This paper studied the characterization and existence of solutions to complex Hessian equations associated with a given weight. We provided a complete characterization of the Radon measures that could be represented as the complex Hessian measure of an ${\mathcal{F}}_{m,\chi }(\mathrm{\Omega })$-function, where $\chi$ was a decreasing weight function. Our main results provided both global and local characterizations of the range of the complex Hessian operator acting on these classes. Specifically, we demonstrated that the solvability of the equation $H_m(\phi )=\mu$ in the class ${\mathcal{F}}_{m,\chi }(\mathrm{\Omega })$ is equivalent to a certain functional inequality involving the measure $\mu$ and the weighted energy ${\delta }_{m,\chi }$. Furthermore, we demonstrate that this global condition could be localized: a solution existed globally if, and only if, for every point in the closure $\overline{\mathrm{\Omega }}$ of the domain, a solution existed in some neighborhood of that point.

In this paper we treat the problem of connection between the convergence in $m-$capacity and the convergence of the Hessian measure for a sequence $f_j$ of $m-$subharmonic functions. We prove first that, under some conditions, the convergence of $f_j$ in capacity $Ca{p}_m$ implies the weak convergence of the Hessian measures $H_m(f_j)$. Then we show that the converse sense of convergence is also true in some particular cases.

This paper investigated the properties of maximal subextensions for $m$-subharmonic ( $m$-sh) functions in the context of complex Hessian operators. We established three main theorems that significantly advance the theory. First, we provided a complete characterization of maximal subextensions in the class ${\mathcal{F}}_m^a(\mathrm{\Omega })$, showing that any subextension satisfying a minimality condition on its Hessian mass must coincide with the maximal subextension. Second, we proved that for functions in ${\mathcal{E}}_m(\mathrm{\Omega })$, the maximal subextension relates to the original function through an $m$-maximal function and preserves Lelong numbers. Third, we demonstrated convergence stability, proving that for sequences with boundary values in ${\mathcal{F}}_m(\mathrm{\Omega },f)$ converging in $L_{\text{loc}}^1$ and uniformly bounded below, their maximal subextensions converged to the maximal subextension of the limit function.