Abstract
We prove that a general (not necessarily symmetric) Lévy process killed on exiting a bounded open set (without regular condition on the boundary) is intrinsically ultracontractive, provided that $B(0,R_{0})\subseteq\operatorname{supp} (\nu)$ for some constant $R_{0}>0$, where $\operatorname{supp} (\nu)$ denotes the support of the associated Lévy measure $\nu$. For a symmetric Lévy process killed on exiting a bounded Hölder domain of order $0$, we also obtain the intrinsic ultracontractivity under much weaker assumption on the associated Lévy measure.
Citation
Xin Chen. Jian Wang. "Intrinsic ultracontractivity for general Lévy processes on bounded open sets." Illinois J. Math. 58 (4) 1117 - 1144, Winter 2014. https://doi.org/10.1215/ijm/1446819305
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