In classical potential theory
one way of defining capacity of a compact K ⊂ Rn puts cap K equal to
the total mass of μ, where μ is the measure associated with the inferior
envelope of the family of nonnegative superharmonic functions majorizing the
characteristic function IK. A second (equivalent) definition puts cap K = 1∕∥γ0∥e
where γ0 is the projection of the null measure onto the set of positive Radon
measures γ supported by K, satisfying ∫dγ ≧ 1 and having finite energy:
∥γ∥e=∫Uγdγ < +∞.
In the axiomatic Hilbert space setting of Dirichlet spaces Beurling and Deny have
shown that equivalence of definitions of the two above types leads to a rich
capacity theory. In this article all of these results are extended to the family of
Banach-Dirichlet (BD) spaces, i.e., uniformly convex Banach spaces of (equivalence
classes of) functions satisfying the Dirichlet space axioms. This is accomplished by
using a capacity of the first type in the BD space D, and of the second type in the
dual space D′.
