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We consider the cubic Nonlinear Schroedinger Equation (NLS) in one spacedimension, either focusing or defocusing. We prove that the solutions satisfya-priori local in time Hs bounds in terms of the Hs size of the initial datafor s >=-1/4 . This improves earlier results of Christ-Colliander-Tao [2] andof the authors [12]. The new ingredients are a localization in space and localenergy decay, which we hope to be of independent interest.

Energy and local energy bounds for the 1-D cubic NLS equation in H^{-1/4}