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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 765685, 9 pages
On Bilipschitz Extensions in Real Banach Spaces
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
Received 11 October 2012; Accepted 20 February 2013
Academic Editor: Beong In Yun
Copyright © 2013 M. Huang and Y. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Suppose that and denote real Banach spaces with dimension at least 2, that and are bounded domains with connected boundaries, that is an -QH homeomorphism, and that is uniform. The main aim of this paper is to prove that extends to a homeomorphism and is bilipschitz if and only if is bilipschitz in . 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 . 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 and denote real Banach spaces with dimension at least . The norm of a vector in is written as , and for every pair of points , in , the distance between them is denoted by , the closed line segment with endpoints and by . We begin with the following concepts following closely the notation and terminology of [4–8] or .
We first recall some definitions.
Definition 1. A domain in is called -uniform in the norm metric, provided there exists a constant with the property that each pair of points in can be joined by a rectifiable arc in satisfying (1) for all , and(2),
where denotes the length of , the part of between and , and the distance from to the boundary of .
Definition 2. Suppose , and . We say that a homeomorphism is -bilipschitz if for all , , and -QH if for all , .
As for the extension of bilipschitz maps in , Ahlfors  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 is a -quasidisk in , that is a Jordan domain in , and that is -bilipschitz. Then there exist -bilipschitz and such that on and depends only on and , where and .
Tukia and Väisälä  dealt with the curious phenomenon that sometimes a quasiconformal property implies the corresponding bilipschitz property.
Theorem B (see [12, Theorem 2.12]). Suppose that is a closed set in , , and that is a -QC map such that is -bilipschitz. Then there is an -bilipschitz map such that(1);(2) for each component of ;(3) depends only on , , and .
In , Gehring raised the following two related problems.
Open Problem 1. Suppose that is a Jordan domain in and that is -bilipschitz. Characterize mappings having -bilipschitz extension to with .
Open Problem 2. Suppose that is a Jordan domain in . For which domains does each -bilipschitz in the have -bilipschitz extension to with ?
Gehring himself discussed these two problems and got the following two results.
Theorem C (see [13, Theorem 2.11]). Suppose that and are Jordan domains in and that if and only if . Suppose also that is a -quasiconformal mapping and that extends to a homeomorphism such that is -bilipschitz. Then there exists an -bilipschitz map with , where .
Theorem D (see [13, Theorem 4.9]). Suppose that and are Jordan domains in . Then each -bilipschitz in has an -bilipschitz extension with if and only if is a -quasidisk, where and .
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 , was applied. In general, this tool is no longer applicable in the context of Banach spaces (see ). A natural problem is whether Theorem C is true or false in Banach spaces. In fact, this problem was raised by Väisälä in  in the following form.
Open Problem 3. Suppose that and are bounded domains with connected boundaries in and . Suppose also that is -QH and that extends to a homeomorphism such that is -bilipschitz. Is it true that -bilipschitz with ?
Our result is as follows.
Theorem 3. Suppose that and are bounded domains with connected boundaries in and , respectively. Suppose also that is -QH and that extends to a homeomorphism such that is -bilipschitz. If is a -uniform domain, then is -bilipschitz with .
For each pair of points , in , the quasihyperbolic distance between and is defined in the usual way: where the infimum is taken over all rectifiable arcs joining to in . For all , in , we have (cf. ) where the infimum is taken over all rectifiable curves in connecting and .
In , Väisälä characterized uniform domains by the quasihyperbolic metric.
Theorem E (see [5, Theorem 6.16]). For a domain , the following are quantitatively equivalent: (1) is a -uniform domain; (2) for all ; (3) for all .
Gehring and Palka  introduced the quasihyperbolic metric of a domain in , and it has been recently used by many authors in the study of quasiconformal mappings and related questions . In the case of domains in , the equivalence of items (1) and (3) in Theorem E is due to Gehring and Osgood  and the equivalence of items (2) and (3) is due to Vuorinen . Many of the basic properties of this metric may be found in [4, 5, 17].
Recall that an arc from to is a quasihyperbolic geodesic if . Each subarc of a quasihyperbolic geodesic is obviously a quasihyperbolic geodesic. It is known that a quasihyperbolic geodesic between every pair of points in exists if the dimension of 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 .
Definition 4. Let be an arc in . The arc may be closed, open, or half open. Let , , be a finite sequence of successive points of . For , we say that is -coarse if for all . Let be the family of all -coarse sequences of . Set with the agreement that if . Then the number is the -coarse quasihyperbolic length of .
In this paper, we will use this concept in the case where is a domain equipped with the quasihyperbolic metric . We always use to denote the -coarse quasihyperbolic length of .
Definition 5. Let be a domain in . An arc is -solid with and if
for all . A -solid arc is said to be a -neargeodesic, that is, an arc is a - neargeodesic if and only if for all .
Obviously, a -neargeodesic is a quasihyperbolic geodesic if and only if .
In , Väisälä got the following property concerning the existence of neargeodesic in .
Theorem F (see [19, Theorem 3.3]). Let and . Then there is a -neargeodesic in joining and .
The following result due to Väisälä is from .
Theorem G (see [5, Theorem 4.15]). For domains and , suppose that is -QH. If is a -neargeodesic in , then the arc is -neargeodesic in with depending only on and .
Let and be metric spaces, and let be a growth function, that is, a homeomorphism with . We say that a homeomorphism is -semisolid if for all , and -solid if both and satisfy this condition.
We say that is fully -semisolid (resp. fully -solid) if f is -semisolid (resp. -solid) on every subdomain of . In particular, when , corresponding subdomains are taken to be proper ones. Fully -solid mapsare also called freely -quasiconformal maps, or briefly -FQC maps.
For convenience, in the following, we always assume that denote points in and the images in of under , respectively. Also we assume that denote curves in and the images in of under , respectively.
3. Bilipschitz Mappings
First we introduce the following Theorems.
Theorem H (see [5, Theorem 7.18]). Let and be domains in and , respectively. Suppose that is a -uniform domain and that is -FQC (see Section 2 for the definition). Then the following conditions are quantitatively equivalent:(1) is a -uniform domain;(2) is -quasimöbius.
Theorem I (see [20, Theorem 1.1]). Suppose that is a -uniform domain and that is -CQH, where and . Then the following conditions are quantitatively equivalent:(1) is a -uniform domain;(2) extends to a homeomorphism and is -QM rel .
The following theorem easily follows from Theorems H and I.
Theorem 6. Suppose that and , that is a -uniform domain, and that is -FQC. Then the following conditions are quantitatively equivalent:(1) is a -uniform domain;(2) is -quasimöbius;(3) extends to a homeomorphism and is -QM rel .
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 and is a -uniform domain, and that is -QH. If is a -uniform domain, then is a -uniform domain with .
Theorem K (see [5, Theorem 6.19]). Suppose that is a -uniform domain and that is a -neargeodesic in with endpoints and . Then there is a constant such that(1) for all , and(2).
Theorem L (see [21, Theorem 1.2]). Suppose that and are convex domains in , where is bounded and is -uniform for some , and that there exist and such that . If there exist constants and such that and , then is a -uniform domain with .
Basic Assumption A. In this paper, we always assume that and are bounded domains with connected boundaries in and , respectively, that is -QH, that extends to a homeomorphism such that is -bilipschitz, and that is a -uniform domain.
Before the proof of Theorem 3, we prove a series of lemmas.
Lemma 7. There is a constant such that if the points satisfies and for sufficiently small , then
Proof. Let , be such that , and for sufficiently small . It follows from “ being -QH in and homeomorphic in ” that (cf. ) for each , where depends only on . Hence,
If , then for each , and so we have which shows that
If , then by the assumption “ being -bilipschitz in ,” The same discussion as the above shows that
Lemma 8. There is a constant such that if the points and satisfies for sufficiently small , then
Proof. Let such that for sufficiently small , and let be the intersection point of with . Then we have
which implies that
Let be a -dimensional linear subspace of which contains and , and we use to denote the circle . Take such that and . Let and denote by .
Claim 1. There must exist a -uniform domain in and satisfying for sufficiently small such that , and .
If , then we take and . Obviously, . Hence Claim 1 holds true in this case.
If , we divide the proof of Claim 1 into two parts.
Case 1. (). Then we take and such that for sufficiently small . It follows from Theorem L that is a -uniform domain and from which we see that Claim 1 is true.
Case 2. (). Obviously, . We let be the first point along the direction from to such that
If , then we take ,and let such that for sufficiently small . Then
It follows from Theorem L that is a -uniform domain, which shows that Claim 1 is true.
If , then we first prove the following subclaim.
Subclaim 1. There exists a simply connected domain in , where or , such that (1), ; (2)for each , ; (3)if , then ; (4), where if or if .
Here , , , and .
To prove this subclaim, we let be such that and let and . Since , we have
Next, we construct a ball denoted by . If , then we let . If , then we let be the intersection of with . Since and for all , we have which implies that We take . Then (26) implies
Now we are ready to construct the needed domain .
If , then we take , , and with , , , and . Obviously, satisfies all the conditions in Subclaim 1. In this case, .
If , then we take with and , with and , and with and . Then Inequalities (24) and (27) show that satisfies all the conditions in Subclaim 1. In this case, .
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 constructed in Subclaim 1 is a -uniform domain.
Let such that for sufficiently small . Then 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 Then it follows from Theorem J that is an -uniform domain, where . Hence, we know from Theorem 6 that is a -Quasimöbius in , where , and so (19), (20), (28), and (29) imply that which, together with (20), shows Thus, the proof of Lemma 8 is complete.
Lemma 10. For all , if such that for sufficiently small , then , where .
Proof. Suppose on the contrary that there exist points and with for sufficiently small such that
We take such that for sufficiently small . From Lemma 7 we know that
Let be a -dimensional linear subspace of determined by , and , and the circle . We take which satisfies and . Let and be the first point along the direction from to such that
Let such that for sufficiently small . Then it follows from Lemma 8 that which, together with Lemmas 7 and 8 and (32), implies that Hence, we infer from (32) that
Since , by the choice of , one has whence which contradicts with (37). The proof of Lemma 10 is complete.
Lemma 11. For and , we have where .
Proof. For , we let such that for sufficiently small . Then it follows from Lemma 8 that
For , if , then by Lemma 7, we have
If , then we have
Hence, by Lemma 7 and (41), from which the proof follows.
Lemma 12. For and , one has where .
Proof. We begin with a claim.
Claim 2. For all , we have .
To prove this claim, we let be such that . It follows from  that By Lemma 8, we have Hence, Lemma 10 implies , whence which shows that Claim 2 is true.
Now we are ready to finish the proof of Lemma 12. For and , if , then by Claim 2,
If , then we take such that for sufficiently small , and so whence Lemmas 7 and 8 imply from which the proof is complete.
By the previous lemmas, we get the following result.
Lemma 13. is a -uniform domain, where .
Proof. We first prove that is -Quasimöbius rel , where , and are the same as in Lemmas 11 and 12, respectively. By definition, it is necessary to prove that for , , , ,
where . Obviously, to prove Inequality (52), we only need to consider the following three cases.
Case 3 (). Since is -bilipschitz in , we have
Case 4 (, , and). It follows from Lemmas 11 and 12 that
Case 5 (, and ). We obtain from Lemmas 11 and 12 that
The combination of Cases shows that Inequality (52) holds, which implies that is a -Quasimöbius rel . Hence, Theorem 6 shows that is a -uniform domain, where depends only on and .
3.1. The Proof of Theorem 3
For any , , it suffices to prove that where depends only on and .
For the other case , we let be a -neargeodesic joining and in . It follows from Theorem G that is a -neargeodesic, where depends only on . Let such that Then we know from and Theorem K that where depends only on and .
We claim that Otherwise, This is the desired contradiction.
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).
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