Proposition: Coefficient normal jets and wavefront bounds

Locally, $\mathcal{A}$ and $\mathcal{T}$ are finite normal jets along $S_{\kappa }$, with smooth tangential coefficients. Their normal orders are at most two and four, respectively. Moreover,

where the zero section is omitted from $N^{\ast }{S}_{\kappa }$. These conclusions do not assert normal-topology convergence of the subtracted family.

Proof. Formula (eq:appAsimple) involves only the normal jets ${\partial }_v\phi ,{\Delta }_X\phi ,\phi$, with smooth coefficients. Its normal order is at most two.

In (eq:appT), the only tangential test derivatives arise from ${\partial }_{b^0}=({\partial }_B+{\partial }_u)/2$ in $F_1,F_2$. Integrate all ${\partial }_B$-terms by parts. Their coefficients are smooth. The remaining derivatives are normal, with order at most four. Thus both distributions have local representations

with $m=2$ or $4$, and the specified tangential density.

For completeness, localize this representation by a compact smooth cutoff. Multiplication by the cutoff produces another finite normal jet with compact smooth tangential coefficients. Its Fourier transform is a finite sum

On a closed cone separated from $\{{\xi }_{\tau }=0,{\xi }_{\nu }\ne 0\}$, $|{\xi }_{\tau }|$ is bounded below by a positive multiple of the total frequency. Rapid tangential Fourier decay dominates the finite normal polynomial. This proves the stated conormal containment. The coordinate change here is linear, so it transports the bound directly to the original joint variables.

This describes the coefficients, not uniform Fourier estimates for the approximating distributions.