JournalsrmiVol. 36, No. 7pp. 2033–2072

Critical weak-LpL^p 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 uLloc1(Ω)u \in L^1_{\rm loc}(\Omega) whose distributional Laplacian Δu\Delta u is a signed Borel measure in an open set Ω\Omega in RN\mathbb{R}^{N}, the distributional gradient u\nabla u is differentiable almost everywhere in Ω\Omega with respect to the weak-LN/(N1)L^{N/(N-1)} Marcinkiewicz norm. We show in addition that the absolutely continuous part of Δu\Delta u with respect to the Lebesgue measure equals zero almost everywhere on the level sets {u=α}\{u= \alpha\} and {u=e}\{\nabla u=e\}, for every αR\alpha \in \mathbb{R} and eRNe \in \mathbb{R}^N. 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-LpL^p differentiability of singular integrals. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2033–2072

DOI 10.4171/RMI/1190