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<−1/2, 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(핋) on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by Killip-Vişan-Zhang to the modulation space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on ℝ is globally well-posed in M^{2,p}(ℝ) for any p<∞, while the renormalized cubic NLS on 핋 is globally well-posed in FL^p(핋) 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.