Ambient support and the initial surface

Let

The coefficient support and (eq:B-support) imply

Both sets on the right are compact in ambient $M$. Indeed, if $K_f$ lies in $\{t\le b,|\mathbf{x}|\le R\}$, then

The causal past of a compact set is closed in Minkowski space, so $C_{\delta }$, and hence $C_0$, is compact. The set $O_{f,g}$ is compact by its definition. The right side of (eq:t-support) is the image of $C_0\times O_{f,g}$ under $(y,x)\mapsto (y,y,x)$.

The support may reach $t=0$. Ambient compactness does not require support bounded away from that surface. The following statements remain separate:

• $t_{\mathrm{D}}$ is defined on ambient $M^3$;

• its support is compact there;

• its regularity or singularity at $t=0$ has not been determined.

Flatness of the switch guarantees smooth coefficient functions; it is not a theorem removing boundary singularities of the resulting kernel.