Operators and the fixed-time response

Let $G_{m,R},G_{m,A}$ be the scalar Green operators for $P$. Their spatial Fourier time kernels are

For componentwise massless waves write $G_{r,R},G_{r,A}$, where $r$ indicates tensor rank, and use

Its time kernel is $S_w(t)=\sin (t|D|)/|D|$.

These formulas define the inverses on compact distributions by transposition. Smooth Cauchy solutions follow from Fourier evolution. Finite propagation follows from the wave energy identity. On finite time intervals, all these maps are continuous in the relevant compact-data smooth topologies.

Fix a real smooth free scalar

with compact smooth Cauchy data, and fix $f\in {C_c}^{\infty }((0,b)\times {\mathbb{R}}^3;\mathbb{R})$. Define the response by

and

The scalar is held fixed in this metric functional.

Put

Then $Pu=-f$, and self-adjointness of the spatial sine propagator gives

The relevant source region is compact: