Lemma: Sum-zero operations actually needed

The following operations are defined:

1. differentiation and multiplication by smooth functions preserve the bound $Z_n$;

2. if $\mathrm{WF}(A(x,y))\subset Z_2$, the insertion

   $$A(x,y)\delta (y-y^{\prime })$$

   exists and has wavefront in $Z_3$;

3. if $A(x,z)$ is compactly supported with wavefront in $Z_2$, and $B(z,y)$ has wavefront in $Z_2$, then $A\circ B$ exists and has wavefront in $Z_2$.

Consequently $N_0$ is defined and

The source argument of $N_0$ need not have a lower time bound.

Proof. Part (a) is the ordinary differential and multiplier wavefront rule.

For (b), the lifted covectors of the two factors have forms

Their sum can vanish only when both are zero, so the product criterion applies. Every possible product covector, including a zero covector from either factor, has zero total sum.

For (c), an opposing pair in the product $A(x,z)B(z,y)$ would have zero external $x$- and $y$-covectors. The $Z_2$ condition then forces its internal covectors to vanish as well. Compact support of $A$ makes the integration in $z$ proper over compact external test sets. For the pushforward, the internal covector is zero; the two edge relations therefore give $p_x+p_y=0$. There are no exceptional one-slot wavefronts because a zero covector in one argument of a $Z_2$ relation forces the other to be zero. Apply this to $D$ and the constant matrix times $R$ in (eq:Delta). The support follows from $y⪯z⪯x$.