operator $\Delta$, see Klainerman-Machedon [@MR1420552]: $$\| u \|_{X^{s,\theta}} = \| (1+|\xi|^2)^{\frac s2}\cdot
(1+||\tau|-|\xi||^2)^{\frac \theta2}\cdot |\hat{u}(\tau,\xi)|
\|_{L^2}
\label{standardxs}$$ where $\hat u=\hat{u}(\tau,\xi)$ is the space-time Fourier transform of function $u=u(t,x)$. Finally, in the most difficult case, $n\geq 5$, this was proved by Tataru [@MR1739207], using a suitable modification of the $X^{s,\theta}$ spaces, needed in order to control the interaction of high and low frequencies in the multiplicative estimates.

For the quasilinear problem (\[qua\]) the sharp result is only known to hold in dimensions $n=2,3$. This was proved by Smith-Tataru [@MR2178963] (see also Lindblad’s counterexample [@MR1666844]). The argument there still requires the use of Strichartz estimates. These are derived from a wave packet parametrix construction for a wave equation with very rough coefficients, which in turn is obtained via a very delicate analysis of the Hamilton flow. A different proof of this result in the special case of the Einstein vacuum equation was independently obtained by Klainerman-Rodnianski [@MR1885093], [@MR2180401], [@MR2052472]. In dimensions $n\geq 4$ it is still unclear which is the optimal threshold, the best results so far being contained in the above mentioned paper of Smith-Tataru [@MR2178963] and in an earlier one, Tataru [@MR1887639]: $$\aligned &n=4,5\quad \quad s>\frac{n}{2}+\frac{1}{2}\\
&n\geq 6\qquad \quad s>\frac{n}{2}+\frac{2}{3}
