Characterization of Function Spaces and Boundedness of Bilinear Pseudodiﬀerential Operators Through Gabor Frames

Characterization of function spaces and boundedness of bilinear pseudodiﬀerential operators through Gabor frames

AThesis Presented to TheAcademicFaculty

by Kasso Akochay´e Okoudjou

In Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy

School of Mathematics Georgia Institute of Technology April 2003 Characterization of function spaces and boundedness of bilinear pseudodiﬀerential operators through Gabor frames

Approved by:

Professor Christopher E. Heil, Adviser Professor Yang Wang

Professor Anthony Yezzi Professor Jeﬀrey Geronimo ECE

Professor Gerd Mockenhaupt

Date Approved To my family.

iii ACKNOWLEDGEMENTS

First and foremost, I would like to thank my advisor, Professor Chris Heil for intro- ducing me to the fascinating theory of time-frequency analysis, and for guiding and supporting me during my studies. I also thank him for setting very high standards for me in research as well as in writing.

I wish to thank Professor Wilfrid Gangbo, not only for oﬀering me the opportunity to apply to Georgia Tech, but also for being so supportive during all the years I have spent here. I would like to thank Professors Jeﬀrey Geronimo, Gerd Mockenhaupt, Yang

Wang, and Anthony Yezzi for serving as members of my defense committee. I am also thankful to Professor Michael Loss for his support while I was working on my thesis.

Over the last five years I have had the opportunity to meet several mathematicians who have helped me in the course of my research. I am especially grateful to Professor Hans G. Feichtinger, Head of the Numerical Harmonic Analysis Group —NuHAG— at the University of Vienna, for inviting me to spend a summer with his group, and to Professor Michael Loss for providing the financial support that made this visit possible. I really appreciated his enthusiasm for sharing his deep understanding of mathematics. I was lucky that Professor Karlheinz Gröchenig was also visiting the

NuHAG at the same time. I thank him for suggesting one of the problems I present in this thesis, and for helping me gain some insights in time-frequency analysis. I also wish to thank all the other members of the NuHAG group who made my stay in

Vienna wonderful and memorable. My thanks in particular go to Massimo Fornasier,

Monika Dörfler, Norbert Kaiblinger, Tobias Werther, as well as to Erik Alapääand

iv Bernard Keville who were visiting the NuHAG at the same time. Several people have made my stay here at Georgia Tech a wonderful one and I

would like to thank them. Ms. Cathy Jacobson, English Language Consultant, has

been especially helpful from the ﬁrst day I came to Georgia Tech till now. I thank her

for helping me improve my oral expression and my writing techniques in English as

well as for the wonderful teaching seminars she organized during the academic year

1998-1999. I wish also to thank my colleague, Brody Johnson, for all his help while I was writing this thesis.

I have been enriched by my friends and colleagues in the School of Mathematics as

we shared so many good moments together. In particular, I am thankful to Martial

Agueh, Claudia Antonini, Gianluigi Del Magno, Jose Enrique Figueroa-Lopez, Luis

Hernandez-Urena, Armel Kélomè, Hamed Maroofi, Victor Morales Duarte, Jose Miguel

Renom and Jorge Viveros for their friendship. I will miss you all. I would like to thank my family for their steady support and love. Last, but not least, I would like to thank my wife, Rookhiyath. My life has positively changed for the better since we got married. Thank you for being so patient and understanding with me, as I sometimes put my research before you.

v TABLE OF CONTENTS

DEDICATION iii

ACKNOWLEDGEMENTS iv

SUMMARY viii

I PRELIMINARIES 1 1.1Introduction...... 1 1.2OutlineoftheThesis...... 5 1.3Notations...... 6

II GABOR FRAMES AND MODULATION SPACES 9 2.1WeightFunctions...... 9 2.1.1 Submultiplicative weights ...... 10 2.1.2 Moderateweights...... 10 2.2 Gabor frames in L2 ...... 11 2.3Modulationspaces...... 15 2.3.1 Deﬁnitionandbasicproperties...... 15 2.3.2 Gaborframesonmodulationspaces...... 19

III GABOR ANALYSIS IN WEIGHTED AMALGAM SPACES 22 3.1Weightedamalgamspaces...... 23 3.1.1 Deﬁnition...... 23 3.1.2 Dualityandconvergence...... 25 3.1.3 Sequencespaces...... 28 3.2 Boundedness of the analysis and synthesis operators ...... 29 3.2.1 Lemmas...... 29 3.2.2 Boundedness of the synthesis operator ...... 32 3.2.3 Boundedness of the analysis operator ...... 35 3.2.4 The Walnut representation of the Gabor frame operator on amalgamspaces...... 38

vi 3.3Gaborexpansionsintheamalgamspaces...... 39 3.4ConvergenceofGaborexpansions...... 42 3.5Necessaryconditionsonthewindow...... 45

IV EMBEDDINGS OF BESOV, TRIEBEL-LIZORKIN SPACES INTO MODULATION SPACES 51 4.1TheBesovandTriebel-LizorkinSpaces...... 52 4.2 Embedding of Besov, Triebel-Lizorkin spaces into modulation spaces 58

V BILINEAR PSEUDODIFFERENTIAL OPERATORS ON MOD- ULATION SPACES 70 5.1 Bilinear operators on modulation spaces ...... 71 5.1.1 Deﬁnitionandbackground...... 71 5.1.2 Bilinear operators ...... 73 5.1.3 Adiscretemodel...... 75 5.1.4 Boundedness of bilinear pseudodiﬀerential operators ..... 78 5.2 Linear Hilbert transform on the modulation spaces ...... 81

REFERENCES 87

VITA 92

vii SUMMARY

A frame in a separable Hilbert space H is a sequence of vectors {fn}n∈I which provides a basis-like expansion for any vector in H. However, this representation is usually not unique, since most useful frames are over-complete systems, and, hence, are not bases. Furthermore, frames with particular structures—wavelet frames, expo- nential frames, or Gabor frames—have proven very useful in numerous applications.

Gabor frames, also known as Weyl-Heisenberg frames, are generated by time- frequency shifts of a single function which is called the window function or the generator. Not only do Gabor frames characterize any square integrable function, but they also provide a precise characterization of a class of Banach spaces called modulation spaces.

One objective of this thesis is to extend the theory of Gabor frames to other

Banach spaces which are not included in the class of the modulation spaces. In particular, we will prove that Gabor frames do characterize a class of Banach spaces called amalgam spaces, which include the Lebesgue spaces and play important roles in sampling theory. Moreover, we will study the behavior of various operators connected to the theory of Gabor frames on the amalgam spaces.

Another objective of this thesis is to formulate and prove sufficient conditions on a function to belong to a particular modulation space. Modulation spaces have a rather implicit definition, yet they are the natural setting for time-frequency analysis. Consequently it is important to give sufficient conditions for membership in them. We will prove that certain classical Banach spaces such as the Besov and Triebel-Lizorkin spaces are embedded in the modulation spaces. These embeddings provide us with sufficient conditions for membership in the modulation spaces.

viii Finally, we will use the theory of Gabor frames to formulate certain boundedness results for bilinear pseudodifferential operators with non-smooth symbols on products of modulation spaces. More precisely, we use the Gabor frame expansions of functions in the modulation spaces to convert the boundedness of these operators to the boundedness of an infinite matrix acting on sequence spaces associated to the modulation spaces. A particular modulation space known as the Feichtinger algebra turns out to be a class of non-smooth symbols that yield the boundedness of the bilinear pseudodifferential operators on products on modulation spaces. Additionally, we use the same decomposition techniques to study the boundedness of the (linear) Hilbert transform on the modulation spaces in the one dimensional case.

ix CHAPTER I

PRELIMINARIES

1.1 Introduction

In 1946 D. Gabor [31] proposed a decomposition of signals that displays simultane-

ously the local time and frequency content of the signal, as opposed to the classical Fourier transform which displays only the global frequency content for the entire sig-

nal. He used building blocks generated by time-frequency shifts of a Gaussian, i.e.,

2πim· −πx2 building blocks of the form gm,n = e g(·−n)whereg(x)=e , and sought an orthonormal basis for L2(R) made up of these elementary functions. However, the

Balian-Low theorem [17, Chapter 2] shows that no orthonormal basis can be obtained

in this fashion with any function g as ”nice” as the Gaussian. However, by relaxing the orthonormal basis requirement, and seeking a representation that preserves the

main features of the signal —such as its energy— and that allows stable reconstruc-

tion, Gabor’s idea turns out to yield positive results. More precisely, one can obtain

using Gabor’s scheme some very good and useful substitutes for orthonormal bases:

Gabor frames.

Frames were introduced by R. J. Duﬃn and A. C. Schaeﬀer [15] in 1952 while working on some problems in nonharmonic Fourier series, but they were used little

until the dawn of the wavelet era. Formally, a frame in a separable Hilbert space H

is a sequence {fn}n∈I for which there exist constants 0

It is remarkable that the above inequalities imply the existence of a (canonical) dual

1 ˜ frame {fn}n∈I , such that the following reconstruction formula holds for every f ∈ H: X ˜ f = hf,fnifn. n∈I In particular, any orthonormal basis for H is a frame. However, in general, a frame need not be a basis and, in fact, most useful frames are over-complete. The redun- dancy that frames carry is what makes them very useful in many applications.

Gabor frames and wavelet frames are examples of “easily constructible” frames, and have played important roles in applications as well as in pure mathematics over the last two decades [14, 46]. From the abstract frame theory of Duffin and Schaeffer and Gabor’s original idea, the theory of Gabor frames has grown to become a field on its own right. Notwithstanding the fact that many questions concerning the existence of Gabor frames remain unsolved, there exist numerous “constructible” examples of such frames whose generators are well-localized in the time-frequency plane. In such cases the frames are necessarily over-complete.

A more remarkable property of Gabor frames lies in the characterizations they provide for a whole class of Banach function spaces. Indeed, a deep result in the area, due to H. G. Feichtinger and K. Gröchenig [23, 24], is the atomic decomposition of the class of Banach spaces known as the modulation spaces via Gabor frames. These spaces were introduced by Feichtinger [21] and can be seen as the proper tools to quantify the time-frequency content of functions. The modulation spaces have since then found numerous applications. In particular, they appear quite naturally in the theory of pseudodifferential operators. A pseudodifferential operator is a formalism that assigns to a distribution σ ∈S0(R2d) —the symbol of the operator— a linear

d 0 d operator Tσ : S(R ) →S(R) in such a way that properties of the symbol can be inferred from properties of the operator. Moreover, pseudodiﬀerential operators are encountered in engineering, where they are known as time-varying ﬁlters, as well as in quantum mechanics, where they appear under the name of quantization rules. We

2 refer to cf. [41, Sect. 14.1] and the references therein for more background on pseudod- iﬀerential operators, as well as for their connections with partial diﬀerential equations.

A natural question one could ask is to ﬁnd conditions on σ under which Tσ can be extended to a bounded operator on L2, or on more general Banach spaces. Sym-

bols in the so-called H¨ormander class are known to yield bounded pseudodiﬀerential

operators on various Banach spaces, cf. [26, Chapter 2]. In particular, Calder´on-

Vaillancourt [10] proved that if σ is smooth enough and has enough decay, then Tσ can be extended to a bounded operator on L2.Gr¨ochenig and Heil [42] recovered

and extended this result using non-smooth symbols with only a mild time-frequency

concentration as measured by a modulation space norm.

In this thesis we consider three problems centered around the theory of Gabor

frames and the modulation spaces. The ﬁrst problem we consider is an extension of the theory of Gabor frames from

its “natural setting” (the modulation spaces) to other spaces. Indeed, because the

Lp spaces are not modulation spaces if p =6 2 [25], it was not known if these spaces could be characterized via Gabor frames. In a joint work with K. Gr¨ochenig and

C. Heil [39], we show that Gabor frames do characterize a class of Banach spaces called the amalgam spaces, which include the Lebesgue spaces. Amalgam spaces are spaces that amalgamate local and global criterion for membership. They appear naturally in sampling theory, where they are the “right” setting for diﬀerent problems

[1]. Additionally, we will prove a weak necessary condition on the Gabor frame’s generator, thereby extending a result due to R. Balan [2].

The second problem we consider is concerned with the modulation spaces. In spite of being the “right” spaces for time-frequency analysis, their rather implicit definition makes it very difficult to decide if a function belongs to a particular modulation space. We formulate sufficient conditions for membership in the modulation spaces by proving embeddings of certain Banach spaces such as the Besov and Triebel-Lizorkin

3 spaces into some modulation spaces [50]. The class of Besov and Triebel-Lirzorkin spaces, includes some well-known Banach spaces such as the Lebesgue, the H¨older-

Lipschitz, the Sobolev spaces, and is equipped with a wide variety of equivalent norms.

We refer to [55, Chapter 4], [58, Chapters 1–2] for background on these spaces. The embedding results we prove can be seen as a comparison among certain properties of functions, i.e., smoothness and decay versus time-frequency concentration.

The last problem we consider can be viewed as an application of the theory of Gabor frames. More precisely, we consider the boundedness of bilinear pseudodifferential operators on modulation spaces. One of the motivation of investigating the bilinear pseudodifferential operators on the modulation spaces comes from the recent developments of their linear counterpart in the realm of the modulation spaces. More precisely, because the Weyl correspondence —which is a particular way of assigning to a symbol a pseudodifferential operator corresponding to the Weyl quantization rule— can be expressed as superposition of time-frequency shifts, which are the main objects used in defining the modulation spaces, it was natural to study the linear pseudodifferential operators on these spaces. Bilinear pseudodifferential operators are defined through their symbols as bilinear operators from S(Rd) ×S(Rd)into

S0(Rd), and are not just generalizations of their linear counterparts, but are important tools in many problems in analysis [49]. A natural question in this context is again to find sufficient conditions on the symbols that guarantee the boundedness of the corresponding operators on products of certain Banach spaces. Smoothness and decay of the symbols are often the conditions needed to prove the boundedness of these operators [11, 49, 35, 36]. In a joint work with A. Bényi [6], we prove that if the symbols are in a particular modulation space —the so-called Feichtinger algebra— then the corresponding bilinear pseudodifferential operators are bounded on products of modulation spaces. As particular cases, we obtain boundedness results on products of certain Lebesgue spaces using non-smooth symbols. Finally, we prove that

4 the Hilbert transform is bounded on the modulation spaces, using a discrete tool via the atomic decomposition of these spaces by Gabor frames and by relying on the L2

theory of the Hilbert transform.

1.2 Outline of the Thesis

The thesis is organized as follows. Chapter 1 contains a brief survey of the notations and deﬁnitions that will be used in the sequel.

Chapter 2 is mostly expository. In particular, it contains the deﬁnition of the

basic tools of time-frequency analysis. Moreover, it contains the deﬁnition of the

Gabor frames, as well as their main properties. Additionally, the deﬁnition of the

modulation spaces and their atomic decompositions by Gabor frames as well as their

main properties is given in the chapter. Chapter 3 is devoted to the ﬁrst main result of the thesis, namely the characteriza-

tion of the weighted amalgam spaces by Gabor frames. To obtain this characterization

we study the behavior of the various operators connected with (Gabor) frames the-

ory. Additionally we prove a weak necessary condition on the generator of the Gabor

frames.

Chapter 4 contains the embedding results of certain Besov and Triebel-Lizorkin spaces into the modulation spaces. These results provide some suﬃcient conditions

for membership in the modulation spaces. To obtain these results, we rely on the

numerous equivalent norms deﬁning the Besov and Triebel-Lizorkin spaces as well as

the properties of the short time Fourier transform (STFT).

Finally, Chapter 5 contains some applications of the theory of Gabor frames to

the study of bilinear pseudodiﬀerential operators. In particular, we present in Section 5.1 a boundedness result for bilinear pseudodiﬀerential operators using a discrete

approach via Gabor frames. Section 5.2 is devoted to prove the boundedness of the

Hilbert transform on the modulation spaces again using a discrete approach.

5 1.3 Notations

d The usual dot product of x, y ∈ R is denoted by x · y = x1y1 + x2y2 + ...+xnyn. The length of x is |x|. We use the notation |a| to denote the magnitude of a complex number a.

We use the notation f ∗(t)=f(−t). R The convolution of f and g is deﬁned formally as (f ∗ g)(x)= f(x−t)g(t)dt.

The Fourier transform of a function f is deﬁned formally by Z Ff(ω)=fˆ(ω)= f(t)e−2πit·ω dt for ω ∈ Rd. Rd

Similarly, the inverse Fourier transform of f is deﬁned formally by Z F −1f(t)=fˇ(t)= f(ω)e2πiω·t dω for t ∈ Rd. Rd

d If E ⊂ R is a measurable set, χE is the characteristic function of E,andwe denote its Lebesgue measure by |E|.

d d If a>0, we deonote Qa thecubeinR with side length a, i.e., Qa =[0,a[ .

Let X be a Banach space, then the norm of u ∈ X will be denoted kukX or simply kuk when the appropriate space is clear from the context. Moreover, if two norms k·k1 and k·k2, are equivalent on a Banach space X we will write kuk1 kuk2 to

mean the existence of two positive constant C1,C2 such that

1 2 1 C1kuk ≤kuk ≤C2kuk ∀u∈X.

The dual of a Banach space X is denoted X∗. We write hf,gi for the action of f ∈ X0 on g ∈ X.

The adjoint of an operator T is denoted by T ∗. ≤ ≤∞ 0 1 1 For 1 p ,p will denote the conjugate of p, i.e., p + p0 =1. Lp(Rd) is the Banach space of complex-valued functions f on Rd with norm Z 1/p p kfkp = kfkLp = |f(x)| dx , Rd

6 for 1 ≤ p<∞.Ifp=∞, the norm is given by

kfk∞ = ess sup |f(x)|. x∈Rd

Similarly, `p(Zd) is the Banach space of complex-valued sequences c on Zd with

norm X 1/p p kckp = kck`p = |cn| , n∈Zd for 1 ≤ p<∞.Ifp=∞, the norm is given by

kck∞ =sup|cn|. n∈Zd

p,q R2d We will also consider weighted mixed-norm spaces Lν ( ), which are Banach spaces of complex-valued functions f on R2d with norm Z Z q/p 1/q k k p,q | |p p f Lν = f(x, y) ν(x, y) dx dy , Rd Rd

for 1 ≤ p, q < ∞, with obvious modiﬁcations if p = ∞,orq=∞.Theweight

function ν will be described in the following chapters.

p,q We deﬁne similarly the discrete weighted mixed-norm spaces `ν˜ as the Banach space of complex-valued sequences c on Z2d with norm X X q/p 1/q k k p,q | |p p c `ν = ck,l ν˜(k, l) , l∈Zd k∈Zd for 1 ≤ p, q < ∞, with usual modiﬁcations if p = ∞ or q = ∞.Theweight˜νis an appropriate sample of the weight function ν, typicallyν ˜(k, l)=ν(αk, βl) for some

α, β > 0.

2 s/2 We use the notation ωs for the function ωs(x)=(1+|x|) for s>0. If E is a measurable subset of Rd,welet

kfkp,E = kfχEkp

denote the norm of the function f restricted to the set E.

7 If T is a bounded linear operator from a Banach space X to a Banach space Y ,we denote the operator norm of T by kT kX→Y ,orsimplybykTkif there is no confusion. P | | d For a multi-index α =(α1,...,αd), we write α = i=1 αi. The diﬀerentiation operator Dα and the multiplication operator Xβ are deﬁned by

Yd Yd α αi β βi D f(x)= (∂xi ) f(x), and X f(x)= xi f(x). i=1 i=1

S(Rd) is the Schwartz space of all infinitely differentiable functions f for which the seminorms X X α β kfk(M,N) = kD X fk∞ |α|≤M |β|≤N are finite for all non-negative integers M,N. Its topological dual, S0(Rd), is the space of tempered distributions.

More details on the basic properties of the Fourier transform and more generally,

on some of the theory from real and functional analysis that we will systematically

used in the sequel can be found in many standard analysis texts, e.g., [27], [48], [52].

8 CHAPTER II

GABOR FRAMES AND MODULATION

SPACES

2πiβn· 2 d A Gabor frame G(g,α,β)={e g(·−αk)}k,n∈Zd for L (R ) provides basis-like series representations of functions in L2, with unconditional convergence of the series.

However, unless the frame is a Riesz basis (and hence, by the Balian–Low theorem has poor time-frequency localization), these representations will not be unique. Still, a canonical and computable representation exists, and Gabor frames have found a wide variety of applications in mathematics, science, and engineering [14, 17, 18, 41].

An important fact is that Gabor frames provide much more than just a means to recognize square-integrability of functions. If the window function g is reasonably well-localized in time and frequency, then Gabor frame expansions are valid not only

2 p,q in L but in an entire range of associated spaces Mν known as the modulation spaces. The frame expansions converge unconditionally in the norm of those spaces,

p,q and membership of a tempered distribution in Mν is characterized by membership p,q of its sequence of Gabor coeﬃcients in a weighted sequence space `ν˜ . We refer to [41] for a recent detailed development of time-frequency analysis and modulation spaces.

In this chapter, we review some of the key results regarding Gabor expansions of

L2 functions. We then deﬁne the modulation spaces and their atomic decompositions by Gabor frames, which will be used often throughout this thesis.

2.1 Weight Functions

Before delving into Gabor analysis per se, we introduce here a class of weight functions that appear in most of the subsequent chapters.

9 2.1.1 Submultiplicative weights

A submultiplicative weight function ω is a positive, symmetric, and continuous func-

tion which satisﬁes ∀ x, y ∈ Rd,ω(x+y)≤ω(x)ω(y).

The prototypical example of a submultiplicative weight is the polynomially-growing

2 s/2 function ωs(x)=(1+|x|) ,wheres>0. We also consider weight functions deﬁned on R2d by making the obvious changes in the deﬁnition.

2.1.2 Moderate weights

A positive, symmetric, and continuous function ν is called ω-moderate function if there exists a constant Cν > 0 such that

d ∀ x, y ∈ R ,ν(x+y)≤Cνω(x)ν(y). (1)

If ν is ω-moderate with respect to ω = ωs, for some s>0, we say that ν is s-moderate.

t 2 s/2 For example, ν (x)=(1+|x|) is moderate with respect to ωs(x)=(1+|x|) , where s>0, exactly for |t|≤s.

If desired, the assumptions of continuity and symmetry of ω and ν could be

removed, but there would be no increase in the generality of the results. For if ω is

a positive, submultiplicative function, then there exists a continuous weight function

ω1 such that 0

If ν is ω-moderate, then by manipulating (1) we see that 1 1 ≤ C ω(x) , ν(x + y) ν ν(y) so 1/ν is also ω-moderate (with the same constant). Thus, the class of ω-moderate

p weights is closed under reciprocals, and consequently the class of spaces Lν using ω-moderate weights is closed under duality (with the usual exception for p = ∞).

This would not be the case if we restricted only to submultiplicative weights.

10 Given an ω-moderate weight ν on Rd, we will often use the notationν ˜ to denote the weight on Zd deﬁned byν ˜(k)=ν(αk), and for a weight ν on R2d we deﬁne

ν˜(k, n)=ν(αk, βn), orν ˜(k)=ν(k/β), the particular choice being clear from context,

2.2 Gabor frames in L2

Before deﬁning Gabor frames we ﬁrst introduce two operators that play important

roles in time-frequency analysis.

Deﬁnition 2.2.1. Given a, b ∈ Rd, the translation and modulation operators are

deﬁned respectively by

2πit·b Taf(t)=f(t−a),Mbf(t)=e f(t) for any function f deﬁned on Rd. Additionally, for c ∈ R,c=6 0 we deﬁne the dilation operator acting on a function

f deﬁned on Rd by

−d/2 Dcf (t)=|c| f(t/c).

It is easily seen that the translation, modulation, and dilation operators are unitary on L2, and that they map S and S0 isomorphically onto themselves. The following

lemma collects some basic facts about the translation and modulation operators.

The proof of the following lemma is immediate from the deﬁnition so we omit it.

Lemma 2.2.2. Let a, b ∈ Rd, c ∈ R,c =06 , and f be a function deﬁned on Rd.The following statements hold.

−2πia·b a. Ta Mb f = e Mb Taf.

b. DcTaf = TcaDcf.

c. D M f = M b D f. c b c c

d ˆ [ ˆ d. Ta f = M−a f, and Mb f = Tbf.

11 d e. D f = D 1 f.ˆ c c

\ 2πia·b ˆ f. Mb Ta f = e M−a Tb f.

We are now in position to deﬁne Gabor frames.

Deﬁnition 2.2.3. Given a window function g ∈ L2(Rd)andgivenα,β>0, we say that

2πiβn· G(g,α,β)={MβnTαkg}k,n∈Zd = {e g(·−αk)}k,n∈Zd

is a Gabor frame for L2(Rd) if there exist constants A, B>0 (called frame bounds) such that for all f ∈ L2(Rd), X k k2 ≤ |h i|2 ≤ k k2 A f L2 f,MβnTαkg B f L2 . (2) k,n∈Zd

We should point out that due to the commutation relations between the transla-

tion and modulation operators, it is trivial to see that the order of the translation and

modulation in the deﬁnition of a Gabor frame is irrelevant. Moreover, the image of a

Gabor frame G(g,α,β) under the Fourier transform is another Gabor frame, namely

G(ˆg, β, α).

Deﬁnition 2.2.4. Consider the collection of time-frequency shifts G(g,α,β) generated by g ∈ L2(Rd), and α, β > 0.

2 d a. The analysis operator associated with G(g,α,β) is the operator Cg : L (R ) → 2 Z2d h i ∈ 2 ` ( ) deﬁned by Cgf = f,MβnTαkg k,n∈Zd, for f L .

2 2d b. The synthesis operator associated with G(g,α,β) is the operator Rg : ` (Z ) → P 2 Rd d ∈ 2 L ( ) deﬁned formally by Rgc = k,n∈Zd ckn MβnTαkg, for c =(ck,n)k,n∈Z ` .

The basic properties of Gabor frames are laid out in the following result; we refer

to [14], [41], or [46] for more extensive treatments of frames and Gabor frames.

12 Theorem 2.2.5. Let G(g,α,β) be a Gabor frame for L2(Rd) with frame bounds A, B. Then the following statements hold. h i a. The analysis operator Cgf = f,MβnTαkg k,n∈Zd is a bounded mapping from 2 2 L into ` , and we have the norm equivalence kfk2 kCgfk`2. P b. The synthesis operator Rgc = k,n∈Zd ckn MβnTαkg is a bounded mapping from

2 2 2 ` into L . The series deﬁning Rgc converges unconditionally in L for every c ∈ `2.

∗ 2 → 2 c. Rg = Cg , and the frame operator Sg = RgCg : L L is strictly positive.

−1 G 2 Rd d. The dual window γ = Sg g generates a Gabor frame (γ,α,β) for L ( ) with frame bounds 1/B, 1/A.

2 d e. Rγ Cg = I on L (R ), i.e., we have the Gabor expansions X f = Rγ Cgf = hf,MβnTαkγi MβnTαkg (3) k,n∈Zd

for f ∈ L2(Rd), with unconditional convergence of the series.

2 2 Proof. a. The fact that Cg : L → ` is bounded follows from the second part of

(2); moreover, (2) is precisely the statement that kfk2 kCgfk`2.

2d 2 b. Let c =(ckn)k,n∈F ,whereF is a ﬁnite subset of Z .Forf∈L,wehave X |hRgc, fi| = | cknhMβnTαkg,fi| k,n∈F X ≤ |ckn||hf,MβnTαkgi| k,n∈F X 1/2 X 1/2 2 2 ≤ |ckn| |hf,MβnTαkgi| kn∈F kn∈Zd √ X 1/2 2 ≤ B kfkL2 |ckn| , kn∈F

13 where we have used the second inequality in (2). By duality we obtain

kRgckL2 =sup|hRgc, fi| kfkL2=1 √ X 1/2 2 ≤ B |ckn| kn∈F for all sequences with ﬁnite support. A standard density argument shows that

2 2 Rg is bounded from ` to L , and that the series deﬁning Rg converges unconditionally.

∈ 2 c. For f L and c =(ckn)k,n∈Zd,wehave X hRgc, fi = cknhf,MβnTαkgihc, Cgfi. k,n∈Zd

∗ 2 Hence, Rg = Cg . The frame operator Sg = RgCg is clearly bounded on L ;

moreover, the ﬁrst part of inequality (2) implies that Sg is strictly positive.

k k2 ≤h i≤ k k2 d. The frame inequality (2) can be rewritten as A f L2 Sgf,f B f L2 for

2 all f ∈ L , or equivalently in operator notation as AI ≤ Sg ≤ BI.Moreover, the above operator inequalities are preserved when multiplied by operators that commute with each of the terms appearing in the inequalities. Thus, we obtain

−1 ≤ −1 ≤ −1 that B I Sg A I. Moreover, by some easy computations, one can

show that Sg commutes with the translation and modulation operators Tαk and

−1 −1 Mβn, and so does Sg . Hence, Sg = Sγ , which together with the last operator inequality concludes the proof of this part.

e. Follows from the fact that f = SgSγf,andthatSg and Sγ commute with the

translation and modulation operators Tαk and Mβn.

In brief, if G(g,α,β) is a frame for L2(Rd) then the `2-norm of the sequence of

2 Gabor coeﬃcients (hf,MβnTαkgi)k,n∈Zd is an equivalent norm for L , and the Gabor

14 expansions given by (3) hold in L2. Moreover, for our purposes it is important to note that once the analysis and synthesis operators are deﬁned, the statement “Gabor

expansions converge in L2” is equivalent to the statement that the identity operator

2 on L factorizes as I = RγCg. In all these statements, and throughout this thesis, the roles of g and γ may be interchanged.

2.3 Modulation spaces 2.3.1 Deﬁnition and basic properties

The modulation spaces introduced by Feichtinger are spaces of tempered distributions deﬁned by imposing some decay condition on their short-time Fourier transforms,

which we next deﬁne.

Deﬁnition 2.3.1. The Short-Time Fourier Transform (STFT) of a function f ∈ L2

with respect to a window g ∈ L2 is Z −2πiy·t Vgf(x, y)=hf,MyTxgi = e g(t − x) f(t) dt. Rd

Remark 2.3.2. From the above definition, it is clear that the STFT can be defined whenever f and g are in dual spaces. In particular, the STFT is well-defined and scalar-valued when f ∈S0 and g ∈S. Moreover, analogously to the Fourier transform, the STFT extends in a distributional sense to f, g in the space of tempered distributions S0, cf. [26, Prop. 1.42].

The next proposition, whose proof is immediate and will be omitted, collects some

diﬀerent deﬁnitions of the STFT that will be used throughout this thesis.

15 Proposition 2.3.3. If f,g ∈ L2(Rd), then the following statements are true:

Vgf(x, y)=hf,MyTxgi

∧ = f · Txg (y) 2πix·y ∗ = e f ∗ Myg (x) ˆ = hf,TyM−xgˆi

−2πix·y ˆ = e hf,M−xTygˆi

−2πix·y ˆ = e Vgˆf(y,−x).

The next result sheds some light on the behavior of the STFT on L2.

Proposition 2.3.4. Let g ∈ L2(Rd), and assume that g =06 . Then for all f ∈ L2(Rd)

we have that

kVgfkL2 = kfkL2 kgkL2.

2 d 2 2d That is, Vg is a multiple of an isometry from L (R ) into L (R ).

k k2 k k2 = f L2 g L2 .

Thus kVgfkL2 = kfkL2 kgkL2 for all f ∈S, and a standard density argument extends the result to all f ∈ L2.

16 The previous proposition shows that on L2 the STFT is an isometry (up to a constant), and hence does not provide any new information beside the conservation

of the energy. However, by imposing other norms on the STFT, we can better quantify

the time-frequency concentration of functions. More precisely, we have the following

deﬁnition.

Deﬁnition 2.3.5. Let ν be an ω-moderate weight on R2d,andlet1≤p, q ≤∞.

∈S p,q Rd Given a window function g , the modulation space Mν ( )isthespaceofall distributions f ∈S0 for which the following norm is ﬁnite: Z Z q/p 1/q k k p,q | |p p k k p,q f Mν = Vgf(x, ξ) ν(x, ξ) dx dξ = Vgf Lν , (4) Rd Rd

with the usual modiﬁcations when p or q is inﬁnite.

For background and information on the basic properties of the modulation spaces

we refer to [21], [23], [24], [41]. The deﬁnition of the modulation space is independent

of the choice of the window g in the sense of equivalent norms. More precisely, the

following result whose prove may be found in [41, Proposition 11.3.1].

d Proposition 2.3.6. Assume that ν is ω-moderate and that g1,g2 ∈S(R)and that

gi g =06 when i =1,2.If1≤p, q ≤∞,letkfk p,q denote the norm of f in the i Mν p,q modulation space Mν as measured by the window gi when i =1,2. Then there exist

two constants C1,C2 >0 such that

1 g2 g1 g2 k k p,q ≤k k p,q ≤ k k 1 k k p,q C1 f f C2 Vg1 g2 Lω f . k k 1 Mν Mν Mν Vg2 g1 Lω

The next theorem collects some basic facts on the modulation spaces, its proof

may be found in [41].

Theorem 2.3.7. Let ν be an ω-moderate weight.

≤ ≤∞ p,q a. For 1 p, q ,Mν is a Banach space.

17 ∞ S p,q p,q b. If p, q < , is a dense subspace of Mν . Moreover, the dual of Mν is the p0,q0 modulation space M1/ν . More precisely we have that

k k p,q |h i| f Mν =sup f,g . kgkp0,q0 =1 M 1/ν

≤ ≤ p1,q1 ⊂ p2,q2 c. If p1 p2, and q1 q2, then Mν Mν .

∞ 1 p,q d. If p, q < , then Mω is a dense subspace of Mν .

p,p p Remark 2.3.8. a. If p = q we denote the modulation space Mν simply by Mν .

p,q p,q Moreover, if ν = 1 we simply denote Mν by M .

b. Among the modulation spaces are certain well-known spaces:

• if ν(x, ξ) = 1, and p = q = 2, it is easy to see that M 2 = L2, • | |2 s/2 2 2 if ν(x, ξ)=(1+ x ) where s>0, and p = q =2,thenMν = Ls,a weighted-L2 space,

• ||2 s/2 2 2 if ν(x, ξ)=(1+ ξ ) where s>0, and p = q =2,thenMν = Hs,the standard Sobolev space.

c. Lp for p =6 2 does not coincide with any modulation space [25].

d. The modulation M 1 has several properties that deserve to be mentioned. It is

a Banach algebra under both pointwise multiplication and convolution. It is

the smallest Banach space that is isometrically invariant under translation and modulation. Moreover, it is a Segal algebra known as the Feichtinger algebra,

and often denoted S0, and plays an important role in time-frequency analysis. We refer to [41] and the references therein for more detail on the Feichtinger

algebra and its weighted version.

The next result, whose proof can be found in [41, Proposition 11.3.1], provides a characterization of S and its dual S0 in terms of the modulation spaces.

18 2d Proposition 2.3.9. Let vs be the weight function deﬁned on R by vs(z)=(1+ |z|)s,z ∈R2d. Then we have \ [ S Rd ∞ S0 Rd ∞ ( )= Mvs and ( )= M1/vs . s≥0 s≥0

The next proposition, due to Feichtinger [21], on complex interpolation of modulation spaces will be used in the proof of our results in the following chapters. For more background on complex interpolation we refer to [8, Chapter 4].

Proposition 2.3.10. Let 1 ≤ p0 < ∞, 1 ≤ q0 < ∞, 1 ≤ p1 ≤∞,1≤q1 ≤∞, and θ ∈ , .If1 1−θ θ 1 1−θ θ (0 1) p= p0 +p1, and q = q0 + q1 then p0,q0 p1,q1 p,q M ,M [θ] = M . (5)

2.3.2 Gabor frames on modulation spaces

Under stronger assumptions on g, the expansions in (3) are valid not only in L2 but in the entire class of the modulation spaces.

The following result summarizes some basic facts on Gabor frames in the modulation spaces, cf. [41, Ch. 12]. The theorem is not stated in its weakest possible form; for example, the boundedness of the analysis and synthesis operators requires only

∈ 1 2 the assumption g Mω, and does not require that g generate a frame for L . Recall p,q that the mixed-norm sequence space `ν˜ consists of all sequences c =(ckn)k,n∈Zd such that X X q/p 1/q k k p,q | |p p ∞ c `ν˜ = ckn ν˜(k, n) < , n∈Zd k∈Zd whereν ˜(k, n)=ν(αk, βn), with the usual adjustments when p = ∞ or q = ∞.

Theorem 2.3.11. Let ν be an ω-moderate weight on R2d, and let 1 ≤ p, q ≤∞.

∈ 1 G 2 Rd Let g Mω be such that (g,α,β) is a Gabor frame for L ( ). Then the following statements hold.

19 h i a. The analysis operator deﬁned by Cgf = f,MβnTαkg k,n∈Zd is a bounded map- p,q p,q ping from Mν to `ν˜ , and we have the norm equivalence

k k p,q k kp,q f Mν Cgf `ν˜ .

P b. The synthesis operator deﬁned by Rgc = k,n∈Zd ckn MβnTαkg is a bounded map-

p,q p,q ping from `ν˜ to Mν . The series deﬁning Rgc converges unconditionally in the p,q ∈ p,q ∞,∞ ∞ norm of Mν for every c `ν˜ (weak* unconditionally in M1/ω if p = or q = ∞).

p,q c. The frame operator Sg = RgCg is a continuously invertible mapping of Mν onto itself.

−1 1 d. The dual window γ = Sg g lies in Mω.

p,q e. Rγ Cg = I on Mν , i.e., we have the Gabor expansions X f = Rγ Cgf = hf,MβnTαkγi MβnTαkg (6) k,n∈Zd

∈ p,q ∞ for f Mν , with unconditional convergence of the series if p, q < , and with unconditional weak* convergence otherwise.

∈ ∞,∞ p,q ∈ p,q ∈S f. A distribution f Mν belongs to Mν if and only if Cgf `ν˜ .Ifg , ∈S0 Rd p,q ∈ p,q then a tempered distribution f ( )belongs to Mν if and only if Cgf `ν˜ .

p,q h i d In brief, the `ν˜ norm of the Gabor coeﬃcients ( f,MβnTαkg )k,n∈Z is an equiva-

p,q p,q lent norm for Mν , and the Gabor expansions (3) are valid in Mν with unconditional

p,q convergence of that series in the norm of Mν . Moreover, there is a strong statement made in part f of Theorem 2.3.11 that is not usually observed in the standard list of

2 k k p,q Gabor frame properties in L (Theorem 2.2.5), namely that Cgf `ν˜ is not only an

p,q equivalent norm for Mν , but membership of f in the modulation space is character- p,q ized by membership of its sequence of Gabor coeﬃcients Cgf in `ν˜ . In particular,

20 only the magnitude of these coeﬃcients is important in determining whether a given

p,q distribution lies in Mν . The proof of Theorem 2.3.11 requires deep analysis. In particular, the invertibility

1 of Sg on Mω for arbitrary values of α, β was only recently proved in [43]. In summary, once the analysis and synthesis operators have been correctly deﬁned, the fact that Gabor expansions converge in the modulation spaces is simply the

p,q statement that the identity operator on Mν factorizes as I = RγCg.

21 CHAPTER III

GABOR ANALYSIS IN WEIGHTED

AMALGAM SPACES

Some results on Gabor analysis outside of the modulation spaces were obtained by

Walnut in [59]. In particular, he introduced what is now known as the Walnut repre-

sentation of the frame operator, and considered the boundedness of the frame operator on Lp. Recently, it was independently observed in [34] and [38] that Gabor expansions actually converge in Lp(Rd)when1

when p =6 2, it was known that Gabor expansions could not converge unconditionally

in Lp [25].

In this chapter we consider a much larger class of spaces than the Lp spaces,

p q namely, we consider the weighted amalgam spaces W (L ,Lν). These spaces amalgamate a local criteria for membership with a global criteria. We will show that not

only do Gabor expansions converge for the special case Lp = W (Lp,Lp), but that

they converge in the entire range of weighted amalgam spaces. Moreover, member-

ship in the amalgam space is characterized by membership of the Gabor coeﬃcients

in an appropriate sequence space. In the course of obtaining these results, we prove

several results of independent interest on the behavior of the analysis and synthesis

operators associated with the Gabor frame, and on the Walnut representation, which is an extremely useful tool in Gabor frame theory. Moreover, we include the cases

p =1,∞or q =1,∞in our consideration. In particular, we show that Gabor ex-

pansions exist even in L1 and in a weak sense in L∞, given the right interpretation

of “expansion.” Additionally, we obtain some necessary conditions on the window

22 g, extending weaker necessary conditions obtained by Balan in [2] for the particular case W (L2,L∞).

3.1 Weighted amalgam spaces

3.1.1 Deﬁnition

Given an ω-moderate weight ν on Rd and given 1 ≤ p, q ≤∞, the weighted amalgam

p q Rd space W (L ,Lν) is the Banach space of all measurable functions on for which the

norm X 1/q q q k k p q k · χ k f W (L ,Lν ) = f Tαk Qα p ν(αk) (7) k∈Zd is ﬁnite, with the usual adjustment if q = ∞.

The ﬁrst use of amalgam spaces was by Wiener, who introduced the spaces

W (L1,L2)andW(L2,L1)in[60]andW(L∞,L1)andW(L1,L∞) in [61], [62], in

connection with his development of the theory of generalized harmonic analysis. The

space W (L∞,L1) is sometimes called the Wiener algebra (although this term is sometimes used to denote FL1), cf. [51]. It was shown in [59] that W (L∞,L1) is a convenient and general class of windows for Gabor analysis within L2.

d Since any cube Qα in R can be covered by a ﬁnite number of translates of a

p q cube Qβ,thespaceW(L,Lν) is independent of the value of α used in (7) in the

p q sense that each diﬀerent choice of α yields an equivalent norm for W (L ,Lν). A wide variety of other equivalent norms is provided by Feichtinger’s theory of amalgam

spaces [20, 19, 22]. We refer to [44] for an exposition of the “continuous” norms on the amalgam spaces.

The following lemma provides some useful inclusions among the amalgam spaces.

Lemma 3.1.1. For each ω-moderate weight ν, we have the following inclusion relations: if p1 ≥ p2, and q1 ≤ q2, then

p1 q1 ⊂ p1 q1 ⊂ p2 q2 ⊂ p2 q2 W (L ,Lω ) W(L ,Lν ) W(L ,Lν ) W(L ,L1/ω).

23 ∞ 1 ⊂ p q ⊂ 1 ∞ In particular, the inclusions W (L ,Lω) W(L ,Lν) W(L ,L1/ω) hold for all ≤ ≤∞ ∞ 1 1 p, q and all ω-moderate weights ν. In this sense W (L ,Lω) is the smallest

1 ∞ and W (L ,L1/ω) is the largest amalgam space in the class of amalgam spaces with ω-moderate weight functions.

Proof. The fact that ν is ω-moderate implies in particular that ν(x) ≤ Cω(x) for some positive constant C (this follows immediately from (1). Hence the inclusion

p1 q1 ⊂ p1 q1 W (L ,Lω ) W(L ,Lν ) follows from the (7).

∈ p1 q1 Now let f W (L ,Lν ). Then X 1/q2 q2 q2 k k p q2 k · χ k f W (L 2 ,Lν ) = f Tαk Qα p2 ν(αk) k∈Zd X 1/q2 ≤ k · χ kq2 q2 C f Tαk Qα p1 ν(αk) k∈Zd X 1/q1 ≤ k · χ kq1 q1 C f Tαk Qα p1 ν(αk) , k∈Zd

q1 d q2 d where we have used the inclusions ` (Z ) ⊂ ` (Z ) (because q1 ≤ q2), as well as

p1 p2 d L (K) ⊂ L (K) for p1 ≥ p2 and K a compact subset of R .Thusweobtain

p1 q1 ⊂ p2 q2 W(L,Lν ) W(L ,Lν ).

p2 q2 ⊂ p2 q2 The last inclusion, W (L ,Lν ) W(L ,L1/ω) follows again from the fact that ν 1 ≤ ∈ Rd is ω-moderate, and so is 1/ν,andso ν(x) Cω(x) for all x .

The last part of the lemma is just an application of the above with p1 = ∞,p2 =

p, q1 =1,and q2 = q.

Remark 3.1.2. For p, q<∞, the Schwartz class S and the space of functions with

p q compact support are dense in W (L ,Lν).

24 3.1.2 Duality and convergence

We will need to be precise about the meaning of convergence of series. For general

references we refer to the text of Singer [54], and for references on Banach function spaces we refer to the text of Bennett and Sharpley [5].

The following lemma characterizing unconditional convergence will be useful.

∗ Lemma 3.1.3. Let X be a Banach space with dual space X , and let fk ∈ X for k ∈ J. Then the following statements are equivalent. P a. k∈J fk converges unconditionally in X, i.e., it converges with respect to every ordering of the index set J.

b. There exists f ∈ X such that for each ε>0, there exists a ﬁnite F0 ⊂ J such that

X ∀ ﬁnite F ⊃ F0, f − fk <ε. X k∈F

c. For every ε>0, there exists a ﬁnite F0 ⊂ J such that X ∗ ∀ ﬁnite F ⊃ F0, sup |hfk,hi| : h ∈ X , khkX∗ =1 <ε. k/∈F

Now let X be a Banach function space in the sense of [5]. In particular, this

p q includes the amalgam spaces W (L ,Lν). The K¨othe dual of X (or the associated space, as it is called in [5]), is the space X˜ consisting of all measurable functions

h such that fh ∈ L1 for each f ∈ X. By [5, Thm. 1.2.9], X˜ is a closed, norm-

fundamental subspace of X∗, so in particular, ∀ ∈ k k |h i| ∈ ˜ k k f X, f X =sup f,h : h X, h X˜ =1 .

By [5, Cor. 1.5.3], X is complete in the σ(X, X˜) topology, i.e., the weak topology on X P ˜ ˜ generated by X. In particular, a series k∈J fk converges in the σ(X, X) topology if P h i ∈ ˜ k∈J fk,h converges for each h X. It converges unconditionally in that topology

25 if the convergence is independent of the ordering of J, and since the terms hfk,hiare scalars, this occurs if and only if X ∀ h ∈ X,˜ |hfk,hi| < ∞. k∈J

Remark 3.1.4. a. If X1 and X2 are two Banach function spaces such that X1 ⊂ ˜ ⊂ ˜ ∈ ˜ ∈ k k X2 then X2 X1. Indeed, let f X2 and g X1 with g X1 = 1, then by the

1 deﬁnition of the K¨othe dual, we have that fg¯ ∈ L , which implies that f ∈ X˜1, and, moreover, Z

|hf,gi| = f(x) g(x) dx Rd

≤k k k k f X˜2 g X1

≤k k f X˜2.

Thus, k k ≤k k f X˜1 f X˜2. (8)

b. For every Banach function space X we have X˜ = X, cf. [5, Theorem 2.7].

The dual and K¨othe dual of the amalgam spaces are given in the next lemma.

Lemma 3.1.5. Let ν be an ω-moderate weight.

0 ≤ ∞ p q p0 q a. For 1 p, q < , the dual space of W (L ,Lν) is W (L ,L1/ν ).

0 ≤ ≤∞ p q p0 q b. For 1 p, q ,theK¨othe dual of W (L ,Lν) is W (L ,L1/ν ).

Proof. a. We refer to [28, 19] for the proof of this part.

b. If 1 ≤ p, q < ∞, the result follows from part a. Now assume that p = ∞ or

q = ∞. We divide the proof in three parts.

26 ≤ ∞ ∞ ∈ p0 1 ∈ p ∞ Case I: 1 p< and q = .Letf W(L ,L1/ν ), and g W (L ,Lν )with k k p ∞ g W(L ,Lν ) =1.Thenwehave Z

|hf,gi| = f(x) g(x) dx d Z R ≤ |f(x)||g(x)|dx Rd X Z = |f(x)||g(x)|dx αk+Qα k∈Zd X ≤ k · k 0 k · k f TαkχQα p g TαkχQα p k∈Zd X 1 = kf · T χ k 0 ν(αk) kg · T χ k αk Qα p ν(αk) αk Qα p k∈Zd X 1 ≤ k · k k · k 0 sup g TαkχQα p ν(αk) f TαkχQα p k∈Zd ν(αk) k∈Zd

= kgk p ∞ kfk p0 1 W (L ,Lν ) W (L ,L1/ν )

= kfk p0 1 . W (L ,L1/ν )

Thus,

kfk ˜ p ∞ ≤kfk p0 1 , W (L ,Lν ) W(L ,L1/ν )

p0 1 ⊂ ˜ p ∞ and consequently, W (L ,L1/ν ) W (L ,Lν ). From part a and (8), as well as Remark 3.1.4, we obtain the reverse inclusion, which completes the proof in this case.

Case II: if p = ∞ and 1 ≤ q<∞then the proof is very similar to the above so we omit it.

∞ 1 ⊂ ∞ ˜ Case III: if p = q = ,weeasilyseethatL1/ν (Lν ), and the proof follows from the same arguments as above.

27 3.1.3 Sequence spaces

Before stating our results, we must deﬁne the sequence spaces that will be associated

with Gabor expansions in the amalgam spaces. We begin by recalling that the Fourier

1 ˆ transform of f ∈ L (Q1/β ) is the sequence f deﬁned by Z fˆ(n)=Ff(n)=βd f(t) e−2πiβn·t dt, n ∈ Zd. Q1/β

p p For 1 ≤ p ≤∞,letFL(Q1/β ) denote the image of L (Q1/β ) under the Fourier

p p transform. Since Fourier coeﬃcients are unique in L ,ifc=(cn)n∈Zd ∈FL(Q1/β )

p then there exists a unique function m ∈ L (Q1/β ) such thatm ˆ (n)=cn for every n,

p and the norm on FL (Q1/β ) is deﬁned by

k k p k k c FL (Q1/β) = m p,Q1/β . (9)

For 1

for (9), cf. [16, Ch. 7]. The ongoing development motivates the following deﬁnition.

p,q q F p Deﬁnition 3.1.6. Let α, β>0begiven.ThenSν˜ = `ν˜( L (Q1/β )) will denote F p q the space of all L (Q1/β )-valued sequences which are `ν˜-summable. That is, a p,q ∈ Zd doubly-indexed sequence c =(ckn)k,n∈Zd lies in Sν˜ if for each k there exists

p mk ∈ L (Q1/β ) such that

d mˆ k(n)=ckn,k,n∈Z,

and such that X 1/q q k k p,q k k q ∞ c Sν˜ = mk p,Q1/β ν˜(k) < , k∈Zd with the usual change if q = ∞.

When 1

28 in the sense that the square partial sums of (10) converge to mk in the norm of

p L (Q1/β ), cf. [48], [63]. Hence, for 1

p,q on Sν˜ as Z X X p q/p 1/q · k k p,q 2πiβn x q c Sν˜ = ckn e dx ν˜(k) . Q k∈Zd 1/β n∈Zd

Remark 3.1.7. A Banach function space X is called a solid space if f ∈ X and

|g|≤|f|implies that g ∈ X and, moreover, kgk≤kfk.

2,q 2,q Note that for p = 2, we have via the Plancherel theorem that Sν˜ = `ν˜ ,thusis 6 p,q a solid space. However, for general p =2,Sν˜ is not a solid space. In particular, changing the phases of the ckn can change the norm of c.

3.2 Boundedness of the analysis and synthesis operators

In this section, we prove the boundedness of the analysis and synthesis operators on

the amalgam spaces. Moreover, we show that the Walnut representation, which is

an extremely useful tool in Gabor analysis, holds on the amalgam spaces. However,

before presenting these results, we give here some Lemmas that will be needed in the sequel.

3.2.1 Lemmas

The following lemmas will be important in the sequel. The ﬁrst lemma is simply a

counting argument.

1 Zd Lemma 3.2.1. Let α, β>0be given. Let Kαβ be the maximum number of β - d translates of Q1/β required to cover any αZ -translate of Qα, i.e., ∈Zd ` ∩ Kαβ =max# ` : (β +Q1/β ) (αk + Qα) > 0 . k∈Zd

29 p Then given 1 ≤ p ≤∞, we have for any 1/β-periodic function m ∈ L (Q1/β ) and any k ∈ Zd that k k ≤ 1/p k k m p,αk+Qα Kαβ m p,Q1/β ,

1/∞ where Kαβ =1.

p Proof. Let m be a 1/β-periodic function in L (Q1/β ), where 1 ≤ p<∞. For any ∈ Zd { ∈ Zd | l ∩ | } k deﬁne Ak = l : ( β + Q1/β ) (αk + Qα) > 0 .Thenwehave: Z k kp | |p mp,αk+Qα = m(x) dx αk+Qα X Z | |p = m(x) TαkχQα (x) dx l +Q l∈Zd β 1/β X Z | |p = m(x) TαkχQα (x) dx l +Q1/β l∈Ak β X Z ≤ |m(x)|p dx l +Q1/β l∈Ak β X Z = |m(x)|p dx Q1/β l∈Ak Z p ≤ Kα,β |m(x)| dx Q1/β

k kp = Kα,β m p,Q1/β .

If p = ∞, then it is easily seen that

k k ≤k k m ∞,αk+Qα m ∞,Q1/β .

The second lemma is a weighted version of an estimate that is useful in the Walnut representation of the Gabor frame operator on L2, see [59, Lemma 2.2].

Lemma 3.2.2. Let ω be a submultiplicative weight, and let α, β>0be given. Then

∈ ∞ 1 there exists a constant C = C(α, β, ω) > 0 such that if g, γ W (L ,Lω) and the

30 functions Gn are deﬁned by (22), then X n k k∞ ≤ k k ∞ 1 k k ∞ 1 Gn ω( β ) C g W (L ,Lω) γ W (L ,Lω). n∈Zd

Proof. It follows from the fact that ω is ω-moderate that kfωkW(L∞,L1) is an equiv- ∞ 1 ∈ ∞ 1 alent norm for W (L ,Lω). In particular, we have gω, γω W(L ,L ), so by [41, Lemma 6.3.1], X d d k ˜ k ≤ 1 k k ∞ 1 k k ∞ 1 Gn ∞ α +1 (2β +1) gω W(L ,L ) γω W(L ,L ), n∈Zd

where G˜n is the analogue of Gn with g replaced by |g|ω and γ replaced by |γ|ω. Hence,

X X X k k n − n − − × Gn ∞ ω( β )= ess sup g(x β αk) γ(x αk) ∈Rd ∈Zd ∈Zd x ∈Zd n n k − − − n − ω (x αk) (x β αk) X X ≤ | − n − | − n − × ess sup g(x β αk) ω(x β αk) x∈Rd n∈Zd k∈Zd

|γ(x − αk)| ω(x − αk) X = kG˜nk∞ n∈Zd

≤ k k ∞ 1 k k ∞ 1 C g W (L ,Lω) γ W (L ,Lω).

Finally, we need an estimate on the eﬀect of translations on the amalgam space

norm.

Lemma 3.2.3. Let ν be an ω-moderate weight. Then for 1 ≤ p, q ≤∞, we have for

∈ p q ∈ Zd each f W (L ,Lν) and ` that

k k p q ≤ k k p q Tα`f W (L ,Lν ) Cν ω(α`) f W (L ,Lν ).

31 ∈ p q ∈ Zd Proof. Let f W (L ,Lν), and l . Then using the fact that ν is ω-moderate, we obtain X 1/q q q k k p q k · χ k Tαlf W (L ,Lν ) = Tαlf Tαk Qα p ν(αk) k∈Zd X 1/q k · kq q = f Tα(k−l)χQα p ν(αk) k∈Zd X 1/q k · kq q = f TαkχQα p ν(α(k + l)) k∈Zd X 1/q ≤ k · kq q Cω(αl) f TαkχQα p ν(αk) k∈Zd

k k p q = Cω(αl) f W (L ,Lν ).

3.2.2 Boundedness of the synthesis operator

Theorem 3.2.4. Let ν be an ω-moderate weight on Rd.Letα,β>0and 1 ≤ p, q ≤

∞ ∈ ∞ 1 be given. Fix g, γ W (L ,Lω). Then the following statement is true. Given ∈ p,q ∈ p c Sν˜ ,letmk L(Qα)be the unique functions satisfying mˆ k(n)=ckn for all k, n ∈ Zd. Then the series X Rgc = mk · Tαkg (11) k∈Zd p q converges unconditionally in W (L ,Lν) (unconditionally in the 0 p q p0 q ∞ ∞ σ(W (L ,Lν),W(L ,L1/ν )) topology if p = or q = ), and Rg is a bounded p,q p q mapping from Sν˜ into W (L ,Lν).

Proof. We divide the proof into cases. First, we consider the case 1 ≤ p, q < ∞.We ∈ p,q are given c Sν˜ , and we must prove that the series (11) deﬁning Rgc converges

p q unconditionally in the norm of W (L ,Lν), and that Rg so deﬁned is a bounded

32 p,q p q mapping from Sν˜ into W (L ,Lν) . To show the convergence we will make use of Lemma 3.1.3.

p,q Fix ε>0. Then, by deﬁnition of the norm in Sν˜ ,wehavethat X k kq q ∞ mk p,Q1/β ν˜(k) < .

Hence there exists a ﬁnite set F0 such that X ∀ ⊃ k kq q q ﬁnite F F0, mk p,Q1/β ν˜(k) <ε. (12) k/∈F

Recall that 1/ν is an ω-moderate weight, and let Kαβ be the constant appearing in

0 ∈ p0 q Lemma 3.2.1. Fix any h W (L ,L1/ν ). Then Z X X

mk ·Tαkg, h ≤ |mk(x)Tαkg(x) h(x)| dx Rd k/∈F k/∈F X X Z | | = mk(x)Tαkg(x) h(x) Tαn+αkχQα (x) dx Qα k/∈F n∈Zd X X ≤ k · k k k × Tαkg Tαn+αkχQα ∞ mk p,αn+αk+Qα k/∈F n∈Zd ν(αk) kh · T χ k 0 αn+αk Qα p ν(αn + αk − αn) X ≤ k · k × g TαnχQα ∞ n∈Zd X 1/p Cν ν(αk) ω(αn) K km k kh · T χ k 0 αβ k p,Q1/β αn+αk Qα p ν(αn + αk) k/∈F X ≤ 1/p k · χ k × CνKαβ g Tαn Qα ∞ ω(αn) n∈Zd X 1/q k kq q × mk p,Q1/β ν(αk) k/∈F 0 X 1/q q0 1 kh · T χ k 0 . (13) αn+αk Qα p ν(αn + αk)q0 k∈Zd Combining (12) and (13), we have that X 1/p · ≤ k k ∞ 1 k k 0 q0 mk Tαkg, h εCνKαβ g W (L ,Lω) h p . W (L ,L1/ν ) k/∈F

33 Therefore, taking the supremum over all h of unit norm and appealing to Lemma 3.1.3, P we see that Rg c = mk · Tαkg converges unconditionally. Further, replacing F by Zd in the calculation in (13) yields X

|hRgc, hi| ≤ mk · Tαkg, h k∈Zd

1/p ≤ k k ∞ 1 k k p,q k k 0 q0 CνKαβ g W (L ,Lω) c Sν˜ h p . (14) W (L ,L1/ν )

0 p0 q p q Since W (L ,L1/ν ) is the dual space of W (L ,Lν), taking the suprema over all h of unit norm in (14) shows that k k p q |h i| k k 0 Rgc W (L ,Lν ) =sup Rgc, h : h p0 q =1 W (L ,L1/ν )

1/p ≤ k k ∞ 1 k k p,q CνKαβ g W (L ,Lω) c Sν˜ , (15)

so Rg is bounded. This completes the proof for the case 1 ≤ p, q < ∞.

0 ∞ ∞ p0 q When p = or q = , we make use of the fact that W (L ,L1/ν)istheKöthe dual p q of W (L ,Lν). The fact that the series defining Rgc converges in the weak topology is given by the same calculations as in (13), (14), and the fact that the Köthe dual is a norm-fundamental subspace of the dual space means that we can again estimate k k p,q Rgc Sν˜ by using (15). Hence Rg is bounded, and the proof is complete.

Remark 3.2.5. When 1

as Fourier series, allowing Rgc to be written as the iterated sum XX 2πiβn·x Rgc(x)= ckn e Tαkg(x), (16) k∈Zd n∈Zd i.e., the same series as appears in the Gabor expansions in (3), or more generally the

Gabor expansions in modulation spaces (6). When p =1orp=∞,thisisnotthe

case. The functions mk are still uniquely determined by c, but cannot be written as Fourier series. When p = q = 2, both the inner and outer sums in the iterated series in (16) converge unconditionally, and then Rgc can also be written as the double sum given by (6), with unconditional convergence of that series.

34 3.2.3 Boundedness of the analysis operator

Theorem 3.2.6. Let ν be an ω-moderate weight on Rd.Letα,β>0and 1 ≤ p, q ≤

∞ ∈ ∞ 1 be given. Fix g, γ W (L ,Lω). Then the analysis operator deﬁned by Cgf = h i p q p,q f,MβnTαkg k,n∈Zd is a bounded mapping from W (L ,Lν)into Sν˜ , Moreover, there p exist unique functions mk ∈ L (Q1/β ) which satisfy mˆ k(n)=Cgf(k, n) for all k, n ∈ Zd, and these are given explicitly by X −d · − n mk(x)=β f Tαkg¯ (x β ) n∈Zd X − d n · n = β T β f Tαk+ β g¯ (x). (17) n∈Zd

p The series on the right side of (17) converges unconditionally in L (Q1/β )(uncondi-

∞ 1 tionally in the σ(L (Q1/β ),L(Q1/β ) topology if p = ∞).

∈ ∞ 1 ≤ ≤∞ ∈ p q Proof. We are given that g W (L ,Lω)andthat1 p, q .Letf W(L,Lν),

1 ∞ which is a subspace of W (L ,L1/ω). First we must show that the functions mk given

by (17) are well-deﬁned. Since mk is the 1/β-periodization of the integrable function

1 f · Tαkg, the series deﬁning mk converges at least in L (Q1/β ). To show that the

p periodization converges unconditionally in L (Q1/β ) (weakly if p = ∞) and to derive

p0 a useful estimate, ﬁx any 1/β-periodic function h ∈ L (Q1/β ). Then for each ﬁxed k,wehave Z X − n − n f(x β ) Tαkg(x β ) h(x) dx Q 1/β n∈Zd Z

≤ |f(x) Tαkg(x) h(x)| dx Rd X Z | | = f(x) Tαkg(x) h(x) Tαk+αnχQα (x) dx Qα n∈Zd X ≤ k · k k · k × Tαkg Tαk+αnχQα ∞ f Tαk+αnχQα p n∈Zd

35 ν(αk + αn − αn) khk 0 p ,αk+αn+Qα ν(αk)

X 0 ≤ k · k k · k 1/p × g TαnχQα ∞ f Tαk+αnχQα p Kαβ n∈Zd

Cνν(αk + αn) ω(αn) khk 0 p ,Q1/β ν(αk) X 1/p0 1 = C K khk 0 kg · T χ k∞ ω(αn) × ν αβ p ,Q1/β ν(αk) αn Qα n∈Zd k · k f Tαk+αnχQα p ν(αk + αn). (18)

This yields the desired convergence, and taking the suprema in (18) over h with unit norm implies the estimate

0 P −d 1/p 1 k k ≤ d k · χ k∞ × mk p,Q1/β β CνKαβ ν(αk) n∈Z g Tαn Qα ω(αn) k · k f Tαk+αnχQα p ν(αk + αn). (19)

2πiβn·x p0 Second, we show thatm ˆ k(n) has the correct form. Since e ∈ L (Q1/β ), we

have by the weak convergence of the series deﬁning mk that

d 2πiβn·x mˆ k(n)=β mk,e Z X 2πiβn·x = T ` f · T ` g,¯ e β αk+ β Q `∈Zd 1/β X Z − ` − ` −2πiβn·(x−`/β) = f(x β ) Tαkg¯(x β ) e dx Q `Z∈Zd 1/β −2πiβn·x = f · Tαkg¯ (x) e dx Rd

= f, MβnTαkg

= Cgf(k, n).

p q p,q Finally, we must show that Cg is a bounded mapping of W (L ,Lν)intoSν˜ . ∈ p q ∈ p,q Given f W (L ,Lν), to show that Cgf Sν˜ we must show that the sequence r given by k k ∈Zd r(k)= mk p,Q1/β ,k ,

36 q ∈ q0 lies in `ν˜. To do this, ﬁx any sequence a `1/ν˜. Then, using (19), we have X |h i| ≤ k k | | r, a mk p,Q1/β a(k) k∈Zd

0 X ≤ −d 1/p k · χ k × β CνKαβ g Tαn Qα ∞ ω(αn) n∈Zd X 1 kf · T χ k ν(αk + αn) |a(k)| αk+αn Qα p ν(αk) k∈Zd

0 X ≤ −d 1/p k · χ k × β CνKαβ g Tαn Qα ∞ ω(αn) n∈Zd X 1/q k · kq q × f Tαk+αnχQα p ν(αk + αn) k∈Zd 0 X 1/q q0 1 |a(k)| 0 ν(αk)q k∈Zd

0 −d 1/p ≤ k k ∞ 1 k k p q k k q0 β CνKαβ g W (L ,Lω) f W (L ,Lν ) a . (20) `1/ν˜

q0 q ∗ ∞ ∞ Since `1/ν˜ equals (`ν˜) when q< and is a norm-fundamental subspace when q = , taking the suprema in (20) over sequences a with unit norm yields the estimate

0 −d 1/p k k p,q k k p ≤ k k ∞ 1 k k p q Cgf Sν˜ = r `ν˜ β CνKαβ g W (L ,Lω) f W (L ,Lν ).

p q p,q Hence Cg is a bounded mapping of W (L ,Lν)intoSν˜ .

Remark 3.2.7. For the case 1

0 0 0 p q → p,q p ,q → p0 q by proving that Cg : W (L ,Lν) Sν˜ is the adjoint of Rg : S1/ν˜ W (L ,L1/ν ), p q and then using the reﬂexivity of the space W (L ,Lν)andthefactthat1/ν is also ω-moderate.

37 3.2.4 The Walnut representation of the Gabor frame operator on amalgam spaces

Theorem 3.2.8. Let ν be an ω-moderate weight on Rd.Letα,β>0and 1 ≤ p, q ≤

∞ ∈ ∞ 1 be given. Fix g, γ W (L ,Lω). Then the Walnut representation X −d Rγ Cgf = β Gn · T n f (21) β n∈Zd

∈ p q holds for f W (L ,Lν), with the series on the right of (21) converging absolutely in

p q W (L ,Lν), and where X −n− − Gn(x)= g(x β αk) γ(x αk) ∈Zd kX n · = Tαk+ g¯ Tαkγ (x). (22) β k∈Zd

∈ ∞ 1 ≤ ≤∞ Proof. We are given g, γ W (L ,Lω)and1 p, q . For this proof, let us use

p q the equivalent norm for W (L ,Lν) obtained by replacing α in (7) by 1/β.Thenby Lemma 3.2.3,

n k n k p q ≤ k k p q T β f W (L ,Lν ) Cν ω( β ) f W (L ,Lν ).

Therefore, using the autocorrelation functions Gn deﬁned in (22), we have for f ∈

p q W (L ,Lν)that X X k · n k p q ≤ k k∞ k n k p q Gn Tβf W(L ,Lν ) Gn T β f W (L ,Lν ) n∈Zd n∈Zd X n ≤ k k p q k k Cν f W (L ,Lν ) Gn ∞ ω( β ) n∈Zd

≤ k k p q k k ∞ 1 k k ∞ 1 Cν f W (L ,Lν ) g W (L ,Lω) γ W (L ,Lω), P · n the last inequality following from Lemma 3.2.2. Hence the series Gn T β f converges

p q absolutely in W (L ,Lν). ∈ p q ∈ p,q Now ﬁx f W (L ,Lν). Then Cgf Sν˜ by Theorem 3.2.6. Letting mk be P deﬁned by (17), we have Cgf(k, n)=m ˆk(n). Further, RγCgf = mk · Tαkγ,this

38 series converging unconditionally if p, q<∞, or unconditionally in the weak topology

0 ∈ p0 q otherwise. In any case, for h W (L ,L1/ν˜)wehave X RγCgf, h = mk · Tαkγ, h k∈Zd X Z ¯ = mk(x)Tαkγ(x) h(x) dx Rd k∈Zd X Z X − d n n ¯ = β T β f(x) Tαk+ β g¯(x) Tαkγ(x) h(x) dx Rd k∈Zd n∈Zd X Z X − d n n ¯ = β T β f(x) Tαk+ β g¯(x) Tαkγ(x) h(x) dx Rd n∈Zd k∈Zd X Z − d n ¯ = β T β f(x) Gn(x) h(x) dx. Rd n∈Zd X − d · n = β Gn T β f, h , n∈Zd from which (21) follows. The interchanges of integration and summation can be justiﬁed by Lemma 3.2.2 and Fubini’s Theorem.

3.3 Gabor expansions in the amalgam spaces

Under the assumption that G(g,α,β) is a frame for L2(Rd), we obtain the following result, which makes precise the characterization of the amalgam spaces in terms of

Gabor frames. In particular, we show in this section that there is an analogue for the amalgam spaces of the Gabor expansions of functions in the modulation spaces

(see Theorem 2.3.11). This is surprising, because the modulation spaces are the natural setting for Gabor analysis. And indeed, while Gabor expansions converge unconditionally in the modulation spaces, the convergence in the amalgam spaces is conditional in general and even the meaning of the term “expansion” must be handled appropriately. Throughout, we will use the notationν ˜(k)=ν(αk).

39 Theorem 3.3.1. Let ν be an ω-moderate weight on Rd, and let α, β>0and 1 ≤

≤∞ ∈ ∞ 1 G p, q be given. Assume that g, γ W (L ,Lω)are such that (g,α,β) is a Gabor frame for L2 with dual frame G(γ,α,β). Then the following statements hold.

p q a. RγCg = I on W (L ,Lν).

k k p q k k p,q b. We have the norm equivalence f W (L ,Lν ) Cgf Sν˜ .

∈ 1 ∞ p q ∈ p,q c. A function f W (L ,L1/ω) belongs to W (L ,Lν) if and only if Cgf Sν˜ .

∈ ∞ 1 G 2 Proof. We are given g, γ W (L ,Lω) such that (g,α,β) is a Gabor frame for L p q → and γ is the dual window to g. By Theorem 3.2.6, we have that Cg, Cγ : W (L ,Lν) p,q p,q → p q ≤ ≤∞ Sν˜ and Rg, Rγ : Sν˜ W (L ,Lν) are bounded mappings for each 1 p, q and each ω-moderate weight ν. Further, for the case p = q =2andν=1,the

2 frame hypothesis implies that the identity RγCg = I holds on L , and the deﬁnition

of Rγ given in Chapter 2 coincides in this case with the deﬁnition of Rγ given in

Theorem 3.2.4. Letting Gn be the autocorrelation functions deﬁned in (22), the fact

2 that RγCg = I holds on L implies by [41, Thm. 7.3.1] that

−d β G0 =1a.e. and Gn =0a.e.for n =06 .

p q Consequently, using the Walnut representation (21) of Rγ Cg on the space W (L ,Lν), ∈ p q we have for f W (L ,Lν)that X − d · n RγCgf = β Gn Tβf = f. n∈Zd

p q Hence Rγ Cg = I holds on W (L ,Lν) as well. This proves part a of Theorem 3.3.1. ∈ p q Next, given f W (L ,Lν), we have

k k p q k k p q f W (L ,Lν ) = RγCgf W (L ,Lν )

≤k kk k p,q Rγ Cgf Sν˜

≤k kk kk k p q Rγ Cg f W(L ,Lν ).

40 k k p,q k k p q Consequently, Cgf Sν˜ f W(L ,Lν ), which proves part b of Theorem 3.3.1. ∈ 1 ∞ Finally, we prove part c of Theorem 3.3.1. Let f W (L ,L1/ω)begiven.We ∈ p q ∈ p,q must show that f W (L ,Lν) if and only if Cgf Sν˜ . The forward direction, ∈ p q ∈ p,q that if f W (L ,Lν)thenCgf Sν˜ , is simply Theorem 3.2.6. For the reverse ∈ p,q ˜ direction, assume that Cgf Sν˜ . Then by Theorem 3.2.6, the function f = Rγ(Cgf)

p q lies in W (L ,Lν). However, the factorization RγCg = I holds on every amalgam

1 ∞ space, including W (L ,L1/ω) in particular, so we also know that f = Rγ Cgf.Thus ˜∈ p q f=f W(L,Lν), which completes the proof.

Remark 3.3.2. a. Theorem 3.3.1 says that, given an appropriate condition on the

window g and its dual window γ, a Gabor frame for L2 extends to the amalgam spaces

and provides “Gabor expansions” for the amalgam spaces in the sense that we have

the factorization of the identity as I = RγCg. The speciﬁc form of these expansions P is that given f, there exist functions mk such that f = Rγ Cgf = mk · Tαkg. When

The inner sum deﬁning mk converges conditionally in general, while the outer sum converges unconditionally.

b. For the case p = 1, the functions mk cannot be written as Fourier series, so we do not have a series expansion of the form (23). A diﬀerent approach to the case p = q =1andν= 1, based on Littlewood–Paley theory, is developed by Gilbert and Lakey in [33], where they show that Gabor frames can be used to characterize a Hardy-type space on the line.

1 ∞ c. Theorem 3.3.1c says that if we use the “largest” amalgam space W (L ,L1/ω)as

p q our “universe,” then membership of a function in an amalgam W (L ,Lν)ischaracter- ized by membership of its sequence of Gabor coeﬃcients in an appropriate sequence

41 space. By imposing additional restrictions on g, γ, we could enlarge the universe on which this characterization is valid. In particular, if we required g, γ to lie in the

Schwartz class S, then the universe on which this characterization was valid would

be the space S0 of tempered distributions.

d. For the case of the modulation spaces, there is a deep result that states that

1 1 if g lies in the Feichtinger algebra Mω, then the dual window γ will lie in Mω as well, [43]. For the case of the amalgam spaces, we do not know if the assumption

∈ ∞ 1 g W (L ,Lω) implies that the dual window γ also lies in that space. This is an interesting and possibly diﬃcult open question.

3.4 Convergence of Gabor expansions

As pointed out above, when 1

general, while the outer series converges unconditionally. Our next result shows that this series can also be written as a double sum as in Theorem 2.3.11, but because the

proof relies on the convergence of Fourier series in Lp, the convergence is conditional

in general. In dealing with Fourier series in higher dimensions, it is important to use

d the maximum norm |x| =max{|x1|,...,|xd|} on R .

Theorem 3.4.1. Let ν be an ω-moderate weight. Let α, β>0and 1

≤ ∞ ∈ ∞ 1 G 1 q< be given. Assume that g, γ W (L ,Lω) are such that (g,α,β) is a Gabor frame for L2 with dual window γ. Then the following statements hold.

∈ p,q a. If c Sν˜ , then the partial sums X X SK,Nc = ckn MβnTαkg, K,N > 0, |k|≤K |n|≤N p q converge to Rgc in the norm of W (L ,Lν), i.e., for each ε>0there exist K0,

N0 > 0 such that

∀ ≥ ∀ ≥ k − k p q K K0, N N0, Rgc SK,N c W (L ,Lν ) <ε.

42 ∈ p q b. If f W (L ,Lν), then the partial sums of the Gabor expansion of f, X X SK,N(Cgf)= hf,MβnTαkgi MβnTαkγ, |k|≤K |n|≤N

p q converge to f in the norm of W (L ,Lν).

∈ ∞ 1 G 2 Proof. We are given g, γ W (L ,Lω) such that (g,α,β) is a Gabor frame for L and γ is the dual window to g, and we ﬁx 1 0, write X 2πiβn·x SN mk(x)= ckn e |n|≤N

2πiβn·x for the partial sums of the Fourier series of mk. The exponentials {e }n∈Zd form

p a basis for L (Q1/β ) [48], [63], so, letting C1 denote the basis constant for this system, we have for each k ∈ Zd that

k − k lim mk SN mk p,Q1/β = 0 (24) N→∞

and k k ≤ k k sup SNmk p,Q1/β C1 mk p,Q1/β . (25) N>0 ∈ p,q Since c Sν˜ ,givenε>0, we can ﬁnd K0 > 0 such that X 1/q ∀ ≥ k kq q K K0, mk p,Q1/β ν˜(k) <ε. (26) |k|≥K

Because of (24) and the fact that K0 is ﬁnite, we can ﬁnd an N0 > 0 such that

∀ ≥ k − k ε N N0, sup mk SN mk p,Q1/β ν˜(k) < d/q . (27) |k|≤K0 (2K0 +1) ∈ p,q ∞ Now, since c Sν˜ and 1

43 Given K ≥ K0 and N ≥ N0, write

− − − − Rgc SK,Nc =(Rgc SK0,∞c)+(SK0,∞c SK0,N c)+(SK0,N c SK,Nc). (28)

p q We will calculate the W (L ,Lν) norm of each of these terms separately.

d For the ﬁrst term, deﬁne a sequence r by rkn = ckn for |k|≤K0 and n ∈ Z , and rkn = 0 otherwise. Then SK0,∞c = Rgr,andRg is a bounded mapping of p,q → p q Sν˜ W (L ,Lν), so using (26) we have

k − k p q k − k p q Rgc SK0,∞c W (L ,Lν ) = Rg(c r) W (L ,Lν )

≤k kk − k p,q Rg c r S ν˜ X 1/q k k k kq q = Rg mk p,Q1/β ν˜(k) |k|>K0

≤kRgkε. (29)

For the second term, deﬁne skn = ckn for |k|≤K0 and |n|≤N,andskn =0 otherwise. Then SK0,N c = Rgs, so using (27), we have

k − k p q SK0,∞c SK0,N c W (L ,Lν )

≤k kk − k p,q Rg r s S ν˜ X 1/q k k k − kq q = Rg mk SN mk p,Q1/β ν˜(k) |k|≤K0

≤kRgkε. (30)

For the third term, deﬁne tkn = ckn for |k|≤Kand |n|≤N,andtkn =0 otherwise. Then SK0,N c = Rgt, so using (25) and (26), we have

k − k p q ≤k kk − k p,q SK0,N c SK,Nc W (L ,Lν ) Rg s t Sν˜ X 1/q k k k kq q = Rg SN mk p,Q1/β ν˜(k) K0<|k|≤K X 1/q ≤ k k k kq q C1 Rg mk p,Q1/β ν˜(k) K0<|k|≤K

44 k − k p q ≤ k k SK0,N c SK,Nc W (L ,Lν ) C1 Rg ε. (31)

k − k p q ≤ k k Applying (29)–(31) to (28), we see that Rgc SK,Nc W (L ,Lν ) (2 + C1) Rg ε, which completes the proof.

3.5 Necessary conditions on the window

In this section we prove a partial converse to Theorem 3.2.6. In particular, Theo-

∈ ∞ 1 p q rem 3.2.6 implies that if g W (L ,Lω), then Cg is bounded on each W (L ,Lν). In the converse direction, if g is a measurable function and 1 ≤ p, q ≤∞are

p q given, then in order for Cg to be well-deﬁned on W (L ,Lν), we must at least have R h i ∈ p q ∈ 1 Cgf(0, 0) = f,g = fg¯ deﬁned for each f W (L ,Lν). Hence fg¯ L for all such

p q f, so we immediately have that g must lie in the K¨othe dual of W (L ,Lν), which is 0 p0 q W (L ,L1/ν ). For the unweighted case, we obtain the following further necessary condition in

p ∞ order that Cg be bounded on W (L ,L ). For the case p = 2, this result was obtained by Balan in [2] and published in [4], [3].

p0 1 Theorem 3.5.1. Let α, β>0and 1

p0 1 Proof. We assume that g ∈ W (L ,L ) is such that Cg is a bounded map from W (Lp,L∞)toSp,∞,where1

Letusshowﬁrstthatg∈L∞. If not, then given any D>0 there would exist a set ` | | J contained in some cube β + Q1/β and with positive measure such that g(x) >D on J. 1 i arg g χ p ∞ Set f = |J|1/p e J . Using the equivalent norm for W (L ,L ) obtained by

replacing α in (7) by 1/β,wehavethatkfkW(Lp,L∞) ≤ 1. By hypothesis, Cgf ∈

p,∞ S ,sothereexist1/β-periodic functions mk such thatm ˆ k(n)=Cgf(k, n). Since

45 1 f · Tαkg¯ ∈ L , it is easy to see that mk is given by (17). In particular, considering k = 0 we have Z X p k kp −pd − n − n m0 p,Q1/β = β f(x β )¯g(x β) dx Q 1/β n∈Zd Z −pd β p = χJ (x) |g(x)| dx |J| ` β +Q1/β

≥ β−pdDp.

Hence

≤ d k k D β sup mk p,Q1/β k∈Zd

d = β kCgfkSp,∞

d d ≤ β kCgkkfkW(Lp,L∞) ≤ β kCgk.

But since D is arbitrary, this contradicts the fact that Cg is a bounded mapping. Hence g must be in L∞. Now we show that g ∈ W (L∞,Lp). Fix ε>0, and for each n ∈ Zd deﬁne n 1 ∈ | |≥ k k∞ n Jn = x β + Q1/β : g(x) 2 g ,β+Q1/β .

0 | |≤ 0 Then set Jn = Jn if Jn ε, otherwise let Jn be a subset of Jn of measure ε.Let

{ ∈N |0|≥ ε | |≤ } Nε =supN :Jn 2 for all n N .

Note that Nε →∞as ε → 0 (and may even be ∞ for some ε). Deﬁne f = P ∞ i arg g χ 0 k k p ∞ ≤ 1/p ∈ p, e |n|≤Nε Jn , and note that f W (L ,L ) ε . Therefore Cgf S ,and

46 letting mk be deﬁned by (17), we have X Z p −pd p k k | | 0 m0 = β g(x) χJn (x) dx p,Q1/β n +Q1/β |n|≤Nε β X k · n k p g T χQ1/β ∞ ≥ β−pd β |J 0 | 2 n |n|≤Nε X − − − ≥ pd p 1 k · n χ kp β 2 ε g T β Q1/β ∞. |n|≤Nε

Hence

X pd p+1 β 2 p k · n χ kp ≤ k k g T β Q1/β ∞ sup mk p,Q1/β ε k∈Zd |n|≤Nε

pd p+1 β 2 p = kC fk p,∞ ε g S pd p+1 β 2 p p ≤ kC k kfk p ∞ ε g W (L ,L )

pd p+1 p ≤ β 2 kCgk .

∞ p Since Nε →∞as ε → 0, this implies that g ∈ W (L ,L ).

0 Remark 3.5.2. As noted above, the hypothesis g ∈ W (Lp ,L1) is not a limitation on the generality of the result, as it is necessary in order that Cg can even be deﬁned.

0 Furthermore, if 1

0 spaces. The result is also true if p = 1, but in this case W (L∞,Lp)=W(Lp,L1)and there is no new information gained.

We now show that, with a mild hypothesis, we obtain a necessary condition on the analysis window. This hypothesis is formulated in terms of the following condition; we refer to [14], [2] for examples.

47 Deﬁnition 3.5.3. A function f : Rd → C has persistency length a if there exists a

δ>0 and a compact set K congruent to Qa mod a such that |f(x)|≥δfor every x ∈ K.

Theorem 3.5.4. Let α, β>0and 1 ≤ p<∞be given. Let g, γ be measurable functions on Rd. Suppose the following: P ∈ p ∞ h i a. for each f W (L ,L ), the series k,n f,MβnTαkg MβnTαkγ converges un- p conditionally in Lloc,

b. the frame operators Sg = RgCg and Sγ = RγCγ are bounded mappings of W (Lp,L∞) onto itself,

c. γ has persistency length 1/β.

Then g ∈ W (L∞,Lp).

P ⊂ Rd 7→ h i Proof. Let F be compact. Then by hypothesis, f k,n f,MβnTαkg MβnTαkγ

p ∞ p is a bounded mapping from W (L ,L )intoLLoc, and the series converges uncondi- P p 7→ h i tionally in Lloc. We ﬁrst show that f Skf = n∈Zd f,MβnTαkg MβnTαkγ is uniformly bounded on W (Lp,L∞), with respect to k ∈ Zd.

Fix k ∈ Zd and any f ∈ W (Lp,L∞). Then the sequence X hf,MβnTαkgiMβnTαkγ |n|≤N

converges in Lp(F )asN→∞, hence is a bounded sequence in Lp(F ). Moreover,

because the operators X SN,k(·):= h·,MβnTαkgiMβnTαkγ, |n|≤N

are bounded, we conclude from the Uniform Boundedness Principle that for each ﬁxed

k the sequence of operators {SN,k}N≥0, are uniformly bounded in N, i.e., for each k

48 there is a constant Ck > 0 such that

kSN,kkW(Lp,L∞)→Lp(F ) ≤ Ck∀N.

p ∞ Next, given >0andf∈W(L,L )withkfkW(Lp,L∞) =1,there exists N0 > 0 P k h i k ≤ such that |n|≥N0 f,MβnTαkg MβnTαkγ p,F .Thus,

kSk(f)kp,F ≤ hf,MβnTαkgiMβnTαkγ + |n|≥N0 p,F

hf,MβnTαkgiMβnTαkγ |n|≤N0 p,F

≤ Ck + .

Letting → 0, we conclude that kSk(f)kp,F ≤ Ck. It then follows that Sk is a bounded P p ∞ p p operator from W (L ,L )intoL(F). Now because k∈Zd Sk(f)convergesinL(F), we know that Sk(f ) is bounded independently of k, so by the Uniform Boundedness

d principle, we conclude that kSkkW (Lp,L∞)→Lp(F ) ≤ C(F ) for all k ∈ Z . Moreover, it is easy to see that

kSkkW (Lp,L∞)→Lp(F +α) = kSk−1kW (Lp,L∞)→Lp(F ) ≤ C(F ).

Thus if we let F = Qα then kSk(f)kp,F ≤ C(F )kfkW (Lp,L∞). On the other hand, taking F and δ>0 as in the deﬁnition of persistency, for any k ∈ Zd we have

hf,MβnTαkgiMβnTαkγ = kSk(f)kp,F +αk ∈Zd p,F +αk n X 2πiβ(·+αk) = γ(·) hf,MβnTαkgie n∈Zd p,F

≥ d k k δβ mk p,Q1/β .

Therefore,

−1 d −1 d d k k ≤ k k ≤ k k p ∞ ∀ ∈ Z mk p,Q1/β δ β Sk(f) p,F +αk Cδ β f W(L ,L ) k .

49 Hence,

−1 d k k p,∞ k k ≤ k k p ∞ Cgf S =supmk p,Q1/β Cδ β f W(L ,L ). k Thus, we conclude by Theorem 3.5.1 that g ∈ W (Lp,L∞), which concludes the proof.

50 CHAPTER IV

EMBEDDINGS OF BESOV,

TRIEBEL-LIZORKIN SPACES INTO

MODULATION SPACES

The apparently simple deﬁnition of the modulation spaces (see Chapter 2) hides the practical problem of how to decide whether or not a distribution belongs to

p,q a given modulation space. In principle one has to estimate the Lν norm of the STFT, which can be a non-trivial task. Therefore it is important to understand the relationship between time-frequency content and other properties of distributions, e.g., smoothness properties. Such relationships may appear in the form of embeddings of certain spaces that measure smoothness and/or decay into modulation spaces.

For example, Gr¨ochenig in [40], Galperin and Gr¨ochenig in [32], and Hogan and

Lakey in [47] derived suﬃcient conditions for membership in the modulation space M 1 from certain uncertainty principles related to the STFT. Another interesting example appears in [45], where Heil, Ramanathan and Topiwala obtained an embedding that is particularly important in relation to pseudodiﬀerential operator theory.

In the present chapter we prove suﬃcient conditions for a tempered distribution to belong to certain (unweighted) modulation spaces by proving some embeddings of classical Banach spaces such as the Besov, Triebel-Lizorkin, or Sobolev spaces into the modulation spaces. As corollaries, we obtain some embeddings which generalize the embedding from [45] mentioned above, and, moreover, we will give an easy suﬃcient condition for membership of a distribution in M 1 in the special case of dimension d =1.

51 Other embeddings results between modulation spaces and Besov spaces were ar- rived at independently and by diﬀerent techniques by P. Gr¨obner [37], and J. Toft

[56].

4.1 The Besov and Triebel-Lizorkin Spaces

Let ψ ∈Sbe a function such that    ≤ ≤  0 ψ(x) 1, | |≤  ψ(x)=1, if x 1,   ψ(x)=0, if |x|≥3/2.

Deﬁne     φ0(x)=ψ(x), φ(x)=ψ(x)−ψ(x),  1 2   −k+1 φk(x)=φ1(2 x),k=2,3, .... { }∞ Then φk k=0 is a partition of unity, and satisﬁes

d k−1 k−1 supp(φk) ⊂{x∈R :2 ≤|x|≤3·2 }.

0 Deﬁnition 4.1.1. Let s ∈ R,1≤q≤∞,andf∈S.

≤ ∞ s (i) For 1 p< the Triebel-Lizorkin space Fp,q is deﬁned by: Z X∞ p/q 1/p s skq −1 q ∈ ⇐⇒ k k s |F ˆ | ∞ f Fp,q f F p,q = 2 (φkf)(x) dx < . (32) Rd k=0 ≤ ≤∞ s (ii) For 1 p , the Besov space Bp,q is deﬁned by: Z X∞ q/p s skq −1 p ∈ ⇐⇒ k k s |F ˆ | ∞ f Bp,q f B p,q = 2 (φkf)(x) dx < . (33) Rd k=0

∞ s (iii) For p = , the Triebel-Lizorkin space F∞,q is deﬁned by: X∞ X∞ 1/q ∈ s ⇐⇒ ∃ { } ∞ F −1 ˆ ksq | |q ∞ f F∞,q f k k =0,f= (φkfk), sup 2 fk(x) < , Rd k=0 k=0 (34)

52 with norm X∞ 1/q ksq q k k s | | f F∞,q =inf sup 2 fk(x) , Rd k=0 the inﬁmum being taken over all admissible representations.

≤ ≤∞ s (iv) For 1 p , the fractional Sobolev space Hp is deﬁned by: Z 1/p s −1 2 s/2 p ∈ ⇐⇒ k k s |F | | ˆ | ∞ f Hp f H p = ((1 + x ) f)(x) dx < . (35) R d

Remark 4.1.2. The classes of Besov and Triebel-Lizorkin spaces comprise many of the spaces encounter in analysis, e.g., we have the following identiﬁcations whose proofs may be found in [57, Sect. 2.3.5].

≤ ≤∞ ∈ R s s a. If 1 p = q and s ,thenBp,p = Fp,p, this follows from the deﬁnition.

∞ 0 p b. If 1

∞ ∈ R s s c. If 1

≤ ≤∞ s ⊂ p Moreover, for 1 p, q and s>0wehavethatBp,q L , additionally if ∞ s ⊂ p p< we also have Fp,q L . We refer to [58, Sect. 2.3.2] for the proof of these last assertions.

More generally, we refer to [57], [58], [53] and [55] for background and information about the Triebel-Lizorkin, Besov, and Sobolev spaces.

Because the Besov and the Tribel-Lizorkin spaces have been rediscovered (under diﬀerent names) by various authors, they have a number of equivalent deﬁnitions. We collect here some of those results that will be needed in the sequel: Propositions 4.1.3

s s and 4.1.4 give equivalent deﬁnitions of Fp,q and Bp,q, respectively, while Proposition 4.1.5 is a result on interpolation of Besov spaces.

The following result is proved in [57, Proposition 1, 2.3.4].

53 ∈ R ≤∞ ≤∞ ∈S0 ∈ s Proposition 4.1.3. Let s , 1

Furthermore, Z X∞ p/q 1/p ksq q inf 2 |fk(x)| dx Rd k=0 s is an equivalent norm on Fp,q, where the inﬁmum is taken over all admissible representations of f.

See [53, Theorem 2, 2.3.2] for a proof of the following result.

≤ ≤∞ ∈S0 ∈ s Proposition 4.1.4. Let 1 p, q and s>0.Iff then f Bp,q if and only if Z X∞ X∞ q/p 1/q ∃{}∞⊂ p kqs | |p ∞ bk=0 L ,f= bk, 2 bk(x) dx < . (37) Rd k=0 k=0

Furthermore, Z X∞ q/p 1/q kqs p inf 2 |bk(x)| dx Rd k=0 s is an equivalent norm on Bp,q, where the inﬁmum is taken over all admissible representations of f.

See [57, Theorem 2.4.7] for a proof of the following result about complex interpolation of Besov spaces.

Proposition 4.1.5. Let s0,s1 ∈R, 1≤p0,q0,p1,q1 ≤∞,and 0 <θ<1. If s −θ s 1 1−θ θ 1 1−θ θ =(1 ) 0 +θs1, p = p0 + p1 , and q = q0 + q1 then s0 s1 s Bp0,q0 ,Bp1,q1 [θ] = Bp,q. (38)

The next proposition collects some of the computations involved in the proofs of

our results. Part (a) computes the STFT of a Gaussian with respect to a dilated

54 Gaussian. The result is essentially the product of two Gaussians (one in time and the other in frequency). Part (b) shows that the inverse Fourier transform of the Bessel

potential

2 −s/2 m−s(x)=(1+|x|)

is in M 1 for s>d. Because M 1 is invariant under Fourier transforms we then conclude

1 that the Bessel potential m−s itself is in M for s>d.

2 −πx2 −πx Proposition 4.1.6. Deﬁne g(x)=e and ga(x)=e a .Let Z ∞ 2 1 1 −d+s − πx t dt G x t 2 e ( t + 4π ) s()= s/2 (4π) Γ(s/2) 0 t

for s, a > 0 and x ∈ Rd, and where Γ refers to the Gamma-function. Then the following hold. d/2 2πi x·ω a a+1 a+1 (a) V a g(x, ω)= e g (x) g (ω). g a+1 a+1 a d/2 1 (b) V mˇ − =(2π) V (D 1 G ) ∈ L for s>d, where D is the unitary dilation g s g 2π s a

−d/2 operator deﬁned by Dag(x)=|a| g(x/a).

1 (c) m−s ∈ M for s>d.

−d/2 Proof. (a) First note that from Lemma 2.2.2 the operator Daf(t)=|a| f(t/a) 2 d ˆ d where a>0 is unitary on L ,andDaf=D1/af. Now for x, ω ∈ R we have: Z 2 − t − · − − 2 π a 2πit ω π(t x) Vga g(x, ω)= e e e dt Rd Z − π 2− · 2 − · = e a (a+1)t 2at x+ax e 2πit ω Rd Z − a+1 − a 2 a 2 − · = e π a (t a+1 x) + a+1 x e 2πt ω dt Rd Z − π 2 − a+1 − a 2 − · = e a+1 x e π a (t a+1 x) e 2πt ω dt Rd

− π 2 ∧ a+1 x a a = e (T a+1 x g a+1 ) (ω)

55 − a da = ga+1(x) M a+1 x g a+1 (ω) a d/2 = ga+1(x) M− a x g a+1 (ω) a +1 a+1 a a d/2 − a · 2πi a+1 x ω = e ga+1(x) g a+1 (ω), a +1 a

2 where we have used the fact that the Fourier transform of the Gaussian g(x)=e−πx

is itself, i.e.,g ˆ = g.

(b) For s>0 it is shown in [55, Proposition 3.1.2] that

2 2 −s/2 Gˆs(ω)=(1+4π |x|) .

1 Notice for future references that Gs ∈ L , see [55, Proposition 3.1.2]. Thus,

2 −s/2 m−s(x)=(1+|x|)

= Gˆs(x/2π)

d/2 =(2π) D2πGˆs(x)

d/2 \ =(2π) D1/2πGs(x).

d/2 Consequently, we have thatm ˇ − (x)=(2π) (D 1 G )(x). Therefore: s 2π s

d/2 V mˇ − (x, ω)=(2π) V (D 1 G )(x, ω) g s g 2π s Z d/2 −2πit·ω =(2π) D 1 Gs(t) e g(t − x) dt, Rd 2π

d/2 =(2π) D 1 G ,MTg 2π s ω x

d/2 =(2π) Gs,D2πMωTxg

ω = Gs,M2πT2πxg2π Z Z ∞ 2 1 s−d − πt u − · u 2 e ( u + 4π ) e 2πit ω/2π× = s/2 (4π) Γ(s/2) Rd 0 2 du e−π(t−2πx) /2π dt u

56 Z Z ∞ 2 1 − s−d − − πt u/4π 2 1 u × = s/2 e u e (4π) Γ(s/2) 0 Rd 2 − · − (t−2πx) e 2πit ω/2π e π 2π dt du

Z ∞ Z 1 − s − − 2 u/4π 2 1 πt × = s/2 e u e (4π) Γ(s/2) 0 Rd √ (t− 2√πx )2 − · u − u e 2πit ω 2π e πu 2π dt du

Z ∞ √ 1 − s − 2πx uω e u/4π u 2 1 V g √ , du. = s/2 g 2π ( ) (4π) Γ(s/2) 0 u u 2π

The last equality follows from some changes of variable and by using part a. Therefore, we have that:

− Z ∞ (d s)/2 4π2x2 u 2 π −u/4π −π −π ω V m− x, ω e e u+2π e 2π(2π+u) × g ˇ s( )= s−d/2 2 Γ(s/2) 0

s − 1 u 2 1 du, (2π + u)d/2

where we have used Lemma 2.2.2 to obtain the last equation. By changing the R −πx2 variables and using the fact that Rd e dx =1,wehave: ZZ

kVgmˇ−skL1 = |Vgmˇ−s(x, ω)| dx dω R2d Z (d−s)/2 ∞ π − s − 1 ≤ e u/4π u 2 1 × 2s−d/2 (2π + u)d/2 0 Z Z 2 2 − 4π x −π u ω2 e π 2π+u e 2π(2π+u) dω dx du Rd Rd Z (d−s)/2 ∞ π − −d+s − u/4π 2 1 d/2 = s−d/2 e u (u +2π) du, (39) 2 0

and the last expression is ﬁnite if s>d.

57 4.2 Embedding of Besov, Triebel-Lizorkin spaces into modulation spaces

There are several embeddings between the Besov or Triebel-Lizorkin and modulation spaces that can easily be derived. Some of these embeddings are summarized in the following result.

s ⊂ p ⊂ p,p0 ≤ ≤ ≤ ≤∞ Proposition 4.2.1. (a) Bp,q L M when s>0,1 p 2, and 1 q . s ⊂ p ⊂ p ≤ ≤∞ ≤ ≤∞ (b) Bp,q L M when s>0,2 p , and 1 q .

Proof. (a) The ﬁrst of these embeddings was mentioned in Remark 4.1.2, and its proof can be found in [58, Remark 3, Sect. 2.3.2]. To prove the second one, let f ∈ Lp,and

d \ p let g ∈S. Then for x, ω ∈ R , Vgf(x, ω)=f·Txg(ω). Note that the f · Txg ∈ L since f ∈ Lp and g ∈S.Moreover,since1≤p≤2≤p0,wehavethatp0/p ≥ 1, and thus using Hausdorﬀ-Young’s inequality and Minkowski’s inequality (for integrals) we have:

0 0 Z Z p /p p/p p p kV fk 0 = |V f(x, ω)| dx dω g Lp,p g Rd Rd

0 Z Z p/p p0 ≤ |Vgf(x, ω)| dω dx Rd Rd

k kp k kp = f Lp g Lp .

Hence,

k k 0 k k 0 ≤k k p k k Vgf Lp,p = f M p,p f L g p, which concludes the proof the second inclusion.

58 (b) We now prove the second of these inclusions. Let f ∈ Lp,since2≤p≤∞, we have that 1 ≤ p0 ≤ 2. Moreover, choosing g ∈Swe see using H¨older’s inequality

0 ∗ p0 d with p/p ≥ 1, that f · Txg ∈ L for almost all x ∈ R . By using Hausdorﬀ-Young’s inequality as well as Young’s inequality we have that ZZ 1/p p kVgfkLp = |Vgf(x, ω)| dx dω R2d ZZ 1/p \ p = |f ·Txg(ω)| dω dx R2d

0 Z Z p/p p0 ≤ |(f · Txg)(t)| dt ,dx Rd Rd Z Z 1/p 0 0 0 = |f(t)|p |g∗(x−t)|p dt p/p dx Rd Rd Z 1/p 0 0 0 = |f|p ∗|g∗|p (x) p/p dx Rd

0 p0 ∗ p0 1/p | | ∗| | 0 = f g Lp/p .

0 0 Now notice that p ≥ p0,andthatf∈Lp ⇐⇒ | f | p ∈ L p/p . Thus, applying Young’s inequality with parameter p/p0 we have that

0 p p0 ∗ p0 k k p ≤ | | ∗| | 0 Vg f L f g Lp/p

p0 ∗ p0 ≤k| | k p/p0 |g | f L L1

p0 p0 = kfk p kgk 0 . L Lp

Consequently,

k k 0 ≤k k 0 k k p f Mp,p g Lp f L ,

which concludes the proof in this case.

Other more subtle embeddings of classical spaces into modulation spaces were

obtained as byproducts of other results. For example, Gr¨ochenig in [40] and Hogan

and Lakey in [47] derived suﬃcient conditions for membership in the modulation

59 space M 1 from certain uncertainty principles related to the STFT. Precisely, it was

p ∩F q ⊂ 1 shown that La Lb M under appropriate conditions on p, q and the weight p p | | p F q parameters a, b,whereLa is a weighted L space (with weight (1 + x ) ), and Lb q is the image of Lb under the Fourier transform. Somewhat more general embeddings involving weighted Lp spaces were given by Galperin and Gr¨ochenig in [32]. Another

interesting example appears in [45]. There, Heil, Ramanathan and Topiwala proved,

in our notation, that Cs(R2d) ⊂ M ∞,1(R2d) for s>2d, while working on a time- frequency approach to pseudodiﬀerential operators.

The embeddings we will prove are more diﬃcult, and require an appropriate norm

on the Besov or Triebel-Lizorkin space in consideration, along with a correct choice

of the form of the STFT. In particular, the following equivalent forms of the STFT

(see Proposition 2.3.3) will be useful:

∧ Vgf(x, ω)= f·Txg (ω)

−2πixω ˆ =e Vgˆf(ω,−x)

−2πixω −1 ˆ = e F (f · Tωgˆ)(x) −2πixω = e f ∗ (Mωg)(x) . (40)

Our ﬁrst main embedding result involves the Besov spaces and is as follows.

≤ ≤ ≤ ≤∞ 2 − Theorem 4.2.2. Let 1 p 2 and 1 q .Ifs>d(p 1) then

s ⊂ p0,p Bp,q M . (41)

P ∈ s ∈ p Proof. Let f Bp,q, and use (37) to write f = bk where bk L ,and X∞ k ksq q 1/q ˆ ⊂{| |≤ } k k s ∼ k k supp(bk) x 2 , and f Bp,q inf 2 bk Lp , k=0 where the inﬁmum is over all possible such representations of f.Giveng∈S,we have using (40) that

60 X∞ −2πixω −1 ˆ Vgf(x, ω)= e F (bk · Tωgˆ)(x). k=0 Hence by the Hausdorﬀ-Young inequality, X∞ − k · k 0 ≤ kF 1 ˆ · k 0 Vgf( ,ω) Lp (bk Tωgˆ) Lp k=0 X∞ ˆ ≤ kbk · TωgˆkLp . k=0 Therefore, by Minkowski’s inequality, Z X∞ 1/p p k k 0 ≤ kˆ · k Vgf Lp ,p bk Tωgˆ Lp dω Rd k=0 X∞ ˆ = kgˆkLp kbkkLp . (42) k=0

ˆ k Now, bk has compact support, which is contained in Ek = {|x|≤2}.Since 1≤p≤2≤p0,wehavethatp0/p ≥ 1, and using H¨older’s inequality and Hausdorﬀ- Young’s inequality we obtain the following estimates

p p kˆ k p kˆ k bk L = bk p,Ek Z ˆ p = |bk(ω)| dω Ek

−p 1 p0 p ≤| | kˆk 0 Ek bk Ek,p

− p ≤ kd(1 p0 ) kˆ kp C 2 bk p0 .

where C is the volume of the ball of center 0 and radius 1. Thus,

1 − p ˆ 1/p kd( p p0 ) ˆ kbkkLp ≤ C 2 kbkkp0

1 − 1 0 kd( p p0 ) ≤ C 2 kbkkLp. (43)

61 Substituting (43) into (42) and applying H¨older’s inequality yields

∞ X 1 1 0 kd( − 0 ) k k 0 ≤ k k p p k k Vgf Lp ,p C gˆ p 2 bk p k=0 X∞ 1 − 1 − s 0 kd( p p0 d ) ks = C kgˆkp 2 2 kbkkp k=0 0 X∞ 1/q X∞ 1/q 0 0 1 − 1 − s ≤ ksq k kq kq d( p p0 d ) C 2 bk Lp 2 k=0 k=0 0 ∞ 1/q X 0 1 1 s kq d( − 0 − ) ≤ k k s p p d C f Bp,q 2 . (44) k=0 1 − 1 The last term in (44) is ﬁnite if and only if s>d(p p0).

The next result recovers and extends the embedding in [45], and follows by iden-

s s tifying Fp,p with Bp,p (see Remark 4.1.2) and by using Proposition 4.1.3 as the appro- s s ⊂ 2 2 priate deﬁnition of Bp,p. However, it does not include the fact that B2,2 L = M for s ≥ 0. This last embedding is obtained as corollary by using complex interpolation

methods.

≤ ≤∞ d Theorem 4.2.3. Let 1 p .Ifs> p0 then

s ⊂ p,p0 Bp,p M . (45)

Proof. We divide the proof in two cases. P ∈ s ∞ F −1 ˆ −πx2 Case 1: p =1.Letf B1,1,iff= k=0 (φkf), then letting g(x)=e ,we have X∞ −1 ˆ −1 Vgf(x, ω)= F (fφk)∗F (Tωgˆ)(x). k=0 Therefore we have the following estimates X∞ −1 ˆ −1 ˆ kVgf(·,ω)kL1 ≤ kF (φk f)k1 kF (δ · Tωgˆ)k1 k=0 X∞ Z −1 ˆ −1 ˆ kVgfkL1,∞ ≤ kF (φk f)k1 sup |F (δ · Tωgˆ)(x)| dx ∈Rd Rd k=0 ω

62 X∞ Z −1 ˆ ≤ kF (φk f)k1 sup |Vgδ(x, ω)| dx. ∈Rd Rd k=0 ω However, Z Z Z

sup |Vgδ(x, ω)| dx =sup |MωTxg(0)| dx = |g(x)| dx < ∞. ω∈Rd Rd ω∈Rd Rd Rd

Thus, X∞ −1 ˆ kVgfkL1,∞ ≤kgk1 kF (φk f)k1 k=0 X∞ −ks −1 ˆ ≤kgk1 2 kF (φk f)k1, k=0 for all s>0. Hence, using (33) we have that

k k 1,∞ ≤k k k k s f M g 1 f B1,1, which concludes the proof for this case.

Case 2: Assume 1

∞ Z 0 X 0 1/p −1 p k k 0 ≤ k k p kF · k 1 Vgf Lp,p fk L (φk Tωgˆ) L dω Rd k=0 ∞ Z 0 X 0 1/p sk −sk −1 p = 2 kfkkLp 2 kF (φk · Tωgˆ)kL1 dω Rd k=0 0 ∞ 1/p ∞ Z 1/p X X 0 ≤ ksp k kp −ksp0 kF −1 · kp 2 fk Lp 2 (φk Tωgˆ) L1 dω , Rd k=0 k=0

63 and therefore

0 ∞ Z 1/p X 0 −ksp0 −1 p k k 0 ≤k k s kF · k f M p,p f Bp,p 2 (φk Tωgˆ) L1 dω . (46) Rd k=0 Now we will estimate the terms in the summation on the right-hand side of (46).

− k−1 k+1 2 Setting gk(x)=g(2 x), i.e., gk =2 D2k−1g,wehave:

0 Z 1/p − p0 kF 1 · k k ˇ k 0 (φk Tωgˆ) L1 dω = Vgφk L1,p . Rd

But, using Lemma 2.2.2, we obtain:

ˇ ˇ Vgφk(x, ω)= φk,MωTxg

= φk,TωM−xgˆ

(k−1)/2 =2 D2k−1 φ1,TωM−xgˆ

(k−1)/2 \ =2 φ1,T21−kωM−2k−1xD2k−1g

(k−1)/2 ˇ =2 φ1,M21−kωT2k−1xD2k−1g

ˇ = φ1,M21−kωT2k−1xgk

ˇ k−1 1−k =Vgkφ1(2 x, 2 ω).

Therefore,

0 Z 1/p − p0 kF 1 · k k ˇ k 0 (φk Tωgˆ) L1 dω = Vgφk L1,p Rd

− − k ˇ k 1· k+1· k 0 = Vgk φ1(2 , 2 ) L1,p

− − 0 d( k+1) d(k 1)/p k ˇ k 0 =2 2 Vgk φ1 L1,p

d/p −kd/p ≤ k k 1,1 k ˇ k 0 C1 2 2 Vgkg L Vgφ1 L1,p , (47)

the last inequality following from the independence of the deﬁnition of the modulation

space with respect to the window used to compute the STFT (see Proposition 2.3.6).

64 Using Proposition 4.1.6 with a =2k−1,wehavethat

2k−2 d/2 k k 1 k k 1 Vgkg L = Vgag L =(1+2 ) . (48)

Combining (46), (47) and (48) yields:

0 X∞ 1/p −kp0(s−d+d/p) k k 0 ≤ k k s f M p,p C2 f Bp,p 2 . (49) k=0

− 1 d The last term in the right-hand side of (49) is ﬁnite if and only if s>d(1 p )= p0.

≤ ≤∞ − 2 Corollary 4.2.4. If 2 p and s>d(1 p ) then

s ⊂ p,p0 Bp,p M . (50)

Proof. We will prove this part by interpolating between the cases p =2,s0 ≥ 0and

p=∞,s1 >d.In particular, we trivially have

s0 s0 s0 ⊂ 2 2 ≥ B2,2 = F2,2 = H2 L = M for s0 0, (51)

and applying Theorem 4.2.3 to p = ∞ yields:

s1 ⊂ ∞,1 B∞,∞ M for s1 >d. (52)

By [57, Remark 4, Sect. 2.4.1] we have that s0 s1 ⊂ 2,2 ∞,1 B2,2,B∞,∞ [θ] M ,M [θ]

for appropriate values of s and θ. We now apply the interpolation result of modulation ∞ 1 1−θ spaces Proposition 2.3.10, with p0 = q0 =2andp1 = ,q1 = 1. Hence p = 2 , 1 1−θ − 1 1 and q = 2 θ. Consequently, p + q = 1. It follows from the above referenced proposition that 2,2 ∞,1 p,q p,p0 M ,M [θ] =M = M .

65 Similarly, if we apply the interpolation of Besov spaces Proposition 4.1.5 with s0 =0, ∞ 1 1−θ 1 s1 >d,p0 =q0 =2,andp1 =q1 = , hence s = θs1, p = 2 = q , i.e., p = q.It follows that 0 s1 s B2,2,B∞,∞ [θ] =Bp,p, and moreover, because 0 <θ=1−2/p < 1, we have that p ≥ 2, hence, s = s1θ> − 2 d(1 p ), which concludes the proof.

Our next result yields an embedding of a fractional Sobolev space (or Bessel potential space) into a modulation space. This can also be seen as an embedding

s s s of the Triebel-Lizorkin space Fp,2 into a modulation space, since Fp,2 = Hp when 1

Theorem 4.2.5. If 1 ≤ p ≤∞, then

s ⊂ p,1 Hp M for s>d. (53)

| |2 −s/2 −πx2 ∈ s Proof. Let m−s(x)=(1+ x ) and g(x)=e . Then for f Hp we have:

−2πixω −1 ˆ Vg f (x, ω)=e F (f · Tωgˆ)(x)

−2πixω −1 ˆ = e F (fms ·m−sTωgˆ)(x) −2πixω −1 ˆ −1 = e F (fms) ∗F (m−s ·Tωgˆ)(x) .

Hence, by Young’s convolution inequality,

−1 ˆ −1 kVgf(·,ω)kLp ≤kF (fms)kLp kF (m−s · Tωgˆ)kL1 , and so Z Z −1 k k p,1 ≤k k s |F · | Vgf L f Hp (m−s Tωgˆ)(x) dx dω d d ZR ZR

k k s | − | = f Hp Vgˆm−s(ω, x) dx dω Rd Rd

k k s k k 1 = f Hp Vgmˇ −s L . (54)

66 Using (39), we have that kVgmˇ −skL1 < ∞ for s>d, which concludes the proof.

The next corollary holds only in dimension one, and gives a useful suﬃcient con-

dition on a function to be in M 1. In particular, (55) below gives a new proof of a

conjecture of Feichtinger, that W 2,1 ⊂ M 1 when d = 1. We point out that another

(unpublished) proof of this conjecture was obtained by Gr¨ochenig. The corollary fol-

2 lows from the identiﬁcation of the Bessel potential space Hp with the Sobolev space W 2,p obtained by imposing that f and its ﬁrst two (distributional) derivatives belong to Lp (p =1orp=∞).

2 Before proving this corollary, we present the proof of the identiﬁcation of Hp and W 2,p when p =1orp=2andd= 1. We refer to [55, Sect. 6.6], and [9, Theorem

16] for more details on these identiﬁcations.

Lemma 4.2.6. If p =1or p = ∞,let

X2 2,p p 0 00 p (k) W (R)={f∈L(R):f,f ∈ L (R), kfkW 2,p = kf kp < ∞}. k=0 Then

2 2,p Hp = W

Proof. Case I: Assume p =1.Letf∈W2,1(R), then f,f0,f00 ∈ L1(R). Moreover,

fˆ00(ω)=−4π2fˆ(ω). Thus,

F −1((1 + |ω|2)fˆ)=F−1(fˆ)+F−1(ω2fˆ)

1 =f− f00. 4π2

Consequently,

F −1((1 + |ω|2)fˆ) ∈ L1(R),

67 and moreover,

−1 2 1 00 kF ((1 + |ω| )fˆ)k 1 ≤kfk 1 + kf k 1 L L 4π2 L X2 (k) ≤ C kf kL1 . k=0 Thus, 2,1 ⊂ 2 W H1 .

For the converse, let h be the function deﬁned on R, such that    e−x : x≥0, h(x)=  0:x<0.

∈ R ˆ 1 We easily obtain that for ω , h(ω)= 1+2πiω . ∈ 2 R ∈ 1 R F−1 2 ˆ ∈ 1 Now let f H1 ( ), then f L ( )and ((1 + ω )f) L .Thus,g= F−1(ω2fˆ)∈L1.Moreover,f∈L1(R) implies that f ∈S0(R), and so f 00 exists as an element of S0(R), hence fˆ00(ω)=−4π2ω2fˆ, where the equality holds in a

distributional sense. Thus, fˆ00 =ˆg,andg∈L1(R), therefore the uniquness of the

Fourier transform implies that f 00 = g ∈ L1(R).

We now show that f,f00 ∈ L1(R) implies that f 0 which exists as a distribution is

in L1(R). To that end, note that from the fact that f,f00,h∈L1(R), we obtain that

f ∗ h,andf00 ∗ h are both L1 functions, and we have the following equalities:

(−2πiω)2 h\∗ f 00(ω)= fˆ(ω) 1+2πiω 1 =(−1+2πiω + ) hatf(ω) 1+2πiω fˆ(ω) = −fˆ(ω)+2πiω fˆ(ω)+ . 1+2πiω

Consequently,

h ∗ f 00 = −f + f ∗ h + k,

68 where k = F −1(2πiω, fˆ). The last equation also implies that k ∈ L1(R), and moreover,

kˆ(ω)=2πiωfˆ(ω)=fˆ0(ω).

Using again the uniqueness of the Fourier transform, we conclude that f 0 = k ∈ L1(R), and moreover,

0 00 kf kL1 ≤ 2kfkL1 + kf kL1 ,

wherewehaveusedthefactthatkhkL1 = 1. Additionally, using the fact that 2 ⊂ 1 H1 L , we conclude that

X2 (k) k k 1 ≤ k k 2 f L C f H1 , k=0

2 ⊂ 2,1 thus, H1 W , and the proof in this case is concluded. The case p = ∞, is identical and so we omit it.

Using the above lemma and Theorem 4.2.5 we have the following result in the

case the dimension is d =1.

Corollary 4.2.7. If d =1and p ∈{1,∞}, then

W 2,p(R) ⊂ M 1(R). (55)

∈{ ∞} 2 2,p Proof. If d =1andp 1, , then from Lemma 4.2.6 we have that Hp = W and so the proof follows from Theorem 4.2.5.

69 CHAPTER V

BILINEAR PSEUDODIFFERENTIAL

OPERATORS ON MODULATION SPACES

In this chapter we present some applications of the modulation spaces. In particu-

lar, we study the boundedness of bilinear pseudodiﬀerential operators on modulation

spaces, as well as the boundedness of the linear Hilbert transform. Modulation spaces have recently been used to formulate and prove boundedness

results of linear pseudodiﬀerential operators, which are formalisms that assign to a

distribution a linear operator in such a way that properties of the distribution can

be inferred from properties of the corresponding operator. The Weyl and the Kohn-

Nirenberg correspondences are well-known examples of pseudodiﬀerential operators,

which can be expressed as a superposition of time-frequency shifts. In particular, if

0 2d d σ ∈S(R ) the Weyl correspondence associate to it the operator Tσ : S(R ) −→ S 0 R d ( ) , such that ZZ −πiξ·u Tσf = σˆ(ξ,u)e T−uMξ du dξ, R2d for f ∈S(Rd). Thus, because the operator can be realized as superposition of time- frequency shifts, the modulation spaces appear to be natural spaces in which to formulate and prove boundedness results of such operators. We refer to [45, 42, 41], for more details on the recent developments of pseudodiﬀerential operators in the realm of the modulation spaces.

In the ﬁrst section of the present chapter, we deal with bilinear integral operators

(deﬁned by a non-smooth kernel) on modulation spaces. This class of operators is large enough to include the bilinear pseudodiﬀerential operators with non-smooth

70 symbols. In particular, we prove that symbols in the Feichtinger algebra give rise to bounded bilinear pseudodiﬀerential operators. We refer to [11, 13, 12, 49] for

background and more detail about these operators.

The second section is devoted to the boundedness of the linear Hilbert transform

on the modulation spaces deﬁned on the real line. We use a discrete approach to

study the Hilbert transform, and rely on its L2 theory to some extent.

5.1 Bilinear operators on modulation spaces 5.1.1 Deﬁnition and background

A bilinear pseudodifferential operator Tσ is á priori defined through its (distributional) symbol σ ∈S0(R3d) as a mapping from S(Rd) ×S(Rd)intoS0(Rd)by: Z Z ˆ 2πix·(ξ+η) Tσ(f,g)(x)= σ(x, ξ, η) f(ξ)ˆg(η)e dξ dη, (56) Rd Rd

for f,g ∈S(Rd). A natural problem then is to ﬁnd suﬃcient (nontrivial) conditions on the symbol that ensure the boundedness of the operator on products of certain

Banach spaces such as Lebesgue, Sobolev, or Besov spaces [11, 13, 12, 35, 36]. For

instance, it is known that the condition

| α β γ |≤ | | | | −|β|−|γ| ∂x ∂ξ ∂η σ(x, ξ, η) Cα,β,γ (1 + ξ + η ) (57) for (x, ξ, η) ∈ R3d and all multi-indices α, β, γ is enough to prove the boundedness

p Rd × q Rd r Rd 1 1 1 of the operator deﬁned by (56) from L ( ) L ( )intoL( )when p +q = r and p, q > 1. This result was ﬁrst obtained by Coifman and Meyer [11], [13], [12],

who noticed that, in general, if the symbol is smooth and has certain decay, then the

boundedness of the corresponding operator can be studied through its decomposition into elementary operators via techniques related to Littlewood-Paley theory. Grafakos

and Torres [35] used the wavelet expansions of the Triebel-Lizorkin spaces that were

proved by Frazier and Jawerth [29, 30] to decompose instead the function on which the

operator acts, and thereby converting the boundedness question into the boundedness

71 of an inﬁnite matrix. By imposing some convenient decay conditions on the entries of the corresponding matrix they obtain some boundedness results on the operator

side on products Triebel-Lizorkin spaces. Here again, the symbols of the operators

are assumed to be suﬃciently smooth and to have decay at inﬁnity.

In this section, we use Gabor expansions of tempered distributions in the modula-

tion spaces to prove the boundedness of bilinear integral operators with non-smooth

kernels, of which (56) will be shown to be a particular case.

Throughout this chapter, ωs will denote the submultiplicative weight function

2d 2 2 s/2 deﬁned on R by ωs(x, y)=(1+|x| +|y|) . Moreover, we let Ωs denote the

6d extension of ωs on R given by

Ωs(X, Y )=(ωs⊗ωs⊗ωs)(X, Y )=ωs(x1,x2)ωs(x3,y1)ωs(y2,y3),

3d 6d where X =(x1,x2,x3),Y =(y1,y2,y3) ∈ R .IfAis an invertible operator on R A we denote Ω˜s the weight function deﬁned by

˜ A Ωs (X, Y )=Ωs(A(X, Y )),

3d 2d where X, Y ∈ R . Additionally, we deﬁneω ˜s on Z byω ˜s(l, k)=ωs(αl, βk) for f α, β > 0, and Ωs is deﬁned similarly.

Before considering general bilinear integral operators, we state a result which

1 R3d characterizes the modulation space MΩs ( ) in terms of Gabor frames using standard tensor product arguments; see [41, p. 272] for further details.

∈ 1 Rd { } d Proposition 5.1.1. Let φ Mωs ( ) be such that MβnTαkφ k,n∈Z is a Gabor frame 2 Rd ∈ 1 Rd K∈ 1 for L ( ) with (canonical) dual γ Mωs ( ). Then MΩs if and only if X K = hK,MβnTαmγ ⊗ MβlTαkγ ⊗ MβjTαiγi MβnTαmφ ⊗ MβlTαkφ ⊗ MβjTαiφ k,m,i,l,n,j∈Zd 1 R3d K with unconditional convergence of the series in MΩs ( ). Moreover, the norm of 1 hK ⊗ in MΩs is equivalent to the norm of its sequence of Gabor coeﬃcients ,MβnTαmγ 1 6d M T γ ⊗ M T γi d in ` (Z ). βl αk βj αi k,m,i,l,n,j∈Z Ω˜ s

72 5.1.2 Bilinear operators

Deﬁnition 5.1.2. A bilinear operator associated with a kernel K ∈S0(R3d), is a

d d 0 d mapping BK deﬁned ´a priori from S(R ) ×S(R )intoS(R)by Z Z

BK(f,g)(x)= K(x, y, z) f(y) g(z) dy dz, (58) Rd Rd for f,g ∈S(Rd).

One of our objectives in this section is to study the boundedness of (58) on

products of modulation spaces, and to derive from such results the boundedness of

(56). The next proposition establishes the relationship between a bilinear integral operator and a bilinear pseudodifferential operator defined by (56). We define an

operator U acting on functions deﬁned on R3d by

Uf(x, y, z)=f(x, y − x, z − x).

It easy to check that U is a unitary operator on L2, is an isomorphism on S,and

extends to an isomorphism on S0.Moreover,

U∗f(x, y, z)=U−1f(x, y, z)=f(x, y + x, z + x).

Proposition 5.1.3. Let Tσ be a bilinear pseudodiﬀerential operator associated to a

0 3d symbol σ ∈S(R )defined by (56). Then Tσ is a bilinear integral operator BK with F−1 F −1 kernel K(x, y, z)=U 1 σˆ(x, y, z), where 1 denotes the inverse Fourier transform in the first variable, and U is the operator defined above.

Proof. For f,g ∈Swe have: Z Z ˆ 2πix·(ξ+η) Tσ (f,g)(x)= σ(x, ξ, η) f(ξ)ˆg(η)e dξ dη d d ZZZZR R = σ(x, ξ, η) f(y) g(z) e−2πiξ·y e−2πiη·z e2πix·(ξ+η) dξ dη dy dz ZZ

= K(x, y, z) f(y) g(z) dy dz = BK(f,g)(x),

73 where ZZ K(x, y, z)= σ(x, ξ, η)e−2πiξ·(y−x) e−2πiη·(z−x) dξ dη

= F2F3σ(x, y − x, z − x)

F −1 = U 1 σˆ(x, y, z).

th Here, Fj denotes the Fourier transform in the j variable.

We show in the next proposition that the symbol of the bilinear pseudodiﬀerential

1 operator is in M B if and only if the corresponding integral kernel as defined in Ωs 1 R6d Proposition 5.1.3 is in MΩs ,whereBis the invertible transformation defined on defined by

B(X, Y )=(x1,x1 +y2,x1 +y3,x2 +x3 +y1 +y3,−x2,−x3),

3d for X =(x1,x2,x3),Y =(y1,y2,y3)∈R .

1 3d −1 1 3d Proposition 5.1.4. σ ∈ M B (R ) if and only if K = UF σˆ ∈ M (R ). Ωs 1 Ωs

3d 3d Proof. Let G ∈S(R ). For u =(u1,u2,u3), and v =(v1,v2,v3) ∈ R we have h i hF −1F −1 ∗ i h F−1F i VGσ(u, v)= σ, MvTuG = 1 U K, MvTuG = K, U 1 MvTuG . Hence,

−2πi(u2·v2+u3·v3) h i VGσ(u, v)=e K, M(v1+u2+u3,−u2,−u3)T(u1,v2+u1,v3+u1)H

−2πi(u2·v2+u3·v3) = e VHK(u1,v2 +u1,v3 +u1,v1 +u2 +u3,−u2,−u3)

−2πi(u2·v2+u3·v3) =e VHK(B(u, v)), (59)

F −1 ˆ where H = U 1 G. Consequently, we have

|VGσ(u, v)| = |VH K(B(u, v))|.

74 Thus

k k 1 k k 1 K MΩ = σ M B , s Ωs and the proof is complete.

5.1.3 A discrete model

Consider φ ∈S(Rd) that generates a Gabor frame for L2 with (canonical) dual

γ ∈S(Rd).Wecanthenexpandf,g and h in S(Rd)asinTheorem3.8, where the series converge unconditionally in every modulation space norm as long as p, q =6 ∞.

Then using (58), we obtain: ZZZ X hBK (f,g),hi= K(x, y, z) hf,MβlTαkγi MβlTαkφ(y)× R3d k,l∈Zd X X hg,MβnTαmγi MβnTαmφ(z) hh, MβjTαiγi MβjTαiφ(x) dx dy dz m,n∈Zd i,j∈Zd X X X = hf,MβlTαkγihg,MβnTαmγi hh, MβjTαiγi× i,j k,l m,n Z Z Z

K(x, y, z) MβjTαiφ(x)MβlTαkφ(y) MβnTαmφ(z) dx dy dz Rd Rd Rd X X X = hf,MβlTαkγihg,MβnTαmγi hh, MβjTαiγi× i,j k,n l,m

hBK(MβlTαkφ, MβnTαmφ),MβjTαiφi. (60)

The exchange of the integrals and summations above is justiﬁed since f,g,h ∈Shave S absolutely summable Gabor coeﬃcients. Moreover, K ∈S0(R3d)= M∞ (see s≥0 1/Ωs Proposition 2.3.9), and φ ∈Simplies that the triple integral in the second equality

is uniformly bounded with respect to i, j, k, l, m, n ∈ Zd. More precisely, deﬁne

Mβ(j,l,n)Tα(i,k,m)Φ(x, y, z)=MβjTαiφ(x) MβlTαkφ(y) MβnTαmφ(z).

∈S R3d ⊂ 1 ≥ ∈ ∞ Clearly, Mβ(j,l,n)Tα(i,k,m)Φ ( ) MΩs for all s 0. Moreover, K M1/Ωs where s>0. Thus, using the fact that the time-frequency shift operator acts isometrically

75 1 on Mωs ,wehave ZZZ

K(x, y, z) MβjTαiφ(x)MβlTαkφ(y) MβnTαmφ(z) dx dy dz R3d ZZZ

≤ |K(x, y, z)||Mβ(j,l,n)Tα(i,k,m)Φ(x, y, z)| dx dy dz R3d

≤kKk∞ kM T Φk 1 M1/Ωs β(j,l,n) α(i,k,m) MΩs

3 = kKk ∞ kφk 1 . M1/Ωs Mωs

Therefore, to study the boundedness of BK on products of modulation spaces, it

suﬃces to analyze the boundedness of the matrix B =(bij,kl,mn) deﬁned by

bij,kl,mn = hBK(MβlTαkφ, MβnTαmφ),MβjTαiφi (61)

on products of appropriate sequence spaces.

The next theorem will be of special importance in proving our main results. In

particular, it shows that, under some mild condition on its entries, an inﬁnite matrix yields a bounded operator on products of sequence spaces associated with the mod-

ulation spaces. For an inﬁnite matrix (amn,ij,kl), let O denote the bilinear operator associated to it, i.e., X (O(fij), (gkl))mn = amn,ij,kl fij gkl, ij,kl

2d where (fij)and(gkl) are sequences deﬁned on Z .

Theorem 5.1.5. Let ν be an s-moderate weight, and let 1 ≤ pi,qi,ri <∞for i =1,2 be such that 1 = 1 + 1 .If(a ) ∈ `1 (Z6d), then O is a bounded operator from r1 p1 q1 mn,ij,kl Ω˜ s p1,p2 Z2d × q1,q2 Z2d r1,r2 Z2d ∈ 1 Z6d O `ν˜ ( ) `ν˜ ( ) into `ν˜ ( ). In particular, if (amn,ij,kl) ` ( ) then is a bounded operator from `p1,p2 (Z2d) × `q1,q2(Z2d) into `r1,r2 (Z2d).

0 0 ∈ p1,p2 Z2d ∈ q1,q2 Z2d ∈ r1,r2 Z2d 0 0 Proof. Let (fij) `ν˜ ( ), (gkl) `ν˜ ( )and(hmn) `1/ν˜ ( )wherer1,r2

76 are the dual indices of r1, respectively r2.Wehave X |hO((fij), (gkl)), (hmn)i| ≤ |amn,kl,ij||fij||gkl||hmn| m,n,i,j,k,l X 1 ν˜(i, j) 1 ν˜(k, l) = |a | p1 |f | |a | q1 |g | × mn,kl,ij ij ν˜(i, j) mn,kl,ij kl ν˜(k, l) m,n,i,j,k,l 1 0 ν˜(m, n) |a | r1 |h | mn,kl,ij mn ν˜(m, n)

X 1 3 p1 ≤ C |amn,kl,ij| ν˜(i, j) |fij| ω˜s(i, j)× m,n,i,j,k,l 1 q1 |amn,ij,kl| ν˜(k, l) |gkl| ω˜s(k, l)×

1 0 1 |a | r1 |h | ω˜ (m, n) mn,ij,kl ν˜(m, n) mn s X 3 1/p1 = C |a˜mn,kl,ij| |fij| ν˜(i, j)× m,n,i,j,k,l 1/q1 |a˜mn,kl,ij| |gkl| ν˜(k, l)×

0 1 |a˜ | 1/r1 |h | , mn,kl,ij mn ν˜(m, n) where

a˜mn,kl,ij = amn,kl,ij Ω˜ s(m, n, k, l, i, j)=amn,kl,ij ω˜s(m, n)˜ωs(k, l)˜ωs(i, j).

1 We have used the fact that ν,and ν are s-moderate with the same constant C.Since 1 1 1 1 1 1 + = ,orequivalently + + 0 , we can apply H¨older’s inequality to obtain p1 q1 r1 p1 q1 r1 the following:

X 1 3 p1 p1 p1 |hO((fij), (gkl)), (hmn)i| ≤ C |a˜mn,ij,kl||fij| ν˜(i, j) × m,n,i,j,k,l X 1 q1 q1 q1 |a˜mn,ij,kl||gkl| ν˜(k, l) × m,n,i,j,k,l

77 X 1 0 1 0 r1 r1 |a˜mn,ij,kl||hmn| 0 ν˜(m, n)r1 m,n,i,j,k,l 3 ≤ C sup |fij| ν˜(i, j) sup |gkl| ν˜(k, l) × i,j k,l 1 X X sup |hmn| |a˜mn,ij,kl| m,n ν˜(m, n) m,i,k n,j,l

XX p2 1 ≤ 3 k k | |p1 p1 p1 p2 × C amn,ij,kl ` ˜ 1 fij ν˜(i, j) Ωs i j

XX q2 1 q1 q1 q1 q2 |gkl| ν˜(k, l) × k l

0 XX r2 1 0 1 0 0 r1 r1 r2 |hmn| 0 ν˜(m, n)r1 m n

3 p ,p q ,q ≤ C kamn,ij,klk 1 kfi,jk 1 2 kgk,lk 1 2 khm,nk r0 ,r0 , `Ω˜ `ν˜ `ν˜ 1 2 s `1/ν˜ wherewehaveusedthefactthat`p,q(Z2d) ⊂ `∞(Z2d), i.e., XX p q/p 1/q sup |xm,n|≤ |xm,n| . m,n n∈Zd m∈Zd p,q Moreover, using the duality of the `ν˜ -spaces i.e., X k k r ,r | || | a ` 1 2 =sup am,n bm,n , ν˜ kk br0,r0 =1 ∈Zd `12 m,n 1/ν˜ we get that

3 kO k r ,r ≤ k k 1 k k p ,p k k q ,q ((fij), (gkl)) ` 1 2 C amn,ij,kl ` (fij) ` 1 2 (gkl) ` 1 2 . ν˜ Ω˜ s ν˜ ν˜

The second part of the theorem follows by choosing ν = ω0 ≡ 1.

5.1.4 Boundedness of bilinear pseudodiﬀerential operators

Our ﬁrst main result of this section shows that a bilinear integral operator with kernel

1 in the modulation space MΩs — in particular, in the Feichtinger algebra — gives rise to a bounded operator.

78 Theorem 5.1.6. Let ν be an s-moderate weight, and let 1 ≤ pi,qi,ri <∞for i =1,2 be such that 1 + 1 = 1 .IfK∈M1(R3d), then the bilinear integral operator B p1 q1 r1 Ωs K p1,p2 Rd × q1,q2 Rd deﬁned by (58) can be extended as a bounded operator from Mν ( ) Mν ( )

r1,r2 Rd into Mν ( ).

Proof. Let f,g,h ∈S(Rd) and expand each of these functions into their Gabor se- P P h i h i ries, i.e., f = i,j f,MβjTαiφ MβjTαiγ, g = k,l g,MβlTαkφ MβlTαkγ,andh= P h i m,n h, MβnTαmφ MβnTαmγ,whereφand γ are dual Gabor frames as in Theo- rem 2.3.11. By Proposition 5.1.1, the matrix deﬁned by (61) belongs to `1 since Ω˜s ∈ 1 K MΩs . Therefore, using Theorem 5.1.5 we have the following estimates: X X X |hBK(f,g),hi| = | amn,ij,kl hf,MβjTαiφihg,MβlTαkφi hh, MβnTαmφi| mn ij kl

≤ k k 1 kh ik p ,p × C amn,ij,kl ` f,MβjTαiφ ` 1 2 Ω˜ s ν˜

khg,M T φik q1,q2 khh, M T φik r0 ,r0 βl αk `ν˜ βn αm 1 2 `1/ν˜

≤ C kKk 1 kfk p1,p2 kgk q1,q2 khk r0 ,r0 , MΩs Mν Mν 1 2 M1/ν by duality we obtain

kB (f,g)k r1,r2 ≤ C kKk 1 kfk p1,p2 kgk q1,q2 . K Mν MΩs Mν Mν

The result then follows by standard density arguments, using the fact that S(Rd)is

p,q ≤ ∞ dense in Mν for 1 p, q < .

The previous result together with Propositions 5.1.3 and 5.1.4 yields our second main result of this chapter, which provides a suﬃcient condition on the symbol so that the operator (56) is bounded on products of modulation spaces. Recall that the invertible transformation B was deﬁned on R6d by

B(X, Y )=(x1,x1 +y2,x1 +y3,x2 +x3 +y1 +y3,−x2,−x3).

79 Theorem 5.1.7. Let ν be an s-moderate weight, and let 1 ≤ pi,qi,ri <∞for i =1,2

1 1 1 1 3d be such that + = .Ifσ∈MB(R), then the bilinear pseudodiﬀerential p1 q1 r1 Ωs p1,p2 Rd × operator Tσ deﬁned by (56) can be extended to a bounded operator from Mν ( )

q1,q2 Rd r1,r2 Rd Mν ( ) into Mν ( ).

1 1 Proof. By Proposition 5.1.4, σ ∈ M B if and only if K ∈ M ,whereKis the kernel Ωs Ωs of the corresponding integral operator, and the result follows from Theorem 5.1.6.

If we assume that ν = ω0 ≡ 1, and that p1 = p2 = p and q1 = q2 = q (hence r1 = r2 = r), we obtain the following.

≤ ∞ ≤ ≤ 1 1 1 Corollary 5.1.8. Let 2 p, q < and 1 r 2 be such that p + q = r .If 1 3d p d q d σ∈M(R), then Tσ can be extended to a bounded operator from L (R ) × L (R )

r d 1 3d into L (R ). In particular, if σ ∈ M (R ), then Tσ has a bounded extension from L2(Rd) × L2(Rd) into L1(Rd).

Proof. If of ≤ p, q < ∞ by Proposition 4.2.1 we have the following embeddings:

Lp ⊂ M p, and Lq ⊂ M q.

Thus, Lp×Lq ⊂ M p×M q.Moreover,since1≤r≤2, we have by the same proposition

that M r ⊂ Lr. These continuous embeddings combined with Theorem 5.1.7 imply then the result.

Remark 5.1.9. It is remarkable that the condition σ ∈ M 1(R3d) does not necessarily

imply any smoothness nor decay on the symbol. In particular, Coifman-Meyer-type conditions (57) are not necessarily satisﬁed by the symbols we consider.

2 2 s/2 Assume that ν(x, y)=ωs(x, y)=(1+|x| +|y|) for some s>0, and that 1 Rd ×{ } pi = qi =2.Letωs be the restriction of ωs to 0 . Then the following holds.

1 Corollary 5.1.10. If σ ∈ M B then T can be extended as a bounded bilinear pseu- Ωs σ 2 2 1 dodiﬀerential operator from M × M into L 1 . ωs ωs ωs

80 1 1 Proof. Notice that M is continuously embedded in L 1 , cf. [41, Prop. 12.1.4]. So, ωs ωs we only need to prove that under the hypotheses of the corollary, the bilinear pseu-

2 × 2 dodiﬀerential operator can be extended to a bounded operator from Mωs Mωs into

1 Mωs . This follows from Theorem 5.1.7 by taking ν = ωs.

Remark 5.1.11. a. If the symbol σ satisﬁes the estimates

| β γ |≤ | | | | −d− ∂ξ ∂η σ(x, ξ, η) Cβ,γ (1 + ξ + η ) (62)

for all (x, ξ, η) ∈ R3d, all multi-indices β and γ,andsome>0, then it

follows from [7, Theorem 1] that the corresponding bilinear pseudodiﬀerential

operator is bounded from L2 × L2 into L1. We wish to point out that, in general, neither that result nor Corollary 5.1.8 in this section imply each other.

2d On one hand, if g ∈S(R )thenσ1(x, ξ, η)=χ[0,1[d (x)g(ξ,η), where χ[0,1[d is the characteristic function of the unit cube in Rd, satisﬁes (62) and hence

2 2 1 it yields a bounded operator from L × L into L . However, because σ1 is not a continuous function, it is not in M 1(R3d). Therefore, our corollary does

not apply. On the other hand, functions in M 1 must be continuous, but there are non-diﬀerentiable functions in M 1, hence they do not satisfy (62), thus [7,

Theorem 1] does not apply here.

b. Notice that (62) requires smoothness of the symbols only in the ξ and η variables

whereas (57) imposes smoothness on all the variables x, ξ and η.Thusthetwo

conditions are diﬀerent.

5.2 Linear Hilbert transform on the modulation spaces

In this section, we consider the boundedness of the one-dimensional (linear) Hilbert transform — which is a prototypical example of a singular integral operator — on

81 the modulation spaces. The Hilbert transform of a function f ∈S(R) is deﬁned by Z Z − 1 Hf(x)= 1 lim f(x t) dt = lim f(t) dt. (63) π → t → x−t 0 |t|> π 0 |x−t|>

The boundedness of the Hilbert transform on the Lp spaces (1

initiated by Besicovitch and Titchmarsh, and further developed by Calder´on and

Zygmund, establishes that the Hilbert transform is of weak-type (1, 1) (in fact, their

theory applies to more general operators). More precisely, the following estimate

holds: Z C |{x ∈ R :(Hf(x)) >α}| ≤ |f(x)| dx, α R where C is a constant independent of f and α>0. Moreover, using a Fourier approach, it is easy to prove the boundedness of the Hilbert transform on L2. Indeed,

it is known that Hfd(ω)=−isign(ω)fˆ(ω), where sign denotes the sign function.

Thus, using Plancherel’s theorem we obtain

d kHfkL2 = kHfkL2

ˆ = kfkL2

= kfkL2,

and this last equality proves the boundedness of H on L2.Theweak-type(1,1) result,

and the L2 boundedness together with duality and interpolation methods can be used to prove that H is bounded on Lp for 1

integrals.

We prove the boundedness of H on the modulation spaces M p,q, for 1

by converting the boundedness question into the boundedness of an inﬁnite matrix

acting on appropriate sequence spaces. We achieve this goal by expanding the function

82 on which H acts into their Gabor expansions, and thus we are reduced to studying the boundedness of an associated inﬁnite matrix on `p,q(Z × Z).

Let α, β > 0 be given, and let φ, γ ∈S(R) be such that supp(ˆγ) ⊂ (0,β)and

γˆ≥0. Assume, moreover, that {MβnTαkφ}k,n,and{MβnTαkγ}k,n are dual Gabor

2 frames for L (R). Then {MβnTαkφ}k,n and {MβnTαkγ}k,n are also dual Gabor frames for all the modulation spaces M p,q(R) (see Theorem 2.3.11).

Proposition 5.2.1. Let f,g ∈S(R), then X X hHf,gi = hf,MβnTαmφi hg,MβlTαkφihHMβnTαmγ,MβlTαkγi. (64) m,n∈Z k,l∈Z

Proof. If f,g ∈S(R), then we can expand them into their Gabor expansions, i.e., X X f = hf,MβnTαmφi MβnTαmγ, and g = hg,MβlTαkφi MβlTαkγ, m,n∈Z k,l∈Z

with unconditional convergence in any modulation space. In particular, the assump-

tions on f,g,φ,andγimply that the Gabor coeﬃcients of f and g are absolutely

summable. Then we have: Z hHf,gi = Hf(x) · g(x)dx R Z X = hg,MβlTαkφi Hf(x) · MβlTαkγ(x) dx. R k,l∈Z

2 1 Because H is bounded on L ,wehavethatforeachk, n ∈ Z, Hf ·MβlTαkγ ∈ L (R).

1 Moreover, since hg,MβlTαkφi∈`(Z×Z), we can apply Fubini’s Theorem to obtain:

X Z hHf,gi = hg,MβlTαkφi Hf(x)· MβlTαkγ(x) dx. (65) R k,l∈Z Further, the adjoint H∗ of H is bounded on L2 (in fact, one can show that H∗ =

83 −H), so we have the following: Z

Hf(x)MβlTαkγ(x) dx = hHf,MβlTαkγi R

∗ = hf,H MβlTαkγi

= −hf,HMβlTαkγi X = hf,MβnTαmφihHMβnTαmγ,MβlTαkγi. (66) m,n∈Z

We can now use (65) and (66) to obtain; XX hHf,gi = hf,MβnTαmφi hg,MβlTαkφihHMβnTαmγ,MβlTαkγi. (67) k,l m,n

2 We denote Aα,β,γ the sequence deﬁned for (k, l) ∈ Z by

Aα,β,γ (k, l)=|Vγˆγˆ(βl,αk)|.

The choice of the window γ implies in particular that γ ∈ M 1, hence X Aα,β,γ(k, l) < ∞. l,k∈Z

We can use the above proposition to prove the following.

Proof. Let S be the sign function deﬁned by    −  1:x<0, S(x)=sign(x)= 0:x=0,   +1 : x>0.

84 We use again the L2-theory of the Hilbert transform. In particular, using the Fourier transformwehave

∧ ∧ hHMβnTαmγ,MβlTαkγi = h(HMβnTαmγ) , (MβlTαkγ) i

∧ ∧ = −ihS · (MβnTαmγ) , (MβlTαkγ) i

2πiαβ(kl−mn) = −ie hS · M−αmTβnγ,Mˆ −αkTβlγˆi

2πiαβk(n−l) = −ie Vγˆ((T−βnS) · γˆ)(β(l − n),α(m−k)). (69)

With that choice of γ it is easy to see that for all n ∈ Z,

T−βnS · γˆ = γ.ˆ (70)

Hence, (69) becomes

2πiαβk(n−l) hHMβnTαmγ,MβlTαkγi = ie Vγˆγˆ(β(l − n),α(m−k)). (71)

By putting all the above together into (65) we have X X 2πiαβk(n−l) hHf,gi = i e Vγˆγˆ(β(l−n),α(m−k)) hf,MβnTαmφi hg,MβlTαkφi. m,n∈Z k,l∈Z (72)

Taking the magnitude of both sides then yields the desired result.

We are now ready to prove the boundedness of the Hilbert transform on all the

modulation spaces M p,q with 1

Theorem 5.2.3. Let 1

bounded linear operator on M p,q. In particular, for any f ∈ M p,q we have the following

estimate:

kHfkMp,q ≤ C kfkM p,q ,

for some positive constant C independent of f.

≤kh ik p,q k ∗|h i|k 0 0 f,Mβ·Tα·φ ` Aα,β,γ g,Mβ·Tα·φ `p ,q

≤k k 1 kh ik p,q kh ik 0 0 Aα,β,γ ` f,Mβ·Tα·φ ` g,Mβ·Tα·φ `p ,q

≤ k k 1 k k p,q k k 0 0 C γ M f M g M p ,q .

We have used Young’s inequality to obtain the fourth inequality. By duality we then obtain

kHfkMp,q ≤ C kγkM 1 kfkM p,q .

The result then follows since S(R) is a dense subspace of each of the modulation

spaces M p,q for 1

Remark 5.2.4. The technique that we use to prove the boundedness of H on the

modulation spaces is diﬀerent from the one used in the Lp theory, but relies heavily

on the L2 theory. We remark that the H cannot be bounded on the Feichtinger algebra M 1.To

seethisnoticethatM1 is a dense subspace of L2, and that functions in M 1 are

continuous. Since M 1 is invariant under the Fourier transform, it is easy to see that

for f ∈ M 1,

Hf ∈ M1 ⇐⇒ Hfd ∈ M1.

However, Hfd = −S ·f,ˆ and this function cannot belong to M 1 since it has a discon-

tinuity at the origin.

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91 VITA

Kasso Akochayé Okoudjou was born on March 30, 1973, in Parakou, Bénin. He attended both Lycée Toffa and Lycée Béhanzin high schools in Porto-Novo, Bénin and graduated from the latter in 1991. Subsequently, he entered the UniversitéNa- tionale du Bénin in Abomey-Calavi, Bénin and completed a Maitrise ès Sciences Mathématiques in 1996. After teaching mathematics at Complexe Scolaire William Ponty high school in Porto-Novo, Bénin from 1996 to 1998, he was admitted to the Ph.D. program in the School of Mathematics at the Georgia Institute of Technology in the fall of 1998. He graduated from Georgia Tech with an M. S. in Electrical and Computer Engineering in May 2003, and with a Ph. D. in Mathematics in August 2003. His thesis was written under the supervision of Professor Christopher Heil. He then moved to Ithaca, NY, to join the Mathematics Department of Cornell University as an H. C. Wang Assistant Professor.