A Kind of Integral Representation on Complex Manifold

Abstract

In this paper, by using the Hermitian metric and Chern connection, we study the case of a strictly pseudoconvex domain $G$ with non-smooth boundaries in a complex manifold. By constructing a new integral kernel, we obtain a new Koppelman-Leray-Norguet formula of type $(p,q)$ on $G$, and get the continuous solutions of $\bar{\partial}$-equations on $G$ under a suitable condition. The new formula doesn't involve integrals on the boundary, thus one can avoid complex estimations of the boundary integrals, and the density of integral may be not defined on the boundary but only in the domain. As some applications, we discuss the Koppelman-Leray-Norguet formula of type $(p,q)$ for general strictly pseudoconvex polyhedrons (unnecessarily non-degenerate) on Stein manifolds, also get the continuous solutions of $\bar{\partial}$-equations under a suitable condition.