DIRICHLET HEAT KERNEL ESTIMATES FOR FRACTIONAL LAPLACIAN WITH GRADIENT PERTURBATION

Suppose that d >= 2 and alpha is an element of (1, 2). Let D be a bounded C-1,C-1 open set in R-d and b an R-d-valued function on R-d whose components are in a certain Kato class of the rotationally symmetric a-stable process. In this paper, we derive sharp two-sided heat kernel estimates for L-b = Delta(alpha/2) + b . del in D with zero exterior condition. We also obtain the boundary Harnack principle for L-b in D with explicit decay rate.