Capacitary differentiability of potentials of finite Radon measures

Abstract

We study differentiability properties of a potential of the type $K\star \mu$, where $\mu$ is a finite Radon measure in $\mathbb{R}^N$ and the kernel $K$ satisfies $|\nabla^j K(x)| \le C\, |x|^{-(N-1+j)}, \quad j=0,1,2.$ We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vall\'ee 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 $L^{N/(N-1)}$ differentiability and thus $L^{p}$ differentiability in the Calder\'on--Zygmund sense for $1\le p < N/(N-1)$. We show that $K\star \mu$ 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 $K\star \mu.$ As an application, we study level sets of newtonian potentials of finite Radon measures.