Local metric control for the coordinate response

Choose once and for all

Let $h_{\theta }(t,x)=\theta (x)h(t,x)$, and let $w_{\theta }$ solve (eq:w) with this coefficient. Every backward causal cone from $K$ down to $t=0$ lies within $B_{R+T}$. Hence finite propagation gives

on an open neighborhood of $K$, including equality of all the derivatives used below. The extra unit in the cutoff radius provides that neighborhood.

Put $F_h=C(h_{\theta })$. It has a fixed compact spatial support, and the product rule gives

The coordinate solution obeys

and therefore

Its higher time derivatives are

Apply the spatial $H^2$ embedding to derivatives of total order at most three. The largest required spatial source norm is $H^5$, from three spatial derivatives of $w_{\theta }$. The other terms are covered by (eq:Fh). Thus

The same estimate holds on the fixed small neighborhood of $K$ used for the product derivatives.

The localization is essential: it permits arbitrary smooth nondecaying $h$ outside $E$, while the estimate uses only $H_6(h)$ on $E$. No global Sobolev assumption on $h$ is made.