Sharp heat kernel estimates for relativistic stable processes in open sets

In this paper, we establish sharp two-sided estimates for the transition densities of relativistic stable processes [i.e., for the heat kernels of the operators m - (m ^2/α - {increment}) ^α/2] in C ^1,1 open sets. Here m > 0 and α ∈ (0, 2). The estimates are uniform in m ∈ (0, M] for each fixed M > 0. Letting m ↓ 0, we recover the Dirichlet heat kernel estimates for {increment} ^α/2 := -(-{increment}) ^α/2 in C ^1,1 open sets obtained in [14]. Sharp two-sided estimates are also obtained for Green functions of relativistic stable processes in bounded C ^1,1 open sets.