Advances in Analysis: The Legacy of Elias M. Stein

478 Pages·2014·3.401 MB·English

by  Charles Fefferman, Alexandru D. Ionescu, D.H. Phong, Stephen Wainger (eds.)

Advances in Analysis
PrincetonMathematicalSeries
Editors: PhillipA.GriffithsandJohnN.Mather
1. TheClassicalGroupsbyHermannWeyl
8. TheoryofLieGroups: IbyClaudeChevalley
9. MathematicalMethodsofStatisticsbyHaraldCramér
14. TheTopologyofFibreBundlesbyNormanSteenrod
17. IntroductiontoMathematicalLogic,Vol. IbyAlonzoChurch
19. HomologicalAlgebrabyHenriCartanandSamuelEilenberg
28. ConvexAnalysisbyR.TyrrellRockafellar
30. SingularIntegralsandDifferentiabilityPropertiesofFunctions
byEliasM.Stein
32. IntroductiontoFourierAnalysisonEuclideanSpacesbyEliasM.Stein
andGuidoWeiss
33. ÉtaleCohomologybyJamesS.Milne
35. Three-DimensionalGeometryandTopology,Volume1byWilliamP.
Thurston. EditedbySilvioLevy
36. RepresentationTheoryofSemisimpleGroups: AnOverviewBased
onExamplesbyAnthonyW.Knapp
38. SpinGeometrybyH.BlaineLawson,Jr.,andMarie-LouiseMichelsohn
43. HarmonicAnalysis: RealVariableMethods,Orthogonality,
andOscillatoryIntegralsbyEliasM.Stein
44. TopicsinErgodicTheorybyYakovG.Sinai
45. CohomologicalInductionandUnitaryRepresentationsbyAnthonyW.
KnappandDavidA.Vogan,Jr.
46. AbelianVarietieswithComplexMultiplicationandModularFunctions
byGoroShimura
47. RealSubmanifoldsinComplexSpaceandTheirMappings
byM.SalahBaouendi,PeterEbenfelt,andLindaPreissRothschild
48. EllipticPartialDifferentialEquationsandQuasiconformalMappings
inthePlanebyKariAstala,TadeuszIwaniec,andGavenMartin
49. APrimeronMappingClassGroupsbyBensonFarbandDanMargalit
50. AdvancesinAnalysis: TheLegacyofEliasM.SteineditedbyCharles
Fefferman,AlexandruD.Ionescu,D.H.Phong,andStephenWainger
Advances in Analysis:
The Legacy of Elias M. Stein
Edited by Charles Fefferman,
Alexandru D. Ionescu, D. H. Phong,
and Stephen Wainger
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright(cid:1)c 2014byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,
Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet,
Woodstock,OxfordshireOX201TW
press.princeton.edu
AllRightsReserved
LibraryofCongressCataloging-in-PublicationData
Advancesinanalysis:thelegacyofEliasM.Stein/editedbyCharlesFefferman,Alexandru
D.Ionescu,D.H.Phong,andStephenWainger.
pagescm.—(Princetonmathematicalseries;50)
Summary: "Princeton University’s Elias Stein was the first mathematician to see the pro-
foundinterconnections thattieclassicalFourieranalysistoseveralcomplexvariablesand
representation theory. His fundamental contributions include the Kunze-Stein phenom-
enon,theconstructionofnewrepresentations,theSteininterpolationtheorem,theideaofa
restrictiontheoremfortheFouriertransform,andthetheoryofHpSpacesinseveralvari-
ables. Throughhisgreatdiscoveries, throughbooksthathavesetthehigheststandardfor
mathematicalexposition,andthroughhisinfluenceonhismanycollaboratorsandstudents,
Steinhaschangedmathematics.DrawinginspirationfromStein’scontributionstoharmonic
analysisandrelatedtopics,thisvolumegatherspapersfrominternationallyrenownedmathe-
maticians, many of whom have been Stein’s students. The book also includes expository
papersonStein’sworkanditsinfluence.ThecontributorsareJeanBourgain,LuisCaffarelli,
MichaelChrist,GuyDavid,CharlesFefferman,AlexandruIonescu,DavidJerison,Carlos
Kenig,SergiuKlainerman,LoredanaLanzani,SanghyukLee,LionelLevine,AkosMagyar,
DetlefMüller,CamilMuscalu,AlexanderNagel,D.H.Phong,MalabikaPramanik,Andrew
Raich, Fulvio Ricci, Keith Rogers, Andreas Seeger, Scott Sheffield, Luis Silvestre,
Christopher Sogge, Jacob Sturm, Terence Tao, Christoph Thiele, Stephen Wainger, and
StevenZelditch”—Providedbypublisher.
Includesbibliographicalreferencesandindex.
ISBN978-0-691-15941-6(hardback)
1. Mathematicalanalysis—Congresses. I. Stein, EliasM., 1931–honouree. II. Feffer-
man,Charles,1949–editorofcompilation. III. Ionescu,AlexandruDan,1973–editorof
compilation. IV. Phong, Duong H., 1953– editor of compilation. V. Wainger, Stephen,
1936–editorofcompilation.
QA299.6.A382014
515—dc23
2013020363
BritishLibraryCataloging-in-PublicationDataisavailable
ThisbookhasbeencomposedinTimes
Printedonacid-freepaper.∞
TypesetbySRNovaPvtLtd,Bangalore,India
PrintedintheUnitedStatesofAmerica
10987654321
To Eli

Contents
Preface ix
Chapter1 SelectedTheoremsbyEliStein 1
CharlesFefferman
Chapter2 Eli’sImpact: ACaseStudy 35
CharlesFefferman
Chapter3 OnOscillatoryIntegralOperatorsinHigherDimensions 47
JeanBourgain
Chapter4 HölderRegularityforGeneralizedMasterEquationswith
RoughKernels 63
LuisCaffarelliandLuisSilvestre
Chapter5 ExtremizersofaRadonTransformInequality 84
MichaelChrist
Chapter6 ShouldWeSolvePlateau’sProblemAgain? 108
GuyDavid
Chapter7 AveragesalongPolynomialSequencesinDiscreteNilpotent
LieGroups: SingularRadonTransforms 146
AlexandruD.Ionescu,AkosMagyar,andStephenWainger
Chapter8 InternalDLAforCylinders 189
DavidJerison,LionelLevine,andScottSheffield
Chapter9 TheEnergyCriticalWaveEquationin3D 215
CarlosKenig
Chapter10 OntheBoundedL2 CurvatureConjecture 224
SergiuKlainerman
Chapter11 OnDiv-CurlforHigherOrder 245
LoredanaLanzaniandAndrewS.Raich
viii CONTENTS
Chapter12 SquareFunctionsandMaximalOperatorsAssociatedwith
RadialFourierMultipliers 273
SanghyukLee,KeithM.Rogers,andAndreasSeeger
Chapter13 ProblemsofHarmonicAnalysisRelatedtoFinite-Type
HypersurfacesinR3,andNewtonPolyhedra 303
DetlefMüller
Chapter14 Multi-LinearMultipliersAssociatedtoSimplexesofArbitrary
Length 346
CamilMuscalu,TerenceTao,andChristophThiele
Chapter15 DiagonalEstimatesforBergmanKernelsinMonomial-Type
Domains 402
AlexanderNagelandMalabikaPramanik
Chapter16 OntheSingularitiesofthePluricomplexGreen’sFunction 419
D.H.PhongandJacobSturm
Chapter17 SmoothnessofSpectralMultipliersandConvolutionKernels
inNilpotentGelfandPairs 436
FulvioRicci
Chapter18 OnEigenfunctionRestrictionEstimatesandL4-Boundsfor
CompactSurfaceswithNonpositiveCurvature 447
ChristopherD.SoggeandSteveZelditch
ListofContributors 463
Index 465
Preface
Thisistheproceedingsoftheconference“AnalysisandApplications,”heldinMay
2011 in honor of the 80th birthday of Elias M. Stein. The reader will find here
an extraordinary range of deep mathematics, covering most of the main trends in
harmonic analysis, partial differential equations, and allied fields. All bear the
stampofStein’sbroadandoriginalviewofthesubject. Itisapleasuretodedicate
thisbooktohim.
WearegratefultotheAmericanInstituteofMathematics,theClayMathematics
Institute, the National Science Foundation, and the Princeton University Mathe-
maticsDepartmentforthefinancialsupportthatmadetheconferencepossible.
Manypeopleworkedhardtomaketheconferenceasuccess. Weareespecially
gratefultoScottKenney,GaleSandor,andCarolDiSantoforsolvingmanypracti-
calproblems.
OurthanksgoalsotoVickieKearn,QuinnFusting,andNathanCarratPrinceton
UniversityPressforinvaluablehelpinproducingthisbook. Wemournthepassing
of Frances Wroblewski of the Princeton Mathematics Department; she also con-
tributedinvaluablehelp.
Wearedeeplygratefultothespeakersandparticipantswhomadetheconference
“AnalysisandApplications”amajorsuccess.
Mostofall,wearegratefultoEliStein.
C.Fefferman,A.D.Ionescu,D.H.Phong,S.Wainger
Fall2012