Closed range of $\bar{\mathrm{\partial }}$ on unbounded domains in ${\mathbb{C}}^n$

In this article, we establish a general sufficient condition for closed range of the Cauchy-Riemann operator $\bar{\mathrm{\partial }}$ 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.