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.

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Correspondence to J. Lebl.

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Gallagher, AK., Lebl, J. & Ramachandran, K. The Closed Range Property for the \({\overline{\partial }}\)-Operator on Planar Domains. J Geom Anal 31, 1646–1670 (2021). https://doi.org/10.1007/s12220-019-00318-9

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Keywords

  • \({\overline{\partial }}\)
  • Closed range
  • Logarithmic capacity
  • Poincaré–Dirichlet inequality

Mathematics Subject Classification

  • 32W05
  • 31A15