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

On a bounded $q$-pseudoconvex domain $\Omega$ in ${ℂ}^n$ with a Lipschitz boundary, we prove that the $\overline{\partial }$-Neumann operator $N$ satisfies a subelliptic $(1/2)$-estimate on $\Omega$ and $N$ can be extended as a bounded operator from Sobolev $(-1/2)$-spaces to Sobolev $(1/2)$-spaces.