Lemma: Zero-Cauchy inverses

For $m\ge 0$, define

There is a unique smooth solution of $P_{m}u=F$ on $\mathcal{D}$ with prescribed smooth Cauchy data. The zero-data inverse ${\mathsf{E}}_m$ is locally represented by

It has finite propagation and is continuous in local smooth topologies on compact truncated cones.

If $u$ has zero Cauchy data, then

For general $u$,

Proof. For compact spatial data and sources, spatial Fourier transformation reduces the equation to forced oscillators. Solving them gives (eq:Em) and the displayed homogeneous solution.

Finite propagation and smooth uniqueness follow from the energy identity on a truncated backward cone: the outward lateral flux is nonnegative, and zero data and zero source give zero energy. Differentiated energy estimates and local Sobolev estimates give smooth dependence.

For a compact target in $\mathcal{D}$, cut off and extend the required data and source from its truncated past to compact spatial data. Different extensions agree on the target by the preceding uniqueness argument. Exhaustion by compact targets defines the smooth local solution. Subtracting $H_m[u]$ proves (eq:leftinverse) and (eq:homogeneous-subtraction).