A Julia-Wolff-Carathéodory theorem for infinitesimal generators in the unit ball. (English) Zbl 1343.32004
Let \(\Delta \subset \mathbb C\) be the unit disk. The authors discuss the problem of generalization of the classical Julia-Wolff-Carathédory theorem for bounded holomorphic functions \(f : \Delta \rightarrow \Delta\) (see, e.g. [R. B. Burckel, An introduction to classical complex analysis. Vol. 1. New York, San Francisco: Academic Press (1979; Zbl 0434.30002)]). In particular, they give a detailed proof of an extension of the theorem to the case of holomorphic self-maps of the unit ball in \(\mathbb C^n\) using results obtained in [F. Bracci and D. Shoikhet, Trans. Am. Math. Soc. 366, No. 2, 1119–1140 (2014; Zbl 1337.32016)].
Reviewer: Aleksandr G. Aleksandrov (Moskva)
MSC:
| 32A40 | Boundary behavior of holomorphic functions of several complex variables |
| 20M20 | Semigroups of transformations, relations, partitions, etc. |
| 37L05 | General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations |
| 32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |
Keywords:
Julia-Wolff-Carathéodory theorem; boundary behaviour; angular derivatives; infinitesimal generators; semigroups of holomorphic mappingsReferences:
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