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

Suppose that d ? 1 and ??(1,2).Let D C(?)Rd and b an Rd-valued function on D which is in a certain Kato class Kd?-1.In this paper,we find out the gradien-t estimate of fractional negative Dirichlet Laplacian-(-?|D)?/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)?/2 + b·?|D).