Lemma: Smooth zero-data inverse

For $\mu \ge 0$, smooth source $s$ on the slab and smooth initial data on $\Sigma$, the equation

has a unique smooth solution. Its value on a compact set depends only on the source and data in the corresponding truncated flat causal past. For zero data denote it by ${\mathsf{E}}_{\mu }s$.

On compact sets this solution map is continuous in the smooth topologies. No spatial decay is required.

Proof. For compactly supported smooth data and source, spatial Fourier transformation gives the solution using

The quotient has value $t-s^{\prime }$ at zero frequency. Differentiation of the Fourier formula gives smoothness. The usual differentiated energy estimate, or the same Fourier formula in Sobolev norms followed by Sobolev embedding, gives finite-seminorm bounds on compact time intervals.

For finite propagation, apply the energy identity with energy density

to a truncated backward cone. Its lateral flux has the required nonnegative sign. Zero source and data on that cone imply zero solution at its vertex. Adding an $u^2$ term and applying Gronwall also covers the massless low-frequency estimate.

For arbitrary smooth data, fix a compact target set, cut off the source and data on a neighbourhood of its compact truncated past, and use the compact-data solution. Finite propagation makes the result independent of the cutoff there. Such solutions agree on overlaps, giving a global smooth solution on the slab. The same argument proves uniqueness and local continuity. This is the standard Cauchy mechanism used on the declared completion, not on the non-globally hyperbolic host BGP.