Critical weak-Lp differentiability of singular integrals

We establish that for every function u ∈ L1loc(Ω) whose distributional Laplacian Δu is a signed Borel measure in an open set Ω in RN, the distributional gradient ∇u is differentiable almost everywhere in Ω with respect to the weak-LN/(N−1) Marcinkiewicz norm. We show in addition that the absolutely continuous part of Δu with respect to the Lebesgue measure equals zero almost everywhere on the level sets {u = α} and {∇u = e}, for every α ∈ R and e ∈ RN. Our proofs rely on an adaptation of Calderón and Zygmund's singular-integral estimates inspired by subsequent work by Haj'lasz.