傅里葉分析

章節目錄

出版說明

序

Preface

Chapter 1. Lp Spaces and Interpolation

1.1. Lp and Weak Lp

1.2. Convolution and Approximate Identities

1.3. Interpolation

1.4. Lorentz Spaces*

Chapter 2. Maximal Functions, Fourier Transform, and Distributions

2.1. Maximal Functions

2.2. The Schwartz Class and the Fourier Transform

2.3. The Class of Tempered Distributions

2.4. More about Distributions and the Fourier Transform*

2.5. Convolution Operators on Lp Spaces and Multipliers

2.6. Oscillatory Integrals

Chapter 3. Fourier Analysis on the Torus

3.1. Fourier Coefficients

3.2. Decay of Fourier Coefficients

3.3. Pointwise Convergence of Fourier Series

3.4. Divergence of Fourier Series and Bochner-Riesz Summability*

3.5. The Conjugate Function and Convergence in Norm

3.6. Multipliers, Transference, and Almost Everywhere Convergence*

3.7. Lacunary Series*

Chapter 4. Singular Integrals of Convolution Type

4.1. The Hilbert Transform and the Riesz Transforms

4.2. Homogeneous Singular Integrals and the Method of Rotations

4.3. The Calder6n-Zygmund Decomposition and Singular Integrals

4.4. Sufficient Conditions for Lp Boundedness

4.5. Vector-Valued Inequalities*

4.6. Vector-Valued Singular Integrals

Chapter 5. Littlewood-Paley Theory and Multipliers

5.1. Littlewood-Paley Theory

5.2. Two Multiplier Theorems

5.3. Applications of Littlewood-Paley Theory

5.4. The Haar System, Conditional Expectation, and Martingales*

5.5. The Spherical Maximal Function*

5.6. Wavelets

Chapter 6. Smoothness and Function Spaces

6.1. Riesz Potentials, Bessel Potentials, and Fractional Integrals

6.2. Sobolev Spaces

6.3. Lipschitz Spaces

6.4. Hardy Spaces*

6.5. Besov-Lipschitz and Triebel-Lizorkin Spaces*

6.6. Atomic Decomposition*

6.7. Singular Integrals on Function Spaces

Chapter 7. BMO and Carleson Measures

7.1. Functions of Bounded Mean Oscillation

7.2. Duality between H1 and BMO

7.3. Nontangential Maximal Functions and Carleson Measures

7.4. The Sharp Maximal Function

7.5. Commutators of Singular Integrals with BMO Functions*

Chapter 8. Singular Integrals of Nonconvolution Type

8.1. General Background and the Role of BMO

8.2. Consequences of L2 Boundedness

8.3. The T(1) Theorem

8.4. Paraproducts

8.5. An Almost Orthogonality Lemma and Applications

8.6. The Cauchy Integral of Calder6n and the T(b) Theorem*

8.7. Square Roots of Elliptic Operators*

Chapter 9. Weighted Inequalities

9.1. The Ap Condition

9.2. Reverse HSlder Inequality for Ap Weights and Consequences

9.3. The Au condition*

9.4. Weighted Norm Inequalities for Singular Integrals

9.5. Further Properties of Ap Weights*

Chapter 10. Boundedness and Convergence of Fourier Integrals

10.1. The Multiplier Problem for the Ball

10.2. Bochner-Riesz Means and the Carleson-SjSlin Theorem

10.3. Kakeya Maximal Operators

10.4. Fourier Transform Restriction and Bochner-Riesz Means

10.5. Almost Everywhere Convergence of Fourier Integrals*

10.6. Lp Boundedness of the Carleson Operator*

Appendix A. Gamma and Beta Functions

A.1. A Useful Formula

A.2. Definitions of F(z) and B(z, w)

A.3. Volume of the Unit Ball and Surface of the Unit Sphere

A.4. A Useful Integral

A.5. Meromorphic Extensions of B(z, w) and F(z)

A.6. Asymptotics of F(x) as x →∞

A.7. The Duplication Formula for the Gamma Function

Appendix B. Bessel Functions

B.I. Definition

B.2. Some Basic Properties

B.3. An Interesting Identity

B.4. The Fourier Transform of Surface Measure on Sn-1

B.5. The Fourier Transform of a Radial Function on Rn

B.6. Asymptotics of Bessel Functions

Appendix C. Rademacher Functions

C.1. Definition of the Rademacher Functions

C.2. Khintchine#s Inequalities

C.3. Derivation of Khintchine#s Inequalities

C.4. Khintchine#s Inequalities for Weak Type Spaces

C.5. Extension to Several Variables

Appendix D. Spherical Coordinates

D.1. Spherical Coordinate Formula

D.2. A useful change of variables formula

D.3. Computation of an Integral over the Sphere

D.4. The Computation of Another Integral over the Sphere

D.5. Integration over a General Surface

D.6. The Stereographic Projection

Appendix E. Some Trigonometric Identities and Inequalities

Appendix F. Summation by Parts

Appendix G. Basic Functional Analysis

Appendix H. The Minimax Lemma

Appendix I. The Schur Lemma

1.1. The Classical Schur Lemma

1.2. Schur#s Lemma for Positive Operators

1.3. An Example

Appendix J. The Whitney Decomposition of Open Sets in Rn

Appendix K. Smoothness and Vanishing Moments

K.I. The Case of No Cancellation

K.2. The Case of Cancellation

K.3. The Case of Three Factors

Bibliography

Index of Notation

Index

教輔材料申請表