In this paper, we first introduce a new function space $MH^{\theta, p}$ whose norm is given by the $\ell^p$-sum of modulated $H^\theta$-norms of a given function. In particular, when $\theta < -\frac 12$, we show that the space $MH^{\theta, p}$ agrees with the modulation space $M^{2, p}(\mathbb R)$ on the real line and the Fourier-Lebesgue space $\mathcal F L^{p}(\mathbb T)$ on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by Killip-Visan-Zhang to the modulation space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlinear Schrodinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on $\mathbb R$ is globally well-posed in $M^{2, p}(\mathbb R)$ for any $p < \infty$, while the renormalized cubic NLS on $\mathbb T$ is globally well-posed in $\mathcal FL^p(\mathbb T)$ for any $p < \infty$. 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.

中文翻译：

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几乎临界空间中一维三次非线性薛定谔方程的全局适定性

在本文中，我们首先介绍了一个新的函数空间 $MH^{\theta, p}$，其范数由给定函数的调制 $H^\theta$-范数的 $\ell^p$-sum 给出。特别地，当 $\theta < -\frac 12$ 时，我们表明空间 $MH^{\theta, p}$ 与调制空间 $M^{2, p}(\mathbb R)$ 在实线和圆上的傅立叶-勒贝格空间 $\mathcal FL^{p}(\mathbb T)$。我们使用范数的这种等价性和伽利略对称性来使 Killip-Visan-Zhang 构造的守恒量适应调制空间设置。通过应用尺度对称性，我们然后证明了一维三次非线性薛定谔方程（NLS）在几乎临界空间中的全局适定性。更准确地说，我们证明 $\mathbb R$ 上的三次 NLS 在 $M^{2, p}(\mathbb R)$ 中全局适定，对于任何 $p < \infty$，而 $\mathbb T$ 上的重整化三次 NLS 在 $\mathcal FL^p(\mathbb T)$ 中全局适定，对于任何 $p < \infty$。在附录中，我们还在实线上的调制空间和圆上的傅立叶-勒贝格空间中为修正的 KdV 方程 (mKdV) 建立了类似的全局时间界限。在这种情况下，证明的另一个关键要素是伽利略变换，它将 mKdV 转换为 mKdV-NLS 方程。