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Let $d \geq 2$, $\alpha \in (0,2)$, and $X$ be the cylindrical$\alpha$-stable process on $\mathbb{R}^d$. We first present a geometriccharacterization of an open subset $D\subset \mathbb{R}^d$ so that the partprocess $X^D$ of $X$ in $D$ is irreducible. We then study the properties of thetransition density functions of $X^D$, including the strict positivity propertyas 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.

Dirichlet heat kernel estimates for cylindrical stable processes