Martingale Inequalities In Rearrangement Invariant Function Space

This thesis is devoted to the consideration and study of martingale space and inequality theory in the background of a rearrangement invariant function space on finite interval. We prove the corresponding Doob's maximal inequality, Burkholder-Gundy-Davis inequality, the greater-smaller operator inequality and Rosenthal's inequality in the framework of rearrangement invariant martingale space on finite interval. In addition, we also give the characteristic of uniformly integrable martingale in the rearrangement invariant martingale space and the relationship between one martingale and the accompanying martingale associated with it which is particular in the rearrangement invariant martingale space under the action of some most important martingale operators such as maximal operator, square operator, the greater operator and the smaller operator. And these results not only extend the classical inequalities of martingale Hardy space, but also point out the close affiliation between the Boyd indices of rearrangement invariant martingale space and the characters of the martingales in it.This thesis consists of five chapters as follow:Chapter 1 is an introduction on the research history of the martingale space and inequality theory relevant to this thesis and the thought and motivation of this thesis.In Chapter 2, considers the definition of rearrangement invariant function space and Boyd indices and some characteristics of them, it also establish the rearrangement invariant martingale space in this chapter.Chapter 3 discusses the fundamentals inequalities in the rearrangement invariant martingale space, such as the corresponding important Doob's maximal inequality, Burkholder-Gundy-Davis inequality and Rosenthal's inequality.In Chapter 4, we prove the corresponding greater-smaller inequality which is stronger than the above Burkholder-Gundy-Davis inequality in the rearrangement invariant martingale space. Furthermore it describes the norm equivalence of therearrangement invariant martingale spaces acted by the greater operator and the smaller operator, and it represents the norm equivalence of these spaces acted by maximal operator and square operator more deeply.Chapter 5 studies the characteristic of uniformly integrable martingale in the rearrangement invariant martingale space and the boundedness of the greater operator, the smaller operator, maximal operator and square operator on rearrangement invariant martingale space and its rearrangement invariant accompanying martingale space.