The compact test and its stability

Write $j=\chi h$. In expanded form,

Taking at most two further spacetime derivatives uses at most three derivatives of $j$ and four derivatives of $u_f$. Since $j$ has support in $L\subset E$,

Taking at most two derivatives of ${\partial }_{\mu }(fw[h{]}^{\mu })$ uses at most three derivatives of $f,w[h]$ on the support of $f$. Equation (eq:w-C3) therefore gives

Adding these inequalities proves

The mass-shell estimate completes Theorem thm:estimate.

The same proof, without a coordinate term, gives for any smooth compact tensor $j$ on a common source support

This is the estimate used for switch differences and for the outside-plateau defect. The support geometry, mass, and time slab remain fixed. Constants are not claimed uniform in support growth, $T\to \infty$, or $m\downarrow 0$.