Elementary quantitative tail bounds

Merely requiring $F_{\delta }(\mathbf{y})\to 0$ at infinity does not imply either estimate. For example,

has a divergent absolute Coulomb integral when $0<q\le 2$. At $q=2$, the divergence is logarithmic. Cancellations can improve particular integrals, but require their own proof.

The classical sources in Proposition prop:rods are compact, so their classical integrals exist. Quantum correlation functions are not compact sources. Their required tail bounds have not been inferred from the classical argument.