Resumen de Critical weak-Lp differentiability of singular integrals

Luigi Ambrosio, Augusto C. Ponce, Rémy Rodiac

• 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 Hajlasz.