On holomorphic maps between Riemann surfaces which preserve BMO
Let \(R(S)\) be an open Riemann surface and \(BMO(R)\) \((BMO(S))\) the set of functions of bounded mean oscillation on \(R(S)\). A nonconstant analytic function \(F:R\to S\) is called a \(BMO\) map if \(g\circ F \in BMO(R)\) for every \(g\in BMO(S)\). Similarly, \(F\) is a \(BMOH\) map if it preserves \(BMOH\), the space of harmonic \(BMO\) functions. The author investigates \(BMO\) maps and \(BMOH\) maps between plane domains and between Riemann surfaces. When \(R\) and \(S\) are plane domains the author presents a number of characterizations for both \(BMO\) maps and \(BMOH\) maps. In particular, a covering map of plane domains is a \(BMO\) map if and only if it is a \(BMOH\) map. For the situation in which \(R\) and \(S\) are Riemann surfaces with \(Y\) noncompact, a characterization of \(BMO\) maps is given, but no characterization is presented for \(Y\) compact. The investigation of \(BMO\) maps between Riemann surfaces makes use of the Hahn metric, so the author establishes a number of properties of this metric. In particular, every closed curve which is homotopically nontrivial has length at least \(\pi/2\) relative to the Hahn metric. The Hahn metric on Riemann surfaces can be regarded as an analogue of the quasihyperbolic metric on plane regions; on plane regions these two metric are equivalent.
scientific article; zbMATH DE number 836415