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 C1,1 exterior open sets are established for all time t>0. These transition densities are also the Dirichlet heat kernels of m−(m2/α−Δ)α/2 with m∈(0,1] in C1,1 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 C1,1 exterior open sets.