Global well-posedness for the L² critical Hartree equation on bbr^n, nge 3

We consider the initial value problem for the L^2-critical defocusing Hartree equation in R^n, n \ge 3. We show that the problem is globally well posed in H^s(R^n) when 1>s> \frac{2(n-2)}{3n-4}$. We use the "I-method" combined with a local in time Morawetz estimate for the smoothed out solution.