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Extension of CR-functions into a wedge. (Russian) Zbl 0714.32005

Let a manifold \(M\subset {\mathbb{C}}^{n+m}\) in a neighbourhood of the origin be given by the equation \(x=h(y,w)\), where \(z=x+iy\in {\mathbb{C}}^ m,\) \(w\in {\mathbb{C}}^ n\), h be a smooth function in a neighbourhood of the origin of \({\mathbb{R}}^ m\times {\mathbb{C}}^ m\) with its values in \({\mathbb{R}}^ m\), \(h(0,0)=0\), \(dh(0,0)=0\). A set \({\mathcal M}_ r(U,{\mathbb{C}})=\{(z,w)\in U\), x-h(y,w)\(\in {\mathbb{C}}\}\) is said to be a wedge of range r with the edge M if U is a non-singular surface of dimension \(2n+m+r\) which contains some neighbourhood of the origin on M and \({\mathbb{C}}\) is an open convex cone in \({\mathbb{R}}^ m\). The manifold M is said to admit a \(W_ r\)-extension in a point \(p\in M\) if one can choose such coordinates (z,w) with the origin in p, a surface M and a cone \({\mathbb{C}}\) that every continuous CR-function on M admits a continuous CR- extension to \({\mathcal M}_ r(U,{\mathbb{C}}).\)
The author studies conditions of the existence of \(W_ r\)-extensions. One of the results obtained is as follows.
Theorem 1. Let M admit no \(W_ r\)-extension in a point \(p\in M\) for some r, \(1\leq r\leq m\). Then there exists a CR-submanifold \(N\subset M\), \(p\in N\), CR-dim N\(=n\), \(\dim N<2n+r\).
Reviewer: L.Ronkin

MSC:

32V40 Real submanifolds in complex manifolds
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