Regularity for the Supercritical Fractional Laplacian with Drift

We consider the linear stationary equation defined by the fractional Laplacian with drift. In the supercritical case, wherein the dominant term is given by the drift instead of the diffusion component, we prove local regularity of solutions in Sobolev spaces employing tools from the theory of pseudo-differential operators. The regularity of solutions in the supercritical case is as expected from the subcritical case. In the subcritical case the diffusion is at least as strong as the drift, and the operator is an elliptic pseudo-differential operator, which is not the case in the supercritical regime. We also compute the leading part of the singularity of the Green's kernel for the supercritical case, which displays some rather unusual behavior.