Capacities from moduli in metric measure spaces

Abstract

The concept of capacity is an indispensable tool for analysis and path families and their moduli play a fundamental role in a metric space $X$. It is shown that the $AM_p(\Gamma )$- and $M_p(\Gamma )$-modulus create the capacities, $\mathrm {Cap}_p^{AM}(E,G)$ and $\mathrm {Cap}_p^{M}(E,G)$, respectively, where Γ is the path family connecting an arbitrary set $E \subset G$ to the complement of a bounded open set $G$. The capacities use Lipschitz functions and their upper gradients. For $p > 1$ the capacities coincide but differ for $p=1$. For $p \geq 1$ it is shown that the $\mathrm {Cap}_p^{AM}(E,G)$-capacity equals to the classical variational Dirichlet capacity of the condenser $(E,G)$ and the $\mathrm {Cap}_p^{M}(E,G)$-capacity to the $M_p(\Gamma )$-modulus.