condition $1 \leq d$ is related to the spatial localization on the unit scale in our problem. The condition $d \leq \lambda$ reflects the fact that at high modulation we use a simpler structure, see e.g. , .

The cutoff at frequency less than $\sqrt{\lambda}$ for the coefficients $\Box_g$ is related to the regularity of the coefficients, $\partial^2 g \in L^2 L^\infty$. This implies that $\Box_{g_{\geq{\sqrt{\lambda}}}}u_{{\lambda},d}$ is an allowable error term.\

We begin our analysis of the $X^{s,\theta}$ spaces with a simple observation, namely that without any restriction in generality one can assume that the functions $u_{{\lambda},d}$ and $f_{{\lambda},d}$ in Definition \[xst\] are localized at frequency ${\lambda}$. Precisely, we have the stronger result:

The following estimate holds: $${\lambda}^{s-1}d^{\theta}\|\nabla S_{\lambda}v\|_{L^2} +
{\lambda}^{s-1}d^{\theta-1}\|\Box_{g_{<{\sqrt{\lambda}}}}S_{\lambda}v \|_{L^2}
\lesssim \|v\|_{X_{{\lambda},d}^{s,\theta}} \label{strongst}$$

We first bound the time derivatives of $v$ in negative Sobolev spaces, $${\lambda}^{s} d^{\theta} (\| {\partial}_t^2 v\|_{L^2 (H^{-2}+{\lambda}^2 L^2)} + \|{\partial}_t
v\|_{L^2(H^{-1}+{\lambda}L^2)}  ) \lesssim \|v\|_{X_{{\lambda},d}^{s,\theta}}
\label{lowv}$$ This follows by Cauchy-Schwartz from the interpolation inequality $$\|{\partial}_t v\|_{L^2 ( H^{-1}+{\lambda}L^2)}^2 \lesssim  (\| {\partial}_t^2 v\|_{L^2 (H^{-2}+{\lambda}^2L^2)}
+\|v\|_{L^2})\|v\|_{L^2}$$ combined with the bound $$\| {\partial}_t^2 v\|_{L^2 (H^{-2}+{\lambda}^2 L^2)} \lesssim {\lambda}^{-2} \|
\Box_{g_{<{\sqrt{\lambda}}}} v\|_{L^2} + \|{\partial}_t v\|_{L^2 ( H^{-1}+{\lambda}L^2)} + \|v\|_{L^2}$$ To prove this last estimate we only use the $L^\infty$ 