Weighted Pluricomplex energy II

TL;DR

This paper advances the understanding of the complex Monge-Ampère operator on weighted pluricomplex energy classes, providing new characterizations, existence, and estimates for solutions to related boundary value problems.

Contribution

It offers new characterizations of the range of the Monge-Ampère operator on weighted energy classes and establishes existence and a priori estimates for solutions with measures dominated by capacity.

Findings

Characterization of measures as Monge-Ampère of functions in energy classes.

Existence of solutions with prescribed boundary data.

A priori bounds for solutions with measures in Orlicz spaces.

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.