Unique continuation in CR manifolds and in hypo-analytic structures
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.
scientific article; zbMATH DE number 4052000