## Abstract

In this paper, we first introduce a new function space

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.

*MH*whose norm is given by the^{θ,p }*l*sum of modulated^{p}-*H*-norms of a given function. In particular, when^{θ}*θ<−12*, we show that the space*MH*agrees with the modulation space^{θ,p}*M*on the real line and the Fourier-Lebesgue space^{2,p}(R)*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*(R)for any^{2,p}*p<∞*, while the renormalized cubic NLS on Tis globally well-posed in*FL*(T)for any^{p}*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.

Original language | English |
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Pages (from-to) | 1-29 |

Number of pages | 29 |

Journal | Journal of Differential Equations |

Volume | 269 |

Issue number | 1 |

Early online date | 16 Jan 2020 |

DOIs | |

Publication status | E-pub ahead of print - 16 Jan 2020 |

## Keywords

- Fourier-Lebesgue space
- complete integrability
- global well-posedness
- modified KdV equation
- modulation space
- nonlinear Schrödinger equation