Cover

Series Page

Title

Copyright

Dedication

Contents

Preface

Chapter 1 Fourier series

1.1 The Laplacian

Exercises

1.2 Some function spaces and sequence spaces

Exercises

1.3 Fourier coefficients

Exercises

1.4 Convolution on T

Exercises

1.5 Young’s inequality

Exercises

Chapter 2 Abel–Poisson means

2.1 Abel–Poisson means of Fourier series

Exercises

2.2 Approximate identities on T

Exercises

2.3 Uniform convergence and pointwise convergence

Exercises

2.4 Weak* convergence of measures

Exercises

2.5 Convergence in norm

Exercises

2.6 Weak* convergence of bounded functions

Exercises

2.7 Parseval’s identity

Exercises

Chapter 3 Harmonic functions in the unit disc

3.1 Series representation of harmonic functions

Exercises

3.2 Hardy spaces on D

Exercises

3.3 Poisson representation of h∞(D) functions

Exercises

3.4 Poisson representation of hp(D) functions (1

Exercises

3.5 Poisson representation of h1(D) functions

Exercises

3.6 Radial limits of hp(D) functions (1 ≤ p≤∞)

Exercises

3.7 Series representation of the harmonic conjugate

Exercises

Chapter 4 Logarithmic convexity

4.1 Subharmonic functions

Exercises

4.2 The maximum principle

Exercises

4.3 A characterization of subharmonic functions

Exercises

4.4 Various means of subharmonic functions

Exercises

4.5 Radial subharmonic functions

Exercises

4.6 Hardy’s convexity theorem

Exercises

4.7 A complete characterization of hp(D) spaces

Exercises

Chapter 5 Analytic functions in theunit disc

5.1 Representation of Hp(D) functions (1 < p ≤ ∞)

Exercises

5.2 The Hilbert transform on T

Exercises

5.3 Radial limits of the conjugate function

5.4 The Hilbert transform of C1(T) functions

Exercises

5.5 Analytic measures on T

Exercises

5.6 Representations of H1(D) functions

Exercises

5.7 The uniqueness theorem and its applications

Exercises

Chapter 6 Norm inequalities for the conjugate function

6.1 Kolmogorov’s theorems

Exercises

6.2 Harmonic conjugate of h2(D) functions

6.3 M. Riesz’s theorem

Exercises

6.4 The Hilbert transform of bounded functions

6.5 The Hilbert transform of Dini continuous functions

Exercises

6.6 Zygmund’s L log L theorem

6.7 M. Riesz’s L log L theorem

Exercises

Chapter 7 Blaschke products and their applications

7.1 Automorphisms of the open unit disc

Exercises

7.2 Blaschke products for the open unit disc

Exercises

7.3 Jensen’s formula

Exercises

7.4 Riesz’s decomposition theorem

Exercises

7.5 Representation of Hp(D) functions (0 < p < 1)

7.6 The canonical factorization in Hp(D) (0 < p ≤ ∞)

Exercises

7.7 The Nevanlinna class

Exercises

7.8 The Hardy and Fej´er–Riesz inequalities

Chapter 8 Interpolating linear operators

8.1 Operators on Lebesgue spaces

Exercises

8.2 Hadamard’s three-line theorem

Exercises

8.3 The Riesz–Thorin interpolation theorem

Exercises

8.4 The Hausdorff–Young theorem

Exercises

8.5 An interpolation theorem for Hardy spaces

Exercises

8.6 The Hardy–Littlewood inequality

Exercises

Chapter 9 The Fourier transform

9.1 Lebesgue spaces on the real line

Exercises

9.2 The Fourier transform on L1(R)

Exercises

9.3 The multiplication formula on L1(R)

Exercises

9.4 Convolution on R

Exercises

9.5 Young’s inequality

Exercises

Chapter 10 Poisson integrals

10.1 An application of the multiplication formula on L1(R)

10.2 The conjugate Poisson kernel

10.3 Approximate identities on R

Exercises

10.4 Uniform convergence and pointwise convergence

Exercises

10.5 Weak* convergence of measures

Exercises

10.6 Convergence in norm

Exercises

10.7 Weak* convergence of bounded functions

Exercises

Chapter 11 Harmonic functions in the upper half plane

11.1 Hardy spaces on C+

11.2 Poisson representation for semidiscs

Exercises

11.3 Poisson representation of h(C+) functions

11.4 Poisson representation of hp(C+) functions (1 ≤ p ≤ ∞)

Exercises

11.5 A correspondence between C+ and D

11.6 Poisson representation of positive harmonic functions

Exercises

11.7 Vertical limits of hp(C+) functions (1 ≤ p ≤ ∞)

Exercises

Chapter 12 The Plancherel transform

12.1 The inversion formula

Exercises

12.2 The Fourier–Plancherel transform

Exercises

12.3 The multiplication formula on Lp(R) (1 ≤ p ≤ 2)

Exercises

12.4 The Fourier transform on Lp(R) (1 ≤ p ≤ 2)

Exercises

12.5 An application of the multiplication formula on Lp(R) (1 ≤ p ≤ 2)

Exercises

12.6 A complete characterization of hp(C+) spaces

Exercises

Chapter 13 Analytic functions in the upper half plane

13.1 Representation of Hp(C+) functions (1

Exercises

13.2 Analytic measures on R

13.3 Representation of H1(C+) functions

13.4 Spectral analysis of Hp(R) (1 ≤ p ≤ 2)

Exercises

13.5 A contraction from Hp(C+) into Hp(D)

Exercises

13.6 Blaschke products for the upper half plane

13.7 The canonical factorization in Hp(C+) (0 < p ≤ ∞)

Exercises

13.8 A correspondence between Hp(C+) and Hp(D)

Chapter 14 The Hilbert transform on R

14.1 Various definitions of the Hilbert transform

14.2 The Hilbert transform of C1c (R) functions

14.3 Almost everywhere existence of the Hilbert transform

14.4 Kolmogorov’s theorem

Exercises

14.5 M. Riesz’s theorem

Exercises

14.6 The Hilbert transform of Lipα(t) functions

Exercises

14.7 Maximal functions

Exercises

14.8 The maximal Hilbert transform

Appendix A Topics from real analysis

A.1 A very concise treatment of measure theory

Exercises

A.2 Riesz representation theorems

A.3 Weak* convergence of measures

A.4 C(T) is dense in Lp(T) (0 < p < ∞)

Exercises

A.5 The distribution function

Exercises

A.6 Minkowski’s inequality

A.7 Jensen’s inequality

Exercises

Appendix B A panoramic view of the representation theorems

B.1 hp(D)

B.1.2 hp(D) (1 < p < ∞)

B.1.3 h∞(D)

B.2 Hp(D)

B.2.2 H∞(D)

B.3 hp(C+)

B.3.1 h1(C+)

B.3.2 hp(C+) (1 < p ≤ 2)

B.3.3 hp(C+) (2 < p < ∞)

B.3.4 h∞(C+)

B.3.5 h+(C+)

B.4 Hp(C+)

B.4.1 Hp(C+) (1 ≤ p ≤ 2)

B.4.2 Hp(C+) (2 < p < ∞)

B.4.3 H∞(C+)

Bibliography

Index