Capacitary differentiability of potentials of finite Radon measures (Q2326511)

This paper is concerned with differentiability properties of potentials of the form \(K\ast \mu \), where \(\mu \) is a finite Radon measure on \(\mathbb{R}^{N}\) and the kernel \(K\) satisfies \(|\nabla ^{j}K(x)|\leq C|x|^{1-N-j}\) for \(j=0,1,2\). It is shown that such potentials are differentiable in a new sense that is formulated in terms of the classical capacity associated with the kernel \(|x|^{1-N}\). This implies differentiability in the \(L^{N/(N-1),\infty}\) sense, and thus differentiability in the \(L^{p}\) sense for \(1 \leq p < N/(N-1)\), and thus approximate differentiability. The paper also offers an alternative route to pointwise Lipschitz estimates for \(K\ast \mu \) that were recently established by \textit{L. Ambrosio}, \textit{A. C. Ponce} and \textit{R. Rodiac} [``Critical weak-\(L^p\) differentiability of singular integrals'', Preprint, \url{arXiv:1810.03924}], and discusses an application to level sets of Newtonian potentials.