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

In this paper we prove that the one-dimensional Schrödinger equation with derivative in the nonlinear term is globally well-posed in H^s for s > 1/2 for data small in L². To understand the strength of this result one should recall that for s 1/2 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 > 2/3. 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 H^s for s > 1/2.