Bilipschitz mappings and quasihyperbolic mappings in real Banach spaces

Authors

  • Yuehui He Shantou University, Department of Mathematics
  • Manzi Huang Hunan Normal University, School of Mathematics and Statistics, MOE-LCSM, and Qufu Normal University, School of Mathematical Science
  • Xiantao Wang Hunan Normal University, School of Mathematics and Statistics, MOE-LCSM

Keywords:

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

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.
Section
Articles

Published

2021-08-03

How to Cite

He, Y., Huang, M., & Wang, X. (2021). Bilipschitz mappings and quasihyperbolic mappings in real Banach spaces. Annales Fennici Mathematici, 46(2), 771–779. Retrieved from https://afm.journal.fi/article/view/110588