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

We establish that for every function $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ whose distributional Laplacian $\mathrm{\Delta }u$ is a signed Borel measure in an open set $\mathrm{\Omega }$ in ${\mathbb{R}}^N$, the distributional gradient $\mathrm{\nabla }u$ is differentiable almost everywhere in $\mathrm{\Omega }$ with respect to the weak-$L^{\frac{N}{N-1}}$ Marcinkiewicz norm. We show in addition that the absolutely continuous part of $\mathrm{\Delta }u$ with respect to the Lebesgue measure equals zero almost everywhere on the level sets $\{u=\alpha \}$ and ${\nabla u = e}$,forevery$\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.