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

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 ${\mathcal{F}}^m$ and ${\mathcal{E}}^m$ by means of the weighted $(m,1)$-capacity. Furthermore, we prove a subsolution theorem for a general complex Hessian equation relatively to $T$.

27 pages

References in corpus (2)

• A priori estimates for the complex Hessian equations
• A variational Approach to complex Hessian equations in ${\mathbb{C}}^n$