Abstract
Let M be an n-dimensional Kahler manifold with positive holomorphic bisectional curvature and let Omega is an element of M be a pseudoconvex domain of order n - q, 1 <= q <= n, with C-2 smooth boundary. Then, we study the (weighted) partial derivative-equation with support conditions in Omega and the closed range property of partial derivative on Omega. Applications to the partial derivative-closed extensions from the boundary are given. In particular, for q = 1, we prove that there exists a number l(0) > 0 such that the partial derivative-Neumann problem and the Bergman projection are regular in the Sobolev space W-l (Omega) for l < l(0).