Characterization of quasi-units in terms of equilibrium potentials

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.