Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation

Suppose that $d\ge 2$ and $\alpha \in (1,2)$. Let D be a bounded $C^{1,1}$ open set in ${\mathbb{R}}^d$ and b an ${\mathbb{R}}^d$-valued function on ${\mathbb{R}}^d$ whose components are in a certain Kato class of the rotationally symmetric \alpha-stable process. In this paper, we derive sharp two-sided heat kernel estimates for ${\mathcal{L}}^b={\mathrm{\Delta }}^{\alpha \mathrm{/}2}+b\cdot \mathrm{\nabla }$ in D with zero exterior condition. We also obtain the boundary Harnack principle for ${\mathcal{L}}^b$ in D with explicit decay rate.