Global heat kernel estimate for relativistic stable processes in exterior open sets

In this paper, sharp two-sided estimates for the transition densities of relativistic α-stable processes with mass m∈(0, 1] in C ^1,1 exterior open sets are established for all time t>0. These transition densities are also the Dirichlet heat kernels of m-(m ^2/α-δ) ^α/2 with m∈(0, 1] in C ^1,1 exterior open sets. The estimates are uniform in m in the sense that the constants are independent of m∈(0, 1]. As a corollary of our main result, we establish sharp two-sided Green function estimates for relativistic α-stable processes with mass m∈(0, 1] in C ^1,1 exterior open sets.