A refined global well-posedness result for Schrodinger equations with derivative

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In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in Hˢ, for s>12 for data small in L². To understand the strength of this result one should recall that for s<12 the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the ``I-method'' used by the same authors to obtain global well-posedness for s>23. The same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in Hˢ, for s>12.