In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces ℱL^p(\({\cal F}{L^p}(\mathbb{T})\)), 1 ≤ p < ∞. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in ℱL^p(\({\cal F}{L^p}(\mathbb{T})\)) for \(1 \leq p \leq {3 \over 2}\).

在本文中，我们研究了圆上的一维三次非线性薛定谔方程（NLS）。特别是，我们开发了一种范式方法来研究几乎临界的傅立叶-勒贝格空间中的 NLS。通过应用第一作者与 Z. Guo 和 S. Kwon (2013) 引入的范式归约的无限迭代，我们推导出了一个范式方程，该方程等效于正则解的重归一化三次 NLS。对于粗糙函数，范式方程的表现优于重归一化的三次 NLS，从而提供了三次 NLS 的进一步重整化。然后我们证明这个范式方程在傅立叶-勒贝格空间ℱL ^p ( \({\cal F}{L^p}(\mathbb{T})\) ), 1 ≤ p <∞。通过反转变换，我们在合适的意义上总结了几乎临界的傅立叶-勒贝格空间中重整化三次 NLS 的全局适定性。这种方法还允许我们证明ℱL ^p ( \({\cal F}{L^p}(\mathbb{T})\) ) for \(1 \leq p \ ) 中（重归一化）三次 NLS 的无条件唯一性leq {3 \over 2}\)。