Global well-posedness for a periodic nonlinear schrödinger equation in 1D and 2D

The initial value problem for the L² critical semilinear Schrödinger equation with periodic boundary data is considered. We show that the problem is globally well-posed in H^s(double-struck T sign^d), for s > 4/9 and s > 2/3 in 1D and 2D respectively, confirming in 2D a statement of Bourgain in [4]. We use the "I- method". This method allows one to introduce a modification of the energy functional that is well defined for initial data below the H¹(double- struck T sign^d) threshold. The main ingredient in the proof is a "refinement" of the Strichartz's estimates that hold true for solutions defined on the rescaled space, double-struck T sign _λ^d = ℝ^d/λℤ^d, d = 1, 2.