A Refined Global Well-Posedness Result for Schrödinger Equations with Derivative

In this paper we prove that the one-dimensional Schrodinger equation with derivative in the nonlinear term is globally well-posed in Hs for $s > \frac12$ for data small in L2 . To understand the strength of this result one should recall that for $s \frac23$. The same argument can be used to prove that any quintic nonlinear defocusing Schrodinger equation on the line is globally well-posed for large data in Hs for $s>\frac12$.