regularity of $g$ together with the condition $g^{00} = 1$. Then we need the fixed time bounds $$\| g_{<{\sqrt{{\lambda}}}} {\partial}_x {\partial}_t v\|_{H^{-2}+{\lambda}^2 L^2} \lesssim
\|{\partial}_t v\|_{H^{-1} + {\lambda}L^2}$$ $$\| g_{<{\sqrt{{\lambda}}}} {\partial}_x^2 v\|_{ H^{-2}+{\lambda}^2 L^2} \lesssim
\|v\|_{L^2}$$ They are similar, so we only discuss the second one. We write $$g_{<{\sqrt{{\lambda}}}} {\partial}_x^2 v = {\partial}_x^2  (g_{<{\sqrt{{\lambda}}}} v) - 2{\partial}_x
 ({\partial}_x g_{<{\sqrt{{\lambda}}}} v) +  {\partial}_x^2 g_{<{\sqrt{{\lambda}}}} v$$ and use the uniform bounds $$|g_{<{\sqrt{{\lambda}}}}| \lesssim 1, \qquad |{\partial}_x g_{<{\sqrt{{\lambda}}}}|
\lesssim {\lambda},
 \qquad |{\partial}_x^2 g_{<{\sqrt{{\lambda}}}}| \lesssim {\lambda}^2$$ This concludes the proof of .

The first term in is directly bounded using . For the second it suffices to prove the commutator estimate $$\|[\Box_{g_{<{\sqrt{\lambda}}}},S_{\lambda}] v\|_{L^2} \lesssim {\lambda}\|v\|_{L^2}+
\|{\partial}_t v\|_{ L^2} \label{smallgcom}$$ We have $$[\Box_{g_{<{\sqrt{\lambda}}}},S_{\lambda}] = [g_{<{\sqrt{{\lambda}}}},S_{\lambda}] {\partial}_t {\partial}_x +
[g_{<{\sqrt{{\lambda}}}},S_{\lambda}] {\partial}_x^2$$ and the commutators are localized at frequency ${\lambda}$ so the spatial derivatives only contribute factors of ${\lambda}$. Hence follows from the standard commutator estimate $$\|[g_{<{\sqrt{{\lambda}}}},S_{\lambda}]\|_{L^2 \to L^2} \lesssim {\lambda}^{-1}
\|\nabla g\|_{L^\infty}$$

Applying the above Lemma with $S_{\lambda}$ replaced by ${{\tilde S}}_{\lambda}$ we obtain

One can replace the $X^{s,\theta}_{{\lambda},d}$ norm in the definition of $X^{s,\theta}$ and $X^{s-1,\theta-1}$ by the norm $$\| v\|_{\tilde{X}^{s,\theta}_{{\lambda},d}} = {\lambda}^{s-1}d^{\theta}\|\nabla
v\|_{L^2} + {\lambda}^{s-1}d^{\theta-1}\|\Box_{g_{<{\sqrt{\lambda}}}} v\|_{L^2}$$ \[tx\]

For the proof of the duality relation 