Existence of Blow-Up

A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND FRACTIONAL VARIATION: EXISTENCE OF BLOW-UP

GIOVANNI E. COMI AND GIORGIO STEFANI

Abstract. We introduce the new space BV α(Rn) of functions with fractional bounded variation in Rn of order α ∈ (0, 1) via a new distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical BV theory, we give a new notion of set E of (locally) finite fractional Caccioppoli α-perimeter and we define its fractional reduced boundary F αE. We are able to show that W α,1(Rn) ⊂ BV α(Rn) continuously and, similarly, that sets with (locally) finite standard fractional α-perimeter have (locally) finite fractional Caccioppoli α-perimeter, so that our theory provides a natural extension of the known fractional framework. Our main result partially extends De Giorgi’s Blow-up Theorem to sets of locally finite fractional Caccioppoli α-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets.

Contents 1. Introduction 2 1.1. Fractional Sobolev spaces and the quest for a fractional gradient 2 1.2. De Giorgi’s distributional approach to perimeter 4 1.3. Fractional variation and perimeter: a new distributional approach 5 1.4. Future developments 6 1.5. Organisation of the paper 7 2. Šilhavý’s fractional calculus 7 2.1. General notation 7 2.2. Definition of ∇α and divα 8 2.3. Equivalent definition of ∇α and divα via Riesz potential 10 2.4. Duality and Leibniz’s rules 12 3. Fractional BV functions 14 3.1. Definition of BV α(Rn) and Structure Theorem 14 3.2. Lower semicontinuity of fractional variation 15 3.3. Approximation by smooth functions 16 3.4. Gagliardo–Nirenberg–Sobolev inequality 18

Date: February 18, 2019. 2010 Mathematics Subject Classiﬁcation. 26A33, 35B44, 47B38, 26B30. Key words and phrases. Function with fractional bounded variation, fractional perimeter, blow- up, fractional Sobolev spaces, fractional calculus, fractional derivative, fractional gradient, fractional divergence. Acknowledgements. The authors thank Professor Luigi Ambrosio for valuable comments and sugges- tions during the realisation of this work. The authors also thank Professor Quoc-Hung Nguyen for having shared with them many useful insights on Bessel potential spaces. This research was partially supported by the PRIN2015 MIUR Project “Calcolo delle Variazioni”. 1 A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 2

3.5. Coarea inequality 19 3.6. A fractional version of the Fundamental Theorem of Calculus 19 3.7. Compactness 20 3.8. The inclusion W α,1(Rn) ⊂ BV α(Rn) 22 3.9. The space Sα,p(Rn) and the inclusion Sα,1(Rn) ⊂ BV α(Rn) 23 3.10. The inclusion Sα,1(Rn) ⊂ BV α(Rn) is strict 26 3.11. The inclusion W α,1(Rn) ⊂ Sα,1(Rn) is strict 29 3.12. The inclusion BV α(Rn) ⊂ W β,1(Rn) for β < α 29 4. Fractional Caccioppoli sets 30 4.1. Deﬁnition of fractional Caccioppoli sets and the Gauss–Green formula 30 4.2. Lower semicontinuity of fractional variation 31 4.3. Fractional isoperimetric inequality 31 4.4. Compactness 32 4.5. Fractional reduced boundary 33 4.6. Sets of ﬁnite fractional perimeter are fractional Caccioppoli sets 34 5. Existence of blow-ups for fractional Caccioppoli sets 38 ∞ n α,1 n Appendix A. Cc (R ) is dense in W (R ) 44 References 45

1. Introduction 1.1. Fractional Sobolev spaces and the quest for a fractional gradient. In the last decades, fractional Sobolev spaces have been given an increasing attention, see [6, Section 1] for a detailed list of references in many directions. If p ∈ [1, +∞) and α ∈ (0, 1), the fractional Sobolev space W α,p(Rn) is the space  1  Z Z p ! p α,p n  p n |u(x) − u(y)|  : : α,p n : W (R ) = u ∈ L (R ) [u]W (R ) = n+pα dx dy < +∞  Rn Rn |x − y|  endowed with the natural norm

kukW α,p(Rn) := kukLp(Rn) + [u]W α,p(Rn). Differently from the standard Sobolev space W 1,p(Rn), the space W α,p(Rn) does not have an evident distributional nature, in the sense that the seminorm [ · ]W α,p(Rn) does not seem to be the Lp-norm of some kind of weakly-defined gradient of fractional order. Recently, the search for a good notion of differential operator in this fractional setting has led several authors to consider the following fractional gradient Z (y − x)(u(y) − u(x)) α : ∇ u(x) = µn,α n+α+1 dy, (1.1) Rn |y − x| α where µn,α is a multiplicative normalising constant controlling the behaviour of ∇ as α → 1−. For a detailed account on the existing literature on this operator, see [19, Section 1]. Here we only refer to [15–20] for the articles tightly connected to the present work. According to [19, Section 1], it is interesting to notice that [9] seems to be the earliest reference for the operator defined in (1.1). A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 3

From its very definition, it is not difficult to see that the fractional gradient ∇α is well defined in L1(Rn) for functions in W α,1(Rn), since we have the simple estimate Z α |∇ u| dx ≤ µn,α [u]W α,1(Rn). (1.2) Rn In analogy with the standard Sobolev space W 1,p(Rn), this observation leads to consider the space k·k α,p n α,p n ∞ n S (R ) S0 (R ) := Cc (R ) , where α kukSα,p(Rn) := kukLp(Rn) + k∇ ukLp(Rn;Rn) ∞ n α,p n for all u ∈ Cc (R ). This is essentially the line followed in [18], where the space S0 (R ) has been introduced (with a different, but equivalent, norm). By [18, Theorem 2.2], it is known that α+ε,p n α,p n α−ε,p n S0 (R ) ⊂ W (R ) ⊂ S0 (R ) (1.3) with continuous embeddings for all α ∈ (0, 1), p ∈ (1, +∞) and 0 < ε < min{α, 1 − α}. In the particular case p = 2, by [18, Theorem 2.2] we actually have that α,2 n α,2 n S0 (R ) = W (R ) (1.4) for all α ∈ (0, 1). In addition, as observed in [21, Section 5.3], we have α,p n α,p n W (R ) ⊂ S0 (R ) (1.5) with continuous embedding for all α ∈ (0, 1) and p ∈ (1, 2]. For further properties of the α,p n space S0 (R ), we refer to Section 3.9 below. The inclusions (1.3) and (1.5), and the identification (1.4) may suggest that the spaces α,p n S0 (R ) can be considered as an interesting distributional-type alternative of the usual fractional Sobolev spaces W α,p(Rn) and thus as a natural setting for the development of a general theory for solutions to PDEs involving the fractional gradient in (1.1), proceeding similarly as in the classical Sobolev framework. This is the point of view pursued in [18,19]. Another important aspect of the fractional gradient in (1.1) is that it satisfies three natural ‘qualitative’ requirements as a fractional operator: invariance under translations and rotations, homogeneity of order α under dilations and some continuity properties in an appropriate functional space, e.g. Schwartz’s space S (Rn). A fundamental result of [20] is that these three requirements actually characterise the fractional gradient in (1.1) (up to multiplicative constants), see [20, Theorem 2.2]. This shows that the definition in (1.1) is well posed not only from a mathematical point of view, but also from a physical point of view. Besides, the same characterisation holds for the following fractional divergence Z (y − x) · (ϕ(y) − ϕ(x)) α : div ϕ(x) = µn,α n+α+1 dy, (1.6) Rn |y − x| see [20, Theorem 2.4]. Moreover, as it is observed in [20, Section 6], the operators ∇α and divα are dual, in the sense that Z Z u divαϕ dx = − ϕ · ∇αu dx (1.7) Rn Rn ∞ n ∞ n n for all u ∈ Cc (R ) and ϕ ∈ Cc (R ; R ). The fractional integration by parts formula in (1.7) can be thus taken as the starting point for the development of a general theory A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 4 of fractional differential operators on the space of Schwartz’s distributions. This is the direction of research pursued in [20]. 1.2. De Giorgi’s distributional approach to perimeter. In the classical framework, the Sobolev space W 1,1(Rn) is naturally continuously embedded in BV (Rn), the space of functions with bounded variation, i.e. the space n n 1 n n o BV (R ) := u ∈ L (R ) : |Du|(R ) < +∞ endowed with the norm n kukBV (Rn) = kukL1(Rn) + |Du|(R ), where Z n ∞ n n |Du|(R ) = sup u divϕ dx : ϕ ∈ Cc (R ; R ), kϕkL∞(Rn;Rn) ≤ 1 Rn is the total variation of the function u ∈ BV (Rn). Thanks to Riesz’s Representation Theorem, one can see that a function u ∈ L1(Rn) belongs to BV (Rn) if and only if there exists a finite vector valued Radon measure Du ∈ M (Rn; Rn) such that Z Z u divϕ dx = − ϕ · dDu Rn Rn ∞ n n for all ϕ ∈ Cc (R ; R ). Functions of bounded variation have revealed to be the perfect tool for the development of a deep geometric analysis of sets with finite perimeter, starting directly from the seminal and profound works of R. Caccioppoli and E. De Giorgi. For a modern exposition of this vast subject and a detailed list of references, see [2,11]. A measurable set E ⊂ Rn has finite Caccioppoli perimeter if n P (E) := |DχE|(R ) < +∞. (1.8) The perimeter functional in (1.8) coincides with the classical surface measure when E has a sufficiently smooth (topological) boundary and, precisely, one can prove that P (E) = H n−1(∂E) (1.9) for all sets E with Lipschitz boundary, where H s denotes the s-dimensional Hausdorff measure for all s ≥ 0. One of the finest De Giorgi’s intuitions is that, for a finite perimeter set E with non- smooth boundary, the right ‘boundary object’ to keep the validity of (1.9) is a special subset of the topological boundary, the so-called measure-theoretic reduced boundary F E. With this notion in hand, a measurable set E ⊂ Rn has finite Caccioppoli perimeter if (and only if) H n−1(F E) < +∞, in which case we have P (E) = H n−1(F E). (1.10) Besides the validity of (1.10), an essential feature of De Giorgi’s reduced boundary is the following blow-up property: if x ∈ F E, then 1 n χ E−x → χHx in Lloc( ) (1.11) r R as r → 0, where

n DχE(Br(x)) Hx := {y ∈ R : (y − x) · νE(x) > 0}, νE(x) = lim . r→0 |DχE|(Br(x)) A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 5

n−1 The function νE : F E → S denotes the so-called measure theoretic inner unit normal of E and coincides with the usual inner unit normal of E when the boundary of E is sufficiently smooth. In other words, the blow-up property in (1.11) shows that, in a neighbourhood of a point x ∈ F E, the finite perimeter set E is infinitesimally close to an half-space oriented by νE(x). 1.3. Fractional variation and perimeter: a new distributional approach. In the fractional framework, an analogue of the space BV (Rn) is completely missing, since no distributional definition of the space W α,1(Rn) is available. Nevertheless, a theory for sets with finite fractional perimeter has been developed in recent years, with a strong interest on minimal fractional surfaces. We refer to [5, Section 7] for a detailed exposition of the most recent results in this direction. A measurable set E ⊂ Rn has finite fractional perimeter if Z Z 1 : α,1 n Pα(E) = [χE]W (R ) = 2 n+α dx dy < +∞. (1.12) Rn\E E |x − y| The fractional perimeter functional in (1.12) has a strong non-local nature in the sense that its value depends also on points which are very far from the boundary of the set E. For this reason, it is not clear if such a perimeter measure may be linked with some kind of fractional analogue of De Giorgi’s reduced boundary (which, a posteriori, cannot be expected to be a special subset of the topological boundary of E). In this paper, we want to combine the functional approach of [18, 19] with the distributional point of view of [20] to develop a satisfactory theory extending De Giorgi’s approach to variation and perimeter in the fractional setting. The natural starting point is the duality relation (1.7), which motivates the definition of the space α n n 1 n α n o BV (R ) := u ∈ L (R ) : |D u|(R ) < +∞ , (1.13) where Z α n α ∞ n n |D u|(R ) = sup u div ϕ dx : ϕ ∈ Cc (R ; R ), kϕkL∞(Rn;Rn) ≤ 1 (1.14) Rn stands for the fractional variation of the function u ∈ BV α(Rn). Note that the fractional α variation in (1.14) is well defined, since one can show that div ϕ ∈ L∞(Rn) for all ϕ ∈ ∞ n n Cc (R ; R ) (see Corollary 2.3). A different approach to fractional variation was developed in [22]. We do not know if the fractional variation defined in (1.14) is linked to the one introduced in [22] and it would be very interesting to establish a connection between the two. With definition (1.13), we are able to show that α,1 n α n W (R ) ⊂ BV (R ) with continuous embedding, in perfect analogy with the classical framework. Thus, emulating the classical definition in (1.8), it is very natural to define the fractional analogue of the Caccioppoli perimeter using the total variation in (1.14). Note that this α definition is well posed, since div ϕ ∈ L1(Rn) for all ϕ ∈ W α,1(Rn; Rn) arguing similarly as in (1.2). With this notion, we are able to show that α n |D χE|(R ) ≤ µn,αPα(E) (1.15) A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 6 for all measurable sets E with finite fractional perimeter, so that our approach naturally incorporates the current notion of fractional perimeter. Following the classical framework, the main results concerning the space BV α(Rn) we are able to prove are the following: • BV α(Rn) is a Banach space and its norm is lower semicontinuous with respect to L1-convergence; • the inclusion W α,1(Rn) ⊂ BV α(Rn) is continuous and strict; ∞ n α n ∞ n α n • the sets C (R ) ∩ BV (R ) and Cc (R ) are dense in BV (R ) with respect to the distance α n α n d(u, v) := ku − vkL1(Rn) + ||D u|(R ) − |D v|(R )|; • a fractional analogue of Gagliardo–Nirenberg–Sobolev inequality holds, i.e. for all n ≥ 2 the embedding α n n n BV (R ) ⊂ L n−α (R ) is continuous; • the natural analogue of the coarea formula does not hold in BV α(Rn), since there are u ∈ BV α( n) such that R |Dαχ |( n) dt = +∞; R R {u>t} R • any uniformly bounded sequence in BV α(Rn) admits limit points in L1(Rn) with 1 respect the Lloc-convergence. Concerning sets with finite fractional Caccioppoli α-perimeter, the main results we are able to prove are the following: 1 • fractional Caccioppoli α-perimeter is lower semicontinuous with respect to Lloc- convergence; • a fractional isoperimetric inequality holds, i.e. n−α α n |E| n ≤ cn,α|D χE|(R ); • any sequence of sets with uniformly bounded fractional Caccioppoli α-perimeter confined in a fixed ball admits limit points with respect to L1-convergence; • a natural analogue of De Giorgi’s reduced boundary, that we call fractional reduced boundary F αE, is well posed for any set E with finite fractional Caccioppoli α- perimeter; • if E has finite fractional Caccioppoli α-perimeter, then its fractional Caccioppoli α n−α α α-perimeter measure satisfies |D χE| H F E; • if E has finite fractional Caccioppoli α-perimeter and x ∈ F αE, then the family E−x 1 ( r )r>0 admits limit points in the Lloc-topology and any such limit point satisfies a rigidity condition. Some of the results listed above are proved similarly as in the classical framework. Since we believe that our approach might be interesting also to researchers that may be not familiar with the theory of functions of bounded variation, we tried to keep the exposition the most self-contained as possible. 1.4. Future developments. Thanks to this new approach, a large variety of classical results might be extended to the context of functions with fractional bounded variations. Here we just list some of the most intriguing open problems: • investigate the case of equality in (1.15); A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 7

• achieve a better characterisation of the blow-ups (possibly, their uniqueness); • prove a Structure Theorem for F αE in the spirit of De Giorgi’s Theorem; • study the fractional isoperimetric inequality and its stability, possibly also for its relative version; • develop a calibration theory for sets of finite fractional Caccioppoli α-perimeter as a useful tool for the study of fractional minimal surfaces; • consider the asymptotics as α → 1− and investigate the Γ-convergence to the classical perimeter; • extend the Gauss–Green and integration by parts formulas to sets of finite fractional Caccioppoli α-perimeter; • give a good definition of BV α functions on a general open set. Some of these open problems will be the subject of a forthcoming paper, see [4]. 1.5. Organisation of the paper. The paper is organised as follows. In Section 2, we introduce and study the fractional gradient and divergence, proving generalised Leibniz’s rules and representation formulas. In Section 3, we define the space BV α(Rn) and we study approximation by smooth functions, embeddings and compactness exploiting a fractional version of the Fundamental Theorem of Calculus. In Section 4, we define sets of (locally) finite fractional Caccioppoli α-perimeter, we prove some compactness results and we introduce the notion of fractional reduced boundary. Finally, in Section 5, we prove existence of blow-ups of sets with locally finite fractional Caccioppoli α-perimeter.

2. Šilhavý’s fractional calculus 2.1. General notation. We start with a brief description of the main notation used in this paper. Given an open set Ω, we say that a set E is compactly contained in Ω, and we write n α E b Ω, if the E is compact and contained in Ω. We denote by L and H the Lebesgue measure and the α-dimensional Hausdorff measure on Rn respectively, with α ≥ 0. Unless otherwise stated, a measurable set is a L n-measurable 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 Br(x) the standard open Euclidean ball with center n n+2 n 2 x ∈ R and radius r > 0. We let Br = Br(0). Recall that ωn := |B1| = π /Γ 2 and n−1 H (∂B1) = nωn, where Γ is Euler’s Gamma function, see [3]. k m For k ∈ N0 ∪ {+∞} and m ∈ N we denote by Cc (Ω; R ) and, respectively, by m k Lipc(Ω; R ), the space of C -regular, respectively, Lipschitz regular, m-vector valued functions defined on Rn with compact support in Ω. For any exponent p ∈ [1, +∞], we denote by p m n m o L (Ω; R ) := u: Ω → R : kukLp(Ω;Rm) < +∞ the space of m-vector valued Lebesgue p-integrable functions on Ω. We denote by 1,p m n p m o W (Ω; R ) := u ∈ L (Ω; R ) : [u]W 1,p(Ω;Rm) := k∇ukLp(Ω;Rn+m) < +∞ the space of m-vector valued Sobolev functions on Ω, see for instance [10, Chapter 10] for its precise definition and main properties. We also let 1,p m n 1 m o w (Ω; R ) := u ∈ Lloc(Ω; R ) : [u]W 1,p(Ω;Rm) < +∞ . A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 8

We denote by m n 1 m o BV (Ω; R ) := u ∈ L (Ω; R ) : [u]BV (Ω;Rm) := |Du|(Ω) < +∞ the space of m-vector valued functions of bounded variation on Ω, see for instance [2, Chapter 3] or [8, Chapter 5] for its precise deﬁnition and main properties. We also let

m n 1 m o bv(Ω; R ) := u ∈ Lloc(Ω; R ) : [u]BV (Ω;Rm) < +∞ . For α ∈ (0, 1) and p ∈ [1, +∞), we denote by  1  Z Z p ! p α,p m  p m |u(x) − u(y)|  W (Ω; R ) := u ∈ L (Ω; R ) : [u]W α,p(Ω;Rm) := dx dy < +∞  Ω Ω |x − y|n+pα  the space of m-vector valued fractional Sobolev functions on Ω, see [6] for its precise deﬁnition and main properties. We also let α,p n 1 m o w (Ω) := u ∈ Lloc(Ω; R ) : [u]W α,p(Ω;Rm) < +∞ . For α ∈ (0, 1) and p = +∞, we simply let ( |u(x) − u(y)| ) α,∞ m : ∞ m : W (Ω; R ) = u ∈ L (Ω; R ) sup α < +∞ , x,y∈Ω, x6=y |x − y|

α,∞ m 0,α m so that W (Ω; R ) = Cb (Ω; R ), the space of m-vector valued bounded α-Hölder continuous functions on Ω. 2.2. Deﬁnition of ∇α and divα. We now recall and study the non-local operators ∇α and divα introduced by Šilhavý in [20]. Let α ∈ (0, 1) and set Γ n+α+1 α − n 2 µ := 2 π 2 . (2.1) n,α 1−α Γ 2 We let Z α zf(x + z) ∇ f(x) := µn,α lim dz (2.2) ε→0 {|z|>ε} |z|n+α+1 ∞ n n be the α-gradient of f ∈ Cc (R ) at x ∈ R . We also let Z α z · ϕ(x + z) div ϕ(x) := µn,α lim dz (2.3) ε→0 {|z|>ε} |z|n+α+1 ∞ n n n α α be the α-divergence of ϕ ∈ Cc (R ; R ) at x ∈ R . The non-local operators ∇ and div are well deﬁned in the sense that the involved integrals converge and the limits exist, see [20, Section 7]. Since Z z dz = 0, ∀ε > 0, (2.4) {|z|>ε} |z|n+α+1 it is immediate to check that ∇αc = 0 for all c ∈ R. Moreover, the cancellation in (2.4) yields Z α (y − x) ∇ f(x) = µn,α lim f(y) dy (2.5a) ε→0 {|y−x|>ε} |y − x|n+α+1 A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 9 Z (y − x)(f(y) − f(x)) = µn,α lim dy (2.5b) ε→0 {|x−y|>ε} |y − x|n+α+1 Z (y − x)(f(y) − f(x)) n = µn,α n+α+1 dy, ∀x ∈ R , (2.5c) Rn |y − x| ∞ n for all f ∈ Cc (R ). Indeed, (2.5a) follows by a simple change of variables and (2.5b) is a consequence of (2.4). To prove (2.5c) it is enough to apply Lebesgue’s Dominated Convergence Theorem. Indeed, we can estimate

Z (y − x)(f(y) − f(x)) Z 1 −α dy ≤ Lip(f) r dr (2.6) |y−x|≤1 |y − x|n+α+1 0 and

Z (y − x)(f(y) − f(x)) Z +∞ −(1+α) dy ≤ 2kfkL∞(Rn) r dr. (2.7) |y−x|>1 |y − x|n+α+1 1 α n As a consequence, the operator ∇ f defined by (2.5c) is well defined for all f ∈ Lipc(R ) and satisfies (2.2), (2.5a) and (2.5b). By [20, Theorem 4.3], ∇α is invariant by translations and rotations and is α-homoge- ∞ n neous. Moreover, for all f ∈ Cc (R ) and λ ∈ R, we have α α α n (∇ f(λ·))(x) = |λ| sgn (λ)(∇ f)(λx), x ∈ R . (2.8) Arguing similarly as above, we can write Z α (y − x) · ϕ(y) div ϕ(x) = µn,α lim dy, (2.9a) ε→0 {|x−y|>ε} |y − x|n+α+1 Z (y − x) · (ϕ(y) − ϕ(x)) = µn,α lim dy, (2.9b) ε→0 {|x−y|>ε} |y − x|n+α+1 Z (y − x) · (ϕ(y) − ϕ(x)) n = µn,α n+α+1 dy, ∀x ∈ R , (2.9c) Rn |y − x| n for all ϕ ∈ Lipc(R ). Exploiting (2.5c) and (2.9c), we can extend the operators ∇α and divα to functions with wα,1-regularity.

α Lemma 2.1 (Extension of ∇α and div to wα,1). Let α ∈ (0, 1). If f ∈ wα,1(Rn) and α ϕ ∈ wα,1(Rn; Rn), then the functions ∇αf(x) and div f(x) given by (2.5c) and (2.9c) α respectively are well deﬁned for L n-a.e. x ∈ Rn. As a consequence, ∇αf(x) and div f(x) satisfy (2.2), (2.5a), (2.5b) and (2.3), (2.9a), (2.9b) respectively for L n-a.e. x ∈ Rn. Proof. Let f ∈ wα,1(Rn). Then

Z Z (y − x)(f(y) − f(x)) α,1 n n+α+1 dy dx ≤ [f]W (R ) Rn Rn |y − x| and thus the function ∇αf(x) given by (2.5c) is well deﬁned for L n-a.e. x ∈ Rn and satisﬁes (2.2), (2.5a) and (2.5b) by (2.4) and by Lebesgue’s Dominated Convergence α,1 n n Theorem. A similar argument proves the result for any ϕ ∈ w (R ; R ). A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 10

2.3. Equivalent deﬁnition of ∇α and divα via Riesz potential. We let Γ n−α Z f(y) I f(x) := 2 dy, x ∈ n, (2.10) α n α n−α R α 2 Rn |x − y| 2 π Γ 2 ∞ n m be the Riesz potential of order α ∈ (0, n) of a function f ∈ Cc (R ; R ). ∞ n m Let α ∈ (0, 1). Note that I1−αf ∈ C (R ; R ). Recalling (2.1), one easily sees that

µn,α Z f(x + y) I1−αf(x) = n+α−1 dy n + α − 1 Rn |y| and

µn,α Z ∇xf(x + y) µn,α Z ∇yf(x + y) ∇I1−αf(x) = n+α−1 dy = n+α−1 dy, n + α − 1 Rn |y| n + α − 1 Rn |y| so that

∇I1−αf = I1−α∇f ∞ n for all f ∈ Cc (R ). A similar argument proves that

divI1−αϕ = I1−αdivϕ ∞ n n for all ϕ ∈ Cc (R ; R ). Thus, accordingly to the approach developed in [9,15–19], we can consider the operators

α ∞ n ∞ n n ∇g := ∇I1−α : Cc (R ) → C (R ; R ) and α ∞ n n ∞ n divg := divI1−α : Cc (R ; R ) → C (R ). We can prove that these two operators coincide with the operators deﬁned in (2.2) and (2.3). See also [18, Theorem 1.2].

α α n Proposition 2.2 (Equivalence). Let α ∈ (0, 1). We have ∇g = ∇ on Lipc(R ) and α α n n divg = div on Lipc(R ; R ). n n Proof. Let f ∈ Lipc(R ) and ﬁx x ∈ R . Integrating by parts, we can compute

µn,α Z ∇yf(x + y) ∇gαf(x) = lim dy n + α − 1 ε→0 {|y|>ε} |y|n+α−1 Z yf(y + x) α = µn,α lim dy = ∇ f(x), ε→0 {|y|>ε} |y|n+α+1 since we can estimate

Z f(x + y) y Z (f(x + y) − f(x)) y n−1 n−1 dH (y) = dH (y) {|y|=ε} |y|n+α−1 |y| {|y|=ε} |y|n+α−1 |y| 1−α ≤ nωnk∇fkL∞(Rn)ε .

α α n n The proof of divg ϕ = div ϕ for all ϕ ∈ Lipc(R ; R ) follows similarly. A useful consequence of the equivalence proved in Proposition 2.2 above is the following result. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 11

Corollary 2.3 (Representation formula for divα and ∇α). Let α ∈ (0, 1). If ϕ ∈ n n α 1 n ∞ n Lipc(R ; R ) then div ϕ ∈ L (R ) ∩ L (R ) with Z α µn,α divϕ(y) div ϕ(x) = n+α−1 dy (2.11) n + α − 1 Rn |y − x| for all x ∈ Rn, α kdiv ϕkL1(Rn) ≤ µn,α[ϕ]W α,1(Rn;Rn) (2.12) and α kdiv ϕkL∞(Rn) ≤ Cn,α,U kdivϕkL∞(Rn) (2.13) for any bounded open set U ⊂ Rn such that supp(ϕ) ⊂ U, where n+α−1 ! nµn,α 1−α nωn n 1−α C := ω diam(U) + |U| n . (2.14) n,α,U (1 − α)(n + α − 1) n n + α − 1 n α 1 n n ∞ n n Analogously, if f ∈ Lipc(R ), then ∇ f ∈ L (R ; R ) ∩ L (R ; R ) with Z α µn,α ∇f(y) ∇ f(x) = n+α−1 dy (2.15) n + α − 1 Rn |y − x| for all x ∈ Rn, α k∇ fkL1(Rn;Rn) ≤ µn,α[f]W α,1(Rn) (2.16) and α k∇ fkL∞(Rn;Rn) ≤ Cn,α,U k∇fkL∞(Rn;Rn) (2.17) n for any bounded open set U ⊂ R such that supp(f) ⊂ U, where Cn,α,U is as in (2.14). Proof. The representation formula (2.11) follows directly from Proposition 2.2. The estimate in (2.12) is a consequence of Lemma 2.1. Finally, if U ⊂ Rn is a bounded open set such that supp(ϕ) ⊂ U, then µ Z |divαϕ(x)| ≤ n,α |y − x|1−n−α |divϕ(y)| dy n + α − 1 Rn µ kdivϕk ∞ n Z ≤ n,α L (R ) |y − x|1−n−α dy n + α − 1 U and (2.13) follows by Lemma 2.4 below. The proof of (2.15), (2.16) and (2.17) is similar and is left to the reader. Lemma 2.4. Let α ∈ (0, 1) and let U ⊂ Rn be a bounded open set. For all x ∈ Rn, we have Z n+α−1 ! 1−n−α n 1−α nωn n 1−α |y − x| dy ≤ ωn diam(U) + |U| n . (2.18) U 1 − α n + α − 1

Proof. For δ > 0, set U δ := {x ∈ Rn : dist(x, U) < δ}. Since clearly δ x ∈ U =⇒ B(diam (U)+δ)(x) ⊃ U, for all x ∈ U δ we can estimate Z Z |y − x|1−n−α dy ≤ |y − x|1−n−α dy U B(diam(U)+δ)(x) Z diam(U)+δ −α = nωn r dr 0 A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 12 nω = n (diam (U) + δ)1−α . 1 − α On the other hand, it is plain that x∈ / U δ, y ∈ U =⇒ |y − x| > δ, so that for all x∈ / U δ we can estimate Z |y − x|1−n−α dy ≤ δ1−n−α|U|. U Thus, for all δ > 0 and x ∈ Rn, we can estimate Z nω |y − x|1−n−α dy ≤ n (diam (U) + δ)1−α + δ1−n−α|U| U 1 − α nω ≤ n diam (U)1−α + δ1−α + δ1−n−α|U| 1 − α since the function s 7→ s1−α is subadditive for all s > 0. Thus (2.18) follows minimising in δ > 0 the right-hand side. 2.4. Duality and Leibniz’s rules. We now study the properties of the operators ∇α and divα. We begin with the following duality relation, see [20, Section 6].

n n n Lemma 2.5 (Duality). Let α ∈ (0, 1). For all f ∈ Lipc(R ) and ϕ ∈ Lipc(R ; R ) it holds Z Z f divαϕ dx = − ϕ · ∇αf dx. (2.19) Rn Rn Proof. Recalling Lemma 2.1 and exploiting (2.5a) and (2.9a), we can write Z Z Z α (y − x) · ϕ(y) f div ϕ dx = µn,α f(x) lim n+α+1 dy dx Rn Rn ε→0 {|x−y|>ε} |y − x| Z Z (y − x) · ϕ(y) = µn,α lim f(x) n+α+1 dy dx ε→0 Rn {|x−y|>ε} |y − x| Z Z (x − y) f(x) = −µn,α lim ϕ(y) · n+α+1 dx dy ε→0 Rn {|x−y|>ε} |x − y| Z = − ϕ(y) · ∇αf(y) dy Rn by the Lebesgue’s Dominated Convergence Theorem and Fubini’s Theorem. We now prove two Leibniz-type rules for the operators ∇α and divα, which in particular show the strong non-local nature of these two operators.

α n Lemma 2.6 (Leibniz’s rule for ∇ ). Let α ∈ (0, 1). For all f, g ∈ Lipc(R ) it holds α α α α ∇ (fg) = f∇ g + g∇ f + ∇NL(f, g), where Z (y − x)(f(y) − f(x))(g(y) − g(x)) α : n ∇NL(f, g)(x) = µn,α n+α+1 dy, ∀x ∈ R , Rn |y − x| with µn,α as in (2.1). Moreover, it holds α 1 n n α α k∇NL(f, g)kL ( ; ) ≤ µn,α[f] ,p n [g] ,q n R R W p (R ) W q (R ) A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 13

1 1 with p, q ∈ (1, ∞) such that p + q = 1 and similarly α k∇NL(f, g)kL1(Rn;Rn) ≤ 2µn,αkfkL∞(Rn)[g]W α,1(Rn). n Proof. Given f, g ∈ Lipc(R ), by Lemma 2.1 and by (2.5c) we have Z α (y − x)(f(y)g(y) − f(x)g(x)) ∇ (fg)(x) = µn,α n+α+1 dy Rn |y − x| Z (y − x)(f(y)g(y) − f(y)g(x) + f(y)g(x) − f(x)g(x)) = µn,α n+α+1 dy Rn |y − x| Z (y − x)f(y)(g(y) − g(x)) α = µn,α n+α+1 dy + g(x)∇ f(x) Rn |y − x| Z (y − x)(f(y) − f(x))(g(y) − g(x)) α α = µn,α n+α+1 dy + f(x)∇ g(x) + g(x)∇ f(x). Rn |y − x| We also have that Z Z α |f(y) − f(x)| |g(y) − g(x)| k∇NL(f, g)kL1(Rn;Rn) ≤ µn,α n+α n+α dy dx, Rn Rn |x − y| p |y − x| q ! 1 ! 1 Z Z |f(y) − f(x)|p p Z Z |g(y) − g(x)|q q ≤ µn,α n+α dy dx n+α dy dx Rn Rn |x − y| Rn Rn |x − y| 1 1 for any p, q ∈ (1, ∞) such that p + q = 1. The case p = ∞, q = 1 follows similarly. α n Lemma 2.7 (Leibniz’s rule for div ). Let α ∈ (0, 1). For all f ∈ Lipc(R ) and ϕ ∈ n n Lipc(R ; R ) it holds α α α α div (fϕ) = fdiv ϕ + ϕ · ∇ f + divNL(f, ϕ), where Z (y − x) · (ϕ(y) − ϕ(x))(f(y) − f(x)) α : n divNL(f, ϕ)(x) = µn,α n+α+1 dy, ∀x ∈ R , (2.20) Rn |y − x| with µn,α as in (2.1). Moreover, it holds α 1 n α α kdivNL(f, ϕ)kL ( ) ≤ µn,α[f] ,p n [ϕ] ,q n n R W p (R ) W q (R ;R ) 1 1 with p, q ∈ (1, ∞) such that p + q = 1 and similarly α kdivNL(f, ϕ)kL1(Rn;Rn) ≤ 2µn,αkfkL∞(Rn)[ϕ]W α,1(Rn;Rn), α kdivNL(f, ϕ)kL1(Rn;Rn) ≤ 2µn,αkϕkL∞(Rn;Rn)[f]W α,1(Rn). n n n Proof. Given f ∈ Lipc(R ) and ϕ ∈ Lipc(R ; R ), by Lemma 2.1 and by (2.5c) we have Z α (y − x) · (f(y)ϕ(y) − f(x)ϕ(x)) div (fϕ)(x) = µn,α n+α+1 dy Rn |y − x| Z (y − x) · (f(y)ϕ(y) − f(y)ϕ(x) + f(y)ϕ(x) − f(x)ϕ(x)) = µn,α n+α+1 dy Rn |y − x| Z (y − x) · (ϕ(y) − ϕ(x))f(y) α = µn,α n+α+1 dy + ϕ(x) · ∇ f(x) Rn |y − x| A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 14

α α Remark 2.8 (Extension of ∇NL and divNL to fractional Sobolev spaces). Thanks to the estimates in Lemma 2.6, for all α ∈ (0, 1) the bilinear operator

α n n 1 n n ∇NL : Lipc(R ) × Lipc(R ) → L (R ; R ) can be continuously extended to a bilinear operator

α α α p ,p n q ,q n 1 n n ∇NL : w (R ) × w (R ) → L (R ; R ) 1 1 for any p, q ∈ [1, ∞] such that p + q = 1, for which we retain the same notation (we α ,∞ ∞ tacitly adopt the the convention w ∞ = L ). Analogously, because of the estimates in Lemma 2.7, the bilinear operator

α n n n 1 n divNL : Lipc(R ) × Lipc(R ; R ) → L (R ) can be continuously extended to a bilinear operator

α α α p ,p n q ,q n n 1 n divNL : w (R ) × w (R ; R ) → L (R ) 1 1 for any p, q ∈ [1, ∞] such that p + q = 1, for which we retain the same notation.

3. Fractional BV functions In this section we introduce and study the fractional BV space naturally induced by the operators ∇α and divα deﬁned in Section 2 following De Giorgi’s distributional approach. In the presentation of the results, we will frequently refer to [8, Chapter 5].

3.1. Definition of BV α(Rn) and Structure Theorem. In analogy with the classical case (see [8, Definition 5.1] for instance), we start with the following definition.

Deﬁnition 3.1 (BV α(Rn) space). Let α ∈ (0, 1). A function f ∈ L1(Rn) belongs to the space BV α(Rn) if Z α ∞ n n sup f div ϕ dx : ϕ ∈ Cc (R ; R ), kϕkL∞(Rn;Rn) ≤ 1 < +∞. Rn We can now state the following fundamental result relating non-local distributional gradients of BV α functions to vector valued Radon measures. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 15

Theorem 3.2 (Structure Theorem for BV α functions). Let α ∈ (0, 1) and f ∈ L1(Rn). Then, f ∈ BV α(Rn) if and only if there exists a finite vector valued Radon measure Dαf ∈ M (Rn; Rn) such that Z Z f divαϕ dx = − ϕ · dDαf (3.1) Rn Rn ∞ n n n for all ϕ ∈ Cc (R ; R ). In addition, for any open set U ⊂ R it holds Z α α ∞ n |D f|(U) = sup f div ϕ dx : ϕ ∈ Cc (U; R ), kϕkL∞(U;Rn) ≤ 1 . (3.2) Rn Proof. If f ∈ L1(Rn) and if there exists a finite vector valued Radon measure Dαf ∈ M (Rn; Rn) such that (3.1) holds, then f ∈ BV α(Rn) by Definition 3.1. If f ∈ BV α(Rn), then the proof is identical to the one of [8, Theorem 5.1], with minor ∞ n n modifications. Define the linear functional L: Cc (R ; R ) → R setting Z α ∞ n n L(ϕ) := − f div ϕ dx ∀ϕ ∈ Cc (R ; R ). Rn Note that L is well defined thanks to Corollary 2.3. Since f ∈ BV α(Rn), we have n ∞ n o C(U) := sup L(ϕ) : ϕ ∈ Cc (U; R ), kϕkL∞(U;Rn) ≤ 1 < +∞ for each open set U ⊂ Rn, so that ∞ n |L(ϕ)| ≤ C(U)kϕkL∞(U;Rn) ∀ϕ ∈ Cc (U; R ). ∞ n n n n Thus, by the density of Cc (R ; R ) in Cc(R ; R ), the functional L can be uniquely ˜ n n extended to a continuous linear functional L: Cc(R ; R ) → R and the conclusion follows by Riesz’s Representation Theorem. 3.2. Lower semicontinuity of fractional variation. Similarly to the classical case, the fractional variation measure given by Theorem 3.2 in (3.2) is lower semicontinuous with respect to L1-convergence. Proposition 3.3 (Lower semicontinuity of fractional variation measure). Let α ∈ (0, 1). α n 1 n α n If fk ∈ BV (R ) for all k ∈ N and fk → f in L (R ) as k → +∞, then f ∈ BV (R ) with α n α n |D f|( ) ≤ lim inf |D fk|( ). R k→+∞ R ∞ n n α ∞ n n Proof. Let ϕ ∈ Cc (R ; R ) with kϕkL∞(Rn;Rn) ≤ 1. Then div ϕ ∈ L (R ; R ) by Corol- lary 2.3 and so we can estimate Z Z Z α α α α n f div ϕ dx = lim fk div ϕ dx = − lim ϕ dD fk ≤ lim inf |D fk|(R ). Rn k→+∞ Rn k→+∞ Rn k→+∞ The conclusion thus follows by Theorem 3.2. From Proposition 3.3 we immediately deduce the following result, whose standard proof is left to the reader. Corollary 3.4 (BV α is a Banach space). Let α ∈ (0, 1). The linear space BV α(Rn) equipped with the norm α n α n kfkBV α(Rn) := kfkL1(Rn) + |D f|(R ), f ∈ BV (R ), where Dαf is given by Theorem 3.2, is a Banach space. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 16

3.3. Approximation by smooth functions. Here and in the following, we let % ∈ ∞ n Cc (R ) be a function such that Z supp % ⊂ B1,% ≥ 0, %(x) dx = 1, (3.3) Rn ∞ n see [8, Section 4.2.1] for an example. We thus let (%ε)ε>0 ⊂ Cc (R ) be deﬁned as 1 x % (x) := % ∀x ∈ n. (3.4) ε εn ε R

We call (%ε)ε>0 be a family of standard molliﬁers. We have the following result.

Lemma 3.5 (Convolution with standard molliﬁers). Let α ∈ (0, 1) and let (%ε)ε>0 as n n in (3.4). If ϕ ∈ Lipc(R ; R ), then α α div (%ε ∗ ϕ) = %ε ∗ div ϕ (3.5) for any ε > 0. Thus, if f ∈ BV α(Rn), then α α n D (%ε ∗ f) = (%ε ∗ D f)L (3.6) for any ε > 0, and α α D (%ε ∗ f) *D f (3.7) in M (Rn; Rn) as ε → 0. n n n Proof. Let ϕ ∈ Lipc(R ; R ) and x ∈ R . Recalling (2.11), we can write α div ϕ = Kn,α ∗ divϕ, where µ K (x) = n,α |x|1−n−α, x ∈ n \{0}. n,α n + α − 1 R n n Since %ε ∗ ϕ ∈ Lipc(R ; R ), we can compute α div (%ε ∗ ϕ) = Kn,α ∗ div(%ε ∗ ϕ)

= Kn,α ∗ (%ε ∗ divϕ)

= %ε ∗ (Kn,α ∗ divϕ) α = %ε ∗ div ϕ α n ∞ n n and (3.5) follows. Now let f ∈ BV (R ) and ϕ ∈ Cc (R ; R ). By (3.1) and (3.5), for all ε > 0 we can compute Z Z α α − (%ε ∗ f) div ϕ dx = − f (%ε ∗ div ϕ) dx Rn Rn Z α = − f div (%ε ∗ ϕ) dx Rn Z α = (%ε ∗ ϕ) dD f Rn Z α = ϕ · (%ε ∗ D f) dx, Rn proving (3.6). The convergence in (3.7) thus follows from standard properties of the molliﬁcation of Radon measures, see [2, Theorem 2.2] for instance. As an immediate application of Lemma 3.5, we can prove that a function in BV α(Rn) can be tested against the fractional divergence of any Lipc-regular vector ﬁeld. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 17

α n Proposition 3.6 (Lipc-regular test). Let α ∈ (0, 1). If f ∈ BV (R ), then (3.1) holds n for all ϕ ∈ Lipc(R ). n n ∞ n Proof. Fix ϕ ∈ Lipc(R ; R ) and let (%ε)ε>0 ⊂ Cc (R ) be as in (3.4). Then %ε ∗ ϕ ∈ ∞ n n Cc (R ; R ) and so, by Lemma 3.5 and (3.1), we have Z Z Z α α α (%ε ∗ f) div ϕ dx = f div (%ε ∗ ϕ) dx = − (%ε ∗ ϕ) · dD f. (3.8) Rn Rn Rn 1 n α ∞ n Since %ε ∗ ϕ → ϕ uniformly and %ε ∗ f → f in L (R ) as ε → 0, and div ϕ ∈ L (R ) by Corollary 2.3, we can pass to the limit as ε → 0 in (3.8) getting Z Z f divαϕ dx = − ϕ · dDαf. Rn Rn n n for any ϕ ∈ Lipc(R ; R ). As in the classical case, we can prove the density of C∞(Rn) ∩ BV α(Rn) in BV α(Rn). Theorem 3.7 (Approximation by C∞ ∩BV α functions). Let α ∈ (0, 1). If f ∈ BV α(Rn), α n ∞ n then there exists (fk)k∈N ⊂ BV (R ) ∩ C (R ) such that 1 n (i) fk → f in L (R ); α n α n (ii) |D fk|(R ) → |D f|(R ). ∞ n α n Proof. Let (%ε)ε>0 ⊂ Cc (R ) be as in (3.4). Fix f ∈ BV (R ) and consider fε := f ∗ %ε 1 n for all ε > 0. Since fε → f in L (R ), by Proposition 3.3 we get that α n α n |D f|( ) ≤ lim inf |D fε|( ). R ε→0 R By Lemma 3.5 we also have that Z α n α α n |D fε|(R ) = |%ε ∗ D f| dx ≤ |D f|(R ) Rn and the proof is complete. ∞ n Let (ηR)R>0 ⊂ Cc (R ) be such that

0 ≤ ηR ≤ 1, ηR = 1 on BR, supp(ηR) ⊂ BR+1, Lip(ηR) ≤ 2. (3.9) ∞ n We call ηR a cut-oﬀ function. As in the classical case, we can prove the density of Cc (R ) in BV α(Rn). ∞ α n Theorem 3.8 (Approximation by Cc functions). Let α ∈ (0, 1). If f ∈ BV (R ), then ∞ n there exists (fk)k∈N ⊂ Cc (R ) such that 1 n (i) fk → f in L (R ); α n α n (ii) |D fk|(R ) → |D f|(R ). ∞ n Proof. Let (ηR)R>0 ⊂ Cc (R ) be as in (3.9). Thanks to Theorem 3.7, it is enough to α n ∞ n α n prove that fηR → f in BV (R ) as R → +∞ for all f ∈ C (R ) ∩ BV (R ). Clearly, 1 n fηR → f in L (R ) as R → +∞. Thus, by Proposition 3.3, we just need to prove that α n α n lim sup |D (fηR)|(R ) ≤ |D f|(R ). (3.10) R→+∞ ∞ n n Fix ϕ ∈ Cc (R ; R ). Then, by Lemma 2.7, we get Z Z Z Z α α α α fηR div ϕ dx = f div (ηRϕ) dx − f ϕ · ∇ ηR dx − f divNL(ηR, ϕ) dx. Rn Rn Rn Rn A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 18

Z Z |ηR(y) − ηR(x)| lim |f(x)| n+α dy dx = 0 R→+∞ Rn Rn |y − x| combining (2.6), (2.7) and (3.9) with Lebesgue’s Dominated Convergence Theorem. 3.4. Gagliardo–Nirenberg–Sobolev inequality. Thanks to Theorem 3.8 we can prove the analogous of the Gagliardo–Nirenberg–Sobolev inequality for the space BV α(Rn). Theorem 3.9 (Gagliardo–Nirenberg–Sobolev inequality). Let α ∈ (0, 1) and n ≥ 2. There exists a constant cn,α > 0 such that α n kfk n ≤ cn,α|D f|( ) (3.11) L n−α (Rn) R for any f ∈ BV α(Rn). As a consequence, BV α(Rn) is continuously embedded in Lq(Rn) n for any q ∈ [1, n−α ]. ∞ n Proof. By [15, Theorem A’], we know that (3.11) holds for any f ∈ Cc (R ). So let α n ∞ n f ∈ BV (R ) and let (fk)k∈N ⊂ Cc (R ) be as in Theorem 3.8. By Fatou’s Lemma and Proposition 3.3, we thus obtain α n α n kfk n ≤ lim inf kfkk n ≤ cn,α lim |D fk|( ) = cn,α|D f|( ) L n−α (Rn) k→+∞ L n−α (Rn) k→+∞ R R and the proof is complete. Remark 3.10. We stress the fact that Theorem 3.9 does not hold for n = 1, as will be shown in Remark 3.27 below. It is worth to notice that an analogous restriction holds for [15, Theorem A], for which the authors provide a counterexample in the case n = 1 (see [15, Counterexample 3.2]). The authors then derive [15, Theorem A’] as a consequence of [15, Theorem A], without proving the necessity of the restriction to n ≥ 2 in this second case, as we do in Remark 3.27. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 19

3.5. Coarea inequality. In analogy with the classical case, we can prove a coarea inequality formula for functions in BV α(Rn). Theorem 3.11 (Coarea inequality). Let α ∈ (0, 1). If f ∈ BV α(Rn) is such that Z α n |D χ{f>t}|(R ) dt < +∞, (3.12) R then Z α α D f = D χ{f>t} dt (3.13) R and Z α α |D f| ≤ |D χ{f>t}| dt. (3.14) R ∞ n n Proof. Let ϕ ∈ Cc (R ; R ). By (3.12) and applying Fubini’s Theorem twice, we can compute Z Z ϕ · dDαf = − f divαϕ(x) dx Rn Rn Z Z α = − div ϕ(x) χ(−∞,f(x))(t) − χ(−∞,0)(t) dt dx Rn R Z Z α = − div ϕ(x) χ{f>t}(x) − χ(−∞,0)(t) dx dt R Rn Z Z α = ϕ · dD χ{f>t} dt R Rn Z Z α = ϕ · d D χ{f>t} dt Rn R proving (3.13). Thus Z Z α α α |D f| = D χ{f>t} dt ≤ |D χ{f>t}| dt R R and the proof is complete. 3.6. A fractional version of the Fundamental Theorem of Calculus. Let α ∈ (0, 1) and let µn,−α be given by (2.1) (note that the expression in (2.1) makes sense for all α ∈ (−1, 1)). We let

n n ∞ n a 1 n n no T (R ) := f ∈ C (R ) : D f ∈ L (R ) ∩ C0(R ) for all multi-index a ∈ N0 (3.15) and n n ∞ n n n T (R ; R ) := {ϕ ∈ C (R ; R ) : ϕi ∈ T (R ), i = 1, . . . , n}. By [20, Section 5], the operator Z z · ϕ(x + z) −α : div ϕ(x) = µn,−α n+1−α dz (3.16) Rn |z| is well deﬁned for any ϕ ∈ T (Rn; Rn). Moreover, by [20, Theorem 5.3], we have the following inversion formula −α α − div ∇ = idT (Rn). (3.17) Exploiting (3.16) and (3.17) we can prove the following fractional version of the Funda- mental Theorem of Calculus. See [18, Theorem 2.1] for a similar approach. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 20

Theorem 3.12 (Fractional Fundamental Theorem of Calculus). Let α ∈ (0, 1). If f ∈ ∞ n Cc (R ), then Z ! z − x z − y α f(y) − f(x) = µn,−α n+1−α − n+1−α · ∇ f(z) dz (3.18) Rn |z − x| |z − y| for any x, y ∈ Rn. ∞ n n α n n Proof. Since clearly Cc (R ) ⊂ T (R ), we have ∇ f ∈ T (R ; R ) by [20, Theorem 4.3]. Applying (3.17), we have f(y) − f(x) = (−div−α∇αf)(y) − (−div−α∇αf)(x) Z z α α = µn,−α n+1−α · ∇ f(x + z) − ∇ f(y + z) dz Rn |z| n for all x, y ∈ R . Then (3.18) follows splitting the integral and changing variables. An easy consequence of Theorem 3.12 is that the distributional α-divergence of the kernel appearing in (3.18) is a diﬀerence of Dirac deltas.

Proposition 3.13. Let α ∈ (0, 1). If x, y ∈ Rn, then · − y · − x ! µ divα − = δ − δ (3.19) n,−α | · −y|n+1−α | · −x|n+1−α y x in the sense of Radon measures.

Proof. It follows immediately from (3.18). 3.7. Compactness. We start with the following Hölder estimate on the L1-norm of trans- ∞ n lations of functions in Cc (R ). 1 ∞ n Proposition 3.14 (L -estimate on translations). Let α ∈ (0, 1). If f ∈ Cc (R ), then Z α α |f(x + y) − f(x)| dx ≤ γn,α |y| k∇ fkL1(Rn;Rn) (3.20) Rn for all y ∈ Rn, where

Z z z − e : 1 γn,α = µn,−α n+1−α − n+1−α dz. (3.21) Rn |z| |z − e1| Proof. By (3.18), we have

Z Z Z z z − y α |f(x + y) − f(x)| dx ≤ µn,−α n+1−α − n+1−α |∇ f(x + z)| dz dx Rn Rn Rn |z| |z − y| Z α z z − y 1 n n = µn,−αk∇ fkL (R ;R ) n+1−α − n+1−α dz. Rn |z| |z − y| Now we notice that the integral appearing in the last term is actually a radial function of y. Indeed, let R ∈ SO(n) be such that Ry = |y|ν, for some ν ∈ Sn−1. Making the change of variable z = |y| tRw, we obtain t t Z z z − y Z Rw R(w − ν) α n+1−α − n+1−α dz = |y| n+1−α − n+1−α dw Rn |z| |z − y| Rn |w| |w − ν| A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 21

Z w (w − ν) α = |y| n+1−α − n+1−α dw. Rn |w| |w − ν| Since ν is arbitrary, we may choose ν = e1, and the proof is complete. Similarly to the classical case, as a consequence of the previous result we can prove the following key estimate of the L1-distance of a function in BV α(Rn) and its convolution with a molliﬁer.

Corollary 3.15 (L1-distance with convolution). Let α ∈ (0, 1). If f ∈ BV α(Rn), then α α n k%ε ∗ f − fkL1(Rn) ≤ γn,α ε |D f|(R ) (3.22) ∞ n for all ε > 0, where (%ε)ε>0 ⊂ Cc (R ) is as in (3.4) and γn,α as in Proposition 3.14. ∞ n Proof. By Theorem 3.8, it is enough to prove (3.22) for f ∈ Cc (R ). By (3.20), we get Z Z k%ε ∗ f − fkL1(Rn) ≤ %(y)|f(x − εy) − f(x)| dy dx Rn Rn Z Z = %(y) |f(x − εy) − f(x)| dx dy Rn Rn Z α α α ≤ γn,α ε k∇ fkL1(Rn;Rn) %(y)|y| dy B1 α α ≤ γn,α ε k∇ fkL1(Rn;Rn) and the proof is complete. We are now ready to prove following compactness result for the space BV α(Rn). α n α n Theorem 3.16 (Compactness for BV (R )). Let α ∈ (0, 1). If (fk)k∈N ⊂ BV (R ) satisﬁes

sup kfkkBV α(Rn) < +∞, k∈N α n 1 n then there exists a subsequence (fkj )j∈N ⊂ BV (R ) and a function f ∈ L (R ) such that 1 n fkj → f in Lloc(R ) as j → +∞.

∞ n Proof. We follow the line of the proof of [2, Theorem 3.23]. Let (%ε)ε>0 ⊂ Cc (R ) be as ∞ n in (3.4) and set fk,ε := %ε ∗ fk. Clearly fk,ε ∈ C (R ) and

kfk,εkL∞(U) ≤ k%εkL∞(Rn)kfkkL1(Rn), k∇fk,εkL∞(U;Rn) ≤ k∇%εkL∞(Rn;Rn)kfkkL1(Rn) n for any open set U b R . Thus (fk,ε)k∈N is locally equibounded and locally equicontinuous for each ε > 0 ﬁxed. By a diagonal argument, we can ﬁnd a sequence (kj)j∈N such that n (fkj ,ε)j∈N converges in C(U) for any open set U b R with ε = 1/p for all p ∈ N. By Corollary 3.15, we thus get Z Z

lim sup |fkh − fkj | dx = lim sup |fkh,1/p − fkj ,1/p| dx h,j→+∞ U h,j→+∞ U Z

+ lim sup |fkh − fkh,1/p| + |fkj − fkj ,1/p| dx h,j→+∞ U

2γn,α α n ≤ α sup |D fk|(R ). p k∈N A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 22

n 1 for all open set U b R . Since p ∈ N is arbitrary and L (U) is a Banach space, this 1 n shows that (fkj )j∈N converges in L (U) for all open set U b R . Up to extract a further subsequence (which we do not relabel for simplicity), we also have that fkj (x) → f(x) for L n-a.e. x ∈ Rn. By Fatou’s Lemma, we can thus infer that

kfkL1( n) ≤ lim inf kfk kL1( n) ≤ sup kfkkBV α( n). R j→+∞ j R R k∈N 1 n Hence f ∈ L (R ) and the proof is complete. Remark 3.17 (Improvement of [18, Theorem 2.1]). The argument presented above can n be used to extend the validity of [18, Theorem 2.1] to all exponents p ∈ [1, α ), since our strategy does not rely on the boundedness of Riesz’s transform but only on the inversion formula (3.17). We leave the details of the proof of this improvement of [18, Theorem 2.1] to the interested reader.

3.8. The inclusion W α,1(Rn) ⊂ BV α(Rn). As in the classical case, fractional BV functions naturally include fractional Sobolev functions.

Theorem 3.18 (W α,1(Rn) ⊂ BV α(Rn)). Let α ∈ (0, 1). If f ∈ W α,1(Rn) then f ∈ BV α(Rn), with α n |D f|(R ) ≤ µn,α[f]W α,1(Rn) (3.23) and Z Z f divαϕ dx = − ϕ · ∇αf dx (3.24) Rn Rn n n α α n for all ϕ ∈ Lipc(R ; R ), so that D f = ∇ f L . Moreover, if f ∈ BV (Rn), then f ∈ W α,1(Rn) for any α ∈ (0, 1), with

kfkW α,1(Rn) ≤ cn,αkfkBV (Rn) (3.25) for some cn,α > 0 and Z α µn,α dDf(y) ∇ f(x) = n+α−1 (3.26) n + α − 1 Rn |y − x| for L n-a.e. x ∈ Rn. α,1 n n n Proof. Let f ∈ W (R ). For any ϕ ∈ Lipc(R ; R ), by Lebesgue’s Dominated Conver- gence Theorem, Fubini’s Theorem and Lemma 2.1, and recalling (2.4), we can compute Z Z Z α (y − x) · ϕ(y) fdiv ϕ dx = µn,α lim f(x) n+α+1 dy dx Rn ε→0 Rn {|x−y|>ε} |y − x| Z Z (y − x)f(x) = µn,α lim ϕ(y) · n+α+1 dx dy ε→0 Rn {|x−y|>ε} |y − x| Z Z (y − x)(f(x) − f(y)) = µn,α lim ϕ(y) · n+α+1 dx dy ε→0 Rn {|x−y|>ε} |y − x| Z = − ϕ(y) · ∇αf(y) dy. Rn This proves (3.24), so that f ∈ BV α(Rn). Inequality (3.23) follows as in Lemma 2.1. n α,1 n ∞ n Now let f ∈ BV (R ). We claim that f ∈ W (R ). Indeed, take (fk)k∈N ⊂ C (R ) ∩ n 1 n n BV (R ) such that fk → f in L (R ) and k∇fkkL1(Rn) → |Df|(R ) as k → +∞ (for A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 23 instance, see [8, Theorem 5.3]). Since W α,1(Rn) ⊂ W 1,1(Rn) (the proof of this inclusion is similar to the one of [6, Proposition 2.2] for example), by Fatou’s Lemma we get that

kfkW α,1( n) ≤ lim inf kfkkW α,1( n) R k→+∞ R

≤ cn,α lim inf kfkkW 1,1( n) k→+∞ R n = cn,α lim (kfkkL1( n) + |Dfk|( )) k→+∞ R R

= cn,αkfkBV (Rn).

n 1 n Since |Df|(R ) < +∞, by Lemma 2.4 the function in (3.26) is well deﬁned in Lloc(R ). ∞ n n Fix ϕ ∈ Cc (R ; R ). By Corollary 2.3, we can write Z Z Z α µn,α divϕ(y) f(x) div ϕ(x) dx = f(x) n+α−1 dy dx. Rn n + α − 1 Rn Rn |y − x| Recalling Lemma 2.4, applying Fubini’s Theorem twice and integrating by parts, we obtain

Z Z divϕ(y) Z Z divyϕ(x + y) f(x) n+α−1 dy dx = f(x) n+α−1 dy dx Rn Rn |y − x| Rn Rn |y|

Z Z divxϕ(x + y) = f(x) n+α−1 dy dx Rn Rn |y| Z Z = |y|1−n−α f(x) divϕ(x + y) dx dy Rn Rn Z Z = − |y|1−n−α ϕ(y + x) · dDf(x) dy Rn Rn Z Z ϕ(y) = − n+α−1 dy · dDf(x) Rn Rn |y − x| Z Z dDf(x) = − ϕ(y) · n+α−1 dy. Rn Rn |x − y| Thus we conclude that Z Z Z α µn,α dDf(x) f(x) div ϕ(x) dx = − ϕ(y) · n+α−1 dy. Rn n + α − 1 Rn Rn |x − y| Recalling (3.24), this proves (3.26) and the proof is complete.

3.9. The space Sα,p(Rn) and the inclusion Sα,1(Rn) ⊂ BV α(Rn). It is now tempting to approach fractional Sobolev spaces from a distributional point of view. Recalling Corollary 2.3, we can give the following deﬁnition.

Deﬁnition 3.19 (Weak α-gradient). Let α ∈ (0, 1), p ∈ [1, +∞], f ∈ Lp(Rn). We say 1 n n α that g ∈ Lloc(R ; R ) is a weak α-gradient of f, and we write g = ∇wf, if Z Z f divαϕ dx = − g · ϕ dx Rn Rn ∞ n n for all ϕ ∈ Cc (R ; R ). A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 24

For α ∈ (0, 1) and p ∈ [1, +∞], we can thus introduce the distributional fractional α,p n Sobolev space (S (R ), k · kSα,p(Rn)) letting α,p n p n α p n n S (R ) := {f ∈ L (R ) : ∃ ∇wf ∈ L (R ; R )} (3.27) and α α,p n kfkSα,p(Rn) := kfkLp(Rn) + k∇wfkLp(Rn;Rn), ∀f ∈ S (R ). (3.28) We omit the standard proof of the following result. Proposition 3.20 (Sα,p is a Banach space). Let α ∈ (0, 1) and p ∈ [1, +∞]. The space α,p n (S (R ), k · kSα,p(Rn)) is a Banach space. We leave the proof of the following interpolation result to the reader.

Lemma 3.21 (Interpolation). Let α ∈ (0, 1) and p1, p2 ∈ [1, +∞], with p1 ≤ p2. Then α,p n α,p n α,q n S 1 (R ) ∩ S 2 (R ) ⊂ S (R ) with continuous embedding for all q ∈ [p1, p2]. Taking advantage of the techniques developed in the study of the space BV α(Rn) above, we are able to prove the following approximation result. Theorem 3.22 (Approximation by C∞ ∩Sα,p functions). Let α ∈ (0, 1) and p ∈ [1, +∞). The set C∞(Rn) ∩ Sα,p(Rn) is dense in Sα,p(Rn). ∞ n α,p n Proof. Let (%ε)ε>0 ⊂ Cc (R ) be as in (3.4). Fix f ∈ S (R ) and consider fε := f ∗ %ε ∞ n α,p n for all ε > 0. By Lemma 3.5, it is easy to check that fε ∈ C (R ) ∩ S (R ) with α α ∇wfε = %ε ∗ ∇wf for all ε > 0, so that the conclusion follows by standard properties of the convolution. ∞ n Given α ∈ (0, 1) and p ∈ [1, +∞], it is easy to see that, if f ∈ Cc (R ), then, by α,p n α α Lemma 2.5, f ∈ S (R ) with ∇wf = ∇ f. In the case p = 1, we can prove that ∞ n α,1 n Cc (R ) is also a dense subset of S (R ). ∞ ∞ n Theorem 3.23 (Approximation by Cc functions). Let α ∈ (0, 1). The set Cc (R ) is dense in Sα,1(Rn). ∞ n Proof. Let (ηR)R>0 ⊂ Cc (R ) be as in (3.9). Thanks to Theorem 3.22, it is enough to α,1 n ∞ n α,1 n prove that fηR → f in S (R ) as R → +∞ for all f ∈ C (R ) ∩ S (R ). Clearly, 1 n fηR → f in L (R ) as R → +∞. We now argue as in the proof of Theorem 3.8. Fix ∞ n n ϕ ∈ Cc (R ; R ). Then, by Lemma 2.7, we get Z Z Z Z α α α α fηR div ϕ dx = f div (ηRϕ) dx − f ϕ · ∇ ηR dx − f divNL(ηR, ϕ) dx. Rn Rn Rn Rn Since f ∈ Sα,1(Rn), we have Z Z α α f div (ηRϕ) dx = − ηRϕ · ∇wf dx. Rn Rn ∞ n Since fηR ∈ Cc (R ), we also have Z Z α α fηR div ϕ dx = − ϕ · ∇ (ηRf) dx. Rn Rn A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 25

Z Z |ηR(y) − ηR(x)| lim |f(x)| n+α dy dx = 0 R→+∞ Rn Rn |y − x| combining (2.6), (2.7) and (3.9) with Lebesgue’s Dominated Convergence Theorem. ∞ n α,p n We do not know if Cc (R ) is a dense subset of S (R ) for α ∈ (0, 1) and p ∈ (1, +∞). In other words, defining k·k α,p n α,p n ∞ n S (R ) S0 (R ) := Cc (R ) , α,p n α,p n we do not know if the (continuous) inclusion S0 (R ) ⊂ S (R ) is strict. α,p n The space (S0 (R ), k·kSα,p(Rn)) was introduced in [18] (with a different, but equivalent, α,p n norm). Thanks to [18, Theorem 1.7], for all α ∈ (0, 1) and p ∈ (1, ∞) we have S0 (R ) = Lα,p(Rn), where Lα,p(Rn) is the Bessel potential space, see [18, Definition 2.1]. It is known that Lα+ε,p(Rn) ⊂ W α,p(Rn) ⊂ Lα−ε,p(Rn) with continuous embeddings for all α ∈ (0, 1), p ∈ (1, +∞) and 0 < ε < min{α, 1 − α}, see [18, Theorem 2.2]. In the particular case p = 2, it holds that Lα,2(Rn) = W α,2(Rn) for all α ∈ (0, 1), see [18, Theorem 2.2]. In addition, W α,p(Rn) ⊂ Lα,p(Rn) with continuous embedding for all α ∈ (0, 1) and p ∈ (1, 2], see [21, Section 5.3]. Proposition 3.24 (Relation between W α,p and Sα,p). The following properties hold. (i) If α ∈ (0, 1) and p ∈ [1, 2], then W α,p(Rn) ⊂ Sα,p(Rn) with continuous embedding. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 26

(ii) If 0 < α < β < 1 and p ∈ (2, +∞], then W β,p(Rn) ⊂ Sα,p(Rn) with continuous embedding. Proof. Property (i) follows from the discussion above for the case p ∈ (1, 2] and from Theorem 3.18 for the case p = 1. Property (ii) follows from the discussion above for the case p ∈ (2, +∞), while for the case p = +∞ it is enough to observe that Z α |f(y) − f(x)| k∇ fkL∞( n; n) ≤ sup dy R R n n+α x∈Rn R |y − x| Z dy Z dy ≤ 2kfkL∞(Rn) + [f]W β,∞(Rn) {|y|>1} |y|n+α {|y|≤1} |y|n+α−β

≤ cn,α,βkfkW β,∞(Rn) β,∞ n for all f ∈ W (R ). As in the classical case, we have Sα,1(Rn) ⊂ BV α(Rn) with continuous embedding. Theorem 3.25 (Sα,1(Rn) ⊂ BV α(Rn)). Let α ∈ (0, 1). If f ∈ BV α(Rn), then f ∈ Sα,1(Rn) if and only if |Dαf| L n, in which case α α n n n D f = ∇wf L in M (R ; R ). Proof. Let f ∈ BV α(Rn) and assume that |Dαf| L n. Then Dαf = g L n for some g ∈ L1(Rn; Rn). But then, by Theorem 3.2, we must have Z Z f divαϕ dx = − g · ϕ dx Rn Rn ∞ n n α,1 n α α,1 n for all ϕ ∈ Cc (R ; R ), so that f ∈ S (R ) with ∇wf = g. Viceversa, if f ∈ S (R ) then Z Z α α f div ϕ dx = − ϕ · ∇wf dx Rn Rn ∞ n n α n α α n n n for all ϕ ∈ Cc (R ; R ), so that f ∈ BV (R ) with D f = ∇wf L in M (R ; R ).

3.10. The inclusion Sα,1(Rn) ⊂ BV α(Rn) is strict. It seems natural to ask whether the inclusion Sα,1(Rn) ⊂ BV α(Rn) is strict as in the classical case. We start to solve this problem in the case n = 1.

Theorem 3.26 (BV α(R) \ Sα,1(R) 6= ∅). Let α ∈ (0, 1). The inclusion Sα,1(R) ⊂ BV α(R) is strict, since for any a, b ∈ R, with a 6= b, the function α−1 α−1 fa,b,α(x) := |x − b| sgn(x − b) − |x − a| sgn(x − a) α satisﬁes fa,b,α ∈ BV (R) with α δb − δa D fa,b,α = (3.29) µ1,−α in the sense of ﬁnite Radon measures.

1 Proof. Let a, b ∈ R be ﬁxed with a 6= b. One can easily check that fa,b,α ∈ L (R). Since α n = 1, we have ∇α = div . Thus, (3.29) follows from (3.19), proving that f ∈ BV α(R). α 1 α,1 But |D fa,b,α| ⊥ L , so that fa,b,α ∈/ S (R) by Theorem 3.25. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 27

α 1 Remark 3.27. Note that fa,b,α ∈ BV (R) \ L 1−α (R), since  |x − a|−1 as x → a, 1  |fa,b,α(x)| 1−α ∼ |x − b|−1 as x → b. Thus, Theorem 3.9 cannot hold for n = 1.

α For the case n > 1, we need to recall the fractional Laplacian operator (−∆) 2 and some of its properties. ∞ n Following [20], for any f ∈ Cc (R ) we set  Z  f(x + h) νn,α dh if α ∈ (−1, 0),  n |h|n+α  R   f(x) if α = 0,  α  (−∆) 2 f(x) := (3.30) Z f(x + h) − f(x)  νn,α dh if α ∈ (0, 1),  n |h|n+α  R   Z  f(x + h) − f(x) νn,α lim dh if α ∈ [1, 2),  ε→0 {|h|>ε} |h|n+α where Γ n+α α − n 2 ν := 2 π 2 . (3.31) n,α α Γ − 2 We stress the fact that this deﬁnition is consistent with the previous deﬁnitions of fractional gradient and divergence in the sense that

α β α+β −div ∇ = (−∆) 2 for any α ∈ (−1, 1) and β ∈ (0, 1) (see [20, Theorem 5.3]), so that, in particular, −divα∇α = (−∆)α for any α ∈ (0, 1). In the case α ∈ (−1, 0), we have

α 2 ∞ n (−∆) = I−α on Cc (R ), where Iα is as in (2.10). In the case α ∈ (0, 1), notice that

α 2 k(−∆) fkL1(Rn) ≤ νn,α[f]W α,1(Rn) (3.32) ∞ n for all f ∈ Cc (R ). Thus the linear operator α 2 ∞ n 1 n (−∆) : Cc (R ) → L (R ) can be continuously extended to a linear operator α α,1 n 1 n (−∆) 2 : W (R ) → L (R ), for which we retain the same notation. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 28

Given α ∈ (0, 1) and ε > 0, for all f ∈ W α,1(Rn) we also set Z α/2 f(x + h) − f(x) (−∆)ε f(x) := νn,α dh. {|h|>ε} |h|n+α By Lebesgue’s Dominate Convergence Theorem, we have that α/2 α lim k(−∆) f − (−∆) 2 fkL1( n) = 0 ε→0 ε R for all f ∈ W α,1(Rn). Thus, arguing as in the proof of [13, Lemma 2.4] (see also [14, Section 25.1]), for all f ∈ W α,1(Rn) we have α 1 n Iα(−∆) 2 f = f in L (R ). (3.33) Taking advantage of the identity in (3.33), we can prove the following result.

Lemma 3.28 (Relation between BV α(Rn) and bv(Rn)). Let α ∈ (0, 1). The following properties hold. α n n α n n (i) If f ∈ BV (R ), then u := I1−αf ∈ bv(R ) with Du = D f in M (R ; R ). n 1−α α n (ii) If u ∈ BV (R ), then f := (−∆) 2 u ∈ BV (R ) with α n n kfkL1(Rn) ≤ cn,αkukBV (Rn) and D f = Du in M (R ; R ). 1−α n α n As a consequence, the operator (−∆) 2 : BV (R ) → BV (R ) is continuous. Proof. We prove the two properties separately. α n 1 n 1 n Proof of (i). Let f ∈ BV (R ). Since f ∈ L (R ), we have I1−αf ∈ Lloc(R ). By ∞ n n Fubini’s Theorem, for any ϕ ∈ Cc (R ; R ) we have Z Z Z α f div ϕ dx = f I1−αdivϕ dx = u divϕ dx, (3.34) Rn Rn Rn n α n n proving that u := I1−αf ∈ bv(R ) with Du = D f in M (R ; R ). Proof of (ii). Let u ∈ BV (Rn). By Theorem 3.18, we know that u ∈ W 1−α,1(Rn), 1−α 2 1 n so that f := (−∆) u ∈ L (R ) with kfkL1(Rn) ≤ cn,αkukBV (Rn) by (3.25) and (3.32). ∞ n n Then, arguing as before, for any ϕ ∈ Cc (R ; R ) we get (3.34), since we have I1−αf = u 1 n in L (R ) by (3.33). The proof is complete. α n 1 n Remark 3.29 (Integrability issues). Note that the inclusion I1−α(BV (R )) ⊂ Lloc(R ) in Lemma 3.28 above is sharp. Indeed, by Tonelli’s Theorem it is easily seen that I1−αχE ∈/ 1 n α,1 n L (R ) whenever χE ∈ W (R ). However, when n ≥ 2, by Theorem 3.9 and by Hardy– Littlewood–Sobolev inequality (see [21, Chapter V, Section 1.2] for instance), the map α p n n n i I1−α : BV → L (R ) is continuous for each p ∈ n−1+α , n−1 . As a consequence of Lemma 3.28, we can prove that the inclusion Sα,1(Rn) ⊂ BV α(Rn) is strict for all α ∈ (0, 1) and n ≥ 1.

Theorem 3.30 (BV α(Rn) \ Sα,1(Rn) 6= ∅). Let α ∈ (0, 1). The inclusion Sα,1(Rn) ⊂ BV α(Rn) is strict. n 1,1 n 1−α Proof. Let u ∈ BV (R ) \ W (R ). By Lemma 3.28, we know that f := (−∆) 2 u ∈ BV α(Rn) with Du = Dαf in M (Rn; Rn). But then |Dαf| is not absolutely continuous n α,1 n with respect to L , so that f∈ / S (R ) by Theorem 3.25. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 29

3.11. The inclusion W α,1(Rn) ⊂ Sα,1(Rn) is strict. By Theorem 3.30, we know that the inclusion W α,1(Rn) ⊂ BV α(Rn) is strict. In the following result we prove that also the inclusion W α,1(Rn) ⊂ Sα,1(Rn) is strict. Theorem 3.31 (Sα,1(Rn) \ W α,1(Rn) 6= ∅). Let α ∈ (0, 1). The inclusion W α,1(Rn) ⊂ Sα,1(Rn) is strict. Proof. We argue by contradiction. If W α,1(Rn) = Sα,1(Rn), then the inclusion map W α,1(Rn) ,→ Sα,1(Rn) is a linear and continuous bijection. Thus, by the Inverse Mapping Theorem, there must exist a constant C > 0 such that

[g]W α,1(Rn) ≤ CkgkSα,1(Rn) (3.35) for all g ∈ Sα,1(Rn). Now let f ∈ BV α(Rn) \ Sα,1(Rn) be given by Theorem 3.30. ∞ n 1 n By Theorem 3.8, there exists (fk)k∈N ⊂ Cc (R ) such that fk → f in L (R ) and α n α n |D fk|(R ) → |D f|(R ) as k → +∞. Up to extract a subsequence (which we do n not relabel for simplicity), we can assume that fk(x) → f(x) as k → +∞ for L -a.e. x ∈ Rn. By (3.35) and Fatou’s Lemma, we have that

[f]W α,1( n) ≤ lim inf[fk]W α,1( n) R k→+∞ R

≤ C lim inf kfkkSα,1( n) k→+∞ R

= C lim kfkkBV α( n) k→+∞ R

= C kfkBV α(Rn) < +∞. Therefore f ∈ W α,1(Rn), in contradiction with Theorem 3.30. We thus must have that α,1 n α,1 n the inclusion map W (R ) ,→ S (R ) cannot be surjective. 3.12. The inclusion BV α(Rn) ⊂ W β,1(Rn) for β < α. Even though the inclusion W α,1(Rn) ⊂ BV α(Rn) is strict, it is interesting to notice that BV α(Rn) ⊂ W β,1(Rn) for all 0 < β < α < 1 with continuous embedding.

Theorem 3.32 (BV α(Rn) ⊂ W β,1(Rn) for β < α). Let α, β ∈ (0, 1) with β < α. Then BV α(Rn) ⊂ W β,1(Rn), with

[f]W β,1(Rn) ≤ Cn,α,β kfkBV α(Rn), (3.36) for all f ∈ BV α(Rn), where α−β α2 β γβ/α C := nω n,α (3.37) n,α,β n β(α − β) and γn,α is as in (3.21). ∞ n Proof. Let f ∈ Cc (R ) and r > 0. By (3.20), we get Z Z |f(x + y) − f(x)| β,1 n [f]W (R ) = n+β dx dy Rn Rn |y| Z 1 α α 1 n n 1 n n ≤ n+β 2kfkL (R )χR \Br (y) + γn,α|y| k∇ fkL (R ;R )χBr (y) dy Rn |y| nωn −β nωn α−β α = 2 r kfk 1 n + γ r k∇ fk 1 n n β L (R ) α − β n,α L (R ;R ) A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 30 ! nωn −β nωn α−β ≤ 2 r + γ r kfk α n , β α − β n,α BV (R ) ∞ n so that both (3.36) and (3.37) are proved by minimising in r > 0 for all f ∈ Cc (R ). α n ∞ n Now let f ∈ BV (R ). By Theorem 3.8, there exists (fk)k∈N ⊂ Cc (R ) such that kfkkBV α(Rn) → kfkBV α(Rn) and fk → f a.e. as k → +∞. Thus, by Fatou’s Lemma, we get that

[f]W β,1( n) ≤ lim inf [fk]W β,1( n) ≤ lim Cn,α,β kfkkBV α( n) = Cn,α,β kfkBV α( n) R k→+∞ R k→+∞ R R and the conclusion follows. Note that the constant in (3.37) satisﬁes

lim Cn,α,β = +∞, β→α− accordingly to the strict inclusion W α,1(Rn) ⊂ BV α(Rn). In particular, the function in β,1 Theorem 3.26 is such that fa,b,α ∈ W (R) for all β ∈ (0, α). As an immediate consequence of Theorem 3.32, we have the following result.

Corollary 3.33. Let 0 < β < α < 1. Then BV α(Rn) ⊂ BV β(Rn) and Sα,1(Rn) ⊂ Sβ,1(Rn) with continuous embeddings.

4. Fractional Caccioppoli sets 4.1. Definition of fractional Caccioppoli sets and the Gauss–Green formula. As in the classical case (see [2, Definition 3.3.5] for instance), we start with the following definition.

Definition 4.1 (Fractional Caccioppoli set). Let α ∈ (0, 1) and let E ⊂ Rn be a measurable set. For any open set Ω ⊂ Rn, the fractional Caccioppoli α-perimeter in Ω is the fractional variation of χE in Ω, i.e. Z α α ∞ n |D χE|(Ω) = sup div ϕ dx : ϕ ∈ Cc (Ω; R ), kϕkL∞(Ω;Rn) ≤ 1 . E α We say that E is a set with finite fractional Caccioppoli α-perimeter in Ω if |D χE|(Ω) < +∞. We say that E is a set with locally finite fractional Caccioppoli α-perimeter in Ω if α |D χE|(U) < +∞ for any U b Ω. We can now state the following fundamental result relating non-local distributional gradients of characteristic functions of fractional Caccioppoli sets and vector valued Radon measures. Theorem 4.2 (Gauss–Green formula for fractional Caccioppoli sets). Let α ∈ (0, 1) and let Ω ⊂ Rn be an open set. A measurable set E ⊂ Rn is a set with finite fractional α n Caccioppoli α-perimeter in Ω if and only if D χE ∈ M (Ω; R ) and Z Z α α div ϕ dx = − ϕ · dD χE (4.1) E Ω ∞ n for all ϕ ∈ Cc (Ω; R ). In addition, for any open set U ⊂ Ω it holds Z α α ∞ n |D χE|(U) = sup div ϕ dx : ϕ ∈ Cc (U; R ), kϕkL∞(U;Rn) ≤ 1 . (4.2) E A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 31

α n Proof. The proof is similar to the one of Theorem 3.2. If D χE ∈ M (Ω; R ) and (4.1) holds, then E has finite fractional Caccioppoli α-perimeter in Ω by Definition 4.1. If E is a set with finite fractional Caccioppoli α-perimeter in Ω, then define the linear ∞ n functional L: Cc (Ω; R ) → R setting Z α ∞ n L(ϕ) := − div ϕ dx ∀ϕ ∈ Cc (Ω; R ). E Note that L is well defined thanks to Corollary 2.3. Since E has finite fractional Cacciop- poli α-perimeter in Ω, we have n ∞ n o C(U) := sup L(ϕ) : ϕ ∈ Cc (U; R ), kϕkL∞(U;Rn) ≤ 1 < +∞ for each open set U ⊂ Ω, so that ∞ n |L(ϕ)| ≤ C(U)kϕkL∞(U;Rn) ∀ϕ ∈ Cc (U; R ). ∞ n n Thus, by the density of Cc (Ω; R ) in Cc(Ω; R ), the functional L can be uniquely extended ˜ n to a continuous linear functional L: Cc(Ω; R ) → R and the conclusion follows by Riesz’s Representation Theorem. 4.2. Lower semicontinuity of fractional variation. As in the classical case, the variation measure of a set with finite fractional Caccioppoli α-perimeter is lower semicontinuous with respect to the local convergence in measure. We also achieve a weak convergence result. Proposition 4.3 (Lower semicontinuity of fractional variation measure). Let α ∈ (0, 1) n and let Ω ⊂ R be an open set. If (Ek)k∈N is a sequence of sets with finite fractional 1 n Caccioppoli α-perimeter in Ω and χEk → χE in Lloc(R ), then α α n D χEk *D χE in M (Ω; R ), (4.3) and α α |D χE|(Ω) ≤ lim inf |D χE |(Ω). (4.4) k→+∞ k

Proof. Up to extract a further subsequence, we can assume that χEk (x) → χE(x) as n n ∞ n k → +∞ for L -a.e. x ∈ R . Now let ϕ ∈ Cc (Ω; R ) be such that kϕkL∞(Ω;Rn) ≤ 1. α Then div ϕ ∈ L1(Rn; Rn) by Corollary 2.3 and so, by Lebesgue’s Dominated Convergence Theorem, we have Z Z Z α α α α div ϕ dx = lim div ϕ dx = − lim ϕ · dD χEk ≤ lim inf |D χEk |(Ω). E k→+∞ Ek k→+∞ Ω k→+∞ By Theorem 4.2, we get (4.4). The convergence in (4.3) easily follows. 4.3. Fractional isoperimetric inequality. As a simple application of Theorem 3.9, we can prove the following fractional isoperimetric inequality. Theorem 4.4 (Fractional isoperimetric inequality). Let α ∈ (0, 1) and n ≥ 2. There exists a constant cn,α > 0 such that n−α α n |E| n ≤ cn,α|D χE|(R ) (4.5) n α n for any set E ⊂ R such that |E| < +∞ and |D χE|(R ) < +∞. α n Proof. Since χE ∈ BV (R ), the result follows directly by Theorem 3.9. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 32

4.4. Compactness. As an application of Theorem 3.16, we can prove the following compactness result for sets with ﬁnite fractional Caccioppoli α-perimeter in Rn (see for instance [11, Theorem 12.26] for the analogous result in the classical case). Theorem 4.5 (Compactness for sets with ﬁnite fractional Caccioppoli α-perimeter). Let

α ∈ (0, 1) and R > 0. If (Ek)k∈N is a sequence of sets with ﬁnite fractional Caccioppoli α-perimeter in Rn such that α n sup |D χEk |(R ) < +∞ and Ek ⊂ BR ∀k ∈ N, k∈N then there exist a subsequence (Ekj )j∈N and a set E ⊂ BR with ﬁnite fractional Caccioppoli α-perimeter in Rn such that χ → χ in L1( n) Ekj E R as j → +∞.

α n n Proof. Since Ek ⊂ BR for all k ∈ N, we clearly have that (χEk )k∈R ⊂ BV (R ). By 1 n Theorem 3.16, there exist a subsequence (Ekj )j∈N and a function f ∈ L (R ) such that χ → f in L1 ( n) as j → +∞. Since again E ⊂ B for all j ∈ , we have that Ekj loc R kj R N χ → f in L1( n) as j → +∞. Up to extract a further subsequence (which we do Ekj R not relabel for simplicity), we can assume that χ (x) → f(x) for n-a.e. x ∈ n as Ekj L R j → +∞, so that f = χE for some E ⊂ BR. By Proposition 4.3 we conclude that E has n finite fractional Caccioppoli α-perimeter in R . Theorem 4.5 can be applied to prove the following compactness result for sets with locally finite fractional Caccioppoli α-perimeter. Corollary 4.6 (Compactness for locally finite fractional Caccioppoli α-perimeter sets).

Let α ∈ (0, 1). If (Ek)k∈N is a sequence of sets with locally finite fractional Caccioppoli α-perimeter in Rn such that α sup |D χEk |(BR) < +∞ ∀R > 0, (4.6) k∈N then there exist a subsequence (Ekj )j∈N and a set E with locally finite fractional Caccioppoli α-perimeter in Rn such that χ → χ in L1 ( n) Ekj E loc R as j → +∞. Proof. We divide the proof into two steps, essentially following the strategy presented in the proof of [11, Corollary 12.27]. Step 1. Let F ⊂ Rn be a set with locally finite fractional Caccioppoli α-perimeter in Rn. We claim that α n α |D χF ∩BR |(R ) ≤ |D χF |(BR) + 3µn,αPα(BR) ∀R > 0. (4.7) 0 ∞ n Indeed, let R < R and, recalling Theorem A.1, let (uk)k∈N ⊂ Cc (R ) be such that supp(u ) B and 0 ≤ u ≤ 1 for all k ∈ and also u → χ in W α,1( n) as k b R k N k BR0 R A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 33

∞ n n k → +∞. If ϕ ∈ Cc (R ; R ) with kϕkL∞(Rn;Rn) ≤ 1, then Z Z Z Z α α α α uk div ϕ dx = div (ukϕ) dx − ϕ · ∇ uk dx − divNL(uk, ϕ) dx F F F F Z α ≤ div (ukϕ) dx + 3µn,α[uk]W α,1(Rn) F α ≤ |D χF |(BR0 ) + 3µn,α[uk]W α,1(Rn) α ≤ |D χF |(BR) + 3µn,α[uk]W α,1(Rn) by Lemma 2.7. Passing to the limit as k → +∞, we conclude that Z α α div ϕ dx ≤ |D χF |(BR) + 3µn,αPα(BR0 ) F ∩BR0 and thus α n α |D χF ∩BR0 |(R ) ≤ |D χF |(BR) + 3µn,αPα(BR) 1 n 0 by Theorem 4.2. Since χF ∩BR0 → χF ∩BR in L (R ) as R → R, the claim in (4.7) follows by Proposition 4.3.

Step 2. By (4.6) and (4.7), we can apply Theorem 4.5 to (Ek ∩ Bj)k∈N for each fixed j ∈ N. By a standard diagonal argument, we find a subsequence (Ekh )h∈N and a sequence (Fj)j∈ of sets with finite fractional Caccioppoli α-perimeter such that χE ∩B → χF in N kh j j 1 n L ( ) as h → +∞ for each j ∈ . Up to null sets, we have Fj ⊂ Fj+1, so that χE → χE R N kh 1 n S in Lloc(R ) with E := j∈N Fj. The conclusion thus follows by Proposition 4.3.

4.5. Fractional reduced boundary. Thanks to the scaling property of the fractional divergence, we have α n−α α D χλE = λ (δλ)#D χE on λΩ, (4.8) n where δλ(x) = λx for all x ∈ R and λ > 0. Indeed, we can compute Z Z Z α n α n−α α div ϕ dx = λ (div ϕ) ◦ δλ dx = λ div (ϕ ◦ δλ) dx λE E E ∞ n for all ϕ ∈ Cc (Ω; R ). In analogy with the classical case, we are thus led to the following deﬁnition.

Deﬁnition 4.7 (Fractional reduced boundary). Let α ∈ (0, 1) and let Ω ⊂ Rn be an open set. If E ⊂ Rn is a set with ﬁnite fractional Caccioppoli α-perimeter in Ω, then we say that a point x ∈ Ω belongs to the fractional reduced boundary of E (inside Ω), and we write x ∈ F αE, if α α D χE(Br(x)) n−1 x ∈ supp(D χE) and ∃ lim α ∈ S . r→0 |D χE|(Br(x)) We thus let Dαχ (B (x)) α : α n−1 α : E r α νE Ω ∩ F E → S , νE(x) = lim α , x ∈ Ω ∩ F E, r→0 |D χE|(Br(x)) be the (measure-theoretic) inner unit fractional normal to E (inside Ω). A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 34

As a consequence of Definition 4.7 and arguing similarly as in the proof of Proposi- tion 3.6, if E ⊂ Rn is a set with finite fractional Caccioppoli α-perimeter in Ω, then the following Gauss–Green formula Z Z α α α div ϕ dx = − ϕ · νE d|D χE|, (4.9) E Ω∩F αE n holds for any ϕ ∈ Lipc(Ω; R ). 4.6. Sets of finite fractional perimeter are fractional Caccioppoli sets. In analogy with the classical case and with the inclusion W α,1(Rn) ⊂ BV α(Rn), we can show that sets with finite fractional α-perimeter have finite fractional Caccioppoli α-perimeter. Recall that the fractional α-perimeter of a set E ⊂ R in an open set Ω ⊂ Rn is defined as Z Z |χ (x) − χ (y)| Z Z |χ (x) − χ (y)| : E E E E Pα(E; Ω) = n+α dx dy + 2 n+α dx dy, Ω Ω |x − y| Ω Rn\Ω |x − y| see [5] for an account on this subject. Proposition 4.8 (Sets of finite fractional perimeter are fractional Caccioppoli sets). Let n n α ∈ (0, 1) and let Ω ⊂ R be an open set. If E ⊂ R satisfies Pα(E; Ω) < +∞, then E is a set with finite fractional Caccioppoli α-perimeter in Ω with α |D χE|(Ω) ≤ µn,αPα(E; Ω) (4.10) and Z Z α α div ϕ dx = − ϕ · ∇ χE dx (4.11) E Ω n α α α α n for all ϕ ∈ Lipc(Ω; R ), so that D χE = νE |D χE| = ∇ χE L . Moreover, if E is such α,1 n that |E| < +∞ and P (E) < +∞, then χE ∈ W (R ) for any α ∈ (0, 1), and Z α µn,α νE(y) ∇ χE(x) = n+α−1 d|DχE|(y) (4.12) n + α − 1 Rn |y − x| for L n-a.e. x ∈ Rn. α 1 n Proof. Note that ∇ χE ∈ L (Ω; R ), because Z Z Z α |χE(y) − χE(x)| |∇ χE| dx ≤ µn,α n+α dy dx Ω Ω Rn |y − x|

Z Z |χE(y) − χE(x)| Z Z |χE(y) − χE(x)| ≤ µn,α n+α dy dx + µn,α n+α dy dx Ω Ω |y − x| Ω Rn\Ω |y − x|

≤ µn,αPα(E; Ω), n Now let ϕ ∈ Lipc(Ω; R ) be ﬁxed. By Lebesgue’s Dominated Convergence Theorem, by (2.4) and by Fubini’s Theorem (applied for each ﬁxed ε > 0), we can compute Z Z Z α (y − x) · ϕ(y) div ϕ dx = µn,α lim dy dx E ε→0 E {|x−y|>ε} |y − x|n+α+1

Z Z (y − x) χE(x) = µn,α lim ϕ(y) · dx dy ε→0 Ω {|x−y|>ε} |y − x|n+α+1

Z Z (y − x)(χE(y) − χE(x)) = −µn,α lim ϕ(y) · dx dy ε→0 Ω {|x−y|>ε} |y − x|n+α+1 A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 35 Z α = − ϕ · ∇ χE dy. Ω Thus (4.10) and (4.11) follow by Theorem 4.2 and Deﬁnition 4.7. Finally, (4.12) follows n from (3.26), since χE ∈ BV (R ). α At the present moment, we do not know if |D χE|(Ω) < +∞ implies that Pα(E; Ω) < +∞. Remark 4.9 (F αE is not L n-negligible in general). It is important to notice that, by Proposition 4.8, we have n α Pα(E; Ω) < +∞ =⇒ L (Ω ∩ F E) > 0 n including even the case χE ∈ BV (R ). This shows a substantial diﬀerence between the standard local De Giorgi’s perimeter measure |DχE| and the non-local fractional α n De Giorgi’s perimeter measure |D χE|: the former is supported on a L -negligible set contained in the topological boundary of E, while the latter, in general, can be supported on a set of positive Lebesgue measure and, for this reason, cannot be expected to be contained in the topological boundary of E. Remark 4.10 (Fractional reduced boundary and precise representative). We let  1 Z lim u(y) dy if the limit exists, r→0 u∗(x) := |Br(x)| Br(x)  0 otherwise,

1 n m ∗ be the precise representative of a function u ∈ Lloc(R ; R ). Note that u is well deﬁned at α α n any Lebesgue point of u. By Proposition 4.8, if Pα(E; Ω) < +∞ then D χE = ∇ χEL α 1 n with ∇ χE ∈ L (Ω; R ). Therefore the set α α ∗ α ∗ RΩE := {x ∈ Ω : |(∇ χE) (x)| = |∇ χE| (x) 6= 0} is such that α α RΩE ⊂ Ω ∩ F E (4.13) and α ∗ α (∇ χE) (x) α νE(x) = α ∗ for all x ∈ RΩE. |∇ χE| (x) The following simple example shows that the inclusion in (4.13) and the inequality in (4.10) can be strict.

Example 4.11. Let n = 1, α ∈ (0, 1) and a, b ∈ R, with a < b. It is easy to see that α,1 χ(a,b) ∈ W (R). By (4.12), for any x 6= a, b we have that Z α µ1,α 1 ∇ χ(a,b)(x) = α d (δa − δb)(y) α R |x − y| 2α Γ 1 + α 1 1 ! = √ 2 − . α π 1−α |x − a|α |x − b|α Γ 2 We claim that (a + b) F α(a, b) = \ (4.14) R 2 A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 36 while ( ) α a + b R n (a, b) = \ a, , b , (4.15) R R 2 so that inclusion (4.13) is strict. Finally, we also claim that α k∇ χ(a,b)kL1(R) < µ1,αPα((a, b)). (4.16) Indeed, notice that α ∇ χ(a,b)(x) ≥ 0 a+b if and only if x ≤ 2 , so that  Z x+r  a + b ∇αχ (y) dy 1 if x < , (a,b)  2 lim x−r = r→0 Z x+r α  a + b |∇ χ(a,b)(y)| dy −1 if x > . x−r  2 a+b If x = 2 , then Z a+b +r 2 ∇αχ (y) dy = 0 ∀r > 0, a+b (a,b) 2 −r and claim (4.14) follows. In particular, we have  a + b 1 if x < ,  α  2 ν(a,b)(x) =  a + b −1 if x >  2 and also claim (4.15) follows. To prove (4.16), note that 4 P ((a, b)) = (b − a)1−α (4.17) α α(1 − α) 1−α since Pα((a, b)) = (b − a) Pα((0, 1)) by the scaling property of the fractional perimeter and Z Z 1 1 Pα((0, 1)) = 2 1+α dy dx R\(0,1) 0 |y − x| 2 Z "sgn(x − y)#y=1 = α dx α R\(0,1) |y − x| y=0 2 Z sgn(x − 1) sgn(x) = α − α dx α R\(0,1) |1 − x| |x| 2 Z ∞ 1 1 2 Z 0 1 1 = − dx + − dx α 1 (x − 1)α xα α −∞ (−x)α (1 − x)α 4 Z ∞ 1 1 4 = − dx = . α 0 xα (1 + x)α α(1 − α) On the other hand, we have 1+α α 2 µ1,α 1−α k∇ χ k 1 = (b − a) . (4.18) (a,b) L (R) α(1 − α) A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 37

α 1−α α Indeed, k∇ χ(a,b)kL1(R) = (b − a) k∇ χ(0,1)kL1(R) by (4.8) and Z α α 1 1 1 k∇ χ(0,1)kL (R) = α − α dx µ1,α R |x| |x − 1|

Z ∞ 1 1 Z 1 1 1

= − dx + − dx 1 xα (x − 1)α 0 xα (1 − x)α

Z 0 1 1

+ − dx −∞ (−x)α (1 − x)α Z ∞ 1 1 Z 1 1 1 = − dx + − dx α α 1 α α 1 (x − 1) x 2 (1 − x) x 1 0 Z 2 1 1 Z 1 1 + − dx + − dx 0 xα (1 − x)α −∞ (−x)α (1 − x)α ∞ 1 Z 1 1 Z 2 1 1 = 2 − dx + 2 − dx 0 xα (1 + x)α 0 xα (1 − x)α 2 21+α = 1 + 2α−1 + 2α−1 − 1 = . 1 − α 1 − α Combining (4.17) and (4.18), we get (4.16). Thanks to Example 4.11 above, we know that inequality (1.15) is strict for E = (a, b) with a, b ∈ R, a < b. We conclude this section proving that this fact holds for all sets α,1 E ⊂ R such that χE ∈ W (R). α,1 α Proposition 4.12. Let α ∈ (0, 1). If χE ∈ W (R), then |D χE|(R) < µ1,αPα(E). α,1 α Proof. We argue by contradiction. Assume χE ∈ W (R) is such that |D χE|(R) = µ1,αPα(E). Then Z Z Z Z

|fE(x, y)| dy dx = fE(x, y) sgn(y − x) dy dx (4.19) R R R R where χ (y) − χ (x) f (x, y) := E E ∀x, y ∈ , x 6= y. E |y − x|1+α R From (4.19) we deduce that Z Z

|fE(x, y)| dy = fE(x, y) sgn(y − x) dy R R for a.e. x ∈ R. If x ∈ E, then fE(x, y) ≤ 0 for all y ∈ R, y 6= x, and thus Z +∞ Z x Z +∞ Z x

|fE(x, y)| dy + |fE(x, y)| dy = |fE(x, y)| dy − |fE(x, y)| dy x −∞ x −∞ for a.e. x ∈ E. Squaring both sides and simplifying, we get that Z +∞ Z x |fE(x, y)| dy |fE(x, y)| dy = 0, x −∞ so that either |Ec ∩ (x, +∞)| = 0 or |Ec ∩ (−∞, x)| = 0 for a.e. x ∈ E, contradicting the fact that |E| < +∞. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 38

5. Existence of blow-ups for fractional Caccioppoli sets In this section we prove existence of blow-ups for sets with locally finite fractional Caccioppoli α-perimeter. We follow the approach presented in [8, Section 5.7]. We start with the following technical preliminary result. Lemma 5.1. Let α ∈ (0, 1). For all ε, r > 0 and x ∈ Rn we define  1 if 0 ≤ |y − x| ≤ r,   r + ε − |y − x| hε,r,x(y) := if r < |y − x| < r + ε,  ε   0 if |y − x| ≥ r + ε. α 1 n n Then ∇ hε,r,x ∈ L (R ; R ) with Z α µn,α x − z 1−n−α ∇ hε,r,x(y) = |z − y| dz (5.1) ε(n + α − 1) Br+ε(x)\Br(x) |x − z| for L n-a.e. y ∈ Rn. n Proof. Clearly hε,r,x ∈ Lipc(R ) and 1 y − x ∇h (y) = − χ (y). ε,r,x ε |y − x| Br+ε(x)\Br(x) Therefore by (3.26) we get Z α 1 µn,α 1 z − x ∇ hε,r,x(y) = − n+α−1 χBr+ε(x)\Br(x)(z) dz ε n + α − 1 Rn |z − y| |z − x| n n α 1 n n for L -a.e. y ∈ R . By Theorem 3.18, we get ∇ hε,r,x ∈ L (R ; R ). We now proceed with the following formula for integration by parts on balls, see [8, Lemma 5.2] for the analogous result in the classical setting. Theorem 5.2 (Integration by parts on balls). Let α ∈ (0, 1). If E ⊂ Rn is a set with locally finite fractional Caccioppoli α-perimeter in Rn, then Z Z Z Z α α α α div ϕ dy+ ϕ ·∇ χBr(x) dy+ divNL(χBr(x), ϕ) dy = − ϕ· dD χE (5.2) E∩Br(x) E E Br(x) n n α 1 for all ϕ ∈ Lipc(R ; R ), x ∈ F E and for L -a.e. r > 0. α n n Proof. Fix ε, r > 0, x ∈ F E and ϕ ∈ Lipc(R ; R ) and let hε,r,x be as in Lemma 5.1. On the one hand, by (4.9) we have Z Z α α div (ϕ hε,r,x) dy = − (hε,r,x ϕ) · dD χE. (5.3) E F αE Since h (y) → χ (y) as ε → 0 for any y ∈ n and |Dαχ |(∂B (x)) = 0 for 1-a.e. ε,r,x Br(x) R E r L r > 0, we can compute Z Z α α lim (hε,r,x ϕ) · dD χE = ϕ · dD χE. α ε→0 F E Br(x) On the other hand, by Lemma 2.1 and Lemma 2.7, we have α α α α div (ϕ hε,r,x) = hε,r,x div ϕ + ϕ · ∇ hε,r,x + divNL(hε,r,x, ϕ). (5.4) A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 39

We deal with each term of the right-hand side of (5.4) separately. For the ﬁrst term, 1 n since 0 ≤ hε,r,x ≤ χBr+1(x) for all ε ∈ (0, 1) and hε,r,x → χBr(x) in L (R ) as ε → 0, by Corollary 2.3 and Lebesgue’s Dominated Convergence Theorem we can compute Z Z α α lim hε,r,x div ϕ dy = div ϕ dy. (5.5) ε→0 E E∩Br(x) For the second term, by (5.1) we have Z Z Z α µn,α x − z 1−n−α ϕ(y) · ∇ hε,r,x(y) dy = ϕ(y) · |z − y| dz dy. E ε(n + α − 1) E Br+ε(x)\Br(x) |x − z| By Fubini’s Theorem, we can compute Z Z x − z ϕ(y) · |z − y|1−n−α dz dy E Br+ε(x)\Br(x) |x − z| Z x − z Z = · ϕ(y) |z − y|1−n−α dy dz Br+ε(x)\Br(x) |x − z| E Z r+ε Z x − z Z = · ϕ(y) |z − y|1−n−α dy dH n−1(z) d%. r ∂B%(x) |x − z| E By Lebesgue’s Diﬀerentiation Theorem, we have 1 Z Z x − z lim ϕ(y) · |z − y|1−n−α dz dy ε→0 ε E Br+ε(x)\Br(x) |x − z| 1 Z r+ε Z x − z Z = lim · ϕ(y) |z − y|1−n−α dy dH n−1(z) d% ε→0 ε r ∂B%(x) |x − z| E Z x − z Z = · ϕ(y) |z − y|1−n−α dy dH n−1(z) ∂Br(x) |x − z| E Z Z x − z = ϕ(y) · |z − y|1−n−α dH n−1(z) dy E ∂Br(x) |x − z| Z Z 1−n−α = ϕ(y) · |z − y| dDχBr(x)(z) dy E Rn for L 1-a.e. r > 0. Therefore, by (3.26), we get that Z α lim ϕ · ∇ hε,r,x dy ε→0 E Z Z µn,α 1−n−α = ϕ(y) · |z − y| dDχBr(x)(z) dy (5.6) n + α − 1 E Rn Z α = ϕ · ∇ χBr(x) dy E for L 1-a.e. r > 0. Finally, for the third term, note that

(z − y) · (ϕ(z) − ϕ(y))(h (z) − h (y)) |ϕ(z) − ϕ(y)| ε,r,x ε,r,x 1 n ≤ 2 ∈ Lz(R ) |z − y|n+α+1 |z − y|n+α for all y ∈ Rn, so that α α lim div (hε,r,x, ϕ)(y) = div (χB (x), ϕ)(y) ε→0 NL NL r A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 40 for L n-a.e. y ∈ Rn by Lebesgue’s Dominated Convergence Theorem. Since Z α |ϕ(z) − ϕ(y)| 1 n |divNL(hε,r,x, ϕ)(y)| ≤ 2 n+α dz ∈ Ly(R ), Rn |z − y| again by Lebesgue’s Dominated Convergence Theorem we can compute Z Z α α lim divNL(hε,r,x, ϕ) dy = divNL(χBr(x), ϕ) dy. (5.7) ε→0 E E Combining (5.3), (5.4), (5.5), (5.6) and (5.7), we obtain (5.2). We can now deduce the following decay estimates for the fractional De Giorgi’s perimeter measure, see [8, Lemma 5.3] for the analogous result in the classical setting.

Theorem 5.3 (Decay estimates). Let α ∈ (0, 1). There exist An,α,Bn,α > 0 with the following property. Let E ⊂ Rn be a set with locally finite fractional Caccioppoli α- n α perimeter in R . For any x ∈ F E, there exists rx > 0 such that α n−α |D χE|(Br(x)) ≤ An,αr (5.8) and α n n−α |D χE∩Br(x)|(R ) ≤ Bn,αr (5.9) for all r ∈ (0, rx). Proof. We divide the proof in two steps, dealing with the two estimates separately. α n n α Step 1: proof of (5.8). Fix x ∈ F E and choose ϕ ∈ Lipc(R ; R ) such that ϕ ≡ νE(x) in B1(x) and kϕkL∞(Rn;Rn) ≤ 1. On the one hand, by Definition 4.7, there exists rx ∈ (0, 1) such that Z α 1 α ϕ · dD χE ≥ |D χE|(Br(x)) (5.10) Br(x) 2 for all r ∈ (0, rx). On the other hand, by (5.2) we have Z Z Z α α α ϕ · dD χ ≤ div ϕ dy + ϕ · dD χ E Br(x) Br(x) E∩Br(x) E (5.11) Z α + divNL(χBr(x), ϕ) dy E 1 for L -a.e. r ∈ (0, rx). We now estimate the three terms in the right-hand side separately. α For the first one, since ϕ(y) ≡ νE(x) in Br(x), we can estimate Z Z Z α |ϕ(z) − ϕ(y)| div ϕ(y) dy ≤ µn,α dz dy n n+α E∩Br(x) E∩Br(x) R |z − y| Z Z α |ϕ(z) − νE(x)| = µn,α dz dy n n+α E∩Br(x) R \Br(x) |z − y| Z Z 1 ≤ 2µn,α dz dy n n+α Br(x) R \Br(x) |z − y|

= 2µn,αPα(Br(x)) so that

Z α n−α div ϕ(y) dy ≤ 2µn,αPα(B1) r . (5.12) E∩Br(x) A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 41

For the second term, by Proposition 4.8 we can estimate Z α α n n−α ϕ · dD χBr(x) ≤ |D χBr(x)|(R ) ≤ µn,αPα(Br(x)) = µn,αPα(B1) r . (5.13) E Finally, by Lemma 2.7, we can estimate Z α α 1 n divNL(χBr(x), ϕ) dy ≤ kdivNL(χBr(x), ϕ)kL (R ) E

α,1 n ≤ 2µn,α[χBr(x)]W (R )

= 2µn,αPα(Br(x)) so that Z α n−α divNL(χBr(x), ϕ) dy ≤ 2µn,αPα(B1) r . (5.14) E Combining (5.10), (5.11), (5.12), (5.13) and (5.14), we conclude that α n−α |D χE|(Br(x)) ≤ 10µn,αPα(B1) r (5.15) 1 for L -a.e. r ∈ (0, rx). Hence (5.8) follows with An,α = 10µn,αPα(B1) for all r ∈ (0, rx) by a simple continuity argument. α n n Step 2: proof of (5.9). Fix x ∈ F E and ϕ ∈ Lipc(R ; R ) with kϕkL∞(Rn;Rn) ≤ 1. Again by (5.2) we can estimate

Z Z α α α n α div ϕ dy ≤ |D χE|(Br(x)) + |D χBr(x)|(R ) + |divNL(χBr(x), ϕ)| dy E∩Br(x) E 1 for L -a.e. r ∈ (0, rx). Using (5.13), (5.14) and (5.15), we conclude that α n n−α |D χE∩Br(x)|(R ) ≤ 13µn,αPα(B1) r 1 for L -a.e. r ∈ (0, rx). Hence (5.9) follows with Bn,α = 13µn,αPα(B1) for all r ∈ (0, rx) by a simple continuity argument. This concludes the proof. As an easy consequence of Theorem 5.3, we can prove that α n−α α |D χE| H F E for any set E with locally ﬁnite fractional Caccioppoli α-perimeter in Rn. α n−α α Corollary 5.4 (|D χE| H F E). Let α ∈ (0, 1). If E is a set with locally ﬁnite fractional Caccioppoli α-perimeter in Rn, then

α n−α An,α n−α α |D χE| ≤ 2 H F E, (5.16) ωn−α where An,α is as in (5.8). Proof. By (5.8), we have that |Dαχ |(B (x)) A ∗ α : E r n,α Θn−α(|D χE|, x) = lim sup n−α ≤ r→0 ωn−αr ωn−α α for any x ∈ F E. Therefore, (5.16) is a simple application of [2, Theorem 2.56]. For any set E of locally ﬁnite fractional Caccioppoli α-perimeter, Corollary 5.4 enables us to obtain a lower bound on the Hausdorﬀ dimension of F αE. A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 42

Proposition 5.5. Let α ∈ (0, 1). If E is a set with locally ﬁnite fractional Caccioppoli α-perimeter in Rn, then α dimH (F E) ≥ n − α. (5.17) α α Proof. Since |D χE|(F E) > 0 by Deﬁnition 4.7, by Corollary 5.4 we conclude that n−α α H (F E) > 0, proving (5.17). As another interesting consequence of Corollary 5.4, we are able to prove that assump- tion (3.12) in Theorem 3.11 cannot be dropped.

Corollary 5.6 (No coarea formula in BV α(R)). Let α ∈ (0, 1). There exist f ∈ BV α(Rn) such that Z α n |D χ{f>t}|(R ) dt = +∞. (5.18) R n n 1−α Proof. Let E ⊂ R be such that χE ∈ BV (R ) and consider f := (−∆) 2 χE. By α n α n−1 Lemma 3.28, we know that f ∈ BV (R ) with |D f| = |DχE| = H F E. If Z α n |D χ{f>t}|(R ) dt < +∞ R then Z α α |D f| ≤ |D χ{f>t}| dt R by Theorem 3.11. Thus |Dαf| H n−α by Corollary 5.4, so that H n−1(F E) = 0, which is clearly absurd. Remark 5.7. If f ∈ W α,1(Rn), then Z Z α n |D χ{f>t}|(R ) dt ≤ µn,α Pα({f > t}) dt = µn,α[f]W α,1(Rn) < +∞ R R by Proposition 4.8 and Tonelli’s Theorem, so that (5.18) does not hold for all f ∈ BV α(Rn). We do not know if (3.14) is an equality for some functions f ∈ BV α(Rn). We can now prove the existence of blow-ups for sets with locally finite fractional Cac- cioppoli α-perimeter in Rn, see [8, Theorem 5.13] for the analogous result in the classical setting. Here and in the following, given a set E with locally finite fractional Caccioppoli α-perimeter and x ∈ F αE, we let Tan(E, x) be the set of all tangent sets of E at x, i.e. 1 n n E−x o the set of all limit points in Lloc(R )-topology of the family r : r > 0 as r → 0. Theorem 5.8 (Existence of blow-up). Let α ∈ (0, 1). Let E be a set with locally finite fractional Caccioppoli α-perimeter in Rn. For any x ∈ F αE we have Tan(E, x) 6= ∅. α Proof. Fix x ∈ F E. Up to a translation, we can assume x = 0. We set Er := E/r = {y ∈ Rn : ry ∈ E} for all r > 0. We divide the proof in two steps. p α Step 1. For each p ∈ N, we define Dr := Er ∩ Bp. By the α-homogeneity of div , we have Z Z Z divαϕ dy = r−n (divαϕ)(r−1z) dz = rα−n divα(ϕ(r−1·)) dz p Dr E∩Brp E∩Brp ∞ n n for all ϕ ∈ Cc (R ; R ). By (5.9), we thus get α n α−n α n n−α p |D χDr |(R ) = r |D χE∩Brp |(R ) ≤ Bn,αp A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 43 for all r > 0 such that rp < r0. Hence, for each fixed p ∈ N, we have α n n−α p sup |D χDr |(R ) ≤ Bn,αp . r

Proof. As in the proof of Theorem 5.8, we assume x = 0 and we set Er = E/r. By Theorem 5.8, there exists (r ) such that r → 0 as k → +∞ and χ → χ in k k∈N k Erk F 1 n Lloc(R ). By Proposition 3.3, it is clear that F has locally ﬁnite fractional Caccioppoli α-perimeter in Rn. By (4.3), we get Dαχ *Dαχ in ( n; n) Erk F Mloc R R as k → +∞. Thus, for L 1-a.e. L > 0, we have Dαχ (B ) → Dαχ (B ) as k → +∞. (5.19) Erk L F L Since α α−n α 1 D χEr = r (δ )#D χE ∀r > 0, r we have that |Dαχ |(B ) = rα−n|Dαχ |(B ) Erk L k E rkL and Dαχ (B ) = rα−nDαχ (B ). Erk L k E rkL Since 0 ∈ F αE, we thus get Dαχ (B ) α Erk L D χE(BrkL) α lim = lim = νE(0). (5.20) k→+∞ |Dαχ |(B ) k→+∞ |Dαχ |(B ) Erk L E rkL Therefore, by Proposition 3.3, (5.19) and (5.20), we obtain that α α |D χF |(BL) ≤ lim inf |D χE |(BL) k→+∞ rk Z = lim να(0) · dDαχ E Erk k→+∞ BL A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 44 Z α α = νE(0) · dD χF BL Z α α α = νE(0) · νF d|D χF | BL α ≤ |D χF |(BL) 1 α α α α for L -a.e. L > 0. We thus get that νF (y) = νE(0) for |D χF |-a.e. y ∈ BL ∩ F F and 1 L -a.e. L > 0, so that the conclusion follows.

∞ n α,1 n Appendix A. Cc (R ) is dense in W (R ) ∞ n α,p n The density of Cc (R ) in W (R ) for all α ∈ (0, 1) and 1 ≤ p < +∞ is stated withouth proof in [6, Theorem 2.4]. For the proof of this result, the authors in [6] refer to [1, Theorem 7.38], where unfortunately the case p = 1 is not explicitly proved. This result is also stated in [7, Proposition 4.27], but the proof is given for the case n = 1. ∞ n For the sake of clarity, we spend some words on the proof of the density of Cc (R ) in W α,1(Rn) for all α ∈ (0, 1). ∞ n α,1 n α,1 n Theorem A.1 (Cc (R ) is dense in W (R )). Let α ∈ (0, 1). If f ∈ W (R ), then ∞ n α,1 n there exists (fk)k∈N ⊂ Cc (R ) such that fk → f in W (R ) as k → +∞. Proof. The proof of the density of C∞(Rn) ∩ W α,1(Rn) in W α,1(Rn) via a standard convolution argument is given in full details in [10, Proposition 14.5] (actually, in the more general setting of Besov spaces, see [10, Section 14.8] for the relation with fractional ∞ n Sobolev spaces). Thus, to conclude, we just need to show the density of Cc (R ) in C∞(Rn) ∩ W α,1(Rn). To this aim, let f ∈ C∞(Rn) ∩ W α,1(Rn) be fixed. For all R > 0, ∞ n ∞ n consider a cut-off function ηR ∈ Cc (R ) defined as in (3.9). Then fηR ∈ Cc (R ) and the conclusion clearly follows if we show that

lim [f(1 − ηR)]W α,1( n) = 0. (A.1) R→+∞ R Indeed, we have Z Z |f(y) − f(x)| α,1 n [f(1 − ηR)]W (R ) ≤ n+α (1 − ηR(y)) dy dx Rn Rn |y − x|

Z Z |ηR(y) − ηR(x)| lim |f(x)| n+α dy dx = 0 R→+∞ Rn Rn |y − x| A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEV SPACES AND VARIATION 45 again by Lebesgue’s Dominated Convergence Theorem, since f ∈ L1(Rn). Thus (A.1) follows and the proof is complete.

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