Semisolidity and locally weak quasisymmetry of homeomorphisms in metric spaces

TL;DR

This paper explores the connection between semisolidity and locally weak quasisymmetry of homeomorphisms in quasiconvex, complete metric spaces, extending previous results and resolving an open problem in the field.

Contribution

It generalizes key results relating semisolidity and locally weak quasisymmetry, and provides a complete solution to an open problem from prior research.

Findings

The composition of two locally weakly quasisymmetric mappings is also locally weakly quasisymmetric.

Such compositions are quasiconformal.

The paper extends previous results to more general metric spaces.

Abstract

In this paper, we investigate the relationship between semisolidity and locally weak quasisymmetry of homeomorphisms in quasiconvex and complete metric spaces. Our main objectives are to (1) generalize the main result in [X. Huang and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in metric spaces, Trans. Amer. Math. Soc. 367 (2015), 6225-6246] together with other related results, and (2) give a complete answer to the open problem given in [X. Huang and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in metric spaces, Trans. Amer. Math. Soc. 367 (2015), 6225-6246]. As an application, we prove that the composition of two locally weakly quasisymmetric mappings is a locally weakly quasisymmetric mapping and that it is quasiconformal.