Composition Operator On Several Function Spaces

This thesis studies the boundedness and compactness of composition operators from?-Bloch spaces to the weighted Bergman spaces on the unit ball and on Bergman spaces of the Dzhrbasian domain.The thesis consists of the following four parts.In the first chapter,we mainly summarize the background,significance and status of the research.And several important theorems associated with the boundedness and compactness of composition operators on Bloch Spaces and Bergman Spaces are given,which greatly inspired the work of this thesis.In the second chapter,the purpose is to study the boundedness and compactness of the composition operators from?-Bloch spaces to the weighted Bergman spaces on the unit ball.In this thesis,we give the sufficient conditions of the boundness and compact-ness of composition operators from?-Bloch spaces to the weighted Bergman spaces on the unit ball by employing the norm equivalence between weighted Bergman spaces and certain weighted Sobolev on the unit ball,which is of great significance for the further study of composition operators from?-Bloch spaces to weighted Bergman spaces.In the third chapter,we aim to characterize the composition operators acting upon L_a~2(?),where?is a Dzhrbasian domain which is an unbounded domain located outside of an angle domain with the vertex at the origin.Let L_a~2(?)denote the Bergman space consisting of analytic functions on the Dzhrbasian domain.Applying the approach of Carleson measure,the boundedness and compactness of the composition operators are characterized on L_a~2(?)in this chapter.In the fourth chapter,we summarize the main work of this thesis and the prospect of the future.