An equality of Monge-Ampère measures

2022/07/27 by Kadiri, Mohamed El
#31C10 #32U05 #32U15 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.13610

Abstract

Let u and v be two plurisubharmonic functions in the domain of definition of the Monge-Ampère operator on a domain Ω⊂ \bf Cⁿ. We prove that if u=v on a plurifinely open set U⊂ Ω that is Borel measurable, then (dd^cu)ⁿ|_U=(dd^cv)ⁿ|_U. This result was proved by Bedford and Taylor in the case where u and v are locally bounded, and by El Kadiri and Wiegerinck when u and v are finite, and by Hai and Hiep when U is of the form U=\bigcup_j=1^m\φ_j>ψ_j\, where φ_j, ψ_j, j=1,...,m, are plurisubharmonic functions on Ω.

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