Classical and modern Fourier analysis / Loukas Grafakos.

Format:

Book

Author/Creator:

Grafakos, Loukas.

Contributor:

Alumni and Friends Memorial Book Fund.

Language:

English

Subjects (All):

Fourier analysis.

Physical Description:

xii, 859, 72 pages : illustrations ; 25 cm

Place of Publication:

Upper Saddle River, N.J. : Pearson/Prentice Hall, [2004]

Contents:

Chapter 1 L[superscript p] Spaces and Interpolation 1

1.1 L[superscript p] and Weak L[superscript p] 1

1.2 Convolution and Approximate Identities 16

1.3 Interpolation 32

1.4 Lorentz Spaces* 45

Chapter 2 Maximal Functions, Fourier Transform, and Distributions 77

2.1 Maximal Functions 78

2.2 The Schwartz Class and the Fourier Transform 94

2.3 The Class of Tempered Distributions 108

2.4 More about Distributions and the Fourier Transform* 122

2.5 Convolution Operators on L[superscript p] Spaces and Multipliers 133

2.6 Oscillatory Integrals 146

Chapter 3 Fourier Analysis on the Torus 157

3.1 Fourier Coefficients 157

3.2 Decay of Fourier Coefficients 172

3.3 Pointwise Convergence of Fourier Series 183

3.4 Divergence of Fourier Series and Bochner-Riesz Summability* 192

3.5 The Conjugate Function and Convergence in Norm 208

3.6 Multipliers, Transference, and Almost Everywhere Convergence* 217

3.7 Lacunary Series* 235

Chapter 4 Singular Integrals of Convolution Type 247

4.1 The Hilbert Transform and the Riesz Transforms 247

4.2 Homogeneous Singular Integrals and the Method of Rotations 264

4.3 The Calderon-Zygmund Decomposition and Singular Integrals 284

4.4 Sufficient Conditions for L[superscript p] Boundedness 299

4.5 Vector-Valued Inequalities* 311

4.6 Vector-Valued Singular Integrals 325

Chapter 5 Littlewood-Paley Theory and Multipliers 337

5.1 Littlewood-Paley Theory 337

5.2 Two Multiplier Theorems 355

5.3 Applications of Littlewood-Paley Theory 368

5.4 The Haar System, Conditional Expectation, and Martingales* 379

5.5 The Spherical Maximal Function* 390

5.6 Wavelets 397

Chapter 6 Smoothness and Function Spaces 413

6.1 Riesz Potentials, Bessel Potentials, and Fractional Integrals 413

6.2 Sobolev Spaces 424

6.3 Lipschitz Spaces 436

6.4 Hardy Spaces* 447

6.5 Besov-Lipschitz and Triebel-Lizorkin Spaces* 477

6.6 Atomic Decomposition* 487

6.7 Singular Integrals on Function Spaces 503

Chapter 7 BMO and Carleson Measures 517

7.1 Functions of Bounded Mean Oscillation 517

7.2 Duality between H[superscript 1] and BMO 530

7.3 Nontangential Maximal Functions and Carleson Measures 535

7.4 The Sharp Maximal Function 545

7.5 Commutators of Singular Integrals with BMO Functions* 557

Chapter 8 Singular Integrals of Nonconvolution Type 569

8.1 General Background and the Role of BMO 569

8.2 Consequences of L[superscript 2] Boundedness 584

8.3 The T(1) Theorem 590

8.4 Paraproducts 608

8.5 An Almost Orthogonality Lemma and Applications 620

8.6 The Cauchy Integral of Calderon and the T(b) Theorem* 634

8.7 Square Roots of Elliptic Operators* 652

Chapter 9 Weighted Inequalities 675

9.1 The A[subscript p] Condition 675

9.2 Reverse Holder Inequality for A[subscript p] Weights and Consequences 685

9.3 The A[subscript infinity] condition* 694

9.4 Weighted Norm Inequalities for Singular Integrals 702

9.5 Further Properties of A[subscript p] Weights* 715

Chapter 10 Boundedness and Convergence of Fourier Integrals 733

10.1 The Multiplier Problem for the Ball 734

10.2 Bochner-Riesz Means and the Carleson-Sjolin Theorem 745

10.3 Kakeya Maximal Operators 762

10.4 Fourier Transform Restriction and Bochner-Riesz Means 780

10.5 Almost Everywhere Convergence of Fourier Integrals* 796

10.6 L[superscript p] Boundedness of the Carleson Operator* 831

Appendix A Gamma and Beta Functions 1

A.1 A Useful Formula 1

A.2 Definitions of [Gamma](z) and B(z, w) 1

A.3 Volume of the Unit Ball and Surface of the Unit Sphere 2

A.4 A Useful Integral 3

A.5 Meromorphic Extensions of B(z, w) and [Gamma](z) 3

A.6 Asymptotics of [Gamma](x) as x to [infinity] 4

A.7 The Duplication Formula for the Gamma Function 5

Appendix B Bessel Functions 7

B.2 Some Basic Properties 7

B.3 An Interesting Identity 9

B.4 The Fourier Transform of Surface Measure on S[superscript n-1] 10

B.5 The Fourier Transform of a Radial Function on R[superscript n] 11

B.6 Asymptotics of Bessel Functions 11

Appendix C Rademacher Functions 15

C.1 Definition of the Rademacher Functions 15

C.2 Khintchine's Inequalities 16

C.3 Derivation of Khintchine's Inequalities 16

C.4 Khintchine's Inequalities for Weak Type Spaces 19

C.5 Extension to Several Variables 19

Appendix D Spherical Coordinates 21

D.1 Spherical Coordinate Formula 21

D.2 A useful change of variables formula 21

D.3 Computation of an Integral over the Sphere 22

D.4 The Computation of Another Integral over the Sphere 23

D.5 Integration over a General Surface 23

D.6 The Stereographic Projection 24

Appendix E Some Trigonometric Identities and Inequalities 25

Appendix F Summation by Parts 27

Appendix G Basic Functional Analysis 29

Appendix H The Minimax Lemma 31

Appendix I The Schur Lemma 35

I.1 The Classical Schur Lemma 35

I.2 Schur's Lemma for Positive Operators 36

Appendix J The Whitney Decomposition of Open Sets in R[superscript n] 41

Appendix K Smoothness and Vanishing Moments 43

K.1 The Case of No Cancellation 43

K.2 The Case of Cancellation 44

K.3 The Case of Three Factors 44.

Notes:

Includes bibliographical references (page B39-B61) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

013035399X

OCLC:

52121527