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

Abstract

Let $\Omega \subset {\mathbb{C}}^n$ be a domain and $1\le q\le n-1$ 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.

23 pages. Comments welcome! arXiv admin note: substantial text overlap with arXiv:1507.06211

References in corpus (1)

• On the weighted $\bar{\partial }$-Neumann problem on unbounded domains

Cited by in corpus (1)

• Boundary invariants and the closed range property for $\bar{\partial }$