and the index $0$ stands for the time variable. To insure hyperbolicity we assume that the matrix $g^{ij}$ has signature $(1,n)$ and the time level sets $x_0=const$ are space-like, i.e. $g^{00}>0$. In effect to simplify some of the computations we make the harmless assumption $g^{00}=1$.

Finally, we consider a corresponding quasilinear equation $$\Box_{g(u)} u\,=\,\Gamma(u)(\nabla u)^2
\label{qua}$$ with similar assumptions on the matrix $g$.

In all three cases we are interested in the local well-posedness of the Cauchy problem in Sobolev spaces $H^s({\mathbb R}^n)\times H^{s-1}({\mathbb R}^n)$ with initial data $$u(0,x)\,=\,u_0(x),\qquad \partial_t u(0,x)\,=\,u_1(x)
\label{id}$$ The first equation is the best understood so far, and is known to be locally well-posed for $s$ in the range $$s>\max\{\frac{n}{2},\frac{n+5}{4}\}$$ This range is sharp. The $\frac{n}{2}$ obstruction comes from scaling, while the $\frac{n+5}{4}$ is related to concentration along light rays, see Lindblad [@MR1375301]. The proof of the positive result is fairly straightforward in dimension $2+1$ and $3+1$, where it suffices to rely on the Strichartz estimates. In $4+1$ dimensions this no longer works and one needs to use instead the $X^{s,\theta}$ spaces, see Foschi-Klainerman [@MR1755116]. These are multiplier weighted $L^2$ spaces associated to the wave operator as the Sobolev spaces $H^s$ are connected to the Laplace 