  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.
