Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces

In this paper, we first introduce a new function space MH^θ,p whose norm is given by the l^p-sum of modulated H^θ-norms of a given function. In particular, when θ<−12, we show that the space MH^θ,p agrees with the modulation space M^2,p(R) on the real line and the Fourier-Lebesgue space FL^p(T) on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quan-tities constructed by Killip-Vi ̧san-Zhang to the modulation space and Fourier-Lebesgue space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlin-ear Schrödinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on Ris globally well-posed in M^2,p(R)for any p<∞, while the renormalized cubic NLS on Tis globally well-posed in FL^p(T)for any p<∞.
In Appendix, we also establish analogous global-in-time bounds for the modified KdV equation (mKdV) in the modulation spaces on the real line and in the Fourier-Lebesgue spaces on the circle. An additional key ingredient of the proof in this case is a Galilean transform which converts the mKdV to the mKdV-NLS equation.