Pluricomplex energy classes associated to a positive closed current

TL;DR

This paper extends the domain of pluricomplex energy operators to broader classes of unbounded plurisubharmonic functions associated with positive closed currents, establishing their properties and the validity of the comparison principle.

Contribution

Introduction of new classes of plurisubharmonic functions for which the pluricomplex energy operator is well-defined, including their quasicontinuity and comparison principle.

Findings

Defined classes _{p}^{T}(\u03a9) and _{p}^{T}(\u03a9) for unbounded psh functions.

Proved these classes are within the domain of the pluricomplex energy operator.

Established quasicontinuity and comparison principle for functions in these classes.

Abstract

The aim of this paper is to extend the domain of definition of $(dd^c\centerdot)^q\wedge T$ on some classes of plurisubharmonic (psh) functions, which are not necessary bounded, where $T$ is a positive closed current of bidimension $(q,q)$ on an open set $\Omega$ of $\Bbb C^n$. We introduce two classes $\mathcal{F}_{p}^{T}(\Omega)$ and $\mathcal{E}_p^T(\Omega)$ and we show that they belong to the domain of definition of the operator $(dd^c\centerdot)^q\wedge T$. We also prove that all functions belong to these classes are $C_T$-quasicontinuous and that the comparison principle is valid in them.