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

TL;DR

This paper develops a comprehensive framework for complex Hessian operators, capacities, and Cegrell's classes associated with positive closed currents, extending previous theories and establishing new comparison principles and potential theories.

Contribution

It introduces new concepts and properties of capacities and Cegrell's classes for $m$-positive closed currents, generalizes the complex Hessian operator, and extends existing potential theories.

Findings

Established properties of capacity and Cegrell's classes for $m$-positive currents

Proved continuity and comparison principles for the complex Hessian operator

Generalized Monge-Ampère operator and Lelong-Skoda potential to broader contexts

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-Amp\`ere operator and the Lelong-Skoda potential associated to a positive closed current.