Complex Hessian Operator, $m$-capacity, Cegrell's classes and $m$-Potential associated to a Positive Closed Current

Abstract

In this paper we firstly introduce the concepts of capacity and Cegrell's classes associated to any $m$-positive closed current $T$. Next, after investigating the most imporant related properties, we study the definition and the continuity of the complex hessian operator in several cases, generalizing then the work of Demailly and Xing in this direction. We also prove a Xing-type comparison principle for the analogous Cegrell class ${\mathcal{F}}^{m,T}$ of negative $m$-subharmonic functions. Finally, we generalize the work of Ben Messaoud-El Mir on the complex Monge-Ampère operator and the Lelong-Skoda potential associated to a positive closed current.

20 pages

Cited by in corpus (1)

• The Dirichlet problem for the complex Hessian operator in the class ${\mathcal{N}}_m(H)$