Hessian operator on weighted m-subharmonic classes

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.