Characteristic properties of Bloch functions (Q1091510)

The authors give several characterizations of Bloch functions, in classical and generated form. M is a finite-dimensional complex manifold and G is a transitive group of holomorphic self-maps of M. Let f be holomorphic on M and \[ O(G,f,z_ 0) = \{f\circ g(z)-(f\circ g)(z_ 0): g\in G\}. \] If O is precompact, then f is called a Bloch function; when \(M=\Delta =\{| z| <1\}\) and G is the full Möbius group on \(\Delta\) we recover the classical Bloch space. They introduce a class of ``admissible'' functions \(\phi\) on the plane \({\mathbb{C}}:\) these are positive, monotone real-valued set functions relative to bounded subsets of \({\mathbb{C}}\), with \(\phi ({\mathbb{C}})=\infty\). The main result is that f is Bloch iff sup \(\phi\) (f\(\circ g(V))<\infty\) (sup over \(g\in G)\), where \(V\subset {\mathbb{C}}\) has compact closure. This result applies to several situations. One is when M is a hyperbolic manifold with the Kobayashi metric; then \({\mathcal B}\) coincides with Lipschitz functions.