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

## TL;DR

This paper characterizes when the ar operator has closed range on planar domains using potential-theoretic conditions, linking functional analysis with complex potential theory.

## Contribution

It establishes necessary and sufficient potential-theoretic conditions for the ar operator to have closed range in L^2 spaces on planar domains.

## Key findings

- ar has closed range iff the Poincare9-Dirichlet inequality holds.
- Provides new potential-theoretic criteria for closed range of ar.
- Characterizes when the Bergman space is infinite dimensional.

## 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.