Complex Analysis
Subextension of plurisubharmonic functions with bounded Monge–Ampère mass
Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 305-308.

Let $\Omega ⋐{ℂ}^{n}$ be a hyperconvex domain. Denote by ${ℰ}_{0}\left(\Omega \right)$ the class of negative plurisubharmonic functions ϕ on $\Omega$ with boundary values 0 and finite Monge–Ampère mass on $\Omega .$ Then denote by $ℱ\left(\Omega \right)$ the class of negative plurisubharmonic functions ϕ on $\Omega$ for which there exists a decreasing sequence (ϕ)j of plurisubharmonic functions in ${ℰ}_{0}\left(\Omega \right)$ converging to ϕ such that ${\mathrm{sup}}_{j}{\int }_{\Omega }{\left({\mathrm{dd}}^{c}{\varphi }_{j}\right)}^{n}+\infty .$

It is known that the complex Monge–Ampère operator is well defined on the class $ℱ\left(\Omega \right)$ and that for a function $\varphi \in ℱ\left(\Omega \right)$ the associated positive Borel measure is of bounded mass on $\Omega .$ A function from the class $ℱ\left(\Omega \right)$ is called a plurisubharmonic function with bounded Monge–Ampère mass on $\Omega .$

We prove that if $\Omega$ and $\stackrel{˜}{\Omega }$ are hyperconvex domains with $\Omega ⋐\stackrel{˜}{\Omega }⋐{ℂ}^{n}$ and $\varphi \in ℱ\left(\Omega \right),$ there exists a plurisubharmonic function $\stackrel{˜}{\varphi }\in ℱ\left(\stackrel{˜}{\Omega }\right)$ such that $\stackrel{˜}{\varphi }⩽\varphi$ on $\Omega$ and ${\int }_{\stackrel{˜}{\Omega }}{\left({\mathrm{dd}}^{c}\stackrel{˜}{\varphi }\right)}^{n}⩽{\int }_{\Omega }{\left({\mathrm{dd}}^{c}\varphi \right)}^{n}.$ Such a function is called a subextension of ϕ to $\stackrel{˜}{\Omega }.$

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

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

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

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

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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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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