in Definition \[xst\], $$u  = \sum_{{\lambda}=1}^\infty \sum_{d=1}^{{\lambda}} S_{\lambda}u_{{\lambda},d}, \qquad
v  = \sum_{{\lambda}=1}^\infty \sum_{d=1}^{{\lambda}} S_{\lambda}v_{{\lambda},d},$$ For the terms in the decomposition we use the $\tilde{X}^{s,\theta}_{{\lambda},d}$ norms, as allowed by Corollary \[tx\]. We denote $$u_{\lambda}= \sum_{d=1}^{{\lambda}} u_{{\lambda},d}, \qquad u_{{\lambda},<d} =  \sum_{h=1}^{d} u_{{\lambda},h}$$ Then we write $$uv = \sum_{\mu =1}^\infty  S_{\mu} (uv) =  \sum_{\mu =1}^\infty
\sum_{{\lambda}_1 =1}^\infty  \sum_{{\lambda}_2 =1}^\infty S_{\mu} (S_{{\lambda}_1}u_{{\lambda}_1} S_{{\lambda}_2}v_{{\lambda}_2})$$ There are two cases when the above summand is nonzero, namely if ${\lambda}_1 \approx {\lambda}_2 \gtrsim \mu$ and if $\max\{{\lambda}_1,{\lambda}_2\} \approx \mu$. We consider them separately.

[**Case 1**]{}, $ {\lambda}_1,{\lambda}_2 \approx {\lambda}\gtrsim \mu$. In this case the summability with respect to ${\lambda}$ is trivial, so it suffices to look at the product $S_{\lambda}u_{\lambda}S_{\lambda}v_{\lambda}$ for fixed ${\lambda}$. This is localized at frequency $\leq {\lambda}$. Combining the $L^\infty L^2$ and the $L^2 L^\infty$ bounds in Corollary \[lessd\] we obtain $$\| S_{\lambda}u_{\lambda}S_{\lambda}v_{\lambda}\|_{L^2} +{\lambda}^{-1} \| {\partial}_t (S_{\lambda}u_{\lambda}S_{\lambda}v_{\lambda})\|_{L^2} \lesssim   {\lambda}^{-2s+\frac32}
 \|u_{\lambda}\|_{\tilde X^{s,\theta}_{{\lambda}}} \|v_{\lambda}\|_{\tilde X^{s,\theta}_{{\lambda}}}
\label{lla}$$ Using the equation we can also bound the second time derivative, $${\lambda}^{-2} \| {\partial}_t^2 (S_{\lambda}u_{\lambda}S_{\lambda}v_{\lambda})\|_{L^2} \lesssim
{\lambda}^{-2s+\frac32}
 \|u_{\lambda}\|_{\tilde X^{s,\theta}_{{\lambda}}} \|v_{\lambda}\|_{\tilde X^{s,\theta}_{{\lambda}}}
\label{llb}$$ The three bounds above allow us to estimate for $\mu \leq {\lambda}$ $$\|S_{\lambda}u_{\lambda}S_{\lambda}v_{\lambda}\|_{X^{s,\theta}_{\mu,\mu}} \lesssim
\mu^{s+\theta-2} {\lambda}^{-2s+\frac72}
 \|u_{\lambda}\|_{\tilde X^{s,\theta}_{{\lambda}}} \|v_{\lambda}\|_{\tilde X^{s,\theta}_{{\lambda}}}$$ This suffices provided that $\theta < s-\frac32$, which is 