Bounded holomorphic functions on bounded symmetric domains
Authors:
Joel M. Cohen and Flavia Colonna
Journal:
Trans. Amer. Math. Soc. 343 (1994), 135156
MSC:
Primary 32A37; Secondary 32M15, 46E15
DOI:
https://doi.org/10.1090/S00029947199411760856
MathSciNet review:
1176085
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Abstract  References  Similar Articles  Additional Information
Abstract: Let D be a bounded homogeneous domain in ${\mathbb {C}^n}$, and let $\Delta$ denote the open unit disk. If $z \in D$ and $f:D \to \Delta$ is holomorphic, then ${\beta _f}(z)$ is defined as the maximum ratio ${\nabla _z}(f)x/{H_z}{(x,\bar x)^{1/2}}$, where x is a nonzero vector in ${\mathbb {C}^n}$ and ${H_z}$ is the Bergman metric on D. The number ${\beta _f}(z)$ represents the maximum dilation of f at z. The set consisting of all ${\beta _f}(z)$ for $z \in D$ and $f:D \to \Delta$ holomorphic, is known to be bounded. We let ${c_D}$, be its least upper bound. In this work we calculate ${c_D}$ for all bounded symmetric domains having no exceptional factors and give indication on how to handle the general case. In addition we describe the extremal functions (that is, the holomorphic functions f for which ${\beta _f} = {c_D}$) when D contains $\Delta$ as a factor, and show that the class of extremal functions is very large when $\Delta$ is not a factor of D.

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Additional Information
Keywords:
Bloch,
bounded symmetric domains,
Lipschitz
Article copyright:
© Copyright 1994
American Mathematical Society