Capacities from moduli in metric measure spaces - Martio, Olli (MATHEMATICA SCANDINAVICA, 2023)
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(Γ )- and M_p(Γ )-modulus create the capacities, Cap_p^AM(E,G) and Cap_p^M(E,G), respectively, where Γ is the path family connecting an arbitrary set E 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 ≥ 1 it is shown that the Cap_p^AM(E,G)-capacity equals to the classical variational Dirichlet capacity of the condenser (E,G) and the Cap_p^M(E,G)-capacity to the M_p(Γ )-modulus.