1 Smoothness and Function Spaces 1
1.1 Smooth Functions and Tempered Distributions 1
1.1.1 Space of Tempered Distributions Modulo Polynomials 2
1.1.2 Calderón Reproducing Formula 5
Exercises 7
1.2 Laplacian,Riesz Potentials,and Bessel Potentials 9
1.2.1 Riesz Potentials 10
1.2.2 Bessel Potentials 13
Exercises 17
1.3 Sobolev Spaces 20
1.3.1 Definition and Basic Properties of General Sobolev Spaces 21
1.3.2 Littlewood-Paley Characterization of Inhomogeneous Sobolev Spaces 25
1.3.3 Littlewood-Paley Characterization of Homogeneous Sobolev Spaces 29
Exercises 32
1.4 Lipschitz Spaces 34
1.4.1 Introduction to Lipschitz Spaces 34
1.4.2 Littlewood-Paley Characterization of Homogeneous Lipschitz Spaces 39
1.4.3 Littlewood-Paley Characterization of Inhomogeneous Lipschitz Spaces 45
Exercises 49
2 Hardy Spaces,Besov Spaces,and Triebel-Lizorkin Spaces 55
2.1 Hardy Spaces 55
2.1.1 Definition of Hardy Spaces 56
2.1.2 Quasi-norm Equivalence of Several Maximal Functions 59
2.1.3 Consequences of the Characterizations of Hardy Spaces 73
2.1.4 Vector-Valued Hp and Its Characterizations 79
2.1.5 Singular Integrals on vector-valued Hardy Spaces 81
Exercises 87
2.2 Function Spaces and the Square Function Characterization of Hardy Spaces 90
2.2.1 Introduction to Function Spaces 91
2.2.2 Properties of Functions with Compactly Supported Fourier Transforms 93
2.2.3 Equivalence of Function Space Norms 97
2.2.4 The Littlewood-Paley Characterization of Hardy Spaces 101
Exercises 105
2.3 Atomic Decomposition of Homogeneous Triebel-Lizorkin Spaces 107
2.3.1 Embeddings and Completeness of Triebel-Lizorkin Spaces 107
2.3.2 The Space of Triebel-Lizorkin Sequences 109
2.3.3 The Smooth Atomic Decomposition of Homogeneous Triebel-Lizorkin Spaces 109
2.3.4 The Nonsmooth Atomic Decomposition of Homogeneous Triebel-Lizorkin Spaces 114
2.3.5 Atomic Decomposition of Hardy Spaces 120
Exercises 124
2.4 Singular Integrals on Function Spaces 127
2.4.1 Singular Integrals on the Hardy Space H1 127
2.4.2 Singular Integrals on Besov-Lipschitz Spaces 130
2.4.3 Singular Integrals on Hp(Rn) 131
2.4.4 A Singular Integral Characterization of H1(Rn) 141
Exercises 148
3 BMO and Carleson Measures 153
3.1 Functions of Bounded Mean Oscillation 153
3.1.1 Definition and Basic Properties of BMO 154
3.1.2 The John-Nirenberg Theorem 160
3.1.3 Consequences of Theorem 3.1.6 164
Exercises 166
3.2 Duality between H1 and BMO 167
Exercises 172
3.3 Nontangential Maximal Functions and Carleson Measures 172
3.3.1 Definition and Basic Properties of Carleson Measures 173
3.3.2 BMO Functions and Carleson Measures 178
Exercises 181
3.4 The Sharp Maximal Function 184
3.4.1 Definition and Basic Properties of the Sharp Maximal Function 184
3.4.2 A Good Lambda Estimate for the Sharp Function 185
3.4.3 Interpolation Using BMO 190
3.4.4 Estimates for Singular Integrals Involving the Sharp Function 191
Exercises 194
3.5 Commutators of Singular Integrals with BMO Functions 196
3.5.1 An Orlicz-Type Maximal Function 197
3.5.2 A Pointwise Estimate for the Commutator 200
3.5.3 Lp Boundedness of the Commutator 203
Exercises 204
4 Singular Integrals of Nonconvolution Type 209
4.1 General Background and the Role of BMO 209
4.1.1 Standard Kernels 210
4.1.2 Operators Associated with Standard Kernels 215
4.1.3 Calderón-Zygmund Operators Acting on Bounded Functions 221
Exercises 223
4.2 Consequences of L2 Boundedness 225
4.2.1 Weak Type(1,1)and Lp Boundedness of Singular Integrals 225
4.2.2 Boundedness of Maximal Singular Integrals 228
4.2.3 H1→L1 and L∞→BMO Boundedness of Singular Integrals 231
Exercises 234
4.3 The T(1)Theorem 236
4.3.1 Preliminaries and Statement of the Theorem 236
4.3.2 The Proof of Theorem 4.3.3 239
4.3.3 An Application 254
Exercises 255
4.4 Paraproducts 257
4.4.1 Introduction to Paraproducts 258
4.4.2 L2 Boundedness of Paraproducts 260
4.4.3 Fundamental Properties of Paraproducts 261
Exercises 267
4.5 An Almost Orthogonality Lemma and Applications 268
4.5.1 The Cotlar-Knapp-Stein Almost Orthogonality Lemma 269
4.5.2 An Application 273
4.5.3 Almost Orthogonality and the T(1)Theorem 275
4.5.4 Pseudodifferential Operators 279
Exercises 281
4.6 The Cauchy Integral of Calderón and the T(b)Theorem 283
4.6.1 Introduction of the Cauchy Integral Operator along a Lipschitz Curve 284
4.6.2 Resolution of the Cauchy Integral and Reduction of Its L2 Boundedness to a Quadratic Estimate 288
4.6.3 A Quadratic T(1)Type Theorem 292
4.6.4 A T(b)Theorem and the L2 Boundedness of the Cauchy Integral 297
Exercises 300
4.7 Square Roots of Elliptic Operators 302
4.7.1 Preliminaries and Statement of the Main Result 303
4.7.2 Estimates for Elliptic Operators on Rn 304
4.7.3 Reduction to a Quadratic Estimate 307
4.7.4 Reduction to a Carleson Measure Estimate 309
4.7.5 The T(b)Argument 315
4.7.6 Proof of Lemma 4.7.9 317
Exercises 322
5 Boundedness and Convergence of Fourier Integrals 327
5.1The Multiplier Problem for the Ball 328
5.1.1 Sprouting of Triangles 328
5.1.2 The counterexample 331
Exercises 338
5.2 Bochner-Riesz Means and the Carleson-Sj？lin Theorem 339
5.2.1 The Bochner-Riesz Kernel and Simple Estimates 339
5.2.2 The Carleson-Sj？lin Theorem 342
5.2.3 The Kakeya Maximal Function 348
5.2.4 Boundedness of a Square Function 350
5.2.5 The Proof of Lemma 5.2.5 352
Exercises 355
5.3 Kakeya Maximal Operators 357
5.3.1 Maximal Functions Associated with a Set of Directions 357
5.3.2 The Boundedness of m∑N on Lp(R2) 359
5.3.3 The Higher-Dimensional Kakeya Maximal Operator 366
Exercises 373
5.4 Fourier Transform Restriction and Bochner-Riesz Means 375
5.4.1 Necessary Conditions for Rp→q(Sn-1)to Hold 376
5.4.2 A Restriction Theorem for the Fourier Transform 378
5.4.3 Applications to Bochner-Riesz Multipliers 381
5.4.4 The Full Restriction Theorem on R2 385
Exercises 391
5.5 Almost Everywhere Convergence of Bochner-Riesz Means 392
5.5.1 A Counterexample for the Maximal Bochner-Riesz Operator 392
5.5.2 Almost Everywhere Summability of the Bochner-Riesz Means 396
5.5.3 Estimates for Radial Multipliers 402
Exercises 410
6 Time-Frequency Analysis and the Carleson-Hunt Theorem 415
6.1 Almost Everywhere Convergence of Fourier Integrals 415
6.1.1 Preliminaries 417
6.1.2 Discretization of the Carleson Operator 421
6.1.3 Linearization of a Maximal Dyadic Sum 425
6.1.4 Iterative Selection of Sets of Tiles with Large Mass and Energy 428
6.1.5 Proof of the Mass Lemma 6.1.8 433
6.1.6 Proof of Energy Lemma 6.1.9 435
6.1.7 Proof of the Basic Estimate Lemma 6.1.10 439
Exercises 446
6.2 Distributional Estimates for the Carleson Operator 449
6.2.1 The Main Theorem and Preliminary Reductions 450
6.2.2 The Proof of Estimate(6.2.18) 457
6.2.3 The Proof of Estimate(6.2.19) 459
6.2.4 The Proof of Lemma 6.2.2 460
Exercises 471
6.3 The Maximal Carleson Operator and Weighted Estimates 472
Exercises 476
7 Multilinear Harmonic Analysis 479
7.1 Multilinear Operators 479
7.1.1 Examples and initial results 480
7.1.2 Kernels and Duality of m-linear Operators 485
7.1.3 Multilinear Convolution Operators with Nonnegative Kernels 486
Exercises 490
7.2 Multilinear Interpolation 493
7.2.1 Real Interpolation for Multilinear Operators 493
7.2.2 Proof of Theorem 7.2.2 500
7.2.3 Proofs of Lemmas 7.2.6 and 7.2.7 508
7.2.4 Multilinear Complex Interpolation 513
7.2.5 Multilinear Interpolation between Adjoint Operators 517
Exercises 520
7.3 Vector-valued Estimates and Multilinear Convolution Operators 523
7.3.1 Multilinear Vector-valued Inequalities 523
7.3.2 Multilinear Convolution and Multiplier Operators 526
7.3.3 Regularizations of Multilinear Symbols and Consequences 528
7.3.4 Duality of Multilinear Multiplier Operators 534
Exercises 535
7.4 Calderón-Zygmund Operators of Several Functions 538
7.4.1 Multilinear Calderón-Zygmund Theorem 541
7.4.2 A Necessary and Sufficient Condition for the Boundedness of Multilinear Calderón-Zygmund Operators 548
Exercises 555
7.5 Multilinear Multiplier Theorems 556
7.5.1 Some Preliminary Facts 557
7.5.2 Coifman-Meyer Method 561
7.5.3 H？rmander-Mihlin Multiplier Condition 564
7.5.4 Proof of Main Result 569
Exercises 575
7.6 An Application Concerning the Leibniz Rule of Fractional Differentiation 577
7.6.1 Preliminary Lemma 578
7.6.2 Proof of Theorem 7.6.1 580
Exercises 584
A The Schur Lemma 589
A.1 The Classical Schur Lemma 589
A.2 Schur's Lemma for Positive Operators 589
A.3 An Example 592
A.4 Historical Remarks 594
B Smoothness and Vanishing Moments 595
B.1 The Case of No Cancellation 595
B.2 One Function has Cancellation 596
B.3 One Function has Cancellation：An Example 597
B.4 Both Functions have Cancellation：An Example 598
B.5 The Case of Three Factors with No Cancellation 599
Glossary 601
References 605
Index 621