Extreme Jensen measures

Abstract

Let Ω be an open subset of R^d, d≥2, and let x∈Ω. A Jensen measure for x on Ω is a Borel probability measure μ, supported on a compact subset of Ω, such that ∫u dμ≤u(x) for every superharmonic function u on Ω. Denote by J_x(Ω) the family of Jensen measures for x on Ω. We present two characterizations of ext(J_x(Ω)), the set of extreme elements of J_x(Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains.

This allows us to relax the local boundedness condition in a previous result of B. Cole and T. Ransford, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29–53.

As an application, we give an improvement of a result by Khabibullin on the question of whether, given a complex sequence {α_n}_n=1^∞ and a continuous function $M\colon\textbf{C}\rightarrow\textbf{R}^+$, there exists an entire function f≢0 satisfying f(α_n)=0 for all n, and |f(z)|≤M(z) for all z∈C.