The Study On The Boundedness And Compactness Of Some Operators On Bloch Type Space And Hardy Type Space

Bloch space β is one of the special situation of Bloch type space. Recently, many researchers argued that the Holomorphic function on the unit disc and the unit ball belongs to the sufficient and necessary conditions of Bloch space β. But there are few literatures on the equivalent characterizations of general Bloch type space .In this paper, Chapter One describes the equivalent characterization of Bloch type space on the unit ball. By using it, a new condition is given, under which the composition operation between Bloch type space on the unit ball will be a bounded operator. The study on the composition operators concerns some of the most fundamental topics in the theory of linear operation by using the theory of analytic functions. Also, it concerns many classical problems of function theory by using the theory of operators. In the long course, many researchers on analytic mathematics have solved many problems in this field. Recently, Mr.Zhang Xuejun systemically explored the boundedness and compactness of composition operators and weighted composition operators between the p-Bloch spaces on the unit disc. He further studied the boundedness and compactness of composition operators between the little p-Bloch spaces, but the result, when p>1,0≤g≤1, has not been given. Also, he has not obtained the relevant results of general weighted composition operators.In this paper, Chapter Two discusses the boundedness of weighted composition operators between Bloch type spaces on the unit disc, when p>1,0≤q≤1. At the same time, it presents the sufficient and necessary condition on the compactness of weighted composition operators between p-Bloch spaces on the unit disc, when P ≥ 0, q ≥ 0.It is very interesting to study the boundedness for the commutators related to some integral operator on the functions spaces. In Chaper Three, the multilinar commutator generaed by the marcinkiwicz operator and the BMO functions are introduced, and also the weighted boundedness for the commutator on the Hardy type.