Prolongement unilateral des fonctions CR. (Unilateral extension of CR functions). (French) Zbl 0609.32012

Sémin. Bony-Sjöstrand-Meyer, Équations Dériv. Partielles 1984-1985, Exp. No. 22, 7 p. (1985).
Let S be a \(C^ 2\)-smooth real hypersurface in \({\mathbb{C}}^ n\), passing through the point \(z_ 0\). Near \(z_ 0\), two sides of the ambient space, \(\Omega_+\) and \(\Omega_-\), are defined. We say that \(\Omega_+\) (resp. \(\Omega_-)\) has the extension property \(\downarrow\!\!z_ 0\) if for any open neighbourhood U of \(z_ 0\), there exists another neighbourhood V of \(z_ 0\), \(V\subset U\), such that any holomorphic function in \(U\cap \Omega_+\) (resp. \(U\cap \Omega_-)\) extends holomorphically to V.
The main result of the paper is the following: Theorem. The following statements are equivalent: (a) there are no germs of complex hypersurfaces through \(z_ 0\) contained in S; (b) at least one among \(\Omega_+\) and \(\Omega_-\) has the extension property \(\downarrow\!\!z_ 0.\)
Of course (b)\(\Rightarrow (a)\) is obvious; the proof of (a)\(\Rightarrow (b)\) is based on the Kontinuitätssatz and the construction of holomorphic disks with boundary on S.


32D15 Continuation of analytic objects in several complex variables
32D10 Envelopes of holomorphy
32V40 Real submanifolds in complex manifolds
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