Weighted Pluricomplex energy II

Abstract

We continue our study of the Complex Monge-Amp\`ere Operator on the Weighted Pluricomplex energy classes. We give more characterizations of the range of the classes $\mathcal E_ \chi$ by the Complex Monge-Amp\`ere Operator. In particular, we prove that a non-negative Borel measure $\mu $ is the Monge-Amp\`ere of a unique function $\varphi \in \mathcal E_\chi$ if and only if $\chi(\mathcal E_\chi ) \subset L^1(d\mu ).$ Then we show that if $\mu = (dd^c \varphi )^n $ for some $\varphi \in \mathcal E_\chi $ then $\mu = (dd^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-Amp\`ere 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.