Sur le prolongement holomorphe des fonctions CR définies sur une hypersurface réelle de classe \(C^ 2\) dans \({\mathbb{C}}^ n\). (Holomorphic continuation of CR-functions defined on a real hypersurface of class \(C^ 2\) in \({\mathbb{C}}^ n)\). (French) Zbl 0586.32016

Let S be a \(C^ 2\) real hypersurface in \({\mathbb{C}}^ n\) and let \(z_ 0\) be a point on S. Nearby \(z_ 0\), S divides the ambient space into 2 domains denoted by \(\Omega_ 1\) and \(\Omega_ 2\). The following result is established: Theorem 1: Let assume that S does not contain any complex hypersurface passing through \(z_ 0\). Then there exists a fundamental system of neighborhoods \(U_ n\) of \(z_ 0\) such that any holomorphic function on \(U_ n\cap \Omega_ i\) can be extended holomorphically to the entire neighborhood \(U_ n\) where \(i=1\) and/or 2.
This result generalizes previous works of Bedford-Fornaess and Baouendi- Treves. The proof relies on the following crucial result: Theorem 2: Let \({\bar \Omega}{}_ i\) be the envelope of holomorphy of \(\Omega_ i\) and let \(\partial {\bar \Omega}_ i\) be their boundaries. Then the set \(E:=S\cap \partial {\bar \Omega}_ 1\cap \partial {\bar \Omega}_ 2\) consists of a union of complex hypersurfaces.
In view of theorem 2, theorem 1 follows by a contradiction argument.
Reviewer: Vo Van Tan


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