with $(s,\theta)$ verifying (\[st\]). The last section of the paper is devoted to proving this bound. The argument is based on a multiscale trilinear wave packet decomposition for linear waves.

The $X^{s,\theta}$ spaces
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We first introduce Littlewood-Paley decompositions. As a general rule, all frequency localizations in the sequel are only with respect to the spatial variables. There is a single exception to this. Precisely, the coefficients $g^{ij}$ are truncated using space-time multipliers. In order for these truncations to work, we need for these coefficients to be defined globally in time. Hence we assume they have been extended to functions with similar properties in all of ${\mathbb R}^{n+1}$.

Let $\phi$ be a smooth function supported in $\{\frac12 \leq |\xi|
\leq 2 \}$ with the property that $$1 = \sum_{j=-\infty}^\infty \phi(2^{-j} \xi)$$ We consider a spatial Littlewood-Paley decomposition, $$1 = \sum_{{\lambda}=1}^\infty S_{{\lambda}}(D_x)$$ where for dyadic ${\lambda}> 1$ we have $$\qquad S_{{\lambda}}(\xi) = \phi\left(\frac{\xi}{{\lambda}}\right)$$ while $S_1$ incorporates the low frequency contribution in $\{|\xi|\leq 1\}$. Set $$S_{<{\lambda}} =  \sum_{\mu=1}^{\frac{\lambda}2} S_\mu$$ We will also use spatial multipliers ${{\tilde S}}_{\lambda}$ with slightly larger support, with $S_{\lambda}{{\tilde S}}_{\lambda}= S_{\lambda}$. We say that a function $u$ is localized at frequency ${\lambda}$ if its Fourier transform is supported 