Bilipschitz mappings and quasihyperbolic mappings in real Banach spaces

Abstract

Suppose that \(G\subsetneq E\) and \(G'\subsetneq E'\) are domains, where \(E\) and \(E'\) denote real Banach spaces with dimension at least 2, and \(f\colon G\to G'\) is a homeomorphism. The aim of this paper is to prove the validity of the implications: \(f\) is \(M\)-bilipschitz \(\Rightarrow f\) is locally \(M\)-bilipschitz \(\Rightarrow f\) is \(M\)-QH \(\Rightarrow f\) is locally \(M\)-QH, and the invalidity of their opposite implications, i.e., \(f\) is locally \(M\)-QH \(\nRightarrow f\) is \(M\)-QH \(\nRightarrow f\) is locally \(M\)-bilipschitz \(\nRightarrow f\) is \(M\)-bilipschitz. Among these results, the relationship that \(f\) is locally \(M\)-QH \(\nRightarrow f\) is \(M\)-QH gives a negative answer to one of the open problems raised by Väisälä in 1999.