Dirichlet Heat kernel estimates for a large class of anisotropic Markov processes

Let $Z=(Z^1,\dots ,Z^d)$ be the d-dimensional Lévy {process} where {$Z^i$’s} are independent 1-dimensional Lévy {processes} with identical jumping kernel ${\nu }^1(r)=r^{-1}\phi (r{)}^{-1}$. Here $\phi$ is {an} increasing function with weakly scaling condition of order $\underset{‾}{\alpha },\bar{\alpha }\in (0,2)$. We consider a symmetric function $J(x,y)$ comparable to \begin{align*} \begin{cases} ν^1(|x^i - y^i|)\qquad&$\text{ if }{x}^i\mathrm{\ne }{y}^i\text{ for some }i\text{ and }{x}^j=y^j\text{ for all }j\mathrm{\ne }i$\ 0\qquad&$\text{ if }{x}^i\mathrm{\ne }{y}^i\text{ for more than one index }i$. \end{cases} \end{align*} Corresponding to the jumping kernel $J$, there exists an anisotropic Markov process $X$, see \cite{KW22}. In this article, we establish sharp two-sided Dirichlet heat kernel estimates for $X$ in $C^{1,1}$ open set, under certain regularity conditions. As an application of the main results, we derive the Green function estimates.