Proposition: The mixed quadratic test

The finite contraction formulas for $B_f{\star }_H{Q}_q$ and $Q_q{\star }_H{B}_f$ define compact distributional polynomial kernels. In particular, the double contraction

is a well-defined distribution.

Proof. Each component of $H_{wh}$ has a direct and a reflected Fourier representation of the form illustrated in (eq:K), with polynomial tensor numerators. In either branch, every nonzero wavefront covector in the $y$-slot has the same temporal orientation as the second slot of ${\Delta }_m^{+}$; it is nonzero and null. This follows directly from the phase $irs-i\mathbf{p}\cdot \mathbf{y}$. The direct branch has the usual null relation and the reflected branch has the relation obtained by reflecting the first time coordinate.

The two factors in the displayed product therefore cannot have opposite total covectors: their $y$-components already have the same orientation in a pointed cone. The multiplication criterion applies.

A single $h$-contraction leaves a distribution multiplied by the smooth classical scalar factors. A single scalar contraction leaves a differentiated Hadamard distribution multiplied by the smooth factors $w[h](x)h(y)$. As in Proposition prop:self, continuity in the smooth configurations and the kernel theorem give the remaining compact polynomial kernels. The reverse product has the reversed, but again common, orientation.