On the modified scattering of 3-d Hartree type fractional Schr\"odinger equations with Coulomb potential

In this paper we study 3-d Hartree type fractional Schr\"odin-ger equations: \begin{equation} i\partial_{t}u-|\nabla|^{\alpha}u = \lambda\left(|x|^{-\gamma} *| u|^{2} \right)u,\;\;1 < \alpha < 2,\;\;0 < \gamma < 3,\;\; \lambda \in \mathbb R \setminus \{0\}. \end{equation} In \cite{cho} it is known that no scattering occurs in $L^2$ for the long range ($0 < \gamma \le 1$). In \cite{c0, chooz2, cho1} the short-range scattering ($1 < \gamma < 3$) was treated for the scattering in $H^s$. In this paper we consider the critical case ($\gamma = 1$) and prove a modified scattering in $L^\infty$ on the frequency to the Cauchy problem with small initial data. For this purpose we investigate the global behavior of $x e^{it\nabla} u$, $x^2 e^{it\nabla} u$ and $\langle\xi\rangle^5 \widehat{e^{it\nabla} u}$. Due to the non-smoothness of $\nabla$ near zero frequency the range of $\alpha$ is restricted to $(\frac{17}{10}, 2)$.