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Bounded holomorphic functions on bounded symmetric domains

By Joel M. Cohen and Flavia Colonna


Abstract. Let D be a bounded homogeneous domain in C, and let A denote the open unit disk. If z e D and /: D — ► A is holomorphic, then ß/(z) is defined as the maximum ratio \Vz(f)x\/Hz(x, 3c)1/2, where x is a nonzero vector in C and Hz is the Bergman metric on D. The number ßf(z) represents the maximum dilation of / at z. The set consisting of all ß/(z), for z e D and /: D — ► A holomorphic, is known to be bounded. We let cr, be its least upper bound. In this work we calculate Cr, 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 / for which ßf = C£>) when D contains A as a factor, and show that the class of extremal functions is very large when A is not a factor of D

Year: 1994
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