# DICHOTOMY OF GLOBAL CAPACITY DENSITY IN METRIC MEASURE SPACES

created by shanmugal on 08 Jan 2017
modified on 19 Jun 2018

[BibTeX]

Accepted Paper

Inserted: 8 jan 2017
Last Updated: 19 jun 2018

The variational capacity $\text{cap}_p$ in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every $E\subset\mathbb{R}^n$, $\inf_{x\in\mathbb{R}^n}\frac{\text{cap}_p(E\cap B(x,r),B(x,2r))}{\text{cap}_p(B(x,r),B(x,2r))}$ is either zero or tends to $1$ as $r \to \infty$. 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 $\mathbb{R}^n$.