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arXiv · 1904.09345

Strong Closed Range Estimates: Necessary Conditions and Applications

Phillip S. Harrington·Andrew Raich
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Abstract

The $L^2$ theory of the $\bar\partial$ operator on domains in $\mathbb{C}^n$ is predicated on establishing a good basic estimate. Typically, one proves not a single basic estimate but a family of basic estimates that we call a family of strong closed range estimates. Using this family of estimates on $(0,q)$-forms as our starting point, we establish necessary geometric and potential theoretic conditions. The paper concludes with several applications. We investigate the consequences for compactness estimates for the $\bar\partial$-Neumann problem, and we also establish a generalization of Kohn's weighted theory via elliptic regularization. Since our domains are not necessarily pseudoconvex, we must take extra care with the regularization.

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Phillip S. Harrington, Andrew Raich. 2019-04-19. Strong Closed Range Estimates: Necessary Conditions and Applications. https://arxiv.org/abs/1904.09345

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