The Study Of Operator Theory On Some Holomorphic Function Spaces

In complex analysis,many problems require in-depth research on characteristics of various function space and properties of some auxiliary operators(such as boundedness,compactness,spectral structure and so on).The research contents of function space and operator theory is closely to Many branches of modern mathematics,such as quasi-conformal mapping([1]),Clifford analysis,complex manifolds,harmonic analysis,functional analysis,complex dynamical systems,partial differential equation,diffusion equation etc.In this paper,we mainly study the operators of several holomorphic function space in high dimensions.The research contents are mainly divided into three parts:(1)The sufficient and necessary conditions that some operators are bounded or compact on normal weight Zygmund space in the unit ball,including weighted composition operator,weighted differential composition operator and Cesaro integral operator.We also give the characteristics that the operators are bounded and compact when the weight function is a specific normal function κ(r)=(1-r)s logt e/1-r.(2)The sufficient and necessary conditions are given that multipliers operators and composition operators is bounded or compact on H(p,q,s)space,and the point-state estimation and inclusion relations of the space involved.(3)The bounded or compact of composition operator on F(p,q,s)space is discussed.The main tools and ideas that we use are the integral estimation of two-variable point,automorphic invariant boundary regular small space theory,equivalent characterization of holomorphic function space,integral representation etc.The research of this paper is not only to improve the basic theory of this subject,At the same time,it also provides some practical means for some related disciplines.The structure of this paper as follows:In chapter 1,we make a comprehensive summary of the research background and conclusions of the paper,mainly introducing the research background,relevant preparatory knowledge,research status and main results.In chapter 2,we describe completely the sufficient and necessary conditions that weighted composition operator ψCφ on normal weight Zygmund space in the unit ball,when φl ∈Zk(B),is bounded or compact.We also describe the new result of weighted composition operator bounded or compact when φl is not in Zκ(B).In chapter 3,we present sufficient and necessary conditions that weighted differential composition operators ψDφ is bounded or compact from normal weight Zygmund space Zκ(B)to the normal weight Bloch space βκ(B),and presents the simple necessary and sufficient conditions that differential composition operator Dφ is compact from Zκ(B)to βκ(B).In chapter 4,we discuss that the sufficient and necessary conditions that composition Cesaro integral operator Tφ,ψ(f)(z)=∫01 f(φ(tz))Rψ(tz)dt/t(f∈H(B),z ∈ B)is bounded or compact on weight Zygmund spaces when φ,ψ ∈ H(B)andψ(0)=0.In chapter 5,we present the necessary and sufficient conditions that multiplier operator Mu and composition operator Cφ on general Hardy space Hp,q,s(B)or from Hpn/q+p(B)to Hp,q,s(B)is bounded or compact.Furthermore,we will discuss the boundedness and compactness of weighted composition operators uCφ from Hp,q,s(B)to Hq+n/p ∞(B).In chapter 6,we discuss what kind of φ makes the composition operator Cφ is bounded or compact on the F(p,q,s)space.In addition,we will give(1)If 1<p<q+n+land φ satisfies certain conditions,Cφ is compact operator of F(p,q,s)If and only If(2)If p≥ 1 and p>q+n+1,then Cφ is a compact operator on F(p,q,s)if and only if ‖φ‖∞<1 and φl ∈ F(p,q,s)for any l ∈ {1,2,…,n}.