New estimates for a class of non-local approximations of the total variation

We consider a class of non-local functionals recently introduced by H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung, which offers a novel way to characterize functions with bounded variation. We give a positive answer to an open question related to these functionals in the case of functions with bounded variation. Specifically, we prove that in this case the liminf of these functionals can be estimated from below by a linear combination in which the three terms that sum up to the total variation (namely the total variation of the absolutely continuous part, of the jump part and of the Cantor part) appear with different coefficients. We prove also that this estimate is optimal in the case where the Cantor part vanishes, and we compute the precise value of the limit in this specific scenario. In the proof we start by showing the results in dimension one by relying on some measure theoretic arguments in order to identify sufficiently many disjoint rectangles in which the difference quotient can be estimated, and then we extend them to higher dimension by a classical sectioning argument.


Introduction
In this paper we consider a class of non-local, non-convex functionals related to the total variation.In order to introduce these functionals, let N ≥ 1 be a positive integer, let Ω ⊂ R N be an open set and let γ ∈ R and λ ∈ (0, +∞) be two real parameters.
The functionals (1.1) were introduced in [11,12] and generalize some other families of functionals previously considered in the literature.In particular, in the case γ = −1, the family {F −1,λ } was first studied in [20,21] (see also the more recent developments in [3,4,5,8]).The main results in this case are that lim where and that, quite surprisingly, where Du M denotes the total variation of u, which is intended to be equal to +∞ if u ∈ L 1 (R N ) \ BV (R N ).
On the other hand, in the case γ = N we recover the quantities considered in [9,10,22], since in this case ν N is equal to the Lebesgue measure L N and where [•] L 1,∞ (Ω) denotes the weak L 1 quasi-norm.
In this paper, motivated by [7, Section 9], we limit ourselves to the case γ > 0. In this case, it is known (see [12,Theorem 1.4]) that there exist two constants c 1 (N, γ) and c 2 (N, γ) such that for every u ∈ L 1 loc (R N ).In particular, it follows that sup λ>0 F γ,λ (u) < +∞ if and only if u belongs to the space ḂV (R N ) of the functions in L 1 loc (R N ) with globally bounded variation.
Moreover, for every γ > 0 it holds that (see [12,Theorem 1.1]) for every u in the space Ẇ 1,1 (R N ) of functions in L 1 loc (R N ) with ∇u ∈ L 1 (R N ), where C N is the constant defined in (1.2).
The extension of (1.4) to the case in which u ∈ ḂV (R N ) is not straightforward.Indeed, it was proved in [12,Lemma 3.6] that if u is the characteristic function of a bounded convex domain with smooth boundary, then for every γ > 0 it holds that This result suggests that the singular part of the derivative contributes to the limit of F γ,λ (u) in a different way with respect to the absolutely continuous part, but anyway it seems that it does add a positive contribution in the limit.
This led to the following questions, which were raised in [7,Section 9] and in [12, Section 7.2].
Question A. Let γ > 0, and let u : R N → R be a measurable function such that Can we conclude that u is constant (almost everywhere)?Question B. Let γ > 0. Is there a positive constant c(N, γ) > 0 such that for every measurable function u : R N → R, with the usual understanding that The main contribution of the present paper is a positive answer to these questions in the case u ∈ ḂV (R N ).
Before stating the result, we recall that for a function u ∈ ḂV (R N ) we can decompose its distributional derivative Du as the sum of three finite R N -valued measures, which are supported on disjoint sets: the absolutely continuous part D a u, the jump part D j u, and the Cantor part D c u (see [2, Section 3.9]).
Our first main result is the following.
Theorem 1.1.For every γ > 0 and every u ∈ ḂV (R N ) it turns out that Our second main result shows that the constants appearing in front of the absolutely continuous part and the jump part are optimal, and unifies (1.4) and (1.5) in the case in which u ∈ ṠBV (R N ), namely u ∈ ḂV (R N ) and D c u = 0. Theorem 1.2.For every γ > 0 and every u ∈ ṠBV (R N ) it turns out that Remark 1.3.For the sake of simplicity, in this paper we consider only the case in which Ω = R N .However, it should not be difficult to extend the main results to the case of bounded regular domains.In particular, in the case of convex domains, this should be almost straightforward, since all their one-dimensional sections are just open intervals.

Overview of the technique
Let us summarize the main ideas in the proofs of our results, which are different from those used in [12,22], since we do not exploit the BBM formula [6,15].
For both theorems, we first establish the result in the one-dimensional case, and then we extend it to the case of higher space dimensions by a sectioning argument.This is a standard but quite effective tool in this kind of problems (see [4,17]).
The proof of the one-dimensional version of Theorem 1.1 relies on some measure-theoretic arguments that allow us to find sufficiently many disjoint rectangles inside E γ,λ on which we can control the difference quotient of u.
As for Theorem 1.2, in view of Theorem 1.1, it is enough to prove an estimate from above for the limsup of F γ,λ (u).To this end we exploit an argument from [12,Section 3.4] which basically shows that it is enough to prove such an estimate for a class of functions u that is dense in ṠBV (R N ) with respect to the strong BV topology.In dimension one such a class is provided by functions with finitely many jump points that are smooth and Lipschitz continuous in every interval that does not contain such points.For functions in this class, the limit can be easily computed.
The case p > 1 The functionals considered in the paper [12] actually depend also on a parameter p ≥ 1, and can be written as The same is true for the various special cases previously considered in [20,21,4,9,10,22].This higher generality allows to obtain characterizations of the Sobolev spaces W 1,p (R N ) or Ẇ 1,p (R N ), and also for more general types of spaces (see [13,14,23,24]), together with estimates on their semi-norms.
In this paper we only consider the case p = 1, which is the most challenging, because the gap between Ẇ 1,1 (R N ) and ḂV (R N ) creates additional difficulties, and we refer to [12] and to the references therein for the numerous interesting results in the case p > 1.
Recent developments After this work was completed, Lahti in [19] proved with different techniques a sharper version of the estimate (1.6), namely for every u ∈ ḂV (R N ) and γ > 0.
He also found a Cantor-type function u ∈ BV (0, 1) for which Du = D c u and thus establishing the optimality of the constant in front of the Cantor part in (1.7).
On the other hand, obtaining good estimates from above for functions with non-vanishing Cantor part (in order to extend Theorem 1.2) is quite complicated, since there is not a nice class of functions which is strongly dense in ḂV (R).Indeed, it is not possible to approximate functions with non-vanishing Cantor part in the strong BV topology with functions without a Cantor part.It is also conceivable that for some functions the liminf might differ from the limsup or that different functions might produce different constants in the limit, since these phenomena occur in some similar contexts (see the pathologies in [8]).
We also mention another very recent development obtained in [18], where it is proved that for every u ∈ L 1 loc (R N ) and γ > 0, thus providing a positive answer to Question A and Question B also for u ∈ L 1 loc (R N ) \ ḂV (R N ), in which case the right-hand side is infinite.

Structure of the paper
The paper is organized as follows.In Section 2, after recalling some basic properties of functions of bounded variation in one dimension, we prove the main results in the one-dimensional case.Then, in Section 3 we show how the problem can be reduced to the one-dimensional setting by a sectioning argument, and we complete the proofs in the higher dimensional case.

The one-dimensional case
In this section we prove our main results in the one-dimensional case N = 1.We observe that in this case in (1.2) we have C 1 = 2 and that we can rewrite the functional in the following more convenient way where E ′ γ,λ (u, Ω) := (x, y) ∈ Ω × Ω : x < y and |u(y) − u(x)| > λ|y − x| 1+γ .We also recall that functions of bounded variation in one dimension have some special properties, that we list in the following lemma.
Lemma 2.1.Let u ∈ ḂV (R).Then we can choose a representative, that we still denote with u, satisfying the following properties.
• u is differentiable in the classical sense at almost every x ∈ R and L 1 is the one-dimensional Lebesgue measure.
• u admits a left limit u(x − ) and a right limit u(x + ) at every x ∈ R, and they coincide for every x outside a set J u that is at most countable.With these notations, it holds that • The Cantor part of the derivative is supported on the set C = C + ∪ C − , where Proof.The first two properties are classical (see, for example, [2, Theorem 3.28]).
As for the third property, we first observe that, on the real line, Besicovitch differentiation theorem [2, Theorem 2.22] holds also with one-sided closed intervals instead of closed balls.Indeed, the proof is based on a covering argument that in the one-dimensional case works also with one-sided intervals.As a consequence, if µ ⊥ ν are two Radon measures supported on disjoint sets, it turns out that lim for µ-almost every x ∈ R. Now let D c + u and D c − u denote respectively the positive and negative part in the Hahn decomposition of D c u.If we take the right continuous representative for u, then for every x ∈ R \ J u and h > 0 it holds that , and , Therefore, Besicovitch differentiation Theorem implies that D c + u is supported on C + .With a similar argument we obtain also that D c − u is supported on C − .
In the proof of Theorem 1.2 we also need a density result for piecewise smooth functions in ṠBV (R).The precise class of functions that we consider is the following.Definition 2.2.We denote with X(R) the set of functions u ∈ ṠBV (R) such that J u is finite, The strong density of X(R) into ṠBV (R) is provided by the following lemma, which is elementary in the one-dimensional case (a similar statement in the higher dimensional case is proved in [16]).
Lemma 2.3.Let u ∈ ṠBV (R).Then there exists a sequence of functions Proof.By [2, Corollary 3.33] we can write u = u a + u j , where u a ∈ Ẇ 1,1 (R) and u j is a pure jump function, namely Du j = i∈I α i δ x i , where I is at most countable, δ x i denotes a Dirac delta in the point x i , and {α i } is a summable sequence (or a finite set) of real numbers.Therefore, we have that Du = (u a ) ′ L 1 + Du j .
We can approximate separately the function (u a ) ′ ∈ L 1 (R) with a sequence {v n } ⊆ C ∞ c (R) and the measure Du j with measures µ n = i∈In α i δ x i that are finite sums of Dirac masses, so that 2.1 Proof of Theorem 1.1 when N = 1 Let J = J u and C = C + ∪ C − be as in Lemma 2.1 and let us set Approximation of the jump part Let us fix a small positive real number ε > 0, and let J ε = {s 1 , . . ., s kε } be a finite subset of J such that |Du|(J \ J ε ) ≤ ε. (2.9) For every i ∈ {1, . . ., k ε } let us fix a positive radius r i > 0 in such a way that the following properties hold.
The monotonicity of r λ,ε (z) with respect to λ and the limit as λ → +∞ are immediate consequences of the first condition in the definition of r λ,ε (z), which is also the only one involving λ.

Approximation of the absolutely continuous part Let us set
For every λ > 0 let us set .
Moreover, if x ∈ A λ,ε , then we have that for every .
As a consequence, it turns out that (2.17) for every λ > 0.
Computation of the functional in the three parts From (2.8), (2.10), (2.15) and (2.17) we deduce that for every λ > λ j ε .The three addenda in the right-hand side can be computed as follows.
Recalling (2.9), we obtain that Now we compute the contribution of the Cantor part.

Proof of Theorem 1.2 when N = 1
First of all, we observe that it is enough to show that lim sup because the opposite inequality is provided by Theorem 1.1, that we have already proved in the case N = 1.We divide the proof of (2.22) into two steps.First we prove the result for functions u ∈ X(R), then we exploit Lemma 2.3 to extend the result to all u ∈ ṠBV (R).
Now we observe that |u(y) − u(x)| ≤ Du M for every (x, y) ∈ R 2 , and therefore . (2.23) Let us set so that for every λ > λ 0 we have that < min{d 0 , 1}. (2.24) As a consequence, setting , for every λ > λ 0 we have that (2.25) We point out that (2.23) and (2.24) imply that points outside [a, b] 2 cannot contribute to E γ,λ (u) when λ > λ 0 , because u is constant in (−∞, a + 1) and in (b − 1, +∞).Now let us fix ε > 0 and let us set We observe that Ω i λ 1 ,ε ⊆ Ω i λ 2 ,ε for every 0 < λ 1 < λ 2 , and that Indeed, if this is not the case, then from (2.23) and the definition of Ω i λ,ε we deduce that which contradicts the fact that (x, y) ∈ E ′ γ,λ (u).On the other hand, since u is Lipschitz continuous with Lipschitz constant bounded by L in each of the intervals Ω i , for every (x, y) ∈ E ′ γ,λ (u) ∩ Ω 2 i it holds that , otherwise we would have that |u(y) − u(x)| ≤ (y − x)L ≤ λ(y − x) 1+γ .Hence we obtain the following estimate Now we need to estimate the contribution of the rectangles R i λ .To this end, let λ j ε > 0 be a positive number large enough so that for every i ∈ {1, . . ., n} and every λ > λ j ε it holds that Then we have that for every (x, y) ∈ R i λ such that . Therefore we deduce that .
Hence we obtain the following estimate and hence lim sup Letting ε → 0 + , we obtain (2.22).
Step 2: u ∈ ṠBV (R) We exploit the very same argument used in [12,Section 3.4] to reduce the case u ∈ Ẇ 1,1 (R N ) to the case in which u is smooth and has compactly supported gradient.So, given u ∈ ṠBV (R), let {u n } be a sequence provided by Lemma 2.3.By the triangle inequality for every n ∈ N and every ε ∈ (0, 1) it holds that Recalling (2.8), it follows that Since u n ∈ X(R), by Step 1 and the second inequality in (1.3) we conclude that lim sup Letting first n → +∞, and then ε → 0 + we obtain (2.22).

The higher dimensional case
In this section we extend the results of Section 2 to the case N > 1, thus establishing Theorem 1.1 and Theorem 1.2 in full generality.
The main tool that we exploit is a representation formula for F γ,λ (u), which allows us to rewrite this functional in terms of its one-dimensional version computed on all one-dimensional sections of the function u.
In order to state the formula, let us introduce some notation.For a function u : R N → R, a unit vector σ ∈ S N −1 and a point z ∈ σ ⊥ let u σ,z : R → R be the one-dimensional function coinciding with the restriction of u to the line parallel to σ passing through z, namely the function u σ,z (t) := u(z + σt) ∀t ∈ R.
The result is the following.Setting first y = x + σr, with σ ∈ S N −1 and r ∈ (0, +∞), and then x = z + σt, with z ∈ σ ⊥ and t ∈ R, we obtain that The second result that we need to perform the sectioning argument is the following wellknown lemma, that follows from the results in [2, Section 3.11].We can now extend the proofs of our main results to the case N > 1.
the jump points of u, namely the finitely many elements of J u , and let (a, b) be an interval such that Du is supported on [a + 1, b − 1].Let us set s 0 := a and s n+1 := b.

Lemma 3 . 2 .
Let u ∈ ḂV (R N ).Then for every σ ∈ S N −1 it holds that u σ,z ∈ ḂV (R) for almost every z ∈ σ ⊥ andS N−1 dσ σ ⊥ D a u σ,z M dz = C N D a u M , S N−1 dσ σ ⊥ D j u σ,z M dz = C N D j u M , S N−1 dσ σ ⊥ D c u σ,z M dz = C N D c u M .