&\  +    \lambda^{2(1-s)} d^{2(1-\theta)} (\|h_{\lambda,2d}\|_{L^2}^2
+\|h_{\lambda,d}\|_{L^2}^2)
\end{split}$$ Finally, for the last term we have $$\begin{split}
 \| h_{\lambda,\lambda}\|_{X_{\lambda,\lambda}^{1-s,1-\theta}}^2
\lesssim  &\ \lambda^{2(1-s)} \lambda^{2(1-\theta)} \| h_{\lambda,\lambda}\|_{L^2}^2
+ \lambda^{-2s} \lambda^{-2\theta}  \| \tilde \Box_{g_{<
\sqrt\lambda}} h_{\lambda,\lambda}\|_{L^2}^2
\\ = &\ \lambda^{2(1-s)} \lambda^{2(1-\theta)} \| h_{\lambda,\lambda}\|_{L^2}^2
+ \lambda^{-2s} \lambda^{-2\theta}  \| S_\lambda u -  f_{\lambda,\lambda}\|_{L^2}^2
\\ \lesssim&\ \lambda^{2(1-s)} \lambda^{2(1-\theta)} \|
h_{\lambda,\lambda}\|_{L^2}^2 + \lambda^{-2s} \lambda^{-2\theta}  \| f_{\lambda,\lambda}\|_{L^2}^2
+ \| S_\lambda u\|_{L^2 H^{-s-\theta}}^2
\end{split}$$ The proof is concluded.

Strichartz estimates and applications. {#secse}
======================================

The Strichartz estimates for the variable coefficient wave equation, as proved in [@MR1887639], have the form:

(Tataru [@MR1887639]) Assume that the coefficients $g^{ij}$ of $\Box_g$ satisfy ${\partial}^2 g^{ij} \in L^1L^\infty$. Then the solutions to the wave equation in $n+1$ dimensions satisfy the bounds $$\| D^{\sigma} \nabla u\|_{L^p L^q} \lesssim \|u(0)\|_{H^1} + \|u_t(0)\|_{L^2} +
\|\Box_g u\|_{L^1 L^2}
\label{se}$$ where $$\sigma = -\frac{n}2 + \frac{1}p +\frac{n}q, \qquad \frac{2}p +\frac{n-1}q
\leq \frac{n-1}2, \qquad 2 \leq p \leq \infty, \ \ 2 \leq q < \infty
\label{pq}$$

Applying this bound on an interval $I$ of size $\epsilon^2$ we obtain by Cauchy-Schwartz $$\| D^{\sigma} \nabla u\|_{L^p(I; L^q)} \lesssim \frac{1}\epsilon \|u\|_{H^1(I
  \times {\mathbb R}^n)} + \epsilon \|\Box_g u\|_{L^2(I \times {\mathbb R}^n)}, \qquad
\epsilon \leq 1$$ Summing up over small intervals this extends to 