Lemma: Separate spectral estimates

The pair coefficient is Hilbert–Schmidt and the mixed coefficient is bounded. There are constants independent of the admitted masks such that

Proof. Write $\ell =t_1-t_0$,

Spatial integration by parts and one temporal integration by parts, retaining the initial endpoint $F(0,k)$, give

For $k=ϵp+\delta q$, transversality gives ${\mathsf{P}}_{\mathrm{TT}}(p)(q\otimes q)={\mathsf{P}}_{\mathrm{TT}}(p)(k\otimes k)$. After absorbing polynomial transfer factors into the arbitrary spatial decay,

For pair signs this implies, after increasing $n$,

The radial integral is finite. The gravitational low and high tails have squared integrals $O(R^{-2})$ and $O(R^{-1})$. The scalar high tail is $O(R^{-1})$, by splitting $|p|<1$ from $|p|\ge 1$ and transferring weights between $p,q,p+q$. This proves the first estimate.

For mixed signs, (eq:master) gives

and, on $|p|\ge 1$, also the bound with $⟨q{⟩}^{-1}$. The low part is Hilbert–Schmidt, with low-tail norm $O(R^{-1})$. The two high-tail Schur integrals are $O(R^{-1})$, using the appropriate $p$- or $q$-weighted bound; the remaining low-$p$, high-$q$ term decays faster. The Schur test proves the second estimate.