The null space of the \(\overline \partial \)-Neumann operator.
type: Item
instance of: scholarly article
title: The null space of the \(\overline \partial \)-Neumann operator.
zbMATH Open document ID: 1083.32033
author: Lars Hörmander
publication date: 2005-04-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116144
review text: In a first part, the author proves results concerning closed range of \(\overline{\partial}\) and invertibility of \(\square\) (and concerning its null space on \((n-1)\)-forms) on spherical shells, by systematic use of spherical harmonics. Some of these results are known, but the author's methods yield more precise constants. In section 3, the author proves a theorem that precises the notion that when the Levi form has the critical signature at some point of the boundary (\((n-q-1,q)\)), then the \(\overline{\partial}\)-Neumann problem is flawed on the domain at the level of \((0,q)\)-forms: the dimension of the null space of \(\square_{q}\) is infinite, or \(\overline{\partial}\) does not have closed range on \(L^{2}_{(0,q)}\) (possibly both). One of the results in section 2 is that the null space of \(\square_{n-1}\) has \(n\) independent multipliers. In section 4, it is shown that this property characterizes shells bounded by two confocal ellipsoids. A description of the null space of \(\square_{n-1}\) is obtained for these domains that allows to discuss the boundary behavior of the kernel of the orthogonal projection on this null space when the range of \(\square_{n-1}\) is closed, at a boundary point where the Levi form has signature \((0,n-1)\).
reviewed by: Emil J. Straube
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32A25
zbMATH DE Number: 2162426
zbMATH Keywords: \(\overline{\partial}\)-Neumann Laplacian
zbMATH Keywords: reproducing kernel
MaRDI profile type: Publication
OpenAlex ID: W1718110812
Wikidata QID: Q110898067
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DOI: 10.5802/AIF.2051
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 2162426