The fractional variation and the precise representative of $BV^{\alpha,p}$ functions

We continue the study of the fractional variation following the distributional approach developed in the previous works arXiv:1809.08575, arXiv:1910.13419 and arXiv:2011.03928. We provide a general analysis of the distributional space $BV^{\alpha,p}(\mathbb{R}^n)$ of $L^p$ functions, with $p\in[1,+\infty]$, possessing finite fractional variation of order $\alpha\in(0,1)$. Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a $BV^{\alpha,p}$ function.

1. Introduction 1.1. The fractional variation. For a parameter α ∈ (0, 1) and an exponent p ∈ [1, +∞], the space of L p functions with bounded fractional variation is where is the (total) fractional variation of the function f ∈ L p (R n ). Here and in the following, for sufficiently smooth functions and vector-fields, we let dy, x ∈ R n , and div α ϕ(x) = µ n,α R n (y − x) · (ϕ(y) − ϕ(x)) |y − x| n+α+1 dy, x ∈ R n , be the fractional gradient and the fractional divergence operators respectively, where µ n,α is a suitable renormalizing constant depending on n and α only. The above fractional operators are dual, in the sense that The fractional variation was considered by the first and the third authors in the work [7] in the geometric framework p = 1, also in relation with the naturally associated notion of fractional Caccioppoli perimeter. The fractional variation of an L p function for an arbitrary exponent p ∈ [1, +∞] was then studied by the same authors in the subsequent paper [8], in connection with some embedding-type results arising from some optimal inequalities proved by the second author [31,32].
Since the first appearance of the fractional gradient [16], the literature around ∇ α and div α has been rapidly growing in various directions, such as the study of PDEs [25,26,28,29] and of functionals [4,5,17] involving these fractional operators, the discovery of new optimal embedding estimates [27,31,32] and the development of a distributional and asymptotic analysis in this fractional framework [6][7][8]30]. We also refer the reader to the survey [33] and to the monograph [24].
At the present stage of the theory, the fine properties of functions having finite fractional variation are not completely understood and, to our knowledge, only some results [7] in the geometric regime p = 1 are available in the literature.
Besides providing a general treatment of the space BV α,p (R n ), in the present paper we aim to develop the existing theory in this direction. On the one side, we study the relation between the fractional variation and the Hausdorff measure. On the other side, we establish the existence of the precise representative of a BV α,p function.
1.2. The Hausdorff dimension of the fractional variation. The natural idea behind the definition of the space BV α,p (R n ) is that a function f ∈ L p (R n ) belongs to BV α,p (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 ), generalizing the integration-by-parts formula (1.3). In the classical integer case α = 1, the variation of a function f ∈ BV (R n ) is known to satisfy |Df | ≪ H n−1 , (1.4) where H s is the s-dimensional Hausdorff measure. If f = χ E for some measurable set E ⊂ R n , then it actually holds that (1.5) where F E is the De Giorgi reduced boundary of E, see the monographs [3,20]. Roughly speaking, formulas (1.4) and (1.5) mean that the variation measure of a BV function in R n lives on sets with Hausdorff dimension n − 1 at least. By the analogy between the integer and the fractional settings, one may expect that a similar phenomenon should occur also for the fractional variation of order α ∈ (0, 1) on a set of Hausdorff dimension n − α at least. In [7,Corollary 5.4], the first and the third authors confirmed this parallelism by showing that, for a measurable set E ⊂ R n such that χ E ∈ BV α (R n ) (or, more generally, for any measurable set having locally finite fractional Caccioppoli perimeter, see [7,Definition 4.1]), it holds that |D α χ E | ≤ c n,α H n−α F α E, (1.6) where c n,α > 0 depends on n and α only and F α E is the fractional analogue of the De Giorgi reduced boundary (1.5), the so-called fractional reduced boundary of E, see [7,Definition 4.7]. However, as shown in [7,Lemma 3.28] by the same authors, if f ∈ BV α (R n ) then the function u = I 1−α f (where I s is the Riesz potential of order s ∈ (0, n), see below for the precise definition) does satisfy |Du|(R n ) < +∞, with In particular, by combining (1.4) with the above (1.7), we immediately get that |D α f | ≪ H n−1 (1.8) for all f ∈ BV α (R n ), thus ruling out the existence of a coarea formula in this fractional setting, see [7,Corollary 5.6].
Equations (1.6) and (1.8) illustrate the richness arising from the innocent-looking definition (1.2) and lead to the idea that the behavior of the fractional variation of a function f ∈ L p (R n ) may depend on its integrability exponent p ∈ [1, +∞]. Our first main result provides a rigorous formulation of this intuitive idea and can be stated as follows. Theorem 1.1 (Absolute continuity properties of the fractional variation). Let α ∈ (0, 1), p ∈ [1, +∞] and assume that f ∈ BV α,p (R n ). We have the following cases: As shown by Theorem 1.1, the fractional variation in the subcritical regime p < n 1−α is comparable with the Hausdorff measure of dimension n − 1, in accordance with (1.8). In fact, we can actually prove a deeper property, in analogy with the relation (1.7). Precisely, the Riesz potential operator is continuous whenever p < n 1−α (see Proposition 4.1(i) below for the detailed statement), from which item (i) in Theorem 1.1 immediately follows. Here and in the following, for any p ∈ [1, +∞], we let be the space of L p functions having finite variation, extending the definition in (1.1) to the integer case α = 1.
In the supercritical regime p ≥ n 1−α instead, the fractional variation is comparable with the Hausdorff measure of dimension n − α − n p , thus recovering (1.6) in the case p = +∞. The proof of item (ii) of Theorem 1.1 is more delicate and requires a finer analysis. The overall idea is to adapt the strategy developed in [7,Section 5] for sets with (locally) finite fractional Caccioppoli perimeter to the present more general L p framework. The key role in this approach is played by the following decay estimate for the fractional variation of a function f ∈ BV α,p (R n ) with p ≥ n n−α , valid for |D α f |-a.e. x ∈ R n and all r > 0 sufficiently small, where c n,α,p > 0 is a constant depending on n, α, and p only (see Theorem 4.3 below for the precise statement). The validity of (1.9) is suggested by the following heuristic argument, valid for all f ∈ BV α,p (R n ) such that which gives (1.9). Without (1.10), the decay estimate (1.9) is a consequence of some new integrability properties in Lorentz spaces of the fractional gradient and of an integrationby-parts formula of BV α,p functions on balls which may be of some independent interest (see Theorem 3.9 and Theorem 3.13, respectively). We note that Theorem 1.1 still holds even in the limit as α → 1 − . Indeed, for all p ∈ [1, +∞] and f ∈ BV 1,p (R n ) we get that |Df | ≪ H n−1 , since point (i) now applies to all p ∈ [1, +∞), while point (ii) refers only to p = +∞, for which we have n−1− n p = n−1. This is in fact a well-known result for functions in BV loc (R n ), see [3,Lemma 3.76] for instance. On the contrary, Theorem 1.1 is not optimal in the limit as α → 0 + . Indeed, in virtue of [6, Theorem 3.3 and Remark A.3], if p ∈ [1, +∞) and f ∈ BV 0,p (R n ), then |D 0 f | ≪ L n (where the space BV 0,p (R n ) is defined as in (1.1) with α = 0, see [6] for a more detailed presentation).
1.3. The precise representative of a BV α,p function. Formulas (1.4) and (1.5) suggest that the set of discontinuity points (in the measure-theoretical sense) of a BV function should have Hausdorff dimension n − 1. In more precise terms, if f ∈ BV (R n ), then the limit exists for H n−1 -a.e. x ∈ R n . In fact, the limit in (1.11) can be strengthened as , then J f is empty). The function f ⋆ defined by (1.11) is the so-called precise representative of the function f (by convention, we set f * (x) = 0 if the limit in (1.11) does not exist). The well-posedness of the precise representative (1.11) of a BV α,p function is not known at the present moment. Our second main result moves in this direction and can be briefly stated as follows (for a more precise statement, we refer the reader to Corollary 5.7 below). Theorem 1.2 (The precise representative of a BV α,p function). Let α ∈ (0, 1), p ∈ [1, +∞] and ε > 0. If f ∈ BV α,p (R n ), then the limit f ⋆ (x) exists for H n−α+ε -a.e. x ∈ R. Moreover, for any such point x ∈ R n , it holds that The idea behind the proof of Theorem 1.2 is very simple and relies on three ingredients naturally arising from our general investigation of the BV α,p (R n ) space. First, we show that C ∞ c functions are dense in energy in BV α,p (R n ) provided that p ∈ 1, n n−α , extending the approximation [7, Theorem 3.8] already proved by the first and the third author in the geometric regime p = 1. Second, by combining this approximation with an optimal embedding inequality [32] due to the second author, we establish a fractional analogue of the Gagliardo-Nirenberg-Sobolev inequality, that is, BV α,p (R n ) ⊂ L n n−α (R n ) with continuous inclusion. Third, we exploit this fractional embedding inequality to prove the continuous inclusion of BV α,p (R n ) into some Bessel potential space of suitable fractional order. At this point, the existence of the precise representative of a BV α,p function for p < n n−α can be inferred from the known theory of Bessel potential spaces, see [1, Section 6.1] for example. The remaining exponents p ≥ n n−α can be recovered from the previous analysis by a simple cut-off argument that may be of some separate interest (see Lemma 5.6 for the detailed statement).
1.4. Future developments. Generally speaking, the precise representative of a function turns out to be the correct object when dealing with the product between the function itself and a sufficiently well-behaved measure.
For example, the precise representative allows to state the general Leibniz rule for the product of two BV functions.
(1.12) Note that the two products appearing in right-hand side of (1.12) are well posed thanks to the combination of the absolute continuity property of the variation (1.4) and the existence of the precise representative (1.11).
With Theorem 1.1 and Theorem 1.2 at hand, the analysis developed in the present work naturally leads to study the interactions between the fractional variation measure and the precise representative of BV α,p functions, aiming at a more general formulation of the Leibniz rule and of the Gauss-Green formula in this fractional setting. These results are the main topic of the subsequent paper [9].
1.5. Organization of the paper. The paper is organized as follows.
In Section 2, we quickly set up the notation used throughout the entire work and recall the elementary features of the fractional operators involved.
In Section 3, we carry out the general analysis of the BV α,p (R n ) space. On the one side, we deal with the approximation in energy by smooth functions and the consequent embedding theorems in Lebesgue and Bessel potential spaces, preparing the ground for the proof of Theorem 1.2. On the other side, we treat some integration-by-parts formulas of BV α,p functions against rough test vector-fields and on balls, developing the tools needed for the proof of the decay estimate (1.9) and thus of Theorem 1.1.
In Section 4, we prove our first main result Theorem 1.1. We divide the proof into two parts, dealing with the subcritical regime (i) and the supercritical regime (ii) separately, see Proposition 4.1(i) and Corollary 4.4 respectively. At the end of this section, we provide two examples to show the sharpness of our result in the one-dimensional case n = 1.
In Section 5, after having recalled some known properties of the fractional capacity in Bessel potential spaces and having proved a localization lemma for BV α,p functions, we end our paper with the proof of our second main result Theorem 1.2.

Preliminaries
2.1. 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 the previous works [6][7][8].
Given an open set Ω ⊂ R n , we say that a set E is compactly contained in Ω, and we write E ⋐ Ω, if the E is compact and contained in Ω. We let L n and H α be the n-dimensional Lebesgue measure and the α-dimensional Hausdorff measure on R n , respectively, with α ∈ [0, n]. 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 B r (x) the standard open Euclidean ball with center x ∈ R n and radius r > 0. We let B r = B r (0). For all β > 0, we set where Γ is Euler's Gamma function, and we recall that |B 1 | = ω n and H n−1 (∂B 1 ) = nω n .
For k ∈ N 0 ∪ {+∞} and m ∈ N, we let C k c (Ω; R m ) and Lip c (Ω; R m ) be the spaces of C k -regular and, respectively, Lipschitz-regular, m-vector-valued functions defined on R n with compact support in the open set Ω ⊂ R n .
For m ∈ N, the total variation on Ω of the m-vector-valued Radon measure µ is defined as We thus let M (Ω; R m ) be the space of m-vector-valued Radon measure with finite total variation on Ω. We say that (µ k ) k∈N ⊂ M (Ω; R m ) weakly converges to µ ∈ M (Ω; R m ), and we write for all ϕ ∈ C 0 c (Ω; R m ). Note that we make a little abuse of terminology, since the limit in (2.1) actually defines the weak*-convergence in M (Ω; R m ).
For any exponent p ∈ [1, +∞], we let L p (Ω; R m ) be the space of m-vector-valued Lebesgue p-integrable functions on Ω.
We let be the space of m-vector-valued Sobolev functions on Ω, see for instance [18,Chapter 11] for its precise definition and main properties. We also let be the space of m-vector-valued functions of bounded variation on Ω, see for instance [3,Chapter 3] or [11,Chapter 5] for its precise definition and main properties. We also let be the space of m-vector-valued fractional Sobolev functions on Ω, see [10] 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 Ω.
In order to avoid heavy notation, if the elements of a function space F (Ω; R m ) are real-valued (i.e., m = 1), then we will drop the target space and simply write F (Ω).
Given α ∈ (0, n), we let be the Riesz potential of order α of f ∈ 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 . As a consequence, the operator in (2.2) 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 (2.3) and (2.4), we refer the reader to [34, Chapter V, Section 1] and to [14, Section 1.2.1].
Given α ∈ (0, 1), we also let Finally, we let for a more detailed exposition. We warn the reader that the definition in (2.6) agrees with the one in [35] and differs from the one in [14,34] for a minus sign, so that R = ∇I 1 on C ∞ c (R n ) in particular. The Riesz transform (2.6) is a singular integral of convolution type, thus in particular it defines a continuous operator R : L p (R n ) → L p (R n ; R n ) for any given p ∈ (1, +∞), see [13,Corollary 5.2.8]. We also recall that its components R i satisfy 2.2. The operators ∇ α and div α . We briefly recall the definitions and the essential features of the non-local operators ∇ α and div α , see [6][7][8]31] and [24,Section 15.2]. Let α ∈ (0, 1) and set We let The non-local operators ∇ α and div α are well defined in the sense that the involved integrals converge and the limits exist. Moreover, since for all ϕ ∈ Lip c (R n ). From the above expressions, it is not difficult to recognize that, given f ∈ Lip c (R n ) and ϕ ∈ Lip c (R n ; R n ), it holds that for all p ∈ [1, +∞], see [7,Corollary 2.3]. Finally, the fractional operators ∇ α and div α are dual, in the sense that With a slight abuse of notation, in the following we let ∇ 1 and div 1 be the usual (local) gradient and divergence. Note that this notation is coherent with the asymptotic behavior of the fractional operators ∇ α and div α when α → 1 − for sufficiently regular functions, see the analysis made in [8].
In this section we study the main properties of the BV α,p functions, following the strategy adopted in [7, Section 3].
3.1. Definition of BV α,p (R n ). Let α ∈ (0, 1] and p ∈ [1, +∞]. We say that a function Section 3] for the case p = 1 and the discussion in [8, Section 3.3] for the case p ∈ (1, +∞]. In the case p = 1, we simply write is a Banach space and that the fractional variation defined in (3.1) is lower semicontinuous with respect to the L p -convergence. Similarly as it was proved in the case p = 1 in [7, Theorem 3.2], it is possible to show the following result relating non-local distributional gradients of BV α,p functions to vector valued Radon measures.

Approximation by smooth functions.
Here and in the rest of the paper, we let (̺ ε ) ⊂ C ∞ c (R n ) be a family of standard mollifiers as in [7, Section 3.3]. The following approximation theorem is the extension to BV α,p functions of [7, Lemma 3.5 and Theorem 3.7]. We leave its proof to the interested reader.
The following result extends the approximation by test functions given in [7, Theorem 3.8] to functions in BV α,p (R n ) for α ∈ (0, 1) and all exponents p ∈ 1, n n−α . In the proof below and in the following, we let 3)

Theorem 3.3 (Approximation by
We can also assume that η R (x) = η 1 ( x R ) for all x ∈ R n and R > 0. The proof now goes as the one of [7, Theorem 3.8] with minor modifications. We simply have to check that Indeed, by Hölder's inequality, we have where 1 p + 1 q = 1, and a simple change of variables shows that for all R > 0. The claim in (3.4) thus follows provided that n q − α < 0, which is equivalent to p ∈ 1, n n−α , and the proof is complete.
For the sake of completeness, we also treat the case α = 1 of the previous result.
and the conclusion immediately follows.

Gagliardo-Nirenberg-Sobolev inequality.
Thanks to the approximation by test functions given by Theorem 3.3, we can extend [7, Theorem 3.9] and prove the analogue of the Gagliardo-Nirenberg-Sobolev inequality for the space BV α,p (R n ) whenever p ∈ 1, n n−α .
Theorem 3.5 (Gagliardo-Nirenberg-Sobolev inequality). Let α ∈ (0, 1) and let p ∈ 1, n n−α . There exists a constant c n,α > 0, depending on n and α only, such that Proof. Assume that f ∈ C ∞ c (R n ) to start. Arguing as in the proof of [8, Theorem 3.8], we can estimate |f | ≤ c n,α I α |∇ α f | for some constant c n,α > 0 depending only on n and α (possibly varying from line to line). Thanks to the Hardy-Littlewood-Sobolev inequality, we immediately deduce that . Moreover, if n ≥ 2, then we can apply [32, Theorem 1.1] to the vector field F = ∇ α f in order to get that and R : L n n−α ,1 (R n ) → L n n−α ,1 (R n ; R n ) strongly (recall the definition in (2.6) and the properties of the Riesz transform), we immediately deduce that The conclusion then follows by combining a standard approximation argument exploiting Theorem 3.3 with Fatou's Lemma.
For α = 1, the previous result can be stated as follows. Proposition 3.6 (Alvino's inequality). Let n ∈ N and p ∈ [1, +∞). If n ≥ 2 and p ≤ n n−1 , then there exists a dimensional constant c n > 0 such that Proof. While the case n = 1 is a well-known property of functions having bounded variation, the case n ≥ 2 follows from Alvino's inequality [2] for functions in BV (R n ) (also see [33,Section 5]) in combination with Proposition 3.4. We leave the details to the interested reader.
3.4. The space S α,p (R n ) and the embedding BV α,p ⊂ S β,q . Let α ∈ (0, 1) and p ∈ [1, +∞]. We define the weak fractional α-gradient of a function f ∈ L p (R n ) as the function for all ϕ ∈ C ∞ c (R n ; R n ). We hence let the linear space be the distributional fractional Sobolev space.
We now want to provide a rigorous formulation of the naïve intuition that if the order of differentiability decreases, then the order of integrability increases, that is to say, if ∇ α f ∈ L p (R n ; R n ) for some α ∈ (0, 1) and p ∈ [1, +∞), then ∇ β f ∈ L q (R n ; R n ) for some lower fractional differentiation order β < α and some higher integrability exponent q > p (depending on β). For p > 1, the above principle is a simple consequence of the known embedding theorems between the fractional Bessel potential spaces, thanks to the aforementioned identification S α,p (R n ) = L α,p (R n ).
The more delicate case p = 1 is covered in Theorem 3.7 below. We refer the reader also to [7,Theorem 3.32] and to [8, Propositions 3.2(i), 3.3 and 3.12] for similar results in this direction.
Theorem 3.7 (BV α,p ⊂ S β,q for p < n n−α ). Let α, β ∈ (0, 1], with β < α, and let p, q ∈ [1, +∞] be such that p ≤ q < n n+β−α . Then BV α,p (R n ) ⊂ S β,q (R n ) continuously. Proof. Assume that f ∈ C ∞ c (R n ) and let R > 0. Arguing as in the proof of [8, Proposition 3.12], we can estimate for all x ∈ R n . On the one side, we can write for all x ∈ R n , so that by Young's inequality. On the other side, arguing as in the proof of [8, Proposition 3.12], we can write by Minkowski's integral inequality, for some constant c n,α,β > 0 depending only on n, α and β. Hence we get that for some constant c n,α,β,q > 0 depending only on n, α, β and q. Choosing The conclusion thus follows from Theorem 3.3 and Theorem 3.5 (Proposition 3.4 and Proposition 3.6 in the case α = 1) via a routine approximation argument, since clearly p < n n−α . 3.5. Generalized integration-by-parts formula for BV α,p functions. The following result is a generalization of the fractional integration-by-parts formula (3.2) (the case p = 1 was actually already analyzed in [8,Proposition 2.7]). This result will be useful for integrating by parts BV α,p functions on balls, see Theorem 3.13 below. Proposition 3.8 (W 1,q ∩ C b -regular test). Let α ∈ (0, 1) and let p, q ∈ [1, +∞] be such Proof. We start by noticing that, under the above assumptions, div α ϕ ∈ L q (R n ), thanks to [ Step 1. Assume ϕ ∈ W 1,q (R n ; R n ) ∩ Lip b (R n ; R n ) ∩ C ∞ (R n ; R n ) and let (η R ) R>0 ⊂ C ∞ c (R n ) be a family of cut-off functions as in [7,Section 3.3]. On the one hand, since for all R > 0, by Lebesgue's Dominated Convergence Theorem we have On the other hand, by [8, Lemmas 2.2 and 2.5] we can write Finally, on the one side we can estimate for all R > 0, while, on the other side, Indeed, f ϕ ∈ L 1 (R n ; R n ) and (3.6) follows by Lebesgue's Dominated Convergence Theorem. We also claim that Indeed, since ϕ ∈ Lip b (R n ; R n ) and η R L ∞ (R n ) ≤ 1 for all R > 0, by Lebesgue's Dominated Convergence Theorem we get that for all x ∈ R n . Moreover, for a.e. x ∈ R n we have Therefore, combining (3.8) and (3.9), again by Lebesgue's Dominated Convergence Theorem we get (3.7). Thus (3.5) is proved whenever ϕ ∈ W 1,q (R n ; R n ) ∩ Lip b (R n ; R n ) ∩ C ∞ (R n ; R n ).
Step 2. Now assume ϕ ∈ W 1,q (R n ; R n ) ∩ C b (R n ; R n ) (if q = 1, then we instead take ϕ ∈ BV (R n ; R n )∩C b (R n ; R n )) and let (̺ ε ) ε>0 ⊂ C ∞ c (R n ) be a family of standard mollifiers as in [7,Section 3.3]. Then ϕ ε = ̺ ε * ϕ ∈ W 1,q (R n ; R n ) ∩ Lip b (R n ; R n ) ∩ C ∞ (R n ; R n ) and so, by Step 1, we can write for all ε > 0. On the one hand, it is not difficult to see that div α ϕ ε = ̺ ε * div α ϕ for all ε > 0, so that lim On the other hand, since ϕ ∈ C b (R n ; R n ), we get by Lebesgue's Dominated Convergence Theorem (with respect to the finite measure |D α f |). This concludes the proof of (3.5).
3.6. Integrability of D α in Lorentz space for BV 1,p functions. We now need to focus on the weak integrability properties of the operator D α defined in (3.3) when applied to functions belonging to BV 1,p (R n ). This analysis will be useful for studying the integrability properties of the fractional gradient of the characteristic function of a ball, see Corollary 3.10 below. This result, in turn, will be useful in the proof of the integrationby-parts formula of BV α,p functions on balls in Theorem 3.13.
The operator D α : BV 1,p (R n ) → L pα,∞ (R n ) is well defined and satisfies for all f ∈ BV 1,p (R n ), where c n,α,p > 0 is a constant depending on n, α and p only.
Proof. The case p = 1 is easy, since (3.10) holds in the following stronger form , whose simple proof is left to the reader (for instance, one can follow the strategy of the proof of [10, Proposition 2.2]). In the following, we thus assume that p > 1. We now divide the proof into three steps. Step for all x ∈ R n . Since f ∈ L p (R n ), by Hölder's inequality in Lorentz spaces (see [23,Theorem 3.4] and [33, Theorem 5.1]) and the well-known continuity properties of the maximal function, we get that where c n,α,p > 0 is a constant depending only on n, α and p.

be a family of cutoff functions as in [7, Section 3.3] and define f
for a.e. x ∈ R n . Inequality (3.10) thus follows again by [13, Exercise 1.
From Theorem 3.9, we immediately deduce the following integrability properties of the fractional gradient of the indicator function of a ball. Corollary 3.10 (Integrability of ∇ α χ Br(x) ). Let α ∈ (0, 1) and let p ∈ 1, 1 α . There exists a constant c n,α,p > 0, depending on n, α and p only, such that ∇ α χ Br(x) L p (R n ; R n ) = c n,α,q r n p −α (3.11) for all x ∈ R n and r > 0.
Proof. Let x ∈ R n and r > 0 be fixed. Since y − x r by the rescaling property of ∇ α , we immediately get that [7,Proposition 4.8] and Theorem 3.9, the conclusion follows by observing that α . This interpolation result can be proved for instance by arguing as in the proof of [13, Proposition 1.1.14] with some minor modifications. We leave the simple details to the interested reader. Remark 3.11 (The case n = 1 in Corollary 3.10). The estimate in (3.11) in the case n = 1 can be obtained by a direct computation from the explicit formula given by [7,Example 4.11]. In particular, we deduce that From Proposition 3.6, we immediately deduce the following improvement of Theorem 3.9. We leave its simple proof to the reader. Corollary 3.12 (Improved weak integrability of D α ). Let α ∈ (0, 1), n ∈ N and p ∈ [1, +∞) be such that p ≤ n n−1 for n ≥ 2.

Integration by parts of BV α,p functions on balls.
We are now ready to state and prove the following integration-by-parts of BV α,p functions on balls, which is a generalization of [7, Theorem 5.2] to BV α,p functions for p ∈ 1 1−α , +∞ . This result will be the central ingredient of the proof of the decay estimates for BV α,p functions in Theorem 4.3 below.

Absolute continuity of the fractional variation
In this section, we prove our first main result Theorem 1.1. We divide the proof into two parts, dealing with the subcritical regime (i) and the supercritical regime (ii) separately, see Proposition 4.1(i) and Corollary 4.4 respectively. At the end of this section, we provide two examples to show the sharpness of our result in the one-dimensional case n = 1.
4.1. The subcritical regime p ∈ 1, n 1−α . Thanks to [7,Lemma 3.28] In the following result, which is a generalization of [7, Lemma 3.28] to the present setting, we show that this phenomenon is typical of the functions belonging to BV α,p (R n ) in the subcritical regime p ∈ 1, n 1−α . Proposition 4.1 (Relation between BV α,p (R n ) and BV 1,p (R n )). Let α ∈ (0, 1), p ∈ 1, n 1−α and q = np As a consequence, we have |D α f | ≪ H n−1 for all f ∈ BV α,p (R n ) and the operator As a consequence, the operator (−∆) Proof. Let s = p p−1 and note that r = ns n+(1−α)s ∈ 1, n 1−α . We prove the two properties separately.
Proof of (i). Let f ∈ BV α,p (R n ). By the Hardy-Littlewood-Sobolev inequality, we immediately get that u = I 1−α f ∈ L q (R n ). Given ϕ ∈ C ∞ c (R n ; R n ), we clearly have I 1−α |divϕ| ∈ L s (R n ), because |divϕ| ∈ L r (R n ). Hence, by Fubini Theorem, we have for all ϕ ∈ C ∞ c (R n ; R n ), proving that D α f = Du in M (R n ; R n ). The remaining part of the statement in (i) follows easily.
Proof of (ii). Let p ∈ 1, n n−α and u ∈ BV 1,p (R n ). Since p < n n−α , we can apply Theorem 3.7 to get that BV 1,p (R n ) ⊂ S 1−α,p (R n ) with continuous inclusion, so that f = (−∆) 1−α 2 u ∈ L p (R n ) (thanks to the identification S 1−α,p (R n ) = L 1−α,p (R n ) following from [6, Corollary 2.1], also see the discussion in [6, Section 2.1]) and thus I 1−α f ∈ L q (R n ) by the Hardy-Littlewood-Sobolev inequality. Since p < n n−α , we also have that p < q < n n−1 and thus BV 1,p (R n ) ⊂ L q (R n ) with continuous inclusion by Proposition 3.6. Hence u ∈ L q (R n ) and we can now claim that I 1−α f = u in L q (R n ). Indeed, this is easily verified if u ∈ C ∞ c (R n ) by applying the Fourier transform (see [19,Lemma 2.3] for instance), so that the claim follows by a plain approximation argument. Therefore, by applying Fubini Theorem again, we can write (4.1) and prove that D α f = Du in M (R n ; R n ), reaching the conclusion. 4.1(ii)). The validity of Proposition 4.1(ii) when p = 1 was already proved in [7,Lemma 3.28]. We also refer the reader to [17,Proposition 3.1], in which the authors prove that, if u ∈ W 1,p (R n ) for some p ∈ [1, +∞], then f = (−∆)

Remark 4.2 (About Proposition
4.2. The supercritical regime p ∈ n 1−α , +∞ . We now focus on the absolute continuity property of the fractional variation with respect to the Hausdorff measure for functions belonging to BV α,p (R n ) in the supercritical regime p ∈ n 1−α , +∞ . The crucial tool in this case is provided by the following important consequence of Theorem 3.13, which extends [7,Theorem 5.3] to the present setting. . Let α ∈ (0, 1) and p ∈ 1 1−α , +∞ . There exist two constants A n,α,p , B n,α,p > 0, depending on n, α and p only, with the following property. If f ∈ BV α,p (R n ) then, for |D α f |-a.e. x ∈ R n , there exists r x > 0 such that and |D Proof. Since f ∈ BV α,p (R n ), by the Polar Decomposition Theorem for Radon measures there exists a Borel vector valued function σ α f : R n → R n such that We divide the proof into two steps, dealing with the two estimates separately.
Step 1: proof of (4.2). Let σ α f : R n → R n be as in (4.4) and let x ∈ R n be such that |σ α f (x)| = 1. Given r > 0, we define the vector field ϕ : R n → R n by setting for all y ∈ R n . We clearly have that ϕ x,r ∈ Lip c (R n ; R n ) with ϕ L ∞ (R n ;R n ) ≤ 1. Thus, on the one hand, we can find r x ∈ (0, 1) such that for all r ∈ (0, r x ). On the other hand, by (3.12) we can write for L 1 -a.e. r ∈ (0, r x ). We now estimate the three terms in the right-hand side of (4.7) separately. For the first term, recalling the definition of ϕ x,r in (4.5), we have After some elementary computations, we get for some constant C n,α > 0 depending only on n and α. Note that the integral appearing in the right-hand side of (4.8) converges if and only if αq < 1, that is, p > 1 1−α . We thus get for some constant C n,α,q > 0 depending only on n, α and q. For the second term in the right-hand side of (4.7), we have R n f ϕ x,r · ∇ α χ Br(x) dy ≤ f L p (B 2r (x)) ∇ α χ Br(x) L q (R n ;R n ) ≤ C n,α,q f L p (B 2r (x)) r n q −α (4.10) thanks to Corollary 3.10, for some constant C n,α,q > 0 depending only on n, α and q. Finally, observing that ϕ x,r (x + ry) = ϕ 0,1 (y) for all y ∈ R n , a simple change of variables gives div α Thus, for the third and last term in the right-hand side of (4.7), we have where C n,α,q = div α NL (χ B 1 , ϕ) L q (R n ) (which is finite thanks to [7, Lemma 2.7 and Remark 2.8]). Combining (4.6) with (4.7), (4.9), (4.10) and (4.12), we get (4.2) with a simple continuity argument.
Thanks to Theorem 4.3 and extending [7,Corollary 5.4] to the present setting, we are now ready to state and prove the following absolute continuity property of the fractional variation for BV α,p functions with p ∈ 1 1−α , +∞ . Note that the result below is truly interesting only for p ∈ n 1−α , +∞ , due to Theorem 1.1(i) (see also Proposition 4.1) and the fact that Inequality (4.13) thus follows from [3, Theorem 2.56].
Remark 4.6 (The limit as α → 1 − ). It is somewhat interesting to observe that Corollary 4.4 still holds true if we send α → 1 − . Indeed, such a limit case would apply only to functions f ∈ BV 1,∞ (R n ), for which it is well known (see [3,Theorem 3.77, Theorem 3.78 and equation (3.90)], for instance) that where J f is the jump set, so that Z 1,∞ f could be any |Df |-negligible subset of R n \ J f .

4.3.
Two examples in one dimension. We conclude this section by discussing the optimality of the absolute continuity properties of the fractional variation stated in Theorem 1.1 in the one-dimensional case n = 1. We begin with the following example, which is borrowed from [7, Theorem 3.26].
Example 4.7 (Proposition 4.1(i) is sharp for n = 1). Let α ∈ (0, 1) and consider with |D α f α | ≪ H ε for all ε > 0. This proves that the absolute continuity property of the fractional variation stated in Theorem 1.1(i) is sharp for n = 1.
We now prove the following result, which combines the properties of the function f α introduced in Example 4.7 with the decay properties of a finite Radon measure.
14) where τ x (y) = y + x for all x, y ∈ R. In addition, if there exist C, ε > 0 such that ν (x − r, x + r) ≤ Cr ε for all x ∈ R and r > 0, (4.15) then (4.16) Proof. We divide the proof into two steps.
Step 1. Let ν ∈ M (R). We start by showing that u α ∈ BV α,p (R) for all p ∈ 1, 1 1−α and that it satisfies (4.14). Indeed, by Young's inequality (for Radon measures) we can estimate u α L 1 (R) ≤ f α L 1 (R) |ν|(R). Moreover, thanks to the translation invariance of div α and exploiting the explicit expression of f α given in Example 4.7, we can write In addition, by Jensen's inequality and Tonelli's Theorem we can estimate 1 1−α , thanks to the integrability properties of f α given in Example 4.7.
Step 2. We prove that (4.15) implies (4.16). To this aim, let δ > 0 and q = p p−1 . Since |f α | = |f α | δ q |f α | 1− δ q , by Hölder's inequality we get for x ∈ R. We now recall the explicit expression of f α in Example 4.7 and write where we have set for all x ∈ R, r ∈ (0, 1) and j ∈ N for brevity. Now, on the one hand, if y ∈ −∞, for all x ∈ R. On the other hand, for all x ∈ R and j ∈ N, we have Reasoning analogously, we obtain for all x ∈ R and j ∈ N. Therefore, inserting (4.18), (4.19) and (4.20) in (4.17), we conclude that for all x ∈ R, where C α,ε,δ > 0 is constant depending on α, ε, and δ which is finite provided that we choose δ < ε 1−α , as we are assuming from now on. We thus get Now, recalling Example 4.7, we immediately see that Since one easily recognizes that < 1 for all α ∈ (0, 1) and δ > 0, the second condition on p in (4.22) can be dropped. As for the first condition on p in (4.22), it is readily seen that Finally, in the case ε ∈ (1 − α, 1], we exploit (4.21) for δ = 1 in order to conclude that for all x ∈ R, which implies that u α ∈ L ∞ (R). The conclusion thus follows.
Therefore, the absolute continuity property of the fractional variation stated in Theorem 1.1(ii) is sharp for n = 1.
The mapping Cap α,p can be extended to more general sets via the following standard routine. If A ⊂ R n is an open set, then we set We now recall the notion of (α, p)-quasievery point, see [1, Definition 2.2.5].
5.2. The precise representative. We now study the precise representatives of BV α,p functions by combining the embedding proved in Theorem 3.7 with the results already known in the literature for the precise representatives of functions in fractional Bessel potential spaces.
We begin by recalling the definition of quasicontinuity. For the integer case α = 1, we refer the reader to [1, Definition 6.1.1] and [11,Definition 4.11].
Here and in the following, the precise representative of a function u ∈ L 1 loc (R n ; R m ) is defined as u ⋆ (x) = lim r→0 + − Br(x) u(y) dy, x ∈ R n , if the limit exists, otherwise u ⋆ (x) = 0 by convention. The following result provides a precise description of the continuity properties of the precise representative of a function in S α,p (R n ) for p ∈ 1, n α . We refer the reader to [1, Theorem 6.2.1] for the proof.
Theorem 5.4 (Quasicontinuity of S α,p functions). Let α ∈ (0, 1) and p ∈ 1, n α . If f ∈ S α,p (R n ), then f ⋆ is an (α, p)-quasicontinuous representative of f and Thanks to the embedding proved in Theorem 3.7, we immediately deduce the following result concerning the quasicontinuity of the functions in BV α,p (R n ).