Unique continuation in CR manifolds and in hypo-analytic structures
type: Item
instance of: scholarly article
title: Unique continuation in CR manifolds and in hypo-analytic structures
zbMATH Open document ID: 0645.32010
DOI: 10.1007/BF02386106
author: Mohamed Salah Baouendi
author: François Treves
published in: Arkiv för Matematik
publication date: 1988-01-01T00:00:00Z
review text: The main result of this paper is an identity Theorem for CR functions on CR submanifolds of complex Euclidean space. Let \({\mathcal M}\) denote a generic submanifold of \({\mathbb{C}}^{n+d}\) of class \({\mathcal C}^ 1 \)and codimension d. Suppose \({\mathcal H}\) is a complex submanifold of \({\mathbb{C}}^{n+d}\) of dimension d such that \(\Sigma ={\mathcal M}\cap {\mathcal H}\) is holomorphic transversal at 0. The authors show how to choose coordinates (z,w) in \({\mathbb{C}}^{n+d}\) so that \({\mathcal H}=\{z=0\}\) and \({\mathcal M}=\{(z,s+i\phi (z,\bar z,s))\}\) for \(\phi\) : \({\mathbb{C}}\) \(n\times {\mathbb{R}}^ d \)a \({\mathcal C}^ 1 \)map with \(\phi (0)=0\) and \(d\phi (0)=0\). Notice then \(\Sigma =\{(0,s+i\phi (0,0,s))\}\). The authors introduce condition (\({\mathcal A}):\) Given any open neighborhood \({\mathcal V}\) of 0 in \(\Sigma\) there is a holomorphic function F in an open subset of \({\mathcal H}\) containing \({\mathcal V}\), such that F(0)\(\neq 0\) and that the connected component of 0 in the set \(\{\) \(p\in {\mathcal V}:\) F(p)\(\neq 0\}\) has compact closure in \({\mathcal V}\). The authors show, for example, if \(\phi\) (0,0,s) is real-analytic near 0 then \(\Sigma\) has property \({\mathcal A}\). The Theorem is as follows: Suppose \(\Sigma\) has property \({\mathcal A}\), U is a neighborhood of 0 in \({\mathcal M}\), and h is a Lipschitz-continuous CR function in U which vanishes to infinite order on \(\Sigma\cap U\). Then \(h\equiv 0\) on an open neighborhood of 0 contained in U. As an interesting aside, the authors raise the question of whether or not there exist such \(\Sigma\) not having property \({\mathcal A}\). They then proceed to obtain analogous results in the general context of hypo- analytic structures.
Mathematics Subject Classification ID: 32V40
zbMATH DE Number: 4052000
zbMATH Keywords: generic CR submanifold
zbMATH Keywords: identity Theorem for CR functions
zbMATH Keywords: holomorphic transversal
zbMATH Keywords: hypo-analytic structures
reviewed by: G. A. Harris
MaRDI profile type: Publication
cites work: Microlocal hypo-analyticity and extension of CR functions
cites work: A property of the functions and distributions annihilated by a locally integrable system of complex vector fields
cites work: Unique continuation of solutions of partial differential equations and inequalities from manifolds of any dimension
cites work: Q3715423
cites work: Q3679621
full work available at URL: https://doi.org/10.1007/bf02386106
OpenAlex ID: W1979832247
description: scientific article; zbMATH DE number 4052000