Weighted energy class of $m$-subharmonic functions

In this paper, we introduce the class $\mathcal{E}_{m,F}(\Omega)$ and solve complex $m$-Hessian equations in the class $\mathcal{E}_{m,F}(\Omega)$. Afterthat, we study subextension in the class $\mathcal{E}_{m,F}(\Omega)$ 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.