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 \(b|x|^{-\beta }\), 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.