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Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS
The $L^2$-critical defocusing nonlinear Schrödinger initial value problem on $\mathbb R^d$ 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 for data in the weighted space $(1+|x|) u_0 \in L^2$. For the $d=2$ problem, it is known that global existence holds for data in $H^s$ and also for data in the weighted space $(1+|x|)^\sigma u_0 \in L^2$ for certain $s, \sigma < 1$. We prove: If global well-posedness holds in $H^s$ then global existence and scattering holds for initial data in the weighted space with $\sigma = s$.