Lp estimates for fractional Schrödinger operators with Kato class potentials

Let α>0, H=(−Δ)α+V(x), V(x) belongs to the higher order Kato class K2α(Rn). For 1≤p≤∞, we prove a polynomial upper bound of ‖e−itH(H+M)−β‖Lp,Lp in terms of time t for all integers α and 2α≥[n2]+1 if α is not an integer. Both the smoothing exponent β and the growth order in t are almost optimal compared to the free case. The main ingredients in our proof are pointwise heat kernel estimates for the semigroup e−tH. We obtain a Gaussian upper bound with sharp coefficient for integral α and a polynomial decay for fractional α.