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Quasihyperbolic metric and Quasisymmetric mappings in metric spaces


Authors: Xiaojun Huang and Jinsong Liu
Journal: Trans. Amer. Math. Soc. 367 (2015), 6225-6246
MSC (2010): Primary 30C65
DOI: https://doi.org/10.1090/S0002-9947-2015-06240-0
Published electronically: February 19, 2015
MathSciNet review: 3356935
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Abstract: In this paper, we prove that the quasihyperbolic metrics are quasi-invariant under a quasisymmetric mapping between two suitable metric spaces. Meanwhile, we also show that quasi-invariance of the quasihyperbolic metrics implies that the corresponding map is quasiconformal. At the end of this paper, as an application of these theorems, we prove that the composition of two quasisymmetric mappings in metric spaces is a quasiconformal mapping.


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Xiaojun Huang
Affiliation: College of Mathematics and Statistics, Chongqing University, Chongqing 401331, People’s Republic of China – and – Mathematical Sciences Research Institute in Chongqing, Chongqing 401331, People’s Republic of China
Email: hxj@cqu.edu.cn

Jinsong Liu
Affiliation: HUA Loo-Keng Key Laboratory of Mathematics, Chinese Academic of Sciences, Beijing 100190, People’s Republic of China – and – Institute of Mathematics, Academic of Mathematics $&$ System Sciences, Chinese Academic of Sciences, Beijing 100190, People’s Republic of China
Email: liujsong@math.ac.cn

DOI: https://doi.org/10.1090/S0002-9947-2015-06240-0
Received by editor(s): June 13, 2012
Received by editor(s) in revised form: January 21, 2013, April 27, 2013, and June 7, 2013
Published electronically: February 19, 2015
Additional Notes: The first author was supported by NSF of China (No. 11471318) and Natural Science Foundation Project of Chongqing (No. CSTC, 2011BB0055)
The second author was supported by NSFC Grant No. 11471318.
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.