Global well-posedness in Sobolev space implies global existence for weighted L^2 initial data for L^2 -critical NLS

Abstract

The L^2 -critical defocusing nonlinear Schrodinger initial value problem on 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|)^σ u_0 in L^2 for certain s, σ< 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 σ= s.