Lemma: Pair and mixed transfer bounds

At each fixed finite $\Lambda$, $g_{\Lambda }$ is a symmetric Hilbert–Schmidt tensor and $h_{\Lambda }$ is bounded self-adjoint. More precisely, on the cutoff support,

and

No separate Newton infrared mask is required.

Proof. For the pair sector, at $q=-p$,

The first expression also has vanishing first derivative there. On bounded momentum sets, smoothness of the massive dispersion relation therefore gives ${C_{00}}^{++}=O(|p+q{|}^2)$ and ${C_{0i}}^{++}=O(|p+q|)$. Thus the initial Newton amplitudes are respectively $O(1)$ and $O(|p+q{|}^{-1})$, and $B_{00}=O(1)$. The wave factors are bounded on the finite time support, proving the pair bound. Its square is integrable in the three-dimensional transfer variable.

In the mixed sector, the bounded numerators give at worst $O(|p-q{|}^{-2})$. The sine-over-$r$ factors are bounded on compact time intervals, so no stronger singularity occurs. On bounded momentum support,

and likewise with $p,q$ interchanged. The Schur test gives boundedness. Reality and polarization give the stated symmetries.