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

Source: [The closed range property for the $\\overline{\\partial}$-operator on planar domains](<https://www.sota2.com/research/paper/68fa57d01e209131ce9a8cc1>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-28T13:55:56.681Z

Authors: A.-K. Gallagher, J. Lebl, K. Ramachandran

Published: 2019-01-14T16:39:56.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1901.04390>)

[PDF](<https://arxiv.org/pdf/1901.04390.pdf>)