The BV-capacity in metric spaces (Q967642)

This paper studies the Sobolev \(p\)-capacity in metric measure spaces for \(p=1\). Here, \(\mathbf{X} = (X, d, \mu)\) is a complete space with metric d, and a positive Borel regular outer measure \(\mu\) such that: (i) for all balls \(B(x, r)\) of \(\mathbf{X}\), \(0 < \mu(B(x, r)) < \infty\); (ii) \(\mu\) is doubling, i.e., there is a doubling constant \(C_D \geq 1\) with \(\mu(B(x,2r))\leq c_D\mu(B(x,r)\). The definition of Sobolev spaces on \(\mathbf{X}\) uses the notion of \(p\)-weak upper gradient of \(\mu\), \(1 \leq p < \infty\), and the fact that \(\mathbf{X}\) supports a weak Poincaré inequality; capacity is defined in terms of functions of bounded variation, called BV-capacity. The authors deal with basic properties of the BV-capacity and Sobolev capacity of order 1. In particular, they show that the BV-capacity is a Choquet capacity and the Sobolev 1-capacity is not. However, these quantities are equivalent by two-sided estimates and they have the same null sets as the Hausdorff measure of co-dimension 1. The theory of functions of bounded variation plays an essential role in their argument. The main tool is a modified version of the boxing inequality [\textit{J. Kinnunen, R. Korte, N. Shanmugalingam} and \textit{H. Tuominen}, Indiana Univ. Math. J. 57, No. 1, 401--430 (2008; Zbl 1146.46018)].