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.

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
     author = {Ahmad K. Al Abdulaali and Hassine El Mir},
     title = {The existence problem of {\protect\emph{S}-plurisubharmonic} currents},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {605--610},
     publisher = {Elsevier},
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     number = {7},
     year = {2015},
     doi = {10.1016/j.crma.2015.04.011},
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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.

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