Several Complex Variables Border Nature Of The Poisson-china Integral With The Cauchy Integral In The Classical Domain

This dissertation deals with boundary behaviour of the Poisson-Hua integrals and the Cauchy integrals on the classical domains of several complex variables. This is an important topic in the function theory of several complex variables. Especially, there is an intimate relation between the Cauchy integrals and the singular integrals of several complex variables. In [3], the result that the Poisson-Hua integrals on the first classical domains Râ… (m,n) converging to the boundary functions when the boundary functions are continuous was obtained. Let ? denote the Hua operator on Râ… . The Dirichlet problem for Hua operator was also researched and the solution was obtained for the continuous boundary functions. In [20], the boundary behaviour of the Poisson-Hua integrals on bounded symmetric domains for continuous boundary functions was obtained. In [24], the case that the boundary functions are integral was investigated for the unit ball, and the result of almost everywhere convergence was obtained. In [7], the singular integrals of several complex variables were widely and deeply researched. In this paper the tool of harmonic analysis on compact homogeneous space is used to research the boundary behaviour of the Poission-Hua integrals and the Cauchy integrals on classical domains of several complex variables for Lp boundary functions. In Chapter 1, a theorem on the relation between the Poission-Hua integrals and the Cauchy integrals on bounded symmetric domains is obtained. The two classes of the integrals are related by a projection operator. This is a very important singular integral operator of several complex variables. In Chapter 2, a different method from [24] namely considering the unit ball as the special case of the first classical domains Râ… (m,n) as m equals 1 to reserch the Poission-Hua integrals and the Cauchy integrals on the unit ball, and the needed results are obtained. Not as in [24], this method can be generalized. In Chapter3, making use of Harish-Chandra model to obtain a decomposition for the Silov boundaries Lâ… (m,n) of the first classical domains Râ… (m,n), and obtain the formula for computating the volume element of L â…  (m,n) under the above decomposition by complicate computation, and obtain a maximal function majorization estimate for the Poission-Hua integrals on the first classical domains Râ… (m,n). The class of the maximal functions in above estimate are defined on a collect of nonconvex sets on the Silov boundaries Lâ… (m,n) by the boundary function. The properties of the maximal functions defined on nonconvex sets even in Euclidean spaces have no general results. Here the needed estimates for this class of the maximal functions are successfully obtained. Combining with Chapter 1, the properties of almost everywhere convergence of the Poission-Hua integrals and the Caunchy integrals on Râ… (m,n) for the Lp boundary functions are gotten. Also the Dirichlet problem for Hua operator on the first classical domains Râ… (m,n) is investigated here, and the solution of the problem for a wider class of functions then that in [3] is obtained. Chapter 4 and 5 deal with the boundary behaviour of the Poission-Hua integrals and the Caunchy integrals on the second and the third classical domains. Making use of the idea used before, the difficulties of dealing with the kernel functions and the estimates for a certain maximal functions defined on nonconvex sets on the Silov boundaries Lâ…¡ and Lâ…¢ and the estimate for th integrals are overtaken, and the results of almost everywhere convergence are gotten. Also the solutions of the Dirichlet problem on the second and the third classical domains for considerably a wider class of boundary functions are obtained. In the paper, a concise proof for the Poission-Hua integrals for continuous boundary functions is also given.