The $L^2$ $\bar \partial $-Cauchy problem on weakly $q$-pseudoconvex domains in Stein manifolds

summary:Let $X$ be a Stein manifold of complex dimension $n\ge 2$ and $\Omega \Subset X$ be a relatively compact domain with $C^2$ smooth boundary in $X$. Assume that $\Omega $ is a weakly \mbox {$q$-pseudoconvex} domain in $X$. The purpose of this paper is to establish sufficient conditions for the closed range of $\bar \partial $ on $\Omega $. Moreover, we study the \mbox {$\bar \partial $-problem} on $\Omega $. Specifically, we use the modified weight function method to study the weighted \mbox {$\bar \partial $-problem} with exact support in $\Omega $. Our method relies on the \mbox {$L^2$-estimates} by Hörmander (1965) and by Kohn (1973)