in the annulus $\{ \frac{\lambda}8 \leq |\xi| \leq 8{\lambda}\}$.

For the paradifferential type calculus we also need to truncate the coefficients of $\Box_g$ in frequency. Given $\Box_g$ in (\[eq\]) we define the modified operators $$\Box_{g_{<{\lambda}}} =(S_{<{\lambda}}(D_x,D_t) g^{\alpha\beta}) \partial_\alpha
\partial_\beta$$ In the sequel we omit the space and time variables in our function space notations, i.e. $L^p:= L^p_{x,t}$, $L^2 H^s:= L^2_t H^s_x$, $L^p
L^q:= L^p_t L^q_x$, etc. We are ready now to define our spaces:

Let $\theta\in (0,1)$ and $s \in {\mathbb R}$. Then $X^{s,\theta}$ is the space of functions $u \in L^2(-1,1; H^s({\mathbb R}^n))$ for which the following norm is finite: $$\|u\|^2_{X^{s,\theta}}\,=\,\inf \left \{ \sum_{{\lambda}=1}^\infty
\sum_{d=1}^\lambda \|u_{\lambda,d}\|^2_{X_{{\lambda},d}^{s,\theta}} ;\ u
=\sum_{{\lambda}=1}^\infty\sum_{d=1}^{{\lambda}}S_{{\lambda}}
u_{{\lambda},d}\right\}
\label{tp}$$ where ${\lambda}$, $d$ take dyadic values and $$\|u_{{\lambda},d}\|_{X_{{\lambda},d}^{s,\theta}}^2 =
{\lambda}^{2s}d^{2\theta}\|u_{{\lambda},d}\|^2_{L^2} +
{\lambda}^{2s-2}d^{2\theta-2}\|\Box_{g_{<{\sqrt{\lambda}}}}u_{{\lambda},d}\|^2_{L^2}
\label{cl}$$

We also define the space $X^{s-1,\theta-1}$ of functions for which the following norm is finite: $$\begin{split}
\| f\|_{X^{s-1,\theta-1}}^2 = \inf \left\{ \|f_0\|_{L^2 H^{s-1}}^2
+ \sum_{{\lambda}=1}^\infty\right.& \sum_{d=1}^\lambda
\|f_{\lambda,d}\|^2_{X_{{\lambda},d}^{s,\theta}} ;\ \\ & \left. f =f_0+
\sum_{{\lambda}=1}^\infty \sum_{d=1}^{{\lambda}}\Box_{g_{<{\sqrt{\lambda}}}} S_{{\lambda}}
f_{{\lambda},d} \right\}
\end{split}
\label{tn}$$ \[xst\]

Intuitively $d$ stands for the modulation of the $u_{{\lambda},d}$ piece. Indeed, in the constant coefficient case one can easily see that $u_{\lambda,d}$ mainly contributes to $u$ in the region where $||\tau|-|\xi|| \approx d$. The 