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A precise result on the boundary regularity of biholomorphic mappings. (English) Zbl 0603.32013

Let \(\Omega_ 1\), \(\Omega_ 2\subset \subset {\mathbb{C}}^ n\) be strongly pseudoconvex domains with \(C^ k\) boundary, with \(k\leq \infty\). There is a sizeable literature devoted to proving that a biholomorphic mapping \(f: \Omega\) \({}_ 1\to \Omega_ 2\) extends to a smooth mapping on \(\overline{\Omega}_ 1\). In the present work, the author defines a notion of smoothness called \(S^ k\), which lies between \(C^ k\) and \(C^{k+1}\). A domain \(\Omega\) has boundary of class \(S^ k\) if \(\partial \Omega\) is \(C^ k\) and the (complex) Gauss mapping taking \(z\in \partial \Omega\) to the complex tangent space \(T_{z}^{{\mathbb{C}}}(\partial \Omega)\) is of class \(C^ k.\)
The main result is: If \(\partial \Omega_ 1\), \(\partial \Omega_ 2\) are strongly pseudoconvex and of class \(S^ k\), then a biholomorphic mapping \(f: \Omega\) \({}_ 1\to \Omega_ 2\) extends to \(\overline{\Omega}_ 1\) to be \(S^ k\), i.e. \(f\in C^ k(\overline{\Omega}_ 1)\) and the complex tangential part of f’ on \(\partial \Omega_ 1\) is also \(C^ k\).
Reviewer: E.Bedford

MSC:

32T99 Pseudoconvex domains
32D15 Continuation of analytic objects in several complex variables
32A40 Boundary behavior of holomorphic functions of several complex variables
32V40 Real submanifolds in complex manifolds
32H99 Holomorphic mappings and correspondences
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References:

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