1 Introduction and Main Results

In this note, we study the behavior of Loewner condition under quasimöbius mappings between metric measure spaces. The class of quasimöbius mappings, under which, in a certain sense, the cross ratio is quasi-invariant, was explicitly defined and investigated in [11]. The main motivation for introducing this class of mappings by Väisälä is that they give a handy tool when studying the relationship between quasisymmetric and quasiconformal mappings. However, this class has appeared before such as in the celebrated Mostow’s work on rigidity. Recently, the motivation in the study of quasimöbius mappings comes from many sources, see e.g. [1, 3, 5, 9, 12,13,14,15,16] and the references therein.

There are two important classes of quasimöbius mappings, known as sphericalization and flattening ([1]), in geometric function theory. The original idea of these deformations comes from the work of Bonk and Kleiner [3] in defining a metric on the one point compactification of an unbounded locally compact metric space. The first class of deformation, sphericalization, is a generalization of the deformation from the Euclidean distance on \(\mathbb {R}^n\) to the chordal distance on the unit sphere \(\mathbb {S}^n\). The second class of flattening deformation is a generalization of inversion on the punctured sphere. It was shown in [5] that these two conformal transformations are dual in the sense that if one starts from a bounded metric space, and then performs a flattening transformation followed by a sphericalization, thus the object space is bilipschitzly equivalent to the original space. This duality comes from the idea that the stereographic projection between the Euclidean space and the Riemann sphere can be realized as a special case of inversion. Sphericalization and flattening have a lot of applications in the area of analysis on metric spaces, such as [1,2,3, 5].

Our motivation comes from Heinonen and Koskela’s celebrated work on the equivalence of quasiconformality and quasisymmetry between metric measure spaces in [8]. They introduced the concept of Loewner spaces, which has many applications in studying Sobolev spaces and quasiconformal theory in metric spaces, see [2, 4, 7]. It should be noted that Tyson [10] answered positively to a conjecture proposed by Heinonen and Koskela [8, Sect. 8.7] in proving that the Q-Loewner condition is preserved under quasisymmetric maps between two Ahlfor Q-regular spaces.

In their recent work [9], Li and Shanmugalingam studied the invariance of bounded geometry under sphericalization and flattening transformations; in particular, doubling measure, Ahlfors regularity and Poincaré inequality. It is natural to ask whether the Loewner condition is preserved under sphericalization and flattening, more general quasimöbius mappings. Indeed, this question has been investigated by Brania and Yang [4] by introducing the notion of controlled modulus condition. They proved that quasimöbius mapping preserves the n-Loewner condition in the extended Euclidean space \(\mathbb {R}^n\cup \{\infty \}\), see [4, Proposition 3.1]. In the setting of metric spaces, they also showed that the Loewner condition is preserved under quasimöbius mappings, provided the spaces are simultaneously bounded or unbounded and the mapping sends infinity to infinity, see [4, Corollary 3.4].

We obtain the following as our main theorem.

Theorem 1.1

Let \((X,d,\mu )\) and \((Y,\sigma ,\nu )\) be locally compact Ahlfors Q-regular metric measure spaces with \(Q>1\). Suppose that \(f:X\rightarrow Y\) is a quasimöbius homeomorphism. If X is Q-Loewner, then Y is also Q-Loewner.

We remark that Theorem 1.1 is a generalization of [4, Proposition 3.1 and Corollary 3.4] and our proof is different. Also, we explain the connection between Theorem 1.1 and Li-Shanmugalingam’s results in [9]:

First, Heinonen and Koskela demonstrated that Q-Loewner condition and (1, Q)-Poincaré inequality are equivalent in proper, Ahlfors Q-regular and \(\varphi \)-convex metric measure spaces, see [8, Corollary 5.13]. In [9, Theorem 1.1], it was shown by Li and Shanmugalingam that for a complete doubling metric measure space which admits a (1, p)-Poincaré inequality with \(1\le p<\infty \), if in addition the sphericalized (or flattened) space is annular quasiconvex, then the deformed space also admits a (1, p)-Poincaré inequality. Notice that we do not need any extra assumptions concerning the connectivity or completeness of the spaces in Theorem 1.1. By using a deformed cross ratio introduced in [2], our proof is direct and simple.

We conclude with reviewing certain examples of Loewner spaces, see [7, 8] and the references therein. For instance, Euclidean spaces and compact Riemannian manifolds, Carnot groups equipped with its Carnot-Carathéodory metrics, Riemannian manifolds of non-negative Ricci curvature, and so on. Indeed, by using Theorem 1.1, we can obtain many new Loewner spaces via sphericalization and flattening transformations.

The rest of this paper is organized as follows. In Sect. 2, we introduce the necessary terminology. The proof of Theorem 1.1 is given in Sect. 3.

2 Preliminary and Notations

Following [4, 5, 7, 11], we introduce certain terminology and recall useful results in this section.

For real numbers s and t, we set

$$\begin{aligned} s\wedge t=\min \{s,t\}\;\;\;\; \text{ and } \;\;\;\; s\vee t=\max \{s,t\}. \end{aligned}$$

Let (X, d) be a metric space. A curve in X means a continuous map \(\gamma :\; I=[a,b]\rightarrow X\) from an interval \(I\subset {\mathbb R}\) to X. We denote the image set \(\gamma (I)\) of \(\gamma \) by \(\gamma \). The length of \(\gamma \) is denoted by

$$\begin{aligned} \ell _d(\gamma )=\sup \Big \{\sum _{i=1}^{n}d(\gamma (t_i),\gamma (t_{i-1}))\Big \}, \end{aligned}$$

where the supremum is taken over all partitions \(a=t_0<t_1<t_2<\ldots <t_n=b\). A curve \(\gamma \) is called rectifiable if its length \(\ell _d(\gamma )<\infty \).

A metric space X is called rectifiably connected if every pair of points in X can be joined with a rectifiable curve \(\gamma \). The length function associated with a rectifiable curve \(\gamma \): \([a,b]\rightarrow X\) is \(z_{\gamma }\): \([a,b]\rightarrow [0, \ell (\gamma )]\), given by \(z_{\gamma }(t)=\ell (\gamma |_{[a,t]})\). For any rectifiable curve \(\gamma :\) \([a,b]\rightarrow X\), there is a unique map \(\gamma _s:\) \([0, \ell (\gamma )]\rightarrow X\) such that \(\gamma =\gamma _s\circ z_{\gamma }\). Obviously, \(\ell (\gamma _s|_{[0,t]})=t\) for \(t\in [0, \ell (\gamma )]\). The curve \(\gamma _s\) is called the arclength parametrization of \(\gamma \).

For a rectifiable curve \(\gamma \) in X, the line integral over \(\gamma \) of each Borel function \(\varrho :\) \(X\rightarrow [0, \infty )\) is

$$\begin{aligned} \int _{\gamma }\varrho \;ds=\int _{0}^{\ell (\gamma )}\varrho \circ \gamma _s(t) \;dt. \end{aligned}$$

2.1 Metric Measure Spaces

Following [8], \((X,d,\mu )\) denotes a metric measure space with X a locally compact, rectifiably connected metric space and \(\mu \) a Borel regular measure with dense support.

Definition 2.1

([6]) Let \((X, d,\mu )\) be a metric measure space. Given \(Q> 1\), we say that X is Ahlfors Q-regular if there exists a constant \(C>0\) such that for each \(x\in X\) and \(0<r\le {{\text {diam}}}(X)\),

$$\begin{aligned} r^Q/C \le \mu (B(x,r))\le Cr^Q, \end{aligned}$$

where \({{\text {diam}}}(X)\) means the diameter of X and \(B(x,r)=\{y\in X;\,d(x,y)<r\}\).

For instance, the Euclidean space \(\mathbb {R}^n\) with Lebesgue measure satisfies the Ahlfors n-regularity.

Definition 2.2

Let \(Q>1\). We define the Q-modulus of a family \(\Gamma \) of curves in a metric measure space \((X, d,\mu )\) by

$$\begin{aligned} mod_Q\Gamma =\inf \int _X \rho ^Q\;d\mu , \end{aligned}$$

where the infimum is taken over all Borel functions \(\rho :X\rightarrow [0,\infty ]\) satisfying

$$\begin{aligned} \int _\gamma \rho \;ds\ge 1 \end{aligned}$$

for all locally rectifiable curves \(\gamma \in \Gamma \). Then the Q-modulus of a pair of disjoint nonempty compact sets \(E,F\subset X\) is

$$\begin{aligned} mod_Q(E,F;X)=mod_Q\Delta (E,F;X), \end{aligned}$$

where \(\Delta (E,F;X)\) is the family of all curves in X joining the sets E and F.

In studying the equivalence of quasiconformality and quasisymmetry, Heinonen and Koskela [8] introduced the notion of Loewner spaces. Moreover, they showed some examples for Loewner, such as Euclidean n-space \(\mathbb {R}^n\), a Carnot group admits a (1, p)-Poincaré inequality for all \(p>1\) and so on. Note that a metric measure space \((X, d,\mu )\) is called Q-Loewner, with \(Q>1\), provided the Loewner control function

$$\begin{aligned} \varphi (t)=\inf \{mod_Q(E,F;X): \Delta _d(E,F)\le t\} \end{aligned}$$

is strictly positive for all \(t>0\); here E and F are non-degenerate disjoint continua in X and

$$\begin{aligned} \Delta _d(E, F)= \frac{{{\text {dist}}}_d(E, F)}{{{\text {diam}}}_d(E)\wedge {{\text {diam}}}_d(F)}, \end{aligned}$$

where \({{\text {dist}}}_d(E, F)\) means the distance between E and F in the metric d.

2.2 Mappings on Metric Spaces

Given a metric space (X, d),  the cross ratio r(x, y, z, w) of each four distinct points \(x, y, z, w \in X\) is defined as

$$\begin{aligned} r(x,y,z,w)=\frac{d(x,z)d(y,w)}{d(x,y)d(z,w)}. \end{aligned}$$

It is often convenient to consider cross ratios also in the extended space \(\dot{X}=X\cup \{\infty \}\). If x, y, z, w are points in \(\dot{X}\) and if one of the points x, y, z, w is \(\infty \), the cross ratio is defined by deleting the distances from \(\infty \). For example,

$$\begin{aligned} r(x,y,z,\infty )=\frac{d(x,z)}{d(x,y)}. \end{aligned}$$

Let \((X_1,d_1)\) and \((X_2,d_2)\) be two metric spaces, and let \(f:X_1\rightarrow X_2\) be a homeomorphism. We say that f is L-bilipschitz if there exists \(L\ge 1\) such that

$$\begin{aligned} d_1(x,y)/L\le d_2(f(x),f(y))\le L d_1(x,y). \end{aligned}$$

Let \(\eta :[0,\infty )\rightarrow [0,\infty )\) be a homeomorphism. We say that f is \(\eta \)-quasisymmetric if for all distinct points \(x,y,z\in X_1\), we have

$$\begin{aligned} \frac{d_2(f(x),f(z))}{d_2(f(x),f(y))}\le \eta \left( \frac{d_1(x,z)}{d_1(x,y)}\right) . \end{aligned}$$

Moreover, we say that f is \(\eta \)-quasimöbius if for all distinct points \(x,y,z,w\in X_1,\) we have

$$\begin{aligned} r(f(x),f(y),f(z),f(w))\le \eta \big (r(x,y,z,w)\big ). \end{aligned}$$

For the properties of quasimöbius and quasisymmetric mappings see [7, 11, 12]. We also need the following result.

Lemma 2.3

([11, Theorem 3.10]) Suppose that X is unbounded and that \(f:X\rightarrow Y\) is \(\theta \)-quasimöbius between two metric spaces. Then f is \(\theta \)-quasisymmetric if and only if \(f(x)\rightarrow \infty \) as \(x\rightarrow \infty \).

Our main tool in this paper is the following useful notation introduced by Bonk and Kleiner in [2]:

$$\begin{aligned} \langle x,y,z,w \rangle =\frac{d(x,z)\wedge d(y,w)}{d(x,y)\wedge d(z,w)}.\end{aligned}$$

They also established a relation between r(x, y, z, w) and \(\langle x,y,z,w \rangle \) as follows.

Lemma 2.4

([2, Lemma 2.3]) For any distinct points x, y, z, w in a metric space X, we have

$$\begin{aligned} \langle x,y,z,w \rangle \le \theta _0(r(x,y,z,w)), \end{aligned}$$

where \(\theta _0(t)=3(t\vee \sqrt{t})\).

3 Proof of Theorem 1.1

We begin the proof of Theorem 1.1 by showing that the sphericalization of an unbounded metric measure space preserves the Q-Loewner condition. Following [5, 9], we first introduce certain terminology.

Bonk and Kleiner introduced sphericalization of an unbounded metric measure space, see [3, Lemma 2.2, p87]. We briefly recall their work. Given an unbounded locally compact metric space (X, d) and a base point \(a\in X\), we consider the one-point compactification \(\dot{X}=X\cup \{\infty \}\) and define the function \(d_a:\dot{X}\times \dot{X}\rightarrow [0,\infty )\) as follows

$$\begin{aligned} d_a(x,y)=d_a(y,x)=\left\{ \begin{array}{ll} \frac{d(x,y)}{[1+d(x,a)][1+d(y,a)]},&{} \text{ if }\;\;x,y\in X,\\ \frac{1}{1+d(x,a)}, &{} \text{ if }\;\;y=\infty \; \text{ and }\;x\in X,\\ 0, &{} \text{ if }\;\; x=\infty =y. \end{array}\right. \end{aligned}$$

Note that, this distance function is an analog of the chordal metric on the Riemann sphere. Unfortunately, \(d_a(x,y)\) will not satisfy the triangle inequality in general. In fact, generally, \(d_a\) is not a metric on \(\dot{X}\) and however a quasimetric. There is a standard procedure, known as chain construction, to construct a metric from a quasimetric as follows. Define

$$\begin{aligned} \widehat{d}_a(x,y):=\inf \sum _{j=0}^k d_a(x_j,x_{j+1}), \end{aligned}$$

where the infimum is taken over all finite sequences \(x=x_0,x_1,\ldots ,x_k,x_{k+1}=y\) from \(\dot{X}\). Then \((\dot{X},\widehat{d}_a)\) is a metric space and called the sphericalization of (X, d) associated to the point \(a\in {X}\). Moreover, by [5, (3.3)] we have for all \(x,y\in \dot{X}\)

$$\begin{aligned} \frac{1}{4}d_a(x,y)\le \widehat{d}_a(x,y)\le d_a(x,y). \end{aligned}$$
(3.1)

If (X, d) is a rectifiably connected unbounded metric space, then the Borel function \(\rho _a:X\rightarrow [0,\infty )\) is defined as

$$\rho _a(x)=\frac{1}{[1+d(a,x)]^2}.$$

A similar argument as [5, (4.1)], we obtain that for any rectifiable curve \(\gamma \) in X joining x and y,

$$\begin{aligned} \ell _{\widehat{d}_a}(\gamma )=\int _\gamma \rho _a(z)\;ds, \end{aligned}$$
(3.2)

where ds means the element of the length. If \((X,d,\mu )\) is Ahlfors Q-regular, then the associated spherical measure \(\mu _a\) on a Borel set \(A\subset X\) is given by

$$\begin{aligned} \mu _a(A)=\int _{A}\rho _a(z)^Q \;d\mu (z). \end{aligned}$$
(3.3)

Lemma 3.1

Suppose that \((X,d,\mu )\) is an unbounded Ahlfors Q-regular Q-Loewner locally compact metric measure space with \(Q>1\) and \(a\in X\). Then, the sphericalized space \((X,\widehat{d_a},\mu _a)\) is a bounded Q-regular Q-Loewner metric measure space.

Proof

Since \((X,d,\mu )\) is Ahlfors Q-regular, we see from [9, Proposition 3.1] that \((X,\widehat{d_a},\mu _a)\) is also Ahlfors Q-regular. We first prove that the sphericalized transformation preserves Q-modulus, that is,

$$\begin{aligned} mod_Q(\Gamma ,d,\mu )=mod_Q(\Gamma ,\widehat{d}_a,\mu _a) \end{aligned}$$
(3.4)

for all rectifiable family of curves \(\Gamma \) in X. Indeed, we know from (3.2) and (3.3) that for all nonnegative Borel function \(\rho :X\rightarrow [0,\infty ]\)

$$\begin{aligned} \int _X\rho ^Q\;d\mu =\int _X\frac{\rho ^Q}{\rho _a^Q} \; d\mu _a, \end{aligned}$$

and for any rectifiable curve \(\gamma \in \Gamma \)

$$\begin{aligned} \int _\gamma \rho \;ds=\int _\gamma \frac{\rho }{\rho _a} \;ds_a, \end{aligned}$$

where \(ds_a\) is the arc-length element with respect to \(\widehat{d}_a\). From the definition of Q-modulus, we know that (3.4) holds true.

In order to show that \((X,\widehat{d_a},\mu _a)\) is Q-Loewner, we only need to find a lower bound of the Q-modulus associated to any pair of disjoint, non-degenerate continua E and F in X. Since \((X,d,\mu )\) is Q-Loewner, by (3.4) we know that it suffices to find an increasing function \(\psi :(0,\infty )\rightarrow (0,\infty )\) satisfying

$$\begin{aligned} \Delta _{\widehat{d}_a}(E,F) \ge \psi \big (\Delta _d(E,F)\big ). \end{aligned}$$
(3.5)

To this end, take \(x\in E\) and \(y\in F\) such that

$$\begin{aligned} {{\text {dist}}}_{\widehat{d}_a}(E,F)= \widehat{d}_a(x,y). \end{aligned}$$

Without loss of generality, we may assume that \({{\text {diam}}}_{\widehat{d}_a}(E)\le {{\text {diam}}}_{\widehat{d}_a}(F)\). Moreover, choose \(z\in E\) such that \({{\text {diam}}}_{\widehat{d}_a}(E)\le 2{\widehat{d}_a}(x,z)\) and choose \(w\in F\) such that \({{\text {diam}}}_{\widehat{d}_a}(F)\le 2{\widehat{d}_a}(y,w)\).

By [3, Lemma 2.2], we see that the identity map \({{\text {id}}}:(X,d)\rightarrow (X,\widehat{d_a})\) is \(\theta \)-quasimöbius with \(\theta (t)=16t\). It thus follows from Lemma 2.4 and (3.1) that

$$\begin{aligned}\frac{d(x,y)\wedge d(z,w)}{d(x,z)\wedge d(y,w)}\le & {} \theta _0\Big (\frac{d(x,y) d(z,w)}{d(x,z) d(y,w)}\Big ) \\\le & {} \theta _0\Big (16\frac{{\widehat{d}_a}(x,y) {\widehat{d}_a}(z,w)}{{\widehat{d}_a}(x,z) {\widehat{d}_a}(y,w)}\Big ) \\\le & {} \eta \Big (\frac{{\widehat{d}_a}(x,y)\wedge {\widehat{d}_a}(z,w)}{{\widehat{d}_a}(x,z)\wedge {\widehat{d}_a}(y,w)}\Big ), \end{aligned}$$

where \(\theta _0(t)=3(t\vee \sqrt{t})\) and \(\eta (t)=\theta _0\big (\frac{16}{\theta _0^{-1}(1/t)}\big )\). Consequently, from the above facts we get

$$\begin{aligned} \Delta _{d}(E,F)= & {} \frac{{{\text {dist}}}_d(E,F)}{{{\text {diam}}}_d(E)\wedge {{\text {diam}}}_d(F)} \\\le & {} \frac{d(x,y)\wedge d(z,w)}{d(x,z)\wedge d(y,w)} \\\le & {} \eta \Big (\frac{{\widehat{d}_a}(x,y)\wedge {\widehat{d}_a}(z,w)}{{\widehat{d}_a}(x,z)\wedge {\widehat{d}_a}(y,w)}\Big ) \\\le & {} \eta \Big (\frac{2{{\text {dist}}}_{\widehat{d}_a}(E,F)}{{{\text {diam}}}_{\widehat{d}_a}(E)\wedge {{\text {diam}}}_{\widehat{d}_a}(F)}\Big ). \end{aligned}$$

Hence, (3.5) is true by letting \(\psi (t)=\frac{1}{2}\eta ^{-1}(t)\). We complete the proof of Lemma 3.1. \(\square \)

Next, we consider the flatting transformations on bounded metric spaces. Given a bounded metric space (X, d) and a base point \(c\in X\), we consider the space \(X^c=X\setminus \{c\}\) and define the function \(d^c:X^c\times X^c\rightarrow [0,\infty )\) as follows

$$\begin{aligned} d^c(x,y)=d^c(y,x)=\frac{d(x,y)}{d(x,c)d(y,c)}. \end{aligned}$$

We also deform the quasimetric \(d^c\) by a chain construction and obtain from [5, (3.1)] that there is a metric \(\widehat{d}^c\) on \(X^c\) satisfying

$$\begin{aligned} \frac{1}{4}d^c(x,y)\le \widehat{d}^c(x,y)\le d^c(x,y). \end{aligned}$$

The metric space \((X^c,\widehat{d}^c)\) is said to be the flattening of (X, d) associated to the point c. In the case that (X, d) is a rectifiably connected bounded metric space, we define the Borel function \(\rho ^c:X^c\rightarrow [0,\infty )\) to be

$$\begin{aligned} \rho ^c(x)=\frac{1}{d(c,x)^2}. \end{aligned}$$

Thus, by [5, (4.1)], we obtain that for any rectifiable curve \(\gamma \) in \(X^c\) joining x and y,

$$\begin{aligned} \ell _{\widehat{d}^c}(\gamma )=\int _\gamma \rho ^c(z)\;ds. \end{aligned}$$

If \((X,d,\mu )\) is Ahlfors Q-regular, then the corresponding flattening measure \(\mu ^c\) is defined by

$$\begin{aligned} \mu ^c(A)=\int _{A} \rho ^c(z)^Q\; d\mu (z), \end{aligned}$$

where \(A\subset X^c\) is a Borel set.

Moreover, we demonstrate that the Q-Loewner condition is preserved under the flattening of a bounded metric measure space. Since the argument for this result is completely similar to the proof of Lemma 3.1, we do not provide the proof.

Lemma 3.2

Suppose that \((X,d,\mu )\) is a bounded Ahlfors Q-regular Q-Loewner metric measure space with \(Q>1\) and \(c\in X\). Then the flattening space \((X^c,\widehat{d}^c,\mu ^c)\) is an unbounded Q-regular Q-Loewner metric measure space.

Now, we are ready to prove Theorem 1.1 by using Lemmas 3.1 and 3.2.

Proof of Theorem 1.1

Let \((X,d,\mu )\) and \((Y,\sigma ,\nu )\) be locally compact Ahlfors Q-regular metric measure spaces with \(Q>1\). Suppose that \(f:X\rightarrow Y\) is a quasimöbius homeomorphism and X is Q-Loewner. We need to show that Y is Q-Loewner.

For this, we only consider the case whenever X and Y are both bounded; for the other cases, it is easy to deal with and the proof is rather similar. No loss of generality we assume that \({{\text {diam}}}_{\sigma }(Y)=1\).

Fix \(c\in X\) and \(c'=f(c)\in Y\). It follows from Lemma 3.2 that the flattening spaces, \((X^c,\widehat{d}^c,\mu ^c)\) and \((Y^{c'},\widehat{\sigma ^{c'}},\nu ^{c'})\), are both unbounded Ahlfors Q-regular metric spaces. By [5, Lemma 3.1], we find that the identity mappings

$$\begin{aligned} \varphi _X:(X^c,d)\rightarrow ({X^c},\widehat{d}^c)\;\;\;\;\text{ and }\;\;\;\;\varphi _Y:(Y^{c'},\sigma )\rightarrow ({Y^{c'}},\widehat{\sigma ^{c'}}) \end{aligned}$$

are both \(\theta \)-quasimöbius with \(\theta (t)=16t\), \(\varphi _X(x)\rightarrow \infty \) as \(x\rightarrow c\) and \(\varphi _Y(y)\rightarrow \infty \) as \(y\rightarrow c'\). Because the inverse and the composition of quasimöbius mappings are also quasimöbius, we obtain a quasimöbius mapping:

$$\begin{aligned} \widehat{f}=\varphi _Y\circ f\circ \varphi _X^{-1}:({X^c},\widehat{d}^c)\rightarrow ({Y^{c'}},\widehat{\sigma ^{c'}}) \end{aligned}$$

with \(\widehat{f}(x)\rightarrow \infty \) as \(x\rightarrow \infty \). Thus \(\widehat{f}\) is quasisymmetric by using Lemma 2.3.

On the one hand, again by Lemma 3.2, we see that \((X^c,\widehat{d}^c,\mu ^c)\) is Q-Loewner because \((X,d,\mu )\) is Q-Loewner. Therefore, appealing to [10, Corollary 1.6] we see that \(({Y^{c'}},\widehat{\sigma ^{c'}},\nu ^{c'})\) is Q-Loewner.

On the other hand, take another point \(p'\in Y\) with \(\sigma (p',c')\ge {{\text {diam}}}(Y)/2=1/2\). Consider the sphericalized space \(({Y^{c'}},\widehat{(\widehat{\sigma ^{c'}})_{p'}},(\nu ^{c'})_{p'})\) of \(({Y^{c'}},\widehat{\sigma ^{c'}},\nu ^{c'})\) with respect to \(p'\). By Lemma 3.1, it follows that \(({Y^{c'}},\widehat{(\widehat{\sigma ^{c'}})_{p'}},(\nu ^{c'})_{p'})\) is also Ahlfors Q-regular and Q-Loewner. According to [5, Proposition 3.5], we see that the identity map \({{\text {id}}}: (Y,\sigma )\rightarrow ({Y},\widehat{(\widehat{\sigma ^{c'}})_{p'}})\) with \(c'\) corresponding to the point \(\infty \) in the extended space \((\dot{Y^{c'}},\widehat{\sigma ^{c'}})\), is 256-bilipschitz. Since Q-Loewner is a bilipschitz invariant, we find that \((Y,\sigma ,\nu )\) is also Q-Loewner. The proof of Theorem 1.1 is complete.

Remark 3.3

As we point out in Sect. 1, by Theorem 1.1, it is easy to get many Loewner spaces via sphericalization and flattening transformations. For instance, we obtain that the spherical surface \(\mathbb {S}^n\) is a Loewner space, since every Euclidean space is a Loewner space. It follows from the space \((\mathbb {B}^2\setminus \{0\}, |\cdot |)\) being a Loewner space that \((\mathbb {R}^2\setminus \mathbb {B}^2, |\cdot |)\) is also a Loewner space by Theorem 1.1 and flattening transformation.