Classifying Triebel–Lizorkin Capacities in Metric Spaces

Abstract We study nonlocal or fractional capacities in metric measure spaces. Our main goal is to clarify the relations between relative Hajłasz–Triebel–Lizorkin capacities, potentional Triebel–Lizorkin capacities, and metric space variants of Riesz capacities. As an application of our results, we obtain a characterization of a Hajłasz–Triebel–Lizorkin capacity density condition, which is based on an earlier characterization of a Riesz capacity density condition in terms of Hausdorff contents.