\partial_x(u^0)\|_{L_T^0L_x^0} \notag \\ & \lesssim T^{\frac{0}{0}}\|u\|_{L_x^0L_T^{\infty}}^0\|\partial_xu\|_{L_x^{\infty}L_T^{0}} \notag \\ & \lesssim T^{\frac{0}{0}}\|u\|_{Z_T}^0. \notag\end{aligned}$$ 