# On Bilipschitz Extensions in Real Banach Spaces

Authors: M. Huang; Y. Li
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Suppose thatE and E' denote real Banach spaces with dimension at least 2, that D≠E and D'≠E' are bounded domains with connected boundaries, that f:D→D' is an M-QH homeomorphism, and that D' is uniform. The main aim of this paper is to prove that f extends to a homeomorphism f¯:D¯→D¯' and f-∣∂D is bilipschitz if and only if f is bilipschitz in D¯. The answer to some open problems of Väisälä is affirmative under a natural additional condition.


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## Body

## 1. Introduction and Main Results

During the past three decades, the quasihyperbolic metric has become an important tool in geometric function theory and in its generalizations to metric spaces and Banach spaces [1]. Yet, some basic questions of the quasihyperbolic geometry in Banach spaces are open. For instance, only recently the convexity of quasihyperbolic balls has been studied in [2, 3] in the setup of Banach spaces.Our study is motivated by Väisälä’s theory of freely quasiconformal maps and other related maps in the setup of Banach spaces [1, 4, 5]. Our goal is to study some of the open problems formulated by him. We begin with some basic definitions and the statements of our results. The proofs and necessary supplementary notation terminology will be given thereafter.Throughout the paper, we always assume thatE and E′ denote real Banach spaces with dimension at least 2. The norm of a vector z in E is written as |z|, and for every pair of points z1, z2 in E, the distance between them is denoted by |z1-z2|, the closed line segment with endpoints z1 and z2 by [z1,z2]. We begin with the following concepts following closely the notation and terminology of [4–8] or [9].We first recall some definitions.Definition 1.
A domainD in E is called c-uniform in the norm metric, provided there exists a constant c with the property that each pair of points  z1,z2 in D can be joined by a rectifiable arc α in D satisfying (1)
minj=1,2ℓ(α[zj,z])≤cdD(z) for all z∈α, and(2)
ℓ(α)≤c|z1-z2|,
whereℓ(α) denotes the length of α, α[zj,z] the part of α between zj and z, and dD(z) the distance from z to the boundary  ∂D of D.Definition 2.
SupposeGE,G′E′, and M≥1. We say that a homeomorphism f:G→G′ is  M-bilipschitz if
(1)1M|x-y|≤|f(x)-f(y)|≤M|x-y|
for all x, y∈G, and M-QH if
(2)1MkG(x,y)≤kG′(f(x),f(y))≤MkG(x,y)
for all x, y∈G.As for the extension of bilipschitz maps inℝ2, Ahlfors [10] proved that if a planar curve through ∞ admits a quasiconformal reflection, it also admits a bilipschitz reflection. Furthermore, Gehring gave generalizations of Ahlfors’ result in the plane.Theorem A (see [11, Theorem  7]).
Suppose thatD is a K-quasidisk in ℝ2, that D′ is a Jordan domain in ℝ2, and that ϕ:∂D→∂D′ is L1-bilipschitz. Then there exist L-bilipschitz f:D¯→D′¯ and f⋆:D⋆¯→D′⋆¯ such that f=f⋆=ϕ on ∂D and L depends only on K and L1, where D⋆=ℝ¯2∖D¯ and D′⋆=ℝ¯2∖D′¯.Tukia and Väisälä [12] dealt with the curious phenomenon that sometimes a quasiconformal property implies the corresponding bilipschitz property.Theorem B (see [12, Theorem  2.12]).
Suppose thatX is a closed set in ℝn, n≠4, and that f:ℝn→ℝn is a K-QC map such that f∣X is L-bilipschitz. Then there is an L1-bilipschitz map g:ℝn→ℝn such that(1)
g∣X=f∣X;(2)
g(D)=f(D) for each component D of ℝn∖X;(3)
L1 depends only on K, L, and n.In [13], Gehring raised the following two related problems.Open Problem 1.
Suppose thatD is a Jordan domain in ℝ¯2 and that f∣∂D is M-bilipschitz. Characterize mappings f having M′-bilipschitz extension to D with M′=M′(c,M).Open Problem 2.
Suppose thatD is a Jordan domain in ℝ¯2. For which domains D does each M-bilipschitz f in the ∂D have M′-bilipschitz extension to D with  M′=M′(c,M)?Gehring himself discussed these two problems and got the following two results.Theorem C (see [13, Theorem  2.11]).
Suppose thatD and D′ are Jordan domains in ℝ¯2 and that ∞∈D′ if and only if ∞∈D. Suppose also that f:D→D′ is a K-quasiconformal mapping and that f extends to a homeomorphism f:D¯→D′¯ such that f∣∂D is M-bilipschitz. Then there exists an M-bilipschitz map g:D¯→D′¯ with g∣∂D=f∣∂D, where M′=M′(M,K).Theorem D (see [13, Theorem  4.9]).
Suppose thatD and D′ are Jordan domains in ℝ¯2. Then each M-bilipschitz f in ∂D has an M′-bilipschitz extension g:D→D′ with g∣∂D=f∣∂D if and only if D is a K-quasidisk, where  M′=M′(M,K) and K=K(M).We remark that Theorem C is a partial answer to Open Problem1 and Theorem D is an affirmative answer to Open Problem 2. In the proof of Theorem C, the modulus of a path family, which is an important tool in the quasiconformal theory in ℝn, was applied. In general, this tool is no longer applicable in the context of Banach spaces (see [4]). A natural problem is whether Theorem C   is true or false in Banach spaces. In fact, this problem was raised by Väisälä in [1] in the following form.Open Problem 3.
Suppose thatD and D′ are bounded domains with connected boundaries in E and E′. Suppose also that f:D→D′ is M-QH and that f extends to a homeomorphism f:D¯→D′¯ such that f∣∂D is M-bilipschitz. Is it true that fM′-bilipschitz with M′=M′(c,M)?Our result is as follows.Theorem 3.
Suppose thatD and D′ are bounded domains with connected boundaries in E and E′, respectively. Suppose also that f:D→D′ is M-QH and that f extends to a homeomorphism f¯:D¯→D′¯ such that f∣∂D is M-bilipschitz. If D′ is a c-uniform domain, then f is M′-bilipschitz with M′=M′(c,M).We see from Theorem3 that the answer to Open Problem 3 is positive by replacing the hypothesis “D′ being bounded" in Open Problem 3 with the one “D′ being bounded and uniform.”The organization of this paper is as follows. The proof of Theorem3 will be given in Section 3.1. In Section 2, some preliminaries are introduced.


## 2. Preliminaries

Thequasihyperbolic length of a rectifiable arc or a path α in the norm metric in D is the number (cf. [14, 15])
(3)ℓk(α)=∫α|dz|dD(z).For each pair of pointsz1, z2 in D, the quasihyperbolic distancekD(z1,z2) between z1 and z2 is defined in the usual way:
(4)kD(z1,z2)=infℓk(α),
where the infimum is taken over all rectifiable arcs α  joining  z1 to z2 in D. For all z1, z2 in D, we have (cf. [15])
(5)kD(z1,z2)≥inf{log(1+ℓ(α)min{dD(z1),dD(z2)})}≥|logdD(z2)dD(z1)|,
where the infimum is taken over all rectifiable curves α in D connecting z1 and z2.In [5], Väisälä characterized uniform domains by the quasihyperbolic metric.Theorem E (see [5, Theorem  6.16]).
For a domainD, the following are quantitatively equivalent: (1)
D is a c-uniform domain;(2)
kD(z1,z2)≤c′log(1+|z1-z2|/min{dD(z1),dD(z2)}) for all z1,z2∈D;(3)
kD(z1,z2)≤c1′log(1+|z1-z2|/min{dD(z1),dD(z2)})+d for all z1,z2∈D.Gehring and Palka [14] introduced the quasihyperbolic metric of a domain in ℝn, and it has been recently used by many authors in the study of quasiconformal mappings and related questions [16]. In the case of domains in ℝn, the equivalence of items (1) and (3) in Theorem E is due to Gehring and Osgood [17] and the equivalence of items (2) and (3) is due to Vuorinen [18]. Many of the basic properties of this metric may be found in [4, 5, 17].Recall that an arcα from z1 to z2 is a quasihyperbolic geodesic if ℓk(α)=kD(z1,z2). Each subarc of a quasihyperbolic geodesic is obviously a quasihyperbolic geodesic. It is known that a quasihyperbolic geodesic between every pair of points in E exists if the dimension of E is finite, see [17, Lemma  1]. This is not true in arbitrary spaces (cf. [19, Example  2.9]). In order to remedy this shortage, Väisälä introduced the following concepts [5].Definition 4.
Letα be an arc in E. The arc may be closed, open, or half open. Let  x¯=(x0,…,xn), n≥1, be a finite sequence of successive points of α. For h≥0, we say that x¯ is  h-coarse if kD(xj-1,xj)≥h for all 1≤j≤n. Let Φk(α,h) be the family of all h-coarse sequences of α. Set
(6)sk(x¯)=∑j=1nkD(xj-1,xj),ℓkD(α,h)=sup{sk(x¯):x¯∈Φk(α,h)}
with the agreement that ℓk(α,h)=0 if Φk(α,h)=∅. Then the number ℓk(α,h) is the h-coarse quasihyperbolic length of α.In this paper, we will use this concept in the case whereD is a domain equipped with the quasihyperbolic metric kD. We always use ℓk(α,h) to denote the h-coarse quasihyperbolic length of α.Definition 5.
LetD be a domain in E. An arc α⊂D is  (ν,h)-solid with ν≥1 and h≥0 if
(7)ℓk(α[x,y],h)≤νkD(x,y)
for all x,y∈α. A (ν,0)-solid arc is said to be a ν-neargeodesic, that is, an arc α⊂D is a ν- neargeodesic if and only if ℓk(α[x,y])≤νkD(x,y) for all x,y∈α.
Obviously, aν-neargeodesic is a quasihyperbolic geodesic if and only if ν=1.
In [19], Väisälä got the following property concerning the existence of neargeodesic in  E.Theorem F (see [19, Theorem  3.3]).
Let{z1,z2}⊂D and ν>1. Then there is a ν-neargeodesic in D joining z1 and z2.The following result due to Väisälä is from [5].Theorem G (see [5, Theorem  4.15]).
For domainsD≠E and D′≠E′, suppose that f:D→D′ is M-QH. If γ is a c-neargeodesic in D, then the arc γ′ is c1-neargeodesic in D′ with c1 depending only on c and M.
LetG≠E and G′≠E′ be metric spaces, and let φ:[0,∞)→[0,∞) be a growth function, that is, a homeomorphism with φ(t)≥t. We say that a homeomorphism  f:G→G′  is φ-semisolid if
(8)kG′(f(x),f(y))≤φ(kG(x,y))
for all x,y∈G, and φ-solidif both fand f-1 satisfy this condition.
We say thatf is fully φ-semisolid (resp. fully φ-solid) if f is φ-semisolid (resp. φ-solid) on every subdomain of G. In particular, when G=E, corresponding subdomains are taken to be proper ones. Fully φ-solid mapsare also called freely φ-quasiconformal maps, or briefly φ-FQC maps.For convenience, in the following, we always assume thatx,y,z,… denote points in D and x′,y′,z′,…  the images in D′ of x,y,z,… under f, respectively. Also we assume that α,β,γ,… denote curves in D and α′,β′,γ′,… the images in  D′ of α,β,γ,… under f, respectively.


## 3. Bilipschitz Mappings

First we introduce the following Theorems.Theorem H (see [5, Theorem  7.18]).
LetD and D′ be domains in E and E′, respectively. Suppose that D is a c-uniform domain and that f:D→D′ is φ-FQC (see Section 2 for the definition). Then the following conditions are quantitatively equivalent:(1)
D′ is a c1-uniform domain;(2)
f is η-quasimöbius.Theorem I (see [20, Theorem  1.1]).
Suppose thatD is a c-uniform domain and that f:D→D′ is (M,C)-CQH, where DE and D′E′. Then the following conditions are quantitatively equivalent:(1)
D′ is a c1-uniform domain;(2)
f extends to a homeomorphism f¯:D¯→D¯′ and f¯ is η-QM rel ∂D.The following theorem easily follows from Theorems H  and  I.Theorem 6.
Suppose thatDE and D′E′, that D is a c-uniform domain, and that f:D→D′ is φ-FQC. Then the following conditions are quantitatively equivalent:(1)
D′ is a c1-uniform domain;(2)
f is θ-quasimöbius;(3)
f extends to a homeomorphism f¯:D¯→D¯′ and f¯ is θ1-QM rel ∂D.Let us recall the following three theorems which are useful in the proof of Theorem3.Theorem J (see [1, Theorem  2.44]).
Suppose thatGE and G′E′ is a c-uniform domain, and that f:G→G′ is M-QH. If D⊂G is a c-uniform domain, then D′=f(D) is a c′-uniform domain with c′=c′(c,M).Theorem K (see [5, Theorem  6.19]).
Suppose thatDE is a c-uniform domain and that γ is a c1-neargeodesic in D with endpoints z1 and z2. Then there is a constant b=b(c,c1)≥1 such that(1)
minj=1,2ℓ(γ[zj,z])≤bdD(z) for all z∈α, and(2)
ℓ(γ)≤b|z1-z2|.Theorem L (see [21, Theorem  1.2]).
Suppose thatD1 and D2 are convex domains in E, where D1 is bounded and D2 is c-uniform for some c>1, and that there exist z0∈D1∩D2 and r>0 such that 𝔹(z0,r)⊂D1∩D2. If there exist constants R1>0 and c0>1 such that R1≤c0r and D1⊂𝔹¯(z0,R1), then D1∪D2 is a c′-uniform domain with c′=(c+1)(2c0+1)+c.Basic Assumption A. In this paper, we always assume that D and D′ are bounded domains with connected boundaries in E and E′, respectively, that f:D→D′ is M-QH, that f extends to a homeomorphism f¯:D¯→D′¯ such that f¯∣∂D is M-bilipschitz, and that D′ is a c-uniform domain.Before the proof of Theorem3, we prove a series of lemmas.Lemma 7.
There is a constantM0=M0(M)>M such that if the points z1,z2∈D satisfies dist(z1,∂D)≤ε and dist(z2,∂D)≤ε for sufficiently small ε>0, then
(9)1M0|z1-z2|≤|z1′-z2′|≤M0|z1-z2|.Proof.
Letx1, x2∈∂D be such that |z1-x1|=(4/3)dist(z1,∂D), |z2-x2|≤(4/3)dist(z2,∂D) and |x1-x2|≤max{|z1-x1|,|z2-x2|}<3|x1-x2| for sufficiently small ε>0. It follows from  “f being M-QH in D and homeomorphic in D¯” that H(x,f)≤K (cf. [1]) for each x∈D, where K depends only on M. Hence,
(10)|z1′-x1′|<32K|x1′-x2′|,|z2′-x2′|<32K|x1′-x2′|.
If|z1-z2|≤(1/4K2M)max{|z1-x1|,|z2-x2|}, then for each z∈[z1,z2],
(11)dD(z)≥3K2M-14K2Mmax{|z1-x1|,|z2-x2|},
and so we have
(12)2|z1′-z2′|min{dD′(z1′),dD′(z2′)}≤log(1+|z1′-z2′|min{dD′(z1′),dD′(z2′)})≤kD′(z1′,z2′)≤MkD(z1,z2)≤M∫[z1,z2]|dz|dD(z)≤4K2M2|z1-z2|(3K2M-1)max{|z1-x1|,|z2-x2|},
which shows that
(13)|z1′-z2′|≤12K3M33K2M-1|z1-z2|.
If|z1-z2|>(1/4K2M)max{|z1-x1|,|z2-x2|}, then by the assumption “f being M-bilipschitz in ∂D,”
(14)|z1′-z2′|≤|z1′-x1′|+|z2′-x2′|+|x1′-x2′|≤(3K+1)|x1′-x2′|≤(3K+1)M|x1-x2|≤(12K+4)K2M2|z1-z2|.
The same discussion as the above shows that
(15)|z1-z2|≤(12K+4)K2M2|z1′-z2′|.Lemma 8.
There is a constantM1=M1(c,M) such that if the points x∈D and z∈𝕊(x,dD(x))∩D¯ satisfies dist(z,∂D)≤ε for sufficiently small ε>0, then
(16)|z′-x′|≤M1dD(x).Proof.
Letx0∈𝕊(x,dD(x))∩D¯ such that dist(x0,∂D)≤ε for sufficiently small ε>0, and let x2 be the intersection point of  𝕊(x0,(1/2)dD(x))  with [x0,x]. Then we have
(17)kD(x2,x)≤log(1+|x-x2|dD(x)-|x-x2|)≤logdD(x)dD(x2)=log2,
which implies that
(18)log|x2′-x′||x2′-x0′|≤kD′(x2′,x′)≤MkD(x2,x)=Mlog2.
Hence,
(19)|x2′-x′|≤2M|x2′-x0′|,
and so
(20)|x′-x0′|≤|x′-x2′|+|x2′-x0′|≤(2M+1)|x2′-x0′|.
LetT be a 2-dimensional linear subspace of E which contains x0 and x2, and we use τ to denote the circle T∩𝕊(x0,(1/2)dD(x)). Take w1∈τ∩∂D such that τ(x2,w1)⊂D and ℓ(τ[x2,w1])≤2dD(x). Let x1∈𝕊(x,dD(x))∩τ[x2,w1]∩D¯ and denote τ(x1,w1) by τ1.
Claim  1. There must exist a 232-uniform domain D1 in D and x3∈∂D1∩D¯ satisfying dist(x3,∂D)≤ε for sufficiently small ε>0 such that x0, x∈D¯1 and (1/12)dD(x)≤|x3-x0|≤(11/12)dD(x).
IfdD(x1)=0, then we take D1=𝔹(x,dD(x)) and x3=x1. Obviously, |x3-x0|=(1/2)dD(x).  Hence Claim 1 holds true in this case.
IfdD(x1)>0, we divide the proof of Claim 1 into two parts.
Case  1. (dD(x1)≤(5/12)dD(x)). Then we take D1=𝔹(x,dD(x))∪𝔹(x1,dD(x1)) and x3∈𝕊(x1,dD(x1))∩D¯ such that dist(x3,∂D)≤ε for sufficiently small ε>0. It follows from Theorem  L that D1 is a 29-uniform domain and
(21)112dD(x)≤|x1-x0|-|x1-x3|≤|x3-x0|≤|x1-x0|+|x1-x3|≤1112dD(x),
from which we see that Claim 1 is true.
Case  2. (dD(x1)>(5/12)dD(x)). Obviously, dD(x1)>(5/6)|x1-x0|. We let w2∈τ1 be the first point along the direction from x1 to w1 such that
(22)dD(w2)=512dD(x).
If|w2-x1|≤(1/3)dD(x), then we take D1=𝔹(x,dD(x))∪𝔹(w2,dD(w2)),and let x3∈𝕊(w2,dD(w2))∩D¯ such that dist(x3,∂D)≤ε for sufficiently small  ε>0. Then
(23)dD(w2)+dD(x)-|w2-x|≥dD(w2)-|w2-x1|≥112dD(x),112dD(x)≤|x3-x0|≤|w2-x0|+|w2-x3|≤1112dD(x).
It follows from Theorem L thatD1 is a  677-uniform domain, which shows that Claim 1 is true.
If|w2-x1|>(1/3)dD(x),  then we first prove the following subclaim.
Subclaim  1. There exists a simply connected domain D1=∪i=0tBi in D, where t=1 or 2, such that  (1)
x0, x∈D¯1;(2)
for eachi∈{0,…,t}, (5/12)dD(x)≤ri≤dD(x);(3)
ift=2, then |x-w2|-r0-r2≥(1/144)dD(x);(4)
ri+ri+1-|vi-vi+1|≥(1/144)dD(x), where i∈{0,1} if t=2 or i=0 if t=1.
HereBi=𝔹(vi,ri),  vi∈τ[x2,w2],  v1∉B0, and v2∉τ[x2,v1].
To prove this subclaim, we lety2∈τ1 be such that |x1-y2|=(1/3)dD(x) and let C0=𝔹(x,dD(x)) and C1=𝔹(y2,dD(y2)). Since dD(y2)>(5/12)dD(x), we have
(24)dD(y2)+dD(x)-|y2-x|≥112dD(x).
Next, we construct a ball denoted byC2.Ifw2∈C¯1, then we let C2=𝔹(w2,dD(w2)).Ifw2∉C¯1, then we let y3 be the intersection of 𝕊(y2,dD(y2)) with τ1[y2,w1]. Since ℓ(τ1)≤2dD(x) and dD(z)≥(5/12)dD(x) for all z∈τ1(x1,x2), we have
(25)|w1-w2|+|w2-y3|+|y3-y2|+|y2-x1|+|x2-x1|≤ℓ(τ1)≤2dD(x),which implies that
(26)|w2-y3|≤13dD(x).
We take C2=𝔹(w2,dD(w2)). Then (26) implies
(27)dD(w2)+dD(x2)-|x2-w2|≥dD(w2)-|w2-y3|≥112dD(x).
Now we are ready to construct the needed domainD1.
IfdD(w2)+dD(x)-|w2-x|≥(1/48)dD(x), then we take B0=C0, B1=C2, and D1=B0∪B1 with v0=x, v1=w2, r0=dD(x), and r1=dD(w2). Obviously, D1 satisfies all the conditions in Subclaim 1. In this case, t=1.
IfdD(w2)+dD(x)-|w2-x|<(1/48)dD(x), then we take B0=𝔹(x,(35/36)dD(x)) with r0=(35/36)dD(x) and v0=x, B1=C1 with r1=dD(y2) and v1=y2, and B2=C2 with r2=dD(w2) and v2=w2. Then Inequalities (24) and (27) show that D1=⋃i=02‍Bi satisfies all the conditions in Subclaim 1. In this case, t=2.
Hence, the proof of Subclaim 1 is complete.
The following  follows from a similar argument as in the proof of [22, Theorem 1.1].
Corollary  9.The domainD1constructed in Subclaim   1  is a232-uniform domain.
Letx3∈𝕊(w2,dD(w2))∩D¯ such that dist(x3,∂D)≤ε for sufficiently small ε>0. Then
(28)112dD(x)≤|x3-x0|≤1112dD(x).
Then the proof of Claim 1 easily follows from (28), Subclaim 1, and Corollary 9.
We come back to the proof of Lemma8. It follows from (28) and Lemma 7 that
(29)|x-x3|≤|x-x0|+|x0-x3|≤2312dD(x),112M0dD(x)≤1M0|x3-x0|≤|x3′-x0′|≤M0|x3-x0|≤11M012dD(x).
Then it follows from Theorem  J that  D1′ is an M′-uniform domain, where M′=M′(c,M). Hence, we know from Theorem 6 that f-1 is a θ-Quasimöbius in D¯1, where θ=θ(c,M), and so (19), (20), (28), and (29) imply that
(30)123≤|x3-x0||x2-x0|·|x2-x||x-x3|≤θ(|x3′-x0′||x2′-x0′|·|x2′-x′||x′-x3′|)≤θ(M02M+1dD(x)|x′-x3′|),
which, together with (20), shows
(31)|x′-x0′|≤|x′-x3′|+|x3′-x0′|≤(2M+1θ-1(1/23)+11M012)dD(x)<2M0+2θ-1(1/23)dD(x).
Thus, the proof of Lemma 8 is complete.Lemma 10.
For allx∈D, if z∈𝕊(x,dD(x))∩D¯ such that  dist(x,∂D)≤ε for sufficiently small ε>0, then |z′-x′|≥(1/e4M0M12)dD(x), where M1=M1(c,M).Proof.
Suppose on the contrary that there exist pointsx1∈D and y1∈𝕊(x1,dD(x1))∩D¯ with dist(y1,∂D)≤ε for sufficiently small ε>0 such that
(32)|x1′-y1′|<1e4M0M12|x1-y1|.
We takey2∈𝕊(y1,dD(x1))∩D¯ such that dist(y2,∂D)≤ε for sufficiently small ε>0. From Lemma 7 we know that
(33)|y1′-y2′|≥1M0|y1-y2|=1M0dD(x1).
LetT1 be a 2-dimensional linear subspace of E determined by x1, y1 and y2, and ω the circle T1∩𝕊(y1,dD(x1)). We take y3∈ω∩∂D which satisfies ω(x1,y3)⊂D and ℓ(ω[x1,y3])≤4dD(x1). Let ω1=ω(x1,y3) and w1 be the first point along the direction from x1 to y3 such that
(34)dD(w1)=14M0M1dD(x1).
Letv1∈𝕊(w1,dD(w1))∩D¯ such that dist(w1,∂D)≤ε for sufficiently small ε>0. Then it follows from Lemma 8 that
(35)dD′(w1′)≤|w1′-v1′|≤M1dD(w1)=14M0dD(x1),
which, together with Lemmas 7 and 8 and (32), implies that
(36)|x1′-w1′|≥|y1′-v1′|-|x1′-y1′|-|v1′-w1′|≥1M0|y1-v1|-1e4M0M12|x1-y1|-M1|v1-w1|≥1M0(dD(x1)-dD(w1))-1e4M0M12|x1-y1|-M1|v1-w1|>12M0dD(x1).
Hence, we infer from (32) that
(37)kD′(x1′,w1′)≥log(1+|x1′-w1′|dD′(x1′))>M12.
Sinceℓ(ω1)≤4dD(x1), by the choice of w1, one has
(38)kD(x1,w1)≤∫ω1[x1,w1]|dx|dD(x)≤16M0M1,
whence
(39)kD′(x1′,w1′)≤MkD(x1,w1)≤16MM0M1,
which contradicts with (37). The proof of Lemma 10 is complete.Lemma 11.
Forx1∈D and x2∈∂D, we have
(40)|x1′-x2′|≤M2|x1-x2|,
where  M2=2M0+M1.Proof.
Forx1∈D, we let y1∈𝕊(x1,dD(x1))∩D¯ such that dist(y1,∂D)≤ε for sufficiently small ε>0. Then it follows from Lemma 8 that
(41)|x1′-y1′|≤M1|x1-y1|.
Forx2∈∂D, if  |y1-x2|≤2|x1-y1|, then by Lemma 7, we have
(42)|x1′-x2′|≤|x1′-y1′|+|y1′-x2′|≤M1|x1-y1|+M0|y1-x2|≤(2M0+M1)|x1-y1|≤(2M0+M1)|x1-x2|.
If|y1-x2|>2|y1-x1|, then we have
(43)|x1-x2|>|y1-x2|-|x1-y1|>12|y1-x2|.
Hence, by Lemma7 and (41),
(44)|x1′-x2′|≤|x1′-y1′|+|y1′-x2′|≤M1|x1-y1|+M0|y1-x2|≤(2M0+M1)|x1-x2|,
from which the proof follows.Lemma 12.
Forx1∈D and x2∈∂D, one has
(45)|x1′-x2′|≥1M3|x1-x2|,
where M3=2M0M1e(5MM0+8M0)M12.Proof.
We begin with a claim.
Claim   2. For all z∈D, we have dD′(z′)≥(1/e(5MM0+8M0)M12)dD(z).
To prove this claim, we letw2∈[z,y1] be such that |w2-y1|=(1/2M1e4M0M12)dD(z). It follows from [18] that
(46)kD(w2,z)≤log(1+|w2-z|dD(z)-|w2-z|)<5M0M12.
By Lemma 8, we have
(47)|w2′-y1′|≤M1|w2-y1|=12e4M0M12dD(z).
Hence, Lemma 10 implies |w2′-z′|≥(1/2e4M0M12)dD(z), whence
(48)log|w2′-z′|dD′(z′)≤kD′(w2′,z′)≤MkD(w2,z)≤5MM0M12,
which shows that Claim 2 is true.
Now we are ready to finish the proof of Lemma12. For x1∈D and x2∈∂D, if |x1-x2|≤2M0M1dD(x1), then by Claim 2,
(49)|x1′-x2′|≥dD′(x1′)≥1e(5MM0+8M0)M12dD(x1)≥12M0M1e(5MM0+8M0)M12|x1-x2|.
If|x1-x2|>2M0M1dD(x1), then we take w3∈𝕊(x1,dD(x1))∩D¯ such that dist(w3,∂D)≤ε for sufficiently small ε>0, and so
(50)|w3-x2|≥|x1-x2|-|x1-w3|≥(1-12M0M1)|x1-x2|,|w3-x2|≥|x1-x2|-|x1-w3|≥(2M0M1-1)|x1-w3|,
whence Lemmas 7 and 8 imply
(51)|x1′-x2′|≥|w3′-x2′|-|x1′-w3′|≥1M0|w3-x2|-M1|x1-w3|≥(1M0-M12M0M1-1)|w3-x2|≥13M0|x1-x2|,
from which the proof is complete.By the previous lemmas, we get the following result.Lemma 13.
D is a c1-uniform domain, where c1=c1(c,M).Proof.
We first prove thatf-1 is θ1-Quasimöbius rel ∂D′, where θ1(t)=(M2M3)2t, M2 and M3 are the same as in Lemmas 11 and 12, respectively. By definition, it is necessary to prove that for x1′, x2′,  x3′,  x4′∈D′¯,
(52)|x4-x1||x4-x2|·|x2-x3||x1-x3|≤(M2M3)2|x4′-x1′||x4′-x2′|·|x2′-x3′||x1′-x3′|,
where x1,x2∈∂D′. Obviously, to prove Inequality (52), we only need to consider the following three cases.
Case  3 (x1′,x2′,x3′,x4′∈∂D′). Since f is M-bilipschitz in ∂D, we have
(53)|x4-x1||x4-x2|·|x2-x3||x1-x3|≤M4|x4′-x1′||x4′-x2′|·|x2′-x3′||x1′-x3′|.
Case  4 (x1′, x2′, x3′∈∂D′ andx4′∈D′). It follows from Lemmas 11 and 12 that
(54)|x4-x1||x4-x2|·|x2-x3||x1-x3|≤M2M3|x4′-x1′||x4′-x2′|·M2|x2′-x3′||x1′-x3′|≤M2M2M3|x4′-x1′||x4′-x2′|·|x2′-x3′||x1′-x3′|.
Case  5 (x1′, x2′∈∂D′ and x3′,x4′∈D′). We obtain from Lemmas 11 and 12 that
(55)|x4-x1||x4-x2|·|x2-x3||x1-x3|≤M2M3|x4′-x1′||x4′-x2′|·M2M3|x2′-x3′||x1′-x3′|≤(M2M3)2|x4′-x1′||x4′-x2′|·|x2′-x3′||x1′-x3′|.
The combination of Cases3~5  shows that Inequality (52) holds, which implies that f-1 is a θ1-Quasimöbius rel ∂D′. Hence, Theorem 6 shows that D is a c1-uniform domain, where c1 depends only on c and M.


### 3.1. The Proof of Theorem3

For anyz1, z2∈D¯, it suffices to prove that
(56)1M′|z1-z2|≤|z1′-z2′|≤M′|z1-z2|,
where M′ depends only on c and M.It follows from the hypothesis “f being M-bilipschitz in ∂D,” Lemmas 11 and 12 that we only need to consider the case z1,z2∈D.If|z1-z2|≤(1/2)max{dD(z1),dD(z2)}, then
(57)kD(z1,z2)≤∫[z1,z2]‍|dx|dD(x)≤2|z1-z2|max{dD(z1),dD(z2)}≤1,
which shows that
(58)log(1+|z1′-z2′|min{dD′(z1′),dD′(z2′)})≤kD′(z1′,z2′)≤MkD(z1,z2)≤M,
and so
(59)|z1′-z2′|eMmin{dD′(z1′),dD′(z2′)}≤log(1+|z1′-z2′|min{dD′(z1′),dD′(z2′)})≤2M|z1-z2|max{dD(z1),dD(z2)}.
We see from Lemma 8 that
(60)min{dD′(z1′),dD′(z2′)}≤M1max{dD(z1),dD(z2)}.
Then (59) implies that
(61)|z1′-z2′|≤2MM1eM|z1-z2|.For the other case|z1-z2|>(1/2)max{dD(z1),dD(z2)}, we let β be a 2-neargeodesic joining z1 and z2 in D. It follows from Theorem G that β′ is a c2-neargeodesic, where c2 depends only on M. Let z′∈β′ such that
(62)|z1′-z′|=12|z1′-z2′|.
Then we know from |z2′-z′|≥(1/2)|z1′-z2′| and Theorem K that
(63)|z1′-z2′|≤2min{|z1′-z′|,|z2′-z′|}≤2min{diam(z1′,z′),diam(z2′,z′)}≤2μdD′(z′),
where μ depends only on c and M.We claim that(64)dD(z)≤3ℓ(β).
Otherwise,
(65)max{dD(z1),dD(z2)}≥dD(z)-max{|z1-z|,|z2-z|}>2ℓ(β)≥2|z1-z2|.
This is the desired contradiction.By Theorem K and Lemma13, we have
(66)dD(z)≤3ℓ(β)≤3b|z1-z2|,
where b=b(c1). Hence, Lemma 8 and (63) show that
(67)|z1′-z2′|≤2μdD′(z′)≤6bM1μ|z1-z2|.By Lemma13, we see that D is a c1-uniform domain. Hence a similar argument as in the proofs of Inequalities (61) and (67) yields that
(68)|z1-z2|≤M4|z1′-z2′|,
where M4=M4(c,M).Obviously, the inequalities (61), (67), and (68) show that (56) holds, and thus the proof of the theorem is complete.


## 3.1. The Proof of Theorem3

For anyz1, z2∈D¯, it suffices to prove that
(56)1M′|z1-z2|≤|z1′-z2′|≤M′|z1-z2|,
where M′ depends only on c and M.It follows from the hypothesis “f being M-bilipschitz in ∂D,” Lemmas 11 and 12 that we only need to consider the case z1,z2∈D.If|z1-z2|≤(1/2)max{dD(z1),dD(z2)}, then
(57)kD(z1,z2)≤∫[z1,z2]‍|dx|dD(x)≤2|z1-z2|max{dD(z1),dD(z2)}≤1,
which shows that
(58)log(1+|z1′-z2′|min{dD′(z1′),dD′(z2′)})≤kD′(z1′,z2′)≤MkD(z1,z2)≤M,
and so
(59)|z1′-z2′|eMmin{dD′(z1′),dD′(z2′)}≤log(1+|z1′-z2′|min{dD′(z1′),dD′(z2′)})≤2M|z1-z2|max{dD(z1),dD(z2)}.
We see from Lemma 8 that
(60)min{dD′(z1′),dD′(z2′)}≤M1max{dD(z1),dD(z2)}.
Then (59) implies that
(61)|z1′-z2′|≤2MM1eM|z1-z2|.For the other case|z1-z2|>(1/2)max{dD(z1),dD(z2)}, we let β be a 2-neargeodesic joining z1 and z2 in D. It follows from Theorem G that β′ is a c2-neargeodesic, where c2 depends only on M. Let z′∈β′ such that
(62)|z1′-z′|=12|z1′-z2′|.
Then we know from |z2′-z′|≥(1/2)|z1′-z2′| and Theorem K that
(63)|z1′-z2′|≤2min{|z1′-z′|,|z2′-z′|}≤2min{diam(z1′,z′),diam(z2′,z′)}≤2μdD′(z′),
where μ depends only on c and M.We claim that(64)dD(z)≤3ℓ(β).
Otherwise,
(65)max{dD(z1),dD(z2)}≥dD(z)-max{|z1-z|,|z2-z|}>2ℓ(β)≥2|z1-z2|.
This is the desired contradiction.By Theorem K and Lemma13, we have
(66)dD(z)≤3ℓ(β)≤3b|z1-z2|,
where b=b(c1). Hence, Lemma 8 and (63) show that
(67)|z1′-z2′|≤2μdD′(z′)≤6bM1μ|z1-z2|.By Lemma13, we see that D is a c1-uniform domain. Hence a similar argument as in the proofs of Inequalities (61) and (67) yields that
(68)|z1-z2|≤M4|z1′-z2′|,
where M4=M4(c,M).Obviously, the inequalities (61), (67), and (68) show that (56) holds, and thus the proof of the theorem is complete.



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*Source: 765685-2013-04-14.xml*