Interpolation Of Lorentz-Orlicz Martingale Spaces And Inequality Of B-valued Quasi-martingale

Since the 1970 s, because of the rich theory and important application value, The martingale theory is gradually becoming the research center of many scholars. In the process of its development, Martingale theory and Banach theory, functional analysis theory in combination with each other, gradually developed into a new research subjec——Martingale space theory.In this paper, we study the interpolation theory of the martingale space and the inequality, and analyze the basic properties of space, and find out the equivalent characterization of the geometric properties of the value space. Specifically,(1)The Interpolation theory of Lorentz-Orlicz martingale spaces. In the theory of martingale space, the Orlicz martingale space and Lorentz martingale space are two very important space.They not only expand classic Lp space, but also have rich theory. On the basis of classical space,we focus on the combination of Orlicz space and Lorentz space,Lorentz-Orlicz martingale space. We apply function parameters to systematically analyze the Interpolation theory of this space,obtained not only the interpolation theory of the classical space, but also the simple method of the geometric properties of the value space by the interpolation theory.(2)Rosenthal type inequality of B-valued quasi-martingale. The inequalities in martingale spaces have always been a concerned research hot spot, 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. So in this paper, we give a generalization of the Rosenthal type inequality of B-valued quasi-martingale. By using good ? inequality,we prove that Rosenthal type inequality of quasi-martingale and geometric properties of Banach spaces are equivalent. As a consequence, we prove the law of large numbers.There are 6 chapters in this academic dissertation.Chapter 1 detailed describes the development of Lorentz space and Orlicz space, as well as the status of martingale difference sequence of various inequalities.Furthermore,we state the significance and motivation of the research.In the second chapter, we mainly study Orlicz space. First, We introduced theresearch history of Orlicz space,Orlicz space theory is constantly developing and Orlicz space is becoming more and more perfect.secondly,based on the important use of real valued function N in Orlicz space when satisfies the2? conditions, we introduce concept of the function meet the2? conditions at one point, and get some basic properties of Orlicz space by using this method.In the third chapter, we mainly study the Lorentz space. we have introduced the change of Lorentz space theory for many years.We have focused on the relevant interpolation theory. An interpolation theorem for Lorentz spaces is presented and proved.On this basis, we strengthen the understanding of the weighted Lorentz space.Using the atomic decomposition of martingale space, we give and prove some similar theorems about the space of weighted Lorentz spaces, which is a good way for us to introduce the interpolation of Lorentz-Orlicz martingale space.In the fourth chapter, we introduce the key research contents of this paper that is the interpolation of Lorentz-Orlicz Martingale Spaces,We first briefly introduce some conclusions about the real interpolation spaceqA?, and spaceqA?,.here we apply the function parameter to the real interpolation of the Lorentz-Orlicz martingale space,using a more general form of function,therefore, more general interpolation space is obtained.Furthermore, some new interpolation theorems are formulated which generalize some known results in Lorentz spaces introduced by Sharpley. And we give a complete proof process.The fifth chapter is the important chapter of the full text.In this chapter, we discuss the Rosenthal type inequality of B-valued quasi-martingale, by introducing various forms inequality of independent random variable sequences,some classical inequalities for extension of the martingale case are studied. For example,it is Rosenthal type inequality,Further, we discuss the Rosenthal type inequality for the quasi martingale.By using the ?inequality,it is proved that the Rosenthal type inequality and the geometric properties of Banach spaces are equivalent.Finally, as the application, we prove the law of large numbers.The sixth chapter is the last part of this paper.We summarize the research work of the full article,and look forward to the future work.