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# Generalized Ponce’s inequality

A Correction to this article was published on 28 April 2021

This article has been updated

## Abstract

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.

## Definitions and preliminaries

Let Ω be an open bounded set in $$\mathbb{R}^{N}$$. We define the family of kernels $$( k_{\delta } ) _{\delta >0}$$ as a set of radial positive functions fulfilling the following properties:

1. (1)
$$\frac{1}{C_{N}} \int _{B ( 0,\delta ) }k_{\delta } \bigl( \vert s \vert \bigr) \,ds=1,$$

where

$$C_{N}=\frac{1}{\operatorname{meas} ( S^{N-1} ) } \int _{S^{N-1}} \vert \sigma \cdot \mathbf{e} \vert ^{p}\,d\mathcal{H}^{N-1} ( \sigma ),$$

$$\mathcal{H}^{N-1}$$ stands for the $$( N-1 )$$-dimensional Hausdorff measure on the unit sphere $$S^{N-1}$$, e is any unit vector in $$\mathbb{R}^{N}$$, $$p>1$$, and $$B(x,\delta )$$ is the ball with center x and radius δ.

2. (2)

$$\operatorname{supp}k_{\delta }\subset B ( 0,\delta )$$.

We define the nonlocal operator $$\mathcal{B}_{h}$$ in $$L^{p} ( \Omega ) \times L^{p} ( \Omega )$$ by

$$\mathcal{B}_{h} ( u,u ) = \int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u \bigl( x^{\prime } \bigr) -u ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx,$$

where $$H ( x^{\prime },x ) = \frac{h ( x^{\prime } ) +h ( x ) }{2}$$, $$h\in \mathcal{H}$$,

$$\mathcal{H}\doteq \bigl\{ h:\Omega \rightarrow \mathbb{R}\mid h ( x ) \in {}[ h_{\min },h_{\max }]\text{ a.e. }x\in \Omega, h=0\text{ in } \mathbb{R}^{N}\setminus \Omega \bigr\} ,$$

and $$0< h_{\min }< h_{\max }$$ are given constants.

For $$h=1$$, the following compactness result is well known (see, e.g.,  and [9, proof of Theorem 1.2, p. 12]).

### Theorem 1

Let $$( u_{\delta } ) _{\delta }$$ be a sequence uniformly bounded in $$L^{p} ( \Omega )$$, and let C be a positive constant such that

$$\int _{\Omega } \int _{\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx\leq C$$
(1.1)

for any δ. Then from $$( u_{\delta } ) _{\delta }$$ we can extract a subsequence, still denoted by $$( u_{\delta } ) _{\delta }$$, and we can find $$u\in W^{1,p} ( \Omega )$$ such that $$u_{\delta }\rightarrow u$$ strongly in $$L^{p} ( \Omega )$$ as $$\delta \rightarrow 0$$ and

$$\lim_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx\geq \int _{\Omega } \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx.$$
(1.2)

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):

$$\lim_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx\geq \int _{\Omega }h ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx,$$
(1.3)

where Ω is an open bounded set, $$H ( x^{\prime },x ) = \frac{h ( x^{\prime })+h(x ) }{2}$$, and $$h\in \mathcal{H}$$.

As we will see, inequality (1.3) is equivalent to (1.2) for measurable sets, that is,

$$\lim_{\delta \rightarrow 0} \int _{E} \int _{E} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx\geq \int _{E} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx$$
(1.4)

for all measurable sets E in Ω.

### 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

$$L_{0}^{p} ( \Omega _{\delta } ) = \bigl\{ u\in L^{p} ( \Omega _{\delta } ):u=0\text{ in }\mathbb{R}^{N} \setminus \Omega \bigr\}$$

and

$$X= \bigl\{ u\in L_{0}^{p} ( \Omega _{\delta } ): \mathcal{B} ( u,u ) < \infty \bigr\} ,$$

where

$$\Omega _{\delta }=\Omega \cup \biggl( \bigcup_{x\in \partial \Omega }B ( x,\delta ) \biggr),$$

$$\mathcal{B}=\mathcal{B}_{1}$$, and $$\mathcal{B}_{h}$$ is the operator defined in $$X\times X$$ by

$$\mathcal{B}_{h} ( u,v ) = \int _{\Omega _{\delta }} \int _{ \Omega _{\delta }}H \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u \bigl( x^{\prime } \bigr) -u ( x ) \bigr\vert ^{p-2} \bigl( u \bigl( x^{\prime } \bigr) -u ( x ) \bigr) \bigl( v \bigl( x^{\prime } \bigr) -v ( x ) \bigr) \,dx^{\prime }\,dx.$$

We define now the following nonlocal variational problem: given $$f\in L^{p^{\prime }} ( \Omega )$$, where $$p^{\prime }=\frac{p}{p-1}$$ and $$p>1$$, find $$u\in X$$ such that

$$\mathcal{B}_{h} ( u,w ) = ( f,w ) _{L^{p^{ \prime }} ( \Omega ) \times L^{p} ( \Omega ) } \quad\text{in }X.$$
(1.5)

Note that (1.5) is equivalent to

\begin{aligned} &\int _{\Omega _{\delta }} \int _{\Omega _{\delta }}H \bigl( x^{\prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u ( x^{\prime } ) -u ( x ) \vert ^{p-2} ( u ( x^{\prime } ) -u ( x ) ) ( w ( x^{\prime } ) -w ( x ) ) }{ \vert x^{\prime }-x \vert ^{p}} \,dx^{\prime }\,dx \\ &\quad= \int _{\Omega _{\delta }}fw\,dx \end{aligned}
(1.6)

for all w X. Since the existence and uniqueness of solution for this problem is a well-known fact, for h fixed and any δ, there exists a solution $$u_{\delta }$$. The aim is to check whether the sequence of solutions $$( u_{\delta } ) _{\delta }$$ converges to the solution of the corresponding local p-Laplacian equation. This convergence (or G-convergence) clearly entails the study of the minimization principle

$$\min_{w\in X} \biggl\{ \frac{1}{p}\mathcal{B}_{h} ( w,w ) - \int _{\Omega }f ( x ) w ( x ) \,dx \biggr\} ,$$

and, consequently, this task inevitably leads us to the study of the problem posed above; [13, 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 Ω (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 $$\{ V_{i} \} _{i\in I}$$ is a Vitali covering for $$\Omega \subset \mathbb{R}^{N}$$ if with any $$x\in \Omega$$ we can associate a number $$\alpha >0$$, a sequence of $$V_{i}$$, and a sequence of balls $$B ( x,\epsilon _{i} )$$ such that $$V_{i}\subset B ( x,\epsilon _{i} )$$ and $$\vert V_{i} \vert \geq \alpha \vert B ( x, \epsilon _{i} ) \vert$$, where $$\epsilon _{i}\rightarrow 0$$ as $$i\rightarrow \infty$$.

### Theorem 2

(Vitali covering theorem)

Let $$\mathcal{A}= \{ V_{i} \} _{k\in K}$$ be a Vitali covering of closed subsets of $$\mathbb{R}^{N}$$ for Ω. There is a sequence of $$( i_{j} ) _{j}\in K$$ such that $$\vert \Omega \setminus \bigcup_{j}V_{i_{j}} \vert =0$$ and the sets $$( V_{i_{j}} ) _{j}$$ are pairwise disjoint.

A particular and useful version of this chief result is the following:

### Proposition 1

Let $$\Omega \subset \mathbb{R}^{N}$$ be an open bounded set, let K be a compact set included in Ω, and let ξ be a nonnegative function in $$L^{1} ( \Omega \times \Omega )$$. Then there is a sequence of pairwise disjoint closed balls $$( \overline{B}_{i} ) \subset \Omega$$ such that $$\vert K\setminus \bigcup_{i=1}^{\infty }\overline{B}_{i} \vert =0$$ and

$$\iint _{K\times K}\xi \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx\geq \sum_{i=1}^{\infty } \iint _{\overline{B}_{i}\times \overline{B}_{i}} \xi \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx.$$

### Proof

Since K is a compact inside Ω and Ω is open, we have $$d\doteq \operatorname{dist} ( K,\mathbb{R}^{N}\setminus \Omega ) >0$$. In particular, any closed ball $$\overline{B}_{i}=\overline{B ( x,r ) }\subset \Omega$$ for any $$r< d$$. Moreover, the family $$\mathcal{F}= \{ \overline{B ( x,s ) }:x\in K, s< r/2 \}$$ is a Vitali covering of K, because every point of K is contained in an arbitrarily small ball belonging to $$\mathcal{F}$$. Consequently, there are disjoint balls $$\overline{B}_{i}$$ such that $$\vert K\setminus \bigcup_{i=1}^{\infty }\overline{B}_{i} \vert =0$$. This covering also serves to approximate $$K\times K$$ because $$\vert ( K\times K ) \setminus ( \bigcup_{i,j=1}^{ \infty } ( \overline{B}_{i}\times \overline{B}_{j} ) ) \vert =0$$, and therefore

$$\iint _{K\times K}\xi \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx= \sum_{i,j=1}^{\infty } \iint _{\overline{B}_{i}\times \overline{B}_{j}} \xi \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx\geq \sum_{i=1}^{ \infty } \iint _{\overline{B}_{i}\times \overline{B}_{i}}\xi \bigl( x^{ \prime },x \bigr) \,dx^{\prime }\,dx.$$

□

In a first step, we assume that h is continuous a.e. in Ω. We adapt [7, Lemma 7.9, p. 129] to prove our key result.

### Proposition 2

Let $$\Omega \subset \mathbb{R}^{N}$$ be an open bounded set such that $$\vert \partial \Omega \vert =0$$, and let f be a positive a.e. continuous function on Ω. Let $$r_{k}:\Omega \setminus N\rightarrow \mathbb{R}^{+}$$ be a sequence of functions, where N is the set of discontinuity points of f. There exist a set of points $$\{ a_{ki} \} _{i}\subset \Omega \setminus N$$ and positive numbers $$\{ \epsilon _{ki} \} _{i}$$ such that for each k, $$\epsilon _{ki}\leq r_{k} ( a_{ki} )$$,

\begin{aligned} & \{ a_{ki}+\epsilon _{ki}\overline{\Omega } \} \quad\textit{are pairwise disjoint}, \\ &\overline{\Omega } =\bigcup_{i} \{ a_{ki}+\epsilon _{ki} \overline{\Omega } \} \cup N_{k},\quad\textit{where } \vert N_{k} \vert =0, \end{aligned}

and

$$\int _{\Omega }f ( x ) \xi ( x ) \,dx=\sum _{i}f ( a_{ki} ) \int _{a_{ki}+\epsilon _{ki}\Omega }\xi ( x ) \,dx+o ( 1 ) \quad\textit{as }k\rightarrow + \infty$$
(2.1)

for all $$\xi \in L^{1} ( \Omega )$$.

### Proof

Let $$C=\Omega \setminus N$$ be the set of points of continuity of f. We define the families

$$\mathcal{F}_{k}= \biggl\{ a+\epsilon \overline{\Omega }\subset \Omega:a \in C, \epsilon \leq r_{k} ( a ), \bigl\vert f ( x ) -f ( a ) \bigr\vert \leq \frac{1}{k} \text{ for any }x\in a+\epsilon \Omega \biggr\} .$$

For each fixed $$k>0$$, the family $$\mathcal{F}_{k}$$ covers C (and Ω) in the sense of Vitali. Thus, Theorem 2 allows us to choose a numerable sequence of disjoints sets $$\{ a_{kj}+\epsilon _{kj}\overline{\Omega } \} _{j}\in \mathcal{F}_{k}$$ such that $$\vert \overline{\Omega }\setminus \bigcup_{j} \{ a_{kj}+ \epsilon _{kj}\overline{\Omega } \} \vert =0$$. Since f is continuous at $$a_{kj}$$, the sequence $$\epsilon _{kj}$$ can be chosen so that

$$\bigl\vert f ( x ) -f ( a_{kj} ) \bigr\vert \leq \frac{1}{k}\quad\text{for any }x\in a_{kj}+\epsilon _{kj} \Omega \text{ and any }j.$$

Consequently,

\begin{aligned} & \biggl\vert \int _{\Omega }\xi ( x ) f ( x ) \,dx- \sum _{j}f ( a_{kj} ) \int _{a_{kj}+\epsilon _{kj}\Omega } \xi ( x ) \,dx \biggr\vert \\ &\quad = \biggl\vert \sum_{j} \int _{a_{kj}+\epsilon _{kj}\Omega } \bigl( f ( x ) -f ( a_{kj} ) \bigr) \xi ( x ) \,dx \biggr\vert \\ &\quad \leq \sum_{j} \int _{a_{kj}+\epsilon _{kj}\Omega } \bigl\vert \bigl( f ( x ) -f ( a_{kj} ) \bigr) \bigr\vert \bigl\vert \xi ( x ) \bigr\vert \,dx \\ &\quad \leq \frac{1}{k}\sum_{j} \int _{a_{kj}+\epsilon _{kj}\Omega } \bigl\vert \xi ( x ) \bigr\vert \,dx \\ &\quad =\frac{1}{k} \Vert \xi \Vert _{L^{1} ( \Omega ) }. \end{aligned}

□

### Application

We apply the previous analysis to the integral

$$I= \int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) \xi _{ \delta } \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx,$$

where

$$\xi _{\delta } \bigl( x^{\prime },x \bigr) = \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert .$$
(2.2)

We consider $$\Omega \times \Omega$$ instead of Ω, and now $$f ( x^{\prime },x )$$ is the symmetric function $$H ( x^{\prime },x ) = \frac{h ( x^{\prime } ) +h ( x ) }{2}$$ with $$h\in \mathcal{H}$$. We assume that h is continuous, and we take $$\bigcup_{i,j} ( a_{ki}+\epsilon _{ki}\Omega ) \times ( a_{kj}+\epsilon _{kj}\Omega )$$, the union of a family of pairwise of disjoint sets covering $$\Omega \times \Omega$$. Then, according to the previous discussion, we trivially deduce

\begin{aligned} I & =\sum_{i,j}H ( a_{ki},a_{kj} ) \int _{a_{ki}+ \epsilon _{ki}\Omega } \int _{a_{kj}+\epsilon _{kj}\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx+o ( 1 ) \\ & \geq \sum_{i}H ( a_{ki},a_{ki} ) \int _{a_{ki}+ \epsilon _{ki}\Omega } \int _{a_{ki}+\epsilon _{ki}\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx+o ( 1 ) \\ & =\sum_{i}h ( a_{ki} ) \int _{a_{ki}+\epsilon _{ki} \Omega }\int _{a_{ki}+\epsilon _{ki}\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx+o ( 1 ). \end{aligned}

We pass to the limit as $$\delta \rightarrow 0$$ in I: we use (1.1), Fatou’s lemma and (1.2) for open sets to derive

\begin{aligned} \liminf_{\delta \rightarrow 0} I \geq{}& \liminf_{\delta \rightarrow 0} \sum_{i}h ( a_{ki} ) \int _{a_{ki}+ \epsilon _{ki}\Omega } \int _{a_{ki}+\epsilon _{ki}\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx+o ( 1 ) \\ \geq{}& \sum_{i}h ( a_{ki} ) \biggl( \liminf_{\delta \rightarrow 0} \int _{a_{ki}+\epsilon _{ki}\Omega } \int _{a_{ki}+ \epsilon _{ki}\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{ \prime }\,dx \biggr)\\ &{} +o ( 1 ) \\ \geq{}& \sum_{i}h ( a_{ki} ) \biggl( \int _{a_{ki}+ \epsilon _{ki}\Omega } \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx \biggr) +o ( 1 ). \end{aligned}

If we take limits as $$k\rightarrow +\infty$$, then this estimate gives

$$\liminf_{\delta \rightarrow 0} I\geq \lim_{k\rightarrow + \infty }\sum _{i}h ( a_{ki} ) \int _{a_{ki}+\epsilon _{ki} \Omega } \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx.$$

By using again Proposition 2 the last inequality clearly provides inequality (1.3).

### Remark 1

The analysis and conclusion we have just arrived at remain valid if we consider any open set $$O\subset \Omega$$ such that $$\vert \partial O \vert =0$$. We can go a step further: we have

$$\liminf_{\delta \rightarrow 0} \int _{O} \int _{O}F \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{ \prime }\,dx\geq \int _{O}F ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx$$
(2.3)

for any symmetric nonnegative continuous function $$F\in L^{\infty } ( O\times O )$$.

### Extension to the case of measurable functions

Let now h be just measurable; without loss of generality, $$\operatorname{supp}H\subset \Omega \times \Omega$$ and $$H=0$$ otherwise. By Luzin’s theorem (see [10, Theorem 2.24, p. 62]), given arbitrary $$\epsilon >0$$, there exists a continuous function $$G\in C_{c} ( \Omega \times \Omega )$$ such that $$\sup G ( x,y ) \leq \sup H ( x,y )$$ and $$G ( x,y ) =H ( x,y )$$ for any $$( x,y ) \in ( \Omega \times \Omega ) \setminus \mathcal{E}$$, where $$\mathcal{E}$$ is a measurable set such that $$\vert \mathcal{E} \vert <\epsilon ^{2}$$. Since H is symmetric, we can assume that $$( \Omega \times \Omega ) \setminus \mathcal{E=} ( \Omega \setminus E ) \times ( \Omega \setminus E )$$, where $$E\subset \Omega$$ is a measurable set such that $$\vert E \vert <\epsilon$$.

At this stage, we consider any compact set $$K\subset \Omega \setminus E\subset \Omega$$. Since Ω is open, we can use Proposition 1: there is a number $$r>0$$ such that the family $$\mathcal{F}= \{ \overline{B ( x,s ) }:x\in K, s< r/2 \}$$ is a Vitali covering of K, and therefore there exists a sequence of pairwise disjoint closed balls $$( \overline{B}_{i} ) _{i=1}^{\infty }\subset \mathcal{F}$$ such that $$\vert K\setminus \bigcup_{i=1}^{\infty }\overline{B}_{i} \vert$$, $$\overline{B}_{i}\subset \Omega$$, and

\begin{aligned} & \int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad \geq \int _{\Omega \setminus E} \int _{\Omega \setminus E}H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad \geq \iint _{ ( \Omega \setminus E ) \times ( \Omega \setminus E ) }G \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad \geq \iint _{K\times K}G \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad \geq \sum_{i} \iint _{\overline{B}_{i}\times \overline{B}_{i}}G \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx. \end{aligned}

We take the limits as $$\delta \rightarrow 0$$ to get

\begin{aligned} & \liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ & \quad\geq \liminf_{\delta \rightarrow 0}\sum_{i} \iint _{\overline{B}_{i}\times \overline{B}_{i}}G \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ & \quad\geq \sum_{i} \int _{B_{i}}G ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx \\ &\quad = \int _{K}G ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx, \end{aligned}

where the second inequality is true because of (2.3) and Fatou’s lemma. Then, since K is any compact set in $$\Omega \setminus E$$, we obtain

\begin{aligned} & \liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad \geq \int _{\Omega \setminus E}G ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx \\ & \quad= \int _{\Omega \setminus E}H ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx \\ &\quad = \int _{\Omega }h ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx- \int _{E}h ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. \end{aligned}

By letting $$\epsilon \downarrow 0$$ and using $$\vert E \vert \leq \epsilon$$, we obtain (1.3), that is,

$$\liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx\geq \int _{\Omega }H ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx.$$
(2.4)

Finally, to circumvent the assumption $$\vert \partial \Omega \vert =0$$, the procedure we follow is identical to that just employed. Take any compact set K included in Ω. Since Ω is assumed to be open, thanks to Proposition 1, K can be exhaustively covered by the union of a numerable sequence of pairwise disjoint closed balls $$\overline{B}_{i}\in \mathcal{F}= \{ \overline{B ( x,s ) }:x\in K, s< r/2 \} \subset \Omega$$, $$i=1,2,\dots$$. Then we realize that

\begin{aligned} &\int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad\geq \sum_{i} \int _{B_{i}} \int _{B_{i}}H \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx. \end{aligned}
(2.5)

By taking into account that $$\vert \partial B_{i} \vert =0$$ we can apply (2.3) and Fatou’s lemma in (2.5) to obtain

$$\liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx\geq \int _{K}H ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx.$$

Since $$K\subset \Omega$$ is arbitrary, we arrive at (2.4) for any open set Ω,

\begin{aligned} &\liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\ &\quad\geq \int _{\Omega }H ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. \end{aligned}
(2.6)

### Corollary

We apply (2.4) to the case $$F ( x^{\prime },x ) =I_{G\times G} ( x^{\prime },x )$$, where G is any measurable set included in Ω: on the one hand, (2.6) guarantees

\begin{aligned} &\liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }F \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx\\ &\quad\geq \int _{\Omega }F ( x,x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx \\ &\quad= \int _{G}I_{G} ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx= \int _{G} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx, \end{aligned}

and, on the other hand, it is obvious that

\begin{aligned} &\int _{\Omega } \int _{\Omega }F \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx\\ &\quad= \int _{G} \int _{G} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{ \prime }\,dx. \end{aligned}

Consequently, (1.4) is proved for any measurable set $$G\subset \Omega$$.

## 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 Ω, then we can find a compact set $$K\subset E$$ such that $$\vert E\setminus K \vert$$ is arbitrarily small. Proposition 1 ensures the existence of a numerable sequence of pairwise disjoint balls $$\overline{B}_{i}\in \mathcal{F}$$ such that $$\vert K\setminus \bigcup_{i=1}^{\infty }\overline{B}_{i} \vert =0$$, $$\overline{B}_{i}\subset \Omega$$ for any i and

\begin{aligned} & \int _{E} \int _{E} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx \\ & \quad\geq \int _{K} \int _{K} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx \\ & \quad\geq \sum_{i} \int _{B_{i}} \int _{B_{i}} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx. \end{aligned}

We apply (1.2) for open sets and Fatou’s lemma in the last chain of inequalities to derive

$$\liminf_{\delta \rightarrow 0} \int _{E} \int _{E} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx\geq \sum_{i} \int _{B_{i}} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx= \int _{K} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx.$$

Since $$K\subset E$$ is arbitrary, we arrive at (1.4), that is,

$$\liminf_{\delta \rightarrow 0} \int _{E} \int _{E} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p} \,dx^{\prime }\,dx\geq \int _{E} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx.$$
(3.1)

### Corollary

We prove (1.3). Let h be a given simple function defined in Ω. Then h can be written as $$h ( x ) =\sum_{i=1}^{m}h_{i}I_{B_{i}} ( x )$$, where $$\{ B_{i} \}$$ is a finite covering of disjoint measurable subsets of Ω, and $$( h_{i} ) _{i}$$ is a set of numbers such that $$h_{\min }\leq h_{i}\leq h_{\max }$$. Consequently, we can easily check that

\begin{aligned} I & \doteq \int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) k_{ \delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{ \prime }\,dx \\ & \geq \sum_{i=1}^{m}h_{i} \int _{B_{i}} \int _{B_{i}}k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{ \prime }\,dx. \end{aligned}

Using inequality (1.4) for measurable sets that we have just proved, we straightforwardly infer

$$\liminf_{\delta \rightarrow 0} I\geq \sum_{i=1}^{m}h_{i} \int _{B_{i}} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx= \int _{ \Omega }h ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx.$$

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

\begin{aligned} & \liminf_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{\prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{\prime }\,dx \\ &\quad =\liminf_{\delta \rightarrow 0} \int _{\Omega }h ( x ) \int _{\Omega }k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{\prime }\,dx \\ &\quad \geq \liminf_{\delta \rightarrow 0} \int _{\Omega }s_{n} ( x ) \int _{\Omega _{\delta }}k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{\prime }\,dx \\ &\quad \geq \int _{\Omega }s_{n} ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. \end{aligned}

It suffices to take the limits in n and apply the monotone convergence theorem to establish (1.3).

## A third proof

The idea is reproducing the arguments from . In a first step, we assume that $$h:\overline{\Omega }\rightarrow [ h_{\min },h_{\max } ]$$ is a continuous function. Moreover, without loss of generality, we suppose that h is a continuous function in the set $$\Omega _{s}=\Omega \cup \{ \bigcup_{p\in \partial \Omega }B ( p,s ) \}$$, where s is a fixed positive number.

Now, for the proof of (1.3), the key idea is extending the Stein inequality (see [8, Lemma 4, p. 245]) in the following sense: by using Jensen’s inequality and performing a change of variables we deduce the inequality

\begin{aligned} & \int _{\Omega } \int _{\Omega }H_{r} \bigl( x^{\prime },x \bigr) k_{ \delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{ \prime }\,dx \\ &\quad \geq \int _{\Omega _{-r}} \int _{\Omega _{-r}}H \bigl( x^{\prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{r,\delta } ( x^{\prime } ) -u_{r,\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{ \prime }\,dx \end{aligned}

for any $$\delta < r$$, where $$u_{r,\delta }=\eta _{r}\ast u_{\delta }$$, $$\eta _{r} ( x ) =\frac{1}{r^{N}}\eta ( \frac{x}{r} )$$, $$x\in \mathbb{R}^{N}$$, η is a nonnegative radial function from $$C_{c}^{\infty } ( B ( 0,1 ) )$$ such that $$\int \eta ( x ) \,dx=1$$,

$$H_{r} \bigl( x^{\prime },x \bigr) = \frac{ ( \eta _{r}\ast h ) ( x^{\prime } ) + ( \eta _{r}\ast h ) ( x ) }{2},$$

and $$\Omega _{-r}= \{ x\in \Omega:\operatorname{dist}(x,\partial \Omega )>r \}$$. By the continuity of H in $$\Omega _{s}\times \Omega _{s}$$ we know that $$H_{r} ( x^{\prime },x ) \rightarrow H ( x^{\prime },x )$$ uniformly on compact sets of $$\Omega _{s}\times \Omega _{s}$$, whereby, for any $$\epsilon >0$$, we can choose $$r_{0}>0$$ such that

$$\biggl\vert \int _{\Omega } \int _{\Omega } \bigl( H \bigl( x^{\prime },x \bigr) -H_{r} \bigl( x^{\prime },x \bigr) \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{ \prime }\,dx \biggr\vert \leq \epsilon C$$

for any $$r< r_{0}$$ and uniformly in $$\delta >0$$. Then

\begin{aligned} & \lim_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{\prime }\,dx \\ & \quad\geq \lim_{\delta \rightarrow 0} \int _{\Omega _{-r}} \int _{\Omega _{-r}}H \bigl( x^{\prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{r,\delta } ( x^{\prime } ) -u_{r,\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{\prime }\,dx-\epsilon C \end{aligned}

for any $$r< r_{0}$$. At this point, we notice that Proposition 1 from [8, p. 242] can be modified by including the term $$H ( x^{\prime },x )$$ within the integrand; this is factually what Remark 1 establishes. Then passing to the limit as $$\delta \rightarrow 0$$ and using the convergence of $$\rho _{r}\ast u_{\delta }\rightarrow \rho _{r}\ast u$$ in $$C^{2} ( \overline{\Omega }_{-r} )$$, we get

\begin{aligned} &\lim_{\delta \rightarrow 0} \int _{\Omega _{-r}} \int _{\Omega _{-r}}H \bigl( x^{\prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{r,\delta } ( x^{\prime } ) -u_{r,\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}} \,dx^{\prime }\,dx\\ &\quad\geq \int _{\Omega _{-r}}h ( x ) \bigl\vert \nabla ( \rho _{r} \ast u ) ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx. \end{aligned}

Consequently, letting $$r\rightarrow 0$$ in this inequality and taking into account that $$\nabla ( \rho _{r}\ast u )$$ strongly converges to u in $$L^{p} ( \Omega )$$, we derive

$$\lim_{\delta \rightarrow 0} \int _{\Omega } \int _{\Omega }H \bigl( x^{ \prime },x \bigr) k_{\delta } \bigl( \bigl\vert x^{\prime }-x \bigr\vert \bigr) \frac{ \vert u_{\delta } ( x^{\prime } ) -u_{\delta } ( x ) \vert ^{p}}{ \vert x^{\prime }-x \vert ^{p}}\,dx^{\prime }\,dx\geq \int _{\Omega }h ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx-\epsilon C.$$

Now, since ϵ is arbitrarily small, the statement is proved under the assumption that h is continuous in $$\Omega _{s}$$.

If $$h:\Omega \rightarrow [ h_{\min },h_{\max } ]$$ is a measurable function, then we extend it by zero to $$\Omega _{s}$$ and then apply Luzin’s theorem to this extended function. The remaining details follow along the lines of Sect. 2.2.

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## Acknowledgements

The author would like to thank the referees and José Carlos Bellido for their helpful comments on the subject studied in this paper.

## Funding

This work was supported by National Spanish Project MTM2017-87912-P, Regional Research Project SBPLY/17/180501/000452 (Junta de Comunidades de Castilla-La Mancha) and by Universidad de Castilla-La Mancha Support for Groups 2018.

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