Subextension of plurisubharmonic functions with bounded Monge–Ampère mass

Abstract

Let Ω⋐ C n be a hyperconvex domain. Denote by E 0(Ω) the class of negative plurisubharmonic functions ϕ on Ω with boundary values 0 and finite Monge–Ampère mass on Ω. Then denote by F(Ω) the class of negative plurisubharmonic functions ϕ on Ω for which there exists a decreasing sequence ( ϕ) j of plurisubharmonic functions in E 0(Ω) converging to ϕ such that sup j∫ Ω(dd cϕ j) n+∞. It is known that the complex Monge–Ampère operator is well defined on the class F(Ω) and that for a function ϕ∈ F(Ω) the associated positive Borel measure is of bounded mass on Ω. A function from the class F(Ω) is called a plurisubharmonic function with bounded Monge–Ampère mass on Ω. We prove that if Ω and Ω are hyperconvex domains with Ω⋐ Ω⋐ C n and ϕ∈ F(Ω), there exists a plurisubharmonic function ϕ ̃ ∈ F( Ω) such that ϕ ̃ ⩽ϕ on Ω and ∫ Ω (dd c ϕ ̃ ) n⩽∫ Ω(dd cϕ) n. Such a function is called a subextension of ϕ to Ω. From this result we deduce a global uniform integrability theorem for the classes of plurisubharmonic functions with uniformly bounded Monge–Ampère masses on Ω. To cite this article: U. Cegrell, A. Zeriahi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).