Weighted energy class of $m$-subharmonic functions

In this paper, we introduce the class ${\mathcal{E}}_{m,F}(\mathrm{\Omega })$ and solve complex $m$-Hessian equations in the class ${\mathcal{E}}_{m,F}(\mathrm{\Omega })$. Afterthat, we study subextension in the class ${\mathcal{E}}_{m,F}(\mathrm{\Omega })$ with the weighted Hessian measure of subextension unchanged. This is an extensive version of the result in [24] in the case when the smaller domain is relatively compact inside the large one.