Lemma: Cauchy identities

The zero-data wave problem has a unique smooth solution for every smooth source on a neighborhood of $M_8$, locally interpreted by finite propagation. It satisfies

For smooth $a$,

where ${\mathsf{H}}_0[a]$ is the homogeneous solution with the Cauchy data of $a$.

Proof. For spatially compact smooth sources, differentiating (eq:Ezero) proves the equation and initial data. The wave energy estimate, after spatial differentiation, bounds the solution on finite slabs by the source and initial-data Sobolev norms. Applied to a difference, it gives uniqueness. Sobolev embedding gives smoothness and continuous dependence.

On a compact observation region, choose compact spatial cutoffs equal to one on its truncated causal past. Finite propagation makes the resulting solutions agree on overlaps. This defines the solution for general smooth sources and preserves uniqueness. These are the standard completion Cauchy arguments BGP.

Finally, $a-{\mathsf{E}}_0{\square }_{\eta }a$ is homogeneous with the Cauchy data of $a$, proving (eq:leftinverse).