Classical and Multilinear Harmonic Analysis -...

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Editorial BoardB. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN,P. SARNAK, B. SIMON, B. TOTARO

Classical and Multilinear Harmonic AnalysisThis two-volume text in harmonic analysis introduces a wealth of analytical results andtechniques. It is largely self-contained and is intended for graduates and researchers inpure and applied analysis. Numerous exercises and problems make the text suitable forself-study and the classroom alike.

The first volume starts with classical one-dimensional topics: Fourier series; harmonicfunctions; Hilbert transforms. Then the higher-dimensional Calderon–Zygmund andLittlewood–Paley theories are developed. Probabilistic methods and their applicationsare discussed, as are applications of harmonic analysis to partial differential equations.The volume concludes with an introduction to the Weyl calculus.

This second volume goes beyond the classical to the highly contemporary and focuseson multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman–Meyer theory; Carleson’s resolution of the Lusin conjecture; Calderon’s commutatorsand the Cauchy integral on Lipschitz curves. The material in this volume has not beencollected previously in book form.

Camil Muscalu is Associate Professor in the Department of Mathematics at CornellUniversity.

Wilhelm Schlag is Professor in the Department of Mathematics at the University ofChicago.

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Classical and MultilinearHarmonic Analysis

Volume II

CAMIL MUSCALUCornell University

WILHELM SCHLAGUniversity of Chicago






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Contents

Preface page ixAcknowledgements xv

1 Leibnitz rules and the generalized Korteweg–de Vriesequation 11.1 Conserved quantities 21.2 Dispersive estimates for the linear equation 51.3 Dispersive estimates for the nonlinear equation 81.4 Wave packets and phase-space portraits 171.5 The phase-space portraits of e2πix2

and e2πix321

1.6 Asymptotics for the Airy function 26Notes 28Problems 28

2 Classical paraproducts 292.1 Paraproducts 302.2 Discretized paraproducts 342.3 Discretized Littlewood–Paley square-function operator 382.4 Dualization of quasi-norms 452.5 Two particular cases of Theorem 2.3 462.6 The John–Nirenberg inequality 512.7 L1,∞ sizes and L1,∞ energies 542.8 Stopping-time decompositions 572.9 Generic estimate of the trilinear paraproduct form 592.10 Estimates for sizes and energies 602.11 Lp bounds for the first discrete model 622.12 Lp bounds for the second discrete model 64

v






vi Contents

2.13 The general Coifman–Meyer theorem 672.14 Bilinear pseudodifferential operators 71

Notes 75Problems 76

3 Paraproducts on polydisks 793.1 Biparameter paraproducts 803.2 Hybrid square and maximal functions 833.3 Biparameter BMO 863.4 Carleson’s counterexample 913.5 Proof of Theorem 3.1; part 1 963.6 Journe’s lemma 1043.7 Proof of Theorem 3.1; part 2 1083.8 Multiparameter paraproducts 1143.9 Proof of Theorem 3.1; a simplification 1163.10 Proof of the generic decomposition 121

Notes 124Problems 124

4 Calderon commutators and the Cauchy integral onLipschitz curves 1264.1 History 1264.2 The first Calderon commutator 1354.3 Generalizations 1554.4 The Cauchy integral on Lipschitz curves 1574.5 Generalizations 183


5 Iterated Fourier series and physical reality 1875.1 Iterated Fourier series 1875.2 Physical reality 1905.3 Generic Lp AKNS systems for 1 ≤ p < 2 1985.4 Generic L2 AKNS systems 201


6 The bilinear Hilbert transform 2086.1 Discretization 2106.2 The particular scale-1 case of Theorem 6.5 218






Contents vii

6.3 Trees, L2 sizes, and L2 energies 2206.4 Proof of Theorem 6.5 2256.5 Bessel-type inequalities 2266.6 Stopping-time decompositions 2326.7 Generic estimate of the trilinear BHT form 2366.8 The 1/2 < r < 2/3 counterexample 2386.9 The bilinear Hilbert transform on polydisks 240


7 Almost everywhere convergence of Fourier series 2447.1 Reduction to the continuous case 2457.2 Discrete models 2487.3 Proof of Theorem 7.2 in the scale-1 case 2547.4 Estimating a single tree 2577.5 Additional sizes and energies 2617.6 Proof of Theorem 7.2 2637.7 Estimates for Carleson energies 2657.8 Stopping-time decompositions 2677.9 Generic estimate of the bilinear Carleson form 2697.10 Fefferman’s counterexample 271


8 Flag paraproducts 2758.1 Generic flag paraproducts 2768.2 Mollifying a product of two paraproducts 2778.3 Flag paraproducts and quadratic NLS 2828.4 Flag paraproducts and U-statistics 2878.5 Discrete operators and interpolation 2888.6 Reduction to the model operators 2918.7 Rewriting the 4-linear forms 2948.8 The new size and energy estimates 2958.9 Estimates for T1 and T1,�0 near A4 2968.10 Estimates for T ∗3

1 and T ∗31,�0

near A31 and A32 2988.11 Upper bounds for flag sizes 2998.12 Upper bounds for flag energies 308







viii Contents

9 Appendix: Multilinear interpolation 311Notes 317

References 318Index 323






Preface

This is the second volume of our textbook devoted to harmonic analysis. Thefirst volume commenced with the one-dimensional theory of Fourier series,harmonic functions, their conjugates, the Hilbert transform and its bounded-ness properties. It then moved on to the higher-dimensional theory of singu-lar integrals, the Calderon–Zygmund and Littlewood–Paley theorems, and therestriction theory of the Fourier transform, as well as a brief introduction topseudodifferential operators via the Weyl calculus. As Vol. I aims for breadth italso includes some basic probability theory and demonstrates how this relatesto Fourier analysis. In addition, some application to PDEs are described. Forexample, in Chapter I.10 we discuss the uncertainty principle and how it allowsfor a simple proof of the Malgrange–Ehrenpreis theorem on the local solvabil-ity of constant-coefficient PDEs. In Chapter 11, which is devoted to restrictionphenomenon for the Fourier transform, applications to dispersive evolutionequations such as the wave and Schrodinger equations appear in the form ofStrichartz estimates.

This second volume is more specialized in the sense that it is entirely devotedto multilinear aspects of singular integrals and pseudodifferential operators.However, at the same time it covers a wide range of topics within that area.By design, each topic is presented within the framework of a few overarchingprinciples. Amongst these the most fundamental notion is that of a paraproduct,and we devote the first three chapters of the present volume to the introduction,motivation, and development of this basic idea. The immediate aim of thesethree chapters is a systematic and unifying treatment of fractional Leibnitzrules. In the later chapters, which analyze more difficult operators such asthe Carleson maximal function, the bilinear Hilbert transform, and Calderoncommutators, it becomes necessary to understand the combinatorics involvedin handling many paraproducts simultaneously.

ix






x Preface

While paraproducts made an appearance in the first volume of our book,in the context of Haar functions and their use, in the proof of the T (1)theorem, in this present volume we delve more deeply into the structure ofthese objects.

The core of Vol. II consists of three main strands, which are very muchinterwoven:(1) Calderon commutators and the Cauchy integral on Lipschitz curves;(2) the bilinear Hilbert transform;(3) Carleson’s theorem on the almost everywhere convergence of L2 Fourier

series.While the relation between topics (1) and (2) was observed many years

ago by Calderon, who also introduced them into harmonic analysis, the closerelation between (2) and (3) is a more recent discovery.

Let us now give a brief synopsis of the history of each of these topics. Inhis thesis of 1915, Lusin conjectured that the Fourier series of any L2 functionconverges almost everywhere. The question of whether this is true proved to bedifficult to answer and, indeed, out of the reach of classical methods. In 1922Kolmogorov famously constructed an example of an L1 function for which theassociated Fourier series diverges almost everywhere. In an attempt to shedsome light on the L2 case, Paley and Zygmund studied random Fourier seriesand established a theorem stating that an L2 series with random coefficientsconverges almost surely almost everywhere; see Chapter 6 of Vol. I for bothKolmogorov’s example and random Fourier series.

It was not until 1966 that Carleson established Lusin’s conjecture as correct.The Lp version of Carleson’s theorem for 1 < p <∞ was obtained shortlythereafter by Hunt. Carleson’s theorem is based on a deep phase-space analysisof L2 functions.

If convergence in L2 of Fourier series is equivalent to the L2-boundednessof each partial-sum operator SNf (x), independently of N (which is an easyconsequence of Plancherel’s theorem), it is natural to expect that proving thealmost everywhere convergence requires the handling of infinitely many such(localized) partial Fourier sums simultaneously. This is a very difficult tasksince in any collection the partial sums may overlap with each other both inspace and in frequency. To prove his theorem, Carleson invented an intricatecombinatorial and analytical way of organizing these partial sums, based ongeometric intuition coming from the phase-space picture and the Heisenbergprinciple, and which rendered these carefully selected partial sums almostorthogonal.

A few years after Carleson’s breakthrough, Fefferman gave a new proof ofthe Carleson–Hunt theorem by building upon, as well as simplifying, some of






Preface xi

these ideas. In addition, he also introduced the modern language of tiles andtrees, which has been used in the field ever since. A proof of the Carleson–Hunttheorem is given in Chapter 7 of this volume.

In the 1960s, Calderon introduced commutators and the Cauchy integral onLipschitz curves as part of his program in the study of PDEs. These interestingand natural singular integrals are no longer of the convolution type and cannotbe treated by means of the earlier Calderon–Zygmund theory. By using acombination of methods from complex and harmonic analysis, Calderon provedthe Lp-boundedness of his first commutator in 1965, but his method could nothandle the second commutator let alone the higher ones. In spite of seriousefforts by the harmonic analysis community, these problems resisted solutionuntil 1975 when Coifman and Meyer proved the desired bounds for the secondcommutator and shortly afterwards for all Calderon commutators. Their proofbuilds on some of Calderon’s ideas, but it was based entirely on harmonicanalysis techniques. These authors were the first to realize in a profound waythat the commutators are in fact multilinear operators, and their method of proofused this observation crucially. Around the same time they proved what is nowcalled the Coifman–Meyer theorem on paraproducts. It is also interesting tonote that all these multilinear operators came out of studies of linear PDEproblems. A few years later, however, Bony realized that paraproducts play animportant role in nonlinear PDEs, and they continue to do so to this day.

After all these developments, the Cauchy integral on Lipschitz curves (whichinitially appeared in complex analysis) was the last operator that remained tobe understood. The Cauchy integral possesses the remarkable feature of beingnaturally decomposable as an infinite sum of all Calderon commutators, inwhich the simplest (linear) term equals the classical Hilbert transform. In orderto prove estimates for it one therefore needed to prove sufficiently good boundsfor the operator norms of all the commutators, so as to be able to sum theircontributions. The initial proof by Coifman and Meyer was very complicatedand it was not clear what type of bounds it yielded for the commutators.Calderon was the first to obtain exponential bounds, thus proving Lp estimatesfor the Cauchy integral under the assumption that the Lipschitz constant is small.It had also been observed that polynomial bounds for the commutators wouldallow for a complete understanding of the Cauchy integral on Lipschitz curvesand many of its natural extensions. The final breakthrough was achieved in 1982by Coifman, McIntosh, and Meyer, who established the desired polynomialbounds for Calderon commutators. Chapter 4 of this volume is devoted to theproofs of these results. The proofs we give there differ from those in the originalliterature.






xii Preface

Prior to the resolution of his conjectures on the commutators, Calderon pro-posed the study of the bilinear Hilbert transform. He viewed this as a steptowards the commutators and eventually to the Cauchy integral on Lipschitzcurves. It had been observed that the first commutator was equal to an averageof such bilinear operators. While the entire Calderon program was settled with-out the use of the bilinear Hilbert transform, the question whether this operatorsatisfies any Lp estimates remained open. It was finally answered affirmativelyby Lacey and Thiele in two papers, in 1997 and 1999. As it turned out, theanalysis of the bilinear Hilbert transform is very closely related to the analysisof the Carleson maximal operator, which appeared implicitly in the proof of thealmost everywhere convergence of Fourier series. A brief explanation for thisis as follows. When viewed as a bilinear multiplier, the symbol of the bilinearHilbert transform is singular along a one-dimensional line. This line regulatesthe one-dimensional modulation symmetry of the bilinear Hilbert transform,which is identical to the modulation symmetry of the Carleson operator. Inparticular, both these objects have precisely the same symmetries: translation,dilation, and modulation invariances. By comparison, paraproducts have clas-sical Marcinkiewicz–Mikhlin–Hormander symbols and these are smooth awayfrom the origin; in other words they have a zero-dimensional singularity. Theyhave only translation and dilation invariance, the usual symmetries of the clas-sical Calderon–Zygmund convolution operators. A proof of the boundednessof the bilinear Hilbert transform appears in Chapter 6 of this volume.

Let us conclude with a few words about Chapter 5, which describes the morerecent theory of iterated Fourier series and integrals. This chapter is almostentirely devoted to motivation and is somewhat speculative in character. Tobe more specific, we describe what appears to be a very natural “physical”problem where both the Carleson maximal operator and the bilinear Hilberttransform appear simultaneously. In fact, they are the simplest operators in aninfinite series that determines the solutions of a certain ODE. This problem goesbeyond multilinear harmonic analysis and may be said to belong to nonlinearharmonic analysis, since the study of its multilinear building blocks aloneis insufficient for its complete understanding. It is towards such a theory ofnonlinear harmonic analysis that Chapter 5 aspires.

How to use this volume. There are several options. For instance, after com-pleting Chapters 7 and 8 of Vol. I on the classical theory of singular integrals,it is natural to move on to some of the more advanced topics presented in thisvolume, such as the theory of Calderon commutators and the Cauchy integralon Lipschitz curves. We would like to emphasize that this is indeed possi-ble since Chapter 4 of the present volume, where these topics are covered,






Preface xiii

assumes familiarity with only basic harmonic analysis such as maximal func-tions, Calderon–Zygmund operators, and Littlewood–Paley square functions.It does not rely on any other material from Vol. I such as, for example, the T (1)theorem, bounded mean oscillation space (BMO), or Carleson measures.

However, for an audience that is more inclined towards learning techniquesuseful for PDEs, it might be advisable to focus on paraproducts and Leibnitzrules. In that case, Chapters 1, 2, 3, and 8 would be the natural order in whichto proceed.

A more mature reader wishing to study the almost everywhere convergence ofFourier series can in principle start with Chapter 7. However, that chapter doesrely on some technical results from the preceding chapter, where the bilinearHilbert transform is presented. We therefore recommend that Chapter 7 shouldbe attempted only after mastering Chapter 6, which can be read independentlyby anyone familiar with paraproducts and the John–Nirenberg inequality (whichare covered in Chapter 2).

Finally, we would like to stress that this volume was designed for the specificpurpose of fitting seemingly disparate objects into a unifying framework. Theclarity and transparency that we hope to have achieved by doing so will onlybe appreciated by the patient reader willing to take the journey from beginningto end.

A few words are in order concerning interpolation. It is used frequentlyeither in the more standard linear form of the Riesz–Thorin and Marcinkiewicztheorems (both in the Banach and the quasi-Banach context) or in the multilinearsetting described in detail in the appendix. For statements of the standardinterpolation theorems, see for example Chapter 1 of Vol. I.

Feedback. The authors welcome comments on this book and ask that theybe sent to [email protected].






Acknowledgements

Wilhelm Schlag expresses his gratitude to Rowan Killip for detailed commentson his old harmonic analysis notes from 2000, from which the first volumeof this book eventually emerged. Furthermore, he thanks Serguei Denissov,Charles Epstein, Burak Erdogan, Patrick Gerard, David Jerison, Carlos Kenig,Andrew Lawrie, Gerd Mockenhaupt, Paul Muller, Casey Rodriguez, BarrySimon, Chris Sogge, Wolfgang Staubach, Eli Stein, and Bobby Wilson formany helpful suggestions and comments on a preliminary version of Vol. I.Finally, he thanks the many students and listeners who attended his lecturesand classes at Princeton University, the California Institute of Technology, theUniversity of Chicago, and the Erwin Schrodinger Institute in Vienna overthe past ten years. Their patience, interest, and helpful comments have led tonumerous improvements and important corrections.

The second volume of the book is partly based on two graduate courses givenby Camil Muscalu at Scoala Normala Superioara, Bucuresti, in the summer of2004 and at Cornell University in the fall of 2007. First and foremost he wouldlike to thank Wilhelm Schlag for the idea of writing this book together. Then,he would like to thank all the participants of those classes for their passionfor analysis and for their questions and remarks. In addition, he would like tothank his graduate students Cristina Benea, Joeun Jung, and Pok Wai Fong fortheir careful reading of the manuscript and for making various corrections andsuggestions and Pierre Germain and Raphael Cote for their meticulous com-ments. He would also like to thank his collaborators Terry Tao and ChristophThiele. Many ideas that came out of this collaboration are scattered throughthe pages of the second volume of the book. Last but not least, he would liketo express his gratitude to Nicolae Popa from the Institute of Mathematics ofthe Romanian Academy for introducing him to the world of harmonic analysisand for his unconditional support and friendship over the years.

xv






xvi Acknowledgements

Many thanks go to our long-suffering editors at Cambridge University Press,Roger Astley and David Tranah, who continued to believe in this project andsupport it even when it might have been more logical not to do so. Theircheerful patience and confidence is gratefully acknowledged. Barry Simon atthe California Institute of Technology deserves much credit for first suggestingto David Tranah roughly ten years ago that Wilhelm Schlag’s harmonic analysisnotes should be turned into a book.

The authors were partly supported by the National Science Foundation duringthe preparation of this book.






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