Proposition: Ramp localization

At fixed smooth $\varphi$, the ghost-slot coefficient in (eq:Wlocal) is a smooth compact kernel on the ambient space, with support contained in

Proof. Transposition of the divergence-form coordinate prescription gives

where $Cr=Dr$. The advanced response is smooth and has compact support in the relevant closed-slab causal past. Multiplication by the ramp makes the kernel flat at $t=0$. Applying the differential operator ${\mathcal{R}}^{\ast }$ gives another smooth compact kernel.

Theorem thm:plateau says that its pairing with every compact ghost supported above $\epsilon$ is zero. Below $t=0$ both terms in (eq:Wlocal) vanish. The asserted support follows.