Fractional Laplacian with supercritical killings

In this paper, we study Feynman-Kac semigroups of symmetric $\alpha$-stable processes with supercritical killing potentials belonging to a large class of functions containing functions of the form $b\mathrm{\mid }x{\mathrm{\mid }}^{-\beta }$, where $b>0$ and $\beta >\alpha$. We obtain two-sided estimates on the densities $p(t,x,y)$ of these semigroups for all $t>0$, along with estimates for the corresponding Green functions.