Heat kernel lower bounds for some singular non-local Dirichlet forms

In this talk, we consider the heat kernels of some singular non-local Dirichlet forms without killing parts on doubling spaces. We give an equivalent characterization of so called localized lower estimate of the heat kernels in terms of Poincare inequality, generalized capacity condition and reverse doubling condition. Moreover, our results also cover some non-singular cases, for instance, the stable-like case.