it is convenient to work with a selfadjoint operator. Thus we consider the selfadjoint counterpart $\tilde{\Box}_g$ of $\Box_g$ $$\tilde{\Box}_g = \partial_i g^{ij} \partial_j$$ Then for $v$ localized at frequency $\lambda$ we commute and estimate the frequency localized difference $$\| \tilde \Box_{g_{<{\sqrt{\lambda}}}} v -  \Box_{g_{<{\sqrt{\lambda}}}} v\|_{L^2}
\lesssim \|\nabla v\|_{L^2}$$ This leads directly to

One can replace the $\Box_{g_{<{\sqrt{\lambda}}}} $ operator in the definition of $X^{s,\theta}$ and $X^{s-1,\theta-1}$ by the similar operator in divergence form $\tilde\Box_{g_{<{\sqrt{\lambda}}}} $. \[self\]

As a consequence of the second part of we have

The following embedding holds for $-1 < \theta < 0$: $$X^{s,\theta} \subset L^2 H^{s+\theta}$$ \[sw\]

Another use of this is to establish energy estimates. A direct application of energy estimates for the wave equation yields the bound $$\| \nabla v\|_{L^\infty L^2}^2 \lesssim \|\nabla v\|_{L^2}^2 +
\|\nabla v\|_{L^2} \| \Box_{g} v\|_{L^2}$$ This leads to $${\lambda}^{s-1} d^{\theta-\frac12} \| \nabla S_{\lambda}v\|_{L^\infty L^2}
\lesssim \| v\|_{\tilde{X}^{s,\theta}_{{\lambda},d}} \label{litwo}$$ Going back to Definition \[xst\], this implies

Assume that $\theta > \frac12$. Then $$\|u\|_{L^\infty H^s} + \|u_t\|_{L^\infty H^{s-1}} \lesssim \|u \|_{X^{s,\theta}}$$ \[trace\]

To prove the estimates and in the context of the $X^{s,\theta}$ spaces we need to switch from the frequency truncated coefficients to the full coefficients $g^{ij}$. The tool needed 