Weighted pluricomplex energy. II. (Q275886)

Summary: 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 nonnegative Borel measure \(\mu\) is the Monge-Ampère 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 \(\mu = (d d^c \varphi)^n\) for some \(\varphi \in \mathcal{E}_\chi\) then \(\mu = (d d^c u)^n\) for some \(\varphi \in \mathcal{E}_\chi\), where \(f\) is given boundary data. If moreover the nonnegative Borel measure \(\mu\) is suitably dominated by the Monge-Ampère capacity, we establish a priori estimates on the capacity of sublevel sets of the solutions. As a 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.