Critical weak-$L^{p}$ differentiability of singular integrals
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2018
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| _version_ | 1866916595781074944 |
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| author | Ambrosio, Luigi Ponce, Augusto C. Rodiac, Rémy |
| author_facet | Ambrosio, Luigi Ponce, Augusto C. Rodiac, Rémy |
| contents | We establish that for every function $u \in L^1_\mathrm{loc}(Ω)$ whose distributional Laplacian $Δu$ is a signed Borel measure in an open set $Ω$ in $\mathbb{R}^{N}$, the distributional gradient $\nabla u$ is differentiable almost everywhere in $Ω$ with respect to the weak-$L^{\frac{N}{N-1}}$ Marcinkiewicz norm. We show in addition that the absolutely continuous part of $Δu$ with respect to the Lebesgue measure equals zero almost everywhere on the level sets $\{u = α\}$ and $\{\nabla u = e\}$, for every $α\in \mathbb{R}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1810_03924 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Critical weak-$L^{p}$ differentiability of singular integrals Ambrosio, Luigi Ponce, Augusto C. Rodiac, Rémy Functional Analysis Analysis of PDEs 26B05, 26D10, 42B20, 42B37, 46E35 We establish that for every function $u \in L^1_\mathrm{loc}(Ω)$ whose distributional Laplacian $Δu$ is a signed Borel measure in an open set $Ω$ in $\mathbb{R}^{N}$, the distributional gradient $\nabla u$ is differentiable almost everywhere in $Ω$ with respect to the weak-$L^{\frac{N}{N-1}}$ Marcinkiewicz norm. We show in addition that the absolutely continuous part of $Δu$ with respect to the Lebesgue measure equals zero almost everywhere on the level sets $\{u = α\}$ and $\{\nabla u = e\}$, for every $α\in \mathbb{R}$ 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. |
| title | Critical weak-$L^{p}$ differentiability of singular integrals |
| topic | Functional Analysis Analysis of PDEs 26B05, 26D10, 42B20, 42B37, 46E35 |
| url | https://arxiv.org/abs/1810.03924 |