Lemma: Infrared finiteness

For every $v\in {C_c}^{\infty }(N;L_{\beta })$, ${B_F}^{-1/4}v\in {\mathcal{H}}_{\beta }$. On a fixed compact support this map is continuous in the compact smooth test topology. There is no $L^2$ zero eigenfunction.

Proof. The spectral theorem gives

For $s\le 1$, contraction bounds the integrand by $s^{-1/2}{\Vert v\Vert }^2$. For $s\ge 1$, Proposition prop:heat bounds it by

Both integrals converge and give continuity. Zero form energy would imply a locally parallel section; no nonzero parallel section is square integrable on $N$.