 \langle D_x \rangle(u\langle \phi_N'(x) \rangle^{\frac{0}{0}}) \|_{L_T^0L_x^0}^0  \notag \\ & \quad + \|u\langle x \rangle ^{s-\frac{0}{0}} \|_{L_T^0L_x^0}^0  \notag \\ & \lesssim \| \langle D_x \rangle(u\langle \phi_N'(x) \rangle^{\frac{0}{0}}) \|_{L_T^0L_x^0}^0  \notag \\ & \quad+ \|u\langle x \rangle ^{s-\frac{0}{0}} \|_{L_T^0L_x^0}^0  \notag \\ & \lesssim \| \langle D_x \rangle^{0s} u \|_{L_T^0L_x^0}^{\frac{0}{s}}  \| 