Dirichlet heat kernel estimates for rectilinear stable processes

Let $d\ge 2$, $\alpha \in (0,2)$, and $X$ be the rectilinear $\alpha$-stable process on ${\mathbb{R}}^d$. We first present a geometric characterization of an open subset $D\subset {\mathbb{R}}^d$ so that the part process $X^D$ of $X$ in $D$ is irreducible. We then study the properties of the transition density functions of $X^D$, including the strict positivity property as well as their sharp two-sided bounds in $C^{1,1}$ domains in ${\mathbb{R}}^d$. Our bounds are shown to be sharp for a class of $C^{1,1}$ domains.