Complex $m$-Hessian type equations in $\mathcal{E}_{m,\chi}(\Omega)$

In this paper, we first concern the existence of solutions of the complex $m$-Hessian type equation $-\chi(u)H_{m}(u)=\mu$ where $\mu$ vanishes on all of $m$-polar sets in the class $\mathcal{E}_{m,\chi}(\Omega)$. Next, we study the existence of solutions of this equation in the class $\mathcal{E}_{m,\chi}(\Omega)$ if there exists subsolution in this class. Using the above results, we study subextension in the class $\mathcal{E}_{m,\chi}(\Omega)$.