Localizing the spatial estimates

Choose a compact neighbourhood of $K$ still inside the retained observation region. Its truncated causal past is compact. Multiply $h,k$ by one fixed spatial cutoff equal to one near that entire past.

Finite propagation leaves $w_{\epsilon },v_{\epsilon }$ unchanged on the observation neighbourhood, including their nested second-order dependence. After this localization all sources and solutions have uniformly bounded spatial support on the fixed slab.

We may therefore use

and return afterwards to the original smooth, not necessarily decaying backgrounds. For spatial smooth control we use

This follows directly from Fourier inversion and Cauchy–Schwarz, since $N+2>N+3/2$.