\endaligned$$ In the same direction but somewhat closer in spirit to the present paper is Bahouri and Chemin’s work [@MR2082388; @MR2003418]. The equation considered there is still quasilinear, but the main estimates are frequency localized versions of the Strichartz estimates for the wave equation on a rough background.

As an intermediate step toward understanding the higher dimensional quasilinear problem, we consider here the semilinear problem on a curved background and we prove the sharp result:

Let $n=4$ and assume that the coefficients $g^{ij}$ satisfy $\partial^2 g \in L^2 L^\infty$. Then the Cauchy problem , is locally well-posed in $H^s\times
H^{s-1}$ for $s>\frac{9}{4}$. \[main\]

Here well-posedness is understood in the strongest sense, i.e. the solutions have Lipschitz dependence on the initial data and they exist on a time interval which only depends on the size of the initial data.

One contribution of the present paper is to introduce variable coefficient versions of the $X^{s,b}$ spaces, study their properties and obtain the corresponding Strichartz type embeddings. However, the main novelty, contained in the last two sections, is a new method, based on a trilinear wave packet decomposition, to prove a key trilinear bound which cannot be obtained directly from the Strichartz estimates.

