\gtrsim N_0^2$

Using Kishimoto’s Lemma 4.1 and Corollary 3.3, $$\begin{aligned}
 &\left[ \sum_{N_1} \left( \sum_{N_0, N_2} \sum_{L_0, L_1, L_2} \left\| \eta_\delta \int_0^t S(t-t') P_{N_1, L_1}[P_{N_2,L_2}(\eta_\delta u) Q_{N_0,L_0}(\eta_\delta w) ] d t' \right\|_{X^{s,\frac12,1}_I} \right)^2 \right]^\frac12 \\
 &\lesssim C_1
 \left[ \sum_{N_1} \left( \sum_{N_0, N_2} \sum_{L_0, L_1, L_2} \delta^{\frac 32 -b } N_1^s L_1^{-b} \overline{L}^\frac12 L_m^{\frac14+} \underline{L}^{\frac14 +} N_1^{\frac12-} N_2^{-1} N_0^{\frac12 - \beta   }  L_0^\frac12 {\langle}\delta L_0 {\rangle}^{-a} \|P_{N_2,L_2}(\eta_\delta u)\|_{L^2L^2}  \right)^2 \right]^\frac12 \\
 &\lesssim C_1
 \left[ \sum_{N_1} \left( \sum_{N_2} \sum_{L_0, L_1, L_2} \delta^{\frac 32 -a -b } N_1^{s+ \frac12 -} L_1^{-b} \overline{L}^\frac12 L_m^{\frac14+} \underline{L}^{\frac14 +} N_2^{-\frac12 - \beta   }  L_0^{\frac12 -a}  \|P_{N_2,L_2}(\eta_\delta u)\|_{L^2L^2}  \right)^2 \right]^\frac12.\end{aligned}$$ If $L_2 \geq L_0$, then $\overline{L} \lesssim L_2$, so we have the bound $$C_1
\left[ \sum_{N_1} \left( \sum_{N_2} \sum_{L_0, L_1, L_2} \delta^{\frac 32 -a -b }  L_0^{\frac34 -a +} L_1^{\frac14 -b+} N_1^{s + \frac12 -}N_2^{-s - \frac12 - \beta   }   \left( N_2^s L_2^\frac12 \|P_{N_2,L_2}(\eta_\delta u)\|_{L^2L^2} \right) \right)^2 \right]^\frac12.$$ Take $a = \frac34 +$ and $b = \frac14 +$ to obtain $$\begin{aligned}
C_1&\left[ \sum_{N_1} \left( \sum_{N_2} \sum_{ L_2} \delta^{\frac12 -} N_1^{s +\frac12 -}N_2^{-s- \frac12 - \beta   }   \left( N_2^s L_2^\frac12 \|P_{N_2,L_2}(\eta_\delta u)\|_{L^2L^2} \right) \right)^2 \right]^\frac12 \\
\lesssim
C_1 &\delta^{\frac12- } \| u\|_{X^{s,\frac12,1}} \left[\sum_{N_2 \gg N_1} N_1^{2s + 1 -}N_2^{-2s - 