Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Bounded holomorphic functions in Siegel domains

Author: Su Shing Chen
Journal: Proc. Amer. Math. Soc. 40 (1973), 539-542
MSC: Primary 32H15
MathSciNet review: 0322211
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A Siegel domain $ D$ of the second kind (not necessarily affine homogeneous) is shown to be complete with respect to the Carathéodory distance. Thus $ D$ is convex with respect to the bounded holomorphic functions, hence is a domain of holomorphy. A Phragmén-Lindelöf theorem for $ D$ is also given. That is, if a holomorphic function $ f$ in $ D$ is continuous in $ \bar D$, bounded on the distinguished boundary $ S$ of $ D$ and not of exponential growth, then $ f$ is bounded in $ \bar D$.

References [Enhancements On Off] (What's this?)

  • [1] Salomon Bochner and William Ted Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. MR 0027863
  • [2] Soji Kaneyuki, Homogeneous bounded domains and Siegel domains, Lecture Notes in Mathematics, Vol. 241, Springer-Verlag, Berlin-New York, 1971. MR 0338467
  • [3] Dong S. Kim, Boundedly holomorphic convex domains, Pacific J. Math. 46 (1973), 441–449. MR 0344520
  • [4] Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
  • [5] Adam Korányi, The Poisson integral for generalized half-planes and bounded symmetric domains, Ann. of Math. (2) 82 (1965), 332–350. MR 0200478,
  • [6] I. I. Pjateckii-Šapiro, Géométrie des domaines classiques et théorie des fonctions automorphes, Dunod, Paris, 1966. MR 33 #5949.
  • [7] E. M. Stein, Boundary behavior of holomorphic functions of several complex variables, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Mathematical Notes, No. 11. MR 0473215
  • [8] E. C. Titchmarsh, Han-shu lun, Translated from the English by Wu Chin, Science Press, Peking, 1964 (Chinese). MR 0197687
  • [9] VasiliĭSergeevič Vladimirov, Methods of the theory of functions of many complex variables, Translated from the Russian by Scripta Technica, Inc. Translation edited by Leon Ehrenpreis, The M.I.T. Press, Cambridge, Mass.-London, 1966. MR 0201669

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32H15

Retrieve articles in all journals with MSC: 32H15

Additional Information

Keywords: Siegel domain, convexity with respect to $ B(M)$, Phragmén-Lindelöf theorem, Kobayashi distance, Carathéodory distance, domain of holomorphy, domain of bounded holomorphy
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society