Vector-Valued Dirichlet Type Function Spaces On The Unit Ball Of C～n

In 1966, Gerald D. Talyer introduces the definition of Dirichlet function space on the unit disc in Trans. Amer. Math. Soc. From 1980 to 2000, Dirichlet function space on the unit disc has undergone a flourishing development. Since Jihuai Shi and Pengyan Hu introduces the definition of Dirichlet type function spaces on the unit ball of C~n in 1999, many people study them and obtain many results. Now, Dirichlet type function spaces have been a concerned research hot spot in the study of fourier analysis.In the course of the development of fourier analysis, the extension from scalar-valued fourier analysis to Banach-valued fourier analysis always be a concerned research hot spot. In the recent years, the development of Banach space geometry promotes the study of Banach-valued fourier analysis.In this dissertation, we introduce vector-valued Dirichlet type function spaces on the unit ball of C~n and study its randomization. We extend the results of scalar-valued Dirichlet type function spaces by function theory in the unit ball of C~n, Banach space geometry, composition operators, theory of H_p martingales and probability. The propery of vector-valued Dirichlet type function spaces is related with the propery of C~n and Banach space.In this dissertation, we also study random power series. In 1954, R. Salem and A. Zygmund studied the real random power series and proveded a famous theorem— Salem-Zygmund theorem. In 1985, Duren P. L. studied the complex random power series. In 1999, Shi Jihuai studied the random power series on the unit ball of C~n. We will introduce vector-valued random power series on the unit ball of C~n and extend their results by theory of martingale.The framework of this dissertation consists of the following four parts:The introduction is an introduction on the background, motivation and the principle results of the dissertation.In chapter 1, we define a new space—vector-valued Dirichlet type function spaces on the unit ball of C~n. The value of Dirichlet type function spaces on the unit ball of C~n is a scalar number and the value of vector-valued Dirichlet type function spaces on the unit ball of C~n belongs to a Banach space. And then, thestudying of vector-valued Dirichlet type function spaces is difficult. So we needs a Banach space—Rademacher p—type space. With the randomaization of function, we study the convergence and the multeriers of V^{X). Lastly, with the theory of Hp martingales, we characterize the Rademacher q—cotype space by sequence spaces.In chapter 2, we study the multipliers of from Vfc(X) to T%(X). First, we study the multipliers from V^(X) to V%(X). And then, we study the multipliers from V^(X) to VP{X). From the relationship of their results, we can know the important of Banach space geometry in Banach-valued fourier analysis.In chapter 3, we study the vector-valued random power series on the unit ball of Cn. With the stopping time of theory of Hp martingales, we extend Salem-Zygmund theorem. By it, we prove the convergence and smoothness of vector-valued random power series. At last, we study the relationship of vector-valued random power series and several vector-valued Hardy spaces.There are mainly three innovations in this dissertation.(1) It is the first time that vector-valued Dirichlet type function spaces on the unit ball of Cn was introduced, and then the study Dirichlet type function spaces on the unit ball of Cn been into vector-valued fourier analysis. It not only widens the range of definition of Dirichlet type function spaces, but also strengths the difficultof studying.(2) We apply the theory of martingale to vector-valued analytic function. In the past, the result of vector-valued analytic function was little, A reason is that the techique of dealing vector-valued analytic function be little. That we applying the theory of martingale to vector-valued analytic function can give a new techique of dealing vector-valued analytic function.(3) In this paper, we studied the influence Banach spaces to vector-valued Dirichlet type function spaces. As we known, the development of Banach space geometry and the study of Banach-valued fourier analysis promotes each other. By our work, we strengthened our impression.