Critical weak- differentiability of singular integrals

  • Luigi Ambrosio

    Scuola Normale Superiore, Pisa, Italy
  • Augusto C. Ponce

    Université Catholique de Louvain, Louvain-la-Neuve, Belgium
  • Rémy Rodiac

    Université Catholique de Louvain, Louvain-la-Neuve, Belgium and Université Paris-Saclay, Orsay, France
Critical weak-$L^p$ differentiability of singular integrals cover
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Abstract

We establish that for every function whose distributional Laplacian is a signed Borel measure in an open set in , the distributional gradient is differentiable almost everywhere in with respect to the weak- Marcinkiewicz norm. We show in addition that the absolutely continuous part of with respect to the Lebesgue measure equals zero almost everywhere on the level sets and , for every and . Our proofs rely on an adaptation of Calderón and Zygmund's singular-integral estimates inspired by subsequent work by Hajlasz.

Cite this article

Luigi Ambrosio, Augusto C. Ponce, Rémy Rodiac, Critical weak- differentiability of singular integrals. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2033–2072

DOI 10.4171/RMI/1190