The Besov Capacity In Metric Spaces

TL;DR

This paper develops a capacity theory based on Haj{}asz-Besov functions in metric spaces with doubling measures, providing bounds related to Netrusov-Hausdorff content and introducing new inequalities involving $tanas.

Contribution

It introduces a new capacity framework in metric spaces and establishes bounds and inequalities, advancing the understanding of Besov capacities in this setting.

Findings

Established lower and upper bounds for the capacity

Connected capacity estimates to Netrusov-Hausdorff content

Proved a new Poincare9 type inequality for $tanas$

Abstract

We study a capacity theory based on a definition of Haj{\l} asz-Besov functions. We prove several properties of this capacity in the general setting of a metric space equipped with a doubling measure. The main results of the paper are lower bound and upper bound estimates for the capacity in terms of a modified Netrusov-Hausdorff content. Important tools are $\gamma$-medians, for which we also prove a new version of a Poincar\'e type inequality.