Heat Kernel Upper Estimate For Fractional Negative Dirichlet Laplacian With Gradient Perturbation

Suppose that d ≥ 1 and α∈(1,2).Let D C(?)Rd and b an Rd-valued function on D which is in a certain Kato class Kdα-1.In this paper,we find out the gradien-t estimate of fractional negative Dirichlet Laplacian-(-△|D)α/2 in a bounded C1,1 open set D.Then,we use the Duhamel formula and induction method to derive the heat kernel upper estimates for fractional negative Dirichlet Laplacian with gradient perturbation(i.e.-(-△|D)α/2 + b·▽|D).