  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 $\theta\in (\alpha/2, 1]$.
