Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Bloch constants of bounded symmetric domains


Author: Genkai Zhang
Journal: Trans. Amer. Math. Soc. 349 (1997), 2941-2949
MSC (1991): Primary 32H02, 32M15
DOI: https://doi.org/10.1090/S0002-9947-97-01518-3
MathSciNet review: 1329540
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $D_{1}$ and $D_{2}$ be two irreducible bounded symmetric domains in the complex spaces $V_{1}$ and $V_{2}$ respectively. Let $E$ be the Euclidean metric on $V_{2}$ and $h$ the Bergman metric on $V_{1}$. The Bloch constant $b(D_{1}, D_{2})$ is defined to be the supremum of $E(f^{\prime }(z)x, f^{\prime }(z)x)^{\frac {1}{2}}/h_{z}(x, x)^{1/2}$, taken over all the holomorphic functions $f: D_{1}\to D_{2}$ and $z\in D_{1}$, and nonzero vectors $x\in V_{1}$. We find the constants for all the irreducible bounded symmetric domains $D_{1}$ and $D_{2}$. As a special case we answer an open question of Cohen and Colonna.


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

  • [CC] J. M. Cohen and F. Colonna, Bounded holomorphic functions on bounded symmetric domains, Trans. Amer. Math. Soc. 343 (1994), 135-156. MR 94g:32007
  • [He] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, London, 1978. MR 80k:53081
  • [Hu] L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Amer. Math. Soc., Providence, Rhode Island, 1963. MR 30:2162
  • [L1] O. Loos, Bounded Symmetric Domains and Jordan Pairs, University of California, Irvine, 1977.
  • [L2] O. Loos, Jordan Pairs, Lecture Notes in Mathematics, No. 460, Springer, 1975. MR 56:3071
  • [R] W. Rudin, Function Theory in the Unit Ball of $\mathbb {C}^{n}$, Springer-Verlag, 1980. MR 82i:32002
  • [Sa] I. Satake, Algebraic structures of symmetric domains, Iwanami Shoten and Priceton Univ. Press, Tokyo and Princeton, NJ, 1980. MR 82i:32003
  • [Up] H. Upmeier, Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics, Regional Conference Series in Mathematics No.67, Amer. Math. Soc., 1987. MR 80h:17032
  • [W] J. Wolf, Fine structure of hermitian symmetric spaces, Symmetric Spaces, Marcel Dekker, New York, 1972, pp. 271-357. MR 53:8516
  • [Y] Z. Yan, Extremal holomorphic mappings between a bounded symmetric domain and the unit ball, preprint, Berkeley, 1993.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 32H02, 32M15

Retrieve articles in all journals with MSC (1991): 32H02, 32M15


Additional Information

Genkai Zhang
Affiliation: School of Mathematics, University of New South Wales, Kensington NSW 2033, Australia
Address at time of publication: Department of Mathematics, University of Karlstad, S-651 88 Karlstad, Sweden
Email: genkai.zhang@hks.se

DOI: https://doi.org/10.1090/S0002-9947-97-01518-3
Keywords: Bounded symmetric domain, holomorphic mapping, Schwarz lemma, Bergman metric, Bloch constant
Received by editor(s): November 21, 1994
Received by editor(s) in revised form: May 10, 1995
Additional Notes: Research sponsored by the Australian Research Council
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society