# A note on $\partial$-bilipschitz mappings in quasiconvex metric spaces

Tiantian Guan  Tiantian Guan, School of Mathematics and Big Data, Foshan
University, Foshan, Guangdong 528000, People’s Republic of China
<EMAIL_ADDRESS>,  Saminathan Ponnusamy  Saminathan Ponnusamy, Department of
Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
Department of Mathematics, Petrozavodsk State University, ul., Lenina 33,
185910 Petrozavodsk, Russia<EMAIL_ADDRESS>and  Qingshan Zhou
${}^{~{}\mathbf{*}}$ Qingshan Zhou, School of Mathematics and Big Data, Foshan
University, Foshan, Guangdong 528000, People’s Republic of China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper focuses on properties of $\partial$-biLipschitz mappings which were
recently introduced by Bulter. We establish several characterizations for the
class of $\partial$-biLipschitz mappings between domains in quasiconvex metric
spaces. As an application, we show that a locally quasisymmetric equivalence
between uniform metric spaces is quasimöbius, quantitatively.

###### Key words and phrases:

$\partial$-biLipschitz mapping, quasihyperbolic metric, quasiconvex metric
space, uniform space, quasimöbius mapping
∗ Corresponding author

###### 2000 Mathematics Subject Classification:

Primary: 30C65, 30F45, 53C23; Secondary: 30C20

File: main.tex, printed: 27-8-2024, 21.24

Qingshan Zhou was supported by NNSF of China (No. 11901090), the Department of
Education of Guangdong Province, China (No. 2021KTSCX116), Guangdong-Hong
Kong-Macao Intelligent Micro-Nano Optoelectronic Technology Joint Laboratory
(Project No. 2020B1212030010), and Guangdong Basic and Applied Basic Research
Foundation (Grant no. 2021A1515012289).

##  1\. Introduction and main results

In order to explain the connections between the uniformizations of Gromov
hyperbolic spaces established by Bonk et al. [1] and Bulter [5] (see also
[19]), Bulter recently introduced the following class of
$\partial$-biLipschitz mappings in [4].

######  Definition 1.1.

Let $L\geq 1$ and $0<\lambda<1$. A homeomorphism
$f:\,\Omega\to\Omega^{\prime}$ between incomplete, locally complete and
connected metric spaces is called $\partial$-Lipschitz with data $(L,\lambda)$
if for each $x\in\Omega$ and for all $y,z\in B(x,\lambda d_{\Omega}(x))$,

$\frac{d^{\prime}(f(y),f(z))}{d_{\Omega^{\prime}}(f(x))}\leq
L\frac{d(y,z)}{d_{\Omega}(x)}.$

Moreover, $f$ is called $\partial$-biLipschitz with data $(L,\lambda)$ if both
$f$ and its inverse $f^{-1}$ are $\partial$-Lipschitz with data $(L,\lambda)$.
Here $d_{\Omega}(x)$ denotes the distance between $x$ and the metric boundary
$\partial\Omega$ for all $x\in\Omega$, and

$B(x,\lambda d_{\Omega}(x))=\\{z\in X:\;d(z,x)<\lambda d_{\Omega}(x)\\}.$

The local completeness ensures that $d_{\Omega}(x)$ does not vanish.

Several important conformal transformations are $\partial$-biLipschitz
mappings (cf. [4]). Examples of such mappings include Riemann mappings from
the unit disc onto simply connected plane domains, sphericalization and
inversion operations in metric spaces setting suggested independently by Bonk
and Kleiner [2] and Buckley et al. [3], and conformal deformations introduced
by Bonk et al. [1] and Bulter [5]. More generally, Bulter proved that a
homeomorphism between uniform spaces is $\partial$-biLipschitz if and only if
it is biLipschitz with respect to the quasihyperbolic metric, see [4, Theorem
1.5].

This work is a continuation of [4]. We investigate new characterizations of
$\partial$-biLipschitz mappings between domains (open and connected nonempty
subsets) in quasiconvex metric spaces. For more investigations concerning
properties of quasiconformal mappings in this setting, one may refer [8, 9,
10, 20].

Obviously, the uniformity property of metric spaces implies the
quasiconvexity. Note that a metric space $(X,d)$ is called quasiconvex if
there is a constant $c\geq 1$ such that for each $x,y\in X$, there is a curve
$\gamma$ in $X$ joining $x$ and $y$ whose length $\ell(\gamma)$ satisfies the
condition $\ell(\gamma)\leq cd(x,y).$ It is worth mentioning that the class of
quasiconvex spaces is very large which includes convex domains, domains
$G\subset\mathbb{R}^{n}$ endowed with its length metric associated to the
Euclidean metric, uniform spaces, Ahlfors’ regular spaces admitting Loewner
conditions introduced by Heinonen and Koskela [7], and more generally doubling
metric measure spaces which satisfy Poincaré inequalities [12].

First, we establish the following characterizations of $\partial$-Lipschitz
mappings. The terminology appeared in Theorem 1.2 and in the rest of this
section will be explained in Section 2.

######  Theorem 1.2.

Suppose that $(X,d)$ and $(X^{\prime},d^{\prime})$ are $c$-quasiconvex and
complete metric spaces, that $G\subsetneq X$ and $G^{\prime}\subsetneq
X^{\prime}$ are domains, and that $f:G\to G^{\prime}$ is a homeomorphism. Then
the following statements are equivalent:

  1. (1)

$f$ is $\partial$-Lipschitz with data $(L,\lambda)$;

  2. (2)

$f$ is $(\theta,t_{0})$-relative with $\theta(t)=c_{1}t$;

  3. (3)

$f$ is $\varphi$-semisolid with $\varphi(t)=c_{2}t$.

Note that the parameters depend only on each other and $c$.

######  Remark 1.3.

The class of relative mappings between domains in $\mathbb{R}^{n}$ was
introduced by Gehring in his study of quasiconformal mappings. Originally,
solid mappings were investigated by Gehring and Osgood [6] in proving that
quasiconformal mappings in Euclidean spaces are solid mappings and the
converse is not true, see also [14]. In [15], Väisälä established the theory
of freely quasiconformal mappings in Banach spaces by using the
quasihyperbolic metric and solid mappings. Recently, Huang et al. [10] studied
the relationships between solid mappings and (weakly) quasisymmetric mappings
in quasiconvex and complete metric spaces.

Using Theorem 1.2, we obtain many characterizations for the class of
$\partial$-biLipschitz mappings between domains in quasiconvex metric spaces.
The following theorem is an improvement of [4, Theorem 1.5], where the domains
are assumed to be uniform.

######  Theorem 1.4.

Suppose that $(X,d)$ and $(X^{\prime},d^{\prime})$ are $c$-quasiconvex and
complete metric spaces, $G\subsetneq X$ and $G^{\prime}\subsetneq X^{\prime}$
are domains, and that $f:G\to G^{\prime}$ is a homeomorphism. Then the
following statements are equivalent:

  1. (1)

$f$ is $\partial$-biLipschitz with data $(L,\lambda)$;

  2. (2)

$f$ is $M$-biLipschitz with respect to the quasihyperbolic metrics;

  3. (3)

$f$ and $f^{-1}$ are $(\theta,t_{0})$-relative with $\theta(t)=c_{1}t$;

  4. (4)

$f$ and $f^{-1}$ are $(L_{1},\lambda_{1})$-locally biLipschitz;

  5. (5)

$f$ and $f^{-1}$ are $(\eta,q)$-locally quasisymmetric with $\eta(t)=c_{2}t$.

Note that the parameters depend only on each other and $c$.

######  Remark 1.5.

The class of locally biLipschitz mappings mentioned in Theorem 1.4(4) was
introduced by Bonk et al. [1], where they demonstrated that there is a one-to-
one correspondence between Gromov hyperbolic space category with biLipschitz
mappings as morphisms and uniform space category with quasisimilarity mappings
as morphisms. For more investigations about these mappings, see [18, 20].

As mentioned before, Bulter used $\partial$-biLipschitz mappings to
investigate the relationships between lots of conformal deformations on
uniform metric spaces, such as the uniformization processes of Gromov
hyperbolic spaces in [1, 5, 19] and the sphericalization and inversion
deformations in [2, 3]. In particular, he obtained the following:

Theorem A. $($[4, Theorem 1.4]$)$ Suppose that $(\Omega,d)$ and
$(\Omega^{\prime},d^{\prime})$ are locally compact and $A$-uniform metric
spaces. If $f:\,\Omega\to\Omega^{\prime}$ is $\partial$-biLipschitz with data
$(L,\lambda)$, then $f$ is $\theta$-quasimöbius with $\theta$ depending only
on $A$, $\lambda$, and $L$.

The class of uniform Euclidean domains was independently introduced by Jones
in [11] and by Martio and Sarvas in [13]. It is known that this notation plays
an important role in geometric function theory and analysis in metric spaces,
see [1, 3, 5, 17, 19] and the references therein. In [15], Väisälä proved that
a quasiconformal mapping between uniform Euclidean domains is quasimöbius. A
well-known fact is that quasiconformal mappings and their inverses are both
locally quasisymmetric.

It is natural to consider whether Theorem A holds or not for the class of
locally quasisymmetric equivalence mappings. We say that a homeomorphism
between two incomplete and locally complete metric spaces is locally
quasisymmetric equivalence if the mapping and its inverse are both locally
quasisymmetric. As an application of Theorem 1.4, we provide an affirmative
answer to this question by showing the following.

######  Theorem 1.6.

Let $A\geq 1$, $0<q<1$ and $\eta:[0,\infty)\to[0,\infty)$ a homeomorphism.
Suppose that $(\Omega,d)$ and $(\Omega^{\prime},d^{\prime})$ are locally
compact and $A$-uniform metric spaces. If a homeomorphism
$f:\Omega\to\Omega^{\prime}$ and $f^{-1}$ are both $(\eta,q)$-locally
quasisymmetric, then $f$ is $\theta$-quasimöbius, where $\theta$ depends only
on $A$, $q$, and $\eta$.

######  Remark 1.7.

$(a)$ Theorem 1.6 is a generalization of Theorem A and the result of Väisälä
in [15]. Indeed, one observes from Theorem 1.4 that $\partial$-biLipschitz
mappings are $(\eta,q)$-locally quasisymmetric mappings with a special control
function $\eta$.

$(b)$ Our method of proof of Theorem 1.6 is also different from that of [4,
15]. In [15], the tools of conformal modulus and $n$-Lebesgue measures are
needed to show that uniform domains satisfy a quasiextremal distance condition
in the sense of Gehring and Martio. In order to establish Theorem A, Bulter
made use of the unbounded uniformizations of Gromov hyperbolic spaces
associated to a class of conformal densities via Busemann functions, see [5,
19]. In particular, he proved that the identity mapping between any two
deformed uniform spaces is quasimöbius.

In this paper we do not use the results established in [5], but apply a result
of quasisymmetric mappings from local to global given in [19, Theorem 1.4] and
also the metric spaces sphericalization transformations introduced in [3].

The rest of this paper is organized as follows. In Section 2, we recall
necessary definitions and preliminary results. The proofs of Theorems 1.2 and
1.4 are given in Section 3. Section 4 is devoted to the proof of Theorem 1.6.

##  2\. Preliminaries and notations

Let $(X,d)$ be a metric space. For a bounded set $E\subset X$, the diameter of
$E$ is denoted by ${\operatorname{diam}}(E)$. The open (resp. closed) metric
ball with center $x\in X$ and radius $r>0$ is denoted by

$B(x,r)=\\{z\in X:\;d(z,x)<r\\}\;\;(\mbox{resp.}\;\;\overline{B}(x,r)=\\{z\in
X:\;d(z,x)\leq r\\}),$

and the metric sphere by $S(x,r)=\\{z\in X:\;d(z,x)=r\\}.$ A geodesic arc
$\gamma$ joining $x$ and $y$ in $X$ is a continuous map $\gamma:\,I=[0,l]\to
X$ from an interval $I$ to $X$ such that $\gamma(0)=x$ and $\gamma(l)=y$ and
$d(\gamma(t),\gamma(t^{\prime}))=|t-t^{\prime}|$ for all $t,t^{\prime}\in I$.
If $I=[0,\infty)$, then $\gamma$ is called a geodesic ray. We say that $X$ is
geodesic if every pair of points can be joined by a geodesic arc. The space
$X$ is called rectifiably connected if each pair of points in $X$ can be
connected by a curve $\gamma$ with $\ell(\gamma)<\infty$, where $\ell(\gamma)$
is the length of $\gamma$.

###  2.1. Quasihyperbolic metric and uniform space

Let $(\Omega,d)$ be a rectifiably connected, locally complete and incomplete
metric space, where the identity mapping
${\operatorname{id}}:(\Omega,d)\to(\Omega,\ell)$ is a homeomorphism and $\ell$
is the length metric of $\Omega$ with respect to $d$. Here $\Omega$ is
incomplete if its metric boundary $\partial\Omega=\bar{\Omega}\setminus\Omega$
is non-empty, where $\bar{\Omega}$ designates the metric completion of
$\Omega$. Note that every domain $G\subsetneq X$ in a quasiconvex complete
space equipped with the metric $d$ is such a space.

For all $x,y\in\Omega$, the number

$k_{\Omega}(x,y)=\inf_{\gamma}\int_{\gamma}\frac{ds}{d_{\Omega}(z)}$

is called the quasihyperbolic distance between $x$ and $y$ in $\Omega$, where
the infimum is taken over all rectifiable curves in $\Omega$ joining $x$ and
$y$.

######  Definition 2.1.

Let $A\geq 1$. The space $\Omega$ is called $A$-uniform if for each
$x,y\in\Omega$, there is a curve $\gamma$ joining $x$ and $y$ such that

  1. (1)

$\ell(\gamma)\leq Ad(x,y)$,

  2. (2)

$\min\\{\ell(\gamma[x,z]),\ell(\gamma[z,y])\\}\leq Ad_{\Omega}(z)$,

where $\gamma[x,z]$ is the subcurve of $\gamma$ between $x$ and $z$. The curve
$\gamma$ is called an $A$-uniform curve.

The following facts about the quasihyperbolic metric are needed for our
purpose.

Lemma B. $($[10, Lemma 3.8]$)$ Let $(X,d)$ be a $c$-quasiconvex and complete
metric space and let $G\subsetneq X$ be a domain.

  1. (1)

For all $x,y\in G$, $d(x,y)\leq(e^{k_{G}(x,y)}-1)d_{G}(x);$

  2. (2)

Suppose that $x,y\in G$ and either $d(x,y)\leq\frac{1}{3c}d_{G}(x)$ or
$k_{G}(x,y)\leq 1$. Then

$\frac{1}{2}\frac{d(x,y)}{d_{G}(x)}\leq k_{G}(x,y)\leq
3c\frac{d(x,y)}{d_{G}(x)}.$

Given a Euclidean domain $G$ with $x\in G$, we know that the ball $B(x,r)$
lies in $G$ for $0<r<d_{G}(x)$. However, this result is not true in the
setting of metric spaces, see [10, Example 3.1]. In [9], Huang et al.
established the following:

Lemma C. $($[10, Lemma 3.4]$)$ Suppose that $X$ is a $c$-quasiconvex and
complete metric space and $G\subsetneq X$ is a domain. Let $x\in G$ and
$0<r\leq\frac{2}{2+c}d_{G}(x)$. Then $B(x,r)\subset G$.

###  2.2. Mappings

######  Definition 2.2.

Let $L\geq 1$, $C\geq 0$ and let $\eta:[0,\infty)\to[0,\infty)$ be a
homeomorphism. A map $f:\,X\to X^{\prime}$ between metric spaces is called

  1. (1)

$(L,C)$-quasi-isometric if for all $x$, $y\in X$,

$L^{-1}d(x,y)-C\leq d^{\prime}(f(x),f(y))\leq Ld(x,y)+C.$

If $C=0$, then $f$ is called $L$-biLipschitz.

  2. (2)

$\eta$-quasisymmetric if $f$ is a homeomorphism and for every distinct points
$x,a,b\in X$, we have

$\frac{d^{\prime}(f(x),f(a))}{d^{\prime}(f(x),f(b))}\leq\eta\left(\frac{d(x,a)}{d(x,b)}\right).$

  3. (3)

$\eta$-quasimöbius if $f$ is a homeomorphism and for any distinct points
$x,y,z,w\in X$, we have

$\frac{d^{\prime}(f(x),f(z))d^{\prime}(f(y),f(w))}{d^{\prime}(f(x),f(y))d^{\prime}(f(z),f(w))}\leq\eta\left(\frac{d(x,z)d(y,w)}{d(x,y)d(z,w)}\right).$

######  Definition 2.3.

Let $L\geq 1$, $0<q<1$ and let $\eta:[0,\infty)\to[0,\infty)$ be a
homeomorphism. A homeomorphism $f:\,\Omega\to\Omega^{\prime}$ between
incomplete, locally complete and rectifiably connected metric spaces is called

  1. (1)

$(\eta,q)$-relative if for each $x\in\Omega$ and for all $y\in
B(x,qd_{\Omega}(x))$, we have

$\frac{d^{\prime}(f(x),f(y))}{d_{\Omega^{\prime}}(f(x))}\leq\eta\left(\frac{d(x,y)}{d_{\Omega}(x)}\right).$

  2. (2)

$\eta$-semisolid if for all $x,y\in\Omega$,
$k_{\Omega^{\prime}}(f(x),f(y))\leq\eta(k_{\Omega}(x,y)).$

  3. (3)

$(L,q)$-locally biLipschitz if for each $x\in\Omega$, there is a constant
$C_{x}$ (depending on the point $x$) such that for all $y,z\in
B(x,qd_{\Omega}(x))$, we have

$L^{-1}C_{x}d(y,z)\leq d^{\prime}(f(y),f(z))\leq LC_{x}d(y,z).$

  4. (4)

$(\eta,q)$-locally quasisymmetric if for every $x\in\Omega$, the restriction
$f|_{B(x,qd_{\Omega}(x))}$ of $f$ on $B(x,qd_{\Omega}(x))$ is
$\eta$-quasisymmetric.

Finally, we record an auxiliary result concerning the quasisymmetry from local
to global between uniform spaces in a quantitative way.

Lemma D. $($[19, Theorem 1.4]$)$ Let $A\geq 1$, $0<q<1$, and let
$\eta:\,[0,\infty)\to[0,\infty)$ be a homeomorphism. Suppose that
$f:\Omega\to\Omega^{\prime}$ is a homeomorphism between bounded, locally
compact, and $A$-uniform spaces. If $f$ and $f^{-1}$ are both
$(\eta,q)$-locally quasisymmetric, then $f$ is $\eta_{0}$-quasisymmetric if
and only if there is a number $C_{0}\geq 1$ and a point $w\in\Omega$ such that
${\rm diam}(\Omega)\leq C_{0}d_{\Omega}(w)$ and ${\rm
diam}(\Omega^{\prime})\leq C_{0}d_{\Omega^{\prime}}^{\prime}(f(w)).$ The
parameters $\eta_{0}$ and $C_{0}$ depend on each other and also on $\eta$,
$q$, and $A$.

##  3\. Proofs of Theorems 1.2 and 1.4

In this section, we assume that $(X,d)$ and $(X^{\prime},d^{\prime})$ are
$c$-quasiconvex and complete metric spaces, that $G\subsetneq X$ and
$G^{\prime}\subsetneq X^{\prime}$ are domains, and that $f:G\to G^{\prime}$ is
a homeomorphism.

###  3.1. Proof of Theorem 1.2

We divide the proof of Theorem 1.2 into several lemmas.

######  Lemma 3.1.

Assume that $L\geq 1,$ $0<\lambda<1$, and $f$ is $\partial$-Lipschitz with
data $(L,\lambda).$ Then $f$ is $(\theta,t_{0})$-relative with $\theta(t)=Lt$
and $t_{0}=\lambda$.

Proof. Let $x\in G$ and $y\in B(x,\lambda d_{G}(x))$. Since $f$ is
$\partial$-Lipschitz, we have

$\frac{d^{\prime}(f(x),f(y))}{d_{G^{\prime}}(f(x))}\leq
L\frac{d(x,y)}{d_{G}(x)},$

which implies that $f$ is $(\theta,t_{0})$-relative with $\theta(t)=Lt$ and
$t_{0}=\lambda$. ∎

######  Lemma 3.2.

Assume that $0<t_{0}\leq 1,$ $c_{1}\geq 1$, and $f$ is
$(\theta,t_{0})$-relative with $\theta(t)=c_{1}t.$ Then $f$ is
$\varphi$-semisolid with $\varphi(t)=c_{2}t$, where $c_{2}$ depends only on
$c_{1}$, $t_{0}$, and $c$.

Proof. Put

$t_{1}=\min\left\\{\log\left(\frac{t_{0}}{2}+1\right),\log\left(1+\frac{1}{3c_{1}c}\right)\right\\}\,\,\,\mbox{and}\,\,\,\,\,\,c_{2}=\frac{24cc_{1}}{t_{1}}.$

We first establish the following:

######  Claim 3.1.

If $x,y\in G$ with $k_{G}(x,y)\leq t_{1}$, then $k_{G^{\prime}}(f(x),f(y))\leq
6cc_{1}k_{G}(x,y)$.

Fix $x,y\in G$ with $k_{G}(x,y)\leq t_{1}$. It follows from Lemma B(1) that

$d(x,y)\leq(e^{t_{1}}-1)d_{G}(x)\leq\min\left\\{\frac{t_{0}}{2},\frac{1}{3cc_{1}}\right\\}d_{G}(x).$

Since $f$ is $(\theta,t_{0})$-relative with $\theta(t)=c_{1}t$, we have

(1)
$\displaystyle\frac{d^{\prime}(f(x),f(y))}{d_{G^{\prime}}(f(x))}\leq\theta\Big{(}\frac{d(x,y)}{d_{G}(x)}\Big{)}=c_{1}\frac{d(x,y)}{d_{G}(x)}\leq\frac{1}{3c}.$

Now we obtain from (1) and Lemma B(2) that

(2) $\displaystyle k_{G^{\prime}}(f(x),f(y))\leq
3c\frac{d^{\prime}(f(x),f(y))}{d_{G^{\prime}}(f(x))}\leq 1.$

Again using Lemma B(2), we see from (1) and (2) that

$k_{G^{\prime}}(f(x),f(y))\leq
3c\frac{d^{\prime}(f(x),f(y))}{d_{G^{\prime}}(f(x))}\leq
3cc_{1}\frac{d(x,y)}{d_{G}(x)}\leq 6cc_{1}k_{G}(x,y),$

and therefore, Claim 3.1 is proved.

Next, we show that $k_{G^{\prime}}(f(x),f(y))\leq c_{2}k_{G}(x,y)$ holds for
all $x,y\in G$.

Fix $x,y\in G$. If $k_{G}(x,y)\leq t_{1}$, then the required assertion follows
from Claim 3.1. So it suffices to consider the case that $k_{G}(x,y)>t_{1}$.
By [10, Lemma 3.9], the space $(G,k_{G})$ is $2$-quasiconvex. Therefore, there
exists a curve $\gamma$ in $G$ joining $x$ and $y$ such that

(3) $\displaystyle\ell_{k_{G}}(\gamma)\leq 2k_{G}(x,y),$

where $\ell_{k_{G}}(\gamma)$ denotes the length of $\gamma$ in the
quasihyperbolic metric. Take a sequence of points $x=x_{0}$, $x_{1}$, $x_{2}$,
$\ldots$, $x_{n-1}$ in $\gamma$ satisfying $k_{G}(x_{i},x_{i-1})=t_{1}$ and
$k_{G}(x_{n-1},y)\leq t_{1},$ for $i=1,2,\ldots,n-1$, and denote $x_{n}=y$.
Then we know from Claim 3.1 that $n\geq 2$ and
$k_{G^{\prime}}(f(x_{i}),f(x_{i-1}))\leq 6cc_{1}t_{1}\leq 6cc_{1}$ for
$i=1,2,\ldots,n$. Thus, one finds that

(4) $\displaystyle
k_{G^{\prime}}(f(x),f(y))\leq\sum\limits_{i=1}^{n}k_{G^{\prime}}(f(x_{i}),f(x_{i-1}))\leq
6cc_{1}n\leq 12cc_{1}(n-1).$

It follows from the inequality (3) that
$(n-1)t_{1}\leq\ell_{k_{G}}(\gamma)\leq 2k_{G}(x,y),$ which, together with
(4), implies $k_{G^{\prime}}(f(x),f(y))\leq 24cc_{1}t_{1}^{-1}k_{G}(x,y).$ The
proof is complete. ∎

######  Lemma 3.3.

If $f$ is $\varphi$-semisolid with $\varphi(t)=c_{2}t$ for some $c_{2}\geq 1$,
then $f$ is $\partial$-Lipschitz with data $(L,\lambda)$, where $L$ and
$\lambda$ depend only on $c_{1}$ and $c$.

Proof. Let $\lambda=1/(36c^{2}c_{2})$ and $L=24cc_{2}$. For all $x\in G$ and
$y,z\in B(x,\lambda d_{G}(x))$, we have by the triangle inequality

$\displaystyle d(y,z)\leq\frac{1}{18c^{2}c_{2}}d_{G}(x),$

and

(5) $\displaystyle d_{G}(y)\geq d_{G}(x)-d(x,y)\geq\frac{1}{2}d_{G}(x).$

Therefore, we have $d(y,z)\leq d_{G}(y)/(9c^{2}c_{2})$. Using Lemma B(2), we
obtain

(6) $\displaystyle k_{G}(y,z)\leq
3c\frac{d(y,z)}{d_{G}(y)}\leq\frac{1}{3cc_{2}}.$

Because $y\in B(x,\lambda d_{G}(x))$, Lemma B(2) leads to

$\displaystyle k_{G}(x,y)\leq
3c\frac{d(x,y)}{d_{G}(x)}\leq\frac{1}{12cc_{2}}.$

Since $f$ is $\varphi$-semisolid, we see from (6) that
$k_{G^{\prime}}(f(x),f(y))\leq c_{2}k_{G}(x,y)\leq 1/(12c)$ and
$k_{G^{\prime}}(f(y),f(z))\leq c_{2}k_{G}(y,z)\leq 1/(3c).$ Together with
Lemma B(2), these two inequalities imply that

(7) $\displaystyle\frac{d^{\prime}(f(y),f(z))}{d_{G^{\prime}}(f(y))}\leq
2k_{G^{\prime}}(f(y),f(z)),$

and

$\displaystyle\frac{d^{\prime}(f(x),f(y))}{d_{G^{\prime}}(f(x))}\leq
2k_{G^{\prime}}(f(x),f(y))\leq\frac{1}{6c}.$

Therefore, we have

$d_{G^{\prime}}(f(y))\leq
d_{G^{\prime}}(f(x))+d^{\prime}(f(x),f(y))\leq(1+1/(6c))d_{G^{\prime}}(f(x))\leq
2d_{G^{\prime}}(f(x)).$

Then it follows from (5), (6), (7) and the semisolidity of $f$ that

$\displaystyle\frac{d^{\prime}(f(y),f(z))}{d_{G^{\prime}}(f(x))}$
$\displaystyle\leq$
$\displaystyle\frac{2d^{\prime}(f(y),f(z))}{d_{G^{\prime}}(f(y))}\leq
4k_{G^{\prime}}(f(y),f(z))$ $\displaystyle\leq$ $\displaystyle
4c_{2}k_{G}(y,z)\leq 12cc_{2}\frac{d(y,z)}{d_{G}(y)}\leq
24cc_{2}\frac{d(y,z)}{d_{G}(x)}.$

The proof of the lemma is complete. ∎

### Proof of Theorem 1.2

The implications (1)$\Rightarrow$ (2)$\Rightarrow$ (3)$\Rightarrow$ (1) in
Theorem 1.2 follow from Lemmas 3.1, 3.2, and 3.3, respectively. ∎

###  3.2. Proof of Theorem 1.4

Before the proof, we establish some results.

######  Lemma 3.4.

Let $0<t_{0}\leq 1$ and $c_{1}\geq 1$. If both $f$ and $f^{-1}$ are
$(\theta,t_{0})$-relative with $\theta(t)=c_{1}t$, then both $f$ and the
inverse $f^{-1}$ are $(L_{1},\vartheta_{1})$-locally biLipschitz, where
$L_{1}$ and $\vartheta_{1}$ depend only on $t_{0}$ and $c_{1}$.

Proof. Let $\vartheta_{1}=t_{0}/(8c_{1})$ and $L_{1}=4c_{1}$. We only need to
show that $f$ is $(L_{1},\vartheta_{1})$-locally Lipschitz, because the other
direction for $f^{-1}$ follows from a similar argument. To this end, for all
$x\in G$ and $y,z\in B(x,\vartheta_{1}d_{G}(x))$, we have

(8) $\displaystyle\frac{1}{2}d_{G}(x)\leq d_{G}(y)\leq 2d_{G}(x),$

and

$d(y,z)\leq
2\vartheta_{1}d_{G}(x)\leq\frac{2\vartheta_{1}}{1-\vartheta_{1}}d_{G}(y)<t_{0}d_{G}(y).$

Since $f$ is $(\theta,t_{0})$-relative, we obtain

(9) $\displaystyle\frac{d^{\prime}(f(y),f(z))}{d_{G^{\prime}}(f(y))}\leq
c_{1}\frac{d(y,z)}{d_{G}(y)}$

and

(10)
$\max\left\\{\frac{d^{\prime}(f(x),f(y))}{d_{G^{\prime}}(x)},\frac{d^{\prime}(f(x),f(z))}{d_{G^{\prime}}(x)}\right\\}\leq
c_{1}\max\left\\{\frac{d(x,y)}{d_{G}(x)},\frac{d(x,z)}{d_{G}(x)}\right\\}\leq
c_{1}\vartheta_{1}\leq\frac{1}{2}.$

It follows that

(11) $\displaystyle\frac{1}{2}d_{G^{\prime}}(f(x))\leq
d_{G^{\prime}}(f(y))\leq 2d_{G^{\prime}}(f(x)).$

By (10) and (11), we get

$\displaystyle d^{\prime}(f(z),f(y))$ $\displaystyle\leq$ $\displaystyle
d^{\prime}(f(x),f(z))+d^{\prime}(f(x),f(y))$ $\displaystyle\leq$
$\displaystyle 2c_{1}\vartheta_{1}d_{G^{\prime}}(f(x))\leq
4c_{1}\vartheta_{1}d_{G^{\prime}}(f(y))<t_{0}d_{G^{\prime}}(f(y)).$

On the other hand, since $f^{-1}$ is $(\theta,t_{0})$-relative, we have

$\frac{d(y,z)}{d_{G}(y)}\leq
c_{1}\frac{d^{\prime}(f(y),f(z))}{d_{G^{\prime}}(f(y))}.$

Hence we obtain from (8) and (11) that

(12) $\displaystyle\frac{d(y,z)}{d_{G}(x)}\leq
4c_{1}\frac{d^{\prime}(f(y),f(z))}{d_{G^{\prime}}(f(x))}.$

By (8), (9), and (11), we have

$\displaystyle\frac{d^{\prime}(f(y),f(z))}{d_{G^{\prime}}(f(x))}\leq
4c_{1}\frac{d(y,z)}{d_{G}(x)},$

which, together with (12), implies

$\frac{1}{4c_{1}}C_{x}d(y,z)\leq d^{\prime}(f(y),f(z))\leq 4c_{1}C_{x}d(y,z),$

where $C_{x}=d_{G^{\prime}}(f(x))/d_{G}(x)$. The proof is complete. ∎

######  Lemma 3.5.

Let $0<\vartheta_{1}<1$ and $L_{1}\geq 1$. If $f$ is
$(L_{1},\vartheta_{1})$-locally biLipschitz, then $f$ is $(\eta,q)$-locally
quasisymmetric with $\eta(t)=c_{2}t$, where $c_{2}$ and $q$ depend only on
$\vartheta_{1}$ and $L_{1}$.

Proof. Let $q=\vartheta_{1}$ and $\eta(t)=L_{1}^{2}t$. Fix $u\in G$ and three
distinct points $x,y,z\in B(u,qd_{G}(u))$. By the assumption, we have

$\frac{C_{u}}{L_{1}}d(x,y)\leq d^{\prime}(f(x),f(y))\leq L_{1}C_{u}d(x,y),$

and

$\frac{C_{u}}{L_{1}}d(x,z)\leq d^{\prime}(f(x),f(z))\leq L_{1}C_{u}d(x,z).$

Hence we obtain

$\frac{d^{\prime}(f(x),f(y))}{d^{\prime}(f(x),f(z))}\leq
L_{1}^{2}\frac{d(x,y)}{d(x,z)},$

which completes the proof. ∎

######  Lemma 3.6.

Let $0<q<1$ and $c_{2}\geq 1$. If both $f$ and $f^{-1}$ are $(\eta,q)$-locally
quasisymmetric with $\eta(t)=c_{2}t$, then $f$ is $\partial$-biLipschitz with
data $L$ and $\lambda$, where $L$ and $\lambda$ depend only on $q$, $c_{2}$
and $c$.

Proof. Let $q_{1}=\min\\{1/(2+c),q/2\\}$, $L=8cc_{2}/q_{1}$ and
$\lambda=q_{1}/(2cc_{2})$. We need to prove that $f$ is $\partial$-Lipschitz
with data $(L,\lambda)$, because the other direction follows from a similar
argument. For all $x\in G$ and $y,z\in B(x,\lambda d_{G}(x))$, we have
$2^{-1}d_{G}(x)\leq d_{G}(y)\leq 2d_{G}(x).$ Let $u^{\prime}\in\partial
G^{\prime}$ be such that $d^{\prime}(f(x),u^{\prime})\leq
2d_{G^{\prime}}(f(x)).$ Since $X^{\prime}$ is $c$-quasiconvex, there exists a
curve $\gamma_{1}^{\prime}$ in $X^{\prime}$ joining $u^{\prime}$ and $f(x)$
such that

(13) $\displaystyle\ell(\gamma_{1}^{\prime})\leq
cd^{\prime}(f(x),u^{\prime})\leq 2cd_{G^{\prime}}(f(x)).$

It follows from Lemma C that

(14) $\displaystyle B(x,q_{1}d_{G}(x))\subset G.$

So there is a point $u_{1}^{\prime}$ belonging to $\gamma_{1}^{\prime}\cap
f(S(x,q_{1}d_{G}(x)))$. We see from (13) that

(15) $\displaystyle
d^{\prime}(f(x),u_{1}^{\prime})\leq\ell(\gamma_{1}^{\prime})\leq
cd^{\prime}(f(x),u^{\prime})\leq 2cd_{G^{\prime}}(f(x)).$

Because $f$ is $(\eta,q)$-locally quasisymmetric, we have

$\frac{d^{\prime}(f(x),f(y))}{d^{\prime}(f(x),u_{1}^{\prime})}\leq
c_{2}\frac{d(x,y)}{d(x,u_{1})}\leq c_{2}\frac{\lambda
d_{G}(x)}{q_{1}d_{G}(x)}=\frac{c_{2}\lambda}{q_{1}},$

where $u_{1}=f^{-1}(u_{1}^{\prime})$. Using (15), we obtain

$d^{\prime}(f(x),f(y))\leq\frac{c_{2}\lambda}{q_{1}}d^{\prime}(f(x),u_{1}^{\prime})\leq\frac{2cc_{2}\lambda}{q_{1}}d_{G^{\prime}}(f(x))\leq\frac{1}{2}d_{G^{\prime}}(f(x)),$

which implies that

(16) $\displaystyle\frac{1}{2}d_{G^{\prime}}(f(x))\leq
d_{G^{\prime}}(f(y))\leq 2d_{G^{\prime}}(f(x)).$

Choose $u_{2}^{\prime}\in\partial G^{\prime}$ such that
$d^{\prime}(f(y),u_{2}^{\prime})\leq 2d_{G^{\prime}}(f(y)).$ The
quasiconvexity of $X^{\prime}$ ensures that there exists a curve
$\gamma_{2}^{\prime}$ in $X^{\prime}$ joining $f(y)$ and $u_{2}^{\prime}$ such
that

(17) $\displaystyle\ell(\gamma_{2}^{\prime})\leq
cd^{\prime}(f(y),u_{2}^{\prime})\leq 2cd_{G^{\prime}}(f(y)).$

By (14), there is a point $u_{3}^{\prime}$ lying on $\gamma_{2}^{\prime}\cap
f(S(x,q_{1}d_{G}(x)))$. Then it follows from (17) that
$d^{\prime}(f(y),u_{3}^{\prime})\leq 2cd_{G^{\prime}}(f(y)).$ Furthermore, by
the local quasisymmetry of $f$ and by (16), we have

$\displaystyle\frac{d^{\prime}(f(y),f(z))}{d_{G^{\prime}}(f(x))}$
$\displaystyle\leq$
$\displaystyle\frac{2d^{\prime}(f(y),f(z))}{d_{G^{\prime}}(f(y))}$
$\displaystyle\leq$ $\displaystyle
4c\frac{d^{\prime}(f(y),f(z))}{d^{\prime}(f(y),u_{3}^{\prime})}\leq
4cc_{2}\frac{d(y,z)}{d(u_{3},y)}$ $\displaystyle\leq$ $\displaystyle
4cc_{2}\frac{d(y,z)}{(q_{1}-\lambda)d_{G}(x)}\leq\frac{8cc_{2}}{q_{1}}\frac{d(y,z)}{d_{G}(x)}.$

Hence we complete the proof of this lemma. ∎

Proof of Theorem 1.4. The implications
(1)$\Rightarrow$(2)$\Rightarrow$(3)$\Rightarrow$(1) follow from Theorem 1.2
directly. From Lemma 3.4 we obtain (3)$\Rightarrow$(4). The implication
(4)$\Rightarrow$(5) follows from Lemma 3.5. By Lemma 3.6, we get the
implication (5)$\Rightarrow$(1). The proof of Theorem 1.4 is complete. ∎

##  4\. Proof of Theorem 1.6

###  4.1. Gromov hyperbolic spaces

Let $(X,d)$ be a metric space. Fix a base point $w\in X$. For $x,y\in X$, the
number

$(x|y)_{w}=\frac{1}{2}\big{(}d(x,w)+d(y,w)-d(x,y)\big{)}$

is called the Gromov product of $x,y$ with respect to $w$. We say that $X$ is
a Gromov hyperbolic space (or a $\delta$-hyperbolic space) if there is a
constant $\delta\geq 0$ such that
$(x|y)_{w}\geq\min\\{(x|z)_{w},(z|y)_{w}\\}-\delta$ for all $x,y,z,w\in X$.
For more information about Gromov hyperbolic spaces see [1, 4, 19]. In
particular, a direct computation gives the following elementary fact: For all
$x,y,z,w,o\in X$, we have

(18)
$(x|y)_{o}+(z|u)_{o}-(x|z)_{o}-(y|u)_{o}=(x|y)_{w}+(z|u)_{w}-(x|z)_{w}-(y|u)_{w}.$

######  Definition 4.1.

Let $X$ be a proper (i.e., all closed balls are compact), geodesic,
$\delta$-hyperbolic space, and $K\geq 0$. Following [1], we say that $X$ is
$K$-roughly starlike with respect to a distinguished point $w\in X$ if for
each $x\in X$, there is a geodesic ray $\gamma$ emanating from $w$ such that
${\rm dist}(x,\gamma)\leq K$.

Lemma E. $($[1]$)$ Let $(\Omega,d)$ be a locally compact and $A$-uniform
metric space and $k$ its quasihyperbolic metric. Then there are constants
$\delta,K\geq 0$ depending only on $A$ so that $(\Omega,k)$ is a proper,
geodesic $\delta$-hyperbolic metric space. If, in addition, $\Omega$ is
bounded, then $(\Omega,k)$ is $K$-roughly starlike with respect to $w$, where
$w$ is a point in $\Omega$ satisfying
$d_{\Omega}(w)=\max_{x\in\Omega}d_{\Omega}(x)$.

###  4.2. Bonk-Heinonen-Koskela’s uniformization

Let $(\Omega,d)$ be a locally compact, rectifiably connected and incomplete
metric space, where the identity mapping
${\operatorname{id}}:(\Omega,d)\to(\Omega,\ell)$ is a homeomorphism. Let $k$
be the quasihyperbolic metric of $\Omega$ and $w\in\Omega$. In [1], Bonk et
al. introduced a family of conformal deformations of $(\Omega,k)$ by the
densities $\rho_{w,\varepsilon}(x)=e^{-\varepsilon k(x,w)}$ $(\varepsilon>0).$
For all $x$, $y\in\Omega$, we define

(19) $d_{w,\varepsilon}(x,y)=\inf\int_{\gamma}\rho_{w,\varepsilon}\;ds_{k},$

where $ds_{k}$ denotes the arc-length element with respect to the metric $k$
and the infimum is taken over all rectifiable curves $\gamma$ in $\Omega$ with
endpoints $x$ and $y$. Then $d_{w,\varepsilon}$ are metrics on $\Omega$, and
we denote the resulting metric spaces by
$\Omega_{w,\varepsilon}=(\Omega,d_{w,\varepsilon})$.

Now, we recall some useful facts concerning Bonk-Heinonen-Koskela’s
uniformization.

Lemma F. $($[1]$)$ If $(\Omega,k)$ is $\delta$-hyperbolic and $K$-roughly
starlike with respect to $w\in\Omega$, then there are constants $A,C\geq
1,\varepsilon\in(0,1)$ that depend only on $\delta$ and a constant
$M=M(\delta,K)\geq 1$ such that

  1. (a)

$\Omega_{w,\varepsilon}$ is $A$-uniform and bounded (with diameter at most
$2/\varepsilon$).

  2. (b)

The identity mapping $(\Omega,k)\to(\Omega,k_{w,\varepsilon})$ is
$M$-biLipschitz, where $k_{w,\varepsilon}$ is the quasihyperbolic metric of
$\Omega_{w,\varepsilon}$.

  3. (c)

For all $x,y\in\Omega$, we have

$C^{-1}d_{w,\varepsilon}(x,y)\leq\varepsilon^{-1}e^{-\varepsilon(x|y)_{w}}\min\\{1,\varepsilon
k(x,y)\\}\leq Cd_{w,\varepsilon}(x,y).$

######  Lemma 4.2.

Let $(\Omega,d)$ be a locally compact and $A$-uniform space. Suppose that
$d_{w_{0},\varepsilon}$ and $d_{w_{1},\varepsilon}$ are metrics defined as in
(19) with respect to the base points $w_{0}$ and $w_{1}$, respectively, where
$\varepsilon=\varepsilon(A)\in(0,1)$ is a parameter. Then the identity mapping
$(\Omega,d_{w_{0},\varepsilon})\to(\Omega,d_{w_{1},\varepsilon})$ is
$\theta_{1}$-quasimöbius, where $\theta_{1}$ depends only on $A$.

Proof. It follows from Lemma E, Lemma F(c) and (18) that there is a constant
$C$ depending only on $A$ such that for any $x,y,z,u\in\Omega$,

$\displaystyle\frac{d_{w_{1},\varepsilon}(x,y)d_{w_{1},\varepsilon}(z,u)}{d_{w_{1},\varepsilon}(x,z)d_{w_{1},\varepsilon}(y,u)}$
$\displaystyle\leq$ $\displaystyle
C^{4}\frac{e^{-\varepsilon(x|y)_{w_{1}}}\min\\{1,\varepsilon
k(x,y)\\}e^{-\varepsilon(z|u)_{w_{1}}}\min\\{1,\varepsilon
k(z,u)\\}}{e^{-\varepsilon(x|z)_{w_{1}}}\min\\{1,\varepsilon
k(x,z)\\}e^{-\varepsilon(y|u)_{w_{1}}}\min\\{1,\varepsilon k(y,u)\\}}$
$\displaystyle=$ $\displaystyle
C^{4}\frac{e^{-\varepsilon(x|y)_{w_{0}}}\min\\{1,\varepsilon
k(x,y)\\}e^{-\varepsilon(z|u)_{w_{0}}}\min\\{1,\varepsilon
k(z,u)\\}}{e^{-\varepsilon(x|z)_{w_{0}}}\min\\{1,\varepsilon
k(x,z)\\}e^{-\varepsilon(y|u)_{w_{0}}}\min\\{1,\varepsilon k(y,u)\\}}$
$\displaystyle\leq$ $\displaystyle
C^{8}\frac{d_{w_{0},\varepsilon}(x,y)d_{w_{0},\varepsilon}(z,u)}{d_{w_{0},\varepsilon}(x,z)d_{w_{0},\varepsilon}(y,u)},$

which implies that the identity mapping
$(\Omega,d_{w_{0},\varepsilon})\to(\Omega,d_{w_{1},\varepsilon})$ is
$\theta_{1}$-quasimöbius, where $\theta_{1}(t)=C^{8}t$. ∎

###  4.3. Proof of Theorem 1.6

In this subsection, we assume that $A\geq 1$, $0<q<1$, and
$\eta:\,[0,\infty)\to[0,\infty)$ is a homeomorphism. Let $\Omega$ and
$\Omega^{\prime}$ be locally compact and $A$-uniform spaces, and let $k$ and
$k^{\prime}$ be the quasihyperbolic metrics of $\Omega$ and $\Omega^{\prime}$,
respectively. Suppose that $f:\Omega\to\Omega^{\prime}$ is a homeomorphism
such that both $f$ and its inverse $f^{-1}$ are $(\eta,q)$-locally
quasisymmetric.

######  Lemma 4.3.

The mapping $f$ is quasi-isometric with respect to the quasihyperbolic
metrics.

Proof. For all $x,y\in\Omega$, we prove that there exist constants $L_{1}$ and
$C_{1}$ such that

$L_{1}^{-1}k(x,y)-C_{1}\leq k^{\prime}(f(x),f(y))\leq L_{1}k(x,y)+C_{1}.$

It suffices to check the right hand side inequality, because the left hand
side inequality follows from a similar argument. Moreover, [16, Theorem 2.2]
tells us that we only need to show that if $x,y\in\Omega$ with $k(x,y)\leq
t_{1}$, then we have

(20) $\displaystyle k^{\prime}(f(x),f(y))\leq 4A^{2}\log 2,$

where

$q_{1}=\frac{q}{2}\,\,\,\;\;\mbox{and}\;\;\,\,\,t_{1}=\min\left\\{\log\left(1+\eta^{-1}\Big{(}\frac{1}{4A}\Big{)}q_{1}\right),\log\Big{(}1+\frac{q}{2}\Big{)}\right\\}.$

To this end, we fix two points $x,y\in\Omega$ with $k(x,y)\leq t_{1}$. By
Lemma B(1), we have

(21) $\displaystyle d(x,y)\leq(e^{t_{1}}-1)d_{\Omega}(x).$

Let $u^{\prime}\in\partial\Omega^{\prime}$ be such that
$d^{\prime}(f(x),u^{\prime})\leq 2d_{\Omega^{\prime}}(f(x)).$ Because
$\Omega^{\prime}$ is $A$-uniform and locally compact, it follows from Lemma E
that $(\Omega^{\prime},k^{\prime})$ is a proper, geodesic and Gromov
hyperbolic space. Applying the Arzela-Ascoli Theorem, we know that there
exists a quasihyperbolic geodesic ray $\gamma^{\prime}$ in $\Omega^{\prime}$
joining $f(x)$ and $u^{\prime}$ that is an $A_{1}$-uniform curve by using [1,
Theorem 2.10], where $A_{1}$ depends only on $A$. For convenience, we assume
that $\gamma^{\prime}$ is an $A$-uniform curve by taking $A$ to be larger.
Hence we obtain

(22) $\displaystyle\ell(\gamma^{\prime})\leq Ad^{\prime}(f(x),u^{\prime})\leq
2Ad_{\Omega^{\prime}}(f(x)).$

Then there is a point $z^{\prime}$ belonging to $\gamma^{\prime}\cap
f(S(x,q_{1}d_{\Omega}(x)))$. Let $z=f^{-1}(z^{\prime})$. Thus, by the local
quasisymmetry of $f$, we see from (21) that

(23)
$\displaystyle\frac{d^{\prime}(f(x),f(y))}{d^{\prime}(f(x),z^{\prime})}\leq\eta\Big{(}\frac{d(x,y)}{d(x,z)}\Big{)}\leq\eta\Big{(}\frac{e^{t_{1}}-1}{q_{1}}\Big{)}\leq\frac{1}{4A},$

where the last inequality follows from the choice of $t_{1}$. By (22), we have

$d^{\prime}(f(x),z^{\prime})\leq\ell(\gamma^{\prime})\leq
2Ad_{\Omega^{\prime}}(f(x)),$

which, together with (23), shows that $d^{\prime}(f(x),f(y))\leq
d_{\Omega^{\prime}}(f(x))/2.$ Then it follows from [1, (2.16)] that

$k^{\prime}(f(x),f(y))\leq
4A^{2}\log\Big{(}1+\frac{d^{\prime}(f(x),f(y))}{\min\\{d_{\Omega^{\prime}}(f(x)),d_{\Omega^{\prime}}(f(y))\\}}\Big{)}\leq
4A^{2}\log 2,$

which implies (20). The proof of Lemma 4.3 is complete. ∎

For proving Theorem 1.6, we consider four cases in terms of the boundedness of
the spaces $\Omega$ and $\Omega^{\prime}$.

######  Lemma 4.4.

Suppose that both $\Omega$ and $\Omega^{\prime}$ are bounded. Then $f$ is
$\theta$-quasimöbius, where $\theta$ depends only on $A$, $q$, and $\eta$.

Proof. Let $w_{0}\in\Omega$ and $w_{1}^{\prime}\in\Omega^{\prime}$ be such
that

$\displaystyle
d_{\Omega}(w_{0})=\max_{x\in\Omega}d_{\Omega}(x)\;\;\mbox{and}\;\;d_{\Omega^{\prime}}(w_{1}^{\prime})=\max_{x^{\prime}\in\Omega^{\prime}}d_{\Omega^{\prime}}(x^{\prime}),$

respectively. We first check that there is a constant $C$ depending on $A$
such that

(24) $\displaystyle{\rm diam}(\Omega)\leq
Cd_{\Omega}(w_{0})\;\;\mbox{and}\;\;{\rm diam}(\Omega^{\prime})\leq
Cd_{\Omega^{\prime}}(w_{1}^{\prime}).$

For this, we only need to prove that ${\rm diam}(\Omega)\leq
Cd_{\Omega}(w_{0})$, because the second estimation follows from a similar
argument. Take $u_{1},u_{2}\in\Omega$ satisfying $d(u_{1},u_{2})\geq{\rm
diam}(\Omega)/2.$ Since $\Omega$ is $A$-uniform, we see that there is an
$A$-uniform curve $\gamma$ joining $u_{1}$ and $u_{2}$. Let $u\in\gamma$ be
such that $\ell(\gamma[u_{1},u])=\ell(\gamma[u,u_{2}])$. Therefore, we have
$\ell(\gamma)\leq 2Ad_{\Omega}(u)\leq 2Ad_{\Omega}(w_{0}),$ which guarantees
that ${\rm diam}(\Omega)\leq 2\ell(\gamma)\leq 4Ad_{\Omega}(w_{0}).$ Hence,
(24) holds.

It follows from Lemma E that $(\Omega,k)$ and $(\Omega^{\prime},k^{\prime})$
are $\delta$-hyperbolic spaces, where $\delta$ depends only on $A$. Let
$\Omega_{w_{0},\varepsilon}=(\Omega,d_{w_{0},\varepsilon})$ be the conformal
deformation of $(\Omega,k)$ based at $w_{0}$ with parameter
$0<\varepsilon\leq\varepsilon_{0}(A)$ which is defined as in (19). Put
$w_{1}=f^{-1}(w_{1}^{\prime})$. Similarly, let
$\Omega_{w_{1},\varepsilon}=(\Omega,d_{w_{1},\varepsilon})$ be the conformal
deformation of $(\Omega,k)$ based at $w_{1}$ with parameter $\varepsilon$. It
follows from Lemma F that $\Omega_{w_{0},\varepsilon}$ and
$\Omega_{w_{1},\varepsilon}$ are $A_{1}$-uniform, where $A_{1}$ depends only
on $A$. Moreover, by [1, (4.6)] and Lemma F(a), we have

(25) $\displaystyle d_{w_{0},\varepsilon}(w_{0})\geq\frac{1}{\varepsilon
e}\rho_{w_{0},\varepsilon}(w_{0})=\frac{1}{\varepsilon e}\geq\frac{1}{2e}{\rm
diam}(\Omega_{w_{0},\varepsilon})$

and similarly,

(26) $\displaystyle d_{w_{1},\varepsilon}(w_{1})\geq\frac{1}{2e}{\rm
diam}(\Omega_{w_{1},\varepsilon}).$

To prove that $f:(\Omega,d)\to(\Omega^{\prime},d^{\prime})$ is
$\theta$-quasimöbius, we need the following:

  1. I:

The identity mapping $\varphi_{1}:(\Omega,d)\to(\Omega,d_{w_{0},\varepsilon})$
is $\eta_{1}$-quasisymmetric.

  2. II:

It follows from Lemma 4.2 that the identity mapping
$\varphi_{2}:(\Omega,d_{w_{0},\varepsilon})\to(\Omega,d_{w_{1},\varepsilon})$
is $\theta_{0}$-quasimöbius with $\theta_{0}$ depending only on $A$.

  3. III:

The induced mapping
$g:=f\circ\varphi_{1}^{-1}\circ\varphi_{2}^{-1}:(\Omega,d_{w_{1},\varepsilon})\to(\Omega^{\prime},d^{\prime})$
is $\eta_{2}$-quasisymmetric.

After that, we see that $f=g\circ\varphi_{2}\circ\varphi_{1}$ is
$\theta$-quasimöbius, because the composition of quasisymmetric mapping and
quasimöbius mapping is also quasimöbius.

So in the remaining part, we only need to prove the assertions I and III.

For I, we check that the identity mapping
$\varphi_{1}:(\Omega,d)\to(\Omega,d_{w_{0},\varepsilon})$ satisfies the
assumptions of Lemma D. By (24) and (25), it suffices to demonstrate that both
$\varphi_{1}$ and $\varphi_{1}^{-1}$ are locally quasisymmetric. Note that
this desired claim follows from Lemma F and Theorem 1.4. Therefore, by Lemma
D, the assertion I is valid.

Finally, it remains to prove III. We shall use Lemma D to show that $g$ is
$\eta_{2}$-quasisymmetric. From (24) and (26), we only need to verify that
both $g$ and $g^{-1}$ are locally quasisymmetric. Since $f$ and $f^{-1}$ are
both $(\eta,q)$-locally quasisymmetric, we are left to demonstrate that the
identity mapping

$\varphi_{2}\circ\varphi_{1}:(\Omega,d)\to(\Omega,d_{w_{1},\varepsilon})$

and its inverse are both locally quasisymmetric. This is because the
composition of locally quasisymmetric mappings is also locally quasisymmetric,
see [10, Theorem 1.12].

This can be seen as follows. Because $\Omega^{\prime}$ is a bounded
$A$-uniform metric space, it follows from Lemma E that
$(\Omega^{\prime},k^{\prime})$ is $K^{\prime}$-roughly starlike with respect
to $w_{1}^{\prime}$, where $K^{\prime}$ depends only on $A$. Moreover, because
the rough starlikeness is preserved under quasi-isometric mappings by [19,
Lemma 4.10], we see from Lemma 4.3 that $(\Omega,k)$ is $K$-roughly starlike
with respect to $w_{1}$, where $K$ depends only on $A$, $q$, and $\eta$. This
fact, combined with Lemma F(b), ensures that the identity mapping
$\varphi_{2}\circ\varphi_{1}:(\Omega,d)\to(\Omega,d_{w_{1},\varepsilon})$ is
$M_{1}$-biLipschitz with respect to the quasihyperbolic metrics, where $M_{1}$
depends only on $A$, $q$, and $\eta$. It follows from Theorem 1.4 that both
$\varphi_{2}\circ\varphi_{1}$ and its inverse are $(\eta_{1},q_{1})$-locally
quasisymmetric, where $q_{1}$ and $\eta_{1}$ depend on $A$, $q$, and $\eta$.
The proof is complete. ∎

Before proceeding, we introduce the sphericalization of metric spaces which
were investigated by Bonk et al. in [2] and Buckley et al. in [3].

Lemma G. Suppose that $\Omega$ is an unbounded $A$-uniform space with
$p\in\partial\Omega$, and that $(\Omega,\hat{d}_{p})$ is the sphericalization
of $(\Omega,d)$ associated to the point $p$. Then the following statements
hold:

  1. (1)

$($[3, Subsection 3.B]$)$ The space $(\Omega,\hat{d}_{p})$ is a bounded metric
space;

  2. (2)

$($[3, Subsection 3.B] and [3, Theorem 4.11]$)$ The identity mapping
$(\Omega,d)\to(\Omega,\hat{d}_{p})$ is $\eta_{0}$-quasimöbius and
$M$-biLipschitz with respect to the quasihyperbolic metrics, where
$\eta_{0}(t)=16t$ and $M=80A$;

  3. (3)

$($[3, Theorem 5.5]$)$ The space $(\Omega,\hat{d}_{p})$ is $A_{1}$-uniform,
where $A_{1}$ depends only on $A$.

######  Lemma 4.5.

Suppose that both $\Omega$ and $\Omega^{\prime}$ are unbounded. Then $f$ is
$\theta$-quasimöbius, where $\theta$ depends only on $A$, $q$, and $\eta$.

Proof. Let $p\in\partial\Omega$ and $p^{\prime}\in\partial\Omega^{\prime}$.
Suppose that $(\Omega,\hat{d}_{p})$ and
$(\Omega^{\prime},\hat{d^{\prime}}_{p^{\prime}})$ are the sphericalizations of
$(\Omega,d)$ and $(\Omega^{\prime},d^{\prime})$ associated to the points $p$
and $p^{\prime}$, respectively. Lemma G(1) ensures that $(\Omega,\hat{d}_{p})$
and $(\Omega^{\prime},\hat{d^{\prime}}_{p^{\prime}})$ are bounded metric
spaces. Since $\Omega$ and $\Omega^{\prime}$ are $A$-uniform, Lemma G(3)
implies that $(\Omega,\hat{d}_{p})$ and
$(\Omega^{\prime},\hat{d^{\prime}}_{p^{\prime}})$ are $A_{1}$-uniform, where
$A_{1}$ depends only on $A$.

By Lemma G(3), we know that the identity mappings
$\phi_{1}:\,(\Omega,d)\to(\Omega,\hat{d}_{p})$ and
$\phi_{2}:\,(\Omega^{\prime},d^{\prime})\to(\Omega^{\prime},\hat{d^{\prime}}_{p^{\prime}})$
are $M$-biLipschitz with respect to the quasihyperbolic metrics, where $M$
depends only on $A$. Moreover, it follows from Theorem 1.4 that $\phi_{1}$,
$\phi_{2}$ and their inverses $\phi_{1}^{-1}$, $\phi_{2}^{-1}$ are
$(\eta_{1},q_{1})$-locally quasisymmetric, where $\eta_{1}$ and $q_{1}$ depend
only on $A$. Thus, $f$ induces a mapping

$g:=\phi_{2}\circ
f\circ\phi_{1}^{-1}:\,(\Omega,\hat{d}_{p})\to(\Omega^{\prime},\hat{d^{\prime}}_{p^{\prime}}).$

Since $f$ and $f^{-1}$ are both $(\eta,q)$-locally quasisymmetric, one finds
that $g$ and its inverse $g^{-1}$ are $(\eta_{2},q_{2})$-locally
quasisymmetric, where $\eta_{2}$ and $q_{2}$ depend only on $\eta$, $q$, and
$A$. This is because the composition of locally quasisymmetric mappings is
also locally quasisymmetric, see [10, Theorem 1.12]. Now, it follows from
Lemma 4.4 that $g$ is $\theta_{1}$-quasimöbius, where $\theta_{1}$ depends
only on $\eta$, $q$ and $A$.

Finally, Lemma G(2) asserts that the identity mappings
$\phi_{1}:\,(\Omega,d)\to(\Omega,\hat{d}_{p})$ and
$\phi_{2}:\,(\Omega^{\prime},d^{\prime})\to(\Omega^{\prime},\hat{d^{\prime}}_{p^{\prime}})$
are $\theta$-quasimöbius with $\eta_{0}(t)=16t$. Hence, $f$ is
$\theta$-quasimöbius with $\theta$ depending only on $\eta$, $q$, and $A$,
because the composition of quasimöbius mappings is also quasimöbius. The proof
of the lemma is complete. ∎

######  Lemma 4.6.

Suppose that $\Omega$ is unbounded and $\Omega^{\prime}$ is bounded. Then $f$
is $\theta$-quasimöbius, where $\theta$ depends only on $A$, $q$, and $\eta$.

Proof. Let $p\in\partial\Omega$. Suppose that $(\Omega,\hat{d}_{p})$ is the
sphericalization of $(\Omega,d)$ associated to the point $p$. Since $\Omega$
is $A$-uniform, one observes from Lemma G(1) and (3) that
$(\Omega,\hat{d}_{p})$ is a bounded $A_{1}$-uniform space, where $A_{1}$
depends only on $A$.

By Lemma G(2), we know that the identity mapping
$\phi_{1}:\,(\Omega,d)\to(\Omega,\hat{d}_{p})$ is $M$-biLipschitz with respect
to the quasihyperbolic metrics, where $M$ depends only on $A$. Then it follows
from Theorem 1.4 that $\phi_{1}$ and its inverse $\phi_{1}^{-1}$ are
$(\eta_{1},q_{1})$-locally quasisymmetric, where $\eta_{1}$ and $q_{1}$ depend
only on $A$. Therefore, $f$ induces a mapping

$g:=f\circ\phi_{1}^{-1}:\,(\Omega,\hat{d}_{p})\to(\Omega^{\prime},d^{\prime}).$

Since $f$ and $f^{-1}$ are both $(\eta,q)$-locally quasisymmetric, we see that
$g$ and its inverse $g^{-1}$ are $(\eta_{2},q_{2})$-locally quasisymmetric,
where $\eta_{2}$ and $q_{2}$ depend only on $\eta$, $q$, and $A$. This is
because the composition of locally quasisymmetric mappings is also locally
quasisymmetric, see [10, Theorem 1.12]. Now, it follows from Lemma 4.4 that
$g$ is $\theta_{1}$-quasimöbius, where $\theta_{1}$ depends only on $\eta$,
$q$ and $A$.

Finally, Lemma G(2) tells us that the identity mapping
$\phi_{1}:\,(\Omega,d)\to(\Omega,\hat{d}_{p})$ is $\theta$-quasimöbius with
$\eta_{0}(t)=16t$. Hence, $f$ is $\theta$-quasimöbius with $\theta$ depending
only on $\eta$, $q$, and $A$, because the composition of quasimöbius mappings
is also quasimöbius. The proof of the lemma is complete. ∎

######  Lemma 4.7.

Suppose that $\Omega$ is bounded and $\Omega^{\prime}$ is unbounded. Then $f$
is $\theta$-quasimöbius, where $\theta$ depends only on $A$, $q$, and $\eta$.

Proof. The proof of the lemma is similar to that of Lemma 4.6 by exchanging
the roles of $\Omega$ and $\Omega^{\prime}$. ∎

Proof of Theorem 1.6. Theorem 1.6 follows from Lemmas 4.4 – 4.7. ∎

## References

  * [1] M. Bonk, J. Heinonen, and P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001), viii+99 pp.
  * [2] M. Bonk and B. Kleiner, Rigidity for quasi-Möbius group actions, J. Differential Geom. 61 (2002), 81–106.
  * [3] S. M. Buckley, D. Herron, and X. Xie, Metric space inversions, quasihyperbolic distance, and uniform spaces, Indiana Univ. Math. J. 57 (2008), 837–890.
  * [4] C. Butler, Uniformization, $\partial$-biLipschitz maps, sphericalization, and inversion, arXiv:2008.06806.
  * [5] C. Butler, Uniformizing Gromov hyperbolic spaces with Busemann functions, arXiv:2007.11143.
  * [6] F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Anal. Math. 36 (1979), 50–74.
  * [7] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry,  Acta Math. 181 (1998), 1–61.
  * [8] X. Huang and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in metric spaces,  Trans. Amer. Math. Soc. 367 (2015), 6225–6246.
  * [9] X. Huang, H. Liu, and J. Liu, Local properties of quasihyperbolic mappings in metric spaces,  Ann. Acad. Sci. Fenn. Ser. A I Math. 41 (2016), 23–40.
  * [10] M. Huang, A. Rasila, X. Wang, and Q. Zhou, Semisolidity and locally weak quasisymmetry of homeomorphisms in metric spaces,  Stud. Math. 242 (2018), 267–301.
  * [11] P. W. Jones, Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980), 41–66.
  * [12] X. Li and N. Shanmugalingam, Preservation of bounded geometry under sphericalization and fattening,  Indiana. Math J. (2015), 1303–1341.
  * [13] O. Martio and J. Sarvas, Injectivity theorems in plane and space,  Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), 383–401.
  * [14] P. Tukia and J. Väisälä, Lipschitz and quasiconformal approximation and extension, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 303–342.
  * [15] J. Väisälä, Quasi-Möbius maps, J. Anal. Math. 44 (1984/85), 218–234.
  * [16] J. Väisälä, Free quasiconformality in Banach spaces. III, Ann. Acad. Sci. Fenn. Ser. A I Math. 17 (1992), 393–408.
  * [17] J. Väisälä, The free quasiworld. Freely quasiconformal and related maps in Banach spaces, Banach Center Publications, Vol. 48, Polish Academy of Science, Warsaw (1999), 55–118.
  * [18] Q. Zhou, Quasihyperbolic mappings in Banach spaces, Ann. Fenn. Math. 46 (2021), 335–344.
  * [19] Q. Zhou, Uniformizing Gromov hyperbolic spaces and Busemann functions, arXiv:2008.01399.
  * [20] Q. Zhou, Y. Li, and Y. He, Quasihyperbolic mappings in length metric spaces, C. R. Math. doi.org/10.5802/crmath.154.