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We prove the global well-posedness and scattering for the defocusing$H^{\frac12}$-subcritical (that is, $2<\gamma<3$) Hartree equation with lowregularity data in $\mathbb{R}^d$, $d\geq 3$. Precisely, we show that a uniqueand global solution exists for initial data in the Sobolev space$H^s\big(\mathbb{R}^d\big)$ with $s>4(\gamma-2)/(3\gamma-4)$, which alsoscatters in both time directions. This improves the result in \cite{ChHKY},where the global well-posedness was established for any$s>\max\big(1/2,4(\gamma-2)/(3\gamma-4)\big)$. The new ingredients in our proofare that we make use of an interaction Morawetz estimate for the smoothed outsolution $Iu$, instead of an interaction Morawetz estimate for the solution$u$, and that we make careful analysis of the monotonicity property of themultiplier $m(\xi)\cdot < \xi>^p$. As a byproduct of our proof, we obtain thatthe $H^s$ norm of the solution obeys the uniform-in-time bounds.

Global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical Hartree equation in $\mathbb{R}^d$