Triebel–Lizorkin capacity and Hausdorff measure in metric spaces with chain condition

In this paper, we improve the lower bound estimate for capacity in Hajłasz–Triebel–Lizorkin spaces in terms of Hausdorff Measure obtained in Karak and Mondal (Math Slovaca 73(4): 937–948, 2023) under an additional assumption that the metric space satisfies a chain condition. With the chain condition, we also improve a similar estimate for capacity in Hajłasz–Triebel–Lizorkin space with generalized smoothness in terms of Hausdorff Measure obtained in [28].