Some results on Complex $m-$subharmonic classes

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 $\mathcal{E}{m,\chi}(\Omega)$.