A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I

We continue the study of the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$BV^\alpha ({\mathbb {R}}^n)$$\end{document}BVα(Rn) of functions with bounded fractional variation in \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}Rn of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document}α∈(0,1) introduced in our previous work (Comi and Stefani in J Funct Anal 277(10):3373–3435, 2019). After some technical improvements of certain results of Comi and Stefani (2019) which may be of some separated insterest, we deal with the asymptotic behavior of the fractional operators involved as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \rightarrow 1^-$$\end{document}α→1-. We prove that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}α-gradient of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,p}$$\end{document}W1,p-function converges in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}Lp to the gradient for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,+\infty )$$\end{document}p∈[1,+∞) as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \rightarrow 1^-$$\end{document}α→1-. Moreover, we prove that the fractional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}α-variation converges to the standard De Giorgi’s variation both pointwise and in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}Γ-limit sense as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \rightarrow 1^-$$\end{document}α→1-. Finally, we prove that the fractional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}β-variation converges to the fractional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}α-variation both pointwise and in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}Γ-limit sense as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \rightarrow \alpha ^-$$\end{document}β→α- for any given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document}α∈(0,1).


A distributional approach to fractional variation
In our previous work [27], we introduced the space BV α (R n ) of functions with bounded fractional variation in R n of order α ∈ (0, 1). Precisely, a function f ∈ L 1 (R n ) belongs to the space BV α (R n ) if its fractional α-variation is the fractional α-divergence of ϕ ∈ C ∞ c (R n ; R n ), where μ n,α := 2 α π − n 2 n+α+1 2 1−α 2 (1.3) for any given α ∈ (0, 1). The operator div α was introduced in [72] as the natural dual operator of the much more studied fractional α-gradient n+α+1 dy, x ∈ R n , (1.4) defined for all f ∈ C ∞ c (R n ). For an account on the existing literature on the operator ∇ α , see [68,Section 1]. Here we only refer to [66][67][68][69][70][72][73][74] for the articles tightly connected to the present work and to [63,Section 15.2] for an agile presentation of the fractional operators defined in (1.2) and in (1.4) and of some of their elementary properties. According to [70,Section 1], it is interesting to notice that [42] seems to be the earliest reference for the operator defined in (1.4).
The operators in (1.2) and in (1.4) are dual in the sense that for all f ∈ C ∞ c (R n ) and ϕ ∈ C ∞ c (R n ; R n ), see [72,Section 6] and [27,Lemma 2.5]. Moreover, both operators have good integrability properties when applied to test functions, namely ∇ α f ∈ L p (R n ) and div α ϕ ∈ L p (R n ; R n ) for all p ∈ [1, +∞] for any given f ∈ C ∞ c (R n ) and ϕ ∈ C ∞ c (R n ; R n ), see [27,Corollary 2.3]. The integration-by-part formula (1.5) represents the starting point for the distributional approach to fractional Sobolev spaces and the fractional variation we developed in [27]. In fact, similarly to the classical case, a function f ∈ L 1 (R n ) belongs to BV α (R n ) if and only if there exists a finite vector-valued Radon measure D α f ∈ M (R n ; R n ) such that for all ϕ ∈ C ∞ c (R n ; R n ), see [27,Theorem 3.2]. Motivated by (1.6) and similarly to the classical case, we can define the weak fractional α-gradient of a function f ∈ L p (R n ), with p ∈ [1, +∞], as the function ∇ α w f ∈ L 1 loc (R n ; R n ) satisfying It is interesting to compare the distributional fractional Sobolev spaces S α, p (R n ) with the well-known fractional Sobolev space W α, p (R n ), that is, the space endowed with the norm If p = +∞, then W α,∞ (R n ) naturally coincides with the space of bounded α-Hölder continuous functions endowed with the usual norm (see [32] for a detailed account on the spaces W α, p ). For the case p = 1, starting from the very definition of the fractional gradient ∇ α , it is plain to see that W α,1 (R n ) ⊂ S α,1 (R n ) ⊂ BV α (R n ) with both (strict) continuous embeddings, see [27,Theorems 3.18 and 3.25].
In the geometric regime p = 1, our distributional approach to the fractional variation naturally provides a new definition of distributional fractional perimeter. Precisely, for any open set ⊂ R n , the fractional Caccioppoli α-perimeter in of a measurable set E ⊂ R n is the fractional α-variation of χ E in , i.e.
Similarly to the aforementioned embedding W α,1 (R n ) ⊂ BV α (R n ), we have the inequality |D α χ E |( ) ≤ μ n,α P α (E; ) (1.9) for any open set ⊂ R n , see [27,Proposition 4.8], where P α (E; ) := |χ E (x) − χ E (y)| |x − y| n+α dx dy is the standard fractional α-perimeter of a measurable set E ⊂ R n relative to the open set ⊂ R n (see [28] for an account on the fractional perimeter P α ). Note that, by definition, the fractional α-perimeter of E in R n is simply P α (E) := P α (E; R n ) = [χ E ] W α,1 (R n ) . We remark that inequality (1.9) is strict in most of the cases, as shown in Sect. 2.6 below. This completely answers a question left open in our previous work [27].

Asymptotics and 0-convergence in the standard fractional setting
The fractional Sobolev space W α, p (R n ) can be understood as an 'intermediate space' between the space L p (R n ) and the standard Sobolev space W 1, p (R n ). In fact, W α, p (R n ) can be recovered as a suitable (real) interpolation space between the spaces L p (R n ) and W 1, p (R n ). We refer to [13,78] for a general introduction on interpolation spaces and to [54] for a more specific treatment of the interpolation space between L p (R n ) and W 1, p (R n ).
One then naturally expects that, for a sufficiently regular function f , the fractional Sobolev seminorm [ f ] W α, p (R n ) , multiplied by a suitable renormalising constant, should tend to f L p (R n ) as α → 0 + and to ∇ f L p (R n ) as α → 1 − . Indeed, for p ∈ [1, +∞), it is known that (1.11) for all f ∈ α∈(0,1) W α, p (R n ), while (1.12) for all f ∈ W 1, p (R n ). Here A n, p , B n, p > 0 are two constants depending only on n, p. The limit (1.11) was proved in [51,52], while the limit (1.12) was established in [14]. As proved in [30], when p = 1 the limit (1.12) holds in the more general case of BV functions, that is, (1.13) for all f ∈ BV (R n ). For a different approach to the limits in (1.11) and in (1.13) based on interpolation techniques, see [54]. The limits (1.12) and (1.13) are special consequences of the celebrated Bourgain-Brezis-Mironescu (BBM, for short) formula for p = 1, (1.14) where C n, p > 0 is a constant depending only on n and p, and ( k ) k∈N ⊂ L 1 loc ([0, +∞)) is a sequence of non-negative radial mollifiers such that R n k (|x|) dx = 1 for all k ∈ N and lim k→+∞ +∞ δ k (r ) r n−1 dr = 0 for all δ > 0.
Concerning the fractional perimeter P α given in (1.10), one has some additional information besides equations (1.11) and (1.13).
On the one hand, thanks to [64,Theorem 1.2], the fractional α-perimeter P α enjoys the following fractional analogue of Gustin's Boxing Inequality (see [41] and [ for all measurable sets E ⊂ R n , where ω n is the volume of the unit ball in R n (it should be noted that in [3] the authors use a slightly different definition of the fractional α-perimeter, since they consider the functional J α (E, ) := 1 2 P α (E, )). For a complete account on -convergence, we refer the reader to the monographs [17,29] (throughout all the paper, with the symbol (X ) -lim we denote the -convergence in the ambient metric space X ). The convergence in (1.16), besides giving aconvergence analogue of the limit in (1.13), is tightly connected with the study of the regularity properties of non-local minimal surfaces, that is, (local) minimisers of the fractional α-perimeter P α .

Asymptotics and 0-convergence for the fractional˛-variation as˛→ 1 −
The main aim of the present work is to study the asymptotic behavior of the fractional α-variation (1.1) as α → 1 − , both in the pointwise and in the -convergence sense.
We provide counterparts of the limits (1.12) and (1.13) for the fractional α-variation. Indeed, we prove that, if f ∈ W 1, p (R n ) for some p ∈ [1, +∞), then f ∈ S α, p (R n ) for all α ∈ (0, 1) and, moreover, In the geometric regime p = 1, we show that if f ∈ BV (R n ) then f ∈ BV α (R n ) for all α ∈ (0, 1) and, in addition, We are also able to treat the case p = +∞. In fact, we prove that if f ∈ W 1,∞ (R n ) then f ∈ S α,∞ (R n ) for all α ∈ (0, 1) and, moreover, and We refer the reader to Theorem 4.9, Theorem 4.11 and Theorem 4.12 below for the precise statements. We warn the reader that the symbol ' ' appearing in (1.18) and (1.20) denotes the weak*-convergence, see Sect. 2.1 below for the notation. Some of the above results were partially announced in [71]. In a similar perspective, we also refer to the work [53], where the authors proved convergence results for nonlocal gradient operators on BV functions defined on bounded open sets with smooth boundary. The approach developed in [53] is however completely different from the asymptotic analysis we presently perform for the fractional operator defined in (1.4), since the boundedness of the domain of definition of the integral operators considered in [53] plays a crucial role.
Notice that the renormalising factor (1 − α) 1 p is not needed in the limits (1.17)- (1.21), contrarily to what happened for the limits (1.12) and (1.13). In fact, this difference should not come as a surprise, since the constant μ n,α in (1.3), encoded in the definition of the operator ∇ α , satisfies and thus plays a similar role of the factor (1 − α) 1 p in the limit as α → 1 − . Thus, differently from our previous work [27], the constant μ n,α appearing in the definition of the operators ∇ α and div α is of crucial importance in the asymptotic analysis developed in the present paper.
Another relevant aspect of our approach is that convergence as α → 1 − holds true not only for the total energies, but also at the level of differential operators, in the strong sense when p ∈ (1, +∞) and in the weak* sense for p = 1 and p = +∞. In simpler terms, the non-local fractional α-gradient ∇ α converges to the local gradient ∇ as α → 1 − in the most natural way every time the limit is well defined.
We also provide a counterpart of (1.16) for the fractional α-variation as α → 1 − . Precisely, we prove that, if ⊂ R n is a bounded open set with Lipschitz boundary, then However, we have 2ω n−1 ω n > 1 for any n ≥ 2 and thus the -lim sup inequality in (1.23) follows from the -lim sup inequality in (1.16) only in the case n = 1. In a similar way, one sees that the -lim inf inequality in (1.23) implies the -lim inf inequality in (1.16) only in the case n = 1.
Besides the counterpart of (1.16), our approach allows to prove that -convergence holds true also at the level of functions. Indeed, if f ∈ BV (R n ) and ⊂ R n is an open set such that either is bounded with Lipschitz boundary or = R n , then (1.24) One can regard the limit (1.24) as an analogue of the -convergence results known in the usual fractional setting, see [57,62]. We refer the reader to Theorems 4.13, 4.14 and 4.17 for the (even more general) results in this direction. Again, as before and thanks to the asymptotic behavior (1.22), the renormalising factor (1 − α) is not needed in the limits (1.23) and (1.24). As a byproduct of the techniques developed for the asymptotic study of the fractional α-variation as α → 1 − , we are also able to characterize the behavior of the fractional β-variation as β → α − , for any given α ∈ (0, 1). On the one hand, if f ∈ BV α (R n ), then see Theorem 5.4. On the other hand, if f ∈ BV α (R n ) and ⊂ R n is an open set such that either is bounded and |D α f |(∂ ) = 0 or = R n , then The operator in (1.26) is well defined (in the principal value sense) for all f ∈ C ∞ c (R n ) and, actually, coincides with the well-known vector-valued Riesz transform R f , see [39,Section 5.1.4] and [76,Chapter 3]. Similarly, the fractional α-divergence in (1.2) is formally converging to the operator 27) which is well defined (in the principal value sense) for all ϕ ∈ C ∞ c (R n ; R n ). In perfect analogy with what we did before, we can introduce the space BV 0 (R n ) as the space of functions f ∈ L 1 (R n ) such that the quantity is finite. Surprisingly (and differently from the fractional α-variation, recall [27, Section 3.10]), it turns out that |D 0 f | L n for all f ∈ BV 0 (R n ). More precisely, one can actually prove that Here is the (real) Hardy space, see [77,Chapter III] for the precise definition. Thus, it would be interesting to understand for which functions f ∈ L 1 (R n ) the fractional α-gradient , then one cannot expect strong convergence in L 1 and, instead, may consider the asymptotic behavior of the rescaled fractional gradient α ∇ α f as α → 0 + , in analogy with the limit in (1.11). This line of research, as well as the identifications BV 0 = H 1 and S α, p = L α, p mentioned above, it is the subject of the subsequent paper [26].

Organization of the paper
The paper is organized as follows.
In Sect. 2, after having briefly recalled the definitions and the main properties of the operators ∇ α and div α , we extend certain technical results of [27].
In Sect. 3, we prove several integrability properties of the fractional α-gradient and two useful representation formulas for the fractional α-variation of functions with bounded De Giorgi's variation. We are also able to prove similar results for the fractional β-gradient of functions with bounded fractional α-variation, see Sect. 3.4.
In Appendix A, for the reader's convenience, we state and prove two known results on the truncation and the approximation of BV functions and sets with finite perimeter that are used in Sect. 3 and in Sect. 4.

General notation
We start with a brief description of the main notation used in this paper. In order to keep the exposition the most reader-friendly as possible, we retain the same notation adopted in our previous work [27].
Given an open set , we say that a set E is compactly contained in , and we write E , if the E is compact and contained in . We denote by L n and H α the n-dimensional Lebesgue measure and the α-dimensional Hausdorff measure on R n respectively, with α ≥ 0. Unless otherwise stated, a measurable set is a L nmeasurable set. We also use the notation |E| = L n (E). All functions we consider in this paper are Lebesgue measurable, unless otherwise stated. We denote by B r (x) the standard open Euclidean ball with center x ∈ R n and radius r > 0. We let B r = B r (0). Recall that ω n := |B 1 | = π n 2 / n+2 2 and H n−1 (∂ B 1 ) = nω n , where is Euler's Gamma function, see [9].
We let GL(n) ⊃ O(n) ⊃ SO(n) be the general linear group, the orthogonal group and the special orthogonal group respectively. We tacitly identify GL(n) ⊂ R n 2 with the space of invertible n × n -matrices and we endow it with the usual Euclidean distance in R n 2 .
For k ∈ N 0 ∪ {+∞} and m ∈ N, we denote by C k c ( ; R m ) and Lip c ( ; R m ) the spaces of C k -regular and, respectively, Lipschitz-regular, m-vector-valued functions defined on R n with compact support in .
For any exponent p ∈ [1, +∞], we denote by L p ( ; R m ) the space of mvector-valued Lebesgue p-integrable functions on . For p ∈ [1, +∞], we say that for all ϕ ∈ L q ( ; R m ), with q ∈ [1, +∞] the conjugate exponent of p, that is, 1 p + 1 q = 1 (with the usual convention 1 +∞ = 0). Note that in the case p = +∞ we make a little abuse of terminology, since the limit in (2.1) actually defines the weak*-convergence in L ∞ ( ; R m ).
We let be the space of m-vector-valued Sobolev functions on , see for instance [46,Chapter 10] for its precise definition and main properties. We also let We let For α ∈ (0, 1) and p ∈ [1, +∞), we let be the space of m-vector-valued fractional Sobolev functions on , see [32] for its precise definition and main properties. We also let For α ∈ (0, 1) and p = +∞, we simply let , the space of m-vector-valued bounded α-Hölder continuous functions on .
We let M ( ; R m ) be the space of m-vector-valued Radon measures with finite total variation, precisely for all ϕ ∈ C 0 c ( ; R m ). Note that we make a little abuse of terminology, since the limit in (2.2) actually defines the weak*-convergence in M ( ; R m ).
In order to avoid heavy notation, if the elements of a function space F( ; R m ) are real-valued (i.e. m = 1), then we will drop the target space and simply write F( ).

Basic properties of ∇˛and divW
e recall the non-local operators ∇ α and div α introduced by Šilhavý in [72] (see also our previous work [27]).
Let α ∈ (0, 1) and set We let be the fractional α-divergence of ϕ ∈ Lip c (R n ; R n ) at x ∈ R n . The non-local operators ∇ α and div α are well defined in the sense that the involved integrals converge and the limits exist, see [72,Section 7] and [27,Section 2]. Moreover, since it is immediate to check that ∇ α c = 0 for all c ∈ R and Given α ∈ (0, n), we let be the Riesz potential of order α ∈ (0, n) of a function u ∈ C ∞ c (R n ; R m ). We recall that, if α, β ∈ (0, n) satisfy α+β < n, then we have the following semigroup property then there exists a constant C n,α, p > 0 such that the operator in (2.3) satisfies for all u ∈ C ∞ c (R n ; R m ). As a consequence, the operator in (2.3) extends to a linear continuous operator from L p (R n ; R m ) to L q (R n ; R m ), for which we retain the same notation. For a proof of (2.4) and (2.5), we refer the reader to [ and and for any bounded open set U ⊂ R n such that supp(ϕ) ⊂ U , where C n,α,U is as in (2.9).

Extension of ∇˛and div˛to Lip b -regular tests
In the following result, we extend the fractional α-divergence to Lip b -regular vector fields.
given by (2.14) and satisfies Proof We split the proof in two steps.
Step 1: proof of (2.13), (2.14) and (2.15). Given x ∈ R n and r > 0, we can estimate Hence the function in (2.13) is well defined for all x ∈ R n and so that (2.14) follows by optimising the right-hand side in r > 0. Moreover, since and {|z|>ε} z |z| n+α+1 dy = 0 for all ε > 0, by Lebesgue's Dominated Convergence Theorem we immediately get the two equalities in (2.15) for all x ∈ R n .
Step 2: proof of (2.16). Assume that I 1−α |divϕ| ∈ L 1 loc (R n ). Then for a.e. x ∈ R n . Hence, by Lebesgue's Dominated Convergence Theorem, we can write for a.e. x ∈ R n . Now let ε > 0 be fixed and let R > 0. Again by (2.17) and Lebesgue's Dominated Convergence Theorem, we have for a.e. x ∈ R n . Moreover, integrating by parts, we get for all R > 0 and for a.e. x ∈ R n . Since ϕ ∈ L ∞ (R n ; R n ), by Lebesgue's Dominated Convergence Theorem we have for all ε > 0 and all x ∈ R n . We can also estimate for all R > 0 and all x ∈ R n . We thus have that for all ε > 0 and a.e. x ∈ R n . Since also {|y|=ε} y |y| We can also extend the fractional α-gradient to Lip b -regular functions. The proof is very similar to the one of Lemma 2.2 and is left to the reader.
given by and satisfies for a.e. x ∈ R n .

Extended Leibniz's rules for ∇˛and divT
he following two results extend the validity of Leibniz's rules proved in [27, Lemmas 2.6 and 2.7] to Lip b -regular functions and Lip b -regular vector fields. The proofs are very similar to the ones given in [27] and to that of Lemma 2.2, and thus are left to the reader.

Extended integration-by-part formulas
We now recall the definition of the space of functions with bounded fractional αvariation. Given α ∈ (0, 1), we let where We refer the reader to [27,Section 3] for the basic properties of this function space. Here we just recall the following result, see [27, Theorem 3.2 and Proposition 3.6] for the proof.
Thanks to Lemma 2.5, we can actually prove that a function in BV α (R n ) can be tested against any Lip b -regular vector field.
Proof We argue as in the proof of [27,Theorem 3.8] On the other hand, by Lemma 2.5 we can write and, similarly, By Lebesgue's Dominated Convergence Theorem, we thus get that and the conclusion follows.
Thanks to Lemma 2.4, we can prove that a function in Lip b (R n ) can be tested against any Lip c -regular vector field. The proof is very similar to the one of Proposition 2.7 and is thus left to the reader.

Comparison between W˛, 1 and BV˛seminorms
In this section, we completely answer a question left open in [27,Section 1.4]. Given α ∈ (0, 1) and an open set ⊂ R n , we want to study the equality cases in the inequalities as long as f ∈ W α,1 (R n ) and P α (E; ) < +∞. The key idea to the solution of this problem lies in the following simple result.
Proof The inequality is well known and it is obvious that it is an equality if F = f ν a.e. in A for some constant direction ν ∈ S m−1 and some scalar function f ∈ L 1 (A) with f ≥ 0 a.e. in A. So let us assume that in A with f = |F| ∈ L 1 (A) and the conclusion follows.
As an immediate consequence of Lemma 2.9, we have the following result.
for all x ∈ U , for some measurable set U ⊂ R n such that L n (R n \ U ) = 0. Now let x ∈ U be fixed. By Lemma 2.9 (applied with A = R n ), (2.20) implies that the (non-identically zero) vector field has constant direction for all y ∈ V x , for some measurable set V x ⊂ R n such that L n (R n \ V x ) = 0. Thus, given y, y ∈ V x , the two vectors y − x and y − x are linearly dependent, so that the three points x, y and y are collinear. If n ≥ 2, then this immediately gives L n (V x ) = 0, a contradiction, so that (2.19) must be strict. If instead n = 1, then we know that We claim that (2.21) implies that the function f is (equivalent to) a (non-constant) monotone function. If so, then f / ∈ L 1 (R), in contrast with the fact that f ∈ W α,1 (R), so that (2.19) must be strict and the proof is concluded. To prove the claim, we argue as follows. Fix x ∈ U and assume that Hence for all x ∈ U (where ess sup and ess inf refer to the essential supremum and the essential infimum respectively) and thus f must be equivalent to a (non-constant) non-decreasing function.
Given an open set ⊂ R n and a measurable set E ⊂ R n , we definẽ It is obvious to see thatP where P α is the fractional perimeter introduced in (1.10). Arguing as in the proof of [27,Proposition 4.8]it is immediate to see that an inequality stronger than that in (1.9). In analogy with Corollary 2.10, we have the following result.
Proof We prove the two statements separately. Proof of (i). Assume n ≥ 2. Since L n (E) > 0, for a given x ∈ \ E the map does not have constant orientation. Similarly, since L n (R n \ E) > 0, for a given x ∈ ∩ E also the map does not have constant orientation. Hence, by Lemma 2.9, we must have and, similarly, We thus get Proof of (ii). Assume n = 1. We argue as in the proof of [27,Proposition 4.12]. Let Then we can writẽ

Hence (2.23) is an equality if and only if
x for a.e. x ∈ . Now, on the one hand, squaring both sides of (2.25) and simplifying, we get that (2.23) is an equality if and only if for a.e. x ∈ . On the other hand, we can rewrite (2.26) as for a.e. x ∈ . Hence (2.27) can be equivalently rewritten as for a.e. x ∈ . Thus (2.23) is an equality if and only if at least one of the two integrals in the left-hand side of (2.28) is zero, and the reader can check that (ii) readily follows.

Remark 2.12
(Half-lines in Corollary 2.11(ii)) In the case n = 1, it is worth to stress that (2.23) is always an equality when the set E ⊂ R is (equivalent to) an half-line, i.e., for any α ∈ (0, 1), any a ∈ R and any open set ⊂ R such thatP α ((a, +∞); ) < +∞. However, the equality cases in (2.23) are considerably richer. Indeed, on the one side, and, on the other side, for any α ∈ (0, 1). We leave the simple computations to the interested reader.

Integrability properties of the fractional˛-gradient
We begin with the following technical local estimate on the W α,1 -seminorm of a function in BV loc .
and f k → f a.e. in B 3R as k → +∞. The conclusion thus follows by a simple application of Fatou's Lemma.
In the following result, we collect several local integrability estimates involving the fractional α-gradient of a function satisfying various regularity assumptions.

Proposition 3.2
The following statements hold.
In addition, for any bounded open set U ⊂ R n , we have for all α ∈ (0, 1), where C n,α,U is as in (2.9). Finally, given an open set A ⊂ R n , we have for all R > 0 and α ∈ (0, 1).
, then the weak fractional α-gradient D α f ∈ M loc (R n ; R n ) exists and satisfies D α f = ∇ α f L n with ∇ α f ∈ L 1 loc (R n ; R n ) and for all R > 0 and α ∈ (0, 1).

Proof
We prove the three statements separately. Proof of (i where C n,α,U is defined as in (2.9). We now prove (3.4) in two steps.
Proof of (3.4), Step 1. Assume f ∈ C ∞ c (R n ) and fix r > 0. We have We estimate the two double integrals appearing in the right-hand side separately. By Tonelli's Theorem, we have Concerning the second double integral, integrating by parts we get for all x ∈ A. Hence, we can estimate Thus (3.4) follows for all f ∈ C ∞ c (R n ) and r > 0. Proof of (3.4), Step 2. Let f ∈ BV (R n ) and fix r > 0. Combining [34,Theorem 5.3] with a standard cut-off approximation argument, we find ( for all k ∈ N. We claim that Indeed, if ϕ ∈ Lip c (R n ; R n ), then div α ϕ ∈ L ∞ (R n ) by (2.12) and thus Thus, (3.9) follows passing to the limit as ε → 0 + . Thanks to (3.9), by [50,Proposition 4.29]we get that for any open set U ⊂ R n by [34, Theorem 5.2], we can estimate Thus, (3.4) follows taking limits as k → +∞ in (3.8). Finally, (3.5) is easily deduced by optimising the right-hand side of (3.4) in the case A = R n with respect to r > 0.
and (3.6) follows. To prove that D α f = ∇ α f L n , we argue as in the proof of [27,Proposition 4.8] Hence, by the definition of div α on Lip c -regular vector fields (see [27,Section 2.2]) and by Lebesgue's Dominated Convergence Theorem, we have for all ε > 0, by Fubini's Theorem we can compute for all y ∈ R n and ε > 0, and by (3.6), again by Lebesgue's Dominated Convergence Theorem we conclude that loc (R n ) for all α ∈ (0, 1), so that D α f ∈ M loc (R n ; R n ) exists by (ii). Hence, inserting (3.1) in (3.6), we find Since for all x ∈ B 1 we have being log-convex on (0, +∞) (see [9]), we can estimate proving (3.7).
Note that Proposition 3.2(i), in particular, applies to any f ∈ W 1,1 (R n ). In the following result, we prove that a similar result holds also for any f ∈ W 1, p (R n ) with p ∈ (1, +∞).
for any r > 0 and any open set A ⊂ R n , where A r := x ∈ R n : dist(x, A) < r . In particular, we have In addition, if p ∈ 1, n 1−α and q = np n−(1−α) p , then ∇ α w f = I 1−α ∇ w f a.e. in R n (3.12) and ∇ α w f ∈ L q (R n ; R n ).
Proof We argue as in the proof of Proposition 3.2(i). Proof of (3.10). The proof of (3.10) for all f ∈ C ∞ c (R n ) is very similar to that of (3.4) and is thus left to the reader. Now let f ∈ W 1, p (R n ) and fix an open set A ⊂ R n and r > 0. Combining [34,Theorem 4.2] with a standard cut-off approximation argument, we find ( for all k ∈ N. Hence, choosing A = R n , we get that the sequence (∇ α f k ) k∈N is uniformly bounded in L p (R n ; R n ). Up to pass to a subsequence (which we do not relabel for simplicity), there exists g ∈ L p (R n ; for all k ∈ N. Passing to the limit as k → +∞, by Proposition 2.1 we get that for any ϕ ∈ C ∞ c (R n ; R n ), so that g = ∇ α w f and hence f ∈ S α, p (R n ) according to [27,Definition 3.19]. We thus have that for any open set A ⊂ R n , since for all ϕ ∈ C ∞ c (A; R n ). Therefore, (3.10) follows by taking limits as k → +∞ in (3.13).
Proof of (3.11). Inequality (3.11) follows by applying (3.10) with A = R n and minimising the right-hand side with respect to r > 0.
Proof of (3.12). Now assume p ∈ 1, n 1−α and let q = np n−(1−α) p . Let ϕ ∈ C ∞ c (R n ; R n ) be fixed. Recalling inequality (2.5), since ϕ ∈ L q q−1 (R n ; R n ) we have that In particular, Fubini's Theorem implies that Since div α ϕ ∈ L p p−1 (R n ) by Proposition 2.1, we also get that Therefore, observing that I 1−α ϕ ∈ Lip b (R n ; R n ) because ∇ I 1−α ϕ = ∇ α ϕ ∈ L ∞ (R n ; R n 2 ) again by Proposition 2.1 and performing a standard cut-off approximation argument, we can integrate by parts and obtain (3.12). In particular, notice that ∇ α w f ∈ L q (R n ; R n ) by inequality (2.5). The proof is complete.
For the case p = +∞, we have the following immediate consequence of Lemma 2.4 and Proposition 2.8. (3.14)

Two representation formulas for the˛-variation
In this section, we prove two useful representation formulas for the α-variation. We begin with the following weak representation formula for the fractional αvariation of functions in BV loc (R n ) ∩ L ∞ (R n ). Here and in the following, we denote by f the precise representative of f ∈ L 1 loc (R n ), see (A.1) for the definition. Proposition 3.5 Let α ∈ (0, 1) and f ∈ BV loc (R n ) ∩ L ∞ (R n ). Then ∇ α f ∈ L 1 loc (R n ; R n ) and for all ϕ ∈ Lip c (R n ; R n ).
Proof By Proposition 3.2(iii), we know that ∇ α f ∈ L 1 loc (R n ; R n ) for all α ∈ (0, 1). By Theorem A.1, we also know that f Moreover, we can split the last integral as For all x ∈ B R/2 , we can estimate and so, since supp ϕ ⊂ B R/2 , we get that Therefore, by (2.11), Lebesgue's Dominated Convergence Theorem, (3.16) and (3.17), we get that and the conclusion follows.
In the following result, we show that for all functions in bv(R n ) ∩ L ∞ (R n ) one can actually pass to the limit as R → +∞ inside the integral in the right-hand side of (3.15).

Corollary 3.6 If either f ∈ BV
Moreover, for all x ∈ R n , we have again by Lebesgue's Dominated Convergence Theorem we get that The conclusion thus follows by combining (3.15) with (3.19).
As a consequence, the operator (− ) We can thus relate functions with bounded α-variation and functions with bounded variation via Riesz potential and the fractional Laplacian. We would like to prove a similar result between functions with bounded α-variation and functions with bounded β-variation, for any couple of exponents 0 < β < α < 1.
However, although the standard variation of a function f ∈ L 1 loc (R n ) is well defined, it is not clear whether the functional is well posed for all ϕ ∈ C ∞ c (R n ; R n ), since div α ϕ does not have compact support. Nevertheless, thanks to Proposition 2.1, the functional in (3.20) is well defined as soon as f ∈ L p (R n ) for some p ∈ [1, +∞]. Hence, it seems natural to define the space for any α ∈ (0, 1) and p ∈ [1, +∞]. In particular, BV α,1 (R n ) = BV α (R n ). Similarly, we let A further justification for the definition of these new spaces comes from the following fractional version of the Gagliardo-Nirenberg-Sobolev embedding: if n ≥ 2 and α ∈ (0, 1), then BV α (R n ) is continuously embedded in L p (R n ) for all p ∈ 1, n n−α , see [27,Theorem 3.9]. Hence, thanks to (3.21), we can equivalently write with continuous embedding for all n ≥ 2, α ∈ (0, 1) and p ∈ 1, n n−α . Incidentally, we remark that the continuous embedding BV α (R n ) ⊂ L n n−α (R n ) for n ≥ 2 and α ∈ (0, 1) can be improved using the main result of the recent work [73] (see also [74]). Indeed, if n ≥ 2, α ∈ (0, 1) and f ∈ C ∞ c (R n ), then, by taking thanks to the boundedness of the Riesz transform R : L n n−α ,1 (R n ) → L n n−α ,1 (R n ; R n ), where c n,α , c n,α > 0 are two constants depending only on n and α, and L n n−α ,1 (R n ) is the Lorentz space of exponents n n−α , 1 (we refer to [39,40] for an account on Lorentz spaces and on the properties of Riesz transform). Thus, recalling [27, Theorem 3.8], we readily deduce the continuous embedding BV α (R n ) ⊂ L n n−α ,1 (R n ) for n ≥ 2 and α ∈ (0, 1) by [39, Exercise 1.1.1(b)] and Fatou's Lemma. This suggests that the spaces defined in (3.21) may be further enlarged by considering functions belonging to some Lorentz space, but we do not need this level of generality here.
In the case n = 1, the space BV α (R) does not embed in L 1 1−α (R) with continuity, see [27,Remark 3.10]. However, somehow completing the picture provided by [73], we can prove that the space BV α (R) continuously embeds in the Lorentz space L 1 1−α ,∞ (R). Although this result is truly interesting only for n = 1, we prove it below in all dimensions for the sake of completeness.  ∈ (0, 1). There exists a constant c n,α > 0 such that for all f ∈ BV α (R n ). As a consequence, BV α (R n ) is continuously embedded in L q (R n ) for any q ∈ [1, n n−α ).

Remark 3.9 (The embedding BV
so that d f α ≥ d g α , where d f α and d g α are the distribution functions of f α and g α . A simple calculation shows that so that, by [39,Proposition 1.4.9], we obtain and thus f α / ∈ L 1 1−α ,q (R) for any q ∈ [1, +∞).
We collect the above continuous embeddings in the following statement.
With Corollary 3.10 at hands, we are finally ready to investigate the relation between α-variation and β-variation for 0 < β < α < 1.
As a consequence, the operator (− ) Proof We begin with the following observation. Let ϕ ∈ C ∞ c (R n ; R n ) and let U ⊂ R n be a bounded open set such that supp ϕ ⊂ U . By Proposition 2.1 and the semigroup property (2.4) of the Riesz potential, we can write Similarly, we also have by [27,Lemma 2.4] We now prove the two statements separately.
We now prove (3.26) in two steps. We argue as in the proof of (3.4). Proof of (3.26), We estimate the two double integrals appearing in the right-hand side separately. By Tonelli's Theorem, we have Concerning the second double integral, we apply [1, Lemma 3.1.1(c)] to each component of the measure D α f ∈ M (R n ; R n ) and get {|h|≥r } Since we can compute Thus (3.4) follows for all f ∈ C ∞ c (R n ) and r > 0. Proof of (3.4), Step 2. Let f ∈ BV α (R n ) and fix r > 0. By [27,Theorem 3.8], we for all k ∈ N. We have that This can be proved arguing as in the proof of (3.9) using (3.25). At this point the proof goes like that of Proposition 3.2(i) and we thus leave the details to the reader.

Convergence of ∇˛and div˛as˛→ 1 −
We begin with the following simple result about the asymptotic behavior of the constant μ n,α as α → 1 − .
In the following technical result, we show that the constant C n,α,U defined in (2.9) is uniformly bounded as α → 1 − in terms of the volume and the diameter of the bounded open set U ⊂ R n . where C n is as in (4.1).
As consequence of Proposition 2.1 and Lemma 4.2, we prove that ∇ α and div α converge pointwise to ∇ and div respectively as α → 1 − .

Proposition 4.3
If f ∈ C 1 c (R n ), then for all x ∈ R n we have As a consequence, if f ∈ C 2 c (R n ) and ϕ ∈ C 2 c (R n ; R n ), then for all x ∈ R n we have and fix x ∈ R n . Writing (2.6) in spherical coordinates, we find Since f ∈ C 1 c (R n ), for each fixed v ∈ ∂ B 1 we can integrate by parts in the variable and get Clearly, we have Thus, by Fubini's Theorem, we conclude that Since f has compact support and recalling (4.2), we can pass to the limit in (4.6) and get proving (4.4). The pointwise limits in (4.5) immediately follows by Proposition 2.1.
In the following crucial result, we improve the pointwise convergence obtained in Proposition 4.3 to strong convergence in L p (R n ) for all p ∈ [1, +∞].

Proposition 4.4 Let p ∈ [1, +∞].
If f ∈ C 2 c (R n ) and ϕ ∈ C 2 c (R n ; R n ), then for all x ∈ R n we can write Therefore, by (2.6), we have for all x ∈ R n . We now distinguish two cases. Case 1: p ∈ [1, +∞). Using the elementary inequality |v + w| p ≤ 2 p−1 (|v| p + |w| p ) valid for all v, w ∈ R n , we have We now estimate the two double integrals appearing in the right-hand side separately. For the first double integral, as in the proof of Proposition 4.3, we pass in spherical coordinates to get for all x ∈ R n . Hence, by (4.2), we find for all x ∈ R n . Therefore, we get for all x ∈ R n . Recalling (4.1), we also observe that |y| n for all α ∈ (0, 1), x ∈ R n and y ∈ B 1 . Moreover, letting R > 0 be such that supp f ⊂ B R , we can estimate for all x ∈ R n , so that In conclusion, applying Lebesgue's Dominated Convergence Theorem, we find For the second double integral, note that for all x ∈ R n . Now let R > 0. Integrating by parts, we have that for all x ∈ R n . Since for all R > 0, we conclude that for all x ∈ R n . Hence, by Minkowski's Integral Inequality (see [76, Section A.1], for example), we can estimate Thus, by (4.2), we get that Case 2: p = +∞. We have Again we estimate the two integrals appearing in the right-hand side separately. We note that so that we can rewrite (4.7) as Hence, we can estimate For the second integral, by (4.8) we can estimate Thus, by (4.2), we get that We can now conclude the proof. Again recalling (4.2), we thus find that for all p ∈ [1, +∞] and the conclusion follows. The L p -convergence of div α ϕ to divϕ as α → 1 − for all p ∈ [1, +∞] follows by a similar argument and is left to the reader.

Remark 4.5
Note that the conclusion of Proposition 4.4 still holds if instead one assumes that f ∈ S (R n ) and ϕ ∈ S (R n ; R n ), where S (R n ; R m ) is the space of m-vector-valued Schwartz functions. We leave the proof of this assertion to the reader.

Weak convergence of˛-variation as˛→ 1 −
In Theorem 4.7 below, we prove that the fractional α-variation weakly converges to the standard variation as α → 1 − for functions either in BV (R n ) or in BV loc (R n ) ∩ L ∞ (R n ). In the proof of Theorem 4.7, we are going to use the following technical result.

Lemma 4.6
There exists a dimensional constant c n > 0 with the following property.
Proof We divide the proof in two steps.
Step 1. Assume f ∈ BV (R n ). By [27,Theorem 3.18], we have for all ϕ ∈ Lip c (R n ; R n ). Thus, given ϕ ∈ C 2 c (R n ; R n ), recalling Proposition 4.3 and the estimates (2.12) and (4.3), by Lebesgue's Dominated Convergence Theorem we get that To achieve the same limit for any ϕ ∈ C 0 c (R n ; R n ), one just need to exploit (3.3) and the uniform estimate (4.3) in Lemma 4.2, and argue as in Step 2 of the proof of (3.4). We leave the details to the reader.
Step 2. Assume f ∈ BV loc (R n ) ∩ L ∞ (R n ). By Proposition 3.2(iii), we know that D α f = ∇ α f L n with ∇ α f ∈ L 1 loc (R n ; R n ). By Proposition 4.4, we get that for all ϕ ∈ C 2 c (R n ; R n ). To achieve the same limit for any ϕ ∈ C 0 c (R n ; R n ), one just need to exploit (4.9) and argue as in Step 1. We leave the details to the reader.
We are now going to improve the weak convergence of the fractional α-variation obtained in Theorem 4.7 by establishing the weak convergence also of the total fractional α-variation as α → 1 − , see Theorem 4.9 below. To do so, we need the following preliminary result.

Lemma 4.8
Let μ ∈ M (R n ; R m ). We have (I α μ)L n μ as α → 0 + . Proof Since Riesz potential is a linear operator and thanks to Hahn-Banach Decomposition Theorem, without loss of generality we can assume that μ is a nonnegative finite Radon measure.
Let now ϕ ∈ C 1 c (R n ) and let U ⊂ R n be a bounded open set such that supp ϕ ⊂ U . We have that I α |ϕ| L ∞ (R n ) ≤ κ n,U ϕ L ∞ (R n ) for all α ∈ (0, 1 2 ) by [ To achieve the same limit for any ϕ ∈ C 0 c (R n ), one just need to exploit [27, Lemma 2.4] and (4.3) and argue as in Step 2 of the proof of (3.4). We leave the details to the reader.
Proof We prove (4.10) and (4.11) separately. Proof of (4.10). By Theorem 4.7, we know that D α f D f as α → 1 − . By [50,Proposition 4.29], we thus have that for any open set A ⊂ R n . Now let K ⊂ R n be a compact set. By the representation formula (3.18) in Corollary 3.6, we can estimate .
Proof of (4.11). Now assume f ∈ BV (R n ). By (3.4) applied with A = R n and r = 1, we have By (4.2), we thus get that lim sup Thus (4.11) follows by combining (4.12) for A = R n with (4.13).
Remark 4. 10 We notice that Theorems 4.7 and 4.9, in the case f = χ E ∈ BV (R n ) with E ⊂ R n bounded, and Theorem 4.11, were already announced in [71,Theorems 16 and 17].
Note that Theorems 4.7 and 4.9 in particular apply to any f ∈ W 1,1 (R n ). In the following result, by exploiting Proposition 3.3, we prove that a stronger property holds for any f ∈ W 1, p (R n ) with p ∈ [1, +∞). (4.14) Proof By Proposition 3.3 we know that f ∈ S α, p (R n ) for any α ∈ (0, 1). We now assume p ∈ (1, +∞) and divide the proof in two steps.
Step 1. We claim that Indeed, on the one hand, by Proposition 4.4, we have for all ϕ ∈ C ∞ c (R n ; R n ). We thus get that On the other hand, applying (3.10) with A = R n and r = 1, we have By (4.2), we conclude that Thus, (4.15) follows by combining (4.17) and (4.18).
For the case p = 1, we argue as follows (we thank Mattia Calzi for this simple argument). Without loss of generality, it is enough to prove the limit in (4.15) with p = 1 for any given sequence (α k ) k∈N such that α k → 1 − as k → +∞. By (4.11), the sequence ( ∇ α k f L 1 (R n ; R n ) ) k∈N is bounded for any f ∈ W 1,1 (R n ) and thus, by Banach-Steinhaus Theorem, the linear operators ∇ α k : W 1,1 (R n ) → L 1 (R n ; R n ), k ∈ N, are uniformly bounded (in the operator norm). The conclusion hence follows by exploiting the density of C ∞ c (R n ) in W 1,1 (R n ) and Proposition 4.4. For the case p = +∞, we have the following result. The proof is very similar to the one of Theorem 4.11 and is thus left to the reader. and (4.21)

0-convergence of˛-variation as˛→ 1 −
In this section, we study the -convergence of the fractional α-variation to the standard variation as α → 1 − . We begin with the -lim inf inequality. holds.
Proof We prove the two statements separately.
Proof of (i).
by Proposition 4.4 we get that and the conclusion follows.
Proof of (ii). Let ϕ ∈ C ∞ c ( ; R n ) be such that ϕ L ∞ ( ;R n ) ≤ 1. Since we can estimate by Proposition 4.4 we get that and the conclusion follows.
We now pass to the -lim sup inequality. (i) If f ∈ BV (R n ) and either is bounded or = R n , then (4.23) (ii) If f ∈ BV loc (R n ) and is bounded, then In addition, if f = χ E , then the recovering sequences ( f α ) α∈(0,1) in (i) and (ii) can be taken such that f α = χ E α for some measurable sets (E α ) α∈(0, 1) .
for any bounded open set ⊂ R n . If = R n , then (4.23) follows immediately from (4.11). This concludes the proof of (i).
Now assume that f ∈ BV loc (R n ) and is bounded. Let (R k ) k∈N ⊂ (0, +∞) be a sequence such that R k → +∞ as k → +∞ and set f k := f χ B R k for all k ∈ N. By Theorem A.1, we can choose the sequence (R k ) k∈N such that, in addition, in L 1 loc (R n ) as k → +∞ and, moreover, since is bounded, |D f k |( ) = |D f |( ) and |D f k |(∂ ) = |D f |(∂ ) for all k ∈ N sufficiently large. By (4.24), we have that This concludes the proof of (ii).
Finally, if f = χ E , then we can repeat the above argument verbatim in the metric spaces {χ F ∈ L 1 (R n ) : F ⊂ R n } for (i) and {χ F ∈ L 1 loc (R n ) : F ⊂ R n } for (ii) endowed with their natural distances.

Remark 4.15
Thanks to (4.23), a recovery sequence in Theorem 4.14(i) is the constant sequence (also in the special case f = χ E ).
Combining Theorems 4.13(i) and 4.14(ii), we can prove that the fractional Caccioppoli α-perimeter -converges to De Giorgi's perimeter as α → 1 − in L 1 loc (R n ). We refer to [3] for the same result on the classical fractional perimeter.
Proof By Theorem 4.13(i), we already know that so we just need to prove the (L 1 loc ) -lim sup inequality. Without loss of generality, we can assume P(E; ) < +∞. Now let (E k ) k∈N be given by Theorem A.4. Since χ E k ∈ BV loc (R n ) and P(E k ; ∂ ) = 0 for all k ∈ N, by Theorem 4.14(ii) we know that for all k ∈ N. Since χ E k → χ E in L 1 loc (R n ) and P(E k ; ) → P(E; ) as k → +∞, by [17, Proposition 1.28] we get that and the proof is complete.
Finally, by combining Theorems 4.13(ii) and 4.14, we can prove that the fractional α-variation -converges to De Giorgi's variation as α → 1 − in L 1 (R n ).
Proof The case = R n follows immediately by [29,Proposition 8.1(c)] combining Theorem 4.13(ii) with Theorem 4.14(i). We can thus assume that is a bounded open set with Lipschitz boundary and argue as in the proof of Theorem 4.16. By Theorem 4.13(ii), we already know that so we just need to prove the (L 1 ) -lim sup inequality. Without loss of generality, we can assume |D f |( ) < +∞. Now let ( f k ) k∈N ⊂ BV (R n ) be given by Theorem A. 6. Since |D f k |(∂ ) = 0 for all k ∈ N, by Theorem 4.14 we know that and the proof is complete.

Remark 4.18
Thanks to Theorem 4.17, we can slightly improve Theorem 4.16. Indeed, if χ E ∈ BV (R n ), then we also have for any open set ⊂ R n such that either is bounded with Lipschitz boundary or = R n .
We begin with the -lim inf inequality.
Proof We argue as in the proof of Theorem 4.13(ii). Let ϕ ∈ C ∞ c ( ; R n ) be such that ϕ L ∞ ( ;R n ) ≤ 1. Let U ⊂ R n be a bounded open set such that supp ϕ ⊂ U . By (2.12), we can estimate for all β ∈ (0, α). Since div β ϕ → div α ϕ in L ∞ (R n ) as β → α − by (5.2), we easily obtain and the conclusion follows.
We now pass to the -lim sup inequality. Theorem 5.6 ( -lim sup inequality for β → α − ) Let α ∈ (0, 1) and let ⊂ R n be an open set. If f ∈ BV α (R n ) and either is bounded or = R n , then for any open set ⊂ R n such that either is bounded or = R n . ∈ (0, 1). For every f ∈ BV α (R n ), we have In particular, the constant sequence is a recovery sequence.
Proof The result follows easily by combining (5.4) and (5.5) in the case = R n .
Remark 5. 8 We recall that, by [27,Theorem 3.25] for any bounded open set ⊂ R n such that L n (∂ ) = 0 (for instance, with Lipschitz boundary). Thus, we can actually obtain the -convergence of the fractional β-variation as β → α − on bounded open sets with Lipschitz boundary for any f ∈ S α,1 (R n ) too. Indeed, it is enough to combine (5.4) and (5.5) and then exploit the fact that |D α f |(∂ ) = 0 to get We were not able to find a reference for the analogue of Corollary 5.7 for the usual fractional Sobolev seminorms. For the sake of completeness, we state and prove it below for all p ∈ [1, +∞) on a general open set. Theorem 5.9 ( (L p ) -lim of W β, p -seminorm as β → α − ) Let ⊂ R n be a nonempty open set, α ∈ (0, 1) and p ∈ [1, +∞). For every f ∈ W α, p ( ), we have In particular, the constant sequence is a recovery sequence.
Given any R > 0, by Fatou's Lemma we thus get that Hence, the set satisfies L 1 ((0, +∞) \ Z ) = 0 and clearly does not depend on the choice of ϕ. Now fix r ∈ Z ∩ W and let (ε k ) k∈N be any sequence realising the lim inf in (A.5). By (A.4), we thus get uniformly for all ϕ satisfying ϕ L ∞ (R n ; R n ) ≤ 1. Passing to the limit along the sequence (ε k ) k∈N as k → +∞ in (A.3), we get that for all ϕ ∈ C ∞ c (R n ; R n ) with ϕ L ∞ (R n ; R n ) ≤ 1. Thus (A.2) follows for all r ∈ W ∩ Z and the proof is concluded.

A.2 Approximation by sets with polyhedral boundary
In this section we state and prove standard approximation results for sets with finite perimeter or, more generally, BV loc (R n ) functions, in a sufficiently regular bounded open set.
We need the following two preliminary lemmas.
Lemma A.2 Let V , W ⊂ S n−1 , with V finite and W at most countable. For any ε > 0, there exists R ∈ SO(n) with |R − I| < ε, where I is the identity matrix, such that R(V ) ∩ W = ∅.
Proof Let N ∈ N be such that V = v i ∈ S n−1 : i = 1, . . . , N . We divide the proof in two steps.
Step 1. Assume that W is finite and set A i := R ∈ SO(n) : R(v i ) / ∈ W for all i = 1, . . . , N . We now claim that A i of SO(n) for all i = 1, . . . , N . Indeed, given any i = 1, . . . , N , since W is finite, the set A c i = SO(n) \ A i is closed in SO(n). Moreover, we claim that int(A c i ) = ∅. Indeed, by contradiction, let us assume that int(A c i ) = ∅. Then there exist ε > 0 and R ∈ A c i such that any S ∈ SO(n) with |S − R| < ε satisfies S ∈ A c i . Now let is an open and dense subset of SO(n). The result is thus proved for any finite set W .
Step 2. Now assume that W is countable, W = {w k ∈ S n−1 : k ∈ N}. For all M ∈ N, set W M := {w k ∈ W : k ≤ M}. By Step 1, we know that A W M is an open and dense subset of SO(n) for all M ∈ N. Since SO(n) ⊂ R n 2 is compact, by Baire's Theorem A := M∈N A W M is a dense subset of SO(n). This concludes the proof.
Since det : GL(n) → R is a continuous map, there exists a dimensional constant δ n ∈ (0, 1) such that det R ≥ 1 2 for all R ∈ GL(n) with |R − I| < δ n .
Proof We divide the proof in two steps.
Step 1. Let r > 0 and let f ∈ C ∞ c (R n ). Setting R t := (1 − t)I + tR for all t ∈ [0, 1], we can estimate Since |R t − I| = t|R − I| < tε < δ n for all t ∈ [0, 1], R t is invertible with det(R −1 t ) ≤ 2 for all t ∈ [0, 1]. Hence we can estimate Step 2. Since χ E ∈ BV (R n ), by combining [34,Theorem 5.3] with a standard cut-off approximation argument, we find ( f k ) k∈N ⊂ C ∞ c (R n ) such that f k → χ E pointwise a.e. in R n and |∇ f k |(R n ) → P(E) as k → +∞. Given any r > 0, by (A.6) in Step 1 we have for all k ∈ N. Passing to the limit as k → +∞, by Fatou's Lemma we get that |(R(E) E) ∩ B r | ≤ 2εr P(E).
Since E ⊂ B r E up to L n -negligible sets, also R(E) ⊂ B r E up to L n -negligible sets. Thus we can choose r = r E and the proof is complete.
We are now ready to prove the main approximation result, see also [3,  → χ E in L 1 loc (R n ), P(E 3 k ; ) → P(E; ) and P(E 3 k ; ∂ ) → 0 as k → +∞. If there exists a subsequence (E 3 k j ) j∈N such that P(E 3 k j ; ∂ ) = 0 for all j ∈ N, then we can set E j := E k j for all j ∈ N and the proof is concluded. If this is not the case, then we need to proceed with the next last step.
Step 4: rotation. We now argue as in the last part of the proof of [3,Proposition 15]. Fix k ∈ N and assume P(E 3 k ; ∂ ) > 0. Since E 3 k has polyhedral boundary, we have H n−1 (∂ E 3 k ∩ ∂ ) > 0 if and only if there exist ν ∈ S n−1 and U ⊂ F such that H n−1 (U ) > 0, ν (x) = ν for all x ∈ U and U ⊂ ∂ H for some (affine) half-space H satisfying ν H = ν. Since P( ) = H n−1 (∂ ) < +∞, the set is at most countable. Since E 3 k has polyhedral boundary, the set is finite. By Lemma A.2, given ε k > 0, there exists R k ∈ SO(n) with |R k − I| < ε k such that R k (V k )∩W = ∅. Hence the set E 4 k := R k (E 3 k ) must satisfy P(E 4 k ; ∂ ) = 0. By Lemma A.3, we can choose ε k > 0 sufficiently small in order to ensure that |E 4 k E 3 k | < 1 k . Now choose η k ∈ 0, 1 2k such that P(E 3 k ; Q k ) ≤ 2P(E 3 k ; ∂ ), where Q k := {x ∈ R n : dist(x, ∂ ) < η k }.
Since is bounded, possibly choosing ε k > 0 even smaller, we can also ensure that R −1 ( ) ⊂ Q k . Hence we can estimate We can thus set E k := E 4 k for all k ∈ N and the proof is complete.

Remark A.5 (A minor gap in the proof of [3, Proposition 15])
We warn the reader that the cut-off and the extension steps presented above were not mentioned in the proof of [3,Proposition 15], although they are unavoidable for the correct implementation of the rotation argument in the last step. Indeed, in general, one cannot expect the existence of a rotation R ∈ SO(n) arbitrarily close to the identity map such that P(R(E); ∂ ) = 0 and, at the same time, the difference between P(R(E); ) and P(E; ) is small. For example, one can consider n = 2, where A = {(x 1 , x 2 ) ∈ R 2 : x 1 > 0, x 2 > 0}. In this case, for any rotation R ∈ SO(2) arbitrarily close to the identity map, we have P(R(E); ) > 2 + P(E; ).
We conclude this section with the following result, establishing an approximation of BV loc functions similar to that given in Theorem A.4. Proof We argue as in the proof of Theorem A.4, in two steps.
Step 1: cut-off at infinity. Since is bounded, we find R 0 > 0 such that ⊂ B R 0 . Given (R k ) k ⊂ (R 0 , +∞), we set g k := f χ B R k for all k ∈ N. By Theorem A.1, we have g k ∈ BV (R n ) for a suitable choice of the sequence (R k ) k∈N , with |Dg k |( ) = |D f |( ) for all k ∈ N and g k → f in L 1 loc (R n ) as k → +∞. If, in addition, f ∈ L 1 (R n ), then g k → f in L 1 (R n ) as k → +∞.
Step 2: extension and cut-off near . Let us define A k := x ∈ R n : dist(x, ) < 1 k for all k ∈ N. Since g k χ ∈ BV ( ) with |Dg k |( ) = |D f |( ) for all k ∈ N, by [4, Definition 3.20 and Proposition 3.21] there exists a sequence (h k ) k∈N ⊂ BV (R n ) such that supp h k ⊂ A 2k , h k = g k in , |Dh k |(∂ ) = 0