On the 1D Cubic Nonlinear Schrödinger Equation in an Almost Critical Space

We consider the Cauchy problem for the one dimensional cubic nonlinear Schrödinger equation iut+uxx-|u|2u=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$iu_t+u_{xx}-|u|^2u=0$$\end{document}. As the first step local well-posedness in the modulation space M2,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{2,p}$$\end{document} (2≤p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\le p<\infty $$\end{document}) is derived (see Theorem 1.4), which covers all the subcritical cases. Afterwards in order to approach the endpoint case, we will prove the almost global well-posedness in some Orlicz type space (see Theorem 1.8), which is a natural generalization of M2,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{2,p}$$\end{document}, and is almost critical from the viewpoint of scaling. The new ingredient is an endpoint version of the two dimensional restriction estimate (see Lemma 3.7).