Table of Contents
International Journal of Partial Differential Equations
Volume 2015 (2015), Article ID 947819, 8 pages
http://dx.doi.org/10.1155/2015/947819
Research Article

Weighted Pluricomplex Energy II

Université de Montréal, Pavillon 3744, Rue Jean-Brillant, Montréal, QC, Canada H3C 3J7

Received 31 August 2014; Revised 29 December 2014; Accepted 9 January 2015

Academic Editor: Antonin Novotny

Copyright © 2015 Slimane Benelkourchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 by the complex Monge-Ampère operator. In particular, we prove that a nonnegative Borel measure is the Monge-Ampère of a unique function if and only if . Then we show that if for some then for some , where is given boundary data. If moreover the nonnegative Borel measure 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.