Sobolev Extension on Lp-quasidisks

In this paper, we study the Sobolev extension property of Lp-quasidisks which are the generalizations of classical quasidisks. After that, we also find some applications of this property.


Introduction
Let ⊂ R 2 be a domain. A homeomorphism h : → R 2 is said to be quasiconformal if h ∈ W 1,2 loc ( , R 2 ) and the inequality |Dh(z)| 2 ≤ KJ h (z) (1.1) holds, where γ z 1 ,z and γ z,z 2 mean the subcurves of γ z 1 ,z 2 from z 1 to z and from z to z 2 respectively. By Jones' result in [24], uniform domains hence quasidisks are Sobolev (p, p)extension domains for arbitrary 1 ≤ p ≤ ∞. We say ⊂ R 2 is a Sobolev (p, q)-extension domain for 1 ≤ q ≤ p ≤ ∞ if for every u ∈ W 1,p ( ), there exists a function E(u) ∈ W 1,q (R 2 ) with E(u) ≡ u and E(u) W 1,q (R 2 ) ≤ C u W 1,p ( ) with a positive constant C that is independent of u.
There are many planar simply connected domains that are not quasidisks, for example inward and outward cuspidal domains, see [16,18,21,22,[25][26][27]. Hence, it is natural to study generalizations of quasiconformal mappings. For instance, we concentrate on homeomorphisms of finite distortion here. A homeomorphism h : → R 2 is said to be a homeomorphism of finite distortion if h ∈ W 1,1 loc ( , R 2 ) and the inequality |Dh(z)| 2 ≤ K(z)J h (z) (1.2) holds for almost every z ∈ with a measurable function K(z) ∈ [1, ∞). For a homeomorphism of finite distortion h, we denote K h to be the optimal distortion function for (1.2) which will be defined in (2.3). If K h ∈ L ∞ ( ), then h is quasiconformal. The inverse of a quasiconformal mapping is still quasiconformal. However, if we relax the regularity of the distortion function from essentially boundedness to some weaker condition, we cannot hope that the distortion of the inverse will attain the same regularity, see [17,18]. Motivated by the Sobolev extension property of quasidisks, we can pose two following interesting problems.
(1) What is the best Sobolev extension property of domains which can be mapped onto the unit disk by homeomorphisms of finite distortion whose distortion functions satisfy some regularity? (2) What is the best Sobolev extension property of domains which are images of the unit disk under homeomorphisms of finite distortion whose distortion functions satisfy some regularity?
In this paper, we mainly concentrate on the first problem with the condition that distortion functions are locally L p -integrable for 1 ≤ p < ∞. Under this condition, the corresponding bounded domains are called L p -quasidisks. The terminology L p -quasidisk was firstly introduced in [22] by the author with Iwaniec and Onninen. In that paper, we gave characterizations of polynomial cuspidal domains which are L p -quasidisks simultaneously. In [22], we showed that every bounded domain with a rectifiable boundary is a L 1 -quasidisk. Hence, locally L 1 -integrablity of distortion function is not enough to permit Sobolev extension.
The main result in [22] tells us that for a fixed 1 < p < ∞, polynomial cuspidal domains with a relatively not very sharp singularities are L p -quasidisks. Hence, for 1 < p < ∞, we can not hope every L p -quasidisk is a Sobolev (k, k)-extension domain for arbitrary 1 ≤ k < ∞, see [32][33][34][35]. The following theorem will imply the Sobolev extension property of L p -quasidisks as a special case. For a simply connected bounded domain ⊂ R 2 , the complementary set R 2 \ is also a domain. Inward cuspidal domains with polynomial-type singularity will shows us the sharpness of this result. Also, the combination of this result with the Sobolev extension property of polynomial inward cuspidal domains due to Maz'ya and Poborchi [32][33][34][35] will give a new and simpler proof to the necessary part of the main result in [22].
By the result due to Hencl and Koskela in [17], for a homeomorphism of finite distortion h : Hence, as an application of Theorem 1.2, we have the following partial answer to the second problem above.
However, we have not obtained the best Sobolev extension property for bounded images of the unit disk under global homeomorphisms of finite distortion whose distortions are locally exponentially integrable.

Preliminarily
The notation always means a domain in the Euclidean plane R 2 . R 2 := R 2 ∪ {∞} is the one-point compactification of the plane R 2 . B(z, r) is a disk with the center z ∈ R 2 and radius 0 < r < ∞. D := B(0, 1) means the unit disk in R 2 . For 0 < r < R < ∞, we denote A(r, R) := B(0, R) \ B(0, r) to be an annulus. Typically, C will be a constant that depends on various parameters and may differ even on the same line of inequalities. For a measurable subset A ⊂ with 0 < |A| < ∞ and a function u ∈ L 1 loc ( ), u A is the integral average defined by setting Let us give the definition of Sobolev spaces first.
for a constant C that is independent of u. Replace W 1,p ( ) byẆ 1,p ( ) and replace W 1,q (R 2 ) byẆ 1,q (R 2 ) in above, we get the definition of homogeneous Sobolev (p, q)extension domains.
In [19], authors proved that for all 1 ≤ p < ∞ a bounded domain is a homogeneous Sobolev (p, p)-extension domain if and only of it is a Sobolev (p, p)-extension domain. Their argument also implies that a bounded homogeneous Sobolev (p, q)-extension domain is also a Sobolev (p, q)-extension domain for 1 ≤ q ≤ p < ∞. For the convenience of readers, we present this observation here. We only give the proof on the plane. However, it can be extended to high dimensional Euclidean spaces trivially.
By (2.1), the triangle inequality and the Hölder inequality, we obtain the desired inequality that Let us define homeomorphisms of finite distortion and L p -quasidisks.

Definition 2.3
We say a homeomorphism h : For a homeomorphism of finite distortion, we define the optimal distortion function by setting and a bounded linear extension operator fromẆ We say a reflection R : R 2 onto − → R 2 over ∂ induces a bounded linear Sobolev (p, q)-extension operator for with 1 ≤ q ≤ p < ∞, if there exists a bounded Lipschitz domain U containing ∂ such that, for every function u ∈Ẇ 1,p ( ), the function v defined by setting v = u on U ∩ and v = u • R on U \ has a representative which belongs to the Sobolev spaceẆ 1,q (U ) and we have with a positive constant C that is independent of u. Similarly, we say the reflection R induces a bounded linear Sobolev (p, q)-extension operator for R 2 \ with 1 ≤ q ≤ p < ∞, if for every u ∈Ẇ 1,p (R 2 \ ), the functionṽ defined by settingṽ = u on U \ and v = u • R on U ∩ has a representative which belongs to the Sobolev spaceẆ 1,q (U ) and we have By using a suitable cut-off function, it is easy to see or R 2 \ is a homogeneous Sobolev (p, q)-extension domain. Here the introduction of the bounded open set U is a convenient way to overcome the non-essential difficulty that functions inẆ 1,p (G) do not necessarily belong toẆ 1,q (G) when 1 ≤ q < p < ∞ and G has infinite volume. The following technical lemma justifies our terminology.

respectively) is a homogeneous Sobolev (p, q)-extension domain with a linear extension operator.
Proof We only consider the case of , since the case of R 2 \ is analogous. Let U ⊂ R 2 be the corresponding Lipschitz domain which contains ∂ . For a given function u ∈Ẇ 1,p ( ), we define a function E R (u) by setting for z ∈ . (2.5) Then E R (u) has a representative that belongs toẆ 1,q (U ) with Since U ⊂ R 2 is a bounded Lipschitz domain, by the result due to Jones [24], for every 1 < q < ∞, there exists a bounded linear extension operator L : The corresponding inward and outward cuspidal domains are defined by setting    Sobolev (1, 1)-extension domain.
By combining Theorem 1.2 and Proposition 2.2, we can give a new and simpler proof to the necessary part of the following proposition which is the main result in [22]. The proof of the sufficient part comes from the construction of desired homeomorphisms of finite distortion in [22].

Proof of Theorem 1.2
Let ⊂ R 2 be a bounded simply connected domain with ⊂⊂ B(0, R) for a large enough R > 0. Suppose there exists a homeomorphism of finite distortion h : (3.1) The circle inversion map R : is an anticonformal reflection over the unit circle ∂D, which means that at every point it preserves angles and reverses orientation. Then a self-homeomorphism R : R 2 onto − → R 2 defined on every z ∈ R 2 by setting is a reflection over the boundary ∂ .
With respect to different domains and R 2 \ , we divide the full proof of Theorem 1.2 into two parts. First, let us prove the Sobolev extension property of the bounded domain . (3.2) By Proposition 2.1, it suffices to prove that E R (u) ∈Ẇ 1, 2q q+1 (B(0, R)) with the following inequality that for a positive constant C independent of u. Since 1 < 2q q+1 ≤ 2 ≤ 2p p−1 < ∞, the Hölder inequality implies By [18,Theorem 4.13], ∂ must be of measure zero. Hence, we only need to prove We define a function v on D by setting v(z) := u • h −1 (z) for every z ∈ D and define an extension function E R (v) on h(B(0, R)) with z = (x, y) by setting (3.5) By the definitions of functions and reflections, for every z ∈ B(0, R), we have We divide the remaining argument into three steps.
Step 1: We would like to show v ∈Ẇ 1,2 (D) with D |Dv(z)| 2 dz for a positive constant C that is independent of u. By [18, Theorem 1.6 and Theorem 2.24], h −1 ∈ W 1,1 (D, R 2 ) and it is differentiable almost everywhere on D. Hence, there exists a measurable subset G ⊂ D with |G| = |D| and h −1 is differentiable on every point z ∈ G. By the chain rule, for every z ∈ G, we have Set A ⊂ G to be the subset such that J h −1 (z) > 0 for every z ∈ A. Then by [18,Theorem 1.6], |Dh −1 (z)| = 0 for almost every z ∈ G \ A. Hence, we have If p = ∞, then h is quasiconformal. By the fact that the inverse of a quasiconformal mapping is also quasiconformal, we have If 1 < p < ∞, the Hölder inequality implies For every 1 < p ≤ ∞, the change of variables formula implies By [18,Lemma A.29], h is differentiable on every w ∈ h −1 (A) with Dh(w) = (Dh −1 (z)) −1 for w = h −1 (z). Hence, for every z ∈ A with w = h −1 (z), we have
Step 2: for almost every z ∈ D, and (3.14) By passing to a subsequence if necessary, we can also assume the sequence of weak gradients {Dv k } converges to Dv almost everywhere on D. h(B(0, R)). By definition (3.5) and the fact that R is anticonformal, it is easy to see that we have with C a positive constant that is independent of u.
Step 3: We prove E R (u) ∈Ẇ 1, 2q q+1 (B(0, R)) and we have for a positive constant C that is independent of u. As we know, for every z ∈ B(0, R). By [18, Theorem 1.7 and Theorem 2.24], h is differentiable almost everywhere on B(0, R)\ with positive determinant value. Hence, there exists a measurable subset G ⊂ B(0, R) \ with | G| = |B(0, R) \ | and h is differentiable on every point in z ∈ G with J h (z) > 0. By the chain rule, for every z ∈ G, we have If q = ∞, then h restricts to B(0, R) \ is quasiconformal. Then we have If 1 < q < ∞, then we have The Hölder inequality implies The change of variables formula implies By the definition of distortion function and the integral condition (3.1), we have The idea of the proof is very similar with the proof of last theorem. The main difference is that R 2 \ is unbounded here. Hence, we give a sketch proof here.
Proof From the geometrical viewpoint, it is easy to see R 2 \ has same Sobolev extension property with the bounded domain B(0, R) \ . Hence, by Lemma 2.1, it suffices to prove is a homeomorphism with h( ) = D, there exist two positive constants 1 > 0 and 2 > 0 such that B(0, 1 + 1 ) ⊂ h(B(0, R)) and To simplify the notation, we still denote u B(0,R)\ by u. Then we define a function E R (u) on U by setting for a positive constant C that is independent of u. By the definition of E R (u), the Hölder inequality implies for a positive constant C that is independent of u. Define an extension function E R (v) on U by setting (3.30) It is easy to see for every z ∈ U, we have By Eq. 3.29, v A(1,1+ 1 ) belongs to the classẆ 1,2 (A(1, 1 + 1 )). To simplify the notation, we still denote v A(1, . With a similar argument to the inequality (3.17), we for a positive constant C that is independent of u. Then, with a similar argument to the inequality (3.18), we obtain the inequality for a positive constant C that is independent of u. Finally, by combining inequalities (3.29), (3.31) and (3.32), we obtain the desired inequality (3.28).

Sharpness of Theorem 1.2
In this section, we discuss the sharpness of Theorem 1.2. Let = i t s ⊂ R 2 be a polynomial inward cuspidal domain with the degree 1 < s < ∞ such that there exists a homeomorphism of finite distortion h : R 2 onto − → R 2 with h( ) = D, and the distortion function satisfies the integral condition (3.1). It is easy to see R 2 \ i t s has a same Sobolev extension property with the polynomial outward cuspidal domain o t s . By combining Propositions 2.2 and 2.3, we obtain that we cannot obtain a better Sobolev extension result than is a Sobolev 2p p−1 , 2q q+1 -extension domain and R 2 \ i t s is a Sobolev 2q q−1 , 2p p+1 -extension domain. It shows the sharpness of Theorem 1.2.

A new proof of the necessity of Proposition 2.3
The necessity claim in Proposition 2.3 is proved by combing Theorem 1.2 and Proposition 2.2.

An application of Theorem 1.2
In this section, we prove Theorem 1.3, which can be regarded as an application of Theorem 1.2.
Funding Open access funding provided by University of Jyväskylä (JYU).

Data Availability Statement
The author declares that the data supporting the findings of this study are available within the article.
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