Vector-valued Riesz potentials: Cartan type estimates and related capacities

TL;DR

This paper provides sharp estimates for the size of the set where the gradient of electrostatic potentials is large, using advanced capacity techniques and extending previous theorems to higher-dimensional spaces.

Contribution

It introduces new relationships between capacities with singular kernels and non-linear capacities, extending prior results to higher dimensions without Menger's curvature.

Findings

Established sharp size estimates for large gradient sets.

Extended capacity theorems to higher-dimensional spaces.

Linked various capacities with singular kernels to non-linear potential capacities.

Abstract

There are many interesting problems about the electrostatic potential of finitely many charges. We consider one of them concerning the intensity of the field, in other words, about the magnitude of the gradient of this potential. We want to give a sharp estimate of the size of the set of points where this gradient is large. Of course we want the estimate to be sharp in number $N$ of charges. The size will be measured by the Hausdorff content with various gauge functions. Such a setting allows us to consider a wide class of measures (not necessarily with finitely many charges). The main technique will be Calder\'on-Zygmund capacities and nonhomogeneous Calder\'on-Zygmund operators. Here we establish a relationship between various types of capacities with singular kernels (e. g. analytic capacity, lipschitz harmonic capacity, etc) and non-linear capacity from the theory of potential \'a la Adams, Hedberg, Havin, Maz'ya, Wolff. "Capacitary" part of the paper extends the theorem of Mateu, Prat and Verdera [J. reine und angew. Math., 578 (2005), 201--223]. "Size estimates" part of the paper extends the theorem of M. Anderson and V. Eiderman [Annals of Math., 163 (2005), 1057--1076]. The difficulty lies in the fact that we cannot use Menger's curvature anymore because we are working in spaces of dimension bigger than two.