Percolation of estimates for \(\bar{\partial}\) by the method of alternating projections
type: Item
instance of: scholarly article
title: Percolation of estimates for \(\bar{\partial}\) by the method of alternating projections
zbMATH Open document ID: 1483.32019
author: Kenneth D. Koenig
author: J. D. McNeal
published in: The Journal of Geometric Analysis
publication date: 2021-08-31T00:00:00Z
review text: On a bounded pseudoconvex domain in \(\mathbb{C}^{n}\), compactness of the \(\overline{\partial}\)-Neumann operator percolates up the Cauchy-Riemann complex. That is, if the property holds at form level \(q\), then it holds at form level \((q+1)\). In this paper, the authors generalize this fact to domains that are not assumed pseudoconvex nor bounded (but with Lipschitz boundary), but are only assumed to have the property that \(\overline{\partial}\) has closed range at the respective form levels. Their approach to these questions is new and very interesting. Rather than relating \textit{estimates} at different form levels, they study the relation(s) between various relevant \textit{operators} at (and between) these different levels. After studying in detail how to relate forms at consecutive levels, and analyzing relations between various naturally occuring operators, they establish the following key fact: the closed range property is equivalent to a geometric rate of convergence in von Neumann's alternating projection algorithm, applied to two basic projection operators. This then fact implies percolation of compactness. The point of view adopted in this paper is original and should prove useful in the future.
reviewed by: Emil J. Straube
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32A70
zbMATH DE Number: 7388713
zbMATH Keywords: percolation of estimates
zbMATH Keywords: alternating projections
zbMATH Keywords: Cauchy-Riemann complex
zbMATH Keywords: closed range
zbMATH Keywords: compactness
zbMATH Keywords: \(\overline{\partial}\)-Neumann problem
MaRDI profile type: Publication
full work available at URL: s12220-020-00532-w
OpenAlex ID: W3092936332
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DOI: 10.1007/S12220-020-00532-W
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description: scientific article; zbMATH DE number 7388713