On Bilipschitz Extensions in Real Banach Spaces

Abstract

Suppose that 𝐸 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.

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 that 𝐸 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 domain 𝐷 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)min_𝑗=1,2ℓ(𝛼[𝑧_𝑗,𝑧])≤𝑐𝑑_𝐷(𝑧) for all 𝑧∈𝛼, and(2)ℓ(𝛼)≤𝑐|𝑧₁−𝑧₂|,
where ℓ(𝛼) denotes the length of 𝛼, 𝛼[𝑧_𝑗,𝑧] the part of 𝛼 between 𝑧_𝑗 and 𝑧, and 𝑑_𝐷(𝑧) the distance from 𝑧 to the boundary 𝜕𝐷 of 𝐷.

Definition 2. Suppose 𝐺⫅_/𝐸,𝐺⫅^′_/𝐸^′, and 𝑀≥1. We say that a homeomorphism 𝑓∶𝐺→𝐺^′ is 𝑀-bilipschitz if 1𝑀𝑥−𝑦≤𝑓(𝑥)−𝑓𝑦≤𝑀𝑥−𝑦 (1) for all 𝑥, 𝑦∈𝐺, and 𝑀-QH if 1𝑀𝑘_𝐺𝑥,𝑦≤𝑘_𝐺′𝑓(𝑥),𝑓𝑦≤𝑀𝑘_𝐺𝑥,𝑦 (2) for all 𝑥, 𝑦∈𝐺.

As for the extension of bilipschitz maps in ℝ², 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 that 𝐷 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^⋆=ℝ²⧵𝐷 and 𝐷^′⋆=ℝ²⧵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 that 𝑋 is a closed set in ℝ^𝑛, 𝑛≠4, and that 𝑓∶ℝ^𝑛→ℝ^𝑛 is a 𝐾-QC map such that 𝑓∣_𝑋 is 𝐿-bilipschitz. Then there is an 𝐿₁-bilipschitz map 𝑔∶ℝ^𝑛→ℝ^𝑛 such that(1)𝑔∣_𝑋=𝑓∣_𝑋;(2)𝑔(𝐷)=𝑓(𝐷) for each component 𝐷 of ℝ^𝑛⧵𝑋;(3)𝐿₁ depends only on 𝐾, 𝐿, and 𝑛.

In [13], Gehring raised the following two related problems.

Open Problem 1. Suppose that 𝐷 is a Jordan domain in ℝ² 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 that 𝐷 is a Jordan domain in ℝ². 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 that 𝐷 and 𝐷^′ are Jordan domains in ℝ² 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 that 𝐷 and 𝐷^′ are Jordan domains in ℝ². 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 Problem 1 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 ℝ^𝑛, 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 that 𝐷 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 that 𝐷 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 Theorem 3 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 Theorem 3 will be given in Section 3.1. In Section 2, some preliminaries are introduced.

2. Preliminaries

The quasihyperbolic length of a rectifiable arc or a path 𝛼 in the norm metric in 𝐷 is the number (cf. [14, 15]) ℓ_𝑘(𝛼)=_𝛼|𝑑𝑧|𝑑_𝐷(𝑧). (3)

For each pair of points 𝑧₁, 𝑧₂ in 𝐷, the quasihyperbolic distance 𝑘_𝐷(𝑧₁,𝑧₂) between 𝑧₁ and 𝑧₂ is defined in the usual way: 𝑘_𝐷𝑧₁,𝑧₂=infℓ_𝑘(𝛼), (4) 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]) 𝑘_𝐷𝑧₁,𝑧₂ℓ(𝛼)≥inflog1+min𝑑_𝐷𝑧₁,𝑑_𝐷𝑧₂≥𝑑log^𝐷𝑧₂𝑑_𝐷𝑧₁, (5) 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 domain 𝐷, the following are quantitatively equivalent: (1)𝐷 is a 𝑐-uniform domain; (2)𝑘_𝐷(𝑧₁,𝑧₂)≤𝑐^′log(1+|𝑧₁−𝑧₂|/min{𝑑_𝐷(𝑧₁),𝑑_𝐷(𝑧₂)}) for all 𝑧₁,𝑧₂∈𝐷; (3)𝑘_𝐷(𝑧₁,𝑧₂)≤𝑐^′₁log(1+|𝑧₁−𝑧₂|/min{𝑑_𝐷(𝑧₁),𝑑_𝐷(𝑧₂)})+𝑑 for all 𝑧₁,𝑧₂∈𝐷.

Gehring and Palka [14] introduced the quasihyperbolic metric of a domain in ℝ^𝑛, 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 ℝ^𝑛, 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 arc 𝛼 from 𝑧₁ to 𝑧₂ is a quasihyperbolic geodesic if ℓ_𝑘(𝛼)=𝑘_𝐷(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. Let 𝛼 be an arc in 𝐸. The arc may be closed, open, or half open. Let 𝑥=(𝑥₀,…,𝑥_𝑛), 𝑛≥1, be a finite sequence of successive points of 𝛼. For ℎ≥0, we say that 𝑥 is ℎ-coarse if 𝑘_𝐷(𝑥_𝑗−1,𝑥_𝑗)≥ℎ for all 1≤𝑗≤𝑛. Let Φ_𝑘(𝛼,ℎ) be the family of all ℎ-coarse sequences of 𝛼. Set 𝑠_𝑘(𝑥)=^𝑛_𝑗=1𝑘_𝐷𝑥_𝑗−1,𝑥_𝑗,ℓ_𝑘𝐷(𝛼,ℎ)=sup𝑠_𝑘(𝑥)∶𝑥∈Φ_𝑘(𝛼,ℎ) (6) with the agreement that ℓ_𝑘(𝛼,ℎ)=0 if Φ_𝑘(𝛼,ℎ)=∅. Then the number ℓ_𝑘(𝛼,ℎ) is the ℎ-coarse quasihyperbolic length of 𝛼.

In this paper, we will use this concept in the case where 𝐷 is a domain equipped with the quasihyperbolic metric 𝑘_𝐷. We always use ℓ_𝑘(𝛼,ℎ) to denote the ℎ-coarse quasihyperbolic length of 𝛼.

Definition 5. Let 𝐷 be a domain in 𝐸. An arc 𝛼⊂𝐷 is (𝜈,ℎ)-solid with 𝜈≥1 and ℎ≥0 if ℓ_𝑘𝛼𝑥,𝑦,ℎ≤𝜈𝑘_𝐷𝑥,𝑦 (7) 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 ℓ_𝑘(𝛼[𝑥,𝑦])≤𝜈𝑘_𝐷(𝑥,𝑦) for all 𝑥,𝑦∈𝛼.
Obviously, a 𝜈-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 domains 𝐷≠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.
Let 𝐺≠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 𝑘_𝐺′𝑓(𝑥),𝑓𝑦≤𝜑𝑘_𝐺𝑥,𝑦 (8) for all 𝑥,𝑦∈𝐺, and 𝜑-solid if both 𝑓and 𝑓⁻¹ satisfy this condition.
We say that 𝑓 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 that 𝑥,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]). Let 𝐷 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 that 𝐷 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 that 𝐷⫅_/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 Theorem 3.

Theorem J (see [1, Theorem  2.44]). Suppose that 𝐺⫅_/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 that 𝐷⫅_/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)min_𝑗=1,2ℓ(𝛾[𝑧_𝑗,𝑧])≤𝑏𝑑_𝐷(𝑧) for all 𝑧∈𝛼, and(2)ℓ(𝛾)≤𝑏|𝑧₁−𝑧₂|.

Theorem L (see [21, Theorem  1.2]). Suppose that 𝐷₁ 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 Theorem 3, we prove a series of lemmas.

Lemma 7. There is a constant 𝑀₀=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 1𝑀₀𝑧₁−𝑧₂≤𝑧^′₁−𝑧^′₂≤𝑀₀𝑧₁−𝑧₂. (9)

Proof. Let 𝑥₁, 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, 𝑧^′₁−𝑥^′₁<32𝐾𝑥^′₁−𝑥^′₂,𝑧^′₂−𝑥^′₂<32𝐾𝑥^′₁−𝑥^′₂. (10)
If |𝑧₁−𝑧₂|≤(1/4𝐾²𝑀)max{|𝑧₁−𝑥₁|,|𝑧₂−𝑥₂|}, then for each 𝑧∈[𝑧₁,𝑧₂], 𝑑_𝐷(𝑧)≥3𝐾²𝑀−14𝐾²𝑀max𝑧₁−𝑥₁,𝑧₂−𝑥₂, (11) and so we have 2𝑧^′₁−𝑧^′₂min𝑑_𝐷′𝑧^′₁,𝑑_𝐷′𝑧^′₂𝑧≤log1+^′1−𝑧^′₂min𝑑_𝐷′𝑧^′₁,𝑑_𝐷′𝑧^′₂≤𝑘_𝐷′𝑧^′₁,𝑧^′₂≤𝑀𝑘_𝐷𝑧₁,𝑧₂≤𝑀_𝑧1,𝑧2|𝑑𝑧|𝑑_𝐷(𝑧)≤4𝐾²𝑀²𝑧₁−𝑧₂3𝐾²𝑀−1max𝑧₁−𝑥₁,𝑧₂−𝑥₂, (12) which shows that 𝑧^′₁−𝑧^′₂≤12𝐾³𝑀³3𝐾²𝑧𝑀−1¹−𝑧₂. (13)
If |𝑧₁−𝑧₂|>(1/4𝐾²𝑀)max{|𝑧₁−𝑥₁|,|𝑧₂−𝑥₂|}, then by the assumption “𝑓 being 𝑀-bilipschitz in 𝜕𝐷,” 𝑧^′₁−𝑧^′₂≤𝑧^′₁−𝑥^′₁+𝑧^′₂−𝑥^′₂+𝑥^′₁−𝑥^′₂≤(3𝐾+1)𝑥^′₁−𝑥^′₂≤(3𝐾+1)𝑀𝑥₁−𝑥₂≤(12𝐾+4)𝐾²𝑀²𝑧₁−𝑧₂. (14) The same discussion as the above shows that 𝑧₁−𝑧₂≤(12𝐾+4)𝐾²𝑀²𝑧^′₁−𝑧^′₂. (15)

Lemma 8. There is a constant 𝑀₁=𝑀₁(𝑐,𝑀) such that if the points 𝑥∈𝐷 and 𝑧∈𝕊(𝑥,𝑑_𝐷(𝑥))∩𝐷 satisfies dist(𝑧,𝜕𝐷)≤𝜀 for sufficiently small 𝜀>0, then 𝑧^′−𝑥^′≤𝑀₁𝑑_𝐷(𝑥). (16)

Proof. Let 𝑥₀∈𝕊(𝑥,𝑑_𝐷(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)𝑑_𝐷(𝑥)) with [𝑥₀,𝑥]. Then we have 𝑘_𝐷𝑥₂,𝑥≤log1+𝑥−𝑥₂𝑑_𝐷(𝑥)−𝑥−𝑥₂𝑑≤log^𝐷(𝑥)𝑑_𝐷𝑥₂=log2, (17) which implies that 𝑥log^′2−𝑥^′𝑥^′₂−𝑥^′₀≤𝑘_𝐷′𝑥^′₂,𝑥^′≤𝑀𝑘_𝐷𝑥₂,𝑥=𝑀log2. (18) Hence, 𝑥^′₂−𝑥^′≤2^𝑀𝑥^′₂−𝑥^′₀, (19) and so 𝑥^′−𝑥^′₀≤𝑥^′−𝑥^′₂+𝑥^′₂−𝑥^′₀≤2^𝑀+1𝑥^′₂−𝑥^′₀. (20)
Let 𝑇 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)𝑑_𝐷(𝑥)). Take 𝑤₁∈𝜏∩𝜕𝐷 such that 𝜏(𝑥₂,𝑤₁)⊂𝐷 and ℓ(𝜏[𝑥₂,𝑤₁])≤2𝑑_𝐷(𝑥). Let 𝑥₁∈𝕊(𝑥,𝑑_𝐷(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∈𝐷₁ and (1/12)𝑑_𝐷(𝑥)≤|𝑥₃−𝑥₀|≤(11/12)𝑑_𝐷(𝑥).
If 𝑑_𝐷(𝑥₁)=0, then we take 𝐷₁=𝔹(𝑥,𝑑_𝐷(𝑥)) and 𝑥₃=𝑥₁. Obviously, |𝑥₃−𝑥₀|=(1/2)𝑑_𝐷(𝑥). Hence Claim 1 holds true in this case.
If 𝑑_𝐷(𝑥₁)>0, we divide the proof of Claim 1 into two parts.
Case 1. (𝑑_𝐷(𝑥₁)≤(5/12)𝑑_𝐷(𝑥)). Then we take 𝐷₁=𝔹(𝑥,𝑑_𝐷(𝑥))∪𝔹(𝑥₁,𝑑_𝐷(𝑥₁)) and 𝑥₃∈𝕊(𝑥₁,𝑑_𝐷(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 1𝑑12^𝐷(𝑥)≤𝑥₁−𝑥₀−𝑥₁−𝑥₃≤𝑥₃−𝑥₀≤𝑥₁−𝑥₀+𝑥₁−𝑥₃≤11𝑑12^𝐷(𝑥), (21) from which we see that Claim 1 is true.
Case 2. (𝑑_𝐷(𝑥₁)>(5/12)𝑑_𝐷(𝑥)). Obviously, 𝑑_𝐷(𝑥₁)>(5/6)|𝑥₁−𝑥₀|. We let 𝑤₂∈𝜏₁ be the first point along the direction from 𝑥₁ to 𝑤₁ such that 𝑑_𝐷𝑤₂=5𝑑12^𝐷(𝑥). (22)
If |𝑤₂−𝑥₁|≤(1/3)𝑑_𝐷(𝑥), then we take 𝐷₁=𝔹(𝑥,𝑑_𝐷(𝑥))∪𝔹(𝑤₂,𝑑_𝐷(𝑤₂)),and let 𝑥₃∈𝕊(𝑤₂,𝑑_𝐷(𝑤₂))∩𝐷 such that dist(𝑥₃,𝜕𝐷)≤𝜀 for sufficiently small 𝜀>0. Then 𝑑_𝐷𝑤₂+𝑑_𝐷(𝑥)−𝑤₂−𝑥≥𝑑_𝐷𝑤₂−𝑤₂−𝑥₁≥1𝑑12^𝐷(𝑥),1𝑑12^𝐷(𝑥)≤𝑥₃−𝑥₀≤𝑤₂−𝑥₀+𝑤₂−𝑥₃≤11𝑑12^𝐷(𝑥). (23)
It follows from Theorem L that 𝐷₁ is a 677-uniform domain, which shows that Claim 1 is true.
If |𝑤₂−𝑥₁|>(1/3)𝑑_𝐷(𝑥), then we first prove the following subclaim.
Subclaim  1. There exists a simply connected domain 𝐷₁=∪^𝑡_𝑖=0𝐵_𝑖 in 𝐷, where 𝑡=1 or 2, such that (1)𝑥₀, 𝑥∈𝐷₁; (2)for each 𝑖∈{0,…,𝑡}, (5/12)𝑑_𝐷(𝑥)≤𝑟_𝑖≤𝑑_𝐷(𝑥); (3)if 𝑡=2, then |𝑥−𝑤₂|−𝑟₀−𝑟₂≥(1/144)𝑑_𝐷(𝑥); (4)𝑟_𝑖+𝑟_𝑖+1−|𝑣_𝑖−𝑣_𝑖+1|≥(1/144)𝑑_𝐷(𝑥), where 𝑖∈{0,1} if 𝑡=2 or 𝑖=0 if 𝑡=1.
Here 𝐵_𝑖=𝔹(𝑣_𝑖,𝑟_𝑖), 𝑣_𝑖∈𝜏[𝑥₂,𝑤₂], 𝑣₁∉𝐵₀, and 𝑣₂∉𝜏[𝑥₂,𝑣₁].
To prove this subclaim, we let 𝑦₂∈𝜏₁ be such that |𝑥₁−𝑦₂|=(1/3)𝑑_𝐷(𝑥) and let 𝐶₀=𝔹(𝑥,𝑑_𝐷(𝑥)) and 𝐶₁=𝔹(𝑦₂,𝑑_𝐷(𝑦₂)). Since 𝑑_𝐷(𝑦₂)>(5/12)𝑑_𝐷(𝑥), we have 𝑑_𝐷𝑦₂+𝑑_𝐷(𝑥)−𝑦₂≥1−𝑥𝑑12^𝐷(𝑥). (24)
Next, we construct a ball denoted by 𝐶₂.  If 𝑤₂∈𝐶₁, then we let 𝐶₂=𝔹(𝑤₂,𝑑_𝐷(𝑤₂)). If 𝑤₂∉𝐶₁, then we let 𝑦₃ be the intersection of 𝕊(𝑦₂,𝑑_𝐷(𝑦₂)) with 𝜏₁[𝑦₂,𝑤₁]. Since ℓ(𝜏₁)≤2𝑑_𝐷(𝑥) and 𝑑_𝐷(𝑧)≥(5/12)𝑑_𝐷(𝑥) for all 𝑧∈𝜏₁(𝑥₁,𝑥₂), we have 𝑤₁−𝑤₂+𝑤₂−𝑦₃+𝑦₃−𝑦₂+𝑦₂−𝑥₁+𝑥₂−𝑥₁≤ℓ𝜏₁≤2𝑑_𝐷(𝑥), (25)which implies that 𝑤₂−𝑦₃≤13𝑑_𝐷(𝑥). (26) We take 𝐶₂=𝔹(𝑤₂,𝑑_𝐷(𝑤₂)). Then (26) implies 𝑑_𝐷𝑤₂+𝑑_𝐷𝑥₂−𝑥₂−𝑤₂≥𝑑_𝐷𝑤₂−𝑤₂−𝑦₃≥1𝑑12^𝐷(𝑥). (27)
Now we are ready to construct the needed domain 𝐷₁.
If 𝑑_𝐷(𝑤₂)+𝑑_𝐷(𝑥)−|𝑤₂−𝑥|≥(1/48)𝑑_𝐷(𝑥), then we take 𝐵₀=𝐶₀, 𝐵₁=𝐶₂, and 𝐷₁=𝐵₀∪𝐵₁ with 𝑣₀=𝑥, 𝑣₁=𝑤₂, 𝑟₀=𝑑_𝐷(𝑥), and 𝑟₁=𝑑_𝐷(𝑤₂). Obviously, 𝐷₁ satisfies all the conditions in Subclaim 1. In this case, 𝑡=1.
If 𝑑_𝐷(𝑤₂)+𝑑_𝐷(𝑥)−|𝑤₂−𝑥|<(1/48)𝑑_𝐷(𝑥), then we take 𝐵₀=𝔹(𝑥,(35/36)𝑑_𝐷(𝑥)) with 𝑟₀=(35/36)𝑑_𝐷(𝑥) and 𝑣₀=𝑥, 𝐵₁=𝐶₁ with 𝑟₁=𝑑_𝐷(𝑦₂) and 𝑣₁=𝑦₂, and 𝐵₂=𝐶₂ with 𝑟₂=𝑑_𝐷(𝑤₂) and 𝑣₂=𝑤₂. Then Inequalities (24) and (27) show that 𝐷₁=⋃²_𝑖=0𝐵_𝑖 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 domain 𝐷₁constructed in Subclaim 1 is a 2³²-uniform domain.
Let 𝑥₃∈𝕊(𝑤₂,𝑑_𝐷(𝑤₂))∩𝐷 such that dist(𝑥₃,𝜕𝐷)≤𝜀 for sufficiently small 𝜀>0. Then 1𝑑12^𝐷(𝑥)≤𝑥₃−𝑥₀≤11𝑑12^𝐷(𝑥). (28) Then the proof of Claim 1 easily follows from (28), Subclaim 1, and Corollary 9.
We come back to the proof of Lemma 8. It follows from (28) and Lemma 7 that 𝑥−𝑥₃≤𝑥−𝑥₀+𝑥₀−𝑥₃≤23𝑑12^𝐷(𝑥),112𝑀₀𝑑_𝐷(𝑥)≤1𝑀₀𝑥₃−𝑥₀≤𝑥^′₃−𝑥^′₀≤𝑀₀𝑥₃−𝑥₀≤11𝑀₀𝑑12^𝐷(𝑥). (29) Then it follows from Theorem J that 𝐷^′₁ is an 𝑀^′-uniform domain, where 𝑀^′=𝑀^′(𝑐,𝑀). Hence, we know from Theorem 6 that 𝑓⁻¹ is a 𝜃-Quasimöbius in 𝐷₁, where 𝜃=𝜃(𝑐,𝑀), and so (19), (20), (28), and (29) imply that 1≤𝑥23³−𝑥₀𝑥₂−𝑥₀⋅𝑥₂−𝑥𝑥−𝑥₃𝑥≤𝜃^′3−𝑥^′₀𝑥^′₂−𝑥^′₀⋅𝑥^′₂−𝑥^′𝑥^′−𝑥^′₃𝑀≤𝜃⁰2^𝑀+1𝑑_𝐷(𝑥)𝑥^′−𝑥^′₃, (30) which, together with (20), shows 𝑥^′−𝑥^′₀≤𝑥^′−𝑥^′₃+𝑥^′₃−𝑥^′₀≤2^𝑀+1𝜃⁻¹()+1/2311𝑀₀𝑑12^𝐷(𝑥)<2^𝑀0+2𝜃⁻¹()𝑑1/23^𝐷(𝑥). (31) Thus, the proof of Lemma 8 is complete.

Lemma 10. For all 𝑥∈𝐷, if 𝑧∈𝕊(𝑥,𝑑_𝐷(𝑥))∩𝐷 such that dist(𝑥,𝜕𝐷)≤𝜀 for sufficiently small 𝜀>0, then |𝑧^′−𝑥^′|≥(1/𝑒^4𝑀0𝑀21)𝑑_𝐷(𝑥), where 𝑀₁=𝑀₁(𝑐,𝑀).

Proof. Suppose on the contrary that there exist points 𝑥₁∈𝐷 and 𝑦₁∈𝕊(𝑥₁,𝑑_𝐷(𝑥₁))∩𝐷 with dist(𝑦₁,𝜕𝐷)≤𝜀 for sufficiently small 𝜀>0 such that 𝑥^′₁−𝑦^′₁<1𝑒^4𝑀0𝑀21𝑥₁−𝑦₁. (32)
We take 𝑦₂∈𝕊(𝑦₁,𝑑_𝐷(𝑥₁))∩𝐷 such that dist(𝑦₂,𝜕𝐷)≤𝜀 for sufficiently small 𝜀>0. From Lemma 7 we know that 𝑦^′₁−𝑦^′₂≥1𝑀₀𝑦₁−𝑦₂=1𝑀₀𝑑_𝐷𝑥₁. (33)
Let 𝑇₁ be a 2-dimensional linear subspace of 𝐸 determined by 𝑥₁, 𝑦₁ and 𝑦₂, and 𝜔 the circle 𝑇₁∩𝕊(𝑦₁,𝑑_𝐷(𝑥₁)). We take 𝑦₃∈𝜔∩𝜕𝐷 which satisfies 𝜔(𝑥₁,𝑦₃)⊂𝐷 and ℓ(𝜔[𝑥₁,𝑦₃])≤4𝑑_𝐷(𝑥₁). Let 𝜔₁=𝜔(𝑥₁,𝑦₃) and 𝑤₁ be the first point along the direction from 𝑥₁ to 𝑦₃ such that 𝑑_𝐷𝑤₁=14𝑀₀𝑀₁𝑑_𝐷𝑥₁. (34)
Let 𝑣₁∈𝕊(𝑤₁,𝑑_𝐷(𝑤₁))∩𝐷 such that dist(𝑤₁,𝜕𝐷)≤𝜀 for sufficiently small 𝜀>0. Then it follows from Lemma 8 that 𝑑_𝐷′𝑤^′₁≤𝑤^′₁−𝑣^′₁≤𝑀₁𝑑_𝐷𝑤₁=14𝑀₀𝑑_𝐷𝑥₁, (35) which, together with Lemmas 7 and 8 and (32), implies that 𝑥^′₁−𝑤^′₁≥𝑦^′₁−𝑣^′₁−𝑥^′₁−𝑦^′₁−𝑣^′₁−𝑤^′₁≥1𝑀₀𝑦₁−𝑣₁−1𝑒^4𝑀0𝑀21𝑥₁−𝑦₁−𝑀₁𝑣₁−𝑤₁≥1𝑀₀𝑑_𝐷𝑥₁−𝑑_𝐷𝑤₁−1𝑒^4𝑀0𝑀21𝑥₁−𝑦₁−𝑀₁𝑣₁−𝑤₁>12𝑀₀𝑑_𝐷𝑥₁. (36) Hence, we infer from (32) that 𝑘_𝐷′𝑥^′₁,𝑤^′₁𝑥≥log1+^′1−𝑤^′₁𝑑_𝐷′𝑥^′₁>𝑀²₁. (37)
Since ℓ(𝜔₁)≤4𝑑_𝐷(𝑥₁), by the choice of 𝑤₁, one has 𝑘_𝐷𝑥₁,𝑤₁≤_{𝜔1𝑥1,𝑤1}|𝑑𝑥|𝑑_𝐷(𝑥)≤16𝑀₀𝑀₁, (38) whence 𝑘_𝐷′𝑥^′₁,𝑤^′₁≤𝑀𝑘_𝐷𝑥₁,𝑤₁≤16𝑀𝑀₀𝑀₁, (39) which contradicts with (37). The proof of Lemma 10 is complete.

Lemma 11. For 𝑥₁∈𝐷 and 𝑥₂∈𝜕𝐷, we have 𝑥^′₁−𝑥^′₂≤𝑀₂𝑥₁−𝑥₂, (40) where 𝑀₂=2𝑀₀+𝑀₁.

Proof. For 𝑥₁∈𝐷, we let 𝑦₁∈𝕊(𝑥₁,𝑑_𝐷(𝑥₁))∩𝐷 such that dist(𝑦₁,𝜕𝐷)≤𝜀 for sufficiently small 𝜀>0. Then it follows from Lemma 8 that 𝑥^′₁−𝑦^′₁≤𝑀₁𝑥₁−𝑦₁. (41)
For 𝑥₂∈𝜕𝐷, if |𝑦₁−𝑥₂|≤2|𝑥₁−𝑦₁|, then by Lemma 7, we have 𝑥^′₁−𝑥^′₂≤𝑥^′₁−𝑦^′₁+𝑦^′₁−𝑥^′₂≤𝑀₁𝑥₁−𝑦₁+𝑀₀𝑦₁−𝑥₂≤2𝑀₀+𝑀₁𝑥₁−𝑦₁≤2𝑀₀+𝑀₁𝑥₁−𝑥₂. (42)
If |𝑦₁−𝑥₂|>2|𝑦₁−𝑥₁|, then we have 𝑥₁−𝑥₂>𝑦₁−𝑥₂−𝑥₁−𝑦₁>12𝑦₁−𝑥₂. (43)
Hence, by Lemma 7 and (41), 𝑥^′₁−𝑥^′₂≤𝑥^′₁−𝑦^′₁+𝑦^′₁−𝑥^′₂≤𝑀₁𝑥₁−𝑦₁+𝑀₀𝑦₁−𝑥₂≤2𝑀₀+𝑀₁𝑥₁−𝑥₂, (44) from which the proof follows.

Lemma 12. For 𝑥₁∈𝐷 and 𝑥₂∈𝜕𝐷, one has 𝑥^′₁−𝑥^′₂≥1𝑀₃𝑥₁−𝑥₂, (45) where 𝑀₃=2𝑀₀𝑀₁𝑒^{(5𝑀𝑀0+8𝑀0)𝑀21}.

Proof. We begin with a claim.
Claim 2. For all 𝑧∈𝐷, we have 𝑑_𝐷′(𝑧^′)≥(1/𝑒^{(5𝑀𝑀0+8𝑀0)𝑀21})𝑑_𝐷(𝑧).
To prove this claim, we let 𝑤₂∈[𝑧,𝑦₁] be such that |𝑤₂−𝑦₁|=(1/2𝑀₁𝑒^4𝑀0𝑀21)𝑑_𝐷(𝑧). It follows from [18] that 𝑘_𝐷𝑤₂𝑤,𝑧≤log1+²−𝑧𝑑_𝐷(𝑧)−𝑤₂−𝑧<5𝑀₀𝑀²₁. (46) By Lemma 8, we have 𝑤^′₂−𝑦^′₁≤𝑀₁𝑤₂−𝑦₁=12𝑒^4𝑀0𝑀21𝑑_𝐷(𝑧). (47) Hence, Lemma 10 implies |𝑤^′₂−𝑧^′|≥(1/2𝑒^4𝑀0𝑀21)𝑑_𝐷(𝑧), whence 𝑤log^′2−𝑧^′𝑑_𝐷′𝑧^′≤𝑘_𝐷′𝑤^′₂,𝑧^′≤𝑀𝑘_𝐷𝑤₂,𝑧≤5𝑀𝑀₀𝑀²₁, (48) which shows that Claim 2 is true.
Now we are ready to finish the proof of Lemma 12. For 𝑥₁∈𝐷 and 𝑥₂∈𝜕𝐷, if |𝑥₁−𝑥₂|≤2𝑀₀𝑀₁𝑑_𝐷(𝑥₁), then by Claim 2, 𝑥^′₁−𝑥^′₂≥𝑑_𝐷′𝑥^′₁≥1𝑒^{(5𝑀𝑀0+8𝑀0)𝑀21}𝑑_𝐷𝑥₁≥12𝑀₀𝑀₁𝑒^{(5𝑀𝑀0+8𝑀0)𝑀21}𝑥₁−𝑥₂. (49)
If |𝑥₁−𝑥₂|>2𝑀₀𝑀₁𝑑_𝐷(𝑥₁), then we take 𝑤₃∈𝕊(𝑥₁,𝑑_𝐷(𝑥₁))∩𝐷 such that dist(𝑤₃,𝜕𝐷)≤𝜀 for sufficiently small 𝜀>0, and so 𝑤₃−𝑥₂≥𝑥₁−𝑥₂−𝑥₁−𝑤₃≥11−2𝑀₀𝑀₁𝑥₁−𝑥₂,𝑤₃−𝑥₂≥𝑥₁−𝑥₂−𝑥₁−𝑤₃≥2𝑀₀𝑀₁−1𝑥₁−𝑤₃, (50) whence Lemmas 7 and 8 imply 𝑥^′₁−𝑥^′₂≥𝑤^′₃−𝑥^′₂−𝑥^′₁−𝑤^′₃≥1𝑀₀𝑤₃−𝑥₂−𝑀₁𝑥₁−𝑤₃≥1𝑀₀−𝑀₁2𝑀₀𝑀₁𝑤−1³−𝑥₂≥13𝑀₀𝑥₁−𝑥₂, (51) 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 that 𝑓⁻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 𝑥^′₁, 𝑥^′₂, 𝑥^′₃, 𝑥^′₄∈𝐷^′, 𝑥₄−𝑥₁𝑥₄−𝑥₂⋅𝑥₂−𝑥₃𝑥₁−𝑥₃≤𝑀₂𝑀₃²𝑥^′₄−𝑥^′₁𝑥^′₄−𝑥^′₂⋅𝑥^′₂−𝑥^′₃𝑥^′₁−𝑥^′₃, (52) where 𝑥₁,𝑥₂∈𝜕𝐷^′. Obviously, to prove Inequality (52), we only need to consider the following three cases.
Case 3 (𝑥^′₁,𝑥^′₂,𝑥^′₃,𝑥^′₄∈𝜕𝐷^′). Since 𝑓 is 𝑀-bilipschitz in 𝜕𝐷, we have 𝑥₄−𝑥₁𝑥₄−𝑥₂⋅𝑥₂−𝑥₃𝑥₁−𝑥₃≤𝑀⁴𝑥^′₄−𝑥^′₁𝑥^′₄−𝑥^′₂⋅𝑥^′₂−𝑥^′₃𝑥^′₁−𝑥^′₃. (53)
Case 4 (𝑥^′₁, 𝑥^′₂, 𝑥^′₃∈𝜕𝐷^′ and𝑥^′₄∈𝐷^′). It follows from Lemmas 11 and 12 that 𝑥₄−𝑥₁𝑥₄−𝑥₂⋅𝑥₂−𝑥₃𝑥₁−𝑥₃≤𝑀₂𝑀₃𝑥^′₄−𝑥^′₁𝑥^′₄−𝑥^′₂⋅𝑀²𝑥^′₂−𝑥^′₃𝑥^′₁−𝑥^′₃≤𝑀²𝑀₂𝑀₃𝑥^′₄−𝑥^′₁𝑥^′₄−𝑥^′₂⋅𝑥^′₂−𝑥^′₃𝑥^′₁−𝑥^′₃. (54)
Case 5 (𝑥^′₁, 𝑥^′₂∈𝜕𝐷^′ and 𝑥^′₃,𝑥^′₄∈𝐷^′). We obtain from Lemmas 11 and 12 that 𝑥₄−𝑥₁𝑥₄−𝑥₂⋅𝑥₂−𝑥₃𝑥₁−𝑥₃≤𝑀₂𝑀₃𝑥^′₄−𝑥^′₁𝑥^′₄−𝑥^′₂⋅𝑀₂𝑀₃𝑥^′₂−𝑥^′₃𝑥^′₁−𝑥^′₃≤𝑀₂𝑀₃²𝑥^′₄−𝑥^′₁𝑥^′₄−𝑥^′₂⋅𝑥^′₂−𝑥^′₃𝑥^′₁−𝑥^′₃. (55)
The combination of Cases 3∼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 Theorem 3

For any 𝑧₁, 𝑧₂∈𝐷, it suffices to prove that 1𝑀^′𝑧₁−𝑧₂≤𝑧^′₁−𝑧^′₂≤𝑀^′𝑧₁−𝑧₂, (56) 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{𝑑_𝐷(𝑧₁),𝑑_𝐷(𝑧₂)}, then 𝑘_𝐷𝑧₁,𝑧₂≤_{[𝑧1,𝑧2]}|𝑑𝑥|𝑑_𝐷(𝑥)≤2𝑧₁−𝑧₂max𝑑_𝐷𝑧₁,𝑑_𝐷𝑧₂≤1, (57) which shows that 𝑧log1+^′1−𝑧^′₂min𝑑_𝐷′𝑧^′₁,𝑑_𝐷′𝑧^′₂≤𝑘_𝐷′𝑧^′₁,𝑧^′₂≤𝑀𝑘_𝐷𝑧₁,𝑧₂≤𝑀, (58) and so 𝑧^′₁−𝑧^′₂𝑒^𝑀min𝑑_𝐷′𝑧^′₁,𝑑_𝐷′𝑧^′₂𝑧≤log1+^′1−𝑧^′₂min𝑑_𝐷′𝑧^′₁,𝑑_𝐷′𝑧^′₂≤2𝑀𝑧₁−𝑧₂max𝑑_𝐷𝑧₁,𝑑_𝐷𝑧₂. (59) We see from Lemma 8 that min𝑑_𝐷′𝑧^′₁,𝑑_𝐷′𝑧^′₂≤𝑀₁max𝑑_𝐷𝑧₁,𝑑_D𝑧₂. (60) Then (59) implies that 𝑧^′₁−𝑧^′₂≤2𝑀𝑀₁𝑒^𝑀𝑧₁−𝑧₂. (61)

For the other case |𝑧₁−𝑧₂|>(1/2)max{𝑑_𝐷(𝑧₁),𝑑_𝐷(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 𝑧^′₁−𝑧^′=12𝑧^′₁−𝑧^′₂. (62) Then we know from |𝑧^′₂−𝑧^′|≥(1/2)|𝑧^′₁−𝑧^′₂| and Theorem K that 𝑧^′₁−𝑧^′₂≤2min𝑧^′₁−𝑧^′,𝑧^′₂−𝑧^′≤2mindiam𝑧^′₁,𝑧^′,diam𝑧^′₂,𝑧^′≤2𝜇𝑑_𝐷′𝑧^′, (63) where 𝜇 depends only on 𝑐 and 𝑀.

We claim that 𝑑_𝐷(𝑧)≤3ℓ𝛽. (64) Otherwise, max𝑑_𝐷𝑧₁,𝑑_𝐷𝑧₂≥𝑑_𝐷(𝑧)−max𝑧₁−𝑧,𝑧₂𝛽𝑧−𝑧>2ℓ≥2₁−𝑧₂. (65) This is the desired contradiction.

By Theorem K and Lemma 13, we have 𝑑_𝐷(𝑧)≤3ℓ𝛽≤3𝑏𝑧₁−𝑧₂, (66) where 𝑏=𝑏(𝑐₁). Hence, Lemma 8 and (63) show that 𝑧^′₁−𝑧^′₂≤2𝜇𝑑_𝐷′𝑧^′≤6𝑏𝑀₁𝜇𝑧₁−𝑧₂. (67)

By Lemma 13, we see that 𝐷 is a 𝑐₁-uniform domain. Hence a similar argument as in the proofs of Inequalities (61) and (67) yields that 𝑧₁−𝑧₂≤𝑀₄𝑧^′₁−𝑧^′₂, (68) 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.

Acknowledgments

The research was partly supported by NSFs of China (No. 11071063 and No. 11101138), the program excellent talent in Hunan Normal University (No. ET11101), the program for excellent young scholars of Department of Education in Hunan Province (No. 12B079) and Hunan Provincial Innovation Foundation For Postgraduate (No. CX2011B199).