Global well-posedness and scattering for the three-dimensional defocusing cubic Schrödinger equation in $H^s$, $s>\frac{1}{2}$

We prove global well-posedness and scattering for the three-dimensional defocusing cubic nonlinear Schrödinger equation with arbitrary initial data in H^s(R^3), $s>\frac{1}{2}$. The proof combines the $I$-method with improved long-time bilinear $L_{t,x}^2$ estimates for frequency-localized components of the solution. The key high–low frequency estimate follows from a directional interaction identity and an induction on frequency.