Intégrabilité uniforme semi-globale d'une classe de fonctions plurisousharmoniques. (Semi-global uniform integrability for a class of pluri-subharmonic functions) (Q1408924)

For \(0\leq s<\exp(-1/2)\) let \(\mathcal G_s\) be the set of all functions \(u\) plurisubharmonic in the unit Euclidean ball \(\mathbb B\subset\mathbb C^n\) such that \(u\leq0\) in \(\mathbb B\) and \(\max_{s\mathbb B}u\geq-1\). The following estimate (which is a generalization of a result by A. Zeriahi) is proved. For every \(\rho<\rho(s):=\frac{1-s\exp(1/2)}{\exp(1/2)-s}\) there exist constants \(C\), \(M>0\) such that \[ \int_{\rho\mathbb B}\exp(-u)d\lambda\leq M\exp \left(C\int_{\rho(s)\mathbb B} | u| \,d\lambda \right),\quad u\in\mathcal G_s. \]