Approximate factorizations for non-symmetric jump processes

Abstract

In this paper, we first extend the approximate factorization for purely discontinuous Markov process established in \cite{CKSV20} by getting rid of some of the conditions imposed in \cite{CKSV20}. Then we apply the approximate factorization to obtain sharp two-sided heat kernel estimates for three classes of processes: stable-like processes with critical killings in $C^{1,\mathrm{Dini}}$ open sets; killed stable-like processes in the setting of \cite{KW24} in $C^{1,\epsilon }$ open sets; and non-symmetric stable processes in what we call $C^{1,2\text{-}\mathrm{Dini}}$ open sets. In particular, we obtain explicit sharp two-sided heat kernel estimates of killed $\hat{I}\pm$-stable processes in $C^{1,\mathrm{Dini}}$ open sets for all $\hat{I}\pm \in (0,2)$ and of censored $\hat{I}\pm$-stable processes in $C^{1,\mathrm{Dini}}$ open sets for all $\hat{I}\pm \in (1,2)$.

56 pages; Minor changes were made