  In this article, we establish a general sufficient condition for closed range
of the Cauchy-Riemann operator $\bar\partial$ in appropriately weighted $L^2$
and $L^2$-Sobolev spaces on $(0,q)$-forms for a fixed $q$ on domains in
$\mathbb{C}^n$. 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.
