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 $L^{\mathrm{\infty }}$ on the frequency to the Cauchy problem with small initial data. For this purpose we investigate the global behavior of $x{e}^{it\mathrm{\nabla }}u$, $x^2{e}^{it\mathrm{\nabla }}u$ and $⟨\xi {⟩}^5\widehat{e^{it\mathrm{\nabla }}u}$. Due to the non-smoothness of $\mathrm{\nabla }$ near zero frequency the range of $\alpha$ is restricted to $(\frac{17}{10},2)$.