Restricted extensions: what agreement does and does not mean

Set

viewed by smooth ambient zero extension, and let $\mathcal{H}$ consist of ambient smooth homogeneous sources $q(t)$ that vanish for $t\le 0$. Their intersection is zero: a nonzero spatially constant source cannot be compactly supported in space.

If $E_{0,\mathrm{int}}^{+}$ is the compact-interior zero-data inverse, define

The direct-sum decomposition makes this well defined. Proposition prop:local-inverse and Lemma lem:C show that it is the restriction of $G_{+}$ to ${\mathcal{D}}_{\mathrm{int}}\oplus \mathcal{H}$. This reproves the mathematical content of the supplied restricted extension CheungXVID (:restricted-extension).