Lemma: Oriented Hadamard smoothing

If $z\in {\mathcal{E}}^{\prime }(M^k)$ is microcausal, then

is a smooth full-shell multisolution. The statement holds for $H^{\prime }$ and for the transposed line orientation as well.

Proof. The input projection is proper on $\mathrm{supp}z$ over every compact output set. The tensor wavefront bound for $H^{\otimes k}$ has entries $(p_j,-p_j)$ with all nonzero $p_j$ in $V_{\sigma }$; zero pairs cover smooth factors. An undefined product would require a nonzero input tuple of $z$ entirely in ${V_{\sigma }}^k$, which microcausality excludes. An output wavefront point would require the same type of input tuple when the input covectors are canceled for pushforward. It is therefore excluded too: $\mathrm{WF}(S)=\varnothing$. The bisolution equations imply the shell equations in each output slot. Adding smooth $d$ does not alter the argument. Transposition reverses the common orientation, and microcausality excludes both orientations.