Subextension of plurisubharmonic functions with bounded Monge–Ampère mass

Urban Cegrell; Ahmed Zeriahi

doi:10.1016/S1631-073X(03)00031-1

Complex Analysis

Subextension of plurisubharmonic functions with bounded Monge–Ampère mass
[Sous-extension des fonctions plurisousharmoniques de masse de Monge–Ampère bornée]

Présenté par : Pierre Lelong

Urban Cegrell ¹ ; Ahmed Zeriahi ²

¹ Department of Mathematics, University of Umeä, 90187 Umeä, Sweden
² Université Paul Sabatier, institut de mathématiques, laboratoire Emile Picard, 118, route de Narbonne, 31062 Toulouse, France

Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 305-308.

Résumé
Abstract

Soit $Ω ⋐ ℂ^{n}$ un domaine hyperconvexe. On désigne par $ℰ_{0} (Ω)$ la classe des fonctions plurisousharmoniques sur $Ω$ avec valeurs au bord nulle et de masse de Monge–Ampère finie sur $Ω .$ On désigne par $ℱ (Ω)$ la classe des fonctions ϕ plurisousharmoniques négatives sur $Ω$ , limite d'une suite décroissante (ϕ_j) de fonctions de $ℰ_{0} (Ω)$ telle que $\sup_{j} \int_{Ω} {({dd}^{c} ϕ_{j})}^{n} < + \infty .$ On sait que l'opérateur de Monge–Ampère est bien défini sur $ℱ (Ω)$ et que pour une fonction $ϕ \in ℱ (Ω),$ la mesure de Monge–Ampère associée est une mesure de Borel sur $Ω$ de masse totale bornée. Une telle fonction sera dite de masse de Monge–Ampère bornée sur $Ω .$

On démontre alors que pour tout domaine hyperconvexe $\tilde{Ω}, Ω ⋐ \tilde{Ω} ⋐ ℂ^{n}$ et tout $ϕ \in ℱ (Ω)$ il existe une fonction $\tilde{ϕ} \in ℱ (\tilde{Ω})$ telle que $\tilde{ϕ} ⩽ ϕ$ sur $Ω$ et $\int_{\tilde{Ω}} {({dd}^{c} \tilde{ϕ})}^{n} ⩽ \int_{Ω} {({dd}^{c} ϕ)}^{n} .$ Une telle fonction $\tilde{ϕ}$ est dite sous-extension de ϕ au domaine $\tilde{Ω} .$ A partir de ce résultat, nous déduisons un théorème d'intégrabilté uniforme global pour les classes de fonction plurisousharmoniques sur $Ω ⋐ ℂ^{n}$ ayant des masses de Monge–Ampère uniformément bornées sur $Ω .$

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 $\sup_{j} \int_{Ω} {({dd}^{c} ϕ_{j})}^{n} + \infty .$

It is known that the complex Monge–Ampère operator is well defined on the class $ℱ (Ω)$ and that for a function $ϕ \in ℱ (Ω)$ 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 $\tilde{Ω}$ are hyperconvex domains with $Ω ⋐ \tilde{Ω} ⋐ ℂ^{n}$ and $ϕ \in ℱ (Ω),$ there exists a plurisubharmonic function $\tilde{ϕ} \in ℱ (\tilde{Ω})$ such that $\tilde{ϕ} ⩽ ϕ$ on $Ω$ and $\int_{\tilde{Ω}} {({dd}^{c} \tilde{ϕ})}^{n} ⩽ \int_{Ω} {({dd}^{c} ϕ)}^{n} .$ Such a function is called a subextension of ϕ to $\tilde{Ω} .$

From this result we deduce a global uniform integrability theorem for the classes of plurisubharmonic functions with uniformly bounded Monge–Ampère masses on $Ω .$

Reçu le : 2002-09-24
Accepté le : 2003-01-13
Publié le : 2003-02-15

DOI : 10.1016/S1631-073X(03)00031-1

Affiliations des auteurs :

Urban Cegrell ¹ ; Ahmed Zeriahi ²

¹ Department of Mathematics, University of Umeä, 90187 Umeä, Sweden
² Université Paul Sabatier, institut de mathématiques, laboratoire Emile Picard, 118, route de Narbonne, 31062 Toulouse, France

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     journal = {Comptes Rendus. Math\'ematique},
     pages = {305--308},
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     doi = {10.1016/S1631-073X(03)00031-1},
     language = {en},
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Urban Cegrell; Ahmed Zeriahi. Subextension of plurisubharmonic functions with bounded Monge–Ampère mass. Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 305-308. doi : 10.1016/S1631-073X(03)00031-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00031-1/

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