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 Eχ by the complex Monge-Ampère operator. In particular, we prove that a nonnegative Borel measure μ is the Monge-Ampère of a unique function φ∈Eχ if and only if χ(Eχ)⊂L1(dμ). Then we show that if μ=(ddcφ)n for some φ∈Eχ then μ=(ddcu)n for some φ∈Eχ, where f is given boundary data. If moreover the nonnegative Borel measure μ 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.
Highlights
Let Ω ⊂ Cn be a bounded hyperconvex domain, that is, a connected, bounded open subset of Cn, such that there exists a negative plurisubharmonic function ρ such that {z ∈ Ω; ρ(z) < −c} ⋐ Ω, for all c > 0
The question of describing the measures which are the Monge-Ampere of bounded psh functions is very important for pluripotential theory, complex dynamic, and complex geometry
In [7], Cegrell introduced the pluricomplex energy classes Ep(Ω) and Fp(Ω) (p ≥ 1) on which the complex Monge-Ampere operator is well defined. He proved that a measure μ is the Monge-Ampere of some function u ∈ Ep(Ω) if and only if it satisfies
Summary
Let Ω ⊂ Cn be a bounded hyperconvex domain, that is, a connected, bounded open subset of Cn, such that there exists a negative plurisubharmonic function ρ such that {z ∈ Ω; ρ(z) < −c} ⋐ Ω, for all c > 0. As known (see [1, 2]), the complex Monge-Ampere operator (ddc ⋅ )n is well defined, as a nonnegative measure, on the set of locally bounded plurisubharmonic functions. In [7], Cegrell introduced the pluricomplex energy classes Ep(Ω) and Fp(Ω) (p ≥ 1) on which the complex Monge-Ampere operator is well defined He proved that a measure μ is the Monge-Ampere of some function u ∈ Ep(Ω) if and only if it satisfies. Such measures dominated by the Monge-Ampere capacity have been extensively studied by Kołodziej in [3,4,5] He proved that if φ : ∂Ω → R is a continuous function and ∫0+∞ ε(t)dt < +∞, μ is the Monge-Ampere of a unique function φ ∈ PSH(Ω) with φ/∂Ω = φ.
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