Plurisubharmonic functions with weak singularities

TL;DR

This paper introduces a new hierarchy of weakly singular plurisubharmonic functions with finite energy, extending previous classes, and provides solutions to related complex Monge-Ampère equations in hyperconvex domains.

Contribution

It generalizes Cegrell's classes by defining a stratification of unbounded plurisubharmonic functions based on energy and capacity decay.

Findings

Defined a scale of weakly singular plurisubharmonic functions

Connected energy classes with Monge-Ampère capacity decay

Solved associated complex Monge-Ampère equations

Abstract

We study the complex Monge-Amp\`ere operator in bounded hyperconvex domains of $\C^n$. We introduce a scale of classes of weakly singular plurisubharmonic functions : these are functions of finite weighted Monge-Amp\`ere energy. They generalize the classes introduced by U.Cegrell, and give a stratification of the space of (almost) all unbounded plurisubharmonic functions. We give an interpretation of these classes in terms of the speed of decreasing of the Monge-Amp\`ere capacity of sublevel sets and solve associated complex Monge-Amp\`ere equations.