Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 1003-1014
Kyushu Sangyo University,
Faculty of Science and Engineering
3-1 Matsukadai, 2-Chome, Higashi-ku,
Fukuoka 813-8503,
Japan; h.hamada 'at' ip.kyusan-u.ac.jp
Abstract. Let BX be a bounded symmetric domain realized as the unit ball of an n-dimensional JB*-triple X = (Cn,‖·‖X). In this paper, we give a new definition of Bloch type mappings on BX and give distortion theorems for Bloch type mappings on BX. When BX is the Euclidean unit ball in Cn, this new definition coincides with that given by Chen and Kalaj or by the author. As a corollary of the distortion theorem, we obtain the lower estimate for the radius of the largest schlicht ball in the image of f centered at f(0) for α-Bloch mappings f on BX. Next, as another corollary of the distortion theorem, we show the Lipschitz continuity of (det B(z,z))1}/2n|det Df(z)|1/n for Bloch type mappings f on BX with respect to the Kobayashi metric, where B(z,z) is the Bergman operator on X, and use it to give a sufficient condition for the composition operator Cφ to be bounded from below on the Bloch type space on BX, where φ is a holomorphic self mapping of BX. In the case BX = Bn, we also give a necessary condition for Cφ to be bounded from below which is a converse to the above result. Finally, as another application of the Lipschitz continuity, we obtain a result related to the interpolating sequences for the Bloch type space on BX.
2010 Mathematics Subject Classification: Primary 32A18, 32M15, 47B38, 30H30.
Key words: Bloch type mapping, bounded from below, bounded symmetric domain, distortion theorem, Lipschitz continuity.
Reference to this article: H. Hamada: Distortion theorems, Lipschitz continuity and their applications for Bloch type mappings on bounded symmetric domains in Cn. Ann. Acad. Sci. Fenn. Math. 44 (2019), 1003-1014.
https://doi.org/10.5186/aasfm.2019.4451
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