High energy plurisubharmonic classes

Abstract

Let $\Omega ⋐{\mathbb{C}}^n$ be a bounded strongly pseudoconvex domain. For any concave increasing weight $\chi :{\mathbb{R}}^{-}\to {\mathbb{R}}^{-}$ such that $\chi (0)=0$, we introduce and study finite energy classes ${\mathcal{E}}_{\chi }(\Omega )$ of plurisubharmonic functions, using the Orlicz space formalism. We investigate the range of the Monge–Ampère operator on these classes, and conjecture that this should lead to an integral characterization of the image of bounded plurisubharmonic functions, an open problem since the birth of pluripotential theory more than forty years ago.