Lemma: Infrared control

Every compact smooth spatial section belongs continuously to $\mathrm{dom}({B_{\beta }}^{-1/4})$ in its fixed-support test topology. Consequently (eq:W) defines a continuous positive covariance on compact spacetime tests.

Proof. With

the heat kernel is

It follows from the Friedrichs Bessel transform and Weber's integral Watson,CheungIBG. On a compact regular spatial set and for $s\ge 1$, the Bessel power series gives

For a compact spatial section $v$,

Use the contraction bound for $0<s<1$, and the displayed compact heat bound with ${\Vert v\Vert }_{L^1}^2$ for $s\ge 1$. The latter integrand is $O(s^{-2-{\nu }_{\ast }})$. The bounds are continuous in fixed-support test seminorms. Time integration then defines ${\mathcal{K}}_{\beta }$, and its Gram form proves positivity.