Non-closed Range Property for the Cauchy-Riemann Operator

In this paper we study the non-closed range of the Cauchy-Riemann operator for relatively compact domains in C-n or in a complex manifold. We give necessary and sufficient conditions for the L-2 closed range property for. on bounded Lipschitz domains in C-2 with connected complement. It is proved for the Hartogs triangle that. 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. might not have closed range in L-2 on a Stein domain in complex manifold.