Some results on Complex $m-$subharmonic classes

TL;DR

This paper investigates the properties of $m$-subharmonic function classes, establishing convergence results and characterizations related to Hessian measures, and extends these results to broader classes with subextension theorems.

Contribution

It proves that convergence in $m$-capacity implies Hessian measure convergence within $ extstyle ext{E}_m( ext{Ω})$, and extends these results to classes depending on an increasing function $ ext{χ}$, providing new characterizations and subextension theorems.

Findings

Convergence in $m$-capacity implies Hessian measure convergence.

Characterization of classes $ ext{E}_{m, ext{χ}}( ext{Ω})$ via Hessian measures.

Extension of results to broader classes with subextension properties.

Abstract

In this paper we study the class $\mathcal{E}_{m}(\Omega)$ of $m-$subharmonic functions introduced by Lu in \cite{L1}. We prove that the convergence in $m-$capacity implies the convergence of the associated Hessian measure for functions that belong to $\mathcal{E}_{m}(\Omega)$. Then we extend those results to the class $\mathcal{E}_{m,\chi}(\Omega)$ that depends on a given increasing real function $\chi$. A complete characterization of those classes using the Hessian measure is given as well as a subextension theorem relative to $\mathcal{E}_{m,\chi}(\Omega)$.