Proposition: Products justified by the global orientation

Let $W_{{\beta }_1},\dots ,W_{{\beta }_m}$ be covariances of the form considered here, with the same time orientation. Their product as kernels on $M\times M$, with the appropriate tensor-product bundle, is defined by the ordinary wavefront product theorem. The same holds after any fixed finite differentiations of the factors.

This assertion does not supply a product between arbitrary oppositely oriented kernels or an interacting algebra.

Proof. Every nonzero wavefront covector of every factor has a future-null first component. A finite sum of nonzero future-causal covectors cannot vanish. Thus no selection of wavefront covectors from the factors can sum to the zero covector on $M\times M$. The product criterion applies HormanderI. Differentiation does not enlarge a wavefront set.