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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., [4] 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 [9]. 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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Muñoz, J. Generalized Ponce’s inequality. J Inequal Appl 2021, 11 (2021). https://doi.org/10.1186/s13660-020-02543-1

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MSC

  • 05C38
  • 15A15
  • 05A15
  • 15A18

Keywords

  • Nonlocal elliptic equations
  • Integral equations