Capacitary differentiability of potentials of finite Radon measures

Joan Verdera Universitat Autònoma de Barcelona

Classical Analysis and ODEs mathscidoc:1912.43046

Arkiv for Matematik, 57, (2), 437 – 450, 2019
We study differentiability properties of a potential of the type K⋆μ, where μ is a finite Radon measure in RN and the kernel K satisfies |∇jK(x)|≤C|x|−(N−1+j),j=0,1,2. We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vallée Poussin sense associated with the kernel |x|−(N−1). We require that the first order remainder at a point is small when measured by means of a normalized weak capacity “norm” in balls of small radii centered at the point. This implies weak LN/(N−1) differentiability and thus Lp differentiability in the Calderón–Zygmund sense for 1≤p<N/(N−1). We show that K⋆μ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for K⋆μ. As an application, we study level sets of newtonian potentials of finite Radon measures.
differentiability, Riesz, Newtonian and logarithmic potentials, capacity, Calderón–Zygmund theory
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  title={Capacitary differentiability of potentials of finite Radon measures},
  author={Joan Verdera},
  booktitle={Arkiv for Matematik},
  pages={437 – 450},
Joan Verdera. Capacitary differentiability of potentials of finite Radon measures. 2019. Vol. 57. In Arkiv for Matematik. pp.437 – 450.
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