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

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 density, establish two-sided Dirichlet heat kernel estimates when $D$ is $C^{1, 1}$. Our result covers case that densities of are regularly varying whose indices equal to Euclidean dimension. This first results such weak lower scaling index not necessarily strictly bigger than