Real-Variable Methods in Harmonic Analysis

by Torchinsky, Alberto

• ISBN13: 9780486435084
• ISBN10: 0486435083
• Format: Paperback
• Copyright: 2004-04-09
• Publisher: Dover Publications

Summary

"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

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