Weighted pluricomplex energy

TL;DR

This paper investigates the complex Monge-Ampère operator on classes of functions with finite pluricomplex energy, providing new insights into their properties and the operator's range in this context.

Contribution

It introduces a novel interpretation of pluricomplex energy classes via capacity decay and characterizes the operator's range on these classes.

Findings

Characterization of energy classes in terms of capacity decay

Complete description of the Monge-Ampère operator's range on these classes

Extension of the theory to cases with infinite total Monge-Ampère mass

Abstract

We study the complex Monge-Ampre operator on the classes of finite pluricomplex energy $\mathcal{E}_\chi (\Omega)$ in the general case ($\chi(0)=0$ i.e. the total Monge-Ampre mass may be infinite). We establish an interpretation of these classes in terms of the speed of decrease of the capacity of sublevel sets and give a complete description of the range of the operator $(dd^c \cdot)^n$ on the classes $\mathcal{E}\chi(\Omega).$