The Study Of Some Problems On Function Spaces Of Several Complex Variables

In this dissertation,our study is about spaces of holomorphic function in several complex variables and operator theory.What questions we investigated are mainly split into three parts.One is the fundamental propositions of spaces of holomorphic functions,such as integral presentation,the dual and predual space,the atomic decomposition and the inclusion relationship between classic function spaces,etc.The another one is to present the boundedness and compactness as well as the essential norm of operators on spaces of holomorphic functions.Teoplitz operator,Hankel operator and(weighted)composition operator are investigated in this dissertation.The last one is the tools and ideals appearing in the process of dealing with the questions above,such as Forelli-Rudin type estimates,the equivalent norm of spaces of holomorphic function,etc.This thesis consists of seven chapters.In Chapter 1,the background,the preliminary knowledge,the research status and the main results of this dissertation are given.In Chapter 2,the bounded or compact weighted composition operator Tφ,ψ,from the normal weight Bergman space Ap(μ)to the normal weight Bloch type space βv in the unit ball are characterized,where p>0 and μ be a normal function on[0.1)and v(r)=(1-r2)1+n/pμ(r)for r ∈[0.1).The briefly sufficient and necessary condition that the composition operator Cφ is compact form Ap(μ)to βv are given.At the same time,the briefly sufficient and necessary condition that is compact on βμ are given for a>1,where a is the constant in the definition of μ.In Chapter 3,we discuss some properties of the normal weight Zygmund space Zμ in several complex variables.Firstly,we establish an integral repre-sentation of function in Zμ.Secondly,we show that Zμ can be identified with the dual space of the normal weight Bergman space A1(u)under the integral pairing where(?)β>max{0,b-1} and b is the constant in the definition of μ.Finally,as an application of the integral representation and the dual,an atomic decomposition for every function in Zμ is given.In Chapter 4,suppose φ is a holomorphic self-map on the unit ball B of Cn.We characterize the conditions that composition operator Cφ is bounded on Zμ,we also estimate the essential norm of Cφ on Zμ when n>1.Thereby we succeed in characterizing the compactness of on Cφ on Zμ.At the same time,we give the sufficient and necessary conditions that Cφ is bounded or compact on Zμ for some spacial cases.It is worth noting that a brief compactness criteria of Cφ on Zμ(or βμ(B))is given,When(?)In Chapter 5,let Ω be a bounded symmetric domain in Cn.The purpose of this Chapter is to define and characterize the general function space F(p,q,s)on Ω.Characterizing functions in the F(p,q,s)space is a work of consid-erable interest nowadays.In this article,the authors give several equivalent descriptions of the functions in the F(p,q,s)space on Ω in terms of fraction-al differential operators.At the same time,the authors give the relationship between F(p,q,s)space and Bloch type space on Ω too.In Chapter 6,given a measure(?) let(?),which is so-called Forelli-Rudin type estimates.We give a bidirectional estimates of typical integral Jw,a,for all cases.As the application of the above integral estimates,we characterize several equivalent norm of F(p,q,s,k)in the unit ball of Cn.In Chapter 7,we study the boundedness of the Toeplitz operator Tφ and the Hankel operator Vφ with symbol φ∈Lipβ(B)on Hp,q,s(FB).Further,we discuss the solvability of Gleason’s problem on Hp,q,s(B).At the same time,the inclusion relations between Hp,q,s(B)and some typical function spaces are also considered.