Fractional Laplacian with supercritical killings

In this paper, we study Feynman-Kac semigroups of symmetric α-stable processes with supercritical killing potentials belonging to a large class of functions containing functions of the form b|x|-β, where $b>0$ and β>α. 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.