Proposition: Fixed-width regularity and dependence

At fixed $\epsilon >0$, the coefficients (eq:Ocoefficients) are continuous polynomial functionals on the stated smooth spaces. Their first- and second-order joint kernels are smooth and compactly supported. Through order ${\kappa }^2$, dependence on $h,k$ is contained in $Q_{\epsilon }$, and dependence on $\varphi$ is contained in $K_f$.

Proof. For a constant symmetric tensor basis $e_A$, put

This is a jointly smooth parameter family: parameter derivatives commute with the linear Cauchy problem, and differentiated local energy estimates give continuous dependence. The polarized second family

is jointly smooth by the same argument and the finite Leibniz rule.

For example, integrating the scalar derivative by parts gives the first kernel

The second coefficient similarly involves finite derivatives of smooth products of $\Lambda$ and $\Omega$, with factors $f(z)$, $g(y)$, and $g(y^{\prime })$. This proves smooth compact support. Pairings with these kernels are bounded by finite products of compact-open smooth seminorms, proving continuity.

A wave source affecting $z\in K_f$ lies in $D_f$. A convolution input affecting such a source lies within Euclidean distance $\epsilon$ and in $\mathrm{supp}g$. In a nested second-order term the earlier wave source remains in $D_f$ by causal transitivity. Each metric input is thickened only by its own single convolution. The scalar derivatives are evaluated on $K_f$.