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

Abstract

We establish that for every function $u\in L_{\mathrm{loc}}^1(\Omega )$ 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.