Four lectures on real H[superscript p] spaces

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.

On t.p. p is superscript.

Ch. 1. Real variable theory of H[symbol] spaces. 1. Definition of H[symbol] spaces. 2. Non-tangential maximal functions. 3. Grand maximal functions -- ch. 2. Decomposition structure theory of H[symbol] spaces. 1. Atom. 2. Dual space of H[symbol]. 3. Atom decomposition. 4. Dual space of H[symbol]. 5. Interpolation of operators. 6. Interpolations of H[symbol] spaces; weak H[symbol] spaces. 7. Molecule; molecule decomposition. 8. Applications to the boundedness of operators -- ch. 3. Applications to Fourier analysis. 1. Fourier transform. 2. The Fourier multiplier. 3. The Riesz potential operators. 4. Singular integral operators. 5. The Bochner-Riesz means. 6. Transference theorems of H[symbol] multipliers -- ch. 4. Applications to approximation theory. 1. K functional. 2. H[symbol] multiplier and Jackson-type inequality. 3. H[symbol] multiplier and Bernstein type inequality. 4. Approximation by Bochner-Riesz means at critical index.