| k_0 + 2k_2 \pm \lambda(k_0)|,$$ where $\lambda(k)$ is the sign of $k$. Since $k_0 \neq 0$, solving this yields $$k_0 = 2k_1 \pm \lambda(k_1) \qquad \qquad k_2 = \mp\lambda(k_1) - k_1,$$ for $k_1 \neq 0.$

On these frequencies, ${\operatorname{W}}^\pm(t) m_0^\pm u$ can be estimated in $X^{s,-\frac12,1}_\delta$ as follows. We use the fact that $|h_k^\pm| \leq C_1 $, and compute $$\begin{aligned}
&\| {\operatorname{W}}^\pm(t) m_0^\pm u \|_{X^{s,-\frac12, 1}_{\delta}} \\ 
\lesssim
&\left\| N_1^s L_1^{-\frac12} \left\| \int  \frac{h^\pm_{2k_1 - \lambda(k_1)}}{{\langle}k_1 {\rangle}^\beta } {\widehat}{u}(\mp\lambda(k_1) - k_1, \tau_2) \delta {\widehat}{\eta}(\delta(\tau_1 - \tau_2 \pm | 2k_1 - \lambda(k_1)|)) {\,\operatorname{d}\!}\tau_2  \right\|_{\ell^2_{k_1}(|k_1| \approx N_1)L^2_{\tau_1}(|\tau_1 - k_1^2| \approx L_1) }\right\|_{\ell^2_{N_1}\ell^1_{L_1}} \\
\lesssim 
& C_1\delta^{\frac12- } \Bigg\| N_1^{-\beta   } L_1^{0-} \Bigg\| \int  {\langle}\tau_1 - k_1^2{\rangle}^{-\frac12+} \frac{{\langle}\tau_2 - (\mp \lambda(k_1) -k_1)^2 {\rangle}^{-\frac12}}{{\langle}\tau_1 -\tau_2 \pm | 2k_1 \pm \lambda(k_1)| {\rangle}^{\frac12+}} \times \\
 & \hspace{2.5in} \left( \sum_{L_2}N_1^s L_2^\frac12 {\widehat}{P_{N_1,L_2}u}(\mp \lambda(k_1) - k_1, \tau_2)  \right) {\,\operatorname{d}\!}\tau_2 \Bigg\|_{\ell^2_{k_1}(|k_1| \approx N_1)L^2_{\tau_1}(|\tau_1 - k_1^2| \approx L_1) } \Bigg\|_{ \ell^2_{N_1} \ell^1_{L_1}} \\
\lesssim 
& C_1\delta^{\frac12- } \left\| N_1^{-\beta   } L_1^{0-} \left\|  {\langle}\tau_1 - k_1^2{\rangle}^{-1+} \left\| \sum_{L_2}N_1^s L_2^\frac12 {\widehat}{P_{N_1,L_2}u}(\mp\lambda(k_1) - k_1, \tau_2)  \right\|_{L^2_{\tau_2}} \right\|_{\ell^2_{k_1}(|k_1| \approx N_1)L^2_{\tau_1}(|\tau_1 - k_1^2| \approx L_1) } \right\|_{ \ell^2_{N_1} \ell^1_{L_1}} \\
\lesssim 
& C_1\delta^{\frac12- } \left\| N_1^{-\beta   } L_1^{0-} \left\|  \left\| \sum_{L_2}N_1^s L_2^\frac12 {\widehat}{P_{N_1,L_2}u}(\mp\lambda(k_1) - 