Table Of ContentAdvanced 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
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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
