Fourier Integrals in Classical Analysis

349 Pages·2017·1.827 MB·English

by  Christopher D. Sogge

CAMBRIDGE TRACTS IN MATHEMATICS
GeneralEditors
B. BOLLOBA´ S, W. FULTON, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
210FourierIntegralsinClassicalAnalysis,SecondEdition
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CAMBRIDGE TRACTS IN MATHEMATICS
GENERAL EDITORS
B. BOLLOBA´S, W. FULTON, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
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Fourier Integrals in Classical
Analysis
Second Edition
CHRISTOPHER D. SOGGE
TheJohnsHopkinsUniversity
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Informationonthistitle:www.cambridge.org/9781107120075
DOI:10.1017/9781316341186
Firstedition(cid:2)c CambridgeUniversityPress1993
Secondedition(cid:2)c ChristopherD.Sogge2017
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Firstpublished1993
Secondedition2017
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Names:Sogge,ChristopherD.(ChristopherDonald),1960–
Title:Fourierintegralsinclassicalanalysis/ChristopherD.Sogge,
TheJohnsHopkinsUniversity.
Description:Secondedition.|Cambridge:CambridgeUniversityPress,2017.|
Series:Cambridgetractsinmathematics;210|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2017004380|ISBN9781107120075(hardback:alk.paper)
Subjects:LCSH:Fourierseries.|Fourierintegraloperators.|Fourieranalysis.
Classification:LCCQA404.S642017|DDC515/.723–dc23
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Contents
PrefacetotheSecondEdition page xi
PrefacetotheFirstEdition xiii
0 Background 1
0.1 FourierTransform 1
0.2 BasicRealVariableTheory 9
0.3 FractionalIntegrationandSobolevEmbeddingTheorems 22
0.4 WaveFrontSetsandtheCotangentBundle 30
0.5 OscillatoryIntegrals 37
Notes 41
1 StationaryPhase 42
1.1 StationaryPhaseEstimates 42
1.2 FourierTransformofSurface-carriedMeasures 51
Notes 56
2 Non-homogeneousOscillatoryIntegralOperators 57
2.1 Non-degenerateOscillatoryIntegralOperators 58
2.2 OscillatoryIntegralOperatorsRelatedtothe
RestrictionTheorem 60
2.3 RieszMeansinRn 70
2.4 NikodymMaximalFunctionsandMaximalRiesz
MeansinR2 76
Notes 96
3 Pseudo-differentialOperators 97
3.1 SomeBasics 97
3.2 EquivalenceofPhaseFunctions 104
3.3 Self-adjointEllipticPseudo-differentialOperators
onCompactManifolds 110
Notes 115
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viii Contents
4 TheHalf-waveOperatorandFunctionsof
Pseudo-differentialOperators 116
4.1 TheHalf-waveOperator 117
4.2 TheSharpWeylFormula 126
4.3 SmoothFunctionsofPseudo-differentialOperators 133
Notes 135
5 LpEstimatesofEigenfunctions 137
5.1 TheDiscreteL2 RestrictionTheorem 138
5.2 EstimatesforRieszMeans 151
5.3 MoreGeneralMultiplierTheorems 155
Notes 160
6 FourierIntegralOperators 162
6.1 LagrangianDistributions 162
6.2 RegularityProperties 170
6.3 SphericalMaximalTheorems:Take1 187
Notes 194
7 PropagationofSingularitiesandRefinedEstimates 195
7.1 WaveFrontSetsRedux 195
7.2 PropagationofSingularities 199
7.3 ImprovedSup-normEstimatesofEigenfunctions 204
7.4 ImprovedSpectralAsymptotics 216
Notes 232
8 LocalSmoothingofFourierIntegralOperators 233
8.1 LocalSmoothinginTwoDimensionsandVariable
CoefficientNikodymMaximalTheorems 234
8.2 LocalSmoothinginHigherDimensions 253
8.3 SphericalMaximalTheoremsRevisited 263
Notes 266
9 Kakeya-typeMaximalOperators 268
9.1 TheKakeyaMaximalOperatorandtheKakeyaProblem 268
9.2 UniversalBoundsforKakeya-typeMaximalOperators 278
9.3 NegativeResultsinCurvedSpaces 285
9.4 Wolff’sBoundsforKakeya-typeMaximalOperators 294
9.5 TheFourierRestrictionProblemandtheKakeyaProblem 310
Notes 314
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Contents ix
Appendix:LagrangianSubspacesofT∗Rn 317
References 319
IndexofNotation 331
Index 333
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