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

Let $Ω\subset\mathbb{C}$ be an open set. We show that $\overline{\partial}$ has closed range in $L^{2}(Ω)$ 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 $L^{2}(Ω)$. We also give a new necessary and sufficient potential-theoretic condition for the Bergman space of $Ω$ to be infinite dimensional.

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