Abstract
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/α t −d/α for C 1,1 open sets and of order t 1/α t −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.
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Acknowledgments
After the first version of this paper, which only contains Theorem 1.1, was finished, the first named author sent it to Professor Bañuelos. Professor Bañuelos encouraged us to work out the Lipschitz case. We thank him for his encouragement and for his helpful comments on a later version of the paper. We also thank the referee for very helpful comments on the first version of this paper.
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Research supported in part by a grant from the Simons Foundation (208236).
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Park, H., Song, R. Trace Estimates for Relativistic Stable Processes. Potential Anal 41, 1273–1291 (2014). https://doi.org/10.1007/s11118-014-9423-8
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DOI: https://doi.org/10.1007/s11118-014-9423-8