Weighted Φ-Inequalities For Martingales

In the development course of martingale theory, the inequalities in martingale spaces have always been a concerned research hot spot. The situation is just as in harmonic analysis, by those inequalities the connections between two operators and the inclusions of martingale spaces are established.People often study the properties of operators via the corresponding martingale inequalities.Weighted inequality and weight function play an important role in martingale theory and are researched extensively. In this paper, we study the maximal operator and the minimal operator on martingale space and obtain some necessary conditions for the strong type and the weak type weighted Φ-inequalities of martingales.This thesis consists of three chapters. In chapter 1, the history background of martingale theory are looked back, and some previous achievements are mentioned. In chapter 2, we summarily describe the basic knowledge needed in the followings. Chapter 3 is the core of the thesis, in which the proofs of the main theorems are given.