Weighted Pluricomplex energy II

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 $\phi \in {\mathcal{E}}_{\chi }$ if and only if $\chi ({\mathcal{E}}_{\chi })\subset L^1(d\mu )\mathrm{.}$ Then we show that if $\mu = (dd^c \varphi )^n $forsome$\varphi \in \mathcal E_\chi $then$\mu = (dd^c u )^n $forsome$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.