\frac{C}{\nu^{1+\delta}}  {\left\lVert M_A \omega_S \right\rVert}_2^2 (1+ {\left\lVert M_A \omega_S \right\rVert}_2)^{2+\delta}, \end{aligned}$$ for any $\delta>0$ and some constant $C\geq 1$. Therefore, for $t \leq 1$ $$\begin{aligned}
 \frac{d}{dt} {\left\lVert M_A \omega_S \right\rVert}_2^2  \leq\frac{C_1}{\nu^{1+\delta}} {\left\lVert M_A \omega_S \right\rVert}_2^2 (1 + {\left\lVert M_A \omega_S \right\rVert}_2^2)^{1+\delta/2}
 \label{eq:step:a:6}\end{aligned}$$ for some constant $C_1 \geq 1$. Letting $$\begin{aligned}
T_1 = \min \left\{ \frac{\nu^{1+\delta}}{2 C_1}, T_{\lambda,\lambda^\prime}  \right\}   = \frac{\nu^{1+\delta}}{2 C_1} \qquad \mbox{(without loss of generality, as }\nu \leq \nu_0\mbox{)}, \label{eq:T1}\end{aligned}$$ we obtain from and the assumption on the $H^\sigma$ norm of $\omega_{S,in}$ that for $\epsilon' \leq 1$ $$\begin{aligned}
{\left\lVert  M_A(t) \omega_S(t) \right\rVert}_2\leq 2 \epsilon^\prime \qquad \mbox{for all} \qquad t \leq T_1.
 \label{eq:step:a:7}\end{aligned}$$

The Gevrey class estimates differ slightly from the real analytic bounds as we need to employ an additional commutator estimate. Consider the Gevrey-$\frac{1}{s}$ Fourier-multiplier $$\begin{aligned}
M_G(t,k,\eta) &= {\left\langle k,\eta \right\rangle}^\sigma e^{\lambda(t) |k,\eta|^{s}}\end{aligned}$$ where the decreasing function $\lambda(t)$ shall be chosen precisely later, and $\sigma>5/2$ is a fixed Sobolev correction. Similarly to we have $$\begin{aligned}
 &\frac{1}{2} \frac{d}{dt} {\left\lVert M_G \omega_S \right\rVert}_2^2 + \nu {\left\lVert  |\nabla| M_G \omega_S \right\rVert}_2^2 = \dot{\lambda}(t) {\left\lVert  |\nabla|^{s/2} M_G \omega_S \right\rVert}_2^2 + T_G + N_G
 \label{eq:step:b:1}\end{aligned}$$ where $T_G$ and $N_G$ are defined precisely as $T_A$ and $N_A$ upon replacing all $M_A$’s by $M_G$’s. It 