Weighted pluricomplex energy

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 $(d{d}^c\cdot {)}^n$ on the classes $\mathcal{E}\chi (\Omega ).$

Contrary to what we claimed in the previous version, in Theorem 5.1 we generalize some Theorem of Urban Cegrell but we do not give a new proof. To appear in Potenial Analysis