  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 for
the closed range of the Cauchy-Riemann operator $\bar\partial$ in appropriately
weighted $L^2$-Sobolev spaces on $(0,q)$-forms. The domains we consider may be
neither bounded nor pseudoconvex, and our condition is a generalization of the
classical $Z(q)$ condition that we call weak $Z(q)$. We provide examples that
explain the necessity of working in weighted spaces both for closed range in
$L^2$ and, even more critically, in $L^2$-Sobolev spaces.
