Triebel-Lizorkin capacity and Hausdorff measure in metric spaces

TL;DR

This paper establishes a relationship between Triebel-Lizorkin capacity and Hausdorff measure in metric spaces, showing bounds and zero-capacity implications for measure.

Contribution

It provides new bounds for Triebel-Lizorkin capacity using Hausdorff measure and characterizes sets of zero capacity via generalized Hausdorff measure.

Findings

Triebel-Lizorkin capacity is bounded by Hausdorff measure in metric spaces

Sets with zero capacity have zero generalized Hausdorff measure

The results extend capacity-measure relationships in metric geometry

Abstract

We provide a upper bound for Triebel-Lizorkin capacity in metric settings in terms of Hausdorff measure. On the other hand, we also prove that the sets with zero capacity have generalized Hausdorff $h$-measure zero for a suitable gauge function $h.$