Lemma: Compact-profile regularity

For real $f\in {C_c}^{\infty }({\mathbb{R}}^2)$, its coefficient $k(f)$ has finite covariance norm and obeys

The instantaneous coefficient

is smooth in the covariance norm and has compact temporal support.

Proof. Here are the specific s-wave threshold facts needed for the proof. The zero-energy Jost solutions are

with Wronskian $-1$. Also

The Jost Volterra equations are therefore continuous at zero. On each compact spatial interval they give

for suitable finite $C_X$ and ${\omega }_c>0$. At zero use the Volterra kernel bound $|\sin (\omega z)/\omega |\le |z|$ and the nonzero Wronskian; above ${\omega }_c$ use $\exp (\Vert V{\Vert }_1/{\omega }_c)$ and flux conservation.

Thus a compact spacetime test has

At high frequency, one time integration by parts gives $O({\omega }^{-3/2})$. Further time integrations are available. Since $n_{\mathrm{H}}\le 1/(\beta \omega )$, these bounds prove all three stated integrability claims, including the extra ${\mathsf{C}}^2$ in the last one.

For the instantaneous coefficient,

The threshold estimate controls its low-frequency covariance. Time derivatives introduce finite powers of $\omega$; compact spatial smoothness and spectral unitarity control those powers by applying powers of $\mathsf{L}$ to $f(t,\cdot )$. This proves smoothness of every order.