the constant coefficient case our definition gives $$\| u \|_{X^{s,\theta}} \approx  \| (1+|\xi|^2)^{\frac s2}
(1+||\tau|-|\xi||^2)^{\frac \theta2}\cdot \hat{u}(\tau,\xi)
\|_{L^2} + \|\Box u\|_{L^2_t H^{s+\theta-2}_x}, \quad 0 < \theta <
1 \label{newxs}$$ and one can see that the second term above alters the behavior at high modulations $|\tau| \gg |\xi|$. Correspondingly, for negative $\theta$ we have $$\| u \|_{X^{s,\theta}} \approx  \| (1+|\xi|^2)^{\frac s2}
(1+||\tau|-|\xi||^2)^{\frac \theta2}\cdot \hat{u}(\tau,\xi)
\|_{L^2} + \| u\|_{L^2_t H^{s+\theta}_x}, \quad -1 < \theta < 0
\label{newxs1}$$ This change is consistent with scaling and simplifies somewhat the study of high modulation interactions.

In Section \[secse\] we discuss the Strichartz estimates for $\Box_g$, which translate into embeddings for the $X^{s,\theta}$ spaces. These turn out to suffice for the proof of the algebra properties - and for the high-high frequency interactions in .

The difficult part is to study the high-low frequency interactions in . For this we first take advantage of the duality relation $$(X^{s,\theta}+ L^2 H^{s+\theta})'\,=\,X^{-s,-\theta} \label{dual}
\qquad s \in {\mathbb R}, \quad 0 < \theta < \frac 12$$ This is consistent with and . Using this duality, after factoring out high modulation interactions, the bound (\[n\]) is transformed into the trilinear estimate: $$\left|\int u\cdot v\cdot w\,\,dx \,dt\right|\lesssim \|u\|_{X^{1-s,1-\theta}}
\|v\|_{X^{s-1,\theta}} \|w\|_{X^{s-1,\theta}}
\label{trilin}$$ 