Small data scattering for semi-relativistic equations with Hartree type nonlinearity

We prove that the initial value problem for the equation−i∂tu+m2−Δu=(e−μ0|x||x|⁎|u|2)uin R1+3,m≥0,μ0>0 is globally well-posed and the solution scatters to free waves asymptotically as t→±∞t→±∞ if we start with initial data which is small in Hs(R3)Hs(R3) for s>12, and if m>0m>0. Moreover, if the initial data is radially symmetric we can improve the above result to m≥0m≥0 and s>0s>0, which is almost optimal, in the sense that L2(R3)L2(R3) is the critical space for the equation. The main ingredients in the proof are certain endpoint Strichartz estimates, L2(R1+3)L2(R1+3) bilinear estimates for free waves and an application of the UpUp and VpVp function spaces.