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arXiv · 1901.04390

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

A. -K. Gallagher·J. Lebl·K. Ramachandran
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Abstract

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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A. -K. Gallagher, J. Lebl, K. Ramachandran. 2019-10-29. The closed range property for the $\overline{\partial}$-operator on planar domains. https://doi.org/10.1007/s12220-019-00318-9

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