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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.

MSC:

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