Global solvability and regularity for $\bar \partial$ on an annulus between two weakly pseudoconvex domains

Abstract:

Let ${M_1}$ and ${M_2}$ be two bounded pseudo-convex domains in ${{\mathbf {C}}^n}$ with smooth boundaries such that ${\overline M _1} \subset {M_2}$. We consider the Cauchy-Riemann operators $\overline \partial$ on the annulus $M = {M_2}\backslash {\overline M _1}$. The main result of this paper is the following: Given a $\overline \partial$-closed $(p,q)$ form $\alpha$, $0 < q < n$, which is ${C^\infty }$ on $\overline M$ and which is cohomologous to zero on $M$, there exists a $(p,q - 1)$ form $u$ which is ${C^\infty }$ on $\overline M$ such that $\overline \partial u = \alpha$.