Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 3-14

NEARLY HYPERHARMONIC FUNCTIONS AND JENSEN MEASURES

Wolfhard Hansen and Ivan Netuka

Universität Bielefeld, Fakultät für Mathematik
33501 Bielefeld, Germany; hansen 'at' math.uni-bielefeld.de

Charles University, Faculty of Mathematics and Physics, Mathematical Institute
Sokolovská 83, 186 75 Praha 8, Czech Republic; netuka 'at' karlin.mff.cuni.cz

Abstract. Let (X,H) be a P-harmonic space and assume for simplicity that constants are harmonic. Given a numerical function φ on X which is locally lower bounded, let

J_φ(x) := sup{∫^*φ dμ : μ ∈ J_x(X)}, x ∈ X,

where J_x(X) denotes the set of all Jensen measures μ for x, that is, μ is a compactly supported measure on X satisfying ∫u dμ ≤ u(x) for every hyperharmonic function u on X. The main purpose of the paper is to show that, assuming quasi-universal measurability of φ, the function J_φ is the smallest nearly hyperharmonic function majorizing φ and that J_φ = φ ∨ \hat J_φ, where \hat J_φ is the lower semicontinuous regularization of J_φ. So, in particular, J_φ turns out to be at least "as measurable as" φ. This improves recent results, where the axiom of polarity was assumed. The preliminaries on nearly hyperharmonic functions in the framework of balayage spaces are closely related to the study of strongly supermedian functions triggered by Mertens more than forty years ago.

2010 Mathematics Subject Classification: Primary 31B05, 31D05, 60J45, 60J75.

Key words: Jensen measure, nearly hyperharmonic function, strongly supermedian function.

Reference to this article: W. Hansen and I. Netuka: Nearly hyperharmonic functions and Jensen measures. Ann. Acad. Sci. Fenn. Math. 44 (2019), 3-14.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4401