Real-variable Methods in Harmonic Analysis (eBook)

Front Cover 1
Real-Variable Methods in Harmonic Analysis 4
Copyright Page 5
Contents 8
Preface 12
Chapter I. Fourier Series 14
1. Fourier Series of Functions 14
2. Fourier Series of Continuous Functions 21
3. Elementary Properties of Fourier Series 26
4. Fourier Series of Functionals 29
5. Notes Further Results and Problems
Chapter II. Cesàro Summability 41
1. (C, 1) Summability 41
2. Fejbér’s Kernel 42
3. Characterization of Fourier Series of Functions and Measures 47
4. A.E. Convergence of (C, 1) Means of Summable Functions 54
5 . Notes Further Results and Problems
Chapter III. Norm Convergence of Fourier Series 61
1. The Case L2( T) Hilbert Space
2. Norm Convergence in Lp(T), 1 < p<
3. The Conjugate Mapping 65
4. More on Integrable Functions 67
5 . Integral Representation of the Conjugate Operator 72
6. The Truncated Hilbert Transform 78
7. Notes Further Results and Problems
Chapter IV. The Basic Principles 87
1. The Calderón–Zygmund Interval Decomposition 87
2. The Hardy–Littlewood Maximal Function 89
3. The Calderón–Zygmund Decomposition 97
4. The Marcinkiewicz Interpolation Theorem 99
5 . Extrapolation and the Zygmund L In L Class 104
6. The Banach Continuity Principle and a.e. Convergence 107
7. Notes Further Results and Problems
Chapter V. The Hilbert Transform and Multipliers 123
1. Existence of the Hilbert Transform of Integrable Functions 123
2. The Hilbert Transform in LP(T), 1< = p <
3. Limiting Results 134
4. Multipliers 139
5. Notes Further Results and Problems
Chapter VI. Paley's Theorem and Fractional Integration 155
1. Paley's Theorem 155
2. Fractional Integration 163
3. Multipliers 169
4. Notes Further Results and Problems
Chapter VII. Harmonic and Subharmonic Functions 180
1. Abel Summability, Nontangential Convergence 180
2. The Poisson and Conjugate Poisson Kernels 184
3. Harmonic Functions 189
4. Further Properties of Harmonic Functions and Subharmonic Functions 194
5 . Harnack's and Mean Value Inequalities 200
6. Notes Further Results and Problems
Chapter VIII. Oscillation of Functions 212
1. Mean Oscillation of Functions 212
2. The Maximal Operator and BMO 217
3. The Conjugate of Bounded and BMO Functions 219
4. Wk-Lp and Kf. Interpolation 222
5 . Lipschitz and Morrey Spaces 226
6. Notes Further Results and Problems
Chapter IX. Ap Weights 236
1. The Hardy–Littlewood Maximal Theorem for Regular Measures 236
2. Ap Weights and the Hardy–Littlewood Maximal Function 238
3. A1 Weights 241
4. Ap Weights, p > 1
5. Factorization of Ap Weights 250
6. Ap and BMO 253
7. An Extrapolation Result 255
8. Notes Further Results and Problems
Chapter X. More about Rn 272
1. Distributions. Fourier Transforms 272
2. Translation Invariant Operators. Multipliers 276
3. The Hilbert and Riesz Transforms 279
4. Sobolev and Poincaré Inequalities 283
Chapter XI. Calderón–Zygmund Singular Integral Operators 293
1. The Bendek–Calderón–Panzone Principle 293
2 . A Theorem of Zó 295
3. Convolution Operators 297
4. Cotlar's Lemma 298
5. Calderón–Zygmund Singular Integral Operators 299
6. Maximal Calderón–Zygmund Singular Integral Operators 304
7. Singular Integral Operators in L00(Rn) 307
8. Notes Further Results and Problems
Chapter XII. The Littlewood–Paley Theory 316
1. Vector–Valued Inequalities 316
2. Vector–Valued Singular Integral Operators 320
3. The Littlewood–Paley g Function 322
4. The Lusin Area Function and the Littlewood-Paley g*. Function 327
5. Hörmander's Multiplier Theorem 331
6. Notes Further Results and Problems
Chapter XIII. The Good ., Principle 341
1 . Good . Inequalities 341
2. Weighted Norm Inequalities for Maximal CZ Singular Integral Operators 343
3. Weighted Weak-Type (1,1) Estimates for CZ Singular Integral Operators 347
4. Notes Further Results and Problems
Chapter XIV. Hardy Spaces of Several Real Variables 353
1. Atomic Decomposition 353
2. Maximal Function Characterization of Hardy Spaces 363
3. Systems of Conjugate Functions 369
4. Multipliers 372
5. Interpolation 376
6. Notes Further Results and Problems
Chapter XV Carleson Measures 385
1. Carleson Measures 385
2. Duals of Hardy Spaces 387
3. Tent Spaces 391
4. Notes Further Results and Problems
Chapter XVI Cauchy Integrals on Lipschitz Curves 405
1 . Cauchy Integrals on Lipschitz Curves 405
2. Related Operators 421
3. The T1 Theorem 425
4. Notes Further Results and Problems
Chapter XVII Boundary Value Problems on C1–Domains 437
1. The Double and Single Layer Potentials on a C1-Domain 437
2. The Dirichlet and Neumann Problems 451
3. Notes 457
Bibliography 459
Index 470