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# Correction to: Generalized Ponce’s inequality

*Journal of Inequalities and Applications*
**volume 2021**, Article number: 80 (2021)

## Introduction

We assume that \(\Omega \subset \mathbb{R}^{N}\) is an open bounded domain with Lipschitz boundary; \(( k_{\delta } ) _{\delta >0}\) is a set of radial positive functions such that \(\operatorname*{supp}k_{\delta }\subset B ( 0,\delta ) \), \(\frac{1}{C_{N}}\int _{B ( 0,\delta ) }k_{\delta } ( \vert s \vert )\,ds=1\), where \(C_{N}=\frac{1}{\operatorname*{meas} ( S^{N-1} ) }\int _{S^{N-1}} \vert \sigma \cdot \mathbf{e} \vert ^{p}\,d\mathcal{H}^{N-1} ( \sigma ) \), \(\mathcal{H}^{N-1}\) is 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 the ball with center *x* and radius *δ*.

In [2], under the assumptions above, the following compactness is recalled (see [2] and references therein):

### Theorem 1

*Assume* Ω *is an open bounded domain with Lipschitz boundary*. *Let* \(( u_{\delta } ) _{\delta }\) *be a sequence uniformly bounded in* \(L^{p} ( \Omega )\), *and let* *C* *be a positive constant such that*

*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*

Even though several authors are involved in the proof, we refer to estimate (1.2) as Ponce’s inequality.

The goal of [2] is to prove the following extension of (1.2):

where Ω is an open bounded set with Lipschitz boundary, \(H ( x^{\prime },x ) = \frac{h ( x^{\prime })+h(x ) }{2}\), and *h* is a nonnegative function from \(L^{\infty } ( \Omega ) \).

Alternatively, the goal is to check the inequality (1.2) for measurable sets, that is,

It must be remarked that both inequalities are true but some basis for the proofs is false. Concretely, Proposition 1 from [2, p. 3] is wrong and, consequently, those parts where it is used have to be modified. Let us go through the steps and distinguish which parts are faulty.

## First proof

Proposition 2 from [2, p. 4] is true and its proof is correct. The analysis application derived that proposition establishes

for any symmetric nonnegative continuous function \(F\in L^{\infty } ( O\times O ) \) and any smooth open set *O* such that \(\vert \partial O \vert =0\). However, the proof extending (1.3) to the case where *H* is a measurable function of \(L^{\infty } ( \Omega ) \) is invalid because it relies on Proposition 1.

The extension to the case of measurable functions is possible because Proposition 2 from [2, p. 4] is also true for the case \(p=\infty \) and \(q=1\). Let us check it. By looking back at the original work where the idea of the proof comes from, we can check that this result is valid for all \(f\in L^{p}\) and \(\xi \in L^{q} ( \Omega ) \), with \(\frac{1}{p}+\frac{1}{q}=1\), even for the case \(p=\infty \) and \(q=1\) (see [3, p. 126]). Namely, in [3, p. 130], given \(f\in L^{p}\), we can select a family of disjoint sets \(\{ a_{kj}+\epsilon _{kj}\overline{\Omega } \} _{j}\) covering Ω such that

for any \(\psi \in L^{q}\).

Now, for simplicity, we assume \(f\in L^{\infty }\) and *ξ*∈ \(L^{1}\) are nonnegative functions. Since \(\xi ^{1/q}\in L^{q}\) for any *q*, and \(f\in L^{p}\) for any *p*, the above inequality for \(\psi =\xi ^{1/q}\) reads as

If we pass to the limit as \(p\uparrow \infty \), then \(q=\frac{p}{p-1}\downarrow 1\) and \(\xi ^{1/q} ( x ) \rightarrow \xi ( x ) \), and, consequently, by monotone and dominated convergence for series and integrals, we infer

Using this inequality and following the previous procedure, then we can conclude that (2.1) remains valid for any symmetric and nonnegative function \(F\in L^{\infty } ( O\times O ) \) and any smooth domain \(O\subset \Omega \) such that \(\vert \partial O \vert =0\).

Finally, in order to circumvent the assumption \(\vert \partial \Omega \vert =0\), we simplify as follows: for the given domain Ω, we consider Ω̃, a regular domain containing Ω whose boundary is a null set, and we extend *H* by zero in \(( \widetilde{\Omega }\times \widetilde{\Omega } ) \setminus ( \Omega \times \Omega ) \). We denote this extended function of *H* by \(H_{0}\), which is measurable, symmetric, and nonnegative. In the same way, we also appropriately extend \(u_{\delta }\) to Ω̃, so that (1.1) still holds. To do that, we first note that Ω is smooth and, therefore, we can extend *u* to \(\widetilde{u}\in W^{1,p} ( \widetilde{\Omega } ) \). Then, we define \(\widetilde{u}_{\delta } ( x ) =u ( x ) \) if *x*∈ \(\widetilde{\Omega }\setminus \Omega \) and \(\widetilde{u}_{\delta } ( x ) =u_{\delta } ( x ) \) if \(x\in \Omega \). It is immediate to check that \(( \widetilde{u}_{\delta } ) _{\delta }\) is uniformly bounded in \(L^{p}\) and

Then, by Theorem 1, we obtain

Now we realize that the above inequality coincides with (2.1) for any open and bounded set Ω.

The analysis performed proving a corollary in Sect. 2.3 in [2, p. 7] is correct and therefore serves to establish that (1.4) is valid for all measurable sets \(G\subset \Omega \).

## A second proof

This part of the paper deserves a stark modification because the proof given in [2] is based entirely on Proposition 1.

We first prove (1.4) and then (1.3). We assume Ω is open and \(\vert \partial \Omega \vert =0\). By hypothesis, \(( \xi _{\delta } ) _{\delta }\) is a sequence uniformly bounded in \(L^{1} ( \Omega \times \Omega )\) and, under these circumstances, we can use Chacon’s biting lemma (see [1]) to ensure the existence of a subsequence of \(\delta ^{\prime }s\), not relabeled, a decreasing sequence of measurable sets \(\mathcal{E}_{n}\subset \Omega \times \Omega \), such that \(\vert \mathcal{E}_{n} \vert \downarrow 0\), and a function \(\xi \in L^{1} ( \Omega \times \Omega ) \) such that \(\xi _{\delta }\rightharpoonup \xi \) weakly in \(L^{1} ( \Omega \times \Omega \setminus \mathcal{E}_{n} ) \) for all *n*. Since we are dealing with a sequence of symmetric functions, we can ensure \(( \Omega \times \Omega ) \setminus \mathcal{E}_{n}= ( \Omega \setminus E_{n} ) \times ( \Omega \setminus E_{n} ) \) where the sequence of sets \(E_{n}\subset \Omega \) is decreasing and \(\vert E_{n} \vert \downarrow 0\) if \(n\rightarrow \infty \).

Let \(O_{n}\) be any open set such that \(E_{n}\subset O_{n}\subset \Omega \), \(\vert \partial O_{n} \vert =0\), \(\vert \overline{O}_{n} \vert \downarrow 0\) if \(n\rightarrow \infty \), and \(\overline{O}_{n}\subset \Omega \) except for a null subset of \(\overline{O}_{n}\). To achieve these properties, we solely need to take \(\overline{O}_{n}\) as the infimum of the unions of open balls containing \(E_{n}\).

We apply Chacon’s biting lemma to guarantee

for any measurable \(A\times A\subset ( \Omega \setminus \overline{O}_{n} ) \times ( \Omega \setminus \overline{O}_{n} ) \). Also, inequality (1.4) for open sets provides

for any measurable set \(A\subset \Omega \setminus \overline{O}_{n}\) (here we are considering the subsequence of \(\delta ^{\prime }s\) for which (1.4) holds).

Now, we first consider \(A=B ( x_{0},r ) \subset \Omega \setminus \overline{O}_{n}\) for any \(x_{0}\in \Omega \setminus \overline{O}_{n}\). Then, on the one hand, by (3.2)we have

On the other hand, since \(B ( x_{0},r ) \times B ( x_{0},r ) \) is a smooth domain, (1.4) can be applied and hence

By using (3.3) and (3.4), we arrive at this crucial inequality for any \(B ( x_{0},r ) \subset \Omega \setminus \overline{O}_{n}\):

Thus, (3.2) holds for any measurable set \(A\subset \Omega \setminus \overline{O}_{n}\).

Finally, we analyze \(\lim_{\delta \rightarrow 0}\iint _{G\times G}\xi _{\delta } ( x^{ \prime },x )\,dx^{\prime }\,dx\), where \(G\subset \Omega \) is any measurable set. We note that

which, thanks to Chacon’s biting lemma, provides the estimate

Since \(G\setminus \overline{O}_{n}\) is a measurable set included in \(\Omega \setminus \overline{O}_{n}\), (3.2) for measurable sets provides the estimate

which straightforwardly implies

By letting \(n\rightarrow \infty \), we finish the proof of (1.4).

To avoid the hypothesis \(\vert \partial \Omega \vert =0\), we proceed as in the previous section.

The analysis performed when proving a corollary in Sect. 3.1 from [2, p. 8] is correct and therefore serves to assert that (1.3) is valid for all measurable functions *h*.

All the changes requested are implemented in this correction.

## References

- 1.
Brooks, J.K., Chacon, R.V.: Continuity and compactness of measures. Adv. Math.

**37**, 16–26 (1980) - 2.
Muñoz, J.: Generalized Ponce’s inequality. J. Inequal. Appl.

**2021**, 11 (2021). https://doi.org/10.1186/s13660-020-02543-1 - 3.
Pedregal, P.: Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997)

## Acknowledgements

The author would like to acknowledge most warmly Anton Egrafov’s comments.

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Muñoz, J. Correction to: Generalized Ponce’s inequality.
*J Inequal Appl* **2021, **80 (2021). https://doi.org/10.1186/s13660-021-02609-8

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