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Proper holomorphic mappings. (English) Zbl 0618.32022

Let \(\Omega,\Omega'\subset \subset {\mathbb{C}}^ n\) be strictly pseudoconvex domains with \(C^{\infty}\) boundaries. The authors prove that if U is a neighborhood of a boundary point p of \(\Omega\) and \(f: U\cap \Omega \to \Omega'\) is holomorphic such that \(f(z_ n)\to \partial \Omega'\) whenever \(z_ n\to \partial \Omega\), then f extends to a Hölder \(1/2\)- continuous function on \(\overline{V\cap \Omega}\) for some neighborhood V of p. If, in addition, f is biholomorphic from \(U\cap \Omega\) onto \(U'\cap \Omega'\) the authors give a simple proof that Condition A of Nirenberg-Webster-Yang is satisfied on an open dense subset \(\Gamma_ 1\) of \(U\cap \partial \Omega\). Hence f is \(C^{\infty}\) on \((U\cap \Omega)\cup \Gamma_ 1\). Finally the authors prove that a proper holomorphic map \(f: \Omega\to \Omega'\) is \(C^{\infty}\) on some dense open subset of \(\partial \Omega\), using an argument of Alexander to show that f is locally biholomorphic on some dense open subset of \(\partial \Omega\).
Reviewer: H.Luk

MSC:

32H35 Proper holomorphic mappings, finiteness theorems
32T99 Pseudoconvex domains
32A40 Boundary behavior of holomorphic functions of several complex variables
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