Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in $C^{1,\eta }$ open sets

Abstract

In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in $C^{1,\eta }$ open sets. The processes are symmetric pure jump Markov processes with jumping intensity $\kappa (x,y){\psi }_1(|x-y|{)}^{-1}|x-y{|}^{-d-\alpha }$, where $\alpha \in (0,2)$. Here, ${\psi }_1$ is an increasing function on $[0,\infty )$, with ${\psi }_1(r)=1$ on $0<r\le 1$ and $c_1{e}^{c_2{r}^{\beta }}\le {\psi }_1(r)\le c_3{e}^{c_4{r}^{\beta }}$ on $r>1$ for $\beta \in [0,\infty ]$, and $\kappa (x,y)$ is a symmetric function confined between two positive constants, with $|\kappa (x,y)-\kappa (x,x)|\le c_5|x-y{|}^{\rho }$ for $|x-y|<1$ and $\rho >\alpha /2$. We establish two-sided estimates for the transition densities of such processes in $C^{1,\eta }$ open sets when $\eta \in (\alpha /2,1]$. In particular, our result includes (relativistic) symmetric stable processes and finite-range stable processes in $C^{1,\eta }$ open sets when $\eta \in (\alpha /2,1]$.

arXiv comment29 pages