Geometric setting

Let

Write $x=(t,\mathbf{x})$, $y=(s,\mathbf{y})$, and $x^{\prime }=(t^{\prime },{\mathbf{x}}^{\prime })$. Set

The operator ${\mathsf{E}}_0$ is the zero-value, zero-first-time-derivative Cauchy inverse at $t=0$. Write $w=Lh$. Configurations $\psi =(h,\varphi )$ and configuration directions are smooth and off shell.

The selected functionals are

Repeated bundle indices are summed.

We use the ordinary smooth mapping properties of these fixed Cauchy and Green operators. Smooth compact sources give smooth solutions; restriction to a fixed compact set is continuous in the source; and finite propagation over bounded time intervals gives compact propagation regions. In particular, applying $L$ in one variable of a smooth joint function is continuous in the smooth topology, including derivatives in its parameter variables. These are smooth linear PDE properties, not singular-extension assumptions.