Manuscript VII-D@1 · VII-D.tex · lemma · ef67c89c

Lemma: Insertion-regulator replacement

Replacement and the complete ultraviolet split › Replacement of the two graviton multipliers · lines 1386–1547 · proof included or linked

VII-D@1#lem:replacement, VII-D@1#eq:Aregerror, VII-D@1#eq:multtelescoping, VII-D@1#eq:meanvalueexact, VII-D@1#eq:replacementexact, VII-D@1#eq:multLipschitz, VII-D@1#eq:replacementcentral, VII-D@1#eq:replacementtail

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For every compact $K \subset X$ ,

⟨ A_{ε}^{ρ} (r) - ϑ (ε r)^{2} A (r), Φ ⟩ \leq C_{K, ρ, m} ε p_{K, 64} (Φ), r \geq 1, 0 < ε \leq 1.

Only boundedness and a global Lipschitz bound for $ϑ$ are needed for this estimate.

Proof. 1. Exact multiplier difference. Before taking absolute values, use the identity

ϑ (εP) ϑ (ε P^{'}) - ϑ (ε r)^{2} = ϑ (ε P^{'}) (ϑ (εP) - ϑ (ε r)) + ϑ (ε r) (ϑ (ε P^{'}) - ϑ (ε r)) .

Each difference has the exact integral remainder

ϑ (εP) - ϑ (ε r) = ε (P - r) \int_{0}^{1} ϑ^{'} (ε (r + τ (P - r))) d τ,

and likewise with $P^{'}$ .

Using the phase notation from the proof of Lemma A, inserting the left-hand side of (eq:multtelescoping) into (eq:Adef) gives exactly

⟨ A_{ε}^{ρ} (r) - ϑ (ε r)^{2} A (r), Φ ⟩ = c_{0} r^{3} ω (r) \int \frac{ϑ ( εP ) ϑ ( ε P ^{'} ) - ϑ ( ε r ) ^{2}}{P ^{2} P ^{'2}} e^{i [φ_{0} + u_{δ} + u_{ω}]} H_{Φ} \times d Ω (ν) d t d s d t^{'} d^{3} y \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}} .

Here the phase is the same exact phase used in Lemma A. No approximation of it is made.

Since

∣ P - r ∣ \leq ∣ α ∣, ∣ P^{'} - r ∣ \leq ∣ β ∣,

equations (eq:multtelescoping)–(eq:meanvalueexact) imply

ϑ (εP) ϑ (ε P^{'}) - ϑ (ε r)^{2} \leq ∥ ϑ ∥_{\infty} ∥ ϑ^{'} ∥_{\infty} ε (∣ α ∣ + ∣ β ∣) .

This uses no decay when $P$ or $P^{'}$ is small.

2. Central region. On the region $C_{r}$ of Lemma A, $r^{3} ω (r) / (P^{2} P^{'2}) \leq C_{m}$ . Taking absolute values in the already formed difference (eq:replacementexact) gives

∣ central part of ∣ \leq C_{K, ρ, m} ε p_{K, 64} (Φ) \int_{R^{6}} \frac{∣ α ∣ + ∣ β ∣}{( 1 + ∣ α ∣ ) ^{12} ( 1 + ∣ β ∣ ) ^{12}} d^{3} α d^{3} β \leq C^{'}_{K, ρ, m} ε p_{K, 64} (Φ) .

{3. Shifted tails, including $P \approx 0$ .} Use the same complement $C_{r}^{c}$ as in Lemma A, and retain $N = 12$ . For the $α$ -tail, apply (eq:shifttail) to the $α$ -integral and (eq:shiftbound) to the full $β$ -integral. The extra momentum factor is absorbed using $∣ α ∣ \leq 1 + ∣ α ∣$ or $∣ β ∣ \leq 1 + ∣ β ∣$ . The following table records the parameters of those two estimates. Every row includes the prefactor $r^{3} ω (r) \leq C_{m} r^{4}$ ; the common factor $C_{K, ρ, m} p_{K, 64} (Φ)$ is suppressed.

{}1.25

Extra factor	$α$ -tail: $(k, ℓ)$ and bound	Full $β$ : $(k, ℓ)$ and bound	Radial majorant
$ε ∣ α ∣$	$(1, 2) : C r^{2 - N}$	$(0, 2) : C r^{- 2}$	$C ε r^{4 - N}$
$ε ∣ β ∣$	$(0, 2) : C r^{1 - N}$	$(1, 2) : C r^{- 2}$	$C ε r^{3 - N}$

Indeed, the tail powers are $3 - 2 + 1 - N = 2 - N$ and $3 - 2 + 0 - N = 1 - N$ . The full-integral estimate with $k = 1, ℓ = 2$ still gives $C r^{- 2}$ , since $N = 12 > 1 + 3$ ; no additional power of $r$ is lost there. The $β$ -tail has the same two bounds after interchanging $α$ and $β$ . The union bound therefore yields

∣ tail part of ∣ \leq C_{K, ρ, m} ε (r^{4 - N} + r^{3 - N}) p_{K, 64} (Φ) = C_{K, ρ, m} ε (r^{- 8} + r^{- 9}) p_{K, 64} (Φ) \leq C^{'}_{K, ρ, m} ε p_{K, 64} (Φ) .

The shifted-tail bounds include the locally integrable singularities at $P = 0$ and $P^{'} = 0$ . Combining (eq:replacementcentral) and (eq:replacementtail) proves the claim.

Displayed formulas in this passage

VII-D@1#eq:Aregerror · line 1389
$⟨ A_{ε}^{ρ} (r) - ϑ (ε r)^{2} A (r), Φ ⟩ \leq C_{K, ρ, m} ε p_{K, 64} (Φ), r \geq 1, 0 < ε \leq 1.$
VII-D@1#eq:multtelescoping · line 1404
$ϑ (εP) ϑ (ε P^{'}) - ϑ (ε r)^{2} = ϑ (ε P^{'}) (ϑ (εP) - ϑ (ε r)) + ϑ (ε r) (ϑ (ε P^{'}) - ϑ (ε r)) .$
VII-D@1#eq:meanvalueexact · line 1416
$ϑ (εP) - ϑ (ε r) = ε (P - r) \int_{0}^{1} ϑ^{'} (ε (r + τ (P - r))) d τ,$
VII-D@1#eq:replacementexact · line 1428
$⟨ A_{ε}^{ρ} (r) - ϑ (ε r)^{2} A (r), Φ ⟩ = c_{0} r^{3} ω (r) \int \frac{ϑ ( εP ) ϑ ( ε P ^{'} ) - ϑ ( ε r ) ^{2}}{P ^{2} P ^{'2}} e^{i [φ_{0} + u_{δ} + u_{ω}]} H_{Φ} \times d Ω (ν) d t d s d t^{'} d^{3} y \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}} .$
VII-D@1#eq:multLipschitz · line 1453
$ϑ (εP) ϑ (ε P^{'}) - ϑ (ε r)^{2} \leq ∥ ϑ ∥_{\infty} ∥ ϑ^{'} ∥_{\infty} ε (∣ α ∣ + ∣ β ∣) .$
VII-D@1#eq:replacementcentral · line 1469
$∣ central part of ∣ \leq C_{K, ρ, m} ε p_{K, 64} (Φ) \int_{R^{6}} \frac{∣ α ∣ + ∣ β ∣}{( 1 + ∣ α ∣ ) ^{12} ( 1 + ∣ β ∣ ) ^{12}} d^{3} α d^{3} β \leq C^{'}_{K, ρ, m} ε p_{K, 64} (Φ) .$
VII-D@1#eq:replacementtail · line 1527
$∣ tail part of ∣ \leq C_{K, ρ, m} ε (r^{4 - N} + r^{3 - N}) p_{K, 64} (Φ) = C_{K, ρ, m} ε (r^{- 8} + r^{- 9}) p_{K, 64} (Φ) \leq C^{'}_{K, ρ, m} ε p_{K, 64} (Φ) .$
unlabeled · line 1448
$∣ P - r ∣ \leq ∣ α ∣, ∣ P^{'} - r ∣ \leq ∣ β ∣,$

Nearby in this section

section: Replacement of the two graviton multipliers