Global regularity for $\bar \partial$ on weakly pseudo-convex manifolds

Abstract:

Let $M’$ be a complex manifold and let $M \subset \subset M’$ be an open pseudo-convex submanifold with a smooth boundary which can be exhausted by strongly pseudo-convex submanifolds. The main result of this paper is the following: Given a $\overline \partial$-closed $(p,q)$-form $\alpha$, which is ${C^\infty }$ on $\overline M$ and which is cohomologous to zero on $M$ then for every $m$ there exists a $(p,q - 1)$-form ${u_{(m)}}$ which is ${C^m}$ on $\overline M$ such that $\overline \partial {u_{(m)}} = \alpha$.