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Let $\Omega\subset\mathbb{C}^n$ be a domain and $1 \leq q \leq n-1$ fixed.Our purpose in this article is to establish a general sufficient condition forthe closed range of the Cauchy-Riemann operator $\bar\partial$ in appropriatelyweighted $L^2$-Sobolev spaces on $(0,q)$-forms. The domains we consider may beneither bounded nor pseudoconvex, and our condition is a generalization of theclassical $Z(q)$ condition that we call weak $Z(q)$. We provide examples thatexplain the necessity of working in weighted spaces both for closed range in$L^2$ and, even more critically, in $L^2$-Sobolev spaces.

Closed range of $\bar\partial$ in $L^2$-Sobolev spaces on unbounded domains in $\mathbb{C}^n$