Capacitary differentiability of potentials of finite Radon measures

© 2019 by Institut Mittag-Leffler. All rights reserved. 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 |∇j K(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.