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

Abstract

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.

19 pages. We split the original paper into two, smaller papers

References in corpus (2)

• Closed Range for $\bar{\partial }$ and ${\bar{\partial }}_b$ on Bounded Hypersurfaces in Stein Manifolds
• On closed range for $\bar{\partial }$

Cited by in corpus (1)

• A remark on boundary estimates on unbounded $Z(q)$ domains in ${\mathbb{C}}^n$