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 Poincar\'e-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.