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Critical weak-$L^p$ differentiability of singular integrals
Revista Matemática Iberoamericana ( IF 1.2 ) Pub Date : 2020-04-03 , DOI: 10.4171/rmi/1190
Luigi Ambrosio 1 , Augusto Ponce 2 , Rémy Rodiac 3
Affiliation  

We establish that for every function $u \in L^1_{\rm loc}(\Omega)$ whose distributional Laplacian $\Delta u$ is a signed Borel measure in an open set $\Omega$ in $\mathbb{R}^{N}$, the distributional gradient $\nabla u$ is differentiable almost everywhere in $\Omega$ with respect to the weak-$L^{N/(N-1)}$ Marcinkiewicz norm. We show in addition that the absolutely continuous part of $\Delta u$ with respect to the Lebesgue measure equals zero almost everywhere on the level sets $\{u= \alpha\}$ and $\{\nabla u=e\}$, for every $\alpha \in \mathbb{R}$ and $e \in \mathbb{R}^N$. Our proofs rely on an adaptation of Calderón and Zygmund's singular-integral estimates inspired by subsequent work by Hajlasz.

中文翻译:

奇异积分的临界弱$ L ^ p $微分

我们确定,对于L ^ 1 _ {\ rm loc}(\ Omega)$中的每个函数$ u \,其分布拉普拉斯算子$ \ Delta u $是$ \ mathbb {R中的开放集$ \ Omega $中的有符号Borel度量。 } ^ {N} $,相对于弱的$ L ^ {N /(N-1)} $ Marcinkiewicz范数,分布梯度$ \ nabla u $几乎在$ \ Omega $中的任何地方都是可微的。此外,我们证明,关于Lebesgue测度,$ \ Delta u $的绝对连续部分几乎在水平集$ \ {u = \ alpha \} $和$ \ {\ nabla u = e \} $,\ mathbb {R} $中的每个$ \ alpha \和\ mathbb {R} ^ N $中的$ e \。我们的证明依赖于Caljón和Zygmund的奇异积分估计值的改编,这些估计值受到Hajlasz后来的工作的启发。
更新日期:2020-04-03
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