Comparisons of relative BV-capacities and Sobolev capacity in metric spaces

TL;DR

This paper explores the relationship between Sobolev and BV capacities in metric spaces, extending known results for p > 1 to the case p = 1, and establishing equivalences between various capacity definitions.

Contribution

It proves the equality of 1-modulus and 1-capacity in metric spaces and introduces alternative definitions for BV-capacities, broadening the understanding of capacity relations.

Findings

Equality of 1-modulus and 1-capacity established

Alternative definitions for BV-capacities proposed

Relations between total 1-capacity and BV-capacity analyzed

Abstract

We study relations between the variational Sobolev 1-capacity and versions of variational BV-capacity in a complete metric space equipped with a doubling measure and supporting a weak $(1,1)$-Poincar\'e inequality. We prove the equality of 1-modulus and 1-capacity, extending the known results for $1 < p < \infty$ to also cover the more geometric case $p = 1$. Then we give alternative definitions for variational BV-capacities and obtain equivalence results between them. Finally we study relations between total 1-capacity and versions of BV-capacity.