Local solvability for \(\square_b\) on degenerate CR manifolds and related systems
type: Item
instance of: scholarly article
title: Local solvability for \(\square_b\) on degenerate CR manifolds and related systems
zbMATH Open document ID: 1200.32023
DOI: 10.1216/RMJ-2009-39-2-605
author: Fabio Nicola
published in: Rocky Mountain Journal of Mathematics
publication date: 2009-06-18T00:00:00Z
review text: In an abstract CR-manifold \(M\), consider the Kohn Laplacian \(\square_ b^{(q)}\) acting on \((0,q)\)-forms \[ \square_ b^{(q)}=\bar\partial_ b\bar\partial_ b^*+\bar\partial_ b^*\bar\partial_ b. \] If the Levi form of \(M\) is degenerate and has a kernel of constant dimension, the author shows that for any point \(x_0\in M\) and any \(\delta>0\), there exists an open neighborhood \(\Omega_\delta\) of \(x_0\) such that \[ \|u\|_0\leq\delta(\square_ b^{(0,q)}u,u) \] for all \(u\in\mathcal{D}_{(0,q)}(\Omega_\delta)\). Here, \((\cdot,\cdot)\) denotes the \(L^2\) inner product, \(\|\cdot\|\) the \(L^2\) norm and \(\mathcal{D}_{(0,q)}(\Omega_\delta)\) is the space of smooth \((0,q)\)-forms with compact support in \(\Omega_\delta\). Moreover, it is also shown that the Kohn Laplacian is locally solvable on such \(M\). This result follows from a more general one concerning a class of systems with double characteristics and scalar principal symbol.
reviewed by: Manuel Flores
Mathematics Subject Classification ID: 32W10
Mathematics Subject Classification ID: 35A01
zbMATH DE Number: 5566190
zbMATH Keywords: CR-manifold
zbMATH Keywords: Kohn Laplacian
zbMATH Keywords: local solvability
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5566190