Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces

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