Linear generators and insertion–linear products

Independently smeared Cauchy fields are admitted using (eq:Cauchymatrix). Their pairings with ordinary spacetime smearings are well defined. This does not define coincident products of boundary fields.

For a compact smooth graviton test tensor $a$, write

Since $H_{hh}a$ is smooth,

Likewise, for $u\in {C_c}^{\infty }(M)$,

The reverse products use the transposed covariance. All displayed coefficients are compact distributional polynomial functionals with the causal support inherited from Proposition prop:reconstruction.