Subextension of plurisubharmonic functions with bounded Monge–Ampère mass

✍ Scribed by Urban Cegrell; Ahmed Zeriahi

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 89 KB
• Volume: 336
• Category: Article
• ISSN: 1631-073X
• DOI: 10.1016/s1631-073x(03)00031-1

✦ Synopsis

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 c ϕ 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 c ϕ) 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.