ZERO VOLUME BOUNDARY FOR EXTENSION DOMAINS FROM SOBOLEV TO BV

<jats:p>In this note, we prove that the boundary of a <jats:inline-formula><jats:alternatives><jats:tex-math>$$(W^{1, p}, BV)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                <mml:mrow>
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                      <mml:mn>1</mml:mn>
                      <mml:mo>,</mml:mo>
                      <mml:mi>p</mml:mi>
                    </mml:mrow>
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                  <mml:mo>,</mml:mo>
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                  <mml:mi>V</mml:mi>
                  <mml:mo>)</mml:mo>
                </mml:mrow>
              </mml:math></jats:alternatives></jats:inline-formula>-extension domain is of volume zero under the assumption that the domain <jats:inline-formula><jats:alternatives><jats:tex-math>$${\Omega }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                <mml:mi>Ω</mml:mi>
              </mml:math></jats:alternatives></jats:inline-formula> is 1-fat at almost every <jats:inline-formula><jats:alternatives><jats:tex-math>$$x\in \partial {\Omega }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                <mml:mrow>
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                  <mml:mi>∂</mml:mi>
                  <mml:mi>Ω</mml:mi>
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              </mml:math></jats:alternatives></jats:inline-formula>. Especially, the boundary of any planar <jats:inline-formula><jats:alternatives><jats:tex-math>$$(W^{1, p}, BV)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                <mml:mrow>
                  <mml:mo>(</mml:mo>
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                  <mml:mo>)</mml:mo>
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              </mml:math></jats:alternatives></jats:inline-formula>-extension domain is of volume zero.</jats:p>


Introduction
Given 1 ≤ q ≤ p ≤ ∞, a bounded domain Ω ⊂ R n , n ≥ 2, is said to be a (W 1,p , W 1,q )extension domain if there exists a bounded extension operator and is said to be a (W 1,p , BV)-extension domain if there exists a bounded extension operator The theory of Sobolev extensions is of interest in several fields in analysis.Partial motivations for the study of Sobolev extensions comes from the thoery of PDEs, for example, see [Maz11].It was proved in [Cal61,Ste70] that for every Lipschitz domain in R n , there exists a bounded linear extension operator E : W k,p (Ω) → W k,p (R n ) for each k ∈ N and 1 ≤ p ≤ ∞.Here W k,p (Ω) is the Banach space of all L p -integrable functions whose distributional derivatives up to order k are L p -integrable.Later, the notion of (ǫ, δ)-domains was introduced by Jones in [Jon81], and it was proved that for every (ǫ, δ)-domain, there exists a bounded linear extension operator E : W k,p (Ω) → W k,p (R n ) for every k ∈ N and 1 ≤ p ≤ ∞.
In [VGL79], a geometric characterization of planar (W 1,2 , W 1,2 )-extension domain was given.By later results in [Kos98, Shv10, KRZ15, KRZ17], we now have geometric characterizations of planar simply connected (W 1,p , W 1,p )-extension domains for all 1 ≤ p ≤ ∞.A geometric characterization is also known for planar simply connected (L k,p , L k,p )-extension domains with 2 < p ≤ ∞, see [SZ16, Whi34, Zob99].Here L k,p (Ω) denotes the homogeneous Sobolev space which contains locally integrable functions whose k-th order distributional derivative is L p -integrable.Beyond the planar simply connected case, geometric characterizations of Sobolev extension domains are still missing.However, several necessary properties have been obtained for general Sobolev extension domains.
In the case where Ω is a planar Jordan W 1,p , W 1,p -extension domain, Ω has to be a so-called John domain when 1 ≤ p ≤ 2 and the complementary domain has to be John when 2 ≤ p < ∞.The John condition implies that the Hausdorff dimension of ∂Ω must be strictly less than 2, see [KR97].Recently, Lu čić, Takanen and the first named author gave a sharp estimate on the Hausdorff dimension of ∂Ω, see [LRT21].In general, the Hausdorff dimension of a (W 1,p , W 1,p )-extension domain can well be n.
The outward cusp domain with a polynomial type singularity is a typical example which is not a (W 1,p , W 1,p )-extension domain for 1 ≤ p < ∞.However, it is a (W 1,p , W 1,q )extension domain, for some 1 ≤ q < p ≤ ∞, see the monograph [MP97] and the references therein.Hence, for 1 ≤ q < p ≤ ∞, it is not necessary for a (W 1,p , W 1,q )-extension domain to be Ahlfors regular.In the absence of Ahlfors regularity, one has to find alternative approaches for proving |∂Ω| = 0.The first approach in [Ukh20, Ukh99] was to generalize the Ahlfors regularity (1.1) to a Ahlfors-type estimate Here Φ is a bounded and quasiadditive set function generated by the (W 1,p , W 1,q )-extension property and defined on open sets U ⊂ R n , see Section 3. By differentiating Φ with respect to the Lebesgue measure, one concludes that |∂Ω| = 0 if Ω is a (W 1,p , W 1,q )-extension domain for n < q < p < ∞.
Recently, Koskela, Ukhlov and the second named author [KUZ21] generalized this result and proved that the boundary of a W 1,p , W 1,q -extension domain must be of volume zero for n − 1 < q < p < ∞ (and for 1 ≤ q < p < ∞ on the plane).For 1 ≤ q < n − 1 and (n − 1)q/(n − 1 − q) < p < ∞, they constructed a W 1,p , W 1,q -extension domain Ω ⊂ R n with |∂Ω| > 0. For the remaining range of exponents where 1 ≤ q ≤ n − 1 and q < p ≤ (n − 1)q/(n − 1 − q), it is still not clear whether the boundary of every W 1,p , W 1,q -extension domain must be of volume zero.
As is well-known, for every domain Ω ⊂ R n , the space of functions of bounded variation BV(Ω) strictly contains every Sobolev space W 1,q (Ω) for 1 ≤ q ≤ ∞.Hence, the class of W 1,p , BV -extension domains contains the class of W 1,p , W 1,q -extension domains for every 1 ≤ q ≤ p < ∞.As a basic example to indicate that the containment is strict when n ≥ 2, we can take the slit disk (the unit disk minus a radial segment) in the plane.The slit disk is a W 1,p , BV -extension domain for every 1 ≤ p < ∞, and even a (BV, BV)extension domain; however it is not a W 1,p , W 1,q -extension domain for any 1 ≤ q ≤ p < ∞.This basic example also shows that it is natural to consider the geometric properties of W 1,p , BV -extension domains.In this paper, we focus on the question whether the boundary of a (W 1,p , BV)-extension domain is of volume zero.Our first theorem tells us that the (BV, BV)-extension property is equivalent to the W 1,1 , BV -extension property.Hence, a (W 1,1 , BV)-extension domain is Ahlfors regular and so its boundary is of volume zero.
The other direction from (W 1,1 , BV)-extension property to (BV, BV)-extension property is not as straightforward, as W 1,1 (Ω) is only a proper subspace of BV(Ω).The essential tool here is the Whitney smoothing operator constructed by García-Bravo and the first named author in [GBR21].This Whitney smoothing operator maps every function in BV(Ω) to a function in W 1,1 (Ω) with the same trace on ∂Ω, so that the norm of the image in W 1,1 (Ω) is uniformly controlled from above by the norm of the corresponding preimage in BV (Ω).
With an extra assumption that Ω is q-fat at almost every point on the boundary ∂Ω, in [KUZ21] it was shown that the boundary of a (W 1,p , W 1,q )-extension domain is of volume zero when 1 ≤ q < p < ∞.The essential point there was that the q-fatness of the domain on the boundary guarantees the continuity of a W 1,q -function on the boundary.Maybe a bit surprisingly, the assumption that the domain is 1-fat at almost every point on the boundary also guarantees that the boundary of a (W 1,p , BV)-extension domain is of volume zero.In particular, every planar domain is 1-fat at every point of the boundary.Hence, we have the following theorem.
Theorem 1.2.Let Ω ⊂ R n be a (W 1,p , BV)-extension domain for 1 ≤ p < ∞, which is 1-fat at almost every x ∈ ∂Ω.Then |∂Ω| = 0.In particular, for every planar (W 1,p , BV)-extension domain In light of the results and example given in [KUZ21], the most interesting open question is what happens in the range 1 < p ≤ (n − 1)/(n − 2) of exponents.For this range, we do not know whether the boundary of a (W 1,p , BV)-extension domain must be of volume zero.If a counterexample exists in this range, it might be easier to construct it in the (W 1,p , BV)-case rather than the (W 1,p , W 1,1 )-case.Hence we leave it as a question here.

Preliminaries
For a locally integrable function u ∈ L 1 loc (Ω) and a measurable subset A ⊂ Ω with 0 < |A| < ∞, we define Definition 2.1.Let Ω ⊂ R n be a domain.
Definition 2.3.We say that a domain Ω ⊂ R n is a W 1,p , BV -extension domain for 1 ≤ p < ∞, if there exists a bounded extension operator E : for a constant C > 1 independent of u.
Let U ⊂ R n be an open set and A ⊂ U be a measurable subset with A ⊂ U.The p-admissible set W p (A; U) is defined by setting Following Lahti [Lah17], we define 1-fatness below.
Definition 2.5.Let A ⊂ R n be a measurable subset.We say that A is 1-thin at the point x ∈ R n , if If A is not 1-thin at x, we say that A is 1-fat at x. Furthermore, we say that a set U is By [Lah17, Lemma 4.2], the collection of 1-finely open sets is a topology on R n .For a function u ∈ BV(R n ), we define the lower approximate limit u ⋆ by setting and the upper approximate limit u ⋆ by setting The set S u := {x ∈ R n : u ⋆ (x) < u ⋆ (x)} is called the jump set of u.By the Lebesgue differentiation theorem, |S u | = 0. Using the lower and upper approximate limits, we define the precise representative ũ := (u ⋆ + u ⋆ )/2.The following lemma was proved in [Lah17, Corollary 5.1].
The following lemma for u ∈ W 1,1 (R n ) was proved in [KUZ21, Lemma 2.6], which is also a corollary of a result in [HKM93].We generalize it to BV(R n ) here.
The following coarea formula for BV functions can be found in [EG15, Section 5.5].See also [GBR21, See [AFP00, Theorem 3.44] for the proof of the following (1, 1)-Poincaré inequality for BV functions.
Proposition 2.9.Let Ω ⊂ R n be a bounded Lipschitz domain.Then there exists a constant C > 0 depending on n and Ω such that for every u ∈ BV(Ω), we have In particular, there exists a constant C > 0 only depending on n so that if (2.1)

A set function arising from the extension
In this subsection, we introduce a set function defined on the class of open sets in R n and taking nonnegative values.Our set function here is a modification of the one originally introduced by Ukhlov [Ukh20, Ukh99].See also [VU04,VU05] for related set functions.The modified version of the set function we use is from [KUZ21], where it was used by Koskela, Ukhlov and the second named author to study the size of the boundary of a (W 1,p , W 1,q )extension domains.Let us recall that a set function Φ defined on the class of open subsets of R n and taking nonnegative values is called quasiadditive (see for example [VU04]), if for all open sets U 1 ⊂ U 2 ⊂ R n , we have and there exists a positive constant C such that for arbitrary pairwise disjoint open sets Then we define the set function Φ by setting The proof of the following lemma is almost the same as the proof of [KUZ21, Theorem 3.1].One needs to simply replace Dv L q (U) by Dv (U) in the proof of [KUZ21, Theorem 3.1] and repeat the argument.

Proofs of the results
In this section we prove Theorems 1.1 and 1.2.
Let us then prove the converse and assume that Ω ⊂ R n is a (W 1,1 , BV)-extension domain with an extension operator E. Let S Ω,Ω be the Whitney smoothing operator defined in [GBR21].Then by [GBR21, Theorem 3.1], for every u ∈ BV(Ω), we have S Ω,Ω (u) for a positive constant C independent of u, and where u − S Ω,Ω (u) is understood to be defined on the whole space R n via a zero-extension. Then Now, define T : BV(Ω) → BV(R n ) by setting for every u ∈ BV(Ω) By (4.1), we have T(u) ∈ BV(R n ) with Hence, Ω is a BV-extension domain.
Proof of Theorem 1.2.Assume towards a contradiction that |∂Ω| > 0. By the Lebesgue density point theorem and Lemma 3.2, there exists a measurable subset U of ∂Ω with |U| = |∂Ω| such that every x ∈ U is a Lebesgue point of ∂Ω and DΦ(x) < ∞.Fix x ∈ U. Since x is a Lebesgue point, there exists a sufficiently small r x > 0, such that for every 0 < r < r x , we have (4.4) We have This contradicts the assumption that x is a Lebesgue point of ∂Ω.Hence, we conclude that |∂Ω| = 0.
Let us then consider the case Ω ⊂ R 2 .By [HK14, Theorem A.29], for every x ∈ ∂Ω and every 0 < r < min 1, 1 4 diam (Ω) , we have Cap 1 (Ω ∩ B(x, r); B(x, 2r)) ≥ cr for a constant 0 < c < 1.This implies that Ω is 1-fat at every x ∈ ∂Ω.Hence, by combining this with the first part of the theorem, we have that the boundary of any planar (W 1,p , BV)extension domain is of volume zero.

Definition 2. 4 .
Let U ⊂ R n be an open set and A ⊂ U with A ⊂ U.The relative p-capacity Cap p (A; U) is defined by setting Cap p (A; U) := inf u∈W p (A;U) U |∇u(x)| p dx.
For every 1 ≤ p ≤ ∞, we define the Sobolev space W 1,p (Ω) to be W 1,p (Ω) := {u ∈ L p (Ω) : ∇u ∈ L p (Ω; R n )} , where ∇u denotes the distributional gradient of u.It is equipped with the nonhomogeneous norm Let Ω ⊂ R n be a domain.A function u ∈ L 1 (Ω) is said to have bounded variation and denoted u ∈ BV(Ω) if u W 1,p (Ω) = u L p (Ω) + ∇u L p (Ω) .Now, let us give the definition of functions of bounded variation.Definition 2.2.
Theorem 2.2].Given a function u ∈ BV(Ω), the superlevel sets u t = {x ∈ Ω : u(x) > t} have finite perimeter in Ω for almost every t ∈ R and