# Critical weak-$L^p$ differentiability of singular integrals

### Luigi Ambrosio

Scuola Normale Superiore, Pisa, Italy### Augusto C. Ponce

Université Catholique de Louvain, Louvain-la-Neuve, Belgium### Rémy Rodiac

Université Catholique de Louvain, Louvain-la-Neuve, Belgium and Université Paris-Saclay, Orsay, France

A subscription is required to access this article.

## Abstract

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.

## Cite this article

Luigi Ambrosio, Augusto C. Ponce, Rémy Rodiac, Critical weak-$L^p$ differentiability of singular integrals. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2033–2072

DOI 10.4171/RMI/1190