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An extension theorem for the CR-functions. (Italian. English summary) Zbl 0569.32009

Let \(\Gamma\) be an orientable, compact, connected, real hypersurface of class \(C^ 1\) in \({\mathbb{C}}^ n\) \((n>2).\)
Assume that the following conditions are satisfied: (i) The boundary \(\partial \Gamma\) of \(\Gamma\) is contained in the zero set M of a pluriharmonic function \(\rho\) : \({\mathbb{C}}^ n\to {\mathbb{R}}\). (ii) \(\Gamma\) \(\setminus \partial \Gamma \subset \{z\in {\mathbb{C}}^ n| \rho (z)>0\}\); (iii) \(\partial \Gamma\) is the boundary of a bounded open set \(A\subset M\). - Let D be the bounded open set in \({\mathbb{C}}^ n\) with boundary \(\Gamma\) \(\cup A\). Main theorem: Every locally Lipschitz CR- function f on \(\Gamma\) \(\setminus \partial \Gamma\) can uniquely be extended to a continuous function F on \(D\cup (\Gamma \setminus \partial \Gamma)\), which is holomorphic on D.
Reviewer: K.Oeljeklaus

MSC:

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