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We prove that a general (not necessarily symmetric) L\'evy process killed onexiting a bounded open set (without regular condition on the boundary) isintrinsically ultracontractive, provided that $B(0,R_0)\subseteq\rm{supp}(\nu)$ for some constant $R_0>0$, where $\rm{supp}(\nu)$ denotes thesupport of the associated L\'evy measure $\nu$. For a symmetric L\'evy processkilled on exiting a bounded H\"older domain of order $0$, we also obtain theintrinsic ultracontractivity under much weaker assumption on the associatedL\'evy measure.

Intrinsic Ultracontractivity for General L\'evy processes on Bounded Open Sets