A comparison principle for the complex Monge-Ampère operator in Cegrell’s classes and applications

Abstract

In this article we will first prove a result about the convergence in capacity. Next we will obtain a general decomposition theorem for complex Monge-Ampère measures which will be used to prove a comparison principle for the complex Monge-Ampère operator.

An inequality for mixed Monge–Ampère measures

“…In particular inequalities (1.3) and (1.4) both fail in this case. One can also show that Demailly inequality fails for these functions, so the result in [21] is sharp too.…”

“…Proof We recall the following known inequality which is a special case of Demailly's inequality (see for example [21]):…”

“…(see [21]). Let us state our main result: [7], or, instead, we can work with germs of functions as in [4]).…”

Weak solutions to the complex Monge-Ampere equation on open subsets of $\mathbb{C}^n$

“…Indeed, let K ⊂ {u > −∞} ∩ {u = w} be a compact set. Since K ⊂ {w + 1 j > u}, by Theorem 4.1 in [18] and using the hypotheses we have ∫…”

Some Weighted Energy Classes of Plurisubharmonic Functions

“…For the convenience of the readers, we recall the following results in [16] which will be used later on.…”