Lemma: Compact mixed equation images

Let $v(x,y)$ be compactly supported with $\mathrm{WF}(v)\subset Z_2$, scalar-density valued in $x$ and covariant tensor valued in $y$. Then the mixed polynomial with kernel

is admitted and belongs to $\mathcal{J}$.

Every element of $\mathcal{J}$ has zero normal symbol on all smooth free pairs $P_{m}u=0,{\mathsf{P}}_{\mathrm{gr}}r=0$. No converse is used.

Proof. Differentiation preserves support and $Z_2$, so the equation image is admitted.

For a finite sum of smooth tensor products $v={\sum }_i{a}_i(x)b_i(y)$, its polynomial is

The cross blocks of $H$ are zero, so this is also the ordinary product. It is in the two-sided equation ideal.

A general compact $v$ is approximated by smooth kernels in the admissible microlocal topology on one slightly larger compact support stage. One may first regularize and then approximate each smooth kernel by finite sums of compact tensor-product tests. Their smooth convergence implies convergence in that microlocal topology. Differentiation and the kernel-to-polynomial map are continuous, and $\mathcal{J}$ is closed on these stages. Thus the same conclusion holds for $v$ Hormander,BDH.

For an equation generator, all Wick contractions with another functional vanish by the bisolution property of $H$. Its product normal symbol therefore vanishes on free pairs. This persists under the specified closure, since pairing compact kernels with fixed smooth fields is continuous.