Calderon-Zygmund capacities and Wolff potentials on Cantor sets

TL;DR

This paper explores the relationship between Calderon-Zygmund capacities and Wolff potentials on specific Cantor sets, revealing comparability in certain cases and highlighting open problems in the field.

Contribution

It establishes a comparability result between Calderon-Zygmund and Wolff capacities for some Cantor sets, advancing understanding of potential theory in fractal geometries.

Findings

Capacities are comparable on certain Cantor sets

Open problems remain for non-integer s and general sets

Discussion of open questions in potential theory

Abstract

We show that, for some Cantor sets in R^d, the capacity g_s associated to the s-dimensional Riesz kernel x/|x|^{s+1} is comparable to the capacity C_{2(d-s)/3,3/2} from non linear potential theory. It is an open problem to show that, when s is positive and non integer, they are comparable for all compact sets in R^d. We also discuss other open questions in the area.