Non-closed Range Property for the Cauchy-Riemann Operator

Abstract

In this paper we study the non-closed range of the Cauchy-Riemann operator for relatively compact domains in \(\mathbb C^{n}\) or in a complex manifold. We give necessary and sufficient conditions for the \(L^{2}\) closed range property for \(\overline{\partial }\) on bounded Lipschitz domains in \(\mathbb C^{2}\) with connected complement. It is proved for the Hartogs triangle that \(\overline{\partial }\) does not have closed range for (0, 1)-forms smooth up to the boundary, even though it has closed range in the weak \(L^{2}\) sense. An example is given to show that \(\overline{\partial }\) might not have closed range in \(L^{2}\) on a Stein domain in complex manifold.