Characterizations Of Some Holmorphic Function And Harmonec Function Spaces And Related Operator Theory

The dissertation deals with some spaces of holomorphic function and harmonic function in several variables, and with some relative linear operators. Our research will be on the characterization of these spaces and property of these linear operators on some function spaces. We will also study the maximal isotropy subgroup at origin for the holomorphic automorphism group of the egg domain closely related with the theory of function space.In the first chapter, we briefly introduce the research background, notations and definitions, and list the main results of this thesis.In chapter2, we study the Schatten(-Herz) class of the extended Cesaro op-erator on the holomorphic weighted Bergman space Aω2(B). Here, B is the unit ball of Cn under the Euclidean metric, and ω is a normal weight function. We obtain some inner product of Aω2(B) in terms of the radial derivative, and find the relation between the extended Cesaro operator and the Toeplitz operator. On this basis, we introduce the Schatten-Herz class of the extended Cesaro operator, and characterize the holomorphic symbol g, for which the extended Cesaro operator Tg acting on Aω2,(B) is in the Schatten (respectively, Schatten-Herz) class. Those results show that the Schatten(-Herz) class of the extended Cesaro operator is close to the holomorphic Besov space.In chapter3, we discuss the boundedness and compactness of the weighted composition operator between some holomorphic function spaces. First, for the holomorphic Bloch-type spaceBu(Dn) and little Bloch-type space Bμ,0(Dn) on the polydisc Dn of Cn, we obtain the characterization on ψ and φ for which the weighted composition operator Tψ,φ:Bμ(Dπ)→(Dn)(respectively, Tψ,φ: Bμ,0(Dn)→Bω,0(Dn)) is bounded and compact. Secondly, for the holomorphic Bloch-type space(B) and little Bloch-type spaceBω,0(B) on the unit ball B, we give the sufficient and necessary conditions such that the weighted composition operator is bounded and compact from BUω(B) to H∞(B)(respectively, Bω,0(B) to H∞(B)). Our work extend some known results about these two operators and lead to new results. In chapter4, we study the equivalent norm for some harmonic function spaces in several variables. Let B be the unit ball of Rn under the Euclidean metric, and let ω be a normal weight function. Suppose Hp,q,ω(B) and BU(M) are the harmonic weighted mixed norm space and Bloch-type space respectively, where0<p, q<+00. we obtain some equivalent norms on Hp,q,ω(B) and Bω(M) with the radial, tangential and partial derivatives. As an application of our approach, the analogous results are also given for the polyharmonic weighted mixed norm space and Bloch-type space on the unit ball B.In chapter5, we discuss the atomic decomposition theorem for harmonic Bergman function in several variables. Let Ω be a bounded domain of Rn with C∞-boundary. For1<p<+∞, we prove the atomic decomposition theorem of the harmonic Bergman space bp(Ω) in terms of some integral operator.In chapter6, we study the maximal isotropy subgroup at origin for the holo-morphic automorphism group of the egg domain. Let0<pi<...<ps<2<q1<...<qt<+∞, where s and t are both positive integers. For egg domainwe use the technique of several complex variables to obtain the maximal isotropy subgroup at origin for the holomorphic automorphism group Aut(Bp,q). On the other hand, from the point of algebra, we give the maximal isotropy subgroup of the holomorphic automorphism group Aut(Bp) at origin for the egg domain and so answer the problem that what is the linear isometry group of the Frechet space (Cn,||·||pp) with0<p<1and the Banach space (Cn,||·||p) with1≤p≤+∞, respectively.Our research enriches the theory of holomorphic function space and operator in several complex variables,and leads to a further understanding about the maximal isotropy subgroup at origin for the holomorphic automorphism group of the egg domain. At the same time, we do some research on the theory of spaces consisting of certain harmonic functions. Some new idea and tools are developed, which we believe have their own interests in other aspects.