Capacities and Hausdorff measures on metric spaces

TL;DR

This paper establishes a relationship between $Q$-capacity zero sets and generalized Hausdorff measures in $Q$-doubling metric spaces that support a Poincaré inequality, extending understanding of measure-theoretic properties in such spaces.

Contribution

It proves that in certain metric measure spaces, sets of zero $Q$-capacity also have zero generalized Hausdorff measure for a specific gauge function.

Findings

Sets of $Q$-capacity zero have measure zero for a specific Hausdorff measure.

The result applies to $Q$-doubling spaces supporting a Poincaré inequality.

The paper extends classical measure theory results to more general metric spaces.

Abstract

In this article, we show that in a $Q$-doubling space $(X,d,\mu),$ $Q>1,$ that supports a $Q$-Poincar\'e inequality and satisfies a chain condition, sets of $Q$-capacity zero have generalized Hausdorff $h$-measure zero for $h(t)=\log^{1-Q-\epsilon}(1/t).$