Weighted Pluricomplex Energy II

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 E by the complex Monge-Ampère operator. In particular, we prove that a nonnegative Borel measure is the Monge-Ampère of a unique function ∈ E if and only if (E ) ⊂ 1 ( ). Then we show that if = ( ) for some ∈ E then = ( ) for some ∈ E , 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.

Degenerate complex Monge-Ampère equations with non-Kähler forms in bounded domains

“…)dµ on a bounded strictly pseudoconvex domain in C n , where ω is a smooth (1, 1)-form, 0 ≤ F is a continuous non-decreasing function, and µ is a positive non-pluripolar measure. Our results extend previous works of Ko lodziej and Nguyen [KN15, KN23a, KN23b] who study bounded solutions, as well as Cegrell [Ceg98,Ceg04,Ceg08], Czyż [Cz09], Benelkourchi [Ben09,Ben15] and others who treat the case when ω = 0 and/or F = 1.…”

“…where µ is a positive Radon measure vanishing on pluripolar sets. The equation (3.1) has been studied by Benelkourchi [Ben09,Ben15] in the case of convex or homogeneous weights. We extend these results to a special type of concave functions χ.…”

“…In the sequel, we denote by φ a psh maximal function in E(Ω) ∩ C 0 ( Ω); the term maximal means that (dd c φ) n = 0. Recall that for K(Ω) ∈ {E(Ω), F (Ω), E p (Ω), E χ (Ω), N (Ω)}, the set K(Ω, φ) is defined by Aha07,Ceg08,Ben15].…”

“…In this series of papers, he gave detailed study of the solutions to the Dirichlet problem. In [Ben09,Ben15] Benelkourchi studied weighted energy classes in the spirit of [GZ07] and provided a complete description of solutions to the Dirichlet problem for weights that are convex or homogeneous. Our first main result provides the same description for concave weights χ that have polynomial-like behavior, i.e.…”

The complex Monge-Ampère equation in the Cegrell’s classes

“…He considered measures that were majorized by the sum of a linear combination of countable numbers of Dirac measures with compact support and a certain regular Monge -Ampère measure. Recently, in [1] and [5], the author study the complex Monge -Ampère equation on Cegrell's classes. In [1], the authors proved that if a non-negative Borel measure is dominated by a complex Monge -Ampère measure, it is a complex Monge -Ampère measure.…”