Advanced Courses in Mathematics CRM Barcelona Feng Dai Yuan Xu Analysis on h -Harmonics and Dunkl Transforms Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Carles Casacuberta More information about this series at http://www.springer.com/series/5038 Feng Dai • Yuan Xu Analysis on h-Harmonics and Dunkl Transforms Editor for this volume: Sergey Tikhonov, ICREA and CRM, Barcelona Feng Dai Yuan Xu Department of Mathematics Department of Mathematics and Statistical Sciences University of Oregon University of Alberta Eugene, OR, USA Edmonton, AB, Canada ISSN 2297-0304 ISSN 2297-0312 (electronic) Advanced Courses in Mathematics - CRM Barcelona ISBN 978-3-0348-0886-6 ISBN 978-3-0348-0887-3 (eBook) DOI 10.1007/978-3-0348-0887-3 Library of Congress Control Number: 2014959869 Mathematics Subject Classification (2010): Primary: 41A10, 42B15; Secondary: 42B25, 42B08, 41A17 S pringer Basel Heidelberg New York Dordrecht London © Springer Basel 2015 This work is subject to copyright. 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Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com) Contents Preface vii 1 Introduction:SphericalHarmonicsandFourierTransform 1 1.1 Sphericalharmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Fouriertransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 DunklOperatorsAssociatedwithReflectionGroups 7 2.1 Weightfunctionsinvariantunderareflectiongroup . . . . . . . . . . . . 7 2.2 Dunkloperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Intertwiningoperator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Notesandfurtherresults . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 h-HarmonicsandAnalysisontheSphere 15 3.1 Dunklh-harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Projectionoperatorandintertwiningoperator . . . . . . . . . . . . . . . 20 3.3 Convolutionoperatorsandorthogonalexpansions . . . . . . . . . . . . . 23 3.4 Maximalfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.5 Convolutionandmaximalfunction . . . . . . . . . . . . . . . . . . . . . 31 3.6 Notesandfurtherresults . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Littlewood–PaleyTheoryandtheMultiplierTheorem 35 4.1 Vector-valuedinequalitiesforself-adjointoperators . . . . . . . . . . . . 35 4.2 TheLittlewood–Paley–Steinfunction . . . . . . . . . . . . . . . . . . . 37 4.3 TheLittlewood–Paleytheoryonthesphere . . . . . . . . . . . . . . . . 39 4.3.1 Acruciallemma . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3.2 ProofofTheorem4.3.3 . . . . . . . . . . . . . . . . . . . . . . . 42 4.4 TheMarcinkiewicztypemultipliertheorem . . . . . . . . . . . . . . . . 45 4.5 ALittlewood–Paleyinequality . . . . . . . . . . . . . . . . . . . . . . . 47 4.6 Notesandfurtherresults . . . . . . . . . . . . . . . . . . . . . . . . . . 50 v vi Contents 5 SharpJacksonandSharpMarchaudInequalities 51 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2 Moduliofsmoothnessandbestapproximation . . . . . . . . . . . . . . . 52 5.3 WeightedSobolevspacesandK-functionals . . . . . . . . . . . . . . . . 54 5.4 ThesharpMarchaudinequality . . . . . . . . . . . . . . . . . . . . . . . 56 5.5 ThesharpJacksoninequality . . . . . . . . . . . . . . . . . . . . . . . . 59 5.6 OptimalityofthepowerintheMarchaudinequality . . . . . . . . . . . . 61 5.7 Notesandfurtherresults . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6 DunklTransform 65 6.1 Dunkltransform:L2theory . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2 Dunkltransform:L1theory . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.3 Generalizedtranslationoperator . . . . . . . . . . . . . . . . . . . . . . 76 6.3.1 Translationoperatoronradialfunctions . . . . . . . . . . . . . . 77 6.3.2 TranslationoperatorforG=Zd . . . . . . . . . . . . . . . . . . 80 2 6.4 Generalizedconvolutionandsummability . . . . . . . . . . . . . . . . . 82 6.4.1 Convolutionwithradialfunctions . . . . . . . . . . . . . . . . . 82 6.4.2 SummabilityoftheinverseDunkltransform . . . . . . . . . . . . 84 6.4.3 ConvolutionoperatorforZd . . . . . . . . . . . . . . . . . . . . 86 2 6.5 Maximalfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.5.1 Boundednessofmaximalfunction . . . . . . . . . . . . . . . . . 87 6.5.2 ConvolutionversusmaximalfunctionforZd . . . . . . . . . . . 90 2 6.6 Notesandfurtherresults . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7 MultiplierTheoremsfortheDunklTransform 95 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.2 ProofofTheorem7.1.1:partI . . . . . . . . . . . . . . . . . . . . . . . 96 7.3 ProofofTheorem7.1.1:partII . . . . . . . . . . . . . . . . . . . . . . . 101 7.4 ProofofTheorem7.1.1:partIII . . . . . . . . . . . . . . . . . . . . . . 105 7.5 Ho¨rmander’smultipliertheoremandtheLittlewood–Paleyinequality. . . 106 7.6 ConvergenceoftheBochner–Rieszmeans . . . . . . . . . . . . . . . . . 108 7.7 Notesandfurtherresults . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Bibliography 111 Index 117 Preface TheselecturenoteswerewrittenasanintroductiontoDunklharmonicsandDunkltrans- forms,whichareextensionsofordinarysphericalharmonicsandFouriertransformswith theusualLebesguemeasurereplacedbyweightedmeasures. ThetheorywasinitiatedbyC.Dunklandsubsequentlydevelopedbymanyauthors inthepasttwodecades.Inthistheory,theroleoforthogonalgroups,whichprovidethe underlinestructurefortheordinaryFourieranalysis,isplayedbyafinitereflectiongroup, thepartialderivativesarereplacedbytheDunkloperators,whichareafamilyofcommut- ingfirstorderdifferentialanddifferenceoperators,andtheLebesguemeasureisreplaced byaweightedmeasurewiththeweightfunctionhκ invariantunderthereflectiongroup, whereκ isaparameter.ThetheoryhasarichstructureparalleltothatofFourieranalysis, whichallowsustoextendmanyclassicalresultstotheweightedsetting,especiallyinthe caseofh-harmonics,whicharetheanaloguesofordinarysphericalharmonics.Thereare stillmanyproblemstobesolvedandthetheoryisstillatitsinfancy,especiallyinthecase ofDunkltransform.Ourgoalistogiveanintroductiontowhathasbeendevelopedsofar. The present notes were written for people working in analysis. Prerequisites on reflection groups are kept to a bare minimum. In fact, even assuming the group is Zd, 2 whichrequiresessentiallynopriorknowledgeofreflectiongroups,areadercanstillgain access to the essence of the theory and to many highly non-trivial results, where the weightfunctionhκ issimply d hκ(x)=∏|xi|κi, κi≥0, 1≤i≤d, i=1 thesurfacemeasuredσ onthesphereSd−1 isreplacedbyh2κdσ,andtheLebesguemea- suredxonRd isreplacedbyh2κdx. Tomotivatetheweightedresults,wegiveabriefrecountofbasicsofordinaryspher- icalharmonicsandtheFouriertransforminthefirstchapter,whichcanbeskippedalto- gether.TheDunkloperatorsandtheintertwiningoperatorbetweenpartialderivativesand theDunkl operators,are introducedand discussedin thesecond chapter. Theintertwin- ing operator plays a key role in the theory as it appears in the concise formula for the reproducingkerneloftheh-sphericalharmonicsandinthedefinitionoftheDunkltrans- form.Thenextthreechaptersaredevotedtoanalysisonthesphere.Thethirdchapteris anintroductiontoh-harmonicsandessentialresultsonharmonicanalysisintheweighted vii viii Preface space.TheLittlewood–Paleytheoryonthesphereisdevelopedinthefourthchapter,and is used to establish a Marcinkiewicz type multiplier theorem on the weighted sphere. As an application, two inequalities, the sharpJackson and sharp Marchaud inequalities, areestablishedinthefifthchapter,whichareusefulforapproximationtheoryandinthe embedding theory of function spaces. The final two chapters are devoted to the Dunkl transform.ThesixthchapterisanintroductiontoDunkltransforms,wherethebasicre- sults are developed in detail. The Littlewood–Paley theory and a multiplier theorem are established in the seventh chapter, using a transference between h-harmonic expansions onthesphereandtheDunkltransforminRd. Thetopicsreflecttheauthors’choice.TherearemanyresultsforDunkltransforms ontherealline(wherethemeasureis|x|κdx)thatwedidnotdiscuss,sincethesettingon thereallineiscloselyrelatedtotheHankeltransformsandoftencannotevenbeextended to the Zd case in Rd. There are also results on partial differential-difference equations, 2 inanalogytoPDE,thatwedidnotdiscuss.Becauseoftheexplicitformulafortheinter- twiningoperator,thecaseZd hasseenfarmore,anddeeper,results,especiallyinthecase 2 ofanalysisonthespheresuchasthoseforCesa`romeans.WechosetheLittlewood–Paley theory and the multiplier theorem, as this part is relatively complete and the results are relatedinthetwosettings,thesphereandtheEuclideanspace. TheselecturenoteswerewrittenfortheadvancedcoursesintheprogramApproxi- mationTheoryandFourierAnalysisattheCentredeRecercaMatema`tica,Barcelona.We aregratefultotheCRMforthewarmhospitalityduringourtwomonthsstay,tothepartic- ipantsinourlectures,andthankespeciallytheorganizeroftheprogram,SergeyTikhonov from CRM, for his great help. We gratefully acknowledge the support received from NSERC Canada under grant RGPIN 311678-2010 (F.D.), from National Science Foun- dationundergrantDMS-1106113(Y.X.),andfromtheSimonsFoundation(#209057to Y.X.). Edmonton,Alberta,andEugene,Oregon FengDai September,2014 YuanXu Chapter 1 Introduction: Spherical Harmonics and Fourier Transform The purpose of these lecture notes is to provide an introduction to two related topics: h-harmonics and the Dunkl transform. These are extensions of the classical spherical harmonicsandtheFouriertransform,inwhichtheunderlyingrotationgroupisreplaced by a finite reflection group. This chapter serves as an introduction, in which we briefly recall classical results on the spherical harmonics and the Fourier transform. Since all resultsareclassical,noproofwillbegiven. 1.1 Spherical harmonics Firstweintroduceseveralnotationsthatwillbeusedthroughouttheselecturenotes. Forx∈Rd,wewritex=(x1,...,xd).Theinnerpro(cid:2)ductofx,y∈Rd isdenotedby (cid:5)x,y(cid:6):=∑d xy andthenormofxisdenotedby(cid:7)x(cid:7):= (cid:5)x,x(cid:6).LetSd−1:={x∈Rd : i=1 i i (cid:7)x(cid:7)=1}denotetheunitsphereofRd,andletN denotethesetofnonnegativeintegers. 0 Forα=(α ,...,α )∈Nd,amonomialxα isaproductxα =xα1···xαd,whichhasdegree 1 d 0 1 d |α|:=α +···+α . 1 d A homogeneous polynomial P of degree n is a linear combination of monomials of degree n, that is, P(x)=∑|α|=ncαxα, where cα are either real or complex numbers. A polynomial of (total) degree at most n is of the form P(x)=∑|α|≤ncαxα. Let Pnd denotethespaceofrealhomogeneouspolynomialsofdegreenandΠd thespaceofreal n polynomials of degree at most n. Counting the cardinalities of {α ∈Nd :|α|=n} and 0 {α ∈Nd :|α|≤n}showsthat 0 (cid:3) (cid:4) (cid:3) (cid:4) n+d−1 n+d dimPd = and dimΠd = . n n n n © Springer Basel 2015 1 F. Dai, Y. Xu, Analysis on h-Harmonics and Dunkl Transforms, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-0887-3_1