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.