Small data scattering of hartree type fractional schrödinger equations in a scaling critical space

In this paper we study the small-data scattering of Hartree type fractional Schrödinger equations in space dimension 2, 3. It has Lévy index a between 1 and 2, and Hartree type nonlinearity F(u) = µ(|x|^-y*|u|²)u with 2d/(2d - 1) < y < 2, y ≥ α > 1. This equation is scaling-critical in H^sc with s_c (y-α)/2. We show that the solution scatters in H^sc,1 where H^sc, 1 is also a scaling critical space of Sobolev type taking in angular regularity with norm defined by. For this purpose we use the recently developed Strichartz estimate which is L²θ -averaged on the unit sphere S^d-1.