The $L^2$-critical defocusing NLS initial value problem on $\Bbb{R}^d$ is known to be locally well-posed for initial data in $L^2$. Hamiltonian conservation and the pseudoconformal transformation show that global well-posedness holds for initial data $u_0$ in Sobolev $H^1$ and in the weighted space $(1+|x|) u_0 \in L^2$. For the $d=2$ problem, it is known that global well-posedess also holds for data in $H^s$ and also for data in the weighted space $(1+|x|)^{\sigma} u_0 \in L ^2$ for certain $s$, $\sigma < 1$. The talk will presents a new result: If global well-posedness holds in $H^s$ then global well-posedness and scattering holds in the weighted space with $\sigma = s$.