Manuscript XVI-H@1 · XVI-H.tex · section · af895076

Imported scalar estimates and resonant evaluations

Partial witness contraction and recovery of the density › Imported scalar estimates and resonant evaluations · lines 1292–1350

XVI-H@1#eq:G-estimate, XVI-H@1#eq:res-cos, XVI-H@1#eq:res-sin

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For a fixed compact set $L \subset M_{T}^{\circ}$ containing $supp f \cup supp g$ , current XVI-G proves

∣ I_{λ} [ϕ] ∣ \leq C_{k, ε} λ^{2 - k} p_{m a x (k, 2), L} (ϕ), k \in N_{0}, λ \geq 1.

The constant is independent of $λ, ϕ$ at fixed observation and background data.

By (eq:t-to-j), this estimate applies to the same witness contraction of $t_{D}^{C}$ . It is imported from XVI-G, not derived from the unspecified finite-order estimate (eq:target).

The established rapid-decay cases are:

one real smooth scalar independent of $λ$ ;
sets bounded in every $p_{r, L}$ , with decay uniform on the stated set;
families satisfying $p_{r, L} (ϕ_{λ}) \leq C_{r} λ^{ρ r + a}$ with $0 \leq ρ < 1$ .

They are current XVI-G, prop:fixeddecay, prop:bounded, and prop:slow XVIG.

For real fixed $b \in C_{c}^{\infty} (M_{T}^{\circ})$ , set

Υ [b] = \int_{M_{T}} σ g b \partial_{t} v_{f} d^{4} x .

The resonant theorem of XVI-G gives

⟨ t_{D}^{C}, γ_{1}^{λ} \otimes γ_{2}^{λ} \otimes b cos (2 λ t) ⟩ ⟨ t_{D}^{C}, γ_{1}^{λ} \otimes γ_{2}^{λ} \otimes b sin (2 λ t) ⟩ = \frac{λ}{2 i} Υ [b] + O (1), = \frac{λ}{2} Υ [b] + O (1) .

The order- $λ$ contribution is scalar-leg only; the coordinate contribution is bounded. No integrals or remainder estimates are recomputed here.

When $Υ [b] \neq = 0$ , these are nonzero leading linear terms. When it vanishes, only $O (1)$ is concluded. The number $Υ [b]$ is not a wavefront covector and does not decide admission.

Displayed formulas in this passage

XVI-H@1#eq:G-estimate · line 1296
$∣ I_{λ} [ϕ] ∣ \leq C_{k, ε} λ^{2 - k} p_{m a x (k, 2), L} (ϕ), k \in N_{0}, λ \geq 1.$
XVI-H@1#eq:res-cos · line 1330
$⟨ t_{D}^{C}, γ_{1}^{λ} \otimes γ_{2}^{λ} \otimes b cos (2 λ t) ⟩ ⟨ t_{D}^{C}, γ_{1}^{λ} \otimes γ_{2}^{λ} \otimes b sin (2 λ t) ⟩ = \frac{λ}{2 i} Υ [b] + O (1), = \frac{λ}{2} Υ [b] + O (1) .$
XVI-H@1#eq:res-sin · line 1330
$⟨ t_{D}^{C}, γ_{1}^{λ} \otimes γ_{2}^{λ} \otimes b cos (2 λ t) ⟩ ⟨ t_{D}^{C}, γ_{1}^{λ} \otimes γ_{2}^{λ} \otimes b sin (2 λ t) ⟩ = \frac{λ}{2 i} Υ [b] + O (1), = \frac{λ}{2} Υ [b] + O (1) .$
unlabeled · line 1325
$Υ [b] = \int_{M_{T}} σ g b \partial_{t} v_{f} d^{4} x .$