}\end{aligned}$$  For the $\phi_N'(x)u^{k+0}$ term, $$\begin{aligned} \left | \int_{[0,T]}\int_{\mathbb R}\phi'_N(x)u^{0}\,dx\,dt \right | & \lesssim  \| \langle x\rangle^{0s-0}|u|^{0} \|_{L_T^0L_x^0} \notag \\ & \leq \|  u \|_{L_T^0L_x^{\infty}} \| \langle x\rangle^{s-\frac{0}{0}}u \|_{L_T^{\infty}L_x^0}^0 \notag \\ & \leq T^{\frac{0}{0}} \|  u \|_{L_T^0L_x^{\infty}} \| \langle x\rangle^{s-\frac{0}{0}}u \|_{L_T^{\infty}L_x^0}^0 \