A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces

We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincar\'e inequality. Compared with previous works, we consider more general functionals. We also give a counterexample in the case $p=1$ demonstrating that unlike in Euclidean spaces, in metric measure spaces the limit of the nonlocal functions is only comparable, not necessarily equal, to the variation measure $\| Df\|(\Omega)$.


Introduction
Consider a sequence {ρ i } ∞ i=1 of nonnegative functions in L 1 (R n ), n ≥ 1, which are radial (i.e.only depend on |x|) and for which ˆRn ρ i (x) dx = 1 for all i ∈ N and lim i→∞ ˆ|x|>δ ρ i (x) dx = 0 for all δ > 0. here K p,n is a constant depending only on p, n.Dávila [14] generalized this result to functions of bounded variation (BV functions) f and their variation measures Df .He showed that when Ω ⊂ R n is a bounded domain with Lipschitz boundary, then for every f ∈ L 1 (Ω) we have where we understand Df (Ω) = ∞ if f / ∈ BV(Ω).To unify the notation, we define the energy E f,p (Ω) := Df (Ω) when p = 1, ´Ω |∇f | p dx when 1 < p < ∞.
Brezis [4,Remark 6] suggested generalizing the theory to more general metric measure spaces (X, d, µ).One generalization was given by Di Marino-Squassina [15], who assumed the measure µ to be doubling and the space to support a (p, p)-Poincaré inequality.Such spaces are often called PI spaces.We will give definitions in Section 2. They considered the mollifiers ρ s (x, y) := (1 − s) 1 d(x, y) ps µ(B(y, d(x, y))) , x, y ∈ X, 0 < s < 1, and showed in [15,Theorem 1.4] that for a constant C ≥ 1 and for every f ∈ L p (X), we have dµ(y) dµ(x) ≤ CE f,p (X). (1.2) A similar result was proved previously in Ahlfors-regular spaces in [29].Górny [18], resp.Han-Pinamonti [21], studied the problem in certain PI spaces that "locally look like" Euclidean spaces, resp.finite-dimensional Banach spaces or Carnot groups, and showed that for every f ∈ N 1,p (X), with 1 < p < ∞, we have These results correspond to certain choices of the mollifiers ρ i satisfying (1.1).In the current paper, our main goal is to study this problem for more general mollifiers ρ i , of which the mollifiers considered in [15,18,29] are special cases.Moreover, we consider domains Ω = X.
Our main result is the following.
Theorem 1.3.Let 1 ≤ p < ∞, and suppose µ is doubling and X supports a (p, p)-Poincaré inequality.Suppose {ρ i } ∞ i=1 is a sequence of mollifiers satisfying conditions (2.8)-(2.11).Suppose Ω ⊂ X is a strong p-extension domain, and let f ∈ L p (Ω).Then for some constants C 1 ≤ C 2 that depend only on p, the doubling constant of the measure, the constants in the Poincaré inequality, and the constant C ρ associated with the mollifiers.
After giving definitions in Section 2 and some preliminary results in Section 3, we prove the two directions of (1.4) in Sections 4 and 5.In Section 6 we give corollaries to our main Theorem 1.3, showing that the mollifiers considered in [15] and [18], as well as other choices, can be handled as special cases.In Section 7 we give a counterexample demonstrating that we do not generally have Acknowledgement: The authors would like to thank Camillo Brena and Enrico Pasqualetto for some useful comments on a preliminary version of the paper.

Notation and definitions
Throughout this paper, we work in a complete and connected metric measure space (X, d, µ) equipped with a metric d and a Borel regular outer measure µ satisfying a doubling property, meaning that there exists a constant for every ball B(x, r) := {y ∈ X : d(y, x) < r}.We assume that 1 ≤ p < ∞ and X consists of at least two points, that is, diam X > 0.
By a curve we mean a rectifiable continuous mapping from a compact interval of the real line into X.The length of a curve γ is denoted by ℓ γ .We will assume every curve to be parametrized by arc-length, which can always be done (see e.g.[19,Theorem 3.2]).A nonnegative Borel function g on X is an upper gradient of a function f where x and y are the end points of γ.We interpret |f (x) − f (y)| = ∞ whenever at least one of |f (x)|, |f (y)| is infinite.Upper gradients were originally introduced in [23].
We always consider 1 ≤ p < ∞.The p-modulus of a family of curves Γ is defined by where the infimum is taken over all nonnegative Borel functions ρ such that ´γ ρ ds ≥ 1 for every curve γ ∈ Γ.A property is said to hold for p-almost every curve if it fails only for a curve family with zero p-modulus.If g is a nonnegative µ-measurable function on X and (2.1) holds for p-almost every curve, we say that g is a p-weak upper gradient of f .By only considering curves γ in a set A ⊂ X, we can talk about a function g being a (p-weak) upper gradient of u in A.
We always let Ω denote an open subset of X.We define the Newton-Sobolev space N 1,p (Ω) to consist of those functions f ∈ L p (Ω) for which there exists a p-weak upper gradient g ∈ L p (Ω) of f in Ω.This space was first introduced in [35].We write f ∈ N 1,p loc (Ω) if for every x ∈ Ω there exists r > 0 such that f ∈ N 1,p (B(x, r)); other local function spaces are defined analogously.For every f ∈ N 1,p loc (Ω) there exists a minimal p-weak upper gradient of f in Ω, denoted by g f , satisfying g f ≤ g µ-almost everywhere (a.e.) in Ω for every p-weak upper gradient g ∈ L p loc (Ω) of f in Ω, see [2,Theorem 2.25].Note that Newton-Sobolev functions are understood to be defined at every x ∈ Ω, whereas the functionals that we consider are not affected by perturbations of f in a set of zero µ-measure.For this reason, we also define For every f ∈ N 1,p (Ω), we can also define g f := g h , where g h is the minimal p-weak upper gradient of any h as above in Ω; this is well defined µ-a.e. in Ω by [2, Corollary 1.49, Proposition 1.59].
Next we define functions of bounded variation.Given an open set Ω ⊂ X and a function f ∈ L 1 loc (Ω), we define the total variation of f in Ω by where each g f i is the minimal 1-weak upper gradient of f i in Ω.We say that a function f ∈ L 1 (Ω) is of bounded variation, and denote f ∈ BV(Ω), if Df (Ω) < ∞.For an arbitrary set A ⊂ X, we define i=1 be a sequence with g i → g weakly in L p (Ω).Then there exist convex combinations g i := N i j=i a i,j g j , for some By convex combinations we mean that the numbers a i,j are nonnegative and that N i j=i a i,j = 1 for every i ∈ N.
Lemma 2.3.Let {g i } ∞ i=1 be a sequence of functions with g i → g in L p (Ω).Then for p-a.e.curve γ in Ω, we have ˆγ g i ds → ˆγ g ds as i → ∞.
We say that X supports a (p, p)-Poincaré inequality, if there exist constants C P > 0 and λ ≥ 1 such that for every ball B(x, r), every f ∈ L p (X), and every p-weak upper gradient g of f , we have where f B(x,r) := ˆB(x,r) f dµ := 1 µ(B(x, r)) ˆB(x,r) f dµ.
In the case p = 1, the following BV version of the Poincaré inequality can be obtained by applying the (1, 1)-Poincaré inequality to the approximating functions in the definition of the total variation: for every f ∈ L 1 (X), we have In the latter case, denote the minimal p-weak upper gradient of f in Ω by g f .For every Borel set A ⊂ Ω, we denote the energy by We can combine (2.4) and (2.5) to give: for every 1 ≤ p < ∞ and every f ∈ L p (X), we have Definition 2.7.We say that an open set Ω ⊂ X is a strong p-extension domain if • in the case p = 1, for every f ∈ BV(Ω) there exists an extension F ∈ BV(X); • in the case 1 < p < ∞, for every f ∈ N 1,p (Ω) there exists an extension F ∈ N 1,p (X); and in both cases, E F,p (∂Ω) = 0.
For example, in Euclidean spaces, a bounded domain with a Lipschitz boundary is a strong p-extension domain for all 1 ≤ p < ∞, see e.g.[1,Proposition 3.21].Now we describe the mollifiers that we will use.We will consider a sequence of nonnegative X ×X-measurable functions {ρ i (x, y)} ∞ i=1 , x, y ∈ X, and a fixed constant 1 ≤ C ρ < ∞ satisfying the following conditions: (1) For every x, y ∈ X with d(x, y) ≤ 1, we have for every i ∈ N where r i ց 0 and each ν i is a positive Radon measure on [0, ∞) for which lim inf i→∞ ˆδ 0 t p dν i ≥ C −1 ρ for all δ > 0.
(2.9) Also for every x, y ∈ X with 0 < d(x, y) ≤ 1, we have (2) For all δ > 0, we have lim i→∞ sup y∈Ω ˆΩ\B(y,δ) Remark 2.12.Conditions (2.10) and (2.11) will be used to prove the upper bound of our main Theorem 1.3, and they are quite close to the Euclidean assumptions (1.1).Since we do not have as many tools at our disposal as in Euclidean spaces, we additionally impose the somewhat stronger conditions (2.8) and (2.9); these will be used to prove the lower bound.In Section 6 we will see that these two conditions are also very natural.
Throughout the paper, we assume that µ is doubling, but we do not always assume that X satisfies a Poincaré inequality.Nonetheless, for convenience we assume that X is connected, from which it follows that µ({x}) = 0 for every x ∈ X, see e.g.[2, Corollary 3.9].Thus, integrating over the set where x = y in the functionals that we consider does not cause any problems.

Preliminary results
First we note the following basic fact: for every f ∈ L p (X) and every ball B(z, r), using the estimate The next lemma is similar to [15, Lemma 3.1(ii)].
Lemma 3.2.For any h(x, y) ≥ 0 that is µ×µ-measurable in X×X and satisfies h(x, y) = 0 for all x, y ∈ X with d(x, y) ≥ δ > 0, we have Proof.For all x, y ∈ X with h(x, y) = 0, we have d(x, y) < δ, and then Note also that χ B(z,2δ) (x) and χ B(z,2δ) (y) are lower semicontinuous functions in the product space X × X × X, and so is µ × µ × µ-measurable, and we can apply Fubini's theorem.We estimate For an open set U ⊂ X and δ > 0, denote Lemma 3.5.For any function h(x, y) ≥ 0 that is µ × µ-measurable on X and satisfies h(x, y) = 0 for all x, y ∈ X with d(x, y) ≥ δ > 0, and for an open set U ⊂ X, we have Proof.Apply Lemma 3.2 with the function h replaced by h(x, y) χ U (y).
4 Upper bound of Theorem 1.3 Recall that we always denote by Ω an open subset of X, and that 1 ≤ p < ∞.
In order to prove the upper bound of our main Theorem 1.3, we first prove the following result.Recall the notation U (R) from (3.4).Proposition 4.1.Suppose X supports the (p, p)-Poincaré inequality (2.4).Let f ∈ L p (X) and 0 < R ≤ 1, and suppose i=1 is a sequence of mollifiers that satisfy (2.10).Then for every i ∈ N and for a constant Proof.We can assume that E p (f, U (8λR)) < ∞.Recall the condition (2.10).Note that on the left-hand side of (4.2) we require d(x, y) < R, but we also know that χ B(y,2 −j+1 )\B(y,2 −j ) (x) can be nonzero only when 2 −j < d(x, y).Then necessarily For every j ∈ Z satisfying (4.3), we estimate µ(B(y, 2 −j+1 )) dµ(x) dµ(y) dµ(z) by Lemma 3.5. ( We estimate further by the Poincaré inequality (2.6). (4.5) Denote the smallest integer at least a ∈ R by ⌈a⌉.Combining the above with (4.4), we get Recalling (2.10), we get by the assumption ∞ j=1 d i,j ≤ C ρ .
Now we can prove one direction of our main theorem.Recall the definition of a strong p-extension domain from Definition 2.7.
Theorem 4.8.Suppose X supports a (p, p)-Poincaré inequality.Suppose Ω ⊂ X is a strong p-extension domain, and let f ∈ BV(Ω) if p = 1, and i=1 is a sequence of mollifiers that satisfy (2.10) and (2.11).Then Proof.Consider 0 < R ≤ 1. Recalling the notation Ω 8λR from (3.4), we have For the first term, we estimate Then we estimate the third term.Since Ω is a strong p-extension domain, we find an extension F ∈ BV(X) in the case p = 1, and F ∈ N 1,p (X) in the case 1 < p < ∞, and in both cases E F,p (∂Ω) = 0. Write U := Ω \ Ω 8λR , and note that U (8λR) ⊂ Ω(8λR) \ Ω 16λR .Thus we can estimate the third term by lim sup i→∞ ˆΩ\Ω 8λR ˆΩ∩B(y,R) by Proposition 4.1.This goes to zero as R → 0, since E F,p (∂Ω) = 0. Combining the three terms, we get lim sup 5 Lower bound of Theorem 1.3 In this section we prove the lower bound of Theorem 1.3.Note that in this section we do not need to assume a Poincaré inequality.As usual, Ω ⊂ X is an open set.Given a ball B = B(x, r) with a specific center x ∈ X and radius r > 0, we denote 2B := B(x, 2r).The distance between two sets A, D ⊂ X is denoted by dist(A, D) := inf{d(x, y) : x ∈ A, y ∈ D}.
Lemma 5.1.Consider an open set U ⊂ Ω with dist(U, X \ Ω) > 0, and a scale 0 < R < dist(U, X \ Ω)/10.Then we can choose an at most countable covering {B j = B(x j , R)} j of U (5R) such that x j ∈ U (5R), each ball 5B j is contained in Ω, and the balls {5B j } ∞ j=1 can be divided into at most C 8 d collections of pairwise disjoint balls.
Proof.Consider a covering {B(x, R/5)} x∈U (5R) .By the 5-covering theorem, see e.g.[24, p. 60], we can choose a countable collection of disjoint balls {B(x j , R/5)} j such that the balls B j = B(x j , R) cover U (5R).Consider a ball B j and denote by I j those k ∈ N for which 5B k ∩ 5B j = ∅.Then and so I j has cardinality at most C 8 d .We can recursively choose maximal collections of pairwise disjoint balls 5B j .After at most C 8 d steps, we have exhausted all of the balls 5B j .
Let C 0 := 3C 8  d .Given such a covering of U (5R), we can take a partition of unity {φ j } ∞ j=1 subordinate to the covering, such that 0 and spt(φ j ) ⊂ 2B j for each j ∈ N; see e.g.[24, p. 104].Finally, we can define a discrete convolution h of any f ∈ L 1 (Ω) with respect to the covering by Clearly h ∈ Lip loc (U ).
for some constant C 1 depending only on C ρ and on the doubling constant of the measure.
Note that here we do not impose any conditions on the open set Ω ⊂ X.
Proof.We can assume that lim inf Fix 0 < ε < 1. Passing to a subsequence (not relabeled), we can assume that Assuming the second option of (2.8), we get It follows that µ(B(y, d(x, y))) dµ(x) dµ(y) t p dν i (t) by Fubini's theorem.By (2.9), given an arbitrarily small 0 < δ < 1, we have and so ´δ 0 t p dν i ≥ (1 − ε)C −1 ρ for all sufficiently large i ∈ N. Then there necessarily exists 0 < t ≤ δ such that ¨{x,y∈Ω: In other words, we find arbitarily small t > 0 such that We obviously obtain this also if the first option of (2.8) holds.Fix a small t > 0. Let U ⊂ Ω with dist(U, X \ Ω) > t, and let R := t/10.Consider a covering {B j } ∞ j=1 of U (5R) at scale R > 0, as described in Lemma 5.1.Then consider the discrete convolution We define the pointwise asymptotic Lipschitz number by Suppose x ∈ U .Then x ∈ B j for some j ∈ N. Consider any other point y ∈ B j .Denote by I j those k ∈ N for which 2B k ∩ 2B j = ∅.We estimate since by Lemma 5.1 we know that I j has cardinality at most C 0 .Letting y → x, we obtain an estimate for Lip h in the ball B j .In total, we conclude (we track the constants for a while in order to make the estimates more explicit) Since the balls {B j } j can be divided into at most C 0 collections of pairwise disjoint balls, we get Thus We know that the minimal p-weak upper gradient g h of h in U satisfies g h ≤ Lip h µ-a.e. in U , see e.g.[2,Proposition 1.14].
Recall that we can do the above for arbitrarily small t > 0 and thus arbitrarily small R > 0. From now on, we can consider any open U ⊂ Ω with dist(U, X \ Ω) > 0. We get a sequence of discrete convolutions {h i } ∞ i=1 corresponding to scales R i ց 0, such that i=1 is a bounded sequence in L p (U ).From the properties of discrete convolutions, see e.g.[22,Lemma 5.3], we know that h i → f in L p (U ).Passing to a subsequence (not relabeled), we also have h i (x) → f (x) for µ-a.e.x ∈ U .When p = 1, we get and so f ∈ BV(U ).In the case 1 < p < ∞, by reflexivity of the space L p (U ), we find a subsequence of {h i } ∞ i=1 (not relabeled) and g ∈ L p (U ) such that g h i → g weakly in L p (U ) (see e.g.[24, Section 2]).By Mazur's lemma (Theorem 2.2), for suitable convex combinations we get the strong convergence N i l=i a i,l g h l → g in L p (U ).We still have Then h = f µ-a.e. in U .Denote N := {x ∈ U : | h(x)| < ∞}, so that µ(N ) = 0.For p-a.e.curve γ in U , denoting the end points by x, y, we have that either x / ∈ N or y / ∈ N ; see [2, Corollary 1.51].For such γ, we obtain l g h l ds = ˆγ g ds by Fuglede's lemma (Lemma 2.3), exclusing another curve family of zero p-modulus.Hence g is a p-weak upper gradient of h in U , and so for the minimal p-weak upper gradient we have by (5.8).Since f = h µ-a.e. in U , we have f ∈ N 1,p (U ).Note that now E f,p is a Radon measure on Ω. Exhausting Ω by sets U , in both cases we obtain Letting ε → 0, this proves (5.4).

Corollaries
The conditions (2.8)-(2.11)that we impose on the mollifiers ρ i are quite flexible, and so we can obtain various existing results in the literature as special cases of our main Theorem 1.3.The following is essentially [15,Theorem 1.4], except that we consider an open set Ω instead of the whole space X.
Corollary 6.1.Suppose X supports a (p, p)-Poincaré inequality.Let Ω ⊂ X be a strong p-extension domain, and let f ∈ L p (Ω).Then for some constant C ≥ 1 depending only on p, the doubling constant of the measure, and the constants in the Poincaré inequality.
Proof.This is obtained from Theorem 1.3 with the choice of mollifiers where s i ր 1 as i → ∞.We only need to check that conditions (2.8)-(2.11)are satisfied.We have and so satisfying the second option of (2.8), and (2.9).For every x, y ∈ X with 0 < d(x, y) ≤ 1, we have and so satisfying (2.10).Finally, we estimate and so (the notation j≤− log 2 δ means that we sum over integers j at most − log 2 δ) ˆX\B(y,δ) and so (2.11) holds.
In particular, in the Euclidean setting, the functional considered in the Corollary 6.1 reduces to the fractional Sobolev seminorm The functional appearing in the following corollary was previously considered by Marola-Miranda-Shanmugalingam [27] as well as Górny [18] and Han-Pinamonti [21] Corollary 6.3.Suppose X supports a (p, p)-Poincaré inequality, and let f ∈ L p (X).Then (6.4) for some constant C depending only on p, the doubling constant of the measure, and the constants in the Poincaré inequality.
Proof.This is obtained from Theorem 1.3 with the choice where r i ց 0 as i → ∞.Now the first option of (2.8) holds.For every x, y ∈ X with 0 < d(x, y) ≤ 1, we have and thus (2.10) is satisfied.The condition (2.11) obviously holds.
The following simple choice of mollifiers, considered in the Euclidean setting e.g. by Brezis [4, Eq. (45)], is also natural.This will be used also in a counterexample in the last section.Corollary 6.5.Suppose X supports a (p, p)-Poincaré inequality.Let f ∈ L p (X).Then for some constant C depending only on p, the doubling constant of the measure, and the constants in the Poincaré inequality.
Proof.This is obtained from Theorem 1.3 with the choice where r i ց 0 as i → ∞.Again the first option of (2.8) holds.We can assume that r i < min{1, diam X/4} for all i ∈ N.For every x, y ∈ X with 0 < d(x, y) ≤ 1, we have where d i,j = µ(B(y, 2 −j+1 ))µ(B(y, r i )) −1 for j ≥ − log 2 r i and d i,j = 0 otherwise.Since X is connected, there exists z ∈ ∂B(y, 3 2 • 2 −j ) for all j ≥ − log 2 r i , and so satisfying (2.10).The condition (2.11) obviously holds.

A counterexample
Recall that the conclusion of our main Theorem 1.3 has the form One natural question to ask is whether C 1 = C 2 might hold.Górny [18] shows that with a suitable choice of the mollifiers ρ i , this holds in the case 1 < p < ∞ if we additionally assume that at µ-a.e. point x ∈ X, the tangent space is Euclidean with fixed dimension.On the other hand, he gives an example where the dimension of the tangent space takes two different values in two different parts of the space, and then it is necessary to choose C 1 < C 2 .In the example below, inspired by [20,Example 4.8], it is easy to check that the tangent space of X is Euclidean with dimension 1 at µ-a.e.x ∈ X (see definitions in [18]), but nonetheless we show in the case p = 1 that C 1 < C 2 .First consider the real line equipped with the Euclidean metric and the one-dimensional Lebesgue measure L 1 .Similarly to Corollary 6.5, consider the sequence of mollifiers Now also f i ∈ Lip(X), and f i → f uniformly.This can be seen as follows.Given i ∈ N, the set A i consists of 2 i intervals of length L i /2 i .If I is one of these intervals, we have 2 −i = ˆI g(s) ds = ˆI g i+1 (s) ds, and also ˆX\A i g dL 1 = 0 = ˆX\A i g i+1 dL 1 .
For a.e.x ∈ A, f is differentiable at x and so we have since both sides are equal to the classical quantities, that is, the quantities obtained when the measure µ is L 1 .This combined with (7.4) shows that we cannot have C 1 = C 2 in (7.1).