Spatial localization

For a fixed compact observation region, its backward causal region in the slab is compact. Multiply $h,k$ by a fixed smooth cutoff equal to one on a neighbourhood of that causal region. Finite propagation, including the nested second coordinate equation, leaves the coordinates on the observation region unchanged.

After this localization the backgrounds and coordinate coefficients have uniformly bounded spatial support on $[0,T]$. We may use spatial Sobolev norms and then return to the original backgrounds. For $\Vert \cdot {\Vert }_r=\Vert \cdot {\Vert }_{H^r({\mathbb{R}}^3)}$, Fourier inversion and Cauchy–Schwarz give

the relevant high-frequency integral converges because $2(N+2)-2N=4>3$.