also frequently use the notation $P_{\neq 0}f = f - f_0$. We denote the standard $L^p$ norms by ${\left\lVert f \right\rVert}_{p}$. We make common use of the Gevrey-$\frac{1}{s}$ norm with Sobolev correction defined by $$\begin{aligned}
{\left\lVert f \right\rVert}_{{\mathcal{G}}^{\lambda,\sigma;s}}^2 = \sum_{k}\int {\left\vert\hat{f}_k(\eta)\right\vert}^2 e^{2\lambda{\left\vertk,\eta\right\vert}^s}{\left\langle k,\eta \right\rangle}^{2\sigma} d\eta. \end{aligned}$$ Since most of the paper we are taking $s$ as a fixed constant, it is normally omitted. Also, if $\sigma =0$, it is omitted. We refer to this norm as the $\mathcal{G}^{\lambda,\sigma;s}$ norm and occasionally refer to the space of functions $$\begin{aligned}
\mathcal{G}^{\lambda,\sigma;s} = {\left\{f \in L^2 :{\left\lVert f \right\rVert}_{{\mathcal{G}}^{\lambda,\sigma;s}}<\infty\right\}}. \end{aligned}$$ See §\[apx:Gev\] for a discussion of the basic properties of this norm and some related useful inequalities.

For $\eta \geq 0$, we define $E(\eta)\in {\mathbb Z}$ to be the integer part. We define for $\eta \in {\mathbb R}$ and $1 \leq {\left\vertk\right\vert} \leq E(\sqrt{{\left\vert\eta\right\vert}})$ with $\eta k \geq 0$, $t_{k,\eta} = {\left\vert\frac{\eta}{k}\right\vert} - \frac{{\left\vert\eta\right\vert}}{2{\left\vertk\right\vert}({\left\vertk\right\vert}+1)} =  \frac{{\left\vert\eta\right\vert}}{{\left\vertk\right\vert}+1} +  \frac{{\left\vert\eta\right\vert}}{2{\left\vertk\right\vert}({\left\vertk\right\vert}+1)}$ and $t_{0,\eta} = 2 {\left\vert\eta\right\vert}$ and the critical intervals $$\begin{aligned}
I_{k,\eta} = \left\{
\begin{array}{lr}
[t_{{\left\vertk\right\vert},\eta},t_{{\left\vertk\right\vert}-1,\eta}] & \textup{ if } \eta k \geq 0 \textup{ and } 1 \leq {\left\vertk\right\vert} \leq E(\sqrt{{\left\vert\eta\right\vert}}), \\ 
\emptyset & otherwise.
\end{array} 
\right. \end{aligned}$$ For minor technical reasons, we define a 