Generalized Ponce’s inequality

We provide a generalization of a remarkable inequality by A. C. Ponce whose consequences are essential in several fields, such as a characterization of Sobolev spaces or nonlocal modelization.

For h = 1, the following compactness result is well known (see, e.g., [4] and [9, proof of Theorem 1.2, p. 12]). Theorem 1 Let (u δ ) δ be a sequence uniformly bounded in L p ( ), and let C be a positive constant such that for any δ. Then from (u δ ) δ we can extract a subsequence, still denoted by (u δ ) δ , and we can find u ∈ W 1,p ( ) such that u δ → u strongly in L p ( ) as δ → 0 and Even though several authors are involved in the proof, we refer to estimate (1.2) as Ponce's inequality.

The objective
Our goal is to prove the following extension of (1.2): where is an open bounded set, H(x , x) = h(x )+h(x) 2 , and h ∈ H. As we will see, inequality (1.3) is equivalent to (1.2) for measurable sets, that is, for all measurable sets E in .

Motivation and organization of the paper
The context in which we locate the present paper is the study of the nonlocal p-Laplacian problem. Before proceeding, we make precise some notation. We define the spaces We define now the following nonlocal variational problem: given f ∈ L p ( ), where p = p p-1 and p > 1, find u ∈ X such that for all w ∈ X. Since the existence and uniqueness of solution for this problem is a wellknown fact, for h fixed and any δ, there exists a solution u δ . The aim is to check whether the sequence of solutions (u δ ) δ converges to the solution of the corresponding local p-Laplacian equation. This convergence (or G-convergence) clearly entails the study of the minimization principle and, consequently, this task inevitably leads us to the study of the problem posed above; [1][2][3]5] are some references where this type of convergence is analyzed. The paper is organized by means of three sections containing different proofs of (1.3) and (1.4).

First proof
Our essential tool in to generalize (1.3) is a convenient Vitali covering of the set (see [11,Chap. 4,Sect. 3,p. 109.] for details or [6, Chap. 2, Sect. 2, p. 26] for an elegant proof in the case of Lebesgue-measurable sets). Recall that the family {V i } i∈I is a Vitali covering for ⊂ R N if with any x ∈ we can associate a number α > 0, a sequence of V i , and a sequence of balls A particular and useful version of this chief result is the following:

Proposition 1 Let ⊂ R N be an open bounded set, let K be a compact set included in
, and let ξ be a nonnegative function in L 1 ( × ). Then there is a sequence of pairwise disjoint closed balls In particular, any closed ball B i = B(x, r) ⊂ for any r < d. Moreover, the family F = {B(x, s) : x ∈ K, s < r/2} is a Vitali covering of K , because every point of K is contained in an arbitrarily small ball belonging to F . Consequently, there are disjoint balls In a first step, we assume that h is continuous a.e. in . We adapt [7, Lemma 7.9, p. 129] to prove our key result.
Proof Let C = \ N be the set of points of continuity of f . We define the families For each fixed k > 0, the family F k covers C (and ) in the sense of Vitali. Thus, Theorem 2 allows us to choose a numerable sequence of disjoints sets {a kj + kj } j ∈ F k such that | \ j {a kj + kj }| = 0. Since f is continuous at a kj , the sequence kj can be chosen so that for any x ∈ a kj + kj and any j. Consequently,

Application
We apply the previous analysis to the integral We consider × instead of , and now with h ∈ H. We assume that h is continuous, and we take i,j (a ki + ki ) × (a kj + kj ), the union of a family of pairwise of disjoint sets covering × . Then, according to the previous discussion, we trivially deduce We pass to the limit as δ → 0 in I: we use (1.1), Fatou's lemma and (1. 2) for open sets to derive for any symmetric nonnegative continuous function F ∈ L ∞ (O × O).

Extension to the case of measurable functions
Let now h be just measurable; without loss of generality, supp H ⊂ × and H = 0 otherwise. By Luzin's theorem (see [10, Theorem 2.24, p. 62]), given arbitrary > 0, there exists a continuous function G ∈ C c ( × ) such that sup G(x, y) ≤ sup H(x, y) and G(x, y) = H(x, y) for any (x, y) ∈ ( × ) \ E, where E is a measurable set such that |E| < 2 . Since H is symmetric, we can assume that ( At this stage, we consider any compact set K ⊂ \ E ⊂ . Since is open, we can use Proposition 1: there is a number r > 0 such that the family F = {B(x, s) : x ∈ K, s < r/2} is a Vitali covering of K , and therefore there exists a sequence of pairwise disjoint closed balls We take the limits as δ → 0 to get where the second inequality is true because of (2.3) and Fatou's lemma. Then, since K is any compact set in \ E, we obtain Consequently, (1.4) is proved for any measurable set G ⊂ .

A second proof
We firstly prove (1.4) and then (1.3). By having a look at the work done in the previous section we will be able to provide a straightforward proof of (1.4). Indeed, if E is a measurable set included in , then we can find a compact set K ⊂ E such that |E \ K| is arbitrarily small. Proposition 1 ensures the existence of a numerable sequence of pairwise disjoint balls We apply (1. 2) for open sets and Fatou's lemma in the last chain of inequalities to derive Since K ⊂ E is arbitrary, we arrive at (1.4), that is, (3.1)

Corollary
We prove (1.3). Let h be a given simple function defined in . Then h can be written as h(x) = m i=1 h i I B i (x), where {B i } is a finite covering of disjoint measurable subsets of , and (h i ) i is a set of numbers such that h min ≤ h i ≤ h max . Consequently, we can easily check that Using inequality (1.4) for measurable sets that we have just proved, we straightforwardly infer Let h be a measurable function. By recalling that any measurable function h can be pointwise approximated by an increasing sequence (s n ) n of simple functions we can write