Weighted Pluricomplex energy II

Abstract

We continue our study of the Complex Monge-Ampère Operator on the Weighted Pluricomplex energy classes. We give more characterizations of the range of the classes ${\mathcal{E}}_{\chi }$ by the Complex Monge-Ampère Operator. In particular, we prove that a non-negative Borel measure $\mu$ is the Monge-Ampère of a unique function $\phi \in {\mathcal{E}}_{\chi }$ if and only if $\chi ({\mathcal{E}}_{\chi })\subset L^1(d\mu ).$ Then we show that if $\mu =(d{d}^c\phi {)}^n$ for some $\phi \in {\mathcal{E}}_{\chi }$ then $\mu =(d{d}^{c}u{)}^n$ for some $u\in {\mathcal{E}}_{\chi }(f)$ where $f$ is a given boundary data. If moreover, the non-negative Borel measure$\mu$ is suitably dominated by the Monge-Ampère capacity, we establish a priori estimates on the capacity of sub-level sets of the solutions. As consequence, we give a priori bounds of the solution of the Dirichlet problem in the case when the measure has a density in some Orlicz space.