Hölder continuity and composition operators in pluriharmonic Bloch spaces

Abstract

We show a Hölder estimate of order $1/n$ for pluriharmonic Bloch mappings in the unit ball ${\mathbb{B}}^n\subset {\mathbb{C}}^n$ with respect to the Bergman metric. We apply this to obtain a sufficient condition for the composition operator on the pluriharmonic Bloch space to be bounded below. As a partial converse, we also give a necessary condition for the boundedness below of the composition operator on the Bloch space of holomorphic mappings in ${\mathbb{B}}^n$.

15 pages