Classical Fourier Analysis

493 Pages·2009·4.537 MB·English

by  Loukas Grafakos

249
Graduate Texts in Mathematics
EditorialBoard
S. Axler
K.A. Ribet
Graduate Texts inMathematics
1 TAKEUTI/ZARING.IntroductiontoAxiomatic 38 GRAUERT/FRITZSCHE.SeveralComplex
SetTheory.2nded. Variables.
2 OXTOBY.MeasureandCategory.2nded. 39 ARVESON.AnInvitationtoC-Algebras.
3 SCHAEFER.TopologicalVectorSpaces. 40 KEMENY/SNELL/KNAPP.Denumerable
2nded. MarkovChains.2nded.
4 HILTON/STAMMBACH.ACoursein 41 APOSTOL.ModularFunctionsandDirichlet
HomologicalAlgebra.2nded. SeriesinNumberTheory.2nded.
5 MACLANE.CategoriesfortheWorking 42 J.-P.SERRE.LinearRepresentationsofFinite
Mathematician.2nded. Groups.
6 HUGHES/PIPER.ProjectivePlanes. 43 GILLMAN/JERISON.RingsofContinuous
7 J.-P.SERRE.ACourseinArithmetic. Functions.
8 TAKEUTI/ZARING.AxiomaticSetTheory. 44 KENDIG.ElementaryAlgebraicGeometry.
9 HUMPHREYS.IntroductiontoLieAlgebras 45 LOE`VE.ProbabilityTheoryI.4thed.
andRepresentationTheory. 46 LOE`VE.ProbabilityTheoryII.4thed.
10 COHEN.ACourseinSimpleHomotopy 47 MOISE.GeometricTopologyinDimensions2
Theory. and3.
11 CONWAY.FunctionsofOneComplex 48 SACHS/WU.GeneralRelativityfor
VariableI.2nded. Mathematicians.
12 BEALS.AdvancedMathematicalAnalysis. 49 GRUENBERG/WEIR.LinearGeometry.
13 ANDERSON/FULLER.RingsandCategories 2nded.
ofModules.2nded. 50 EDWARDS.Fermat’sLastTheorem.
14 GOLUBITSKY/GUILLEMIN.StableMappings 51 KLINGENBERG.ACourseinDifferential
andTheirSingularities. Geometry.
15 BERBERIAN.LecturesinFunctionalAnalysis 52 HARTSHORNE.AlgebraicGeometry.
andOperatorTheory. 53 MANIN.ACourseinMathematicalLogic.
16 WINTER.TheStructureofFields. 54 GRAVER/WATKINS.Combinatoricswith
17 ROSENBLATT.RandomProcesses.2nded. EmphasisontheTheoryofGraphs.
18 HALMOS.MeasureTheory. 55 BROWN/PEARCY.IntroductiontoOperator
19 HALMOS.AHilbertSpaceProblemBook. TheoryI:ElementsofFunctionalAnalysis.
2nded. 56 MASSEY.AlgebraicTopology:An
20 HUSEMOLLER.FibreBundles.3rded. Introduction.
21 HUMPHREYS.LinearAlgebraicGroups. 57 CROWELL/FOX.IntroductiontoKnotTheory.
22 BARNES/MACK.AnAlgebraicIntroduction 58 KOBLITZ.p-adicNumbers,p-adicAnalysis,
toMathematicalLogic. andZeta-Functions.2nded.
23 GREUB.LinearAlgebra.4thed. 59 LANG.CyclotomicFields.
24 HOLMES.GeometricFunctionalAnalysisand 60 ARNOLD.MathematicalMethodsinClassical
ItsApplications. Mechanics.2nded.
25 HEWITT/STROMBERG.RealandAbstract 61 WHITEHEAD.ElementsofHomotopyTheory.
Analysis. 62 KARGAPOLOV/MERIZJAKOV.Fundamentals
26 MANES.AlgebraicTheories. oftheTheoryofGroups.
27 KELLEY.GeneralTopology. 63 BOLLOBAS.GraphTheory.
28 ZARISKI/SAMUEL.CommutativeAlgebra. 64 EDWARDS.FourierSeries.Vol.I.2nded.
Vol.I. 65 WELLS.DifferentialAnalysisonComplex
29 ZARISKI/SAMUEL.CommutativeAlgebra. Manifolds.2nded.
Vol.II. 66 WATERHOUSE.IntroductiontoAffineGroup
30 JACOBSON.LecturesinAbstractAlgebraI. Schemes.
BasicConcepts. 67 SERRE.LocalFields.
31 JACOBSON.LecturesinAbstractAlgebraII. 68 WEIDMANN.LinearOperatorsinHilbert
LinearAlgebra. Spaces.
32 JACOBSON.LecturesinAbstractAlgebraIII. 69 LANG.CyclotomicFieldsII.
TheoryofFieldsandGaloisTheory. 70 MASSEY.SingularHomologyTheory.
33 HIRSCH.DifferentialTopology. 71 FARKAS/KRA.RiemannSurfaces.2nded.
34 SPITZER.PrinciplesofRandomWalk.2nded. 72 STILLWELL.ClassicalTopologyand
35 ALEXANDER/WERMER.SeveralComplex CombinatorialGroupTheory.2nded.
VariablesandBanachAlgebras.3rded. 73 HUNGERFORD.Algebra.
36 KELLEY/NAMIOKAETAL.Linear 74 DAVENPORT.MultiplicativeNumberTheory.
TopologicalSpaces. 3rded.
37 MONK.MathematicalLogic. (continuedafterindex)
Loukas Grafakos
Classical Fourier Analysis
Second Edition
123
Loukas Grafakos
Department of Mathematics
University of Missouri
Columbia, MO 65211
USA
[email protected]
Editorial Board
S. Axler K.A. Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California at Berkeley
San Francisco, CA 94132 Berkeley, CA 94720-3840
USA USA
[email protected] [email protected]
ISSN: 0072-5285
ISBN: 978-0-387-09431-1 e-ISBN: 978-0-387-09432-8
DOI: 10.1007/978-0-387-09432-8
Library of Congress Control Number: 2008933456
Mathematics Subject Classifi cation (2000): 42-xx 42-02
© 2008 Springer Science+Business Media, LLC
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ToSuzanne
Preface
The great response to the publication of the book Classical and Modern Fourier
Analysishasbeenverygratifying.IamdelightedthatSpringerhasofferedtopublish
the second edition of this book in two volumes: Classical Fourier Analysis, 2nd
Edition,andModernFourierAnalysis,2ndEdition.
These volumes are mainly addressed to graduate students who wish to study
Fourieranalysis.Thisfirstvolumeisintendedtoserveasatextforaone-semester
courseinthesubject.Theprerequisiteforunderstandingthematerialhereinissatis-
factorycompletionofcoursesinmeasuretheory,Lebesgueintegration,andcomplex
variables.
The details included in the proofs make the exposition longer. Although it will
behoovemanyreaderstoskimthroughthemoretechnicalaspectsofthepresenta-
tion and concentrate on the flow of ideas, the fact that details are present will be
comforting to some. The exercises at the end of each section enrich the material
ofthecorrespondingsectionandprovideanopportunitytodevelopadditionalintu-
itionanddeepercomprehension.Thehistoricalnotesofeachchapterareintendedto
provideanaccountofpastresearchbutalsotosuggestdirectionsforfurtherinvesti-
gation.Theappendixincludesmiscellaneousauxiliarymaterialneededthroughout
thetext.
Awebsiteforthebookismaintainedat
http://math.missouri.edu/∼loukas/FourierAnalysis.html
I am solely responsible for any misprints, mistakes, and historical omissions in
thisbook.Pleasecontactmedirectly([email protected])ifyouhavecor-
rections,comments,suggestionsforimprovements,orquestions.
Columbia,Missouri, LoukasGrafakos
April2008
vii
Acknowledgments
Iamveryfortunatethatseveralpeoplehavepointedouterrors,misprints,andomis-
sionsinthefirsteditionofthisbook.OthershaveclarifiedissuesIraisedconcerning
thematerialitcontains.Alltheseindividualshaveprovidedmewithinvaluablehelp
that resulted in the improved exposition of the present second edition. For these
reasons, I would like to express my deep appreciation and sincere gratitude to the
followingpeople:
MarcoAnnoni,PascalAuscher,AndrewBailey,DmitriyBilyk,MarcinBownik,
Leonardo Colzani, Simon Cowell, Mita Das, Geoffrey Diestel, Yong Ding, Jacek
Dziubanski,WeiHe,PetrHonz´ık,HeidiHulsizer,PhilippeJaming,SvanteJanson,
Ana Jime´nez del Toro, John Kahl, Cornelia Kaiser, Nigel Kalton, Kim Jin My-
ong,DoowonKoh,ElenaKoutcherik,EnricoLaeng,SungyunLee,QifanLi,Chin-
ChengLin,LiguangLiu,Stig-OlofLonden,DiegoMaldonado,Jose´Mar´ıaMartell,
Mieczyslaw Mastylo, Parasar Mohanty, Carlo Morpurgo, Andrew Morris, Mihail
Mourgoglou, Virginia Naibo, Hiro Oh, Marco Peloso, Maria Cristina Pereyra,
Carlos Pe´rez, Humberto Rafeiro, Maria Carmen Reguera Rodr´ıguez, Alexander
Samborskiy, Andreas Seeger, Steven Senger, Sumi Seo, Christopher Shane, Shu
Shen, Yoshihiro Sawano, Vladimir Stepanov, Erin Terwilleger, Rodolfo Torres,
SuzanneTourville,IgnacioUriarte-Tuero,KunyangWang,HuoxiongWu,Takashi
Yamamoto,andDachunYang.
Fortheirvaluablesuggestions,corrections,andotherimportantassistanceatdif-
ferentstagesinthepreparationofthefirsteditionofthisbook,Iwouldliketooffer
mydeepestgratitudetothefollowingindividuals:
Georges Alexopoulos, Nakhle´ Asmar, Bruno Calado, Carmen Chicone, David
Cramer, Geoffrey Diestel, Jakub Duda, Brenda Frazier, Derrick Hart, Mark Hoff-
mann,StevenHofmann,HelgeHolden,BrianHollenbeck,PetrHonz´ık,Alexander
Iosevich,TundeJakab,SvanteJanson,AnaJime´nezdelToro,GregoryJones,Nigel
Kalton,EmmanouilKatsoprinakis,DennisKletzing,StevenKrantz,DouglasKurtz,
George Lobell, Xiaochun Li, Jose´ Mar´ıa Martell, Antonios Melas, Keith Mers-
man,StephenMontgomety-Smith,AndreaNahmod,NguyenCongPhuc,Krzysztof
Oleszkiewicz,CristinaPereyra,CarlosPe´rez,DanielRedmond,JorgeRivera-Nori-
ega,DmitriyRyabogin,ChristopherSansing,LynnSavinoWendel,Shih-ChiShen,
ix
x Acknowledgments
Roman Shvidkoy, Elias Stein, Atanas Stefanov, Terence Tao, Erin Terwilleger,
ChristophThiele,RodolfoTorres,DeanieTourville,NikolaosTzirakisDonVaught,
IgorVerbitsky,BrettWick,JamesWright,andLinqiaoZhao.
I would also like to thank all reviewers who provided me with an abundance
of meaningful remarks, corrections, and suggestions for improvements. Finally, I
wouldliketothankSpringereditorMarkSpencer,Springer’sdigitalproductsupport
personnelFrankGanzandFrankMcGuckin,andcopyeditorDavidKramerfortheir
invaluableassistanceduringthepreparationofthisedition.
Contents
1 LpSpacesandInterpolation ..................................... 1
1.1 LpandWeakLp............................................ 1
1.1.1 TheDistributionFunction ............................. 2
1.1.2 ConvergenceinMeasure .............................. 5
1.1.3 AFirstGlimpseatInterpolation........................ 8
Exercises........................................... 10
1.2 ConvolutionandApproximateIdentities ....................... 16
1.2.1 ExamplesofTopologicalGroups ....................... 16
1.2.2 Convolution......................................... 18
1.2.3 BasicConvolutionInequalities ......................... 19
1.2.4 ApproximateIdentities ............................... 24
Exercises........................................... 28
1.3 Interpolation............................................... 30
1.3.1 RealMethod:TheMarcinkiewiczInterpolationTheorem... 31
1.3.2 ComplexMethod:TheRiesz–ThorinInterpolationTheorem.. 34
1.3.3 InterpolationofAnalyticFamiliesofOperators ........... 37
1.3.4 ProofsofLemmas1.3.5and1.3.8 ...................... 39
Exercises........................................... 42
1.4 LorentzSpaces............................................. 44
1.4.1 DecreasingRearrangements ........................... 44
1.4.2 LorentzSpaces ...................................... 48
1.4.3 DualsofLorentzSpaces .............................. 51
1.4.4 TheOff-DiagonalMarcinkiewiczInterpolationTheorem ... 55
Exercises........................................... 63
2 MaximalFunctions,FourierTransform,andDistributions.......... 77
2.1 MaximalFunctions ......................................... 78
2.1.1 TheHardy–LittlewoodMaximalOperator ............... 78
2.1.2 ControlofOtherMaximalOperators.................... 82
2.1.3 ApplicationstoDifferentiationTheory .................. 85
Exercises........................................... 89
xi
xii Contents
2.2 TheSchwartzClassandtheFourierTransform.................. 94
2.2.1 TheClassofSchwartzFunctions ....................... 95
2.2.2 TheFourierTransformofaSchwartzFunction ........... 98
2.2.3 TheInverseFourierTransformandFourierInversion ......102
2.2.4 TheFourierTransformonL1+L2 ......................103
Exercises...........................................106
2.3 TheClassofTemperedDistributions ..........................109
2.3.1 SpacesofTestFunctions ..............................109
2.3.2 SpacesofFunctionalsonTestFunctions.................110
2.3.3 TheSpaceofTemperedDistributions ...................112
2.3.4 TheSpaceofTemperedDistributionsModuloPolynomials..121
Exercises...........................................122
2.4 MoreAboutDistributionsandtheFourierTransform.............124
2.4.1 DistributionsSupportedataPoint ......................124
2.4.2 TheLaplacian.......................................125
2.4.3 HomogeneousDistributions ...........................127
Exercises...........................................133
2.5 ConvolutionOperatorsonLpSpacesandMultipliers.............135
2.5.1 OperatorsThatCommutewithTranslations ..............135
2.5.2 TheTransposeandtheAdjointofaLinearOperator .......138
2.5.3 TheSpacesMp,q(Rn)................................139
2.5.4 CharacterizationsofM1,1(Rn)andM2,2(Rn)............141
2.5.5 TheSpaceofFourierMultipliersM (Rn) ...............143
p
Exercises...........................................146
2.6 OscillatoryIntegrals ........................................148
2.6.1 PhaseswithNoCriticalPoints .........................149
2.6.2 SublevelSetEstimatesandtheVanderCorputLemma.....151
Exercises...........................................156
3 FourierAnalysisontheTorus ...................................161
3.1 FourierCoefficients.........................................161
3.1.1 Then-TorusTn ......................................162
3.1.2 FourierCoefficients ..................................163
3.1.3 TheDirichletandFeje´rKernels ........................165
3.1.4 ReproductionofFunctionsfromTheirFourierCoefficients ..168
3.1.5 ThePoissonSummationFormula.......................171
Exercises...........................................173
3.2 DecayofFourierCoefficients ................................176
3.2.1 DecayofFourierCoefficientsofArbitraryIntegrable
Functions...........................................176
3.2.2 DecayofFourierCoefficientsofSmoothFunctions........179
3.2.3 FunctionswithAbsolutelySummableFourierCoefficients..183
Exercises...........................................185
3.3 PointwiseConvergenceofFourierSeries.......................186
3.3.1 PointwiseConvergenceoftheFeje´rMeans...............186