Closed range of ∂¯ in L2-Sobolev spaces on unbounded domains in Cn

Let Ω⊂Cn be a domain and 1≤q≤n−1 fixed. Our purpose in this article is to establish a general sufficient condition for the closed range of the Cauchy-Riemann operator ∂¯ in appropriately weighted L2-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 L2 and, even more critically, in L2-Sobolev spaces.