Let $Z=(Z^{ 1}, \ldots, Z^{ d})$ be the d-dimensional L\'evy { process} where { $Z^i$'s} are independent 1-dimensional L\'evy { 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 $\underline \alpha, \overline \alpha\in (0, 2)$. We consider a symmetric function $J(x,y)$ comparable to \begin{ align*} \begin{ cases} \nu^1(|x^i - y^i|)\qquad&\text{ if $x^i \ne y^i$ for some $i$ and $x^j = y^j$ for all $j \ne i$}\\ 0\qquad&\text{ if $x^i \ne y^i$ 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.
Categories: math.PR