CAPACITIES FROM MODULI IN METRIC MEASURE SPACES

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, Cap(p)(AM) (E, G) and Cap(p)(M) (E, G), respectively, where Gamma is the path family connecting an arbitrary set E subset of G to the complement of a bounded open set G. The capacities use Lipschitz functions and their upper gradients. Forp > 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 CapM p (E, G)-capacity to the M-p(Gamma)-modulus.