Abstract
Let X be a complex manifold of dimension n >= 2 and let Omega subset of X be a weakly q-convex domain with smooth boundary. Assume that E is a holomorphic line bundle over X and E-circle times m is the m-times tensor product of E for positive integer m. If there exists a strongly plurisubharmonic function on a neighborhood of b Omega, then we solve the (partial derivative) over bar -problem with support condition in Omega for forms of type (r, s), s >= q with values in E-circle times m. Moreover, the solvability of the (partial derivative) over bar (b)-problem on boundaries of weakly q-convex domains with smooth boundary in Kahler manifolds are given. Furthermore, we shall establish an extension theorem for the (partial derivative) over barb-closed forms.