Arkiv för Matematik

Volume 57 (2019)

Number 2

Capacitary differentiability of potentials of finite Radon measures

Pages: 437 – 450



Joan Verdera (Departament de Matemàtiques, Barcelona Graduate School of Mathematics, Universitat Autònoma de Barcelona, Catalonia, Spain)


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.


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

2010 Mathematics Subject Classification

26B05, 31B15, 42B20

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