Geodesic
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 Cet article a éte moissonné depuis la source Journal.fi

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

Yuehui He 1 ; Manzi Huang 2 ; Xiantao Wang 3

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 
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     title = {Bilipschitz mappings and quasihyperbolic mappings in real {Banach} spaces},
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Yuehui He; Manzi Huang; Xiantao Wang. Bilipschitz mappings and quasihyperbolic mappings in real Banach spaces. Annales Fennici Mathematici, Tome 46 (2021) no. 2, pp. 771-779. http://geodesic.mathdoc.fr/item/AFM_2021_46_2_a11/