Boundary regularity for Monge-Amp\`ere equations with unbounded right hand side

We consider Monge-Amp`ere equations with right hand side $f$ that degenerate to $\mathrm{\infty }$ near the boundary of a convex domain $\mathrm{\Omega }$, which are of the type $$\mathrm{d}\mathrm{e}\mathrm{t}{D}^{2}u=f\mathrm{i}\mathrm{n}\mathrm{\Omega },f\sim d_{\mathrm{\partial }\mathrm{\Omega }}^{-\alpha }\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{\partial }\mathrm{\Omega },$$ where $d_{\mathrm{\partial }\mathrm{\Omega }}$ represents the distance to $\mathrm{\partial }\mathrm{\Omega }$ and $-\alpha$ is a negative power with $\alpha \in (0,2)$. We study the boundary regularity of the solutions and establish a localization theorem for boundary sections.