Lemma: Cauchy properties

For every smooth source on a neighborhood of $M_T$, the zero-data wave problem has a unique smooth solution. It depends continuously on the source on each compact truncated causal region. In particular,

For smooth $a$,

where $H_0[a]$ is the homogeneous wave solution with the Cauchy data of $a$. Thus ${\mathsf{E}}_0{\square }_{\eta }a=a$ when those data vanish.

Proof. For spatially compact sources, differentiation of (eq:Ezero) gives the equation and the data. On a finite time interval, differentiated energy estimates give

The same estimates give uniqueness. Increasing $s$ and using Sobolev embedding give smooth dependence. Local energy estimates give finite propagation, allowing compatible compact cutoffs for general smooth sources. These arguments are the standard smooth wave Cauchy theory BGP.

Finally, $a-{\mathsf{E}}_0{\square }_{\eta }a$ solves the homogeneous equation with the data of $a$, proving the last assertion.