Atomic Decompositions For Several Classes Of B-valued Martingale Spaces With Small Index

As a powerful technique, atomic decomposition has played an important role in harmonic analysis and martingale thoery, it can not only deal with the case of one-parameter and the case of multi-parameter in the same way, but also give some results which are difficult to treat by other means. The basic thought is that the functions or martingales among the spaces are made linearly up with a kind of very simple functions or martingales. The technique of atomic decomposition is of special interest in the martingale spaces with small index. In this paper, we will disscuss the atomic decompositions of Banach-space-valued martingale spaces with small index, and establish their inter-embedding relationships by using atomic decompositions. It differs from the scalar-valued case that the existence of the atomic decompositions of Banach-space-valued martingales is closely connected with the geometrical properties of the Banach spaces.This thesis consists of three chapters. The history backgroud of Banach-space-valued martingale space theory and atomic decomposition are introduced in chapter one. In chapter two , we provide the definitions of Banach-space-valued martingale spaces , the related kowledge and theorems. In last chapter, our main theorems are established , and corresponding proofs are given.