Small data scattering of hartree type fractional schrödinger equations in dimension 2 and 3

In this paper we study the small-data scattering of the d dimensional fractional Schrödinger equations with d = 2, 3, Lévy index 1 < α < 2 and Hartree type nonlinearity F (u) = µ(|x|^−γ ∗ |u|²)u with max (Formula presented) < γ ≤ 2, γ < d. This equation is scaling-critical in Ḣ^sc, (Formula presented). We show that the solution scatters in H^s,1 for any s > s_c, where H^s,1 is a space of Sobolev type taking in angular regularity with norm defined by (Formula presented). For this purpose we use the recently developed Strichartz estimate which is L² -averaged on the unit sphere S^d−1 and utilize U^p -V^p space argument.