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Local characterization of holomorphic automorphisms of Siegel domains. (English. Russian original) Zbl 0572.32018
Funct. Anal. Appl. 17, 285-294 (1983); translation from Funkts. Anal. Prilozh. 17, No. 4, 49-61 (1983).
Extending previous work, the authors investigate the Bergman-Shilov boundary of an arbitrary ”nondegenerate” Siegel domain and prove the following basic theorem: Let D and D’ be nondegenerate Siegel domains in \({\mathbb{C}}^{n+m}\) and M and M’ their Bergman-Shilov boundary respectively. Let \(\phi\) : \(S\to S'\) be a homeomorphism of connected open subsets \(S\subset M\) and S’\(\subset M'\), satisfying (componentwise in the sense of distribution theory) the tangential Cauchy-Riemann equations on S. Then the map \(\phi\) can be extended to a biholomorphic map \(\phi\) : \(D\to D'\). In section 1 simple facts about the Shilov boundary are proved. In section 2 deep results (e.g. of Naruki and Alexander) are used to investigate the ”analyticity of a CR-homeomorphisms of the skeleton of a Siegel domain”. Section 3 contains the proof of the basic theorem and the last section indicates ”additional results”.
Reviewer: J.Dorfmeister

MSC:
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32D15 Continuation of analytic objects in several complex variables
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32H99 Holomorphic mappings and correspondences
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