## Sharp heat kernel estimates for relativistic stable processes in open sets.(English)Zbl 1235.60101

Summary: We establish sharp two-sided estimates for the transition densities of relativistic stable processes (i.e., for the heat kernels of the operators $$m - (m^{2/\alpha } - \Delta )^{\alpha /2}$$) in $$C^{1,1}$$ open sets. Here, $$m > 0$$ and $$\alpha \in$$ (0, 2). The estimates are uniform in $$m \in (0, M$$] for each fixed $$M > 0$$. Letting $$m \downarrow 0$$, we recover the Dirichlet heat kernel estimates for $$\Delta ^{\alpha /2} := - ( - \Delta )^{\alpha /2}$$ in $$C^{1,1}$$ open sets obtained in [the authors, J. Eur. Math. Soc. (JEMS) 12, No. 5, 1307–1329 (2010; Zbl 1203.60114)]. Sharp two-sided estimates are also obtained for Green functions of relativistic stable processes in bounded $$C^{1,1}$$ open sets.

### MSC:

 60J35 Transition functions, generators and resolvents 47G20 Integro-differential operators 60J75 Jump processes (MSC2010) 47D07 Markov semigroups and applications to diffusion processes

Zbl 1203.60114
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### References:

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