Critical weak$L^{p}$ differentiability of singular integrals
Abstract
We establish that for every function $u \in L^1_\mathrm{loc}(\Omega)$ 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$L^{\frac{N}{N1}}$ 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 singularintegral estimates inspired by subsequent work by Hajlasz.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 DOI:
 10.48550/arXiv.1810.03924
 arXiv:
 arXiv:1810.03924
 Bibcode:
 2018arXiv181003924A
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Analysis of PDEs;
 26B05;
 26D10;
 42B20;
 42B37;
 46E35
 EPrint:
 Accepted for publication in Revista Matem\'atica Iberoamericana