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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

MSC:

32D15 Continuation of analytic objects in several complex variables
32V40 Real submanifolds in complex manifolds
32D10 Envelopes of holomorphy
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[1] Baouendi, M.S., Tr?ves, F.: About the holomorphic extension ofCR functions on real hypersurfaces in complex space. Duke Math. J.51, 77-107 (1984) · Zbl 0564.32011 · doi:10.1215/S0012-7094-84-05105-6
[2] Bedford, E., Fornaess, J.E.: Local extension ofCR functions from weakly pseudoconvex boundaries. Michigan Math. J.25, 259-262 (1978) · Zbl 0401.32007 · doi:10.1307/mmj/1029002109
[3] Bishop, E.: Differentiable manifolds in complex Euclidean space. Duke Math. J.32, 1-22 (1965) · Zbl 0154.08501 · doi:10.1215/S0012-7094-65-03201-1
[4] Bony, J.M.: Principe du maximum, in?galit? de Harnack.... Ann. Inst. Fourier19, 277-304 (1969) · Zbl 0176.09703
[5] Denson Hill, C., Taiani, G.: Families of analytic discs in ? n with boundaries on a prescribedCR submanifold. Ann. Sc. Norm. Super. Pisa5, 327-380 (1978) · Zbl 0399.32008
[6] H?rmander, L.: An introduction to complex analysis in several variables (2?me ?dition). North-Holland Math. Library (1973)
[7] Polking, C., Wells, Jr., R.O.: Hyperfunction boundary values and a generalized Bochner-Hartogs theorem. Proc. Symp. Pure Math.30, 187-193 (1977) · Zbl 0358.32014
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