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In this paper, we first introduce a new function space $MH^{\theta, p}$ whosenorm is given by the $\ell^p$-sum of modulated $H^\theta$-norms of a givenfunction. 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 thereal line and the Fourier-Lebesgue space $\mathcal F L^{p}(\mathbb T)$ on thecircle. We use this equivalence of the norms and the Galilean symmetry to adaptthe conserved quantities constructed by Killip-Vi\c{s}an-Zhang to themodulation space setting. By applying the scaling symmetry, we then proveglobal well-posedness of the one-dimensional cubic nonlinear Schr\"odingerequation (NLS) in almost critical spaces. More precisely, we show that thecubic NLS on $\mathbb R$ is globally well-posed in $M^{2, p}(\mathbb R)$ forany $p < \infty$, while the renormalized cubic NLS on $\mathbb T$ is globallywell-posed in $\mathcal FL^p(\mathbb T)$ for any $p < \infty$.  In Appendix, we also establish analogous global-in-time bounds for themodified KdV equation (mKdV) in the modulation spaces on the real line and inthe Fourier-Lebesgue spaces on the circle. An additional key ingredient of theproof in this case is a Galilean transform which converts the mKdV to themKdV-NLS equation.

Global well-posedness of the one-dimensional cubic nonlinear Schr\"odinger equation in almost critical spaces