The closed range property for the $\overline{\partial }$-operator on planar domains

Abstract

Let $\Omega \subset \mathbb{C}$ be an open set. We show that $\overline{\partial }$ has closed range in $L^2(\Omega )$ if and only if the Poincaré-Dirichlet inequality holds. Moreover, we give necessary and sufficient potential-theoretic conditions for the $\overline{\partial }$-operator to have closed range in $L^2(\Omega )$. We also give a new necessary and sufficient potential-theoretic condition for the Bergman space of $\Omega$ to be infinite dimensional.

Part (iv) of Proposition 2.2 in the previous version was stated prematurely. To correct this, some changes in section 2 were necessary, see Prop. 2.9 and its corollaries in the current version