Real-Variable Methods in Harmonic Analysis

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  • Format: Paperback
  • Copyright: 2004-04-09
  • Publisher: Dover Publications

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"A very good choice."--MathSciNet, American Mathematical Society An exploration of the unity of several areas in harmonic analysis, this self-contained text emphasizes real-variable methods. Appropriate for advanced undergraduate and graduate students, it starts with classical Fourier series and discusses summability, norm convergence, and conjugate function. An examination of the Hardy-Littlewood maximal function and the Calderoacute;n-Zygmund decomposition is followed by explorations of the Hilbert transform and properties of harmonic functions. Additional topics include the Littlewood-Paley theory, good lambda inequalities, atomic decomposition of Hardy spaces, Carleson measures, Cauchy integrals on Lipschitz curves, and boundary value problems. 1986 ed.

Table of Contents

Prefacep. xi
Fourier Series
Fourier Series of Functionsp. 1
Fourier Series of Continuous Functionsp. 8
Elementary Properties of Fourier Seriesp. 13
Fourier Series of Functionalsp. 16
Notes; Further Results and Problemsp. 22
Cesaro Summability
(C, 1) Summabilityp. 28
Fejer's Kernelp. 29
Characterization of Fourier Series of Functions and Measuresp. 34
A.E. Convergence of (C, 1) Means of Summable Functionsp. 41
Notes; Further Results and Problemsp. 43
Norm Convergence of Fourier Series
The Case L[superscript 2](T); Hilbert Spacep. 48
Norm Convergence in L[superscript p] (T), 1 [less than or equal] p [less than or equal] [infinity]p. 51
The Conjugate Mappingp. 52
More on Integrable Functionsp. 54
Integral Representation of the Conjugate Operatorp. 59
The Truncated Hilbert Transformp. 65
Notes; Further Results and Problemsp. 68
The Basic Principles
The Calderon-Zygmund Interval Decompositionp. 74
The Hardy-Littlewood Maximal Functionp. 76
The Calderon-Zygmund Decompositionp. 84
The Marcinkiewicz Interpolation Theoremp. 86
Extrapolation and the Zygmund L ln L Classp. 91
The Banach Continuity Principle and a. e. Convergencep. 94
Notes; Further Results and Problemsp. 100
The Hilbert Transform and Multipliers
Existence of the Hilbert Transform of Integrable Functionsp. 110
The Hilbert Transform in L[superscript p](T), 1 [less than or equal] p [less than sign] [infinity]p. 115
Limiting Resultsp. 121
Multipliersp. 126
Notes; Further Results and Problemsp. 132
Paley's Theorem and Fractional Integration
Paley's Theoremp. 142
Fractional Integrationp. 150
Multipliersp. 156
Notes; Further Results and Problemsp. 158
Harmonic and Subharmonic Functions
Abel Summability, Nontangential Convergencep. 167
The Poisson and Conjugate Poisson Kernelsp. 171
Harmonic Functionsp. 176
Further Properties of Harmonic Functions and Subharmonic Functionsp. 181
Harnack's and Mean Value Inequalitiesp. 187
Notes; Further Results and Problemsp. 191
Oscillation of Functions
Mean Oscillation of Functionsp. 199
The Maximal Operator and BMOp. 204
The Conjugate of Bounded and BMO Functionsp. 206
Wk-L[superscript p] and K[subscript f]. Interpolationp. 209
Lipschitz and Morrey Spacesp. 213
Notes; Further Results and Problemsp. 216
A[subscript p] Weights
The Hardy-Littlewood Maximal Theorem for Regular Measuresp. 223
A[subscript p] Weights and the Hardy-Littlewood Maximal Functionp. 225
A[subscript 1] Weightsp. 228
A[subscript p] Weights, p [greater than sign] 1p. 233
Factorization of A[subscript p] Weightsp. 237
A[subscript p] and BMOp. 240
An Extrapolation Resultp. 242
Notes; Further Results and Problemsp. 247
More about R[superscript n]
Distributions, Fourier Transformsp. 259
Translation Invariant Operators. Multipliersp. 263
The Hilbert and Riesz Transformsp. 266
Sobolev and Poincare Inequalitiesp. 270
Calderon-Zygmund Singular Integral Operators
The Benedek-Calderon-Panzone Principlep. 280
A Theorem of Zop. 282
Convolution Operatorsp. 284
Cotlar's Lemmap. 285
Calderon-Zygmund Singular Integral Operatorsp. 286
Maximal Calderon-Zygmund Singular Integral Operatorsp. 291
Singular Integral Operators in L[superscript infinity] (R[superscript n])p. 294
Notes; Further Results and Problemsp. 295
The Littlewood-Paley Theory
Vector-Valued Inequalitiesp. 303
Vector-Valued Singular Integral Operatorsp. 307
The Littlewood-Paley g Functionp. 309
The Lusin Area Function and the Littlewood-Paley g*[subscript lambda] Functionp. 314
Hormander's Multiplier Theoremp. 318
Notes; Further Results and Problemsp. 321
The Good [lambda] Principle
Good [lambda] Inequalitiesp. 328
Weighted Norm Inequalities for Maximal CZ Singular Integral Operatorsp. 330
Weighted Weak-Type (1,1) Estimates for CZ Singular Integral Operatorsp. 334
Notes; Further Results and Problemsp. 337
Hardy Spaces of Several Real Variables
Atomic Decompositionp. 340
Maximal Function Characterization of Hardy Spacesp. 350
Systems of Conjugate Functionsp. 356
Multipliersp. 359
Interpolationp. 363
Notes; Further Results and Problemsp. 366
Carleson Measures
Carleson Measuresp. 372
Duals of Hardy Spacesp. 374
Tent Spacesp. 378
Notes; Further Results and Problemsp. 383
Cauchy Integrals on Lipschitz Curves
Cauchy Integrals on Lipschitz Curvesp. 392
Related Operatorsp. 408
The T1 Theoremp. 412
Notes; Further Results and Problemsp. 416
Boundary Value Problems on C[superscript 1]-Domains
The Double and Single Layer Potentials on a C[superscript 1]-Domainp. 424
The Dirichlet and Neumann Problemsp. 438
Notesp. 444
Bibliographyp. 446
Indexp. 457
Table of Contents provided by Rittenhouse. All Rights Reserved.

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