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

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)(1,1)-Poincaré inequality. We prove the equality of 1-modulus and the continuous 1-capacity, extending the known results for 1

► We prove comparison results for capacities in metric spaces. ► We show that the 1-modulus and the continuous 1-capacity are equal. ► We consider different pointwise conditions in the definition of BV-capacity. ► We show that the different notions of BV-capacity are comparable. ► Furthermore, the BV-capacities we consider are comparable to the 1-capacity.