Solution to Hessian type equations with prescribed singularity on compact Kahler manifold

Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix an integer $m$ such that $1\leq m\leq n$. We reformulate most relative pluripotential results of Darvas-DiNezza-Lu's survey \cite{DNL23} to the Hessian setting. As an application, we use a slightly different method and give an characterization of finite energy range of the Hessian operator, which cannot be directly reformulated by \cite{DNL23}. Given a model potential $\phi$, we also study degenerate complex Hessian equations of the form $(\omega+dd^c \varphi)^m\wedge\omega^{n-m}=F(x,\varphi)\omega^n$. Under some natrual conditions on $F$, we prove that the solution of this type equation has the same singularity type as $\phi$.