1 Introduction

To simplify the definition of extension domains, we always assume \(\Omega \subset {{{{\mathbb {R}}}}^n}\) is a bounded domain. Given \(1\le q\le p\le \infty \), a domain \({\Omega }\subset {{{{\mathbb {R}}}}^n}\), \(n\ge 2\), is said to be a \((W^{1, p}, W^{1, q})\)-extension domain if there exists a bounded extension operator

$$\begin{aligned} E:W^{1,p}({\Omega })\mapsto W^{1,q}({{{{\mathbb {R}}}}^n}), \end{aligned}$$

and is said to be a \((W^{1, p}, BV)\)-extension domain if there exists a bounded extension operator

$$\begin{aligned} E:W^{1, p}({\Omega })\mapsto BV({{{{\mathbb {R}}}}^n}). \end{aligned}$$

The theory of Sobolev extensions is of interest in several fields in analysis. Partial motivations for the study of Sobolev extensions comes from the theory of PDEs, for example, see [18]. It was proved in [2, 22] that for every Lipschitz domain in \({{{{\mathbb {R}}}}^n}\), there exists a bounded linear extension operator \(E:W^{k, p}({\Omega })\mapsto W^{k, p}({{{{\mathbb {R}}}}^n})\) for each \(k\in {\mathbb {N}}\) and \(1\le p\le \infty \). Here \(W^{k, p}({\Omega })\) 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 \((\epsilon , \delta )\)-domains was introduced by Jones in [9], and it was proved that for every \((\epsilon , \delta )\)-domain, there exists a bounded linear extension operator \(E:W^{k, p}({\Omega })\mapsto W^{k, p}({{{{\mathbb {R}}}}^n})\) for every \(k\in {\mathbb {N}}\) and \(1\le p\le \infty \).

In [26], a geometric characterization of planar \((L^{1, 2}, L^{1, 2})\)-extension domain was given. Here \(L^{k, p}({\Omega })\) denotes the homogeneous Sobolev space which contains locally integrable functions whose k-th order distributional derivative is \(L^p\)-integrable. By later results in [11, 13, 14, 21], we now have geometric characterizations of planar simply connected \((W^{1,p}, W^{1, p})\)-extension domains for all \(1\le p\le \infty \). A geometric characterization is also known for planar simply connected \((L^{k, p}, L^{k, p})\)-extension domains with \(2<p\le \infty \), see [23, 29, 30]. 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.

For a measurable subset \(F\subset {{{{\mathbb {R}}}}^n}\), we use |F| to denote its Lebesgue measure. In [7, 8], Hajłasz, Koskela and Tuominen proved for \(1\le p<\infty \) that a \(\left(W^{1, p}, W^{1,p}\right)\)-extension domain \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) must be Ahlfors regular which means that there exists a positive constant \(C>1\) such that for every \(x\in {\overline{{\Omega }}}\) and \(0<r<\min \left\rbrace 1, \frac{1}{4}\,\textrm{diam}\,{\Omega }\right\lbrace \), we have

$$\begin{aligned} |B(x, r)|\le C|B(x, r)\cap {\Omega }|. \end{aligned}$$
(1.1)

From the results in [4, 10], we know that also (BV, BV)-extension domains are Ahlfors regular. For Ahlfors regular domains, the Lebesgue differentiation theorem then easily implies \(|\partial {\Omega }|=0\).

In the case where \({\Omega }\) is a planar Jordan \(\left(W^{1, p}, W^{1,p}\right)\)-extension domain, \({\Omega }\) has to be a so-called John domain when \(1\le p\le 2\) and the complementary domain has to be John when \(2\le p<\infty \). The John condition implies that the Hausdorff dimension of \(\partial {\Omega }\) must be strictly less than 2, see [12]. Recently, Lu\(\check{c}\)ić, Takanen and the first named author gave a sharp estimate on the Hausdorff dimension of \(\partial {\Omega }\), see [17]. 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 \(1\le p<\infty \). However, it is a \((W^{1, p}, W^{1, q})\)-extension domain, for some \(1\le q<p\le \infty \), see the monograph [19] and the references therein. Hence, for \(1\le q<p\le \infty \), 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 \(|\partial {\Omega }|=0\). The first approach in [24, 25] was to generalize the Ahlfors regularity (1.1) to a Ahlfors-type estimate

$$\begin{aligned} |B(x, r)|^p\le C\Phi ^{p-q}(B(x, r))|B(x, r)\cap {\Omega }|^q \end{aligned}$$
(1.2)

for \((W^{1, p}, W^{1, p})\)-extension domains with \(n<q<p<\infty \). Here \(\Phi \) is a bounded and quasiadditive set function generated by the \((W^{1, p}, W^{1, q})\)-extension property and defined on open sets \(U\subset {{{{\mathbb {R}}}}^n}\), see Sect. 3. By differentiating \(\Phi \) with respect to the Lebesgue measure, one concludes that \(|\partial {\Omega }|=0\) if \({\Omega }\) is a \((W^{1, p}, W^{1, q})\)-extension domain for \(n< q< p < \infty \). Recently, Koskela, Ukhlov and the second named author [15] generalized this result and proved that the boundary of a \(\left(W^{1, p}, W^{1, q}\right)\)-extension domain must be of volume zero for \(n-1<q< p<\infty \) (and for \(1\le q< p<\infty \) on the plane). For \(1\le q<n-1\) and \((n-1)q/(n-1-q)<p<\infty \), they constructed as a counterexample a \(\left(W^{1, p}, W^{1, q}\right)\)-extension domain \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) with \(|\partial {\Omega }|>0\). For the remaining range of exponents where \(1\le q\le n-1\) and \(q<p\le (n-1)q/(n-1-q)\), it is still not clear whether the boundary of every \(\left(W^{1, p}, W^{1, q}\right)\)-extension domain must be of volume zero.

As is well-known, for every domain \({\Omega }\subset {{{{\mathbb {R}}}}^n}\), the space of functions of bounded variation \(BV({\Omega })\) strictly contains every Sobolev space \(W^{1, q}({\Omega })\) for \(1\le q\le \infty \). Hence, the class of \(\left(W^{1, p}, BV\right)\)-extension domains contains the class of \(\left(W^{1, p}, W^{1, q}\right)\)-extension domains for every \(1\le q\le p<\infty \). As a basic example to indicate that the containment is strict when \(n \ge 2\), we can take the slit disk (the unit disk minus a radial segment) in the plane. The slit disk is a \(\left(W^{1, p}, BV\right)\)-extension domain for every \(1\le p<\infty \), and even a (BV, BV)-extension domain; however it is not a \(\left(W^{1, p}, W^{1, q}\right)\)-extension domain for any \(1\le q\le p<\infty \). This basic example also shows that it is natural to consider the geometric properties of \(\left(W^{1, p}, BV\right)\)-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 \(\left(W^{1, 1}, BV\right)\)-extension property. Hence, a \((W^{1, 1}, BV)\)-extension domain is Ahlfors regular and so its boundary is of volume zero.

Theorem 1.1

A domain \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) is a (BV, BV)-extension domain if and only if it is a \(\left(W^{1, 1}, BV\right)\)-extension domain.

Since, \(W^{1,1}({\Omega })\) is a proper subspace of \(BV({\Omega })\) with \(\Vert u\Vert _{W^{1,1}({\Omega })}=\Vert u\Vert _{BV({\Omega })}\) for every \(u\in W^{1, 1}({\Omega })\), (BV, BV)-extension property implies \((W^{1, 1}, BV)\)-extension property immediately. The other direction from \((W^{1, 1}, BV)\)-extension property to (BV, BV)-extension property is not as straightforward, as \(W^{1, 1}({\Omega })\) is only a proper subspace of \(BV({\Omega })\). The essential tool here is the Whitney smoothing operator constructed by García-Bravo and the first named author in [4]. This Whitney smoothing operator maps every function in \(BV({\Omega })\) to a function in \(W^{1, 1}({\Omega })\) with the same trace on \(\partial {\Omega }\), so that the norm of the image in \(W^{1, 1}({\Omega })\) is uniformly controlled from above by the norm of the corresponding preimage in \(BV({\Omega })\).

With an extra assumption that \({\Omega }\) is q-fat at almost every point on the boundary \(\partial {\Omega }\), in [15] it was shown that the boundary of a \((W^{1,p}, W^{1, q})\)-extension domain is of volume zero when \(1\le q<p<\infty \). 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 \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) be a \((W^{1, p}, BV)\)-extension domain for \(1\le p<\infty \), which is 1-fat at Lebesgue almost every \(x\in \partial {\Omega }\). Then \(|\partial {\Omega }|=0\). In particular, for every planar \((W^{1, p}, BV)\)-extension domain \({\Omega }\) with \(1\le p<\infty \), we have \(|\partial {\Omega }|=0\).

In light of the results and example given in [15], the most interesting open question is what happens in the range \(1<p\le (n-1)/(n-2)\) of exponents, without the assumption of 1-fatness. 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.

Question 1.3

For \(1<p\le (n-1)/(n-2)\), is the boundary of a \((W^{1, p}, BV)\)-extension domain of volume zero?

2 Preliminaries

For a locally integrable function \(u\in L^1_\textrm{loc}({\Omega })\) and a measurable subset \(A\subset {\Omega }\) with \(0<|A|<\infty \), we define

Definition 2.1

Let \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) be a domain. For every \(1\le p\le \infty \), we define the Sobolev space \(W^{1, p}({\Omega })\) to be

$$\begin{aligned} W^{1, p}({\Omega }):=\left\{ u\in L^p({\Omega }): \nabla u\in L^p({\Omega };{{{{\mathbb {R}}}}^n})\right\} , \end{aligned}$$

where \(\nabla u\) denotes the distributional gradient of u. It is equipped with the nonhomogeneous norm

$$\begin{aligned}\Vert u\Vert _{W^{1, p}({\Omega })}=\Vert u\Vert _{L^p({\Omega })}+\Vert \nabla u\Vert _{L^p({\Omega })}.\end{aligned}$$

Now, let us give the definition of functions of bounded variation.

Definition 2.2

Let \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) be a domain. A function \(u\in L^1({\Omega })\) is said to have bounded variation and denoted \(u\in BV({\Omega })\) if

$$\begin{aligned}\Vert Du\Vert ({\Omega }):=\sup \left\rbrace \int _{\Omega }f\textrm{div}(\phi ) dx:\phi \in C^1_o({\Omega };{{{{\mathbb {R}}}}^n}), |\phi |\le 1\right\lbrace <\infty .\end{aligned}$$

The space \(BV({\Omega })\) is equipped with the norm

$$\begin{aligned}\Vert u\Vert _{BV({\Omega })}:=\Vert u\Vert _{L^1({\Omega })}+\Vert Du\Vert ({\Omega }).\end{aligned}$$

Note that \(\Vert Du\Vert \) is a Radon measure defined on \(\Omega \) that is defined for every set \(F\subset \Omega \) as

$$\begin{aligned}\Vert Du\Vert (F):=\inf \left\{ \Vert Du\Vert (U): F\subset U\subset \Omega , U\ \textrm{open}\right\} .\end{aligned}$$

Definition 2.3

We say that a domain \(\Omega \subset {{{{\mathbb {R}}}}^n}\) is a \(\left(W^{1, p}, BV\right)\)-extension domain for \(1\le p<\infty \), if there exists a bounded extension operator \(E:W^{1, p}({\Omega })\mapsto BV({{{{\mathbb {R}}}}^n})\) i.e. for every \(u\in W^{1, p}({\Omega })\), we have \(E(u)\in BV({{{{\mathbb {R}}}}^n})\) with \(E(u)\big |_{\Omega }\equiv u\) and

$$\begin{aligned}\Vert E(u)\Vert _{BV({{{{\mathbb {R}}}}^n})}\le C\Vert u\Vert _{W^{1, p}({\Omega })}\end{aligned}$$

for a constant \(C>1\) independent of u.

Let \(U\subset {{{{\mathbb {R}}}}^n}\) be an open set and \(K\subset U\) be a compact subset. The p-admissible set \({\mathcal {W}}_p(K; U)\) is defined by setting

$$\begin{aligned}{\mathcal {W}}_p(K; U):=\left\rbrace u\in W^{1, p}_0(U)\cap C(U):u\big |_{K}\ge 1\right\lbrace .\end{aligned}$$

Definition 2.4

Let \(U\subset {{{{\mathbb {R}}}}^n}\) be an open set and \(K\subset U\) be compact. The relative p-capacity \(Cap_p(K; U)\) is defined by setting

$$\begin{aligned}Cap_p(K; U):=\inf _{u\in {\mathcal {W}}_p(K;U)}\int _{U}|\nabla u(x)|^p\,dx.\end{aligned}$$

For an open subset \(A\subset U\), we define the relative p-capacity \(Cap_p(K; U)\) by setting

$$\begin{aligned}Cap_p(A; U):=\sup \left\{ Cap_p(K; U): K\subset A\subset U, K\ \textrm{compact}\right\} .\end{aligned}$$

For arbitrary Borel measurable subset \(E\subset U\), we define the relative p-capacity \(Cap_p(E; U)\) by setting

$$\begin{aligned}Cap_p(E; U):=\inf \left\{ Cap_p(A; U): E\subset A\subset U, A\ \textrm{open}\right\} .\end{aligned}$$

Following Lahti [16], we define 1-fatness below.

Definition 2.5

Let \(A\subset {{{{\mathbb {R}}}}^n}\) be a measurable subset. We say that A is 1-thin at the point \(x\in {{{{\mathbb {R}}}}^n}\), if

$$\begin{aligned}\lim _{r\rightarrow 0^+}r\frac{Cap_1\left(A\cap B(x, r); B(x, 2r)\right)}{\left|B(x, r)\right|}=0.\end{aligned}$$

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 1-finely open, if \({{{{\mathbb {R}}}}^n}\setminus U\) is 1-thin at every \(x\in U\).

By [16, Lemma 4.2], the collection of 1-finely open sets is a topology on \({{{{\mathbb {R}}}}^n}\). For a function \(u\in BV({{{{\mathbb {R}}}}^n})\), we define the lower approximate limit \(u_\star \) by setting

$$\begin{aligned}u_\star (x):=\sup \left\rbrace t\in \overline{{{\mathbb {R}}}}:\lim _{r\rightarrow 0^+}\frac{\left|B(x, r)\cap \{u<t\}\right|}{\left|B(x, r)\right|}=0\right\lbrace \end{aligned}$$

and the upper approximate limit \(u^\star \) by setting

$$\begin{aligned}u^\star (x):=\inf \left\rbrace t\in {\overline{{{\mathbb {R}}}}}:\lim _{r\rightarrow 0^+}\frac{\left|B(x, r)\cap \{u>t\}\right|}{\left|B(x, r)\right|}=0\right\lbrace .\end{aligned}$$

The set

$$\begin{aligned}S_u:=\left\rbrace x\in {{{{\mathbb {R}}}}^n}: u_\star (x)<u^\star (x)\right\lbrace \end{aligned}$$

is called the jump set of u. By the Lebesgue differentiation theorem, \(\left|S_u\right|=0\). Using the lower and upper approximate limits, we define the precise representative \(\tilde{u}:=(u^\star +u_\star )/2\). The following lemma was proved in [16, Corollary 5.1].

Lemma 2.6

Let \(u\in BV({{{{\mathbb {R}}}}^n})\). Then \({\tilde{u}}\) is 1-finely continuous at \({\mathcal {H}}^{n-1}\)-almost every \(x\in {{{{\mathbb {R}}}}^n}\setminus S_u\).

The following lemma for \(u\in W^{1, 1}({{{{\mathbb {R}}}}^n})\) was proved in [15, Lemma 2.6], which is also a corollary of a result in [6]. We generalize it to \(BV({{{{\mathbb {R}}}}^n})\) here.

Lemma 2.7

Let \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) be a domain which is 1-fat at almost every point \(x\in \partial {\Omega }\). If \(u\in BV({{{{\mathbb {R}}}}^n})\) with \(u\big |_{B(x, r)\cap {\Omega }}\equiv c\) for some \(x\in \partial {\Omega }\), \(0<r<1\) and \(c\in {{\mathbb {R}}}\). Then \(u(y)=c\) for almost every \(y\in B(x, r)\cap \partial {\Omega }\).

Proof

Let \(u\in BV({{{{\mathbb {R}}}}^n})\) satisfy the assumptions. Then the precise representative \({\tilde{u}}\big |_{B(x, r)\cap {\Omega }}\equiv c\). Since \(|S_u|=0\), by Lemma 2.6, there exists a subset \(N_1\subset {{{{\mathbb {R}}}}^n}\) with \(|N_1|=0\) such that \({\tilde{u}}\) is 1-finely continuous on \({{{{\mathbb {R}}}}^n}\setminus N_1\). By the assumption, there exists a measure zero set \(N_2\subset \partial {\Omega }\) such that \({\Omega }\) is 1-fat on \(\partial {\Omega }\setminus N_2\). By Definition 2.5, one can see that \(B(x, r)\cap {\Omega }\) is also 1-fat on \((B(x, r)\cap \partial {\Omega }){\setminus } N_2\). For every \(y\in (B(x, r)\cap \partial {\Omega }){\setminus }(N_1\cup N_2)\), since \({\tilde{u}}\) is 1-finely continuous on it and any 1-fine neighborhood of y must intersect \(B(x, r)\cap {\Omega }\), we have \({\tilde{u}}(y)=c\). Hence \(u(y)=c\) for almost every \(y\in B(x, r)\cap \partial {\Omega }\). \(\square \)

We say a set \(E\subset \Omega \) has finite perimeter in \(\Omega \), if \(\chi _E\in BV(\Omega )\), where \(\chi _E\) means the characteristic function of E. We set \(P(E, \Omega ):=\Vert D\chi _E\Vert (\Omega )\) and call it the perimeter of E in \(\Omega \). To simplify the notation, P(E) is set to be \(P(E, {{{{\mathbb {R}}}}^n})\). For every Borel subset \(F\subset \Omega \), define

$$\begin{aligned}P(E, F):=\inf \left\rbrace P(E, U): F\subset U\subset \Omega , U\ \textrm{open}\right\lbrace .\end{aligned}$$

The following coarea formula for BV functions can be found in [3, Section 5.5]. See also [4, Theorem 2.2].

Proposition 2.8

Given a function \(u\in BV({\Omega })\), the superlevel sets \(u_t=\{x\in {\Omega }:u(x)>t\}\) have finite perimeter in \({\Omega }\) for almost every \(t\in {{\mathbb {R}}}\) and

$$\begin{aligned}\Vert Du\Vert (F)=\int _{-\infty }^\infty P(u_t, F)\,dt\end{aligned}$$

for every Borel set \(F\subset {\Omega }\). Conversely, if \(u\in L^1({\Omega })\) and

$$\begin{aligned}\int _{-\infty }^\infty P(u_t,{\Omega })\,dt<\infty \end{aligned}$$

then \(u\in BV({\Omega })\).

See [1, Theorem 3.44] for the proof of the following (1, 1)-Poincaré inequality for BV functions. For a cube \(Q\subset {{{{\mathbb {R}}}}^n}\), we denote by l(Q) its side-length.

Proposition 2.9

Let \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) be a bounded Lipschitz domain. Then there exists a constant \(C>0\) depending on n and \({\Omega }\) such that for every \(u\in BV({\Omega })\), we have

$$\begin{aligned}\int _{\Omega }|u(y)-u_{\Omega }|\,dy\le C\Vert Du\Vert ({\Omega }).\end{aligned}$$

In particular, there exists a constant \(C>0\) only depending on n so that if \(Q, Q'\subset {{{{\mathbb {R}}}}^n}\) are two closed dyadic cubes with \(\frac{1}{4}l(Q')\le l(Q)\le 4\,l(Q')\) and \({\Omega }:=\textrm{int}(Q\cup Q')\) connected, then for every \(u\in BV({\Omega })\),

$$\begin{aligned} \int _{\Omega }|u(y)-u_{\Omega }|\,dy\le Cl(Q)\Vert Du\Vert ({\Omega }). \end{aligned}$$
(2.1)

3 A set function arising from the extension

In this subsection, we introduce a set function defined on the class of open sets in \({{{{\mathbb {R}}}}^n}\) and taking nonnegative values. Our set function here is a modification of the one originally introduced by Ukhlov [24, 25]. See also [27, 28] for related set functions. The modified version of the set function we use is from [15], 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 \(\Phi \) defined on the class of open subsets of \({{{{\mathbb {R}}}}^n}\) and taking nonnegative values is called quasiadditive (see for example [27]), if for all open sets \(U_1\subset U_2\subset {{{{\mathbb {R}}}}^n}\), we have

$$\begin{aligned}\Phi (U_1)\le \Phi (U_2),\end{aligned}$$

and there exists a positive constant C such that for arbitrary pairwise disjoint open sets \(\{U_i\}_{i=1}^\infty \), we have

$$\begin{aligned} \sum _{i=1}^\infty \Phi (U_i)\le C\Phi \left(\bigcup _{i=1}^\infty U_i\right). \end{aligned}$$
(3.1)

Let \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) be a \((W^{1, p}, BV)\)-extension domain for some \(1<p<\infty \). For every open set \(U\subset {{{{\mathbb {R}}}}^n}\) with \(U\cap {\Omega }\ne \emptyset \), we define

$$\begin{aligned}W^p_0(U, {\Omega }):=\left\rbrace u\in W^{1, p}({\Omega })\cap C({\Omega }): u\equiv 0\ \textrm{on}\ {\Omega }\setminus U\right\lbrace .\end{aligned}$$

For every \(u\in W^p_0(U, {\Omega })\), we define

$$\begin{aligned}\Gamma (u):=\inf \left\rbrace \Vert Dv\Vert (U): v\in BV({{{{\mathbb {R}}}}^n}), v\big |_{{\Omega }}\equiv u\right\lbrace .\end{aligned}$$

Then we define the set function \(\Phi \) by setting

$$\begin{aligned} \Phi (U):={\left\{ \begin{array}{ll}\sup _{u\in W^p_0(U, {\Omega })}\left(\frac{\Gamma (u)}{\Vert u\Vert _{W^{1, p}(U\cap {\Omega })}}\right)^{k}, \ \ \text {with } \frac{1}{k}=1-\frac{1}{p}, &{} \text {if }U\cap {\Omega }\ne \emptyset ,\\ 0, &{} \text {otherwise.} \end{array}\right. }\nonumber \\ \end{aligned}$$
(3.2)

In [7], Hajłasz, Koskela and Tuominen proved that for an arbitrary \((W^{1, p}, W^{1, p})\)-extension domain with \(1<p<\infty \), there always exists a bounded linear extension operator. For \(q<p\), the existence of a bounded linear \((W^{1, p}, W^{1, q})\)-extension operator is still open. However, in [15, Lemma 2.1], the authors proved that for \((W^{1, p}, W^{1, q})\)-extension domains there always exists a bounded, positively homogeneous \((W^{1, p}, W^{1, q})\)-extension operator. The next lemma is a version of this result in our setting of \((W^{1, p}, BV)\)-extensions that follows similarly to the proof of [15, Lemma 2.1].

Lemma 3.1

Let \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) be a \((W^{1, p}, BV)\)-extension domain. Then every bounded extension operator \(E:W^{1, p}({\Omega })\rightarrow BV({{{{\mathbb {R}}}}^n})\) promotes to a bounded, positively homogeneous extension operator \(E_h:W^{1, p}({\Omega })\rightarrow BV({{{{\mathbb {R}}}}^n})\) with the operator norm inequality \(\Vert E_h\Vert \le \Vert E\Vert \).

The proof of the following lemma is almost the same as the proof of [15, Theorem 3.1]. One needs to simply replace \(\Vert Dv\Vert _{L^q(U)}\) by \(\Vert Dv\Vert (U)\) in the proof of [15, Theorem 3.1] and repeat the argument.

Lemma 3.2

Let \(1<p<\infty \) and let \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) be a bounded \((W^{1, p}, BV)\)-extension domain. Then the set function defined in (3.2) for all open subsets of \({{{{\mathbb {R}}}}^n}\) is bounded and quasiadditive.

The upper and lower derivatives of a quasiadditive set function \(\Phi \) are defined by setting

$$\begin{aligned}\overline{D\Phi }(x):=\limsup _{r\rightarrow 0^+}\frac{\Phi (B(x, r))}{|B(x, r)|}\quad \textrm{and}\quad \underline{D\Phi }(x) = \liminf _{r\rightarrow 0^+}\frac{\Phi (B(x,r))}{|B(x, r)|}.\end{aligned}$$

By [20, 27], we have the following lemma. See also [15, Lemma 3.1].

Lemma 3.3

Let \(\Phi \) be a bounded and quasiadditive set function defined on open sets \(U\subset {{{{\mathbb {R}}}}^n}\). Then \(\overline{D\Phi }(x)<\infty \) for almost every \(x\in {{{{\mathbb {R}}}}^n}\).

The following lemma immediately comes from the definition (3.2) for the set function \(\Phi \).

Lemma 3.4

Let \(1<p<\infty \) and let \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) be a bounded \((W^{1, p}, BV)\)-extension domain. Then, for a ball B(x, r) with \(x\in \partial {\Omega }\) and every function \(u\in W^p_0(B(x, r), {\Omega })\), there exists a function \(v\in BV(B(x, r))\) with \(v\big |_{B(x, r)\cap {\Omega }}\equiv u\) and

$$\begin{aligned} \Vert Dv\Vert (B(x, r))\le 2\Phi ^{\frac{1}{k}}(B(x, r))\Vert u\Vert _{W^{1, p}(B(x, r)\cap {\Omega })}, \ \ \textrm{where}\ \ \frac{1}{k}=1-\frac{1}{p}. \end{aligned}$$
(3.3)

4 Proofs of the results

c 1.1 and 1.2.

Proof of Theorem 1.1

Let us first assume that \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) is a (BV, BV)-extension domain with the extension operator E. Since \(W^{1,1}({\Omega })\subset BV({\Omega })\) with \(\Vert u\Vert _{BV({\Omega })}=\Vert u\Vert _{W^{1, 1}({\Omega })}\) for every \(u\in W^{1, 1}({\Omega })\), we have

$$\begin{aligned} \Vert E(u)\Vert _{BV({\Omega })}\le C\Vert u\Vert _{BV({\Omega })}\le C\Vert u\Vert _{W^{1, 1}({\Omega })}. \end{aligned}$$

This implies that \({\Omega }\) is a \(\left(W^{1,1}, BV\right)\)-extension domain with the same operator E restricted to \(W^{1,1}(\Omega )\).

Let us then prove the converse and assume that \({\Omega }\subset {{{{\mathbb {R}}}}^n}\) is a \((W^{1,1}, BV)\)-extension domain with an extension operator E. Let \(S_{{\Omega },{\Omega }}\) be the Whitney smoothing operator defined in [4]. Then by [4, Theorem 3.1], for every \(u\in BV({\Omega })\), we have \(S_{{\Omega },{\Omega }}(u)\in W^{1, 1}({\Omega })\) with

$$\begin{aligned}\Vert S_{{\Omega },{\Omega }}(u)\Vert _{W^{1, 1}({\Omega })}\le C\Vert u\Vert _{BV({\Omega })}\end{aligned}$$

for a positive constant C independent of u, and

$$\begin{aligned} \Vert D(u-S_{{\Omega }, {\Omega }}(u))\Vert (\partial {\Omega })=0, \end{aligned}$$
(4.1)

where \(u-S_{{\Omega },{\Omega }}(u)\) is understood to be defined on the whole space \({{{{\mathbb {R}}}}^n}\) via a zero-extension. Then \(E(S_{{\Omega }, {\Omega }}(u))\in BV({{{{\mathbb {R}}}}^n})\) with

$$\begin{aligned}\Vert E(S_{{\Omega }, {\Omega }}(u))\Vert _{BV({{{{\mathbb {R}}}}^n})}\le C\Vert S_{{\Omega }, {\Omega }}(u)\Vert _{W^{1, 1}({\Omega })}\le C\Vert u\Vert _{BV({\Omega })}.\end{aligned}$$

Now, define \(T:BV(\Omega ) \rightarrow BV({{{{\mathbb {R}}}}^n})\) by setting for every \(u \in BV(\Omega )\)

$$\begin{aligned}T(u)(x):={\left\{ \begin{array}{ll} u(x),\ \ \textrm{if}\ \ x\in {\Omega }\\ E(S_{{\Omega }, {\Omega }}(u))(x),\ \ \textrm{if}\ \ x\in {{{{\mathbb {R}}}}^n}\setminus {\Omega }. \end{array}\right. }\end{aligned}$$

By (4.1), we have \(T(u)\in BV({{{{\mathbb {R}}}}^n})\) with

$$\begin{aligned}\Vert T(u)\Vert _{BV({{{{\mathbb {R}}}}^n})}\le \Vert E(S_{{\Omega },{\Omega }}(u))\Vert _{BV({{{{\mathbb {R}}}}^n})}+\Vert u\Vert _{BV({\Omega })}\le C\Vert u\Vert _{BV({\Omega })}.\end{aligned}$$

Hence, \({\Omega }\) is a BV-extension domain. \(\square \)

Proof of Theorem 1.2

Assume towards a contradiction that \(|\partial {\Omega }|>0\). By the Lebesgue density point theorem and Lemma 3.3, there exists a measurable subset U of \(\partial {\Omega }\) with \(|U|=|\partial {\Omega }|\) such that every \(x\in U\) is a Lebesgue point of \(\partial {\Omega }\) and \(\overline{D\Phi }(x)<\infty \). Fix \(x\in U\). Since x is a Lebesgue point, there exists a sufficiently small \(r_x>0\), such that for every \(0<r<r_x\), we have

$$\begin{aligned}\left|B(x, r)\cap {\overline{{\Omega }}}\right|\ge \frac{1}{2^{n-1}}\left|B(x, r)\right|.\end{aligned}$$

Let \(r\in (0, r_x)\) be fixed. Since \(|\partial B(x, s)|=0\) for every \(s\in (0, r)\), we have

$$\begin{aligned} \left|B\left(x, \frac{r}{4}\right)\cap {\overline{{\Omega }}}\right|\ge \frac{1}{2^{n-1}}\left|B\left(x, \frac{r}{4}\right)\right|\ge \frac{1}{2^{3n-1}}\left|B(x, r)\right| \end{aligned}$$
(4.2)

and

$$\begin{aligned} \left|\left(B(x,r)\setminus B\left(x, \frac{r}{2}\right)\right)\cap {\overline{{\Omega }}}\right|\ge |B(x, r)\cap {\overline{{\Omega }}}|-\left|B\left(x, \frac{r}{2}\right)\right|\ge \frac{1}{2^n}|B(x, r)|.\nonumber \\ \end{aligned}$$
(4.3)

Define a test function \(u\in W^{1, p}({\Omega })\cap C({\Omega })\) by setting

$$\begin{aligned} u(y):={\left\{ \begin{array}{ll} 1,\ \ {} &{}\textrm{if}\ y\in B\left(x, \frac{r}{4}\right)\cap {\Omega },\\ \frac{-4}{r}|y-x|+2,\ \ {} &{}\textrm{if}\ y\in \left(B\left(x, \frac{r}{2}\right)\setminus B\left(x, \frac{r}{4}\right)\right)\cap {\Omega },\\ 0,\ \ {} &{}\textrm{if}\ y\in {\Omega }\setminus B\left(x, \frac{r}{2}\right). \end{array}\right. } \end{aligned}$$
(4.4)

We have

$$\begin{aligned} \left(\int _{B(x, r)\cap {\Omega }}|u(y)|^p+|\nabla u(y)|^p\,dx\right)^{\frac{1}{p}}\le \frac{C}{r}|B(x, r)\cap {\Omega }|^{\frac{1}{p}}. \end{aligned}$$
(4.5)

Since \(u\equiv 0\) on \({\Omega }{\setminus } B(x, r/2)\), we have \(u\in W^p_0(B(x, r), {\Omega })\). Then, by the definition (3.2) of the set function \(\Phi \) and by Corollary 3.4, there exists a function \(v\in BV(B(x, r))\) with \(v\big |_{B(x, r)\cap {\Omega }}\equiv u\) and

$$\begin{aligned} \Vert Dv\Vert (B(x, r))\le 2\Phi ^{\frac{1}{k}}(B(x, r))\Vert u\Vert _{W^{1, p}(B(x, r)\cap {\Omega })}. \end{aligned}$$
(4.6)

By the Poincaré inequality of BV functions stated in Proposition 2.9, we have

$$\begin{aligned} \int _{B(x, r)}|v(y)-v_{B(x, r)}|\,dy\le C r\Vert Dv\Vert (B(x, r)). \end{aligned}$$
(4.7)

Since \({\Omega }\) is 1-fat on almost every \(z\in \partial {\Omega }\), by Lemma 2.7, \(v(z)=1\) for almost every \(z\in B\left(x,\frac{r}{4}\right)\cap \partial {\Omega }\) and \(v(z)=0\) for almost every \(z\in \left(B(x,r){\setminus } B\left(x, \frac{r}{2}\right)\right)\cap \partial {\Omega }\). Hence, on one hand, if \(v_{B(x, r)}\le \frac{1}{2}\), we have

$$\begin{aligned}\int _{B(x, r)}|v(y)-v_{B(x, r)}|\,dy\ge \frac{1}{2}\left|B\left(x, \frac{r}{4}\right)\cap {\overline{{\Omega }}}\right|\ge c|B(x, r)|.\end{aligned}$$

On the other hand, if \(v_{B(x, r)}>\frac{1}{2}\), we have

$$\begin{aligned}\int _{B(x, r)}|v(y)-v_{B(x, r)}|\,dy\ge \frac{1}{2}\left|\left(B(x, r)\setminus B\left(x, \frac{r}{2}\right)\right)\cap {\overline{{\Omega }}}\right|>c|B(x, r)|.\end{aligned}$$

All in all, we always have

$$\begin{aligned} \int _{B(x, r)}|v(y)-v_{B(x, r)}|\,dy\ge c|B(x, r)| \end{aligned}$$
(4.8)

for a sufficiently small constant \(c>0\). Thus, by combining inequalities (4.5)–(4.8), we obtain

$$\begin{aligned}\Phi (B(x, r))^{p-1}|B(x, r)\cap {\Omega }|\ge c|B(x, r)|^p\end{aligned}$$

for a sufficiently small constant \(c>0\). This gives

$$\begin{aligned}|B(x, r)\cap \partial {\Omega }|\le |B(x, r)|-|B(x, r)\cap {\Omega }|\le |B(x, r)|-C\frac{|B(x, r)|^p}{\Phi (B(x, r))^{p-1}}.\end{aligned}$$

Since \(\overline{D\Phi }(x)<\infty \), we have

$$\begin{aligned} \limsup _{r\rightarrow 0^+}\frac{|B(x, r)\cap \partial {\Omega }|}{|B(x, r)|}\le & {} \limsup _{r\rightarrow 0^+}\left(1-\frac{|B(x, r)\cap {\Omega }|}{|B(x, r)|}\right)\\\le & {} \limsup _{r\rightarrow 0^+}\left(1-\frac{|B(x, r)|^{p-1}}{\Phi (B(x, r))^{p-1}}\right)\le 1-c\overline{D\Phi }(x)^{1-p}<1. \end{aligned}$$

This contradicts the assumption that x is a Lebesgue point of \(\partial {\Omega }\). Hence, we conclude that \(|\partial {\Omega }|=0\).

Let us then consider the case \({\Omega }\subset {{\mathbb {R}}}^2\). By [5, Theorem A.29], for every \(x\in \partial {\Omega }\) and every \(0<r<\min \left\rbrace 1, \frac{1}{4}\,\textrm{diam}\,({\Omega })\right\lbrace \), we have

$$\begin{aligned}Cap_1({\Omega }\cap B(x, r); B(x, 2r))\ge cr\end{aligned}$$

for a constant \(0<c<1\). This implies that \({\Omega }\) is 1-fat at every \(x\in \partial {\Omega }\). 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. \(\square \)