribution is the continuity of the map (f1 , f2 ) �→ W(f1 , f2 ) in the relevant Banach spaces. The basic result in this connection is the Moyal’s identity (1.82) 󵄩󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩L2 (ℝ2d ) = ‖f1 ‖L2 (ℝd ) ‖f2 ‖L2 (ℝd ) ,

f1 , f2 ∈ L2 (ℝd ).

Beside L2 , the time–frequency concentration of signals is often measured by the modp,q ulation space norm Mm , 1 ≤ p, q ≤ ∞, for a suitable weight function m. Although several particular sufficient conditions have appeared in this connection in the literature, the general problem was solved in the paper [75], by providing the full range of modulation spaces in which the continuity of the cross-Wigner distribution W(f , g) holds,

4.4 Continuity results for the (cross-)Wigner distribution

| 215

as a function of f , g. The consequences of these results are manifold: new bounds for the short-time Fourier transform and the ambiguity function, boundedness results for pseudodifferential (in particular, localization) operators, and properties of the Cohen class (see the next sections). Theorem 4.4.1. Assume pi , qi , p, q ∈ [1, ∞], s ∈ ℝ, are such that pi , qi ≤ q,

i = 1, 2,

(4.41)

and 1 1 1 1 + ≥ + , p1 p2 p q p ,q

1 1 1 1 + ≥ + . q1 q2 p q

p ,q

(4.42)

p,q Then if f1 ∈ Mv|s|1 1 (ℝd ) and f2 ∈ Mvs2 2 (ℝd ), we have W(f1 , f2 ) ∈ M1⊗v (ℝ2d ) and s

󵄩󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≲ ‖f1 ‖Mvp1 ,q1 ‖f2 ‖Mvp2 ,q2 . 1⊗vs s |s|

(4.43)

Conversely, assume that there exists a constant C > 0 such that 󵄩󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≤ C‖f1 ‖M p1 ,q1 ‖f2 ‖M p2 ,q2 ,

∀f1 , f2 ∈ 𝒮 (ℝ2d ).

(4.44)

Then (4.41) and (4.42) must hold. In this section we prove Theorem 4.4.1. We will focus separately on the sufficient and necessary part in the statement. The sufficient conditions were proved by Toft in [287, Theorem 4.2] (cf. also [286, Theorem 4.1] for modulation spaces without weights) under the conditions p ≤ pi , qi ≤ q,

i = 1, 2,

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

(4.45)

Theorem 4.4.1 shows that the sufficient conditions can be widened and such an extension is sharp. Theorem 4.4.2 (Sufficient conditions). If p1 , q1 , p2 , q2 , p, q ∈ [1, ∞] are indices which p ,q p ,q satisfy (4.41) and (4.42), s ∈ ℝ, f1 ∈ Mv|s|1 1 (ℝd ) and f2 ∈ Mvs2 2 (ℝd ), then W(f1 , f2 ) ∈ p,q M1⊗v (ℝ2d ), and the estimate (4.43) holds true. s

Proof. We first study the case p, q < ∞. Let g ∈ 𝒮 (ℝd ) and set Φ = W(g, g) ∈ 𝒮 (ℝ2d ). If ζ = (ζ1 , ζ2 ) ∈ ℝ2d , we write ζ ̃ = (ζ2 , −ζ1 ). Then from Lemma 1.3.37, 󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨VΦ (W(f1 , f2 ))(z, ζ )󵄨󵄨󵄨 = 󵄨󵄨󵄨Vg f2 (z + 󵄨󵄨

ζ ̃ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 )󵄨󵄨V f (z − 2 󵄨󵄨󵄨󵄨󵄨󵄨 g 1

ζ ̃ 󵄨󵄨󵄨󵄨 )󵄨. 2 󵄨󵄨󵄨

(4.46)

216 | 4 Pseudodifferential operators Consequently, q

p p 1/q 󵄨󵄨 p ζ ̃ 󵄨󵄨󵄨 󵄨󵄨󵄨 ζ ̃ 󵄨󵄨󵄨 󵄨 󵄩 󵄩󵄩 sq 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≍ ( ∫ ( ∫ 󵄨󵄨󵄨Vg f2 (z + )󵄨󵄨󵄨 󵄨󵄨󵄨Vg f1 (z − )󵄨󵄨󵄨 dz) ⟨ζ ⟩ dζ ) . 1⊗vs 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 2d 2d ℝ

ℝ

After the change of variables z �→ z − ζ ̃ /2, the integral over z becomes the convolution (|Vg f2 |p ∗ |(Vg f1 )∗ |p )(ζ ̃ ), and observing that (1 ⊗ vs )(z, ζ ) = ⟨ζ ⟩s = vs (ζ ) = vs (ζ ̃ ), we obtain 1/p

q 󵄩󵄩 󵄩 q p 󵄨 ∗ 󵄨p 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≍ (∬(|Vg f2 | ∗ 󵄨󵄨󵄨(Vg f1 ) 󵄨󵄨󵄨 ) p (ζ ̃ )vs (ζ ̃ ) dζ ) 1⊗vs

ℝ2d

1

󵄩 󵄨 󵄨p 󵄩 = 󵄩󵄩󵄩|Vg f2 |p ∗ 󵄨󵄨󵄨(Vg f1 )∗ 󵄨󵄨󵄨 󵄩󵄩󵄩 p q . Lvpps

Hence 󵄩󵄩 󵄩p 󵄩 p 󵄨 ∗ 󵄨p 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≍ 󵄩󵄩󵄩|Vg f2 | ∗ 󵄨󵄨󵄨(Vg f1 ) 󵄨󵄨󵄨 󵄩󵄩󵄩 1⊗vs

q

Lvpps

.

(4.47)

Case p ≤ q < ∞. Step 1. Here we prove the desired result in the case p ≤ pi , qi , i = 1, 2. Suppose first that (4.45) are satisfied (and hence pi , qi ≤ q, i = 1, 2). Since q/p ≥ 1, we can apply Young’s inequality for mixed-normed spaces and majorize (4.47) as follows: 󵄩󵄩 󵄩 󵄨 p p ∗ 󵄨p 󵄩󵄩W(f1 , f2 )‖M p,q ≲󵄩󵄩󵄩|Vg f2 | ‖Lrv2 ,s2 ‖󵄨󵄨󵄨(Vg f1 ) 󵄨󵄨󵄨 ‖Lrv1 ,s1 ps p|s| 1⊗vs 󵄩󵄩 󵄩 p p = 󵄩󵄩|Vg f1 | ‖Lrv1 ,s1 󵄩󵄩󵄩|Vg f2 | ‖Lr2 ,s2 vps p|s| = ‖Vg f1 ‖ppr1 ,ps1 ‖Vg f2 ‖ppr2 ,ps2 ,

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Lv|s|

Lvs

for every 1 ≤ r1 , r2 , s1 , s2 ≤ ∞ such that 1 1 1 p 1 + = + =1+ . r1 r2 s1 s2 q

(4.48)

Choosing ri = pi /p ≥ 1, si = qi /p ≥ 1, i = 1, 2, the indices’ relation (4.48) becomes (4.45), and we obtain 󵄩󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≲ ‖Vg f1 ‖Lpv 1 ,q1 ‖Vg f2 ‖Lpv 2 ,q2 ≍ ‖f1 ‖Mvp1 ,q1 ‖f2 ‖Mvp2 ,q2 . 1⊗vs s s |s| |s| Now, still assume p ≤ pi , qi , i = 1, 2 but 1 1 1 1 + ≥ + , p1 p2 p q

1 1 1 1 + = + , q1 q2 p q

4.4 Continuity results for the (cross-)Wigner distribution

| 217

(hence pi , qi ≤ q, i = 1, 2). We set u1 = tp1 , and look for t ≥ 1 (hence u1 ≥ p1 ) such that 1 1 1 1 + = + , u1 p2 p q which gives 0
0 since p ≤ p2 . Hence the previous part of the proof gives 󵄩󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≲ ‖f1 ‖Mvu1 ,q1 ‖f2 ‖Mvp2 ,q2 1⊗vs s |s| ≲ ‖f1 ‖Mvp1 ,q1 ‖f2 ‖M p2 ,q2 |s|

vs

where the last inequality follows by inclusion relations for modulations spaces p ,q u ,q Mvs1 1 (ℝd ) ⊆ Mvs1 1 (ℝd ) for p1 ≤ u1 . The general case, when 1 1 1 1 + ≥ + , p1 p2 p q

1 1 1 1 + ≥ + , q1 q2 p q

can be treated analogously. Step 2. Assume now that pi , qi ≤ q, i = 1, 2, and they satisfy relation (4.42). If at least one of the indices p1 , p2 is less than p, assume, for instance, p1 ≤ p, whereas p ≤ q1 , q2 , then we proceed as follows. We choose u1 = p, u2 = q, and deduce by the results in Step 1 (with p1 = u1 and p2 = u2 ) that

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󵄩󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≲ ‖f1 ‖Mvu1 ,q1 ‖f2 ‖Mvu2 ,q2 ≲ ‖f1 ‖Mvp1 ,q1 ‖f2 ‖Mvp2 ,q2 1⊗vs s s |s| |s| where the last inequality follows by inclusion relations for modulation spaces, since p1 ≤ u1 = p and p2 ≤ u2 = q. Similarly, we argue when at least one of the indices q1 , q2 is less than p and p ≤ p1 , p2 or when at least one of the indices q1 , q2 is less than p and at least one of the indices p1 , p2 is less than p. The remaining case p ≤ pi , qi ≤ q is treated in Step 1. Case p < q = ∞. The argument is similar to the case p ≤ q < ∞. Case p = q = ∞. We use (4.46) and the submultiplicative property of the weight vs to get 󵄨󵄨 ζ ̃ 󵄨󵄨󵄨󵄨󵄨󵄨 ζ ̃ 󵄨󵄨󵄨 󵄩󵄩 󵄩 󵄨 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M ∞ = sup 󵄨󵄨󵄨Vg f2 (z + )󵄨󵄨󵄨󵄨󵄨󵄨Vg f1 (z − )󵄨󵄨󵄨vs (ζ ) 1⊗vs 󵄨 2 󵄨󵄨󵄨󵄨 2 󵄨󵄨 z,ζ ∈ℝ2d 󵄨 󵄨󵄨 󵄨󵄨 󵄨 = sup 󵄨󵄨󵄨󵄨󵄨󵄨Vg f2 (z)󵄨󵄨󵄨󵄨󵄨󵄨(Vg f1 )∗ (z − ζ ̃ )󵄨󵄨󵄨vs (ζ ) 2d z,ζ ∈ℝ

218 | 4 Pseudodifferential operators 󵄨 󵄨󵄨 󵄨󵄨 = sup 󵄨󵄨󵄨󵄨󵄨󵄨Vg f2 (z)󵄨󵄨󵄨󵄨󵄨󵄨(Vg f1 )∗ (z − ζ ̃ )󵄨󵄨󵄨vs (ζ ̃ ) 2d z,ζ ∈ℝ

󵄨 󵄨 ≤ sup (‖Vg f1 v|s| ‖∞ 󵄨󵄨󵄨Vg f2 (z)vs (z)󵄨󵄨󵄨) = ‖Vg f1 v|s| ‖∞ ‖Vg f2 vs ‖∞ z∈ℝ2d

≍ ‖f ‖Mv∞ ‖g‖Mv∞ ≤ ‖f ‖Mvp1 ,q1 ‖f ‖M p2 ,q2 , |s|

s

|s|

vs

for every 1 ≤ pi , qi ≤ ∞, i = 1, 2. Notice that in this case conditions (4.41) and (4.44) are trivially satisfied. Case p > q. Using the inclusion relations for modulation spaces, we majorize 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≲ 󵄩󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M q,q ≲ ‖f1 ‖Mvp1 ,q1 ‖f2 ‖Mvp2 ,q2 1⊗vs 1⊗vs s |s| for every 1 ≤ pi , qi ≤ q, i = 1, 2. Here we have applied the case p ≤ q with p = q. Notice that in this case condition (4.44) is trivially satisfied, since from p1 , qi ≤ q we infer 1/p1 + 1/p2 ≥ 1/q + 1/q, 1/q1 + 1/q2 ≥ 1/q + 1/q. This concludes the proof. Remark 4.4.3. (i) The result of Theorem 4.4.2 can be extended to more general weights. In particular, it holds for polynomial weights satisfying relation (4.10) in [287]. Hence our result extends Toft’s result [287, Theorem 4.2] (cf. also [286, Theorem 4.1] for modulation spaces without weights). Other examples of suitable weights are given by β (sub-)exponential weights of the type v(z) = eα|z| for α > 0 and 0 < β ≤ 1. (ii) The particular case p = 1, 1 ≤ q ≤ ∞, p1 = q1 = 1, p2 = q2 = q, s ≥ 0, was already proved in [66, Proposition 2.2]. (iii) Among modulation spaces, the following well-known function spaces occur: Weighted L2 -spaces: if vs (x) = ⟨x⟩s , then Mv2s ⊗1 (ℝd ) = L2s (ℝd ) = {f : f (x)⟨x⟩s ∈ L2 (ℝd )};

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Sobolev spaces: for s ∈ ℝ, vs (ξ ) = ⟨ξ ⟩s , 2 M1⊗v (ℝd ) = H s (ℝd ) = {f : f ̂(ξ )⟨ξ ⟩s ∈ L2 (ℝd )}; s

Shubin–Sobolev spaces [265] (see their definition in (4.66) below): if vs (x, ξ ) = ⟨(x, ξ )⟩s , then Mv2s (ℝd ) = L2s (ℝd ) ∩ H s (ℝd ) = Qs (ℝd ). For pi = qi = p = q = 2, i = 1, 2, we obtain the following continuity result for the cross-Wigner distribution acting between Shubin spaces and Sobolev spaces: For s ≥ 0, f1 , f2 ∈ Qs (ℝd ), the cross-Wigner distribution W(f1 , f2 ) is in H s (ℝ2d ) with 󵄩󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩H s (ℝ2d ) ≲ ‖f1 ‖Qs (ℝd ) ‖f2 ‖Qs (ℝd ) .

4.4 Continuity results for the (cross-)Wigner distribution

| 219

(iv) Continuity properties of the cross-Wigner distribution on modulation spaces with different weight functions can be inferred using the techniques of Theorem 4.4.2 and the Young-type inequalities for weighted spaces shown by Toft et al. in [292, Theorems 2.2–2.5]. The estimate in (4.43) can be slightly improved if s ≥ 0. Precisely, we have the following result. Theorem 4.4.4. If p1 , q1 , p2 , q2 , p, q ∈ [1, ∞] are indices which satisfy (4.41) and (4.42), p ,q p ,q p,q s ≥ 0, f1 ∈ Mvs1 1 (ℝd ) and f2 ∈ Mvs2 2 (ℝd ), then W(f1 , f2 ) ∈ M1⊗v (ℝ2d ), with s

󵄩󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≲ ‖f1 ‖M p1 ,q1 ‖f2 ‖Mvp2 ,q2 + ‖f1 ‖Mvp1 ,q1 ‖f2 ‖M p2 ,q2 . 1⊗vs s s Proof. The proof is similar to that of Theorem 4.4.2, but in this case from the estimate (4.47) we proceed by using vs (z) ≲ vs (z − w) + vs (w),

s≥0

(with sp in place of s) instead of vs (z) ≲ v|s| (z − w)vs (w). If, in particular, we consider the Wigner distribution W(f , f ), then Theorem 4.4.4 can be rephrased as follows. Corollary 4.4.5. Assume s ≥ 0, p1 , q1 , p, q ∈ [1, ∞] are such that 2 min{

1 1 1 1 , }≥ + . p1 q1 p q

p ,q

p,q If f ∈ Mvs1 1 (ℝd ), then the Wigner distribution W(f , f ) is in M1⊗v (ℝ2d ), with s

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󵄩󵄩 󵄩 󵄩󵄩W(f , f )󵄩󵄩󵄩M p,q (ℝ2d ) ≲ ‖f ‖M p1 ,q1 (ℝd ) ‖f ‖Mvp1 ,q1 (ℝd ) . 1⊗vs s

(4.49)

We now prove the sharpness of Theorem 4.4.2 (and Corollary 4.4.5) in the unweighted case s = 0. We need some preliminaries, concerning the modulation norm of Gaussians. The following result is an easy consequence of Proposition 1.3.33, and extends Lemma 1.8 in [286]. Corollary 4.4.6. Consider the generalized Gaussian f defined in (1.126) and the window 2 2 function Φ(x, ξ ) = e−π(x +ξ ) . Then for every 1 ≤ p, q ≤ ∞ we have d

‖f ‖M p,q ≍ ‖VΦ f ‖Lp,q ≍

[(a + 1)(b + 1) + c2 ] p

+ dq − d2

d

[(c2 + ab + a)(c2 + ab + b)] 2q d

d − 2p d

[b2 (a + 1) + b(c2 + a + 1)] 2q [a2 (b + 1) + a(c2 + b + 1)] 2q

.

(4.50)

The cases when p = ∞ or q = ∞ can be obtained by using the rule 1/∞ = 0 in formula (4.50).

220 | 4 Pseudodifferential operators Proof. By Proposition 1.3.33, we can write 1

󵄨 󵄨󵄨 󵄨󵄨VΦ f (z, ζ )󵄨󵄨󵄨 =

d

[(a + 1)(b + 1) + c2 ] 2 ×e

−π

[a(b+1)+c2 ]z12 +[(a+1)b+c2 ]z22 +(b+1)ζ12 +(a+1)ζ22 −2c(z1 ⋅ζ2 +z2 ⋅ζ1 ) (a+1)(b+1)+c2

.

It remains to compute the mixed Lp,q -norm of the previous function. We treat the cases 1 ≤ p, q < ∞. The cases when either p = ∞ or q = ∞ are obtained with obvious modifications. For simplicity, we set c2 + a(b + 1) , (a + 1)(b + 1) + c2 (a + 1) δ= , (a + 1)(b + 1) + c2

c2 + (a + 1)b , (a + 1)(b + 1) + c2 c σ= . (a + 1)(b + 1) + c2

α=

β=

(b + 1) , (a + 1)(b + 1) + c2

γ=

(4.51) (4.52)

Hence ‖VΦ f ‖Lp,q

q p

=(∫ I e

d − 2p

[(a + 1)(b + 1) + c2 ] 2

−πq(γζ12 +δζ22 )

1 q

dζ1 dζ2 ) =: A,

(4.53)

ℝ2d

2

where I := ∫ℝ2d e−πpαz1 −πpβz2 e2πpσ(z1 ⋅ζ2 +z2 ⋅ζ1 ) dz1 dz2 . Now straightforward computations and change of variables yield 2

2

I = ∫ ( ∫ e−πp(αz1 −2σζ2 ⋅z1 ) dz1 )e−πβz2 +2πpσz2 ⋅ζ1 dz2 ℝd ℝd σ2

2

= eπp α ζ2 e

2

πp σβ ζ12

∫e

σ −πp(√αz1 − √α ζ2 )2

dz1 ∫ e

ℝd Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

d

d

d

σ √β

ζ1 )2

dz2

ℝd σ2

d

−πp(√βz2 −

2

= p− 2 α− 2 p− 2 β− 2 eπp α ζ2 e

2

πp σβ ζ12

.

Substituting the value of the integral I in (4.53), we obtain A=p

d d − dp − 2p − 2p

α

β

(∫ e

2

πq σβ ζ12 −πqγζ12

ℝd

=p

d d − dp − 2p − 2p

α

β

ℝd

(∫ e

2

−πq(γ− σβ )ζ12

dζ1 ∫ e

ℝd

=p

d d − dp − 2p − 2p − dq

α

β

q

dζ1 ∫ e

2

πq σα ζ22 −πqδζ22

ℝd d − 2q

σ2 (γ − ) β

1 q

2

−πq(δ− σα )ζ22

d − 2q

σ2 (γ − ) α

.

)

dζ2 )

1 q

4.4 Continuity results for the (cross-)Wigner distribution

| 221

Finally, the goal is attained by substituting in A the values of the parameters α, β, γ, δ, σ in (4.51) and (4.52) and observing that − d2

‖f ‖M p,q ≍ ‖VΦ f ‖Lp,q = A[(a + 1)(b + 1) + c2 ]

.

This concludes the proof. We have now all the tools to compute the modulation norm of the (cross-)Wigner distribution W(φ, φ√λ ) in (1.129). Precisely, setting in formula (4.50) the values a = aλ , b = bλ , c = cλ , where aλ , bλ , cλ are defined as aλ =

4λ 1+λ

bλ =

4 1+λ

cλ =

2(1 − λ) , 1+λ

and making easy simplifications we attain the following result. Corollary 4.4.7. For λ > 0 consider the (cross-)Wigner distribution W(φ, φ√λ ) defined in (1.129) (cf. Lemma 1.3.34). Then d

[(2λ + 1)(λ + 2)] 2q 󵄩󵄩 󵄩 󵄩󵄩W(φ, φ√λ )󵄩󵄩󵄩M p,q ≍ d d d − λ 2q (1 + λ) 2 p

d − 2p

.

(4.54)

The cases when p = ∞ or q = ∞ can be obtained by using the rule 1/∞ = 0 in formula (4.54). Theorem 4.4.8 (Necessary conditions). Consider p1 , p2 , q1 , q2 , p, q ∈ [1, ∞]. Assume that there exists a constant C > 0 such that 󵄩󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≤ C‖f1 ‖M p1 ,q1 ‖f2 ‖M p2 ,q2 ,

∀f1 , f2 ∈ 𝒮 (ℝ2d ),

(4.55)

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

then (4.41) and (4.42) must hold. Proof. Let us first demonstrate the necessity of (4.42). We consider the dilated Gaus2 sians φ(√λx), with φ(x) = e−πx . An easy computation (see also [160, formula (4.20)]) shows that d

d

W(φ√λ , φ√λ )(x, ξ ) = 2 2 λ− 2 φ√2λ (x)φ√2/λ (ξ ), hence d 󵄩󵄩 󵄩 −d 󵄩󵄩W(φ√λ , φ√λ )󵄩󵄩󵄩M p,q (ℝ2d ) = 2 2 λ 2 ‖φ√2λ ‖M p,q (ℝd ) ‖φ√2/λ ‖M p,q (ℝd ) .

Now, from Lemma 2.6.1, d

‖φ√λ ‖M r,s ≍ λ− 2r (λ + 1)

− d2 (1− q1 − p1 )

.

222 | 4 Pseudodifferential operators The assumption (4.55) in this case becomes − d2 (1− q1 − p1 )

d

λ− 2 (λ + 1) ≲λ

− 2pd

1

− d2 (1− q1 − p1 )

(λ−1 + 1)

− d2 (1− q1 − p1 ) − 2pd

(1 + λ)

1

λ

1

2

− d2 (1− q1 − p1 )

(1 + λ)

2

2

,

and, letting λ → +∞, we obtain 1 1 1 1 + ≤ + p q q1 q2 whereas, for λ → 0+ , 1 1 1 1 + ≤ + , p q p1 p2 so that (4.42) must hold. It remains to prove the sharpness of (4.41). We first show the conditions p2 , q2 ≤ q. We test (4.55) on the (cross-)Wigner distribution W(φ, φ√λ ) defined in (1.129), that is, 󵄩󵄩 󵄩 󵄩󵄩W(φ, φ√λ )󵄩󵄩󵄩M p,q (ℝ2d ) ≲ ‖φ‖M p1 ,q1 (ℝd ) ‖φ√λ ‖M p2 ,q2 (ℝd ) . Using Corollary 4.4.7, the previous estimate can be rephrased as d

[(2λ + 1)(λ + 2)] 2q d 2q

λ (1 + λ)

d d − 2 p

d − 2p

≲λ

− 2pd

2

λ

− d2 (1− q1 − p1 ) 2

2

,

∀λ > 0.

Letting λ → +∞, we obtain q2 ≤ q

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

whereas, for λ → 0+ , p2 ≤ q. The conditions p1 , q1 ≤ q then follow by using the cross-Wigner property, W(φ√λ , φ)(x, ξ ) = W(φ, φ√λ )(x, ξ ), so that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩W(φ√λ , φ)󵄩󵄩󵄩M p,q (ℝ2d ) = 󵄩󵄩󵄩W(φ, φ√λ )󵄩󵄩󵄩M p,q (ℝ2d ) = 󵄩󵄩󵄩W(φ, φ√λ )󵄩󵄩󵄩M p,q (ℝ2d ) , and applying the same argument as before.

4.4 Continuity results for the (cross-)Wigner distribution

| 223

4.4.1 Continuity results for the short-time Fourier transform and the ambiguity function The bounds obtained for the Wigner distribution can be transferred into new estimates for other time–frequency representations such that the STFT or the ambiguity function. Precisely, given f , g ∈ L2 (ℝd ), we recall the definition of the (cross-)ambiguity function in Definition 1.3.1: x x A(f1 , f2 )(x, ξ ) = ∫ e−2πit⋅ξ f1 (t + )f2 (t − ) dt. 2 2 ℝd

In Lemma 1.3.11 it was shown that the Wigner distribution is the symplectic Fourier transform of the ambiguity function, in other words, we can write W(f1 , f2 )(x, ξ ) = ℱ𝒰 A(f1 , f2 )(x, ξ ),

f1 , f2 ∈ L2 (ℝd ),

(4.56)

where the operator 𝒰 is the rotation 𝒰 F(x, ξ ) = F(ξ , −x) of a function F on ℝ2d . We need the following norm equivalence. Lemma 4.4.9. For s ∈ ℝ, 1 ≤ p, q ≤ ∞, the following equivalence holds: 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q = 󵄩󵄩󵄩A(f1 , f2 )󵄩󵄩󵄩W(ℱ Lp ,Lqv ) ≍ ‖Vℐf2 f1 ‖W(ℱ Lp ,Lqv ) , s 1⊗vs s where ℐ f2 (x) = f2 (−x). Proof. Let us observe that the weight vs , s ∈ ℝ, is symmetric in each coordinate: vs (x, ξ ) = vs (x, −ξ ) = vs (−x, ξ ) = vs (−x, −ξ ).

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Using (4.56), the connection between modulation spaces and the symmetry of the weights vs , we can write 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q = 󵄩󵄩󵄩ℱ𝒰 A(f1 , f2 )󵄩󵄩󵄩M p,q = 󵄩󵄩󵄩𝒰 A(f1 , f2 )󵄩󵄩󵄩W(ℱ Lp ,Lqv ) 1⊗vs 1⊗vs s 󵄩 󵄩 = 󵄩󵄩󵄩A(f1 , f2 )󵄩󵄩󵄩W(ℱ Lp ,Lqv ) . s Now, Lemma 1.3.5 lets us write A(f1 , f2 )(x, ξ ) = 2d eπix⋅ξ Vℐf2 f1 (x, ξ ). It was proved in Proposition 4.3.3 that the function F(x, ξ ) = eπix⋅ξ is in the Wiener amalgam space W(ℱ L1 , L∞ )(ℝ2d ). This means that, by the product properties for Wiener amalgam spaces, for every s ∈ ℝ, 󵄩󵄩 󵄩 󵄩󵄩A(f1 , f2 )󵄩󵄩󵄩W(ℱ Lp ,Lqv ) ≲ ‖F‖W(ℱ L1 ,L∞ ) ‖Vℐf2 f1 ‖W(ℱ Lp ,Lqv ) s s

224 | 4 Pseudodifferential operators ̄ ξ ) = e−πix⋅ξ ∈ W(ℱ L1 , L∞ ) as well, with ‖F‖ ̄ and, since F(x, W(ℱ L1 ,L∞ ) = ‖F‖W(ℱ L1 ,L∞ ) , we can analogously write 󵄩 󵄩 ‖Vℐf2 f1 ‖W(ℱ Lp ,Lqv ) ≲ ‖F‖W(ℱ L1 ,L∞ ) 󵄩󵄩󵄩A(f1 , f2 )󵄩󵄩󵄩W(ℱ Lp ,Lqv ) . s

s

This proves the desired result. These observations, together with the Wigner property W(f1 , f2 )(x, ξ ) = W(f2 , f1 ), let us translate the sufficient and necessary conditions for Wigner distributions in terms of STFT acting from modulation spaces to Wiener amalgam spaces. Notice that the following two corollaries also hold for the ambiguity function A(f1 , f2 ) in place of the STFT. Corollary 4.4.10. Consider s ∈ ℝ and assume that p1 , p2 , q1 , q2 , p, q ∈ [1, ∞] satisfy conp ,q p ,q ditions (4.41) and (4.42). Then if f1 ∈ Mv|s|1 1 (ℝd ) and f2 ∈ Mvs2 2 (ℝd ), we have Vf1 f2 ∈ W(ℱ Lp , Lqvs )(ℝ2d ) with

‖Vf1 f2 ‖W(ℱ Lp ,Lqv )(ℝ2d ) ≲ ‖f1 ‖Mvp1 ,q1 (ℝd ) ‖f2 ‖M p2 ,q2 (ℝd ) . s

vs

s

Conversely, assume that there exists a constant C > 0 such that ∀f1 , f2 ∈ 𝒮 (ℝ2d ).

‖Vf1 f2 ‖W(ℱ Lp ,Lq )(ℝ2d ) ≤ C‖f1 ‖M p1 ,q1 (ℝd ) ‖f2 ‖M p2 ,q2 (ℝd ) , Then (4.41) and (4.42) must hold.

The previous result has many special and interesting cases. Let us just give a flavor of the main important ones. Corollary 4.4.11. Assume that p1 , p2 , p, q ∈ [1, ∞] satisfy p1 , p2 ≤ q, p

1 1 1 1 + ≥ + . p1 p2 p q

(4.57)

p

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

For s ∈ ℝ, if f1 ∈ Mv|s|1 (ℝd ) and f2 ∈ Mvs2 (ℝd ), we have Vf1 f2 ∈ W(ℱ Lp , Lqvs )(ℝ2d ) with ‖Vf1 f2 ‖W(ℱ Lp ,Lqv )(ℝ2d ) ≲ ‖f1 ‖Mvp1 s

|s|

(ℝd ) ‖f2 ‖Mvs2 (ℝd ) . p

Conversely, assume that there exists a constant C > 0 such that ‖Vf1 f2 ‖W(ℱ Lp ,Lq )(ℝ2d ) ≤ C‖f1 ‖M p1 (ℝd ) ‖f2 ‖M p2 (ℝd ) ,

∀f1 , f2 ∈ 𝒮 (ℝ2d ).

Then (4.57) must hold. Remark 4.4.12. The previous result holds also for the cross-Wigner distribution if we replace ‖Vf1 f2 ‖W(ℱ Lp ,Lqv )(ℝ2d ) by ‖W(f1 , f2 )‖M p,q (ℝ2d ) . s

1⊗vs

The unweighted case s = 0 leads to a surprising refinement of Lemma 4.4.9.

4.4 Continuity results for the (cross-)Wigner distribution

| 225

Lemma 4.4.13. For 1 ≤ p, q ≤ ∞, we have the equivalence 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩M p,q ≍ 󵄩󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩W(ℱ Lp ,Lq ) , for every f1 , f2 ∈ 𝒮 󸀠 (ℝd ) such that the norms above are bounded. Proof. We use again the connection between cross-Wigner distribution and STFT in Lemma 1.3.5. For a > 0, define the dilation operator Da on ℝd by Da f (x) = ad/2 f (ax), x ∈ ℝd , so that the cross-Wigner distribution W(f1 , f2 ) can be written as W(f1 , f2 )(x, ξ ) = D2 (eπix⋅ξ Vℐf2 f1 )(x, ξ ). Using the dilation properties for modulation spaces in Corollary 2.6.4, we estimate, similarly as in the proof of Lemma 4.4.9, 󵄩󵄩 󵄩 2d(1/p+1/q−1) 󵄩 󵄩󵄩eπix⋅ξ Vℐf f1 󵄩󵄩󵄩 󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩W(ℱ Lp ,Lq ) ≤ 2 p q 󵄩 2 󵄩W(ℱ L ,L ) 󵄩󵄩 πix⋅ξ 󵄩󵄩 ≲ 󵄩󵄩e 󵄩󵄩W(ℱ L1 ,L∞ ) ‖Vℐf2 f1 ‖W(ℱ Lp ,Lq ) ≲ ‖Vℐf2 f1 ‖W(ℱ Lp ,Lq ) . Conversely, x ξ Vℐf2 f1 (x, ξ ) = 2−d e−πix⋅ξ W(f1 , f2 )( , ), 2 2 and the same arguments as before with the dilation D1/2 give 󵄩 󵄩 󵄩 󵄩 ‖Vℐf2 f1 ‖W(ℱ Lp ,Lq ) ≲ 󵄩󵄩󵄩e−πix⋅ξ 󵄩󵄩󵄩W(ℱ L1 ,L∞ ) 󵄩󵄩󵄩W(f1 , f2 )󵄩󵄩󵄩W(ℱ Lp ,Lq ) , hence ‖Vℐf2 f1 ‖W(ℱ Lp ,Lq ) ≍ ‖W(f1 , f2 )‖W(ℱ Lp ,Lq ) . Finally, it is immediate to check that ‖Vℐf2 f1 ‖W(ℱ Lp , Lq ) = ‖Vf2 f1 ‖W(ℱ Lp , Lq ) . The result is then a consequence of Lemma 4.4.9.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

If we choose s = 0, p = q󸀠 in Lemma 4.4.9 and apply the previous result, we can refine some Lieb’s integral bounds for the ambiguity function showed in [218]. Corollary 4.4.14. Assume 1 ≤ p1 , p2 , q1 , q2 ≤ q are such that 1 1 + ≥1 p1 p2

1 1 + ≥ 1. q1 q2

(4.58)

If fi ∈ M pi ,qi (ℝd ), i = 1, 2, then the ambiguity function satisfies A(f1 , f2 ) ∈ Lq (ℝ2d ), with 󵄩󵄩 󵄩 󵄩󵄩A(f1 , f2 )󵄩󵄩󵄩Lq (ℝ2d ) ≲ ‖f1 ‖M p1 ,q1 (ℝd ) ‖f2 ‖M p2 ,q2 (ℝd ) . Conversely, assume that there exists a constant C > 0 such that 󵄩󵄩 󵄩 󵄩󵄩A(f1 , f2 )󵄩󵄩󵄩Lq (ℝ2d ) ≤ C‖f1 ‖M p1 ,q1 (ℝd ) ‖f2 ‖M p2 ,q2 (ℝd ) . Then we must have 1 ≤ p1 , p2 , q1 , q2 ≤ q and (4.58).

226 | 4 Pseudodifferential operators Proof. The results in Lemmas 4.4.9 and 4.4.13 imply 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩A(f1 , f2 )󵄩󵄩󵄩M q󸀠 ,q (ℝ2d ) ≍ 󵄩󵄩󵄩A(f1 , f2 )󵄩󵄩󵄩W(ℱ Lq󸀠 ,Lq )(ℝ2d ) .

(4.59)

Using the inclusion relations between Lebesgue and modulation spaces [286, Propo󸀠 󸀠 sition 1.7], we have, for q ≤ 2, M q ,q (ℝd ) ⊆ Lq (ℝd ) and for q ≥ 2, Lq (ℝd ) ⊆ M q ,q (ℝd ). 󸀠 For q ≤ 2, Lq (ℝd ) ⊆ ℱ Lq (ℝd ), and the inclusion relations for Wiener amalgam spaces 󸀠 give, for q ≤ 2, Lq (ℝd ) = W(Lq , Lq )(ℝd ) ⊆ W(ℱ Lq , Lq )(ℝd ), so that M q ,q (ℝd ) ⊆ Lq (ℝd ) ⊆ W(ℱ Lq , Lq )(ℝd ). 󸀠

󸀠

Since ℱ Lq (ℝd ) ⊆ Lq (ℝd ) if q ≥ 2, the inclusion relations for Wiener amalgam spaces 󸀠 let us write W(ℱ Lq , Lq )(ℝ2d ) ⊆ W(Lq , Lq )(ℝ2d ) = Lq (ℝ2d ). Hence, for q ≥ 2, 󸀠

W(ℱ Lq , Lq )(ℝd ) ⊆ Lq (ℝd ) ⊆ M q ,q (ℝd ). 󸀠

󸀠

The sufficient and necessary conditions then follow from the equivalence (4.59), Lemma 4.4.9 and Corollary 4.4.10. Observe that for pi = qi = 2, i = 1, 2, M pi ,qi (ℝd ) = L2 (ℝd ), and we recapture Lieb’s bound, see [218, Theorem 1]. 4.4.2 Pseudodifferential operators on modulation spaces, conclusions

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

In this section we apply Theorem 4.4.1 to the study of pseudodifferential operators on modulation spaces. The key tool is the weak definition of a Weyl operator (4.6). Theorem 4.4.15 below extends [89, Theorem 1.1] to weighted modulation spaces, thus widening the sufficient boundedness conditions presented by Toft in [287, Theorem 4.3]. It is stated for pseudodifferential operators in the Weyl form, but the same result holds for any τ-pseudodifferential operator by the previous observations. Using Theorem 4.4.1, the proof of Theorem 4.4.15 is decidedly simple. Theorem 4.4.15. Assume s ≥ 0, pi , qi , p, q ∈ [1, ∞], i = 1, 2, are such that min{

1 1 1 1 1 1 + , + }≥ 󸀠 + 󸀠 p1 p󸀠2 q1 q2󸀠 p q

(4.60)

q ≤ min{p󸀠1 , q1󸀠 , p2 , q2 }.

(4.61)

and

Then the pseudodifferential operator Opw (a), from 𝒮 (ℝd ) to 𝒮 󸀠 (ℝd ), having symbol a ∈ p ,q p ,q p,q M1⊗v (ℝ2d ), extends uniquely to a bounded operator from ℳvs1 1 (ℝd ) to ℳvs2 2 (ℝd ), with s the estimate 󵄩󵄩 󵄩 p,q ‖f ‖ p ,q 󵄩󵄩Opw (a)f 󵄩󵄩󵄩ℳpv 2 ,q2 ≲ ‖a‖M1⊗v ℳvs1 1 . s s

(4.62)

4.4 Continuity results for the (cross-)Wigner distribution

| 227

Conversely, if (4.62) holds for s = 0 and for every f ∈ 𝒮 (ℝd ), σ ∈ 𝒮 󸀠 (ℝ2d ), then (4.60) and (4.61) must be satisfied. p ,q

p,q Proof. Assume a ∈ M1⊗v (ℝ2d ), f ∈ ℳvs1 1 (ℝd ) are such that (4.60) and (4.61) are sat-

isfied. For g ∈ ℳ

p󸀠2 ,q2󸀠 v−s

s

(ℝd ), Theorem 4.4.1 says that the cross-Wigner distribution is in

p ,q M1⊗v (ℝ2d ), provided that p1 , q1 , p󸀠2 , q2󸀠 ≤ q󸀠 and 󸀠

󸀠

−s

min{1/p1 + 1/p󸀠2 , 1/q1 + 1/q1󸀠 } ≥ 1/p󸀠 + 1/q󸀠 , which are conditions (4.61) and (4.60), respectively. Thereby, there exists a constant C > 0 such that 󵄨󵄨 󵄨 󵄩󵄩 󵄩󵄩 󸀠 󸀠 p,q 󵄨󵄨⟨a, W(g, f )⟩󵄨󵄨󵄨 ≤ ‖a‖M1⊗v p ,q (ℝ2d ) 󵄩 󵄩W(g, f )󵄩󵄩M1⊗v (ℝ2d ) s ≤ C‖f ‖

−s

p ,q ℳvs1 1 (ℝd )

‖g‖

p󸀠 ,q󸀠 2 2 (ℝd ) ℳv−s

.

Since |⟨Opw (a)f , g⟩| = |⟨a, W(g, f )⟩|, this concludes the proof of the sufficient conditions. The arguments for the necessity are similar, by using Theorem 4.4.8. Taking p = ∞, q = 1, p1 = p2 , q1 = q2 , we deduce in particular that for symbols a ∈ M ∞,1 (ℝ2d ) we have boundedness of Opw (a) on M r,s (ℝd ) for any r, s ∈ [1, ∞], and we recapture [163].

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

4.4.3 Localization operators on modulation spaces Time–frequency localization operators are a mathematical instrument to define a restriction of functions to a region in time–frequency plane that is compatible with the uncertainty principle and to extract time–frequency features. In this sense they have been introduced and studied by Daubechies [99] and Ramanathan and Topiwala [246], and they have been extensively investigated as an important mathematical tool in signal analysis and other applications [23, 66, 75, 139, 141, 142, 143, 318, 319]. Localization operators in connection with the Donoho–Stark uncertainty principle were considered in [21, 22]. In other mathematical contexts, time–frequency localization operators have already been used as a quantization procedure (“anti-Wick operators”) by Berezin [16, 265] or as an approximation of pseudodifferential operators (“wave packets”) by Cordoba and Fefferman [94, 145]. Given a function or distribution a on ℝ2d and windows φ1 , φ2 , we define the genφ ,φ eralized anti-Wick operator Aa 1 2 by the (formal) integral Aφa 1 ,φ2 f := ∫ a(x, ξ )Vφ1 f (x, ξ )Mξ Tx φ2 dx dξ , ℝ2d

(4.63)

228 | 4 Pseudodifferential operators whenever this vector-valued integral makes sense. In most cases it is preferable to interpret the integral in a weak sense as ⟨Aφa 1 ,φ2 f , g⟩ = ∫ a(x, ξ )Vφ1 f (x, ξ )⟨Mξ Tx φ2 , g⟩ dx dξ = ⟨a, Vφ1 f Vφ2 g⟩

for f , g ∈ 𝒮 (ℝd ).

(4.64)

Here the brackets ⟨⋅, ⋅⟩ express the duality on a suitable pair of dual spaces B󸀠 × B and extend the inner product on L2 (ℝ2d ). 2 φ ,φ If φ1 (t) = φ2 (t) = e−πt , then Aa = Aa 1 2 is the classical anti-Wick operator and the mapping a → Aa is interpreted as a quantization rule [16, 318]. In this case Aa makes sense as a continuous operator from 𝒮 to 𝒮 󸀠 whenever a ∈ 𝒮 󸀠 (ℝ2d ). For general winφ ,φ dows, Aa 1 2 is often viewed as a localization operator. Taking a to be a characteristic φ ,φ function a = χΩ for Ω ⊆ ℝ2d , then Aa 1 2 evaluates “essentially” the restriction of Vφ1 f to Ω. See [99, 246, 318] for investigations in this direction. φ,φ In the literature only operators of the form Aa are studied because they satisfy certain additional properties required by a quantization rule; for instance, if a ≥ 0, φ,φ φ,φ then Aa is a positive operator, and if a is real-valued, then Aa is self-adjoint. On the φ,φ other hand, if we want to study how Aa depends on φ, then it seems more appropriate to deal with anti-Wick operators with two different windows. Specifically, if φ = φ1 + φ,φ φ ,φ φ ,φ φ ,φ φ ,φ φ ,φ φ2 , then Aa = Aa 1 1 + Aa 2 2 + Aa 1 2 + Aa 2 1 , and it is natural to study Aa 1 2 . In addition, some application in signal analysis suggest the use of windows with different φ ,φ regularities for the analysis f → Vφ1 f and for the synthesis Vφ1 f → Aa 1 2 . In this section we will therefore always work with this simple and natural generalization of the classical anti-Wick operators, presenting some results in [23, 66, 75]. Lemma 4.4.16 (Connection between Weyl and localization operators). If a ∈ 𝒮 󸀠 (ℝ2d ), φ ,φ φ1 , φ2 ∈ 𝒮 (ℝd ), then Aa 1 2 possesses the Weyl symbol σ = a ∗ W(φ2 , φ1 ), in other words, Aφa 1 ,φ2 = Opw (σ),

(4.65)

with

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

σ = a ∗ W(φ2 , φ1 ), where W(φ2 , φ1 ) is the cross-Wigner distribution of φ2 , φ1 . φ ,φ2

Proof. Let f , g ∈ 𝒮 (ℝd ), and a ∈ 𝒮 (ℝ2d ), then Aa 1

is the integral operator with kernel

K(x, y) = ∫ a(z)(π(z)φ2 (x)π(z)φ1 (y)) dz = ∫ a(z)(π(z)φ2 ⊗ π(z)φ1 )(x, y) dz. The connection between the distributional (or integral) kernel K of an operator and its Weyl symbol σ is given by the formula σ = ℱ2 𝒯s K, see (4.5). Since the integrals above are absolutely convergent, Lemma 1.3.9 implies that σ = ℱ2 𝒯s (∫ a(z)π(z)φ2 ⊗ π(z)φ1 dz) = ∫ a(z)ℱ2 𝒯s (π(z)φ2 ⊗ π(z)φ1 ) dz

4.4 Continuity results for the (cross-)Wigner distribution

| 229

= ∫ a(z)W(π(z)φ2 , π(z)φ1 ) dz = ∫ a(z)W(φ2 , φ1 )(⋅ − z) dz = a ∗ W(φ2 , φ1 ). Using a standard approximation argument, we then obtain the equivalence of the operators for all a ∈ 𝒮 󸀠 (ℝ2d ). Lemma 4.4.16 allows us to study localization operators by means of their Weyl symbol. We now present sharp boundedness results for localization operators. Theorem 4.4.17. Assume s ≥ 0, the indices pi , qi , p, q ∈ [1, ∞], i = 1, 2, fulfill relap,q tions (4.60) and (4.61), and consider r ∈ [1, 2]. If a ∈ M1⊗v (ℝ2d ) and φ1 , φ2 ∈ Mvr2s (ℝd ), φ ,φ2

then the localization operator Aa 1

−s

p ,q

p ,q

is continuous from ℳvs1 1 (ℝd ) to ℳvs2 2 (ℝd ) with

󵄩󵄩 φ1 ,φ2 󵄩󵄩 󵄩󵄩Aa 󵄩󵄩op ≲ ‖a‖M p,q ‖φ1 ‖M2sr ‖φ2 ‖Mvr . 1⊗v−s

2s

1,∞ Proof. Using Theorem 4.4.1 for φ1 , φ2 ∈ Mvr2s (ℝd ), we obtain that W(φ2 , φ1 ) ∈ M1⊗v , 2s for every r ∈ [1, 2]. Now the convolution relations in Proposition 2.4.19, in the form p,q p,q p,q 1,∞ M1⊗v ∗ M1⊗v ⊆ M1⊗v , yield that the Weyl symbol σ = a ∗ W(φ2 , φ1 ) belongs to M1⊗v . −s 2s s s The result now follows from Theorem 4.4.15.

Remark 4.4.18. Using the same techniques as in the proof Theorem 4.4.17, one can φ ,φ study conditions on symbols and window functions such that the operator Aa 1 2 is in the Schatten class Sp , cf., e. g., [66, Theorem 3.4] and [283, Theorem 3.9]. 4.4.3.1 Relations between modulation and Shubin–Sobolev spaces 2 Shubin–Sobolev spaces [265]. Let φ(t) = 2n/4 e−πt and a(z) = ⟨z⟩s for s ∈ ℝ, and set φ,φ As = Aa . Then the Shubin–Sobolev space Qs for s ∈ ℝ is defined by

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

2 d Qs (ℝd ) := {f ∈ 𝒮 󸀠 (ℝd ) : As f ∈ L2 (ℝd )} = A−1 s L (ℝ ),

(4.66)

with norm ‖u‖Qs := ‖As u‖2 . Lemma 4.4.19 (Characterization of Shubin–Sobolev spaces). For all s ∈ ℝ, we have Mv2s (ℝd ) = Qs (ℝd )

(4.67)

with equivalent norms. Proof. (a) We first show the inclusion Mv2s ⊆ Qs for all s ∈ ℝ. Recall that the adjoint of the STFT f → Vg f is the operator Vg∗ F = ∫ F(z)π(z)g dz, ℝ2d

230 | 4 Pseudodifferential operators which is a bounded mapping from L2 (ℝ2d ) to L2 (ℝd ) by Theorem 2.3.7 (i), with 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ∗ 󵄩󵄩 󵄩󵄩Vg F 󵄩󵄩2 = 󵄩󵄩󵄩 ∫ F(z)π(z)g dz 󵄩󵄩󵄩 ≤ ‖g‖2 ‖F‖2 󵄩󵄩2 󵄩󵄩 2d

for F ∈ L2 (ℝ2d ).

(4.68)

ℝ

Since As is defined explicitly by the vector-valued integral As f = ∫ ⟨z⟩s Vφ f (z)π(z)φ dz, ℝ2d

(4.68) implies that 󵄩 󵄩 ‖As f ‖2 ≤ ‖φ‖2 󵄩󵄩󵄩⟨⋅⟩s Vφ f 󵄩󵄩󵄩2 ≍ ‖f ‖Mv2 , s

and thus Mv2s is continuously embedded in Qs for all s ∈ ℝ. (b) To obtain the reverse inclusion, we argue by duality. Since Q󸀠s = Q−s [265] and 2 󸀠 (Mvs ) = Mv2−s by Theorem 2.3.10, we have Qs = (Q−s )󸀠 ⊆ (Mv2−s ) = Mv2s 󸀠

for all s ∈ ℝ. The equivalence of the norms follows from the inverse mapping theorem. Remark 4.4.20. (i) In other words, the previous result says that the localization operφ,φ ator Aa provides an isomorphism between L2 and Qs = Mv2s . The lifting property between Sobolev–Shubin spaces has been extended to more general modulation spaces. For this topic, we refer the interested reader to the contributions [26, 169, 170]. (ii) The equality in (4.67) was first observed in [180, Lemma 7.3.1].

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4.5 Opτ (a) operators with symbols in Wiener amalgam spaces First we focus on the members of the Cohen class, defined in (1.96) as 𝒬σ f = W(f , f ) ∗ σ,

f ∈ 𝒮 (ℝd ),

where σ ∈ 𝒮 󸀠 (ℝ2d ) is called the Cohen kernel. If σ = δ, then 𝒬σ f = W(f , f ) and we come back to the Wigner distribution. More generally, if σ = στ in (1.112), then 𝒬στ f = Wτ (f , f ), the τ-Wigner distribution. In what follows we study the properties of the Cohen kernels στ in the realm of modulation spaces. Proposition 4.5.1. We have, for every τ ∈ [0, 1] \ {1/2}, στ ∈ W(ℱ L1 , L∞ )(ℝ2d ) ∩ M 1,∞ (ℝ2d ).

4.5 Opτ (a) operators with symbols in Wiener amalgam spaces | 231

Proof. We exploit the dilation properties in Corollary 2.6.5: for A = λI, λ > 0, d/2 󵄩 󵄩󵄩 d( 1 − 1 −1) 2 󵄩󵄩f (A⋅)󵄩󵄩󵄩W(ℱ Lp ,Lq ) ≤ Cλ p q (λ + 1) ‖f ‖W(ℱ Lp ,Lq ) .

Using them for λ = √t, 0 < t < 1/2, p = 1, q = ∞, we obtain 󵄩󵄩 ±2πiζ1 ⋅ζ2 t 󵄩󵄩 󵄩 ±2πiζ1 ⋅ζ2 󵄩󵄩 󵄩󵄩e 󵄩󵄩W(ℱ L1 ,L∞ ) ≤ C 󵄩󵄩󵄩e 󵄩󵄩W(ℱ L1 ,L∞ ) 2 , when τ > 1/2 and with constant C > 0 independent on the parameter t. For t = 2τ−1 2 t = − 2τ−1 , when 0 ≤ τ < 1/2, we obtain that στ ∈ W(ℱ L1 , L∞ )(ℝ2d ). Now, an easy computation gives

ℱ στ (ζ1 , ζ2 ) = e

−πi(2τ−1)ζ1 ⋅ζ2

,

so that, using ℱ M 1,∞ (ℝ2d ) = W(ℱ L1 , L∞ )(ℝ2d ) and repeating the previous argument, we obtain στ ∈ M 1,∞ (ℝ2d ) for every τ ∈ [0, 1] \ {1/2}. The case τ = 1/2 is different. Indeed, σ1/2 = δ and, for any fixed g ∈ 𝒮 (ℝ2d ) \ {0}, the STFT Vg δ is given by Vg δ(z, ζ ) = ⟨δ, Mζ Tz g⟩ = g(−z), yielding δ ∈ M 1,∞ (ℝ2d ) \ W(ℱ L1 , L∞ )(ℝ2d ). As already mentioned before, the τ-Wigner distribution comes into play in the weak definition the pseudodifferential operator Opτ (a) in (4.35), since

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

⟨Opτ (a)f , g⟩ = ⟨a, Wτ (f , g)⟩, cf. (4.36). The continuity properties on modulation spaces M p,q for Opτ (a) operators with symbols a ∈ M p,q (ℝ2d ) are the same as for Weyl operators, see Theorem 4.4.15. In the following two subsections we exhibit the continuity properties on modulation spaces for τ-pseudodifferential operators with symbols a in Wiener amalgam spaces W(ℱ Lp , Lq )(ℝ2d ) contained in [65, 110]. We will obtain boundedness results for τ ∈ (0, 1) whereas, at the end-points τ = 0 and τ = 1, we shall show that the corresponding operators are in general unbounded. Furthermore, for τ ∈ (0, 1), we will exhibit a function of τ which is an upper bound for the operator norm. For 1 ≤ r1 , r2 ≤ ∞, we introduce the function α(r1 ,r2 ) (τ) =

τ

d( 1󸀠 + r1 ) r

1

2

1

d( r1 + 1󸀠 )

(1 − τ)

1

r

,

τ ∈ (0, 1).

(4.69)

2

Observe that the function α(r1 ,r2 ) (τ) is unbounded on (0, 1). Indeed, for (r1 , r2 ) ∈ ̸ {(1, ∞), (∞, 1)}, lim α(r1 ,r2 ) (τ) = lim− α(r1 ,r2 ) (τ) = +∞.

τ→0+

τ→1

232 | 4 Pseudodifferential operators For (r1 , r2 ) = (1, ∞), we have limτ→1− α(1,∞) (τ) = +∞ whereas, for (r1 , r2 ) = (∞, 1), limτ→0+ α(∞,1) (τ) = +∞. In the sequel we shall heavily use the following symplectic matrix: 𝒜=(

0d×d

τ 1/2 −( 1−τ ) Id×d

( 1−τ )1/2 Id×d τ 0d×d

τ ∈ (0, 1).

),

(4.70)

The main properties of 𝒜 are detailed below. Their proof is attained by easy computations. Lemma 4.5.2. For any τ ∈ (0, 1), the matrix 𝒜 in (4.70) enjoys the following properties: (i) 𝒜τ ∈ Sp(d, ℝ); in particular, 𝒜1/2 = J. −1 (ii) 𝒜⊤ τ = −𝒜1−τ , 𝒜τ = −𝒜τ . ⊤ −1 (iii) 𝒜1−τ 𝒜τ = 𝒜τ 𝒜τ = I2d×2d − ℬτ , where ℬτ = (

1 I 1−τ d×d

0d×d

0d×d 1 I τ d×d

(4.71)

).

(iv) √τ(1 − τ)(𝒜τ + 𝒜1−τ ) = √τ(1 − τ)ℬτ 𝒜τ = J. 4.5.1 Boundedness properties of τ-Wigner distributions This section is devoted to investigating the continuity properties of τ-Wigner distributions in the realm of Wiener and modulation spaces. For a submultiplicative weight v, we set vJ (z) = v(Jz),

(4.72)

where J denotes the canonical symplectic matrix (1.1). A particular instance of the Young’s inequality for mixed LP spaces in (2.12) reads as follows. Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

p ,q

Lemma 4.5.3. For 1 ≤ pi , qi , r, s ≤ ∞, i = 1, 2, m ∈ ℳv (ℝ2d ), F ∈ Lv 1 1 (ℝ2d ), G ∈ p ,q 2d Lm2 2 (ℝ2d ), we have F ∗ G ∈ Lr,s m (ℝ ), with 1/p1 + 1/p2 = 1 + 1/r, 1/q1 + 1/q2 = 1 + 1/s, and ‖F ∗ G‖Lr,s ≤ ‖F‖Lpv 1 ,q1 ‖G‖Lp2 ,q2 . m m

(4.73)

1 4d We say that a measurable function f on ℝ4d is in the space L∞ z (Lζ ,m )(ℝ ), with m

weight function on ℝ2d , if

1 ‖f ‖L∞ z (L

ζ ,m

)

󵄨 󵄨 = sup ∫ 󵄨󵄨󵄨f (z, ζ )󵄨󵄨󵄨m(ζ ) dζ < ∞. z∈ℝ2d

(4.74)

ℝ2d

When working on the STFT of τ-Wigner distribution, we will use the following Youngtype inequality:

4.5 Opτ (a) operators with symbols in Wiener amalgam spaces | 233

1 4d Lemma 4.5.4. If m ∈ ℳv (ℝ2d ), f ∈ L11⊗v (ℝ4d ) and g ∈ L∞ z (Lζ ,m )(ℝ ), then f ∗ g ∈ 1 4d L∞ z (Lζ ,m )(ℝ ), with

1 ‖f ∗ g‖L∞ z (L

ζ ,m

)

1 ). ≤ ‖f ‖L11⊗v ‖g‖L∞ z (L ζ ,m

1 Proof. Using the definition of L∞ z (Lζ ,m )-norm in (4.74), 1 I := ‖f ∗ g‖L∞ z (L

ζ ,m

)

= sup ∫ |f ∗ g|(z, ζ )m(ζ ) dζ z∈ℝ2d

ℝ2d

󵄨󵄨 󵄨󵄨 󵄨 󵄨 = sup ∫ 󵄨󵄨󵄨 ∫ f (y, η)g(z − y, ζ − η) dy dη󵄨󵄨󵄨m(ζ ) dζ 󵄨 󵄨󵄨 2d 󵄨 z∈ℝ 2d 4d ℝ

ℝ

≤ sup ∫ ∫ ( ∫ |f |(y, η)|g|(z − y, ζ − η) dη)m(ζ ) dy dζ z∈ℝ2d

ℝ2d ℝ2d ℝ2d

= sup ∫ ∫ (|f |(y, ⋅) ∗ |g|(z − y, ⋅))(ζ )m(ζ ) dy dζ . z∈ℝ2d

ℝ2d ℝ2d

By Young’s inequality (4.73), 󵄩 󵄩 󵄩 󵄩 I = sup ∫ 󵄩󵄩󵄩|f |(y, ⋅)󵄩󵄩󵄩L1 󵄩󵄩󵄩|g|(z − y, ⋅)󵄩󵄩󵄩L1 dy z∈ℝ2d

v

ℝ2d

m

󵄩 󵄩 󵄩 󵄩 ≤ ∫ 󵄩󵄩󵄩|f |(y, ⋅)󵄩󵄩󵄩L1 sup 󵄩󵄩󵄩|g|(z − y, ⋅)󵄩󵄩󵄩L1 dy v

ℝ2d

m

z∈ℝd

1 ) ‖f ‖L1 , = ‖g‖L∞ z (L 1⊗v ζ ,m

as claimed.

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A particular case of Lemma 4.5.3 gives: Lemma 4.5.5. Suppose m ∈ ℳv (ℝ2d ), f ∈ L11⊗v (ℝ4d ) and g ∈ L21⊗m (ℝ4d ). Then f ∗ g ∈ L21⊗m (ℝ4d ), with ‖f ∗ g‖L2

1⊗m

≤ ‖f ‖L11⊗v ‖g‖L2 . 1⊗m

p ,p

Lemma 4.5.6. Assume that m ∈ ℳv (ℝ2d ), 1 ≤ p1 , p2 ≤ ∞, f ∈ Mm1 2 (ℝd ), g ∈

p󸀠1 ,p󸀠2 M1/m (ℝd ). ∞ 2d

Then for every τ ∈ (0, 1), the τ-Wigner distribution Wτ (g, f ) is in W(ℱ L11/vJ ,

L )(ℝ ), with

󵄩󵄩 󵄩 󵄩󵄩Wτ (g, f )󵄩󵄩󵄩W(ℱ L1

1/vJ

,L∞ )

≤ Cα(p1 ,p2 ) (τ)‖f ‖M p1 ,p2 ‖g‖ m

p󸀠 ,p󸀠

1 2 M1/m

,

where the function α(p1 ,p2 ) (τ) is defined in (4.69) and C > 0 is independent of τ.

(4.75)

234 | 4 Pseudodifferential operators Proof. We compute the STFT of Wτ (g, f ) with respect to the window function Φτ ∈ 𝒮 (ℝ2d ) defined in Lemma 1.3.38. Using that lemma and the properties of the matrix 𝒜 in Lemma 4.5.2, by performing the change of variables √τ(1 − τ)𝒜ζ = η, we deduce 1 󵄨 󵄨 dζ ∫ 󵄨󵄨󵄨VΦτ Wτ (g, f )󵄨󵄨󵄨(z, ζ ) v(Jζ )

ℝ2d

1 󵄨 󵄨󵄨 󵄨 dζ = ∫ 󵄨󵄨󵄨Vφ1 g(z + √τ(1 − τ)𝒜T ζ )󵄨󵄨󵄨󵄨󵄨󵄨Vφ2 f (z + √τ(1 − τ)𝒜ζ )󵄨󵄨󵄨 √τ(1 − τ)ℬτ 𝒜ζ ) v( 2d ℝ

=

1 󵄨 󵄨󵄨 󵄨 1 dη. ∫ 󵄨󵄨Vφ g(z + 𝒜1−τ 𝒜η)󵄨󵄨󵄨󵄨󵄨󵄨Vφ2 f (z + η)󵄨󵄨󵄨 v(ℬτ η) [τ(1 − τ)]d 󵄨 1 ℝ2d

Since m is a v-moderate weight, we can find a positive constant C, independent of τ, such that m(z + η) 1 ≤C , v(ℬτ η) m(z + 𝒜1−τ 𝒜η)

(4.76)

so 1 󵄨 󵄨 dζ ∫ 󵄨󵄨󵄨VΦτ Wτ (g, f )󵄨󵄨󵄨(z, ζ ) v(Jζ )

ℝ2d

≤C

1 󵄨 󵄨󵄨 󵄨 m(z + η) dη. ∫ 󵄨󵄨Vφ g(z + 𝒜1−τ 𝒜η)󵄨󵄨󵄨󵄨󵄨󵄨Vφ2 f (z + η)󵄨󵄨󵄨 m(z + 𝒜1−τ 𝒜η) [τ(1 − τ)]d 󵄨 1 ℝ2d

Consequently,

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󵄩󵄩 󵄩 󵄩󵄩Wτ (g, f )󵄩󵄩󵄩W(ℱ L1 ,L∞ ) 1/vJ 1 󵄨󵄨 󵄨 󵄨 dζ ≍ sup ∫ 󵄨󵄨󵄨Vφ1 g(z + √τ(1 − τ)𝒜T ζ )󵄨󵄨󵄨󵄨󵄨󵄨Vφ2 f (z + √τ(1 − τ)𝒜ζ )󵄨󵄨󵄨 v(√τ(1 − τ)ℬτ 𝒜ζ ) z∈ℝ2d 2d ℝ

≤C

1 󵄨 󵄨󵄨 󵄨 m(z + η) sup ∫ 󵄨󵄨Vφ g(z + 𝒜1−τ 𝒜η)󵄨󵄨󵄨󵄨󵄨󵄨Vφ2 f (z + η)󵄨󵄨󵄨 dη m(z + 𝒜1−τ 𝒜η) [τ(1 − τ)]d z∈ℝ2d 󵄨 1 ℝ2d

≤C

󵄩󵄩 󵄩󵄩 1 1 󵄩 󵄩󵄩 p1 ,p2 󵄩 ‖V (z + 𝒜 𝒜 ⋅) fm‖ V g 󵄩󵄩 p󸀠 ,p󸀠 󵄩 1−τ φ L φ 󵄩󵄩 2 m 1 󵄩󵄩L 1 2 [τ(1 − τ)]d 󵄩 d( p1 − p1 )

1−τ 1 ( ) ≲ d τ [τ(1 − τ)] The claim is proved.

2

1

‖f ‖M p1 ,p2 ‖g‖ m

p󸀠 ,p󸀠

1 2 M1/m

.

4.5 Opτ (a) operators with symbols in Wiener amalgam spaces | 235

The previous estimate is not uniform with respect to τ, in the sense that the W(ℱ L11/vJ , L∞ )-norm of the τ-Wigner distribution has been calculated by using a window function Φτ depending on τ. The next goal is to find an upper bound of this norm independent of τ. We will need some preliminaries. The τ-Wigner distribution of the 2 Gaussian function φ(t) = e−πt is in turn a generalized Gaussian function, as showed in the next lemma. 2

Lemma 4.5.7. Consider φ1 (t) = φ2 (t) = φ(t) = e−πt , t ∈ ℝd , and τ ∈ [0, 1]. Then Wτ φ(x, ξ ) =

(2τ2

1 1 1 −π x2 −π ξ 2 2πi 2τ−1 x⋅ξ e 2τ2 −2τ+1 e 2τ2 −2τ+1 e 2τ2 −2τ+1 , d/2 − 2τ + 1)

for all (x, ξ ) ∈ ℝ2d . Proof. Using the definition of the τ-Wigner distribution in (1.109), 2

2

Wτ φ(x, ξ ) = ∫ e−2πiξ ⋅t e−π(x+τt) e−π(x−(1−τ)t) dt ℝd 2

2

= ∫ e−2πiξ ⋅t e−2πx e−π[(2τ −2τ+1)t

2

+2(2τ−1)x⋅t]

dt

ℝd

=e

−2πx 2 +π(

2τ−1

√2τ2 −2τ+1

)2 x2

∫ e−2πiξ ⋅t e

−π(√2τ2 −2τ+1t+

2τ−1

√2τ2 −2τ+1

x)2

dt.

ℝd

We perform the following change of variables: √2τ2 − 2τ + 1t +

2τ − 1 x = y, √2τ2 − 2τ + 1

so that, letting c(τ) = 2τ2 − 2τ + 1 > 0,

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Wτ φ(x, ξ ) = =

1 c(τ) 1

d 2

e−π(2−

(2τ−1)2 c(τ)

c(τ)

∫e

y −2πiξ ⋅ √c(τ) ξ ⋅x −πy2 2πi 2τ−1 c(τ)

e

e

dy

ℝd 1

d 2

)x2

2τ−1

2

1

2

e−π c(τ) x e2πi c(τ) ξ ⋅x e−π c(τ) ξ ,

as desired. 2

2

Lemma 4.5.8. Consider Φ(x, ξ ) = e−π(x +ξ ) , (x, ξ ) ∈ ℝ2d , and Φτ = Wτ (φ, φ), where 2 φ(t) = e−πt , t ∈ ℝd . Then for vJ in (4.72) there exists a constant C > 0 such that ‖VΦ Φτ ‖L11⊗v ≤ C, J

∀τ ∈ [0, 1].

(4.77)

Consequently, 1 ‖Φτ ‖M1⊗v ≤ C, J

∀τ ∈ [0, 1].

(4.78)

236 | 4 Pseudodifferential operators Proof. Using Lemma 4.5.7 and formula (1.127), with z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ ℝ2d , we compute |VΦ Φτ |(z, ζ ) =

(2τ2 − 2τ + 1)d/2 1 (2τ2 − 2τ + 1)d/2 (2τ2 − 2τ + 5)d/2 ×e

2 3 (z 2 +z 2 )+ 2τ −2τ+2 (ζ 2 +ζ 2 )−2 22τ−1 (z1 ⋅ζ2 +z2 ⋅ζ1 ) 2τ2 −2τ+1 1 2 2τ2 −2τ+1 1 2 2τ −2τ+1 2τ2 −2τ+5 2τ2 −2τ+1

−π

2

=

2

2

2

2

3(z +z )+(2τ −2τ+2)(ζ1 +ζ2 )+(2−4τ)(z1 ⋅ζ2 +z2 ⋅ζ1 ) 1 −π 1 2 2τ2 −2τ+5 e . 2 d/2 (2τ − 2τ + 5)

Observing that d/2

1 2 1 ≤ max =( ) 2 d/2 2 d/2 τ∈(0,1) 9 (2τ − 2τ + 5) (2τ − 2τ + 5)

,

by Lemma 2.1.4, we have d/2

2 ‖VΦ Φτ ‖L11⊗v ≤ ( ) J 9

× ∫ ∫e ℝ2d

−π

3(z12 +z22 )+(2τ2 −2τ+2)(ζ12 +ζ22 )+(2−4τ)(z1 ⋅ζ2 +z2 ⋅ζ1 ) 2τ2 −2τ+5

vJ (ζ ) dζ1 dζ2 dz1 dz2

ℝ2d

≤C ∫ ∫ e

−π

3(z12 +z22 )+(2τ2 −2τ+2)(ζ12 +ζ22 )+(2−4τ)(z1 ⋅ζ2 +z2 ⋅ζ1 ) 2τ2 −2τ+5

ea|Jζ | dζ1 dζ2 dz1 dz2

ℝ2d ℝ2d

=C ∫ e

−π

3(z12 +z22 ) 2τ2 −2τ+5

I1 dz1 dz2 ,

ℝ2d

where

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

I1 := ∫ e

−π

(2τ2 −2τ+2)(ζ12 +ζ22 )+(2−4τ)(z1 ⋅ζ2 +z2 ⋅ζ1 ) 2τ2 −2τ+5

ea|Jζ | dζ1 dζ2 .

ℝ2d

The integral I1 can be computed as follows: I1 = ∫ e

−π

(2τ2 −2τ+2)(ζ12 +ζ22 )+(2−4τ)(z1 ⋅ζ2 +z2 ⋅ζ1 ) 2τ2 −2τ+5

ea|Jζ | dζ1 dζ2

ℝ2d

≤ ∫e

−π

(2τ2 −2τ+2)(ζ12 +ζ22 )+(2−4τ)(z1 ⋅ζ2 +z2 ⋅ζ1 ) 2τ2 −2τ+5

ea(|ζ1 |+|ζ2 |) dζ1 dζ2

ℝ2d

= (∫ e ℝd

−π

(2τ2 −2τ+2)ζ12 +(2−4τ)z2 ⋅ζ1 2τ2 −2τ+5

ea|ζ1 | dζ1 )( ∫ e ℝd

−π

(2τ2 −2τ+2)ζ22 +(2−4τ)z1 ⋅ζ2 2τ2 −2τ+5

ea|ζ2 | dζ2 ).

4.5 Opτ (a) operators with symbols in Wiener amalgam spaces | 237

We calculate the integral with respect to the variable ζ1 (the other integral is analogous): ∫e

−π

(2τ2 −2τ+2)ζ12 +(2−4τ)z2 ⋅ζ1 2τ2 −2τ+5

ea|ζ1 | dζ1

ℝd

= ∫e

−π

(1−2τ)2 z22 (1−2τ)2 z22 − (2τ2 −2τ+2)ζ12 +(2−4τ)z2 ⋅ζ1 + 2 2τ −2τ+2 2τ2 −2τ+2 2τ2 −2τ+5

ea|ζ1 | dζ1

ℝd

=e

π

(1−2τ)2 z22 (2τ2 −2τ+2)(2τ2 −2τ+5)

π

(1−2τ)2 z22 (2τ2 −2τ+2)(2τ2 −2τ+5)

(√2τ2 −2τ+2ζ1 +

∫e

−π

1−2τ z2 )2 √2τ2 −2τ+2

2τ2 −2τ+5

ea|ζ1 | dζ1

ℝd

=e

∫e

−π

((2τ2 −2τ+2)ζ1 +(1−2τ)z2 )2 (2τ2 −2τ+5)(2τ2 −2τ+2)

ea|ζ1 | dζ1 .

d ℝ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ :=I3

In I3 we perform the following change of variables: (2τ2 − 2τ + 2)ζ1 + (1 − 2τ)z2 = η1 , so that I3 =

2

η1 a 1 −π 2 |η −(1−2τ)z2 | (2τ −2τ+5)(2τ2 −2τ+2) e 2τ2 −2τ+2 1 e dη1 ∫ 2 d (2τ − 2τ + 2)

ℝd

a|1−2τ| |z | 2τ2 −2τ+2 2

≤ C1d e

2

∫ e−πC2 η1 eaC1 |η1 | dη1 , ℝd

where C1 = max Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

τ∈[0,1]

2 1 = , (2τ2 − 2τ + 2) 3

Using lim|η1 |→∞ e−π e

−π

C1 2

η21 aC2 |η1 |

e

C1 2

η21 aC2 |η1 |

e

C2 = min

τ∈[0,1]

1 1 = . (2τ2 − 2τ + 5)(2τ2 − 2τ + 2) 10

= 0, for every ϵ > 0 there exists R > 0 such that

≤ ϵ, for all |η1 | with |η1 | > R. Hence a|1−2τ|

I3 ≤ C1d e 2τ2 −2τ+2

|z2 |

2

∫ e−πC2 η1 eaC1 |η1 | dη1 ℝd

a|1−2τ|

= C1d e 2τ2 −2τ+2

|z2 |

(

∫

2

e−πC2 η1 eaC1 |η1 | dη1

{η1 ∈ℝd :|η1 |≤R}

+

∫ {η1 ∈ℝd :|η1 |>R}

2

e−πC2 η1 eaC1 |η1 | dη1 )

238 | 4 Pseudodifferential operators a|1−2τ|

≤ C1d e 2τ2 −2τ+2

|z2 |

2

(eaC1 R ∫ e−πC2 η1 dη1 + ϵ ∫ e−π ℝd

a|1−2τ|

̃ 2τ2 −2τ+2 = Ce

|z2 |

C2 2

η21

dη1 )

ℝd

< ∞,

where C̃ is a constant independent of τ. In conclusion, the integral I1 can be majorized as (1−2τ)2 z22

(1−2τ)2 z12

a|1−2τ|

a|1−2τ|

̃ π (2τ2 −2τ+2)(2τ2 −2τ+5) + 2τ2 −2τ+2 |z2 | eπ (2τ2 −2τ+2)(2τ2 −2τ+5) + 2τ2 −2τ+2 |z1 | . I1 ≤ 2Ce Thus, there exists a constant M1 > 0 independent of τ such that ‖VΦ Φτ ‖L11⊗v ≤ M1 ∫ e J

−π

3z12 2τ2 −2τ+5

e

π

(1−2τ)2 z12 + a|1−2τ| |z | (2τ2 −2τ+2)(2τ2 −2τ+5) 2τ2 −2τ+2 1

dz1

ℝd

×∫e

−π

3z22 2τ2 −2τ+5

e

π

(1−2τ)2 z22 + a|1−2τ| |z | (2τ2 −2τ+2)(2τ2 −2τ+5) 2τ2 −2τ+2 2

dz2

ℝd

= 2M1 ∫ e

−π

3z12 2τ2 −2τ+5

e

π

(1−2τ)2 z12 + a|1−2τ| |z | (2τ2 −2τ+2)(2τ2 −2τ+5) 2τ2 −2τ+2 1

dz1 .

ℝd

The integral with respect the variable z1 is computed analogously to that for ζ1 above. The estimate (4.78) follows by 1 ‖Φτ ‖M1⊗v ≍ ‖VΦ Φτ ‖L11⊗v ≤ C, J

J

as desired.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Proposition 4.5.9. Under the assumptions of Lemma 4.5.6, there exists a constant C > 0 independent of τ such that 󵄩󵄩 󵄩 󵄩󵄩Wτ (g, f )󵄩󵄩󵄩W(ℱ L1 ,L∞ ) ≤ Cα(p1 ,p2 ) (τ)‖f ‖Mmp1 ,p2 ‖g‖ p󸀠1 ,p󸀠2 , 1/vJ M 1/m

τ ∈ (0, 1).

(4.79)

Proof. Changing window in the computation of the STFT as in Lemma (1.2.29), using Lemmas 4.5.4, 4.5.8 and Moyal’s formula (1.115), we have 1 󵄩󵄩 󵄩 󵄩󵄩󵄩󵄨󵄨󵄨V W (g, f )󵄨󵄨󵄨 ∗ |VΦ Φτ |󵄩󵄩󵄩 ∞ 1 󵄩󵄩VΦ Wτ (g, f )󵄩󵄩󵄩L∞ (L1 ) ≤ 󵄨 󵄩Lz (Lζ ,1/vJ ) z ζ ,1/vJ |⟨Φτ , Φτ ⟩| 󵄩󵄨 Φτ τ 1 󵄩󵄩 󵄩 ≤ 󵄩V W (g, f )󵄩󵄩󵄩L∞ (L1 ) ‖VΦ Φτ ‖L11⊗v z ζ ,1/vJ J ‖φ‖2 ‖φ‖2 󵄩 Φτ τ ≤ Cα(p1 ,p2 ) (τ)‖f ‖M p1 ,p2 ‖g‖ m

This completes the proof.

p󸀠 ,p󸀠

1 2 M1/m

.

4.5 Opτ (a) operators with symbols in Wiener amalgam spaces | 239

Repeating the pattern of Lemma 4.5.6 and Proposition 4.5.9 in the Wiener amalgam space W(ℱ L21/vJ , L2 )(ℝ2d ), we can state the following. 2 2 Proposition 4.5.10. Let m ∈ ℳv (ℝ2d ), f ∈ Mm (ℝd ) and g ∈ M1/m (ℝd ). For τ ∈ (0, 1), the 2 2 2d τ-Wigner distribution Wτ (g, f ) is in W(ℱ L1/vJ , L )(ℝ ), with the uniform estimate

󵄩 󵄩󵄩 󵄩󵄩Wτ (g, f )󵄩󵄩󵄩W(ℱ L2

1/vJ

,L2 )

≤ C‖f ‖Mm2 ‖g‖M 2 ,

(4.80)

1/m

where the positive constant C is independent of τ. Proof. First step. We use Lemma 1.3.38, Young’s inequality L1 ∗L1 ⊂ L1 , and the change of variables ℬτ η → η, to compute 󵄩󵄩 󵄩 󵄩󵄩Wτ (g, f )󵄩󵄩󵄩W(ℱ L2 ,L2 ) 1/vJ 1

2 󵄨 󵄨2 󵄨 󵄨2 1 ≍ ( ∫ ∫ 󵄨󵄨󵄨Vφ1 g(z + √τ(1 − τ)𝒜T ζ )󵄨󵄨󵄨 󵄨󵄨󵄨Vφ2 f (z + √τ(1 − τ)𝒜ζ )󵄨󵄨󵄨 2 dζ dz) v (Jζ )

ℝ2d ℝ2d

1

2 2 1 󵄨󵄨 󵄨󵄨2 󵄨󵄨 󵄨󵄨2 m (z + η) ≤C dη dz) ( V g(z + η − ℬ η) V f (z + η) ∫ ∫ 󵄨 󵄨 󵄨 󵄨 φ τ φ 󵄨 1 󵄨󵄨 2 󵄨 m2 (z + η − ℬ η) [τ(1 − τ)]d τ ℝ2d ℝ2d

1

=C

2 1 1 ( ∫ (|Vφ2 f |2 m2 ) ∗ (|Vφ1 g|2 2 )(ℬτ η) dη) d m [τ(1 − τ)]

ℝ2d

󵄩 1 󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 ≲ 󵄩󵄩󵄩|Vφ f |2 m2 󵄩󵄩󵄩1 󵄩󵄩󵄩|Vφ g|2 2 󵄩󵄩󵄩 󵄩󵄩 m 󵄩󵄩1 ≲ ‖f ‖Mm2 ‖g‖M 2 . 1/m

Second step. Consider now Φ ∈ 𝒮 (ℝ2d ). Then the same pattern as in the proof of Proposition 4.5.9, with Lemma 4.5.4 replaced by Lemma 4.5.5, gives the uniform estimate (4.80).

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

The previous issue can be rephrased in terms of modulation spaces as follows. 2 2 Corollary 4.5.11. For τ ∈ (0, 1), m ∈ ℳv (ℝ2d ), f ∈ Mm (ℝd ), g ∈ M1/m (ℝd ), the τ-Wigner 2 distribution belongs to M1/v (ℝ2d ) with J ⊗1

󵄩󵄩 󵄩 ≤ C‖f ‖Mm2 ‖g‖M 2 , 󵄩󵄩Wτ (g, f )󵄩󵄩󵄩M 2 1/m 1/vJ ⊗1 with C > 0 independent of τ. 4.5.2 Continuity properties of Opτ (a) operators This section is devoted to the proof of Theorem 4.5.14. We will start with two preliminary results about τ-pseudodifferential operators acting on modulation spaces and

240 | 4 Pseudodifferential operators 1 2d 2 2 2d having symbols in W(ℱ L∞ 1/vJ , L )(ℝ ) and W(ℱ L1/vJ , L )(ℝ ), respectively. Then by means of complex interpolation between Wiener amalgam spaces, we shall reach our goal.

Proposition 4.5.12. Suppose that m ∈ ℳv (ℝ2d ) and consider a symbol function a ∈ 1 2d W(ℱ L∞ vJ , L )(ℝ ). Then for every τ ∈ (0, 1), the τ-pseudodifferential operator Opτ (a) is p ,p

bounded on ℳm1 2 (ℝd ), for every 1 ≤ p1 , p2 ≤ ∞, with

󵄩󵄩 󵄩 p ,p 1 ‖f ‖ 󵄩󵄩Opτ (a)f 󵄩󵄩󵄩ℳpm1 ,p2 ≤ Cα(p1 ,p2 ) (τ)‖a‖W(ℱ L∞ ℳm1 2 vJ ,L )

(4.81)

(C > 0 does not depend on τ). p󸀠 ,p󸀠

p ,p

1 2 Proof. For every f ∈ ℳm1 2 (ℝd ) and g ∈ ℳ1/m (ℝd ), we can write

󵄨󵄨 󵄨 󵄨 󵄨 󵄩󵄩 󵄩󵄩 . 󵄨󵄨⟨Opτ (a)f , g⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨a, Wτ (g, f )⟩󵄨󵄨󵄨 ≤ ‖VΦ a‖L1z (L∞ 1 ) 󵄩VΦ Wτ (g, f )󵄩 󵄩L∞ vJ ,ζ 󵄩 z (L1/vJ ,ζ ) Observing that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩Wτ (g, f )󵄩󵄩󵄩W(ℱ L1 ,L∞ ) ≍ 󵄩󵄩󵄩VΦ Wτ (g, f )󵄩󵄩󵄩L∞ (L1 ) z 1/vJ 1/vJ ,ζ and using Proposition 4.5.9, we conclude the proof. Proposition 4.5.13. Let m ∈ ℳv (ℝ2d ), a ∈ W(ℱ L2vJ , L2 )(ℝ2d ) and τ ∈ (0, 1). Then the

2 operator Opτ (a) is bounded on Mm with

󵄩󵄩 󵄩 󵄩󵄩Opτ (a)f 󵄩󵄩󵄩M 2 ≤ C‖a‖W(ℱ L2v ,L2 ) ‖f ‖Mm2 , m J

(4.82)

where the constant C > 0 is independent of τ. Proof. The proof is similar to that of Proposition 4.5.12, where Proposition 4.5.9 is replaced by Proposition 4.5.10.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Propositions 4.5.12 and 4.5.13 are the main ingredients in the proof of Theorem 4.5.14. Theorem 4.5.14. Suppose that 1 ≤ p, q, r1 , r2 ≤ ∞ satisfy q ≤ p󸀠

(4.83)

max{r1 , r2 , r1󸀠 , r2󸀠 } ≤ p.

(4.84)

and

Let m ∈ ℳv (ℝ2d ) and a ∈ W(ℱ LpvJ , Lq )(ℝ2d ). For τ ∈ (0, 1), every τ-pseudodifferential r ,r

operator Opτ (a) is a bounded operator on ℳm1 2 (ℝd ). Moreover, there exists a constant C > 0 independent of τ such that 󵄩󵄩 󵄩 󵄩󵄩Opτ (a)f 󵄩󵄩󵄩ℳrm1 ,r2 ≤ Cα(r1 ,r2 ) (τ)‖a‖W(ℱ Lpv ,Lq ) ‖f ‖ℳrm1 ,r2 , J

τ ∈ (0, 1).

(4.85)

4.5 Opτ (a) operators with symbols in Wiener amalgam spaces | 241

Proof. The key tool is the complex interpolation between Wiener amalgam and modulation spaces. We regard Opτ as the bilinear map (a, f ) �→ Opτ (a)f . Propositions 4.5.12 and 4.5.13 give the continuity of the τ-pseudodifferential operator Opτ on the following function spaces: 1 2d p1 ,p2 d p1 ,p2 d W(ℱ L∞ vJ , L )(ℝ ) × ℳm (ℝ ) → ℳm (ℝ ), 2 2 W(ℱ L2vJ , L2 )(ℝ2d ) × Mm (ℝd ) → Mm (ℝd ),

for 1 ≤ p1 , p2 ≤ ∞. Using the complex interpolation between Wiener amalgam and modulation spaces, for θ ∈ [0, 1], we have 1 2 2 p p [W(ℱ L∞ vJ , L ), W(ℱ LvJ , L )]θ = W(ℱ LvJ , L ), 󸀠

p ,p

r ,r

2 with 2 ≤ p ≤ ∞, and [ℳm1 2 , Mm ]θ = ℳm1 2 , with

1−θ θ 1−θ 1 1 = + = + r1 p1 2 p1 p

(4.86)

1 1−θ θ 1−θ 1 = + = + , r2 p2 2 p2 p

(4.87)

and

so that r1 , r2 ≤ p. Similarly, we obtain r1󸀠 , r2󸀠 ≤ p, and thus the relation (4.84). Due to inclusion relations for Wiener amalgam spaces, we relax the assumptions on symbols, so that the symbol a may belong to W(ℱ LpvJ , Lq )(ℝ2d ), with q ≤ p󸀠 , which gives (4.83). Finally, the norm is provided by ‖Opτ ‖B(W(ℱ Lp ,Lq )×ℳr1 ,r2 ,ℳr1 ,r2 ) ≤ ‖Opτ ‖1−θ p ,p p ,p B(W(ℱ L∞ ,L1 )×ℳ 1 2 ,ℳ 1 2 ) vJ

m

m

× ≤C Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

m

vJ

≤C

m

‖Opτ ‖θB(W(ℱ L2 ,L2 )×M 2 ,M 2 ) v m m J

τ

d(1−θ)(1− p1 + p1 )

τ

d(1− p1 + p1 )

1

1

2

2

1

1

d(1−θ)(1+ p1 − p1 )

(1 − τ)

1

d(1+ p1 − p1 )

(1 − τ)

1

2

,

2

since 1 − θ ≤ 1. This concludes the proof. We finally consider the end-points τ = 0 and τ = 1, for which the boundedness results stated above do not hold, in general. We remark that the modulation space M 2 (ℝd ) is simply the Lebesgue space L2 (ℝd ). Hence Proposition 4.5.13 matches with what we observed after Proposition 4.2.4, namely Opw (a) is not L2 -bounded in general for a ∈ W(ℱ L2 , Lq ) if q > 2, and L2 -continuity fails also if a ∈ W(ℱ L2 , L2 ) = L2 for Op1 (a) and Op0 (a). Concerning Proposition 4.5.12, the following example generalizes a 1-dimensional example exhibited by Boulkhemair in [34].

242 | 4 Pseudodifferential operators Proposition 4.5.15. There exists a symbol a ∈ W(ℱ L∞ , L1 )(ℝ2d ) such that the corresponding Kohn–Nirenberg operator a(x, D) and Op1 (a) (so-called anti-Kohn–Nirenberg operator) are not bounded on L2 (ℝd ). Proof. Consider the symbol function 2

a(x1 , . . . , xd , ξ1 , . . . , ξd ) = x1−1/2 ⋅ ⋅ ⋅ xd−1/2 χ(0,1] (x1 ) ⋅ ⋅ ⋅ χ(0,1] (xd )e−πξ ,

(4.88)

with ξ 2 = ξ12 + ⋅ ⋅ ⋅ + ξd2 . An easy computation shows that a ∈ L1 (ℝ2d ) = W(L1 , L1 )(ℝ2d ) ⊂ W(ℱ L∞ , L1 )(ℝ2d ). Let us show that the Kohn–Niremberg operator a(x, D) is unbounded on L2 (ℝd ). Con2 sider the Gaussian function φ(t) = eπt ∈ L2 (ℝd ), then a(x, D)f ∉ L2 (ℝd ). Indeed, by a tensor product argument, we reduce to compute the following one-dimensional integral: 2

2

∫ e2πix⋅ξ x−1/2 χ(0,1] (x)e−πξ e−πξ dξ = ℝ

x2 1 −1/2 x χ(0,1] (x)e−π 2 , √2

whose result is a function that does not belong to L2 (ℝ). To prove that the anti-Kohn–Nirenberg operator Op1 (a), where a is defined in (4.88), is unbounded on L2 (ℝd ), it is sufficient to observe that its adjoint operator is the Kohn–Niremberg one: (Op1 (a))∗ = a(x, D), as detailed below: ⟨Op1 (a)f , g⟩ = ⟨a, ℛ∗ (g, f )⟩ = ⟨a, ℛ(f , g)⟩ = ⟨ℛ(f , g), a⟩ = ⟨f , Op0 (a)g⟩. This proves our claim.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

4.6 Born–Jordan quantization and related pseudodifferential calculus Roughly speaking, quantization is the process of associating to a function or distribution defined on phase space an operator. Historically, this notion appears explicitly for the first time in Born and Jordan’s foundational paper [33] where they set out to give a firm mathematical basis to Heisenberg’s matrix mechanics. Born and Jordan’s quantization scheme was strictly limited to polynomials in the variables x and p; it was soon superseded by another rule due to Weyl, and whose extension is nowadays the preferred quantization in physics. However, it turns out that there is a recent regain in interest in an extension of Born and Jordan’s initial rule, both in quantum physics and time–frequency analysis. In fact, on the one hand, it is the correct rule if one wants matrix and wave mechanics to be equivalent quantum theories (see the discussion in [106]). On the other hand, as a time–frequency representation the Born–Jordan distri-

4.6 Born–Jordan quantization and related pseudodifferential calculus | 243

bution has been proved to be surprisingly successful, because it allows damping very well the unwanted “ghost frequencies”, as shown in [24], see also [59]. The difference between Born–Jordan and Weyl quantization is most easily apprehended on the level of monomial quantization: in dimension d = 1, for any integers r, s ≥ 0, we have Opw (xr ξ s ) =

1 s s ̂ s−ℓ r ̂ ℓ 1 r r ℓ ̂ s r−ℓ ̂ ( ) ξ x ξ = ∑ ∑ ( )x̂ ξ x̂ 2s ℓ=0 ℓ 2r ℓ=0 ℓ

(4.89)

1 r ℓ ̂ s r−ℓ 1 s ̂ s−ℓ r ̂ ℓ ∑ ξ x̂ ξ = ∑ x̂ ξ x̂ s + 1 ℓ=0 r + 1 ℓ=0

(4.90)

(see [224]) and OpBJ (xr ξ s ) =

(see [33]). As usual, here ξ̂ = −iℎ𝜕/𝜕x and x̂ is the multiplication operator by x. More generally, in this section we shall use the notation x̂j for the operator of multiplication by xj and ξ̂j = −iℎ𝜕/𝜕xj . These operators satisfy Born’s canonical commutation relations [x̂j , ξ̂j ] = iℎ, where ℎ denotes the Planck constant. Here we shall treat the general case of quantization depending on ℎ. Notice that the particular choice ℎ = 1/(2π) gives the quantization and related pseudodifferential calculus in the framework of time–frequency analysis, recapturing the definition of pseudodifferential operators considered before. The Born–Jordan scheme appears as being an equally-weighted quantization, as opposed to the Weyl scheme: OpBJ (xr ξ s ) is the average of all possible permutations of the product x̂r ξ̂ s . One can extend the Weyl and Born–Jordan quantizations to arbitrary symbols a ∈ 󸀠 ̂ W = Opw (a) and A ̂ BJ = OpBJ (a): 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) as 𝒮 (ℝ2d ) by defining the operators A d

̂ ̂ W ψ = ( 1 ) ∫ aσ (z)T(z)ψ dz, A 2πℏ

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

d

(4.91)

ℝ2d

̂ BJ ψ = ( 1 ) ∫ aσ (z)Θ(z)T(z)ψ ̂ A dz 2πℏ ℝ2d

where ψ ∈ 𝒮 (ℝd ) and the integrals are to be understood in the distributional sense; ̂ 0 ) = e−i(x0 ⋅ξ̂−ξ0 ⋅̂x)/ℏ , z0 = (x0 , ξ0 ), is the Heisenberg operator, aσ (z) is the symhere T(z plectic Fourier transform of a,2 and Θ is Cohen [52] kernel function, defined by Θ(z) =

sin(x ⋅ ξ /2ℏ) , x ⋅ ξ /2ℏ

z = (x, ξ ) ≠ 0,

2 The ℎ symplectic Fourier transform of a function a(z) in phase space ℝ2d is n

ℱσ a(z) = aσ (z) = (

i 󸀠 1 ) ∫ e− ℎ σ(z,z ) a(z 󸀠 ) dz 󸀠 . 2πℎ

ℝ2d

244 | 4 Pseudodifferential operators Θ(z) = 1 for z = 0, see [24, 108] and the references therein. The presence of the function Θ produces a smoothing effect (in comparison with the Weyl quantization) which is responsible of the superiority of the Born–Jordan quantization in several respects. However, although this effect is numerically evident, it has remained a challenging open problem to quantify it analytically till the recent works [63, 59], which provide a mathematical explanation of the smoothing effects. Now, it readily follows from the Schwartz kernel theorem that for every linear con̂ : 𝒮 (ℝd ) �→ 𝒮 󸀠 (ℝd ) there exists a unique b ∈ 𝒮 󸀠 (ℝ2d ) such that tinuous operator A ̂ A = Opw (b). In other terms, dequantization can always be performed, and in a unique way. Instead, the situation is more complicated for Born–Jordan operators. In fact, to ̂ = OpBJ (a), one has to solve a division prove that there exists a ∈ 𝒮 󸀠 (ℝ2d ) such that A problem, namely to find a distribution a such that its symplectic Fourier transform aσ satisfies bσ = Θaσ . The existence of such a symbol a is far from being obvious because of the zeroes of Θ. Moreover, it turns out, as we shall see, that the solution is not even unique. In the following we shall investigate these issues, contained in [60]. The problem of the division of temperate distributions by smooth functions is in general a very subtle one, even in the presence of simple zeros. The basic idea here is, of course, that the space 𝒮 󸀠 (ℝd ) contains (generalized) functions rough enough to absorb the singularities and the loss of decay arising in the division by Θ, and we have, in fact, the following result:

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Theorem 4.6.1. For every b ∈ 𝒮 󸀠 (ℝ2d ), there exists a symbol a ∈ 𝒮 󸀠 (ℝ2d ) such that OpBJ (a) = Opw (b). ̂ : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) can be written in Hence, every linear continuous operator A ̂ = OpBJ (a). Born–Jordan form, i. e., there exists a symbol a ∈ 𝒮 󸀠 (ℝ2d ) such that A We provide two proofs of this result. One is completely elementary and constructive. The other is shorter and based on the machinery of a priori estimates developed in [190] to prove that the division by a (nonidentically zero) polynomial is always possible in 𝒮 󸀠 (ℝd ). Actually, the reader familiar with [190] will notice that the tools used there are excessively sophisticated for our purposes: after all the function Θ(z) is the composition of the harmless “sinc” function with the monomial xξ , and, in fact, we will show that a suitable change of variables will reduce matters to the problem of division by the “sinc” function. One can also rephrase this result as follows. The map 𝒮 󸀠 (ℝ2d ) → 𝒮 󸀠 (ℝ2d ) a �→ a ∗ Θσ ,

(4.92)

which gives the Weyl symbol of an operator with Born–Jordan symbol a, is surjective.

4.6 Born–Jordan quantization and related pseudodifferential calculus | 245

̂ is never It is important to observe that the above representation of the operator A i 2d ̂ = OpBJ (a) and z0 ∈ ℝ verifies Θ(z0 ) = 0 then the symbol a(z) + e ℎ σ(z0 ,z) unique: if A ̂ see Example 4.6.6 below. gives rise to the same operator A; As one may suspect, imaginary-exponential symbols play an important role in the i discussion. In fact, the function e ℎ σ(z0 ,z) turns out to be the Weyl symbol of the oper̂ 0 ) and, in general, any operator can be regarded as a superposition of T(z)’s; ̂ ator T(z

cf. (4.91). This suggests the study of the map (4.92) in spaces of smooth temperate functions which extend to entire functions in ℂ2n , with a growth at most exponential in the imaginary directions. To be precise, for r ≥ 0, let 𝒜r be the space of smooth functions a in ℝ2n that extend to entire functions a(ζ ) in ℂ2n and satisfying the estimate r N 󵄨󵄨 󵄨 󵄨󵄨a(ζ )󵄨󵄨󵄨 ≤ C(1 + |ζ |) exp( | Im ζ |), ℎ

ζ ∈ ℂ2n ,

i

for some C, N > 0. For example, the symbol e ℎ σ(z0 ,z) belongs to 𝒜r with r = |z0 |, whereas 𝒜0 is nothing but the space of polynomials in phase space. Then we have the following result (Proposition 4.6.9 and Theorem 4.6.10). The map 𝒜r → 𝒜r in (4.92) is surjective for every r ≥ 0. It is also one-to-one (and therefore a bijection) if and only if 0 ≤ r < √4πℎ. We conjecture that the above threshold r = √4πℎ should have an interesting physical interpretation in terms of the Heisenberg uncertainty principle/symplectic capacity. In what follows we present the tools we need to prove Theorem 4.6.1.

4.6.1 Division of distributions

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We recall the Paley–Wiener–Schwartz theorem (see, e. g., [194, Theorem 7.3.1]). Theorem 4.6.2. For r ≥ 0, let Br be the closed ball |x| ≤ r in ℝd . If u is a distribution with compact support in Br then its (symplectic) Fourier transform ℱσ u extends to an entire analytic function in ℂd and satisfies r N 󵄨󵄨 󵄨 󵄨󵄨ℱσ u(ζ )󵄨󵄨󵄨 ≤ C(1 + |ζ |) exp( | Im ζ |) ℎ for some C, N > 0. Conversely, every entire analytic function satisfying an estimate of this type is the (symplectic) Fourier transform of a distribution supported in Br . We begin with a technical result, inspired by [261, Theorem VII, page 123], which will be used in the sequel. The Schwartz seminorms ‖ψ‖N , for ψ ∈ 𝒮 (ℝd ), we are going

246 | 4 Pseudodifferential operators to use here are ‖ψ‖N :=

󵄨 󵄨 sup sup 󵄨󵄨󵄨x α 𝜕xβ ψ(x)󵄨󵄨󵄨 < ∞,

|α|+|β|≤N x∈ℝd

(4.93)

for every N ≥ 0. Such seminorms on 𝒮 (ℝd ) are equivalent to those recalled in (1.57) (Chapter 1). Proposition 4.6.3. Let v ∈ 𝒮 󸀠 (ℝd ) and χ ∈ 𝒟(ℝ). For every t ∈ ℝ, there exists a distribution ut ∈ 𝒮 󸀠 (ℝd ) satisfying (xd − t)ut = χ(xd − t)v.

(4.94)

Moreover, ut can be chosen so that 󵄨󵄨 󵄨 󵄨󵄨ut (φ)󵄨󵄨󵄨 ≤ C‖φ‖N ,

∀φ ∈ 𝒮 (ℝd )

(4.95)

for some constants C, N > 0 independent of t. Proof. We define ut as follows. Write x = (x 󸀠 , xd ), and let φ ∈ 𝒮 (ℝd ). We have φ(x) = φ(x󸀠 , t) + (xd − t)φ̃ t (x) with 1

φ̃ t (x) = ∫ 𝜕xd φ(x󸀠 , t + τ(xd − t)) dτ. 0

Observe that χ(xd − t)φ̃ t ∈ 𝒮 (ℝd ) (whereas φ̃ t is not Schwartz in the xd direction, in general) and define ut (φ) = v(χ(xd − t)φ̃ t ). It is easy to see that ut is, in fact, a temperate distribution: since v ∈ 𝒮 󸀠 (ℝd ), we have

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󵄨󵄨 󵄨 󵄩 󵄩 󵄨󵄨ut (φ)󵄨󵄨󵄨 ≤ C 󵄩󵄩󵄩χ(xd − t)φ̃ t 󵄩󵄩󵄩N for some C, N > 0 independent of t. On the other hand, on the support of χ(xd − t), we have |xd − t| ≤ C1 and 󵄨 󵄨 1 + |xd | ≤ C2 (1 + |t|) ≤ C3 (1 + 󵄨󵄨󵄨t + τ(xd − t)󵄨󵄨󵄨), for every τ ∈ [0, 1], so that 1

󵄨󵄨 αd β ̃ 󵄨 󵄨 α 󵄨󵄨 β 󵄨 󸀠 󵄨󵄨xd 𝜕x φt (x)󵄨󵄨󵄨 ≤ ∫󵄨󵄨󵄨xdd 󵄨󵄨󵄨󵄨󵄨󵄨𝜕x [𝜕xd φ(x , t + τ(xd − t))]󵄨󵄨󵄨 dτ 0

1

󵄨 󵄨α 󵄨 󵄨 ≤ C 󸀠 ∫(1 + 󵄨󵄨󵄨t + τ(xd − t)󵄨󵄨󵄨) d 󵄨󵄨󵄨𝜕xβ [𝜕xd φ(x󸀠 , t + τ(xd − t))]󵄨󵄨󵄨 dτ, 0

4.6 Born–Jordan quantization and related pseudodifferential calculus | 247

which gives 󵄩 󵄩󵄩 󸀠󸀠 󵄩󵄩χ(xd − t)φ̃ t 󵄩󵄩󵄩N ≤ C ‖φ‖N+1 with constants C 󸀠󸀠 , N independent of t. This gives (4.95). Formula (4.94) is easily verified: for φ ∈ 𝒮 (ℝd ), (xd − t)ut (φ) = ut ((xd − t)φ) = v(χ(xd − t)φ) = χ(xd − t)v(φ). We emphasize that the point in the above result is the control of the constants C and N with respect to t; in fact, the existence of a temperate distribution solution ut for every fixed t already follows from a variant of the arguments in [261, page 127]. We also need the following division result with control of the support. This result could probably be proved by extracting and combining several arguments disseminated in [219], but we prefer to provide a self-contained and more accessible proof. As above we split the variable x in ℝd into x = (x󸀠 , xd ). Proposition 4.6.4. Let B󸀠 = {x 󸀠 ∈ ℝd−1 : |x 󸀠 | < 1}, B = B󸀠 × ℝ ⊂ ℝd and f : B󸀠 → ℝ be a smooth function. Let K := {x = (x󸀠 , xd ) ∈ B : xd ≥ f (x󸀠 )}. Suppose that K0 := {x󸀠 ∈ B󸀠 : f (x 󸀠 ) = 0} = {x 󸀠 ∈ B󸀠 : x1 = ⋅ ⋅ ⋅ = xk = 0} for some 1 ≤ k ≤ d − 1, and N 󵄨󵄨 󸀠 󵄨󵄨 󸀠 󵄨󵄨f (x )󵄨󵄨 ≥ C0 dist(x , K0 )

x󸀠 ∈ B

(4.96)

for some C0 , N > 0. Then for every v ∈ ℰ 󸀠 (B) with supp v ⊂ K the equation xd u = v

(4.97)

admits a solution u ∈ ℰ 󸀠 (B) with supp u ⊂ K.

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The condition (4.96) is known as Lojasiewicz’ inequality and is automatically satisfied if f is real-analytic ([219, Theorem 17]). Proof. As a preliminary remark, it is clear that we can limit ourselves to constructing a solution u ∈ 𝒟󸀠 (B) with supp u ⊂ K, because one can then multiple u by a cut-off function, equal to 1 in a neighborhood of the support of v, and get another solution in ℰ 󸀠 (B), still satisfying supp u ⊂ K. Now, it is easy to see that a solution of (4.97) is given by the distribution 𝒟(B) ∋ φ �→ v(φ)̃

where 1

̃ φ(x) = ∫ 𝜕xd φ(x󸀠 , τxd ) dτ. 0

(4.98)

248 | 4 Pseudodifferential operators More generally, any distribution of the form ̃ u(φ) = w ⊗ δ + v(φ),

(4.99)

where w is an arbitrary distribution in B󸀠 and δ = δ(xd ), is solution of (4.97). Hence we have reduced the situation to prove that w can be chosen so that supp u ⊂ K, i. e., u(φ) = 0 for every φ supported in the open set Ω := {x ∈ B : xd < f (x)}. This condition, as we will see, forces the values of w on the test functions φ1 ∈ 𝒟(Ω). We construct w as follows. Fix once for all a function φ2 (xd ) in 𝒟(ℝ), with φ2 (0) = 1, supported in the interval [−1, 1]. Let Ω󸀠 = Ω ∩ {xd = 0}. Let φ1 (x 󸀠 ) be any function in 𝒟(Ω󸀠 ) and ϵ > 0 such that dist(supp φ1 , K0 ) > ϵ. Then we define w(φ1 ) := −v(φ)̃ where φ(x) = φ1 (x󸀠 )φ2 (xd /(C0 ϵN )),

(4.100)

with the constants C0 , N appearing in (4.96), and φ̃ is constructed from φ as in (4.98): 1

̃ φ(x) =

ϕ1 (x󸀠 ) ∫ φ󸀠2 (τxd /(C0 ϵN )) dτ C0 ϵ N

(4.101)

0

ϕ (x󸀠 ) = 1 xd

xd /(C0 ϵN )

∫ 0

φ󸀠2 (τ) dτ;

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see Figure 4.2. xd

K

xd = f (x󸀠 ) Ω Ω󸀠

ϵ

x󸀠

Figure 4.2: The box contains the support of φ in (4.100).

Observe that the function φ in (4.100) is supported in Ω by (4.96), but this is not the case for φ.̃ We now prove the following facts.

4.6 Born–Jordan quantization and related pseudodifferential calculus | 249

(i) w is well defined. Let us verify that the definition of w(φ1 ) does not depend on the choice of ϵ. Let dist(supp φ1 , K0 ) > ϵ > ϵ󸀠 > 0; then the difference function N

φ(x) = ϕ1 (x󸀠 )[φ2 (xd /(C0 ϵN )) − φ2 (xd /(C0 ϵ󸀠 ))] is obviously still supported in Ω but, in addition, it vanishes at xd = 0, so the corresponding function φ̃ in (4.98) has compact support contained in Ω. Since v is supported in K, we have v(φ)̃ = 0. (ii) w ∈ 𝒟󸀠 (Ω󸀠 ). This is easy to verify and is also a consequence of the next point. (iii) w extends to a distribution in 𝒟󸀠 (B󸀠 ). It is sufficient to prove an estimate of the type 󵄨󵄨 󵄨 󵄨 α 󸀠 󵄨 󵄨󵄨w(φ1 )󵄨󵄨󵄨 ≤ C sup sup 󵄨󵄨󵄨𝜕x󸀠 φ1 (x )󵄨󵄨󵄨 󸀠 󸀠 |α|≤M x ∈B

for some constants C, M > 0 and every φ1 ∈ 𝒟(Ω󸀠 ). In fact, by the Hahn–Banach theorem, one can then extend the linear functional w : 𝒟(Ω󸀠 ) → ℂ, which is continuous when 𝒟(Ω󸀠 ) is endowed with the norm in the above right-hand side, to a functional on 𝒟(B󸀠 ), continuous with respect to the same norm and therefore a fortiori for the usual topology of this space. Now, by the definition of w and since v has compact support in K, we have (cf. [194, Theorem 2.3.10]) 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 β ̃ 󵄨󵄨 󵄨󵄨w(φ1 )󵄨󵄨󵄨 = 󵄨󵄨󵄨v(φ)̃ 󵄨󵄨󵄨 ≤ C sup sup 󵄨󵄨󵄨𝜕x φ(x) 󵄨󵄨 |β|≤M

x∈K xd ≤C1

(4.102)

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for some C, C1 , M > 0. To estimate the last term, we observe that in the expression for φ̃ in (4.101) the integral is, in fact, constant if xd ≥ C0 ϵN . In particular, this happens if x ∈ K and x󸀠 ∈ supp φ1 , because (4.96) implies that for such x it turns out N

xd ≥ f (x) ≥ C0 dist(x󸀠 , K0 ) ≥ C0 ϵN . Hence for x ∈ K, xd ≤ C1 , we have 󵄨 ̃ 󵄨󵄨 󵄨 α 󸀠 󵄨 −M sup 󵄨󵄨󵄨𝜕xβ φ(x) 󵄨󵄨 ≤ C sup 󵄨󵄨󵄨𝜕x󸀠 φ1 (x )󵄨󵄨󵄨 ⋅ xd |α|≤M

|β|≤M

−NM 󵄨 󵄨 ≤ C sup 󵄨󵄨󵄨𝜕xα󸀠 φ1 (x 󸀠 )󵄨󵄨󵄨 ⋅ C0−M dist(x󸀠 , K0 ) . |α|≤M

On the other hand, we have 1/2

dist(x󸀠 , K0 ) = (|x1 |2 + ⋅ ⋅ ⋅ + |xk |2 ) ,

250 | 4 Pseudodifferential operators so that a Taylor expansion (with remainder of order NM) of 𝜕xα󸀠 φ1 with respect to x1 , . . . , xk (taking into account that φ1 vanishes in a neighborhood of K0 ) gives 󵄨 ̃ 󵄨󵄨 󸀠 sup sup 󵄨󵄨󵄨𝜕xβ φ(x) 󵄨󵄨 ≤ C x∈K

|β|≤M

xd ≤C1

sup

󵄨 󵄨 sup 󵄨󵄨󵄨𝜕xα󸀠 φ1 (x 󸀠 )󵄨󵄨󵄨.

|α|≤M+NM x󸀠 ∈B󸀠

This, together with (4.102), gives the desired conclusion. (iv) With the choice of w in (4.99), we have supp u ⊂ K. Let φ ∈ 𝒟(Ω), and write φ(x) = [φ(x) − φ(x󸀠 , 0)φ2 (xd /(C0 ϵN ))] + φ(x󸀠 , 0)φ2 (xd /(C0 ϵN )), where we choose ϵ < dist(supp φ(⋅, 0), K0 ). Now the distribution u vanishes when applied to the second term of this sum just by the definition of w (with φ(x󸀠 , 0) playing the role of φ1 (x 󸀠 )). On the other hand, the function φ(x) − φ(x󸀠 , 0)φ2 (xd /(C0 ϵN )) is supported in Ω and vanishes at xd = 0, so that one sees from the definition of u in (4.99) that its pairing with u is 0.

4.6.2 Changes of coordinates for temperate distributions In the sequel we will perform changes of coordinates which preserve Schwartz functions and temperate distributions in the following sense. Let ϕ be a smooth diffeomorphism of the semispace {x1 > 0} ⊂ ℝd into itself. Suppose that ϕ is positively homogeneous for some positive order, say, r > 0, i. e., ϕ(λx) = λr ϕ(x) for every x ∈ ℝd with x1 > 0, λ > 0. It follows that the image of every truncated cone

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U = {x ∈ ℝd : x1 ≥ ϵ|x|, |x| ≥ ϵ}, with 0 < ϵ ≤ 1 (see Figure 4.3 below) is contained in another truncated cone of the same type. In fact, if y = ϕ(x), then for some ϵ󸀠 > 0 we have |y| ≥ ϵ󸀠 when x ∈ U and |x| = ϵ by compactness, and therefore for every x ∈ U by homogeneity. The same argument implies that y1 /|y| ≥ ϵ󸀠 > 0 for x ∈ U.

Figure 4.3: A truncated cone in the semispace x1 > 0.

4.6 Born–Jordan quantization and related pseudodifferential calculus | 251

Moreover, the same applies to the inverse function ϕ−1 (x), which will be homogeneous of degree 1/r > 0. Now, let 𝒮 (ℝd )cone , 𝒮 󸀠 (ℝd )cone be the spaces of Schwartz functions and temperate distributions in ℝd , respectively, with support contained in some truncated cone, as above. Then ϕ induces bijections ϕ∗ : 𝒮 (ℝd )cone → 𝒮 (ℝd )cone ,

ϕ∗ : 𝒮 󸀠 (ℝd )cone → 𝒮 󸀠 (ℝd )cone

defined as follows. If ψ ∈ 𝒮 (ℝd )cone we define ϕ∗ ψ(x) = ψ(ϕ−1 (x)) for x1 > 0 and = 0 otherwise, and it is easy to see that ϕ∗ ψ ∈ 𝒮 (ℝd )cone using the homogeneity of ϕ−1 and the support condition on ψ. If u ∈ 𝒮 󸀠 (ℝd )cone we define the distribution ϕ∗ u by 󵄨 󵄨 ϕ∗ u(φ) = u(χ ⋅ φ ∘ ϕ󵄨󵄨󵄨det ϕ󸀠 󵄨󵄨󵄨), for every φ ∈ 𝒮 (ℝd ), where χ : ℝd → [0, 1] is a smooth function, positively homogeneous of degree 0 for large |x|, χ(x) = 1 on a truncated cone slightly larger than one containing the support of u and χ is supported in a truncated cone. It is easy to see that χ(x)φ(ϕ(x))| det ϕ󸀠 (x)| is a function in 𝒮 (ℝd )cone because on every truncated cone the Jacobian determinant | det ϕ󸀠 | is smooth and has an at most polynomial growth, together with its derivatives. The maps ϕ∗ : 𝒮 (ℝd )cone → 𝒮 (ℝd )cone and ϕ∗ : 𝒮 󸀠 (ℝd )cone → 𝒮 󸀠 (ℝd )cone are bijections, the inverses being given by (ϕ−1 )∗ on 𝒮 (ℝd )cone and 𝒮 󸀠 (ℝd )cone . Observe that, more generally, if ψ is a smooth function with an at most polynomial growth together with its derivatives, we can similarly define ϕ∗ ψ(x) = ψ(ϕ−1 (x)) for x1 > 0 and have the formula

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ϕ∗ (ψu) = ϕ∗ ψϕ∗ u

(4.103)

if u ∈ 𝒮 󸀠 (ℝd )cone (the formula makes sense even if ϕ∗ ψ is defined only for x1 > 0, because u and therefore ϕ∗ u is supported in a truncated cone). 4.6.3 Born–Jordan pseudodifferential operators We recall the Grossmann–Royer operator, defined by, for z0 = (x0 , ξ0 ), 2i

T̂GR (z0 )ψ(x) = e ℎ (x−x0 )⋅ξ0 ψ(2x0 − x). It is a unitary operators on L2 (ℝd ) and an involution: T̂GR (z0 )T̂GR (z0 ) = I.

252 | 4 Pseudodifferential operators The two following important formulas hold: ̂ 0 )R∨ T(z ̂ 0 )−1 T̂GR (z0 ) = T(z

(4.104)

̂ 0 ) is the Heisenberg operator defined above, R∨ = T̂GR (0) is the reflection where T(z operator, R∨ ψ(x) = ψ(−x), and ̂ T̂GR (z0 )ψ(x) = 2−n ℱσ [T(⋅)ψ(x)](−z 0)

(4.105)

where recall ℱσ is the symplectic Fourier transform. ̂ W = Opw (a) can be defined as the image of Let a ∈ 𝒮 󸀠 (ℝ2d ). The Weyl operator A ̂ ̂ W defined by the quantization mapping a �→ AW , with A n

̂ W ψ = ( 1 ) ∫ a(z0 )T̂GR (z0 )ψ dz0 , A πℎ

(4.106)

ℝ2d

which is equivalent, using (4.105), to n

̂ W ψ = ( 1 ) ∫ aσ (z0 )T(z ̂ 0 )ψ dz0 . A 2πℎ

(4.107)

ℝ2d

Notice that when, for instance, a ∈ 𝒮 (ℝ2d ), formula (4.106) can be rewritten in the familiar form n

i ̂ W ψ(x) = Opw (a)ψ(x) = ( 1 ) ∫ e ℎ (x−y)⋅ξ a( 1 (x + y), p)ψ(y) dp dy. A 2πℎ 2

ℝ2d

̂ BJ = OpBJ (a) is constructed as follows: one first defines The Born–Jordan operator A ̂ τ = Opτ (a) by the Shubin τ-operator A ̂ τ ψ = ∫ aσ (z0 )T̂τ (z0 )ψ dz0 A

(4.108)

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ℝ2d

where T̂τ (z) is the unitary operator on L2 (ℝd ) defined by i

1

̂ 0 ). T̂τ (z0 ) = e ℎ (τ− 2 )x0 ⋅ξ0 T(z

(4.109)

One thereafter defines 1

̂ BJ ψ = ∫ A ̂ τ ψ dτ. A 0

Using the obvious formula 1

i

1

Θ(z0 ) := ∫ e ℎ (τ− 2 )x0 ⋅ξ0 dτ = { 0

sin(x0 ⋅ξ0 /2ℎ) x0 ⋅ξ0 /2ℎ

1

for x0 ⋅ ξ0 ≠ 0,

for x0 ⋅ ξ0 = 0,

4.6 Born–Jordan quantization and related pseudodifferential calculus | 253

one thus has ̂ BJ ψ = ∫ aσ (z0 )T̂BJ (z0 )ψ dz0 A ̂ 0 ). where T̂BJ (z0 ) = Θ(z0 )T(z ̂ It follows that ABJ (a) is the Weyl operator with covariant symbol (aW )σ = Θaσ .

(4.110)

̂ BJ (a) is thus (taking the symplectic Fourier transform) The Weyl symbol of A aW = (

d

1 ) a ∗ Θσ . 2πℎ

(4.111)

̂ = Opw (aW ). Then A ̂ = OpBJ (a), provided that a satisfies Conversely, assume that A (aW )σ = Θaσ . 4.6.4 Invertibility of the Born–Jordan quantization In this section we investigate the injectivity and surjectivity of the map 𝒮 󸀠 (ℝ2d ) → 𝒮 󸀠 (ℝ2d ) given by a ��→ (

d

1 ) a ∗ Θσ , 2πℎ

(4.112)

namely the map which gives the Weyl symbol of an operator with Born–Jordan symbol a, cf. (4.111). We have the following result.

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Theorem 4.6.5. The equation d

(

1 ) a ∗ Θσ = b 2πℎ

admits a solution a ∈ 𝒮 󸀠 (ℝ2d ), for every b ∈ 𝒮 󸀠 (ℝ2d ). Proof. Taking the symplectic Fourier transform, we have reduced the situation to proving that the equation Θa = b admits at least a solution a ∈ 𝒮 󸀠 (ℝ2d ), for every b ∈ 𝒮 󸀠 (ℝ2d ). This is a problem of division of temperate distributions. We provide two proofs.

254 | 4 Pseudodifferential operators First proof. We localize the problem by considering a finite and smooth partition of unity ψ0 (z), ψ±j (z), j = 1, . . . , 2d, in phase space, where ψ0 has support in a ball |z| ≤ r, with r < √4πℎ, and ψ+j for j = 1, . . . , 2d, is supported in a truncated cone (cf. Section 4.6.2) of the type Uj+ = {z ∈ ℝ2d : zj ≥ ϵ|z|, |z| ≥ ϵ} contained in the semispace zj > 0, and similarly ψ−j is supported in a similar truncated cone contained in the semispace zj < 0, with ψ±j homogeneous of degree 0 for large z (it is easy to see that such a partition of unity can be constructed if ϵ is small enough, e. g., if ϵ < r/2 and ϵ < 1/(2√d)). It is clear that if a0 and a±j , j = 1, . . . , 2d, solve in 𝒮 󸀠 (ℝ2d ) the equations Θa0 = ψ0 b

and Θa±j = ψ±j b, then a := a0 + ∑dj=1 (a+j + a−j ) solves Θa = b. The equation Θa0 = ψ0 b is easily solved as in the proof of Proposition 4.6.9, using the fact that ψ0 b is supported in a closed ball where Θ ≠ 0. Let us now solve the equation Θa = b

where b ∈ 𝒮 󸀠 (ℝ2d ) is supported in a truncated cone in the semispace, say, z1 > 0 in phase space. We look for a ∈ 𝒮 󸀠 (ℝ2d ) supported in a truncated cone as well. We apply the following algebraic change of variables in phase space: y1 = z12 ,

y2 = z1 z2 , . . . ,

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yd+1 = z1 zd+2 , . . . ,

yd = z1 zd ,

y2d−1 = z1 z2d ,

d

y2d = x ⋅ ξ = ∑ zj zj+d j=1

where z = (x, ξ ). It is easy to check that the map z �→ y is a diffeomorphism of the semispace z1 > 0 into itself (y1 > 0),3 and moreover it is homogeneous of degree 2. We now apply the remarks in Section 4.6.2 (where the dimension of the space is now 2d), in particular (4.103), and we have reduced the situation to solve (in the new coordinates) the equation sin(y2d /2ℎ) a=b y2d /2ℎ 3 The inverse change of variables is given by z1 = √y1 , z2 = y2 /√y1 , . . . , zd = yd /√y1 , d

zd+1 = (y2d − ∑ yj yd+j /y1 )/√y1 , zd+2 = yd+1 /√y1 , . . . , z2d = y2d−1 /√y1 . j=2

(4.113)

4.6 Born–Jordan quantization and related pseudodifferential calculus | 255

where b ∈ 𝒮 󸀠 (ℝ2d ) is supported in a truncated cone in the semispace y1 > 0, and we look for a ∈ 𝒮 󸀠 (ℝ2d ) similarly supported in a truncated cone in the same semispace. We now consider a partition of unity in ℝ obtained by translation of a fixed function, of the type χ(y2d − 2πkℎ), k ∈ ℤ, where χ ∈ 𝒟(ℝ) is a fixed function supported in the interval [−(3/2)πℎ, (3/2)πℎ]. Observe that on the support of χ(y2d − 2πkℎ) the sin(y /2ℎ) function y 2d/2ℎ has only a simple zero at y2d = 2πkℎ, for k ≠ 0, whereas it does not 2d

vanish on the support of χ (case k = 0). We now solve, for every k ∈ ℤ, the equation

sin(y2d /2ℎ) ak = χ(y2d − 2πkℎ)b. y2d /2ℎ

(4.114)

We suppose k ≠ 0, the case k = 0 being easier. Since the function sin(y2d /2ℎ) (y2d − 2kπℎ)y2d /2ℎ is smooth and does not vanish on the support of χ(y2d − 2πkℎ), it is sufficient to solve the equation (y2d − 2kπℎ)ak = χ(y2d − 2πkℎ)[

(y2d − 2kπℎ)y2d /2ℎ b]. sin(y2d /2ℎ)

y −2kπℎ

2d Observe that the function sin(y has derivatives uniformly bounded with respect 2d /2ℎ) to k (in fact, we have sin(y2d /2ℎ) = ± sin((y2d − 2πkℎ)/2ℎ), according to the parity of k, and the “sinc” function has bounded derivatives of any order). It follows from this remark and Proposition 4.6.3 that there exists a solution ak ∈ 𝒮 󸀠 (ℝ2d ) of the above equation, satisfying the estimate

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󵄨󵄨 󵄨 󵄨󵄨ak (φ)󵄨󵄨󵄨 ≤ C‖φ‖N

∀φ ∈ 𝒮 (ℝ2d )

for some constants C, N > 0 independent of k. Moreover, we can suppose that all the ak ’s are supported in a fixed truncated cone (by multiplying by a cut-off function in phase space with the same property as ψ+1 (z) above, and = 1 on a truncated cone slightly larger than one containing the support ̃ 2d − 2πkℎ), where χ̃ ∈ 𝒟(ℝ), and of b). We can also multiply ak by a cut-off function χ(y χ̃ = 1 in a neighborhood of the support of χ, and obtain new solutions ̃ 2d − 2πkℎ)ak ã k := χ(y to (4.114), satisfying 󵄨󵄨 ̃ 󵄨 󵄩 ̃ 󵄩󵄩 󵄨󵄨ak (φ)󵄨󵄨󵄨 ≤ C 󵄩󵄩󵄩χ(y 2d − 2πkℎ)φ󵄩 󵄩N for every φ ∈ 𝒮 (ℝ2d ); see Figure 4.4 below.

(4.115)

256 | 4 Pseudodifferential operators y2 2πkℎ y1

Figure 4.4: The distribution ã k is supported in the strip (d = 1 in this figure).

Now we claim that a := ∑k∈ℤ ã k solves (4.113). We have only to check that the series converges in 𝒮 󸀠 (ℝ2d ). Let us verify that, given φ ∈ 𝒮 (ℝ2d ), the series ∑ ã k (φ)

n∈ℤ

converges absolutely. ̃ 2d −2πkℎ), we have 1+|y| ≥ 1+|y2d | ≥ C(1+|k|), and |yα 𝜕yβ φ(y)| ≤ On the support of χ(y

CN󸀠 󸀠 (1 + |k|)−N for every N 󸀠 and α, β ∈ ℕ2d . Hence by (4.115) we obtain 󸀠

−N 󸀠 󵄨󵄨 ̃ 󵄨 󸀠󸀠 󵄨󵄨ak (φ)󵄨󵄨󵄨 ≤ CN 󸀠 (1 + |k|) ,

and it is sufficient to take N 󸀠 = 2 for the above series to converge absolutely.

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Second proof. In [190] it was proved that the equation Pu = v is always solvable in 𝒮 󸀠 (ℝd ) if P is a nonidentically zero polynomial. As observed there (page 556), that proof continues to hold if the polynomial P is replaced by a smooth function which satisfies the estimates (4.3) and (4.10) in that paper. Here we are interested in the division by the function Θ(z), which has only simple zeros, and those estimates read −μ󸀠󸀠 󵄨󵄨 󵄨 μ󸀠 󵄨󵄨Θ(z)󵄨󵄨󵄨 ≥ C dist(z, Z) (1 + |z|)

∀z ∈ ℝ2d

(4.116)

and −μ󸀠󸀠 󵄨󵄨 󵄨 󵄨󵄨∇Θ(z)󵄨󵄨󵄨 ≥ C(1 + |z|)

∀z ∈ Z

(4.117)

for some C, μ󸀠 , μ󸀠󸀠 > 0, where Z = {z ∈ ℝ2d : Θ(z) = 0}. To check that these estimates are satisfied, let sinc(t) = sin t/t (sinc(0) = 1), so that Θ(z) = Θ(x, ξ ) = sinc(x ⋅ ξ /2ℎ), z = (x, ξ ). Observe that at points t where sinc(t/2ℎ) = 0 we have 󵄨󵄨 d 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 sinc(t/2ℎ)󵄨󵄨󵄨 = 1/|t| 󵄨󵄨 󵄨󵄨 dt

4.6 Born–Jordan quantization and related pseudodifferential calculus | 257

so that 2 󵄨 |(x, ξ )| 󵄨󵄨 ≥ , 󵄨󵄨∇Θ(z)󵄨󵄨󵄨 = |x ⋅ ξ | |z|

∀z ∈ Z,

which implies (4.117) with μ󸀠󸀠 = 1. Concerning (4.116), observe first of all that, setting Z0 = {2πkℎ : k ∈ ℤ, k ≠ 0} ⊂ ℝ, we have, for |t| > πℎ, 2 󵄨 2ℎ 1 󵄨󵄨 󵄨 2ℎ 󵄨 dist(t, Z0 ) = dist(t, Z0 ) 󵄨󵄨sinc(t/2ℎ)󵄨󵄨󵄨 = 󵄨󵄨󵄨sin(t/2ℎ)󵄨󵄨󵄨 ≥ |t| |t| πℎ π|t| whereas if |t| ≤ πℎ, 1 󵄨󵄨 󵄨 2 󵄨󵄨sinc(t/2ℎ)󵄨󵄨󵄨 ≥ ≥ 2 dist(t, Z0 ). π π ℎ In both cases we have −1 󵄨󵄨 󵄨 󵄨󵄨sinc(t/2ℎ)󵄨󵄨󵄨 ≥ C0 (1 + |t|) dist(t, Z0 )

(4.118)

for some C0 > 0. Now, (4.116) is clearly satisfied in a neighborhood of 0, so that it is sufficient to prove it in any truncated cone contained in the semispaces zj > 0 or zj < 0, j = 1, . . . , 2d. Consider for example a truncated cone U where z1 > 0. We perform the change of coordinates y = y(z) in this semispace, exactly as in the previous proof, and we observe that by (4.118) we have

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−1 −1 󵄨󵄨 󵄨 󵄨󵄨sinc(y2d /2ℎ)󵄨󵄨󵄨 ≥ C0 (1 + |y2d |) |y2d − y2d | ≥ C0 (1 + |y|) |y2d − y2d |

where y2d ∈ Z0 is such that |y2d − y2d | = dist(y2d , Z0 ). Now, for z ∈ U we have 0 < ϵ ≤ |y| ≤ C|z|2 and moreover the inverse map z = z(y) is Lipschitz in any truncated cone U 󸀠 ⊃ z(U), because the derivatives 𝜕zj /𝜕yk are positively homogeneous of degree −1/2 < 0, and therefore bounded in U 󸀠 . Using these facts and setting y = (y1 , . . . , y2d−1 , y2d ), z = z(y) ∈ Z, we conclude that for every z ∈ U, −1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Θ(z)󵄨󵄨󵄨 = 󵄨󵄨󵄨sinc(y2d /2ℎ)󵄨󵄨󵄨 ≥ C0 (1 + |y|) |y − y|

≥ C(1 + |z|) |z − z|, −2

≥ C(1 + |z|) dist(z, Z). −2

This concludes the proof. From the previous theorem, we obtain at once the proof of Theorem 4.6.1 above. Concerning the injectivity of the map (4.112), we begin with a simple example, which shows that the map (4.112) is not one-to-one, even when restricted to real analytic functions which extend to entire functions in ℂ2d .

258 | 4 Pseudodifferential operators Example 4.6.6. Consider the Born–Jordan symbol i

i

a(z) = e ℎ σ(z0 ,z) = e ℎ (x⋅ξ0 −x0 ⋅ξ ) ,

(4.119)

where z0 = (x0 , ξ0 ) is any point on the zero set Θ(z) = 0. The symplectic Fourier transform of a is aσ (z) = (2πℎ)d δ(z − z0 ) and therefore Θaσ = 0, because Θ(z0 ) = 0. Hence the corresponding Weyl symbol is aW = 0 by (4.110). Observe that the symbol a in (4.119) extends to an entire function a(ζ1 , ζ2 ) = i e ℎ (ξ0 ⋅ζ1 −x0 ⋅ζ2 ) in ℂ2d , satisfying the estimate r 󵄨󵄨 󵄨 󵄨󵄨a(ζ )󵄨󵄨󵄨 ≤ exp( | Im ζ |), ℎ

ζ ∈ ℂ2d ,

where r = |(ξ0 , −x0 )| = |z0 |. For future reference we observe that the minimum value of r is reached for the points z0 at which the hypersurface Θ(z) has minimum distance from 0, which turns out to be r = √4πℎ. For example, one can consider x0 = ξ0 = (2πℎ/d)1/2 (1, . . . , 1), so that x0 ⋅ ξ0 = 2πℎ and |x0 |2 + |ξ0 |2 = 4πℎ (see Figure 4.5). ξ

z0 √4πℎ

x ⋅ ξ = 2πℎ x

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Figure 4.5: The point z0 minimizes the distance of the zero set Θ(z) = 0 from 0.

Inspired by the above example, we now exhibit a nontrivial class of functions on which the map (4.112) is injective. Definition 4.6.7. For r ≥ 0, let 𝒜r be the space of smooth functions a in ℝ2d that extend to entire functions a(ζ ) in ℂ2d and satisfying the estimate r N 󵄨󵄨 󵄨 󵄨󵄨a(ζ )󵄨󵄨󵄨 ≤ C(1 + |ζ |) exp( | Im ζ |), ℎ

ζ ∈ ℂ2d ,

for some C, N > 0. Equivalently (by the Paley–Wiener–Schwartz theorem), 𝒜r is the space of temperate distributions in ℝ2d whose (symplectic) Fourier transform is supported in the closed ball |z| ≤ r.

4.6 Born–Jordan quantization and related pseudodifferential calculus | 259

Remark 4.6.8. Observe that the space 𝒜0 is just the space of polynomials in phase space. We have the following result. Proposition 4.6.9. The map (4.112) is a bijection 𝒜r → 𝒜r if and only if 0 ≤ r < √4πℎ. Proof. The “only if” part follows at once from the Remark 4.6.6 because the symbol a(z) in (4.119) belongs to 𝒜r for r ≥ √4πℎ and is mapped to 0. Consider now the “if” part. Taking the symplectic Fourier transform in (4.112) and by the Paley–Wiener–Schwartz theorem, we have reduced the situation to prove that, when 0 ≤ r < √4πℎ, the map a �→ Θa is a bijection ℰ 󸀠 (Br ) → ℰ 󸀠 (Br ), where ℰ 󸀠 (Br ) is the space of distributions on ℝ2d supported in the closed ball Br given by |z| ≤ r. Now, it is clear that if a ∈ ℰ 󸀠 (Br ) then Θa ∈ ℰ 󸀠 (Br ). On the other hand, since the function Θ(z) does not vanish for |z| < √4πℎ, hence in a neighborhood of Br (by assumption r < √4πℎ), the equation Θa = b, for every b ∈ ℰ 󸀠 (Br ), has a unique solution a ∈ ℰ 󸀠 (Br ) obtained simply by multiplying by Θ−1 : a = Θ−1 b. We now study the surjectivity of the map (4.112) on the spaces 𝒜r when r ≥ √4πℎ. Theorem 4.6.10. Given r ≥ 0, for every b ∈ 𝒜r there exists a ∈ 𝒜r such that d

(

1 ) a ∗ Θσ = b. 2πℎ

Proof. As above we have to prove that the equation

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Θa = b admits at least a solution a ∈ ℰ 󸀠 (Br ), for every b ∈ ℰ 󸀠 (Br ). Since all the distributions here are compactly supported, the problem is local and we can solve the equation Θa = b in ℰ 󸀠 (Uz0 ) for a sufficiently small open neighborhood Uz0 of any given point z0 and conclude with a finite smooth partition of unity. If |z0 | > r and Uz0 ⊂ {|z| > r} one can choose a = 0 in Uz0 . When |z0 | < r and Uz0 ⊂ {|z| < r}, we apply the classical division theorem valid for smooth functions with at most simple zeros [261, page 127]: for every b ∈ ℰ 󸀠 (Uz0 ) there therefore exists a solution a ∈ ℰ 󸀠 (Uz0 ). Of course, if Θ(z0 ) ≠ 0 the division is trivial, so that we now suppose that z0 = (x0 , ξ0 ) belongs to both |z| = r and x ⋅ ξ = 2πkℎ for some k ∈ ℤ, k ≠ 0. Then necessarily we have r ≥ √4π|k|ℎ, because this is the distance of the hypersurface x ⋅ ξ = 2πkℎ from the origin. We therefore distinguish two cases.

260 | 4 Pseudodifferential operators First case: r > √4π|k|ℎ. Then the hypersurfaces |z| = r and x ⋅ ξ = 2πkℎ cut transversally at z0 , i. e., their normal vectors are linearly independent and the intersection Σ is therefore a submanifold of codimension 2. In fact, one sees easily that the vector normals to these two hypersurfaces at z0 are linearly dependent if and only if ξ0 = sign(k)x0 and |x0 |2 = 2π|k|ℎ. In that case we must have r = √4π|k|ℎ. Second case: r = √4π|k|ℎ. Then the hypersurfaces |z| = r and x ⋅ ξ = 2πkℎ touch along the submanifold Σ of codimension n having equations ξ = sign(k)x, |x|2 = 2π|k|ℎ. In both cases by the implicit function theorem we can take analytic coordinates y = (y󸀠 , y2d ) near z0 so that z0 has coordinates y = 0, the hypersurface x ⋅ ξ = 2πkℎ is straightened to y2d = 0 and moreover the above submanifold Σ has equations y1 = y2d = 0 (in the first case) or y1 = ⋅ ⋅ ⋅ = yn−1 = y2n = 0 (in the second case). The portion of ball |z| ≤ r near z0 is defined now by the inequality y2d ≥ f (y󸀠 ) for some real-analytic function f (y󸀠 ) defined in a neighborhood of 0, and vanishing on Σ ∋ 0. We have reduced the situation to solving the equation y2d a = b in a neighborhood of 0, where b is supported in the set y2d ≥ f (y󸀠 ), and we look for a supported in the same set. This is exactly the situation of Proposition 4.6.4 (possibly after a rescaling). As already observed, condition (4.96) is satisfied by every real-analytic function and therefore Proposition 4.6.4 gives the desired conclusion. Example 4.6.11. We want to find a Born–Jordan symbol of the Heisenberg operator ̂ 0 ), z0 = (x0 , ξ0 ) ∈ ℝ2d . First of all, we observe that T(z ̂ 0 ) has Weyl symbol T(z i

b(z) = e ℎ σ(z0 ,z) ; 1 d ) a∗ see [105, Proposition 198]. Hence we are looking for a ∈ 𝒮 󸀠 (ℝ2d ) such that ( 2πℎ Θσ = b, or equivalently, taking the symplectic Fourier transform, Θaσ = bσ , that is,

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Θ(z)aσ (z) = (2πℎ)d δ(z − z0 ).

(4.120)

Now, if Θ(z0 ) ≠ 0, we can take aσ (z) = Θ(z0 )−1 (2πℎ)d δ(z − z0 ), namely a(z) = i

Θ(z0 )−1 e ℎ σ(z0 ,z) . If instead Θ(z0 ) = 0, we look for aσ in the form 2d

aσ (z) = (2πℎ)d ∑ cj 𝜕j δ(z − z0 ) j=1

for unknown cj ∈ ℂ, j = 1, . . . , 2d. Since Θ(z0 ) = 0, we have 2d

Θ(z)aσ (z) = (2πℎ)d (− ∑ cj 𝜕j Θ(z0 ))δ(z − z0 ), j=1

(4.121)

4.7 Born–Jordan pseudodifferential operators on modulation spaces | 261

so that equation (4.120) reduces to 2d

− ∑ cj 𝜕j Θ(z0 ) = 1, j=1

which has infinitely many solutions because ∇Θ(z0 ) ≠ 0 if Θ(z0 ) = 0 (Θ(z) has only simple zeros). For any solution c := (c1 , . . . , c2d ), taking the inverse symplectic Fourier transform in (4.121) (using the formulas (aσ )σ = a, aσ (z) = ℱ a(Jz), ℱ (𝜕j a) = ℎi zj ℱ a), we find a Born–Jordan symbol a(z) =

i i σ(z, c)e ℎ σ(z0 ,z) . ℎ

Observe that b ∈ 𝒜r with r = |z0 | and a ∈ 𝒜r as well.

4.7 Born–Jordan pseudodifferential operators on modulation spaces The main result concerning the sufficient boundedness conditions of Born–Jordan operators on modulation spaces shows that they behave similarly to Weyl pseudodifferential operators or any other τ-form of pseudodifferential operators. The necessary boundedness conditions still contain some open problems, as shown in the following Theorem 4.7.1. These results are contained in [62]. Since we apply the theory of the previous section in the framework of time– frequency analysis, we set ℎ=

1 2π

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and consider the corresponding Cohen kernel sin(πx⋅ξ ) πx⋅ξ

Θ(z) := sinc(x ⋅ ξ ) = { 1

for x ⋅ ξ ≠ 0, for x ⋅ ξ = 0.

(4.122)

The (cross-)Born–Jordan distribution Q(f , g) is then defined by Q(f , g) = W(f , g) ∗ Θσ ,

f , g ∈ 𝒮 (ℝd ),

(4.123)

where Θσ is the symplectic Fourier transform of Θ recalled in (1.17) below. Likewise, the Weyl operator, a Born–Jordan operator with symbol a ∈ 𝒮 󸀠 (ℝ2d ), can be defined as ⟨OpBJ (a)f , g⟩ = ⟨a, Q(g, f )⟩

f , g ∈ 𝒮 (ℝd ).

(4.124)

262 | 4 Pseudodifferential operators Now, the first relevant feature of this section is to have the Cohen kernel Θσ computed explicitly (cf. the subsequent Proposition 4.7.5). Namely, (ζ1 , ζ2 ) ∈ ℝ2 , d = 1, {−2Ci(4π|ζ1 ζ2 |), Θσ (ζ1 , ζ2 ) = { d−2 2d {ℱ (χ{|s|≥2} |s| )(ζ1 ⋅ ζ2 ), (ζ1 , ζ2 ) ∈ ℝ , d ≥ 2, where χ{|s|≥2} is the characteristic function of the set {s ∈ ℝ : |s| ≥ 2} and where +∞

Ci(t) = − ∫ t

cos s ds, s

t ∈ ℝ,

(4.125)

is the cosine integral function. This expression of Θσ shows that this kernel behaves badly in general: it does not even belong to L∞ loc (see Corollary 4.7.6) and has no decay at infinity (see Corollary 4.7.7). Despite these undesirable aspects, some directional smoothing effect is still present, as proved in [63], see the following section. Our main result below refers to continuity properties on the same modulation space M r1 ,r2 (ℝd ). Observe that a similar result for Weyl operators contained in Theorem 4.4.15 above studied the action of Opw (a) : ℳp1 ,q1 (ℝd ) → ℳp2 ,q2 (ℝd ), but this is only a straightforward generalization. Theorem 4.7.1. Consider 1 ≤ p, q, r1 , r2 ≤ ∞, such that p ≤ q󸀠

(4.126)

q ≤ min{r1 , r2 , r1󸀠 , r2󸀠 }.

(4.127)

and

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Then the Born–Jordan operator OpBJ (a), from 𝒮 (ℝd ) to 𝒮 󸀠 (ℝd ), having symbol a ∈ M p,q (ℝ2d ), extends uniquely to a bounded operator on ℳr1 ,r2 (ℝd ), with the estimate 󵄩󵄩 󵄩 󵄩󵄩OpBJ (a)f 󵄩󵄩󵄩ℳr1 ,r2 ≲ ‖a‖M p,q ‖f ‖ℳr1 ,r2

f ∈ ℳr1 ,r2 .

(4.128)

Conversely, if this conclusion holds true, the constraint (4.126) is satisfied and it must hold 1 1 1 1 1 1 max{ , , 󸀠 , 󸀠 } ≤ + , r1 r2 r1 r2 q p

(4.129)

which is (4.127) when p = ∞. Notice that condition (4.129) is weaker than (4.127) when p < ∞. Condition (4.129) is obtained by working with rescaled Gaussians which provide the best localization in terms of Wigner distribution (cf. [218]). On the Fourier side, the Born–Jordan distribution is the pointwise multiplication of the Wigner distribution with the kernel Θ. This

4.7 Born–Jordan pseudodifferential operators on modulation spaces | 263

reasoning conduces to conjecture that condition (4.129) should be optimal so that the sufficient boundedness conditions for Born–Jordan operators might be weaker than the corresponding ones for Weyl and τ-pseudodifferential operators. But the matter is really subtle and requires a new and most refined analysis of the kernel Θ. In particular, the zeroes of the Θ function should play a key role for a thorough understanding of such operators, which certainly deserve further study. 4.7.1 Analysis of the Cohen kernel Θ Consider the Cohen kernel Θ defined in (4.122). Obviously, Θ ∈ 𝒞 ∞ (ℝ2d ) ∩ L∞ (ℝ2d ), but it displays a vary bad decay at infinity, as clarified in what follows. Proposition 4.7.2. For 1 ≤ p < ∞, the function Θ ∉ Lp (ℝ2d ). Proof. Observe that, for t ∈ ℝ, |t| ≤ 1/2, the function sinct satisfies |sinc(t)| ≥ 2/π. Hence, for any 1 ≤ p < ∞, 󵄨 󵄨 󵄨p 󵄨p ∫ 󵄨󵄨󵄨Θ(x, ξ )󵄨󵄨󵄨 dx dξ = ∫ 󵄨󵄨󵄨sinc(x ⋅ ξ )󵄨󵄨󵄨 dx dξ ℝ2d

ℝ2d

≥

󵄨󵄨 󵄨p 󵄨󵄨sinc(x ⋅ ξ )󵄨󵄨󵄨 dx dξ

∫ |x⋅ξ |≤1/2 p

2 ≥( ) π

∫

dx dξ

|x⋅ξ |≤1/2 p

2 = ( ) meas{(x, ξ ) : |x ⋅ ξ | ≤ 1/2} = +∞. π

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This concludes the proof. We continue our investigation of the function Θ by looking for the right Wiener amalgam and modulation spaces containing this function. For this reason, we first reckon explicitly the STFT of the Θ function, with respect to the Gaussian window 2 2 g(x, ξ ) = e−πx e−πξ ∈ 𝒮 (ℝ2d ). Proposition 4.7.3. For z1 , z2 , ζ1 , ζ2 ∈ ℝd , Vg Θ(z1 , z2 , ζ1 , ζ2 ) 1/2

= ∫ −1/2

(4.130)

2 1 −2πi[ 1t ζ1 ⋅ζ2 + 2t (z1 − 1t ζ2 )⋅(z2 − 1t ζ1 )] −π 2t [(z1 − 1t ζ2 )2 +(z2 − 1t ζ1 )2 ] t +1 t +1 e dt. e (t 2 + 1)d/2

(4.131)

1/2

Proof. We write Θ(z1 , z2 ) = F1 (z1 , z2 ) + F2 (z1 , z2 ), where F1 (z1 , z2 ) = ∫0 e2πiz1 ⋅z2 t dt and

F2 (z) = F1 (Jz), z = (z1 , z2 ). Let us first reckon Vg F1 (z, ζ ), z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ ℝ2d ,

264 | 4 Pseudodifferential operators where g is the Gaussian function above. For t > 0, we define the function Ht (z1 , z2 ) = e2πitz1 ⋅z2 and observe that 1 −2πi 1t ζ1 ⋅ζ2 (4.132) ℱ Ht (ζ1 , ζ2 ) = d e t (cf. [145, Appendix A, Theorem 2]). By the dominated convergence theorem, 1/2

1/2

̂ 1 , ζ2 ) dt Vg F1 (z, ζ ) = ∫ ℱ (Ht Tz g)(ζ ) dt = ∫ (ℱ (Ht ) ∗ M−z g)(ζ 0

1/2

= ∫ 0

0

1 2 2 1 ∫ e−2πi t (ζ1 −y1 )⋅(ζ2 −y2 ) e−2πi(z1 ,z2 )⋅(y1 ,y2 ) e−πy1 e−πy2 dy1 dy2 dt d t

ℝ2d

1/2

= ∫ 0

1 1 2 2 1 −2πi 1t ζ1 ⋅ζ2 e ∫ e−2πi t y1 ⋅y2 +2πi t (ζ2 ⋅y1 +ζ1 ⋅y2 )−2πi(z1 ⋅y1 +z2 ⋅y2 ) e−π(y1 +y2 ) dy1 dy2 dt d t

ℝ2d

1/2

= ∫ 0

1 2 1 −2πi 1t ζ1 ⋅ζ2 e ∫ e2πi( t ζ1 ⋅y2 −z2 ⋅y2 ) e−πy2 d t

ℝd

1

1

2

× ( ∫ e−2πiy1 ⋅( t y2 − t ζ2 +z1 ) e−πy1 dy1 ) dy2 dt ℝd 1/2

= ∫ 0

ℝd

1/2

= ∫ 0

1 1 −2πi 1t ζ1 ⋅ζ2 −π(z1 − 1t ζ2 )2 −π((1+ 12 )y22 −2(z1 − 1t ζ2 )⋅ 1t y2 ) t e dy2 dt e ∫ e−2πiy2 ⋅(z2 − t ζ1 ) e d t

ℝd

1/2

= ∫ 0 Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

1 1 1 2 2 1 −2πi 1t ζ1 ⋅ζ2 e ∫ e−2πiy2 ⋅(z2 − t ζ1 ) e−πy2 e−π( t y2 − t ζ2 +z1 ) dy2 dt d t

1 1 −2πi 1t ζ1 ⋅ζ2 −π 2t2 (u1 − 1t ζ2 )2 e e t +1 ∫ e−2πiy2 ⋅(u2 − t ζ1 ) d t

ℝd

×e 1/2

= ∫ 0

−π(

√t 2 +1 t

y2 −

t

√t 2 +1

(z1 − 1t ζ2 ))2

dy2 dt

t2 1 2 1 −2πi 1t ζ1 ⋅ζ2 −π t 2 +1 (z1 − t ζ2 ) e e 2 d/2 (t + 1)

×∫e

−2πi(

t

√t 2 +1

w+

t (z − 1 ζ ))⋅(z2 − 1t ζ1 ) t 2 +1 1 t 2

2

e−πw dw dt

ℝd 1/2

= ∫ 0

t2 t2 1 1 2 2 1 −2πi 1t ζ1 ⋅ζ2 −π t 2 +1 (z1 − t ζ2 ) −π t 2 +1 (z2 − t ζ1 ) e e e (t 2 + 1)d/2

×e

−2πi

t (z − 1 ζ )⋅(z2 − 1t ζ1 ) t 2 +1 1 t 2

dt.

4.7 Born–Jordan pseudodifferential operators on modulation spaces | 265

Now, an easy computation shows Vg F2 (z, ζ ) = Vg F1 (Jz, Jζ ) so that Vg Θ = Vg F1 + Vg F2 , and we obtain (4.131). Proposition 4.7.4. The function Θ in (4.122) belongs to W(ℱ L1 , L∞ )(ℝ2d ). Proof. We simply have to calculate 󵄨 󵄨 sup ∫ 󵄨󵄨󵄨Vg Θ(z, ζ )󵄨󵄨󵄨 dζ .

z∈ℝ2d

ℝ2d

From (4.131) we observe that 1/2

2 2 1 −π 2t (z1 − 1t ζ2 )2 −π 2t (z2 − 1t ζ1 )2 󵄩󵄩 󵄩 t +1 t +1 e e dζ1 dζ2 dt 󵄩󵄩Vg Θ(z, ⋅)󵄩󵄩󵄩1 ≤ ∫ ∫ 2 (t + 1)d/2

−1/2 ℝ2d 1/2

= ∫ ∫ −1/2 ℝ2d 1/2

(t 2

1 −π 1 (tz −ζ )2 −π 1 (tz −ζ )2 e t2 +1 1 2 e t2 +1 2 1 dζ1 dζ2 dt d/2 + 1) d/2 −π(v12 +v22 )

= ∫ ∫ (t 2 + 1) −1/2

e

dv1 dv2 dt = C < ∞,

ℝ2d

from which the claim follows. Using the STFT of the function Θ in (4.131), we observe that 1/2

2 2 1 −π 2t (u1 − 1t ζ2 )2 −π 2t (u2 − 1t ζ1 )2 󵄩󵄩 󵄩 t +1 t +1 e e du1 du2 dt = +∞ 󵄩󵄩Vg Θ(⋅, ζ )󵄩󵄩󵄩1 ≤ ∫ ∫ 2 (t + 1)d/2

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−1/2 ℝ2d

and conjecture that Θ ∉ M 1,∞ (ℝ2d ). The previous claim will follow if we prove that Θσ ∉ W(ℱ L1 , L∞ )(ℝ2d ). Note that Θσ (ζ ) = ℱ Θ(Jζ ) = ℱ Θ(ζ ). Furthermore, the distributional Fourier transform of Θ can be computed explicitly as follows. First, recall the definition of the cosine integral function (4.125). Proposition 4.7.5. For d ≥ 1, the distribution symplectic Fourier transform Θσ of the function Θ is provided by (ζ1 , ζ2 ) ∈ ℝ2 , d = 1, {−2Ci(4π|ζ1 ζ2 |), Θσ (ζ1 , ζ2 ) = { d−2 2d {ℱ (χ{|s|≥2} |s| )(ζ1 ⋅ ζ2 ), (ζ1 , ζ2 ) ∈ ℝ , d ≥ 2,

(4.133)

where χ{|s|≥2} is the characteristic function of the set {s ∈ ℝ : |s| ≥ 2}. The case d = 1 can be recaptured by the case d ≥ 2 using (4.125).

266 | 4 Pseudodifferential operators Proof. We carry out the computations of Θσ by studying first the case in dimension d = 1 and then, inspired by the former case, d > 1. First step: d = 1. By Proposition 4.7.2, the function Θ is in L∞ (ℝ2 ) \ Lp (ℝ2 ) ⊂ 𝒮 󸀠 (ℝ2 ),

1 ≤ p < ∞,

so that the Fourier transform is meant in 𝒮 󸀠 (ℝ2 ). Observe that ℱ Θ(ζ1 , ζ2 ) = ℱ2 ℱ1 Θ(ζ1 , ζ2 ),

where ℱ1 (resp. ℱ2 ) is the partial Fourier transform with respect to the first (resp. second) variable. Indeed, for every test function φ ∈ 𝒮 (ℝ2 ), ⟨ℱ Θ, φ⟩ = ⟨Θ, ℱ −1 φ⟩ and ℱ −1 φ(x, ξ ) = ℱ1−1 ℱ2−1 φ(x, ξ ) = ℱ2−1 ℱ1−1 φ(x, ξ ), by Fubini’s theorem. Using ℱ1 sinc(y2 ⋅)(ζ1 ) =

1 p (ζ /y ), |y2 | 1/2 1 2

y2 ≠ 0,

where p1/2 (t) is the box function defined by p1/2 (t) = 1 for |t| ≤ 1/2 and p1/2 (t) = 0 otherwise, we obtain, for ζ1 ζ2 ≠ 0 (hence in particular |ζ1 | > 0), ℱ Θ(ζ1 , ζ2 ) = ∫ e

−2πiζ2 y2

ℝ

=

e

∫ |s|≥2|ζ1 ζ2 |

=

∫ |s|≥2|ζ1 ζ2 |

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=

∫ |s|≥2|ζ1 ζ2 | +∞

=2 ∫

2|ζ1 ζ2 |

1 p (ζ /y ) dy2 = |y2 | 1/2 1 2

−2πis

1 ds |s|

∫

e−2πiζ2 y2

|y2 |≥2|ζ1 |

1 dy |y2 | 2

cos(2πs) − i sin(2πs) ds |s| cos 2πs ds |s|

cos 2πs ds = −2Ci(4π|ζ1 ζ2 |), s

by (4.125), so that, since ζ1 ζ2 = 0 is a set of Lebesgue measure zero on ℝ2 , we can write Θσ (ζ1 , ζ2 ) = −2Ci(4π|ζ1 ζ2 |),

(ζ1 , ζ2 ) ∈ ℝ2 .

(4.134)

Second step: d > 1. This is a simple generalization on the former step. For (z1 , z2 ), (ζ1 , ζ2 ) ∈ ℝ2d , d > 1, we write zi = (zi󸀠 , zi,d ), ζi = (ζi󸀠 , ζi,d ),

zi󸀠 , ζi󸀠 ∈ ℝd−1 ,

zi,d , ζi,d ∈ ℝ,

i = 1, 2.

(4.135)

4.7 Born–Jordan pseudodifferential operators on modulation spaces | 267

We decompose ℱ Θ = ℱ2d ℱ 󸀠 ℱ1 Θ where, for Θ = Θ(z1 , z2 ), ℱ1 is the partial Fourier transform with respect to the variable z1,d , ℱ 󸀠 is the partial Fourier transform with respect to the 2d−2 variables (z1󸀠 , z2󸀠 ) ∈ ℝ2d−2 , and ℱ2d is the partial Fourier transform with respect to the last variable z2,d . We start by computing the partial Fourier transform ℱ1 : ℱ1 Θ(z1 , ⋅, z2 , z2,d )(ζ1,d ) = ℱ1 (T −z1󸀠 ⋅z2󸀠 sinc(z2,d ⋅))(ζ1,d ) 󸀠

󸀠

z2,d

=e =e

1

ζ

2πi z1,d z1󸀠 ⋅z2󸀠 2,d

ζ1,d

2πi z

2,d

|z2,d | 1

z1󸀠 ⋅z2󸀠

|z2,d |

ℱ1 (sinc)(

p1/2 (

ζ1,d ) z2,d

ζ1,d ). z2,d

Using the Gaussian integrals in [145, Appendix A, Theorem 2], we calculate ζ

󸀠

ℱ (e

2πi z1,d z1󸀠 ⋅z2󸀠 2,d

󵄨󵄨 z 󵄨󵄨d−1 −2πi z2,d ζ 󸀠 ⋅ζ 󸀠 󵄨 2,d 󵄨󵄨 ζ1,d 1 2 )(ζ1󸀠 , ζ2󸀠 ) = 󵄨󵄨󵄨 , 󵄨 e 󵄨󵄨 ζ1,d 󵄨󵄨󵄨

so that ℱ Θ(ζ1 , ζ2 ) = ℱ2d (e

= ζ

z 󵄨 󵄨d−1 −2πi ζ2,d ζ1󸀠 ⋅ζ2󸀠 󵄨󵄨 z2,d 󵄨󵄨 1,d

󵄨󵄨 󵄨 󵄨󵄨 ζ 󵄨󵄨󵄨 󵄨 1,d 󵄨

ζ1,d 1 p ( ))(ζ2,d ) |z2,d | 1/2 z2,d

󵄨󵄨 z 󵄨󵄨d−1 1 −2πi z2,d ζ ⋅ζ 󵄨 2,d 󵄨󵄨 ζ1,d 1 2 e dz2,d ∫ 󵄨󵄨󵄨 󵄨 󵄨󵄨 ζ1,d 󵄨󵄨󵄨 |z2,d |

| z1,d |≤ 21 2,d

= ∫ e−2πis(ζ1 ⋅ζ2 ) |s|d−2 ds, |s|≥2

as claimed. Notice that the second equation (4.133) can be written as

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Θσ (ζ1 , ζ2 ) = ∫ e−2πis(ζ1 ⋅ζ2 ) |s|d−2 ds. |s|≥2

Corollary 4.7.6. We have 2d Θσ ∉ L∞ loc (ℝ ).

Proof. For the case d = 1, recall that the cosine integral Ci(x) has the series expansion +∞

Ci(x) = γ + log x + ∑

k=1

(−x2 )k , 2k(2k)!

x>0

where γ is the Euler–Mascheroni constant, from which our claim easily follows. For d ≥ 2, Θσ is only defined as a tempered distribution.

268 | 4 Pseudodifferential operators Corollary 4.7.7. The function Θσ ∉ Lp (ℝ2d ), for any 1 ≤ p ≤ ∞. Proof. The case p = ∞ is already treated in Corollary 4.7.6. For d ≥ 2, again we observe that Θσ is not defined as a function but only as a tempered distribution. For d = 1, 1 ≤ p < ∞, the claim follows by the expression (4.134). Indeed, choose 0 < ϵ < π/2, then |Ci(x)| ≥ |Ci(ϵ)|, for 0 < x < ϵ, so that 󵄨p 󵄨 ∫ 󵄨󵄨󵄨Θσ (ζ1 , ζ2 )󵄨󵄨󵄨 dζ1 dζ2 ≥ 2

∫

󵄨p 󵄨󵄨 󵄨󵄨Ci(4π|ζ1 ζ2 |)󵄨󵄨󵄨 dζ dζ2

ϵ |ζ1 ζ2 |< 4π

ℝ2

≥ Cp meas{(ζ1 , ζ2 ) : |ζ1 ζ2 |
0. Since ℱ L1 ⊂ L∞ , the Wiener amalgam space W(ℱ L1 , L∞ ) is included in L∞ loc . This proves our claim: Corollary 4.7.8. The function Θσ ∉ W(ℱ L1 , L∞ )(ℝ2d ) or, equivalently, Θ ∉ M 1,∞ (ℝ2d ).

4.7.2 Proof of Theorem 4.7.1 This section is focused on the proof of Theorem 4.7.1. This requires some preliminaries. Lemma 4.7.9. Let χ ∈ 𝒟(ℝ). Then for ζ1 , ζ2 ∈ ℝd the function χ(ζ1 ζ2 ) belongs to W(ℱ L1 , L∞ )(ℝ2d ). Proof. We write χ(ζ1 ζ2 ) = ∫ e2πitζ1 ⋅ζ2 χ̂(t) dt,

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ℝ

and the desired result then follows as in proof of Proposition 4.7.4 because χ̂(t) lies in 𝒮 (ℝ) ⊂ L1 (ℝ). Lemma 4.7.10. Let ψ ∈ 𝒟(ℝd ) \ {0} and λ > 0. Then we have the following dilation properties: 󵄩󵄩 󵄩 −d/p󸀠 as λ → +∞, 󵄩󵄩ψ(λ⋅)󵄩󵄩󵄩W(ℱ Lp ,Lq ) ≍ λ 󵄩󵄩 󵄩󵄩 −d/q as λ → 0. 󵄩󵄩ψ(λ⋅)󵄩󵄩W(ℱ Lp ,Lq ) ≍ λ

(4.136) (4.137)

Proof. Formula (4.136) follows by observing that for, say, λ ≥ 1, the functions ψ(λ⋅) are supported in a fixed compact set, so that their M p,q -norm is equivalent to their norm in ℱ Lp , which is easily estimated.

4.7 Born–Jordan pseudodifferential operators on modulation spaces | 269

Let us now prove (4.137). Observe, first of all, that for f ∈ 𝒮 (ℝd ), g ∈ 𝒟(ℝd ), q ≥ 1, we have 󵄩 󵄩 󵄩 󵄩 ‖fTx g‖ℱ L1 ≲ ∑ 󵄩󵄩󵄩𝜕α (fTx g)󵄩󵄩󵄩L1 ≲ ∑ 󵄩󵄩󵄩𝜕α (fTx g)󵄩󵄩󵄩Lq , |α|≤d+1

|α|≤d+1

so that 󵄩 󵄩 ‖f ‖W(ℱ Lp ,Lq ) ≲ ‖f ‖W(ℱ L1 ,Lq ) ≲ ∑ 󵄩󵄩󵄩𝜕α f 󵄩󵄩󵄩Lq . |α|≤d+1

Applying this formula with f = ψ(λx), 0 < λ ≤ 1, we obtain 󵄩󵄩 󵄩 −d/q . 󵄩󵄩ψ(λ⋅)󵄩󵄩󵄩W(ℱ Lp ,Lq ) ≲ λ To obtain the lower bound, we observe that 󵄩 󵄩2 λ−d ‖ψ‖2L2 = 󵄩󵄩󵄩ψ(λ⋅)󵄩󵄩󵄩L2 󵄩 󵄩 󵄩 󵄩 ≲ 󵄩󵄩󵄩ψ(λ⋅)󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) 󵄩󵄩󵄩ψ(λ⋅)󵄩󵄩󵄩W(ℱ Lp ,Lq ) 󸀠 󵄩 󵄩 ≲ λ−d/q 󵄩󵄩󵄩ψ(λ⋅)󵄩󵄩󵄩W(ℱ Lp ,Lq ) ,

which implies 󵄩󵄩 󵄩 −d/q . 󵄩󵄩ψ(λ⋅)󵄩󵄩󵄩W(ℱ Lp ,Lq ) ≳ λ A straightforward generalization of the previous arguments is as follows: Lemma 4.7.11. Consider 1 ≤ p, q ≤ ∞, ψ ∈ 𝒟(ℝd ) \ {0} and λ > 0. Then − d 󵄩󵄩 √ 󵄩󵄩 󵄩󵄩ψ( λ⋅)󵄩󵄩W(ℱ Lp ,Lq ) ≍ λ 2p󸀠 as λ → +∞, 󵄩󵄩 √ 󵄩󵄩 −d + 󵄩󵄩ψ( λ⋅)󵄩󵄩W(ℱ Lp ,Lq ) ≍ λ 2q as λ → 0 .

(4.138) (4.139)

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The same conclusion holds uniformly with respect to λ if ψ varies in bounded subsets of 𝒟(ℝd ). We first demonstrate the sufficient boundedness conditions. Theorem 4.7.12. Assume that 1 ≤ p, q, r1 , r2 ≤ ∞. Then the pseudodifferential operator OpBJ (a), from 𝒮 (ℝd ) to 𝒮 󸀠 (ℝd ), having symbol a ∈ M p,q (ℝ2d ), extends uniquely to a bounded operator on M r1 ,r2 (ℝd ), with the estimate (4.128) and the indices’ conditions (4.126) and (4.127). The result relies on a thorough understanding of the action of the mapping A : a �→ a ∗ Θσ ,

(4.140)

which gives the Weyl symbol of an operator with Born–Jordan symbol a, on modulation spaces.

270 | 4 Pseudodifferential operators Proposition 4.7.13. For every 1 ≤ p, q ≤ ∞, the mapping A in (4.140), defined initially on 𝒮 󸀠 (ℝ2d ), restricts to a linear continuous map on M p,q (ℝ2d ), i. e., there exists a constant C > 0 such that ‖Aa‖M p,q ≤ C‖a‖M p,q .

(4.141)

Proof. By Proposition 4.7.4, the function Θ ∈ W(ℱ L1 , L∞ )(ℝ2d ). Its symplectic Fourier transform Θσ belongs to ℱσ W(ℱ L1 , L∞ )(ℝ2d ) = M 1,∞ (ℝ2d ). Now, for every 1 ≤ p, q ≤ ∞, the convolution relations for modulation spaces (2.81) give M p,q (ℝ2d ) ∗ M 1,∞ (ℝ2d ) �→ M p,q (ℝ2d ), and this shows the claim (4.141). Proof of Theorem 4.7.12. Assume a ∈ M p,q (ℝ2d ), then Proposition 4.7.13 proves that Aa = a ∗ Θσ ∈ M p,q (ℝ2d ) as well. We next write OpBJ (a) = OpW (Aa) and use the sufficient conditions for Weyl operators in Theorem 4.7.1: if the Weyl symbol Aa is in M p,q (ℝ2d ), then OpW (Aa) extends to a bounded operator on M r1 ,r2 (ℝd ), with 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩OpBJ (a)f 󵄩󵄩󵄩M r1 ,r2 = 󵄩󵄩󵄩OpW (Aa)f 󵄩󵄩󵄩M r1 ,r2 ≲ ‖Aa‖M p,q ‖f ‖M r1 ,r2 where the indices r1 , r2 , p, q satisfy (4.126) and (4.127). The inequality (4.141) then provides the claim. The necessary conditions of Theorem 4.7.1 require some preliminaries. We reckon the adjoint operator OpBJ (a)∗ of a Born–Jordan operator OpBJ (a) using the connection with Weyl operators. An easy computation shows OpW (b)∗ = OpW (b)̄

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(see also [193]), so that OpBJ (a)∗ = OpW (a ∗ Θσ )∗ = OpW (a ∗ Θσ ) = OpW (ā ∗ Θσ̄ ) = OpW (ā ∗ Θσ ) = OpBJ (a)̄ because Θ is an even real-valued function. Hence the adjoint of a Born–Jordan operator OpBJ (a) with symbol a is the Born–Jordan operator having symbol ā (the complexconjugate of a). This nice property is the key argument for the following auxiliary result, already obtained for the case of Weyl operators in Lemma 4.1.7. The proof uses the same pattern as the former result and hence is omitted. Lemma 4.7.14. Suppose that, for some 1 ≤ p, q, r1 , r2 ≤ ∞, the following estimate holds: 󵄩󵄩 󵄩 󵄩󵄩OpBJ (a)f 󵄩󵄩󵄩M r1 ,r2 ≤ C‖a‖M p,q ‖f ‖M r1 ,r2 ,

∀a ∈ 𝒮 (ℝ2d ), ∀f ∈ 𝒮 (ℝd ).

Then the same estimate is satisfied with r1 , r2 replaced by r1󸀠 , r2󸀠 (even if r1 = ∞ or r2 = ∞).

4.7 Born–Jordan pseudodifferential operators on modulation spaces | 271

The above instruments let us show the necessity of (4.126) and (4.129). However, we first need a few preliminaries. Proposition 4.7.15. The function F(ζ1 , ζ2 ) = e2πiζ1 ⋅ζ2 belongs to W(ℱ L1 , L∞ )(ℝ2d ). 2

2

Proof. Consider the Gaussian function g(ζ1 , ζ2 ) = e−πζ1 e−πζ2 as a window function to compute the W(ℱ L1 , L∞ )-norm. Then we have 󵄩 󵄩 ‖F‖W(ℱ L1 ,L∞ )(ℝ2d ) = sup 󵄩󵄩󵄩ℱ (FTu g)󵄩󵄩󵄩L1 (ℝ2d ) . u∈ℝ2d

Let us compute ℱ (FTu g)(z). For z = (z1 , z2 ), ̂ 1 , z2 ) ℱ (FTu g)(z1 , z2 ) = (ℱ (F) ∗ M−u g)(z 2

2

= ∫ e−2πi(z1 −y1 )⋅(z2 −y2 ) e−2πi(u1 ,u2 )⋅(y1 ,y2 ) e−πy1 e−πy2 dy1 dy2 ℝ2d 2

2

= e−2πiz1 ⋅z2 ∫ e−2πiy1 ⋅y2 +2πi(z2 ⋅y1 +z1 ⋅y2 )−2πi(u1 ⋅y1 +u2 ⋅y2 ) e−πy1 e−πy2 dy1 dy2 ℝ2d 2

2

= e−2πiz1 ⋅z2 ∫ e2πi(z1 ⋅y2 −u2 ⋅y2 ) e−πy2 ( ∫ e−2πiy1 ⋅(y2 −z2 +u1 ) e−πy1 dy1 ) dy2 ℝd

ℝd

= e−2πiz1 ⋅z2 ∫ e−2πiy2 ⋅(u2 −z1 ) e

−πy22

2

e−π(y2 −z2 +u1 ) dy2

ℝd

=e

−2πiz1 ⋅z2 −π(u1 −z2 )2

=e

−2πiz1 ⋅z2 −π(u1 −z2 )2 + π2 (u1 −z2 )2

e

2

∫ e−2πiy2 ⋅(u2 −z1 ) e−2π(y2 +(u1 −z2 )⋅y2 ) dy2 ℝd

e

∫ e−2πiy2 ⋅(u2 −z1 ) e−2π(y2 + ℝd

π

2

2

= e−2πiz1 ⋅z2 e− 2 (u1 −z2 ) ℱ (T− u1 −z2 e−2π|⋅| )(u2 − z1 ) Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

2

=2

−d/2 −2πiz1 ⋅z2 − π2 (u1 −z2 )2 − π2 (u2 −z1 )2 πi(u1 −z2 )⋅(u2 −z1 )

=2

−d/2 πi(u1 ⋅u2 −z1 ⋅z2 −u1 ⋅z1 −u2 ⋅z2 ) − π2 (u1 −z2 )2 − π2 (u2 −z1 )2

e

e

e

e

e

e

e

.

Hence π 2 󵄩󵄩 󵄩 −d/2 󵄩 󵄩󵄩e− 2 |⋅| 󵄩󵄩󵄩 1 = C, 󵄩󵄩ℱ (FTu g)󵄩󵄩󵄩L1 = 2 󵄩L 󵄩

for a constant C independent of the variable u, and, consequently, 󵄩 󵄩 sup 󵄩󵄩󵄩ℱ (FTu g)󵄩󵄩󵄩L1 < ∞,

u∈ℝ2d

as desired.

u1 −z2 2 ) 2

dy2

272 | 4 Pseudodifferential operators Corollary 4.7.16. For ζ = (ζ1 , ζ2 ), consider the function FJ (ζ ) = F(Jζ ) = e−2πiζ1 ⋅ζ2 . Then FJ ∈ W(ℱ L1 , L∞ )(ℝ2d ). Proof. The result immediately follows by Proposition 4.7.15 and by the dilation properties for Wiener amalgam spaces (2.107). Theorem 4.7.17. Suppose that, for some 1 ≤ p, q, r1 , r2 ≤ ∞, C > 0, the estimate 󵄩 󵄩󵄩 󵄩󵄩OpBJ (a)f 󵄩󵄩󵄩M r1 ,r2 ≤ C‖a‖M p,q ‖f ‖M r1 ,r2

∀a ∈ 𝒮 (ℝ2d ), f ∈ 𝒮 (ℝd )

(4.142)

holds. Then the constraints in (4.126) and (4.129) must hold. Proof. The estimate (4.142) can be written as ∀a ∈ 𝒮 (ℝ2d ), f , g ∈ 𝒮 (ℝd ),

󵄨󵄨 󵄨 󵄨󵄨⟨a, Q(f , g)⟩󵄨󵄨󵄨 ≤ C‖a‖M p,q ‖f ‖M r1 ,r2 ‖g‖M r1󸀠 ,r2󸀠 which is equivalent to

∀f , g ∈ 𝒮 (ℝd ).

󵄩󵄩 󵄩 󵄩󵄩Q(f , g)󵄩󵄩󵄩M p󸀠 ,q󸀠 ≤ C‖f ‖M r1 ,r2 ‖g‖M r1󸀠 ,r2󸀠

Now, one should test this estimate on families of functions fλ , gλ such that Q(fλ , gλ ) is concentrated inside the hyperbola |x ⋅ ξ | < 1 (say), see Figure 4.6, where θ ≍ 1, so that the left-hand side is comparable to ‖W(fλ , gλ )‖M p󸀠 ,q󸀠 and can be estimated from below. ξ

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x

Figure 4.6: The region |x ⋅ ξ | < 1 (d = 1). 2

The choice fλ (x) = gλ (x) = φ√λ (x) = e−πλx provides the estimate (4.126) when λ → +∞. By (2.103), we obtain the estimate ‖φ√λ ‖M r1 ,r2 ‖φ√λ ‖

󸀠 󸀠 M r1 ,r2

≲λ

−

d 2r 󸀠 2

λ

− 2rd

2

.

(4.143)

We gauge from below the norm ‖Q(φ√λ , φ√λ )‖M p󸀠 ,q󸀠 as follows. By taking the symplectic Fourier transform and using Lemma 4.7.9 and the product property (4.103), we

4.7 Born–Jordan pseudodifferential operators on modulation spaces | 273

have 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩Q(φ√λ , φ√λ )󵄩󵄩󵄩M p󸀠 ,q󸀠 = 󵄩󵄩󵄩Θσ ∗ W(φ√λ , φ√λ )󵄩󵄩󵄩M p󸀠 ,q󸀠 󵄩 󵄩 ≍ 󵄩󵄩󵄩Θℱσ [W(φ√λ , φ√λ )]󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) 󵄩 󵄩 ≳ 󵄩󵄩󵄩Θ(ζ1 , ζ2 )χ(ζ1 ⋅ ζ2 )ℱσ [W(φ√λ , φ√λ )]󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) for any χ ∈ 𝒟(ℝ). Choosing χ supported in the interval [−1/4, 1/4] and equal to 1 in the interval [−1/8, 1/8], we write ̃ 1 ⋅ ζ2 ), χ(ζ1 ⋅ ζ2 ) = χ(ζ1 ⋅ ζ2 )Θ(ζ1 , ζ2 )Θ−1 (ζ1 , ζ2 )χ(ζ with χ̃ ∈ 𝒟(ℝ) supported in [−1/2, 1/2] and χ̃ = 1 on [−1/4, 1/4], therefore on the sup̃ 1 ⋅ ζ2 ) belongs to W(ℱ L1 , L∞ ), port of χ. Since, by Lemma 4.7.9, the function Θ−1 (ζ1 , ζ2 )χ(ζ again by the product property the last expression is estimated from below as 󵄩 󵄩 ≳ 󵄩󵄩󵄩χ(ζ1 ⋅ ζ2 )ℱσ [W(φ√λ , φ√λ )]󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) . Consider a function ψ ∈ 𝒟(ℝd ) \ {0}, supported in [−1/4, 1/4]. Using 1 󵄨 −1 󵄨2 |ζ1 ⋅ ζ2 | ≤ (|√λζ1 |2 + 󵄨󵄨󵄨√λ ζ2 󵄨󵄨󵄨 ), 2 we see that χ(ζ1 ⋅ ζ2 ) = 1 on the support of ψ(√λζ1 )ψ(√λ ζ2 ), for every λ > 0. Then we can write −1

−1

−1

ψ(√λζ1 )ψ(√λ ζ2 ) = χ(ζ1 ⋅ ζ2 )ψ(√λζ1 )ψ(√λ ζ2 ) and, by Lemma 4.7.11, also infer

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

−1 󵄩󵄩 √ 󵄩 󵄩󵄩ψ( λζ1 )ψ(√λ ζ2 )󵄩󵄩󵄩W(ℱ L1 ,L∞ ) ≲ 1,

so that we can continue the above estimate as −1 󵄩 󵄩 ≳ 󵄩󵄩󵄩ψ(√λζ1 )ψ(√λ ζ2 )ℱσ [W(φ√λ , φ√λ )]󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) .

Using (see, e. g., [160, formula (4.20)]) 2 W(φ√λ , φ√λ )(x, ω) = 2 2 λ− 2 φ(√2λx)φ(√ ω), λ d

d

we calculate d

ℱσ [W(φ√λ , φ√λ )](ζ1 , ζ2 ) = 2 2 λ

− d2

λ φ((√2λ)−1 ζ2 )φ(√ ζ1 ), 2

(4.144)

274 | 4 Pseudodifferential operators so that −1 󵄩󵄩 √ 󵄩 󵄩󵄩ψ( λζ1 )ψ(√λ ζ2 )ℱσ [W(φ√λ , φ√λ )]󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) d 󵄩 󵄩 = 2d/2 λ− 2 󵄩󵄩󵄩ψ(√λζ1 )φ((1/√2)√λζ1 )󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) −1 󵄩 󵄩 × 󵄩󵄩󵄩ψ(√λ ζ2 )φ((√2λ)−1 ζ2 )󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) .

By Lemma 4.7.11, we can estimate the last expression and obtain −d+ d + d 󵄩󵄩 󵄩 󵄩󵄩Q(φ√λ , φ√λ )󵄩󵄩󵄩M p󸀠 ,q󸀠 ≳ λ 2p󸀠 2q󸀠

as λ → +∞.

Finally, using the estimate (4.143), we infer (4.126). We now prove that max{1/r1 , 1/r1󸀠 } ≤ 1/q + 1/p. If we show the estimate 1/r1 ≤ 1/q + 1/p, then the constraint 1/r1󸀠 ≤ 1/q + 1/p follows by the duality argument of Lemma 4.7.14. To reach this goal, we consider fλ = φ (independent of the parameter λ) and g = φ√λ as before and use the previous pattern for these families of functions, in the case λ → 0+ . By (2.103), the upper estimate becomes ‖φ‖M r1 ,r2 ‖φ√λ ‖

≲λ

M r1 ,r2 󸀠 󸀠

−

d 2r 󸀠 1

.

(4.145)

The same arguments as before let us write −1 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩Q(φ, φ√λ )󵄩󵄩󵄩M p󸀠 ,q󸀠 ≳ 󵄩󵄩󵄩ψ(√λζ1 )ψ(√λ ζ2 )ℱσ [W(φ, φ√λ )]󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) ,

where ℱσ [W(φ, φ√λ )] is computed in (1.131). Observe that, given any F ∈ W(ℱ Lp , Lq ), 󸀠

󸀠

λ−1 λ−1 λ−1 󵄩󵄩 πi λ+1 󵄩 󵄩 󵄩 ζ1 ⋅ζ2 F(ζ1 , ζ2 )󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) ≳ 󵄩󵄩󵄩e−πi λ+1 ζ1 ⋅ζ2 eπi λ+1 ζ1 ⋅ζ2 F(ζ1 , ζ2 )󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) 󵄩󵄩e 󵄩 󵄩 = 󵄩󵄩󵄩F(ζ1 , ζ2 )󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) ,

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

λ−1

because ‖e−πi λ+1 ζ1 ⋅ζ2 ‖W(ℱ L1 ,L∞ ) ≤ C, for every λ > 0 by Corollary 4.7.16. So writing cλ =

1

d

(λ + 1) 2

(notice cλ → 1 for λ → 0+ ), we have reduced the situation to −1 󵄩󵄩 󵄩 󵄩 󵄩 − π ζ2󵄩 − πλ ζ 2 󵄩 󵄩󵄩Q(φ, φ√λ )󵄩󵄩󵄩M p󸀠 ,q󸀠 ≳ cλ 󵄩󵄩󵄩ψ(√λζ1 )e λ+1 1 󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) 󵄩󵄩󵄩ψ(√λ ζ2 )e λ+1 2 󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) .

By Lemma 4.7.11, we can estimate, for λ → 0+ , − d󸀠 󵄩󵄩 √ 󵄩 − πλ ζ 2 󵄩 − π (√λζ1 )2 󵄩 󵄩󵄩 2q , 󵄩󵄩ψ( λζ1 )e λ+1 1 󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) = 󵄩󵄩󵄩ψ(√λζ1 )e λ+1 󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) ≍ λ

4.7 Born–Jordan pseudodifferential operators on modulation spaces | 275

whereas 󵄩󵄩 √ −1 − π ζ2󵄩 󵄩󵄩ψ( λ ζ2 )e λ+1 2 󵄩󵄩󵄩W(ℱ Lp󸀠 ,Lq󸀠 ) 󵄩 d d󵄩 2 󵄩󵄩 ̂ √λ(ζ − η))e−π(λ+1)η dη󵄩󵄩󵄩󵄩 ≳ λ 2 (λ + 1) 2 󵄩󵄩󵄩 ∫ ψ( 2 󵄩󵄩 p󸀠 󵄩󵄩 󵄩L 󵄩 d − d 󵄩 d 2 󵄩 󵄩 ̂ − √λη)e−π(λ+1)η dη󵄩󵄩󵄩󵄩 = λ 2 (λ + 1) 2 λ 2p󸀠 󵄩󵄩󵄩 ∫ ψ(ζ 2 󵄩󵄩 p󸀠 󵄩󵄩 󵄩L 󵄩 󵄩 λ+1 2 d − d 󵄩 󵄩 󵄩 ̂ 󵄩 −π λ t = (λ + 1) 2 λ 2p󸀠 󵄩󵄩󵄩∫ ψ(ζ dt 󵄩󵄩󵄩 󸀠 2 − t)e 󵄩󵄩 󵄩󵄩Lp d

= λ2

d

∼ λ2

−

d 2p󸀠

−

d 2p󸀠

d

‖ψ̂ ∗ K1/√λ ‖Lp󸀠 ̂ 󸀠, ‖ψ‖ p d

where K1/√λ (ζ2 ) = λ− 2 (λ + 1) 2 e− yields

as λ → 0+

π(λ+1) 2 ζ2 λ

λ

−

d 2r 󸀠 1

, λ → 0+ , is an approximate identity. This

≳λ

−

d 2q󸀠

d

λ 2p ,

and, for λ → 0+ , we obtain 1 1 1 ≤ + , r1 q p as desired. It remains to prove that max{1/r2 , 1/r2󸀠 } ≤ 1/q + 1/p. Again, it is enough to show that 1/r2 ≤ 1/q + 1/p and invoke Lemma 4.7.14 for 1/r2󸀠 ≤ 1/q + 1/p. An explicit computation shows that ℱ OpW (σ)ℱ = OpW (σ ∘ J),

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

−1

(4.146)

where we recall J(x, ω) = (ω, −x) as defined in (1.1) (this is also a consequence of the intertwining property of the metaplectic operator ℱ with the Weyl operator OpW (σ) [105, Corollary 221]). Now, observing that Θσ ∘ J = Θσ , we obtain (a ∗ Θσ )(Jz) = ∫ a(u)Θσ (Jz − u) du = ∫ a(u)Θσ (J(z − J −1 u)) du ℝ2d

ℝ2d

= ∫ a(u)Θσ (z − J −1 u) du = ∫ a(Ju)Θσ (z − u) du ℝ2d

ℝ2d

= (a ∘ J) ∗ Θσ (z). The previous computations, together with (4.146), give ℱ OpBJ (a)ℱ = OpBJ (a ∘ J). −1

276 | 4 Pseudodifferential operators On the other hand, the map a �→ a ∘ J is an isomorphism of M p,q , so that (4.142) is, in fact, equivalent to ∀a ∈ 𝒮 (ℝ2d ), f ∈ 𝒮 (ℝd ).

󵄩 󵄩󵄩 󵄩󵄩OpBJ (a)f 󵄩󵄩󵄩W(ℱ Lr1 ,Lr2 ) ≲ ‖a‖M p,q ‖f ‖W(ℱ Lr1 ,Lr2 )

(4.147)

The estimate (4.147) can be written as ∀a ∈ 𝒮 (ℝ2d ), f , g ∈ 𝒮 (ℝd ),

󵄨 󵄨󵄨 󵄨󵄨⟨a, Q(f , g)⟩󵄨󵄨󵄨 ≤ C‖a‖M p,q ‖f ‖W(ℱ Lr1 ,Lr2 ) ‖g‖W(ℱ Lr1󸀠 ,Lr2󸀠 ) which is equivalent to

󵄩󵄩 󵄩 󵄩󵄩Q(f , g)󵄩󵄩󵄩M p󸀠 ,q󸀠 ≤ C‖f ‖W(ℱ Lr1 ,Lr2 ) ‖g‖W(ℱ Lr1󸀠 ,Lr2󸀠 )

∀f , g ∈ 𝒮 (ℝd ).

Now, taking f = φ and g = φ√λ as before, we observe that, for λ → 0+ , by (2.103), ‖φ√λ ‖

d

󸀠 󸀠 W(ℱ Lr1 ,Lr2 )

≍ λ− 2 ‖φ1/√λ ‖

󸀠 󸀠 M r1 ,r2

≍λ

− d2 + 2rd

2

=λ

−

d 2r 󸀠 2

.

Arguing as in the previous case, we obtain 1/r2 ≤ 1/q + 1/p. This concludes the proof. We end up this chapter by observing that the Born–Jordan distribution can be generalized as follows. Definition 4.7.18. For n ∈ ℕ, the nth Born–Jordan kernel is the function Θn on ℝ2d defined by Θn (x, ω) = sincn (x ⋅ ω),

(x, ω) ∈ ℝ2d .

(4.148)

The Born–Jordan distribution of order n (BJDn) is given by Qn f = Wf ∗ ℱσ (Θn ),

f ∈ L2 (ℝd ).

(4.149)

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The cross-BJDn is given by Qn (f , g) = W(f , g) ∗ ℱσ (Θn ),

f , g ∈ L2 (ℝd ).

(4.150)

We write Qn (f , f ) = Qn f for every f ∈ L2 (ℝd ). Note that Θ0 ≡ 1, hence ℱσ (Θ0 ) = δ and Q0 f = Wf , the Wigner distribution of f . The nth Born–Jordan pseudodifferential operator with symbol a ∈ 𝒮 󸀠 (ℝd ) is defined by ⟨OpBJ,n (a)f , g⟩ = ⟨a, Qn (g, f )⟩,

f , g ∈ 𝒮 (ℝd ).

(4.151)

(Observe that n = 1 is the standard Born–Jordan operator, whereas n = 0 gives the Weyl operator.) For a complete study of such operators, we refer to [59].

5 Time–frequency analysis of constant-coefficient partial differential equations Here we apply time–frequency analysis tools in the study of constant-coefficient partial differential equations (PDEs). In the first part of the chapter we study the dispersive properties of the constantcoefficient Schrödinger equation, presenting results contained in [70, 71, 72, 91]. Precisely, we look for estimates which give a control on the local regularity and decay at infinity separately. The Banach spaces that allow such a treatment are the Wiener amalgam spaces, which improve the classical Strichartz estimates in the case of large time. Attention is limited to the free particle equation i𝜕t u + Δu = 0 with initial datum u(0) = u0 . We prove in the homogeneous case that

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󵄩󵄩 itΔ 󵄩󵄩 󵄩󵄩e u0 󵄩󵄩W(Lq1 ,Lq2 )t W(Lr1 ,Lr2 )x ≲ ‖u0 ‖L2x , under conditions on q1 , q2 , r1 , r2 which are sharp, as obtained by testing the problem on Gaussian initial data. In the last part of the chapter we show that Gabor frames provide optimally sparse decompositions for representing solutions to hyperbolic and parabolic-type differential equations with constant coefficients. In fact, the Gabor matrix representation of the corresponding propagator displays superexponential decay away from the diagonal. To achieve this goal, we first perform a time–frequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultraanalytic type. These ideas stem from [83]. Let us remark a few aspects we shall use below and in the following chapter. Given a linear continuous operator T : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ), providing the solution of a wellposed problem for a PDE, we expect T to be a pseudodifferential or a Fourier integral operator. For some problems, the space 𝒮 (ℝd ) can be replaced by other spaces of test functions, and 𝒮 󸀠 (ℝd ) by the respective duals. In any case we can compute the Gabor matrix of T. Consider a window function g ∈ L2 (ℝd ), a lattice Λ = αℤd × βℤd , with α, β > 0 such that 𝒢 (g, α, β) (defined in (3.18)) is a frame for L2 (ℝd ) with dual window γ ∈ L2 (ℝd ). For simplicity, define the related Gabor atoms gm,n := Mβn Tαm g,

(m, n) ∈ Λ,

(5.1)

and similarly γm,n . Any signal f ∈ L2 (ℝd ) admits the representations f =

∑ ⟨f , gm,n ⟩γm,n = (m,n)∈Λ

∑ ⟨f , γm,n ⟩gm,n .

(5.2)

(m,n)∈Λ

The analysis extends to f in 𝒮 (ℝd ), 𝒮 󸀠 (ℝd ) or other distributional setting, and we can choose its representation f = ∑(m,n)∈Λ ⟨f , γm,n ⟩gm,n . Applying to f the operator T, we https://doi.org/10.1515/9783110532456-006

278 | 5 Time–frequency analysis of constant-coefficient partial differential equations can decompose Tf as Tf (x) = =

∑ (m󸀠 ,n󸀠 )∈Λ

∑

⟨Tf , gm󸀠 ,n󸀠 ⟩γm󸀠 ,n󸀠 , g⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ∑ ⟨Tg m,n⏟⏟⏟ m󸀠 ,n󸀠⏟⟩⏟cm,n γm󸀠 ,n󸀠 , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(m󸀠 ,n󸀠 )∈Λ (m,n)∈Λ

Tm󸀠 ,n󸀠 ,m,n

where cm,n = ⟨f , γm,n ⟩. The infinite Gabor matrix Tm󸀠 ,n󸀠 ,m,n is defined by

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Tm󸀠 ,n󸀠 ,m,n = ⟨Tgm,n , gm󸀠 ,n󸀠 ⟩,

(m, n), (m󸀠 , n󸀠 ) ∈ Λ

(5.3)

(by abuse of notation, we identify the matrix with its general entry) and the properties of T can be equivalently studied via those of the matrix above. We remark that this way of building the Gabor matrix involves only the function g. The dual window γ does not play any role. This has the advantage of working only with the Gabor frame {gm,n }(m,n)∈Λ , no information about its dual is needed. Of course, one can build up the Gabor matrix of an operator T considering both frames, as was already done in Chapter 3, see the Gabor matrix in (3.37). In what follows we will limit ourselves to the Gabor matrix defined in (5.3). Hyperbolic problems were treated by many authors using frames. In particular, curvelet and shearlet representations of wave propagators were proved to be optimally sparse, cf. [41, 43, 173, 174]. Very recently new issues in shearlet theory have been shown in [1, 2, 5]. Our concern is limited to constant coefficient hyperbolic and evolution-type equations. The reason is simple: the example of the transport equation with nonconstant coefficients, standard test for numerical schemes, is somewhat discouraging. Let us denote by T the operator providing the solution of the related Cauchy problem. For a fixed time t > 0, the operator T turns out to be a change of variables, and Gabor frames are not suited to follow the corresponding nonlinear propagation of singularities. Heuristically this can be seen as follows. Consider a window function g such that the related Gabor system 𝒢 (g, α, β) is a frame. Decomposing a function or distribution f in terms of Gabor atoms gm,n , that is, f = ∑ cm,n (f )gm,n , m,n∈ℤ

corresponds geometrically to a uniform partition of the time–frequency plane (or phase plane, in PDEs terminology) into boxes, each atom occupying a box, loosely speaking; see Figure 5.1. The correspondence principle in quantum mechanics suggests following the PDE’s evolution in terms of its classical analog, namely the Hamiltonian flow in the phase plane. The propagator has therefore a sparse Gabor matrix if its Hamiltonian flow moves the above mentioned boxes, but introduces only a controlled number of overlaps. However, this is not the case for changes of variables, as one sees easily by

5 Time–frequency analysis of constant-coefficient partial differential equations | 279

ξ

2β β α

2α

x

Figure 5.1: Decomposition of the time–frequency plane in uniform boxes.

direct inspection. More rigorously, it follows from the results in [78] that Hörmander’s type Fourier integral operators, in particular, the changes of variables, do not have a sparse Gabor representation. Because of this, the study of hyperbolic equations via Gabor frames will be limited to operators with constant coefficients. With respect to [41, 43], such results are stronger in the following aspects: – Gabor frames allow treating hyperbolic equations, not necessarily strictly hyperbolic, of any order and in an arbitrary number of space variables; – Gabor frames allow also considering parabolic equations; – The decay of the Gabor matrix corresponding to the evolution operator is of superexponential type. As a simple example, let us consider the Cauchy problem for the wave equation: 𝜕t2 u − Δx u = 0,

u(0, x) = u0 (x),

(t, x) ∈ ℝ × ℝd ,

(5.4)

𝜕t u(0, x) = u1 (x).

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We may express the solution in the form u(t, x) = Tt u1 (x) + 𝜕t Tt u0 (x),

(5.5)

where Tt is the Fourier multiplier Tt f (x) = ∫ e2πix⋅ξ σt (ξ )̂f (ξ ) dξ ,

(5.6)

with symbol σt (ξ ) =

sin(2π|ξ |t) , 2π|ξ |

ξ ∈ ℝd .

(5.7)

The operator T = Tt can be viewed in terms of its infinite Gabor matrix Tm󸀠 ,n󸀠 ,m,n in (5.3).

280 | 5 Time–frequency analysis of constant-coefficient partial differential equations 2

Taking the Gaussian window function g(x) = e−πx and discretizing consequently Tt , we shall prove that the Gabor matrix satisfies 󸀠 2

󸀠 2

|Tm󸀠 ,n󸀠 ,m,n | ≤ Ce−ϵ(|m−m | +|n−n | ) .

(5.8)

We address to the sequel of the chapter for the dependence of C and ϵ on t > 0. Let us describe in short how (5.8) can be obtained in the general context of symbols depending on z = (x, ξ ). The proof will be largely based on the properties of Gelfand–Shilov spaces introduced below. For s ≥ 0, assume that the symbol σ satisfies 󵄨󵄨 α 󵄨 |α|+1 (α!)s , 󵄨󵄨𝜕 σ(z)󵄨󵄨󵄨 ≤ C

z ∈ ℝ2d ,

(5.9)

for a constant C > 0 depending on σ. We say that the symbol σ is Gevrey when s > 1, analytic when s = 1, and ultraanalytic if s < 1. Let us observe that, when s = 1, σ extends to an analytic function in the strip {z + iζ ∈ ℂ2d : |ζ | < 1/C}, whereas when s < 1, σ extents to an entire function σ(z + iζ ) satisfying |σ(z + iζ )| ≤ C̃ exp(δ|ζ |1/(1−s) ), for some constants C,̃ δ > 0 (see, e. g., [236, Section 6.1]). As we will show, such estimates are satisfied by the symbols of the fundamental solutions for a large class of evolution operators with constant coefficients, with 0 ≤ s < 1. In particular, we have s = 0 for the wave equation (5.4). Let T be a pseudodifferential operator with symbol σ satisfying (5.9) for some s ≥ 0, and define r = min{2, 1/s}. Fix as a window the Gaussian, or more generally any function g belonging to the Gelfand–Shilov classes Srr (ℝd ). Then we shall prove that 󸀠

󸀠 r

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|Tm󸀠 ,n󸀠 ,m,n | ≤ Ce−ϵ(|m−m |+|n−n |) .

(5.10)

Accordingly, every column or row of the Gabor matrix, rearranged in decreasing order, displays a similar decay, i. e., we obtain an exponential-type sparsity. The contents of the chapter are organized in the following way. Section 5.1 is devoted to Strichartz estimates, in particular, the proof of the main result is given in Sections 5.1.1 and 5.1.2, the necessity of the conditions is proved in Section 5.1.3. Section 5.2 consists of six subsections. Namely, in Section 5.2.1 we recall the basics of Gelfand–Shilov spaces, in Section 5.2.2 we treat almost diagonalization of pseudodifferential operators in this setting, i. e., estimates (5.10) for operators with symbols satisfying (5.9). Sparsity of the corresponding Gabor matrix is proved in Section 5.2.3. Section 5.2.4 presents some applications to the boundedness of pseudodifferential operators on Gelfand–Shilov spaces. Finally, Sections 5.2.5 and 5.2.6 concern the applications to evolution partial differential equations, with numerics for wave equation, as mentioned before, and heat equation.

5.1 Strichartz estimates for the Schrödinger equation in Wiener amalgam spaces | 281

5.1 Strichartz estimates for the Schrödinger equation in Wiener amalgam spaces Consider the Cauchy problem for the homogeneous Schrödinger equation (free particle) i𝜕t u + Δu = 0, { u(0, x) = u0 (x),

(5.11)

with (t, x) ∈ ℝ × ℝd , d ≥ 1. Applying the Fourier transform with respect to the space variable x, one easily sees that the solution u(t, x) to the Cauchy problem in (5.11) can be formally written as u(t, x) = ∫ e2πix⋅ξ e−4π

2

itξ 2

û0 (ξ ) dξ ,

ℝd

that is, using the Fourier multiplier eitΔ with symbol at (ξ ) = e−4π u(t, x) = (eitΔ u0 )(x),

2

itξ 2

,

for every fixed t ∈ ℝ.

(5.12)

The study of space-time integrability properties for the solution of (5.11) has been pursued by many authors in the last 40 years. We recall the important fixed-time estimates 󵄩󵄩 itΔ 󵄩󵄩 −d( 1 − 1 ) 󵄩󵄩e u0 󵄩󵄩Lr (ℝd ) ≲ |t| 2 r ‖u0 ‖Lr󸀠 (ℝd ) ,

2 ≤ r ≤ ∞,

(5.13)

as well as the celebrated homogeneous Strichartz estimates [157, 206, 208, 322]:

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󵄩󵄩 itΔ 󵄩󵄩 q 󵄩󵄩e u0 󵄩󵄩L Lr ≲ ‖u0 ‖L2x , t x

(5.14)

for q ≥ 2, r ≥ 2, with 2/q + d/r = d/2, (q, r, d) ≠ (2, ∞, 2), i. e., for (q, r) Schrödinger admissible. Here we set 1/q

󵄩 󵄩q ‖F‖Lq Lrx = (∫󵄩󵄩󵄩F(t, ⋅)󵄩󵄩󵄩Lr dt) t

x

.

These estimates express a gain of local x-regularity of the solution u(t, ⋅), and a decay of its Lrx -norm, both in some Lqt -averaged sense. Recently several authors have turned their attention to fixed-time and space-time estimates of the Schrödinger propagator, see [306, 14, 15, 264, 303, 304, 305]. In what follows we study similar estimates in Wiener amalgam spaces. One of the main motivations for considering estimates in these spaces is that they control the local regularity of a function and its decay at infinity separately. Hence, they highlight and distinguish between the local properties and the global behavior of the solution

282 | 5 Time–frequency analysis of constant-coefficient partial differential equations u(t, x) = eitΔ u0 and therefore, as far as Strichartz estimates are concerned, they are natural candidates to perform an analysis of the solution which is finer than the classical one. Actually, Wiener amalgam spaces have already appeared as a technical tool in PDEs. In particular, the space W(Lp , Lq ) coincides with the space X0q,p introduced in [276]. For comparison, the classical estimates (5.14) can be rephrased in terms of Wiener amalgam spaces as follows: 󵄩󵄩 itΔ 󵄩󵄩 󵄩󵄩e u0 󵄩󵄩W(Lq ,Lq )t W(Lr ,Lr )x ≲ ‖u0 ‖L2x .

(5.15)

Let us start with a brief discussion of the results. As usual we first establish an estimate of dispersive type. Namely, in our setting, we show that it is possible to move local regularity in (5.14) from the space variable to the time variable. As a result, we obtain new estimates involving the Wiener amalgam spaces W(Lp , Lq ) that generalize (5.14). This requires some preliminary steps. First, we show the fixed-time estimates d 1 1 󵄩󵄩 itΔ 󵄩󵄩 2 − ( − ) 󵄩󵄩e u0 󵄩󵄩W(L2 ,Lr ) ≲ (1 + t ) 2 2 r ‖u0 ‖W(L2 ,Lr󸀠 ) ,

2 ≤ r ≤ ∞.

The related Strichartz estimates are achieved in Theorem 5.1.8 below. Finally, the complex interpolation with the classical estimates (5.14) yields our main result: Theorem 5.1.1. Let 1 ≤ q1 , r1 ≤ ∞, 2 ≤ q2 , r2 ≤ ∞ be such that r1 ≤ r2 , d d 2 + ≥ , q1 r1 2 2 d d + ≤ , q2 r2 2

(5.16) (5.17)

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(r1 , d) ≠ (∞, 2), (r2 , d) ≠ (∞, 2), and, if d ≥ 3, r1 ≤ 2d/(d − 2). The same for q̃ 1 , q̃ 2 , r1̃ , r2̃ . Then, we have the homogeneous Strichartz estimates 󵄩󵄩 itΔ 󵄩󵄩 󵄩󵄩e u0 󵄩󵄩W(Lq1 ,Lq2 )t W(Lr1 ,Lr2 )x ≲ ‖u0 ‖L2x ,

(5.18)

the dual homogeneous Strichartz estimates 󵄩󵄩 󵄩󵄩 󵄩󵄩 −isΔ 󵄩 󵄩󵄩∫ e F(s) ds󵄩󵄩󵄩 ≲ ‖F‖W(Lq̃1󸀠 ,Lq̃2󸀠 ) W(Lr1󸀠̃ ,Lr2󸀠̃ ) , 󵄩󵄩 󵄩󵄩L2 t x

(5.19)

and the retarded Strichartz estimates 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 i(t−s)Δ 󸀠̃ 󸀠̃ F(s) ds󵄩󵄩󵄩 ≲ ‖F‖ q̃1󸀠 q̃2󸀠 . 󵄩󵄩 ∫ e W(L ,L )t W(Lr1 ,Lr2 )x 󵄩󵄩 󵄩󵄩W(Lq1 ,Lq2 )t W(Lr1 ,Lr2 )x s q2 .

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Figure 5.2: When d ≥ 3, (5.18) holds for all pairs (1/q1 , 1/r1 ) ∈ I1 , (1/q2 , 1/r2 ) ∈ I2 , with 1/r2 ≤ 1/r1 .

Since there are no relations between the pairs (q1 , r1 ) and (q2 , r2 ) other than r1 ≤ r2 , these estimates tell us that the analysis of the local regularity of the Schrödinger propagator is quite independent of its decay at infinity. We will then discuss the sharpness of the results above. Let us start with the fixedtime estimates. 5.1.1 Fixed-time estimates In this section we study estimates for the solution u(t, x) to the Cauchy problem (5.11), for fixed t. We take advantage of the explicit formula for the solution u(t, x) = (Kt ∗ u0 )(x),

(5.21)

ix 2 1 e 4t . d/2 (4πit)

(5.22)

where Kt (x) =

284 | 5 Time–frequency analysis of constant-coefficient partial differential equations Precisely, we will first show that the function in (5.22) is in the Wiener amalgam space W(ℱ L1 , L∞ ) and we shall compute its norm. This goal is attained thanks to the nice 2 choice of the Gaussian function e−πx ∈ 𝒮 (ℝd ), as a fitting window function. Proposition 5.1.2. For a ∈ ℝ, a ≠ 0, let fa (x) = (ai)−d/2 e−πx with

2

/(ai)

. Then fa ∈ W(ℱ L1 , L∞ ),

d

‖fa ‖W(ℱ L1 ,L∞ )

1 + a2 4 =( 4 ) . a

(5.23)

Proof. By definition, ‖fa ‖W(ℱ L1 ,L∞ ) = sup ‖fa Tx g‖ℱ L1 , x∈ℝd

for some nonzero window g ∈ W(ℱ L1 , L1 ) (different windows give equivalent norms). 2 We then choose g = e−πx , and, observing that ĝ = g, can write ̂ ‖fa Tx g‖ℱ L1 = ‖f? a Tx g‖L1 = ‖fa ∗ M−x g‖L1 . Using (0.28), we compute the Fourier transform of f which happens to be 2

2

fâ (ξ ) = (ai)−d/2 (ai)d/2 e−πaiξ = e−πaiξ . Thereby, 󵄨󵄨 󵄨󵄨 2 2 󵄨 󵄨 ‖fâ ∗ M−x g‖L1 = ∫ 󵄨󵄨󵄨 ∫ e−πai(ξ −y) e−2πix⋅y e−πy dy󵄨󵄨󵄨 dξ 󵄨󵄨 󵄨󵄨 d d ℝ ℝ

󵄨󵄨 󵄨󵄨 2 2 󵄨 󵄨 = ∫ 󵄨󵄨󵄨e−πaiξ ∫ e−2πi(x−aξ )⋅y e−π(1+ai)y dy󵄨󵄨󵄨 dξ 󵄨󵄨 󵄨󵄨 d d ℝ

ℝ

2

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󵄨 󵄨 = ∫ 󵄨󵄨󵄨ℱ (e−π(1+ai)y )(x − aξ )󵄨󵄨󵄨 dξ ℝd 2

󵄨 󵄨 = ∫ 󵄨󵄨󵄨(1 + ai)−d/2 e−π(x−aξ ) /(1+ai) 󵄨󵄨󵄨 dξ , ℝd

where in the last equality we use (0.28). Performing the change of variables x − aξ = z, hence dξ = a−d dz and observing that |(1 + ai)−d/2 | = (1 + a2 )−d/4 , we can write ‖fâ ∗ M−x g‖L1 = (1 + a2 )

a

−d/4 −d

2 󵄨 󵄨 ∫ 󵄨󵄨󵄨e−πz /(1+ai) dz 󵄨󵄨󵄨

ℝd

= (1 + a2 )

a

−d/4 −d

∫ e−πz ℝd

2

/(1+a2 )

dz

5.1 Strichartz estimates for the Schrödinger equation in Wiener amalgam spaces | 285

= (1 + a2 )

d/2

a (1 + a2 )

−d/4 −d d/4

=(

1 + a2 ) a4

.

Since the right-hand side does not depend on x, taking the supremum on ℝd with respect to the x-variable, we attain the desired estimate. The space W(ℱ L1 , L∞ ) is the finest Wiener amalgam space-norm for (5.22) which, consequently, gives the worst behavior in the time variable. We aim at improving the latter, at the expense of a rougher x-norm. This is achieved in the following result. Indeed, W(ℱ L1 , L∞ ) ⊂ W(ℱ Lp , L∞ ) for 1 ≤ p ≤ ∞ by the inclusion relations of these spaces (cf. Theorem 2.4.9). Proposition 5.1.3. For a ∈ ℝ, a ≠ 0, let fai (x) = (ai)−d/2 e−πx fai ∈ W(ℱ Lp , L∞ ) and

2

/(ai)

‖fai ‖W(ℱ Lp ,L∞ ) ≍ |a|−d/p (1 + a2 )

(d/2)(1/p−1/2)

. Then, for 1 ≤ p ≤ ∞,

(5.24)

.

Proof. It follows the footsteps of Proposition 5.1.2. We consider the Gaussian g(t) = 2 e−πt as a window function to compute ‖fai ‖W(ℱ Lp ,L∞ ) ≍ sup ‖faî ∗ M−x g‖Lp . x∈ℝd

2

−πaiξ Using f̂ , we have, for p < ∞, ai (ξ ) = e 1/p 󵄨󵄨 󵄨󵄨p 2 2 2 󵄨 󵄨 ‖fâ ∗ M−x g‖Lp = ( ∫ 󵄨󵄨󵄨 ∫ e−πai(ξ −y) e−2πix⋅y e−π y dy󵄨󵄨󵄨 dξ ) 󵄨󵄨 󵄨󵄨 d d ℝ ℝ

= (1 + a2 )

−d/4

|a|−d/p ( ∫ e−πpz

2

/(1+a2 )

1/p

dz)

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ℝd 2 (d/2)(1/p−1/2)

= (1 + a )

|a|−d/p p−d/(2p) .

Since the right-hand side does not depend on x, taking the supremum on ℝd with respect to the x-variable we attain the desired estimate. If p = ∞, 2

−d/4 󵄨 󵄨 ‖fa ‖W(ℱ L∞ ,L∞ ) ≍ sup 󵄨󵄨󵄨(1 + ai)−d/2 e−π(x−aξ ) /(1+ai) 󵄨󵄨󵄨 = (1 + a2 ) , d ξ ∈ℝ

and we are done. For a = 4πt, t ≠ 0, we infer:

286 | 5 Time–frequency analysis of constant-coefficient partial differential equations Corollary 5.1.4. Let Kt (x) be the kernel in (5.22). Then, if 1 ≤ p ≤ ∞, ‖Kt ‖W(ℱ Lp ,L∞ ) ≍ |t|−d/p (1 + t 2 )

(d/2)(1/p−1/2)

(5.25)

.

Lemma 5.1.5. Let 1 ≤ p, q, r ≤ ∞, with 1/r = 1/p + 1/q, then W(ℱ Lp , L∞ ) ∗ W(ℱ Lq , L1 ) �→ W(ℱ Lr , L∞ ).

(5.26)

Proof. This is a consequence of the convolution relations for Wiener amalgam spaces (cf. Theorem 2.4.9), namely ℱ Lp ∗ ℱ Lq = ℱ (Lp ⋅ Lq ) �→ ℱ Lr by Hölder’s inequality with 1/r = 1/p + 1/q, and L∞ ∗ L1 �→ L∞ . Proposition 5.1.6. It turns out, for 2 ≤ q ≤ ∞, that d(1/4−1/q) 󵄩󵄩 itΔ 󵄩󵄩 d(2/q−1) (1 + t 2 ) ‖u0 ‖W(ℱ Lq ,L1 ) . 󵄩󵄩e u0 󵄩󵄩W(ℱ Lq󸀠 ,L∞ ) ≲ |t|

(5.27)

1 2 + = 1. p q

(5.28)

Proof. We use the explicit representation of the Schrödinger evolution operator eitΔ u0 (x) = (Kt ∗ u0 )(x). Let 1 ≤ p, q ≤ ∞ satisfy

Then the kernel norm (5.25) and the convolution relations (5.26) yield the desired result. Theorem 5.1.7. For 2 ≤ q, r, s ≤ ∞ such that 1 1 2 1 1 = + ( − ), s r q 2 r we have

(5.29)

d( 1 − 1 )(1− 2r ) 󵄩󵄩 itΔ 󵄩󵄩 d( 2 −1)(1− 2r ) (1 + t 2 ) 4 q ‖u0 ‖W(ℱ Ls ,Lr󸀠 ) . 󵄩󵄩e u0 󵄩󵄩W(ℱ Ls󸀠 ,Lr ) ≲ |t| q

In particular, for 2 ≤ r ≤ ∞,

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d 1 1 󵄩󵄩 itΔ 󵄩󵄩 2 − ( − ) 󵄩󵄩e u0 󵄩󵄩W(L2 ,Lr ) ≲ (1 + t ) 2 2 r ‖u0 ‖W(L2 ,Lr󸀠 ) .

(5.30)

Proof. Estimates (5.30) follow by complex interpolation of (5.27), which corresponds to r = ∞, with the L2 conservation law 󵄩󵄩󵄩eitΔ u0 󵄩󵄩󵄩 2 = ‖u0 ‖L2 , (5.31) 󵄩 󵄩L which corresponds to r = 2. Indeed, L2 = W(ℱ L2 , L2 ) = W(L2 , L2 ). By complex interpolation for Wiener amalgam spaces (2.64), for 0 < θ = 2/r < 1, and 1/s󸀠 = (1−2/r)/q󸀠 +1/r, so that relation (5.29) holds, [W(ℱ Lq , L∞ ), W(ℱ L2 , L2 )][θ] = W([ℱ Lq , ℱ L2 ][θ] , [L∞ , L2 ][θ] ) = W(ℱ Ls , Lr ) 󸀠

󸀠

󸀠

and [W(ℱ Lq , L1 ), W(ℱ L2 , L2 )][θ] = W([ℱ Lq , ℱ L2 ][θ] , [L1 , L2 ][θ] ) = W(ℱ Ls , Lr ), 󸀠

so that the estimate (5.30) is attained.

5.1 Strichartz estimates for the Schrödinger equation in Wiener amalgam spaces | 287

5.1.2 Strichartz estimates In this section we prove Theorem 5.1.1. Precisely, we first study the case q1 = q̃ 1 = ∞, r1 = r1̃ = 2. In view of the inclusion relation of Wiener amalgam spaces, it suffices to prove it for the pairs (q2 , r2 ) which are scale invariant (i. e., satisfying (5.17) with equality), namely we have the following result. Theorem 5.1.8. Let 2 ≤ q ≤ ∞, 2 ≤ r ≤ ∞, be such that 2 d d + = , q r 2 (q, r, d) ≠ (2, ∞, 2), and similarly for q,̃ r.̃ Then we have the homogeneous Strichartz estimates 󵄩󵄩 itΔ 󵄩󵄩 󵄩󵄩e u0 󵄩󵄩W(L∞ ,Lq )t W(L2 ,Lr )x ≲ ‖u0 ‖L2x ,

(5.32)

the dual homogeneous Strichartz estimates 󵄩󵄩 󵄩󵄩 󵄩󵄩 −isΔ 󵄩 󵄩󵄩∫ e F(s) ds󵄩󵄩󵄩 ≲ ‖F‖W(L1 ,Lq̃󸀠 ) W(L2 ,Lr󸀠̃ ) , 󵄩󵄩 󵄩󵄩L2 t x and the retarded Strichartz estimates 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 i(t−s)Δ F(s) ds󵄩󵄩󵄩 ≲ ‖F‖W(L1 ,Lq̃󸀠 ) W(L2 ,Lr󸀠̃ ) . 󵄩󵄩 ∫ e 󵄩󵄩 󵄩󵄩W(L∞ ,Lq )t W(L2 ,Lr )x t x

(5.33)

(5.34)

s 2, q̃ > 2. We first show the estimate (5.32). The case q = ∞, r = 2, follows 2 2 at once from the conservation law (5.31). Indeed, W(L∞ , L∞ )t = L∞ t and W(L , L )x = 2 itΔ Lx , so that, taking the supremum over t in ‖e u0 ‖L2x = ‖u0 ‖L2x , we obtain the claim. To prove the remaining cases, we can apply the usual TT ∗ method (or “orthogonality principle”, see [157, Lemma 2.1] or [268, page 353]), because of Hölder-type inequality 󵄨󵄨 󵄨 󵄨󵄨⟨F, G⟩L2t L2x 󵄨󵄨󵄨 ≤ ‖F‖W(L∞ ,Lq )t W(L2 ,Lr )x ‖G‖W(L1 ,Lq󸀠 ) W(L2 ,Lr󸀠 ) , t x

(5.35)

which can be proved directly from the definition of these spaces. As a consequence, it suffices to prove the estimate 󵄩󵄩 󵄩󵄩 󵄩󵄩 i(t−s)Δ 󵄩 F(s) ds󵄩󵄩󵄩 ≲ ‖F‖W(L1 ,Lq󸀠 ) W(L2 ,Lr󸀠 ) . 󵄩󵄩∫ e 󵄩󵄩 󵄩󵄩W(L∞ ,Lq )t W(L2 ,Lr )x t x

(5.36)

Recall the Hardy–Littlewood–Sobolev fractional integration theorem (see, e. g., [268], page 119) in dimension 1: Lp (ℝ) ∗ L1/α,∞ (ℝ) �→ Lq (ℝ),

(5.37)

for 1 ≤ p < q < ∞, 0 < α < 1, with 1/p = 1/q + 1 − α (here L1/α,∞ is the weak L1/α 󸀠 space, see, e. g., [270]). Now, set α = d(1/2 − 1/r) = 2/q (q > 2) so that, for p = q󸀠 , Lq ∗

288 | 5 Time–frequency analysis of constant-coefficient partial differential equations L1/α,∞ �→ Lq . Moreover, observe that (1 + |t|)−α ∈ W(L∞ , L1/α,∞ )(ℝ). The convolution relations for Wiener amalgam spaces in Theorem 2.4.9 then give W(L1 , Lq )(ℝ) ∗ W(L∞ , L1/α,∞ )(ℝ) �→ W(L∞ , Lq )(ℝ). 󸀠

The preceding relations, together with the fixed-time estimates (5.30) and Minkowski’s inequality, allow writing 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 i(t−s)Δ F(s) ds󵄩󵄩󵄩 󵄩󵄩∫ e 󵄩󵄩W(L∞ ,Lq )t W(L2 ,Lr )x 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩 ∫ 󵄩󵄩󵄩ei(t−s)Δ F(s)v‖W(L2 ,Lr )x ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩W(L∞ ,Lq )t 󵄩 󵄩󵄩󵄩 󵄩 −α 󵄩 󵄩 󵄩 ≲ 󵄩󵄩󵄩 ∫ 󵄩󵄩󵄩F(s)󵄩󵄩󵄩W(L2 ,Lr󸀠 ) (1 + |t − s|) ds󵄩󵄩󵄩 x 󵄩󵄩 󵄩󵄩W(L∞ ,Lq )t ≲ ‖F‖W(L1 ,Lq󸀠 ) W(L2 ,Lr󸀠 ) . t

x

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The estimate (5.33) follows from (5.32) by duality. Consider now the retarded Strichartz estimate (5.34). By complex interpolation, in ̃ (1/∞, 1/2) collinear, it suffices to prove (5.34) order to get (5.34) with (1/q, 1/r), (1/q,̃ 1/r), in the three cases (q,̃ r)̃ = (q, r), (q, r) = (∞, 2), (q,̃ r)̃ = (∞, 2), as shown in Figure 5.3.

Figure 5.3: The complex interpolation method is applied in these two cases.

Now, the case (q,̃ r)̃ = (q, r) follows from (5.36) with χ{s 0. Choose g = χ[−1,1] as a window function in the definition of the space W(Lq1 , Lq2 )t . Then, for y ∈ ℝ, λ ≥ 1, 󵄩 󵄩󵄩 2 󵄩󵄩u(λ tλ, ⋅)󵄩󵄩󵄩W(Lr1 ,Lr2 ) Ty g(t)‖Lq1 d r2

y+1

≍ λ ( ∫ (λ

−4

2

+t )

q1 d 1 (r 2 1

− 21 )

(λ

−2

2

+t )

q1 d 1 (r 2 2

− r1 ) 1

1 q1

dt) .

(5.68)

y−1

To estimate this expression from below, we apply the easily verified formula 1 2

γ

γ

∫ (μ2 + t 2 ) 1 (μ + t 2 ) 2 dt ≳ μ1+2γ1 +γ2 ,

0 < μ ≤ 1,

− 21

with μ = λ−2 . Thereby, − 2 −d 󵄩󵄩󵄩󵄩 2 󵄩 󵄩 󵄩󵄩󵄩󵄩u(λ t, λ⋅)󵄩󵄩󵄩W(Lr1 ,Lr2 ) Ty g(t)󵄩󵄩󵄩Lq1 ≳ λ q1 r1

1 for |y| ≤ , λ ≥ 1. 2

Taking the Lq2 -norm of this expression yields − 2 −d 󵄩󵄩 2 󵄩 󵄩󵄩u(λ ⋅, λ⋅)󵄩󵄩󵄩W(Lr1 ,Lr2 )t W(Lr1 ,Lr2 )x ≳ λ q1 r1 ,

λ ≥ 1. d

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On the other hand, the right-hand side of (5.65) is equal to λ− 2 ; therefore, letting λ → +∞, we obtain the index relation (5.66). We now prove (5.67). If 0 < λ ≤ 1, it follows from the norm estimate (5.62) that d 1 1 d −d −4 󵄩󵄩 2 󵄩 2 ( − ) 󵄩󵄩u(λ t, λ⋅)󵄩󵄩󵄩W(Lr1 ,Lr2 ) ≍ λ r2 (λ + t ) 2 r2 2 .

Hence, we have d 1 1 d −d −4 󵄩󵄩󵄩󵄩 2 󵄩 󵄩 2 ( − ) 󵄩󵄩󵄩󵄩u(λ t, λ⋅)󵄩󵄩󵄩W(Lr1 ,Lr2 ) Ty g 󵄩󵄩󵄩Lq1 ≍ λ r2 (λ + y ) 2 r2 2 ,

0 < λ ≤ 1.

Taking the Lq2 -norm of this expression and applying the formula +∞

γ

1

∫ (μ + y2 ) dy = cγ μ 2 +γ ,

−∞

μ > 0,

298 | 5 Time–frequency analysis of constant-coefficient partial differential equations (for every γ, and for a convenient cγ ∈ (0, ∞], independent of μ (cγ < ∞ if γ < − 21 )) with μ = λ−4 , we obtain the minorization − 2 −d 󵄩 󵄩󵄩 2 󵄩󵄩u(λ ⋅, λ⋅)󵄩󵄩󵄩W(Lq1 ,Lq2 )t W(Lr1 ,Lr2 )x ≳ λ q2 r2 ,

0 < λ ≤ 1.

d

Since the right-hand side of (5.65) is equal to λ− 2 , the estimate (5.67) follows by letting λ → 0+ . We are left to prove the condition q2 ≥ 2. We follow the pattern outlined in [277, Exercise 2.42]. Precisely, we take as initial datum N

u0 (x) = ∑ e−itj Δ f , j=1

where f is a fixed test function, normalized so that the function v(t, x), defined by v(t, x) = (eitΔ f )(x), satisfies ‖v‖W(Lq1 ,Lq2 )t W(Lr1 ,Lr2 )x = 1.

(5.69)

Here t1 , . . . , tN are widely separated times that will be chosen later on. Notice that the corresponding solution will be N

u(t, x) = (eitΔ u0 )(x) = ∑ v(t − tj , x). j=1

We claim that

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󵄩󵄩 N 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 ‖u0 ‖L2 = 󵄩󵄩󵄩∑ e−itj Δ f 󵄩󵄩󵄩 ≤ (N + 1) 2 ‖f ‖L2 , 󵄩󵄩 󵄩󵄩 2 󵄩j=1 󵄩L

(5.70)

if t1 , . . . , tN are suitable separated. Indeed, 󵄩󵄩 N 󵄩󵄩2 N 󵄩󵄩 󵄩 󵄩󵄩∑ e−itj Δ f 󵄩󵄩󵄩 = ∑󵄩󵄩󵄩e−itj Δ f 󵄩󵄩󵄩2 2 + ∑⟨ei(tj −tk )Δ f , f ⟩ 󵄩󵄩 󵄩󵄩 󵄩 󵄩L 󵄩󵄩j=1 󵄩󵄩L2 j=1 j=k̸ d

≤ N‖f ‖2L2 + C ∑ |tj − tk |− 2 ‖f ‖2L1 , j=k̸

where we used Cauchy–Schwarz inequality and the classical dispersive estimate ‖eitΔ f ‖L∞ ≤ C|t|−d/2 ‖f ‖L1 . Hence (5.70) follows if |tj − tk |−d/2 ≤ [C(N 2 − N)‖f ‖2L1 ] ‖f ‖2L2 . −1

5.1 Strichartz estimates for the Schrödinger equation in Wiener amalgam spaces | 299

̃ x) = We now estimate from below the left-hand side of (5.65). To this end, let v(t, v(t, x)χR (t), where χR (t) is the characteristic function of the interval [−R, R]. Moreover, we assume q2 < ∞ and choose R large enough so that ‖v − v‖̃ W(Lq1 ,Lq2 )t W(Lr1 ,Lr2 )x ≤

1 . N

(5.71)

We claim that, if |tj − tk | ≥ 2R + 2, for every j ≠ k, then 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩u(t, x)󵄩󵄩W(Lq1 ,Lq2 )t W(Lr1 ,Lr2 )x = 󵄩󵄩∑ v(t − tj , x)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 q1 q2 󵄩j=1 󵄩W(L ,L )t W(Lr1 ,Lr2 )x 1

≥ N q2 (1 −

1 ) − 1. N

(5.72)

This, together with the assumption (5.65) and the L2 -estimate of the initial datum (5.70), gives the condition q2 ≥ 2, for N large enough. In order to prove the minorization (5.72), observe that, by assumption (5.71), 󵄩󵄩 N 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 󵄩󵄩∑ v(t − tj , x) − ∑ v(t 󵄩󵄩 ̃ − t , x) ≤ 1. j 󵄩󵄩󵄩 󵄩󵄩󵄩 q1 q2 j=1 󵄩j=1 󵄩W(L ,L )t W(Lr1 ,Lr2 )x Hence it suffices to prove 󵄩󵄩 N 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 1 󵄩󵄩∑ v(t 󵄩󵄩 ̃ − t , x) ≥ N q2 (1 − ). j 󵄩󵄩󵄩 󵄩󵄩󵄩 q1 q2 N 󵄩j=1 󵄩W(L ,L )t W(Lr1 ,Lr2 )x

(5.73)

Now,

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󵄩󵄩 N 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 󵄩󵄩 ̃ 󵄩 󵄩󵄩∑ v(t 󵄩 ̃ − t , ⋅) = ∑ 󵄩 󵄩󵄩v(t − tj , ⋅)󵄩󵄩󵄩W(Lr1 ,Lr2 ) , j 󵄩 󵄩󵄩 󵄩󵄩 r1 r2 󵄩󵄩j=1 󵄩W(L ,L ) j=1 since, for every fixed t, there is at most one function in the sum which is not identically ̃ − tj , ⋅)‖W(Lr1 ,Lr2 ) , we have zero. Hence, upon setting hj (t) := ‖v(t 󵄩󵄩 N 󵄩󵄩 󵄩󵄩󵄩󵄩 N 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩∑ v(t 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 . ̃ − t , ⋅) = h T g ∑ 󵄩 󵄩 󵄩 j 󵄩 j y 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 q 󵄩󵄩 q1 q2 󵄩󵄩 󵄩󵄩j=1 󵄩󵄩Lq1 󵄩󵄩Ly2 󵄩W(L ,L )t W(Lr1 ,Lr2 )x 󵄩󵄩󵄩󵄩j=1 Choosing the window function g supported in [0, 1], since the hj ’s are supported in intervals separated by a distance ≥ 2, we see that the last expression is equal to 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ ‖hj Ty g‖Lq1 󵄩󵄩󵄩 . 󵄩󵄩 q 󵄩󵄩󵄩 󵄩󵄩Ly2 󵄩 j=1

300 | 5 Time–frequency analysis of constant-coefficient partial differential equations In turn, since the functions y �→ ‖hj Ty g‖Lq1 , j = 1, . . . , N, have disjoint supports, the norm above can be written as N

󵄩q 󵄩 (∑ 󵄩󵄩󵄩‖hj Ty g‖Lq1 󵄩󵄩󵄩L2q2 ) y

1 q2

j=1

1

= N q2 ‖v‖̃ W(Lq1 ,Lq2 )t W(Lr1 ,Lr2 )x .

Hence, the minorization (5.73) follows from the assumptions (5.69) and (5.71). One can also widen this theory to the study of partial differential equations with variable coefficients. Precisely, consider the Cauchy problem for the Schrödinger equation with a quadratic Hamiltonian, namely i 𝜕u + Hu = 0, { 𝜕t u(0, x) = u0 (x),

(5.74)

where H is the Weyl quantization of a quadratic form on ℝd × ℝd . The most interesting case is certainly the Schrödinger equation with a quadratic potential. In this case the solution u(t, x) to (5.74) is given by u(t, x) = eitH u0 , where the operator eitH is a metaplectic operator, see Section (1.1.2) and the work [70], where the authors show fixed-time estimates for the solution u(t, x) in terms of the initial datum u0 . An example is provided by the harmonic oscillator H=−

1 Δ + πx2 4π

(5.75)

(see, e. g., [145]), for which they deduce the dispersive estimate

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󵄩󵄩 itH 󵄩󵄩 −d 󵄩󵄩e u0 󵄩󵄩W(ℱ L1 ,L∞ ) ≲ | sin t| ‖u0 ‖W(ℱ L∞ ,L1 ) . 1 Δ − π|x|2 . In this case, the Another Hamiltonian that has been considered is H = − 4π dispersive estimate reads d

1 + | sinh t| 2 󵄩󵄩 itH 󵄩󵄩 ) ‖u0 ‖W(ℱ L∞ ,L1 ) . 󵄩󵄩e u0 󵄩󵄩W(ℱ L1 ,L∞ ) ≲ ( sinh2 t Using the fixed-time estimates above, one can infer suitable Strichartz estimates. For instance, the homogeneous Strichartz estimates achieved for the harmonic oscil1 lator H = − 4π Δ + π|x|2 read 󵄩󵄩 itH 󵄩󵄩 󵄩󵄩e u0 󵄩󵄩Lq/2 ([0,T])W(ℱ Lr󸀠 ,Lr ) ≲ ‖u0 ‖L2x , x for every T > 0, 4 < q, q̃ ≤ ∞, 2 ≤ r, r ̃ ≤ ∞, such that 2/q + d/r = d/2, and, similarly, for q,̃ r.̃ In the endpoint case (q, r) = (4, 2d/(d − 1)), d > 1, the same estimate can be

5.2 Gabor representations of evolution operators | 301

proved with ℱ Lr replaced by the slightly larger space ℱ Lr ,2 , where Lr ,2 is a Lorentz space. We refer the interested reader to the original paper [70]. We end up this section by recalling that Strichartz estimates for Schrödinger equations can be found in the recent contributions [209, 210]. Similar estimates for the vibrating plate equation were studied in [92]. For the time–frequency analysis of the Dirac equation, we refer to the contribution [294]. 󸀠

󸀠

󸀠

5.2 Gabor representations of evolution operators In this section we shall apply Gabor frames to the study of PDEs, following the plan in the introduction of the chapter. The functional frame is given by Gelfand–Shilov spaces. Modulation spaces are not abandoned in this presentation, but play a minor role. In fact, we have here to relax the assumption of algebraic growth (2.17) for the weight functions and use exponential weights b

v(z) = ea|z| ,

a > 0, b > 0.

p,q As we observed in Chapter 2, the main properties of Mm , m ∈ ℳv , extend to this case, provided v(z) is submultiplicative. Recall that submultiplicativity is valid for exponential weights if and only if b ≤ 1, cf. Lemma 2.1.4 in Chapter 2. This technical requirement restricts the application of modulation spaces, see the next Definition 5.2.6 and the subsequent Proposition 5.2.16, concerning boundedness of pseudodifferential p,q operators on Mm . For recent results on the subject, let us refer to [45, 45, 82, 289].

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5.2.1 Gelfand–Shilov spaces The Schwartz class 𝒮 (ℝd ) does not give enough information about how fast a function f ∈ 𝒮 (ℝd ) and its derivatives decay at infinity. This is the main motivation to use subspaces of the Schwartz class, the so-called Gelfand–Shilov-type spaces, introduced in [155, 156] and now available also in a textbook [236]. Let us recall their definition and main properties; for proofs, we refer the reader to [155, 156, 236, 242]. Definition 5.2.1. Let there be given s, r ≥ 0 and A, B > 0. The Gelfand–Shilov-type s,A space Sr,B (ℝd ) is defined by 󵄨 󵄨 s,A Sr,B (ℝd ) = {f ∈ 𝒮 (ℝd ) : 󵄨󵄨󵄨xα 𝜕β f (x)󵄨󵄨󵄨 ≲ A|α| B|β| (α!)r (β!)s , α, β ∈ ℕd }. Their projective and inductive limits are respectively denoted by s,A ; Σsr = proj lim Sr,B A>0,B>0

s,A Srs = ind lim Sr,B . A>0,B>0

(5.76)

302 | 5 Time–frequency analysis of constant-coefficient partial differential equations The space Srs (ℝd ) is nontrivial if and only if r + s > 1, or r + s = 1 and r, s > 0. So the 1/2 smallest nontrivial space with r = s is provided by S1/2 (ℝd ). Every function of the type 2

1/2 P(x)e−a|x| , with a > 0 and P(x) polynomial on ℝd , is in the class S1/2 (ℝd ). We observe s

s

the trivial inclusions Sr11 (ℝd ) ⊂ Sr22 (ℝd ) for s1 ≤ s2 and r1 ≤ r2 . Moreover, if f ∈ Srs (ℝd ), also xδ 𝜕γ f belongs to the same space for every fixed δ, γ. The action of the Fourier transform on Srs (ℝd ) interchanges the indices s and r, as explained in the following theorem. Theorem 5.2.2. For f ∈ 𝒮 (ℝd ), we have f ∈ Srs (ℝd ) if and only if f ̂ ∈ Ssr (ℝd ).

Therefore, for s = r, the spaces Sss (ℝd ) are invariant under the action of the Fourier transform. Theorem 5.2.3. Assume s > 0, r > 0, s + r ≥ 1. For f ∈ 𝒮 (ℝd ), the following conditions are equivalent: (a) f ∈ Srs (ℝd ). (b) There exist constants A, B > 0 such that 󵄩󵄩 α 󵄩󵄩 |α| r 󵄩󵄩x f 󵄩󵄩L∞ ≲ A (α!)

󵄩 󵄩 and 󵄩󵄩󵄩ξ β f ̂󵄩󵄩󵄩L∞ ≲ B|β| (β!)s ,

α, β ∈ ℕd .

(c) There exist constants A, B > 0 such that 󵄩󵄩 α 󵄩󵄩 |α| r 󵄩󵄩x f 󵄩󵄩L∞ ≲ A (α!)

󵄩 󵄩 and 󵄩󵄩󵄩𝜕β f 󵄩󵄩󵄩L∞ ≲ B|β| (β!)s ,

α, β ∈ ℕd .

(d) There exist constants h, k > 0 such that 󵄩󵄩 h|x|1/r 󵄩󵄩 󵄩󵄩fe 󵄩󵄩L∞ < ∞

and

󵄩󵄩 ̂ k|ξ |1/s 󵄩󵄩 󵄩󵄩f e 󵄩󵄩L∞ < ∞.

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A suitable window class for weighted modulation spaces (see the subsequent Definition 5.2.6) is the Gelfand–Shilov-type space Σ11 (ℝd ), consisting of functions f ∈ 𝒮 (ℝd ) such that for every constant A > 0 and B > 0, 󵄨󵄨 α β 󵄨 |α| |β| 󵄨󵄨x 𝜕 f (x)󵄨󵄨󵄨 ≲ A B α!β!,

α, β ∈ ℕd .

(5.77)

We have Sss (ℝd ) ⊂ Σ11 (ℝd ) ⊂ S11 (ℝd ) for every s < 1. Observe that the characterization of Theorem 5.2.3 can be adapted to Σ11 (ℝd ) by replacing the words “there exist” by “for every” and taking r = s = 1. Let us underline the following property, which exhibits two equivalent ways of expressing the decay of a continuous function f on ℝd . This follows immediately from [236, Proposition 6.1.5], where the property was formulated for f ∈ 𝒮 (ℝd ). For the sake of clarity, we shall detail the proof showing the mutual dependence between the constants ϵ and C below. Proposition 5.2.4 ([236, Proposition 6.1.5]). Consider r > 0 and let h be a continuous function on ℝd . Then the following conditions are equivalent:

5.2 Gabor representations of evolution operators | 303

(i) There exists a constant ϵ > 0 such that 1

x ∈ ℝd ,

󵄨 󵄨󵄨 −ϵ|x| r , 󵄨󵄨h(x)󵄨󵄨󵄨 ≲ e

(5.78)

(ii) There exists a constant C > 0 such that 󵄨 󵄨󵄨 α |α| r 󵄨󵄨x h(x)󵄨󵄨󵄨 ≲ C (α!) ,

x ∈ ℝd , α ∈ ℕd .

(5.79)

Proof. We rewrite (5.78) in the form 1

󵄨󵄨 󵄨1 − ϵ |x| r , 󵄨󵄨h(x)󵄨󵄨󵄨 r ≲ e r

x ∈ ℝd .

(5.80)

In turn, (5.80) can be rewritten as n

∞ n ϵ 󵄨 󵄨1 sup ∑ ( ) (n!)−1 |x| r 󵄨󵄨󵄨h(x)󵄨󵄨󵄨 r < ∞. x∈ℝd n=0 r

(5.81)

Hence the sequence of the terms of the series is uniformly bounded, as well as the sequence of the rth powers: ϵrn 󵄨 󵄨 (n!)−r |x|n 󵄨󵄨󵄨h(x)󵄨󵄨󵄨, r rn

n ∈ ℕ,

and we obtain rn 󵄨 󵄨 r |x|n 󵄨󵄨󵄨h(x)󵄨󵄨󵄨 ≲ rn (n!)r , ϵ

x ∈ ℝd , n = 0, 1, . . .

Writing |α| = n and applying (0.2) gives r|α|

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rd 󵄨󵄨 α 󵄨 󵄨󵄨x h(x)󵄨󵄨󵄨 ≲ ( ) ϵ

(α!)r = C |α| (α!)r ,

x ∈ ℝd , α ∈ ℕd ,

where C = ( rd )r . Therefore (5.79) is proved. ϵ Conversely, let (5.79) be satisfied. Using the following inequalities: n! 󵄨󵄨 α 󵄨󵄨 󵄨󵄨x 󵄨󵄨, α! |α|=n

|x|n ≤ ∑

n! = dn , α! |α|=n ∑

and

α! ≤ |α|!,

and assumption (5.79), we obtain n! 󵄨󵄨 α n! |α| 󵄨 󵄨 󵄨 |x|n 󵄨󵄨󵄨h(x)󵄨󵄨󵄨 ≤ ∑ C (α!)r ≤ C n (n!)r dn = (dC)n (n!)r , 󵄨󵄨x h(x)󵄨󵄨󵄨 ≲ ∑ α! α! |α|=n |α|=n for every x ∈ ℝd , n ∈ ℕ. Therefore the sequence 󵄨 󵄨 (dC)−n (n!)−r |x|n 󵄨󵄨󵄨h(x)󵄨󵄨󵄨,

n ∈ ℕ,

304 | 5 Time–frequency analysis of constant-coefficient partial differential equations is uniformly bounded for x ∈ ℝd , as well as the sequence n n 󵄨1 󵄨 (dC)− r (n!)−1 |x| r 󵄨󵄨󵄨h(x)󵄨󵄨󵄨 r ,

n ∈ ℕ.

1

If we choose ϵ = q(dC)− r , for a fixed q ∈ (0, 1), we conclude 1

∞

∞

n=0

n=0

n n r 󵄨 󵄨1 󵄨1 󵄨 eϵ|x| 󵄨󵄨󵄨h(x)󵄨󵄨󵄨 r = ∑ qn (dC)− r (n!)−1 |x| r 󵄨󵄨󵄨h(x)󵄨󵄨󵄨 r ≲ ∑ qn .

This is (5.80), hence the proof is complete. The precise relation between the constants ϵ and C follows from this proof. In)r . Conversely, (5.79) imdeed, assuming (5.78), then (5.79) is satisfied with C = ( rd ϵ 1

plies (5.78) for any ϵ < r(dC)− r . The bound is sharp for d = 1. Furthermore, it follows from the proof that the constant implicit in the notation ≲ in (5.78) depends only on the corresponding one in (5.79), and vice versa. The strong dual spaces of Srs (ℝd ) and Σ11 (ℝd ) are called spaces of tempered ultradistributions and denoted by (Srs )󸀠 (ℝd ) and (Σ11 )󸀠 (ℝd ), respectively. Notice that they contain the space of tempered distributions 𝒮 󸀠 (ℝd ). The spaces Srs (ℝd ) are nuclear spaces [226]. This property yields a kernel theorem for Gelfand–Shilov spaces, cf. [295]. Theorem 5.2.5. There exists an isomorphism between the space of linear continuous maps T from Srs (ℝd ) to (Srs )󸀠 (ℝd ) and (Srs )󸀠 (ℝ2d ), which associates to every T a kernel KT ∈ (Srs )󸀠 (ℝ2d ) such that ̄ ⟨Tu, v⟩ = ⟨KT , v ⊗ u⟩,

∀u, v ∈ Srs (ℝd ).

This KT is called the kernel of T.

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We now list some results about time–frequency representations of Gelfand–Shilov functions, cf. [58, 87, 171, 282]: f , g ∈ Sss (ℝd ),

f , g ∈ Sss (ℝd ), f,g ∈

s ≥ 1/2 �⇒ Vg f ∈ Sss (ℝ2d ),

(5.82)

s ≥ 1/2 �⇒ W(f , g) ∈ Sss (ℝ2d ),

(5.83)

�⇒ W(f , g) ∈

Σ11 (ℝ2d ).

(5.84)

for some ϵ > 0.

(5.85)

Σ11 (ℝd )

If g ∈ Sss (ℝd ), s ≥ 1/2, then 1/s

󵄨 󵄨 f ∈ Sss (ℝd ) ⇐⇒ 󵄨󵄨󵄨Vg (f )(z)󵄨󵄨󵄨 ≲ e−ϵ|z|

We can extend modulation spaces to the realm of ultradistributions as follows. Note that we permit now exponential submultiplicative weights v, and they represent, in fact, the more interesting case here.

5.2 Gabor representations of evolution operators | 305

Definition 5.2.6. Given g ∈ Σ11 (ℝd ), a weight function m ∈ ℳv (ℝ2d ), and 1 ≤ p, q ≤ ∞, p,q the modulation space Mm (ℝd ) consists of all tempered ultradistributions f ∈ (Σ11 )󸀠 (ℝd ) 2d p,q d such that Vg f ∈ Lp,q m (ℝ ). The norm on Mm (ℝ ) is as usual (cf. (2.20)) q/p

󵄨p 󵄨 ‖f ‖Mmp,q = ‖Vg f ‖Lp,q = ( ∫ ( ∫ 󵄨󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 m(x, ξ )p dx) m ℝd

1/q

dξ )

(5.86)

ℝd

(obvious changes if p = ∞ or q = ∞). We observe that for f , g ∈ Σ11 (ℝd ) the above integral is convergent and thus p,q Σ11 (ℝd ) ⊂ Mm (ℝd ), 1 ≤ p, q ≤ ∞, cf. [87], with dense inclusion when p, q < ∞, cf. [58]. p p,p p,q When p = q, we simply write Mm (ℝd ) instead of Mm (ℝd ). The spaces Mm (ℝd ) are 1 d Banach spaces, and every nonzero g ∈ Mv (ℝ ) yields an equivalent norm in (5.86), so p,q Mm (ℝd ) is independent of the choice of g ∈ Mv1 (ℝd ). In this section we characterize the smoothness and growth of a function f on ℝd in terms of the decay of its STFT Vg f , for a suitable window g. In the proofs we shall detail the relations among the constants Cf , Cg , Cf ,g and ϵ which come into play, since they could be useful for constructing efficient algorithms related to these objects. 1 Theorem 5.2.7. Consider s > 0, m ∈ ℳv (ℝd ), g ∈ Mv⊗1 (ℝd ) \ {0} such that there exists Cg > 0 satisfying

󵄩󵄩 α 󵄩󵄩 |α| s 󵄩󵄩𝜕 g 󵄩󵄩L1 (ℝd ) ≲ Cg (α!) , v

α ∈ ℕd .

(5.87)

For f ∈ 𝒞 ∞ (ℝd ) the following conditions are equivalent: (i) There exists a constant Cf > 0 such that

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󵄨󵄨 α 󵄨 s |α| 󵄨󵄨𝜕 f (x)󵄨󵄨󵄨 ≲ m(x)Cf (α!) ,

x ∈ ℝd , α ∈ ℕd .

(5.88)

(x, ξ ) ∈ ℝ2d , α ∈ ℕd .

(5.89)

(ii) There exists a constant Cf ,g > 0 such that 󵄨󵄨 α 󵄨 |α| s 󵄨󵄨ξ Vg f (x, ξ )󵄨󵄨󵄨 ≲ m(x)Cf ,g (α!) , (iii) There exists a constant ϵ > 0 such that 1

󵄨󵄨 󵄨 −ϵ|ξ | s , 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 ≲ m(x)e

(x, ξ ) ∈ ℝ2d , α ∈ ℕd .

(5.90)

If equivalent conditions (5.88), (5.89), and (5.90) are satisfied, we say that f is an m-Gevrey symbol when s > 1, an m-analytic symbol if s = 1, and an m-ultraanalytic symbol when s < 1.

306 | 5 Time–frequency analysis of constant-coefficient partial differential equations Proof. (i) �⇒ (ii). Since f , Tx g ∈ 𝒞 ∞ (ℝd ), for every x ∈ ℝd , we can use Leibniz formula and write α 𝜕α (fTx g)(t) = ∑ ( )𝜕α−β f (t)Tx (𝜕β g)(t), β β≤α

t ∈ ℝd , α ∈ ℕd .

Let us estimate the L1 (ℝd )-norm of the products 𝜕α−β fTx (𝜕β g). Using the positivity and v-moderateness of the weight m, Hölder’s inequality, and assumptions (5.88) and (5.87), 󵄩󵄩 α−β 󵄩 α−β 󵄩 󵄩 β 󵄩 β 󵄩 󵄩󵄩𝜕 fTx (𝜕 g)󵄩󵄩󵄩L1 ≤ 󵄩󵄩󵄩𝜕 f 󵄩󵄩󵄩L∞ 󵄩󵄩󵄩mTx (𝜕 g)󵄩󵄩󵄩L1 1/m ≲ Cf

s󵄩 󵄩 ((α − β)!) 󵄩󵄩󵄩v(⋅ − x)𝜕β g(⋅ − x)󵄩󵄩󵄩L1 m(x)

≲ Cf

((α − β)!) Cg|β| (β!)s m(x)

|α−β| |α−β|

s

≤ C |α| (α!)s m(x) where C := max{Cf , Cg } and we have used (α − β)!β! ≤ α!. These estimates tell us, in particular, that Vg f (x, ξ ) = ℱ (fTx g)(ξ ) is well-defined. We can exchange partial derivatives and Fourier transform as follows: ξ α Vg f (x, ξ ) =

1 α 1 α ℱ (𝜕 (fTx g))(ξ ) = ∑ ( )ℱ (𝜕α−β fTx (𝜕β g))(ξ ). |α| |α| (2πi) (2πi) β≤α β

Using 󵄨󵄨 󵄨 󵄩 󵄩 󵄩 α−β α−β β α−β β β 󵄩 󵄨󵄨ℱ (𝜕 fTx (𝜕 g))(ξ )󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩ℱ (𝜕 fTx (𝜕 g))󵄩󵄩󵄩L∞ ≤ 󵄩󵄩󵄩𝜕 fTx (𝜕 g)󵄩󵄩󵄩L1 , the majorizations above, and (0.1), α m(x) m(x) 󵄨󵄨 α 󵄨 ∑ ( )C |α| (α!)s = |α| C |α| (α!)s . 󵄨󵄨ξ Vg f (x, ξ )󵄨󵄨󵄨 ≲ (2π)|α| β≤α β π

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max{C ,C }

f g This proves (5.89), with constant Cf ,g := . π (ii) �⇒ (i). We use the inversion formula (1.55), observing that (5.89) implies ∞,1 f ∈ Mm⊗1 (ℝd ), so that the equality in (1.55) holds a. e. and we can assume that it holds everywhere since f is smooth. Let us consider the partial derivatives of f and exchange them with the integrals, the estimates below will provide a justification of this operation. So formally we can write

𝜕α f (t) =

1 ∫ Vg f (x, ξ )𝜕α (Mξ Tx g)(t) dx dξ , ‖g‖22

t ∈ ℝd .

ℝ2d

Using Leibniz formula 𝜕α (Mξ Tx g) = ∑β≤α (αβ)(2πiξ )β Mξ Tx (𝜕α−β g), we estimate α 1 󵄨󵄨 α 󵄨 󵄨 󵄨 󵄨 󵄨 ∑ ( )(2π)|β| ∫ 󵄨󵄨󵄨Vg f (x, ξ )ξ β 󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨Tx 𝜕α−β g(t)󵄨󵄨󵄨 dx dξ , 󵄨󵄨𝜕 f (t)󵄨󵄨󵄨 ≤ ‖g‖22 β≤α β ℝ2d

t ∈ ℝd .

5.2 Gabor representations of evolution operators | 307

We set 󵄨 󵄨 󵄨 󵄨 Iα,β (t) := ∫ 󵄨󵄨󵄨Vg f (x, ξ )ξ β 󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨Tx 𝜕α−β g(t)󵄨󵄨󵄨 dx dξ ℝ2d

and prove that, for every fixed t ∈ ℝd , Iα,β (t) are absolutely convergent integrals. Using 1 + |ξ |d+1 ≤ cd ∑|γ|≤d+1 |ξ γ |, where cd depends only on the dimension d, and assumption (5.89), d+1

󵄨 󵄨 1 + |ξ | 󵄨󵄨 α−β 󵄨 Iα,β (t) = ∫ 󵄨󵄨󵄨Vg f (x, ξ )ξ β 󵄨󵄨󵄨 󵄨Tx 𝜕 g(t)󵄨󵄨󵄨 dx dξ 1 + |ξ |d+1 󵄨 ℝ2d

≲ ∫ ℝ2d

1 󵄨󵄨 α−β 󵄨 󵄨 󵄨 ∑ 󵄨󵄨󵄨Vg f (x, ξ )ξ β+γ 󵄨󵄨󵄨 󵄨Tx 𝜕 g(t)󵄨󵄨󵄨 dx dξ 1 + |ξ |d+1 󵄨

|γ|≤d+1

s

≲ ∑ Cf ,g ((β + γ)!) ( ∫ |β+γ|

|γ|≤d+1

ℝd

1 󵄨 󵄨 dξ )( ∫ m(x)󵄨󵄨󵄨Tx 𝜕α−β g(t)󵄨󵄨󵄨 dx). 1 + |ξ |d+1 ℝd

Now, (β + γ)! ≤ 2|β|+|γ| (β!)(γ!) ≤ 2d+1 (d + 1)!2|β| β! and Cf ,g the constants depending only on d, f , and g,

|β+γ|

= Cfd+1 ,g Cf ,g so forgetting about |β|

|β| 󵄨 󵄨 Iα,β (t) ≲ Cf ,g 2sβ (β!)s ( ∫ m(x)󵄨󵄨󵄨Tx 𝜕α−β g(t)󵄨󵄨󵄨 dx) ℝd

|β| 󵄨 󵄨 ≲ Cf ,g 2sβ (β!)s m(t)( ∫ v(x − t)󵄨󵄨󵄨𝜕α−β g(t − x)󵄨󵄨󵄨 dx) ℝd

|β| 󵄩 󵄩 = Cf ,g 2sβ (β!)s m(t)󵄩󵄩󵄩𝜕α−β g 󵄩󵄩󵄩L1 . v

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The estimate above and assumption (5.87) on g allow the following majorization: α |β| s 󵄨󵄨 α 󵄨 m(t) ∑ ( )(2π)|β| (2s Cf ,g ) (β!)s Cg|α−β| ((α − β)!) 󵄨󵄨𝜕 f (t)󵄨󵄨󵄨 ≲ ‖g‖22 β≤α β ≤

m(t) (α!)s max{Cf ,g , Cg }|α| 2s|α| (2π)|α| 2|α| ‖g‖22

≲ m(t)Cfα (α!)s , where Cf := 2s+2 π max{Cf ,g , Cg }, and we used (β)!(α − β)! ≤ α! and (0.1). (ii) ⇐⇒ (iii). The equivalence is an immediate consequence of Proposition 5.2.4 and the subsequent remarks on the constants, where, for every fixed x ∈ ℝd , we choose h(ξ ) := Vg f (x, ξ )/m(x) and r = s. A natural question is whether we may find window functions satisfying (5.87). To this end, we recall the following characterization of Gelfand–Shilov spaces.

308 | 5 Time–frequency analysis of constant-coefficient partial differential equations Proposition 5.2.8. Let g ∈ 𝒮 (ℝd ). We have g ∈ Srs (ℝd ), with s, r > 0, r + s ≥ 1, if and only if there exist constants A > 0, ϵ > 0 such that 1

󵄨 󵄨󵄨 α |α| s −ϵ|x| r , 󵄨󵄨𝜕 g(x)󵄨󵄨󵄨 ≲ A (α!) e

x ∈ ℝd , α ∈ ℕd .

We have g ∈ Σ11 (ℝd ) if and only if, for every A > 0, ϵ > 0, 󵄨 󵄨󵄨 α |α| −ϵ|x| , 󵄨󵄨𝜕 g(x)󵄨󵄨󵄨 ≲ A α!e

x ∈ ℝd , α ∈ ℕd .

Proof. The first part of the statement is in [236, Proposition 6.1.7]. For the second part, the assumption g ∈ Σ11 (ℝd ) means that for every A > 0, B > 0, 󵄨󵄨 β α 󵄨 |α| |β| 󵄨󵄨x 𝜕 g(x)󵄨󵄨󵄨 ≲ A B α!β!,

x ∈ ℝd , α, β ∈ ℕd .

α

Therefore the function h = A𝜕|α|gα! satisfies (5.79) in Proposition 5.2.4 for every C > 0, and the estimate (5.78) is then satisfied for every ϵ > 0 (see the observations after the proof of Proposition 5.2.4 for the uniformity of the constants). This gives the claim. Hence every g ∈ Srs (ℝd ), with s > 0, 0 < r < 1, s + r ≥ 1, satisfies (5.87) for every submultiplicative weight v. The same holds true if g ∈ Σ11 (ℝd ) and s ≥ 1.

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5.2.2 Almost diagonalization for pseudodifferential operators Some results of [165] can be extended to the case of pseudodifferential operators having (ultra)analytic symbols. The results below cover also the Gevrey-analytic case which was already discussed in [165]. The almost diagonalization of different classes of pseudodifferential operators by using Gabor frames is given in [97, 243, 252, 275]. In view of the kernel theorem in 𝒮 − 𝒮 󸀠 , every linear continuous map from 𝒮 (ℝd ) 󸀠 to 𝒮 (ℝ2d ) can be represented by means of a kernel K ∈ 𝒮 󸀠 (ℝ2d ) and, consequently, in the form of a Weyl pseudodifferential operator, cf. (4.5). Same arguments are valid in Gelfand–Shilov classes with s = r ≥ 1/2 (see [226, 295]). Namely, considering σ ∈ (Sss )󸀠 (ℝ2d ), f , g ∈ Sss (ℝd ), we have Opw (σ) : Sss (ℝd ) → (Sss )󸀠 (ℝd ) continuously, in view of (5.83). The kernel K of Opw (σ) belongs to (Sss )󸀠 (ℝ2d ). In the opposite direction, in view of Theorem 5.2.5, every linear continuous map from Sss (ℝd ) to (Sss )󸀠 (ℝd ) can be represented in the Weyl form Opw (σ), with σ ∈ (Sss )󸀠 (ℝ2d ). The same holds for the couple of spaces Σ11 , (Σ11 )󸀠 . The crucial relation between the action of the Weyl operator Opw (σ) on time– frequency shifts and the short-time Fourier transform of its symbol, contained in [162, Lemma 3.1], can now be extended to Gelfand–Shilov spaces and their dual spaces as follows.

5.2 Gabor representations of evolution operators | 309

Lemma 5.2.9. Consider s ≥ 1/2, g ∈ Sss (ℝd ), Φ = W(g, g). Then, for σ ∈ (Sss )󸀠 (ℝ2d ), 󵄨 󵄨󵄨 z+w 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 , J(w − z))󵄨󵄨󵄨 = 󵄨󵄨󵄨VΦ σ(u, v)󵄨󵄨󵄨 󵄨󵄨⟨Opw (σ)π(z)g, π(w)g⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨VΦ σ( 󵄨󵄨 2 󵄨󵄨

(5.91)

󵄨 󵄨󵄨 1 −1 1 −1 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨VΦ σ(u, v)󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨Opw (σ)π(u − J (v))g, π(u + J (v))g⟩󵄨󵄨󵄨, 󵄨󵄨 2 2 󵄨󵄨

(5.92)

and

where we recall J(z1 , z2 ) = (z2 , −z1 ). Moreover, the same results hold true if we replace the space Sss (ℝd ) with the space Σ11 (ℝd ). Proof. Since Φ = W(g, g) ∈ Sss (ℝ2d ) for g ∈ Sss (ℝd ) the duality ⟨σ, π(u, v)Φ⟩(Sss )󸀠 ×Sss is well defined so that the short-time Fourier transform VΦ σ(u, v) makes sense. The same pattern applies to the case g ∈ Σ11 (ℝd ). Using the weak definition of the Weyl operator in (4.6) and the intertwining property in Proposition 1.3.6 (iv), we can write ⟨Opw (σ)π(z)g, π(w)g⟩ = ⟨σ, W(π(w)g, π(z)g)⟩

= ⟨σ, cMJ(w−z) T w+z W(g, g)⟩ 2

w+z ̄ W(g,g) σ( , J(w − z)), = cV 2

(5.93)

with c being the phase factor (|c| = 1). Setting u = (w+z)/2, v = J(w−z), we obtain (5.91). Furthermore, with w = u + J −1 (v)/2 and z = u − J −1 (v)/2, reading formula (5.93) backwards yields (5.92).

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We now exhibit a characterization of smooth symbols σ ∈ 𝒞 ∞ (ℝ2d ) for Opw (σ) in terms of the continuous Gabor matrix. Theorem 5.2.10. Let s ≥ 1/2 and m ∈ ℳv (ℝ2d ). If 1/2 ≤ s < 1, consider a window function g ∈ Sss (ℝd ); otherwise, if s ≥ 1, assume either g ∈ Σ11 (ℝd ), or g ∈ Sss (ℝd ) and the following growth condition on the weight v: 1/s

v(z) ≲ eϵ|z| ,

z ∈ ℝ2d ,

for every ϵ > 0. Then the following properties are equivalent for σ ∈ 𝒞 ∞ (ℝ2d ): (i) The symbol σ satisfies 󵄨󵄨 α 󵄨 |α| s 󵄨󵄨𝜕 σ(z)󵄨󵄨󵄨 ≲ m(z)C (α!) ,

∀z ∈ ℝ2d , ∀α ∈ ℕ2d .

(5.94)

(ii) There exists ϵ > 0 such that w + z −ϵ|w−z| s 󵄨󵄨 󵄨 )e , 󵄨󵄨⟨Opw (σ)π(z)g, π(w)g⟩󵄨󵄨󵄨 ≲ m( 2 1

∀z, w ∈ ℝ2d .

(5.95)

310 | 5 Time–frequency analysis of constant-coefficient partial differential equations Proof. (i) �⇒ (ii). Proposition 5.2.8 used to the window Φ = W(g, g) in Lemma 5.2.9, which lives in the space Sss (ℝ2d ) since g ∈ Sss (ℝd ) by (5.83), and the assumptions on v imply that Φ satisfies the assumptions of Theorem 5.2.7. Hence, using the equivalence (5.88) ⇐⇒ (5.90), the assumption (5.94) is equivalent to the following decay estimate of the corresponding short-time Fourier transform: 1

󵄨 󵄨󵄨 −ϵ|v| s , 󵄨󵄨VΦ σ(u, v)󵄨󵄨󵄨 ≲ m(u)e

u, v ∈ ℝ2d ,

for a suitable ϵ > 0, which, combined with (5.91), yields 󵄨󵄨 󵄨󵄨 z+w w + z −ϵ|w−z| s1 w + z −ϵ|J(w−z)| s1 󵄨󵄨 󵄨 , J(w − z))󵄨󵄨󵄨 ≲ m( )e = m( )e , 󵄨󵄨VΦ σ( 󵄨󵄨 󵄨󵄨 2 2 2 that is, (ii) holds. (ii) �⇒ (i). Relation (5.92) and the decay assumption (5.95) give 󵄨󵄨 󵄨 1 −1 1 −1 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨VΦ σ(u, v)󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨Opw (σ)π(u − J (v))g, π(u + J (v))g⟩󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 2 ≲ m(u)e−ϵ|J

−1

1

(v)| s

1 s

= m(u)e−ϵ|v| ,

and, using the equivalence (5.88) ⇐⇒ (5.90), we obtain the claim. Of course, from (5.95) we deduce the discrete Gabor matrix decay in (5.98) below. The converse requires some preliminaries. We apply a result obtained by Gröchenig and Lyubarskii in [164]. They find sufficient conditions on the lattice Λ = Aℤ2 , A ∈ GL(2, ℝ), such that g = ∑nk=0 ck Hk , with Hk Hermite function, forms a so-called Gabor (super)frame 𝒢 (g, Λ). Besides, they prove the existence of dual windows γ that belong 1/2 to the space S1/2 (ℝ) (cf. [164, Lemma 4.4]). This theory transfers to the d-dimensional

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1/2 (ℝd ) of windows as above, case by taking a tensor product g = g1 ⊗ ⋅ ⋅ ⋅ ⊗ gd ∈ S1/2 which defines a Gabor frame on the lattice Λ1 × ⋅ ⋅ ⋅ × Λd and possesses a dual window 1/2 γ = γ1 ⊗⋅ ⋅ ⋅⊗γd in the same space ∈ S1/2 (ℝd ). Let us simply call Gabor superframe 𝒢 (g, Λ)

for L2 (ℝd ) a Gabor frame with the above properties. Gabor superframes are the key for the discretization of the kernel in (5.95). First, we need the preliminary result below, which reflects the algebra property of Gelfand–Shilov spaces. For s ≥ 1/2, ϵ > 0, we define the following weight functions: 1 s

ws,ϵ (z) := e−ϵ|z| ,

z ∈ ℝ2d .

(5.96)

Lemma 5.2.11. Let Λ ⊂ ℝ2d be a lattice of ℝ2d . Then the sampling {ws,ϵ (λ)}λ∈Λ of (5.96) (defined on ℝ2d ) satisfies (ws,ϵ ∗ ws,ϵ )(λ) := ∑ ws,ϵ (λ − ν)ws,ϵ (ν) ≲ ws,ϵ2−1/s (λ). ν∈Λ

(5.97)

5.2 Gabor representations of evolution operators | 311

Proof. We use the arguments of [160, Lemma 11.1.1(c)]. For λ ∈ Λ, we divide the lattice Λ into the subsets Nλ = {ν ∈ Λ : |λ − ν| ≤ |λ|/2} and Nλc = {ν ∈ Λ : |λ − ν| > |λ|/2}. For ν ∈ Nλ , |ν| ≥ |λ|/2 and |ν|1/s ≥ (|λ|/2)1/s , so − s1

(ws,ϵ ∗ ws,ϵ )(λ) ≤ e−(ϵ2

1

)|λ| s

1 s

1 s

− s1

( ∑ e−ϵ|λ−ν| + ∑ e−ϵ|ν| ) ≲ e−(ϵ2 ν∈Nλ

1

)|λ| s

ν∈Nλc

.

This concludes the proof. Theorem 5.2.12. Let 𝒢 (g, Λ) be a Gabor superframe for L2 (ℝd ). Consider m ∈ ℳv (ℝ2d ), s ≥ 1/2, and a symbol σ ∈ 𝒞 ∞ (ℝ2d ). Then the following properties are equivalent: (i) There exists ϵ > 0 such that the estimate (5.95) holds. (ii) There exists ϵ > 0 such that λ + μ −ϵ|λ−μ| s 󵄨󵄨 󵄨 , )e 󵄨󵄨⟨Opw (σ)π(μ)g, π(λ)g⟩󵄨󵄨󵄨 ≲ m( 2 1

∀λ, μ ∈ Λ.

(5.98)

Proof. It remains to show that (ii) �⇒ (i). The arguments of [162, Theorem 3.2] can be adapted to this proof by using a Gabor superframe 𝒢 (g, Λ), with a dual window 1/2 γ ∈ S1/2 (ℝd ). Let 𝒬 be a symmetric relatively compact fundamental domain of the lattice Λ ⊂ ℝ2d . Given w, z ∈ ℝ2d , we can write them uniquely as w = λ + u, z = μ + u󸀠 , for λ, μ ∈ Λ and u, u󸀠 ∈ 𝒬. Using the Gabor reproducing formula for the time–frequency shift 1/2 π(u)g ∈ S1/2 (ℝd ), we can write π(u)g = ∑ ⟨π(u)g, π(ν)γ⟩π(ν)g. ν∈Λ

Inserting the prior expansions in the assumption (5.98) gives 󵄨󵄨 󵄨 󸀠 󵄨󵄨⟨Opw (σ)π(μ + u )g, π(λ + u)g⟩󵄨󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨 ≤ ∑ 󵄨󵄨󵄨⟨Opw (σ)π(μ + ν󸀠 )g, π(λ + ν)g⟩󵄨󵄨󵄨󵄨󵄨󵄨⟨π(u󸀠 )g, π(ν󸀠 )γ⟩󵄨󵄨󵄨󵄨󵄨󵄨⟨π(u)g, π(ν)γ⟩󵄨󵄨󵄨

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ν,ν󸀠 ∈Λ

≲ ∑ m( ν,ν󸀠 ∈Λ

λ + μ + ν + ν󸀠 −ϵ|λ+ν−μ−ν󸀠 | s1 󵄨󵄨 󵄨 󸀠 󸀠 󵄨󵄨 )e 󵄨󵄨Vγ g(ν − u )󵄨󵄨󵄨󵄨󵄨󵄨Vγ g(ν − u)󵄨󵄨󵄨. 2

(5.99)

1/2 1/2 Since the window functions g, γ are both in S1/2 (ℝd ), the STFT Vγ g is in S1/2 (ℝ2d ) in 2

view of (5.82). Thus there exists h > 0 such that |Vγ g(z)| ≲ e−h|z| , for every z ∈ ℝ2d . In particular, being 𝒬 relatively compact and u ∈ 𝒬, 2 2 󵄨󵄨 󵄨 −h|ν−u|2 ≤ sup e−h|ν−u| ≲ e−h|ν| . 󵄨󵄨Vγ g(ν − u)󵄨󵄨󵄨 ≲ e

u∈Q

The assumption m ∈ ℳv (ℝ2d ) yields m(

λ+μ λ + μ + ν + ν󸀠 ν ν󸀠 ) ≲ m( )v( )v( ) 2 2 2 2

312 | 5 Time–frequency analysis of constant-coefficient partial differential equations and, for every 0 < h̃ < h, 2 ̃ 2 ν v( )e−h|ν| ≲ e−h|ν| , 2

∀ν ∈ Λ.

Inserting these estimates in (5.99) yields 󵄨 󵄨󵄨 󸀠 󵄨󵄨⟨Opw (σ)π(μ + u )g, π(λ + u)g⟩󵄨󵄨󵄨 󸀠 s1 2 󸀠 2 λ+μ ν ν󸀠 ≲ m( ) ∑ e−ϵ|λ+ν−μ−ν | v( )e−h|ν| v( )e−h|ν | 2 2 2 ν,ν󸀠 ∈Λ ≲ m( ≤ m(

󸀠 s1 ̃ 2 ̃ 󸀠2 λ+μ ) ∑ e−ϵ|λ+ν−μ−ν | e−h|ν| e−h|ν | 2 ν,ν󸀠 ∈Λ

1 󸀠 s1 󸀠 s1 λ+μ s ) ∑ e−b(|λ+ν−μ−ν | +|ν| +|ν | ) , 2 ν,ν󸀠 ∈Λ

(5.100)

with b = ϵ for s > 1/2 whereas b = min{ϵ, h}̃ for s = 1/2 (there may be a loss of decay). We observe that the row (5.100) can be rewritten as m(

λ+μ )(ws,b ∗ ws,b ∗ ws,b )(λ − μ), 2

1/s

where ws,b (λ) = e−b|λ| . Now we apply Lemma 5.2.11 twice and obtain (5.100) ≲ m(

1 λ + μ −ϵ|λ−μ| s ̃ )e 2

(5.101)

with ϵ̃ = b2−2/s . If w, z ∈ ℝ2d and w = λ + u, z = μ + u󸀠 , λ, μ ∈ Λ, u, u󸀠 ∈ 𝒬, then λ − μ = w − z + u󸀠 − u and u󸀠 − u ∈ 𝒬 − 𝒬, which is a relatively compact set, thus ̃

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1 s

1 s

1 s

e−ϵ|λ−μ| ≲ sup e−ϵ|w−z+u| ≲ e−ϵ|w−z| . ̃

̃

u∈𝒬−𝒬

(5.102)

Finally, the v-moderateness of the weight m ∈ ℳv (ℝ2d ), together with the fact that v is continuous and the set 𝒬 + 𝒬 is relatively compact, let us write m(

λ+ν w + z u + u󸀠 w+z u + u󸀠 ) = m( − ) ≲ m( )v(− ) 2 2 2 2 2 w+z u w+z ≲ m( ) sup v( ) ≲ m( ). 2 2 2 u∈𝒬+𝒬

(5.103)

Combining estimates (5.102) and (5.103) with (5.101), we obtain (5.95), with the parameter ϵ = ϵ̃ which appears in (5.101).

5.2 Gabor representations of evolution operators | 313

5.2.3 Sparsity of the Gabor matrix The operators Opw (σ) which satisfy Theorem 5.2.10, say, with m = v = 1, enjoy a fundamental sparsity property. Let 𝒢 (g, Λ) be a Gabor frame for L2 (ℝd ), with g ∈ Sss (ℝd ), s ≥ 1/2. Then, as we saw, 1

󵄨󵄨 󵄨 −ϵ|λ−μ| s , 󵄨󵄨⟨Opw (σ)π(μ)g, π(λ)g⟩󵄨󵄨󵄨 ≤ Ce

∀λ, μ ∈ Λ,

(5.104)

with suitable constants C > 0, ϵ > 0. This gives at once an exponential-type sparsity, in the sense specified by Proposition 5.2.15 (cf. [41, 173] for the more standard notion of superpolynomial sparsity). We need first preliminary results on sequences. Lemma 5.2.13. Let a = (an )n∈ℕ+ be a nonincreasing sequence with an ≥ 0, for every n ∈ ℕ+ . If ‖a‖ℓ1 < ∞, then an ≤

‖a‖ℓ1 , n

n ∈ ℕ+ .

(5.105)

Proof. Since an ≥ 0 for every n ∈ ℕ+ , for any N ∈ ℕ, we can write N

‖a‖ℓ1 ≥ ∑ an ≥ NaN , n=1

and the claim follows. Corollary 5.2.14. If a = (an )n∈ℕ+ ∈ ℓp , for 0 < p < ∞, then

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|an | ≤

‖a‖ℓp 1

np

n ∈ ℕ+ .

,

(5.106)

∞ p p Proof. By assumption, ‖a‖pℓp = ∑∞ n=1 |an | < ∞. Since the series ∑n=1 |an | is absolutely convergent, it is also unconditionally convergent, and we can rearrange the entries |an |p of the sequence (|an |p )n∈ℕ+ in a decreasing order. The claim then follows by applying Lemma 5.2.13.

Proposition 5.2.15. Let the Gabor matrix ⟨Opw (σ)π(μ)g, π(λ)g⟩ satisfy (5.104). Then it is sparse in the following sense. Let a be any column or row of the matrix, and let |a|n be the nth largest entry of the sequence a. Then |a|n satisfies |a|n ≤ Ce−ϵn

1 2ds

,

n ∈ ℕ,

for some constants C > 0, ϵ > 0. Proof. By a discrete analog of Proposition 5.2.4, it suffices to prove that nα |a|n ≤ C α+1 (α!)2ds ,

α ∈ ℕ.

314 | 5 Time–frequency analysis of constant-coefficient partial differential equations On the other hand, applying the estimate in (5.106) to the nth largest entry |a|n , we have 1

n p ⋅ |a|n ≤ ‖a‖ℓp , for every 0 < p < ∞. Hence by (5.104) and setting p = 1/α, we obtain α

n |a|n ≤ ( ∑ e

1 p

1

−ϵp|λ−μ| s

) = (∑ e

1 p

1

−ϵp|λ| s

) .

λ∈Λ

λ∈Λ

Let 𝒬 be a fundamental domain of the lattice Λ. Then if x ∈ λ + 𝒬, λ ∈ Λ, we have |x| ≤ |λ| + C0 , therefore |x|1/s ≤ C1 (|λ|1/s + 1). Hence ∑e

λ∈Λ

1

−ϵp|λ| s

≤ C2 ∫ e

1

−ϵp|x| s

+∞

1 s

dx = ∫ dσ ∫ e−ϵpρ ρ2d−1 dρ

ℝ2d

�2d−1

0

+∞

=

C sΓ(2ds) C3 s C4 ∫ e−t t 2ds−1 dt = 3 2ds = 2ds (ϵp)2ds (ϵp) p 0

Finally, by Stirling’s formula, nα |a|n ≤

C41/p p

2ds p

≤ C5α+1 (α!)2ds .

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5.2.4 Boundedness of pseudodifferential operators on modulation and Gelfand–Shilov spaces Although not necessary for our numerical applications, we list some continuity results on modulation and Gelfand–Shilov spaces which follow easily from Theorem 5.2.10. Proposition 5.2.16. Let s ≥ 1/2 and consider a symbol σ ∈ 𝒞 ∞ (ℝ2d ) satisfying the estimates 󵄨󵄨 α 󵄨 |α| s 󵄨󵄨𝜕 σ(z)󵄨󵄨󵄨 ≲ C (α!) ,

(5.107)

for some C > 0. Let m ∈ ℳv (ℝ2d ) and, if s ≥ 1, assume the weight v satisfies 1/s

v(z) ≲ eϵ|z| ,

z ∈ ℝ2d

for every ϵ > 0. Then the Weyl operator Opw (σ) extends to a bounded operator on p,q Mm (ℝd ).

5.2 Gabor representations of evolution operators | 315

Proof. Let g ∈ Σ11 (ℝd ) with ‖g‖L2 = 1. From the inversion formula (1.55), Vg (Opw (σ)f )(u) = ∫ ⟨Opw (σ)π(z)g, π(u)g⟩Vg f (z) dz. ℝ2d

The desired result thus follows if we can prove that the map M(σ) defined by M(σ)G(u) = ∫ ⟨Opw (σ)π(z)g, π(u)g⟩G(z) dz

(5.108)

ℝ2d 2d p,q 2d is continuous from Lp,q m (ℝ ) into Lm (ℝ ). The characterization of Theorem 5.2.10 assures the existence of an ϵ > 0 such that

󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Vg (Opw (σ)f )󵄨󵄨󵄨 = 󵄨󵄨󵄨M(σ)Vg (f )󵄨󵄨󵄨 ≲ (ws,ϵ ∗ |Vg f |)(u),

u ∈ ℝ2d ,

where the weight function ws,ϵ on ℝ2d is defined in (5.96). The desired conclusion then 1 p,q 1 follows from the relation Lp,q m ∗ Lv �→ Lm , for ws,ϵ ∈ Lv by the growth assumption on v. Proposition 5.2.17. Let s ≥ 1/2, and consider a symbol σ ∈ 𝒞 ∞ (ℝ2d ) that satisfies (5.107). Then the Weyl operator Opw (σ) is bounded on Sss (ℝd ). 1/2 Proof. Fix a window function g ∈ S1/2 (ℝd ). For f ∈ Sss (ℝd ), we have Vg f ∈ Sss (ℝ2d ) in 1/s

view of (5.82). Hence there exists a constant h > 0 such that |Vg f |(z) ≲ e−h|z| , for every z ∈ ℝ2d . Thus, taking ϵ̃ = min{ϵ, h} and using a continuous version of Lemma 5.2.11, for every u ∈ ℝ2d , 󵄨󵄨 󵄨 󵄨󵄨Vg (Opw (σ)f )(u)󵄨󵄨󵄨 ≲ (ws,ϵ ∗ ws,h )(u) ≲ (ws,ϵ̃ ∗ ws,ϵ̃ )(u) ≲ ws,ϵ2̃ −1/s (u). Using (5.85), we obtain the claim.

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The reader interested in further boundedness results for pseudodifferential operators on Gelfand–Shilov spaces can look at the paper by J. Toft [289]. 5.2.5 Evolution equations In this section we apply the almost diagonalization result to the propagators of Cauchy problems for constant coefficient hyperbolic and parabolic-type equations. We consider operators m

P(𝜕t , Dx ) = 𝜕tm + ∑ ak (Dx )𝜕tm−k , k=1

t ∈ ℝ, x ∈ ℝd ,

(5.109)

where ak (ξ ), 1 ≤ k ≤ m, are polynomials. They may be nonhomogeneous, and their 1 degree may be arbitrary. Here we set Dxj = 2πi 𝜕xj , j = 1, . . . , d.

316 | 5 Time–frequency analysis of constant-coefficient partial differential equations We are interested in the forward Cauchy problem (t, x) ∈ ℝ+ × ℝd , {P(𝜕t , Dx )u = 0, { k {𝜕t u(0, x) = uk (x), 0 ≤ k ≤ m − 1, where uk ∈ 𝒮 (ℝd ), 0 ≤ k ≤ m − 1. Suppose that the forward Hadamard–Petrowsky condition is satisfied: There exists a constant C > 0 such that (τ, ξ ) ∈ ℂ × ℝd ,

P(iτ, ξ ) = 0 �⇒ Im τ ≥ −C.

(5.110)

This is a sufficient and necessary condition for the above Cauchy problem with Schwartz data to be well posed [247, Section 3.10]. By taking the Fourier transform with respect to x and using classical results for the fundamental solution to ordinary differential operators [261, pp. 126–127], one sees that the solution is then given by m−1

m−k−1

k=0

j=1

u(t, x) = ∑ 𝜕tk E(t, ⋅) ∗ (um−1−k + ∑ aj (Dx )um−k−1−j ). Here E(t, x) = ℱξ−1→x σ(t, ξ ), where σ(t, ξ ) is the unique solution to m

(𝜕tm + ∑ ak (ξ )𝜕tm−k )σ(t, ξ ) = δ(t) k=1

supported in [0, +∞) × ℝd . The distribution E(t, x) is therefore the fundamental solution of P supported in [0, +∞) × ℝd . The study of the Cauchy problem is therefore reduced to that of the Fourier multiplier

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Opw (σ)f = σ(t, Dx )f = ℱ −1 σ(t, ⋅)ℱ f = E(t, ⋅) ∗ f .

(5.111)

(For Fourier multipliers the Weyl and Kohn–Nirenberg quantizations give the same operator.) Example 5.2.18. Here are some classical operators and symbols σ(t, ξ ) of the corresponding propagators for t ≥ 0 (σ(t, ξ ) = 0 for t < 0): sin(2π|ξ |t) . 2π|ξ |

–

Wave operator 𝜕t2 − Δ; σ(t, ξ ) =

–

Klein–Gordon operator 𝜕t2 − Δ + m2 , with m > 0; σ(t, ξ ) =

–

Heat operator 𝜕t − Δ; σ(t, ξ ) = e−4π

2

|ξ |2 t

.

sin(t √4π 2 |ξ |2 +m2 ) √4π 2 |ξ |2 +m2

.

We now introduce an assumption under which the multiplier σ(t, Dx ) falls in the class of pseudodifferential operators considered above.

5.2 Gabor representations of evolution operators | 317

Assume that there are constants C > 0, r ≥ 1 such that (τ, ζ ) ∈ ℂ × ℂd ,

r

P(iτ, ζ ) = 0 �⇒ Im τ ≥ −C(1 + | Im ζ |) .

(5.112)

Clearly, this condition is stronger than the forward Hadamard–Petrowsky condition. Theorem 5.2.19. Assume P satisfies (5.112) for some C > 0, r ≥ 1. Then the symbol σ(t, ξ ) of the corresponding propagator σ(t, Dx ) in (5.111) satisfies the following estimates: 󵄨󵄨 α 󵄨 (t+1)|α|+t (α!)μ , 󵄨󵄨𝜕ξ σ(t, ξ )󵄨󵄨󵄨 ≤ C

ξ ∈ ℝd , t ≥ 0, α ∈ ℕd ,

(5.113)

with μ = 1 − 1/r, for a new constant C > 0. Proof. It is well known (see, e. g., [48, Proposition 1.3.2, Lemma 1.3.3], [247, Section 3.10]) that σ(t, ξ ) extends to an entire analytic function in the second variable, and that the estimates (5.112) imply the bound 󵄨󵄨 󵄨 Ct(1+| Im ζ |)r , 󵄨󵄨σ(t, ζ )󵄨󵄨󵄨 ≤ e

ζ ∈ ℂd .

(5.114)

Now, given ξ ∈ ℝd , we consider the polydisk B(ξ , R) = ∏dj=1 Bj (ξj , R) = {ζ ∈ ℂd :

|ζj − ξj | ≤ R, 1 ≤ j ≤ d}, with R = (1 + |α|)1/r , α ∈ ℕd . Observe that (5.114) implies r √ 󵄨 󵄨 sup 󵄨󵄨󵄨σ(t, ζ )󵄨󵄨󵄨 ≤ eCt(1+ dR) .

(5.115)

ζ ∈B(ξ ,R)

The Cauchy’s generalized integral formula 𝜕ξα σ(t, ξ ) =

α! ∫⋅⋅⋅ (2πi)d

∫ 𝜕B1 (ξ1 ,R)×⋅⋅⋅×𝜕Bd (ξd ,R)

σ(t, ζ1 , . . . , ζd ) dζ ⋅ ⋅ ⋅ dζd (ζ1 − ξ1 )α1 +1 ⋅ ⋅ ⋅ (ζd − ξd )αd +1 1

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and estimate (5.115) yield √dR)r

Ct(1+ 󵄨󵄨 α 󵄨 α!e 󵄨󵄨𝜕ξ σ(t, ξ )󵄨󵄨󵄨 ≤ R|α|

≤ C1t(|α|+1)

α! (1 + |α|)|α|/r

.

(5.116)

Using Stirling formula and 1/r = 1 − μ gives C2|α| 1 ≤ , (1 + |α|)|α|/r (α!)1−μ which, combined with (5.116), provides the desired majorization (5.113). Observe that the assumption r ≥ 1 in the above theorem implies 0 ≤ μ < 1. Combining Theorems 5.2.19 and 5.2.10, we obtain at once our main application.

318 | 5 Time–frequency analysis of constant-coefficient partial differential equations Theorem 5.2.20. Assume P satisfies (5.112) for some C > 0, r ≥ 1, and set s = max{1/2, 1 − 1/r}. If g ∈ Sss (ℝd ) then σ(t, Dx ) in (5.111) satisfies 1

󵄨󵄨 󵄨 −ϵ|w−z| s , 󵄨󵄨⟨σ(t, Dx )π(z)g, π(w)g⟩󵄨󵄨󵄨 ≤ Ce

∀z, w ∈ ℝ2d ,

(5.117)

for some ϵ > 0 and for a new constant C > 0. The constants ϵ and C are uniform when t lies in bounded subsets of [0, +∞). Notice that in the above theorem 1/2 ≤ s < 1, so that we always obtain superexponential decay, i. e., 1/s > 1 in (5.117). We now show that, if P(𝜕t , Dx ) is any hyperbolic operator, then (5.112) is satisfied with r = 1, and hence the above theorem applies with Gaussian decay (s = 1/2 1/2 in (5.117)), for windows g ∈ S1/2 (ℝd ). We recall that the operator P(𝜕t , Dx ) is called hyperbolic with respect to t if the direction N = (1, 0, . . . , 0) ∈ ℝ × ℝd is noncharacteristic for P (i. e., its principal symbol – the highest order homogeneous part in the symbol – does not vanish at N) and P satisfies the forward Hadamard–Petrowsky condition (5.110). It follows then that the operators aj (Dx ) in (5.109) must have degree ≤ j and P has order m. The wave and Klein–Gordon operators are, of course, the most important examples of hyperbolic operators. We emphasize, however, that P is not required to be strictly hyperbolic, namely the roots of the principal symbol are allowed to coincide. For example, the operator P = 𝜕t2 − ∑dj,k=1 aj,k 𝜕xj 𝜕xk is hyperbolic if the matrix aj,k is real, symmetric, and positive semidefinite. Proposition 5.2.21. Assume P(𝜕t , Dx ) is hyperbolic with respect to t. Then the condition (5.112) is satisfied with r = 1 for some C > 0, and hence 󵄨󵄨 󵄨 −ϵ|w−z|2 , 󵄨󵄨⟨σ(t, Dx )π(z)g, π(w)g⟩󵄨󵄨󵄨 ≤ Ce

∀z, w ∈ ℝ2d ,

(5.118)

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1/2 (ℝd ), for some ϵ > 0 and for a new constant C > 0. if g ∈ S1/2

Proof. Denote by Pm the principal symbol of P and Γ(P, N) the component of N in {θ ∈ ℝ × ℝd : Pm (θ) ≠ 0}. It follows from the hyperbolicity assumption (see, e. g., [193, Proposition 12.4.4]) that the symbol Q(τ, ξ ) = P(iτ, ξ ) of P(𝜕t , Dx ) satisfies Q(Ξ + τN + σθ) ≠ 0

if Ξ ∈ ℝd+1 , Im τ < −C, Im σ ≤ 0, θ ∈ Γ(P, N),

for the same constant C which appears in (5.110). We then deduce that (τ, ζ ) ∈ ℂ × ℂd ,

Q(τ, ζ ) = P(iτ, ζ ) = 0 �⇒ (Im τ, Im ζ ) ∈ ̸ −CN − Γ(P, N).

Indeed, if (τ, ζ ) = Ξ + i(−CN − θ), with θ ∈ Γ(P, N), Ξ ∈ ℝd+1 , then θ − ϵN ∈ Γ(P, N) is ϵ is small enough because Γ(P, N) is open, and then P(τ, ζ ) = P(Ξ − i(C + ϵ)N − i(θ − ϵN)) ≠ 0.

5.2 Gabor representations of evolution operators | 319

Hence −(C + Im τ, Im ζ ) ∈ ̸ Γ(P, N), and since the cone Γ(P, N) is open, this implies −(C + Im τ, Im ζ ) ⋅ N −(C + Im τ) = ≤ C1 , |(C + Im τ, Im ζ )| |(C + Im τ, Im ζ )| for some constant C1 < 1, which gives Im τ ≥ −C −

C1 | Im ζ |. 1 − C1

Example 5.2.22. Consider the wave operator P = 𝜕t2 −Δ in ℝ×ℝd , with symbol σ(t, ξ ) = sin(2π|ξ |t) . Using some further information on the fundamental solution, we can give a 2π|ξ | 2

very precise result on the matrix decay in dimension d ≤ 3. We take g(x) = 2d/4 e−πx 1/2 as a window function, which is allowed because g ∈ S1/2 (ℝd ), and moreover ‖g‖L2 = 1 (Gaussian functions minimize the Heisenberg uncertainty so that they are, generally speaking, a natural choice for wave-packet decompositions). We claim that 󵄨󵄨 󵄨 − π [|ξ 󸀠 −ξ |2 +(|x󸀠 −x|−t)2+ ] , 󵄨󵄨⟨σ(t, Dx )Mξ Tx g, Mξ 󸀠 Tx󸀠 g⟩󵄨󵄨󵄨 ≤ te 2

x, x 󸀠 , ξ , ξ 󸀠 ∈ ℝd , d ≤ 3,

where (⋅)+ denotes the positive part. This agrees with Proposition 5.2.21, with the constants made explicit. In fact, in dimension d ≤ 3, we know that σ(t, Dx )f (x) = ∫ f (x − y) dμt (y)

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where μt is a positive Borel measure, supported in the ball Bt (0), with total mass = t (see, e. g., [247, Chapter 4]). To be precise, we have dμt (y) = (1/2)χ[−t,t] dy in dimen−1 sion 1, dμt (y) = (2π)−1 (t 2 − |y|2 )−1/2 + dy in dimension 2, and dμt = (4πt) dσ𝜕Bt (0) in dimension 3 (surface measure). Hence, using Ty Mξ = e−2πiy⋅ξ Mξ Ty , ⟨σ(t, Dx )Mξ Tx g, Mξ 󸀠 Tx󸀠 g⟩ = ∫ e−2πiy⋅ξ ⟨Mξ Tx+y g, Mξ 󸀠 Tx󸀠 g⟩ dμt (y). An explicit computation shows that 󵄨󵄨 󵄨 − π |w−z|2 , 󵄨󵄨⟨π(z)g, π(w)g⟩󵄨󵄨󵄨 = e 2

z, w ∈ ℝ2d ,

so that 󵄨󵄨 󵄨 − π [|ξ 󸀠 −ξ |2 +|x 󸀠 −x−y|2 ] dμt (y) 󵄨󵄨⟨σ(t, Dx )Mξ Tx g, Mξ 󸀠 Tx󸀠 g⟩󵄨󵄨󵄨 ≤ ∫ e 2 π

≤ te− 2 [|ξ which gives the claim.

󸀠

−ξ |2 +(|x 󸀠 −x|−t)2+ ]

,

320 | 5 Time–frequency analysis of constant-coefficient partial differential equations 2

Consider now the Gabor frame 𝒢 (g, Λ), with g(x) = 2d/4 e−πx , lattice Λ = ℤd × (1/2)ℤd ([160, Theorem 7.5.3]), and the Gabor matrix Tm󸀠 ,n󸀠 ,m,n = ⟨σ(t, Dx )Mn Tm g, Mn󸀠 Tm󸀠 g⟩, We have

π

󸀠

2

(m, n), (m󸀠 , n󸀠 ) ∈ Λ. 2

󸀠

|Tm󸀠 ,n󸀠 ,m,n | ≤ T̃ m󸀠 ,n󸀠 ,m,n := te− 2 [|n −n| +(|m −m|−t)+ ] ,

(m, n), (m󸀠 , n󸀠 ) ∈ Λ,

with d ≤ 3. Figure 5.5 shows the magnitude of the entries, rearranged in decreasing order, of a generic column, e. g., T̃ m󸀠 ,n󸀠 ,0,0 (obtained for m = n = 0), at time t = 0.75, in dimension d = 2. In fact, the same figure applies to all columns, for T̃ m󸀠 ,n󸀠 ,m,n = T̃ m󸀠 −m,n󸀠 −n,0,0 . This figure should be compared with [43, Figure 15], where a similar investigation was carried out for the curvelet matrix of the wave propagator on the unit square (d = 2) with periodic boundary conditions. It turns out that the Gabor decay is even better, despite the fact that we consider here the wave operator in the whole ℝ2 .

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Figure 5.5: Decay of a generic column of the Gabor matrix for time t = 0.75, with window g(x) = √2e

−πx

2

2

sin(2π|D|t) 2π|D| 2

in dimension d = 2 and at

and lattice ℤ × (1/2)ℤ .

We now present a class of examples of operators which satisfy Theorem 5.2.20 and are not hyperbolic. Example 5.2.23. Consider the operator P(𝜕t , Dx ) = 𝜕t + (−Δ)k ,

(5.119)

with k ≥ 1 integer. When k = 1, we have the heat operator. Its symbol is the polynomial k

P(iτ, ζ ) = iτ + (4π 2 ζ 2 ) . We claim that it satisfies (5.112) with r = 2k.

5.2 Gabor representations of evolution operators | 321

If τ ∈ ℂ, ξ , η ∈ ℝd , and P(iτ, ξ + iη) = 0 then k

Im τ = (2π)2k Re(|ξ |2 + 2iξ ⋅ η − |η|2 ) = (2π)2k |ξ |2k + Q(ξ , η), where Q(ξ , η) is a homogeneous polynomial of degree 2k, with Q(ξ , 0) = 0. Hence, for every ϵ > 0 and some constant Cϵ > 0, 󵄨 󵄨󵄨 󵄨󵄨Q(ξ , η)󵄨󵄨󵄨 ≲

∑ j+l=2k,j≥0,l≥1

where we applied the inequality ab ≤

|ξ |j |η|l ≤ ϵ|ξ |2k + Cϵ |η|2k ,

(ϵa)p p

+

(b/ϵ)q q

(with a = |ξ |j , b = |η|l , p = 2k/j and

q = 2k/l) to the terms of the sum with j ≥ 1. Taking ϵ = (2π)2k /2, we get Im τ ≥

(2π)2k 2k |ξ | − C|η|2k ≥ −C|η|2k , 2

which proves the claim. Hence, for the operator P in (5.119), Theorem 5.2.20 applies with r = 2k and s = 1 − 1/(2k). Remark 5.2.24. Theorems 5.2.10 and 5.2.12 allow also some applications to equations with variable coefficients. As an elementary example consider the transport equation d d {𝜕t u − i(∑j=1 aj Dxj + ∑j=1 bj xj )u = 0, { {u(0, x) = u0 (x),

with aj , bj ∈ ℝ. The fundamental solution of the problem has symbol d

σ(t, x, ξ ) = exp[it ∑(aj ξj + bj xj )] j=1

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satisfying (5.94) with m = 1, s = 0, and the preceding arguments apply. More generally, we conjecture that pseudodifferential problems of the form 𝜕t − iaw (x, D)u = 0,

u(0, x) = u0 (x),

where a(x, ξ ) is real-valued and dx,ξ a(x, ξ ), as well as higher order derivatives, are bounded in ℝ2d , can be treated similarly. 5.2.6 Some numerics for the heat equation From Example 5.2.23 and Theorem 5.2.20, we infer that the heat propagator σ(t, Dx ) = 2 2 e−4π t|D| satisfies the estimate 2 󵄨󵄨 −4π 2 t|D|2 󵄨 π(z)g, π(w)g⟩󵄨󵄨󵄨 ≤ Ce−ϵ|w−z| , 󵄨󵄨⟨e

∀z, w ∈ ℝ2d ,

322 | 5 Time–frequency analysis of constant-coefficient partial differential equations 1/2 for some ϵ > 0, C > 0, if g ∈ S1/2 (ℝd ). Namely, the same decay as in the case of hyperbolic equations occurs. In the following figures we summarized some numeric information about its Gabor discretization. Namely, Figure 5.6 shows the decay of a column of the Gabor matrix for the heat propagator, i. e.,

Tm󸀠 ,n󸀠 ,0,0 = ⟨e−4π

2

t|D|2

g, Mn󸀠 Tm󸀠 g⟩

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for a Gaussian window, at different instant times t and in dimension d = 2. For t = 0, we get the identity operator, and therefore its matrix decay is optimal, compatibly with the uncertainty principle. As one sees from the other figures, the decay remains extremely good as time evolves. Also for t = 0.75, the decay matches that of the wave equation displayed in Figure 5.5, in spite that we no longer have here the finite speed propagation property.

Figure 5.6: Decay of the column corresponding to m = n = 0, of the Gabor matrix for the heat propa2 2 2 gator e−4π t|D| in dimension d = 2 at different instant times, with window g(x) = √2e−πx and lattice

ℤ2 × (1/2)ℤ2 .

5.2 Gabor representations of evolution operators | 323

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Figure 5.7 instead shows, in dimension d = 1, the magnitude of the STFT of the solu2 2 2 tion u(t, ⋅) = e−4π t|D| u0 of the heat equation, with initial datum u0 (x) = √2e−πx and 2 window g(x) = √2e−πx . The dissipative effect of the equation in the spatial domain is evident. A similar analysis for the Schrödinger equation will be carried on in the next chapter.

Figure 5.7: Magnitude of the STFT of the solution u = e−4π

2

t|D|2

u0 in dimension d = 1 and at different 2

instant times, with initial datum and window u0 (x) = g(x) = √2e−πx .

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6 Fourier integral operators and applications to Schrödinger equations In this chapter we define the Fourier integral operators, the object of our study, exhibiting their main features, mainly contained in [79]; see also [80, 88, 291]. Then we show that Gabor frames provide optimally sparse representations of such operators. Numerical examples for the Schrödinger case demonstrate the fast computation of these representations (cf. [77]). We address to the introduction of each section in the sequel for motivations and bibliographical remarks, and try to give here a flavor of the contents. The general idea is the same as in the second part of the preceding chapter, namely we express a linear continuous map T : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) in terms of the infinite Gabor matrix Tμ,λ , λ, μ ∈ Λ, defined by Tμ,λ = ⟨Tgλ , gμ ⟩, with Gabor atoms gλ = π(λ)g, see (0.15) for the notation and fix Λ = αℤd × βℤd , α, β > 0 such that 𝒢 (g, α, β) is a frame. The operator T is understood to provide the solution of a well-posed problem for a PDE, and the window g is assumed to belong to a suitable function space. In the preceding chapter we proved for a class of problems, and very smooth g, that the Gabor matrix is almost diagonal, with exponential decay of Tμ,λ for ⟨λ − μ⟩ → ∞. This provided exponential sparsity of the matrix with applications to numerics. In the present chapter we shall content ourselves with an algebraic off-diagonal decay, namely an estimate of the type |Tμ,λ | ≤ C⟨λ − μ⟩−s , for some C > 0, s ≥ 0. On the other hand, the analysis will be much more general, since the diagonal will be replaced by other submanifolds of Λ × Λ. In fact, we shall consider the graph in Λ × Λ of a symplectic map χ : ℝ2d → ℝ2d . Assumptions on χ will be those natural in the global setting, including the linear symplectic map of Section 1.1 as basic examples. Consequently, we shall assume |Tμ,λ | ≤ C⟨λ − χ(μ)⟩ ,

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−s

λ, μ ∈ Λ.

(6.1)

Such estimates represent our very definition of Fourier integral operators, and grant sparsity properties and related numerics. Applications concern Schrödinger equations, under general assumptions on the Hamiltonians, including in particular quadratic Hamiltonians in the phase-space variables. In fact, the Gabor matrix of the corresponding Schrödinger propagator satisfies estimates of the form in (6.1), χ = χt being the Hamiltonian flow at a given time t ∈ ℝ. Let us describe in short the content of each section. Section 6.1 is devoted to the so-called naive theory of FIOs. It includes the case when the symplectomorphism χ is a small perturbation of the identity, as we have in the case of Schrödinger propagators T = Tt for small values of the time t. We get then a simple expression for T of pseudodifferential type Tf (x) = ∫ e2πiΦ(x,η) f ̂(η) dη ℝd https://doi.org/10.1515/9783110532456-007

326 | 6 Fourier integral operators and applications to Schrödinger equations which leads to estimates as in (6.1) and boundedness properties on modulation spaces. We shall refer to them as FIOs of type I. In Section 6.2 we show that Gabor frames provide optimally sparse representations of such operators, with applications to numerics for the free particle Schrödinger equation. Section 6.3 presents our theory of FIOs in full generality, basing on (6.1). They form a Wiener algebra, containing as subset the FIOs of type I as shown in Section 6.4 (note that the space of the FIOs of type I is not closed under composition). In Section 6.5 we treat the relevant examples given by the generalized metaplectic operators, corresponding to the case when χ in (6.1) is a linear symplectic map, cf. Section 1.1. Section 6.6 is devoted to the applications to Schrödinger equations with Hamiltonians having principal part homogeneous of degree 2 in z = (x, ξ ). The propagator is expressed by the general FIOs of Section 6.3. With respect to the classical theory, novelty of our approach is represented by the fact that in terms of Gabor matrices we are able to provide a global-in-time representation. Besides, we introduce the notion of Gabor wave front set, which allows a precise description of propagation of microsingularities, see the conclusive examples, concerning in particular the quantum harmonic oscillator.

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6.1 Fourier integral operators Fourier integral operators (FIOs) are a mathematical tool to study a variety of problems arising in partial differential equations. Originally introduced by Lax [217] for the construction of parametrices in the Cauchy problem for hyperbolic equations, they have been widely employed to represent solutions in the framework of both pure and applied mathematics (see, e. g., the papers [34, 41, 114, 115, 175, 191], books [193, 269, 295] and references therein). In particular, a class of FIOs in ℝd was applied by Helffer and Robert [187, 188] to study the spectral properties of a class of globally elliptic operators, generalizing the harmonic oscillator of the quantum mechanics. The Fourier integral operators we work with possess a phase function similar to those of [187, 188]. A simple example is the solution operator of the Cauchy problem for the Schrödinger equation with a quadratic Hamiltonian. Following first the elementary presentation, given a function f on ℝd , the Fourier integral operator (FIO) T with symbol σ and phase Φ on ℝ2d can be formally defined by Tf (x) = ∫ e2πiΦ(x,η) σ(x, η)f ̂(η) dη. ℝd

Later we shall refer to such T as FIO of type I. The phase function Φ(x, η) is smooth on ℝ2d , fulfills the estimates 󵄨󵄨 α 󵄨 󵄨󵄨𝜕z Φ(z)󵄨󵄨󵄨 ≤ Cα ,

|α| ≥ 2, z ∈ ℝ2d ,

(6.2)

6.1 Fourier integral operators | 327

and the nondegeneracy condition 󵄨󵄨 󵄨 2 󵄨󵄨det 𝜕x,η Φ(x, η)󵄨󵄨󵄨 ≥ δ > 0,

(x, ξ ) ∈ ℝ2d .

(6.3)

|α| ≤ N, a. e. z ∈ ℝ2d ,

(6.4)

The symbol σ on ℝ2d satisfies 󵄨󵄨 α 󵄨 󵄨󵄨𝜕z σ(z)󵄨󵄨󵄨 ≤ Cα ,

for a fixed N > 0 (in the sequel we shall work also with rougher symbols). We denote as usual the Gabor frame 𝒢 (g, α, β) = {gm,n }(m,n) , as defined in (5.1). Our goal will be to show that the Gabor matrix representation of an FIO T with respect to a Gabor frame with g ∈ 𝒮 (ℝd ) (defined in (5.3)) is well-organized (similarly to frames of curvelets and shearlets [41, 175]), provided that the symbol σ is smooth and satisfies the decay estimate (6.4) for every N > 0. Namely, we shall prove For each N > 0, there exists a constant CN > 0 such that |Tm󸀠 ,n󸀠 ,m,n | ≤ CN ⟨χ(m, n) − (m󸀠 , n󸀠 )⟩

−2N

,

(6.5)

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where χ is the canonical transformation generated by Φ (defined below). In the special case of pseudodifferential operators, such an almost diagonalization was already obtained in [162, 252]. Indeed, pseudodifferential operators correspond to the phase Φ(x, η) = xη and canonical transformation χ(y, η) = (y, η). One should mention that the use of almost diagonal estimates in proving continuity results goes back to the pioneering work [151], where the Calderón–Zygmund class of operators was studied via such a technique, and this was achieved by working with wavelets. Also observe that one simple case of a pseudodifferential operator is the Hilbert transform, and the usefulness of Gabor frames was first exhibited for this operator in [12]. A relevant byproduct of the results above are the boundedness properties of the operator T on modulation spaces. Indeed, the boundedness of the FIO T on M p,q (ℝd ) is equivalent to that of the related Gabor matrix ⟨Tgm,n , gm󸀠 ,n󸀠 ⟩ on the spaces of sequences ℓp,q . Hence, the estimates (6.5) readily give For 1 ≤ p < ∞, the Fourier integral operator T, with symbol σ and phase Φ as above, extends to a continuous operator on M p (ℝd ). (In the case p = ∞, the space M ∞ (ℝd ) is replaced by the closure of the Schwartz functions with respect to ‖ ⋅ ‖M ∞ .) The continuity property of an FIO T on M p,q , with p ≠ q, fails in general. Indeed, 2 an example is provided by the operator Tf (x) = eπix f (x), corresponding to Φ(x, η) = xη + x2 /2, σ ≡ 1, which is bounded on M p,q if and only if p = q (see Proposition 6.1.17 below).

328 | 6 Fourier integral operators and applications to Schrödinger equations To have a simple idea of the possible applications of the previous results, consider the Cauchy problem in (5.74) where H is the Weyl quantization of a quadratic form on ℝd × ℝd . A simple example is the harmonic oscillator already defined in (5.75). The solution to (5.74) can be viewed as a one-parameter family of FIOs, u(t, x) = eitH u0 , with symbol σ ≡ 1 and a phase given by a quadratic form Φ(x, ξ ), satisfying trivially the preceding assumptions (6.2) and (6.3). Our approach provides mathematical tools to solving Cauchy problems like (5.74) numerically, as we shall show in the sequel. We remark that the L2 boundedness of FIOs of the type above was first investigated Boulkhemair in [35], where he considered a more general situation compared to the condition (6.2). Namely, his assumption was (𝜕zα Φ) ∈ M ∞,1 (ℝ2d ),

|α| = 2

(6.6)

(observe that he did not assume that |α| ≥ 2, merely equality). In particular, no second order derivative of Φ needs to be smooth with this assumption. In [56] Concetti and Toft extend the results by Boulkhemair, using some basic arguments in [35] to show some continuity and Schatten properties of such FIOs. In particular, it is the first time where the authors prove that an FIO with symbol in M ∞,1 (ℝ2d ) is continuous on M p (ℝd ). Finally, in [291], the authors extend the results in [56] to weighted modulation spaces. 6.1.1 Phase functions and canonical transformations

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From now onward we shall work with a phase function satisfying the following assumptions. Definition 6.1.1. A real phase function Φ on ℝ2d is called tame if the following three properties are satisfied: A1. Φ ∈ 𝒞 ∞ (ℝ2d ); A2. For z = (x, η), 󵄨󵄨 α 󵄨 󵄨󵄨𝜕z Φ(z)󵄨󵄨󵄨 ≤ Cα ,

|α| ≥ 2;

(6.7)

A3. There exists δ > 0 such that 󵄨󵄨 󵄨 2 󵄨󵄨det 𝜕x,η Φ(x, η)󵄨󵄨󵄨 ≥ δ.

(6.8)

If we set {

y = ∇η Φ(x, η), ξ = ∇x Φ(x, η),

(6.9)

6.1 Fourier integral operators | 329

we can solve with respect to (x, ξ ) by the global inverse function theorem (see, e. g., [214]) and obtain a mapping χ defined by (x, ξ ) = χ(y, ξ ). The canonical transformation χ enjoys the following properties: B1. χ : ℝ2d → ℝ2d is smooth, invertible, and preserves the symplectic form in ℝ2d , i. e., dx ∧ dξ = dy ∧ dη; that is, χ is a symplectomorphism. B2. For z = (y, η), 󵄨󵄨 α 󵄨 󵄨󵄨𝜕z χ(z)󵄨󵄨󵄨 ≤ Cα ,

󵄨󵄨 α −1 󵄨󵄨 󵄨󵄨𝜕ζ χ (ζ )󵄨󵄨 ≤ Cα , |α| ≥ 1;

(6.10)

B3. There exists δ > 0 such that, for (x, ξ ) = χ(y, η), 󵄨󵄨 󵄨󵄨 𝜕x 󵄨󵄨 󵄨 󵄨󵄨det (y, η)󵄨󵄨󵄨 ≥ δ. 󵄨󵄨 󵄨󵄨 𝜕y

(6.11)

Conversely, to every transformation χ satisfying B1, B2, B3 (called tame canonical transformation in the sequel) corresponds a tame phase Φ, uniquely determined up to a constant. This can be easily proved by (6.11) and the global inverse function theorem [214]. When necessary and to avoid ambiguity, we shall denote by Φχ the phase function (up to constants) corresponding to the canonical transformation χ. Observe that B1 and B2 imply that χ and χ −1 are globally Lipschitz. This property implies that ⟨w − χ(z)⟩ ≍ ⟨χ −1 (w) − z⟩ w, z ∈ ℝ2d ,

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which we will use frequently. Moreover, if χ and χ̃ are two transformations satisfying B1 and B2, the same is true for χ ∘ χ,̃ whereas the additional property B3 is not necessarily preserved, even if χ and χ̃ are linear. This reflects the lack of the algebra property of the corresponding FIOs; see Section 6.5 below.

6.1.2 Almost diagonalization of FIOs For a given function f on ℝd , the FIO T with symbol σ and phase Φ can be formally defined by Tf (x) = ∫ e2πiΦ(x,η) σ(x, η)f ̂(η) dη.

(6.12)

ℝd

To avoid technicalities, we initially take f ∈ 𝒮 (ℝd ) or, more generally, f ∈ M 1 (ℝd ). If σ ∈ L∞ (ℝ2d ) and the phase Φ is real then the integral converges absolutely and defines a function in L∞ (ℝd ).

330 | 6 Fourier integral operators and applications to Schrödinger equations Assume that the phase function is tame as in Definition 6.1.1. We shall prove an almost diagonalization result for FIOs as above, with respect to a Gabor frame. Let us first consider the case of regular symbols. Precisely, for a given N ∈ ℕ, we consider symbols σ on ℝ2d satisfying, for z = (x, η), 󵄨 󵄨󵄨 α 󵄨󵄨𝜕z σ(z)󵄨󵄨󵄨 ≤ Cα ,

a. e. z ∈ ℝ2d , |α| ≤ 2N

(6.13)

(here 𝜕zα denotes distributional derivatives). Our goal is to study the decay properties of the matrix of the FIO T with respect to a Gabor frame {gm,n }(m,n) , with g ∈ 𝒮 (ℝd ). Theorem 6.1.2. Consider a tame phase function and a symbol satisfying (6.13). There exists CN > 0 such that −2N 󵄨󵄨 󵄨 󸀠 󸀠 󵄨󵄨⟨Tgm,n , gm󸀠 ,n󸀠 ⟩󵄨󵄨󵄨 ≤ CN ⟨∇z Φ(m , n) − (n , m)⟩ ,

(6.14)

with (m, n), (m󸀠 , n󸀠 ) ∈ Λ. Proof. Recall that the time-frequency shifts interchange under the action of the Fourier transform, (Tx f )∧ = M−x f ̂ and (Mη f )∧ = Tη f ̂, besides they fulfill the commutation relations Tx Mη = e−2πix⋅η Mη Tx . Using these properties, we compute the Gabor matrix entries as follows: ⟨Tgm,n , gm󸀠 ,n󸀠 ⟩ = ∫ Tgm,n (x)Mn󸀠 Tm󸀠 g(x) dx ℝd

̂ ̄ dx dη = ∫ ∫ e2πiΦ(x,η) σ(x, η)Tn M−m g(η)M −n󸀠 Tm󸀠 g(x) ℝd ℝd

̂ ̄ dx dη = ∫ ∫ M(0,−m) T(0,−n) (e2πiΦ(x,η) σ(x, η))g(η)M −n󸀠 Tm󸀠 g(x) ℝd ℝd

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̄ g(η) ̂ dx dη = ∫ ∫ T(−m󸀠 ,0) M−(n󸀠 ,0) M(0,−m) T(0,−n) (e2πiΦ(x,η) σ(x, η))g(x) ℝd ℝd

̄ g(η) ̂ dx dη. = ∫ ∫ e2πi[Φ(x+m ,η+n)−(n ,m)⋅(x+m ,η)] σ(x + m󸀠 , η + n)g(x) 󸀠

󸀠

󸀠

ℝd ℝd

Since Φ is smooth, we may expand Φ(x, η) into a Taylor series around (m󸀠 , n) and obtain Φ(x + m󸀠 , η + n) = Φ(m󸀠 , n) + ∇z Φ(m󸀠 , n) ⋅ (x, η) + Φ2,(m󸀠 ,n) (x, η) where the remainder is given by 1

Φ2,(m󸀠 ,n) (x, η) = 2 ∑ ∫(1 − t)𝜕α Φ((m󸀠 , n) + t(x, η)) dt |α|=2 0

(x, η)α . α!

(6.15)

6.1 Fourier integral operators | 331

Whence, we can write 󵄨 󵄨󵄨 󵄨󵄨⟨Tgm,n , gm󸀠 ,n󸀠 ⟩󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󸀠 󸀠 󵄨 ̄ g(η) ̂ dx dη󵄨󵄨󵄨󵄨. = 󵄨󵄨󵄨 ∫ ∫ e2πi{[∇z Φ(m ,n)−(n ,m)]⋅(x,η)} e2πiΦ2,(m󸀠 ,n) (x,η) σ(x + m󸀠 , η + n)g(x) 󵄨󵄨 󵄨󵄨 ℝd ℝd

For N ∈ ℕ, using the identity (1 − Δz )N e2πi{[∇z Φ(m ,n)−(n ,m)]⋅(x,η)} 󸀠

󸀠

2N

= ⟨2π(∇z Φ(m󸀠 , n) − (n󸀠 , m))⟩ e2πi{[∇z Φ(m ,n)−(n ,m)]⋅(x,η)} , 󸀠

󸀠

we integrate by parts and obtain 󵄨󵄨 󵄨 󵄨󵄨⟨Tgm,n , gm󸀠 ,n󸀠 ⟩󵄨󵄨󵄨 =

󵄨󵄨 1 󵄨󵄨 2πi{[∇z Φ(m󸀠 ,n)−(n󸀠 ,m)]⋅(x,η)} 󵄨󵄨 ∫ ∫ e 󸀠 󸀠 2N 󵄨 ⟨2π(∇z Φ(m , n) − (n , m))⟩ 󵄨 ℝd ℝd

󵄨󵄨 󵄨 ̄ g(η)] ̂ × (1 − Δz )N [e2πiΦ2,(m󸀠 ,n) (x,η) σ(x + m󸀠 , η + n)g(x) dx dη󵄨󵄨󵄨. 󵄨󵄨 By means of Leibniz’s formula, the factor ̄ g(η)] ̂ (1 − Δz )N [e2πiΦ2,(m󸀠 ,n) (x,η) σ(x + m󸀠 , η + n)g(x) can be expressed as e2πiΦ2,(m󸀠 ,n) (z)

∑

̂ Cα,βγ p(𝜕|α| Φ2,(m󸀠 ,n) )(z)(𝜕zβ σ)(z + (m󸀠 , n))𝜕zγ (ḡ ⊗ g)(z),

|α|+|β|+|γ|≤2N

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where p(𝜕|α| Φ2,(m󸀠 ,n) )(z) is a polynomial made of derivatives of Φ2,(m󸀠 ,n) of order at most |α|. As a consequence of (6.7), we have 𝜕zα Φ2,(m󸀠 ,n) (z) = O(⟨z⟩2 ), which combined with the assumption (6.13) and the hypothesis g ∈ 𝒮 (ℝd ) yields the desired estimate. Remark 6.1.3. More generally, one can consider symbols satisfying estimates of the form 󵄨󵄨 α 󵄨 󵄨󵄨𝜕z σ(z)󵄨󵄨󵄨 ≤ Cα μ(z),

a. e. z ∈ ℝ2d , |α| ≤ 2N,

(6.16)

with μ ∈ ℳv and also more general windows g. Indeed, by arguing as above and using 󵄨 󵄨󵄨 β 󸀠 󸀠 󸀠 󸀠 󵄨󵄨𝜕z σ(z + (m , n))󵄨󵄨󵄨 ≤ Cβ μ(z + (m , n)) ≤ C Cβ v(z)μ(m , n), one deduces the decay estimates μ(m󸀠 , n) 󵄨󵄨 󵄨 , 󵄨󵄨⟨Tgm,n , gm󸀠 ,n󸀠 ⟩󵄨󵄨󵄨 ≤ CN ⟨∇z Φ(m󸀠 , n) − (n󸀠 , m)⟩2N

332 | 6 Fourier integral operators and applications to Schrödinger equations provided the integral 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ̂ Cα,β,γ p(𝜕|α| Φ2,(m󸀠 ,n) )(z)v(z)𝜕zγ (ḡ ⊗ g)(z) ∑ ∫ 󵄨󵄨󵄨 󵄨󵄨 dz 󵄨󵄨 󵄨󵄨 2d |α|+|β|+|γ|≤2N

(6.17)

ℝ

γ

̂ converges. This is guaranteed if, e. g., ⟨z⟩2N v(z)𝜕z (ḡ ⊗ g)(z) ∈ L1 . We are going to rewrite (6.14) in a form convenient for the applications to the continuity of FIOs in the next subsection. We need the following lemma. Lemma 6.1.4. Consider a tame phase function Φ. Then 󵄨󵄨 󵄨 󵄨 󸀠 󸀠󵄨 󵄨 󸀠 󸀠󵄨 󵄨 󸀠󵄨 󵄨󵄨∇x Φ(m , n) − n 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇η Φ(m , n) − m󵄨󵄨󵄨 ≳ 󵄨󵄨󵄨x(m, n) − m 󵄨󵄨󵄨 + 󵄨󵄨󵄨ξ (m, n) − n 󵄨󵄨󵄨,

(6.18)

where (y, η) �→ (x, ξ ) is the canonical transformation generated by Φ. Proof. It suffices to prove the following inequalities: 󵄨󵄨 󵄨 󵄨 󸀠 󸀠󵄨 󵄨󵄨∇η Φ(m , n) − m󵄨󵄨󵄨 ≳ 󵄨󵄨󵄨x(m, n) − m 󵄨󵄨󵄨, 󵄨󵄨 󵄨 󵄨 󵄨 󸀠 󸀠󵄨 󸀠󵄨 󸀠 󵄨󵄨∇x Φ(m , n) − n 󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨ξ (m, n) − n 󵄨󵄨󵄨 − C 󵄨󵄨󵄨∇η Φ(m , n) − m󵄨󵄨󵄨.

(6.19) (6.20)

We observe that, by (6.9), we have y = ∇η Φ(x(y, η), η)

∀(y, η) ∈ ℝ2d

(6.21)

and ∇x Φ(x, η) = ξ (∇η Φ(x, η), η)

∀(x, η) ∈ ℝ2d .

(6.22)

Hence, we have m = ∇η Φ(x(m, n), n), so that

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󵄨󵄨 󵄨 󵄨 󵄨 󸀠 󸀠 󵄨󵄨∇η Φ(m , n) − m󵄨󵄨󵄨 = 󵄨󵄨󵄨∇η Φ(m , n) − ∇η Φ(x(m, n), n)󵄨󵄨󵄨 󵄨 󵄨 ≳ 󵄨󵄨󵄨x(m, n) − m󸀠 󵄨󵄨󵄨, where the last inequality follows from the fact that, for every fixed η, the map x �→ ∇Φη (x, η) has a Lipschitz inverse, with Lipschitz constant uniform with respect to η. This proves (6.19). In order to prove (6.20), we observe that, in view of (6.22), one has ∇x Φ(m󸀠 , n) − n󸀠 = ξ (∇η Φ(m󸀠 , n), n) − n󸀠 = ξ (m + ∇η Φ(m󸀠 , n) − m, n) − n󸀠 = ξ (m, n) − n󸀠 + O(∇η Φ(m󸀠 , n) − m), where the last inequality follows from the Taylor formula for the function y �→ ξ (y, n), taking into account that the function ξ has bounded derivatives. This proves (6.20). Hence the proof is completed.

6.1 Fourier integral operators | 333

Combining the previous lemma with (6.14), we obtain the following result. Theorem 6.1.5. Consider a tame phase function Φ and a symbol satisfying (6.13). Let g ∈ 𝒮 (ℝd ). There exists a constant CN > 0 such that 󵄨 󵄨󵄨 󸀠 󸀠 −2N 󵄨󵄨⟨Tgm,n , gm󸀠 ,n󸀠 ⟩󵄨󵄨󵄨 ≤ CN ⟨χ(m, n) − (m , n )⟩ ,

(6.23)

where χ is the canonical transformation generated by Φ. This result shows that the matrix representation of an FIO with respect a Gabor frame is well-organized, similarly to the results recently obtained by [41, 173] in terms 0 of shearlets and curvelets frames. More precisely, if σ ∈ S0,0 , namely if (6.13) is satisfied for every N ∈ ℕ, then the Gabor matrix of T is highly concentrated along the graph of χ.

p

6.1.3 Continuity of FIOs on Mμ (ℝd ) In this section we study the continuity of FIOs on the modulation spaces Mμp associated

with a weight function μ ∈ ℳvs (ℝ2d ), s ≥ 0. We first state some preliminaries. d p,q d Recall that ℳp,q μ (ℝ ) is the closure of the Schwartz class in Mμ (ℝ ) (cf. Definip,q d p,q d ̃ for the tion 2.3.15), so that ℳμ (ℝ ) = Mμ (ℝ ) if p < ∞ and q < ∞. Also write ℓμp,q p,q p,q p,q ̃ closure of the space of eventually zero sequences in ℓ . Similarly, ℓ = ℓ if p < ∞ μ

and q < ∞.

μ

μ

Theorem 6.1.6. Let μ ∈ ℳv (ℝ2d ), 𝒢 (g, α, β) be a Parseval Gabor frame for L2 (ℝd ), with lattice Λ = αℤd × βℤd , and g ∈ Mv1 (ℝd ). For every 1 ≤ p, q ≤ ∞, the coefficient operd p,q ̃ ator Cg is continuous from ℳp,q μ (ℝ ) into ℓμ (Λ), whereas the synthesis operator Dg is p,q p,q d continuous from ℓ̃ (Λ) into ℳ (ℝ ). Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

μ

μ

Proof. Since Cg is continuous from Mμp,q (ℝd ) into ℓμp,q (Λ) by Theorem 3.2.32, it suffices ̃ (Λ). This follows from the fact to verify that, if f is a Schwartz function then Cg (f ) ∈ ℓμp,q 1 that Cg (f ) ∈ ℓμ (Λ). Similarly, by Theorem 3.2.33, the synthesis operator Dg is bounded

from ℓμp,q (Λ) into Mμp,q (ℝd ). For Dg it suffices to verify that, if c is any eventually zero d 1 d sequence, then Dg (c) ∈ ℳp,q μ (ℝ ). This is true because Dg (c) ∈ Mμ (ℝ ).

We need here the classical version of the so-called Schur’s test (other versions of the Schur’s test are given by Propositions 3.3.6 and 3.3.12 in Chapter 3). Lemma 6.1.7. Consider a lattice Λ and an operator K defined on sequences as (Kc)λ = ∑ Kλ,ν cν . ν∈Λ

334 | 6 Fourier integral operators and applications to Schrödinger equations (i) For 1 ≤ p ≤ ∞, assume sup ∑ |Kλ,ν | < ∞,

sup ∑ |Kλ,ν | < ∞.

ν∈Λ λ∈Λ

λ∈Λ ν∈Λ

Then K is continuous on ℓp (Λ) and, moreover, maps the space c0 (Λ) of sequences vanishing at infinity into itself. (ii) If 0 < p ≤ 1 and supν ∑λ |Kλ,ν |p < ∞, then 1/p

‖Kc‖ℓp ≤ (sup ∑ |Kλ,ν |p ) ν

λ

‖c‖ℓp ,

that is, K is continuous on ℓp (Λ). Proof. (i) Let (cν )ν be a sequence in ℓp (Λ), 1 ≤ p ≤ ∞. Applying Hölder’s inequality to 1 󸀠

1

p p Kλ,ν cν , (Kc)λ = ∑ Kλ,ν

ν∈Λ

we infer 1 󸀠

1

p p 󵄨󵄨 󵄨 p 󵄨󵄨(Kc)λ 󵄨󵄨󵄨 ≤ ( ∑ |Kλ,ν |) ( ∑ |Kλ,ν ||cν | ) .

ν∈Λ

ν∈Λ

By summing over λ, we obtain the boundedness of the operator K ‖Kc‖pℓp

p p󸀠

≤ ∑ ( ∑ |Kλ,ν |) ( ∑ |Kλ,ν ||cν |p ) ν∈Λ

λ∈Λ ν∈Λ

≲ ∑ ( ∑ |Kλ,ν |)|cν |p

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≲

ν∈Λ λ∈Λ ‖c‖pℓp .

Let us show the second part. Since we know that K is continuous on ℓ∞ and the space of eventually zero sequences is dense in c0 , it suffices to verify that K maps every eventually zero sequence in c0 . This follows from the fact that any eventually zero sequence belongs to ℓ1 and therefore, since K is continuous on ℓ1 , is mapped in ℓ1 �→ c0 . (ii) Since, for any sequence b = (bλ ), we have ‖b‖ℓ1 ≤ ‖b‖ℓp if 0 < p ≤ 1, it turns out that p

‖Kc‖pℓp ≤ ∑(∑ |Kλ,ν ||cν |) ≤ ∑ ∑ |Kλ,ν |p |cν |p ν

λ

λ

ν

= ∑ ∑ |Kλ,ν |p |cν |p ≤ (sup ∑ |Kλ,ν |p )‖c‖pℓp . ν

λ

ν

λ

6.1 Fourier integral operators | 335

We can now state our result. Theorem 6.1.8. Consider a tame phase function and a symbol satisfying (6.13). Let 0 ≤ s < 2N − 2d, and μ ∈ ℳvs (ℝ2d ). For every 1 ≤ p < ∞, T extends to a continuous operator p (ℝd ) into Mμp (ℝd ), and for p = ∞ it extends to a continuous operator from from Mμ∘χ d ∞ d ℳ∞ μ∘χ (ℝ ) into ℳμ (ℝ ).

Moreover, observe that μ ∘ χ ∈ ℳvs (ℝ2d ). Indeed, vs ∘ χ ≍ vs , due to the bi-Lipschitz property of χ. Proof. We first prove that p , ‖Tf ‖Mμp ≤ C‖f ‖Mμ∘χ

for every f ∈ 𝒮 (ℝd ). This proves the theorem in the case p < ∞, since 𝒮 (ℝd ) is dense in Mμp (ℝd ).

We see at once that, since σ ∈ L∞ (ℝ2d ), T defines a bounded operator from M 1 (ℝd ) into L∞ (ℝd ) �→ M ∞ (ℝd ). Hence, for all f ∈ 𝒮 (ℝd ), we have Tf ∈ M ∞ (ℝd ), and Theop p and ‖Tf ‖ p ≲ ‖C (Tf )‖ p . On the other hand, rem 6.1.6 shows that ‖f ‖Mμ∘χ ≳ ‖Cg (f )‖ℓμ∘χ g Mμ ℓμ

the expansion (3.29) (with γ = g) holds for f with convergence in M 1 (ℝd ). Therefore, Tf = ∑ ⟨f , gm,n ⟩Tgm,n m,n

with convergence in M ∞ (ℝd ). Hence, Cg (Tf )m󸀠 ,n󸀠 = ⟨Tf , gm󸀠 ,n󸀠 ⟩ = ∑ ⟨Tgm,n , gm󸀠 ,n󸀠 ⟩⟨f , gm,n ⟩ m,n

= ∑ ⟨Tgm,n , gm󸀠 ,n󸀠 ⟩Cg (f )m,n . m,n

Therefore we are reduced to proving that the matrix operator

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{cm,n } �→

∑ ⟨Tgm,n , gm󸀠 ,n󸀠 ⟩cm,n

m,n∈ℤd

(6.24)

p is bounded from ℓμ∘χ into ℓμp . This follows from Schur’s test (Lemma 6.1.7) if we prove that, upon setting

Km󸀠 ,n󸀠 ,m,n = ⟨Tgm,n , gm󸀠 ,n󸀠 ⟩μ(m󸀠 , n󸀠 )/μ(χ(m, n)), we have ∞ 1 Km󸀠 ,n󸀠 ,m,n ∈ ℓm,n ℓm󸀠 ,n󸀠

(6.25)

∞ 1 Km󸀠 ,n󸀠 ,m,n ∈ ℓm 󸀠 ,n󸀠 ℓm,n .

(6.26)

and

336 | 6 Fourier integral operators and applications to Schrödinger equations In view of (6.23), we have |Km󸀠 ,n󸀠 ,m,n | ≲ ⟨χ(m, n) − (m󸀠 , n󸀠 )⟩

−2N+s

μ(m󸀠 , n󸀠 ) . ⟨χ(m, n) − (m󸀠 , n󸀠 )⟩s μ(χ(m, n))

(6.27)

Now, the last quotient in (6.27) is bounded because μ is vs -moderate, so we deduce (6.25). Finally, since χ is a bi-Lipschitz function, we have 󵄨󵄨 󵄨 󸀠 󸀠 󵄨 −1 󸀠 󸀠 󵄨 󵄨󵄨χ(m, n) − (m , n )󵄨󵄨󵄨 ≍ 󵄨󵄨󵄨(m, n) − χ (m , n )󵄨󵄨󵄨,

(6.28)

so that (6.26) follows as well. The case p = ∞ is proved analogously by using Theorem 6.1.6 (with p = q = ∞), and the last part of the statement of Lemma 6.1.7. Remark 6.1.9. Theorem 6.1.8 with μ ≡ 1 gives, in particular, continuity on the unweighted modulation spaces M p (ℝd ). If, moreover, p = 2, we recapture the classical L2 -continuity result by Asada and Fujiwara [3]. Also Theorem 6.1.8 applies to μ = vt with |t| ≤ s. In that case we obtain continuity on Mvpt (ℝd ), because vt ∘ χ ≍ vt . 6.1.4 Continuity of FIOs on Mp,q

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Now let us study the continuity of FIOs on modulation spaces M p,q (ℝd ) possibly with p ≠ q. As we shall see in Section 6.1.6, under the assumptions of Theorem 6.1.8 such operators may fail to be bounded when p ≠ q. A counterexample is given by the phase Φ(x, η) = xη + x2 /2, and symbol σ = 1, which does not yield a bounded operator on M p,q (ℝd ), except for the case p = q. Here the obstruction is essentially due to the fact that the map x �→ ∇x Φ(x, η) has unbounded range. Indeed, we will show (for general phases) that if this map has range of finite diameter, uniformly with respect to η, then the corresponding operator is bounded on all M p,q (ℝd ). Theorem 6.1.10. Consider a tame phase function Φ and a symbol satisfying (6.13), with N > d. Suppose, in addition, that 󵄨 󵄨 sup 󵄨󵄨󵄨∇x Φ(x, η) − ∇x Φ(x󸀠 , η)󵄨󵄨󵄨 < ∞.

x,x 󸀠 ,η

(6.29)

Then the corresponding Fourier integral operator T extends to a bounded operator on M p,q (ℝd ) for every 1 ≤ p, q < ∞ and on ℳp,q (ℝd ) if p = ∞ or q = ∞. Proof. By arguing as in the proof of Theorem 6.1.8, it suffices to prove the continuity p ̃ if p = ∞ or q = ∞, of the operator on ℓp,q = ℓnq ℓm if p < ∞ and q < ∞, or ℓp,q {cm,n } �→

∑ Tm󸀠 ,n󸀠 ,m,n cm,n ,

m,n∈ℤd

6.1 Fourier integral operators | 337

where Tm󸀠 ,n󸀠 ,m,n = ⟨Tgm,n , gm󸀠 ,n󸀠 ⟩. By applying Proposition 3.3.6, it suffices to verify that ∞ 1 {Tm󸀠 ,n󸀠 ,m,n } ∈ ℓn∞ ℓn1 󸀠 ℓm 󸀠 ℓm ,

(6.30)

∞ 1 {Tm󸀠 ,n󸀠 ,m,n } ∈ ℓn∞󸀠 ℓn1 ℓm ℓm󸀠 ,

(6.31)

because we have already seen from (6.23) and (6.28) that ∞ 1 ∞ 1 {Tm󸀠 ,n󸀠 ,m,n } ∈ ℓm,n ℓm󸀠 ,n󸀠 ∩ ℓm 󸀠 ,n󸀠 ℓm,n .

Let us now prove (6.30). It follows from (6.14) and (6.20) that 󵄨 󵄨2 󵄨 󵄨2 −N |Tm󸀠 ,n󸀠 ,m,n | ≲ (1 + 󵄨󵄨󵄨∇x Φ(m󸀠 , n) − n󸀠 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇η Φ(m󸀠 , n) − m󵄨󵄨󵄨 ) 󵄨 󵄨2 󵄨 󵄨2 −N ≲ (1 + 󵄨󵄨󵄨ξ (m, n) − n󸀠 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇η Φ(m󸀠 , n) − m󵄨󵄨󵄨 )

󵄨 󵄨 −N 󵄨 󵄨 −N ≲ (1 + 󵄨󵄨󵄨ξ (m, n) − n󸀠 󵄨󵄨󵄨) (1 + 󵄨󵄨󵄨∇η Φ(m󸀠 , n) − m󵄨󵄨󵄨) . By (6.9), we have ξ (y, η) = ∇x Φ(x(y, η), η),

∀(y, η) ∈ ℝ2d ,

so that the hypothesis (6.29) yields ξ (m, n) = ξ (0, n) + O(1). Hence (6.30) follows. We now prove (6.31). As above, it follows from (6.23) and (6.19) that 󵄨 󵄨2 󵄨 󵄨2 −N |Tm󸀠 ,n󸀠 ,m,n | ≲ (1 + 󵄨󵄨󵄨∇x Φ(m󸀠 , n) − n󸀠 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇η Φ(m󸀠 , n) − m󵄨󵄨󵄨 )

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󵄨 󵄨2 󵄨 󵄨2 −N ≲ (1 + 󵄨󵄨󵄨∇x Φ(m󸀠 , n) − n󸀠 󵄨󵄨󵄨 + 󵄨󵄨󵄨x(m, n) − m󸀠 󵄨󵄨󵄨 )

󵄨 󵄨 −N 󵄨 󵄨 −N ≲ (1 + 󵄨󵄨󵄨∇x Φ(m󸀠 , n) − n󸀠 󵄨󵄨󵄨) (1 + 󵄨󵄨󵄨x(m, n) − m󸀠 󵄨󵄨󵄨) . By (6.29), we have ∇x Φ(m󸀠 , n) = ∇x Φ(0, n) + O(1), so that 󵄨 󵄨 󵄨 󵄨 1 + 󵄨󵄨󵄨∇x Φ(m󸀠 , n) − n󸀠 󵄨󵄨󵄨 ≳ 1 + 󵄨󵄨󵄨∇x Φ(0, n) − n󸀠 󵄨󵄨󵄨 󵄨 󵄨 ≳ 1 + 󵄨󵄨󵄨n − ψ(n󸀠 )󵄨󵄨󵄨,

(6.32)

where ψ is the inverse function of the bi-Lipschitz function η �→ ∇x Φ(0, η). Therefore we obtain (6.31). This concludes the proof.

338 | 6 Fourier integral operators and applications to Schrödinger equations Example 6.1.11. Theorem 6.1.10 applies, in particular, to phases of the type 󵄨 󵄨 Φ(x, η) = x ⋅ η + a(x, η) where 󵄨󵄨󵄨𝜕xα 𝜕ηβ a(x, η)󵄨󵄨󵄨 ≤ Cα,β ,

2|α| + |β| ≥ 2.

In the special case when a(x, η) = a(η) is independent of x and the symbol σ ≡ 1, the FIO reduces to a Fourier multiplier Tf (x) = ∫ e2πix⋅η e2πia(η) f ̂(η) dη ℝd

and we reobtain the result of [14, Theorem 1] on the continuity of T on all ℳp,q (ℝd ), 1 ≤ p, q ≤ ∞. 6.1.5 Modulation spaces as symbol classes In what follows we shall rephrase the quantity |⟨Tgm,n , gm󸀠 ,n󸀠 ⟩| in terms of the STFT of the symbol σ, without assuming the existence of derivatives of σ. This will be applied to prove the continuity of FIOs with symbols in M ∞,1 (ℝ2d ) on modulation spaces M p (ℝd ). The same arguments as in the proof of Theorem 6.1.2 yield the equality ̂ ⟨Tgm,n , gm󸀠 ,n󸀠 ⟩ = e2πim⋅n ∫ e2πiΦ(x,η) σ(x, η)M(−n󸀠 ,−m) T(m󸀠 ,n) (ḡ ⊗ g)(x, η) dx dη. ℝd

Expanding the phase Φ into a Taylor series around (m󸀠 , n), we obtain Φ(x, η) = Φ(m󸀠 , n) + ∇z Φ(m󸀠 , n) ⋅ (x − m󸀠 , η − n) + T(m󸀠 ,n) Φ2,(m󸀠 ,n) (x, η) where the remainder Φ2,(m󸀠 ,n) is given by (6.15). Inserting this expansion in the integrals above, we can write ⟨Tgm,n , gm󸀠 ,n󸀠 ⟩ = e2πi(m⋅n+Φ(m ,n)−∇z Φ(m ,n)⋅(m ,n)) ∫ e2πi∇z Φ(m ,n)(x,η) σ(x, η)

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󸀠

󸀠

󸀠

󸀠

ℝd

̂ × M(−n󸀠 ,−m) T(m󸀠 ,n) e2πiΦ2,(m󸀠 ,n) (x,η) (ḡ ⊗ g)(x, η) dx dη.

(6.33)

Defining ̂ Ψ(m󸀠 ,n) (x, η) := e2πiΦ2,(m󸀠 ,n) (x,η) (ḡ ⊗ g)(x, η),

(6.34)

and computing the modulus of the left-hand side of (6.33), we are led to 󵄨󵄨 󵄨 󵄨 󵄨 󸀠 󸀠 󸀠 󸀠 󵄨󵄨⟨Tgm,n , gm󸀠 ,n󸀠 ⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨VΨ(m󸀠 ,n) σ((m , n), (n − ∇x Φ(m , n), m − ∇ξ Φ(m , n)))󵄨󵄨󵄨.

(6.35)

Observe that the window Ψ(m󸀠 ,n) of the STFT above depends on the pair (m󸀠 , n).

6.1 Fourier integral operators | 339

We now study the continuity problem for T when the symbol σ is in the modulation space M ∞,1 (ℝ2d ). Theorem 6.1.12. Consider a tame phase function Φ and a symbol σ ∈ M ∞,1 (ℝ2d ). For every 1 ≤ p < ∞, T extends to a continuous operator on M p (ℝd ), for p = ∞ it extends to a continuous operator on ℳ∞ (ℝd ). By arguing as in the proof of Theorem 6.1.8, it suffices to prove the continuity ̃ = c0 . In view of Schur’s test of the operator (6.24) on ℓp , if 1 ≤ p < ∞, and on ℓ∞ (Lemma 6.1.7) and (6.35), it suffices to prove the following result. Proposition 6.1.13. Consider a tame phase function Φ and a symbol σ ∈ M ∞,1 (ℝ2d ). If we set zm,n,m󸀠 ,n󸀠 := ((m󸀠 , n), (n󸀠 − ∇x Φ(m󸀠 , n), m − ∇ξ Φ(m󸀠 , n))),

m, m󸀠 ∈ αℤd , n, n󸀠 ∈ βℤd , (6.36)

then sup

∑

󵄨󵄨 󵄨 󵄨󵄨VΨ(m󸀠 ,n) σ(zm,n,m󸀠 ,n󸀠 )󵄨󵄨󵄨 ≲ ‖σ‖M ∞,1 ,

(6.37)

sup

󵄨 󵄨 ∑ 󵄨󵄨󵄨VΨ 󸀠 σ(zm,n,m󸀠 ,n󸀠 )󵄨󵄨󵄨 ≲ ‖σ‖M ∞,1 . (m ,n)

(6.38)

(m,n)∈Λ (m󸀠 ,n󸀠 )∈Λ (m󸀠 ,n󸀠 )∈Λ

(m,n)∈Λ

We need the following lemma. Lemma 6.1.14. Let Ψ0 ∈ 𝒮 (ℝ2d ) with ‖Ψ0 ‖L2 = 1 and Ψ(m󸀠 ,n) be defined by (6.34), with (m󸀠 , n) ∈ Λ = αℤd × βℤd , and g ∈ 𝒮 (ℝd ). Then ∫

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ℝ4d

󵄨 󵄨 sup 󵄨󵄨󵄨VΨ 󸀠 Ψ0 (w)󵄨󵄨󵄨 dw < ∞. (m ,n)

(m󸀠 ,n)∈Λ

(6.39)

Proof of Lemma 6.1.14. We shall show that 󵄨󵄨 󵄨 −(4d+1) , 󵄨󵄨VΨ(m󸀠 ,n) Ψ0 (w)󵄨󵄨󵄨 ≤ C⟨w⟩

∀(m󸀠 , n) ∈ Λ.

(6.40)

Using the switching property of the STFT (Vf g)(x, η) = e−2πiη⋅x (Vg f )(−x, −η), we observe that |VΨ

(m󸀠 ,n)

Ψ0 (w1 , w2 )| = |VΨ0 Ψ(m󸀠 ,n) |(−w1 , −w2 ), and since the weight ⟨⋅⟩

is even, relation (6.40) is equivalent to

󵄨󵄨 󵄨 −(4d+1) , 󵄨󵄨VΨ0 Ψ(m󸀠 ,n) (w)󵄨󵄨󵄨 ≤ C⟨w⟩

∀(m󸀠 , n) ∈ Λ.

(6.41)

340 | 6 Fourier integral operators and applications to Schrödinger equations Now, the mapping VΨ0 is continuous from 𝒮 (ℝ2d ) to 𝒮 (ℝ4d ) (see [160, Chapter 11]). This means that there exist M ∈ ℕ and K > 0 such that 4d+1 󵄨 ⟨w⟩ 󵄨 󵄨 󵄨󵄨 󵄨󵄨VΨ0 Ψ(m󸀠 ,n) (w)󵄨󵄨󵄨 = 󵄨󵄨󵄨VΨ0 Ψ(m󸀠 ,n) (w)󵄨󵄨󵄨 4d+1 ⟨w⟩

󵄩 󵄩 ≤ 󵄩󵄩󵄩VΨ0 Ψ(m󸀠 ,n) ⟨⋅⟩4d+1 󵄩󵄩󵄩L∞ (ℝ4d ) ⟨w⟩−(4d+1) 󵄩 󵄩 ≤ K ∑ 󵄩󵄩󵄩Dγ X δ Ψ(m󸀠 ,n) 󵄩󵄩󵄩L∞ (ℝ2d ) ⟨w⟩−(4d+1) , |γ|+|δ|≤M

for every (m󸀠 , n) ∈ Λ. We now claim that Ψ(m󸀠 ,n) ∈ 𝒮 (ℝ2d ) uniformly with respect to (m󸀠 , n). This is proved as follows: the function e2πiΦ2,(m󸀠 ,n) (x,η) is in 𝒞 ∞ (ℝ2d ) and possesses derivatives dominated by powers ⟨(x, η)⟩k , k ∈ ℕ, uniformly with respect to (m󸀠 , n), due to (6.7); since (ḡ ⊗ g)̂ ∈ 𝒮 (ℝ2d ), it follows that Ψ(m󸀠 ,n) ∈ 𝒮 (ℝ2d ), with seminorms uniformly bounded pM (Ψ(m󸀠 ,n) ) :=

∑ |γ|+|δ|≤M

󵄩󵄩 γ δ 󵄩 󵄩󵄩D X Ψ(m󸀠 ,n) 󵄩󵄩󵄩L∞ ≤ CM ,

∀(m󸀠 , n) ∈ Λ.

Consequently, 󵄨󵄨 󵄨 󵄨󵄨VΨ0 Ψ(m󸀠 ,n) (w)󵄨󵄨󵄨 ≤ K

∑ |γ|+|δ|≤M

󵄩󵄩 γ δ 󵄩 −(4d+1) 󵄩󵄩D X Ψ(m󸀠 ,n) 󵄩󵄩󵄩L∞ (ℝ2d ) ⟨w⟩

≤ KCM ⟨w⟩−(4d+1) , for every (m󸀠 , n) ∈ Λ, as desired.

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We exhibit a slight generalization of [160, Proposition 11.1.4] and its subsequent remark. This will be also essential in the following proof of Proposition 6.1.13. Proposition 6.1.15. Let 𝒳 be a separated sampling set in ℝ2d , that is, there exists δ > 0 such that infx,y∈𝒳 :x=y̸ |x − y| ≥ δ. Then there exists a constant C > 0 such that, if F ∈ 2d W(L∞ , Lp,q μ ) is any function everywhere defined on ℝ and lower semicontinuous, then p,q the restriction F|𝒳 is in ℓμ with ‖F|𝒳 ‖ℓμp,q ≤ C‖F‖W(L∞ ,Lp,q . μ ) Proof. One uses the arguments of [160, Proposition 11.1.4] and its subsequent remark. We just shall highlight the key points that make those arguments work under our assumptions. First of all, if (r, s) ∈ ℤ2d and 𝒳 is separated, then the number of sampling points of 𝒳 in (r, s) + [0, 1]2d is bounded independently of (r, s). Secondly, for x ∈ 𝒳 such that x ∈ (r, s) + [0, 1]2d , 󵄨󵄨 󵄨 󵄨󵄨F(x)󵄨󵄨󵄨μ(x) ≤ C‖F ⋅ T(r,s) χ[0,1]2d ‖L∞ μ(r, s).

6.1 Fourier integral operators | 341

Indeed, since F is everywhere defined on ℝ2d and lower semicontinuous, we have sup |f | = ess sup|f | on every box, whereas μ(x) ≤ Cμ(r, s) because μ is v-moderate and v is bounded on [0, 1]2d , see (2.7). Proof of Proposition 6.1.13. We shall prove (6.37). First, Lemma 1.2.29 for h = γ = Ψ0 , yields 󵄨 󵄨󵄨 󵄨󵄨VΨ(m󸀠 ,n) σ(z)󵄨󵄨󵄨 ≤ (|VΨ0 σ| ∗ |VΨ(m󸀠 ,n) Ψ0 |)(z),

z ∈ ℝ4d ,

so that 󵄨󵄨 󵄨󵄨VΨ

∑

(m󸀠 ,n)

(m󸀠 ,n󸀠 )∈Λ

≤ ∫ ℝ4d

≤ ∫ ℝ4d

󵄨 σ(zm,n,m󸀠 ,n󸀠 )󵄨󵄨󵄨

∑

󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨VΨ0 σ(zm,n,m󸀠 ,n󸀠 − w)󵄨󵄨󵄨󵄨󵄨󵄨VΨ(m󸀠 ,n) Ψ0 (w)󵄨󵄨󵄨 dw

∑

󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨VΨ0 σ(zm,n,m󸀠 ,n󸀠 − w)󵄨󵄨󵄨 sup 󵄨󵄨󵄨VΨ(m󸀠 ,n) Ψ0 (w)󵄨󵄨󵄨 dw 󸀠

(m󸀠 ,n󸀠 )∈Λ

(m󸀠 ,n󸀠 )∈Λ

≤ sup

∑

w∈ℝ4d (m󸀠 ,n󸀠 )∈Λ

≤ C sup

(m ,n)∈Λ

󵄨󵄨 󵄨 󵄨󵄨VΨ0 σ(zm,n,m󸀠 ,n󸀠 − w)󵄨󵄨󵄨 ∫

∑

ℝ4d

w∈ℝ4d (m󸀠 ,n󸀠 )∈Λ

󵄨 󵄨 sup 󵄨󵄨󵄨VΨ 󸀠 Ψ0 (w)󵄨󵄨󵄨 dw (m ,n)

(m󸀠 ,n)∈Λ

󵄨󵄨 󵄨 󵄨󵄨VΨ0 σ(zm,n,m󸀠 ,n󸀠 − w)󵄨󵄨󵄨,

where the last majorization is due to Lemma 6.1.14. Since ∑

(m󸀠 ,n󸀠 )∈Λ

󵄨󵄨 󵄨 󵄨󵄨VΨ0 σ(zm,n,m󸀠 ,n󸀠 − w)󵄨󵄨󵄨 ≤

∑

(m󸀠 ,n󸀠 )∈Λ

󵄨 󵄨 ̃ 󸀠 ,n󸀠 ,w )󵄨󵄨, sup 󵄨󵄨󵄨VΨ0 σ(u1 , zm,n,m 2 󵄨

u1 ∈ℝ2d

with 󸀠 󸀠 󸀠 ̃ 󸀠 ,n󸀠 ,w := (n − ∇x Φ(m , n), m − ∇ξ Φ(m , n)) − w2 , zm,n,m 2

w = (w1 , w2 ) ∈ ℝ4d ,

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we shall prove that ∑

(m󸀠 ,n󸀠 )∈Λ

󵄨 󵄨 ̃ 󸀠 ,n󸀠 ,w )󵄨󵄨 ≲ ‖σ‖ ∞,1 , sup 󵄨󵄨󵄨VΨ0 σ(u1 , zm,n,m M 2 󵄨

u1 ∈ℝ2d

(6.42)

uniformly with respect to (m, n) ∈ Λ, w2 ∈ ℝ2d . For every fixed (m, n), the set 𝒳 = 𝒳m,n,w2 , given by 2d

̃ 󸀠 ,n󸀠 ,w ; (m , n ) ∈ Λ, w2 ∈ ℝ 𝒳m,n,w2 = {zm,n,m }, 2 󸀠

󸀠

is separated, uniformly with respect to (m, n), w2 . Indeed, given (m󸀠1 , n󸀠1 ) ≠ (m󸀠2 , n󸀠2 ), if m󸀠1 ≠ m󸀠2 , 󵄨 󵄨 󵄨 󸀠 󸀠 󸀠 󸀠󵄨 ̃ ̃ 󸀠 ,n󸀠 ,w − zm,n,m 󸀠 ,n󸀠 ,w | ≥ 󵄨󵄨∇ξ Φ(m , n) − ∇ξ Φ(m , n)󵄨󵄨 ≥ C 󵄨󵄨m − m 󵄨󵄨 ≥ αC, |zm,n,m 1 2 2󵄨 󵄨 󵄨 󵄨 1 1 1 2 2 2 2

342 | 6 Fourier integral operators and applications to Schrödinger equations uniformly with respect to (m, n), w2 , because the mapping x �→ ∇ξ Φ(x, ξ ) has an inverse that is Lipschitz continuous, thanks to (6.7) and (6.8). On the other hand, if m󸀠1 = m󸀠2 , 󵄨 󸀠 󸀠󵄨 ̃ ̃ 󸀠 ,n󸀠 ,w − zm,n,m 󸀠 ,n󸀠 ,w | ≥ 󵄨󵄨n − n 󵄨󵄨 ≥ β, |zm,n,m 2󵄨 󵄨 1 1 1 2 2 2 2

∀(m, n) ∈ Λ, w2 ∈ ℝ2d .

Hence, 𝒳 is separated uniformly with respect to (m, n), w2 . Now, we apply Proposition 6.1.15 (with p = q = 1) to the function 󵄨 󵄨 F(u2 ) = sup 󵄨󵄨󵄨VΨ0 σ(u1 , u2 )󵄨󵄨󵄨, 2d u1 ∈ℝ

u2 ∈ ℝ2d ,

which is lower semicontinuous due to VΨ0 σ being continuous. We obtain ‖F|𝒳 ‖ℓ1 ≤ C‖F‖W(L∞ ,L1 ) = ‖VΨ0 σ‖W(𝒞,L∞,1 ) .

(6.43)

If the symbol σ is in M ∞,1 , by Lemma 2.4.15, the STFT VΨ0 σ belongs to the Wiener amalgam space W(ℱ L1 , L∞,1 ), and ‖VΨ0 σ‖W(𝒞,L∞,1 ) ≲ ‖VΨ0 σ‖W(ℱ L1 ,L∞,1 ) ≲ ‖σ‖M ∞,1 ‖Ψ0 ‖M 1 . The first inequality is due to ℱ L1 �→ 𝒞 and the inclusion relations between Wiener amalgam spaces. Combining this inequality with (6.43), we obtain (6.42), uniformly with respect to (m, n) and w2 , that is, (6.37) holds. The estimate (6.38) is obtained by similar arguments. Remark 6.1.16. We observe that the continuity on M 2 (ℝd ) = L2 (ℝd ) of FIOs as above, with symbols in M ∞,1 (ℝ2d ), is also proved in [35] by other methods.

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6.1.6 The case of quadratic phases: metaplectic operators In this section we briefly discuss the particular case of quadratic phases, namely phases of the type 1 1 Φ(x, η) = Ax ⋅ x + Bx ⋅ η + Cη ⋅ η + η0 ⋅ x − x0 ⋅ η, 2 2

(6.44)

where x0 , η0 ∈ ℝd , A, C are real symmetric d × d matrices and B is a real d × d nondegenerate matrix. It is easy to see that, if we take the symbol σ ≡ 1 and the phase (6.44), the corresponding FIO T is (up to a constant factor) a metaplectic operator (cf. Section 1.1.2). This can be seen by means of the easily verified factorization T = Mη0 UA DB ℱ −1 UC ℱ Tx0 ,

(6.45)

6.1 Fourier integral operators | 343

where UA and UC are the multiplication operators by eπiAx⋅x and eπiCη⋅η , respectively, and DB is the dilation operator f �→ f (B⋅). Each of the factors is (up to a constant factor) a metaplectic operator (see (1.15) and (1.16)), and so is T. The corresponding canonical map, defined by (6.9), is now an affine symplectic map. For the benefit of the reader, some important special cases are detailed in the table below. Operator

Phase Φ(x, η)

Canonical transformation

Tx0

(x − x0 ) ⋅ η

χ(y, η) = (y + x0 , η)

DB

Bx ⋅ η

χ(y, η) = (B−1 y,t Bη)

Mη0 UA

(η + η0 ) ⋅ x x ⋅ η + 21 Ax ⋅ x

χ(y, η) = (y, η + η0 )

χ(y, η) = (y, η + Ax)

We observe that there are metaplectic operators, as the Fourier transform, which cannot be expressed as FIOs of the type (6.12). Metaplectic operators are known to be bounded on Mvps (ℝd ), see, e. g., [160, Proposition 12.1.3]. In our case, this also follows from Theorem 6.1.8. Indeed, since χ is a bi-Lipschitz function, we have vs ∘ χ ≍ vs . Theorem 6.1.10 applies to quadratic phases whose affine symplectic map χ is (up to translations on the phase space) defined by an upper-triangular matrix, which happens precisely when A = 0. Indeed, we obtain the map χ by solving {

y = Bx + Cη − x0 , ξ = Ax + Bη + η0 .

The phase condition (6.8) here becomes det B ≠ 0,

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so that B is an invertible matrix and x = B−1 y − B−1 Cη + B−1 x0 . Whence the mapping χ : (y, η) �→ (x, ξ ) is given by x B−1 ( ) = ( −1 ξ AB

−B−1 C y B−1 x0 ) ( ) + ( ). B − AB−1 C η AB−1 x0 + η0

When A = 0, the phase Φ satisfies (6.29) and, consequently, the corresponding operators are bounded on all M p,q (ℝd ). This can also be verified by means of the factorization (6.45) (with A = 0). Indeed, the continuity of the operators Mη0 , Tx0 , and DB is easily seen, whereas that of the Fourier multiplier ℱ −1 UC ℱ was shown, e. g., in [163, Lemma 2.1]. On the other hand, generally the metaplectic operators are not bounded on p,q M (ℝd ) if p ≠ q. An example is given by the Fourier transform itself (see [120]). An instance which instead falls in the class of FIOs considered here is the following one.

344 | 6 Fourier integral operators and applications to Schrödinger equations Proposition 6.1.17. The multiplication UId is unbounded on M p,q (ℝd ), for every 1 ≤ p, q ≤ ∞, with p ≠ q. 2

Proof. We have UId f (x) = eπix f (x). For λ > 0, we consider the one-parameter family 2

2

−d/2 −π(1/λ)ξ ̂ e . For of Gaussian functions φ√λ (x) = e−πλx ∈ M p,q (ℝd ), so that φ √λ (ξ ) = λ every 1 ≤ p, q ≤ ∞, by Lemma 2.7.1, we have

‖φ√λ ‖M p,q

̂ = ‖φ √λ ‖W(ℱ Lp ,Lq ) ≍

d( p1 − 21 )

(λ + 1)

d

d

λ 2q (λ2 + λ) 2

( p1 − q1 )

.

2 ̂ (η) = (λ − i)−d/2 e−(π/(λ−i))ξ 2 , the same formula as Since Uφ√λ (x) = e−π(λ−i)x so that Uf above yields d

‖Uφ√λ ‖M p,q ≍

[(λ + 1)2 + 1] 2 d

( p1 − 21 ) d

λ 2q (λ2 + λ + 1) 2

( p1 − q1 )

.

As λ → 0, we have ‖Uφ√λ ‖M p,q ≍ λ

d − 2q

,

‖f√λ ‖M p,q ≍ λ

d − 2q − d2 ( p1 − q1 )

so that, if we assume ‖Uφ√λ ‖M p,q ≤ C‖φ√λ ‖M p,q , then 1/p − 1/q ≥ 0, that is, p ≤ q. Moreover, the same argument applies to the adjoint operator 2

U ∗ f (x) = e−πix f (x). Now we show that p = q. By contradiction, if U were bounded on M p,q (ℝd ), with p < q, its adjoint U ∗ would satisfy 󵄩󵄩 ∗ 󵄩󵄩 󸀠 󸀠 󵄩󵄩U f 󵄩󵄩M p ,q ≤ C‖f ‖M p󸀠 ,q󸀠 ,

∀f ∈ 𝒮 (ℝd ),

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with q󸀠 < p󸀠 , which is a contradiction to what we have just proved. 6.1.6.1 Boundedness of metaplectic operators on W (ℱ Lp , Lq ) spaces Let us continue the study of metaplectic operators, briefly defined in Section 1.1.2. We shall prove boundedness properties of the operator μ(𝒜) (with 𝒜 symplectic matrix, cf. (1.3)) on the Wiener amalgam spaces (or modulation spaces) W(ℱ Lp , Lq ), with 1 ≤ p, q ≤ ∞. These issues were initially proved in the work [70]. Explicit integral representations for classical metaplectic operators, extending the results already contained in the literature [105, 145], were given by Morsche and Oonincx in [284] and applied to energy localization problems and to fractional Fourier transforms. The novelty of [284], with respect to the classical works [105, 145], is the explicit integral representation of metaplectic operators, covering all possible cases of symplectic matrices. Indeed, the integral representation of metaplectic operators in [105, 145] covers only the cases of nonsingular upper-left or upper-right component

6.1 Fourier integral operators | 345

of the parameterizing matrix. For details, we refer the interested reader to the original work [284]. Here we also limit ourselves to metaplectic operators related to a symplectic matrix 𝒜 having nonsingular upper-left or upper-right component, following the d × d block decomposition in (1.4). The subsequent formulae for metaplectic operators can be found in [145, Theorems 4.51 and 4.53]. Proposition 6.1.18. Let 𝒜 be a symplectic matrix with the d × d block decomposition in (1.4) A C

𝒜=(

B ). D

(6.46)

(i) If det B ≠ 0 then μ(𝒜)f (x) = id/2 (det B)−1/2 ∫ e−πix⋅DB

−1

x+2πiy⋅B−1 x−πiy⋅B−1 Ay

f (y) dy.

(6.47)

(ii) If det A ≠ 0, −1

μ(𝒜)f (x) = (det A)−1/2 ∫ e−πix⋅CA

x+2πiξ ⋅A−1 x+πiξ ⋅A−1 Bξ

f ̂(ξ ) dξ .

(6.48)

The following hybrid formula will be also used in the sequel. Proposition 6.1.19. If 𝒜 is in Sp(d, ℝ), having block decomposition in (6.46) with det B ≠ 0 and det A ≠ 0, then −1

μ(𝒜)f (x) = (−i det B)−1/2 e−πix⋅CA

x

−1

(e−πiy⋅B

Ay

∗ f )(A−1 x).

(6.49)

Proof. By (6.48), we can write −1

μ(𝒜)f (x) = (det A)−1/2 e−πix⋅CA

x

∫ e2πiξ ⋅A

−1

−1

= (−i det B)−1/2 e−πix⋅CA

x

x

∫ e2πiξ ⋅A

−1 πiξ ⋅A−1 Bξ

ℱ (ℱ e −1

x

ℱ (e

−πiy⋅B−1 Ay

)f ̂(ξ ) dξ ∗ f )(ξ ) dξ ,

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where we used the formula (see [145, Theorem 2, page 257]) −1

ℱ (e

iπξ ⋅A−1 Bξ

−1/2 −πiy⋅B−1 Ay

)(y) = (−i det A−1 B)

e

.

Hence, from the Fourier inversion formula we obtain (6.49). The first continuity result is shown below. Theorem 6.1.20. Consider 𝒜 ∈ Sp(d, ℝ) with the block decomposition in (6.46) and 1 ≤ p ≤ q ≤ ∞. (i) If det B ≠ 0, then 󵄩󵄩 󵄩 󵄩󵄩μ(𝒜)f 󵄩󵄩󵄩W(ℱ Lp ,Lq ) ≲ α(𝒜, p, q)‖f ‖W(ℱ Lq ,Lp ) ,

(6.50)

󵄨 󵄨1/2 α(𝒜, p, q) = | det B|1/q−1/p−3/2 󵄨󵄨󵄨det(I + B∗ B)(B + iA)(B + iD)󵄨󵄨󵄨 .

(6.51)

where

346 | 6 Fourier integral operators and applications to Schrödinger equations (ii) If det A ≠ 0 and det B ≠ 0, then we have the estimate 󵄩 󵄩󵄩 󵄩󵄩μ(𝒜)f 󵄩󵄩󵄩W(ℱ L1 ,L∞ ) ≲ β(𝒜)‖f ‖W(ℱ L∞ ,L1 ) ,

(6.52)

󵄨1/2 󵄨 β(𝒜) = | det A|−3/2 | det B|−1 󵄨󵄨󵄨det(I + A∗ A)(B + iA)(A + iC)󵄨󵄨󵄨 .

(6.53)

with

Thanks to the previous theorem, we have all the tools to give the proof of Proposition 4.2.5 in Chapter 4. We recall its statement for the convenience of the reader. Proposition 6.1.21 (Proposition 4.2.5). Let 1 ≤ p, q ≤ ∞. If p ≠ q, the operator 𝒰 in (4.8) does not map W(ℱ Lp , Lq ) into itself. Proof. Consider the symmetric matrix B =

I B ( 02d I2d

1 2

0 Id 0

( Id

), and the symplectic matrix 𝒜 =

). It follows from the formula (6.48) that the operator 𝒰 is exactly the metaplectic operator associated with the matrix 𝒜. We write 𝒰 = μ(𝒜). Hence, as a consequence of Theorem 6.1.20, we deduce that 𝒰 and 𝒰 −1 = μ(𝒜−1 ) map W(ℱ Lq , Lp ) into W(ℱ Lp , Lq ) continuously for every 1 ≤ p ≤ q ≤ ∞. Now, assume p < q. Let f be any distribution in W(ℱ Lq , Lp ); therefore by the boundedness result we have just recalled, 𝒰 −1 f ∈ W(ℱ Lp , Lq ). Suppose, by contradiction, that 𝒰 maps W(ℱ Lp , Lq ) into itself. Then one would obtain f = 𝒰𝒰 −1 f ∈ W(ℱ Lp , Lq ), and therefore the inclusion W(ℱ Lq , Lp ) ⊆ W(ℱ Lp , Lq ), which is false. Suppose now p > q. Assume, by contradiction, that for every f ∈ W(ℱ Lp , Lq ) it turns out that 𝒰 f ∈ W(ℱ Lp , Lq ). Then we would have f = 𝒰 −1 𝒰 f ∈ W(ℱ Lq , Lp ), and therefore the inclusion W(ℱ Lp , Lq ) ⊆ W(ℱ Lq , Lp ), which is false. If the matrices A or B are symmetric, Theorem 6.1.20 can be sharpened as follows.

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Theorem 6.1.22. Consider 𝒜 ∈ Sp(d, ℝ) with the block decomposition in (6.46). (i) If det B ≠ 0, B∗ = B, with eigenvalues λ1 , . . . , λd , 1 ≤ p ≤ q ≤ ∞, then 󵄩󵄩 󵄩 󸀠 󵄩󵄩μ(𝒜)f 󵄩󵄩󵄩W(ℱ Lp ,Lq ) ≲ α (𝒜, p, q)‖f ‖W(ℱ Lq ,Lp ) ,

(6.54)

where 󵄨 󵄨1/2 α󸀠 (𝒜, p, q) = 󵄨󵄨󵄨det(B + iA)(B + iD)󵄨󵄨󵄨 d

μ1 (p,q)−1/2

× ∏(max{1, |λj |}) j=1

μ2 (p,q)−1/2

(min{1, |λj |})

,

(6.55)

with μ1 and μ2 defined in (2.108) and (2.109), respectively. (ii) If det A ≠ 0, det B ≠ 0, and A∗ = A with eigenvalues ν1 , . . . , νd , then 󵄩󵄩 󵄩 󸀠 󵄩󵄩μ(𝒜)f 󵄩󵄩󵄩W(ℱ L1 ,L∞ ) ≲ β (𝒜)‖f ‖W(ℱ L∞ ,L1 ) ,

(6.56)

6.1 Fourier integral operators | 347

with 󵄨1/2 󵄨 β󸀠 (𝒜) = | det B|−1 󵄨󵄨󵄨det(B + iA)(A + iC)󵄨󵄨󵄨 d

−1/2

× ∏(max{1, |νj |}) j=1

−3/2

(min{1, |νj |})

.

(6.57)

We now prove Theorems 6.1.20 and 6.1.22. We need the following preliminary result, improving Proposition 4.3.3. Lemma 6.1.23. Let R be a d × d real symmetric matrix, and f (y) = e−πiRy⋅y . Then 󵄨 󵄨1/2 ‖f ‖W(ℱ L1 ,L∞ ) = 󵄨󵄨󵄨det(I + iR)󵄨󵄨󵄨 .

(6.58)

Proof. We first compute the short-time Fourier transform of f , with respect to the win2 dow g(y) = e−πy . We have 2

Vg f (x, ξ ) = ∫ e−2πiy⋅ξ e−iπRy⋅y e−πy dy 2

= e−πx ∫ e−2πiy⋅(ξ +ix)−π(I+iR)y⋅y dy 2

−1/2 −π(I+iR)−1 (ξ +ix)⋅(ξ +ix)

= e−πx (det(I + iR))

e

,

where we used [145, Theorem 1, page 256]. Hence 󵄨󵄨 󵄨 󵄨 󵄨−1/2 −π(I+R2 )−1 (ξ +Rx)⋅(ξ +Rx) , 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 = 󵄨󵄨󵄨det(I + iR)󵄨󵄨󵄨 e and, performing the change of variables (I + R2 )

−1/2

(ξ + Rx) = y,

with dξ = | det(I + R2 )|1/2 dy, we obtain

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2 1/2 󵄨 󵄨−1/2 󵄨 󵄨1/2 ∫ Vg f (x, ξ ) dξ = 󵄨󵄨󵄨det(I + iR)󵄨󵄨󵄨 (det(I + R2 )) ∫ e−πy dy = 󵄨󵄨󵄨det(I + iR)󵄨󵄨󵄨 .

ℝd

(6.59)

ℝd

The last equality follows from (I + iR) = (I + R2 )(I − iR)−1 , so that det(I + iR)−1 = det(I + R2 )−1 det(I − iR). Now, relation (6.58) is proved by taking the supremum with the respect to x ∈ ℝd in (6.59). Proof of Theorem 6.1.20. (i) We use the expression of μ(𝒜)f in formula (6.47). The estimates below are obtained by using (in this order): Proposition 2.4.21 with Lemma 6.1.23, the estimate (2.107), Proposition 2.4.13, and, finally, Proposition 2.4.21 combined with Lemma 6.1.23 again: 󵄩󵄩 󵄩 󵄩 −1/2 󵄩 −πix⋅DB−1 x −1 −πiy⋅B−1 Ay ℱ (e f )(B−1 x)󵄩󵄩󵄩W(ℱ Lp ,Lq ) 󵄩󵄩μ(𝒜)f 󵄩󵄩󵄩W(ℱ Lp ,Lq ) = | det B| 󵄩󵄩󵄩e −1 󵄩 󵄩 ≤ | det B|−1/2 󵄩󵄩󵄩e−πix⋅DB x 󵄩󵄩󵄩W(ℱ L1 ,L∞ )

348 | 6 Fourier integral operators and applications to Schrödinger equations −1 󵄩 󵄩 × 󵄩󵄩󵄩(ℱ −1 (e−πiy⋅B Ay f ))B−1 󵄩󵄩󵄩W(ℱ Lp ,Lq )

1/2 󵄨 󵄨1/2 ≲ | det B|1/q−1/p−1/2 (det(B∗ B + I)) 󵄨󵄨󵄨det(I + iDB−1 )󵄨󵄨󵄨 −1 󵄩 󵄩 × 󵄩󵄩󵄩ℱ −1 (e−πiy⋅B Ay f )󵄩󵄩󵄩W(ℱ Lp ,Lq ) 1/2

≲ | det B|1/q−1/p−1/2 (det(B∗ B + I)) −1 󵄩 󵄩 × 󵄩󵄩󵄩e−πiy⋅B Ay f 󵄩󵄩󵄩W(ℱ Lq ,Lp )

det(I + iDB−1 )|1/2

≲ α(𝒜, p, q)‖f ‖W(ℱ Lq ,Lp )

with α(𝒜, p, q) given by (6.51). (ii) In this case, we use formula (6.49). Then proceeding like in case (i), we majorize as follows: 󵄩󵄩 󵄩 󵄩 −1/2 󵄩 −πix⋅CA−1 x −πiy⋅B−1 Ay (e ∗ f )(A−1 x)󵄩󵄩󵄩W(ℱ L1 ,L∞ ) 󵄩󵄩μ(𝒜)f 󵄩󵄩󵄩W(ℱ L1 ,L∞ ) = | det B| 󵄩󵄩󵄩e −1 󵄩 󵄩 ≤ | det B|−1/2 󵄩󵄩󵄩e−πix⋅CA x 󵄩󵄩󵄩W(ℱ L1 ,L∞ ) −1 󵄩 󵄩 × 󵄩󵄩󵄩(e−πiy⋅B Ay ∗ f )A−1 󵄩󵄩󵄩W(ℱ L1 ,L∞ ) 1/2 󵄨 󵄨1/2 ≲ | det B|−1/2 | det A|−1 (det(A∗ A + I)) 󵄨󵄨󵄨det(I + iCA−1 )󵄨󵄨󵄨 −1 󵄩 󵄩 × 󵄩󵄩󵄩e−πiy⋅B Ay ∗ f 󵄩󵄩󵄩W(ℱ L1 ,L∞ ) ≲ β(𝒜)‖f ‖W(ℱ L∞ ,L1 ) ,

where the last row is due to (2.83), with β(𝒜) defined in (6.53). Proof of Theorem 6.1.22. The proof uses the same arguments as in Theorem 6.1.20. Here, the estimate (2.107) is replaced by (2.130). Besides, the index relation (2.127) is applied in the final step. In details, 󵄩󵄩 󵄩 󵄩󵄩μ(𝒜)f 󵄩󵄩󵄩W(ℱ Lp ,Lq ) −1 −1 󵄩 󵄩 󵄩 󵄩 ≤ | det B|−1/2 󵄩󵄩󵄩e−πix⋅DB x 󵄩󵄩󵄩W(ℱ L1 ,L∞ ) 󵄩󵄩󵄩(ℱ −1 (e−πiy⋅B Ay f ))B−1 󵄩󵄩󵄩W(ℱ Lp ,Lq ) Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

d

󵄨 󵄨1/2 ≲ ∏ |λj |−1/2 󵄨󵄨󵄨det(I + iDB−1 )(I + iB−1 A)󵄨󵄨󵄨 j=1

d

μ1 (p󸀠 ,q󸀠 )

× ∏(max{1, |λj |−1 }) j=1

μ2 (p󸀠 ,q󸀠 )

(min{1, |λj |−1 })

‖f ‖W(ℱ Lq ,Lp )

󵄨 󵄨1/2 = 󵄨󵄨󵄨det(B + iD)(B + iA)󵄨󵄨󵄨 d

μ1 (p,q)−1/2

× ∏(max{1, |λj |}) j=1

μ2 (p,q)−1/2

(min{1, |λj |})

‖f ‖W(ℱ Lq ,Lp ) ,

that is, case (i) is proved. Case (ii) indeed is not an improvement of (6.52), but is just (6.52) rephrased in terms of the eigenvalues of A.

6.2 Sparsity of Gabor representation of Schrödinger propagators | 349

Remark 6.1.24. The above theorems require the condition det B ≠ 0. However, in some special cases with det B = 0, the previous results stay valid and can be used to obtain estimates between Wiener amalgam spaces. For example, if 𝒜 = ( CI 0I ), with C = C ∗ , then μ(𝒜)f (x) = ±e−πiCx⋅x f (x) (see (1.16)), so that, for every 1 ≤ p, q ≤ ∞, Proposition 2.4.21 and identity (6.58) give d

󵄩 󵄩󵄩 2 1/4 󵄩󵄩μ(𝒜)f 󵄩󵄩󵄩W(ℱ Lp ,Lq ) ≲ ∏(1 + λj ) ‖f ‖W(ℱ Lp ,Lq ) , j=1

where the λj ’s are the eigenvalues of C. This estimate was also shown in [306, 14, 72].

6.2 Sparsity of Gabor representation of Schrödinger propagators In this section we shall apply Fourier integral operators to study problems arising in partial differential equations. In this framework we list some of the many contributions [35, 41, 43, 95, 114, 115, 173, 187, 188, 190] (see also the references therein). Our attention will be initially fixed on numerical aspects. The Fourier integral operator T object of this study is of the type (6.12), with a tame phase function Φ. The symbol σ is a function on ℝ2d satisfying condition (6.13). We recall that an important example of such an FIO T is the solution operator to the free particle Schrödinger equation i𝜕t u + Δu = 0 with initial condition u(0, x) = u0 (x), x ∈ ℝd , d ≥ 1. The solution u(t, x) is 2

u(t, x) = ∫ e2πi(x⋅η−2πtη ) û0 (η) dη,

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

ℝd

that is, u(t, x) = Tt u0 (x), where T = Tt is the FIO with phase Φ(x, η) = x ⋅ η − 2πtη2 and symbol σ ≡ 1. More generally, one can replace the Laplacian Δ by an operator H given by the 1 Δ + πx2 or H = Weyl quantization of a quadratic form on ℝ2d . For instance, H = − 4π 1 2 − 4π Δ − πx . Let us emphasize that, as opposed to Hörmander’s FIOs [114, 115, 190] where the phase function is assumed homogeneous of degree 1 with respect to the dual variables, these FIOs possess quite different properties of the phase, basic example being a quadratic form in x and ξ . Consider a Gabor frame {gm,n }m,n , with gm,n (x) = e2πin⋅x g(x − m), (m, n) ∈ Λ = d αℤ × βℤd , α, β > 0 and window g ∈ 𝒮 (ℝd ). In Theorem 6.1.2 it is shown that an FIO T with the features mentioned above has Gabor matrix satisfying −2N 󵄨 󵄨 |Tm󸀠 ,n󸀠 ,m,n | = 󵄨󵄨󵄨⟨Tgm,n , gm󸀠 ,n󸀠 ⟩󵄨󵄨󵄨 ≤ CN ⟨χ(m, n) − (m󸀠 , n󸀠 )⟩ ,

∀N ∈ ℕ,

350 | 6 Fourier integral operators and applications to Schrödinger equations where χ is the canonical transformation generated by the phase function Φ. Using this result, we shall exhibit the sparsity of the matrix Tm󸀠 ,n󸀠 ,m,n . Namely, let a be any column or row of the matrix, and let |a|n be the nth largest entry of the sequence a. Then for each M > 0, |a|n satisfies |a|n ≤ CM n−M . Notice that Hörmander’s FIOs, arising as solution operators to Cauchy problems for hyperbolic differential equations, have been widely studied from the numerical point of view in the works [41, 43, 42, 173], where the authors prove that curvelet and shearlet frames almost diagonalize such operators and provide their sparsest possible representations. Our results show that Gabor frames are the right choice of frames when studying Schrödinger-type operators. Furthermore, differently from the frames studied in [41, 43, 42, 173], Gabor frames may be employed to characterize (quasi-)Banach and Hilbert spaces different from the L2 case. Consequently, boundedness properties of FIOs acting on these spaces may be obtained via the continuity properties of the infinite matrix Tm󸀠 ,n󸀠 ,m,n . Before detailing these issues, we give a flavor of the related results. Given f ∈ M p (ℝd ), 0 < p ≤ ∞ (see Definition 2.7.6 for quasi-Banach modulation spaces), we infer a formula to compute Tf by nonlinear approximation, namely a formula which recovers Tf via a finite sum of Gabor atoms gm,n and an error term. Setting ν := (m, n), bν := ⟨f , gm,n ⟩, we consider a non-increasing rearrangement |bν1 | ≥ |bν2 | ≥ ⋅ ⋅ ⋅. For N ∈ ℕ and B ≥ 1, introduce the index sets JN = {ν1 , . . . , νN },

󵄨 󵄨 IN,B = {λ ∈ Λ : ∃ν ∈ JN , 󵄨󵄨󵄨λ − χ(ν)󵄨󵄨󵄨 ≤ B},

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where χ is the canonical transformation generated by Φ. Then we show that (Theorem 6.2.5) Tf = ∑ ∑ Tλ,ν ⟨f , gν ⟩gλ + eN,B , λ∈IN,B ν∈JN

where the error eN,B satisfies the estimate ‖eN,B ‖M q ≤ CBd max{1,1/q}−N ‖f ‖M q + N 1/q−1/p ‖f ‖M p , 󸀠

for every 0 < p < q ≤ ∞, N 󸀠 ∈ ℕ, with N 󸀠 > d max{1, 1/q} and the constant C > 0 is independent of f , B, N. Hence, the error term decays faster than any negative (fixed) power of N and B, as N, B → +∞. This is easily seen by fixing N 󸀠 conveniently large and p small enough, provided that f ∈ M p (ℝd ). As a side preliminary result, boundedness properties of the FIO T on M p (ℝd ), 0 < p < 1 (Theorem 6.2.1) are obtained.

6.2 Sparsity of Gabor representation of Schrödinger propagators | 351

The last part of the section focuses on the solution u(t, x) to the Schrödinger equa2 tion for the free particle case with initial datum given by u0 (x) = e2πim⋅x e−π|x−n| , where (m, n) ∈ αℤd × βℤd . Choosing a Gabor frame {gm,n }m,n , (m, n) ∈ αℤd × βℤd , αβ < 1, gen2

erated by the Gaussian g(x) = e−πx , the entries of {Tm󸀠 ,n󸀠 ,m,n }m󸀠 ,n󸀠 ,m,n can be computed explicitly, see (6.67). Using Maple software, several numerical examples concerning the time–frequency concentration of the solution u(t, x) varying in time are exhibited, according to the law (6.43). Indeed, they demonstrate the accuracy of (6.43). Moreover, numerical experiments on the coefficient decay, achieved in dimension d = 1 and d = 2, display the same results as for the curvelet case in [42]. We conclude with further open problems related to this theory. First, the offdiagonal decay of the matrix Tm󸀠 ,n󸀠 ,m,n could be used in the finite section method [166] or in the adaptive application of operators [51]. Secondly, an interesting issue is the resolution of the wave front set of a distribution using Gabor frames (see [205]), instead of shearlets frames as done in [215]. We shall come back to Gabor wave front sets in Section 6.6. p

6.2.1 Continuity of FIOs on Mμ (ℝd ), 0 < p < 1 The continuity of FIOs on the modulation spaces Mμp , 1 ≤ p ≤ ∞, associated with a weight function μ ∈ ℳvs (ℝ2d ), s ≥ 0, was studied in Section 6.1.3. Here we study the remaining cases of the quasi-Banach spaces Mμp (ℝd ), 0 < p < 1. The outcome will be used in the subsequent part to deal with the nonlinear approximation problem.

Theorem 6.2.1. Consider a tame phase function Φ and a symbol satisfying (6.13). Let 0 ≤ s < 2N − 2d/p, 0 < p < 1, and μ ∈ ℳvs (ℝ2d ). Then T extends to a continuous p operator from Mμ∘χ (ℝd ) into Mμp (ℝd ).

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Proof. We set Tm󸀠 ,n󸀠 ,m,n = ⟨Tgm,n , gm󸀠 ,n󸀠 ⟩ and use Theorem 3.2.37. For T = Cγ ∘ Tm󸀠 ,n󸀠 ,m,n ∘ Dγ , the following diagram is commutative: p Mμ∘χ

T-

Mμp Dγ

Cγ

6 ?

p ℓμ∘χ

Tm󸀠 ,n󸀠 ,m,n

- ℓμp

where T is viewed as an operator with dense domain 𝒮 (ℝd ). Whence it is enough to p prove the continuity of the infinite matrix Tm󸀠 ,n󸀠 ,m,n from ℓμ∘χ into ℓμp . 󸀠 󸀠 We apply Lemma 6.1.7 (ii) (with λ = (m , n ), ν = (m, n)) to the matrix Km󸀠 ,n󸀠 ,m,n = Tm󸀠 ,n󸀠 ,m,n

μ(m󸀠 , n󸀠 ) . μ(χ(m, n))

352 | 6 Fourier integral operators and applications to Schrödinger equations That is, we need to prove sup ∑ |Km󸀠 ,n󸀠 ,m,n |p < ∞. m,n

m󸀠 ,n󸀠

In view of (6.23), we have |Km󸀠 ,n󸀠 ,m,n |p ≲ ⟨χ(m, n) − (m󸀠 , n󸀠 )⟩

(−2N+s)p

≲ ⟨χ(m, n) − (m󸀠 , n󸀠 )⟩

(−2N+s)p

(μ(m󸀠 , n󸀠 ))p ⟨χ(m, n) − (m󸀠 , n󸀠 )⟩sp (μ(χ(m, n)))p ,

where the quotient in the second-to-last row above is bounded because μ is vs -moderate. Since, by assumption, (−2N + s)p < −2d, the series ∑ ⟨χ(m, n) − (m󸀠 , n󸀠 )⟩

(−2N+s)p

m󸀠 ,n󸀠

converges, with sum uniformly bounded with respect to (m, n), yielding the desired result.

6.2.2 Sparsity of the Gabor matrix and nonlinear approximation Given an FIO T as above and f ∈ M q (ℝd ), we are going to present a suitable formula to compute Tf by nonlinear approximation, namely we shall expresses Tf as a finite sum of Gabor atoms, modulo an error whose size is estimated in terms of its time–frequency concentration. Consider g ∈ 𝒮 and 𝒢 (g, α, β) = {gm,n = Tm Mn g, (m, n) ∈ Λ = αℤd × βℤd }, a Gabor frame for L2 (ℝd ). Assume the symbol σ fulfills (6.13) for every N ∈ ℕ. Set λ = (m󸀠 , n󸀠 ) ∈ Λ and ν = (m, n) ∈ Λ. Define

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Tλ,ν = ⟨Tgν , gλ ⟩. Then Theorem 6.1.5 shows that, for every N 󸀠 ∈ ℕ, −N 󸀠

|Tλ,ν | ≤ CN 󸀠 ⟨λ − χ(ν)⟩

.

(6.60)

This gives at once that the Gabor matrix Tλ,ν is sparse, in the sense clarified by the following proposition Proposition 6.2.2. The Gabor matrix Tλ,ν is sparse. Namely, let a be any column or row of the matrix, and let |a|n be the nth largest entry of the sequence a. Then for each M > 0, |a|n satisfies |a|n ≤ CM n−M .

6.2 Sparsity of Gabor representation of Schrödinger propagators | 353

Proof. We have n1/p ⋅ |a|n ≤ ‖a‖ℓp , for every 0 < p ≤ ∞. Hence it suffices to prove that every column or row is in ℓp for arbitrarily small p. This follows immediately from (6.60) (and was also already verified in the proof of Theorem 6.1.8). We now turn to the nonlinear approximation problem. First, some preliminary results are required. Given a nonincreasing sequence bj , j = 1, 2, . . ., and 0 < p ≤ ∞, define 1/p

σN,p (b) = ( ∑ |bj |p ) j≥N+1

.

The quantity above satisfies the following estimate. Lemma 6.2.3. For every 0 < p < q ≤ ∞, it turns out that σN,p (b) ≤ CN 1/p−1/q ‖b‖ℓq ,

∀b ∈ ℓq ,

for a constant C = Cq depending only on q. Proof. It follows from [111], see also [167, Lemma 5], that q1

∞

(∑[j1/q−1/p σj,p (b)] j=1

j

1/q

)

≤ Cq ‖b‖ℓq .

Hence the desired estimate is a consequence of this elementary fact: if a numerical series ∑∞ j=1 aj converges and the sequence aj is nonincreasing and nonnegative, then aj ≤ C/j for some constant C > 0.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Let us show the easy estimate below. Lemma 6.2.4. For γ > d, ∑

λ∈Λ:|λ−y|≥B

⟨λ − y⟩−γ ≲ (1 + B)d−γ ,

uniformly with respect to B ≥ 0, y ∈ ℝd . Precisely, the constant implicit in the notation ≲ depends only on d, γ, and the Gabor constants α, β. Proof. By shifting the index in the summation, we can suppose that y ∈ α[0, 1)d × β[0, 1)d . Then ⟨λ − y⟩ ≍ ⟨λ⟩, and it is enough to prove that ∑

λ∈Λ:|λ|≥B

⟨λ⟩−γ ≲ (1 + B)d−γ .

354 | 6 Fourier integral operators and applications to Schrödinger equations Now, we can consider the partition of ℝd defined by the boxes ∏dj=1 [kj , kj + 1), kj ∈ ℤ, (having diameter √d). The number of points of Λ which fall into any such a box is uniformly bounded and we also have ⟨λ⟩ ≍ ⟨x⟩, if λ, x belong to the same box. Hence, let {Q󸀠 } be the boxes that intersect the set |x| ≥ R. Then ∑

λ∈Λ:|λ|≥B

⟨λ⟩−γ = ∑ ∑ ⟨λ⟩−γ ≲ ∑ inf󸀠 ⟨x⟩−γ Q󸀠 λ∈Λ∩Q󸀠

Q󸀠

≲ ∑ ∫ ⟨x⟩−γ dx ≤ Q󸀠 Q󸀠

x∈Q

∫

⟨x⟩−γ dx ≤ (1 + B)d−γ ,

|x|≥B−√d

as an explicit computation shows. Given N ∈ ℕ and B ≥ 1, we associate to any function f ∈ Mμq (ℝd ) two index sets JN and IN,B defined as follows. Let bν be the sequence of weighted Gabor coefficients of f , namely bν = ⟨f , gν ⟩μ(ν), and let |bν1 | ≥ |bν2 | ≥ ⋅ ⋅ ⋅ be a nonincreasing rearrangement. We then define JN = {ν1 , . . . , νN } and 󵄨 󵄨 IN,B = {λ ∈ Λ : ∃ν ∈ JN , 󵄨󵄨󵄨λ − χ(ν)󵄨󵄨󵄨 ≤ B}, where χ is the canonical transformation generated by Φ. The approximation result can be formulated as follows. p Theorem 6.2.5. Let 0 < p < q ≤ ∞, and f ∈ Mμ∘χ (ℝd ), with μ ∈ ℳvs (ℝ2d ), s ≥ 0. Let JN and IN,B be the index sets associated to f as above. Then for every N 󸀠 , N ∈ ℕ, with N 󸀠 > d max{1, 1/q} + s, and B ≥ 1, we have

Tf = ∑ ∑ Tλ,ν ⟨f , gν ⟩gλ + eN,B , λ∈IN,B ν∈JN

where the error eN,B satisfies the estimate 1/q−1/p p . q + N ‖f ‖Mμ∘χ ‖eN,B ‖Mμq ≲ Bd max{1,1/q}+s−N ‖f ‖Mμ∘χ

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󸀠

Here the constant implicit in the notation ≲ is independent of f , B and N. Hence, given f , the error term decays faster of any negative fixed power of N and B, as N, B → +∞. This is seen by fixing N 󸀠 conveniently large and p small enough. Proof. Let cν = ⟨f , gν ⟩. We have (1) (2) eN,B = eN,B + eN,B

where (1) eN,B = ∑ ∑ Tλ,ν cν gλ λ∈I̸ N,B ν∈JN

6.2 Sparsity of Gabor representation of Schrödinger propagators | 355

and (2) eN,B = ∑ ∑ Tλ,ν cν gλ . λ ν∈J̸ N

We shall show the estimates 󵄩󵄩 (1) 󵄩󵄩 q d max{1,1/q}+s−N 󸀠 q ‖f ‖Mμ∘χ 󵄩󵄩eN,B 󵄩󵄩Mμ ≲ B

(6.61)

󵄩󵄩 (2) 󵄩󵄩 q 1/q−1/p p , ‖f ‖Mμ∘χ 󵄩󵄩eN,B 󵄩󵄩Mμ ≲ N

(6.62)

and

where, as in the statement, the constant implicit in the notation ≲ is independent of f , B and N. Let us prove (6.61). We have 1/q 󵄨󵄨 󵄨󵄨q 󵄩󵄩 (1) 󵄩󵄩 q 󵄨 󵄨 q 󵄩󵄩eN,B 󵄩󵄩Mμ ≲ ( ∑ 󵄨󵄨󵄨 ∑ Tλ,ν cν 󵄨󵄨󵄨 μ(λ) ) . 󵄨 󵄨󵄨 λ∈I̸ N,B 󵄨ν∈JN

Now, we want to apply Lemma 6.1.7 with Kλ,ν := Tλ,ν μ(λ)/μ(χ(ν)). To this end, we observe, as in the proof of Theorem 6.1.8, that |Tλ,ν |μ(λ) −N 󸀠 +s ≲ ⟨λ − χ(ν)⟩ . μ(χ(ν)) Hence, Lemma 6.1.7, together with Lemma 6.2.4 (applied to γ = q(N 󸀠 − s), if 0 < q ≤ 1, and γ = N 󸀠 − s, if 1 ≤ q ≤ ∞), and the equivalence |λ − χ(ν)| ≍ |χ −1 (λ) − ν| (which follows from the bi-Lipschitz continuity of χ) give d/q+s−N q , ‖c‖ℓμ∘χ 󵄩󵄩 (1) 󵄩󵄩 q {B 󵄩󵄩eN,B 󵄩󵄩Mμ ≲ { d+s−N 󸀠 q , B ‖c‖ℓμ∘χ { 󸀠

0 < q ≤ 1, 1 ≤ q ≤ ∞.

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q q , we deduce (6.61). Since ‖c‖ℓμ∘χ ≲ ‖f ‖Mμ∘χ

q Let us now verify (6.62). From the continuity of the matrix Tλ,ν from ℓμ∘χ into ℓμq it follows that 1/q 󵄨󵄨 󵄨󵄨q 󵄩󵄩 (2) 󵄩󵄩 q 󵄨 󵄨 q 󵄩󵄩eN,B 󵄩󵄩Mμ ≲ (∑󵄨󵄨󵄨 ∑ Tλ,ν cν 󵄨󵄨󵄨 μ(λ) ) 󵄨 󵄨󵄨 λ 󵄨ν∈J̸ N q

1/q

≲ ( ∑ |cν |q μ(χ(ν)) ) ν∈J̸ N

.

By Lemma 6.2.3, applied to the nonincreasing rearrangement of {cν μ(χ(ν))}, this last expression is p . ≲ N 1/q−1/p ‖c‖ℓμ∘χ p p , we deduce (6.62). Since ‖c‖ℓμ∘χ ≲ ‖f ‖Mμ∘χ

356 | 6 Fourier integral operators and applications to Schrödinger equations 6.2.3 Application to the solution of the Schrödinger equation We now turn our attention to some important examples of the class of FIOs studied here, represented by solution operators to the Cauchy problem for Schrödinger equations. 6.2.3.1 The free particle Consider the Cauchy problem for the Schrödinger equation in (5.11) The explicit formula for the solution is given by (5.21) and (5.22). The solution u(t, x) can be expressed as an FIO u(t, x) = ∫ e2πix⋅η ℱ (Kt ∗ u0 )(η) dη ℝd

= ∫ e2πix⋅η ℱ (Kt )(η)û0 (η) dη ℝd

2

= ∫ e2πi(x⋅η−2πtη ) û0 (η) dη, ℝd 2

2

since, as an easy computation shows, ℱ ((ai)−d/2 e−πx /(ai) )(η) = e−πaiη , a ∈ ℝ \ {0}. Whence u(t, x) = Tu0 (x), where T is the FIO with phase Φ(x, ξ ) = x ⋅ η − 2πtη2 and symbol σ ≡ 1. The smooth bi-Lipschitz canonical transformation χ associated with the phase Φ is given by χ(y, η) = (y + 4πtη, η). Hence, the matrix decay (6.60), for λ = (m󸀠 , n󸀠 ), ν = (m, n), is given by

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󵄨 󵄨 −N 󸀠 |Tλ,ν | ≤ CN 󸀠 (1 + 󵄨󵄨󵄨(m󸀠 − (m + 4πtn), n󸀠 − n)󵄨󵄨󵄨) .

(6.63)

In what follows we shall compute the matrix entries explicitly. 2

Lemma 6.2.6. For c ∈ ℂ, c ≠ 0, consider the function ϕ(c) (x) = e−πcx , x ∈ ℝd . For every c1 , c2 ∈ ℂ, with Re c1 ≥ 0, Re c2 > 0, n ∈ ℤd , we have ϕ(c1 ) ∗ Mn ϕ(c2 ) (x) = (c1 + c2 )−d/2 ϕ

(c

1

1 +c2

)

(n)M

c1 n c1 +c2

ϕ

c c

( c 1+c2 ) 1

2

(x).

(6.64)

Proof. This is a straightforward computation. Set z 2 = z12 + ⋅ ⋅ ⋅ + zn2 , for z = (z1 , . . . , zn ) ∈ ℂd . Then 2

2

(ϕ(c1 ) ∗ Mn ϕ(c2 ) )(x) = ∫ e−πc1 (x−t) +2πit⋅n−πc2 t dt 2

= e−πc1 x ∫ e−π[(c1 +c2 )t

2

−2(c1 x+in)⋅t]

dt

6.2 Sparsity of Gabor representation of Schrödinger propagators | 357

=e

−π(c1 x2 −

(c1 x+in)2 c1 +c2

= (c1 + c2 )−d/2 e

)

−π

∫e

−π((c1 +c2 )1/2 t−

c1 c2 x2 +n2 c1 +c2

e

c1 1 +c2

2πi c

c1 x+in (c1 +c2 )1/2

n⋅x

2

)

dt

,

as desired. 2

Corollary 6.2.7. Let Kt (x) be the kernel in (5.22), then for g0 = e−πx , n ∈ ℤd , we have (Kt ∗ Mn g0 )(x) =

2 n2 it 1 − 4π − π x2 1+4πit M n e 1+4πit e . 1+4πit (1 + 4πit)d/2

(6.65)

Proof. Since (Kt ∗ Mn g0 )(x) =

−1 1 (ϕ((4πit) ) ∗ Mn ϕ(1) )(x), d/2 (4πit)

the result follows at once from Lemma 6.2.6. Corollary 6.2.8. Let u(t, x) be the solution of the Cauchy problem (5.22), with u0 (x) = 2 Mn Tm e−πx . Then u(t, x) = (1 + 4πit)−d/2 e−

4π 2 tn⋅(2m+in) 1+4πit

M

n 1+4πit

π

2

Tm e− 1+4πit x .

(6.66)

Proof. The convolution operator commutes with translations and Mn Tm e2πim⋅n Tm Mn , so that

=

u(t, x) = (Kt ∗ Mn Tm g0 )(x) = e2πim⋅n Tm (Kt ∗ Mn g0 )(x). Using (6.65), 4π 2 n2 it

u(t, x) = (1 + 4πit)−d/2 e2πim⋅n e− 1+4πit Tm M 4π 2 n2 it

n 1+4πit

π

e− 1+4πit x

n

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= (1 + 4πit)−d/2 e2πim⋅n e− 1+4πit e−2πim⋅ 1+4πit M = (1 + 4πit)−d/2 e−

4π 2 tn⋅(2m+in) 1+4πit

M

n 1+4πit

π

n 1+4πit

2

π

Tm e− 1+4πit x

2

2

Tm e− 1+4πit x ,

that is, (6.66) holds. Hence, we have a more explicit formula for the matrix entries {Tm󸀠 ,n󸀠 ,m,n }m󸀠 ,n󸀠 ,m,n = ⟨T(Mn Tm g0 ), Mn󸀠 Tm󸀠 g0 ⟩. In fact, Tm󸀠 ,n󸀠 ,m,n = (1 + 4πit)−d/2 e−

4π 2 tn⋅(2m+in) 1+4πit

⟨M

n 1+4πit

π

2

Tm e− 1+4πit |⋅| , Mn󸀠 Tm󸀠 g0 ⟩.

(6.67)

We now present some numerical examples concerning (5.11). We choose the Gabor 2 frame {Mn Tm g0 }, with (m, n) ∈ ℤd × (1/2)ℤd and g0 (x) = e−πx .

358 | 6 Fourier integral operators and applications to Schrödinger equations First, we study the one-dimensional case (d = 1), taking the initial datum u0 (x) = 2 M1 T2 g0 (x) = e2πix e−π(x−2) . Figures 6.1, 6.2, 6.3, and 6.4 represent the magnitude of the STFT of the solution u(t, x) for t = 0, 1, 2, 10, respectively. Figures 6.5, 6.6, 6.7, and 6.8 exhibit the contour plots of the STFT magnitude related to the same instant time as before.

Figure 6.1: Time t = 0.

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Figure 6.2: Time t = 1.

Figure 6.3: Time t = 2.

6.2 Sparsity of Gabor representation of Schrödinger propagators | 359

Figure 6.4: Time t = 10.

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Figure 6.5: Time t = 0.

Figure 6.6: Time t = 1.

360 | 6 Fourier integral operators and applications to Schrödinger equations

Figure 6.7: Time t = 2.

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Figure 6.8: Time t = 10.

Indeed, the propagator eitΔ moves the time–frequency concentration of the solution u(t, x) when t increases, according to the law in (6.63) (with ν = (m, n) = (2, 1)). Figures 6.9, 6.10, 6.11, and 6.12 represent the magnitude of the coefficients (sorted in decreasing order) of one column of the matrix Tm󸀠 ,n󸀠 ,m,n , namely that obtained by fixing m = n = 0. Notice that choosing the column corresponding to m = n = 0 does not represent a loss of generality. Indeed, as one can verify by a direct computation, Tm󸀠 ,n󸀠 ,m,n = Tm󸀠 −m−4πtn,n󸀠 −n,0,0 , for every (m, n), (m󸀠 , n󸀠 ) ∈ ℤd × (1/2)ℤd , so that for all columns essentially the same figures as those presented can be drawn. Finally, Figure 6.13 shows a similar analysis in dimension d = 2, for the column corresponding to m = (0, 0), n = (0, 0). Notice that we need about 106 coefficients to reach the threshold 10−25 . The same type of decay was obtained in [42, Figure 15] for the curvelet representation of the wave propagator.

6.2 Sparsity of Gabor representation of Schrödinger propagators | 361

Figure 6.9: Time t = 0.

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Figure 6.10: Time t = 1.

6.2.3.2 The harmonic oscillator This is the case of the Cauchy problem for the Schrödinger equation below 𝜕u 1 2 {i 𝜕t − 4π Δu + π|x| u = 0, { {u(0, x) = u0 (x).

(6.68)

The solution is the metaplectic operator (hence FIO) given by [70, 145] 1

u(t, x) = (cos t)−d/2 ∫ e2πi[ cos t x⋅η+ ℝd

tan t 2 (x +η2 )] 2

f ̂(η) dη,

t ≠

π + kπ, k ∈ ℤ. 2

(6.69)

362 | 6 Fourier integral operators and applications to Schrödinger equations

Figure 6.11: Time t = 2.

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Figure 6.12: Time t = 10.

Here u(t, x) = Tt , where Tt is an FIO with symbol σ ≡ 1 and phase Φt (x, η) =

tan t 2 1 x⋅η+ (x + η2 ). cos t 2

The canonical transformation χ = χt is then obtained by solving (6.9), so that (cos t)I (sin t)I

χt (y, η) = ( Since

(− sin t)I y )( ). (cos t)I η

1 󵄨󵄨 󵄨 2 > 0, 󵄨󵄨det 𝜕x,η Φt (x, η)󵄨󵄨󵄨 = | cos t|d

t ≠

π + kπ, k ∈ ℤ, 2

(6.70)

6.3 Wiener algebras of Fourier integral operators | 363

Figure 6.13: Coefficient magnitude at time t = 1 for d = 2.

the assumptions of Theorem 6.1.5 are fulfilled, and we get the estimate 󵄨󵄨 󵄨 󵄨 󸀠 󸀠 󵄨 −N 󵄨󵄨⟨Tgm,n , gm󸀠 ,n󸀠 ⟩󵄨󵄨󵄨 ≤ CN (1 + 󵄨󵄨󵄨(cos t)m + (sin t)n − m , −(sin t)m + (cos t)n − n 󵄨󵄨󵄨) ,

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for every N ∈ ℕ, if t ≠

π 2

+ kπ, k ∈ ℤ.

Remark 6.2.9. Gabor frames have also been successfully employed to study the semiclassical analysis of linear Schrödinger equations. In this context, time-dependent smooth Hamiltonians with at most quadratic growth have been considered and higher order parametrices for the corresponding Schrödinger equations have been constructed. Nonlinear parametrices, in the spirit of the nonlinear approximation, are presented and numerical experiments exhibited in [18]. A follow-up of the previous work is contained in [61]. Such papers represent an attempt to create a systematic common ground for semiclassical and time–frequency analysis. These two different areas combined together provide interesting outcomes for Schrödinger-type equations. In fact, continuity results of both Schrödinger propagators and their asymptotic solutions are obtained on ℏ-dependent Banach spaces, the semiclassical version of modulation spaces. Continuity properties of more general Fourier integral operators (FIOs) and their sparsity are also investigated. We refer the interested reader to [61]. Further applications of Gabor decompositions to Schrödinger and semilinear parabolic equations are contained in [85, 234].

6.3 Wiener algebras of Fourier integral operators In this section we will consider algebras of Fourier integral operators and their properties, in particular Wiener algebras.

364 | 6 Fourier integral operators and applications to Schrödinger equations The original Wiener’s lemma is a classical statement about absolutely convergent series, see [311] and [312]. Let us first recall this statement for the benefit of the reader. Let A([0, 1]) denote the space of functions defined on the interval [0, 1] having absolutely convergent Fourier series expansion. That is, every f in A([0, 1]) is given by f (x) = ∑ ck e2πikx k∈ℤ

where ∑ |ck | < ∞.

k∈ℤ

Recall that every f in A([0, 1]) is continuous. Moreover, the Fourier coefficients (ck ) corresponding to a given f are unique. This allows one to impose a norm on A([0, 1]) by setting ‖f ‖A([0,1]) = ∑ |ck |. k∈ℤ

It can be shown that A([0, 1]) is complete with respect to this norm, making this a Banach space. Moreover, if f and g belong to A([0, 1]), then so does fg, and ‖fg‖A([0,1]) ≤ ‖f ‖A([0,1]) ‖g‖A([0,1]) . This multiplicative structure, along with its usual interaction with addition and scalar multiplication, makes A([0, 1]) a Banach algebra. Wiener’s lemma states that the inverse of each nonzero element in A([0, 1]) is again in A([0, 1]).

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Theorem 6.3.1 (Wiener’s lemma). Suppose that f is a function in A([0, 1]) and is nonzero on [0, 1]. Then 1/f belongs to A([0, 1]). In a more general setting, Wiener’s property represents nowadays one of the driving forces in the development of Banach algebra theory. Let us first fix our attention on pseudodifferential operators, which we may express in the Kohn–Nirenberg form σ(x, D)f (x) = ∫ e2πix⋅η σ(x, η)f ̂(η) dη. The best known result about Wiener’s property for pseudodifferential operators is maybe that in [9], see also the subsequent contributions of [32, 299]. It concerns 0 symbols σ in the Hörmander’s class S0,0 (ℝ2d ), i. e., smooth functions on ℝ2d such that, 2d for every multiindex α and every z ∈ ℝ , 󵄨󵄨 α 󵄨 󵄨󵄨𝜕z σ(z)󵄨󵄨󵄨 ≤ Cα .

(6.71)

6.3 Wiener algebras of Fourier integral operators | 365

The corresponding pseudodifferential operators form a subalgebra of ℒ(L2 (ℝd )), usually denoted by L00,0 . The standard symbolic calculus concerning the principal part of symbols of products does not hold, nevertheless, Wiener’s lemma is still valid: If σ(x, D) is invertible in ℒ(L2 (ℝd )), then its inverse is again a pseudodifferential operator 0 with symbol in S0,0 (ℝ2d ), hence belonging to L00,0 . In the absence of a symbolic calculus, such a version of Wiener’s lemma seems to be the minimal property required of any reasonable algebra of pseudodifferential operators. A subalgebra 𝒜 of operators in ℒ(L2 (ℝd )) that satisfies Wiener’s lemma and is thus closed under inversion is usually called spectrally invariant or inverse-closed, or sometimes a Wiener algebra. We take the point of view of time–frequency analysis and signal processing, for which Wiener’s lemma provides an important justification of the engineering practice to model σ(x, D)−1 as an almost diagonal matrix (this is a peculiar property of pseudodifferential operators, see Theorem 6.3.3 in the sequel). Actually, in the applications to signal processing, the symbol σ(x, η) is not always 0 smooth, and it is convenient to use some generalized version of S0,0 (ℝ2d ) [271]. The spaces that do this job are the modulation classes M ∞,1 (ℝ2d ) (the Sjöstrand class) and ∞ ∞,∞ M1⊗v (ℝ2d ) = M1⊗v (ℝ2d ), s s

vs (z) = ⟨z⟩s = (1 + |z|2 )

s/2

, z ∈ ℝ2d ,

0 with the parameter s ∈ [0, ∞). The Hörmander symbol class S0,0 (ℝ2d ) can be characterized by means of modulation spaces as follows (see, for example, [165]): 0 ∞ S0,0 = ⋂ M1⊗v (ℝ2d ). s s≥0

(6.72)

∞ As s → 2d, the symbols in M1⊗v (ℝ2d ) have less regularity, until in the maximal s

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space M ∞,1 (ℝ2d ) even differentiability is lost.

Theorem 6.3.2 ([266]). The pseudodifferential operators with a symbol in M ∞,1 (ℝ2d ) form a Wiener subalgebra of ℒ(L2 (ℝd )), the so-called Sjöstrand algebra. The symbol ∞ classes M1⊗v (ℝ2d ) with s > 2d provide a scale of Wiener subalgebras of the Sjöstrand s algebra, and their intersection coincides with L00,0 .

In this section we construct and investigate Wiener subalgebras consisting of Fourier integral operators and generalize Sjöstrand’s theory in [266, 267] to FIOs. We will consider FIOs in (6.12), that we call FIO of type I. When necessary, we will also write the FIO T as Tf (x) = TI,Φ,σ f (x) = ∫ e2πiΦ(x,η) σ(x, η)̂f (η) dη. ℝd

(6.73)

366 | 6 Fourier integral operators and applications to Schrödinger equations 0 We first assume σ ∈ S0,0 (ℝ2d ) and consider a tame phase function Φ. If Φ(x, η) = xη then χ = Id, and we recapture the pseudodifferential operators in the Kohn–Nirenberg form. It is an easy exercise to show that the L2 -adjoint of an FIO of type I is an FIO of type II

Tf (x) = TII,Φ,τ f (x) = ∫ e−2πi[Φ(y,η)−x⋅η] τ(y, η)f (y) dy dη.

(6.74)

ℝ2d

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For the L2 -boundedness of such FIOs of type I and II, see, for example, [3]. It is worth to observe that in contrast to the standard setting of Hörmander [191], we argue globally on ℝd , our basic examples being the propagators for Schrödingertype equations. The second remark is that operators of the reduced form (6.73) or (6.74) do not form an algebra, quite in line with the calculus of Hörmander in [191]. The composition of FIOs requires heavier machinery and is addressed, for example, in [187], in the case when symbol and phase belong to the more restrictive Shubin class of [265]. As a minimal objective, we will present a cheap definition of a subalgebra of ℒ(L2 (ℝd )) containing FIOs of type I, II, and hence L00,0 , and prove the Wiener property for this class. As a more ambitious objective, we will extend our analysis to the case when the ∞ symbol σ belongs to the symbol class M1⊗v (ℝ2d ), and we will define a corresponds ing scale of Wiener algebras of FIOs. The new algebras of FIOs will be constructed by means of Gabor frames and the decay properties of the corresponding Gabor matrix outside the graph of a symplectic map χ. Let us recall that for the case of pseudodifferential operators the Gabor discretization provides an equivalent characterization of the Sjöstrand algebra in Theorem 6.3.2. Theorem 6.3.3 ([162, 165]). Assume that 𝒢 (g, Λ) is a frame for L2 (ℝd ) with g ∈ 𝒮 (ℝd ) and fix s > 2d. Then the following statements are equivalent for a distribution σ ∈ 𝒮 󸀠 (ℝd ): ∞ (i) σ ∈ M1⊗v (ℝ2d ). s (ii) There exists C > 0 such that 󵄨󵄨 󵄨 −s 󵄨󵄨⟨σ(x, D)π(λ)g, π(μ)g⟩󵄨󵄨󵄨 ≤ C⟨μ − λ⟩ ,

∀λ, μ ∈ Λ.

(6.75)

0 Hence, the assumption σ ∈ S0,0 (ℝd ) is equivalent to (6.75) being satisfied for all s ≥ 0.

Moreover, σ ∈ M ∞,1 (ℝ2d ) if and only if there exists a sequence h ∈ ℓ1 (Λ) such that |⟨σ(x, D)π(λ)g, π(μ)g⟩| ≤ h(λ − μ). The above theorem gives a precise meaning to the statement that the Gabor matrix of a pseudodifferential operators is almost diagonal, or that pseudodifferential operators are almost diagonalized by Gabor frames.

6.3 Wiener algebras of Fourier integral operators | 367

Definition 6.3.4. Let χ be a transformation satisfying B1 and B2, and s ≥ 0. Fix g ∈

𝒮 (ℝd ) \ {0} and let 𝒢 (g, Λ) be a Parseval frame for L2 (ℝd ). We say that a continuous linear operator T : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) is in the class FIO(χ, s) if its Gabor matrix satisfies

the decay condition

−s 󵄨 󵄨󵄨 󵄨󵄨⟨Tπ(λ)g, π(μ)g⟩󵄨󵄨󵄨 ≤ C⟨μ − χ(λ)⟩ ,

∀λ, μ ∈ Λ.

(6.76)

The class FIO(Ξ, s) = ⋃χ FIO(χ, s) is the union of these classes where χ runs over the set of all transformations satisfying B1, B2. Note that we do not require assumption B3. We shall show that this definition does not depend on the choice of the Gabor frame (cf. Lemma 6.3.6). If χ = Id, the identity operator, then the corresponding Fourier integral operators are simply pseudodifferential operators. The decomposition of an FIO with respect to a Gabor frame provides a technique to settle the following issues: (i) Boundedness of T on L2 (ℝd ) (Theorem 6.3.7). If s > 2d and T ∈ FIO(χ, s), then T can be extended to a bounded operator on L2 (ℝd ). (ii) The algebra property (Theorem 6.3.9). For i = 1, 2, si > 2d, s = min(s1 , s2 ), T (i) ∈ FIO(χi , si ) ⇒ T (1) T (2) ∈ FIO(χ1 ∘ χ2 , s). (iii) Wiener property (Theorem 6.3.10). If s > 2d, T ∈ FIO(χ, s), and T is invertible on L2 (ℝd ), then T −1 ∈ FIO(χ −1 , s). These three properties can be summarized neatly by saying that the union ⋃ FIO(χ, s)

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χ

is a Wiener subalgebra of ℒ(L2 (ℝd )) consisting of FIOs. Although it is impossible to do justice to the vast literature on Fourier integral operators, let us mention some of the contributions that are most related to these ideas. From the formal point of view, this approach is very similar to that in [30, 31] and [279], where FIO(χ, ∞) was treated. Instead of Gabor frames, in [30, 31] partitions of unity of the Weyl–Hörmander calculus are used, whereas in [279] the Bargmann transform is the main tool. The boundedness and composition of FIOs are treated in [35, 94, 291, 187]. Here we follow strictly [68]. 6.3.1 Wiener algebra properties We first present an equivalence between continuous decay conditions and the decay of the discrete Gabor matrix for a linear operator 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ).

368 | 6 Fourier integral operators and applications to Schrödinger equations Theorem 6.3.5. Let T be a continuous linear operator 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) and χ a canonical transformation which satisfies properties B1 and B2 following Definition 6.1.1. Let 𝒢 (g, Λ) be a Parseval frame with g ∈ 𝒮 (ℝd ) and s ≥ 0. Then the following properties are equivalent: (i) There exists C > 0 such that −s 󵄨 󵄨󵄨 󵄨󵄨⟨Tπ(z)g, π(w)g⟩󵄨󵄨󵄨 ≤ C⟨w − χ(z)⟩ ,

∀z, w ∈ ℝ2d .

(6.77)

∀λ, μ ∈ Λ.

(6.78)

(ii) There exists C > 0 such that −s 󵄨󵄨 󵄨 󵄨󵄨⟨Tπ(λ)g, π(μ)g⟩󵄨󵄨󵄨 ≤ C⟨μ − χ(λ)⟩ ,

Proof. The implication (i) �⇒ (ii) is obvious. (ii) �⇒ (i). The argument is borrowed from the proof of a similar result for pseudodifferential operators in [162, Theorem 3.2]. Let C be a relatively compact fundamental domain of the lattice Λ. Given z, w ∈ ℝ2d , we can write w = λ + u, z = μ + u󸀠 for unique λ, μ ∈ Λ, u, u󸀠 ∈ C, and 󵄨󵄨 󵄨 󵄨 󵄨 󸀠 󵄨󵄨⟨Tπ(z)g, π(w)g⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨Tπ(μ)π(u )g, π(λ)π(u)g⟩󵄨󵄨󵄨. Now we expand π(u)g = ∑ν∈Λ ⟨π(u)g, π(ν)g⟩π(ν)g, and likewise π(u󸀠 )g. Since Vg g ∈ 𝒮 (ℝ2d ), the coefficients in this expansion satisfy 󵄨 󵄨 󵄨 󵄨 sup󵄨󵄨󵄨⟨π(u)g, π(ν)g⟩󵄨󵄨󵄨 = sup󵄨󵄨󵄨Vg g(ν − u)󵄨󵄨󵄨 ≲ sup⟨ν − u⟩−N ≲ (sup⟨u⟩N )⟨ν⟩−N ≲ ⟨ν⟩−N u∈C

u∈C

u∈C

u∈C

for every N. Using (6.78), we now obtain that 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󸀠 󸀠 󸀠 󵄨󵄨⟨Tπ(z)g, π(w)g⟩󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨⟨Tπ(μ + ν )g, π(λ + ν)g⟩󵄨󵄨󵄨󵄨󵄨󵄨⟨π(u )g, π(ν )g⟩󵄨󵄨󵄨󵄨󵄨󵄨⟨π(u)g, π(ν)g⟩󵄨󵄨󵄨 ν,ν󸀠 ∈Λ

≲ ∑ ⟨λ + ν − χ(μ + ν󸀠 )⟩ ⟨ν󸀠 ⟩ −s

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ν,ν󸀠 ∈Λ

−N

⟨ν⟩−N .

Since vs (z)−1 = ⟨z⟩−s is vs -moderate, we majorize the main term of the sum as ⟨λ + ν − χ(μ + ν󸀠 )⟩

−s

= ⟨λ − χ(μ) + ν − χ(μ + ν󸀠 ) + χ(μ)⟩

−s s

s

≤ ⟨λ − χ(μ)⟩ ⟨χ(μ + ν󸀠 ) − χ(μ) − ν⟩ ≲ ⟨λ − χ(μ)⟩ ⟨ν󸀠 ⟩ ⟨ν⟩s , −s

−s

since χ(μ + ν󸀠 ) − χ(μ) = 𝒪(ν󸀠 ) by the Lipschitz property of χ. Hence, −s −s 󵄨󵄨 󵄨 󸀠 s−N s−N ≲ ⟨λ − χ(μ)⟩ , 󵄨󵄨⟨Tπ(z)g, π(w)g⟩󵄨󵄨󵄨 ≲ ⟨λ − χ(μ)⟩ ∑ ⟨ν ⟩ ⟨ν⟩ ν,ν󸀠 ∈Λ

for N ∈ ℕ large enough.

6.3 Wiener algebras of Fourier integral operators | 369

Finally, with λ = w − u, μ = z − u󸀠 , we apply the above estimate again and obtain ⟨λ − χ(μ)⟩

−s

= ⟨w − u − χ(z − u󸀠 ) − χ(z) + χ(z)⟩

−s

≤ ⟨w − χ(z)⟩ ⟨u + χ(z − u󸀠 ) − χ(z)⟩ −s

s

s

≲ ⟨w − χ(z)⟩ ⟨u⟩s ⟨u󸀠 ⟩ ≲ ⟨w − χ(z)⟩ , −s

−s

since supu∈C ⟨u⟩s < ∞. Thus we have proved that |⟨Tπ(z)g, π(w)g⟩| ≲ ⟨w − χ(z)⟩−s . In view of the equivalence between (6.77) and (6.78), we observe that the class of FIOs in Definition 6.3.4 can be defined either using (6.78) or (6.77). We first prove that Definition 6.3.4 does not depend on the choice of the Gabor frame. Lemma 6.3.6. The definition of FIO(χ, s) is independent of the Gabor frame 𝒢 (g, Λ).

Proof. Let 𝒢 (φ, Λ󸀠 ) be a Gabor frame with a window φ ∈ 𝒮 (ℝd ) and a possibly different lattice Λ󸀠 . As in the proof of Theorem 6.3.5, we expand π(λ)φ = ∑ν∈Λ ⟨π(λ)φ, π(ν)g⟩π(ν)g with convergence in 𝒮 (ℝd ) and likewise π(μ)φ, where λ, μ ∈ Λ󸀠 . Consequently, Tπ(λ)φ = ∑ν∈Λ ⟨π(λ)φ, π(ν)g⟩Tπ(ν)g converges weak∗ in 𝒮 󸀠 (ℝd ) and the following identity is well-defined: ⟨Tπ(λ)φ, π(μ)φ⟩ = ∑ ⟨π(λ)φ, π(ν)g⟩⟨Tπ(ν)g, π(μ)φ⟩ ν∈Λ

= ∑ ∑ ⟨π(λ)φ, π(ν)g⟩⟨Tπ(ν)g, π(ν󸀠 )g⟩⟨π(μ)φ, π(ν󸀠 )g⟩. ν∈Λ ν󸀠 ∈Λ

Since φ, g ∈ 𝒮 (ℝd ), the characterization (3.32) and the covariance property (1.50) imply that |⟨π(λ)φ, π(ν)g⟩| = |⟨φ, π(ν−λ)g⟩| ≲ ⟨ν−λ⟩−N for ν ∈ Λ, λ ∈ Λ󸀠 and every N ≥ 0. After substituting these estimates and choosing N large enough, we obtain the majorization −s 󸀠 −N 󵄨󵄨 󵄨 −N 󸀠 󵄨󵄨⟨Tπ(λ)φ, π(μ)φ⟩󵄨󵄨󵄨 ≲ ∑ ∑ ⟨ν − λ⟩ ⟨ν − χ(ν)⟩ ⟨ν − μ⟩ ν∈Λ ν󸀠 ∈Λ

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≲ ∑ ⟨ν − λ⟩−N ⟨μ − χ(ν)⟩ ν∈Λ −1

≲ ⟨χ (μ) − λ⟩

−s

−s

≍ ∑ ⟨ν − λ⟩−N ⟨χ −1 (μ) − ν⟩

−s

ν∈Λ −s

≍ ⟨μ − χ(λ)⟩ .

As in [165], the definition of classes of operators by their Gabor matrices facilitates the investigation of their basic properties. In line with Sjöstrand’s original program, we next derive the boundedness, composition rules, and properties of the inverse operator in the classes FIO(χ, s). Theorem 6.3.7. Let s > 2d and T ∈ FIO(χ, s). Then T extends to a bounded operator on ℳp (ℝd ), 1 ≤ p ≤ ∞, and in particular on L2 (ℝd ) = M 2 (ℝd ).

Proof. Let 𝒢 (g, Λ) be a Parseval frame with g ∈ 𝒮 (ℝd ). Since the frame operator Sg = Dg Cg is the identity operator, we can write T as T = Dg Cg TDg Cg , where Dg and Cg

370 | 6 Fourier integral operators and applications to Schrödinger equations are the synthesis and coefficient operators, respectively. Since 𝒢 (g, Λ) is a frame and g ∈ 𝒮 (ℝd ), Cg is bounded from M p (ℝd ) to ℓp (Λ) (cf. Theorem 3.2.32) and Dg = Cg∗ is

bounded from ℓp (Λ) to ℳp (ℝd ) (cf. Theorem 3.2.33). The operator Cg TDg maps sequences to sequences, and its matrix K is precisely the Gabor matrix of T, namely, Kμ,λ = ⟨Tπ(λ)g, π(μ)g⟩. Since by assumption |⟨Tπ(λ)g, π(μ)g⟩| ≲ ⟨μ − χ(λ)⟩−s and s > 2d, Schur’s test implies that the matrix K representing Cg TDg is bounded on ℓp (Λ). Consequently, T is bounded on ℳp (ℝd ) for 1 ≤ p ≤ ∞.

Remark 6.3.8. The boundedness of the operator T ∈ FIO(χ, s), s > 2d, fails in general on the modulation spaces M p,q (ℝ2d ), with p ≠ q. A concrete counterexample is the FIO of type I in (6.92) below, cf. Proposition 6.1.17. Next we show that the class FIO(Ξ, s), for s > 2d, is an algebra. Theorem 6.3.9. If T (i) ∈ FIO(χi , si ) with si > 2d, i = 1, 2, then the composition T (1) T (2) is in FIO(χ1 ∘ χ2 , s) with s = min(s1 , s2 ). Consequently, the class FIO(Ξ, s) = ⋃χ FIO(χ, s) is an algebra with respect to the composition of operators. Proof. We write the product T (1) T (2) as T (1) T (2) = Dg Cg T (1) T (2) Dg Cg = Dg (Cg T (1) Dg )(Cg T (2) Dg )Cg . Then Cg T (1) T (2) Dg is the Gabor matrix of T (1) T (2) with entries Kμ,λ = ⟨T (1) T (2) π(λ)g, π(μ)g⟩ and Cg T (i) Dg , i = 1, 2, is the Gabor matrix of T (i) with entries (i) Kμ,λ = ⟨T (i) π(λ)g, π(μ)g⟩.

Thus the composition of operators corresponds to the multiplication of their Gabor (i) matrices. Using the decay estimates for Kμ,λ and s = min(s1 , s2 ), we estimate the size

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of the Gabor matrix of T (1) T (2) as follows:

(1) (2) |Kμ,λ | = ∑ Kμ,ν Kν,λ ≲ ∑ ⟨μ − χ1 (ν)⟩ ν∈Λ

≲

∑ ⟨χ1−1 (μ) ν∈Λ

ν∈Λ −s

−s1

⟨ν − χ2 (λ)⟩

−s2

− ν⟩ ⟨ν − χ2 (λ)⟩ . −s

Since vs−1 = ⟨ν⟩−s ∈ ℓ1 (Λ) is subconvolutive for s > 2d, that is, vs−1 ∗ vs−1 ≤ Cvs−1 , cf. [160, Lemma 11.1.1 (d)], the last expression is dominated by ⟨χ1−1 (μ) − χ2 (λ)⟩−s ≍ ⟨μ − χ1 (χ2 (λ))⟩−s . By Proposition 6.3.7, T (1) and T (2) extend to bounded operators on L2 (ℝd ), so that the product T (1) T (2) is well-defined and bounded on L2 (ℝd ).

6.4 FIOs of type I

| 371

Finally, we consider the invertibility in the class FIO(χ, s) and show that FIO(Ξ, s) is inverse-closed in ℬ(L2 (ℝd )) for s > 2d. Theorem 6.3.10. Let T ∈ FIO(χ, s) with s > 2d. If T is invertible on L2 (ℝd ), then T −1 ∈ FIO(χ −1 , s). Consequently, the algebra FIO(Ξ, s) is inverse-closed in ℒ(L2 (ℝd )). Proof. We first show that the adjoint operator T ∗ belongs to the class FIO(χ −1 , s). Indeed, since χ is bi-Lipschitz, we have 󵄨󵄨 ∗ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨⟨T π(λ)g, π(μ)g⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨π(λ)g, T(π(μ)g)⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨T(π(μ)g, π(λ)g)⟩󵄨󵄨󵄨 ≲ ⟨λ − χ(μ)⟩

−s

≍ ⟨χ −1 (λ) − μ⟩ . −s

Hence, by Theorem 6.3.9, the operator P := T ∗ T is in FIO(Id, s) and satisfies the estimate |⟨Pπ(λ)g, π(μ)g⟩| ≲ ⟨λ − μ⟩−s , ∀λ, μ ∈ Λ. We now exploit the characterization for pseudodifferential operators contained in Theorem 6.3.3 and deduce that P is a pseudodifferential operator with a symbol ∞ in M1⊗v (ℝ2d ). Since T and therefore T ∗ are invertible on L2 (ℝd ), P is also invertible s on L2 (ℝd ). Now we apply Theorem 6.3.2 and conclude that the inverse P −1 is again ∞ a pseudodifferential operator with a symbol in M1⊗v (ℝ2d ). Hence P −1 is in FIO(Id, s). s Finally, using the algebra property of Theorem 6.3.9 once more, we obtain that T −1 = P −1 T ∗ is in FIO(χ −1 , s) and thus satisfies the estimate |⟨T −1 π(λ)g, π(μ)g⟩| ≲ ⟨χ −1 (λ) − μ⟩−s , ∀λ, μ ∈ Λ.

Combining Theorems 6.3.7, 6.3.9, and 6.3.10, we see that FIO(Ξ, s) with s > 2d is a Wiener subalgebra of ℒ(L2 (ℝd )) consisting of FIOs.

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6.4 FIOs of type I In this section we relate the Wiener algebra of FIOs in the preceding Section 6.3 to the FIOs of type I in Section 4.1. An evident connection is given by Theorem 6.1.5. Namely, assume T of the form (6.12) with symbol satisfying (6.13), for a given N ∈ ℕ, and a phase function Φ which fulfills A1, A2, and A3 in Definition 6.1.1 (hence associated to a canonical transformation χ satisfying B1, B2, and B3). Then from (6.23) we have that T ∈ FIO(χ, 2N); applying Theorem 6.3.7, we obtain that T is bounded on ℳp (ℝd ), 1 ≤ p ≤ ∞, provided s > 2N > 2d, that is, N > d, and we recapture Theorem 6.1.8 in the case of weight μ = 1. A natural question is whether an operator T ∈ FIO(χ, s), s ≥ 0, with χ satisfying B1, B2, and B3 can be written as an FIO of type I. A positive answer will be given in Theorem 6.4.10, where we shall characterize precisely the corresponding class of ∞ symbols, given by M1⊗v (ℝ2d ). When s = 2N, this includes strictly the class of symbols s satisfying (6.13). Note that by the Schwartz’ kernel theorem every continuous linear operator T : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) can be written as an FIO of type I with a given phase Φ(x, η) for some

372 | 6 Fourier integral operators and applications to Schrödinger equations symbol σ(x, η) in 𝒮 󸀠 (ℝ2d ). Hence, if T is a continuous linear operator 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) and χ satisfies B1, B2, and B3, then T = TI,Φχ ,σ is an FIO of type I with symbol σ and phase Φχ . As the first step we formulate the following result. Proposition 6.4.1. Let T = TI,Φ,σ be an FIO of type I with symbol σ ∈ 𝒮 󸀠 (ℝ2d ) and a phase Φ satisfying A1 and A2. If the Gabor matrix of T satisfies −s 󵄨󵄨 󵄨 󸀠 󸀠 󸀠 󸀠 󸀠 󵄨󵄨⟨Tπ(x, η)g, π(x , η )g⟩󵄨󵄨󵄨 ≤ C⟨∇x Φ(x , η) − η , ∇η Φ(x , η) − x⟩ ,

x, x󸀠 , η, η󸀠 ∈ ℝd , (6.79)

∞ for some C > 0 and s ≥ 0, then σ is in the generalized Sjöstrand class M1⊗v (ℝ2d ). s

In particular, if s > 2d, then we have σ ∈ M ∞,1 (ℝ2d ).

Proof. To set up notation, let Φ2,z be the remainder in the second order Taylor expansion of the phase Φ computed in (6.15) centered at z, that is, 1

Φ2,z (w) = 2 ∑ ∫(1 − t)𝜕α Φ(z + tw) dt |α|=2 0

wα α!

z, w ∈ ℝ2d ,

and set Ψz (w) = e2πiΦ2,z (w) g ⊗ ĝ (w).

(6.80)

With the notation above, the fundamental relation between the Gabor matrix of an FIO and the STFT of its symbol in (6.35), for g ∈ 𝒮 (ℝd ), becomes 󵄨󵄨 󵄨 󵄨 󵄨 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 󵄨󵄨⟨Tπ(x, η)g, π(x , η )g⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨VΨ(x󸀠 ,η) σ((x , η), (η − ∇x Φ(x , η), x − ∇η Φ(x , η)))󵄨󵄨󵄨. Writing u = (x󸀠 , η), v = (η󸀠 , x), (6.79) translates into

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−s 󵄨󵄨 󵄨 󵄨󵄨VΨu σ(u, v − ∇Φ(u))󵄨󵄨󵄨 ≤ C⟨v − ∇Φ(u)⟩ ,

and then into the estimate sup

(u,w)∈ℝ2d ×ℝ2d

󵄨 󵄨 ⟨w⟩s 󵄨󵄨󵄨VΨu σ(u, w)󵄨󵄨󵄨 < ∞.

Now, setting G = g ⊗ ĝ ∈ 𝒮 (ℝ2d ), we can write VG2 σ(u, v) = ∫ e−2πit⋅v σ(t)G2 (t − u) dt = ∫ e−2πit⋅v σ(t)e−2πiΦ2,u (t−u) G(t − u)e2πiΦ2,u (t−u) G(t − u) dt = ∫ e−2πit⋅v σ(t)Ψu (t − u)e2πiΦ2,u (t−u) G(t − u) dt

(6.81)

6.4 FIOs of type I

| 373

= ℱ (σTu Ψu ) ∗v ℱ (Tu (e2πiΦ2,u G))(v) = VΨu σ(u, ⋅) ∗ ℱ (Tu (e2πiΦ2,u G))(v).

(6.82)

2d 1 2d ∞ 2d Using (6.82) and the weighted Young’s inequality L∞ s (ℝ ) ∗ Ls (ℝ ) �→ Ls (ℝ ), we get

‖σ‖M ∞

2d )

1⊗vs (ℝ

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≍ sup󵄩󵄩󵄩VG2 σ(u, ⋅)󵄩󵄩󵄩L∞ ≲ sup󵄩󵄩󵄩VΨu σ(u, ⋅)󵄩󵄩󵄩L∞ sup󵄩󵄩󵄩ℱ (e2πiΦ2,u G)󵄩󵄩󵄩L1 . u

s

u

s

u

s

The first factor on the right-hand side is finite by (6.81). The second is finite because the set {e2πiΦ2,u G : u ∈ ℝ2d } is bounded in 𝒮 (ℝ2d ), and the embedding 𝒮 �→ ℱ L1s is ∞ continuous. This gives σ ∈ M1⊗v (ℝ2d ). s The last statement follows from the inclusion relations for modulation spaces in ∞,∞ Proposition 2.4.18. Namely, M1⊗v (ℝ2d ) �→ M ∞,1 (ℝ2d ) if and only if s > 2d. s

The next lemma clarifies further the relation between the phase Φ and the canonical transformation χ. Lemma 6.4.2. Consider a phase function Φ satisfying A1, A2, and A3. Then 󵄨󵄨 󵄨 󵄨 󸀠 󸀠󵄨 󵄨 󸀠 󸀠󵄨 󵄨 󸀠󵄨 󵄨󵄨∇x Φ(x , η)−η 󵄨󵄨󵄨+󵄨󵄨󵄨∇η Φ(x , η)−x󵄨󵄨󵄨 ≍ 󵄨󵄨󵄨χ1 (x, η)−x 󵄨󵄨󵄨+󵄨󵄨󵄨χ2 (x, η)−η 󵄨󵄨󵄨,

∀x, x 󸀠 , η, η󸀠 ∈ ℝd . (6.83)

Proof. The estimate ≳ was already proved in Lemma 6.1.4. For the converse estimate observe that x = ∇η Φ(χ1 (x, η), η) by definition of χ1 , hence 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󸀠 󵄨 󸀠 󸀠 󵄨󵄨∇η Φ(x , η) − x󵄨󵄨󵄨 = 󵄨󵄨󵄨∇η Φ(x , η) − ∇η Φ(χ1 (x, η), η)󵄨󵄨󵄨 ≤ C 󵄨󵄨󵄨x − χ1 (x, η)󵄨󵄨󵄨,

(6.84)

because of assumption (A2) on Φ. Since ∇x Φ(x󸀠 , η) = χ2 (∇η Φ(x󸀠 , η), η), the first term on the left-hand side of (6.83) can be estimated as 󵄨󵄨 󵄨 󵄨 󵄨 󸀠 󸀠󵄨 󸀠 󸀠󵄨 󵄨󵄨∇x Φ(x , η) − η 󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨χ2 (∇η Φ(x , η), η) − χ2 (x, η)󵄨󵄨󵄨 + 󵄨󵄨󵄨χ2 (x, η) − η 󵄨󵄨󵄨. Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Finally, the Lipschitz continuity of χ and (6.84) imply that 󵄨󵄨 󵄨 󸀠 󵄨 󵄨 󸀠 󵄨 󸀠 󸀠󵄨 󵄨󵄨∇x Φ(x , η) − η 󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨η − χ2 (x, η)󵄨󵄨󵄨 + C 󵄨󵄨󵄨x − χ1 (x, η)󵄨󵄨󵄨. We shall identify the abstract class FIO(χ, s) with a class of concrete Fourier integral operators. To chase this goal, we use an new characterization of the symbol ∞,1 ∞ classes M1⊗m and M1⊗m that is perfectly adapted to the investigation of Fourier integral operators with a tame phase Φ and was first introduced in [67]. The main technical work relies on showing that the set of windows Ψz in (6.80) possesses a joint time–frequency envelope as shown in Lemma 6.4.7 below. This property will allow us to replace the z-dependent family of windows Ψz by a single window in many estimates. Before proving the existence of a time–frequency envelope in Lemma 6.4.7 below, we first look at the phase factor e2πiΨ2,z occurring in (6.80).

374 | 6 Fourier integral operators and applications to Schrödinger equations Lemma 6.4.3. For every s ∈ ℝ, N ∈ ℕ, N >

s 2

+ d, and Ψ ∈ 𝒮 (ℝ2d ), we have

󵄨 󵄨 4d sup 󵄨󵄨󵄨VΨ e2πiΦ2,z 󵄨󵄨󵄨 ∈ L∞,1 v−4N ⊗vs (ℝ ),

z∈ℝ2d

(6.85)

with v−4N and vs being weight functions on ℝ2d , defined in (2.5). Proof. Let Ψ ∈ 𝒮 (ℝ2d ), then 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 2πiΦ2,z ̄ )e−2πiζ ⋅w dζ 󵄨󵄨󵄨. (u, w)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ e2πiΦ2,z (ζ ) Tu Ψ(ζ 󵄨󵄨VΨ e 󵄨󵄨 󵄨 󵄨󵄨 2d ℝ

Using the identity (1 − Δζ )N e−2πiζ ⋅w = ⟨2πw⟩2N e−2πiζ ⋅w , we integrate by parts and obtain 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨 󵄨󵄨 2πiΦ2,z N 2πiΦ2,z (ζ ) ̄ ))e−2πiζ ⋅w dζ 󵄨󵄨󵄨. (u, w)󵄨󵄨󵄨 = Tu Ψ(ζ 󵄨󵄨VΨ e 󵄨󵄨 ∫ (1 − Δζ ) (e 󵄨󵄨 2N 󵄨 ⟨2πw⟩ 󵄨 󵄨 ℝ2d

̄ )) can be expressed By means of Leibniz’s formula, the factor (1 − Δζ )N (e2πiΦ2,z (ζ ) Tu Ψ(ζ further as e2πiΦ2,z (ζ )

∑ |α|+|β|≤2N

β ̄ ), pα (𝜕Φ2,z (ζ ))(Tu 𝜕ζ Ψ)(ζ

where pα (𝜕Φ2,z (ζ )) is a polynomial of derivatives of Φ2,z of degree at most |α|. As a consequence of condition (6.7), we have |pα (𝜕Φ2,z (ζ ))| ≤ Cα ⟨ζ ⟩2|α| for every z ∈ ℝ2d with a constant independent of z. Moreover, the assumption Ψ ∈ 𝒮 (ℝ2d ) yields that β 󵄨 󵄨 sup 󵄨󵄨󵄨Tu 𝜕ζ Ψ̄ 󵄨󵄨󵄨 ≤ CN,s ⟨ζ − u⟩−ℓ

|β|≤2N

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for every ℓ ≥ 0. Consequently,

1 󵄨󵄨 󵄨 2πiΦ2,z (u, w)󵄨󵄨󵄨 ≲ ∫ 󵄨󵄨VΨ e ⟨2πw⟩2N

ℝ2d

≲

∑

⟨ζ ⟩2|α| ⟨ζ − u⟩−ℓ dζ

|α|+|β|≤2N

1 ∫ ⟨ζ ⟩4N ⟨ζ − u⟩−ℓ dζ ⟨2πw⟩2N ℝ2d

1 ≲ ⟨u⟩4N , ⟨2πw⟩2N

whenever ℓ > 4N + 2d. Since −2N + s < −2d by assumption, we obtain 󵄨 󵄨 ∫ sup sup 󵄨󵄨󵄨VΨ e2πiΦ2,z 󵄨󵄨󵄨(u, w)⟨u⟩−4N ⟨w⟩s dw < ∞, 2d 2d

ℝ2d

u∈ℝ

whence (6.85) follows.

z∈ℝ

6.4 FIOs of type I

| 375

Remark 6.4.4. Notice that the two weights v−4N and vs compensate each other in (6.85). It is easy to see that some form of compensation is necessary. For example, if Φ(ζ ) = ζ 2 /2, then Φ2,z (ζ ) = ζ 2 /2 independent of z. A direct computation shows 2

that the STFT of e2πiΦ2,z (ζ ) with Gaussian window Ψ(ζ ) = e−πζ has modulus, up to 2 4d constants, e−π(u1 −u2 ) /2 , u1 , u2 ∈ ℝ2d . This function belongs to L∞,1 v−4N ⊗vs (ℝ ) if and only if −4N + s < −2d. This is, in fact, better than the condition −2N + s < −2d in the assumptions, due to the fact that here the derivatives of order ≥ 1 of Φ2,z (ζ ) are bounded above by ⟨ζ ⟩, rather than ⟨ζ ⟩2 , as in the general case. This explains the presence of 2N instead of 4N. In the general case the best upper bound for the derivatives of Φ2,z (ζ ) is ⟨ζ ⟩2 , so that we conjecture the condition −2N + s < −2d to be sharp. Lemma 6.4.5. Let m ∈ ℳv (ℝd ), v, ν, w be weight functions on ℝd , and {fz : z ∈ ℝ2d } ⊆ 𝒮 󸀠 (ℝd ) be a set of distributions in 𝒮 󸀠 . If sup |Vφ fz | ∈ L∞,1 (ℝ2d ), ν−1 ⊗m

z∈ℝd

(6.86)

1 for given φ ∈ 𝒮 (ℝd ), then for every h ∈ Mwν⊗v (ℝd )

󵄨 󵄨 sup 󵄨󵄨󵄨Vφ2 (fz h)󵄨󵄨󵄨 ∈ L1w⊗m (ℝ2d ).

z∈ℝd

(6.87)

̄ ) by (1.24), we obtain the identity Proof. Since Vg f (x, ξ ) = (f ? ⋅ Tx g)(ξ Vφ2 (fz h)(x, ξ ) = ((fz Tx φ)(hTx φ))̂(ξ ) = (fz Tx φ)̂ ∗ξ (hTx φ)̂(ξ ), and consequently, 󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ⋅ Tx φ)̄ 󵄨󵄨󵄨(ξ ), ⋅ Tx φ)̄ 󵄨󵄨󵄨 ∗ξ 󵄨󵄨󵄨(h? 󵄨󵄨Vφ2 (fz h)(x, ξ )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨(fz? where the convolution is in the second variable ξ . Now set 󵄨 󵄨 󵄨 󵄨 ̄ F(x, ξ ) = sup 󵄨󵄨󵄨Vφ fz (x, ξ )󵄨󵄨󵄨 = sup 󵄨󵄨󵄨(fz ⋅ Tx φ)̂(ξ )󵄨󵄨󵄨 2d 2d z∈ℝ

z∈ℝ

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and H(x, ξ ) = |Vφ h(x, ξ )|. Then 󵄨 󵄨 sup 󵄨󵄨󵄨Vφ2 (fz h)(x, ξ )󵄨󵄨󵄨 ≤ (F ∗ξ H)(x, ξ ).

z∈ℝ2d

Finally, ‖F ∗ξ H‖L1w⊗m = ∫ ∫ ∫ F(x, ξ − ζ )H(x, ζ ) dζw(x)m(η) dx dξ ℝd ℝd ℝd

≤ ∫ ∫ ∫ F(x, ξ − ζ )ν(x)−1 m(ξ − ζ )H(x, ζ )w(x)ν(x)v(ζ ) dx dξ dζ ℝd ℝd ℝd

≤ ‖F‖L∞,1 ‖H‖L1νw⊗v . ν−1 ⊗m

In the last expression both norms are finite by assumption.

376 | 6 Fourier integral operators and applications to Schrödinger equations Corollary 6.4.6. Let s ≥ 0, N ∈ ℕ, N > 2s + d and g ∈ Mv14N ⊗v4N (ℝd ) and Ψ ∈ 𝒮 (ℝ2d ). Then the function Ψz (w) = e2πiΦ2,z (w) g ⊗ ĝ (w) defined as in (6.80) satisfies 󵄨 󵄨 sup 󵄨󵄨󵄨VΨz Ψ2 󵄨󵄨󵄨 ∈ L11⊗vs (ℝ4d ).

z∈ℝ2d

Proof. The symmetry property of the weight ⟨x⟩4N ⟨η⟩4N implies that Mv14N ⊗v4N (ℝd ) is invariant under the Fourier transform, and thus ĝ ∈ Mv1 ⊗v (ℝd ). A tensor product argu4N

4N

1 ment then shows that ḡ ⊗ ĝ ∈ MW (ℝ2d ) with W(x1 , x2 , ξ1 , ξ2 ) = ⟨x1 ⟩4N ⟨x2 ⟩4N ⟨ξ1 ⟩4N ⟨ξ2 ⟩4N . 4N Since W(x1 , x2 , η1 , η2 ) ≥ ⟨(x1 , x2 )⟩ ⟨(η1 , η2 )⟩4N , we also obtain that ḡ ⊗ ĝ ∈ Mv14N ⊗v4N (ℝ2d ).

4d By Lemma 6.4.3, we have supz∈ℝ2d |VΨ e2πiΦ2,z | ∈ L∞,1 v−4N ⊗vs (ℝ ). 2πiΦ2,z We now apply Lemma 6.4.5 with fz = e and h = ḡ ⊗ g,̂ ν = v4N , m = vs , w ≡ 1, and v = v4N (observe that vs is v4N -moderate, since s < 4N). As a conclusion, we obtain that supz∈ℝ2d |VΨ2 Ψz | ∈ L11⊗vs (ℝ4d ). The desired conclusion follows from (1.32).

Lemma 6.4.7. Under the assumptions of Corollary 6.4.6, we have sup |VΨz Ψ| ∈ W(L∞ , L11⊗vs )(ℝ4d ).

(6.88)

z∈ℝ2d

Proof. Using Lemma 1.2.29 with h = γ = Ψ2 , g = Ψz , f = Ψ, we have 󵄨󵄨−2 󵄨 󵄨 |VΨz Ψ|(u1 , u2 ) ≤ ‖Ψ2 󵄨󵄨󵄨󵄨󵄨󵄨2 |VΨ2 Ψ|∗󵄨󵄨󵄨VΨz Ψ2 󵄨󵄨󵄨(u1 , u2 )

󵄨󵄨−2 󵄨 󵄨 󵄨 󵄨 ≤ ‖Ψ2 󵄨󵄨󵄨󵄨󵄨󵄨2 󵄨󵄨󵄨VΨ2 Ψ󵄨󵄨󵄨 ∗ ( sup 󵄨󵄨󵄨VΨz Ψ2 󵄨󵄨󵄨)(u1 , u2 ). z∈ℝ2d

Since VΨ2 Ψ ∈ 𝒮 (ℝ4d ) ⊂ W(L∞ , L11⊗vs ) and ‖ supz∈ℝ2d |VΨz Ψ2 |‖L11⊗v

s

< ∞ by Corol-

lary 6.4.6, the convolution relations for Wiener amalgam spaces L11⊗vs ∗W(L∞ , L11⊗vs ) W(L∞ , L11⊗vs ) (recall that L11⊗vs = W(L1 , L11⊗vs ), cf. (2.67)) implies that

�→

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󵄩󵄩 󵄩 󵄩󵄩 󵄨󵄨 󵄩󵄩 2 󵄨󵄨󵄩 󵄩󵄩 sup |VΨ Ψ|󵄩󵄩󵄩 󵄩 󵄩󵄩 2d 󵄩󵄩W(L∞ ,L1 ) ≤ C‖VΨ2 Ψ‖W(L∞ ,L11⊗vs ) 󵄩󵄩󵄩 sup2d 󵄨󵄨VΨz Ψ 󵄨󵄨󵄩󵄩󵄩L1 < ∞, z 1⊗vs 1⊗vs z∈ℝ z∈ℝ and so supz∈ℝ2d |VΨz Ψ| ∈ W(L∞ , L11⊗vs ). ∞,1 We can now formulate the new characterization of the symbol classes M1⊗m and mentioned above.

∞ M1⊗m

Proposition 6.4.8. Let s ≥ 0, m ∈ ℳvs (ℝ2d ), g ∈ 𝒮 (ℝd ) and σ ∈ 𝒮 󸀠 (ℝ2d ). ∞,1 (i) Then the symbol σ is in M1⊗m (ℝ2d ) if and only if 󵄩󵄩 󵄩 󵄨󵄨 󵄨󵄨 󵄩󵄩 sup |VΨ σ|󵄩󵄩󵄩 󵄩󵄩 2d 󵄩󵄩L∞,1 (ℝ4d ) = ∫ supd sup2d 󵄨󵄨VΨz σ(u1 , u2 )󵄨󵄨m(u2 ) du2 < ∞, z 1⊗m z∈ℝ u ∈ℝ z∈ℝ ℝ2d

(6.89)

1

with ‖σ‖M ∞,1 ≍ ‖ supz∈ℝ2d |VΨz σ|‖L∞,1 . In this case the function H(u2 ) = supu1 ∈ℝd × 1⊗m

1⊗m

supz∈ℝ2d |VΨz σ(u1 , u2 )| is in W(L∞ , L1m )(ℝ2d ).

6.4 FIOs of type I

| 377

4d ∞ ∞ ≍ (ii) Likewise, σ ∈ M1⊗m (ℝ2d ) if and only if supz∈ℝ2d |VΨz σ| ∈ L∞ 1⊗m (ℝ ) with ‖σ‖M1⊗m . ‖ supz∈ℝ2d |VΨz σ|‖L∞ 1⊗m

Proof. We detail the proof of case (i). Case (ii) is obtained similarly. Let Ψ ∈ 𝒮 (ℝ2d ) with ‖Ψ‖2 = 1. ∞,1 Assume first that σ ∈ M1⊗m (ℝ2d ). Then, by Lemma 1.2.29, we have 󵄨 󵄨󵄨 󵄨󵄨VΨz σ(u1 , u2 )󵄨󵄨󵄨 ≤ |VΨ σ| ∗ |VΨz Ψ|(u1 , u2 ) ≤ |VΨ σ| ∗ sup |VΨz Ψ|(u1 , u2 ). z∈ℝ2d

Set F(u1 , u2 ) = supz∈ℝ2d |VΨz Ψ(u1 , u2 )|, so that 󵄨󵄨 󵄨 󵄨󵄨VΨz σ(u1 , u2 )󵄨󵄨󵄨 ≤ (|VΨ σ| ∗ F)(u1 , u2 ). Finally, 󵄩󵄩 󵄩 󵄨󵄨 󵄨󵄨 󵄩󵄩 sup |VΨ σ|󵄩󵄩󵄩 󵄩󵄩 2d 󵄩󵄩L∞,1 = ∫ supd sup2d 󵄨󵄨VΨz σ(u1 , u2 )󵄨󵄨m(u2 ) du2 z 1⊗m z∈ℝ u ∈ℝ z∈ℝ ℝ2d

1

󵄩 󵄩 ≤ 󵄩󵄩󵄩|VΨ σ| ∗ F 󵄩󵄩󵄩L∞,1 1⊗m ≤ ‖VΨ σ‖L∞,1 ‖F‖L11⊗v ≲ ‖σ‖M ∞,1 ‖F‖L11⊗v < ∞, 1⊗m

s

1⊗m

s

where in the last step we used the independence of the weighted M ∞,1 -norm of the window and Corollary 6.4.6. To obtain a sharper estimate, set H(u2 ) = supu1 ∈ℝd supz∈ℝ2d |VΨz σ(u1 , u2 )|, F1 (u2 ) = ∞,1 (ℝ2d ) ∫ℝ2d F(u1 , u2 ) du1 , and G1 (u2 ) = supu1 ∈ℝ2d |VΨ σ(u1 , u2 )|. Then the definition of M1⊗m

implies that G1 ∈ L1m (ℝ2d ), and Lemma 6.4.7 implies that F1 ∈ W(L∞ , L1vs )(ℝ2d ). With these definitions, we obtain a pointwise estimate for H, namely 󵄨 󵄨 H(w) = sup sup 󵄨󵄨󵄨VΨz σ(u1 , w)󵄨󵄨󵄨 d 2d u1 ∈ℝ z∈ℝ

󵄨 󵄨 ≤ ∫ sup󵄨󵄨󵄨VΨ σ(u1 , u2 )󵄨󵄨󵄨F(z − u1 , w − u2 ) du1 du2 u ℝ2d

1

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= (G1 ∗ F1 )(w). The convolution relations for Wiener amalgam spaces (recall that L1m = W(L1 , L1m ) by (2.67)) now yield H ≤ G1 ∗ F1 ∈ L1m ∗ W(L∞ , L1vs ) �→ W(L∞ , L1vs )(ℝ2d ), as claimed. Conversely, assume (6.89). Using Lemma 1.2.29 again, we deduce that 1 󵄨󵄨 󵄨 |V σ| ∗ |VΨ Ψz |(u1 , u2 ), 󵄨󵄨VΨ σ(u1 , u2 )󵄨󵄨󵄨 ≤ ⟨Ψz , Ψz ⟩ Ψz and 󵄨 󵄨2 ⟨Ψz , Ψz ⟩ = ∫ 󵄨󵄨󵄨e2πiΦ2,z (w) (g ⊗ ĝ )(w)󵄨󵄨󵄨 dw ℝ2d

󵄨 󵄨2 = ∫ 󵄨󵄨󵄨(g ⊗ ĝ )(w)󵄨󵄨󵄨 dw = ‖g‖22 ‖ĝ ‖22 = ‖g‖42 ℝ2d

378 | 6 Fourier integral operators and applications to Schrödinger equations is, in fact, a constant (depending on g). Using the involution f ∗ (z) = f (−z), we continue with |VΨ σ|(u1 , u2 ) ≲ |VΨz σ| ∗ |VΨ Ψz |(u1 , u2 ) 󵄨 󵄨 = |VΨz σ| ∗ 󵄨󵄨󵄨(VΨz Ψ)∗ 󵄨󵄨󵄨(u1 , u2 ) 󵄨 󵄨 ≤ ( sup |VΨz σ|) ∗ ( sup 󵄨󵄨󵄨(VΨz Ψ)∗ 󵄨󵄨󵄨)(u1 , u2 ). 2d 2d z∈ℝ

z∈ℝ

Since, by Corollary 6.4.6, 󵄨 󵄨 sup 󵄨󵄨󵄨(VΨz Ψ)∗ 󵄨󵄨󵄨 = sup |VΨz Ψ| ∈ L11⊗vs (ℝ4d ), z∈ℝ2d

z∈ℝ2d

we get 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ‖VΨ σ‖L∞,1 ≲ 󵄩󵄩󵄩 sup |VΨz σ|󵄩󵄩󵄩 ∞,1 󵄩󵄩󵄩 sup |VΨz Ψ|󵄩󵄩󵄩 1 < ∞, 1⊗m 󵄩L1⊗vs 󵄩z∈ℝ2d 󵄩L1⊗m 󵄩z∈ℝ2d and the proof is complete. Theorem 6.4.9. Let s ≥ 0, N, N >

s 2

+ d, g ∈ Mv14N ⊗v4N (ℝd ) and assume that 𝒢 (g, Λ) is a

∞ Parseval frame for L2 (ℝd ). If the phase Φ is tame and σ ∈ M1⊗v (ℝ2d ), then there exists s C > 0 such that ∞ ‖σ‖M1⊗v 󵄨󵄨 󵄨 s , 󵄨󵄨⟨Tπ(μ)g, π(λ)g⟩󵄨󵄨󵄨 ≤ C ⟨χ(μ) − λ⟩s

∀λ, μ ∈ ℝ2d .

(6.90)

Proof. Using again the fundamental relation between the Gabor matrix of an FIO and the STFT of its symbol in (6.35), for g ∈ 𝒮 (ℝd ), with the function Ψz defined in (6.80), we can write 󵄨󵄨 󵄨 󵄨 󵄨 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 󵄨󵄨⟨Tπ(x, ξ )g, π(x , ξ )g⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨VΨ(x󸀠 ,ξ ) σ((x , ξ ), (ξ − ∇x Φ(x , ξ ), x − ∇η Φ(x , ξ )))󵄨󵄨󵄨 󵄨 󵄨 ≤ sup 󵄨󵄨󵄨VΨz σ((x 󸀠 , ξ ), (ξ 󸀠 − ∇x Φ(x󸀠 , ξ ), x − ∇η Φ(x󸀠 , ξ )))󵄨󵄨󵄨. (6.91) 2d Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

z∈ℝ

4d If H(u1 , u2 ) = supz∈ℝ2d |VΨz σ(u1 , u2 )|, by Proposition 6.4.8 (ii), H ∈ L∞ 1⊗vs (ℝ ) and ∞ . Now, using (6.91), ‖H‖L∞ ≲ ‖σ‖M1⊗v 1⊗v s

s

󵄨 󵄨󵄨 󸀠 󸀠 󸀠 󸀠 󸀠 󵄨󵄨⟨Tπ(x, ξ )g, π(x , ξ )g⟩󵄨󵄨󵄨 ≤ sup H(u1 , ξ − ∇x Φ(x , ξ ), x − ∇η Φ(x , ξ )) 2d u1 ∈ℝ

≲ sup H(u1 , ξ 󸀠 − ∇x Φ(x󸀠 , ξ ), x − ∇η Φ(x󸀠 , ξ )) u1 ∈ℝ2d

× ≲

⟨ξ 󸀠 − ∇x Φ(x󸀠 , ξ ), x − ∇η Φ(x 󸀠 , ξ )⟩s

⟨ξ 󸀠 − ∇x Φ(x󸀠 , ξ ), x − ∇η Φ(x 󸀠 , ξ )⟩s

supu2 supu1 ∈ℝ2d H(u1 , u2 )vs (u2 )

⟨ξ 󸀠 − ∇x Φ(x󸀠 , ξ ), x − ∇η Φ(x 󸀠 , ξ )⟩s

6.4 FIOs of type I

= ≲

| 379

‖H‖L∞ 1⊗v

s

⟨ξ 󸀠 − ∇x Φ(x󸀠 , ξ ), x − ∇η Φ(x 󸀠 , ξ )⟩s ∞ ‖σ‖M1⊗v

s

⟨ξ 󸀠 − χ2 (x, ξ ), x 󸀠 − χ1 (x, ξ )⟩s

,

where in the last inequality we have used Lemma 6.1.4. Theorem 6.4.10. Fix 𝒢 (g, Λ) be a Parseval frame with g ∈ 𝒮 (ℝd ) and let s ≥ 0. Let T be a continuous linear operator 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) and χ a canonical transformation which satisfies B1, B2, and B3. Then the following properties are equivalent: ∞ (i) T = TI,Φχ ,σ is a FIO of type I for some σ ∈ M1⊗v (ℝ2d ). s (ii) F ∈ FIO(χ, s). Proof. The implication (i) �⇒ (ii) is proved in Theorem 6.4.9. The implication (ii) �⇒ (i) follows immediately from Proposition 6.4.1 and Lemma 6.4.2, since ⟨(x, η󸀠 ) − ∇Φ(x, η)⟩−s ≲ ⟨(x󸀠 , η󸀠 ) − χ(x, η)⟩−s . Corollary 6.4.11. Under the same assumptions as in Theorem 6.4.10, the following statements are equivalent: 0 (i) T = TI,Φχ ,σ is a FIO of type I for some σ ∈ S0,0 . (ii) T ∈ FIO(χ, ∞) = ⋂s≥0 FIO(χ, s).

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Remark 6.4.12. The latter corollary should be juxtaposed to Tataru’s characterization of FIO(χ, ∞) in [279, Theorem 4]. Tataru assumes only conditions B1 and B2, but not B3, on the canonical transformation χ and therefore obtains a larger class of FIOs satisfying the decay condition (6.77). The new insight of Corollary 6.4.11 is that, under the additional assumption B3, every FIO admits the classical representation (6.12). As a byproduct, we see that the integral operator in (14) of [279] possesses the classical representation, provided that χ also satisfies B3. This observation might be useful for the solution of the Cauchy problem in [279, Section 5], where, for small time, χ is a small perturbation of the identity transformation and therefore certainly satisfies B3. ∞ Thanks to Theorem 6.3.7, the FIOs of type I with symbol in M1⊗v (ℝ2d ), with s > 2d, s

are bounded on M p (ℝd ), 1 ≤ p ≤ ∞. Consider now the multiplication operator 2

Tf (x) = eπix f (x).

(6.92)

Then T is an FIO of type I having phase Φ(x, ξ ) = x 2 /2 + x ⋅ ξ and symbol σ(x, ξ ) = 1, 0 for every (x, ξ ) ∈ ℝ2d . Observe that σ = 1 ∈ S0,0 , hence σ ∈ M ∞,1 (ℝ2d )s , for every s ≥ 0. However, the multiplication operator T is not bounded on M p,q , when p ≠ q, as proved in Proposition 6.1.17. We now look at the Wiener property of the class of FIOs of type I with symbol in ∞ M1⊗v (ℝ2d ), with s > 2d. As we will see in Section 6.5, this class is not closed under s

380 | 6 Fourier integral operators and applications to Schrödinger equations composition, therefore the Wiener property must necessarily involve FIOs of type II (see (6.74)). We first need to recall the following result, originally contained in [73, Lemma 2.10], though with different weights. p,q Lemma 6.4.13. Let σ ∈ M1⊗v (ℝ2d ), s ∈ ℝ, and consider its adjoint σ ∗ defined by s

σ ∗ (x, η) = σ(η, x),

(x, η) ∈ ℝ2d .

(6.93)

p,q Then σ ∗ ∈ M1⊗v (ℝ2d ). s

Proof. Fix a window function g ∈ 𝒮 (ℝd ). By a direct computation, Vg σ ∗ (z1 , z2 ; ζ1 , ζ2 ) = ∫ e−2πi(y1 ⋅ζ1 +y2 ⋅ζ2 ) σ(y2 , y1 )g(y1 − z1 , y2 − z2 ) dy1 dy2 = ∫ e−2πi(v2 ⋅ζ1 +v1 ⋅ζ2 ) σ(v1 , v2 )g(v2 − z1 , v1 − ζ2 ) dv1 dv2 = Vt g σ(z2 , z1 ; ζ2 , ζ1 ) p,q (ℝ2d ), the result imwhere we used the notation t g(y1 , y2 ) := g(y2 , y1 ). Since σ ∈ M1⊗v s mediately follows by the independence of the window function for the computation of the modulation space norm and the symmetry of the weight vs , that is, vs (ζ1 , ζ2 ) = vs (ζ2 , ζ1 ).

Theorem 6.4.14. Let T be an FIO of type I with a tame phase Φ and a symbol σ ∈ ∞ M1⊗v (ℝ2d ), with s > 2d. If T is invertible on L2 (ℝd ), then T −1 is an FIO of type II with s ∞ same phase Φ and a symbol τ ∈ M1⊗v (ℝ2d ). s

Proof. Let χ be the canonical transformation associated to Φ. Then by Theorem 6.4.10 T belongs to FIO(χ, s). As in the proof of Theorem 6.3.10, we consider P = T ∗ T and write T −1 = P −1 T ∗ = (T(P −1 ) ) = (TP −1 ) . Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

∗ ∗

∗

We have already shown that P is in FIO(Id, s) and a pseudodifferential operator with ∞ a symbol in M1⊗v (ℝ2d ) and that also P −1 ∈ FIO(Id, s) by the spectral invariance of s pseudodifferential operators of Theorem 6.3.2. Now Theorem 6.3.9 implies that TP −1 ∈ FIO(χ, s) and Theorem 6.4.10 implies that TP −1 is an FIO of type I with tame phase Φ ∞ and a symbol ρ ∈ M1⊗v (ℝ2d ). Since T −1 (TP −1 )∗ , T −1 is an FIO of type II with phase Φ s ∞ and the symbol τ(x, η) = ρ(η, x). Lemma 6.4.13 shows that τ ∈ M1⊗v (ℝ2d ). s

6.5 Generalized metaplectic operators As an example, we will consider the class of FIO(χ, s) whose phase is a linear transformation χ(z) = 𝒜z for some invertible matrix 𝒜 ∈ GL(2d, ℝ). Since χ must preserve

6.5 Generalized metaplectic operators | 381

the symplectic form (assumption B2), 𝒜 must be a symplectic matrix, 𝒜 ∈ Sp(d, ℝ), cf. Definition 1.1.2. Definition 6.5.1. Let 𝒜 ∈ Sp(d, ℝ) and s ≥ 0. Fix a Parseval frame 𝒢 (g, Λ) with g ∈ 𝒮 (ℝd ). We say that a continuous linear operator T : 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ) is a generalized metaplectic operator, in short, T ∈ FIO(𝒜, s), if its Gabor matrix satisfies the decay condition 󵄨 󵄨󵄨 −s 󵄨󵄨⟨Tπ(λ)g, π(μ)g⟩󵄨󵄨󵄨 ≤ C⟨μ − 𝒜λ⟩ ,

∀λ, μ ∈ Λ.

The union ⋃𝒜∈Sp(d,ℝ) FIO(𝒜, s) is called the class of generalized metaplectic operators and denoted by FIO(Sp, s). Since Sp(d, ℝ) is a group, Theorems 6.3.9 and 6.3.10 imply the following statement. Theorem 6.5.2. For s > 2d, FIO(Sp, s) is a Wiener subalgebra of FIO(Ξ, s). The main examples in FIO(Sp, s) are the operators of the metaplectic representation of Sp(d, ℝ), see Section 1.1.2 for details. Let 𝒜 = ( AC DB ) ∈ Sp(d, ℝ) with d × d blocks A, B, C, D, cf. (1.4). Then condition B3 of Definition 6.1.1 is equivalent to det A ≠ 0. In this case, μ(𝒜) is explicitly given by the FIO of type I in (6.48). Namely, μ(𝒜)(x) = (det A)−1/2 ∫ e2πiΦ(x,η) f ̂(η) dη ℝd

with the phase Φ given by 1 1 Φ(x, η) = x ⋅ CA−1 x + η ⋅ A−1 x − η ⋅ A−1 Bη. 2 2

(6.94)

Even without the explicit form of μ(𝒜), Definition 6.5.1 yields some interesting information about the metaplectic representation.

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Proposition 6.5.3. If 𝒜 ∈ Sp(d, ℝ), then μ(𝒜) ∈ ⋂s≥0 FIO(𝒜, s). Proof. According to Theorem 6.3.5, it is enough to prove that μ(𝒜) satisfies the continuous decay condition (6.77). Using the definition of μ(𝒜) in (1.10) and the covariance property of the STFT (1.50), we write the Gabor matrix of μ(𝒜) as 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨⟨μ(𝒜)π(z)g, π(w)g⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨π(𝒜z)μ(𝒜)g, π(w)g⟩󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨Vg (π(𝒜z)μ(𝒜)g)(w)󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨Vg (μ(𝒜)g)(w − 𝒜z)󵄨󵄨󵄨.

(6.95)

Since both g ∈ 𝒮 (ℝd ) and μ(𝒜)g ∈ 𝒮 (ℝd ), we also have Vg (μ(𝒜)g) ∈ 𝒮 (ℝ2d ), by Corollary 1.2.24. This gives 󵄨󵄨 󵄨 −s 󵄨󵄨⟨μ(𝒜)π(z)g, π(w)g⟩󵄨󵄨󵄨 ≤ C⟨w − 𝒜z⟩ , for every s ≥ 0, as desired.

(6.96)

382 | 6 Fourier integral operators and applications to Schrödinger equations The following theorem shows that every generalized metaplectic operator is a product of a metaplectic operator and a classical pseudodifferential operator. Theorem 6.5.4. Let 𝒜 ∈ Sp(d, ℝ) and T ∈ FIO(𝒜, s) with s > 2d. Then there exist sym∞ bols σ1 , σ2 ∈ M1⊗v (ℝ2d ) with the corresponding pseudodifferential operators σ1 (x, D) s and σ2 (x, D), such that T = σ1 (x, D)μ(𝒜) and

T = μ(𝒜)σ2 (x, D).

(6.97)

Proof. We prove the factorization T = σ1 (x, D)μ(𝒜), the other factorization is obtained analogously. Since μ(𝒜)−1 = μ(𝒜−1 ) is in FIO(𝒜−1 , s) by Proposition 6.5.3, the algebra property of Theorem 6.3.9 implies that Tμ(𝒜−1 ) ∈ FIO(Id, s). The fundamental characterization of pseudodifferential operators of Theorem 6.3.3 implies that existence of a symbol ∞ σ1 ∈ M1⊗v (ℝ2d ), such that Tμ(𝒜)−1 = σ1 (x, D), which is what we wanted to show. s Finally, we check the counterpart of FIO I and II for generalized metaplectic operators. Let 𝒜 = ( AC DB ) ∈ Sp(d, ℝ) with det A ≠ 0. As proved in Theorem 6.4.10, every generalized metaplectic operator T ∈ FIO(𝒜, s) with s > 2d is an FIO of type I, Tf (x) = (det 𝒜)−1/2 ∫ e2πiΦ(x,η) σ(x, ξ )f ̂(η) dη,

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∞ with a symbol σ ∈ M1⊗v (ℝ2d ) and phase Φ(x, η) = 21 x ⋅ CA−1 x + η ⋅ A−1 x − 21 η ⋅ A−1 Bη. s We obtain examples of FIOs of type II by taking adjoints. If T ∈ FIO(𝒜, s), then T ∗ is an FIO of type II. As it is well known, the generalized metaplectic operators of type I do not enjoy the algebra property. In fact, consider, for instance, the operators T1 = μ(𝒜1 ) and T2 = μ(𝒜2 ), with

Id 0

𝒜1 = (

Id ), Id

Id −Id

𝒜2 = (

Id ). 0

Then both T1 and T2 are FIOs of type I but their product T1 T2 = μ(𝒜1 )μ(𝒜2 ) = μ(𝒜1 𝒜2 ) = μ(−J) = ℱ cannot be an FIO of type I. Indeed, the Fourier transform ℱ = μ(−J) is an example of a metaplectic operator that is an FIO of neither type I nor of type II. Note that in this case assumption B3 is not satisfied for 𝒥 . As in Remark 6.4.12, we see again that there is a crucial difference between FIOs satisfying all axioms B1, B2, and B3, or only B1 and B2.

6.6 Propagation of Gabor wave front set for Schrödinger equations | 383

6.6 Propagation of Gabor wave front set for Schrödinger equations In the previous sections we proposed a new approach to the calculus of Fourier integral operators in terms of time–frequency localization, using Gabor analysis. The FIOs under consideration were of the type of those appearing in the study of the Schrödinger equations, typically the phase function being a homogeneous function of degree 2 in the whole of the phase space variables. With respect to the standard representations of FIOs, the time–frequency representation looks more involved, since old and new phase-space variables appear simultaneously, and everything depends on the choice of the window function. On the other hand, the problem of the caustics is automatically solved in this new setting and the expression provides an excellent tool for numerical analysis, as we have seen above. In this last section we mainly report the outcomes of the paper [84], where the aforesaid results are applied to the analysis of Schrödinger equations. With respect to the enormous existing literature, such results are new in the following aspects. For a fixed a real-valued Hamiltonian which is homogeneous of degree 2, a pseudodifferential perturbation (also called potential in the following) is allowed with a bounded, complex-valued, nonsmooth symbol, for which even differentiability may be lost. A global-in-time propagator is constructed in the class of the FIOs in Definition 6.3.4, and well-posedness of the Cauchy problem is deduced in suitable modulation spaces. Regarding propagation of singularities, which is our main concern in this section, the known results do not apply to such a situation. We are then led to the new definition of the Gabor wave front set, which allows the expression of optimal results of propagation in our context. To be precise, we aim at studying the representation in terms of time–frequency analysis of the propagator eitH , H = a(x, D) + σ(x, D),

(6.98)

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

providing the solution to the Cauchy problem {i 𝜕t + a(x, D)u + σ(x, D)u = 0, { {u(0, x) = u0 (x). 𝜕u

(6.99)

The Hamiltonian a(x, D) is a pseudodifferential operator in the Kohn–Nirenberg form Af (x) = a(x, D)f (x) = ∫ e2πix⋅ξ a(x, ξ )f ̂(ξ ) dξ ,

(6.100)

ℝd

where the symbol a(z), z = (x, ξ ), is real-valued and positively homogeneous of degree 2, i. e., a(λz) = λ2 a(z) for λ > 0, with a ∈ 𝒞 ∞ (ℝ2d \ 0). This implies that a(x, D) is formally self-adjoint modulo 0th order perturbations. Basic examples are real-valued

384 | 6 Fourier integral operators and applications to Schrödinger equations quadratic forms a(z), including the cases when i𝜕t + a(x, D) is the free particle or the harmonic oscillator operator. When a(z) is not a polynomial, we shall assume a(z) to be modified in a bounded neighborhood of the origin in such a way that we have a ∈ 𝒞 ∞ (ℝ2d ) keeping real values. As we shall see, cf. Example 6.6.28 below, the singularity at the origin of a(z) can be admitted as well, by absorbing it in a nonsmooth potential. The pseudodifferential operator a(x, D) enters the classes of [265], see also [187], to which we refer for the symbolic calculus and other properties, see also the next Section 6.6.1. Concerning the potential σ(x, D), the regularity assumptions will be expressed in ∞ terms of the modulation spaces M1⊗v (ℝ2d ). We shall not treat the case σ ∈ M ∞,1 (ℝ2d ), s the interested reader can refer to [69], where quadratic Hamiltonians with a Sjöstrand potential are studied. We want to study the Gabor matrix k(t, w, z) of the propagator eitH . Its structure will be linked, as expected, to the Hamiltonian field of a(x, ξ ). Namely, consider 2π ẋ = −∇ξ a(x, ξ ), { { { 2π ξ ̇ = ∇x a(x, ξ ), { { { {x(0) = y, ξ (0) = η, (the factor 2π depends on our normalization of the STFT). Under our assumptions, the solution χt (y, η) = (x(t, y, η), ξ (t, y, η)) exists for all t ∈ ℝ and defines a symplectic 2d diffeomorphism χt : ℝ2d y,η → ℝx,ξ which is homogeneous of degree 1 with respect to w = (y, η) for large |w|, for every fixed t ∈ ℝ. We shall show the following result. Theorem 6.6.1. Let the preceding assumptions be satisfied, in particular, let σ ∈ ∞ M1⊗v (ℝ2d ), s > 2d, and k(t, w, z) = ⟨eitH π(z)g, π(w)g⟩ be the Gabor matrix of the s Schrödinger propagator eitH . Then there exists C = C(t, s) > 0 such that

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−s 󵄨󵄨 󵄨 󵄨󵄨k(t, w, z)󵄨󵄨󵄨 ≤ C⟨z − χt (w)⟩ ,

z = (x, ξ ), w = (y, η) ∈ ℝ2d .

(6.101)

According to the notations of Definition 6.3.4, this can be rephrased as eitH ∈ (t, y, η) ≠ 0 in the exFIO(χt , s). For t sufficiently small, our assumptions yield det 𝜕x 𝜕y pression of χt , and (6.101) is then equivalent to ̂ 0 (ξ ) dξ , (eitH u0 )(t, x) = ∫ e2πiΦ(t,x,ξ ) b(t, x, ξ )u

(6.102)

ℝd ∞ with the phase Φ linked to χt as standard and b(t, ⋅) ∈ M1⊗v (ℝ2d ), see Theorem 6.4.10. s In the classical approach, cf. [3], the occurrence of caustics makes the validity of (6.102) local in time. So for t ∈ ℝ, one is led to multiple compositions of local representations, with unbounded number of variables possibly appearing in the expression whereas k(t, w, z) obviously remains defined for every t ∈ ℝ, and the estimates (6.101) hold for χt with t ∈ ℝ. Let us now define the Gabor wave front set WFp,r G (f ) under our consideration.

6.6 Propagation of Gabor wave front set for Schrödinger equations | 385

Definition 6.6.2. Let g ∈ 𝒮 (ℝd ), g ≠ 0, r > 0, 1 ≤ p ≤ ∞. For f ∈ Mvp−r (ℝd ), z0 ∈ ℝ2d , 2d z0 ≠ 0, we say that z0 ∉ WFp,r G (f ) if there exists an open conic neighborhood Γz0 ⊂ ℝ of z0 such that

󵄨p 󵄨 ∫ 󵄨󵄨󵄨Vg f (z)󵄨󵄨󵄨 ⟨z⟩pr dz < ∞

(6.103)

Γz0

(with obvious changes for p = ∞). 2d Then WFp,r G (f ) is well defined as a conic closed subset of ℝ \ {0}. The main result of this last section can be summarized as follows. ∞ Theorem 6.6.3. Consider σ ∈ M1⊗v (ℝ2d ), s > 2d, 1 ≤ p ≤ ∞. Then s

eitH : Mvpr (ℝd ) → Mvpr (ℝd )

(6.104)

continuously, for |r| < s − 2d. Moreover, for u0 ∈ Mvp−r (ℝd ), p,r itH WFp,r G (e u0 ) = χt (WFG (u0 )),

(6.105)

provided 0 < 2r < s − 2d. Observe the more restrictive assumption on r for (6.105) compared to that for (6.104). As an elementary example, consider the perturbed harmonic oscillator (studied in Example 6.6.28 in the sequel)

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1 2 2 μ {i𝜕t u − 4π 𝜕x u + πx u + | sin x| u = 0, { {u(0, x) = u0 (x),

(6.106)

∞ with μ > 1. We shall prove that | sin x|μ ∈ M1⊗v and then, from Theorem 6.6.3, we have μ+1 that the Cauchy problem is well-posed for u0 ∈ Mvpr (ℝ), |r| < μ − 2 and the propagation of WFp,r G (u(t, ⋅)) for t ∈ ℝ takes place as in Theorem 6.6.3 for 0 < r < μ/2 − 1, where χt is defined in (6.70). Using (6.72) and (6.105), we may recapture results for the propagation in the case 0 of a smooth potential, i. e., σ ∈ S0,0 . We define the wave front set WFG (f ) by stating z0 ∉ WFG (f ) if there exists an open conic set Γz0 ⊂ ℝ2d containing z0 such that for every r > 0,

󵄨󵄨 󵄨 −r 󵄨󵄨Vg f (z)󵄨󵄨󵄨 ≤ Cr ⟨z⟩ ,

z ∈ Γz0 ,

(6.107)

for a suitable Cr > 0. Then the estimate (6.101) is satisfied for every s and, from Theorem 6.6.3, we recapture for u0 ∈ 𝒮 󸀠 (ℝd ), WFG (eitH u0 ) = χt (WFG (u0 )).

(6.108)

386 | 6 Fourier integral operators and applications to Schrödinger equations Although it is impossible to do justice to the vast literature in this connection, let us mention some of the related contributions. The pioneering work is that of Hörmander [195], who defined the wave front set in (6.107), as well as its analytic version, and proved (6.108) in the case of the metaplectic operators. For subsequent results providing (6.108) and its analytic-Gevrey version for general smooth symbols, let us refer to [28, 178, 200, 201, 222, 223, 227, 232, 233, 321]. The wave front sets introduced there under different names actually coincide with those of Hörmander, cf. [253], see also [44, 47, 245, 259, 260, 301]. Still concerning propagation of singularities in the case of smooth or analytic symbols, we refer to [96, 225, 250, 251, 307]. Besides, concerning global-in-time representations of eitH , solving the problem of the caustics for smooth symbols, see [3, 30, 31, 158, 279]. The general result in Theorem 6.6.3 is due to [84]. 6.6.1 Preliminaries and Shubin classes We shall use the following properties. ∞ Lemma 6.6.4. Consider μ > 0. Then the function f (x) = | sin x|μ belongs to M1⊗v (ℝ). μ+1

Proof. Consider a window function g ∈ 𝒟(ℝ), with supp g ⊂ [−π/4, π/4] to compute the STFT Vg f with f (x) = | sin x|μ . Then |Vg f (x, ξ )| is a periodic function of period π in the x variable. So ∞ ‖f ‖M1⊗v

μ+1

= sup sup⟨ξ ⟩μ+1 |Vg f |(x, ξ ). |x|≤π/2 ξ ∈ℝ

Now observe that supp Tx g ⊂ [−3π/4, 3π/4], for x ∈ [−π/2, π/2], and on that interval f (x) = |x|μ φ(x), with φ ∈ 𝒟(ℝ). We can write +∞

0

0

−∞

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Vg f (x, ξ ) = ∫ e−2πitξ t μ φ(t)g(t − x) dt + ∫ e−2πitξ (−t)μ φ(t)g(t − x) dt := A + B. So it suffices to estimate integral A, the estimate of B is analogous. Setting Fx (t) = et φ(t)g(t − x) ∈ 𝒮 (ℝ), we observe that the family {Fx }x∈[−π/2,π/2] belongs to a bounded subset of 𝒮 (ℝ). Now +∞

A = ∫ e−2πitξ t μ e−t et φ(t)g(t − x) dt 0

=

Γ(μ + 1) ∗ F̂x (ξ ), (1 + 2πiξ )μ+1

and this yields ⟨ξ ⟩μ+1 |A| ≲

⟨ξ ⟩μ+1 ∗ (⟨ξ ⟩μ+1 F̂x (ξ )) ∈ L∞ (ℝ), (1 + 2πiξ )μ+1

6.6 Propagation of Gabor wave front set for Schrödinger equations | 387

by Young’s inequality, since the first factor of the convolution product is bounded and the second lies in a bounded subset of 𝒮 (ℝ) ⊂ L1 (ℝ). Corollary 6.6.5. Consider the symbol σ(x, ξ ) = | sin x|μ on ℝ2 . Then we have σ ∈ ∞ M1⊗v (ℝ2 ). μ+1 Proof. It is an immediate consequence of Lemma 6.6.4. Indeed, taking ψ(x, ξ ) = g(x)φ(ξ ), with g being the 1-dimensional window of the previous proof and φ ∈ 𝒮 (ℝ), we have Vψ σ((x1 , x2 ), (ξ1 , ξ2 )) = Vg (| sin(⋅)|μ )(x1 , ξ1 )Vφ 1(x2 , ξ2 ), and the claim follows 0 ∞ since ⟨(ξ1 , ξ2 )⟩ ≤ ⟨ξ1 ⟩⟨ξ2 ⟩ and 1 ∈ S0,0 ⊂ M1⊗v (ℝ), for every s ≥ 0. s Proposition 6.6.6. Let h ∈ 𝒞 ∞ (ℝd \ {0}) be positively homogeneous of degree r > 0, i. e., h(λx) = λr h(x) for x ≠ 0, λ > 0, and χ ∈ 𝒟(ℝd ). Set f = hχ. Then for ψ ∈ 𝒮 (ℝd ) there exists a constant C > 0 such that −r−d 󵄨󵄨 󵄨 , 󵄨󵄨Vψ f (x, ξ )󵄨󵄨󵄨 ≤ C(1 + |ξ |)

x, ξ ∈ ℝd .

Proof. We know that the Fourier transform of h is a homogeneous distribution of degree −r − d, smooth in ℝd \ {0} [194, Theorems 7.1.16, 7.1.18]. Hence, if χ 󸀠 ∈ 𝒟(ℝd ), χ 󸀠 = 1 in a neighborhood of the origin, we have 󵄨󵄨 󸀠 ̂ )󵄨󵄨󵄨 ≤ C(1 + |ξ |)−r−d , 󵄨󵄨(1 − χ (ξ ))h(ξ 󵄨

ξ ∈ ℝd .

(6.109)

On the other hand, by the very definition of the STFT, we have ̂ ̂ 󵄨 󵄨󵄨 󵄨 󵄨 󸀠̂ 󵄨 ̂ ∗ χ̂󵄨󵄨󵄨 ∗ |ψ|. ∗ξ χ̂)󵄨󵄨󵄨 ∗ξ |ψ| + 󵄨󵄨󵄨((1 − χ 󸀠 )h) 󵄨󵄨Vψ f (x, ξ )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨((χ h) ξ 󵄨 ξ

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̂ Since ψ, χ̂ ∈ 𝒮 (ℝd ), the first term on the right-hand side has rapid decay because ℰ 󸀠 ∗ 𝒮 ⊂ 𝒮 , whereas the second term is estimated using (6.109), as at the end of the proof of Lemma 6.6.4. Here we are interested in operators with symbols in the Shubin classes (cf. [265], Helffer [187]); indeed, we shall use them as symbol and phase spaces for the unperturbed initial value problem for Schrödinger equations. Definition 6.6.7. For m ∈ ℝ, the class Γm (ℝ2d ) is the set of functions a ∈ 𝒞 ∞ (ℝ2d ) such that for every α ∈ ℤ2d + there exists a constant Cα > 0 such that 󵄨󵄨 α 󵄨 󵄨󵄨𝜕z a(z)󵄨󵄨󵄨 ≤ Cα vm−|α| (z),

z ∈ ℝ2d ,

where we recall vs (z) = ⟨z⟩s is defined in (2.5). Consider aj ∈ Γmj (ℝ2d ) with mj being a decreasing sequence tending to −∞. Then a function a ∈ 𝒞 ∞ (ℝ2d ) satisfies ∞

a ∼ ∑ aj j=1

(6.110)

388 | 6 Fourier integral operators and applications to Schrödinger equations if ∀r ≥ 2

r−1

a − ∑ aj ∈ Γmr (ℝ2d ). j=1

Namely, our symbol class will be a subclass of Γm (ℝ2d ), defined as follows [187, Section 1.5]. Definition 6.6.8. A function a is in the class Γm,cl (ℝ2d ) if a ∈ Γm (ℝ2d ) and admits an asymptotic expansion ∞

a ∼ ∑ am−j , j=0

(6.111)

where am−j ∈ 𝒞 ∞ (ℝ2d ) satisfies am−j (λz) = λm−j am−j (z), for |z| ≥ 1, λ ≥ 1. The function am corresponding to j = 0 in expansion (6.111) is called the principal symbol of the symbol a. For a ∈ Γm (ℝ2d ), the corresponding pseudodifferential operator a(x, D) is defined by the Kohn–Nirenberg form (6.100). Definition 6.6.9. We say that A ∈ Gm (resp. A ∈ Gm,cl ) if its symbol satisfies a ∈ Γm (ℝ2d ) (resp. a ∈ Γm,cl (ℝ2d )). A pseudodifferential operator A ∈ Gm,cl is called globally elliptic if there exist R > 0 and C > 0 such that 󵄨󵄨 󵄨 m 󵄨󵄨am (z)󵄨󵄨󵄨 ≥ C⟨z⟩ ,

for z ∈ ℝ2d , |z| ≥ R,

(6.112)

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where am is the principal symbol. Let a ∈ Γ2,cl (ℝ2d ) with real principal symbol a2 . The related classical evolution, given by the linear Hamilton–Jacobi system, following our normalization can be written as 2π𝜕t x(t, y, η) = −∇ξ a2 (x(t, y, η), ξ (t, y, η)), { { { { { {2π𝜕t ξ (t, y, η) = ∇x a2 (x(t, y, η), ξ (t, y, η)), { { x(0, y, η) = y, { { { { {ξ (0, y, η) = η.

(6.113)

The solution (x(t, y, η), ξ (t, y, η)) exists for every t ∈ ℝ. Indeed, setting u := (x, ξ ), F(u) := (−∇ξ a(u), ∇x a(u)), the initial value problem (6.113) can be rephrased as u󸀠 (t) = F(u(t)),

u(t0 ) = u0 ,

(6.114)

in the particular case t0 = 0. Observe that a ∈ Γ2,cl (ℝ2d ) implies Fj ∈ Γ1,cl (ℝ2d ), for j = 1, . . . , 2d and 𝜕α Fj ∈ Γ0,cl (ℝ2d ), for every |α| > 0, j = 1, . . . , 2d, hence in particular

6.6 Propagation of Gabor wave front set for Schrödinger equations | 389

F : ℝ2d → ℝ2d is a Lipschitz continuous mapping. Thus the previous ODE is an autonomous ODE with a mapping F ∈ 𝒞 ∞ (ℝ2d → ℝ2d ) having at most linear growth, hence ‖F(u)‖ ≲ 1 + ‖u‖. This implies that for each u0 ∈ ℝ2d and t0 ∈ ℝ there exists a unique classical global solution u : ℝ → ℝ2d (in this case u ∈ 𝒞 ∞ (ℝ → ℝ2d ) since F ∈ 𝒞 ∞ (ℝ2d → ℝ2d )) to (6.114). Moreover, the solution maps St0 (t) : ℝ2d → 𝒞 ∞ (ℝ → ℝ2d ), defined by St0 (t)u0 = u(t), and St0 (t0 ) = Id, the identity operator on ℝ2d , are Lipschitz continuous mappings, obey the time translation invariance St0 (t) = S0 (t − t0 ) and the group laws S0 (t)S0 (t 󸀠 ) = S0 (t + t 󸀠 ),

S0 (0) = Id.

(6.115)

Observe that S0 (t) is a bi-Lipschitz diffeomorphism with S0−1 (t) = S0 (−t). To be consistent with our notations, we call the bi-Lipschitz diffeomorphism χt (y, η) := S0 (t)(y, η),

(y, η) ∈ ℝ2d .

(6.116)

The theory of Hamilton–Jacobi allows finding a T > 0 such that for t ∈ ]−T, T[ there exists a phase function Φ(t, x, η), solution of the eiconal equation (cf. [187, (3.2.12),(3.2.13)]) 2π𝜕t Φ + a2 (x, ∇x Φ) = 0, { Φ(0, x, η) = xη.

(6.117)

The phase Φ(t, x, η) ∈ 𝒞 ∞ (]−T, T[, Γ2 (ℝ2d )) is real-valued since the principal symbol a2 (x, ξ ) is real-valued; moreover, Φ fulfills the condition of nondegeneracy 󵄨󵄨 󵄨 2 󵄨󵄨det 𝜕x,η Φ(t, x, η)󵄨󵄨󵄨 ≥ c > 0,

(t, x, η) ∈ ]−T, T[ × ℝ2d ,

(6.118)

after possibly shrinking T > 0 (cf. [187, pages 142–143] and [81]). The relation between the phase Φ and the canonical transformation χ is given by

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(x, ∇x Φ(t, x, η)) = χt (∇η Φ(t, x, η), η),

t ∈ ]−T, T[.

(6.119)

In particular, {

y(t, x, η) = ∇η Φ(t, x, η), ξ (t, x, η) = ∇x Φ(t, x, η),

(6.120)

and there exists δ > 0 such that 󵄨󵄨 󵄨󵄨 𝜕x 󵄨󵄨 󵄨 󵄨󵄨det (t, y, η)󵄨󵄨󵄨 ≥ δ, 󵄨󵄨 󵄨󵄨 𝜕y

t ∈ ]−T, T[.

(6.121)

Observe that each component of χt is a function in 𝒞 ∞ (]−T, T[, Γ1 (ℝ2d )), positively homogeneous of degree 1 for (y, η) large. Moreover, using (6.115), we observe that the same holds, in fact, for every t ∈ ℝ.

390 | 6 Fourier integral operators and applications to Schrödinger equations For t ∈ ]−T, T[, the phase function Φ(t, ⋅) above is a tame phase, and similarly for the canonical transformation χt , according to Definition 6.1.1. Note that in this general context we have no assumption of homogeneity for large (x, η), nevertheless, the mapping χ is well defined by the global inverse function theorem; moreover, χ is a smooth bi-Lipschitz canonical transformation and satisfies, for (x, ξ ) = χ(y, ξ ), 󵄨 󵄨󵄨 α 󵄨 󵄨 α 󵄨󵄨𝜕z xi (z)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜕z ξi (z)󵄨󵄨󵄨 ≤ Cα ,

|α| ≥ 1, z = (y, η), i = 1, . . . , d.

(6.122)

Finally, the mapping χ fulfills condition (6.11), which allows us to uniquely determine (up to a constant) the related tame phase function Φχ (see the comments after Definition 6.1.1). We shall refine and apply results for tame canonical transformations in Definition 6.1.1 to the special case of the canonical transformations coming from (6.113). First, we need to show a few properties of the class of FIOs introduced in Definition 6.3.4 which occur in this study. Lemma 6.6.10. For s > 2d, T (i) ∈ FIO(χi , s), i = 1, 2, the continuous Gabor matrix of the composition T (1) T (2) is controlled by −s 󵄨󵄨 (1) (2) 󵄨 󵄨󵄨⟨T T π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C0 C1 C2 ⟨z − χ1 ∘ χ2 (w)⟩ ,

w, z ∈ ℝ2d ,

(6.123)

where Ci > 0 is the constant of T (i) in (6.77), i = 1, 2, whereas C0 > 0 depends only on s and on the Lipschitz constants of χ1 and χ1−1 . Proof. Consider g ∈ 𝒮 (ℝd ) with ‖g‖2 = 1. We write the product T (1) T (2) as

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T (1) T (2) = Vg∗ Vg T (1) T (2) Vg∗ Vg = Vg∗ (Vg T (1) Vg∗ )(Vg T (2) Vg∗ )Vg . Thus the composition of operators corresponds to the multiplication of their (continuous) Gabor matrices. Using the decay estimates for the continuous Gabor matrices of T (i) , i = 1, 2, 󵄨󵄨 (1) (2) 󵄨 󵄨 (1) 󵄨󵄨 (2) 󵄨 󵄨󵄨⟨T T π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨󵄨⟨T π(w)g, π(y)g⟩󵄨󵄨󵄨󵄨󵄨󵄨⟨T π(y)g, π(z)g⟩󵄨󵄨󵄨 dy ℝ2d

≤ C1 C2 ∫ ⟨z − χ1 (y)⟩ ⟨y − χ2 (w)⟩ dy −s

−s

ℝ2d

≤ C1 C2 C(χ1 ) ∫ ⟨χ1−1 (z) − y⟩ ⟨y − χ2 (w)⟩ dy −s

−s

ℝ2d

≤ C1 C2 C(χ1 )Cs ⟨χ1−1 (z) − χ2 (w)⟩

−s

≤ C1 C2 C(χ1 )Cs C(χ1−1 )⟨z − χ1 ∘ χ2 (w)⟩

−s

(6.124)

6.6 Propagation of Gabor wave front set for Schrödinger equations | 391

for every z, w ∈ ℝ2d , s > 2d, where C1 and C2 are the controlling constants in (6.77) of the operators T (1) and T (2) , and the bi-Lipschitz property of χ1 gives ⟨z − χ1 (y)⟩

−s

≤ C(χ1 )⟨χ1−1 (z) − y⟩ ,

∀y, z ∈ ℝ2d ,

≤ C(χ1−1 )⟨y − χ1 (z)⟩ ,

∀y, z ∈ ℝ2d .

−s

and ⟨χ1−1 (y) − z⟩

−s

−s

Furthermore, we used that v−s is subconvolutive for s > 2d, i. e., v−s ∗ v−s ≤ Cs v−s [160, Lemma 11.1.1(d)]. If we call C0 = C(χ1 )Cs C(χ1−1 ), the claim is proved. By induction, we immediately obtain Corollary 6.6.11. For n ∈ ℕ, n ≥ 2, s > 2d, T (i) ∈ FIO(χi , s), i = 1, . . . , n, we have −s 󵄨󵄨 (1) (2) 󵄨 (n) 󵄨󵄨⟨T T ⋅ ⋅ ⋅ T π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C0 C1 ⋅ ⋅ ⋅ Cn ⟨z − χ1 ∘ χ2 ∘ ⋅ ⋅ ⋅ ∘ χn (w)⟩ ,

(6.125)

where C0 depends on s and on the Lipschitz constants of the mappings χ1 , χ1−1 , χ1 ∘ χ2 , (χ1 ∘ χ2 )−1 , . . . , χ1 ∘ χ2 ∘ ⋅ ⋅ ⋅ ∘ χn−1 , (χ1 ∘ χ2 ∘ ⋅ ⋅ ⋅ ∘ χn−1 )−1 . Observe that using Schur’s test and the same techniques as in the proof of Theorem 6.3.7, it is straightforward to obtain the following weighted version of Theorem 6.3.7. We omit the proof.

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Theorem 6.6.12. Let 0 ≤ r < s − 2d, and μ ∈ ℳvr (ℝ2d ). For every 1 ≤ p ≤ ∞, T ∈ p FIO(χ, s) extends to a continuous operator from Mμ∘χ (ℝd ) into Mμp (ℝd ). Let us underline that μ ∘ χ ∈ ℳvr (ℝ2d ), since vr ∘ χ ≍ vr , due to the bi-Lipschitz property of χ. The characterization below is written for pseudodifferential operators in the Kohn–Nirenberg form σ(x, D), but it works the same for any τ-form (in particular, Weyl form σ w (x, D)) in which a pseudodifferential operator is written. Gluing together the results of Theorems 6.4.9 and 6.4.10, we observe that the constant C in (6.77) satisfies ∞ . C ≍ ‖σ‖M1⊗v s

(6.126)

For χ = Id, we recapture the characterization for pseudodifferential operators of Theorem 6.3.3. Since we shall apply our results to the mappings χt (x, η) coming from the Hamilton–Jacobi system (6.113), we need to be more precise in the estimate (6.126): it is important to see how the constants involved in the equivalence depend on the time variable t. This amounts to rewriting the proofs of the results cited above for the special case of a phase function Φ ∈ 𝒞 ∞ (]−T, T[, Γ2 (ℝ2d )) and following the time variable t. We state the result here, sketching the main points of the proofs, and leaving the details to the interested reader.

392 | 6 Fourier integral operators and applications to Schrödinger equations Theorem 6.6.13. Consider g ∈ 𝒮 (ℝd ), s ≥ 0, and T > 0 such that in ]−T, T[ equation (6.117) is solved by the tame phase Φ ∈ 𝒞 ∞ (]−T, T[, Γ2 (ℝ2d )). Let χt be the related tame canonical transformation in (6.119). Let I be a continuous linear operator 𝒮 (ℝd ) → 𝒮 󸀠 (ℝd ). Then the following statements are equivalent: ∞ (i) I = I(σt , Φχt ) is a FIO of type I for some σt ∈ M1⊗v (ℝ2d ) such that s ∞ ≤ H(t) ∈ 𝒞 (]−T, T[). ‖σt ‖M1⊗v

(6.127)

s

(ii) I ∈ FIO(χt , s) and the constant C = C(t) in (6.77) is in 𝒞 (]−T, T[). Proof. First we prove (ii) �⇒ (i). Assume I ∈ FIO(χt , s) and |⟨Iπ(w)g, π(z))g⟩| ≤ C(t)⟨z − χt (w)⟩,

(6.128)

with C(t) positive continuous function on ]−T, −T[. Setting w = (x, η) and z = (x 󸀠 , η󸀠 ), using the fact that each component of the mapping χt (y, η) and its inverse is in 𝒞 ∞ (]−T, T[, Γ1 (ℝ2d )), we can control the Lipschitz constants of χt and χt−1 by continuous constants of t so that the equivalence of Lemma 6.4.2 becomes 󵄨󵄨 󵄨 󵄨 󸀠 󸀠󵄨 󵄨 󸀠 󸀠󵄨 󵄨 󸀠󵄨 󵄨󵄨∇x Φ(t, x , η) − η 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇η Φ(x , η) − x󵄨󵄨󵄨 ≍t 󵄨󵄨󵄨χ1 (t, x, η) − x 󵄨󵄨󵄨 + 󵄨󵄨󵄨χ2 (t, x, η) − η 󵄨󵄨󵄨 for every x, x󸀠 , η, η󸀠 ∈ ℝd and the implicit constants in the equivalence ≍t are continuous with respect to t ∈ ]−T, T[. This reduces the study to showing that if the operator I(σt , Φ(t, ⋅)), with Φ(t, ⋅) being the phase related to χt in (6.119) and satisfying (6.117), fulfills the estimate −s 󵄨󵄨 󵄨 󸀠 󸀠 󸀠 󸀠 󸀠 󵄨󵄨⟨I(σt , Φ(t, ⋅))π(x, η)g, π(x , η )g⟩󵄨󵄨󵄨 ≤ C(t)⟨∇x Φ(t, x , η) − η , ∇η Φ(t, x , η) − x⟩ (6.129)

with x, x󸀠 , η, η󸀠 ∈ ℝd , t ∈ ]−T, T[, then ∞ ‖σt ‖M1⊗v ≤ C(t),

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s

t ∈ ]−T, T[.

(6.130)

For z, w ∈ ℝ2d , let Φ2,z (t, ⋅) be the remainder in the second order Taylor expansion of the phase Φ(t, ⋅), i. e., 1

Φ2,z (t, w) = 2 ∑ ∫(1 − τ)𝜕α Φ(t, z + τw) dτ |α|=2 0

wα . α!

For a given window g ∈ 𝒮 (ℝd ), we set Ψz (t, w) = e2πiΦ2,z (t,w) (g ⊗ ĝ )(w). Then the fundamental relation between the Gabor matrix of an FIO and the STFT of its symbol in (6.35) can be rephrased in this framework as 󵄨󵄨 󵄨 󵄨 󵄨 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 󵄨󵄨⟨Iπ(x, η)g, π(x , η )g⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨VΨ(x󸀠 ,η) σt ((x , η), (η − ∇x Φ(t, x , η), x − ∇η Φ(t, x , η)))󵄨󵄨󵄨.

6.6 Propagation of Gabor wave front set for Schrödinger equations | 393

Writing u = (x󸀠 , η), v = (η󸀠 , x), (6.129) translates into −s 󵄨 󵄨󵄨 󵄨󵄨VΨu (t,⋅) σt (u, v − ∇Φ(t, u))󵄨󵄨󵄨 ≤ C(t)⟨v − ∇Φ(t, u)⟩ ,

and then into the estimate

󵄨 󵄨 ⟨w⟩s 󵄨󵄨󵄨VΨu (t,⋅) σt (u, w)󵄨󵄨󵄨 ≤ C(t).

sup

(u,w)∈ℝ2d ×ℝ2d

(6.131)

We have already shown for the time independent case Ψu (t, ⋅) = Ψu (⋅) that the set of windows Ψu possesses a joint time–frequency envelope, cf. Proposition 6.4.8. This ∞ 4d property allows writing σ ∈ M1⊗v (ℝ2d ) if and only if supu∈ℝ2d |VΨu σ| ∈ L∞ 1⊗vs (ℝ ) with s 󵄩󵄩 󵄩󵄩 ∞ ‖σ‖M1⊗v ≍ 󵄩󵄩󵄩 sup |VΨu σ|󵄩󵄩󵄩 ∞ 󵄩u∈ℝ2d 󵄩L1⊗vs s

(6.132)

(cf. Proposition 6.4.8 (ii)). The proof of the previous equivalence passes through several lemmas. We point out that the crucial element of the equivalence is a control of |VΨ e2πiΦ2,z (u, w)|, with Ψ ∈ 𝒮 (ℝ2d ) \ {0} fixed, by a polynomial pα (𝜕Φ2,z (ζ )) of derivatives of Φ2,z of degree at most |α| times a factor that does not depend on t. Since Φ(t, ⋅) ∈ 𝒞 ∞ (]−T, T[, Γ2 (ℝ2d )), we can control the polynomial by a continuous function of t and in the end obtain that the equivalence (6.132) depends continuously on t, which together with (6.131) gives (6.130). (i) �⇒ (ii) If I = I(σt , Φχ ) is an FIO of type I for Φ(t, ⋅) and χt in (6.119) and some σt ∈ ∞ M1⊗vs (ℝ2d ) which satisfies (6.127), then essentially reading backwards the arguments above gives I(σt , Φχ ) ∈ FIO(χt , s) with C(t) being a continuous function of t. 6.6.2 Unperturbed Schrödinger equations The previous theory applies in the study of the Cauchy problem for linear Schrödinger equations. First, consider the unperturbed case i𝜕t u + Au = 0,

{ Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

u(0, x) = u0 (x),

(6.133)

with x ∈ ℝd , u0 ∈ 𝒮 (ℝd ). The operator A = a(x, D) ∈ G2,cl is a formally self-adjoint pseudodifferential operator in the Kohn–Nirenberg form. This means that the symbol a ∈ Γ2,cl (ℝ2d ) has the expansion ∞

a(x, ξ ) ∼ ∑ a2−j (x, ξ ), j=0

where the principal symbol a2 (x, ξ ) is real-valued, since A is self-adjoint. The problem (6.133) is forward and backward well-posed in 𝒮 (ℝd ) and the corresponding evolution operator eitA , acting from 𝒮 (ℝd ) into 𝒮 (ℝd ), extends to L2 -isometries [187]. The classical evolution (6.113) has the solution (x(t), ξ (t)) = χt (y, η) in (6.116) and, for a suitable T > 0 and t ∈ ]−T, T[, the evolution operator eitA can be well approxi-

394 | 6 Fourier integral operators and applications to Schrödinger equations mated by an FIO of type I, as expressed in [187, Proposition 3.1.1] for the special case of elliptic operators (that is, operators whose corresponding principal symbols satisfy (6.112)), but still valid without the assumption (6.112), as observed in [81, Section 5.3]. In our framework the result [187, Proposition 3.1.1] can be rephrased as follows. Proposition 6.6.14. Given the Cauchy problem (6.133) with a(x, D) as above, then there exists a T > 0, a symbol σ(t, x, η) ∈ 𝒞 ∞ (]−T, T[, Γ0 (ℝ2d )), and a real-valued phase function Φ ∈ 𝒞 ∞ (]−T, T[, Γ2 (ℝ2d )) satisfying (6.117) and (6.118) such that the evolution operator can be written as (eitA u0 )(t, x) = (Ft u0 )(t, x) + (Rt u0 )(t, x),

(6.134)

where Ft is the FIO of type I (Ft u0 )(t, x) = ∫ e2πiΦ(t,x,η) σ(t, x, η)û0 (η) dη

(6.135)

ℝd

and the operator Rt has kernel in 𝒞 ∞ (]−T, T[, 𝒮 (ℝ2d )) (thus Rt is regularizing, i. e., Rt : 𝒮 󸀠 (ℝd ) → 𝒮 (ℝd )). This result says that in an interval ]−T, T[ the propagator eitA can be represented by a type I FIO Ft up to an error, which, however, is a regularizing operator. Remark 6.6.15. We observe that the function Φ(t, ⋅) of Proposition (6.6.14) and the related canonical transformation χt in (6.116) are tame, with Lipschitz constants of χt and its inverse that can be controlled by a continuous function of t on the interval ]−T, T[ and so can be chosen uniform with respect to t on ]−T, T[. We will show that if we replace the type I FIO Ft by a more general operator in the classes FIO(χt , s), we will be able to remove the error Rt in (6.134). Precisely, we can state the following result.

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Proposition 6.6.16. Under the assumptions of Proposition 6.6.14, we have eitA ∈ ∩s≥0 FIO(χt , s),

t ∈ ]−T, T[,

where χt is defined in (6.116). Moreover, for every s ≥ 0, there exists C(t) = Cs (t) ∈

𝒞 (]−T, T[) such that, for every g ∈ 𝒮 (ℝd ), the Gabor matrix satisfies −s 󵄨󵄨 itA 󵄨 󵄨󵄨⟨e π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C(t)⟨z − χt (w)⟩ ,

w, z ∈ ℝ2d .

(6.136)

Proof. By Proposition 6.6.14, there exists a T > 0 such that the evolution eitA can be written as (6.134), where Ft is a type I FIO with symbol in σ(t, x, η) ∈ 𝒞 ∞ (]−T, T[, Γ0 ) and phase Φ(t, ⋅) in (6.117). Since 0

0

𝒞 (]−T, T[, Γ ) ⊂ 𝒞 (]−T, T[, S0,0 ) = 𝒞 (]−T, T[, ⋂ M1⊗vs ), ∞

∞

∞

∞

s≥0

6.6 Propagation of Gabor wave front set for Schrödinger equations | 395

we can find C > 0 and k1 = k1 (s) ∈ ℕ such that 󵄩 󵄩 󵄩 α 󵄩󵄩 󵄩󵄩σ(t, ⋅)󵄩󵄩󵄩M ∞ ≤ C ∑ 󵄩󵄩󵄩𝜕 σ(t, ⋅)v|α| 󵄩󵄩󵄩∞ 1⊗vs |α|≤k1

where ∑|α|≤k1 ‖𝜕α σ(t, ⋅)v−|a| ‖∞ ∈ 𝒞 (]−T, T[) by assumption. Hence the characterization of Theorem 6.6.13 gives Ft ∈ ∩s≥0 FIO(χt , s), t ∈ ]−T, T[, where the canonical transformation χt is defined in (6.116) and related with Φ(t, ⋅) by (6.119) and −s 󵄨󵄨 󵄨 󵄨󵄨⟨Ft π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C(t)⟨z − χt (w)⟩

with C(t) ∈ 𝒞 (]−T, T[). Fix now g ∈ 𝒮 (ℝd ) with ‖g‖2 = 1 so that the inversion formula (1.56) becomes Id = Vg∗ Vg and we can write Rt = Vg∗ Vg Rt Vg∗ Vg . Since Rt is a regularizing operator, for Tt := Vg Rt Vg∗ , the following diagram is commutative: 󸀠

2d

𝒮 (ℝ )

Tt

- 𝒮 (ℝ2d ) 6 Vg

Vg∗

? d 𝒮 (ℝ ) 󸀠

Rt

- 𝒮 (ℝd )

This means that the linear operator Tt : 𝒮 󸀠 (ℝ2d ) → 𝒮 (ℝ2d ) is regularizing as well, and so its kernel kt (w, z) = ⟨Rt π(w)g, π(z)g⟩ ∈ 𝒮 (ℝ4d ) satisfies 󵄨󵄨 󵄨 󵄨 󵄨 −N −N 󵄨󵄨kt (w, z)󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨Rt π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ ⟨z⟩ ⟨w⟩ ,

∀z, w ∈ ℝ2d , ∀N ∈ ℕ.

(6.137)

The previous estimates yield Rt ∈ FIO(χ, s), for every bi-Lipschitz mapping χ and every s ≥ 0. Indeed,

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

⟨z − χ(w)⟩ ≤ ⟨z⟩⟨χ(w)⟩ ≍ ⟨z⟩⟨w⟩, and, choosing N ≥ s in (6.137), we obtain −s 󵄨󵄨 󵄨 −N −N 󵄨󵄨kt (w, z)󵄨󵄨󵄨 ≲ ⟨z⟩ ⟨w⟩ ≲ ⟨z − χ(w)⟩ . 0 Finally, if σ(Rt )(z) is the Kohn–Nirenberg symbol of Rt , using the fact that 𝒮 (ℝ2d ) ⊂ S0,0 with continuous embedding for every s ≥ 0, we find C > 0 and k2 ∈ ℕ such that

󵄩󵄩 󵄩 󵄩󵄩σ(Rt )󵄩󵄩󵄩M ∞ ≤ C 1⊗vs

󵄩 󵄩 ∑ 󵄩󵄩󵄩z α 𝜕zβ σ(Rt )(z)󵄩󵄩󵄩∞ ∈ 𝒞 (]−T, T[).

|α+β|≤k2

Using Theorem 6.6.13 with χt in (6.116), which is tame for t ∈ ]−T, T[, we find C(t) ∈ 𝒞 (]−T, T[) such that −s 󵄨󵄨 󵄨 󵄨󵄨⟨Rt π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C(t)⟨z − χt (w)⟩ .

396 | 6 Fourier integral operators and applications to Schrödinger equations Finally, the claim follows since FIO(χ, s) are linear spaces 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 itA 󵄨󵄨⟨e π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨⟨Ft π(w)g, π(z)g⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨⟨Rt π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C(t)⟨z − χt (w)⟩ , −s

which gives (6.136). The previous proposition gives an approximation of eitA for |t| < T. Using the group property of the propagator eitA Helffer in [187, page 139] describes how to obtain an approximation of eitA for every t ∈ ℝ. Indeed, a classical trick, jointly with the group property of eitA , applies. We consider T0 < T/2 and define Ih = ]hT0 , (h + 2)T0 [,

h ∈ ℤ.

For t ∈ Ih , by the group property of eitA , eitA = ei(t−hT0 )A (ei(hT0 )A/|h| ) , |h|

(6.138)

and, using Proposition 6.6.14, one can write eitA − Ft−hT0 (F h T )|h| ∈ 𝒞 ∞ (Ih , ℒ(𝒮 󸀠 , 𝒮 )). |h|

0

(6.139)

In general, eitA or even the composition Ft−hT0 (F h T )|h| cannot be represented as a type |h|

0

I FIO in the form (6.135). We shall prove below that the evolution eitA is in the class ⋂s≥0 FIO(χt , s) for every t ∈ ℝ, with χ defined in (6.116), so that this class is proven to be the right framework for describing the evolution eitA . Theorem 6.6.17. Given the Cauchy problem (6.133) with A = a(x, D) as above, consider the mapping χt defined in (6.116). Then eitA ∈ ⋂ FIO(χt , s), Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

s≥0

t ∈ ℝ,

and for every s > 2d there exists C(t) ∈ 𝒞 (ℝ) such that −s 󵄨󵄨 itA 󵄨 󵄨󵄨⟨e π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C(t)⟨z − χt (w)⟩ ,

w, z ∈ ℝ2d , t ∈ ℝ.

(6.140)

Proof. We fix T0 < T/2 as above. For t ∈ ℝ, there exists an h ∈ ℤ such that t ∈ Ih . Using Proposition 6.6.16, for t1 = t − hT0 ∈ ]−T, T[, we have that eit1 A ∈ FIO(χt1 , s) and, for

h T0 ∈ ]−T, T[, eit2 A ∈ FIO(χt2 , s), for every s ≥ 0, and there exists a continuous t2 = |h| function C(t) on ]−T, T[ such that (6.136) is satisfied for t = t1 and t = t2 . Using the algebra property (6.123), for every s > 2d,

eit1 A (eit2 A )

|h|

∈ FIO(χt1 ∘ (χt2 )|h| , s),

6.6 Propagation of Gabor wave front set for Schrödinger equations | 397

and the group law (6.115) for χt (y, η) = S0 (t)(y, η) gives χt1 ∘ (χt2 )|h| = χt1 +|h|t2 = χt , as expected, hence, using (6.125), we obtain that the Gabor matrix of the product eit1 A (eit2 A )|h| is controlled by a continuous function Ch (t) on Ih . Finally, from the estimate −s 󵄨󵄨 itA 󵄨 󵄨󵄨⟨e π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ Ch (t)⟨z − χt (w)⟩ ,

t ∈ Ih ,

with Ch ∈ 𝒞 (Ih ), it is easy to construct a new continuous controlling function C(t) on ℝ such that (6.140) is satisfied. 6.6.3 Schrödinger equations with bounded perturbations We now study the Cauchy problem for linear Schrödinger equations of the type {i 𝜕t + Hu = 0, { {u(0, x) = u0 (x), 𝜕u

(6.141)

with t ∈ ℝ and the initial condition u0 ∈ 𝒮 (ℝd ). We consider a Hamiltonian of the form

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H = a(x, D) + σ(x, D),

(6.142)

where A = a(x, D) is the pseudodifferential operator satisfying (6.133), whose corresponding propagator eitA ∈ ⋂s≥0 FIO(χt , s), for t ∈ ℝ, as shown in the preceding subsection. The perturbation B = σ(x, D) is a pseudodifferential operator with a symbol σ ∈ ∞ M1⊗v (ℝ2d ), s > 2d. This last requirement implies the boundedness of B on Mμ (ℝd ) for s a weight μ as in the assumptions of Theorem 6.6.12 (with χ = Id) (see also [162] using ∞ M1⊗v (ℝ2d ) ⊂ M ∞,1 (ℝ2d ), s > 2d), and in particular on L2 (ℝd ). Hence, H = A + B is s a bounded perturbation of the generator A of a unitary group by [249], and H is the generator of a well-defined (semi)group. We shall heavily use the theory of operator semigroups, referring to the textbooks [249] and [116] for an introduction on the topic. Our result, containing Theorem 6.6.1, is as follows. Theorem 6.6.18. Let s > 2d. Consider the Cauchy problem (6.141) with A = a(x, D) and B = σ(x, D) as above. Let χt be the mapping defined in (6.116). Then the solution can be written as itA ̃ eitH = eitA Q(t) = Q(t)e ∈ FIO(χt , s),

t ∈ ℝ,

398 | 6 Fourier integral operators and applications to Schrödinger equations ̃ are pseudodifferential operators with symbols in M ∞ (ℝ2d ) and the where Q(t) and Q(t) 1⊗vs continuous Gabor matrix satisfies −s 󵄨 󵄨󵄨 itH 󵄨󵄨⟨e π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C(t)⟨z − χt (w)⟩ ,

w, z ∈ ℝ2d ,

for a suitable positive continuous function C(t) on ℝ. Proof. The pattern is similar to [69, Theorem 4.1]. We show the result on the interval [0, +∞[, for the interval ] − ∞, 0] the result is obtained by the previous case by replacing t with −t. The operator A is the generator of a strongly continuous one-parameter group on L2 (ℝd ) and T(t) = eitA is the corresponding (semi)group that solves the evolution equation i dT(t) = AT(t). Then eitA is a strongly continuous one-parameter group on L2 (ℝd ). dt As already observed, by the assumptions on the symbol of B, it follows that B is a bounded operator on L2 (ℝd ), hence H = A + B is the generator of a strongly continuous one-parameter group S(t) [116]. The perturbed semigroup S(t) = eitH satisfies an abstract Volterra equation t

t

S(t)f = T(t)f + ∫ T(t − s)BS(s)f ds = T(t)(Id + ∫ T(−s)BT(s)T(−s)S(s) ds)f 0

(6.143)

0

for every f ∈ L2 (ℝd ) and t ≥ 0. If we define Q(t) = T(−t)S(t), then, by (6.143), Q(t) satisfies the Volterra equation t

Q(t) = Id + ∫ T(−s)BT(s)Q(s) ds,

(6.144)

0

where the integral is to be understood in the strong sense. Now write B(s) = T(−s)BT(s), then the solution of (6.144) can be written as a so-called Dyson–Phillips expansion ([249, X.69] or [116, Chapter 3, Theorem 1.10])

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∞

t t1

tn−1

∞

Q(t) = Id + ∑ (−i)n ∫ ∫⋅ ⋅ ⋅ ∫ B(t1 )B(t2 ) ⋅ ⋅ ⋅ B(tn ) dt1 ⋅ ⋅ ⋅ dtn := ∑ Qn (t). n=1

0 0

0

n=0

(6.145)

∞ We shall show that Q(t) is a pseudodifferential operator with symbol in M1⊗v (ℝ2d ). s For τ ∈ [0, t], the algebra property (6.123) gives

B(τ) = ei(−τ)A BeiτA ∈ FIO(χ−τ ∘ Id ∘ χτ , s) = FIO(Id, s) since χ−τ ∘ Id ∘ χτ = χ−τ ∘ χτ = S0 (0) = Id by (6.115). Moreover, ei(±τ)A satisfies (6.140), so that, using (6.125) with n = 3, T (1) = ei(−τ)A , T (2) = B, T (3) = eiτA , and χ = Id, we can write 󵄨󵄨 󵄨 −s 󵄨󵄨⟨B(τ)π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C(τ)⟨z − w⟩ ,

(6.146)

6.6 Propagation of Gabor wave front set for Schrödinger equations | 399

for a new continuous function C(τ) on ℝ. Using (6.125) again for the composition of pseudodifferential operators ∏nj=1 B(tj ), we obtain 󵄨󵄨 󵄨󵄨 n 󵄨 󵄨󵄨 󵄨󵄨⟨∏ B(tj )π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C0 C(t1 ) ⋅ ⋅ ⋅ C(tn )⟨z − w⟩−s , 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 j=1 with C(t) ∈ 𝒞 (ℝ) in (6.146). ∞ We now show that Qn (t) is a pseudodifferential operator with symbol in M1⊗v (ℝ2d ). s We control the Gabor matrix of Qn (t) as follows: tn−1

t t1

󵄨󵄨 󵄨 −s 󵄨󵄨⟨Qn (t)π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C0 ∫ ∫⋅ ⋅ ⋅ ∫ C(t1 ) ⋅ ⋅ ⋅ C(tn )dt1 ⋅ ⋅ ⋅ dtn ⟨z − w⟩ . 0 0

0

Defining H(t) = max C(τ) ∈ 𝒞 (ℝ), τ∈[0,t]

we obtain n 󵄨󵄨 󵄨 nt ⟨z − w⟩−s . 󵄨󵄨⟨Qn (t)π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ C0 H(t) n!

̃ Finally, setting H(t) = tH(t) ∈ 𝒞 (ℝ), ∞

󵄨 󵄨 󵄨󵄨 󵄨 −s 󵄨󵄨⟨Q(t)π(w)g, π(z)g⟩󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨⟨Qn (t)π(w)g, π(z)g⟩󵄨󵄨󵄨⟨z − w⟩ n=0

̃ n H(t) ⟨z − w⟩−s n! n=0 ∞

≤ C0 ∑

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= C(t)⟨z − w⟩−s , for a new function C(t) ∈ 𝒞 (ℝ). This gives, by Theorem 6.6.13, that Q(t) ∈ FIO(Id, s), ∞ that is, Q is a pseudodifferential operator with symbol in M1⊗v (ℝ2d ). Finally, the alges bra property again gives eitH = T(t)C(t) ∈ FIO(χt , s), and the estimate (6.123) gives that the Gabor matrix of eitH is controlled by a continuous function C(t) on ℝ. Consequently, the Schrödinger equation preserves the phase-space concentration, as expressed by the following result. p Corollary 6.6.19. Let 0 ≤ r < s − 2d, and μ ∈ ℳvr . If the initial condition u0 ∈ Mμ∘χ , 1 ≤ p ≤ ∞, then u(t, ⋅) = eitH u0 ∈ Mμp , for all t ∈ ℝ.

400 | 6 Fourier integral operators and applications to Schrödinger equations Proof. It follows immediately from Theorems 6.6.18 and 6.6.12. Using vr ∘ χ ≍ vr , we observe that the Schrödinger evolution preserves the phasespace concentration Mvpr of the initial condition u0 . In other words, the time evolution leaves Mvpr invariant. Corollary 6.6.20. Let |r| < s − 2d. If the initial condition u0 ∈ Mvpr , 1 ≤ p ≤ ∞, then

u(t, ⋅) = eitH u0 ∈ Mvpr , for all t ∈ ℝ.

Proof. The result is a special case of Corollary 6.6.19 once we prove, for r > 0, that vq is vr -moderate if and only if |q| ≤ r. But this is an easy consequence of Peetre’s inequality ⟨z + ζ ⟩q ≤ ⟨z⟩|q| ⟨ζ ⟩q . From Corollaries 6.6.19 and 6.6.20, we recapture (6.104) in Theorem 6.6.3. 6.6.4 Propagation of singularities We have reached the last topic of this book concerning the Gabor wave front set and its generalization in Definition 6.6.2. Proposition 6.6.21. Let f ∈ 𝒮 󸀠 (ℝd ), r > 0. Then (i) The definitions of WFp,r G (f ) and WFG (f ) do not depend on the choice of the window g. d (ii) f ∈ Mvpr (ℝd ) if and only if WFp,r G (f ) = 0. Similarly, f ∈ 𝒮 (ℝ ) if and only if WFG (f ) = 0. The proof of (i) will be given later, as a consequence of more general arguments. The proof of (ii) follows easily from the compactness of the sphere 𝒮 2d−1 and the characterization of the Schwartz class using modulation spaces in (2.28). The following statement gives the second part of Theorem 6.6.3.

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Theorem 6.6.22. Under the assumptions of Theorem 6.6.18, for u0 ∈ Mvp−r (ℝd ), 1 ≤ p ≤ ∞, 0 < 2r < s − 2d, we have p,r itH WFp,r G (e u0 ) = χt (WFG (u0 )).

(6.147)

p,r itH Proof. We shall prove that WFp,r G (e u0 ) ⊂ χt (WFG (u0 )) for any t ∈ ℝ. Then, by applying the inclusion to v0 = e−itH u0 , the opposite inclusion will follow, and (6.147) will be

proved. Fixing t ∈ ℝ, we assume z0 ∉ χt (WFp,r G (u0 )). Since χt is a homogeneous diffeomorphism for large |z|, this is equivalent to saying that w0 = χt−1 (z0 ) does not belong to 2d WFp,r G (u0 ). Therefore, for a sufficiently small open conic neighborhood Γw0 ⊂ ℝ \ 0 of w0 , we have 󵄨 󵄨p ∫ 󵄨󵄨󵄨Vg u0 (w)󵄨󵄨󵄨 ⟨w⟩pr < ∞.

Γw0

(6.148)

6.6 Propagation of Gabor wave front set for Schrödinger equations | 401

Note also that, in view of the assumption u0 ∈ Mvp−r (ℝd ), we have 󵄨p 󵄨 ∫ 󵄨󵄨󵄨Vg u0 (w)󵄨󵄨󵄨 ⟨w⟩−pr < ∞.

(6.149)

ℝd

Now using Theorem 6.6.18, we can write Vg (eitH u0 )(z) = ∫ k(t, w, z)Vg u0 (w) dw

(6.150)

ℝ2d

with −s 󵄨󵄨 󵄨 󵄨󵄨k(t, w, z)󵄨󵄨󵄨 ≤ C(t)⟨z − χt (w)⟩ ,

w, z ∈ ℝ2d .

(6.151)

itH We have to show that z0 ∉ WFp,r G (e u0 ). To this end, take an open conic neighborhood

Γ󸀠z0 of z0 , such that Γ󸀠z0 ⊂ χt (Γw0 ). This implies that, for z ∈ Γ󸀠z0 and w ∉ Γw0 , we have ⟨z − χt (w)⟩ ≳ max{⟨z⟩, ⟨w⟩},

(6.152)

since χt is a Lipschitz diffeomorphism. Using (6.150) and (6.151), we estimate 󵄨󵄨 r 󵄨 itH 󵄨󵄨⟨z⟩ Vg (e u0 )(z)󵄨󵄨󵄨 ≲ ∫ I(z, w) dw,

(6.153)

−s 󵄨 󵄨 I(z, w) = ⟨z⟩r ⟨z − χt (w)⟩ 󵄨󵄨󵄨Vg u0 (w)󵄨󵄨󵄨.

(6.154)

ℝ2d

with

To show z0 ∉ WFrG (eitH u0 ), it will be sufficient to establish that 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 < ∞. 󵄩󵄩 ∫ I(⋅, w) dw󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩Lp (Γ󸀠 ) z0 2d Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

ℝ

First, we estimate ∫ℝ2d I(z, w) dw for z ∈ Γ󸀠z0 . We split the domain of integration into

two domains Γw0 and ℝ2d \ Γw0 . In ℝ2d \ Γw0 , we use (6.152) to obtain I(z, w) dw ≤

∫ ℝ2d \Γ

ℝ2d \Γ

w0

≲

−s |Vg u0 (w)| ⟨w⟩r

∫ ⟨z⟩r ⟨w⟩r ⟨z − χt (w)⟩ w0

2r−s |Vg u0 (w)| ⟨w⟩r

∫ ⟨z − χt (w)⟩ ℝ2d \Γw0

≲ (⟨⋅⟩2r−s ∗

|Vg u0 (⋅)| ⟨⋅⟩r

)(z).

dw

dw

402 | 6 Fourier integral operators and applications to Schrödinger equations So by (6.149) and using 2r − s < −2d, 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∫ 󵄩󵄩 2d

ℝ \Γw0

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≲ 󵄩󵄩⟨⋅⟩2r−s 󵄩󵄩󵄩L1 (ℝ2d ) 󵄩󵄩󵄩|Vg u0 |⟨⋅⟩−r 󵄩󵄩󵄩Lp (ℝ2d ) < ∞. I(⋅, w) dw󵄩󵄩󵄩 󵄩󵄩Lp (Γ󸀠 ) 󵄩 z0

In the domain Γw0 , we have ∫ I(z, w) dw ≤ ∫ ⟨z⟩r ⟨w⟩−r ⟨z − χt (w)⟩ ⟨z − χt (w)⟩ −r

Γw0

Γw0

r−s 󵄨

󵄨󵄨Vg u0 (w)󵄨󵄨󵄨⟨w⟩r dw 󵄨 󵄨

r−s 󵄨

≲ ∫ ⟨z − χt (w)⟩ Γw0

r−s

≲ ⟨χt−1 (⋅)⟩

󵄨󵄨Vg u0 (w)󵄨󵄨󵄨⟨w⟩r dw 󵄨 󵄨

∗ (CharΓw ⋅ |Vg u0 |⟨⋅⟩r )(z) 0

where CharΓw is the characteristic function of the set Γw0 . The assumption (6.148) 0 yields the estimate 󵄩󵄩 󵄩󵄩 r−s 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≲ 󵄩󵄩⟨χ −1 (⋅)⟩ 󵄩󵄩󵄩L1 (ℝ2d ) 󵄩󵄩󵄩|Vg u0 |⟨⋅⟩r 󵄩󵄩󵄩Lp (Γ ) 󵄩󵄩 ∫ I(⋅, w) dw󵄩󵄩󵄩 w0 󵄩󵄩 󵄩󵄩Lp (Γ󸀠 ) 󵄩 t z0 Γw0

󵄩 󵄩 󵄩 󵄩 ≍ 󵄩󵄩󵄩⟨⋅⟩r−s 󵄩󵄩󵄩L1 (ℝ2d ) 󵄩󵄩󵄩Vg u0 ⟨⋅⟩r 󵄩󵄩󵄩Lp (Γ

w0 )

< ∞,

for χt is a bi-Lipschitz diffeomorphism and r − s < 2r − s < −2d. This concludes the proof.

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The preceding arguments apply with small changes in the proof of (6.108). Let us detail the proof for sake of clarity. Proof of (6.108). As in the previous proof, it is enough to show WFG (eitH u0 ) ⊂ χt (WFG (u0 )) for any t ∈ ℝ. We have to prove that for every u0 ∈ 𝒮 󸀠 (ℝd ) and z0 ∈ ℝd , z0 ≠ 0, the assumption z0 ∉ χt (WFG u0 ) implies z0 ∉ WFG (eitH u0 ). Arguing as before, we have that the estimates (6.148) are satisfied for every r > 0 in a cone Γw0 independent of r. Now recall from (2.28) that 𝒮 󸀠 (ℝd ) = ⋃s≥0 Mv∞−s (ℝd ). Therefore,

0 ∞ u0 ∈ Mv∞−r (ℝd ) for some r0 ≥ 0. Since σ ∈ S0,0 = ⋂s≥0 M1⊗v (ℝ2d ) by (6.72), we have s 0

∞ σ ∈ M1⊗v (ℝ2d ) for every s ≥ 0. We may then apply the arguments in the preceding s proof with s > 2r + 2d > 2r0 + 2d and obtain the expected estimates (6.107) for any r > 0. By observing that the choice of the cone Γ󸀠z0 does not depend on r, the proof is concluded.

Proof of Proposition 6.6.21 (i). We prove the independence of the definition of WFp,r G (f ) on the choice of the window g. The independence of WFG (f ) is attained similarly. We assume the estimate for Vg f in (6.103) is satisfied for some fixed g ∈ 𝒮 (ℝd ) \ {0} and some conic neighborhood Γz0 , and we want to prove that the estimate holds for

6.6 Propagation of Gabor wave front set for Schrödinger equations | 403

Vh f , where h ∈ 𝒮 (ℝd ) \ {0} is fixed arbitrary, after possibly shrinking Γz0 . To this end, we use Lemma 1.2.29 which gives 󵄨 󵄨󵄨 󵄨󵄨Vh f (z)󵄨󵄨󵄨 ≲ (|Vg f | ∗ |Vh g|)(z). Since Vh g ∈ 𝒮 (ℝ2d ) for g, h ∈ 𝒮 (ℝd ), we have that for every s ≥ 0, 󵄨 󵄨󵄨 −s 󵄨󵄨Vh f (z)󵄨󵄨󵄨 ≲ ∫ ⟨z − w⟩ |Vg f |(w) dw. ℝ2d

We know that f ∈ Mvp−r (ℝd ), for some r0 ≥ 0. Taking then s > max{r, r0 + 2d}, the 0 arguments in the proof of Theorem 6.6.22 apply with χt = Id, w0 = z0 . ∞ Proposition 6.6.23. Let σ ∈ M1⊗v (ℝ2d ), s > 2d and 0 < 2r < s − 2d. Then for every s

f ∈ Mvp−r (ℝd ), we have

p,r WFp,r G (σ(x, D)f ) ⊂ WFG (f ).

(6.155)

0 If σ ∈ S0,0 , then for every f ∈ 𝒮 󸀠 (ℝd ),

WFG (σ(x, D)f ) ⊂ WFG (f ).

(6.156)

∞ Proof. If σ ∈ M1⊗v (ℝ2d ), then, from Theorems 6.3.3 and 6.3.5, we have that the Gabor s matrix k(w, z) of σ(x, D) satisfies

󵄨󵄨 󵄨 −s 󵄨󵄨k(w, z)󵄨󵄨󵄨 ≲ ⟨z − w⟩ ,

w, z ∈ ℝ2d ,

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so that σ(x, D) ∈ FIO(χ, s) with χ = Id. The arguments of the proof of the Theorem 6.6.22 then apply with w0 = z0 . The proof of (6.156) is similar. We end this subject with some examples of Schrödinger equations. Addressing first nonexpert readers, we present some properties of WFG (f ) and treat in this framework the free particle and the harmonic oscillator with smooth potentials, cf. Examples 6.6.25, 6.6.26, and 6.6.27. The final Example 6.6.28 concerns nonsmooth potentials. Proposition 6.6.24. Let f ∈ 𝒮 󸀠 (ℝd ). Then (i) WFG (π(z0 )f ) = WFG (f ) for every z0 = (x0 , ξ0 ) ∈ ℝ2d . (ii) Let δx0 be the Dirac distribution at the point x0 ∈ ℝd . Then WFG (δx0 ) = {z = (x, ξ ), x = 0, ξ ≠ 0} independently of x0 . (iii) Let ξ0 be fixed in ℝd . Then WFG (e2πix⋅ξ0 ) = {z = (x, ξ ), x ≠ 0, ξ = 0} independently of ξ0 .

404 | 6 Fourier integral operators and applications to Schrödinger equations (iv) Let c ∈ ℝ, c ≠ 0, be fixed. Then 2

WFG (eπicx ) = {z = (x, ξ ), x ≠ 0, ξ = cx}. Proof. The proof of (i) is a consequence of Proposition 6.6.23, since π(z0 ) = Mξ0 Tx0 = σ(x, D), with σ(x, D) being a pseudodifferential operator with symbol 0 σ(x, ξ ) = e2πi(x⋅ξ0 −x0 ⋅ξ ) ∈ S0,0 .

Concerning (ii), we are left to compute WFG (δ) since WFG (δx0 ) = WFG (Tx0 δ) = WFG (δ) by claim (i). On the other hand, Vg (δ)(x, ξ ) = g(−x). Hence in a small conic neighborhood Γ ⊂ ℝ2d of the ray x = tξ , t ∈ ℝ, ξ ≠ 0, we have rapid decay of g(−tξ ) except for t = 0, giving the claim. To prove (iii), we proceed similarly as before. From claim (i), we obtain that WFG (e2πix⋅ξ0 ) = WFG (Mξ0 1) = WFG (1). ̂ ̂ On the other hand, |Vg 1(x, ξ )| = |M−x g(−ξ )| so that |Vg 1(x, ξ )| = |g(−ξ )| and the arguments when proving claim (ii) give the desired result. 2 We now prove (iv). We use the Gaussian g(x) = e−πx as a window for the STFT Vg f 2

with f (x) := eπicx . Then standard computations (see also [14, Theorem 14]) give 2

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(ξ −cx) 󵄨󵄨 󵄨 2 −d/4 −π 1+c2 . e 󵄨󵄨Vg f (x, ξ )󵄨󵄨󵄨 = (1 + c )

The right-hand side is rapidly decaying in any open cone of ℝ2d excluding the line ξ − cx = 0. This concludes the proof of the proposition. Example 6.6.25 (The free particle). Consider the Cauchy problem for the Schrödinger equation in (5.11) (cf. Section 6.2.3.1). Namely, i𝜕t u + Δu = 0,

{

u(0, x) = u0 (x),

with x ∈ ℝd , d ≥ 1. The explicit formula for the solution in terms of the kernel is given by (5.21), with Kt in (5.22). In terms of classical FIO, 2

u(t, x) = ∫ e2πi(x⋅η−2πtη ) û0 (η) dη, ℝd

6.6 Propagation of Gabor wave front set for Schrödinger equations | 405

2

where the Gabor matrix with window function g(x) = e−πx is controlled by 󵄨 󵄨󵄨 −ϵ(z−χt (w))2 , 󵄨󵄨k(w, z)󵄨󵄨󵄨 ≤ Ce for suitable constants C > 0 and ϵ > 0 and where, for w = (y, η), (x, ξ ) = χt (y, η) = (y + 4πtη, η).

(6.157)

Beside the effectiveness in numerical analysis already detailed in Section 6.2.3.1, this expression emphasizes the microlocal properties of the propagator. Let us test the propagator of the Gabor wave front set on some particular initial data. If u0 = δ then u(t, x) = Kt (x) by (5.21). This is coherent with (6.108) and (6.157), since from Proposition 6.6.24, (iv) and (ii), we have WFG (u(t, x)) = WFG (Kt ) = {(x, ξ ), x = 4πtξ , ξ ≠ 0}

= χt (WFG (δ)) = χt ({(y, η), y = 0, η ≠ 0}).

We remark a similar propagation for the initial datum u0 = K−1 (t) = (−4πi)−d/2 e−ix

2

/4

,

for which we have ut=1 = δ. Instead, for u0 = e2πix⋅ξ0 , with ξ0 ∈ ℝd , we have u(t, x) = e−4π

2

itξ02 2πix⋅ξ0

e

,

and in this case the Gabor wave front set is stuck WFG (u(t, x)) = WFG (u0 ) = {(x, 0), x ≠ 0}, by Proposition 6.6.24 (iii) and (6.157).

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Example 6.6.26 (The harmonic oscillator). Consider the Cauchy problem for the harmonic oscillator in (6.68), with solution in terms of an FIO type I detailed in (6.69). 2 The Gabor matrix with Gaussian window g(x) = e−πx can be explicitly computed as 󵄨󵄨 󵄨 − d − π (z−χt (w))2 , 󵄨󵄨k(w, z)󵄨󵄨󵄨 = 2 2 e 2

(6.158)

where the canonical transformation is defined in (6.70). Observe that the expression (6.158) is meaningful for every t ∈ ℝ. We may test (6.108) on the initial datum u0 (x) = 1, giving for t < π/2, 2

u(t, x) = (cos t)−d/2 eπi tan tx . From Proposition 6.6.24, (iii) and (iv), we have coherently with (6.70) that WFG (u(t, x)) = {(x, ξ ), x = (cos t)y, ξ = (sin t)y, y ≠ 0} = χt (WFG (1)) = χt ({(y, η), y ≠ 0, η = 0}).

406 | 6 Fourier integral operators and applications to Schrödinger equations Example 6.6.27 (Smooth potentials). We now consider the presence in Example 6.6.25 0 of a potential with symbol in the class S0,0 . Consider the case e−2πix0 ⋅ξ ,

x0 ∈ ℝd fixed.

The related pseudodifferential operator σ(D) is the translation operator σ(D)f (x) = Tx0 f (x) = f (x − x0 ), which does not preserve the singular support. Consider first the equation i𝜕t u + σ(D)u = 0, { u(0, x) = u0 (x). The solution is given by u(t, x) = eitTx0 u0 (x) = ∫ e2πix⋅ξ exp(ite−2πix0 ⋅ξ )û0 (ξ ) dξ . ℝd 0 Despite the nasty oscillations, the symbol of the solution operator belongs to S0,0 and, from Proposition 6.6.23, we have for every fixed t ∈ ℝ,

WFG (eitTx0 u0 ) = WFG (u0 ), the identity being granted by the fact that Tx−10 = T−x0 . Note that the singular support can be expanded. In fact, taking u0 = δ, we have (it)n δnx0 ∈ 𝒮 󸀠 (ℝd ) n! n=0 ∞

eitTx0 δ = ∑

so that sing supp eitTx0 δ = {nx0 }n∈ℤ+ as soon as t ≠ 0, whereas

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WFG (eitTx0 δ) = WFG (δ) = {(0, ξ ), ξ ≠ 0}. Adding now the potential σ(D) to the free particle in Example 6.6.25, we have the Schrödinger equation with space-delay i𝜕t u + Δu + Tx0 u = 0, { u(0, x) = u0 (x). Since the operators eitΔ and Tx0 commute, the arguments of Section 6.6.3 provide as propagator eitTx0 eitΔ , that is, the convolution with (it)n Kt (x − nx0 ) ∈ 𝒮 󸀠 (ℝd ), n! n=0 ∞

∑

where Kt is defined in (5.22). The Gabor propagation is the same as in Example 6.6.25.

6.6 Propagation of Gabor wave front set for Schrödinger equations | 407

From a physical point of view, it is perhaps more natural to consider the case when the potential depends on x alone, for example, i𝜕t u + Δu + Mξ0 u = 0,

{

u(0, x) = u0 (x),

with Mξ0 u0 = e2πix⋅ξ0 u0 , ξ0 fixed in ℝd . Notice that now the operators eitΔ and Mξ0 do not commute and, proceeding as in Section 6.6.3 with the perturbation Bu = Mξ0 u, we have first to consider B(t) = e−itΔ e2πix⋅ξ0 eitΔ . Omitting further explicit computations, we obtain B(t) = e4π

2

iξ02 t

Mξ0 T−4πtξ0 .

(6.159)

In principle, one could then continue the computation of the pseudodifferential operator Q(t) in (6.145) explicitly, and the solution operator will be eitΔ Q(t). Observe in (6.159) the presence of the translation factor T4πtξ0 , providing the same phenomena as before. Example 6.6.28 (Nonsmooth potentials). As examples of admissible nonsmooth potentials, consider first a nonpolynomial homogeneous function h(z), z = (x, ξ ), h(λz) = λr h(z) for z ≠ 0, λ > 0, r > 0, with h ∈ 𝒞 ∞ (ℝ2d \ {0}), and take then as potential any 0 function σ(z) = h(z), for |z| ≤ 1, and h(z) ∈ S0,0 for |z| ≥ 1. This potential satisfies ∞ 2d σ ∈ M1⊗vr+2d (ℝ ). In fact, we may limit the analysis to the singularity at the origin. From Proposition 6.6.6 we have, for ψ ∈ 𝒮 (ℝ2d ),

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󵄨󵄨 󵄨 −r−2d , 󵄨󵄨Vψ σ(z, ζ )󵄨󵄨󵄨 ≤ C⟨ζ ⟩

z, ζ ∈ ℝ2d .

(6.160)

We may now return to the discussion about the smoothness at the origin of the Hamiltonian a(z) in the Introduction. Consider h(z) real-valued nonpolynomial homogeneous of degree 2, h ∈ 𝒞 ∞ (ℝ2d \ {0}), just to give an example 1/2

h(x, ξ ) = (|x|4 + |ξ |4 ) . We can include in our analysis the equation i𝜕t u + h(x, D)u = 0, { u(0, x) = u0 (x),

(6.161)

by absorbing the singularity at the origin into the potential. Namely, take φ ∈ 𝒞0∞ (ℝd ), 0 ≤ φ(z) ≤ 1, φ(z) = 1 for |z| ≤ 1, φ(z) = 0 for |z| ≤ 2, and split h(z) = a(z) + σ(z),

a(z) = (1 − φ(z))h(z),

σ(z) = φ(z)h(z).

408 | 6 Fourier integral operators and applications to Schrödinger equations At this moment a(z) satisfies the assumptions in the Introduction and the potential σ ∞ belongs to M1⊗v (ℝ2d ), in view of (6.160). We may then apply Theorem 6.6.3 to the 2+2d Cauchy problem (6.161). Note that the result of propagation should be limited to u0 ∈ itH Mvp−r (ℝd ) and WFp,r G (e u0 ) with 0 < r < 1. Finally, we present an example of a nonsmooth potential depending on x alone, namely in dimension d = 1, σ(x, ξ ) = | sin x|μ ,

μ > 1, x, ξ ∈ ℝ.

∞ By Corollary 6.6.5, σ ∈ M1⊗v (ℝ2 ). So consider, for instance, the perturbed harmonic μ+1 oscillator in (6.106). From Theorem 6.6.3 we have that the Cauchy problem is wellposed for u0 ∈ Mvpr (ℝ), |r| < μ − 2, and the propagation of WFp,r G (u(t, ⋅)) for t ∈ ℝ takes place as in Example 6.6.26 for 0 < r < μ/2 − 1.

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For further applications to the Gabor wave front set to regularity problems and propagation of singularities, we refer to [237, 238]. Finally, we underline that time–frequency analysis has been successfully applied to the theory of path integrals. This subject is out of the scope of this book, but it has many challenging open questions. We refer the interested reader to the works [235, 239].

A Appendix In this section we list some basic results from functional analysis, we have used throughout this book. Most of the topics are from the text [183]. Additional references are the textbooks [37, 57, 146, 160, 254, 268]. Theorem A.0.1 (Minkowski’s integral inequality). Suppose that (S1 , ℳ1 , μ1 ) and (S2 , ℳ2 , μ2 ) are two σ-finite measure spaces and F : S1 × S2 → ℝ is a (ℳ1 ⊗ ℳ2 )-measurable function. (i) If F ≥ 0, 1 ≤ p < ∞, then

1 p

p

1 p

(∫(∫ F(x, y) dμ2 (y)) dμ1 (x)) ≤ ∫(∫ F(x, y)p dμ1 (x)) dμ2 (y). S1 S2

S2 S1

(ii) If 1 ≤ p ≤ ∞, F(⋅, y) ∈ Lp (S1 ) for a. e. y, and the function y → ‖F(⋅, y)‖p is in L1 (S2 ), then F(x, ⋅) ∈ L1 (S2 ) for a. e. x, the function x �→ ∫S F(x, y) dμ2 (y) is in Lp (S1 ), and 2

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩∫ F(⋅, y) dμ2 (y)󵄩󵄩󵄩 ≤ ∫󵄩󵄩󵄩F(⋅, y)󵄩󵄩󵄩p dμ2 (y). 󵄩󵄩 󵄩󵄩p S2

S2

The closed graph theorem provides a convenient means to test whether a linear operator acting between Banach spaces is continuous. Theorem A.0.2 (Closed graph theorem). Let X and Y be Banach spaces. If A : X → Y is linear, then the following are equivalent: (i) A is continuous. (ii) The graph A = {(f , Af ) : f ∈ X} is a closed subset of X × Y. (iii) If fn → f in X and Afn → g in Y, then g = Af .

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Theorem A.0.3 (Uniform boundedness principle). Let X be a Banach space and Y a normed space. If {Ai } is a collection of linear bounded operators from X to Y such that ∀f ∈ X

sup ‖Ai f ‖ < ∞, i

then sup ‖Ai ‖ < ∞. i

Theorem A.0.4. Let H, K be Hilbert spaces and A ∈ ℬ(H, K). We denote by R(A) the range of the operator A. Then the following are equivalent: (i) There exists C > 0 such that ‖x‖H ≤ C‖Ax‖K , https://doi.org/10.1515/9783110532456-008

∀x ∈ H.

(A.1)

410 | A Appendix (ii) ker(A) = {0} and R(A) is closed. (iii) A∗ : K → H is surjective, i. e., R(A∗ ) = H. Proof. (i) �⇒ (ii) If Ax = 0 then by (A.1), x = 0, that is, ker(A) = {0}. Let us prove that R(A) is closed. Consider {yn } ⊂ R(A) such that yn → y, then there exists a sequence {xn } ⊂ H such that Axn = yn . By (A.1), ‖xm − xn ‖ ≤ C‖Axm − Axn ‖ = ‖ym − yn ‖, so that, since {yn } is a Cauchy sequence, so is {xn }. By the completeness of H, xn → x and Axn → Ax for A is bounded. So Ax = y, as desired. (ii) �⇒ (iii) Observe that R(A) is closed if and only if ker(A)⊥ = R(A∗ ) (cf. the next Definition A.1.8 and [37, Theorem 2.19]). Using the assumption ker(A) = {0}, we infer R(A∗ ) = H, that is, A∗ is surjective. (iii) �⇒ (i) Since ker(A) = R(A∗ )⊥ (see [37, Corollary 2.18]), by the surjectivity of ∗ A we obtain that A is injective. Since R(A∗ ) is closed also R(A) is so (see [37, Theorem 2.19]). Hence A : H → R(A) is a topological isomorphism; this means that if we call A−1 : R(A) → H its pseudoinverse, then A−1 is bounded, that is, (A.1) holds. Recall that a linear bounded self-adjoint operator on a Hilbert space H is called positive if ⟨Tx, x⟩ ≥ 0,

∀x ∈ H.

Lemma A.0.5 (Generalized Cauchy–Schwarz inequality). If T : H → H is a linear bounded self-adjoint positive operator on a Hilbert space H, then 󵄨󵄨 󵄨2 󵄨󵄨⟨Tx, y⟩󵄨󵄨󵄨 ≤ ⟨Tx, x⟩⟨Ty, y⟩,

∀x, y ∈ H.

(A.2)

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The following result defines the square root of a positive operator (for a proof, see, e. g., [183, Theorem 2.18]). Theorem A.0.6. Let T be a linear bounded self-adjoint operator on a Hilbert space H. If T is positive, then there exists a unique positive operator, denoted by T 1/2 , such that (T 1/2 )2 = T. Moreover, T 1/2 commutes with any operator which commutes with T. Remark A.0.7. If a positive operator T is a topological isomorphism, then it is easy to check that T −1 is a positive operator as well. Hence, by Theorem A.0.6, the operator T −1 has a positive square root T −1/2 . Let us check that (T 1/2 )−1 = T −1/2 . First, observe that, since T and T −1 commute, by Theorem A.0.6, also T 1/2 and T −1/2 commute. Denoting by I the identity operator, 2

2

I = TT −1 = (T 1/2 ) (T −1/2 ) = T 1/2 T −1/2 T 1/2 T −1/2 2

= (T 1/2 T −1/2 ) . Since I 2 = I and by the uniqueness of the square-root (cf. Theorem A.0.6), we deduce T 1/2 T −1/2 = I. Analogously, it can be shown that T −1/2 T 1/2 = I.

A Appendix | 411

Theorem A.0.8. If T ∈ B(H) (that is, T is a bounded linear operator on the Hilbert space H) is a self-adjoint operator, then 󵄨 󵄨 ‖T‖ = sup 󵄨󵄨󵄨⟨Tx, x⟩󵄨󵄨󵄨. ‖x‖=1

If T is positive then ‖T‖ = sup ⟨Tx, x⟩. ‖x‖=1

Theorem A.0.9 (The density principle). Let B1 , B2 be Banach spaces. Let X be a dense subspace of B1 . Assume that A : X → B2 is a (conjugate-)linear operator that satisfies the inequality ‖Af ‖B2 ≤ C‖f ‖B1 ,

∀f ∈ X,

(A.3)

for some C > 0. Then A extends to a bounded operator from B1 to B2 and (A.3) holds for all f in B1 . Likewise, A : X → B2 extends to a bounded operator on B1 if it satisfies 󵄨󵄨 󵄨 󵄨󵄨⟨Af , h⟩󵄨󵄨󵄨 ≤ C‖f ‖B1 ‖h‖B∗2 ,

∀f ∈ X,

∀h ∈ Y

(A.4)

where Y is a dense subset of B∗2 . Proof. Assume (A.3). For every f ∈ B1 there exists a sequence {fn } ⊂ X such that limn→∞ ‖f − fn ‖B1 = 0. So {fn } is a Cauchy sequence and 󵄩 󵄩 ‖Afm − Afn ‖B2 = 󵄩󵄩󵄩A(fm − fn )󵄩󵄩󵄩B ≤ C‖fm − fn ‖B1 , 2

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by (A.3). Thus {Afn } ⊂ B2 is a Cauchy sequence and, by the completeness of B2 , converges, therefore we may define Af := limn→∞ Afn . As a consequence, relation (A.3) is satisfied for every f ∈ B1 . The definition does not depend on the choice of the approximating sequence. Indeed, choose {gn } ⊂ X such that gn → f and set Af̃ := limn→∞ Agn , then ‖Af̃ − Af ‖B2 = ‖Af̃ − Agn + Agn − Afn + Afn − Af ‖B2 󵄩 󵄩 ≤ ‖Af̃ − Agn ‖B2 + 󵄩󵄩󵄩A(gn − fn )󵄩󵄩󵄩B + ‖Afn − Af ‖B2 2

≤ ‖Af̃ − Agn ‖B2 + C‖gn − fn ‖B1 + ‖Afn − Af ‖B2 → 0

for n → +∞, hence Af̃ = Af . The extended mapping A is linear: for every λ, μ ∈ ℂ, f , g ∈ B1 there exist sequences {fn }, {gn } ⊂ X such that fn → f , gn → g, for n → +∞ and so A(λfn + μgn ) = λA(fn ) + μA(gn ), and, taking the limit as n → +∞, we have A(λf + μg) = λAf + μAg.

412 | A Appendix Now, fix f ∈ X and consider the mapping h �→ ⟨Af , h⟩. This mapping is (conjugate-)linear and by assumption (A.4) is bounded on Y, a dense subset of B∗2 . Hence, by the previous part of the proof, it extends to a (conjugate-)linear mapping on B∗2 and fulfills (A.4) for every h ∈ B∗2 . Finally, using ‖Af ‖B2 = sup‖h‖B∗ =1 |⟨Af , h⟩| and (A.4), we 2

obtain ‖Af ‖B2 ≤ C‖f ‖B1 for every f ∈ X; the first part of the proof allows extending the previous estimate to every f ∈ B1 , yielding the claim. Theorem A.0.10. Let Q : H × H → ℂ be a bounded sesquilinear form on a Hilbert space H, in the sense that Q(f , g) is linear in the first component, conjugate linear in the second, and verifies the estimate 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Q(f , g)󵄨󵄨󵄨 ≤ C 󵄨󵄨󵄨|f ||H ||g|󵄨󵄨󵄨H ,

∀f , g ∈ H.

Then there exists a unique bounded operator A with the operator norm ||A|| ≤ C such that Q(f , g) = ⟨Af , g⟩ for all f , g ∈ H.

A.1 Basis theory and series expansions Here we shall recall definitions and basic properties of this subject, borrowed from the textbook [183]. A.1.1 Bases and orthonormal sequences Definition A.1.1. A sequence {xn } of vectors in a Banach space X is a basis for X if every vector x ∈ X can be written as Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

∞

x = ∑ cn (x)xn , n=1

(A.5)

for a unique choice of scalars (cn (x)). Observe that this definition implies that the coefficients cn (x) are linear functionals, also called coefficient functionals. Definition A.1.2. A sequence {xn } of vectors in a Banach space X is complete if N

span{xn } = { ∑ cn xn , cn ∈ ℂ}, n=1

that is, the set of all finite linear combinations of elements of {xn } is dense in X.

A.1 Basis theory and series expansions | 413

If {xn } is a basis, for every x, we can find a sequence of scalars (cn ) such that SN = ∑Nn=1 cn xn → x, for N → +∞, and this means that the span of {xn } is dense in X. Therefore every basis is a complete sequence. The converse is not true in general, as shown in the following example. Example A.1.3. For a, b ∈ ℝ, a < b, consider the space 𝒞 ([a, b]) with uniform norm. Weierstrass approximation theorem says that for every f ∈ 𝒞 ([a, b]) and ϵ > 0 there exists a polynomial p(x) = ∑kn=0 cn xn such that ‖f − p‖∞ < ϵ. So the set of monomials {xn } is a complete sequence in 𝒞 ([a, b]), but not a basis. Indeed, not every function in ∞ 𝒞 ([a, b]) can be written as f (x) = ∑n=0 cn x n with uniform convergence, cf. [183]. Example A.1.4. A standard basis for the space c0 (ℕ) = {(an ) : limn→+∞ |an | = 0}, the space of sequences that decay to zero at infinity, with the uniform norm, is given by {δn }n∈ℕ , where δn = (0, 0, . . . , 0, 1, 0, . . . , 0, . . . ), with the 1 in the nth component. The same sequence {δn } is also a basis for the space of sequences ℓp (ℕ), 1 ≤ p ≤ ∞, where p 1/p ‖(an )‖ℓp = (∑∞ n=0 |an | ) . Let H be a Hilbert space with inner product ⟨⋅, ⋅⟩. Then we recall the following definitions. Definition A.1.5. Let {xn } be a sequence in a Hilbert space H. (i) {xn } is an orthogonal sequence if ⟨xm , xn ⟩ = 0 whenever m ≠ n. (ii) {xn } is an orthonormal (in short, o. n.) sequence if ⟨xm , xn ⟩ = δm,n , that is, {xn } is an orthogonal sequence and ‖xn ‖ = 1 for every n. (iii) An orthonormal sequence {xn } is an orthonormal basis if it is both orthonormal and a basis.

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When ⟨x, y⟩ = 0, we write x⊥y. Proposition A.1.6. Let H be a Hilbert space and let x, y ∈ H. Then (i) (Polar identity) ‖x + y‖2 = ‖x‖2 + 2ℜ(⟨x, y⟩) + ‖y‖2 . (ii) (Pythagorean theorem) If ⟨x, y⟩ = 0, then ‖x + y‖2 = ‖x‖2 + ‖y‖2 . (iii) (Parallelogram law) ‖x + y‖2 + ‖x − y‖2 = 2(‖x‖2 + ‖y‖2 ). (iv) (Cauchy–Schwarz inequality) |⟨x, y⟩| ≤ ‖x‖‖y‖, for all x, y ∈ H. (v) ‖x‖ = ⟨x, x⟩1/2 is a norm on H. (vi) ‖x‖ = sup‖y‖=1 |⟨x, y⟩|. Corollary A.1.7. Let H be a Hilbert space. Then we have (i) (Continuity of the inner product) If xn → x ∈ H and yn → y ∈ H, then ⟨xn , yn ⟩ → ⟨x, y⟩. (ii) If the series ∑n xn converges to x ∈ H, then for any y ∈ H we have ⟨x, y⟩ = ⟨∑ xn , y⟩ = ∑⟨xn , y⟩. n

n

414 | A Appendix Definition A.1.8. Let A be a subset (not necessarily closed) of a Hilbert space H. The orthogonal complement of A is A⊥ = {x ∈ H : x⊥A} = {x ∈ H : ⟨x, y⟩ = 0 ∀y ∈ A}. Remark A.1.9. Recall that A⊥ is always a closed subset of H, even if A is not. The following theorem summarizes some basic results related to convergence of series of orthonormal vectors (for the proof we refer, for instance, to [183, Theorem 1.49]). Theorem A.1.10. If {xn } is an o. n. sequence in a Hilbert space H, then we have (i) (Bessel inequality) ∑n |⟨x, xn ⟩|2 ≤ ‖x‖2 for every x ∈ H. (ii) If H ∋ x = ∑n cn xn , then cn = ⟨x, xn ⟩, for every n. (iii) ∑n cn xn converges ⇐⇒ ∑n |cn |2 < ∞. (iv) x ∈ span{xn } ⇐⇒ x = ∑n ⟨x, xn ⟩xn . (v) If x ∈ H then p = ∑n ⟨x, xn ⟩xn is the orthogonal projection of x onto span{xn }. Hence, if an o. n. sequence {xn } in H is complete, that is, span{xn } = H, then every x ∈ H can be uniquely written as x = ∑n ⟨x, xn ⟩xn , by Theorem A.1.10(ii)–(iv), that is, {xn } is a basis. In what follows we present several equivalent conditions for a sequence {xn } to be an o. n. basis (see [183, Theorem 1.50]).

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Theorem A.1.11. If {xn } is an o. n. sequence in a Hilbert space H, then the following conditions are equivalent: (i) {xn } is complete in H. (ii) {xn } is a basis for H, that is, for every x ∈ H there exists a unique sequence of scalars (cn ) such that x = ∑n cn xn . (iii) x = ∑n ⟨x, xn ⟩xn , for every x ∈ H. (iv) (Plancherel’s equality) ‖x‖2 = ∑n |⟨x, xn ⟩|2 . (v) (Parseval’s equality) ⟨x, y⟩ = ∑n ⟨x, xn ⟩⟨xn , y⟩. A.1.2 Convergence of series We recall the definition of convergent series in normed vector spaces. In what follows we shall use the notation {xn } for a sequence in a normed vector space, whereas (cn ) for a sequence of scalars. Definition A.1.12. Let {xn } be a sequence in a normed linear space X. N (i) The series ∑∞ n=1 xn is convergent to x ∈ X if the partial sums XN = ∑n=1 xn converge to x in the X-norm: ∀ϵ > 0, ∃N0 > 0, ∀N ≥ N0 ,

󵄩󵄩 N 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ xn − x 󵄩󵄩󵄩 < ϵ. 󵄩󵄩 󵄩󵄩 󵄩󵄩n=1 󵄩󵄩

A.1 Basis theory and series expansions | 415

∞ (ii) The series ∑∞ n=1 xn is unconditionally convergent if ∑n=1 xσ(n) is convergent for every permutation σ of ℕ. ∞ (iii) The series ∑∞ n=1 xn is absolutely convergent if ∑n=1 ‖xn ‖ < ∞. ∞ (iv) If a series ∑∞ n=1 xn converges but not unconditionally, we say that ∑n=1 xn is conditionally convergent.

The following result is a simplified version of the celebrated Dirichlet and Riemann–Dini theorems. Lemma A.1.13. If (cn ) is a sequence of real or complex scalars, then ∞

∞

n=1

n=1

∑ cn is absolutely convergent ⇐⇒ ∑ cn is unconditionally convergent.

Proof. “�⇒” Suppose ∑∞ n=1 |cn | < ∞. Choose any ϵ > 0, then there exists N0 > 0 such that for every N > M ≥ N0 , ∑Nn=M+1 |cn | < ϵ (the sequence of partial sums is a Cauchy sequence). Note that, since | ∑Nn=M+1 cn | ≤ ∑Nn=M+1 |cn |, this implies that ∑∞ n=1 cn is convergent. For any permutation σ of ℕ, set N1 = max{σ −1 (1), . . . , σ −1 (N0 )}. Then for every N > M ≥ N1 , take n = M + 1, . . . , N and observe that n > N1 implies n ≠ σ −1 (1), . . . , σ −1 (N0 ) so σ(n) > N0 . In particular, K = min{σ(M + 1), . . . , σ(N)} > N0 ,

L = max{σ(M + 1), . . . , σ(N)} ≥ K > N0 .

Hence

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󵄨󵄨 N 󵄨󵄨 N L 󵄨󵄨 󵄨 󵄨󵄨 ∑ cσ(n) 󵄨󵄨󵄨 ≤ ∑ |cσ(n) | ≤ ∑ |cn | < ϵ, 󵄨󵄨 󵄨󵄨 󵄨󵄨n=M+1 󵄨󵄨 n=M+1 n=K that is, ∑Nn=1 cσ(n) is a Cauchy sequence and, by the completeness of ℂ (or ℝ), ∑ cσ(n) converges. “⇐�” Assume first that ∑∞ n=1 cn is a sequence of real scalars that does not converge absolutely. Define by (pn ) and (qn ) the sequences of nonnegative and negative terms in the given order, respectively (note that (pn ) or (qn ) may also be a finite sequence). If both ∑n pn and ∑n qn converge, then ∑n |cn | = ∑n pn − ∑n qn < ∞, yielding a contradiction. So either ∑n pn or ∑n qn must diverge. Suppose that ∑n pn diverges. Since pn ≥ 0 for every n, there exists m1 ∈ ℕ such that p1 + ⋅ ⋅ ⋅ + pm1 > 1.

416 | A Appendix There exists m2 > m1 such that p1 + ⋅ ⋅ ⋅ + pm1 + q1 + pm1 +1 + ⋅ ⋅ ⋅ + pm2 > 2; there exists m3 > m2 such that p1 + ⋅ ⋅ ⋅ + pm1 + q1 + pm1 +1 + ⋅ ⋅ ⋅ + pm2 + q2 + pm2 +1 + ⋅ ⋅ ⋅ + pm3 > 3, and so on. We have constructed a rearrangement of ∑n cn that does not converge. Since ∑n cn converges unconditionally, this is a contradiction. A similar proof applies if ∑n qn is divergent. Thus we have shown that if ∑n cn is a series of real scalars that converges unconditionally, then it must converge absolutely. Finally, let ∑n cn be a series of complex scalars cn = an + ibn that converges unconditionally. Since ∑ cσ(n) = ∑ aσ(n) + i ∑ bσ(n) n

n

n

is convergent, both ∑n aσ(n) and ∑n bσ(n) converge. So ∑n an and ∑n bn converge unconditionally and, by the previous step, absolutely. Finally, ∑ |cn | = ∑ |an + ibn | ≤ ∑ |an | + ∑ |bn | < ∞. n

n

n

n

So ∑n cn is absolutely convergent, and we are done. n Example A.1.14. The alternating harmonic series ∑∞ n=1 (−1) /n converges to log(1/2). However, it does not converge absolutely, so it cannot converge unconditionally.

Lemma A.1.15. Let {xn } be a sequence in a Banach space X. If ∑ xn converges absolutely �⇒ ∑ xn converges unconditionally.

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n

n

Proof. Assume ∑n ‖xn ‖ < ∞. Then for every ϵ > 0 there exists an N0 such that for every N > M ≥ N0 , 󵄩󵄩 N 󵄩󵄩 N 󵄩󵄩 󵄩 󵄩󵄩 ∑ xn 󵄩󵄩󵄩 ≤ ∑ ‖xn ‖ < ϵ, 󵄩󵄩 󵄩󵄩 󵄩󵄩n=M+1 󵄩󵄩 n=M+1 hence the sequence of partial sums {SN }, with SN = ∑Nn=1 xn , is a Cauchy sequence and converges to x ∈ X by the completeness of X. Take now any permutation σ of ℕ. We can repeat the previous argument for ∑n ‖xσ(n) ‖ < ∞, by Lemma A.1.13. Therefore ∑n xn is unconditionally convergent. However, unconditionally convergence does not imply absolutely convergence, as shown in the following example.

A.1 Basis theory and series expansions | 417

Example A.1.16. Consider the Hilbert space of 1-periodic, square integrable functions L2 ([0, 1]). An orthonormal basis is the Fourier sequence {e2πint }, n ∈ ℤ. We consider 2πint the series ∑∞ , with cn = 1/n. Observe that n=1 cn e 󵄩󵄩2 ∞ 󵄩󵄩 ∞ 󵄩󵄩 󵄩 󵄩󵄩 ∑ cn e2πint 󵄩󵄩󵄩 = ∑ 1 < ∞. 󵄩󵄩 󵄩󵄩 󵄩󵄩n=1 󵄩󵄩2 n=1 n2 1 n2

Since the series of scalars ∑∞ n=1

is absolutely convergent, it is also unconditionally

2πint convergent by Lemma A.1.13. This implies that ∑∞ is unconditionally convern=1 cn e gent. However, the previous series is not absolutely convergent; indeed, ∞ ∞ ∞ 1 󵄩 󵄩 ∑ 󵄩󵄩󵄩cn e2πint 󵄩󵄩󵄩2 = ∑ |cn | = ∑ = +∞, n n=1 n=1 n=1

since ‖e2πint ‖2 = 1. Definition A.1.17. A directed set is a nonempty set I with a relation ≤ on I such that (i) ≤ is reflexive: i ≤ i, for i ∈ I; (ii) ≤ is transitive: i ≤ j and j ≤ k ⇒ i ≤ k; and (iii) for any i, j ∈ I there exists some k ∈ I such that i ≤ k and j ≤ k (every pair of elements has an upper bound). Definition A.1.18. A net in X is a sequence {xi }i∈I of elements xi in X, indexed by a directed set I. The set of natural numbers ℕ is a directed set and hence every sequence {xn }n∈ℕ is an example of net. A more interesting example is as follows. Consider the set of all finite subset of ℕ: I = {F ⊂ ℕ; F is finite},

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ordered by inclusion. Then I is a directed set. Definition A.1.19 (Convergence of a net). Let X be a topological space and {xi }i∈I a net in X. We say that the sequence {xi }i∈I converges to x ∈ X if for every open set U containing x there exists i0 ∈ I such that for every i ∈ I, i0 ≤ i, we have xi ∈ U. In this case we write x = limi xi . Observe that we shall work with normed vector spaces X, so that it is enough to check the previous convergence on the open balls Bϵ (x) = {y ∈ X : ‖y − x‖ < ϵ}. Given a formal series ∑n xn (there is no requirement that the series converges in any sense), the associated net of all possible partial sums is given by { ∑ xn } n∈F

F∈I

= { ∑ xn , F ⊂ ℕ, F finite}, n∈F

(A.6)

with I being finite subsets of a index set. Note that the index set is not always ℕ but could be a lattice (for example, in Gabor analysis). Definition A.1.19 in this case can be rephrased as follows.

418 | A Appendix Definition A.1.20. Let X be a Banach space and xn , x ∈ X. We say that the series ∑n xn converges to x with respect to the directed set of finite subsets of ℕ if ∀ϵ > 0 ∃ finite F0 ⊂ ℕ, ∀ finite F ⊇ F0 ,

󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩x − ∑ xn 󵄩󵄩󵄩 < ϵ. 󵄩󵄩 󵄩󵄩 n∈F

In this case we write x = limF ∑n∈F xn . We now present several equivalent formulations for unconditional convergence in Banach spaces. Theorem A.1.21. Given a sequence {xn } in a Banach space X, the following are equivalent: (i) ∑n xn converges unconditionally. (ii) limF ∑n∈F xn exists. (iii) For every ϵ > 0, there exists N ∈ ℕ+ such that for every finite F ⊂ ℕ with min(F) > N, ‖ ∑n∈F xn ‖ < ϵ. Proof. (i) �⇒ (ii) Suppose x = ∑n xn and let us prove that x = limF ∑n∈F xn . We argue by contradiction. Assume that there exists ϵ > 0 such that ∀ finite F0 ⊂ ℕ, ∃F1 ⊇ F0 :

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩x − ∑ xn 󵄩󵄩󵄩 ≥ ϵ. 󵄩󵄩 󵄩󵄩 n∈F1

(A.7)

Since ∑n xn converges, there exists M1 ∈ ℕ such that for every N ≥ M1 ,

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󵄩󵄩 󵄩󵄩 N 󵄩󵄩 󵄩 󵄩󵄩x − ∑ xn 󵄩󵄩󵄩 < ϵ/2. 󵄩󵄩 󵄩󵄩 󵄩󵄩 n=1 󵄩 󵄩 Define F1 = {1, . . . , M1 }, then by (A.7) there exists G1 ⊇ F1 such that ‖x − ∑n∈G1 xn ‖ ≥ ϵ. Consider M2 = max(G1 ) and define F2 = {1, . . . , M2 }, then by (A.7) there exists G2 ⊇ F2 such that ‖x − ∑n∈G2 xn ‖ ≥ ϵ. Proceeding this way, we construct a sequence of finite sets such that F1 ⊆ G1 ⊆ F2 ⊆ G2 ⊆ ⋅ ⋅ ⋅ ⊆ FN ⊆ GN ⊂ ⋅ ⋅ ⋅ and 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩x − ∑ xn 󵄩󵄩󵄩 < ϵ/2, 󵄩󵄩 󵄩󵄩 n∈FN

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩x − ∑ xn 󵄩󵄩󵄩 ≥ ϵ. 󵄩󵄩 󵄩󵄩 n∈GN

Hence 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ∑ xn 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ xn − ∑ xn 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 n∈FN n∈GN n∈GN \FN 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≥ 󵄩󵄩󵄩 ∑ xn − x 󵄩󵄩󵄩 − 󵄩󵄩󵄩x − ∑ xn 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 n∈FN n∈GN > ϵ − ϵ/2 = ϵ/2.

A.1 Basis theory and series expansions | 419

In particular, FN must be a proper subset of GN . Let σ be a permutation of ℕ obtained by enumerating in turn the elements of F1 , then G1 \ F1 , then F2 \ G1 , then G2 \ F2 , etc. Then, for each N, we have 󵄩󵄩 󵄩 󵄩󵄩 |GN | 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 ∑ xσ(n) 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ xn 󵄩󵄩󵄩 > ϵ/2. 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩n∈GN \FN 󵄩󵄩 󵄩󵄩n=|FN |+1 Since |FN |, |GN | → +∞ as N increases, we see that ∑Nn=1 xσ(n) is not a Cauchy sequence and cannot converge, and this is a contradiction. (ii) �⇒ (iii) Assume that limF ∑n∈F xn = x ∈ X. Choose any ϵ > 0. Then there exists a finite set F0 such that for every finite F ⊆ F0 , ‖x − ∑n∈F xn ‖ < ϵ/2. Set N = max(F0 ). For any finite set G with min(G) > N, we have F0 ∩ G = 0 and 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ∑ xn 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩x − ∑ xn 󵄩󵄩󵄩 + 󵄩󵄩󵄩 ∑ xn − x 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 n∈F0 n∈G n∈F0 ∪G < ϵ/2 + ϵ/2 = ϵ. (iii) �⇒ (i) Choose any permutation σ of ℕ. We have to prove that ∑n xσ(n) converges in X. So we shall show that the sequence of partial sums {∑Nn=1 xσ (n)} is a Cauchy sequence. Consider an ϵ > 0, and let N be the number in claim (iii). Define N0 = max{σ −1 (1), . . . , σ −1 (N)}. Choose L > K ≥ N0 and define F = {σ(K + 1), . . . , σ(L)}. Then for every k ≥ K + 1, we have k > N0 and hence σ(k) ≠ 1, . . . , N, so σ(k) > N and, finally, min(F) > N. Thus 󵄩󵄩 L 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ∑ xσ(n) 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ xn 󵄩󵄩󵄩 < ϵ. 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩n=K+1 󵄩󵄩 󵄩n∈F 󵄩󵄩

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We can now show that if ∑n xn is unconditionally convergent, then its limit does not depend on the permutation σ. Corollary A.1.22. Let {xn } be a sequence in a Banach space. If ∑n xn is unconditionally convergent, then ∑ xσ(n) = ∑ xn , n

n

for every permutation σ of ℕ. Proof. Suppose ∑n xn unconditionally convergent. Then by Theorem A.1.21 (ii), limF ∑n∈F xn = x ∈ X. Consider any permutation σ and any ϵ > 0. Then, by Definition A.1.20, there exists a finite set F0 ⊂ ℕ such that ∀F finite,

󵄩󵄩 󵄩󵄩 󵄩 󵄩 F ⊇ F0 �⇒ 󵄩󵄩󵄩x − ∑ xn 󵄩󵄩󵄩 < ϵ. 󵄩󵄩 󵄩󵄩 n∈F

(A.8)

420 | A Appendix Choose N0 ∈ ℕ large enough such that F0 ⊂ {σ(1), . . . , σ(N0 )}. For N ≥ N0 , define F = {σ(1), . . . , σ(N)}. Then F ⊇ F0 and, by equation (A.8), 󵄩󵄩 N 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩x − ∑ xσ(n) 󵄩󵄩󵄩 = 󵄩󵄩󵄩x − ∑ xn 󵄩󵄩󵄩 < ϵ, 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩 n=1 n∈F 󵄩 󵄩 as desired. We shall present a necessary condition for unconditional convergence of series in Hilbert spaces. This is expressed by Orlicz’s theorem. For the proof we refer, e. g., to [183, Theorem 3.16]. Theorem A.1.23 (Orlicz’s theorem). If {xn } is a sequence in a Hilbert space, then ∑ xn converges unconditionally �⇒ ∑ ‖xn ‖2 < ∞. n

n

(A.9)

Orlicz’s theorem does not extend to Banach spaces, in general. Further, its converse is not true, in general. An example is provided below. Example A.1.24. Let H be a Hilbert space and fix x ∈ H with ‖x‖ = 1. Then 󵄩󵄩 N 󵄩󵄩 󵄨󵄨 N 󵄨󵄨 󵄨󵄨 N 󵄨󵄨 󵄩󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 󵄩󵄩 ∑ cn x󵄩󵄩󵄩 = 󵄨󵄨󵄨 ∑ cn 󵄨󵄨󵄨 ‖x‖ = 󵄨󵄨󵄨 ∑ cn 󵄨󵄨󵄨. 󵄩󵄩 󵄩󵄩 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄩󵄩n=M+1 󵄩󵄩 󵄨󵄨n=M+1 󵄨󵄨󵄨 󵄨󵄨n=M+1 󵄨󵄨󵄨 So ∑n cn x converges (unconditionally) in H if and only if ∑n cn converges (unconditionally) as series of scalars. Therefore, if (cn ) ∈ ℓ2 is such that ∑n cn converges conditionally, then also ∑n cn x converges conditionally. However, we have 󵄩 󵄩 ∑ ‖cn x‖2 = ∑ |cn |2 = 󵄩󵄩󵄩(cn )󵄩󵄩󵄩ℓ2 < ∞. n

n

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For example, take cn = (−1)n /n. We now come back to the concept of basis in a Banach space. Some types of bases with extra properties are collected in the following definition. Definition A.1.25. Let {xn } be a basis for a Banach space X. (i) {xn } is an unconditional basis if the series in (A.5) is unconditionally convergent, for each x ∈ X. If this is not the case, we call {xn } a conditional basis. (ii) {xn } is an absolutely convergent basis if the series in (A.5) converges absolutely, for each x ∈ X. (iii) {xn } is a bounded basis if {xn } is norm-bounded, both above and below, that is, 0 < inf ‖xn ‖ ≤ sup ‖xn ‖ < ∞. (iv) {xn } is a normalized basis if ‖xn ‖ = 1, for every n.

A.2 Some results concerning real interpolation theory | 421

An example of unconditional basis is as follows. Consider L2 ([0, 1]), with the o. n. basis {e2πint }n∈ℤ . Then, for every f ∈ L2 ([0, 1]), 󵄩󵄩2 󵄩󵄩 󵄩 󵄩 = ∑ |c |2 , ‖f ‖2L2 ([0,1]) = 󵄩󵄩󵄩 ∑ cn e2πint 󵄩󵄩󵄩 󵄩󵄩L2 ([0,1]) n∈ℤ n 󵄩󵄩n∈ℤ where cn = ⟨f , e2πint ⟩ is the nth Fourier coefficient. Since ∑n∈ℤ |cn |2 is absolutely convergent, and hence unconditionally convergent, it follows that {e2πint }n∈ℤ is an unconditional basis. More generally, every o. n. basis {xn }n ⊂ H, with H a Hilbert space, is an unconditional basis. We now provide an example of absolutely convergent basis. Consider the Banach space ℓ1 (ℤ) with the basis {δn }n∈ℤ , where we recall that δn is the sequence having elements all equal zero but the nth element that is one. Take any sequence c = (cn )n ∈ ℓ1 , hence c = ∑n cn δn and ‖c‖ℓ1 = ∑n |cn |. Now, ∑ ‖cn δn ‖ℓ1 = ∑ |cn |‖δn ‖ℓ1 = ∑ |cn |. n

n

n

That is, {δn }n∈ℤ is an absolutely convergent basis. An example of normalized basis (hence, in particular, a bounded basis) is provided by every o. n. basis in a Hilbert space. In particular, if we restrict our attention to Hilbert spaces, than we recall the following definition. Definition A.1.26 (Riesz basis). A sequence {xn } for a Hilbert space H is a Riesz basis if it is equivalent to some (and hence every) orthonormal basis for H. That is, there exists a topological isomorphism T : H → H and there exists an o. n. basis {en } for H such that

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Ten = xn

∀n.

A.2 Some results concerning real interpolation theory Here we collected some results of real interpolation theory which are used in the proof of the Strichartz estimates (see Chapter 5). We use the notation and terminology of [296]. Let (X, ℬ, μ) be a measure space, where X is a set, ℬ a σ-algebra, and μ a positive σ-finite measure. If A is a Banach space, 1 ≤ p ≤ ∞, then we shall write Lp (A) for the usual vector-valued Lp spaces in the sense of the Bochner integral. Proposition A.2.1. Let {A0 , A1 } be an interpolation couple. For every 1 ≤ p0 , p1 < ∞, 0 < θ < 1, 1/p = (1 − θ)/p0 + θ/p1 . and p ≤ q, we have ℓp ((A0 , A1 )θ,q ) �→ (ℓp0 (A0 ), ℓp1 (A1 ))θ,q .

(A.10)

422 | A Appendix Proof. Set η = pθ/p1 and q = pr, r ≥ 1. It follows from [296, Theorem 1.4.2, page 29], that, given c = {cj } ∈ ℓp (A0 ) + ℓp (A1 ), we have ‖c‖p(ℓp0 (A

p1 0 ),ℓ (A1 ))θ,q

󵄩󵄩 ≍ 󵄩󵄩󵄩t −η 󵄩

inf

a+b=c a∈ℓp0 (A0 ), b∈ℓp1 (A1 )

󵄩󵄩 p p ‖a‖ℓp00 (A ) + t‖b‖ℓp11 (A ) 󵄩󵄩󵄩 r . dt 0 1 󵄩L (ℝ+ , ) t

Hence, ‖c‖p(ℓp0 (A

0

),ℓp1 (A

1 ))θ,q

󵄩󵄩 󵄩 ≍ 󵄩󵄩󵄩t −η 󵄩󵄩

inf

a+b=c a∈ℓp0 (A0 ), b∈ℓp1 (A1 )

󵄩󵄩 󵄩 = 󵄩󵄩󵄩t −η ∑ 󵄩󵄩 j≥0

inf

aj +bj =cj aj ∈A0 , bj ∈A1

󵄩 p p 󵄩 󵄩 ∑ (‖aj ‖A0 + t‖bj ‖A1 )󵄩󵄩󵄩 0 1 󵄩 r 󵄩L (ℝ+ , dtt ) j≥0

󵄩 p p 󵄩 󵄩 . (‖aj ‖A0 + t‖bj ‖A1 )󵄩󵄩󵄩 0 1 󵄩 r 󵄩L (ℝ+ , dtt )

(A.11)

By Minkowski’s inequality, we deduce ‖c‖p(ℓp0 (A

p1 (A

0 ),ℓ

1 ))θ,q

󵄩󵄩 ≲ ∑ 󵄩󵄩󵄩t −η 󵄩 j≥0

inf

aj +bj =cj aj ∈A0 , bj ∈A1

≍ ‖c‖pℓp ((A

0 ,A1 )θ,q )

p p 󵄩 󵄩 (‖aj ‖A0 + t‖bj ‖A1 )󵄩󵄩󵄩 r 0 1 󵄩L (ℝ+ , dt ) t

(A.12)

.

Consider now a partition of unity1 given by functions ϕα ∈ 𝒟(ℝd ), α ∈ ℤd , with ϕα = Tα ϕ, supp ϕ ⊂ [−2, 2]d . Thus supp ϕα ⊂ α + [−2, 2]d . Let then ψ ∈ 𝒟(ℝd ), ψ = 1 on [−2, 2]d and ψ = 0 away from [−4, 4]d , and set ψα = Tα ψ. Observe that there is a constant Cd such that ∀α ∈ ℤd , #{β ∈ ℤd : supp ψβ ∩ supp ϕα ≠ 0} ≤ Cd

(A.13)

∀α ∈ ℤd , #{β ∈ ℤd : supp ψα ∩ supp ϕβ ≠ 0} ≤ Cd .

(A.14)

and

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The linear operators ℤd

S : L1loc → (L1loc ) ,

R : (L1loc )

ℤd

→ L1loc ,

defined by Sf = {fϕα }α ,

R({uα }α ) = ∑ uα ψα , α

(A.15)

enjoy the following properties (see [122, Remark 2.2]). 1 Such a partition of unity can be constructed as follows. Take χ ∈ 𝒟(ℝd ), 0 ≤ χ ≤ 1, χ = 1 on [−1, 1]d , χ = 0 away from [−2, 2]d . Set Φ(x) = ∑α∈ℤd χ(x − α). Since the sum is locally finite, Φ is well defined

and smooth. Moreover, Φ(x + β) = Φ(x) for every β ∈ ℤd , and also Φ ≥ 1. Hence it suffices to take ϕα = Tα χ/Φ = Tα (χ/Φ).

A.2 Some results concerning real interpolation theory | 423

Proposition A.2.2. We have RS = Id on L1loc and, for every local component B, as at the beginning of this section, and every p ≥ 1, we have S : W(B, Lp ) → ℓp (B)

(A.16)

R : ℓp (B) → W(B, Lp )

(A.17)

and

continuously. Proof. The equality RS = Id on L1loc is clear, whereas (A.16) follows at once from Remark 4 of [122]. To prove (A.17), we observe that 󵄩󵄩 󵄩󵄩p 󵄩󵄩 󵄩p 󵄩 󵄩 󵄩󵄩R({uα }α )󵄩󵄩󵄩W(B,Lp ) ≍ ∑󵄩󵄩󵄩∑ uα ψα ϕβ 󵄩󵄩󵄩 󵄩󵄩 α 󵄩󵄩B β

≲ ∑ ∑ ‖uα ψα ϕβ ‖pB β

α

α

β

(by Remark 4 of [122])

(A.18)

(by (A.13))

(A.19)

≲ ∑ ∑ ‖ψα ϕβ ‖pℱ L1 ‖uα ‖pB

(A.20)

󵄩 󵄩p ≲ 󵄩󵄩󵄩{uα }α 󵄩󵄩󵄩ℓp (B) .

(A.21)

In the last inequality we used the fact that, by (A.14), ∑ ‖ψα ϕβ ‖pℱ L1 ≲ β

∑

β:supp ψα ∩supp ϕβ =0̸

‖ψα ‖pℱ L1 ‖ϕβ ‖pℱ L1 ≤ Cd ‖ψ0 ‖pℱ L1 ‖ϕ0 ‖pℱ L1 .

Proposition A.2.3. Given two local components B0 , B1 as at the beginning of this section, for every 1 ≤ p0 , p1 < ∞, 0 < θ < 1, 1/p = (1 − θ)/p0 + θ/p1 , and p ≤ q, we have W((B0 , B1 )θ,q , Lp ) �→ (W(B0 , Lp0 ), W(B1 , Lp1 ))θ,q . Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Proof. Let R and S be the operators defined above. Then, by Proposition A.2.2, ‖f ‖(W(B0 ,Lp0 ),W(B1 ,Lp1 ))θ,q = ‖RSf ‖(W(B0 ,Lp0 ),W(B1 ,Lp1 ))θ,q ≲ ‖Sf ‖(ℓp0 (B0 ),ℓp1 (B1 ))θ,q ≲ ‖Sf ‖ℓp ((B0 ,B1 )θ,q ) ,

(A.22)

≲ ‖f ‖W((B0 ,B1 )θ,q ,Lp ) , where for (A.22) we used (A.10). This concludes the proof. Proposition A.2.4. Let {A0 , A1 } be an interpolation couple. For every 1 ≤ p0 , p1 < ∞, 0 < θ < 1, 1/p = (1 − θ)/p0 + θ/p1 , and p ≤ q, we have Lp ((A0 , A1 )θ,q ) �→ (Lp0 (A0 ), Lp1 (A1 ))θ,q .

(A.23)

424 | A Appendix Proof. We use the fact that the bounded functions with values in A0 ∩ A1 , vanishing outside a set with finite measure, are dense in Lp0 (A0 ) ∩ Lp1 (A1 ) and may be approximated in Lp0 (A0 ) ∩ Lp1 (A1 ) by simple functions: N

c(x) = ∑ cj χFj (x),

cj ∈ A0 ∩ A1 ,

μ(Fj ) < ∞,

j=1

Fj ∩ Fk = 0 if j ≠ k.

Set η = pθ/p1 and q = pr, r ≥ 1. If c(x) is such a function, it follows from [296, Theorem 1.4.2, page 29] that ‖c‖p(Lp0 (A

0 ),L

󵄩󵄩 ≍ 󵄩󵄩󵄩t −η 󵄩

p1 (A

1 ))θ,q

󵄩󵄩 󵄩 = 󵄩󵄩󵄩t −η 󵄩󵄩

inf

󵄩󵄩 p p ‖a‖Lp00 (A ) + t‖b‖Lp1 1 (A ) 󵄩󵄩󵄩 r dt 0 1 󵄩L (ℝ+ , ) t

inf

󵄩 󵄩 󵄩p 󵄩 󵄩p 󵄩󵄩 ∫(󵄩󵄩󵄩a(x)󵄩󵄩󵄩A0 + t 󵄩󵄩󵄩b(x)󵄩󵄩󵄩A1 )󵄩󵄩󵄩 0 1 󵄩 r 󵄩L (ℝ+ , dtt )

a+b=c a∈Lp0 (A0 ), b∈Lp1 (A1 )

a+b=c a∈Lp0 (A0 ), b∈Lp1 (A1 ) X

󵄩󵄩 󵄩 = 󵄩󵄩󵄩t −η ∫ 󵄩󵄩

inf

a(x)+b(x)=c(x) X a(x)∈A0 , b(x)∈A1

󵄩 󵄩 󵄩p 󵄩 󵄩p 󵄩󵄩 . (󵄩󵄩󵄩a(x)󵄩󵄩󵄩A0 + t 󵄩󵄩󵄩b(x)󵄩󵄩󵄩A1 )󵄩󵄩󵄩 0 1 󵄩 r 󵄩L (ℝ+ , dtt )

(A.24)

In order to justify the last equality, we observe that the inequality “≥” is obvious. As far as the converse inequality “≤” is concerned, we take advantage of the fact that c(x) is a simple function, so that one can find minimizing sequences an (x), bn (x) given by simple functions which are constants where c(x) is, and zero where c(x) = 0. Finally, by Minkowski’s integral inequality, ‖c‖p(Lp0 (A

0 ),L

p1 (A

1 ))θ,q

󵄩󵄩 ≲ ∫󵄩󵄩󵄩t −η 󵄩 X

inf

a(x)+b(x)=c(x) a(x)∈A0 , b(x)∈A1

≍ ‖c‖pLp ((A

0 ,A1 )θ,q )

󵄩 󵄩p 󵄩 󵄩p 󵄩󵄩 (󵄩󵄩󵄩a(x)󵄩󵄩󵄩A0 + t 󵄩󵄩󵄩b(x)󵄩󵄩󵄩A1 )󵄩󵄩󵄩 r 0 1 󵄩L (ℝ , dt ) + t

.

(A.25)

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We finally recall [17, Section 3.13.5(b)]. Lemma A.2.5. If Ai , Bi , Ci , i = 0, 1, are Banach spaces and T is a bilinear operator bounded from T : A0 × B0 → C0 , T : A0 × B1 → C1 ,

T : A1 × B0 → C1 , then, if 0 < θi < θ < 1, i = 0, 1, θ = θ0 + θ1 , one has T : (A0 , A1 )θ0 ,2 × (B0 , B1 )θ1 ,2 → (C0 , C1 )θ,1 .

Bibliography [1]

[2]

[3] [4] [5] [6] [7] [8]

[9] [10] [11] [12] [13] [14]

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

[15]

[16] [17] [18] [19] [20] [21]

G. S. Alberti, S. Dahlke, F. De Mari, E. De Vito, and H. Führ. Recent progress in shearlet theory: systematic construction of shearlet dilation groups, characterization of wavefront sets, and new embeddings. In Frames and Other Bases in Abstract and Function Spaces, Appl. Numer. Harmon. Anal., pages 127–160. Birkhäuser/Springer, Cham, 2017. G. S. Alberti, S. Dahlke, F. De Mari, E. De Vito, and S. Vigogna. Continuous and discrete frames generated by the evolution flow of the Schrödinger equation. Anal. Appl. (Singap.), 15(6):915–937, 2017. K. Asada and D. Fujiwara. On some oscillatory integral transformations in L2 (Rn ). Jpn. J. Math. New Ser., 4(2):299–361, 1978. R. Balian. Un principe d’incertitude fort en théorie du signal ou en mécanique quantique. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 292(20):1357–1362, 1981. F. Bartolucci, F. De Mari, E. De Vito, and F. Odone. Shearlets as multi-scale Radon transforms. Sampl. Theory Signal Image Process., 17(1):1–15, 2018. F. Bastianoni, E. Cordero, and F. Nicola. Decay and smoothness for eigenfunctions of localization operators. Preprint, arXiv:1902.03413, 2019. G. Battle. Heisenberg proof of the Balian-Low theorem. Lett. Math. Phys., 15(2):175–177, 1988. D. Bayer, E. Cordero, K. Gröchenig, and S. I. Trapasso. Linear perturbations of the Wigner transform and the Weyl quantization. In Advances in Microlocal and Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, pages 79–120. Birkhäuser/Springer, Cham, 2020. R. Beals. Characterization of pseudodifferential operators and applications. Duke Math. J., 44(1):45–57, 1977. A. Benedek and R. Panzone. The space Lp , with mixed norm. Duke Math. J., 28:301–324, 1961. J. J. Benedetto, C. Heil, and D. F. Walnut. Differentiation and the Balian-Low theorem. J. Fourier Anal. Appl., 1(4):355–402, 1995. A. Bényi, L. Grafakos, K. Gröchenig, and K. Okoudjou. A class of Fourier multipliers for modulation spaces. Appl. Comput. Harmon. Anal., 19(1):131–139, 2005. A. Bényi, K. Gröchenig, C. Heil, and K. Okoudjou. Modulation spaces and a class of bounded multilinear pseudodifferential operators. J. Oper. Theory, 54(2):387–399, 2005. A. Bényi, K. Gröchenig, K. A. Okoudjou, and L. G. Rogers. Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal., 246(2):366–384, 2007. A. Bényi and K. A. Okoudjou. Time-frequency estimates for pseudodifferential operators. In Harmonic Analysis, Partial Differential Equations, and Related Topics, volume 428 of Contemp. Math., pages 13–22. Amer. Math. Soc., Providence, RI, 2007. F. A. Berezin. Wick and anti-Wick symbols of operators. Mat. Sb. (N.S.), 86(128):578–610, 1971. J. Bergh and J. Löfström. Interpolation Spaces. An Introduction, volume 223 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin-New York, 1976. M. Berra, I. M. Bulai, E. Cordero, and F. Nicola. Gabor frames of Gaussian beams for the Schrödinger equation. Appl. Comput. Harmon. Anal., 43(1):94–121, 2017. S. Bishop. Mixed modulation spaces and their application to pseudodifferential operators. J. Math. Anal. Appl., 363(1):255–264, 2010. W. R. Bloom. Strict local inclusion results between spaces of Fourier transforms. Pac. J. Math., 99(2):265–270, 1982. P. Boggiatto, E. Carypis, and A. Oliaro. Two aspects of the Donoho-Stark uncertainty principle. J. Math. Anal. Appl., 434(2):1489–1503, 2016.

https://doi.org/10.1515/9783110532456-009

426 | Bibliography

[22]

[23] [24] [25] [26] [27]

[28] [29] [30]

[31]

[32] [33] [34] [35] [36]

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

[37] [38] [39] [40] [41] [42] [43] [44]

P. Boggiatto, E. Carypis, and A. Oliaro. Cohen class of time-frequency representations and operators: boundedness and uncertainty principles. J. Math. Anal. Appl., 461(1):304–318, 2018. P. Boggiatto, E. Cordero, and K. Gröchenig. Generalized anti-Wick operators with symbols in distributional Sobolev spaces. Integral Equ. Oper. Theory, 48(4):427–442, 2004. P. Boggiatto, G. De Donno, and A. Oliaro. Time-frequency representations of Wigner type and pseudo-differential operators. Trans. Am. Math. Soc., 362(9):4955–4981, 2010. P. Boggiatto and G. Garello. Pseudo-differential operators and existence of Gabor frames. J. Pseudo-Differ. Oper. Appl., 11(1):93–117, 2020. P. Boggiatto and J. Toft. Embeddings and compactness for generalized Sobolev-Shubin spaces and modulation spaces. Appl. Anal., 84(3):269–282, 2005. C. Boiti, D. Jornet, and A. Oliaro. Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms. J. Math. Anal. Appl., 446(1):920–944, 2017. C. Boiti, D. Jornet, and A. Oliaro. The Gabor wave front set in spaces of ultradifferentiable functions. Monatshefte Math., 188(2):199–246, 2019. C. Boiti, D. Jornet, and A. Oliaro. Real Paley-Wiener theorems in spaces of ultradifferentiable functions. J. Funct. Anal., 278(4):108348, 45, 2020. J.-M. Bony. Opérateurs intégraux de Fourier et calcul de Weyl-Hörmander (cas d’une métrique symplectique). In Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1994). Exp. No. IX, 14. École Polytech., Palaiseau, 1994. J.-M. Bony. Evolution equations and generalized Fourier integral operators. In Advances in Phase Space Analysis of Partial Differential Equations, volume 78 of Progr. Nonlinear Differential Equations Appl., pages 59–72. Birkhäuser Boston, Inc., Boston, MA, 2009. J.-M. Bony and J.-Y. Chemin. Espaces fonctionnels associés au calcul de Weyl-Hörmander. Bull. Soc. Math. Fr., 122(1):77–118, 1994. M. Born and P. Jordan. Zur quantenmechanik. Z. Phys., 34:858–888, 1925. A. Boulkhemair. L2 estimates for pseudodifferential operators. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 22(1):155–183, 1995. A. Boulkhemair. Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators. Math. Res. Lett., 4(1):53–67, 1997. M. Bownik, M. S. Jakobsen, J. Lemvig, and K. A. Okoudjou. On Wilson bases in L2 (ℝd ). SIAM J. Math. Anal., 49(5):3999–4023, 2017. H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. R. C. Busby and H. A. Smith. Product-convolution operators and mixed-norm spaces. Trans. Am. Math. Soc., 263(2):309–341, 1981. E. Buzano and A. Oliaro. Global regularity of second order twisted differential operators. J. Differ. Equ., 268(12):7364–7416, 2020. A.-P. Calderón and R. Vaillancourt. On the boundedness of pseudo-differential operators. J. Math. Soc. Jpn., 23:374–378, 1971. E. J. Candès and L. Demanet. The curvelet representation of wave propagators is optimally sparse. Commun. Pure Appl. Math., 58(11):1472–1528, 2005. E. J. Candès, L. Demanet, and L. Ying. Fast computation of Fourier integral operators. SIAM J. Sci. Comput., 29(6):2464–2493, 2007. E. J. Candès and D. L. Donoho. Continuous curvelet transform. I. Resolution of the wavefront set. Appl. Comput. Harmon. Anal., 19(2):162–197, 2005. M. Cappiello, R. Schulz, and P. Wahlberg. Conormal distributions in the Shubin calculus of pseudodifferential operators. J. Math. Phys., 59(2):021502, 18, 2018.

Bibliography | 427

[45] [46] [47]

[48]

[49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60]

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

[61] [62] [63] [64] [65] [66] [67] [68]

M. Cappiello and J. Toft. Pseudo-differential operators in a Gelfand-Shilov setting. Math. Nachr., 290(5-6):738–755, 2017. D. Cardona and M. Ruzhansky. Multipliers for Besov spaces on graded Lie groups. C. R. Math. Acad. Sci. Paris, 355(4):400–405, 2017. E. Carypis and P. Wahlberg. Propagation of exponential phase space singularities for Schrödinger equations with quadratic Hamiltonians. J. Fourier Anal. Appl., 23(3):530–571, 2017. L. Cattabriga. Alcuni problemi per equazioni differenziali lineri con coefficienti costanti. In Quaderni dell’Unione Matematica Italiana, Pitagora, Bologna, volume 24, arXiv:1902.08940, 1983. O. Christensen. Frame Decomposition in Hilbert Spaces. PhD thesis. Arhus Univ. and Univ. of Vienna, DK/Austria, 1993. O. Christensen. An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Cham, second edition, 2016. A. Cohen, W. Dahmen, and R. Devore. Adaptive wavelet schemes for nonlinear variational problems. SIAM J. Numer. Anal., 41(5):1785–1823, 2003. L. Cohen. Generalized phase-space distribution functions. J. Math. Phys., 7:781–786, 1966. L. Cohen. Time-Frequency Analysis, Electrical Engineering Signal Processing. Prentice Hall PTR, 1995. R. R. Coifman, G. Matviyenko, and Y. Meyer. Modulated Malvar-Wilson bases. Appl. Comput. Harmon. Anal., 4(1):58–61, 1997. R. R. Coifman and Y. Meyer. Au delà des opérateurs pseudo-différentiels, volume 57 of Astérisque. Société Mathématique de France, Paris, 1978. With an English summary. F. Concetti and J. Toft. Schatten-von Neumann properties for Fourier integral operators with non-smooth symbols. I. Ark. Mat., 47(2):295–312, 2009. J. B. Conway. Functions of One Complex Variable, volume 11 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, second edition, 1978. E. Cordero. Gelfand-Shilov window classes for weighted modulation spaces. Integral Transforms Spec. Funct., 18(11-12):829–837, 2007. E. Cordero, M. de Gosson, M. Dörfler, and F. Nicola. Generalized Born-Jordan distributions and applications. Adv. Comput. Math., 46:51, 2020. E. Cordero, M. de Gosson, and F. Nicola. On the invertibility of Born-Jordan quantization. J. Math. Pures Appl. (9), 105(4):537–557, 2016. E. Cordero, M. de Gosson, and F. Nicola. Semi-classical time-frequency analysis and applications. Math. Phys. Anal. Geom., 20(4):26, 23, 2017. E. Cordero, M. de Gosson, and F. Nicola. Time-frequency analysis of Born-Jordan pseudodifferential operators. J. Funct. Anal., 272(2):577–598, 2017. E. Cordero, M. de Gosson, and F. Nicola. On the reduction of the interferences in the Born-Jordan distribution. Appl. Comput. Harmon. Anal., 44(2):230–245, 2018. E. Cordero, M. de Gosson, and F. Nicola. A characterization of modulation spaces by symplectic rotations. J. Funct. Anal., 278(11):108474, 2020. E. Cordero, L. D’Elia, and S. I. Trapasso. Norm estimates for τ-pseudodifferential operators in Wiener amalgam and modulation spaces. J. Math. Anal. Appl., 471(1-2):541–563, 2019. E. Cordero and K. Gröchenig. Time-frequency analysis of localization operators. J. Funct. Anal., 205(1):107–131, 2003. E. Cordero, K. Gröchenig, and F. Nicola. Approximation of Fourier integral operators by Gabor multipliers. J. Fourier Anal. Appl., 18(4):661–684, 2012. E. Cordero, K. Gröchenig, F. Nicola, and L. Rodino. Wiener algebras of Fourier integral operators. J. Math. Pures Appl. (9), 99(2):219–233, 2013.

428 | Bibliography

[69]

[70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80]

[81]

[82] [83]

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

[84] [85] [86]

[87] [88] [89] [90]

E. Cordero, K. Gröchenig, F. Nicola, and L. Rodino. Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class. J. Math. Phys., 55(8):081506, 17, 2014. E. Cordero and F. Nicola. Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation. J. Funct. Anal., 254(2):506–534, 2008. E. Cordero and F. Nicola. Some new Strichartz estimates for the Schrödinger equation. J. Differ. Equ., 245(7):1945–1974, 2008. E. Cordero and F. Nicola. Strichartz estimates in Wiener amalgam spaces for the Schrödinger equation. Math. Nachr., 281(1):25–41, 2008. E. Cordero and F. Nicola. Boundedness of Schrödinger type propagators on modulation spaces. J. Fourier Anal. Appl., 16(3):311–339, 2010. E. Cordero and F. Nicola. Pseudodifferential operators on Lp , Wiener amalgam and modulation spaces. Int. Math. Res. Not., (10):1860–1893, 2010. E. Cordero and F. Nicola. Sharp integral bounds for Wigner distributions. Int. Math. Res. Not., (6):1779–1807, 2018. E. Cordero and F. Nicola. Kernel theorems for modulation spaces. J. Fourier Anal. Appl., 25(1):131–144, 2019. E. Cordero, F. Nicola, and L. Rodino. Sparsity of Gabor representation of Schrödinger propagators. Appl. Comput. Harmon. Anal., 26(3):357–370, 2009. E. Cordero, F. Nicola, and L. Rodino. On the global boundedness of Fourier integral operators. Ann. Glob. Anal. Geom., 38(4):373–398, 2010. E. Cordero, F. Nicola, and L. Rodino. Time-frequency analysis of Fourier integral operators. Commun. Pure Appl. Anal., 9(1):1–21, 2010. E. Cordero, F. Nicola, and L. Rodino. Time-frequency analysis of Schrödinger propagators. In Evolution Equations of Hyperbolic and Schrödinger Type, volume 301 of Progr. Math., pages 63–85. Birkhäuser/Springer Basel AG, Basel, 2012. E. Cordero, F. Nicola, and L. Rodino. Schrödinger equations in modulation spaces. In Studies in Phase Space Analysis with Applications to PDEs, volume 84 of Progr. Nonlinear Differential Equations Appl., pages 81–99. Birkhäuser/Springer, New York, 2013. E. Cordero, F. Nicola, and L. Rodino. Exponentially sparse representations of Fourier integral operators. Rev. Mat. Iberoam., 31(2):461–476, 2015. E. Cordero, F. Nicola, and L. Rodino. Gabor representations of evolution operators. Trans. Am. Math. Soc., 367(11):7639–7663, 2015. E. Cordero, F. Nicola, and L. Rodino. Propagation of the Gabor wave front set for Schrödinger equations with non-smooth potentials. Rev. Math. Phys., 27(1):1550001, 33, 2015. E. Cordero, F. Nicola, and L. Rodino. Wave packet analysis of Schrödinger equations in analytic function spaces. Adv. Math., 278:182–209, 2015. E. Cordero, F. Nicola, and S. I. Trapasso. Almost diagonalization of τ-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces. J. Fourier Anal. Appl., 25(4):1927–1957, 2019. E. Cordero, S. Pilipović, L. Rodino, and N. Teofanov. Localization operators and exponential weights for modulation spaces. Mediterr. J. Math., 2(4):381–394, 2005. E. Cordero and L. Rodino. Time-frequency analysis: function spaces and applications. Note Mat., 31(1):173–189, 2011. E. Cordero, A. Tabacco, and P. Wahlberg. Schrödinger-type propagators, pseudodifferential operators and modulation spaces. J. Lond. Math. Soc. (2), 88(2):375–395, 2013. E. Cordero and S. I. Trapasso. Linear perturbations of the Wigner distribution and the Cohen class. Anal. Appl. (Singap.), 18(3):385–422, 2020.

Bibliography | 429

[91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101]

[102] [103] [104] [105]

[106]

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

[107] [108] [109] [110]

[111] [112] [113]

E. Cordero and D. Zucco. Strichartz estimates for the Schrödinger equation. CUBO, 12(3):213–239, 2010. E. Cordero and D. Zucco. Strichartz estimates for the vibrating plate equation. J. Evol. Equ., 11(4):827–845, 2011. H. O. Cordes. On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators. J. Funct. Anal., 18:115–131, 1975. A. Córdoba and C. Fefferman. Wave packets and Fourier integral operators. Commun. Partial Differ. Equ., 3(11):979–1005, 1978. S. Coriasco. Fourier integral operators in SG classes. II. Application to SG hyperbolic Cauchy problems. Ann. Univ. Ferrara, Sez. 7: Sci. Mat., 44:81–122 (1999), 1998. W. Craig, T. Kappeler, and W. Strauss. Microlocal dispersive smoothing for the Schrödinger equation. Commun. Pure Appl. Math., 48(8):769–860, 1995. W. Czaja and Z. Rzeszotnik. Pseudodifferential operators and Gabor frames: spectral asymptotics. Math. Nachr., 233/234:77–88, 2002. A. Dasgupta and M. Ruzhansky. The Gohberg lemma, compactness, and essential spectrum of operators on compact Lie groups. J. Anal. Math., 128:179–190, 2016. I. Daubechies. Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inf. Theory, 34(4):605–612, 1988. I. Daubechies. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory, 36(5):961–1005, 1990. I. Daubechies. Ten Lectures on Wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 27(5):1271–1283, 1986. I. Daubechies, S. Jaffard, and J.-L. Journé. A simple Wilson orthonormal basis with exponential decay. SIAM J. Math. Anal., 22(2):554–573, 1991. I. Daubechies and A. J. E. M. Janssen. Two theorems on lattice expansions. IEEE Trans. Inf. Theory, 39(1):3–6, 1993. M. A. de Gosson. Symplectic Methods in Harmonic Analysis and in Mathematical Physics, volume 7 of Pseudo-Differential Operators. Theory and Applications. Birkhäuser/Springer Basel AG, Basel, 2011. M. A. de Gosson. Born-Jordan quantization and the equivalence of the Schrödinger and Heisenberg pictures. Found. Phys., 44(10):1096–1106, 2014. M. A. de Gosson. The canonical group of transformations of a Weyl-Heisenberg frame; applications to Gaussian and Hermitian frames. J. Geom. Phys., 114:375–383, 2017. M. A. de Gosson and F. Luef. Preferred quantization rules: Born-Jordan versus Weyl. The pseudo-differential point of view. J. Pseudo-Differ. Oper. Appl., 2(1):115–139, 2011. J. Delgado and M. Ruzhansky. Lp -nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups. J. Math. Pures Appl. (9), 102(1):153–172, 2014. L. D’Elia and S. I. Trapasso. Boundedness of pseudodifferential operators with symbols in Wiener amalgam spaces on modulation spaces. J. Pseudo-Differ. Oper. Appl., 9(4):881–890, 2018. R. A. DeVore and V. N. Temlyakov. Some remarks on greedy algorithms. Adv. Comput. Math., 5(2-3):173–187, 1996. T. Dobler. Wiener amalgam spaces over locally compact groups. Master Thesis. University of Vienna, Austria, 1989. R. J. Duffin and A. C. Schaeffer. A class of nonharmonic Fourier series. Trans. Am. Math. Soc., 72:341–366, 1952.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

430 | Bibliography

[114] J. J. Duistermaat and V. W. Guillemin. The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math., 29(1):39–79, 1975. [115] J. J. Duistermaat and L. Hörmander. Fourier integral operators. II. Acta Math., 128(3-4):183–269, 1972. [116] K.-J. Engel and R. Nagel. A Short Course on Operator Semigroups, Universitext. Springer, New York, 2006. [117] H. G. Feichtinger. Modulation spaces on locally compact Abelian groups. Technical report, University of Vienna, 1983. [118] H. G. Feichtinger. Segal algebras that are not character invariant. Chin. J. Math., 7(1):55–59, 1979. [119] H. G. Feichtinger. Un espace de Banach de distributions tempérées sur les groupes localement compacts abéliens. C. R. Acad. Sci. Paris Sér. A-B, 290(17):A791–A794, 1980. [120] H. G. Feichtinger. Banach spaces of distributions of Wiener’s type and interpolation. In Functional Analysis and Approximation (Oberwolfach, 1980), volume 60 of Internat. Ser. Numer. Math., pages 153–165. Birkhäuser, Basel-Boston, Mass., 1981. [121] H. G. Feichtinger. A characterization of minimal homogeneous Banach spaces. Proc. Am. Math. Soc., 81(1):55–61, 1981. [122] H. G. Feichtinger. Banach convolution algebras of Wiener type. In Functions, Series, Operators, Vol. I, II (Budapest, 1980), pages 509–524. North-Holland, Amsterdam, 1983. [123] H. G. Feichtinger. Atomic characterizations of modulation spaces through Gabor-type representations. Rocky Mt. J. Math., 19(1):113–125, 1989. Constructive Function Theory–86 Conference (Edmonton, AB, 1986). [124] H. G. Feichtinger. Generalized amalgams, with applications to Fourier transform. Can. J. Math., 42(3):395–409, 1990. [125] H. G. Feichtinger. Wiener amalgams over Euclidean spaces and some of their applications. In Function Spaces (Edwardsville, IL, 1990), volume 136 of Lecture Notes in Pure and Appl. Math., pages 123–137. Dekker, New York, 1992. [126] H. G. Feichtinger. Modulation spaces on locally compact Abelian groups. In Krishna, M., Radha, R., & Thangavelu, S. (eds.), Wavelets and Their Applications, pages 99–140, Chennai, Jan. 2002, Allied Publishers (New Delhi), 2003. [127] H. G. Feichtinger. Modulation spaces: looking back and ahead. Sampl. Theory Signal Image Process., 5(2):109–140, 2006. [128] H. G. Feichtinger. Elements of postmodern harmonic analysis. In Operator-Related Function Theory and Time-Frequency Analysis, volume 9 of Abel Symp., pages 77–105. Springer, Cham, 2015. [129] H. G. Feichtinger, C. C. Graham, and E. H. Lakien. Nonfactorization in commutative, weakly selfadjoint Banach algebras. Pac. J. Math., 80(1):117–125, 1979. [130] H. G. Feichtinger and P. Gröbner. Banach spaces of distributions defined by decomposition methods. I. Math. Nachr., 123:97–120, 1985. [131] H. G. Feichtinger and K. Gröchenig. Banach spaces related to integrable group representations and their atomic decompositions. I. J. Funct. Anal., 86(2):307–340, 1989. [132] H. G. Feichtinger and K. Gröchenig. Banach spaces related to integrable group representations and their atomic decompositions. II. Monatshefte Math., 108(2-3):129–148, 1989. [133] H. G. Feichtinger and K. Gröchenig. Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view. In Wavelets, volume 2 of Wavelet Anal. Appl., pages 359–397. Academic Press, Boston, MA, 1992. [134] H. G. Feichtinger and K. Gröchenig. Gabor frames and time-frequency analysis of distributions. J. Funct. Anal., 146(2):464–495, 1997.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Bibliography | 431

[135] H. G. Feichtinger, K. Gröchenig, and D. Walnut. Wilson bases and modulation spaces. Math. Nachr., 155:7–17, 1992. [136] H. G. Feichtinger and A. J. E. M. Janssen. Validity of WH-frame bound conditions depends on lattice parameters. Appl. Comput. Harmon. Anal., 8(1):104–112, 2000. [137] H. G. Feichtinger and N. Kaiblinger. Varying the time-frequency lattice of Gabor frames. Trans. Am. Math. Soc., 356(5):2001–2023, 2004. [138] H. G. Feichtinger and W. Kozek. Quantization of TF lattice-invariant operators on elementary LCA groups. In Gabor Analysis and Algorithms, Appl. Numer. Harmon. Anal., pages 233–266. Birkhäuser Boston, Boston, MA, 1998. [139] H. G. Feichtinger and K. Nowak. A first survey of Gabor multipliers. In Advances in Gabor Analysis, Appl. Numer. Harmon. Anal., pages 99–128. Birkhäuser Boston, Boston, MA, 2003. [140] H. G. Feichtinger and G. Zimmermann. A Banach space of test functions for Gabor analysis. In Gabor Analysis and Algorithms, Appl. Numer. Harmon. Anal., pages 123–170. Birkhäuser Boston, Boston, MA, 1998. [141] C. Fernández and A. Galbis. Compactness of time-frequency localization operators on L2 (ℝd ). J. Funct. Anal., 233(2):335–350, 2006. [142] C. Fernández, A. Galbis, and J. Martínez. Multilinear Fourier multipliers related to time-frequency localization. J. Math. Anal. Appl., 398(1):113–122, 2013. [143] C. Fernández, A. Galbis, and E. Primo. Compactness of Fourier integral operators on weighted modulation spaces. Trans. Am. Math. Soc., 372(1):733–753, 2019. [144] V. Fischer and M. Ruzhansky. Quantization on Nilpotent Lie Groups, volume 314 of Progress in Mathematics. Birkhäuser/Springer, Cham, 2016. [145] G. B. Folland. Harmonic Analysis in Phase Space, volume 122 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1989. [146] G. B. Folland. Real Analysis, Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, second edition, 1999. Modern techniques and their applications, A Wiley-Interscience Publication. [147] J. J. F. Fournier. Local complements to the Hausdorff-Young theorem. Mich. Math. J., 20:263–276, 1973. [148] J. J. F. Fournier and J. Stewart. Amalgams of Lp and lq . Bull., New Ser., Am. Math. Soc., 13(1):1–21, 1985. [149] M. Frank, V. I. Paulsen, and T. R. Tiballi. Symmetric approximation of frames and bases in Hilbert spaces. Trans. Am. Math. Soc., 354(2):777–793, 2002. [150] M. Frazier and B. Jawerth. Decomposition of Besov spaces. Indiana Univ. Math. J., 34(4):777–799, 1985. [151] M. Frazier and B. Jawerth. A discrete transform and decompositions of distribution spaces. J. Funct. Anal., 93(1):34–170, 1990. [152] D. Gabor. Theory of communication. J. IEE, 93(III):429–457, 1946. p,q [153] Y. V. Galperin and S. Samarah. Time-frequency analysis on modulation spaces Mm , 0 < p, q ≤ ∞. Appl. Comput. Harmon. Anal., 16(1):1–18, 2004. [154] C. Garetto and M. Ruzhansky. Wave equation for sums of squares on compact Lie groups. J. Differ. Equ., 258(12):4324–4347, 2015. [155] I. M. Gelfand and G. E. Shilov. Generalized Functions. Vol. 3: Theory of Differential Equations. Translated from the Russian by Meinhard E. Mayer. Academic Press, New York-London, 1967. [156] I. M. Gelfand and G. E. Shilov. Generalized Functions, volume 2. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1968 [1977]. Spaces of fundamental and generalized functions, Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

432 | Bibliography

[157] J. Ginibre and G. Velo. Smoothing properties and retarded estimates for some dispersive evolution equations. Commun. Math. Phys., 144(1):163–188, 1992. [158] S. Graffi and L. Zanelli. Geometric approach to the Hamilton-Jacobi equation and global parametrices for the Schrödinger propagator. Rev. Math. Phys., 23(9):969–1008, 2011. [159] K. Gröchenig. Describing functions: atomic decompositions versus frames. Monatshefte Math., 112(1):1–42, 1991. [160] K. Gröchenig. Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA, 2001. [161] K. Gröchenig. A pedestrian’s approach to pseudodifferential operators. In Harmonic Analysis and Applications, Appl. Numer. Harmon. Anal., pages 139–169. Birkhäuser Boston, Boston, MA, 2006. [162] K. Gröchenig. Time-frequency analysis of Sjöstrand’s class. Rev. Mat. Iberoam., 22(2):703–724, 2006. [163] K. Gröchenig and C. Heil. Modulation spaces and pseudodifferential operators. Integral Equ. Oper. Theory, 34(4):439–457, 1999. [164] K. Gröchenig and Y. Lyubarskii. Gabor (super)frames with Hermite functions. Math. Ann., 345(2):267–286, 2009. [165] K. Gröchenig and Z. Rzeszotnik. Banach algebras of pseudodifferential operators and their almost diagonalization. Ann. Inst. Fourier (Grenoble), 58(7):2279–2314, 2008. [166] K. Gröchenig, Z. Rzeszotnik, and T. Strohmer. Convergence analysis of the finite section method and Banach algebras of matrices. Integral Equ. Oper. Theory, 67(2):183–202, 2010. [167] K. Gröchenig and S. Samarah. Nonlinear approximation with local Fourier bases. Constr. Approx., 16(3):317–331, 2000. [168] K. Gröchenig and T. Strohmer. Pseudodifferential operators on locally compact abelian groups and Sjöstrand’s symbol class. J. Reine Angew. Math., 613:121–146, 2007. [169] K. Gröchenig and J. Toft. Isomorphism properties of Toeplitz operators and pseudo-differential operators between modulation spaces. J. Anal. Math., 114:255–283, 2011. [170] K. Gröchenig and J. Toft. The range of localization operators and lifting theorems for modulation and Bargmann-Fock spaces. Trans. Am. Math. Soc., 365(8):4475–4496, 2013. [171] K. Gröchenig and G. Zimmermann. Spaces of test functions via the STFT. J. Funct. Spaces Appl., 2(1):25–53, 2004. [172] A. Grossmann, G. Loupias, and E. M. Stein. An algebra of pseudodifferential operators and quantum mechanics in phase space. Ann. Inst. Fourier (Grenoble), 18(fasc. 2):343–368 viii (1969), 1968. [173] K. Guo and D. Labate. Optimally sparse multidimensional representation using shearlets. SIAM J. Math. Anal., 39(1):298–318, 2007. [174] K. Guo and D. Labate. Sparse shearlet representation of Fourier integral operators. Electron. Res. Announc. Math. Sci., 14:7–19, 2007. [175] K. Guo and D. Labate. Representation of Fourier integral operators using shearlets. J. Fourier Anal. Appl., 14(3):327–371, 2008. [176] W. Guo, J. Chen, D. Fan, and G. Zhao. Characterizations of some properties on weighted modulation and Wiener amalgam spaces. Mich. Math. J., 68(3):451–482, 2019. [177] W. Guo, H. Wu, and G. Zhao. Inclusion relations between modulation and Triebel-Lizorkin spaces. Proc. Am. Math. Soc., 145(11):4807–4820, 2017. [178] A. Hassell and J. Wunsch. The Schrödinger propagator for scattering metrics. Ann. Math. (2), 162(1):487–523, 2005. [179] C. Heil. Wiener amalgam spaces in generalized harmonic analysis and wavelet theory. ProQuest LLC, Ann Arbor, MI, 1990. Thesis (Ph.D.)–University of Maryland, College Park.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Bibliography | 433

[180] C. Heil. Integral operators, pseudodifferential operators, and Gabor frames. In Advances in Gabor Analysis, Appl. Numer. Harmon. Anal., pages 153–169. Birkhäuser Boston, Boston, MA, 2003. [181] C. Heil. An introduction to weighted Wiener amalgams. In Krishna, M., Radha, R., & Thangavelu, S. (eds.), Wavelets and Their Applications, pages 183–216, Chennai, Jan. 2002, Allied Publishers (New Delhi), 2003. [182] C. Heil. History and evolution of the density theorem for Gabor frames. J. Fourier Anal. Appl., 13(2):113–166, 2007. [183] C. Heil. A Basis Theory Primer, Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York, expanded edition, 2011. [184] C. Heil, J. Ramanathan, and P. Topiwala. Linear independence of time-frequency translates. Proc. Am. Math. Soc., 124(9):2787–2795, 1996. [185] C. Heil, J. Ramanathan, and P. Topiwala. Singular values of compact pseudodifferential operators. J. Funct. Anal., 150(2):426–452, 1997. [186] C. Heil and D. F. Walnut. Continuous and discrete wavelet transforms. SIAM Rev., 31(4):628–666, 1989. [187] B. Helffer. Théorie spectrale pour des opérateurs globalement elliptiques, volume 112 of Astérisque. Société Mathématique de France, Paris, 1984. With an English summary. [188] B. Helffer and D. Robert. Comportement asymptotique précisé du spectre d’opérateurs globalement elliptiques dans Rn . C. R. Acad. Sci. Paris Sér. I Math., 292(6):363–366, 1981. [189] F. Holland. Harmonic analysis on amalgams of Lp and 1q . J. Lond. Math. Soc. (2), 10:295–305, 1975. [190] L. Hörmander. On the division of distributions by polynomials. Ark. Mat., 3:555–568, 1958. [191] L. Hörmander. Fourier integral operators. I. Acta Math., 127(1-2):79–183, 1971. [192] L. Hörmander. The Weyl calculus of pseudodifferential operators. Commun. Pure Appl. Math., 32(3):360–444, 1979. [193] L. Hörmander. The Analysis of Linear Partial Differential Operators. III, volume 274 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1985. Pseudodifferential operators. [194] L. Hörmander. The Analysis of Linear Partial Differential Operators. I, Springer Study Edition. Springer-Verlag, Berlin, second edition, 1990. Distribution theory and Fourier analysis. [195] L. Hörmander. Quadratic hyperbolic operators. In Microlocal Analysis and Applications (Montecatini Terme, 1989), volume 1495 of Lecture Notes in Math., pages 118–160. Springer, Berlin, 1991. [196] L. Hörmander. The Analysis of Linear Partial Differential Operators. II, Classics in Mathematics. Springer-Verlag, Berlin, 2005. Differential Operators with Constant Coefficients. [197] R. Howe. Quantum mechanics and partial differential equations. J. Funct. Anal., 38(2):188–254, 1980. [198] R. L. Hudson. When is the Wigner quasi-probability density non-negative? Rep. Math. Phys., 6(2):249–252, 1974. [199] K. Husimi. Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn., 22(2):264–314, 1940. [200] K. Ito. Propagation of singularities for Schrödinger equations on the Euclidean space with a scattering metric. Commun. Partial Differ. Equ., 31(10-12):1735–1777, 2006. [201] K. Ito and S. Nakamura. Singularities of solutions to the Schrödinger equation on scattering manifold. Am. J. Math., 131(6):1835–1865, 2009. [202] M. S. Jakobsen. On a (no longer) new Segal algebra: a review of the Feichtinger algebra. J. Fourier Anal. Appl., 24(6):1579–1660, 2018.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

434 | Bibliography

[203] A. J. E. M. Janssen. Bargmann transform, Zak transform, and coherent states. J. Math. Phys., 23(5):720–731, 1982. [204] A. J. E. M. Janssen. The Zak transform: a signal transform for sampled time-continuous signals. Philips J. Res., 43(1):23–69, 1988. [205] K. Johansson, S. Pilipović, N. Teofanov, and J. Toft. Gabor pairs, and a discrete approach to wave-front sets. Monatshefte Math., 166(2):181–199, 2012. [206] T. Kato. Linear evolution equations of “hyperbolic” type. J. Fac. Sci. Univ. Tokyo Sect. I, 17:241–258, 1970. [207] T. Kato. Boundedness of some pseudo-differential operators. Osaka J. Math., 13(1):1–9, 1976. [208] M. Keel and T. Tao. Endpoint Strichartz estimates. Am. J. Math., 120(5):955–980, 1998. [209] S. Kim and Y. Koh. Strichartz estimates for the magnetic Schrödinger equation with potentials V of critical decay. Commun. Partial Differ. Equ., 42(9):1467–1480, 2017. [210] S. Kim, Y. Koh, and I. Seo. Strichartz estimates for the Schrödinger propagator in Wiener amalgam spaces. J. Math. Anal. Appl., 478(1):236–248, 2019. [211] M. Kobayashi. Modulation spaces Mp,q for 0 < p, q ≤ ∞ . J. Funct. Spaces Appl., 4(3):329–341, 2006. [212] M. Kobayashi. Dual of modulation spaces. J. Funct. Spaces Appl., 5(1):1–8, 2007. [213] J. J. Kohn and L. Nirenberg. An algebra of pseudo-differential operators. Commun. Pure Appl. Math., 18:269–305, 1965. [214] S. G. Krantz and H. R. Parks. The Implicit Function Theorem. Modern Birkhäuser Classics. Birkhäuser/Springer, New York, 2013. History, theory, and applications, Reprint of the 2003 edition. [215] G. Kutyniok and D. Labate. Resolution of the wavefront set using continuous shearlets. Trans. Am. Math. Soc., 361(5):2719–2754, 2009. [216] D. Labate. Pseudodifferential operators on modulation spaces. J. Math. Anal. Appl., 262(1):242–255, 2001. [217] P. D. Lax. Asymptotic solutions of oscillatory initial value problems. Duke Math. J., 24:627–646, 1957. [218] E. H. Lieb. Integral bounds for radar ambiguity functions and Wigner distributions. J. Math. Phys., 31(3):594–599, 1990. [219] S. Lojasiewicz. Sur le problème de la division. Stud. Math., 18:87–136, 1959. [220] F. Low. Complete sets of wave packets. In A Passion for Physics-Essays in Honor of Geoffrey Chew, pages 17–22. World Scientific, Singapore, 1985. [221] Y. I. Lyubarskii. Frames in the Bargmann space of entire functions. In Entire and Subharmonic Functions, volume 11 of Adv. Soviet Math., pages 167–180. Amer. Math. Soc., Providence, RI, 1992. [222] A. Martinez, S. Nakamura, and V. Sordoni. Analytic smoothing effect for the Schrödinger equation with long-range perturbation. Commun. Pure Appl. Math., 59(9):1330–1351, 2006. [223] A. Martinez, S. Nakamura, and V. Sordoni. Analytic wave front set for solutions to Schrödinger equations. Adv. Math., 222(4):1277–1307, 2009. [224] N. H. McCoy. On the function in quantum mechanics which corresponds to a given function in classical mechanics. Proc. Natl. Acad. Sci., 18(11):674–676, 1932. [225] R. B. Melrose. Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces. In Spectral and Scattering Theory (Sanda, 1992), volume 161 of Lecture Notes in Pure and Appl. Math., pages 85–130. Dekker, New York, 1994. [226] B. S. Mitjagin. Nuclearity and other properties of spaces of type S. Trudy Moskov. Mat. Obšč., 9:317–328, 1960. [227] R. Mizuhara. Microlocal smoothing effect for the Schrödinger evolution equation in a Gevrey class. J. Math. Pures Appl. (9), 91(2):115–136, 2009.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Bibliography | 435

[228] S. Molahajloo, K. A. Okoudjou, and G. E. Pfander. Boundedness of multilinear pseudo-differential operators on modulation spaces. J. Fourier Anal. Appl., 22(6):1381–1415, 2016. [229] S. Molahajloo and G. E. Pfander. Boundedness of pseudo-differential operators on Lp , Sobolev and modulation spaces. Math. Model. Nat. Phenom., 8(1):175–192, 2013. [230] M. Măntoiu and M. Ruzhansky. Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups. Doc. Math., 22:1539–1592, 2017. [231] M. Nagase. The Lp -boundedness of pseudo-differential operators with non-regular symbols. Commun. Partial Differ. Equ., 2(10):1045–1061, 1977. [232] S. Nakamura. Propagation of the homogeneous wave front set for Schrödinger equations. Duke Math. J., 126(2):349–367, 2005. [233] S. Nakamura. Semiclassical singularities propagation property for Schrödinger equations. J. Math. Soc. Jpn., 61(1):177–211, 2009. [234] F. Nicola. Phase space analysis of semilinear parabolic equations. J. Funct. Anal., 267(3):727–743, 2014. [235] F. Nicola. Convergence in Lp for Feynman path integrals. Adv. Math., 294:384–409, 2016. [236] F. Nicola and L. Rodino. Global Pseudo-Differential Calculus on Euclidean Spaces, volume 4 of Pseudo-Differential Operators. Theory and Applications. Birkhäuser Verlag, Basel, 2010. [237] F. Nicola and L. Rodino. Global regularity for ordinary differential operators with polynomial coefficients. J. Differ. Equ., 255(9):2871–2890, 2013. [238] F. Nicola and L. Rodino. Propagation of Gabor singularities for semilinear Schrödinger equations. Nonlinear Differ. Equ. Appl., 22(6):1715–1732, 2015. [239] F. Nicola and S. I. Trapasso. On the pointwise convergence of the integral kernels in the Feynman-Trotter Formula. Commun. Math. Phys., 376:2277–2299, 2020. [240] J. Peetre. New Thoughts on Besov Spaces, volume 1 of Duke University Mathematics Series. Mathematics Department, Duke University, Durham, N.C., 1976. [241] G. E. Pfander. Sampling of operators. J. Fourier Anal. Appl., 19(3):612–650, 2013. [242] S. Pilipović. Tempered ultradistributions. Boll. Unione Mat. Ital., B (7), 2(2):235–251, 1988. [243] S. Pilipović and N. Teofanov. Pseudodifferential operators on ultra-modulation spaces. J. Funct. Anal., 208(1):194–228, 2004. [244] S. Pilipović, N. Teofanov, and J. Toft. Micro-local analysis with Fourier Lebesgue spaces. Part I. J. Fourier Anal. Appl., 17(3):374–407, 2011. [245] K. Pravda-Starov, L. Rodino, and P. Wahlberg. Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians. Math. Nachr., 291(1):128–159, 2018. [246] J. Ramanathan and P. Topiwala. Time-frequency localization via the Weyl correspondence. SIAM J. Math. Anal., 24(5):1378–1393, 1993. [247] J. Rauch. Partial Differential Equations, volume 128 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. [248] H. Rauhut. Coorbit space theory for quasi-Banach spaces. Stud. Math., 180(3):237–253, 2007. [249] M. Reed and B. Simon. Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. [250] L. Robbiano and C. Zuily. Microlocal analytic smoothing effect for the Schrödinger equation. Duke Math. J., 100(1):93–129, 1999. [251] L. Robbiano and C. Zuily. Analytic theory for the quadratic scattering, wave front set and application to the Schrödinger equation. In Hyperbolic Problems and Related Topics, Grad. Ser. Anal., pages 319–339. Int. Press, Somerville, MA, 2003. [252] R. Rochberg and K. Tachizawa. Pseudodifferential operators, Gabor frames, and local trigonometric bases. In Gabor Analysis and Algorithms, Appl. Numer. Harmon. Anal., pages 171–192. Birkhäuser Boston, Boston, MA, 1998.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

436 | Bibliography

[253] L. Rodino and P. Wahlberg. The Gabor wave front set. Monatshefte Math., 173(4):625–655, 2014. [254] W. Rudin. Real and Complex Analysis. McGraw-Hill Book Co., New York, third edition, 1987. [255] M. Ruzhansky, M. Sugimoto, J. Toft, and N. Tomita. Changes of variables in modulation and Wiener amalgam spaces. Math. Nachr., 284(16):2078–2092, 2011. [256] M. Ruzhansky and D. Suragan. Uncertainty relations on nilpotent Lie groups. Proc. A., 473(2201):20170082, 12, 2017. [257] M. Ruzhansky and V. Turunen. Pseudo-Differential Operators and Symmetries, volume 2 of Pseudo-Differential Operators. Theory and Applications. Birkhäuser Verlag, Basel, 2010. Background analysis and advanced topics. [258] M. Ruzhansky and J. Wirth. Lp Fourier multipliers on compact Lie groups. Math. Z., 280(3-4):621–642, 2015. [259] R. Schulz and P. Wahlberg. Microlocal properties of Shubin pseudodifferential and localization operators. J. Pseudo-Differ. Oper. Appl., 7(1):91–111, 2016. [260] R. Schulz and P. Wahlberg. Equality of the homogeneous and the Gabor wave front set. Commun. Partial Differ. Equ., 42(5):703–730, 2017. [261] L. Schwartz. Théorie des Distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée. Hermann, Paris, 1966. [262] K. Seip. Density theorems for sampling and interpolation in the Bargmann-Fock space. Bull., New Ser., Am. Math. Soc., 26(2):322–328, 1992. [263] K. Seip and R. Wallstén. Density theorems for sampling and interpolation in the Bargmann-Fock space. II. J. Reine Angew. Math., 429:107–113, 1992. [264] K. Seongyeon, K. Youngwoo, and S. Ihyeok. Strichartz estimates for the Schrödinger Propagator in Wiener Amalgam Spaces. Preprint, arXiv:1902.08940, 2019. [265] M. A. Shubin. Pseudodifferential Operators and Spectral Theory, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1987. Translated from the Russian by Stig I. Andersson. [266] J. Sjöstrand. An algebra of pseudodifferential operators. Math. Res. Lett., 1(2):185–192, 1994. [267] J. Sjöstrand. Wiener type algebras of pseudodifferential operators. In Séminaire sur les Équations aux Dérivées Partielles, 1994–1995, pages Exp. No. IV, 21. École Polytech., Palaiseau, 1995. [268] E. M. Stein. Singular Integrals and Differentiability Properties of Functions, volume 30 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1970. [269] E. M. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. [270] E. M. Stein and G. Weiss. Introduction to Fourier Analysis on Euclidean Spaces, volume 32 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1971. [271] T. Strohmer. Pseudodifferential operators and Banach algebras in mobile communications. Appl. Comput. Harmon. Anal., 20(2):237–249, 2006. [272] M. Sugimoto. Lp -Boundedness of pseudodifferential operators satisfying Besov estimates. I. J. Math. Soc. Jpn., 40(1):105–122, 1988. [273] M. Sugimoto. Lp -Boundedness of pseudodifferential operators satisfying Besov estimates. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 35(1):149–162, 1988. [274] M. Sugimoto and N. Tomita. The dilation property of modulation spaces and their inclusion relation with Besov spaces. J. Funct. Anal., 248(1):79–106, 2007. [275] K. Tachizawa. The pseudodifferential operators and Wilson bases. J. Math. Pures Appl. (9), 75(6):509–529, 1996.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

Bibliography | 437

[276] T. Tao. Low regularity semi-linear wave equations. Commun. Partial Differ. Equ., 24(3-4):599–629, 1999. [277] T. Tao. Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics. Amer. Math. Soc., 2006. [278] T. Tao. Lecture Notes 2 for 247 A: Fourier analysis. www.math.ucla.edu/˜tao, 2007. [279] D. Tataru. Phase space transforms and microlocal analysis. In Phase Space Analysis of Partial Differential Equations. Vol. II, Pubbl. Cent. Ric. Mat. Ennio De Giorgi, pages 505–524. Scuola Norm. Sup., Pisa, 2004. [280] M. E. Taylor. Pseudodifferential Operators. Princeton Univ. Press, Princeton, NJ, 1981. [281] M. E. Taylor. Tools for PDE, volume 81 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000. Pseudodifferential operators, paradifferential operators, and layer potentials. [282] N. Teofanov. Ultradistributions and time-frequency analysis. In Pseudo-Differential Operators and Related Topics, volume 164 of Oper. Theory Adv. Appl., pages 173–192. Birkhäuser, Basel, 2006. [283] N. Teofanov. Continuity and Schatten–von Neumann properties for localization operators on modulation spaces. Mediterr. J. Math., 13(2):745–758, 2016. [284] H. ter Morsche and P. J. Oonincx. On the integral representations for metaplectic operators. J. Fourier Anal. Appl., 8(3):245–257, 2002. [285] E. C. Titchmarsh. Introduction to the Theory of Fourier Integrals. Chelsea Publishing Co., New York, third edition, 1986. [286] J. Toft. Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I. J. Funct. Anal., 207(2):399–429, 2004. [287] J. Toft. Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II. Ann. Glob. Anal. Geom., 26(1):73–106, 2004. [288] J. Toft. Continuity and Schatten properties for pseudo-differential operators on modulation spaces. In Modern Trends in Pseudo-Differential Operators, volume 172 of Oper. Theory Adv. Appl., pages 173–206. Birkhäuser, Basel, 2007. [289] J. Toft. The Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators. J. Pseudo-Differ. Oper. Appl., 3(2):145–227, 2012. [290] J. Toft. Matrix parameterized pseudo-differential calculi on modulation spaces. In Generalized Functions and Fourier Analysis, volume 260 of Oper. Theory Adv. Appl., pages 215–235. Birkhäuser/Springer, Cham, 2017. [291] J. Toft, F. Concetti, and G. Garello. Schatten-von Neumann properties for Fourier integral operators with non-smooth symbols II. Osaka J. Math., 47(3):739–786, 2010. [292] J. Toft, K. Johansson, S. Pilipović, and N. Teofanov. Sharp convolution and multiplication estimates in weighted spaces. Anal. Appl. (Singap.), 13(5):457–480, 2015. [293] J. Toft, R. Üster, E. Nabizadeh, and S. Öztop. Continuity and Bargmann mapping properties of quasi-Banach Orlicz modulation spaces. Preprint, arXiv:1907.02331, 2020. [294] S. I. Trapasso. Time-frequency analysis of the Dirac equation. J. Differ. Equ., 269(3):2477–2502, 2020. [295] F. Trèves. Topological Vector Spaces, Distributions and Kernels. Academic Press, New York-London, 1967. [296] H. Triebel. Interpolation Theory, Function Spaces, Differential Operators, volume 18 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam-New York, 1978. [297] H. Triebel. Spaces of Besov-Hardy-Sobolev Type. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1978. Teubner-Texte zur Mathematik, with German, French and Russian summaries.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

438 | Bibliography

[298] H. Triebel. Modulation spaces on the Euclidean n-space. Z. Anal. Anwend., 2(5):443–457, 1983. [299] J. Ueberberg. Zur Spektralinvarianz von Algebren von Pseudodifferentialoperatoren in der Lp -Theorie. Manuscr. Math., 61(4):459–475, 1988. [300] J. Ville. Theorie et applications de la notion de signal analytique. Câbles et Transmissions, 2(2):61–74, 1948. [301] P. Wahlberg. Propagation of polynomial phase space singularities for Schrödinger equations with quadratic Hamiltonians. Math. Scand., 122(1):107–140, 2018. [302] D. F. Walnut. Continuity properties of the Gabor frame operator. J. Math. Anal. Appl., 165(2):479–504, 1992. [303] B. Wang and C. Huang. Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations. J. Differ. Equ., 239(1):213–250, 2007. [304] B. Wang and H. Hudzik. The global Cauchy problem for the NLS and NLKG with small rough data. J. Differ. Equ., 232(1):36–73, 2007. [305] B. Wang, Z. Huo, C. Hao, and Z. Guo. Harmonic Analysis Method for Nonlinear Evolution Equations. i. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. λ [306] B. Wang, Z. Lifeng, and G. Boling. Isometric decomposition operators, function spaces Ep,q and applications to nonlinear evolution equations. J. Funct. Anal., 233(1):1–39, 2006. [307] A. Weinstein. A symbol class for some Schrödinger equations on Rn . Am. J. Math., 107(1):1–21, 1985. [308] F. Weisz. Inversion of the short-time Fourier transform using Riemannian sums. J. Fourier Anal. Appl., 13(3):357–368, 2007. [309] H. Weyl. Gruppentheorie und Quantenmechanik. Wissenschaftliche Buchgesellschaft, Darmstadt, second edition, 1977. [310] N. Wiener. On the representation of functions by trigonometrical integrals. Math. Z., 24(1):575–616, 1926. [311] N. Wiener. Tauberian theorems. Ann. Math. (2), 33(1):1–100, 1932. [312] N. Wiener. Generalized Harmonic Analysis and Tauberian Theorems. The M.I.T. Press, Cambridge, Mass.-London, 1966. [313] N. Wiener. The Fourier Integral and Certain of Its Applications, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988. Reprint of the 1933 edition, With a foreword by Jean-Pierre Kahane. [314] E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749–759, Jun 1932. [315] P. Wojdyllo. Characterization of Wilson systems for general lattices. Int. J. Wavelets Multiresolut. Inf. Process., 6(2):305–314, 2008. [316] T. H. Wolff. Lectures on Harmonic Analysis, volume 29 of University Lecture Series. American Mathematical Society, Providence, RI, 2003. [317] M. W. Wong. Weyl Transforms. Springer-Verlag, New York, 1998. [318] M. W. Wong. Localization Operators. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1999. [319] M. W. Wong. Wavelet Transforms and Localization Operators, volume 136 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2002. [320] P. M. Woodward. Probability and Information Theory, with Applications to Radar. McGraw-Hill Book Co., Inc./Pergamon Press Ltd., New York/London, 1953. [321] J. Wunsch. Propagation of singularities and growth for Schrödinger operators. Duke Math. J., 98(1):137–186, 1999. [322] K. Yajima. Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys., 110(3):415–426, 1987.

Index τ-Wigner distribution see Wigner distribution Ambiguity function 38 – Moyal’s identity 39 Bessel potential operator 94 Bessel sequence 148 Cohen class 46 Commutation relations 6

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Evolution equations 315 Formula – Leibniz 2 – Taylor 3 Fourier integral operator – FIO(χ, s) 366 – type I 365 – type II 366 Fourier integral operator (FIO) 326 Fourier transform 4 – partial 23 Frame – alternative dual 155 – analysis operator 149 – canonical dual frame 155 – definition 145 – exact 146 – frame bounds 145 – frame operator 152 – Gabor frame 162 – Gram operator 152 – Parseval 145 – reproducing formulae 153 – synthesis operator 152 – tight 145 Fundamental identity of time–frequency analysis 19 Gabor frames – characterization of modulation spaces 179 – density 171 – for L2 (ℝd ) 162 – Gaussian frames 173 – Wexler–Raz biorthogonality relations 173 Gabor matrix 183 – evolution operator 318

– FIO of type I 372, 378 – Fourier integral operator 330, 333, 338 – free particle 356 – generalized metaplectic operator 381 – harmonic oscillator 362 – heat propagator 321 – perturbed Schrödinger equation 397 – pseudodifferential operator 366 – Schrödinger propagator 384 – sparsity 352 – unperturbed Schrödinger equation 394 – Weyl operator 311, 313 Gabor wave front set 385 Gaussian function 7 Gelfand–Shilov space 301 Kernel theorem – Gelfand–Shilov spaces 304 – modulation spaces 181, 185 – Schwartz 5 Lebesgue – discrete mixed norm spaces 76 – mixed norm spaces 76 – spaces 3 Metaplectic group 14 Metaplectic operator 345, 346 – generalized 381 Modulation spaces – Banach space 86 – complex interpolation 92 – convolution relations 110 – density 86 – dilation properties 118, 120, 131 – duality 87 – equivalent norms 96 – inclusion relations 110 d – MP,Q m (ℝ ) 79 p,q d – Mm (ℝ ) 79 p,q – ℳm (ℝd ) 92 p,q – Mm (ℝd ), ultradistributions 304 – mixed modulation spaces 80 – multiplication relations 112 Multi-index 1 Operator – dilation Dλ 7

440 | Index

– Fourier integral see Fourier integral operator – frequency-uniform decomposition operator ◻k 96 – involution 7 – modulation Mξ 6 – pseudodifferential see Pseudodifferential operator – reflection ℐ 6 – symmetric coordinate change Ts 42 – Ta 23 – Tb 192 – time-frequency shift π(z) 6 – translation Tx 6 – 𝒰 195 Peetre’s inequality 2 Phase function – quadratic 342 – tame 328 p Potential Sobolev spaces Ws (ℝd ) 93 Pseudodifferential operator – τ-representation Opτ (a) 211 – Born–Jordan 243, 261 – Fourier multiplier 192 – Kohn–Nirenberg 191 – localization operator 227 – Shubin Gm 388 – Weyl 194

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Riesz basis 159 Rihaczek distribution 58 Schauder estimates 6 Schrödinger representation 14 Schwartz function 4 Short-time Fourier transform – adjoint 28 – definition 18 – inversion formula 27 – orthogonality relations 24

– symplectic covariance formula 25 Shubin class Γm (ℝ2d ) 387 Shubin–Sobolev space – characterization as modulation space 229 – Qs 229 Sobolev space – embedding 6 Sobolev space Hs (ℝd ) 5 Strichartz estimates – Schrödinger equation 281, 282 Symplectic – covariance formula for the STFT 25 – form 12 – Fourier transform 17 – group 12 – standard symplectic matrix 12 symplectomorphism 329 Temperate distribution 4 Tensor product f ⊗ g 23 Weight function – definition 72 – equivalent 72 – submultiplicative 72 – v-moderate 72 – vs (z) = ⟨z⟩s 74 Wiener – algebra of FIO 370 – algebra of pseudodifferential operators 365 – lemma 364 Wiener amalgam spaces – dilation 113, 119 – properties 104 – W (B, C) 103 Wigner distribution 39 – τ-Wigner distribution 53 – covariance property 41 – marginal densities 44 – Moyal’s identity 40

Index of Notation Function spaces p ,...,p – ℓm1 d (X ) 76 p – ℓ (ℤd ) 4 – Γm (ℝ2d ) 387 – 𝒞0m (ℝd ) 3 – c00 (ℤd ) 4 – 𝒞0 (ℝd ) 3 – c0 (ℤd ) 4 – 𝒞bm (ℝd ) 3 – 𝒞cm (ℝd ) 3 – 𝒞 ∞ (ℝd ) 3 – 𝒞 m (ℝd ) 3 – 𝒟(ℝd ) 3 – Hs (ℝd ) 5 p ,p ,...,p – Lm1 2 d 76 p – Lloc (ℝd ) 3 – LPm (ℝd ) 77 – Lp (ℝd ) 3 – ℳv (ℝ2d ) 72 0 – S0,0 333

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s,A – Sr,B (ℝd ) 301

– Srs (ℝd ) 302 – 𝒮(ℝd ) 4 – 𝒮 󸀠 (ℝd ) 4 – W0 (L∞ , L1 )(ℝd ) 175 p – Ws (ℝd ) 93 Fundamental operators – ◻k 95 – π(z) 6 – Aτ 57 – Dλ 7 –f∗ 7 –ℐ 7 – Mξ 6 – Ta 23 – Ts 42 – Tx 6 Inequalities up to a constant – f (t) ≍ g(t) 1 – f (t) ≲ g(t) 1 LCA 79 Modulation spaces p,q – Mm (ℝd ) 80 d – MP,Q m (ℝ ) 79

p,q

– ℳm (ℝd ) 92 p – Mm (ℝd ) 80 d – M(c)P,Q m (ℝ ) 80 Number sets – ℕ, ℕ+ , ℤ, ℝ, ℝ+ , ℝ− , ℂ 1 Operators – ℓp (ℤd ) 4 – μ(𝒜) 14, 345 – σ(t, Dx ) 318 φ ,φ – Aa 1 2 227 – a(x, D) 193 – eitΔ 281 – Opτ (a) 211 – OpBJ (a) 243 – Opw (a) 194 – P(𝜕t , Dx ) 315 – Tm󸀠 ,n󸀠 ,m,n 280 – UId 343 Special symbols – α! 1 –α ≤ β 1 – (αβ ) 1 – 𝜕j 2 – 𝜕α 2 – ρk 95 – σk 95 – Gs 94 – |α| 1 – 𝒬k 95 Time-frequency representations – Af 38 – A(f , g) 38 – 𝒬σ f 46 – Qn f 276 – Q(f , g) 261 – R(f , g) 58 – Vg f 18 – W0 (f , g) 58 – Wτ (f , g) 53 – Wf 39 – W (f , g) 39 Weights – ⟨z⟩ 2

442 | Index of Notation

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– ma,b,s,t (z) 73 – va,b,s,t (z) 73 – vs (z) 74 – ws,ϵ (z) 310

Wiener amalgam spaces – W (B, C) 103 p,q – W (ℱ L1 , Lm )(ℝ2d ) 107 p q – W (L , Lm ) 100

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