Scattering for 2d semi-relativistic Hartree equations with short range potential

We study the long time behavior of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is convolved with the singular potential $\mathrm{\mid }x{\mathrm{\mid }}^{-\gamma }$ for $1<\gamma <2$, which is referred to as short-range interaction potential in the sense of scattering phenomenon. We establish the scattering results for small solutions in a weighted space, in other words, we prove that the nonlinear solutions exist globally and behave asymptotically like a linear solution whenever the initial data is sufficiently small. To achieve this, we should obtain time decay estimates for the nonlinear term which is integrable. A key observation is that the loss in time in the course of weighted energy estimates can be recovered by the space resonance method with the help of null structure.