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In this paper, we establish several characterizations for quasihyperbolic mappings in Banach spaces. As an application, we provide a partial solution to a problem left open by Vaisala regarding the local properties of quasihyperbolic mappings.

In this paper, we provide partial solutions to a problem raised by Vaisala on local properties of free quasiconformal mappings. In particular, we show that a locally free quasiconformal mapping is a globally free quasiconformal mapping under the condition of locally relative quasisymmetry.

The main aim of this paper is to investigate the invariant properties of uniform domains under flattening and sphericalization in nonlocally compact complete metric spaces. Moreover, we show that quasi-Möbius maps preserve uniform domains in nonlocally compact spaces as well.

The aim of this paper is to investigate the equivalence conditions for uniform perfectness of quasimetric spaces. We also obtain the invariance property of uniform perfectness under quasim?bius mappings in quasimetric spaces. In the end, two applications are given.

By introducing new deformations on symbolic Cantor sets and ultrametric spaces, we prove that doubling ultrametric spaces admit bilipschitz embedding into Cantor sets. If in addition the spaces are uniformly perfect, we show that they are quasisymmetrically equivalent to Cantor sets. As an application, we provide a new proof for a recent work of Heer (Anal. Geom. Metr. Spaces, 2017) regarding quasim\"obius uniformization of Cantor set., Comment: 8 pages, BMMS revised

Let $\Omega \subset \mathbb{R}^n$ be a Gromov hyperbolic, $\varphi$-length John domain. We show that there is a uniformly continuous identification between the inner boundary of $\Omega$ and the Gromov boundary endowed with a visual metric, By using this result, we prove the boundary continuity not only for quasiconformal homeomorphisms, but also for more generally rough quasi-isometries between the domains equipped with the quasihyperbolic metrics.

In this paper, we introduce a class of mappings related to the distance ratio metric and study its connection to the freely quasiconformal mapping in Banach spaces. Moreover, we investigate the invariance of Gromov hyperbolic, natural and uniform domains under this class of mappings.

The main purpose of this paper is to investigate the properties of a mapping which is required to be roughly bilipschitz with respect to the Apollonian metric (roughly Apollonian bilipschit...

The main purpose of this paper is to study the characterizations of John spaces. We obtain five equivalent characterizations for length John spaces. As an application, we establish a dimension-free quasisymmetric invariance of length John spaces. This result is new also in the case of the Euclidean space.

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that $G\subset X$ and $G'\subset Y$ are domains, and that $f: G\to G'$ 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 $G'$ 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 $G'$ is an uniform domain, then the images of the short arcs in $G$ under $f$ are uniform arcs in the sense of diameter., Comment: Comput. Methods Funct. Theory;2018. arXiv admin note: substantial text overlap with arXiv:1501.07375

The aim of this note is to construct a ψ-uniform domain G in the complex plane C such that the identity mapping id: ( G , j G ) → ( G , k G ) is not an η-quasi-symmetric mapping for any homeomorphism η: [ 0 , ∞ ) → [ 0 , ∞ ) . This result shows that the answer to the related open problem, posed by Hasto, Klen, Sahoo and Vuorinen, is negative.

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., 34 pages, 1 figures