Correction to: Generalized Ponce’s inequality,Journal of Inequalities and Applications

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Correction to: Generalized Ponce’s inequality
Journal of Inequalities and Applications ( IF 1.6 ) Pub Date : 2021-04-28 , DOI: 10.1186/s13660-021-02609-8
Julio Muñoz

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

$$ \int _{\Omega } \int _{\Omega } \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx\leq C $$(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 goal of [2] 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 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,

$$\begin{aligned}& \lim_{\delta \rightarrow 0} \int _{E} \int _{E} \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \\& \quad \geq \int _{E} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx,\quad \text{for any measurable }E\subset \Omega . \end{aligned}$$ (1.4)

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.

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

$$ \liminf_{\delta \rightarrow 0} \int _{O} \int _{O}H \bigl( x^{\prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert u_{\delta } \bigl( x^{\prime } \bigr) -u_{\delta } ( x ) \bigr\vert ^{p}\,dx^{ \prime } \,dx\geq \int _{O}h ( x ) \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx $$(2.1)

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

$$ \int _{\Omega }f ( x ) \psi ( x )\,dx\leq \sum _{i}f ( a_{ki} ) \int _{a_{ki}+\epsilon _{ki}\Omega }\psi ( x )\,dx-\frac{1}{k} \vert \Omega \vert ^{1/p} \Vert \psi \Vert _{L^{q} ( \Omega ) }$$

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

$$ \int _{\Omega }f ( x ) \xi ^{1/q} ( x )\,dx\leq \sum _{i}f ( a_{ki} ) \int _{a_{ki}+\epsilon _{ki}\Omega } \xi ^{1/q} ( x )\,dx- \frac{1}{k} \vert \Omega \vert ^{1/p} \Vert \xi \Vert _{L^{1} ( \Omega ) }^{1/q}. $$

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

$$ \int _{\Omega }f ( x ) \xi ( x )\,dx\leq \sum _{i}f ( a_{ki} ) \int _{a_{ki}+\epsilon _{ki}\Omega }\xi ( x )\,dx-\frac{1}{k} \Vert \xi \Vert _{L^{1} ( \Omega ) }. $$

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

$$ \int _{\widetilde{\Omega }} \int _{\widetilde{\Omega }}H_{0} \bigl( x^{ \prime },x \bigr) \frac{k_{\delta } ( \vert x^{\prime }-x \vert ) }{ \vert x^{\prime }-x \vert ^{p}} \bigl\vert \widetilde{u}_{\delta } \bigl( x^{\prime } \bigr) - \widetilde{u}_{\delta } ( x ) \bigr\vert ^{p}\,dx^{\prime }\,dx \leq C. $$

Then, by Theorem 1, we obtain

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

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 $.

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

$$ \lim_{\delta \rightarrow 0} \iint _{A\times A}\xi _{\delta } \bigl( x^{ \prime },x \bigr)\,dx^{\prime }\,dx= \iint _{A\times A}\xi \bigl( x^{ \prime },x \bigr) \,dx^{\prime }\,dx $$(3.1)

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

$$ \lim_{\delta \rightarrow 0} \iint _{A\times A}\xi _{\delta } \bigl( x^{ \prime },x \bigr)\,dx^{\prime }\,dx\geq \int _{A} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx, $$(3.2)

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

$$ \lim_{\delta \rightarrow 0} \iint _{B ( x_{0},r ) \times B ( x_{0},r ) }\xi _{\delta } \bigl( x^{\prime },x \bigr)\,dx^{ \prime }\,dx= \iint _{B ( x_{0},r ) \times B ( x_{0},r ) } \xi \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx. $$(3.3)

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

$$ \lim_{\delta \rightarrow 0} \iint _{B ( x_{0},r ) \times B ( x_{0},r ) }\xi _{\delta } \bigl( x^{\prime },x \bigr)\,dx^{ \prime }\,dx\geq \int _{B ( x_{0},r ) } \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. $$(3.4)

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

$$ \iint _{B ( x_{0},r ) \times B ( x_{0},r ) } \xi \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx\geq \int _{B ( x_{0},r ) } \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx\,dx^{ \prime }. $$(3.5)

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

$$ \iint _{G\times G}\xi _{\delta } \bigl( x^{\prime },x \bigr)\,dx^{ \prime }\,dx\geq \iint _{ ( G\setminus \overline{O}_{n} ) \times ( G\setminus \overline{O}_{n} ) }\xi _{\delta } \bigl( x^{ \prime },x \bigr)\,dx^{\prime }\,dx $$

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

$$ \lim_{\delta \rightarrow 0} \iint _{G\times G}\xi _{\delta } \bigl( x^{ \prime },x \bigr)\,dx^{\prime }\,dx\geq \iint _{ ( G\setminus \overline{O}_{n} ) \times ( G\setminus \overline{O}_{n} ) }\xi \bigl( x^{\prime },x \bigr) \,dx^{\prime }\,dx. $$

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

$$ \iint _{ ( G\setminus \overline{O}_{n} ) \times ( G \setminus \overline{O}_{n} ) }\xi \bigl( x^{\prime },x \bigr) \,dx^{ \prime }\,dx\geq \int _{G\setminus \overline{O}_{n}} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx, $$

which straightforwardly implies

$$ \lim_{\delta \rightarrow 0} \iint _{G\times G}\xi _{\delta } \bigl( x^{ \prime },x \bigr)\,dx^{\prime }\,dx\geq \int _{G\setminus \overline{O}_{n}} \bigl\vert \nabla u ( x ) \bigr\vert ^{p}\,dx. $$

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.

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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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Pedregal, P.: Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997)
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The author would like to acknowledge most warmly Anton Egrafov’s comments.

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Departamento de Matemáticas, Universidad de Castilla-La Mancha, Toledo, Spain
Julio Muñoz

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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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Published: 28 April 2021
DOI: https://doi.org/10.1186/s13660-021-02609-8

中文翻译：

更正为：广义Ponce不等式

我们假设\（\ Omega \ subset \ mathbb {R} ^ {N} \）是一个具有Lipschitz边界的开放域；\（（k _ {\ delta}）_ {\ delta> 0} \）是一组径向正函数，例如\（\ operatorname * {supp} k _ {\ delta} \ subset B（0，\ delta）\ ），\（\ frac {1} {C_ {N}} \ int _ {B（0，\ delta）} k _ {\ delta}（\ vert s \ vert）\，ds = 1 \），其中\（ 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} \）是\（（N-1 ）\）-单位球面上\（S ^ {N-1} \）的尺寸Hausdorff测度，e是\（\ mathbb {R} ^ {N} \），\（p> 1 \）和\（B（x，\ delta）\）是中心为x且半径为δ的球。

在[2]中，在上述假设下，召回了以下紧凑性（请参见[2]及其参考）：

定理1

假设Ω是一个具有Lipschitz边界的开界域。令 \（（u _ {\ delta}）_ {\ delta} \）是在 \（L ^ {p}（\ Omega）\）中均匀有界的序列，令 C 为一个正常数，使得

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

对于任何 δ。然后，从 \（（u _ {\ delta}）_ {\ delta} \） 中提取一个子序列，仍然用 \（（u _ {\ delta}）_ {\ delta} \）表示，我们可以找到 \ （u \ in W ^ {1，p}（\ Omega）\）使得 \（u _ {\ delta} \ rightarrow u \）在 \（L ^ {p}（\ Omega）\）中 强烈地像 \（\ delta \ rightarrow 0 \）和

$$ \ lim _ {\ delta \ rightarrow 0} \ int _ {\ Omega} \ int _ {\ Omega} \ frac {k _ {\ delta}（\ vert x ^ {\ prime} -x \ vert）} {\ vert x ^ {\ prime} -x \ vert ^ {p}} \ bigl \ vert u _ {\ delta} \ bigl（x ^ {\ prime} \ bigr）-u _ {\ delta}（x）\ bigr \ vert ^ {p} \，dx ^ {\ prime} \，dx \ geq \ int _ {\ Omega} \ bigl \ vert \ nabla u（x）\ bigr \ vert ^ {p} \，dx。$$（1.2）

即使有几位作者参与了证明，我们也将估计（1.2）称为庞塞不等式。

[2]的目标是证明（1.2）的以下扩展：

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

其中Ω是具有Lipschitz边界的开放边界集\（H（x ^ {\ prime}，x）= \ frac {h（x ^ {\ prime}）+ h（x）} {2} \），并且h是\（L ^ {\ infty}（\ Omega）\）中的非负函数。

或者，目标是检查可测集合的不等式（1.2），即

$$ \ begin {aligned}＆\ lim _ {\ delta \ rightarrow 0} \ int _ {E} \ int _ {E} \ frac {k _ {\ delta}（\ vert x ^ {\ prime} -x \ vert ）} {\ vert x ^ {\ prime} -x \ vert ^ {p}} \ bigl \ vert u _ {\ delta} \ bigl（x ^ {\ prime} \ bigr）-u _ {\ delta}（x） \ bigr \ vert ^ {p} \，dx ^ {\ prime} \，dx \\＆\ quad \ geq \ int _ {E} \ bigl \ vert \ nabla u（x）\ bigr \ vert ^ {p} \，dx，\ quad \ text {用于任何可测量的} E \ subset \ Omega。\ end {aligned} $$（1.4）

必须指出，两个不等式都是正确的，但证明的某些依据是错误的。具体来说，[2，p。1]中的命题1。3]是错误的，因此，必须修改使用它的那些部分。让我们逐步执行这些步骤，并区分出哪些部分是有缺陷的。

来自[2，p。2的命题2。4]是正确的，其证明是正确的。命题建立的分析应用

$$ \ liminf _ {\ delta \ rightarrow 0} \ int _ {O} \ int _ {O} H \ bigl（x ^ {\ prime}，x \ bigr）\ frac {k _ {\ delta}（\ vert x ^ {\ prime} -x \ vert）} {\ vert x ^ {\ prime} -x \ vert ^ {p}} \ bigl \ vert u _ {\ delta} \ bigl（x ^ {\ prime} \ bigr） -u _ {\ delta}（x）\ bigr \ vert ^ {p} \，dx ^ {\ prime} \，dx \ geq \ int _ {O} h（x）\ bigl \ vert \ nabla u（x） \ bigr \ vert ^ {p} \，dx $$（2.1）

对于任何对称的非负连续函数\（F \ in L ^ {\ infty}（O \ times O）\）和任何平滑开集O使得\（\ vert \ partial O \ vert = 0 \）。但是，将（1.3）扩展到H是\（L ^ {\ infty}（\ Omega）\）的可测量函数的情况是无效的，因为它依赖于命题1。

由于[2，p。1中的命题2]可以扩展到可测量函数的情况。4]对于\（p = \ infty \）和\（q = 1 \）的情况也是如此。让我们检查一下。通过回顾证明思想起源的原始工作，我们可以检查该结果对于所有\（f \ in L ^ {p} \）和\（\ xi \ in L ^ {q} （\ Omega）\）和\（\ frac {1} {p} + \ frac {1} {q} = 1 \），即使对于\（p = \ infty \）和\（q = 1 \）（请参阅[3，第126页]）。即，在[3，p。130]，在给定\（f \ in L ^ {p} \）的情况下，我们可以选择不相交集族\（\ {a_ {kj} + \ epsilon _ {kj} \ overline {\ Omega} \} _ { j} \）覆盖Ω，使得

$$ \ int _ {\ Omega} f（x）\ psi（x）\，dx \ leq \ sum _ {i} f（a_ {ki}）\ int _ {a_ {ki} + \ epsilon _ {ki } \ Omega} \ psi（x）\，dx- \ frac {1} {k} \ vert \ Omega \ vert ^ {1 / p} \ Vert \ psi \ Vert _ {L ^ {q}（\ Omega） } $$

对于任何\（\ psi \ in L ^ {q} \）。

现在，为了简单起见，我们假设\（F \在L ^ {\ infty} \）和ξ ＆Element; \（L ^ {1} \）是非负函数。由于\（在L ^ {Q} \ XI ^ {1 / Q} \ \）对于任何q，和\（F \在L ^ {P} \）对于任何p，对于上面的不等式\（\ PSI = \ xi ^ {1 / q} \）读为

$$ \ int _ {\ Omega} f（x）\ xi ^ {1 / q}（x）\，dx \ leq \ sum _ {i} f（a_ {ki}）\ int _ {a_ {ki} + \ epsilon _ {ki} \ Omega} \ xi ^ {1 / q}（x）\，dx- \ frac {1} {k} \ vert \ Omega \ vert ^ {1 / p} \ Vert \ xi \垂直_ {L ^ {1}（\ Omega）} ^ {1 / q}。$$

如果我们将极限传递为\（p \ uparrow \ infty \），则\（q = \ frac {p} {p-1} \ downarrow 1 \）和\（\ xi ^ {1 / q}（x ）\ rightarrow \ xi（x）\），因此，通过级数和积分的单调和占优收敛，我们可以得出

$$ \ int _ {\ Omega} f（x）\ xi（x）\，dx \ leq \ sum _ {i} f（a_ {ki}）\ int _ {a_ {ki} + \ epsilon _ {ki } \ Omega} \ xi（x）\，dx- \ frac {1} {k} \ Vert \ xi \ Vert _ {L ^ {1}（\ Omega）}。$$

使用该不等式并遵循前面的过程，我们可以得出结论：（2.1）对于任何对称和非负函数\（F \ in L ^ {\ infty}（O \ times O）\）和任何平滑域\（ O \ subset \ Omega \）使得\（\ vert \ partial O \ vert = 0 \）。

最后，为了规避假设\（\ vert \ partial \ Omega \ vert = 0 \），我们简化如下：对于给定的域Ω，我们考虑Ω̃，这是一个包含Ω的常规域，其边界是空集，然后在\（（\ widetilde {\ Omega} \ times \ widetilde {\ Omega}} \ setminus（\ Omega \ times \ Omega} \中将H扩展为零。我们用\（H_ {0} \）表示H的扩展函数，它是可测量的，对称的和非负的。同样，我们也将\（u _ {\ delta} \）适当地扩展到Ω̃，以便（1.1）仍然成立。为此，我们首先注意到Ω是平滑的，因此，我们可以将u扩展为\（\ widetilde {u} \ in W ^ {1，p}（\ widetilde {\ Omega}）\）。然后，我们定义\（\ widetilde【U} _ {\增量}（X）= U（x）的\）如果X ∈ \（\ widetilde {\欧米茄} \ setminus \欧米茄\）和\（\ widetilde【U } _ {\ delta}（x）= u _ {\ delta}（x）\）如果\（x \ in \ Omega \）。立即检查\（（\ widetilde {u} _ {\ delta}）_ {\ delta} \）是否均匀地限制在\（L ^ {p} \）和

$$ \ int _ {\ widetilde {\ Omega}} \ int _ {\ widetilde {\ Omega}} H_ {0} \ bigl（x ^ {\ prime}，x \ bigr）\ frac {k _ {\ delta} （\ vert x ^ {\ prime} -x \ vert）} {\ vert x ^ {\ prime} -x \ vert ^ {p}} \ bigl \ vert \ widetilde {u} _ {\\ delta} \ bigl（ x ^ {\ prime} \ bigr）-\ widetilde {u} _ {\ delta}（x）\ bigr \ vert ^ {p} \，dx ^ {\ prime} \，dx \ leq C. $$

然后，根据定理1，我们得到

$$ \ liminf _ {\ delta \ rightarrow 0} \ int _ {\ widetilde {\ Omega}} \ int _ {\ widetilde {\ Omega}} H_ {0} \ bigl（x ^ {\ prime}，x \ bigr ）\ frac {k _ {\ delta}（\ vert x ^ {\ prime} -x \ vert）} {\ vert x ^ {\ prime} -x \ vert ^ {p}} \ bigl \ vert \ widetilde {u } _ {\ delta} \ bigl（x ^ {\ prime} \ bigr）-\ widetilde {u} _ {\ delta}（x）\ bigr \ vert ^ {p} \，dx ^ {\ prime} \， dx \ geq \ int _ {\ widetilde {\ Omega}} H_ {0}（x，x）\ bigl \ vert \ nabla \ widetilde {u}（x）\ bigr \ vert ^ {p} \，dx。$$

现在我们意识到，对于任何开放集和有界集Ω，上述不等式都与（2.1）一致。

该分析证明了Sect中的必然结果。2.3 in [2，p。7]是正确的，因此可以证明（1.4）对所有可测集合\（G \ subset \ Omega \）有效。

本文的这一部分应该进行明显的修改，因为[2]中给出的证明完全基于命题1。

我们首先证明（1.4），然后证明（1.3）。我们假设Ω是开路的并且\（\ vert \ partial \ Omega \ vert = 0 \）。根据假设，\（（\ xi _ {\ delta} _ {\ delta} \）是一个均匀地以\（L ^ {1}（\ Omega \ times \ Omega）\）为边界的序列，在这种情况下，我们可以使用Chacon的咬住引理（请参见[1]）来确保存在\（\ delta ^ {\ prime} s \）的子序列（未重新标记），并减少了可测集\（\ mathcal {E} _ {n} \ subset \ Omega \ times \ Omega \），例如\（\ vert \ mathcal {E} _ {n} \ vert \ downarrow 0 \）和一个函数\（\ xi \ in L ^ {1 }（\ Omega \ times \ Omega）\）使得\（\ xi _ {\ delta} \ rightharpoonup \ xi \）弱\（L ^ {1}（\ Omega \ times \ Omega \ setminus \ mathcal {E} _ {n}）\）对于所有n。由于我们要处理一系列对称函数，因此我们可以确保\（（\ Omega \ times \ Omega）\ setminus \ mathcal {E} _ {n} =（\ Omega \ setminus E_ {n}）\ times（\ Omega \ setminus E_ {n}）\）其中集\（E_ {n} \ subset \ Omega \）的序列递减，而\（\ vert E_ {n} \ vert \ downarrow 0 \）如果\（n \ rightarrow \ infty \）。

令\（O_ {n} \）为任何开放集，使得\（E_ {n} \ subset O_ {n} \ subset \ Omega \），\（\ vert \ partial O_ {n} \ vert = 0 \），\（\ vert \ overline {O} _ {n} \ vert \ downarrow 0 \）如果\（n \ rightarrow \ infty \）和\（\ overline {O} _ {n} \ subset \ Omega \）\（\ overline {O} _ {n} \）的空子集除外。要获得这些属性，我们只需要将\（\ overline {O} _ {n} \）作为包含\（E_ {n} \）的开球并集的最小值即可。

我们应用Chacon的咬人引理来保证

$$ \ lim _ {\ delta \ rightarrow 0} \ iint _ {A \ times A} \ xi _ {\ delta} \ bigl（x ^ {\ prime}，x \ bigr）\，dx ^ {\ prime} \ ，dx = \ iint _ {A \ times A} \ xi \ bigl（x ^ {\ prime}，x \ bigr）\，dx ^ {\ prime} \，dx $$（3.1）

对于任何可测量的\（A \ times A \ subset（\ Omega \ setminus \ overline {O} _ {n}）\ times（\ Omega \ setminus \ overline {O} _ {n}）\）。另外，开放集的不等式（1.4）提供了

$$ \ lim _ {\ delta \ rightarrow 0} \ iint _ {A \ times A} \ xi _ {\ delta} \ bigl（x ^ {\ prime}，x \ bigr）\，dx ^ {\ prime} \ ，dx \ geq \ int _ {A} \ bigl \ vert \ nabla u（x）\ bigr \ vert ^ {p} \，dx，$$（3.2）

对于任何（1.4）成立的可测量集合\（A \ subset \ Omega \ setminus \ overline {O} _ {n} \）（这里我们考虑的是\（\ delta ^ {\ prime} s \）的子序列）。

现在，我们首先考虑\ Omega \ setminus \中的任何\（x_ {0} \\\ Omega \ setminus \ overline {O} _ {n} \）\（A = B（x_ {0}，r）\ subset \ Omega \ setminus \ overline {O} _ {n} \上划线{O} _ {n} \）。然后，一方面，通过（3.2）

$$ \ lim _ {\ delta \ rightarrow 0} \ iint _ {B（x_ {0}，r）\ times B（x_ {0}，r）} \ xi _ {\ delta} \ bigl（x ^ {\素数}，x \ bigr）\，dx ^ {\ prime} \，dx = \ iint _ {B（x_ {0}，r）\倍B（x_ {0}，r）} \ xi \ bigl（x ^ {\ prime}，x \ bigr）\，dx ^ {\ prime} \，dx。$$（3.3）

另一方面，由于\（B（x_ {0}，r）\ times B（x_ {0}，r）\）是一个平滑域，因此可以应用（1.4）

$$ \ lim _ {\ delta \ rightarrow 0} \ iint _ {B（x_ {0}，r）\ times B（x_ {0}，r）} \ xi _ {\ delta} \ bigl（x ^ {\质数}，x \ bigr）\，dx ^ {\ prime} \，dx \ geq \ int _ {B（x_ {0}，r）}} \ bigl \ vert \ nabla u（x）\ bigr \ vert ^ { p} \，dx。$$（3.4）

通过使用（3.3）和（3.4），我们得出任何\（B（x_ {0}，r）\ subset \ Omega \ setminus \ overline {O} _ {n} \）的关键不等式：

$$ \ iint _ {B（x_ {0}，r）\ times B（x_ {0}，r）}} \ xi \ bigl（x ^ {\ prime}，x \ bigr）\，dx ^ {\ prime } \，dx \ geq \ int _ {B（x_ {0}，r）} \ bigl \ vert \ nabla u（x）\ bigr \ vert ^ {p} \，dx \，dx ^ {\ prime}。$$（3.5）

因此，（3.2）对于任何可测量的集合\（A \ subset \ Omega \ setminus \ overline {O} _ {n} \）成立。

最后，我们分析\（\ lim _ {\ delta \ rightarrow 0} \ iint _ {G \ times G} \ xi _ {\ delta}（x ^ {\ prime}，x）\，dx ^ {\ prime} \ ，dx \），其中\（G \ subset \ Omega \）是任何可测量的集合。我们注意到

$$ \ iint _ {G \ times G} \ xi _ {\ delta} \ bigl（x ^ {\ prime}，x \ bigr）\，dx ^ {\ prime} \，dx \ geq \ iint _ { G \ setminus \ overline {O} _ {n}）\ times（G \ setminus \ overline {O} _ {n}）} \ xi _ {\ delta} \ bigl（x ^ {\ prime}，x \ bigr ）\，dx ^ {\ prime} \，dx $$

由于Chacon咬人的引理，它提供了估计

$$ \ lim _ {\ delta \ rightarrow 0} \ iint _ {G \ times G} \ xi _ {\ delta} \ bigl（x ^ {\ prime}，x \ bigr）\，dx ^ {\ prime} \ ，dx \ geq \ iint _ {（G \ setminus \ overline {O} _ {n}）\ times（G \ setminus \ overline {O} _ {n}）} \ xi \ bigl（x ^ {\ prime} ，x \ bigr）\，dx ^ {\ prime} \，dx。$$

由于\（G \ setminus \ overline {O} _ {n} \）是包含在\（\ Omega \ setminus \ overline {O} _ {n} \）中的可测量集合，因此，针对可测量集合的（3.2）提供了估计

$$ \ iint _ {（G \ setminus \ overline {O} _ {n}）\ times（G \ setminus \ overline {O} _ {n}）} \ xi \ bigl（x ^ {\ prime}，x \ bigr）\，dx ^ {\ prime} \，dx \ geq \ int _ {G \ setminus \ overline {O} _ {n}} \ bigl \ vert \ nabla u（x）\ bigr \ vert ^ {p } \，dx，$$

直接暗示

$$ \ lim _ {\ delta \ rightarrow 0} \ iint _ {G \ times G} \ xi _ {\ delta} \ bigl（x ^ {\ prime}，x \ bigr）\，dx ^ {\ prime} \ ，dx \ geq \ int _ {G \ setminus \ overline {O} _ {n}} \ bigl \ vert \ nabla u（x）\ bigr \ vert ^ {p} \，dx。$$

通过让\ {n \ rightarrow \ infty \），我们完成了（1.4）的证明。

为了避免假设\（\ vert \ partial \ Omega \ vert = 0 \），我们按照上一节中的步骤进行操作。

证明Sect中的推论时执行的分析。3.1来自[2，p。[8]是正确的，因此可以断言（1.3）对所有可测函数h有效。

在此更正中实现了所有要求的更改。

1。
Brooks，JK，Chacon，RV：措施的连续性和紧凑性。进阶数学。37，16-26（1980）
MathSciNet文章Google学术搜索
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Muñoz，J .：广义Ponce不等式。J.不等。应用 2021、11（2021）。https://doi.org/10.1186/s13660-020-02543-1
MathSciNet文章Google学术搜索
3。
Pedregal，P .：参数化度量和变分原理。Birkhäuser，巴塞尔（1997）
预订Google Scholar

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作者要最热烈地感谢安东·埃格拉夫（Anton Egrafov）的评论。

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卡斯蒂利亚-拉曼恰大学，西班牙托莱多
胡里奥·穆尼奥斯

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