Comparisons of relative BV-capacities and Sobolev capacity in metric spaces

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We study relations between the variational Sobolev 1-capacity and versions of variational BV-capacity in a complete metric space equipped with a doubling measure and supporting a weak (1,1)-Poincaré inequality. We prove the equality of 1-modulus and 1-capacity, extending the known results for 1 < p < ∞ to also cover the more geometric case p = 1. Then we give alternative definitions for variational BV-capacities and obtain equivalence results between them. Finally we study relations between total 1-capacity and versions of BV-capacity.