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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Characterization of quasi-units in terms of equilibrium potentials


Authors: Maynard Arsove and Heinz Leutwiler
Journal: Proc. Amer. Math. Soc. 89 (1983), 267-272
MSC: Primary 31D05
DOI: https://doi.org/10.1090/S0002-9939-1983-0712635-7
MathSciNet review: 712635
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Abstract: In the cone of nonnegative superharmonic functions on a bounded euclidean region $ \Omega $, quasi-units were introduced as those elements invariant under the infinitesimal generator of the fundamental one-parameter semigroup of operators $ {S_\lambda }(\lambda \geqslant 0)$. All harmonic measures and capacitary potentials are quasi-units, but the latter class has more extensive closure properties. Quasi-units arise naturally under various operations of classical potential theory and have important applications, for example in proving that the convex set of Green's potentials $ u$ of positive mass distributions on $ \Omega $ with $ u \leqslant 1$ has as its extreme points precisely the capacitary potentials. Some new properties of quasi-units are developed here. In particular, it is shown that quasi-units can be characterized as limits of increasing sequences of continuous equilibrium potentials for which the equilibrium values tend to 1.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0712635-7
Article copyright: © Copyright 1983 American Mathematical Society