6 Smoothness and Function Spaces 1
6.1 Riesz and Bessel Potentials,Fractional Integrals 1
6.1.1 Riesz Potentials 2
6.1.2 Bessel Potentials 6
Exercises 9
6.2 Sobolev Spaces 12
6.2.1 Definition and Basic Properties of General Sobolev Spaces 13
6.2.2 Littlewood-Paley Characterization of Inhomogeneous Sobolev Spaces 16
6.2.3 Littlewood-Paley Characterization of Homogeneous Sobolev Spaces 20
Exercises 22
6.3 Lipschitz Spaces 24
6.3.1 Introduction to Lipschitz Spaces 25
6.3.2 Littlewood-Paley Characterization of Homogeneous Lipschitz Spaces 27
6.3.3 Littlewood-Paley Characterization of Inhomogeneous Lipschitz Spaces 31
Exercises 34
6.4 Hardy Spaces 37
6.4.1 Definition of Hardy Spaces 37
6.4.2 Quasinorm Equivalence of Several Maximal Functions 40
6.4.3 Consequences of the Characterizations of Hardy Spaces 53
6.4.4 Vector-Valued Hp and Its Characterizations 56
6.4.5 Singular Integrals on Hardy Spaces 58
6.4.6 The Littlewood-Paley Characterization of Hardy Spaces 63
Exercises 66
6.5 Besov-Lipschitz and Triebel-Lizorkin Spaces 68
6.5.1 Introduction of Function Spaces 68
6.5.2 Equivalence of Definitions 71
Exercises 76
6.6 Atomic Decomposition 78
6.6.1 The Space of Sequences Fα,q p 78
6.6.2 The Smooth Atomic Decomposition of Fα,q p 78
6.6.3 The Nonsmooth Atomic Decomposition of Fα,q p 82
6.6.4 Atomic Decomposition of Hardy Spaces 86
Exercises 90
6.7 Singular Integrals on Function Spaces 93
6.7.1 Singular Integrals on the Hardy Space H1 93
6.7.2 Singular Integrals on Besov-Lipschitz Spaces 96
6.7.3 Singular Integrals on Hp(Rn) 96
6.7.4 A Singular Integral Characterization of H1(Rn) 104
Exercises 111
7 BMO and Carleson Measures 117
7.1 Functions of Bounded Mean Oscillation 117
7.1.1 Definition and Basic Properties of BMO 118
7.1.2 The John-Nirenberg Theorem 124
7.1.3 Consequences of Theorem 7.1.6 128
Exercises 129
7.2 Duality between H1 and BMO 130
Exercises 135
7.3 Nontangential Maximal Functions and Carleson Measures 135
7.3.1 Definition and Basic Properties of Carleson Measures 136
7.3.2 BMO Functions and Carleson Measures 141
Exercises 144
7.4 The Sharp Maximal Function 146
7.4.1 Definition and Basic Properties of the Sharp Maximal Function 146
7.4.2 A Good Lambda Estimate for the Sharp Function 148
7.4.3 Interpolation Using BMO 151
7.4.4 Estimates for Singular Integrals Involving the Sharp Function 152
Exercises 155
7.5 Commutators of Singular Integrals with BMO Functions 157
7.5.1 An Orlicz-Type Maximal Function 158
7.5.2 A Pointwise Estimate for the Commutator 161
7.5.3 Lp Boundedness of the Commutator 163
Exercises 165
8 Singular Integrals of Nonconvolution Type 169
8.1 General Background and the Role of BMO 169
8.1.1 Standard Kernels 170
8.1.2 Operators Associated with Standard Kernels 175
8.1.3 Calderón-Zygmund Operators Acting on Bounded Functions 179
Exercises 181
8.2 Consequences of L2 Boundedness 182
8.2.1 Weak Type(1,1)and Lp Boundedness ofSingular Integrals 183
8.2.2 Boundedness of Maximal Singular Integrals 185
8.2.3 H1→L1 and L∞→BMO Boundedness of Singular Integrals 188
Exercises 191
8.3 The T(1)Theorem 193
8.3.1 Preliminaries and Statement of the Theorem 193
8.3.2 The Proof of Theorem 8.3.3 196
8.3.3 An Application 209
Exercises 211
8.4 Paraproducts 212
8.4.1 Introduction to Paraproducts 212
8.4.2 L2 Boundedness of Paraproducts 214
8.4.3 Fundamental Properties of Paraproducts 216
Exercises 222
8.5 An Almost Orthogonality Lemma and Applications 223
8.5.1 The Cotlar-Knapp-Stein Almost Orthogonality Lemma 224
8.5.2 An Application 227
8.5.3 Almost Orthogonality and the T(1)Theorem 230
8.5.4 Pseudodifferential Operators 233
Exercises 236
8.6 The Cauchy Integral of Calderón and the T(b) Theorem 238
8.6.1 Introduction of the Cauchy Integral Operator along a Lipschitz Curve 239
8.6.2 Resolution of the Cauchy Integral and Reduction of Its L2 Boundednessto a Quadratic Estimate 242
8.6.3 A Quadratic T(1)Type Theorem 246
8.6.4 A T(b) Theorem and the L2 Boundedness of the Cauchy Integral 250
Exercises 253
8.7 Square Roots of Elliptic Operators 256
8.7.1 Preliminaries and Statement of the Main Result 256
8.7.2 Estimates for Elliptic Operators on Rn 257
8.7.3 Reduction to a Quadratic Estimate 260
8.7.4 Reduction to a Carleson Measure Estimate 261
8.7.5 The T(b) Argument 267
8.7.6 The Proof of Lemma 8.7.9 270
Exercises 275
9 Weighted Inequalities 279
9.1 The Ap Condition 279
9.1.1 Motivation for the Ap Condition 280
9.1.2 Properties of Ap Weights 283
Exercises 291
9.2 Reverse H？lder Inequality and Consequences 293
9.2.1 The Reverse H？lder Property of Ap Weights 293
9.2.2 Consequences of the Reverse H？lder Property 297
Exercises 299
9.3 The A∞ Condition 302
9.3.1 The Class of A∞ Weights 302
9.3.2 Characterizations of A∞ Weights 304
Exercises 308
9.4 Weighted Norm Inequalities for Singular Integrals 309
9.4.1 A Review of Singular Integrals 309
9.4.2 A Good Lambda Estimate for Singular Integrals 310
9.4.3 Consequences of the Good Lambda Estimate 316
9.4.4 Necessity of the Ap Condition 321
Exercises 322
9.5 Further Properties of Ap Weights 324
9.5.1 Factorization of Weights 324
9.5.2 Extrapolation from Weighted Estimates on a Single Lp0 325
9.5.3 Weighted Inequalities Versus Vector-Valued Inequalities 332
Exercises 335
10 Boundedness and Convergence of Fourier Integrals 339
10.1 The Multiplier Problem for the Ball 340
10.1.1 Sprouting of Triangles 340
10.1.2 The counterexample 343
Exercises 350
10.2 Bochner-Riesz Means and the Carleson-Sj？lin Theorem 351
10.2.1 The Bochner-Riesz Kernel and Simple Estimates 351
10.2.2 The Carleson-Sj？lin Theorem 354
10.2.3 The Kakeya Maximal Function 359
10.2.4 Boundedness of a Square Function 361
10.2.5 The Proof of Lemma 10.2.5 363
Exercises 366
10.3 Kakeya Maximal Operators 368
10.3.1 Maximal Functions Associated with a Set of Directions 368
10.3.2 The Boundedness of ？ΣN on Lp(R2) 370
10.3.3 The Higher-Dimensional Kakeya Maximal Operator 378
Exercises 384
10.4 Fourier Transform Restriction and Bochner-Riesz Means 387
10.4.1 Necessary Conditions for Rp→q(Sn-1) to Hold 388
10.4.2 A Restriction Theorem for the Fourier Transform 390
10.4.3 Applications to Bochner-Riesz Multipliers 393
10.4.4 The Full Restriction Theoremon R2 396
Exercises 402
10.5 Almost Everywhere Convergence of Bochner-Riesz Means 403
10.5.1 A Counterexample for the Maximal Bochner-Riesz Operator 404
10.5.2 Almost Everywhere Summability of the Bochner-Riesz Means 407
10.5.3 Estimates for Radial Multipliers 411
Exercises 419
11 Time-Frequency Analysis and the Carleson-Hunt Theorem 423
11.1 Almost Everywhere Convergence of Fourier Integrals 423
11.1.1 Preliminaries 424
11.1.2 Discretization of the Carleson Operator 428
11.1.3 Linearization of a Maximal Dyadic Sum 432
11.1.4 Iterative Selection of Sets of Tiles with Large Mass and Energy 434
11.1.5 Proof of the Mass Lemma 11.1.8 439
11.1.6 Proof of Energy Lemma 11.1.9 441
11.1.7 Proof of the Basic Estimate Lemma 11.1.10 446
Exercises 452
11.2 Distributional Estimates for the Carleson Operator 456
11.2.1 The Main Theorem and Preliminary Reductions 456
11.2.2 The Proof of Estimate(11.2.8) 460
11.2.3 The Proof of Estimate(11.2.9) 462
11.2.4 The Proof of Lemma 11.2.2 463
Exercises 474
11.3 The Maximal Carleson Operator and Weighted Estimates 475
Exercises 479
Glossary 483
References 487
Index 501