Integral-type Operators On Holomorphic Function Spaces

In recent decades, many scholars are interested in the research of the propertiesof diferent operators in function spaces. Since the study objects are the properties ofdiferent operators on holomorphic spaces, so this kind of problems attract people moreand more. In order to understand the properties of the research objects further,we needto find some good tools and methods to make the problem easier to solve.I also findthere are some good regular ways to study the properties of operators on holomorphicspaces when I learned the literatures. Thus I also use these regular ways to solve somenew problems about the properties of integral-type operators,which the predecessorshave not referred to.This paper mainly characterizes the sufcient and necessary condi-tions for the boundedness and compactness of integral-type operators on some commonholomorphic spaces.This paper is mainly divided into five chapters.The first chapter is the introduction of the whole paper.In this part,we mainlyintroduce some recent results about the integral-type operators which are contained inthis paper.Thereafter we will describe the proof in this paper under the basis of existingresults.The second chapter is some prior knowledge section. In this part,we mainly intro-duce some basic notations and properties which will be used in the later part of thispaper. This section includes some related contents about complex analysis, the theoryof function spaces, real analysis, functional analysis, geometric function theory andoperator theory.The third chapter mainly gives the sufcient and necessary conditions for theboundedness and compactness of the integral-type operator ThC from the Lipthchizspace L_αto F（p, q, s） on the unit ball B_n of Cⁿ. Thus we can easily obtain the bound-edness and compactness of the Extended Cesa`ro operator.The fourth chapter mainly gives the sufcient and necessary conditions for bound-edness and compactness of the integral-type operator P_ψ^g from the Zygmund space Z（Z₀）to Q_p on the unit ball B_n of Cⁿ. The fifth chapter mainly gives the sufcient and necessary conditions for the bound-edness and compactness of the integral-type operator L_hbetween the mixed-norm spaceH（p, q, φ）) and the iterated logarithmic B_logk(B_logk,0) on the unit ball B_nofCⁿ. Fromthis part we can grasp the properties of this integral-type operator from two directions.