Critical nonlinear Schrِdinger equations with data in homogeneous weighted spaces

We consider the initial value problem for nonlinear Schrödinger equations with the critical nonlinearities λ 1 | u | 2 n u , where Im λ 1 ≤ 0 , when the space dimension n = 1 , 2 . We prove the global existence of small solutions in homogeneous weighted L 2 ( R n ) spaces. It is shown that the small solutions decay uniformly like t − n 2 for t > 1 if Im λ 1 = 0 . The higher uniform time decay rates t − n 2 ( log t ) − n 2 for t > 1 are obtained if Im λ 1 < 0 .