Authorization, descent, and separate microlocal interfaces

For a finite supported correction

the tests $(-1{)}^{|\gamma |}{n}^{\gamma }\vartheta (n)\psi (s)/\gamma !$, with $\vartheta =1$ near zero, extract the separate coefficients. Thus coherence is coefficientwise. A finite normal order does not make the tangential ambiguity finite-dimensional.

Restriction is strongly continuous, being the transpose of extension by zero of compact tests. Limits inherit a separately proved reference. Equal references on $O$ imply difference support in $X\setminus O$, not specifically in $N$ unless that stronger support conclusion is proved.

Compatible local maps glue by a locally finite partition of unity. Only finitely many terms meet each compact test support. Overlap compatibility is a hypothesis, not supplied by a local limit.

Finally fix a closed conic $\Gamma \subset T^{\ast }X\setminus 0$. Suppose $u_{\delta }\in {{\mathcal{D}}^{\prime }}_{\Gamma }(X;E)$, $u_{\delta }\to u$ strongly, and for every compact cutoff $\theta$, closed frequency cone $\Omega$ disjoint from $\Gamma$ over $\mathrm{supp}\theta$, and every $N$,

is uniformly bounded. Passing to the limit gives rapid decay outside $\Gamma$, hence $\mathrm{WF}(u)\subset \Gamma$. For convergence of an order-$N$ Fourier seminorm, the uniform order-$N+1$ bound controls frequencies above $R$ by $2C/(1+R)$; below $R$, strong convergence is uniform on the bounded family of localized exponential tests. This is a separate Fourier certificate, not a consequence of the finite-order Dini estimate.