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

We prove that the initial value problem for the equation $$-i{\mathrm{\partial }}_{t}u+\sqrt{m^2-\mathrm{\Delta }}u=(\frac{e^{-{\mu }_0\mathrm{\mid }x\mathrm{\mid }}}{\mathrm{\mid }x\mathrm{\mid }}\ast \mathrm{\mid }u{\mathrm{\mid }}^2)u\text{ in }{\mathbb{R}}^{1+3},m\ge 0,{\mu }_0>0$$ is globally well-posed and the solution scatters to free waves asymptotically as $t\to \pm \mathrm{\infty }$ if we start with initial data which is small in $H^s({\mathbb{R}}^3)$ for $s>\frac{1}{2}$, and if $m>0$. Moreover, if the initial data is radially symmetric we can improve the above result to $m\ge 0$ and $s>0$, which is almost optimal, in the sense that $L^2({\mathbb{R}}^3)$ is the critical space for the equation. The main ingredients in the proof are certain endpoint Strichartz estimates, $L^2({\mathbb{R}}^{1+3})$ bilinear estimates for free waves and the application of the $U^p$ and $V^p$ function spaces.