Global existence and scattering for rough solutions of a nonlinear Schroedinger equation on R^3

We prove global existence and scattering for the defocusing, cubic nonlinear Schr"odinger equation in $H^s({\text{rr}}^3)$ for $s>4\mathrm{/}5$. The main new estimate in the argument is a Morawetz-type inequality for the solution $\varphi$. This estimate bounds $\mathrm{\parallel }\varphi (x,t){\mathrm{\parallel }}_{L_{x,t}^4({\text{rr}}^3\times \text{rr})}$, whereas the well-known Morawetz-type estimate of Lin-Strauss controls $\int_0^{\infty}\int_{\rr^3}\frac{(\phi(x,t))^4}{|x|} dx dt