\Bigr) {\,\operatorname{d}\!}t' \right\|_{X^{s,\frac12,1}_\delta} \lesssim C_1 \delta^{\frac12-}  {N_{\text{HL}}}^{s-\frac12-\beta  } \| u\|_{X^{s,\frac12,1}_\delta} \lesssim C_0 C_1 \delta^{\epsilon_0 -} {N_{\text{HL}}}^{s- \frac12 -\beta   }.$$ The remainder of the paper is devoted to obtaining this bound, relying heavily on estimates previously established by Kishimoto.

The requirement that $\delta$ be small exists because we will require $$\begin{aligned}
 C_0 \delta^{\epsilon_0 -} \lesssim 1 \\
 C_0 C_1 \delta^{\epsilon_0-} \lesssim 1\end{aligned}$$ to close the contraction.

In the following, we employ dyadic decompositions. We always use $P_{N_1,L_1}  {\operatorname{Re}}({\operatorname{W}}^\pm(t) m_0^\pm)u$, $P_{N_2,L_2}u$, and $Q^\pm_{N_0,L_0} {\operatorname{W}}^\pm m_0^\pm$; i.e. the product ${\operatorname{Re}}({\operatorname{W}}^\pm(t) m_0^\pm)u$ is associated with the dyadic variables $N_1$ and $L_1$, the linear wave flow is associated the dyadic variables $N_0$ and $L_0$, etc.

We also use bars to denote maxima and minima of the dyadic variables, i.e. $$\overline{L}_{jk} := \max\{ L_j, L_k\}, \qquad \overline{L} := \overline{L}_{012} = \max\{ L_0, L_1, L_2\},$$ $$\underline{L}_{jk} = \min\{ L_j, L_k\}, \qquad \underline{L} = \underline{L}_{012} = \min\{ L_0, L_1, L_2\}.$$ We also define $$L_m := \operatorname{median}\{ L_0, L_1, L_2\}.$$

Resonant case
-------------

Here we confine our attention to the resonant frequencies. These occur when $$0= |L_1-L_2-L_0| \lesssim \overline{L}.$$ We have $$|L_1-L_2-L_0| =|\tau_0 + \tau_2 - (k_0 + k_2)^2 - \tau_2 + k_2^2 - \tau_0 \mp |k_0||  = |k_0| 