Proposition: The finite-time smooth inverse

For every smooth source $F$, define

Then $G_{+}F$ is the unique smooth zero-data solution of ${\square }_{+}u=F$. It requires no spatial decay and agrees with the local compact-cutoff construction. Moreover,

is the corresponding zero-data inverse of ${\square }_{-}$. For spatially homogeneous $F(t)$,

Proof. Let $M_{r}F(s,\mathbf{x})$ denote the normalized spherical mean. For smooth $F$, the spherical-mean identity is

It follows by the divergence theorem, equivalently from ${{\partial }_r}^2{M}_{r}F+(2/r){\partial }_r{M}_{r}F={\Delta }_{\mathbf{x}}{M}_{r}F$, with its smooth value at $r=0$.

Differentiate

The first endpoint term vanishes. The second derivative has endpoint term $F(t,\mathbf{x})$, because ${\partial }_r(r{M}_{r}F){|}_{r=0}=F$. The spherical-mean identity gives

The position and first time derivative vanish at $0$. Smoothness follows by differentiation on the compact spheres and bounded time intervals. Lemma lem:C proves uniqueness.

For a compact observation set, the integration points in (eq:Kirchhoff) lie in a compact truncated causal past. Multiplication of $F$ by a smooth spatial cutoff equal to one on that region leaves the answer unchanged there. This proves the local-cutoff statement. The sign relation follows from ${\square }_{-}=-{\square }_{+}$. A homogeneous source has spherical mean $F$, which proves (eq:homogeneous-inverse).