Manuscript XV-D@1 · XV-D.tex · section · 967136eb

What the record says about v α and c D,β

Coefficient and extension calculations › What the record says about v α and c D,β · lines 1032–1091

XV-D@1#eq:reference, XV-D@1#eq:referenceWF, XV-D@1#eq:collected, XV-D@1#eq:jetbound

The supplied supported-sector reference is Jreference

t_{D}^{\circ} = ∣ α ∣ \leq 3 β \leq α \sum (- 1)^{∣ β ∣} (β α) E_{α - β} (v_{α} c_{D, β}) on U ∖ N_{U},

with wavefront bound

WF (t_{D}^{\circ}) \subseteq Γ_{supp}^{\circ} = N^{*} S_{resp}_{U ∖ N_{U}} ∖ 0 \cup i = 1 ⋃^{2} {(w, λ d q_{i} + ω) : w \in S_{resp} ∖ N_{U}, q_{i} (w) = 0, λ \neq = 0, ω \in N_{w}^{*} S_{resp}} ∖ 0.

XV-C collected the unique coefficients

b_{D, γ}^{\circ} = β \geq 0 ∣ γ + β ∣ \leq 3 \sum (- 1)^{∣ β ∣} (γ γ + β) v_{γ + β} c_{D, β}

and proved that the unrestricted supported realization exists if and only if each $b^{\circ}_{D, γ}$ satisfies the compact-uniform estimate

∣ ⟨ b_{D, γ}^{\circ}, ψ ⟩ ∣ \leq C_{K, γ} p_{k_{K, γ}} (ψ), ψ \in D (S_{0} ∖ Z), supp ψ \subset K,

where $S_{0}$ is the base of the normal chart of $S_{resp}$ and $Z = {x : (x, 0) \in N_{U}}$ XVC.

The defining formulas of $v_{α}$ and $c_{D, β}$ are not in the record (D0.18). The supplied proof of the reference theorem contains the sentence “the normal traces have the tangential bounds in that lemma; multiplication by $v_{α}$ adds no wavefront” Jreference. We record the following reading, not a supplied definition: $c_{D, β}$ are normal traces of the cross-line factors along $S_{resp}$ , and $v_{α}$ are multipliers whose multiplication adds no wavefront, as a smooth function would. The next result is stated under an explicit hypothesis matching that reading, so that it can be applied or discarded once the formulas are supplied.

Displayed formulas in this passage

XV-D@1#eq:reference · line 1035
$t_{D}^{\circ} = ∣ α ∣ \leq 3 β \leq α \sum (- 1)^{∣ β ∣} (β α) E_{α - β} (v_{α} c_{D, β}) on U ∖ N_{U},$
XV-D@1#eq:referenceWF · line 1045
$WF (t_{D}^{\circ}) \subseteq Γ_{supp}^{\circ} = N^{*} S_{resp}_{U ∖ N_{U}} ∖ 0 \cup i = 1 ⋃^{2} {(w, λ d q_{i} + ω) : w \in S_{resp} ∖ N_{U}, q_{i} (w) = 0, λ \neq = 0, ω \in N_{w}^{*} S_{resp}} ∖ 0.$
XV-D@1#eq:collected · line 1061
$b_{D, γ}^{\circ} = β \geq 0 ∣ γ + β ∣ \leq 3 \sum (- 1)^{∣ β ∣} (γ γ + β) v_{γ + β} c_{D, β}$
XV-D@1#eq:jetbound · line 1072
$∣ ⟨ b_{D, γ}^{\circ}, ψ ⟩ ∣ \leq C_{K, γ} p_{k_{K, γ}} (ψ), ψ \in D (S_{0} ∖ Z), supp ψ \subset K,$