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's 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,C-theta for some theta is an element of (alpha/2,1].