Trace estimates for relativistic stable processes

In this paper, we study the asymptotic behavior, as the time t goes to zero, of the trace of the semigroup of a killed relativistic α-stable process in bounded C^1,1 open sets and bounded Lipschitz open sets. More precisely, we establish the asymptotic expansion in terms of t of the trace with an error bound of order t^2/αt^−d/α for C^1,1 open sets and of order t^1/αt^−d/α for Lipschitz open sets. Compared with the corresponding expansions for stable processes, there are more terms between the orders t^−d/α and t^(2−d)/α for C^1,1 open sets, and, when α∈(0,1], between the orders t^−d/α and t^(1−d)/α for Lipschitz open sets.