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We prove global well-posedness for the L^{2}-critical cubic defocusingnonlinear Schr\"odinger equation on R^{2} with data u_{0} \in H^{s}(R^{2}) fors > {1/3}.

Bootstrapped Morawetz Estimates And Resonant Decomposition For Low Regularity Global Solutions Of Cubic NLS On R^{2}