Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite L norm

We prove global wellposedness for the one-dimensional cubic non-linear Schrödinger equation in a space of distributions which is invariant under Galilean transformations and includes L². This space arises naturally in the study of the restriction properties of the Fourier transform to curved surfaces. The L^p bounds, p≠2, for the extension operator, dual to the restricition one, plays a fundamental role in our approach.