Lemma: Smoothness after compact kernel smearing

Let $k(y,y^{\prime })$ be a distribution with $\mathrm{WF}(k)\subset {\Sigma }_2$, where

For a compact smooth $a(y)$, the properly supported pairing $\int k(y,y^{\prime })a(y)\mathrm{d}y$ is smooth in $y^{\prime }$.

Proof. The pushforward wavefront bound can only produce an output covector ${\eta }^{\prime }$ from a kernel covector with $\eta =0$. The sum-zero condition then forces ${\eta }^{\prime }=0$, which is not a wavefront covector.