Four Lectures on Real Hp̳ Spaces

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World Scientific, 1995 - Mathematics - 217 pages
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This book introduces the real variable theory of HP spaces briefly and concentrates on its applications to various aspects of analysis fields. It consists of four chapters. Chapter 1 introduces the basic theory of Fefferman-Stein on real HP spaces. Chapter 2 describes the atomic decomposition theory and the molecular decomposition theory of real HP spaces. In addition, the dual spaces of real HP spaces, the interpolation of operators in HP spaces, and the interpolation of HP spaces are also discussed in Chapter 2. The properties of several basic operators in HP spaces are discussed in Chapter 3 in detail. Among them, some basic results are contributed by Chinese mathematicians, such as the decomposition theory of weak HP spaces and its applications to the study on the sharpness of singular integrals, a new method to deal with the elliptic Riesz means in HP spaces, and the transference theorem of HP-multipliers etc. The last chapter is devoted to applications of real HP spaces to approximation theory.
 

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Contents

Preface Chapter 1 Real Variable Theory of HpRn Spaces 1 Definition of HpRn spaces
1
2 Nontangential maximal functions
3
3 Grand maximal functions
13
Decomposition Structure Theory of HpRn Spaces 1 Atom
20
2 Dual space of H 1Wl
25
3 Atom decomposition
34
4 Dual space of HpRn
54
5 Interpolation of operators
63
2 The Fourier multiplier
98
3 The Riesz potential operators
105
4 Singular integral operators
108
5 The BochnerRiesz means
121
6 Transference theorems of Hp multipliers
144
Applications to Approximation Theory 1 K functional
173
2 Hp multiplier and Jacksontype inequality
175
3 Hp multiplier and Bernstein type inequality
182

6 Interpolations of Hp spaces weak Hp spaces
74
7 Molecule molecule decomposition
83
8 Applications to the boundedness of operators
90
Applications to Fourier Analysis 1 Fourier transform
95
4 Approximation by BochnerRiesz means at critical index
189
References
214
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