现代傅里叶分析 第3版PDF|Epub|txt|kindle电子书版本网盘下载

现代傅里叶分析 第3版PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

1 Smoothness and Function Spaces1

1.1 Smooth Functions and Tempered Distributions1

1.1.1 Space of Tempered Distributions Modulo Polynomials2

1.1.2 Calderón Reproducing Formula5

Exercises7

1.2 Laplacian,Riesz Potentials,and Bessel Potentials9

1.2.1 Riesz Potentials10

1.2.2 Bessel Potentials13

Exercises17

1.3 Sobolev Spaces20

1.3.1 Definition and Basic Properties of General Sobolev Spaces21

1.3.2 Littlewood-Paley Characterization of Inhomogeneous Sobolev Spaces25

1.3.3 Littlewood-Paley Characterization of Homogeneous Sobolev Spaces29

Exercises32

1.4 Lipschitz Spaces34

1.4.1 Introduction to Lipschitz Spaces34

1.4.2 Littlewood-Paley Characterization of Homogeneous Lipschitz Spaces39

1.4.3 Littlewood-Paley Characterization of Inhomogeneous Lipschitz Spaces45

Exercises49

2 Hardy Spaces,Besov Spaces,and Triebel-Lizorkin Spaces55

2.1 Hardy Spaces55

2.1.1 Definition of Hardy Spaces56

2.1.2 Quasi-norm Equivalence of Several Maximal Functions59

2.1.3 Consequences of the Characterizations of Hardy Spaces73

2.1.4 Vector-Valued Hp and Its Characterizations79

2.1.5 Singular Integrals on vector-valued Hardy Spaces81

Exercises87

2.2 Function Spaces and the Square Function Characterization of Hardy Spaces90

2.2.1 Introduction to Function Spaces91

2.2.2 Properties of Functions with Compactly Supported Fourier Transforms93

2.2.3 Equivalence of Function Space Norms97

2.2.4 The Littlewood-Paley Characterization of Hardy Spaces101

Exercises105

2.3 Atomic Decomposition of Homogeneous Triebel-Lizorkin Spaces107

2.3.1 Embeddings and Completeness of Triebel-Lizorkin Spaces107

2.3.2 The Space of Triebel-Lizorkin Sequences109

2.3.3 The Smooth Atomic Decomposition of Homogeneous Triebel-Lizorkin Spaces109

2.3.4 The Nonsmooth Atomic Decomposition of Homogeneous Triebel-Lizorkin Spaces114

2.3.5 Atomic Decomposition of Hardy Spaces120

Exercises124

2.4 Singular Integrals on Function Spaces127

2.4.1 Singular Integrals on the Hardy Space H1127

2.4.2 Singular Integrals on Besov-Lipschitz Spaces130

2.4.3 Singular Integrals on Hp(Rn)131

2.4.4 A Singular Integral Characterization of H1(Rn)141

Exercises148

3 BMO and Carleson Measures153

3.1 Functions of Bounded Mean Oscillation153

3.1.1 Definition and Basic Properties of BMO154

3.1.2 The John-Nirenberg Theorem160

3.1.3 Consequences of Theorem 3.1.6164

Exercises166

3.2 Duality between H1 and BMO167

Exercises172

3.3 Nontangential Maximal Functions and Carleson Measures172

3.3.1 Definition and Basic Properties of Carleson Measures173

3.3.2 BMO Functions and Carleson Measures178

Exercises181

3.4 The Sharp Maximal Function184

3.4.1 Definition and Basic Properties of the Sharp Maximal Function184

3.4.2 A Good Lambda Estimate for the Sharp Function185

3.4.3 Interpolation Using BMO190

3.4.4 Estimates for Singular Integrals Involving the Sharp Function191

Exercises194

3.5 Commutators of Singular Integrals with BMO Functions196

3.5.1 An Orlicz-Type Maximal Function197

3.5.2 A Pointwise Estimate for the Commutator200

3.5.3 Lp Boundedness of the Commutator203

Exercises204

4 Singular Integrals of Nonconvolution Type209

4.1 General Background and the Role of BMO209

4.1.1 Standard Kernels210

4.1.2 Operators Associated with Standard Kernels215

4.1.3 Calderón-Zygmund Operators Acting on Bounded Functions221

Exercises223

4.2 Consequences of L2 Boundedness225

4.2.1 Weak Type(1,1)and Lp Boundedness of Singular Integrals225

4.2.2 Boundedness of Maximal Singular Integrals228

4.2.3 H1→L1 and L∞→BMO Boundedness of Singular Integrals231

Exercises234

4.3 The T(1)Theorem236

4.3.1 Preliminaries and Statement of the Theorem236

4.3.2 The Proof of Theorem 4.3.3239

4.3.3 An Application254

Exercises255

4.4 Paraproducts257

4.4.1 Introduction to Paraproducts258

4.4.2 L2 Boundedness of Paraproducts260

4.4.3 Fundamental Properties of Paraproducts261

Exercises267

4.5 An Almost Orthogonality Lemma and Applications268

4.5.1 The Cotlar-Knapp-Stein Almost Orthogonality Lemma269

4.5.2 An Application273

4.5.3 Almost Orthogonality and the T(1)Theorem275

4.5.4 Pseudodifferential Operators279

Exercises281

4.6 The Cauchy Integral of Calderón and the T(b)Theorem283

4.6.1 Introduction of the Cauchy Integral Operator along a Lipschitz Curve284

4.6.2 Resolution of the Cauchy Integral and Reduction of Its L2 Boundedness to a Quadratic Estimate288

4.6.3 A Quadratic T(1)Type Theorem292

4.6.4 A T(b)Theorem and the L2 Boundedness of the Cauchy Integral297

Exercises300

4.7 Square Roots of Elliptic Operators302

4.7.1 Preliminaries and Statement of the Main Result303

4.7.2 Estimates for Elliptic Operators on Rn304

4.7.3 Reduction to a Quadratic Estimate307

4.7.4 Reduction to a Carleson Measure Estimate309

4.7.5 The T(b)Argument315

4.7.6 Proof of Lemma 4.7.9317

Exercises322

5 Boundedness and Convergence of Fourier Integrals327

5.1The Multiplier Problem for the Ball328

5.1.1 Sprouting of Triangles328

5.1.2 The counterexample331

Exercises338

5.2 Bochner-Riesz Means and the Carleson-Sj？lin Theorem339

5.2.1 The Bochner-Riesz Kernel and Simple Estimates339

5.2.2 The Carleson-Sj？lin Theorem342

5.2.3 The Kakeya Maximal Function348

5.2.4 Boundedness of a Square Function350

5.2.5 The Proof of Lemma 5.2.5352

Exercises355

5.3 Kakeya Maximal Operators357

5.3.1 Maximal Functions Associated with a Set of Directions357

5.3.2 The Boundedness of m∑N on Lp(R2)359

5.3.3 The Higher-Dimensional Kakeya Maximal Operator366

Exercises373

5.4 Fourier Transform Restriction and Bochner-Riesz Means375

5.4.1 Necessary Conditions for Rp→q(Sn-1)to Hold376

5.4.2 A Restriction Theorem for the Fourier Transform378

5.4.3 Applications to Bochner-Riesz Multipliers381

5.4.4 The Full Restriction Theorem on R2385

Exercises391

5.5 Almost Everywhere Convergence of Bochner-Riesz Means392

5.5.1 A Counterexample for the Maximal Bochner-Riesz Operator392

5.5.2 Almost Everywhere Summability of the Bochner-Riesz Means396

5.5.3 Estimates for Radial Multipliers402

Exercises410

6 Time-Frequency Analysis and the Carleson-Hunt Theorem415

6.1 Almost Everywhere Convergence of Fourier Integrals415

6.1.1 Preliminaries417

6.1.2 Discretization of the Carleson Operator421

6.1.3 Linearization of a Maximal Dyadic Sum425

6.1.4 Iterative Selection of Sets of Tiles with Large Mass and Energy428

6.1.5 Proof of the Mass Lemma 6.1.8433

6.1.6 Proof of Energy Lemma 6.1.9435

6.1.7 Proof of the Basic Estimate Lemma 6.1.10439

Exercises446

6.2 Distributional Estimates for the Carleson Operator449

6.2.1 The Main Theorem and Preliminary Reductions450

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 6.2.2460

Exercises471

6.3 The Maximal Carleson Operator and Weighted Estimates472

Exercises476

7 Multilinear Harmonic Analysis479

7.1 Multilinear Operators479

7.1.1 Examples and initial results480

7.1.2 Kernels and Duality of m-linear Operators485

7.1.3 Multilinear Convolution Operators with Nonnegative Kernels486

Exercises490

7.2 Multilinear Interpolation493

7.2.1 Real Interpolation for Multilinear Operators493

7.2.2 Proof of Theorem 7.2.2500

7.2.3 Proofs of Lemmas 7.2.6 and 7.2.7508

7.2.4 Multilinear Complex Interpolation513

7.2.5 Multilinear Interpolation between Adjoint Operators517

Exercises520

7.3 Vector-valued Estimates and Multilinear Convolution Operators523

7.3.1 Multilinear Vector-valued Inequalities523

7.3.2 Multilinear Convolution and Multiplier Operators526

7.3.3 Regularizations of Multilinear Symbols and Consequences528

7.3.4 Duality of Multilinear Multiplier Operators534

Exercises535

7.4 Calderón-Zygmund Operators of Several Functions538

7.4.1 Multilinear Calderón-Zygmund Theorem541

7.4.2 A Necessary and Sufficient Condition for the Boundedness of Multilinear Calderón-Zygmund Operators548

Exercises555

7.5 Multilinear Multiplier Theorems556

7.5.1 Some Preliminary Facts557

7.5.2 Coifman-Meyer Method561

7.5.3 H？rmander-Mihlin Multiplier Condition564

7.5.4 Proof of Main Result569

Exercises575

7.6 An Application Concerning the Leibniz Rule of Fractional Differentiation577

7.6.1 Preliminary Lemma578

7.6.2 Proof of Theorem 7.6.1580

Exercises584

A The Schur Lemma589

A.1 The Classical Schur Lemma589

A.2 Schur's Lemma for Positive Operators589

A.3 An Example592

A.4 Historical Remarks594

B Smoothness and Vanishing Moments595

B.1 The Case of No Cancellation595

B.2 One Function has Cancellation596

B.3 One Function has Cancellation：An Example597

B.4 Both Functions have Cancellation：An Example598

B.5 The Case of Three Factors with No Cancellation599

Glossary601

References605

Index621