+ d^{-1} \|f_t\|_{L^\infty}
 + d^{-\frac12} \|f_t\|_{L^2 L^\infty} + d^{-\frac32} \|f_{tt}\|_{L^2 L^\infty}$$ $$\|f\|_{M_{<d}} = \|f\|_{M_d} + d^{\frac12} \| f\|_{L^2  L^\infty}$$ Then we have the multiplicative properties

Assume that $f$ is localized at frequency $d \leq {\lambda}$. Then we have $$\|f S_{\lambda}u\|_{\tilde X^{s,\theta}_{{\lambda},d}} \lesssim \|f\|_{M_d} \|u\|_{\tilde
  X^{s,\theta}_{{\lambda},d}}$$ respectively $$\begin{split}
\|f S_{\lambda}u\|_{\tilde X^{s,\theta}_{{\lambda},d}} & \lesssim \|f\|_{M_{<d}} \|u\|_{\tilde
  X^{s,\theta}_{{\lambda},<d}}, \qquad \theta < \frac12 \\
\|f S_{\lambda}u\|_{\tilde X^{s,\theta}_{{\lambda},d}} & \lesssim
d^{\theta-\frac12} \|f\|_{M_{<d}} \|u\|_{\tilde
  X^{s,\theta}_{{\lambda},<d}}, \qquad \theta > \frac12
\end{split}$$ \[md\]

The proof is straightforward, using Leibnitz’s rule and the energy estimate . To bound functions in the $M_d$, respectively $M_{<d}$ norms we use Corollary \[lessd\] with $d = {\lambda}$ to obtain:

a\) Assume that $\theta > \frac12$. Then $$\| S_{\lambda}u\|_{M_{<{\lambda}}} \leq {\lambda}^{2-s} \|u\|_{\tilde
X^{s,\theta}_{{\lambda}}}\qquad \| S_{<{\lambda}} u\|_{M_{{\lambda}}} \leq
\max\{1,{\lambda}^{2-s}\} \|u\|_{X^{s,\theta}}$$ $$\| S_{<{\lambda}} u\|_{M_{<{\lambda}}} \leq
\max\{{\lambda}^ \frac12,{\lambda}^{2-s}\} \|u\|_{X^{s,\theta}}$$

b\) Assume that $\theta < \frac12$. Then $$\| S_{\lambda}u\|_{M_{<{\lambda}}} \leq {\lambda}^{\frac52-\theta-s} \|u\|_{\tilde
X^{s,\theta}_{{\lambda}}}\qquad \| S_{<{\lambda}} u\|_{M_{{\lambda}}} \leq
\max\{1,{\lambda}^{\frac52-\theta-s}\} \|u\|_{X^{s,\theta}}$$ $$\| S_{<{\lambda}} u\|_{M_{<{\lambda}}} \leq
\max\{{\lambda}^ \frac12,{\lambda}^{\frac52-\theta-s}\} \|u\|_{X^{s,\theta}}$$ \[xmd\]

Using the above property we prove the algebra property for the space $X$.

Assume that $s > 2$ and $\frac12 < \theta < s-\frac32$ . Then $X^{s,\theta}$ is an algebra. \[xxx\]

Let $u,v \in X^{s,\theta}$. For both we consider the decomposition 