Research On Generalized Martingale Spaces And Applications Of Martingale Methods In Dyadic Harmonic Analysis

Since the beginning of the last century70's, martingale theory has become an interesting research content, the reason of that is martingale theory has its own rich theory, and has higher application value. In its development process, martingale and the theory of Banach space, the theory of harmonic analysis and the functional analysis theory combined with each other, mutually permeated, gradually became the new re-search subject-the martingale space theory. Since probability space has no topological structure and algebraic structure, research methods in the Euclidean space no longer hold for the martingale space theory.Because the structure of martingale itself more regularity, people created many new research tools and methods in the study of martingales such as martingale transforms, stopping times, atomic decompositions, good ? inequality. The martingale methods not only provide a concise proof for many important conclusions, but also lead to the discoveries and solutions of many new problems by martingale method, such as the singular integral operator theory in T (b) theorem and arising of UMD space.The research of this paper is about several classes of generalized martingale spaces and we analyse the basic properties of the spaces as above. Moreover we try to find out the new characterizations of geometric properties of spaces. Furthermore we discuss the applications of martingale method in the dyadic harmonic analysis. Concretely:(1).BMO-Lorentz martingale space. Lorentz spaces and BMO spaces are two kinds of important spaces in martingale Spaces. With the help of rearrangement function, we define the Lp,q norm, which not only establishes contact between strong and weak space, but also extends the classic Lp space. BMO space is called bounded mean oscillation space, which can be as the dual spaces of H1spaces. Moreover there is close relation between Carleson measure and BMO space. We study the combination of Lorentz spaces and BMO martingale spaces, which can deepen the understanding of the two types of Martingale Spaces (2) Banach Lattice valued martingale space. As everyone knows, the atomic de-composition is not only one of the important research contents of martingale theory, but also a very important research tool. With the help of atomic decomposition method, we can conveniently discuss the small index martingale space. Moreover the single param-eter and multi parameter spaces can be studied together. By the atomic decomposition theory, we can easily build some inequalities of martingale and duality theory. Note that for the study of vector valued martingale, the results depend on the geometric prop-erties of space. In this paper we will discuss atomic decompositions and interpolations of Banach Lattice valued martingales.(3) Generalized Orlicz martingale space. Orlicz space is a generalization of the clas-sical Lp space. The previous research on Orlicz space focus on the convex function, and some good results are obtained. While for the concave function, the research is short. We find that the concave function is closely lined with small index martingale spaces, and atomic decomposition is very important research content and research tool in small index martingale spaces. In this paper we mainly discuss the atomic decomposition of Orlicz martingale spaces generated by concave function.(4) The application of martingale methods in dyadic harmonic analysis. There are some difference between dyadic harmonic analysis and classical harmonic analysis. Some results in the classical harmonic analysis do not hold in dyadic harmonic analysis, such as Newton-Leibniz formula of derivative function. The martingale methods provide a bridge between classical harmonic analysis and dyadic harmonic analysis. Moreover the martingale method mainly specializing in the treatment of integral can be made up. Since there are closely relation between dyadic harmonic analysis and regular martingales. In view of this, we will analyse the application of martingale methods in dyadic harmonic analysis which can strengthen our understanding of martingale theory. In this paper we mainly study application of martingale methods in dyadic harmonic analysis.The paper is divided into five chapters:In the first chapter, we review the historical background and development of mar-tingale space theory and of dyadic harmonic analysis. We also explain the research motivation. In second chapter, we mainly research BMO-Lorentz martingale space. The basic properties of the space will be discussed. Moreover we will discuss the geometric prop-erties of space, such as p smoothness and q convexity, and we try to find the equivalent characterizations of geometric properties.In third chapter, we mainly study the Banach Lattice valued martingale space. We give the atomic decompositions of several kinds of martingale space in the normal case and the weighted case, respectively.In fourth chapter, we mainly research Orlicz martingale space generated by concave functions and we analyse the properties of the space. Moreover we give the atomic decomposition theorems of several martingale spaces when some conditions of concave functions are satisfied.In last chapter, we mainly research the application of martingale methods in dyadic harmonic analysis:in the first aspect, we discuss the boundedness of maximal operator of dyadic derivative in two parameter case by the atomic decomposition method, and boundedness of strict Sunouchi operator in Kaczmarz rearrangement. In the second aspects, we discuss the convergence properties of hole series in two-dimensional case with help of the stopping time and the martingale convergence properties.