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Transactions of the American Mathematical Society

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



Bounded holomorphic functions on bounded symmetric domains

Authors: Joel M. Cohen and Flavia Colonna
Journal: Trans. Amer. Math. Soc. 343 (1994), 135-156
MSC: Primary 32A37; Secondary 32M15, 46E15
MathSciNet review: 1176085
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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 $ \vert{\nabla _z}(f)x\vert/{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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Keywords: Bloch, bounded symmetric domains, Lipschitz
Article copyright: © Copyright 1994 American Mathematical Society