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Comptes Rendus Mathématique
Volume 353, n° 7
pages 605-610 (juillet 2015)
Doi : 10.1016/j.crma.2015.04.011
Received : 23 February 2015 ;  accepted : 21 April 2015
The existence problem of S -plurisubharmonic currents
Le problème de l'existence de courants S -plurisousharmoniques
 

Ahmad K. Al Abdulaali , Hassine El Mir
 The Department of Mathematics, College of Science, King Faisal University, P.O. Box 380, Post Code 31982, Al-Ahsaa, Saudi Arabia 

Abstract

In this paper, we prove the existence of the trivial extension of S -plurisubharmonic currents of bidimension   defined outside an obstacle A of Hausdorff measure  . Furthermore, a valid definition of the current   is achieved for every positive closed current T and plurisubharmonic function g . The above results rely on an improvement of a classical result due to Demailly on the Monge–Ampère operator with a sharp condition on the Hausdorff measure.

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Résumé

Dans cette Note, nous montrons l'existence de l'extension triviale des courants S -plurisousharmoniques de bi-dimension  , définis en-dehors d'un obstacle A de mesure de Hausdorff  . De plus, nous montrons que le courant   est bien défini, pour tout courant positif fermé T et toute fonction plurisousharmonique g . Ces résultats reposent sur un relâchement de la condition de nullité d'une mesure de Hausdorff, dans un résultat classique de Demailly sur l'opérateur de Monge–Ampère.

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