# The Quasisymmetric Mappings under the Quasihyperbolic Metric in Real Banach Spaces

Authors: Caicai Hu; Liang Guo
Journal: Journal of Mathematics
 (2022)

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## Abstract

Suppose thatE,E′ are Q-regular Banach spaces, that G⊂E and G′⊂E′ are domains, and that f:G⟶G′ is a homeomorphism. The aim of this study is to prove that the quasisymmetric mapping under the quasihyperbolic metric is equivalent to the freely quasiconformal mapping under the condition E,E′ being Q-regular Banach spaces. This result is a partial solution to an open problem raised by Väisälä.


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## Body

## 1. Introduction and Main Result

Quasiconformal mappings in the planeℝ2 were introduced by Grötzsch in 1928 when he studied the most nearly conformal mappings of a square onto a rectangle, not a square, which map vertices on vertices [1]. After that, this class of mappings attracted much attention (see, e.g., [2, 3] and the references therein). The concept was later generalized to the setting of ℝn by several authors (see [4–8]). From the late 1980’s onwards, by using quasihyperbolic metrics, Väisälä established the theory of freely quasiconformal mappings in Banach spaces [9–13].The class of quasisymmetric mappings on the real axis was first introduced by Beurling and Ahlfors [14], who found a way to obtain a quasiconformal extension of a quasisymmetric self-mapping of the real axis to a self-mapping of the upper half plane. This concept was later generalized by Tukia and Väisälä when they studied quasisymmetric mappings among metric spaces [15]. As we know, every quasisymmetric mapping is freely quasiconformal, but the converse is not true in general. As in the definition of freely quasiconformal mappings, by using quasihyperbolic metrics, in [9], Väisälä introduced a new class of mappings, that is, quasisymmetric mappings in the quasihyperbolic metric, and got the following relationship between freely quasiconformal mappings and quasisymmetric mappings in the quasihyperbolic metric.Theorem 1.
(see [9], Theorem 5.14). Suppose that E and E′ are Banach spaces and that G⊂E and G′⊂E′ are domains (i.e., nonempty open and connected sets). If f:G⟶G′ is freely φ-quasiconformal, then f is a fully η-quasisymmetric mapping in the quasihyperbolic metric with η=ηφ, where the symbol “ηφ” means that the control function η depends only on the control function φ.Inℝn, the converse of Theorem 1 is also true, which means that quasisymmetric mappings in the quasihyperbolic metric and freely quasiconformal mappings are quantitatively equivalent (cf. [9], Remark 3.5). (For two conditions, we say that Condition A implies Condition B quantitatively if Condition A implies Condition B, and the data γB of Condition B depend only on the one γA of Condition A and other given quantities. If also Condition B implies Condition A quantitatively, then we say that Condition A is quantitatively equivalent to Condition B).Naturally, one will ask whether the converse of Theorem1 is true in Banach spaces. In fact, the problem was raised by Väisälä in the following form.Problem 1.
(see [13], Problem 13.2). Suppose that E and E′ are Banach spaces, that G⊂E and G′⊂E′ are domains, and that f:G⟶G′ is an η-quasisymmetric mapping in the quasihyperbolic metric. Is f freely φ-quasiconformal with φ=φη?
In 1998, Heinonen and Koskela [6] showed that quasiconformality and quasisymmetry are quantitatively equivalent in a large class of metric spaces, which includes Euclidean spaces, but not all Banach spaces. The spaces satisfy some controlled geometry conditions, for instance, regularity. A metric measure space X,d,μ is called Q-regular, Q>1, if there is a constant C≥1 so that(1)C−1RQ≤μB≤CRQ,for all balls B=Bx,R in X with radius 0<R<diamX, where “diam” means “diameter.” Also, we say that a metric space X is called Q-regular if there is a measure μ on X such that (1) holds for some constants Q>1 and C≥1.
With the aid of regularity, we get the following result.Theorem 2.
Suppose thatE and E′ are Q-regular Banach spaces and that G⊂E and G′⊂E′ are domains. Then, f is freely φ-quasiconformal if and only if f is an η-quasisymmetric mapping in the quasihyperbolic metric with φ and η depending only on each other and the constant Q.Remark 1.
Note that Banach spaces need not to beQ-regular; hence, the sufficiency of Theorem 2 can be regarded as a partial answer to Open Problem 1.
The rest of this paper is organized as follows. In Section2, necessary definitions and results will be recalled, and in Section 3, Theorem 2 will be proved.


## 2. Preliminaries

### 2.1. Notation

Throughout the study,E and E′ denote real Banach spaces with dimension at least 2, G⊂E and G′⊂E′ are domains, the norm of a vector x in E is written as x, and for every pair of points x, y in E, the distance between them is denoted by x−y. Moreover, Bx,r denotes the open ball with the center x∈E and radius r>0, and Sx,r and B¯x,r denote the boundary and the closure of Bx,r, respectively. Let clusf,x0=∩clfU∩G denote the cluster set of f at a point x0∈∂G over all neighborhoods U of x0 in E˙=E∪∞. For x, y in E, let x,y denote the segment or the closed interval in E with the endpoints x and y, respectively. And we remark x,y=x,y/y and x,y=x,y/x,y.


### 2.2. Quasihyperbolic Metric

LetG⊂E be a domain. For a rectifiable curve γ in G, that is, the arclength ℓγ<∞, its quasihyperbolic length is defined by(2)ℓkGγ=∫γdxδGx,where δGx denotes the distance from x to the boundary ∂G of G.For each pair of pointsx, y in G, the quasihyperbolic distance kGx,y between x and y is defined in the usual way:(3)kGx,y=infγℓkGγ,where the infimum is taken over all rectifiable curves γ joining x to y in G. It is known that kG is a metric in G, which is called quasihyperbolic metric (abbreviated QH metric).The QH metric of a domain inℝn was introduced by Gehring and Palka [16]. This metric was later generalized to the setting of Banach spaces and metric spaces and has become an important tool in the study of geometric function theory [9, 10, 13, 17–20].An arcγ is c-quasiconvex if(4)ℓγx,y≤cx−y,for all x,y∈γ, where γx,y is the subarc of γ between x and y, and ℓγ is the arclength of γ.Theorem 3.
(see [9], Lemma 2.2)(1)
For allz1,z2∈G, we have(5)kGz1,z2≥log1+z1−z2minδGz1,δGz2≥logδGz1δGz2,where “min” means “minimum,” and(6)z1−z2≤expkGz1,z2−1δGz1.(2)
Forw∈G, 0<t<1 and z1, z2∈B¯w,tδGw,(7)kGz1,z2≤11−tz1−z2δGw.In addition, ift≤1/2, then(8)kGz1,z2≥11+2tz1−z2δGw.Theorem 4.
(see [9], Lemma 2.6). For every r>0, there is a constant μ=μr≥1 such that(9)1μrz1−z2δGw≤kGz1,z2≤μrz1−z2δGw,for all w∈G and z1,z2∈B¯kGw,r, where QH balls BkGw,r=x∈G:kGx,w<r with center w∈G and radius r>0.Theorem 5.
(see [10], Lemma 2.2). For each x0∈E and r>0, let S=Sx0,r be a sphere in E. Then, each pair of points in S can be joined by a 2-quasiconvex arc in S.


### 2.3. Freely Quasiconformal Mappings and Quasisymmetric Mappings in theQH Metric

Definition 1.
Letf: G⟶G′ be a homeomorphism, where G⊂E and G′⊂E′ are domains, and let φ:0,∞⟶0,∞ be a growth function, that is, a homeomorphism of 0,∞ with φt≥t for t≥0. We say that f is(1)
φ-semisolid if(10)kG′x′,y′≤φkGx,y,for all x,y∈G. Here and hereafter, the primes are used to denote the images, e.g., x′=fx;(2)
φ-solid if both f and f−1 are φ-semisolid(3)
M-quasihyperbolic (abbreviated M-QH) iff is φ-solid with φt=Mt(4)
Fullyφ-solid if f is φ-solid on every subdomain of G. Fully φ-solid mappings are also said to be freely φ-quasiconformal (abbreviated φ-FQC) mappings.Theorem 6.
(see [9], Theorem 4.7). Suppose that G≠E, G′≠E′, and that f: G⟶G′ is M-QH. Then, f is fully 4M2-QH.Theorem6 shows that if f is M-QH, then f is φ-FQC with φt=4M2t. To further illustrate the preciseness of Theorem 6, an example of QH mapping is given below.Inversion 1(see [9], Inversion 4.9). Let G=E/0 and let u: G⟶G′ be the inversion ux=x/x2. Then, u is 3-QH and fully 36-QH.Definition 2.
Let0<q<1. For domains G⊂E and G′⊂E′, a homeomorphism f: G⟶G′ is called(1)
η-quasisymmetric (abbreviated η-QS) if there exists a homeomorphism η:0,∞⟶0,∞ with η0=0 such that(11)x−a≤tx−bimpliesx′−a′≤ηtx′−b′,for each t>0 and each triple x, a, b of distinct points in G(2)
q-locally η-QS if the restriction f|Bz,qr is η-QS whenever Bz,qr⊂G(3)
η-QHQS if f is η-QS in the QH metric in GRemark 2.
We see from ([15], Theorem 2.2) and ([9], 3.10) that the inverse of an η-QS (res. QHQS) is η′-QS (res. QHQS) with η′t=η−1t−1−1, and also, the inverse of an q-locally η-QS may not be q′-locally η′-QS. By [9] (3.4), we know that M-QH is η-QHQS with ηt=M2t.
In [9], Väisälä shows that FQC is quantitatively equivalent to local QS. This result has been widely used to study the properties of freely quasiconformal mappings, see [21] and the references therein.Theorem 7.
(see [9], Theorem 5.10). For a homeomorphism f: G⟶G′, the following conditions are quantitatively equivalent:(1)
f is φ-FQC(2)
f and f−1 are q-locally η-QS


## 2.1. Notation

Throughout the study,E and E′ denote real Banach spaces with dimension at least 2, G⊂E and G′⊂E′ are domains, the norm of a vector x in E is written as x, and for every pair of points x, y in E, the distance between them is denoted by x−y. Moreover, Bx,r denotes the open ball with the center x∈E and radius r>0, and Sx,r and B¯x,r denote the boundary and the closure of Bx,r, respectively. Let clusf,x0=∩clfU∩G denote the cluster set of f at a point x0∈∂G over all neighborhoods U of x0 in E˙=E∪∞. For x, y in E, let x,y denote the segment or the closed interval in E with the endpoints x and y, respectively. And we remark x,y=x,y/y and x,y=x,y/x,y.


## 2.2. Quasihyperbolic Metric

LetG⊂E be a domain. For a rectifiable curve γ in G, that is, the arclength ℓγ<∞, its quasihyperbolic length is defined by(2)ℓkGγ=∫γdxδGx,where δGx denotes the distance from x to the boundary ∂G of G.For each pair of pointsx, y in G, the quasihyperbolic distance kGx,y between x and y is defined in the usual way:(3)kGx,y=infγℓkGγ,where the infimum is taken over all rectifiable curves γ joining x to y in G. It is known that kG is a metric in G, which is called quasihyperbolic metric (abbreviated QH metric).The QH metric of a domain inℝn was introduced by Gehring and Palka [16]. This metric was later generalized to the setting of Banach spaces and metric spaces and has become an important tool in the study of geometric function theory [9, 10, 13, 17–20].An arcγ is c-quasiconvex if(4)ℓγx,y≤cx−y,for all x,y∈γ, where γx,y is the subarc of γ between x and y, and ℓγ is the arclength of γ.Theorem 3.
(see [9], Lemma 2.2)(1)
For allz1,z2∈G, we have(5)kGz1,z2≥log1+z1−z2minδGz1,δGz2≥logδGz1δGz2,where “min” means “minimum,” and(6)z1−z2≤expkGz1,z2−1δGz1.(2)
Forw∈G, 0<t<1 and z1, z2∈B¯w,tδGw,(7)kGz1,z2≤11−tz1−z2δGw.In addition, ift≤1/2, then(8)kGz1,z2≥11+2tz1−z2δGw.Theorem 4.
(see [9], Lemma 2.6). For every r>0, there is a constant μ=μr≥1 such that(9)1μrz1−z2δGw≤kGz1,z2≤μrz1−z2δGw,for all w∈G and z1,z2∈B¯kGw,r, where QH balls BkGw,r=x∈G:kGx,w<r with center w∈G and radius r>0.Theorem 5.
(see [10], Lemma 2.2). For each x0∈E and r>0, let S=Sx0,r be a sphere in E. Then, each pair of points in S can be joined by a 2-quasiconvex arc in S.


## 2.3. Freely Quasiconformal Mappings and Quasisymmetric Mappings in theQH Metric

Definition 1.
Letf: G⟶G′ be a homeomorphism, where G⊂E and G′⊂E′ are domains, and let φ:0,∞⟶0,∞ be a growth function, that is, a homeomorphism of 0,∞ with φt≥t for t≥0. We say that f is(1)
φ-semisolid if(10)kG′x′,y′≤φkGx,y,for all x,y∈G. Here and hereafter, the primes are used to denote the images, e.g., x′=fx;(2)
φ-solid if both f and f−1 are φ-semisolid(3)
M-quasihyperbolic (abbreviated M-QH) iff is φ-solid with φt=Mt(4)
Fullyφ-solid if f is φ-solid on every subdomain of G. Fully φ-solid mappings are also said to be freely φ-quasiconformal (abbreviated φ-FQC) mappings.Theorem 6.
(see [9], Theorem 4.7). Suppose that G≠E, G′≠E′, and that f: G⟶G′ is M-QH. Then, f is fully 4M2-QH.Theorem6 shows that if f is M-QH, then f is φ-FQC with φt=4M2t. To further illustrate the preciseness of Theorem 6, an example of QH mapping is given below.Inversion 1(see [9], Inversion 4.9). Let G=E/0 and let u: G⟶G′ be the inversion ux=x/x2. Then, u is 3-QH and fully 36-QH.Definition 2.
Let0<q<1. For domains G⊂E and G′⊂E′, a homeomorphism f: G⟶G′ is called(1)
η-quasisymmetric (abbreviated η-QS) if there exists a homeomorphism η:0,∞⟶0,∞ with η0=0 such that(11)x−a≤tx−bimpliesx′−a′≤ηtx′−b′,for each t>0 and each triple x, a, b of distinct points in G(2)
q-locally η-QS if the restriction f|Bz,qr is η-QS whenever Bz,qr⊂G(3)
η-QHQS if f is η-QS in the QH metric in GRemark 2.
We see from ([15], Theorem 2.2) and ([9], 3.10) that the inverse of an η-QS (res. QHQS) is η′-QS (res. QHQS) with η′t=η−1t−1−1, and also, the inverse of an q-locally η-QS may not be q′-locally η′-QS. By [9] (3.4), we know that M-QH is η-QHQS with ηt=M2t.
In [9], Väisälä shows that FQC is quantitatively equivalent to local QS. This result has been widely used to study the properties of freely quasiconformal mappings, see [21] and the references therein.Theorem 7.
(see [9], Theorem 5.10). For a homeomorphism f: G⟶G′, the following conditions are quantitatively equivalent:(1)
f is φ-FQC(2)
f and f−1 are q-locally η-QS


## 3. Proof of Theorem 2

The purpose of this section is to prove Theorem2. To reach this goal, we only need to prove the sufficiency of Theorem 2 since the necessity of Theorem 2 follows from Theorem 1.Assume that bothE and E′ are Q-regular Banach spaces with Q>1, and the homeomorphism f: G⟶G′ is η-QHQS. By performing auxiliary similarities, we may assume that 0∈∂G and clusf,0 are either 0 or ∞. By Theorem 3 and Remark 2, we know that the inversion is FQC and QHQS, and the composition of two FQC (resp. QHQS) mappings is also FQC (resp. QHQS). Hence, we can use auxiliary inversions to normalize the map so that both G and G′ are unbounded and ∞∈∂G, ∞∈∂G′, clusf,∞=∞.Now, we begin to prove the sufficiency of Theorem2, i.e., f is φ-FQC.To provef is φ-FQC, by Theorem 5, it suffices to show that both f and f−1 are 1/8-locally η1-QS (η1 will be given in Lemma 3). Since f−1 is also η′-QHQS with η′t=η−1t−1−1, by symmetry, we only need to prove that f is 1/8-locally η1-QS; i.e., for each x∈G, the restriction f|Bx,δGx/8 is η1-QS.Before proving Theorem2, we need some preparation. Without loss of generality, we assume that η1>1. For convenience, denote(12)c0=2Q4Q+53QC2Q+1η−156−Q2,c0′=2Q4Q+53QC2Q+1η65Q2.In order to simplify the proof process, we make some simple statements.For eachx∈G, let x′=fx. We take w′∈∂G′ such that(13)x′−w′≤min1+εδG′x′,δG′x′+16ε,where ε is a small constant satisfying(14)0<ε≤112mindistfB¯x,δGx8,∂G′,1.Fori∈1,2,…, we denote(15)Bi=Bx,δGx2i+2,Si=Sx,δGx2i+2.We denote the images ofBi and Si under f by Bi′ and Si′, respectively. Obviously, Bi′⊂G′ and Si′⊂G′. By Theorem 7, we know that Si and Si′ are connected. Now, the proof of Theorem 2 is mainly discussed on Bi and Si.Sincew′∈∂G′, we have x′,w′∩Si′≠∅. We take ui′∈x′,w′∩Si′ to be the first point starting from x′ to w′. Then, x′,ui′∈G′. Obviously, ui=f−1ui′∈Si∩f−1x′,ui′. Therefore, we get two sequences uii=1⊂G and ui′i=1⊂G′ which satisfy(16)ui∈Si∩f−1x′,ui′,ui′∈Si′∩x′,w′,for any i∈1,2,….LetM1=2c0, where 2c0 means the greatest integer part of 2c0. Under these preparations, now, we are ready to establish several lemmas. The first lemma will show that u1′ does not too close to the boundary, which is the main lemma to prove Theorem 2. Denote(17)M0=maxexp5η22η2M1+2+2,24Q+3C2η−156−Q.First, we come to compare the distances of pointsx′ and u1′ to the boundary G′.Lemma 1.
δG′u1′≥1/M0δG′x′.Proof.
We prove this lemma by a contradiction. Suppose on the contrary that(18)δG′u1′<1M0δG′x′.
Under the assumption “δG′u1′<1/M0δG′x′,” we first prove the following claim.Claim 1.
For eachi∈1,2,…,M1 and each z′∈x′,ui′, we have(19)δG′z′≥56δG′ui′+z′−ui′.
For this claim, we first show that for any pointz′∈x′,ui′,(20)δG′z′≥56z′−w′.
We will prove this inequality by a contradiction. Suppose on the contrary that there exist an integeri∈1,2,…,M1 and one point z0′∈x′,ui′ such that the inequality(21)δG′z0′<56z0′−w′,holds.
Sinceu1′∈ui′,w′∩fB¯1 and w′∈∂G′, we have(22)ui′−w′≥u1′−w′≥distB¯1′,∂G′≥12ε.
This, together with the assumption “δG′z0′<5/6z0′−w′,” implies(23)δG′x′≤x′−z0′+δG′z0′<x′−z0′+56z0′−w′=x′−z0′+z0′−w′−16z0′−w′<x′−w′−16ui′−w′≤x′−w′−2ε,and so(24)x′−w′>δG′x′+2ε,which contradicts with (13). Hence, (20) is true.
Again, since (20) and w′∈∂G′, we have(25)δG′z′≥56z′−w′=56z′−ui′+ui′−w′≥56z′−ui′+δG′ui′.
Then, Claim1 holds.
Next, we come to prove the following inequality:(26)δG′uM1′<1M2δG′x′.
Becauseu1∈S1 and uM1∈SM1, we see from Theorem 3 that(27)kGx,u1≤8x−u17δGx=17,kGx,uM1≥2M1+22M1+2+2⋅x−uM1δGx=12M1+2+2.
From the above two inequalities, we get(28)kGx,u1kGx,uM1≤2M1+2+27.
SinceδG′u1′<1/M0δG′x′, we see from Theorem 3 that(29)kG′x′,u1′≥logδG′x′δG′u1′>logM0.
For any pointξ1′∈x′,uM1′, by Claim 1, we get(30)δG′ξ1′≥56δG′uM1′+ξ1′−uM1′.
Meanwhile, we also get from Claim1 that(31)δG′uM1′+x′−uM1′≤65δG′x′.
The above two inequalities show that(32)kG′x′,uM1′≤∫x′,uM1′dzδG′z≤∫0x′−uM1′65⋅duδG′uM1′+u=65log1+x′−uM1′δG′uM1′≤65log6δG′x′5δG′uM1′.
It follows from the condition “f being η-QHQS” that(33)kG′x′,u1′kG′x′,uM1′≤ηkGx,u1kGx,uM1.
The inequality (34) is obtained by (28), (29) and (32)(34)δG′uM1′<1M2δG′x′,where M2=5/6exp56η2M1+2+2/7−1logM0. Therefore, (26) holds.
By (26), for any i∈1,2,…,M1, we have(35)uM1′−ui′≤uM1′−w′=x′−w′−x′−uM1′≤x′−w′−δG′x′−δG′uM1′≤1+εδG′x′−δG′x′+1M2δG′x′=ε+1M2δG′x′.
Obviously,(36)ui′∈BuM1′,δG′x′.
And so(37)ui′∈Si′∩BuM1′,δG′x′.
This implies(38)Si′∩BuM1′,δG′x′≠∅,for any i∈1,2,…,M1. By this fact, we have the following claim.Claim 2.
Suppose thati≠j∈3,4,…,M1, and points u′∈Si′∩BuM1′,δG′x′ and v′∈Sj′∩BuM1′,δG′x′. Then,(39)Bu′,δG′u′8∩Bv′,δG′v′8=∅.
We shall prove this claim by a contradiction. Suppose on the contrary that there exist two integersi0≠j0∈3,4,…,M1, such that there are two points u0′∈Si0′∩BuM1′,δG′x′ and v0′∈Sj0′∩BuM1′,δG′x′ which satisfy(40)Bu0′,δG′u0′8∩Bv0′,δG′v0′8≠∅.
Because pointsu0,ui0∈Si0, and by Theorem 3, we get(41)kGx,u0≥2i0+22i0+2+2⋅x−u0δGx=12i0+2+2,kGx,ui0≤2i0+22i0+2−1⋅x−ui0δGx=12i0+2−1.
These show that(42)kGx,ui0kGx,u0≤2i0+2+22i0+2−1≤3431.
The choice of pointui0′, (20) and (26) and Theorem 3, implies(43)δG′ui0′≤ui0′−w′≤uM1′−w′≤65δG′uM1′<65M2δG′x′.
Thus,(44)kG′x′,ui0′≥logδG′x′δG′ui0′>log5M26.
It follows from the condition “f being η-QHQS” that(45)kG′x′,ui0′kG′x′,u0′≤ηkGx,ui0kGx,u0.
Then, (42) and (44) imply(46)kG′x′,u0′≥kG′x′,ui0′⋅ηkGx,ui0kGx,u0−1>3η2.
On the other hand, we come to prove the following inequality:(47)kG′x′,u0′<27η2063.
To this end, without loss of generality, we may assume thati0>j0. The choice of points u0∈Si0 and v0∈Sj0 yields that(48)u0−v0≥u0−x−x−v0=12j0+2−12i0+2δGx=2i0−j0−12i0+2δGx.
Therefore, by Theorem3, we get(49)kGu0,v0≥2j0+22j0+2+2⋅u0−v0δGx=2i0+2−2j0+22i0+j0+2+2i0+3,kGx,u0≤2i0+22i0+2−1⋅x−u0δGx=12i0+2−1.
These imply(50)kGx,u0kGu0,v0≤2i0+2i0−j0+12i0+2−12i0−j0−1<2i0+42i0+2−1<2063.
According to the assumptionBu0′,δG′u0′/8∩Bv0′,δG′v0′/8≠∅, we can find one point ξ2′ such that(51)ξ2′∈Bu0′,δG′u0′8∩Bv0′,δG′v0′8.
Again, we see from Theorem3 that(52)kG′ξ2′,v0′≤8v0′−ξ2′7δG′v0′<17,kG′u0′,ξ2′≤8u0′−ξ2′7δG′u0′<17.
Then, we have(53)kG′u0′,v0′≤kG′u0′,ξ2′+kG′ξ2′,v0′<27.
It follows from (50) and (53) and the condition “f being η-QHQS” that(54)kG′x′,u0′≤ηkGx,u0kGu0,v0⋅kG′u0′,v0′<27η2063.
Thus, (47) holds. But it contradicts with (46). Hence, we complete the proof of Claim 2.Claim 3.
There must exist an integeri1∈3,4,…,M1 such that(55)δG′u′≤δG′x′4M3,for each point u′∈Si1′∩BuM1′,δG′x′, where M3=24Q+3C2η−15/6−Q.
Also, we will prove this claim by a contradiction. Suppose on the contrary that for any integeri∈3,4,…,M1, there are some points xi′∈Si′∩BuM1′,δG′x′ such that(56)δG′xi′>δG′x′4M3.
DenoteBxi′=Bxi′,δG′x′/8M3. For any i≠j∈3,4,…,M1, by Claim 2, we have(57)Bxi′∩Bxj′=∅.
Now, we will prove that all ofBxi′ are contained in BuM1′,3/2δG′x′.
For any pointz′∈Bxi′, we have(58)uM1′−z′≤uM1′−xi′+xi′−z′<δG′x′+δG′x′8M3<32δG′x′.
Then,(59)Bxi′⊂BuM1′,32δG′x′.
SinceE′ is a Q-regular Banach space, we have(60)∑i=3M1μBxi′≤μBuM1′,32δG′x′≤C32δG′x′Q,∑i=3M1μBxi′≥M1−2CδG′x′8M3Q>C232δG′x′Q>C32δG′x′Q.
This is the desired contradiction, and hence, we complete the proof of Claim3.
Claim3 shows the relationship between the distance of point x′ to the boundary G′ and the distance of point on Si1′∩BuM1′,δG′x′ to the boundary G′. Now, we come to compare the distance of point x′ to the boundary G′ with the diameter of Si1′∩BuM1′,δG′x′.
LetD′=S2′∩BuM1′,δG′x′ and Di1′=Si1′∩BuM1′,δG′x′. Then, we will show that(61)diamDi1′≥1M3δG′x′.
To this end, we illustrate the proof of (61) in two cases.
IfSi1′⊂B¯uM1′,δG′x′, then Di1′=Si1′. Since x and ∞ are separated by Si1, and f is a homeomorphism and clusf,∞=∞, we know that x′ and ∞ are separated by Si1′. Let L be a line which contains x′,w′. Then, x′ divides L into two parts L1 and L2. No loss of generality, we assume that L1 contains w′. Since x′ and ∞ are separated by Si1′, we know L2∩Si1′≠∅. And so there exists one point z′∈L2∩Si1′. Therefore, the condition z′,x′,uM1′,ui1′⊂L implies(62)diamDi1′=diamSi1′≥z′−ui1′=z′−x′+x′−uM1′+uM1′−ui1′≥x′−uM1′≥δG′x′−δG′uM1′≥1−1M2δG′x′>1M3δG′x′.
IfSi1′⊂B¯uM1′,δG′x′, then there exists some point p∈Si1′ with p∉BuM1′,δG′x′. Since ui1′∈Di1′ and Si1′ is connected, we get from (35) that there exists some point. z′∈Si1′∩SuM1′,SG′x′. Again by (35), we have(63)diamDi1′≥z′−ui1′≥z′−uM1′−uM1′−ui1′≥δG′x′−ε+1M2δG′x′>1M3δG′x′.
Hence, (61) holds.
For eachk∈1,2,3,…,M3/4, it follows from (61) that there exists some point yk′∈Di1′ such that(64)yk′−ui1′=kδG′x′M3.
Takingwk′∈∂G′ with(65)yk′−wk′≤1+εδG′yk′.
Next, letzk′ be the first intersection point of yk′,vk′ with S2′ starting from yk′ to wk′.
From the above construction, we get two sequencesyk′k=1M3/4 and zk′k=1M3/4 in G′. Furthermore, we also get two corresponding sequences ykk=1M3/4 and zkk=1M3/4 in G satisfying zk∈S2 and yk∈Si1 for all k∈1,2,3,…,M3/4. Based on the above facts, we have the following claims.Claim 4.
For anyk≠t∈1,2,3,…,M3/4, we have(66)kGzk,zt≥η−11113⋅kGzk,yk.
To prove this claim, without loss of generality, we may assume thatk>t.
Based on the construction of two sequencesyk′k=1M3/4 and zk′k=1M3/4, we get(67)yk′−yt′≥yk′−y0′−y0′−yt′=k−tM3δG′x′>δG′x′M3.
It follows from Claim3 that(68)yk′−zk′<yk′−wk′<1+εδG′yk′<1+εδG′x′4M3.
The above two inequalities show that(69)zk′−zt′≥yk′−yt′−zk′−yk′−yt′−zt′>1−εδG′x′2M3.
By Theorem2, we have(70)kG′yk′,zk′≤4M34M3−1−ε⋅yk′−zk′δG′x′<1+ε4M3−1−ε<1348M3,kG′zk′,zt′≥log1+zk′−zt′δG′zk′≥log1+1−ε2M3>1−ε4M3>1148M3.
These, together with the condition “f being η-QHQS,” imply(71)kGzk,ztkGyk,zk≥η−1kG′zk′,zt′kG′yk′,zk′≥η−11113.
Hence, Claim4 holds.Claim 5.
For anyk≠t∈1,2,3,…,M3/4, we have(72)Bzk,r1∩Bzt,r1=∅,where r1=η−111/13/27⋅δGx.
According to the condition pointszk,zt∈S2 and yk∈Si1, by Theorem 2, we get(73)kGzk,yk≥89⋅zk−ykδGx≥89⋅124−12i1+2=2i1−2−118⋅2i1−2≥136,kGzk,zt≤16zk−zt15δGx.
These, together with Claim4, imply(74)zk−zt≥5η−111/133⋅26δGx>2r1,where r1=η−111/13/27⋅δGx.
Therefore,(75)Bzk,r1∩Bzt,r1=∅.
So Claim5 holds.
Combined with the above claims, now we continue to complete the proof of Lemma1.
Suppose thatk∈1,2,…,M3/4, and points z∈Bzk,r1 and zk∈S2, respectively. Then, we have(76)z−x≤z−zk+zk−x≤r1+δGx16<18δGx.
The last inequality holds sinceη1>1, and so(77)Bzk,r1⊂B1.
SinceE is a Q-regular Banach space, and by Claim 5, we get(78)μB1≤CδGx8Q,μB1≥∑k=1M3/4μBzk,r1≥M34C−1r1Q>CδGx8Q.
This is the desired contradiction, and hence, we complete the proof of Lemma1.
Next, we come to estimate the quasihyperbolic distances between the pointx′ and any point in B1′.Lemma 2.
For any pointz′∈B1′, we have(79)kG′x′,z′<c1,where c1=η10/7log72M0/71.Proof.
We see from Theorem2 and u1∈S1 that(80)kGx,z≤8x−z7δGx<17,kGx,u1≥4x−u15δGx=110.
The above two inequalities show that(81)kGx,zkGx,u1<107.
Sincew′∈∂G′, and by Lemma 1, we get(82)x′−u1′=x′−w′−w′−u1′≤δG′x′+16ε−w′−u1′≤δG′x′+172w′−u1′−w′−u1′=δG′x′−7172w′−u1′≤δG′x′−7172δG′u1′≤1−7172M0δG′x′.
This shows that(83)kG′x′,u1′≤∫x′,u1′dzδG′z≤∫0x′−u1′dsδG′x′−s=logδG′x′δG′x′−x′−u1′≤log72M071.
It follows from (81) and (83) and the condition “f being η-QHQS” that(84)kG′x′,z′≤ηkGx,zkGx,u1⋅kG′x′,u1′<η107log72M071=c1.
Finally, we will discuss thatf is 1/8-locally η1-QS.Lemma 3.
The restrictionf|B1 is η1-QS with η1t=μ22⋅ημ12t, where μ1=μ1/7 and μ2=μc1 are the constants of Theorem 4.Proof.
We take three pointsv1, v2, and v3 in B1. By Theorem 2, we have(85)kGx,vi≤8x−v17δGx<17,for each i∈1,2,3. Then, v1,v2,v3∈B¯kGx,1/7.
It follows from Theorem4 that there is a constant μ1=μ1/7 such that(86)kGv1,v3<μ1⋅v1−v3δGx,kGv2,v3>1μ1⋅v2−v3δGx.
These show that(87)kGv1,v3kGv2,v3<μ12v1−v3v2−v3.
Moreover, we infer fromvi∈B1 and Lemma 2 that(88)kG′x′,vi′<c1,for i∈1,2,3. Then, v1′,v2′,v3′∈B¯kG′x′,c1.
It follows from Theorem4 and the condition “f being η-QHQS” that there is another constant μ2=μc1 such that(89)v1′−v3′≤μ2⋅kG′v1′,v3′⋅δG′x′≤μ2⋅ηkGv1,v3kGv2,v3⋅kG′v2′,v3′⋅δG′x′≤μ22⋅ηkGv1,v3kGv2,v3⋅v2′−v3′.
Thus, (87) and (89) imply(90)v1′−v3′v2′−v3′≤μ22⋅ηkGv1,v3kGv2,v3≤μ22⋅ημ12⋅v1−v3v2−v3.
Therefore,f|B1 is η1-QS with η1t=μ22⋅ημ12t. Lemma 3 holds.
Hence, we complete the proof of Theorem2.



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