Atomic Decomposition Of Martingale Spaces With Variable Exponents And Their Applications In Dyadic Harmonic Analysis

Martingale spaces with variable exponents is a generalization of martingale spaces with constant exponents,but for martingale spaces with variable exponents,many conclusions in martingale spaces with constant exponents do not hold in variable exponents case.Atomic decomposition is an analytical tool for studying spatial structure,so in this paper,atomic decomposition is a tool for studying martingale spaces with variable exponents and their application in dyadic harmonic analysis.The structure of this paper is divided into three parts.The first part corresponds the chapter 1 and chapter 2,we summarize the research status and preparatory knowledges of the theory of martingale with variable exponents,mainly introduce the tools of_?·?martingale,the stop time and some conclusions.The second part is the main content of this paper.In the third chapter,by defining the stopping time,we establish the atomic decomposition of several kinds of quasi-martingale spaces with variable exponents,and prove the boundedness of sublinear operators and some quasi-martingale inequalities on the quasi-martingale spaces with variable exponents.In Chapter 4 and Chapter 5,we study dyadic martingale spaces with variable exponents and two-dimension martingale spaces with variable exponents and their applications.In this paper,we first give the equivalent characterization of log-H¨older continuity in dyadic case,which makes it the key to prove the Doob's maximal inequality,then we study the atomic decomposition of one-dimensional and two-dimensional martingale spaces with variable exponents and their applications in dyadic harmonic analysis.The third part is corresponding to the sixth chapter.On the one hand,it makes a general summary of the research content of this paper.On the other hand,it is the future outlook part.Mainly for the problems encountered in the research process,hope to be solved in the future research,and hope to do further promotion research on the conclusions obtained in this paper.