Capacities and density conditions in metric spaces

TL;DR

This paper investigates relationships between different capacities in metric measure spaces, establishing comparability and equivalence results under various geometric and kernel conditions.

Contribution

It proves a comparability between Riesz and Hajlasz capacities and shows capacity density conditions are equivalent in geodesic spaces without kernel assumptions.

Findings

Riesz and Hajlasz capacities are comparable under kernel estimates.

Capacity density conditions are equivalent in geodesic spaces.

Hajlasz and variational p-capacities are compared.

Abstract

We examine the relations between different capacities in the setting of a metric measure space. First, we prove a comparability result for the Riesz $(\beta,p)$-capacity and the relative Hajlasz $(\beta,p)$-capacity, for $1<p<\infty$ and $0<\beta \le 1$, under a suitable kernel estimate related to the Riesz potential. Then we show that in geodesic spaces the corresponding capacity density conditions are equivalent even without assuming the kernel estimate. In the last part of the paper, we compare the relative Hajlasz $(1,p)$-capacity to the relative variational $p$-capacity.