Critical weak-Lp differentiability of singular integrals

We establish that for every function u is an element of L-loc(1)(Omega) whose distributional Laplacian Delta u is a signed Borel measure in an open set Omega in R-N, the distributional gradient del u is differentiable almost everywhere in Omega with respect to the weak-LN/(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 {del u = e}, for every alpha is an element of R and e is an element of R-N. Our proofs rely on an adaptation of Calderon and Zygmund's singular-integral estimates inspired by subsequent work by Hajlasz.