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-$L^{\frac{N}{N-1}}$ 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 Calder\'on and Zygmund's singular-integral estimates inspired by subsequent work by Hajlasz.