Capacities and density conditions in metric spaces

We examine the relations between different capacities in the setting of a metric measure space. First, we prove a comparability result for the Riesz $(\beta ,p)$-capacity and the relative Hajlasz $(\beta ,p)$-capacity, for $1<p<\mathrm{\infty }$ and $0<\beta \le 1$, under a suitable kernel estimate related to the Riesz potential. Then we show that in geodesic spaces the corresponding capacity density conditions are equivalent even without assuming the kernel estimate. In the last part of the paper, we compare the relative Hajlasz $(1,p)$-capacity to the relative variational $p$-capacity.