Estimates of Dirichlet heat kernels for unimodal Lévy processes with low intensity of small jumps

Abstract: In this paper, we study transition density functions for pure jump unimodal L\'evy processes killed upon leaving an open set $D$. Under some mild assumptions on the L\'evy density, we establish two-sided Dirichlet heat kernel estimates when the open set $D$ is $C^{1, 1}$. Our result covers the case that the L\'evy densities of unimodal L\'evy processes are regularly varying functions whose indices are equal to the Euclidean dimension. This is the first results on two-sided Dirichlet heat kernel estimates for L\'evy processes such that the weak lower scaling index of the L\'evy densities is not necessarily strictly bigger than the Euclidean dimension.