Besov capacity and Hausdorff measures in metric measure spaces

Abstract

This paper studies Besov $p$-capacities as well as their relationship to Hausdorff measures in Ahlfors regular metric spaces of dimension $Q$ for $1<Q<p<\infty$. Lower estimates of the Besov $p$-capacities are obtained in terms of the Hausdorff content associated with gauge functions~$h$ satisfying the decay condition $\int_0^1 h(t)^{1/(p-1)} \frac{dt}{t}<\infty$.