Critical weak-Lp differentiability of singular integrals

• Luigi Ambrosio ^[2] ; Augusto C. Ponce ^[1] ; Rémy Rodiac ^[1]
  1. [1] Université Catholique de Louvain

     Université Catholique de Louvain

     Arrondissement de Nivelles, Bélgica

     
  2. [2] Scuola Normale Superiore, Pisa
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 7, 2020, págs. 2033-2072
• Idioma: inglés
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• Resumen

  • 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.