$C=C(\|u\|_{L^\infty})$ is a constant that depends solely on $\|u\|_{L^\infty}$. In the flat case (\[sem\]), for dimension $n=4$, one can make this argument work by choosing $$X\,=\,X^{s,\theta}\qquad Y\,=\,X^{s-1,\theta-1}$$ with $$s\,=\,\theta\,+\frac 32\qquad \theta\,>\,\frac 34 \label{st}$$

For our problem the challenge is twofold: first we need to find suitable variable coefficient versions for the $X^{s,\theta}$ spaces and then, in this new context, prove the corresponding estimates (\[h\])-(\[n\]).

Such spaces were previously introduced by Tataru [@MR1391526], where they are used in the context of a unique continuation problem. There, for a hyperbolic operator $P$ one defines $$X^{s,0}\,=\,H^s,\quad X^{s,1}\,=\,\{u\in H^s| Pu\in H^{s-1}\}$$ Then all the other spaces are defined through interpolation and duality.

In this article we choose to follow a different path based on dyadic decompositions with respect to the spatial frequency and the distance to the characteristic cone. Likely one should be able to prove that the two approaches are equivalent, but we choose not to pursue this here.

Our article is structured as follows. In the next section we define the $X^{s,\theta}$ spaces and prove that they satisfy the linear estimates , . Our definition of the $X^{s,\theta}$ is slightly different from the standard one in the constant coefficient case. Precisely, in 