Heat kernel estimates for Dirichlet fractional Laplacian with gradient perturbation

We give a direct proof of the sharp two-sided estimates, recently established in [4,9], for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1,1}$ open sets by using Duhamel formula. We also obtain a gradient estimate for the Dirichlet heat kernel. Our assumption on the open set is slightly weaker in that we only require $D$ to be $C^{1,\theta }$ for some $\theta \in (\alpha \mathrm{/}2,1]$.