Weakly quasisymmetric maps and uniform spaces

Abstract

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that $G\subset X$ and $G^{\prime }\subset Y$ are domains, and that $f:G\to G^{\prime }$ is a homeomorphism. In this paper, we first give some basic properties of short arcs, and then we show that: if $f$ is a weakly quasisymmetric mapping and $G^{\prime }$ is a quasiconvex domain, then the image $f(D)$ of every uniform subdomain $D$ in $G$ is uniform. As an application, we get that if $f$ is a weakly quasisymmetric mapping and $G^{\prime }$ is an uniform domain, then the images of the short arcs in $G$ under $f$ are uniform arcs in the sense of diameter.

arXiv commentComput. Methods Funct. Theory;2018. arXiv admin note: substantial text overlap with arXiv:1501.07375