Bilipschitz mappings and quasihyperbolic mappings in real Banach spaces

Yuehui He¹ ; Manzi Huang² ; Xiantao Wang³

¹ Shantou University, Department of Mathematics
² Hunan Normal University, School of Mathematics and Statistics, MOE-LCSM, and Qufu Normal University, School of Mathematical Science
³ Hunan Normal University, School of Mathematics and Statistics, MOE-LCSM

Annales Fennici Mathematici, Tome 46 (2021) no. 2, pp. 771-779.

Résumé

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.

Détail

Keywords: Quasihyperbolic metric, (locally) bilipschitz mapping, (locally) QH mapping, fully QH mapping