Heat kernel for fractional diffusion operators with perturbations

Let L be an elliptic differential operator on a complete connected Riemannian manifold M such that the associated heat kernel has two-sided Gaussian bounds as well as a Gaussian type gradient estimate. Let L (α) L^α be the α-stable subordination of L for α (1,2). We found some classes K_α^γ,^β (β,γ [0,α)) of time-space functions containing the Kato class, such that for any measurable functions b:[0,∞)×M→TM and c:[0,∞) with |b|,c Kα1,1, the operator [EQUATION PRESENTED] for some constant C > 1, where ρ is the Riemannian distance. The estimate of (α){∇yp{α}_b,c and the Hölder continuity of (α) ∇_xp_b,cα are also considered. The resulting estimates of the gradient and its Hölder continuity are new even in the standard case where L=Δon ^d and b, c are time-independent.