Bilipschitz mappings and quasihyperbolic mappings in real Banach spaces

Abstract

Suppose that $G⊊E$ and $G^{\prime }⊊E^{\prime }$ are domains, where $E$ and $E^{\prime }$ denote real Banach spaces with dimension at least 2, and $f:G\to G^{\prime }$ 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.