Capacitary differentiability of potentials of finite Radon measures

We study differentiability properties of a potential of the type $K \star \mu$, where $\mu$ is a finite Radon measure in $\mathrm{R}^N$ and the kernel $K$ satisfies ${\lvert \nabla^j K(x) \rvert \leq C \lvert x \rvert}^{-(N-1+j)} , j={0, 1, 2}$. 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 ${\lvert x \rvert}^{-(N-1)}$. 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 Calderón–Zygmund sense for $1 \leq p \lt N / (N-1)$. 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.

Keywords

differentiability, Riesz, Newtonian and logarithmic potentials, capacity, Calderón–Zygmund theory

2010 Mathematics Subject Classification

26B05, 31B15, 42B20

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This research was partially supported by the grants 2017SGR395 (Generalitat de Catalunya), MTM2016–75390 (Ministerio de Educación y Ciencia) and María de Maeztu Programme for Units of Excellence in R&D, MDM-2014-0445 (Spanish Ministry of Economy and Competitiveness).

Received 13 December 2018

Received revised 29 December 2018

Accepted 8 May 2019

Published 7 October 2019