arXiv subjects

Yaxiang Li

Publications and source records attributed to Yaxiang Li.

At least 19 recordsLinked to original sources

arXiv2026

Equivalent characterizations of John and uniform domains in doubling metric spaces

In this paper, we characterize John and uniform domains in doubling metric spaces. Specifically, we show that a locally quasiconvex domain in a doubling metric space is length John if and only if it is diameter John. For uniform domains, we prove that a domain in a doubling metric space is length uniform if and only if it is diameter uniform (or distance uniform) and locally quasiconvex. Moreover, in a doubling length metric space, we refine this result by showing that a domain is length uniform (resp. John) if and only if it is diameter uniform (resp. John).

math.CV↗

arXiv2025

Dimension-free inner uniform estimates for quasigeodesics

In this paper, we establish a dimension-free inner uniform estimate for quasigeodesics. More precisely, we prove that a $c_0$-quasigeodesic in a $\delta$-Gromov hyperbolic $c$-John domain in $\mathbb{R}^n$ is $b$-inner uniform, for some constant $b$ depending only on $c_0$, $\delta$ and $c$, but not on the dimension $n$. The proof relies crucially on the techniques introduced by Guo-Huang-Wang in their recent work [arXiv:2502.02930, 2025]. In particular, we actually show that the above result holds in general Banach spaces, which answers affirmatively an open question of J. V\"ais\"al\"a in [Analysis, 2004] and partially addresses the open question of Bonk-Heinonen-Koskela in [Asterisque, 2001]. As a byproduct of our main result, we obtain that a $c_0$-quasigeodesic in a $\delta$-Gromov hyperbolic $c$-John domain in $\mathbb{R}^n$ is a $b$-cone arc with a dimension-free constant $b=b(c_0,\delta,c)$. This resolves an open problem of J. Heinonen in [Rev. Math. Iberoam., 1989].

math.CV↗

arXiv2023

On Schwarz-Pick type inequality and Lipschitz continuity for solutions to nonhomogeneous biharmonic equations

The purpose of this paper is to study the Schwarz-Pick type inequality and the Lipschitz continuity for the solutions to the nonhomogeneous biharmonic equation: $\Delta(\Delta f)=g$, where $g:$ $\overline{\ID}\rightarrow\mathbb{C}$ is a continuous function and $\overline{\ID}$ denotes the closure of the unit disk $\ID$ in the complex plane $\mathbb{C}$. In fact, we establish the following properties for these solutions: Firstly, we show that the solutions $f$ do not always satisfy the Schwarz-Pick type inequality $$\frac{1-|z|^2}{1-|f(z)|^2}\leq C, $$ where $C$ is a constant. Secondly, we establish a general Schwarz-Pick type inequality of $f$ under certain conditions. Thirdly, we discuss the Lipschitz continuity of $f$, and as applications, we get the Lipschitz continuity with respect to the distance ratio metric and the Lipschitz continuity with respect to the hyperbolic metric.

math.CV↗

arXiv2020

A note on Väisälä's problem concerning free quasiconformal mappings

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

math.CV↗

arXiv2020

Sphericalization with its applications in Gromov hyperbolic spaces

In this paper, we study certain applications of sphericalization in Gromov hyperbolic metric spaces. We first show that the doubling property regarding two classes of metrics on the Gromov boundary of hyperbolic spaces are coincided. Next, we obtain a characterization of unbounded Gromov hyperbolic domains via metric spaces sphericalization. Finally, we investigate the topological equivalence of Gromov hyperbolic $φ$-uniform domains between the Gromov boundary and the inner metric boundary.

math.MG↗

arXiv2020

Sphericalization and flattening with their applications in quasimetric measure spaces

The main purpose of the note is to explore the invariant properties of sphericalization and flattening and their applications in quasi-metric spaces. We show that sphericalization and flattening procedures on a quasimetric spaces preserving properties such as Ahlfors regular and doubling property. By using these properties, we generalize a recent result in \cite{WZ}. We also show that the Loewner condition can be preserved under quasimöbius mapping between two $Q$-Ahlfors regular spaces. Finally, we prove that the $Q$-regularity of $Q$-dimensional Hausdorff measure of Bourdon metric are coincided with Hausdorff measure of Hamenstädt metric defined on the boundary at infinity of a Gromov hyperbolic space.

math.CV↗

arXiv2019

Gromov hyperbolicity, John spaces and quasihyperbolic geodesics

We show that every quasihyperbolic geodesic in a John space admitting a roughly starlike Gromov hyperbolic quasihyperbolization is a cone arc. This result provides a new approach to the elementary metric geometry question, formulated in \cite[Question 2]{Hei89}, which has been studied by Gehring, Hag, Martio and Heinonen. As an application, we obtain a simple geometric condition connecting uniformity of the space with the existence of Gromov hyperbolic quasihyperbolization.

math.CV↗

arXiv2019

Deformations on symbolic Cantor sets and ultrametric spaces

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 quasimöbius uniformization of Cantor set.

math.CV↗

arXiv2019

Uniform perfectness for quasi-metric spaces

The aim of this paper is to investigate the equivalence conditions for uniform perfectness of quasi-metric spaces. We also obtain the invariant property of uniform perfectness under quasimöbius maps in quasi-metric spaces. In the end, two applications are given.

math.CV↗

arXiv2018

Weakly quasisymmetric maps and uniform spaces

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.

math.CV↗

arXiv2018

Characterizations of John spaces

The main purpose of this paper is to study the characterizations of John spaces. We obtain five equivalence characteristics 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.

math.CV↗

arXiv2018

Apollonian metric, uniformity and Gromov hyperbolicity

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 bilipschitz) of its domain. We prove that under these mappings the uniformity, $φ$-uniformity and $δ$-hyperbolicity (in the sense of Gromov with respect to quasihyperbolic metric) of proper domains of $\mathbb{R}^n$ are invariant. As applications, we give four equivalent conditions for a quasiconformal mapping which is defined on a uniform domain to be roughly Apollonian bilipschitz, and we conclude that $φ$-uniformity is invariant under quasimöbius mappings.

math.CV↗

arXiv2015

Balls in the triangular ratio metric

We consider the triangular ratio metric and estimate the radius of convexity for balls in some special domains and prove the inclusion relations of metric balls defined by the triangular ratio metric, the quasihyperbolic metric and the $j$-metric.

math.MG↗

arXiv2015

On the subinvariance of uniform domains in metric spaces

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. Our main result is the following subinvariance property of the class of uniform domains: Suppose both $f$ and $f^{-1}$ are weakly quasisymmetric mappings and $G'$ is a quasiconvex domain. Then the image $f(D)$ of every uniform subdomain $D$ in $G$ under $f$ is uniform. The subinvariance of uniform domains with respect to freely quasiconformal mappings or quasihyperbolic mappings is also studied with the additional condition that both $G$ and $G'$ are locally John domains.

math.CV↗

arXiv2014

Near geodesics in John domains in Banach spaces

Let $E$ be a real Banach space with dimension at least 2. In this paper, we prove that if $D\subset E$ is a John domain which is homeomorphic to an inner uniform domain via a CQH map, then each neargeodesic in $D$ is a cone arc.

math.CV↗

arXiv2013

Inner uniform domains and the Apollonian inner metric

In this paper, we characterize inner uniform domains in $\IR^n$ in terms of Apollonian inner metric and the metric $j'_D$ when $D$ are Apollonian. As an application, a new characterization for $A$-uniform domains is obtained.

math.CV↗

arXiv2012

Subdomain geometry of hyperbolic type metrics

Given a domain $G \subsetneq \Rn$ we study the quasihyperbolic and the distance ratio metrics of $G$ and their connection to the corresponding metrics of a subdomain $D \subset G$. In each case, distances in the subdomain are always larger than in the original domain. Our goal is to show that, in several cases, one can prove a stronger domain monotonicity statement. We also show that under special hypotheses we have inequalities in the opposite direction.

math.MG↗

arXiv2012

On bilipschitz extensions in real Banach spaces

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\not=E$ and $D'\not=E'$ are bounded domains with connected boundaries, that $f: D\to D'$ is an $M$-QH homeomorphism, and that $D'$ is uniform. The main aim of this paper is to prove that $f$ extends to a homeomorphism $\bar \bar{D}\to \bar{D}'$ and $\bar{f}|\partial D$ is bilipschitz if and only if $f$ is bilipschitz in $\bar{D}$. The answer to some open problem of Väisälä is affirmative under an natural additional condition.

math.CV↗