On some weighted energy classes of plurisubharmonic functions

TL;DR

This paper explores the relationships between weighted energy classes of plurisubharmonic functions, introduces a generalized comparison principle, and demonstrates existence results for Monge-Ampère equations within these classes.

Contribution

It establishes connections between new weighted energy classes and classical classes, and proves a generalized comparison principle and existence results for Monge-Ampère equations.

Findings

Established relations between weighted energy classes and classical classes.

Proved a generalized comparison principle for the operator M_χ.

Demonstrated existence of solutions for Monge-Ampère type equations in weighted classes.

Abstract

In this paper we study the relation between the weighted energy class $\mathcal{E}_{\chi}$ introduced by S. Benelkouchi, V. Guedj and A. Zeriahi recently with the classes $\mathcal{E}$ and $\mathcal{N}$ studied by Cegrell. Moreover, we establish a generalized comparison principle for the operator $\text{M}_{\chi}$ and, as an application, we prove a slight version of existence of solutions of Monge-Amp\`ere type equation in the class $\mathcal{E}_{\chi}(H,\O)$.