Global theory for NLS in the weighted spaces I: finite pseudo conformal energy

We intend to prove that for defocusing nonlinear Schrödinger equations, solutions with finite pseudo conformal energy must be global and scatter. This fact is first proved by Bourgain when $0<s_c<1$: the equation is locally well-posed in $H_x^{s_c}$, and if one further assumes that the initial data satisfies $x{u}_0\in L_x^2$, then the solution is global and scatters. The $s_c<0$ case remains not studied, and the best result is given by Beceanu, Deng, Soffer, and Wu: If the initial data is in $H_x^{s_c}$, radial, and compactly supported, then the solution is globally well-posed. In this paper, we extend Bourgain’s result to the $s_c<0$ case. We prove that if the initial data satisfies $\mathrm{\mid }x{\mathrm{\mid }}^{-s_c}{u}_0\in L_x^2$ and $x{u}_0\in L_x^2$, or if the radial initial data satisfies $u_0\in {\dot{H}}_x^{s_c}$ and $x{u}_0\in L_x^2$, then the solution is global and scatters. Bourgain’s result is based on the subcritical a priori control on $L_x^{p+2}$, while our argument relies on the supercritical bound on $\mathcal{F}{\dot{H}}_x^1$.