The Koppelman-Leray-Norguet Formula Without Boundary Integrals On Stein Manifolds And Uniform Estimates Of Solutions Of (?)-Equations

It is well known that a Stein manifold is a very important manifold on which there are a lot of nonconstant holomorphic functions. C~n is just a Stein manifold, so it is very natural to research into complex analysis in several variables on Stein manifolds. The integral representation method is one of main methods of complex analysis in several variables , because one of its main virtues is that it is easy to estimate like the Cauchy integral formula in one complex variable .In this paper, by using the Hermitian metric and Chern connection and constructing new integral kernels with respect to (p, q) differential forms under the invariant metric, we give some modifications of this classical formula replacing the boundary integrals by the volume integrals, and obtain a new Koppelman-Leray-Norguet formula of type (p, q) and solutions of (?)-equations for a strictly pseudoconvex domain with not necessarily smooth boundary on a Stein manifold. Furthermore, the author gives the uniform estimate of the solution of the (?)-equation.The whole dissertation includes three chapters. In the chapter 1 we introduce some preliminaries,including some definitions and notations. In the chapter 2 we construct new kernel for a strictly pseudoconvex domain with not necessarily smooth boundary on a Stein manifold, and obtain the new Koppelman-Leray-Norguet formula of type (p,q). In the last chapter the author gives the solution of the (?)-equation and the uniform estimate.