Decomposition And Analytic Properties Of Some Holomorphic Function Spaces In The Unit Ball C~n

Complex harmonic analysis and theory of function space are significant research field of fundamental mathematics.Great achievements have been received in this field since the 1960s.By means of functional analysis and real harmonic analysis,we give decomposition and analytic properties of some holomorphic function spaces in the unit ball of Cn.The main content is decomposition theorem for F(p,q,s)spaces in the unit ball of Cn.F(p,q,s)spaces contain many classical function spaces,such as Besov spaces,the weighted Bergman spaces,the weighted Dirichlet spaces,the?-Bloch spaces,BMOA and the Qs spaces.Decomposition theorem is a very important tool in the study of function space theory.It can be used to solve the related problems of Toeplitz operator and Hankel operator and rational function approximation.At the same time,we also discuss the equivalent conditions of the Besov-Sobolev type spaces functions with hadamard gaps.In addition,we study a sufficient condition of F(p,q,s)spaces functions with hadamard gaps.Since the argument in the case of several variables is more difficult than that in the case of one variable,some new approach and techniques have been used,which are different from the case of one variable.In Chapter 1 we summarize the research background of holomorphic function space theory and space operator theory,and review some preliminaries about function spaces and operators needed later.In Chapter 2 we study the decomposition theorem for F(p,q,s)spaces in the unit ball of Cn.Because F(p,q,s)spaces contains many classical function spaces,we could get many important results of other function spaces through studying F(p,q,s)spaces.A decomposition theorem is established for F(p,q,s)spaces in the case of 1<p<?,by means of s-Carleson measure and Schur theorem in the unit ball of Cn.In Chapter 3 we give a necessary and sufficient condition for Besov-Sobolev spaces functions with hadamard gaps,by means of the high-order radial derivative of Besov-Sobolev spaces B??(B).Besov-Sobolev spaces contain a number of important function spaces,when p = 2 and ? = 0,it is Dirichlet type spaces B2(B)(= B20(B));when p = 2 and 0<?<1/2,it is weighted Dirichlet type spaces;when p = 2 and ?=1/2,it is Drury-Arveson Hardy spaces Hn2(=B21/2(B));when p = 2 and ?=n/2,it is Hardy spaces H2(=B2n/2(B));when p = 2 and?>n/2,it is weighted Bergman spaces So we could get many important results of other function spaces through studying Besov-Sobolev spaces Bp?(B).In Chapter 4 we give a sufficient condition for F(p,q,s)spaces functions with hadamard gaps in the case of o<p<?,-1<q<?,0?s<?,by means of the radial derivative description of F(p,q,s)spaces is given in[28].