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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 Lévy processes killed upon leaving an open set $D$. Under some mild assumptions on the Lévy 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 Lévy densities of unimodal Lévy 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 Lévy processes such that the weak lower scaling index of the Lévy densities is not necessarily strictly bigger than the Euclidean dimension.