Equivalent Characterizations And Differentiation Composition Operators Of μ-Bergman Space In C~n

In this dissertation, we will discuss several equivalent characterizations of the μ-Bergman space and the conditions that the differentiation composition operator in μ-Bergman space is bounded or compact on the unit ball in Cn. At the same time, the briefly sufficient and necessary conditions are given for which the weighted composition is bounded or compact between different weighted Bloch-type spaces in the disc.It consists of three chapters.In the first chapter, we will carry on a comprehensive summary about the research background as well as the conclusions that we will given.In the second chapter, we will discuss the equivalent characterizations and the differentiation composition operators of the μ-Bergman space on the unit ball in Cn. Fist, several equivalent characterizations of the μ-Bergman space on the unit ball in Cn are given, that is the theorem 2.3.2. Next, we will characterize the conditions that the differentiation composition operator Dφ is bounded or compact from Ap(μ) to Ap(μ1), that is theorem 2.3.3. Finally, we will give a kind of briefly sufficient condition and necessary condition that Dφ is compact from Ap(μ) to Ap(μ1) for p> 1, that is the theorem 2.3.4.In the third chapter, we will discuss two questions about the weighted composition operator between different weighted Bloch-type spaces in the disc: the one is the briefly sufficient and necessary conditions for which the weighted operator is bounded, that is the theorem 3.2.1. The second is different sufficient and necessary conditions for which the weighted operator is compact, that is the theorem 3.2.2.