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