Heat kernel estimates for critical fractional diffusion operator
  In this work we construct the heat kernel of the 1/2-order Laplacian perturbed by the first-order gradient term in H\"older space and the zero-order potential term in generalized Kato's class, and obtain sharp two-sided estimates as well as the gradient estimate of the heat kernel. 
Longjie Xie and Xicheng Zhang
math.AP