Proposition: Normal-jet wavefront upper bound

For the extension in Theorem thm:jetextension,

Proof. Each summand is $b_{\gamma }\otimes {{\partial }_z}^{\gamma }{\delta }_0$. The tensor-product wavefront bound allows tangential covectors in $\mathrm{WF}(b_{\gamma })$, or the zero tangential covector over $\mathrm{supp}{b}_{\gamma }$, together with normal covectors; the total covector must be nonzero. This gives the displayed set.

Equivalently, after product localization, the Fourier transform is the localized Fourier transform of $b_{\gamma }$ multiplied by a polynomial in the normal frequency. Rapid decay in a tangential cone is preserved where the tangential frequency controls the total frequency. The remaining potentially singular directions are the pure normal ones. Taking a finite sum gives the asserted inclusion.