Manuscript Em01-D@2 · Em01-D-v2.tex · proposition · d3b49151

Proposition: An explicit microlocal certificate

Support, profiles, admissibility, and output › Fourier bounds and the output operation · lines 1758–1796 · proof included or linked

Em01-D@2#prop:Fourier

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Let

v_{δ} = t_{δ} - S_{δ}^{ρ}, v_{δ} ⟶ T^{ρ}

strongly as proved above. Fix a closed joint cone $Γ \subset T^{*} U ∖ 0$ . Suppose, for each localization $ζ$ and closed frequency cone $V_{0}$ with

(supp ζ \times V_{0}) \cap Γ = \emptyset,

and each integer $r \geq 0$ , that

ξ \in V_{0} sup (1 + ∣ ξ ∣)^{r} ∣ ζ v_{δ} (ξ) ∣ \leq C_{r, ζ, V_{0}}

uniformly for small $δ$ .

Then $WF (T^{ρ}) \subset Γ$ , and $v_{δ}$ converges in the normal topology associated with $Γ$ .

Proof. For each fixed $ξ$ , strong convergence gives convergence of the localized Fourier transform. Passing to the limit in every weighted bound gives rapid decay off $Γ$ , hence the wavefront inclusion.

For convergence of an order- $r$ Fourier seminorm, use the uniform order- $r + 1$ bound to control frequencies above a radius $R$ by $C / (1 + R)$ . Below that radius the localized exponential tests form a bounded test set, so strong convergence is uniform. First choose $R$ large, then take $δ$ small.

Displayed formulas in this passage

unlabeled · line 1761
$v_{δ} = t_{δ} - S_{δ}^{ρ}, v_{δ} ⟶ T^{ρ}$
unlabeled · line 1769
$(supp ζ \times V_{0}) \cap Γ = \emptyset,$
unlabeled · line 1773
$ξ \in V_{0} sup (1 + ∣ ξ ∣)^{r} ∣ ζ v_{δ} (ξ) ∣ \leq C_{r, ζ, V_{0}}$

Proposition: An explicit microlocal certificate

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