Introduction to function spaces on the disk

Pavlovic M.8680593370

This text contains some facts, ideas, and techniques that can help or motivate the reader to read books and papers on various classes of functions on the disk and the circle. The reader will find several well known, fundamental theorems as well as a number of the author’s results, and new proofs or extensions of known results. Most of assertions are proved, although sometimes in a rather concise way. A number of assertions are named by Exercise, while certain assertions are collected in Miscellaneous or Remarks: most of them can be treated by the reader as exercises.The reader is assumed to have good foundation in Lebesgue integration, complex analysis, functional analysis, and Fourier series, which means in particular that he/she had a good training through these areas. It is of some importance that the reader can accept the following:Throughout this text, constants are often given without computing their exact values. In the course of a proof, the value of a constant С may change from one occurrence to the next. Thus, the inequality 2C 0.

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Table of contents :
Preface……Page 6
Quasinorm and p-norm……Page 7
Linear operators……Page 10
Open mapping, closed graph……Page 12
F-spaces……Page 14
The spaces p……Page 16
The Riesz/Thorin theorem……Page 18
Weak Lp-spaces and Marcinkiewicz’s theorem……Page 22
Maximal function and Lebesgue points……Page 25
The Rademacher functions……Page 28
Nikishin’s theorem……Page 30
Nikishin and Stein’s theorem……Page 34
Banach’s principle……Page 36
Harmonic functions……Page 38
Borel measures and the space h1……Page 42
Radial limits of the Poisson integral……Page 45
The spaces hp and Lp(T)……Page 49
The Littlewood/Paley theorem……Page 51
Harmonic Schwarz lemma……Page 53
Basic properties……Page 55
Properties of the mean values……Page 60
Integral means of univalent functions……Page 62
The subordination principle……Page 64
The Riesz measure……Page 67
A Littlewood/Paley theorem……Page 70
Basic properties……Page 73
The space H1……Page 78
Blaschke product……Page 80
Inner and outer functions……Page 85
Composition with inner functions……Page 88
Harmonic conjugates……Page 92
Riesz projection theorem……Page 96
Applications of the projection theorem……Page 99
Aleksandrov’s theorem……Page 100
Strong convergence in H1……Page 101
Quasiconformal harmonic homeomorphisms……Page 104
Maximal theorems……Page 110
Maximal characterization of Hp……Page 114
“Smooth” Cesàro means……Page 116
Interpolation of operators on Hardy spaces……Page 119
On the Hardy/Littlewood inequality……Page 123
On the dual of H1……Page 127
Bergman spaces……Page 129
Reproductive kernels……Page 130
The Coifman/Rochberg theorem……Page 133
Coefficients of vector-valued functions……Page 137
Subharmonic behavior and Bergman spaces……Page 143
The space hp, p