# Article

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Keywords:
$\bar \partial$ operator; $\bar \partial$-Neumann operator; $q$-convex domain; Stein manifold
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).
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