Morrey–Sobolev Extension Domains

We show that every uniform domain of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\mathbb {R}}}^n}$$\end{document} with n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document} is a Morrey–Sobolev W1,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {W}}^{1,\,p}$$\end{document}-extension domain for all p∈[1,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\,n)$$\end{document}, and moreover, that this result is essentially the best possible for each p∈[1,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\,n)$$\end{document} in the sense that, given a simply connected planar domain or a domain of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\mathbb {R}}}^n}$$\end{document} with n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 3$$\end{document} that is quasiconformal equivalent to a uniform domain, if it is a W1,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {W}}^{1,\,p} $$\end{document}-extension domain, then it must be uniform.

where ∇u is the distributional derivative of u. Modulo constant functions, W 1, p ( ) forms a Banach space. We say that is a W 1, p -extension domain if there exists a bounded (linear) operator : W 1, p ( ) → W 1, p (R n ) such that u| = u for all u ∈ W 1, p ( ). Regarding the issue of linearity in our definition we refer the reader to [4]. Notice that our definition of W 1, p and the corresponding norm only deal with ∇u. The extension problem for the usual norm is equivalent to our definition when is bounded [5].
Jones showed in his seminal paper [6] that every uniform domain of R n is a W 1, pextension domain for all p ∈ [1, ∞). Let us recall the definition. Jones's idea was to construct an extension operator via a reflection technique for Whitney cubes, motivated by "quasiconformal reflections"; it was well known that a simply connected planar domain is uniform if and only if it is the image of the unit disk under some quasiconformal mapping on R 2 ; see [2]. In the case p = n, Jones's result is essentially the best possible: a simply connected planar domain is a W 1, 2 -extension domain if and only if it is a uniform domain; see [13] and also [2] for a higher dimensional analog.
In the case p > n, the correct geometric condition is p−n p−1 -subhyperbolicity: each such domain is a W 1, p -extension domain and a simply connected planar domain is a W 1, p -extension domain if and only if it is a p−2 p−1 -subhyperbolic domain; see [11] and also [8]. This class of domains is strictly larger than the class of uniform domains.
In the case 1 ≤ p < n, it is also known that even a simply connected planar W 1, pextension domain is not necessarily uniform (see [10] and related examples in [7,9]), and it is still open to find a geometric characterization.
The geometric characterization for the planar case for p = n can be obtained via the concept of linear local connectivity.

Definition 1.2 A domain
is linearly locally connected (for short, LLC) if there exists a constant b ∈ (0, 1] such that for all z ∈ R n and r > 0, LLC(1) points in ∩ B(z, r ) can be joined by a rectifiable curve in ∩ B(z, r/b); LLC(2) points in \B(z, r ) can be joined by a rectifiable curve in \B(z, br ).
Linear local connectivity is intimately related to uniformity in the sense that each domain that is quasiconformally equivalent to a uniform domain is uniform if and only if LLC; see, e.g., [2]. Hence a simply connected planar domain is LLC if and only if it is uniform. Regarding Sobolev extensions, one knows that each W 1, n -extension domain is LLC; see [2]. In fact, W 1, p -extension domains are known to be LLC(1) for p ≥ n (see [14]) and LLC(2) for n − 1 < p ≤ n and also for p = 1 when n = 2; see [3,7].
As a byproduct of the proof of our main result Theorem 1.5, we are able to remove the restriction that p > n − 1 for concluding LLC (2 where the supremum is taken over all balls B ⊂ R n , r B is the radius of B and ∇u is the distributional gradient of u. The above formula only gives a seminorm, but modulo constant functions, W 1, p ( ) is a Banach space.
Observe that W 1, n ( ) = W 1, n ( ), and that (via the Hölder inequality), for every p ∈ [1, n), we always have W 1, n ( ) ⊂ W 1, p ( ). Moreover, for every p ∈ [1, n), The main aim of this paper is to establish the following result for W 1, p -extension domains. Below is a (Morrey-Sobolev) W 1, p -extension domain if there exists a bounded (linear) operator : The proof of Theorem 1.5(i) employs a slight modification to the extension operator constructed by Jones and several properties of uniform domains. The proof of Theorem 1.5(ii) relies on a lower bound on the Morrey-Sobolev capacity established in Lemma 5.2, a measure density property obtained in Theorem 4.1, and some ideas from [2]. We stress here that (ii) is not an obvious consequence of the ideas in [2]: for p ≤ n − 1 and n ≥ 3, it may well happen that the Morrey-Sobolev capacity of a pair of continua is zero; see [15]. Up to now, this phenomenon for the usual variational capacity has been the obstacle for establishing Proposition 1.3. This paper is organized as follows. In Sect. 2, we recall several properties of uniform domains. In Sect. 3, we prove Theorem 1.5(i). In Sect. 4, we obtain a strong measure density property of Morrey-Sobolev and Sobolev extension domains. In Sect. 5, we establish a simple lower bound on the Morrey-Sobolev capacity. In Sect. 6, we prove Proposition 1.3 and Theorem 1.5(ii).
The notation used in what follows is standard. We denote by C a positive constant which is independent of the main parameters, but which may vary from line to line. Constants with subscripts, such as 0 , do not change in different occurrences. The symbol A B or B A means that A ≤ C B. If A B and B A, we then write A ∼ B. For any locally integrable function u and measurable set X , we denote by -X u the average of f on X , namely, -X f ≡ 1 |X | X f dx. For a set and x ∈ R n , we use d(x, ) to denote inf z∈ |x − z|, the distance from x to . Here λQ denotes the cube concentric with Q, with sides parallel to the axes, and with edge length (λQ) = λ (Q).

Uniform Domains
We recall several facts about uniform domains and establish some lemmas which will be used to prove Theorem 1.5(i).
Let be a uniform domain. It is well known that |∂ | = 0; see [6]. Since is open it admits a Whitney decomposition, that is, there exists a collection W 1 = {S j } j∈N of countably many dyadic (closed) cubes such that which is the collection of all neighbor cubes of Q in W i . It is easy to see that there exists an integer N depending only on n such that for all Q ∈ W i and i = 1, 2, 3. Jones [6] derived from (1.1) that for each cube in W 3 , we can pick a "reflected" cube in W 1 in the following sense: for every Q j ∈ W 3 , we can find at least one S k ∈ W 1 satisfying that where C 1 is a constant depending on 0 and n but not on Q j and S k . Usually, for each Q j ∈ W 3 , there may exist more than one such S k , but if S k ∈ W 1 is another cube as above, we have Below we fix one of these cubes and denote it by Q * j . Then there is an integer N 1 depending only on C 1 and n such that for all S k ∈ W 1 , there are at most N 1 cubes Q j ∈ W 3 such that Q * j = S k , that is, For every pair of Q j , Q k ∈ W 3 with Q j ∩ Q k = ∅, by (2.1), we have Moreover, based on (1.1), we have consists of an (n − 1)-dimensional (closed) cube for all s = 1, . . . , m − 1, and m ≤ N 2 , where N 2 is an integer depending only on n and 0 but not on Q j and Q k .
We call such a family F j,k a chain of length m joining Q * j and Q * k . Obviously, For each pair of Q j , Q k ∈ W 3 with Q j ∩ Q k = ∅, we fix one family F j,k as in Lemma 2.1 (notice that there may have more than one such family), and set where and in the sequel, by abuse of notation, we also denote the union ∪ m k=1 S i k by F j, k . One can easily derive from Lemma 2.1, (2.1) and (2.2) that Lemma 2.2 There exists a positive constant N 3 depending only on 0 and n such that for each Q j ∈ W 3 , we have To prove Theorem 1.5(i), we need further properties of uniform domains.

Lemma 2.3
There exists C 3 depending only on n and 0 such that for every ball as desired. This finishes the proof of Lemma 2.3.

Lemma 2.4
There exists a positive constant C 2 depending only on 0 and n such that for all Q j ∈ W 3 and u ∈ W 1, p ( ), we have It then suffices to show that Hence applying Lemma 2.2, we have as desired. Towards (2.5), let R s be the cube with (R s ) = 1 32 (S i s ) and with the same center as the (n − 1)-dimensional cube S i 1 ∩ S i s+1 . Also let R + s ⊂ S i s be the cube with same edge length as R s and |R + A similar estimate holds for u S i s+1 − u R s . Combining these estimates gives the desired results, and hence completes the proof of Lemma 2.4.
Jones further established the following density result.

Lemma 2.5 Suppose that is a bounded uniform domain. Then C
Finally, we need a Sobolev-Poincaré inequality for suitable subdomains of uniform domains. This result follows by combining the main result in [1] with a localization result; see 1.9 in [12].

Lemma 2.6
Suppose that is a uniform domain and 1 ≤ p < n. Set p * = pn/(n − p). Then there exist positive constants C 4 , C 5 depending only on n and so that the following holds. Given any x ∈ and r > 0, there is a subdomain r ⊂ with and such that for all u ∈ W 1, p ( r ), we have u ∈ L 1 ( r ) and

Proof of Theorem 1.5(i)
Let be a uniform domain. We divide the proof into the following 4 steps.
Step 1 We employ Jones's construction of extension operator on uniform domains (with a slight modification when diam < ∞). Let W 1 , W 2 and W 3 be as in Section 2.
where C is a constant independent of Q j .
For u ∈ W 1, p ( ), we define an extension u as: where Q * j ∈ W 1 corresponds to Q j as in Sect. 2, and u = 1 Notice that is linear and independent of p, and u is defined on R n \∂ . Now we have to show that u induces another function It suffices to show that u ∈ W 1, p loc ( ) and for every ball B = B(x 0 , r ), we have Without loss of generality, we may assume that ( ) = ∅.
We first show that u ∈ W as a finite sum. Hence u is differentiable on B and |∇ u| Observe that if diam = ∞, then W 3 = W 2 and hence H 2 = 0. Recall that on each Q i ∈ W 3 , we have Q j ∈W 3 φ j = 1. Thus, by (3.1), we have By Lemmas 2.2 and 2.3, we obtain To estimate H 1, 2 , we may assume that B ∩ Q i 0 = ∅ for some Q i 0 ∈ W 3 with 32r < (Q i 0 ). Observe that 32r < (Q i 0 ) implies that B ⊂ 9 8 Q i 0 , and hence, applying Lemma 2.3 to the ball B = B(x Q i 0 , (Q i 0 )), we have If diam = ∞, by H 2 = 0, the proof of (3.2) is finished. Assume that diam < ∞. Suppose that u = 0 on Q i for some Q i ∈ W 2 \W 3 . Observing that φ j = 0 on Q i only happens when Q j ∈ W 3 (Q i ), we have and hence, by Lemma 2.6, If r ≥ 1 2nC 0 diam , then Combining the estimates of H 1 and H 2 , we obtain the desired result.
Step 3 For u ∈ W 1, p ( ) ∩ C 1 0 (R n ), we define u = u on R n \ and u = u on ∂ . We claim that u ∈ W 1, p loc (R n ). To see that u ∈ W 1, p (B) for every ball B ⊂ R n , it suffices to show that u is locally Lipschitz on R n , that is, . Case x ∈ and y ∈ R n \ . Then y ∈ Q i for some

Thus (3.3) follows.
Step 4 Finally, for arbitrary u ∈ W 1, p ( ), we define with u = u on R n \∂ as above.
Since |∂ | = 0, for every ball B ⊂ R n , we have By this and the estimate from Step 2, our claim would follow if we knew that u ∈ W 1, p loc (R n ). We show that u has this degree of regularity relying on Step 3. We separate our argument into two cases.
Case of diam < ∞. Then u ∈ W 1, p ( ) implies that u ∈ W 1, p ( ) and from Lemma 2.6 we further deduce that u ∈ L p ( ). From Lemma 2.5 we conclude that C 1 0 (R n ) is dense in W 1, p ( ) and since u ∈ L p ( ) it easily follows via the Poincaré inequality that there exists a sequence {u k } k∈N ⊂ C 1 0 (R n ) such that u −u k W 1, p ( ) + u − u k L p ( ) 2 −k . By Step 3, u k is well defined, and u k − u = (u k − u ) for all k, ∈ N. Moreover, by Jones [6], By the Poincaré inequality, we know that there exists a function still denoted by u such that u k converges to u in L p loc (R n ) and in W 1, p (R n ). Obviously on , we have with x 0 ∈ ∂ and λ ∈ N. Fix λ and choose η λ ∈ C 1 0 (B λ ) so that η λ = 1 on B λ/2 . Applying Lemma 2.6 for the ball B(x 0 , C 8 λ) we see that uη λ ∈ W 1, p ( ) ∩ L p ( ).
Fix a ball B ⊂ R n . Observe that whenever λ, τ ∈ N with λ < τ, we have There exists a positive integer N such that for all τ ≥ N λ, F(Q i ) ⊂ B τ/2 , and hence The claim follows by noticing from above that we may define u via the extensions (uη N λ )

Measure Density of W 1, p -and W 1, p -Extension Domains
We first show the following strong measure density property for W 1, p -extension domains, which will be employed to prove Theorem 1.5(ii).
Theorem 4.1 means that the components of \K for an arbitrary closed set K of R n are Ahlfors n-regular to some extent. To prove this, we use ideas from [4].

Lemma 4.2
Let p ∈ [1, n) and be a W 1, p -extension domain of R n . There exist positive constants C 6 and C 7 such that for each B ⊂ R n and for all u ∈ W 1, p ( ), Proof Since is a W 1, p -extension domain of R n , denoting the extension operator by , for every u ∈ W 1, p ( ), we have that u ∈ W 1, p (R n ) with where C 0 is a fixed constant. Applying Poincaré's inequality, we have u ∈ B M O(R n ) with where C 0 is a fixed constant. Moreover, by the John-Nirenberg theorem for BMOfunctions, for each ball B, where C 0 is a fixed constant. Thus, choosing as desired.
Proof of Theorem 4.1 Without loss of generality, we may assume that r < ∞. Let b 0 = 1 and b j ∈ (0, 1) for j ∈ N be such that Define a function u on by setting Since B(x, r )∩ 0 ∩K = ∅ and B(x, r ) has empty intersection with other components of \K , we know that u is well defined. Then By Lemma 4.2 and (4.1), we have Observe that, for each c ∈ R, |u − c| ≥ 1/2 either on (B(x, r )\B(x, b 1 r )) ∩ 0 or on B(x, b 2 r ) ∩ 0 . By (4.1), we conclude that which implies that Similar inequalities also hold for b j r − b j+1 r with j ≥ 2. This leads to If b 1 ≥ 1/10, observing that t log 1 t ≥ 1 implies that t 1, we have |B(x, r ) ∩ 0 | 1/n r as desired. If b 1 ≤ 1/10, we choose R = 2r/5 and a point y ∈ B(x, r ) ∩ 0 such that |y − x| = b 1 r + R/2. Then we have b 1 ≥ 1/2. Applying the result when b 1 ≥ 1/10, we conclude that which implies |B(x, r ) ∩ 0 | r n as desired.
Theorem 4.1 implies the following result.

Corollary 4.3
Let p ∈ [1, n) and be a W 1, p -extension domain of R n . Then has the measure density property, that is, there exists a constant C > 0 such that for all x ∈ and 0 < r < diam , we have For W 1, p -extension domains with p ∈ [1, n), a result similar to Corollary 4.3 was established in [4]. But to show the LLC(2)-property of Sobolev W 1, p -extension domains when p ∈ [1, n −1], we need the following stronger version, which is similar to Theorem 4.1; see Sect. 6 below.

Theorem 4.4 Let p ∈ [1, n) and
be a W 1, p -extension domain of R n . Then there exists a constant C > 0 such that for every component 0 of \K , where K is an arbitrary closed set of R n , and for all x ∈ 0 and 0 < r ≤ min{ dist (x, ∂ 0 \∂ ), diam } we have Proof of Theorem 4.4 Without loss of generality, we may assume that r < ∞. The proof is similar to that of Theorem 4.1; we sketch it. Let b j for j ∈ N ∪ {0} and the function u be exactly as in Theorem 4.1. Then Since is a W 1, p -extension domain, we may extend u to u ∈ W 1, p (R n ) with a norm bound. Applying the Sobolev-Poincaré inequality, we have Observe that, for each c ∈ R, |u − c| ≥ 1/2 either on (B(x, r )\B(x, b 1 r )) ∩ 0 or on B(x, b 2 r ) ∩ 0 . By (4.1), we have Similar inequalities also hold for b j r − b j+1 r with j ≥ 2. This leads to Applying the same argument on b 1 as Theorem 4.1, we have |B(x, r ) ∩ 0 | 1/n r as desired.

Morrey-Sobolev Capacity
We obtain a lower bound for the Morrey-Sobolev capacity, which will be used in the proof of Theorem 1.5(ii). Since we will only deal with estimates in terms of the usual Lebesgue measure, the following definition will be sufficient for our purposes. Given a pair of disjoint set E, F ⊂ , define the W 1, p -capacity as where (E, F, ) denotes the class of all functions u ∈ W 1, p ( ) such that 0 ≤ u ≤ 1 on , u = 0 a.e. on E, and u = 1 a.e. on F. Obviously, if E, F ⊂ R n are disjoint sets, E ⊂ E and F ⊂ F, then Moreover, we have the following trivial estimates.
Proof By Lemma 5.1, we may assume that = R n . Let u ∈ (E, F, R n ). Assume first that u B(x 0 , r ) ≥ 1/2. Then, by the Poincaré inequality, , and the desired estimate follows. If u B(x 0 , r ) < 1/2, we obtain our estimate by replacing E in the above string of inequalities by F.
Define the Sobolev W 1, p -capacity Cap W 1, p (E, F, ) similarly to (5.1). We have the following well-known consequence of the above proof.

Proofs of Proposition 1.3 and Theorem 1.5(ii)
Proof of Theorem 1.5 (ii) Case LLC (2). Let x 1 , x 2 ∈ \B(x 0 , r ) for some x 0 ∈ and r > 0. Without loss of generality, we may assume that x 1 , x 2 ∈ ∂ B(x 0 , r ) ∩ . Indeed, the general case can easily be reduced to this one since is a domain and hence pathwise connected.
Assume that x 1 and x 2 are not in the same component of \B(x 0 , br ) for some b ∈ (0, 1). We may assume that b < 1/16. It then suffices to prove that b is bounded away from zero, independent of x 1 , x 2 , x 0 , r . Denote by i the component of \B(x 0 , br ) containing x i for i = 1, 2. Then 1 ∩ 2 = ∅. Let and F i ∩ B(x 0 , r/2) = ∅ for i = 1, 2. By Theorem 4.1, |F i | r n for i = 1, 2. Applying Lemma 5.2, we obtain The lower bounded on b then will follow from the upper bound Indeed, if this holds, then we have log 1 b 1 n −1 1, which gives b 1. To obtain (6.2), for all x ∈ R n , define x ∈ 2 \B(x 0 , r/2); otherwise.
Case LLC (1). Let x 1 , x 2 be a pair of points in and x 1 , x 2 ∈ B(z, r ) for some z ∈ R n and r > 0. We are going to show the existence of a rectifiable curve γ ⊂ ∩ B(z, r/b) joining x 1 and x 2 for some constant b < 1.
Proof of Proposition 1. 3 The proof is similar to the Case LLC (2) in the proof of Theorem 1.5(ii). Indeed, assume that that x 1 and x 2 are not in the same component of \B(x 0 , br ) for some 0 < b < 1/16. It then suffices to prove that b is bounded away from zero, independently of x 1 , x 2 , x 0 , r .
Let i be the component of \B(x 0 , br ) containing x i and let F i = i ∩B(x i , r/2) for i = 1, 2. Since is pathwise connected, Theorem 4.4 easily implies that |F i | r n for i = 1, 2. Applying Lemma 5.3, similarly to (6.1), we obtain Cap W 1, p (F 1 , F 2 , ) r n− p .
The lower bounded of b then will follow from the upper bound
Remark 6.1 Finally, if p ∈ [1, n), one cannot prove the condition LLC(1) for W 1, pextension domains by using the argument in the proof of the case LLC(1), Theorem 1.5 (ii). The point is that if p ∈ [1, n), the seminorm of W 1, p is not scaling invariant, while the seminorms for W 1, n and W 1, p are. With the aid of scaling property, W 1, p is imbedded into the Sobolev space W 1,n and one has the estimate Cap W 1, p ( 1 , 2 , ) 1. whenever 1 , 2 ⊂ satisfy | 1 | ∼ | 2 | ∼ r n and dist ( 1 , 2 ) r n . In fact, for p ∈ [1, n), a W 1, p -extension domain may fail to satisfy LLC(1); see, e.g., [7,9,10].