Lecture Notes in Mathematics 1924 Editors: J.-M.Morel,Cachan F.Takens,Groningen B.Teissier,Paris Michael Wilson Weighted Littlewood-Paley Theory and Exponential-Square Integrability ABC Author MichaelWilson DepartmentofMathematics UniversityofVermont Burlington,Vermont05405 USA e-mail:[email protected] LibraryofCongressControlNumber:2007934022 MathematicsSubjectClassification(2000):42B25,42B20,42B15,35J10 ISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 ISBN978-3-540-74582-2SpringerBerlinHeidelbergNewYork DOI10.1007/978-3-540-74587-7 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2008 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. TypesettingbytheauthorandSPiusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:12114160 41/SPi 543210 I dedicate this book to my parents, James and Joyce Wilson. Preface Littlewood-Paleytheorycanbethoughtofasaprofoundgeneralizationofthe Pythagorean theorem.(cid:1)If x∈Rd—say, x=(x1, x2, ... , xd)—then we define x’s norm, (cid:2)x(cid:2), to be ( dx2)1/2. This norm has the good property that, if 1 n y =(y1, y2, ... , yd)isanyothervectorinRd,and|yn|≤|xn|foreachn,then (cid:2)y(cid:2) ≤ (cid:2)x(cid:2). In other words, the size of x, as measured by the norm function, is determined entirely by the sizes of x’s components. This remains true if we let the dimension d increase to infinity, and define the no(cid:1)rm of a vector (actually, an infinite sequence) x=(x , x , ...) to be (cid:2)x(cid:2)≡( ∞x2)1/2. 1 2 1 n In analysis it is often convenient (and indispensable) to decompose func- tions f into infinite series, (cid:2) f(x)= λnφn(x), (0.1) where the functions φn come from some standard family (such as the Fourier system) and the λn’s are complex numbers. (For the time being we will not specify how the series 0.1 is supposed to converge.) Typically the coefficients λn are defined by integrals of f against some other functions ψn. If we are interested about convergence in the sense of L2 (or “mean-square”), and if the φn’s comprise a complete orthonormal family, then each ψn can be taken to be φ¯n, the complex conjugate of φn; i.e., (cid:3) λn = f(x)φ¯n(x)dx, and we have (cid:3) (cid:2) |f(x)|2dx= |λn|2. (For the time being we will not specify the domain on which f and the φn’s are defined.) If we are only interested inL2 functions, then the natural norm, (cid:4)(cid:3) (cid:5)1/2 (cid:6)(cid:2) (cid:7)1/2 (cid:2)f(cid:2)2 ≡ |f(x)|2dx = |λn|2 , VIII Preface has th(cid:1)e same domination property possessed by the Euclidean norm on Rd: if g = γnφn and |γn|≤|λn| for all n, then (cid:2)g(cid:2)2 ≤(cid:2)f(cid:2)2. Even better, if, for some (cid:5)>0, we have |γn|≤(cid:5)|λn| for all n, then (cid:2)g(cid:2)2 ≤(cid:5)(cid:2)f(cid:2)2. Unfortunately, L2 is not always the most useful function space for a given problem. We might want to work in L4, with its norm defined by (cid:4)(cid:3) (cid:5) 1/4 (cid:2)f(cid:2) ≡ |f(x)|4dx . 4 To make things specific, let’s suppose that our functions are defined on [0,1). The collection {exp(2πinx)}∞ defines a complete orthonormal family in −∞ L2[0,1). Now, if f ∈L4[0,1), then the coefficients (cid:3) 1 λn ≡ f(x) exp(−2πinx)dx 0 are defined, and the infinite series, (cid:2)∞ λnexp(2πinx), −∞ converges to f in the L4 sense, if we sum it up right. But the domination property fails in a very strong sense. Given f ∈ L4, and given an integrable function g such |γn|≤|λn| for all n, where (cid:3) 1 γn = g(x) exp(−2πinx)dx, 0 there is no reason to expect that (cid:2)g(cid:2) is even finite, let alone controlled by 4 (cid:2)f(cid:2) . 4 Littlewood-Paley theory provides a way to almost preserve the domina- tion property. To each function f, one associates something called the square function of f, denoted S(f). (Actually, the square function comes in many guises, but we will not go into that now.) Each square function is defined via innerproductswithafixedcollectionoffunctions.Sometimesthiscollectionis a complete orthonormal family for L2, but it doesn’t have to be. The square function S(f)(x) is defined as a weighted sum (or integral) of the squares of the inner products, |(cid:5)f,φ(cid:6)|2, where φ belongs to the fixed collection. The function S(f)(x) varies from point to point, but, if f and g are two functions such that |(cid:5)g,φ(cid:6)|≤|(cid:5)f,φ(cid:6)| for all φ, then S(g)(x)≤S(f)(x) everywhere. The square function S(f) also has the property that, if 1 < p < ∞, and f ∈ Lp, the Lp norms of S(f) and f are comparable. Thecombinationofthesetwofacts—dominationpluscomparablility—lets us, in many situations, reduce the analysis of infinite series of functions, (cid:2) f(x)= λiφi(x), Preface IX to the analysis of infinite series of non-negative functions, (cid:6)(cid:2) (cid:7) 1/2 S(f)(x)= |γi|2|ψi(x)|2 ; and that greatly simplifies things. We have already mentioned the practice, common in analysis, of cutting a function into infinitely many pieces. Typi- cally we do this to solve a problem, such as a PDE. We break the data into infinitely many pieces, solve the problem on each piece, and then sum the “piece-wise” solutions. The sums encountered this way are likely to contain a lot of complicated cancelations. Littlewood-Paley theory lets us control them by means of sums that have no cancelations. The mutual control between |f| and S(f) is very tight. We will soon show that,iff isaboundedfunctiondefinedon[0,1),thereisapositiveαsuchthat exp(α(S(f))2) is integrable on [0,1)—and vice versa. (This is not quite like saying that |f| and S(f) are pointwise comparable, but in many applications they might aswell be.)This tight control isexpressed quantitatively in terms of weighted norm inequalities. The reader will learn some sufficient (and not terribly restrictive) conditions on pairs of weights which ensure that (cid:3) (cid:3) |f(x)|pvdx≤ (S(f)(x))pwdx (0.2) or (cid:3) (cid:3) (S(f)(x))pvdx≤ |f(x)|pwdx (0.3) holdsforallf insuitabletestclasses,forvariousrangesofp(usually,1<p< ∞). He will also learn some necessary conditions for such inequalities. The usefulness of the square function (in its many guises) comes chiefly from the fact that, for many linear operatorsT, S(T(f)), the square function of T(f), is bounded pointwise by a function S˜(f), where S˜(·) is an operator similar to—and satisfying estimates similar to—S(·). This makes it possible to understand the behavior of T, because one can say: |T(f)| is controlled by S(T(f)), which is controlled by S˜(f), which is controlled by |f|. Obviously, theclosertheconnectionbetween|f|andS(f),themoreefficientthisprocess willbe.Theexponential-squareresults(andthecorrespondingweightednorm inequalities) imply that this connection is pretty close. We have tried to make this book self-contained, not too long, and ac- cessible to non-experts. We have also tried to avoid excessive overlap with other books on weighted norm inequalities. Therefore we have not treated every topic of relevance to weighted Littlewood-Paley theory. We have not touched on multi-parameter analysis at all, and we have dealt only briefly with vector-valued inequalities. We discuss Ap weights mainly with reference tothesquarefunctionandsingularintegraloperators.Weprovethebounded- ness of the Hardy-Littlewood operator on Lp(w) for w ∈Ap and we prove an extrapolation result—because we need both—but we don’t prove Ap factor- ization or the Rubio de Francia extrapolation theorem, excellent treatments of which can be found in many books (e.g., [16] and [24]). X Preface The book is laid out this way. Chapter 1 covers some basic facts from harmonicanalysis.Mostofthematerialtherewillbereviewformanypeople, butwehavetriedtopresentitsoasnottointimidatethenon-experts.Chapter 2introducestheone-dimensionaldyadicsquarefunctionandprovessomeofits properties;italsointroducesafewmoretechniquesfromharmonicanalysis.In chapter3weprovetheexponential-squareestimatesmentionedabove(inone dimensiononly).Theseleadtoanin-depthlookatweightednorminequalities. In chapter 4 we extend the results of the preceding chapters to d dimensions and to continuous analogues of the dyadic square function. Chapters 5, 6, and 7 are devoted to the Caldero´n reproducing formula. The Caldero´n formula provides a canonical way of expressing “arbitrary” functions as linear sums of special, smooth, compactly supported functions. It is the foundation of wavelet theory. Aside from some casual remarks1, we don’t talk about wavelets. The expert will see the close connections between wavelets and the material in chapters 5–7. The non-expert doesn’t have to worry about them to understand the material; but, should he ever encounter wavelets, a good grasp of the Caldero´n formula will come in very handy. We havedevotedthreechapterstoitbecausewebelievethereaderwillgainmore by seeing essentially the same problem (the convergence of the Calder´on in- tegral formula) treated in increasing levels of generality, than in having one big portmanteau theorem dumped onto his lap. The portmanteau theorem (Theorem 7.1) does come; but we trust that, when it does, the reader is more than able to bear its weight. Chapters 8 and 9 give some straightforward applications of weighted Littlewood-Paley theory to the analysis of Schro¨dinger and singular integral operators. This material could easily have come after that in chapter 10, but we felt that, where it is, it gave the reader a well-earned break from purely theoretical discussions. In chapter 10 we return to theory. The scale of Orlicz spaces (which in- cludes that of Lp spaces for 1 ≤ p ≤ ∞) provides a flexible way of keeping track of the integrability properties of functions. It is very useful in the study of weighted norm inequalities. The material here could have come at the very beginning, but we felt that the reader would understand this theory better if he first saw the need for it. As an application of Orlicz space theory, chapter 11 presents a different proof of Theorem 3.8 from chapter 3. This ingenious argument, due to Fedor Nazarov,completelyavoidstheuseofgood-λinequalities(whichweintroduce in chapter 2). These have been a mainstay of analysis since the early 1970s. In the opinion of some researchers, they have also become a crutch. We are neutral on this issue, but please see our note at the end of chapter 2. Chapter 12 applies the theory from the preceding chapters to give a new (and,wehope,accessible)proofoftheHo¨rmander-Mihlinmultipliertheorem. Chapter 13 extends the main weighted norm results from earlier chapters 1 Like this one. Preface XI to the (cid:8)2-valued setting. In chapter 14 we prove one theorem (Khinchin’s Inequalities), but our discussion there is mainly philosophical. We look at what Littlewood-Paley theory can tell us about pointwise summation errors of Haar function expansions. We have put exercises at the end of almost every chapter. Some of them expand on topics treated in the text; some tie up loose ends in proofs; some are referred to later in the book. We encourage the reader to understand all of them and to attempt at least a few. (We have supplied hints for the more difficult ones.) The author wishes to thank the many colleagues who have offered sug- gestions, helped him track down references, and steered him away from blunders. These colleagues include David Cruz-Uribe, SFO (of Trinity Uni- versity in Hartford, Connecticut), Doug Kurtz (of New Mexico State Univer- sity), Jos´e Martell (of the Universidad Auto´noma de Madrid), Fedor Nazarov (of Michigan State University), Carlos P´erez Moreno (of the Universidad de Sevilla), and Richard Wheeden (of Rutgers University, New Brunswick). I must particularly thank Roger Cooke, now retired from the University of Vermont, who read early drafts of the first chapters, and whose insightful criticisms have made them much more intelligible and digestible. Theauthorcouldnothavewrittenthisbookwithoutthegeneroussupport of the Spanish Ministerio de Educacio´n, Cultura, y Deporte, which provided himwitharesearchgrant(SAB2003-0003)duringhis2004-2005sabbaticalat theUniversidaddeSevilla.MyfamilyandIareindebtedtosomanymembers of the Facultad de Matematicas for their hospitality, that I hesitate to try to name them, for fear of omitting some. However, I must especially point out the kindness of my friend and colleague, Carlos P´erez Moreno. Without his tireless efforts, our visit to Sevilla would never have taken place. I do not have adequate words to express how much my family and I owe to him for everything he did for us, both before and after we arrived in Spain. Carlos, Sevilla, y Espan˜a se quedara´n siempre en nuestros corazones. Contents 1 Some Assumptions ........................................ 1 2 An Elementary Introduction............................... 9 3 Exponential Square........................................ 39 4 Many Dimensions; Smoothing ............................. 69 5 The Caldero´n Reproducing Formula I ..................... 85 6 The Caldero´n Reproducing Formula II ....................101 7 The Caldero´n Reproducing Formula III....................129 8 Schro¨dinger Operators.....................................145 9 Some Singular Integrals ...................................151 10 Orlicz Spaces ..............................................161 11 Goodbye to Good-λ .......................................189 12 A Fourier Multiplier Theorem .............................197 13 Vector-Valued Inequalities.................................203 14 Random Pointwise Errors .................................213 References.....................................................219 Index..........................................................223