Global well-posedness for the defocusing Hartree equation with radial data in ${\mathbb{R}}^4$

By $I$-method, the interaction Morawetz estimate, long time Strichartz estimate and local smoothing effect of Schr"odinger operator, we show global well-posedness and scattering for the defocusing Hartree equation $$\{\begin{array}{ll}i{u}_t+\mathrm{\Delta }u& =F(u),(t,x)\in \mathbb{R}\times {\mathbb{R}}^{4}u(0)\\ & =u_0(x)\in H^s({\mathbb{R}}^4),\end{array}$$ where $F(u)=(V\ast \mathrm{\mid }u{\mathrm{\mid }}^2)u$, and $V(x)=\mathrm{\mid }x{\mathrm{\mid }}^{-\gamma }$, $3<\gamma <4$, with radial data in $H^s({\mathbb{R}}^4)$ for $s>s_c:=\gamma \mathrm{/}2-1$. It is a sharp global result except of the critical case $s=s_c$, which is a very difficult open problem.