Vol. 48, No. 2, 1973

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Capacity theory in Banach spaces

Peter A. Fowler

Vol. 48 (1973), No. 2, 365–385

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 = 1γ0e where γ0 is the projection of the null measure onto the set of positive Radon measures γ supported by K, satisfying 1 and having finite energy: γe = Uγ dγ < +.

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.

Mathematical Subject Classification 2000
Primary: 31C25
Received: 5 October 1971
Revised: 26 April 1973
Published: 1 October 1973
Peter A. Fowler