k_1, \tau_2)  \right\|_{L^2_{\tau_2}} \right\|_{\ell^2_{k_1}(|k_1| \approx N_1 } \right\|_{ \ell^2_{N_1} \ell^1_{L_1}} \\
\lesssim 
& C_1\delta^{\frac12- } \left\| N_1^{-\beta   } L_1^{0-}  \sum_{L_2}N_1^s L_2^\frac12 \left\| P_{N_1,L_2}u \right\|_{L^2L^2}   \right\|_{ \ell^2_{N_1} \ell^1_{L_1}} \\
\lesssim 
& C_1\delta^{\frac12- } \left\| N_1^{-\beta   }  \sum_{L_2}N_1^s L_2^\frac12 \left\| P_{N_1,L_2}u \right\|_{L^2L^2}   \right\|_{ \ell^2_{N_1}} \\
\lesssim 
& C_1\delta^{\frac12- } {N_{\text{HL}}}^{-\beta   } \| u \|_{X^{s,\frac12,1}_I} \lesssim C_0 C_1 \delta^{\epsilon_0-} {N_{\text{HL}}}^{-\beta   }.\end{aligned}$$

High Schrödinger frequencies
----------------------------

We decompose dyadically in frequency space as follows. For general $m$, $$\begin{aligned}
&\| \eta_\delta \int_0^t S(t - t')(mu) d t' \|_{X^{s,\frac12, 1}_I} \\
&\approx
\left[ \sum_{N_1} \| P_{N_1} \eta_\delta \int_0^t S(t - t')(mu) d t' \|^2_{X^{s,\frac12,1}_I} \right]^\frac12 \\
&\lesssim 
\left[ \sum_{N_1} \left( \sum_{N_0, N_2} \sum_{L_0, L_1, L_2} \left\| \eta_\delta \int_0^t S(t-t') P_{N_1, L_1}[P_{N_2,L_2}(\eta_\delta u) Q_{N_0,L_0}(\eta_\delta m) ] d t' \right\|_{X^{s,\frac12,1}_I} \right)^2 \right]^\frac12.\end{aligned}$$

Furthermore, we calculate that $$\begin{aligned}
\| Q_{N_0,L_0}( \eta_\delta {\operatorname{W}}(t) m_0) \|_{L^2L^2}  &= 
\left\| \frac{h_k}{{\langle}k {\rangle}^\beta}  \delta {\widehat}{\eta}(\delta(\tau \pm |k|)) \right\| _{L^2_\tau \ell^2_k(|k| \approx N_0, |\tau \pm |k| \approx L_0)}\\
&= 
\delta \left\| \frac{h_k}{{\langle}k {\rangle}^\beta}  \right\|_{\ell^2(|k| \approx N_0)} \left\| {\widehat}{\eta}(\delta \tau) \right\|_{L^2(|\tau| \approx L_0)} \\
&\lesssim
C_1 \delta \left( N_0^{\frac12 - \beta   } \right) L_0^\frac12 {\langle}\delta L_0 {\rangle}^{-a}  = C_1 \delta N_0^{\frac12 - \beta   }  L_0^\frac12 {\langle}\delta L_0 {\rangle}^{-a}.\end{aligned}$$

### $N_0 \approx N_2 \gg N_1$ and $\overline{L_{02}} 