Large data mass-subcritical NLS: critical weighted bounds imply scattering

We consider the mass-subcritical nonlinear Schrödinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity sc∈(max{-1,-d2},0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_c\in (\max \{-1,-\frac{d}{2}\},0)$$\end{document}, we prove that any solution satisfying |x||sc|e-itΔuLt∞Lx2<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\| \, |x|^{|s_c|}e^{-it\Delta } u\right\| _{L_t^\infty L_x^2} <\infty \end{aligned}$$\end{document}on its maximal interval of existence must be global and scatter.