1- 2\beta } \right]^\frac12 \\
\lesssim 
C_1 &\delta^{\frac12- } {N_{\text{HL}}}^{-\beta   } \| u\|_{X^{s,\frac12,1}} \lesssim C_0C_1 \delta^{\epsilon_0-} {N_{\text{HL}}}^{-\beta   }.\end{aligned}$$

If $L_2 \lesssim L_0$, then $\overline{L} \lesssim L_0$ and we have the bound $$C_1 \left[ \sum_{N_1} \left( \sum_{N_2} \sum_{L_0, L_1, L_2} \delta^{\frac 32 -a -b } L_0^{1-a} L_1^{\frac14 - b +}  N_1^{s +\frac12 -}  N_2^{-s -\frac12 - \beta   }  \left( N_2^s L_2^{\frac14+} \|P_{N_2,L_2}(\eta_\delta u)\|_{L^2L^2} \right) \right)^2 \right]^\frac12.$$ Take $a = 1+$ and $b = \frac14+$ to obtain $$\begin{aligned}
C_1 \delta^{\frac14- } {N_{\text{HL}}}^{-\beta  } \| u\|_{X^{s,\frac14+,1}} \lesssim \delta^{\frac12- } {N_{\text{HL}}}^{-\beta    } \| u\|_{X^{s,\frac12,1}} \lesssim C_0C_1 \delta^{\epsilon_0-} {N_{\text{HL}}}^{-\beta   }.\end{aligned}$$

### $N_0 \approx N_2 \gg N_1$ and $\overline{L_{02}} \ll N_0^2$

Using the same results as in the previous case, and noting that $L_1 = \overline{L} \approx N_0^2$, we have the bound

$$\begin{aligned}
&C_1\left[ \sum_{N_1} \left( \sum_{N_2} \sum_{L_0, L_1, L_2} \delta^{\frac 32- a-b } L_0^{\frac34 - a +} L_1^{\frac12 -b}   N_1^{s +\frac12 - }N_2^{-s-\frac12 - \beta   }   \left( N_2^s L_2^{\frac14+} \|P_{N_2,L_2}(\eta_\delta u)\|_{L^2L^2} \right) \right)^2 \right]^\frac12.\end{aligned}$$

Take $b = \frac12$ and $a = \frac34 +$ and proceed as above to obtain the bound $$\begin{aligned}
C_1 \delta^{\frac14- } {N_{\text{HL}}}^{-\beta  } \| u\|_{X^{s,\frac14+,1}} \lesssim C_1 \delta^{\frac12- } {N_{\text{HL}}}^{-\beta   } \| u\|_{X^{s,\frac12,1}} \lesssim C_0C_1 \delta^{\epsilon_0 -} {N_{\text{HL}}}^{-\beta   } .\end{aligned}$$

### 