Vol. 48, No. 2, 1973

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
Capacity theory in Banach spaces

Peter A. Fowler

Vol. 48 (1973), No. 2, 365–385
Abstract

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
Milestones
Received: 5 October 1971
Revised: 26 April 1973
Published: 1 October 1973
Authors
Peter A. Fowler