Small Data Scattering of Hartree Type Fractional Schrödinger Equations in a Scaling Critical Space

Abstract

In this paper we study the small-data scattering of Hartree type fractional Schrödinger equations in space dimension 2, 3. It has Lévy index α between 1 and 2, and Hartree type nonlinearity F(u) = μ(|x|-γ * |u|2)u with 2d/(2d-1) < γ < 2, γ ≥ α > 1. This equation is scaling-critical in Ḣsc with sc = (γ-α)/2. We show that the solution scatters in Ḣsc,1, where Ḣsc,1 is also a scaling critical space of Sobolev type taking in angular regularity with norm defined by ||ϕ||Ḣsc,1 = ||ϕ||Ḣsc + ||∇Sϕ||Ḣsc. For this purpose we use the recently developed Strichartz estimate which is Lθ2-averaged on the unit sphere Sd-1.