Jensen measures in product harmonic spaces

Abstract

Let \(\Omega _j \subset \mathbf {R}^{n_j}\,(j=1,2)\) be an open connected set; here if \(n_j=2\), it is assumed that \(\Omega \) has a Green function. The concept of Jensen measure is extended to classes of multiply superharmonic functions on \(\Omega _1\times \Omega _2\). It is proved that product of extreme Jensen measures on the component space is an extreme Jensen measure in the product space. The article is finished by raising a question that is left as an open problem for further investigation.