Theorem: Correct supports and microcausal admission

All three kernels are compact distributions, with

Before normalized scalar symmetrization, the source points of each cubic agree: $y=y^{\prime }$.

Their wavefronts satisfy

Consequently the three tree coefficients belong to $\mathcal{W}$. All their external points lie in $J_{\eta }^{-}(K_f)$.

Proof. The distribution $g(y)N_0(x,y)$ is compactly supported in $C\times S$. The distribution $\sigma (y)N_0(x,y)$ is compactly supported in $C\times D_f$: its source time is bounded below by zero, and $y⪯x$ with $x\in C$. This argument includes the support closure at $t=0$; smooth flatness of $\sigma$ introduces no surface distribution.

The smooth multipliers and differential operators in (eq:treekernels) preserve $Z_2$. Lemma lem:sumzero(b) handles the external coincidence of the two source fields and gives $Z_3$. It retains conormal covectors of that coincidence. It is not an appeal to a two-point chain as a substitute for the cubic test. Differential operators and permutations do not enlarge the stated supports.

Finally $Z_n$ avoids the two co-oriented causal cones, so these compact kernels meet the defining microcausal criterion FR,Hormander. Their partially contracted derivatives meet the same criterion.