The L-2 partial derivative-Cauchy problem on pseudoconvex domains and applications

Let X be a complex manifold of dimension n >= 2 and Omega (sic) X be a weakly pseudoconvex domain with smooth boundary in X. Let E be a holomorphic line bundle over X which is positive on a neighborhood of b Omega. Let E-circle times m be the m-times tensor product of E for positive integer m. The purpose of this paper is to study the partial derivative-problem with support conditions in Omega for forms of type (r, s), s >= 1 with values in E-circle times m . Applications to the partial derivative(b)-problem for smooth forms on boundaries of Omega are given.