Cauchy operators

Write ${\mathsf{E}}_{\mu }s$ for the solution with zero data; ${\mathsf{E}}_0$ is the wave case and ${\mathsf{E}}_m$ the Klein–Gordon case. In spatial Fourier variables, with ${\Omega }_{\mu }=\sqrt{-\Delta +{\mu }^2}$,

interpreted locally by finite propagation; the sine quotient has its continuous value $t-s^{\prime }$ at zero frequency when $\mu =0$. For a compactly supported smooth $a$ on $M$, ${\mathsf{E}}_{\mathrm{A}}a$ denotes the advanced solution of $(\square +{\mu }^2)b=a$, smooth and supported in $J^{-}(\mathrm{supp}a)$ BGP (Chapter 3).