Subharmonicity without upper semicontinuity
Let \(\Omega\subseteq \mathbb{R}^d\) be open and let \(x\in\Omega\). There are many probability measures \(\mu\) with compact support in \(\Omega\) which have the following property: \(u(x)\leq\int u d\mu\) for every subharmonic function \(u\) on \(\Omega\). (Such a measure is called a Jensen measure for \(x\).) Familiar examples are normalized surface measure on the sphere \(\partial B(x,r)\) and normalized volume measure on the ball \(B(x,r)\) whenever \(\overline {B(x,r)}\subset \Omega\). Further, if \(\omega\) is any bounded open set such that \(x\in\omega\) and \(\overline {\omega}\subset\Omega\), then harmonic measure for \(\omega\) and \(x\) is also a Jensen measure for \(x\). The idea of this paper is to study functions \(u\) (satisfying mild measurability conditions) on \(\Omega\) with the property that \(u(x)\leq\int u d\mu\) for each pair \((x,\mu)\) such that \(\mu\) is a Jensen measure for \(x\). Such functions are called quasi-subharmonic. The first main result of the paper is that, if \(u\) is quasi-subharmonic on \(\Omega\), then the u.s.c. regularization \(u^*\) is subharmonic on \(\Omega\) and \(u^*=u\) outside a polar set. (The converse is obvious.) The potential usefulness of this result is shown by the ease with which one can deduce the classical Cartan theorem concerning the supremum of a family of subharmonic functions which is locally uniformly bounded above. Another result is a converse to the Cartan theorem: under very mild assumptions, any quasi-subharmonic function \(u\) which is locally bounded above can be expressed as the supremum of all subharmonic minorants of \(u\). This leads to an interesting duality theorem. The final section discusses corresponding results for plurisubharmonic functions. The paper is very well written.
scientific article; zbMATH DE number 1046932