Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals

Title page......Page 1
Date-line......Page 2
Dedication......Page 3
Contents......Page 5
Preface......Page 9
Guide to the Reader......Page 11
Title......Page 13
Prologue......Page 15
I. Real-Variable Theory......Page 19
1. Basic assumptions......Page 20
2. Examples......Page 21
3. Covering lemmas and the maximal function......Page 24
4. Generalization of the Calderon-Zygmund decomposition......Page 28
5. Singular integrals......Page 30
6. Examples of the general theory......Page 35
7. Appendix: Truncation of singular integrals......Page 42
8. Further results......Page 49
II. More about Maximal Functions......Page 61
1. Vector-valued maximal functions......Page 62
2. Nontangential behavior and Carleson measures......Page 68
3. Two applications......Page 77
4. Singular approximations of the identity......Page 83
5. Further results......Page 87
III. Hardy Spaces......Page 99
1. Maximal characterization of $H^p$......Page 100
2. Atomic decomposition for $H^p$......Page 113
3. Singular integrals......Page 125
4. Appendix: Relation with harmonic functions......Page 130
5. Further results......Page 139
IV. $H^1$ and BMO......Page 151
1. The space of functions of bounded mean oscillation......Page 152
2. The sharp function......Page 158
3. An elementary approach and a dyadic version......Page 161
4. Further properties of BMO......Page 167
5. An interpolation theorem......Page 185
6. Further results......Page 189
V. Weighted Inequalities......Page 205
1. The class $A_p$......Page 206
2. Two further characterizations of $A_p$......Page 210
3. The main theorem about $A_p$......Page 213
4. Weighted inequalities for singular integrals......Page 216
5. Further properties of $A_p$ weights......Page 224
6. Further results......Page 230
VI. Pseudo-Differential and Singular Integral Operators: Fourier Transform......Page 240
1. Pseudo-differential operators......Page 242
2. An $L^2$ theorem......Page 246
3. The symbolic calculus......Page 249
4. Singular integral realization of pseudo-differential operators......Page 253
5. Estimates in $L^p$, Sobolev, and Lipschitz spaces......Page 262
6. Appendix: Compound symbols......Page 270
7. Further results......Page 273
VII. Pseudo-Differential and Singular Integral Operators: Almost Orthogonality......Page 281
1. Exotic and forbidden symbols......Page 282
2. Almost orthogonality......Page 290
3. $L^2$ theory of operators with Calderon-Zygmund kernels......Page 301
4. Appendix: The Cauchy integral......Page 322
5. Further results......Page 329
VIII. Oscillatory Integrals of the First Kind......Page 341
1. Oscillatory integrals of the first kind, one variable......Page 342
2. Oscillatory integrals of the first kind, several variables......Page 353
3. Fourier transforms of measures supported on surfaces......Page 359
4. Restriction of the Fourier transform......Page 364
5. Further results......Page 367
IX. Oscillatory Integrals of the Second Kind......Page 387
1. Oscillatory integrals related to the Fourier transform......Page 388
2. Restriction theorems and Bochner-Riesz summability......Page 398
3. Fourier integral operators: $L^2$ estimates......Page 406
4. Fourier integral operators: $L^p$ estimates......Page 414
5. Appendix: Restriction theorems in two dimensions......Page 424
6. Further results......Page 426
X. Maximal Operators: Some Examples......Page 445
1. The Besicovitch set......Page 446
2. Maximal functions and counterexamples......Page 452
3. Further results......Page 466
XI. Maximal Averages and Oscillatory Integrals......Page 479
1. Maximal averages and square functions......Page 481
2. Averages over a $k$-dimensional submanifold of finite type......Page 488
3. Averages on variable hypersurfaces......Page 505
4. Further results......Page 523
XII. Introduction to the Heisenberg Group......Page 539
1. Geometry of the complex ball and the Heisenberg group......Page 540
2. The Cauchy-Szego integral......Page 544
3. Formalism of quantum mechanics and the Heisenberg group......Page 559
4. Weyl correspondence and pseudo-differential operators......Page 565
5. Twisted convolution and singular integrals on $\\mathbb{H}^n$......Page 569
6. Appendix: Representations of the Heisenberg group......Page 580
7. Further results......Page 586
XIII. More about the Heisenberg Group......Page 599
1. The Cauchy-Riemann complex and its boundary analogue......Page 601
2. The operators $\\bar\\partial_b$ and $\\square_b$ on the Heisenberg group......Page 606
3. Applications of the fundamental solution......Page 617
4. The Lewy operator......Page 623
5. Homogeneous groups......Page 630
6. Appendix: The $\\bar\\partial$-Neumann problem......Page 639
7. Further results......Page 644
Bibliography......Page 657
Author Index......Page 691
Subject Index......Page 697