Theory of H[superscript p] spaces, Volume 38

Dedication

Preface

Chapter 1: Harmonic and Subharmonic Functions

1.1. Harmonic Functions

1.2. Boundary Behavior of Poisson–Stieltjes Integrals

1.3. Subharmonic Functions

1.4. Hardy’s Convexity Theorem

1.5. Subordination

1.6. Maximal Theorems

Exercises

Notes

Chapter 2: Basic Structure of Hp Functions

2.1. Boundary Values

2.2. Zeros

2.3. Mean Convergence to Boundary Values

2.4. Canonical Factorization

2.5. The Class N+

2.6. Harmonic Majorants

Exercises

Notes

Chapter 3: Applications

3.1. Poisson Integrals and H1

3.2. Description of the Boundary Functions

3.3. Cauchy and Cauchy–Stieltjes Integrals

3.4. Analytic Functions Continuous in |z| ≤ 1

3.5. Applications to Conformal Mapping

3.6. Inequalities of Fejér–Riesz, Hilbert, and Hardy

3.7. Schlicht Functions

Exercises

Notes

Chapter 4: Conjugate Functions

4.1. Theorem of M. Riesz

4.2. Kolmogorov’s Theorem

4.3. Zygmund’s Theorem

4.4. Trigonometric Series

4.5. The Conjugate of an h1 Function

4.6. The Case p < 1: A Counterexample

Exercises

Notes

Chapter 5: Mean Growth and Smoothness

5.1. Smoothness Classes

5.2. Smoothness of the Boundary Function

5.3. Growth of a Function and its Derivative

5.4. More on Conjugate Functions

5.5. Comparative Growth of Means

5.6. Functions with Hp Derivative

Exercises

Notes

Chapter 6: Taylor Coefficients

6.1. Hausdorff–Young Inequalities

6.2. Theorem of Hardy and Littlewood

6.3. The Case p ≤ 1

6.4. Multipliers

Exercises

Notes

Chapter 7: Hp as a Linear Space

7.1. Quotient Spaces and Annihilators

7.2. Representation of Linear Functionals

7.3. Beurling’s Approximation Theorem

7.4. Linear Functionals on Hp, 0 < p < 1

7.5. Failure of the Hahn-Banach Theorem

7.6. Extreme Points

Exercises

Notes

Chapter 8: Extremal Problems

8.1. The Extremal Problem and its Dual

8.2. Uniqueness of Solutions

8.3. Counterexamples in the Case p = 1

8.4. Rational Kernels

Chapter 9: Interpolation Theory

9.1. Universal Interpolation Sequences

9.2. Proof of the Main Theorem

9.3. The Proof for p < 1

9.4. Uniformly Separated Sequences

9.5. A Theorem of Carleson

Exercises

Notes

Chapter 10: Hp Spaces Over General Domains

10.1. Simply Connected Domains

10.2. Jordan Domains With Rectifiable Boundary

10.3. Smirnov Domains

10.4. Domains Not Of Smirnov Type

10.5. Multiply Connected Domains

Exercises

Notes

Chapter 11: Hp Spaces Over A Half-Plane

11.1 Subharmonic Functions

11.2 Boundary Behavior

11.3 Canonical Factorization

11.4. Cauchy Integrals

11.5 Fourier Transforms

Exercises

Notes

Chapter 12: The Corona Theorem

12.1. Maximal Ideals

12.2. Interpolation And The Corona Theorem

12.3. Harmonic Measures

12.4. Construction Of The Contour Γ

12.5. Arclength Of Γ

Exercises

Notes

Appendix A: Rademacher Functions

Notes

Appendix B: Maximal Theorems

Notes

References

Author Index

Pure and Applied Mathematics

The theory of HP spaces has its origins in discoveries made forty or fifty years ago by such mathematicians as G. H. Hardy, J. E. Littlewood, I. I. Privalov, F. and M. Riesz, V. Smirnov, and G. Szego. Most of this early work is concerned with the properties of individual functions of class HP, and is classical in spirit. In recent years, the development of functional analysis has stimulated new interest in the HP classes as linear spaces. This point of view
has suggested a variety of natural problems and has provided new methods of attack, leading to important advances in the theory.

This book is an account of both aspects of the subject, the classical and the modern. It is intended to provide a convenient source for the older parts of the theory (the work of Hardy and Littlewood, for example), as well as to give a self-contained exposition of more recent developments such as Beurling’s theorem on invariant subspaces, the Macintyre-RogosinskiShapiro-Havinson theory of extremal problems, interpolation theory, the dual space structure of HP with p < 1, HP spaces over general domains, and Carleson’s proof of the corona theorem. Some of the older results are proved by modern methods. In fact, the dominant theme of the book is the interplay of “ hard” and “ soft” analysis, the blending of classical and modern techniques and viewpoints.