In this talk, I will present some recent results on sharp two-sided estimates on the heat kernels of ( fractional) Laplacians with supercritical killing potentials, that is, heat kernels of operators of the form $$-(-\Delta)^{\alpha/2}-\kappa (x) $$ where $\alpha\in (0, 2]$ and $\kappa$ belongs to a class of supercritical potentials including$\kappa(x)=c|x|^{-\beta}$ with $\beta>\alpha$. This talk is based on a joint paper with Soobin Cho, and a joint paper with Soobin Cho and Panki Kim.
