Comptes Rendus
Complex analysis
The existence problem of S-plurisubharmonic currents
Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 605-610.

In this paper, we prove the existence of the trivial extension of S-plurisubharmonic currents of bidimension (p,p) defined outside an obstacle A of Hausdorff measure H2p(A)=0. Furthermore, a valid definition of the current dgdcgT 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.

Dans cette Note, nous montrons l'existence de l'extension triviale des courants S-plurisousharmoniques de bi-dimension (p,p), définis en-dehors d'un obstacle A de mesure de Hausdorff H2p(A)=0. De plus, nous montrons que le courant dgdcgT 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.04.011

Ahmad K. Al Abdulaali 1; Hassine El Mir 1

1 The Department of Mathematics, College of Science, King Faisal University, P.O. Box 380, Post Code 31982, Al-Ahsaa, Saudi Arabia
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Ahmad K. Al Abdulaali; Hassine El Mir. The existence problem of S-plurisubharmonic currents. Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 605-610. doi : 10.1016/j.crma.2015.04.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.04.011/

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