FRACTIONAL DIVERGENCE-MEASURE FIELDS, LEIBNIZ RULE AND GAUSS–GREEN FORMULA

. Given α ∈ (0 , 1] and p ∈ [1 , + ∞ ], we deﬁne the space DM α,p ( R n ) of L p vector ﬁelds whose α -divergence is a ﬁnite Radon measure, extending the theory of divergence-measure vector ﬁelds to the distributional fractional setting. Our main results concern the absolute continuity properties of the α -divergence-measure with respect to the Hausdorﬀ measure and fractional analogues of the Leibniz rule and the Gauss–Green formula. The sharpness of our results is discussed via some explicit examples.

The basic definition goes as follows (see Section 2.1 for the notation).Given p ∈ [1, +∞], we say that a vector field F ∈ L p (R n ; R n ) has divergence-measure, and we write F ∈ DM 1,p (R n ), if there exists a finite Radon measure divF ∈ M(R n ) such that for all ξ ∈ C ∞ c (R n ).The integration-by-parts formula (1.1) clearly generalizes the usual Divergence Theorem.In fact, if the vector field F is sufficiently regular, say F ∈ Lip loc (R n ; R n ), then divF = divF L n in (1.1), where L n is the n-dimensional measure.
As for the analogous case of functions with bounded variation, the two principal questions regarding DM 1,p vector fields concern the absolute continuity properties of the divergence-measure with respect to the Hausdorff measure H s , for s ∈ [0, n], and the well-posedness of a Leibniz rule with suitable scalar functions.
The absolute continuity properties of the divergence-measure of a DM 1,p vector field with respect to the Hausdorff measure hold as follows, see [49,Th. 3.2 and Exam. 3.3].
Theorem 1.1 (Absolute continuity properties of the divergence-measure).Assume that F ∈ DM 1,p The Leibniz rule involving DM 1,p vector fields and Sobolev functions is stated in Theorem 1.2 below, for which we refer to [7,Prop. 3.1], [8,Th. 3.1], [13,Th. 3.2.3]and [34,Th. 2.1].Here and in the following, for x ∈ R n , we let g(y) dy if the limit exists, 0 otherwise, (1.2) be the precise representative of g ∈ L 1 loc (R n ).For the notion of (Anzellotti's) pairing measure briefly recalled in the statement, we refer the reader to [3,Def. 1.4], [8,Th. 3.2], or to [23,Sec. 2.5] for a more general formulation.Theorem 1.2 (Leibniz rule for DM 1,p vector fields and weakly differentiable functions).Let p, q ∈ [1, +∞] be such that 1  p 3) is the pairing measure between F and Dg, where (F, Dg) is the unique weak limit 3) for p < +∞).For p < +∞, the function g ⋆ appearing in (1.3) can be defined in a more specific way.For p ∈ 1, n n−1 , g ⋆ can be taken as the continuous representative of g.Instead, for p ∈ n n−1 , +∞ , g ⋆ can be taken as the q-quasicontinuous representative of g.See [13,Sec. 3.2] for a more detailed discussion.
1.2.Fractional divergence-measure fields.The aim of the present note is to introduce a fractional analogue of the theory of divergence-measure fields, following the distributional approach to fractional spaces recently introduced and studied by the authors and collaborators in the series of papers [4,[17][18][19][20][21][22].For results close to the main topic of this paper, we also refer to [42,53,54], even though our point of view is different.
In the fractional setting, for α ∈ (0, 1), one has the integration-by-parts formula for all functions ξ ∈ Lip c (R n ) and vector fields where is the fractional α-gradient, is the fractional α-divergence, and is a renormalization constant, see [18, Sec.2.2] for a detailed exposition.According to the main results of [4,19], with a slight (but justified) abuse of notation, we may identify (1.5) with the usual gradient ∇ for α = 1, and with the vector-valued Riesz transform ∇ 0 = R for α = 0 (see Section 2.1 for the definition).
As already done by the authors in the case of scalar functions, the basic idea is now to use formula (1.4) to define a fractional analogue of the divergence-measure (1.1).

Definition 1.4 (DM
The case α = 1 in Definition 1.4 corresponds to classical divergence-measure fields.Without loss of generality, we always assume n ≥ 2, since for n = 1 one clearly identifies DM α,p (R) = BV α,p (R), the space of L p functions with finite totale fractional α-variation, see the aforementioned [17,20,21] for an extensive presentation of BV α,p functions on R n .We also observe that BV α,p (R n ; R n ) ⊂ DM α,p (R n ) for n ≥ 2, with strict inclusion at least in the case p ∈ 1, n n−α , due to the fact that the vector fields in Example 3.1 below cannot belong to BV α,p (R n ; R n ), in the light of [17,Th. 1].
Similarly to the case of BV α,p functions (see [18,Th. 3.2] and [4, Th. 5]), we can state the following structural result for DM α,p vector fields.The proof is very similar to the one of [18,Th. 3.2] and is therefore omitted.Theorem 1.5 (Structure Theorem for DM α,p vector fields).Let α ∈ (0, 1] and p ∈ If the vector field is sufficiently regular, say F ∈ Lip c (R n ; R n ) for instance, then the fractional divergence-measure given by Theorem 1.5 is div α F = div α F L n , where div α F is as in (1.6).Moreover, thanks to Theorem 1.5, the linear space is a Banach space, and the fractional divergence-measure in (1.8) is lower semicontinuous with respect to the L p convergence.Remark 1.6 (On the space DM 0,p ).Although not strictly necessary for the purposes of the present paper, let us briefly comment on the case α = 0 in Definition 1.4.By exploiting [4,Lem. 26], if F ∈ DM 0,p (R n ) for some p ∈ (1, +∞), then , where R = ∇ 0 the vector-value Riesz transform (see Section 2.1 for the definition).Therefore, for p ∈ (1, +∞), we can write Hence, if F ∈ DM 0,p (R n ) for some p ∈ (1, +∞), then |div 0 F | ≪ L n .The limiting cases p ∈ {1, +∞} seem more intricate and we leave them for future investigations.
1.3.Main results.Our first main result deals with the absolute continuity properties of DM α,p vector fields with respect to the Hausdorff measure, extending Theorem 1.1.
Theorem 1.7 (Absolute continuity properties of the fractional divergence-measure).Let α ∈ (0, 1), p ∈ [1, +∞] and assume that F ∈ DM α,p (R n ).We have the following cases: In particular, Theorem 1.7 tells that, if F ∈ DM α,∞ (R n ), then |div α F | ≪ H n−α , exactly as in Theorem 1.1 for p = +∞.For p < +∞, instead, the properties of the fractional divergence-measure are different from the corresponding ones in the classical setting.Indeed, as for the fractional variation of BV α,p functions (see [17,Th. 1] for the corresponding result), the threshold p = n 1−α imposes an interesting change of dimension of the Hausdorff measure.This is quite customary in the distributional fractional framework, and is essentially due to the mapping properties of Riesz potential I 1−α , see [18,Sec. 2.3].
Our second main result concerns Leibniz rules for DM α,p -fields and Besov functions, see [20,Th 1.1] for the corresponding result for BV α,p functions.We refer to Section 2.1 for the definitions of fractional Sobolev and Besov spaces.
Theorem 1.8 (Leibniz rule for DM α,p vector fields with Besov functions).Let α ∈ (0, 1) and let p, q ∈ [1, +∞] be such that 1  p where is the non-local fractional divergence of the couple (g, F ), and satisfies Theorem 1.8, besides providing an extention of Theorem 1.2, provides a Gauss-Green formula for DM α,∞ vector fields on W α,1 sets.For the definitions of the fractional reduced boundary F α E and of the inner fractional normal ν α E : , we refer the reader to [18,Def. 4.7].
Corollary 1.9 provides the most general version known so far of the fractional Gauss-Green formula proved in [18,Th. 4.2].Unfortunately, we do not know if the assumption χ E ∈ W α,1 (R n ) can be replaced with the weaker one χ E ∈ BV α,1 (R n ) in Corollary 1.9.In fact, as observed in [17], we do not know whether the precise representative g ⋆ defined in (1.2) of g ∈ BV α,∞ (R n ) is well defined up to H n−α -negligible sets.We plan to tackle this and other strictly-connected challenging open questions in future works.
1.4.Organization of the paper.In Section 2, we collect all the needed intermediate results to prove our main theorems.In particular, Section 2.4 and Section 2.5 contain the proofs of points (ii) and (iii) of Theorem 1.7, respectively.The proof of Theorem 1.8, instead, can be found in Section 2.6.Section 3 collects several examples.In Section 3.1 we show point (i) of Theorem 1.7, while in Section 3.2 we discuss the sharpness of the other two points (ii) and (iii) of Theorem 1.7.

Proofs of the main results
In this section, we provide the proofs of our main results Theorem 1.7 and Theorem 1.8.The proof of Theorem 1.7 is split across Sections 3.1, 2.4 and 2.5, while the proof of Theorem 1.8 is given in Section 2.6.

General notation.
We start with a brief description of the main notation used in this paper.In order to keep the exposition the most reader-friendly as possible, we retain the same notation adopted in our works [4,[17][18][19][20][21][22].
Lebesgue and Hausdorff measures.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].We denote by B r (x) the standard open Euclidean ball with center x ∈ R n and radius r > 0. We let Radon measures.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 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 ).
Lebesgue, Sobolev and BV spaces.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 [41,Ch. 11], and be the space of m-vector-valued functions of bounded variation on Ω, see [2,Ch. 3].
Besov spaces.For α ∈ (0, 1) and p, q ∈ [1, +∞], we let < +∞ be the space of m-vector-valued Besov functions on R n , see [41,Ch. 17], where Shorthand for scalar function spaces.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).
Riesz potential.Given α ∈ (0, n), we let We recall that, if α, β ∈ (0, n) satisfy α + β < n, then we have the following semigroup property In addition, if 1 < p < q < +∞ satisfy 1 q = 1 p − α n , then there exists a constant C n,α,p > 0 such that the operator in (2.2) satisfies 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

Riesz transform. We let
be the (vector-valued) Riesz transform of a (sufficiently regular) function f .We refer the reader to [36, Sec.2.1 and 2.4.4],[55, Ch.III, Sec.1] and [56, Ch.III] for a more detailed exposition.We warn the reader that the definition in (2.5) agrees with the one in [56] and differs from the one in [36,55] for a minus sign.The Riesz transform (2.5) is a singular integral of convolution type, thus in particular it defines a continuous operator [35,Cor. 5.2.8].We also recall that its components R i satisfy

Approximation by smooth vector fields.
Here and in the rest of the paper, we let (̺ ε ) ⊂ C ∞ c (R n ) be a family of standard mollifiers as in [18,Sec. 3.3].The following approximation result is the natural generalization to DM α,p vector fields of [17,Th. 4].We leave its proof to the reader.

Integration-by-parts with Sobolev tests.
For future convenience, we note that the integration-by-parts formula (1.7) actually holds for a wider class of test functions.To this aim, let us recall the notion of non-local fractional gradient The operator ∇ α NL can be continuously extended to Lebesgue and Besov spaces, see [20,Cor. 2.7] for the precise statement.Proposition 2.2 (W 1,q ∩ C b -regular test).Let α ∈ (0, 1) and let p, q ∈ [1, +∞] be such that 1  p ) Proof.The proof is analogous to the one of [17,Prop. 3], so we only sketch it for the reader's convenience.By a routine regularization-by-convolution argument, it is not restrictive to assume that ξ the conclusion follows by passing to the limit as R → +∞ in (2.7).
To this aim, we study the relationship between DM 1,p and DM α,p vector fields.As one may expect, DM 1,p vector fields can be regarded as DM α,p vector fields, but only locally with respect to the divergence-measure.For α ∈ (0, 1) and p ∈ [1, +∞], we write Consequently, the Radon measure div α F ∈ M loc (R n ) given by (1.7) may be such that This issue is quite normal, and essentially due to the properties of Riesz potential, in view of the representation Hence, for any bounded open set U ⊃ supp ξ, by [18,Lem. 2.4] we can find a constant C n,α,U > 0, depending only on n, α and diam(U), such that This implies that F ∈ DM α,p loc (R n ), as desired.The inclusion given by Lemma 2.3 can be somewhat reversed, as done in Lemma 2.4 below.Note that this result, besides providing analogues of [18,Lem. 3.28], [19,Lem. 3.7] and [17,Prop. 4], proves point (ii) of Theorem 1.7 Lemma 2.4 (Relation between DM α,p and DM 1,p ).Let α ∈ (0, 1), p ∈ 1, n As a consequence, the operator By the Hardy-Littlewood-Sobolev inequality, we immediately get that Hence, by Fubini Theorem, we can write The remaining part of the statement easily follows from Theorem 1.1 (also see [49,Th. 3.2]).
2.5.Decay estimates.We now deal with point (iii) of Theorem 1.7.To this aim, we prove some decay estimates of the fractional divergence-measure on balls.
Let us begin with the following result, which may be considered as a toy case for the more general result in Theorem 2.8 below.

Lemma 2.5 (Decay estimate for div
(2.9) for all x ∈ A and r > 0 such that B 2r (x) ⊂ A.
) be such that ξ ≥ 0 and ξ ≡ 1 on B 1 .Then, for x ∈ A and r > 0 such that B 2r (x) ⊂ A, we can estimate Thus we easily get from which the conclusion immediately follows.
Lemma 2.5, despite its simplicity, allows to recover the following rigidity result, which may be seen as the natural fractional analogue of [44,Th. 3.1].
for all r > 0 by Lemma 2.5 in the case x = 0. Hence the conclusion follows by taking the limit as r → +∞.If instead p = n n−α , then and thus The proof is complete.
Proof.The proof is very similar to that of [17, Th. 9], so we only sketch it for the reader's convenience.Fix x ∈ R n and ξ ∈ Lip c (R n ) be fixed.
In the case p = +∞, we consider h ε,r,x ∈ Lip c (R n ) for ε > 0 and r > 0 defined as for L n -a.e.y ∈ R n .
Since h ε,r,x (y) → χ Br(x) (y) as ε → 0 + for all y ∈ R n and |div α F |(∂B r (x)) = 0 for L 1 -a.e.r > 0, we can use h ε,r,x to approximate χ Br(x) in (2.10).On the one hand, since (2.12) On the other hand, by [18, Lem.2.6], we can compute One then has to deal with each term of the right-hand side of (2.13) separately.The most difficult term is the second one, for which one has to observe that, by (2.11), Hence, by Lebesgue's Differentiation Theorem, for L 1 -a.e.r > 0. Thus, by [18,Th. 3.18,Eq. (3.26)], we get that for L 1 -a.e.r > 0. The other terms are easier and hence left to the reader.In the case p ∈ convolution to reduce to the previous case p = +∞.The conclusion then follows by exploiting the convergence properties given by Theorem 2.1 and recalling that, thanks to [17, Cor.1], ∇ α χ Br(x) ∈ L q (R n ; R n ) for any p ∈ 1 1−α , ∞ , where q = p p−1 , and that ∇ α NL (χ Br(x) , ξ) ∈ L q (R n ; R n ) as well, thanks to [20,Cor. 2.7].We leave the details to the reader.
As a consequence of Theorem 2.8, we get the following result, in particular proving the validity of point (iii) in Theorem 1.7.Note that Corollary 2.9 below is actually relevant only in the case of point (iii) of Theorem 1.7, since n − 1−α and p ≤ n n−α , but in this second case both exponents are negative.

Corollary 2.9 (|div
where A n,α,p is as in (2.15) and q ∈ [1, +∞) is such that 1 p + 1 q = 1.Proof.By Theorem 2.8, there exists a set Z α,p F ⊂ R n , which we can assume to be Borel without loss of generality, such that |div α F |(Z α,p F ) = 0 and (2.15) holds for any x / ∈ Z α,p F .Hence, for all x ∈ R n \ Z α,p F , we have Inequality (2.21) thus follows from [2, Th. 2.56].
) be such that η ≡ 1 on B 1 and set η k (x) = η x k for k ∈ N and x ∈ R n .By (1.7) and the α-homogeneity of the fractional gradient, we have Hence, by the Dominated Convergence Theorem with respect to the measure |div α F |, we get that We can now deal with the Leibniz rule for DM α,p vector fields and bounded continuous Besov functions, in analogy with [20,Th. 3.1].To this purpose, we need to recall the notion of non-local fractional divergence The operator div α NL can be continuously extended to Lebesgue and Besov spaces, see [20,Cor. 2.7].Theorem 2.12 (Leibniz rule for DM α,p with C b ∩ B α q,1 for 1 p + 1 q = 1).Let α ∈ (0, 1) and let p, q ∈ [1, +∞] be such that In addition, and Proof.We mimic the proof of [20,Th. 3 ) by Hölder's inequality.In addition, [20,Cor. 2.7] implies that div α NL (g, F ) ∈ L 1 (R n ).We now divide the proof in two steps.

Examples
In this section, we illustrate some examples concerning Theorem 1.7.

3.1.
Example for point (i) of Theorem 1.7.Example 3.1 below shows that, if p ∈ 1, n n−α , the fractional divergence-measure of DM α,p vector fields is not absolutely continuous with respect to H ε for any ε > 0, in general, proving point (i) of Theorem 1.7.
Hence F y,z,1 ∈ DM 1,p (R n ) for all p ∈ 1, n n−1 .Actually, we have F y,z,1 ∈ DM Proof.We divide the proof into two steps.
Step 1.Let ν ∈ M (R n ).We claim that G α ∈ DM α,p (R n ) for all p ∈ 1, n n−α and that G α satisfies (3.2).Indeed, by Young's inequality (for Radon measures), we can estimate Moreover, thanks to the translation invariance of ∇ α and exploiting the explicit expression of F α given in Example 3.1, we can write 2).In addition, by Jensen's inequality and Tonelli's Theorem, we can estimate for all x ∈ R n .On the other hand, for all x ∈ R n and j ≥ 1, we have for all x ∈ R n , where C α,ε,δ > 0 is constant depending on α, ε, and δ which is finite provided that we choose δ < ε n−α , as we are assuming from now on.We thus have |F α (x)| p(1− δ q ) dx.Now, recalling Example 3.1, we immediately see that Hence, since the function δ → n−δn+αδ (n−α)(1−δ) is monotone increasing, we easily see that Finally, in the case ε ∈ (n − α, n], we exploit (3.9) for δ = 1 in order to conclude that |F α (x − y)| d|ν|(y) = C α,ε < +∞ for all x ∈ R n , which implies that G α ∈ L ∞ (R n ).The conclusion thus follows.
Thanks to Proposition 3.3, we can now give the following example.
Therefore, these lower bounds on ε imply that, for p ∈ n n−α , +∞ , we have |div α G α | ≪ H s for all s > n − αq.

2 and H n− 1 (
∂B 1 ) = nω n , where Γ is Euler's Gamma function.Regular maps.Let Ω ⊂ R n be an open (non-empty) set.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 .Analogously, we let C k b (Ω; R m ) and Lip b (Ω; R m ) be the spaces of C kregular and, respectively, Lipschitz-regular, m-vector-valued bounded functions defined on the open set Ω ⊂ R n .In the case k = 0, we drop the superscript and simply write C c (Ω; R m ) and C b (Ω; R m ).

Partial sharpness of Theorem 1.7.
[17,ing as in [49, Exam.3.3 and Prop.6.1],we can exploit the properties of the vector field (3.1) in Example 3.1 to construct additional examples proving a partial sharpness of Theorem 1.7.The following result is the analogue of[17, Prop.5].