Theory of H[superscript p] spaces
- 1st Edition, Volume 38 - July 31, 1970
- Latest edition
- Editor: Peter L. Duren
- Language: English
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. Sm… Read more
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Description
Description
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 viewhas 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.
Table of contents
Table of contents
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
Product details
Product details
- Edition: 1
- Latest edition
- Volume: 38
- Published: July 31, 1970
- Language: English
About the editor
About the editor
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