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Weighted Pluricomplex Energy II


Affiliations
1 Universite de Montreal, Pavillon 3744, Rue Jean-Brillant, Montreal, QC, H3C 3J7, Canada
 

We continue our study of the complex Monge-Ampere operator on the weighted pluricomplex energy classes. We give more characterizations of the range of the classes ξχ by the complexMonge-Ampere operator. In particular, we prove that a nonnegative Borelmeasure μ is theMonge-Ampere of a unique function φεξχ if and only if χ(ξχ)CL1 (δμ). Then we show that if μ = (ddcφ)η for some φεξχ then μ = (ddcu)η for some φεξχ, where f is given boundary data. If moreover the nonnegative Borel measure μ is suitably dominated by the Monge-Ampere 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 problemin the case when the measure has a density in some Orlicz space.
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  • Weighted Pluricomplex Energy II

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Authors

Slimane Benelkourchi
Universite de Montreal, Pavillon 3744, Rue Jean-Brillant, Montreal, QC, H3C 3J7, Canada

Abstract


We continue our study of the complex Monge-Ampere operator on the weighted pluricomplex energy classes. We give more characterizations of the range of the classes ξχ by the complexMonge-Ampere operator. In particular, we prove that a nonnegative Borelmeasure μ is theMonge-Ampere of a unique function φεξχ if and only if χ(ξχ)CL1 (δμ). Then we show that if μ = (ddcφ)η for some φεξχ then μ = (ddcu)η for some φεξχ, where f is given boundary data. If moreover the nonnegative Borel measure μ is suitably dominated by the Monge-Ampere 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 problemin the case when the measure has a density in some Orlicz space.