Complex Hessian Operator associated to an $m$-positive closed current and weighted $m$-capacity

TL;DR

This paper develops the theory of complex Hessian operators associated with $m$-positive currents, introduces weighted capacities, and characterizes certain function classes, advancing the understanding of complex Hessian equations.

Contribution

It defines and analyzes the complex Hessian operator for unbounded $m$-subharmonic functions and introduces weighted capacities linked to $m$-positive currents.

Findings

Characterization of Cegrell classes via weighted capacity

Continuity results for the complex Hessian operator

A subsolution theorem for complex Hessian equations

Abstract

In this paper, we first study the definition and the continuity of the complex Hessian operator associated to an $m$-positive closed current $T$, for some classes of unbounded $m$-subharmonic functions as well as when we consider a regularization sequence of $T$. Next, we introduce the notion of weighted $(m,T)$-capacity in the complex Hessian setting and we investigate the link with the weighted $m$-extremal function. As an application we give a characterization of the Cegrell classes ${\mathscr F}^m$ and ${\mathscr E}^m$ by means of the weighted $(m,1)$-capacity. Furthermore, we prove a subsolution theorem for a general complex Hessian equation relatively to $T$.