欧几里得空间的傅里叶分析

Preface

CHAPTERⅠTheFourierTransform

1.ThebasicL1theoryoftheFouriertransform

2.TheL2theoryandthePlanchereltheorem

3.Theclassoftempereddistributions

4.Furtherresults

CHAPTERⅡBoundaryValuesofHarmonicFunctions

1.Basicpropertiesofharmonicfunctions

2.ThecharacterizationofPoissonintegrals

3.TheHardy-Littlewoodmaximalfunctionandnontangentialconvergenceofharmonicfunctions

4.Subharmonicfunctionsandmajorizationbyharmonicfunctions

5.Furtherresults

CHAPTERⅢTheTheoryofHpSpacesonTubes

1.Introductoryremarks

2.TheH2theory

3.Tubesovercones

4.ThePaley-Wienertheorem

5.TheHptheory

6.Furtherresults

CHAPTERⅣ.SymmetryPropertiesoftheFourierTransform

1.DecompositionofL2(Ez)intosub,pacesinvariantundertheFouriertransform

2.Sphericalharmonics

3.TheactionoftheFouriertransformonthespaces

4.Someapplications

5.Furtherresults

CHAPTERⅤInterpolationofOperators

1.TheM.RieszconvexitytheoremandinterpolationofoperatorsdefinedonLpspaces

2.TheMarcinkiewiczinterpolationtheorem

3.L(p,q)spaces

4.Interpolationofanalyticfamiliesofoperators

5.Furtherresults

CHAPTERⅥSingularIntegralsandSystemsofConjugateHarmonicFunctions

1.TheHilberttransform

2.Singularintegraloperatorswithoddkernels

3.Singularintegraloperatorswithevenkernels

4.Hpspacesofconjugateharmonicfunctions

5.Furtherresults

CHAPTERⅦMultipleFourierSeries

1.Elementaryproperties

2.ThePoissonsummationformula

3.Multipliertransformations

4.Summabilitybelowthecriticalindex(negativeresults)

5.Summabilitybelowthecriticalindex

6.Furtherresults

Bibliography

Index

　　ThisbookisdesignedtobeanintroductiontoharmonicanalysisinEuclideanspaces.Thesubjecthasseenaconsiderablefloweringduringthepasttwentyyears.Wehavenottriedtocoverallphasesofthisdevelopment.Rather,ourchiefconcernwastoillustratevariousmethodsusedinthisaspectofFourieranalysisthatexploitthestructureofEuclideanspaces.Inparticular,wetrytoshowtheroleplayedbytheactionoftranslations,dilations,androtations.Anotherconcern,notindependentofthischiefone,istomotivatethestudyofharmonicanalysisonmoregeneralspaceshavingananalogousstructure(suchasarisesinsymmetricspaces).ItisourfeelingthatthestudyofFourieranalysisinthatcontextand,also,inothergeneralsettings,ismoremeaningfuloncethespecialEuclideancaseisunderstood.