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 $\alpha$-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 $t^{2\mathrm{/}\alpha }{t}^{-d\mathrm{/}\alpha }$ for $C^{1,1}$ open sets and of order $t^{1\mathrm{/}\alpha }{t}^{-d\mathrm{/}\alpha }$ for Lipschitz open sets. Compared with the corresponding expansions for stable processes, there are more terms between the orders $t^{-d\mathrm{/}\alpha }$ and $t^{(2-d)\mathrm{/}\alpha }$ for $C^{1,1}$ open sets, and, when $\alpha \in (0,1]$, between the orders $t^{-d\mathrm{/}\alpha }$ and $t^{(1-d)\mathrm{/}\alpha }$ for Lipschitz open sets.