u\|_{L^2
  L^3} + {\lambda}^{s-1} \| S_{\lambda}\nabla u\|_{L^4 L^2} \lesssim
d^{\frac14-\theta} \|u\|_{\tilde X^{s,\theta}_{{\lambda},d}}$$ \[d\]

The reason we include the gradient is to have also bounds for $u_t$. Because of the frequency localization, if we drop the gradient the same bounds hold with one less power of ${\lambda}$.

In our estimates later on we also need to work with $X^{s,b}$ functions which are concentrated into a smaller modulation range. For this we introduce the additional norm $$\| u\|_{\tilde X^{s,\theta}_{{\lambda},<d}}^2 = \inf \left\{
\sum_{h=1}^d \|u_h\|_{\tilde X^{s,\theta}_{{\lambda},h}}^2;\ u = \sum_{h=1}^d u_h\right\}$$ If $d = {\lambda}$ we simply write $\tilde X^{s,\theta}_{{\lambda}}$. A simple argument leads to $$\|u\|^2_{X^{s,\theta}}\,=\,\inf \left \{ \sum_{{\lambda}=1}^\infty
 \|S_{{\lambda}} u_{{\lambda}}\|_{\tilde
X^{s,\theta}_{{\lambda}}}^2 ;\ u =\sum_{{\lambda}=1}^\infty S_{{\lambda}}
u_{{\lambda}}\right\}$$ We also have

a\) Assume that $\theta > \frac12$. Then $${\lambda}^{s-1} \|S_{\lambda}\nabla u\|_{L^\infty L^2} + {\lambda}^{s-\frac52} \|S_{\lambda}\nabla u\|_{L^2
  L^\infty} + {\lambda}^{s-3} \| S_{\lambda}\nabla u\|_{L^\infty} \lesssim  \|u\|_{{{\tilde{X}}}^{s,\theta}_{{\lambda},<d}}$$

b\) Assume that $\theta < \frac12$. Then $${\lambda}^{s-1} \|S_{\lambda}\nabla u\|_{L^\infty L^2} + {\lambda}^{s-\frac52} \|S_{\lambda}\nabla u\|_{L^2
  L^\infty} + {\lambda}^{s-3} \| S_{\lambda}\nabla u\|_{L^\infty} \lesssim
d^{\frac12-\theta} \|u\|_{\tilde X^{s,\theta}_{{\lambda},<d}}$$ \[lessd\]

In preparation for proving bilinear estimates for the $X^{s,\theta}$ spaces we first investigate which multiplications leave the ${{\tilde{X}}}^{s,\theta}_{{\lambda},d}$ space unchanged. For this we define the algebras $M_d$, $M_{<d}$ with the norms $$\|f\|_{M_d} = \| f\|_{L^\infty} 