Dichotomy of global capacity density in metric measure spaces

TL;DR

This paper investigates the density dichotomy of variational capacity in various metric measure spaces, showing it holds in certain geodesic spaces but can fail in nongeodesic spaces, with implications for understanding capacity behavior.

Contribution

It extends the understanding of the density dichotomy of variational capacity to unbounded geodesic metric spaces and identifies conditions under which it fails or holds.

Findings

Density dichotomy holds in unbounded complete geodesic metric spaces with doubling measure and Poincaré inequality.

The property can fail in nongeodesic metric spaces and for Sobolev capacity in Euclidean spaces.

Certain set families, like John domains, satisfy the density dichotomy even in more general spaces.

Abstract

The variational capacity cap_p in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every subset E of R^n, inf_{x in R^n} (cap_p(E \cap B(x,r),B(x,2r)) / cap_p(B(x,r),B(x,2r))) is either zero or tends to 1 as r tends to infinity. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincar\'e inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in R^n. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.