Manuscript Em03-F@1 · Em03-F-v1.tex · lemma · 0d4d148b

Lemma: Weighted compact-test continuity

Exterior field, scattering coordinates, and infrared control › A compact-test estimate · lines 489–520 · proof included or linked

Em03-F@1#lem:compact, Em03-F@1#eq:energybound

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For a real compact smooth test supported in $∣ x ∣ \leq X$ ,

∥ k (f) ∥_{U} \leq B_{X} (f) .

Furthermore,

\int_{0}^{\infty} ω j \sum ∣ k_{j} (f) (ω) ∣^{2} d ω \leq \frac{C _{X} ^{2}}{2 π} (\frac{ω _{c}}{3} ∥ f ∥_{1}^{2} + \frac{1}{ω _{c}} ∥ \partial_{t} f ∥_{1}^{2}) .

Proof. At low frequency,

∣ k_{j} (f) ∣ \leq \frac{C _{X} ω}{4 π ω _{c}} ∥ f ∥_{1} .

At high frequency, one time integration by parts gives

∣ k_{j} (f) ∣ \leq \frac{C _{X}}{4 π ω ^{3/2}} ∥ \partial_{t} f ∥_{1} .

Use $1 + 2 n_{H} \leq 1 + 2/ (β ω)$ , sum both channels, and integrate the two intervals. This gives (eq:Bbound). Integrating the same estimates with weight $ω$ gives (eq:energybound).

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Em03-F@1#eq:energybound · line 496
$\int_{0}^{\infty} ω j \sum ∣ k_{j} (f) (ω) ∣^{2} d ω \leq \frac{C _{X} ^{2}}{2 π} (\frac{ω _{c}}{3} ∥ f ∥_{1}^{2} + \frac{1}{ω _{c}} ∥ \partial_{t} f ∥_{1}^{2}) .$
unlabeled · line 492
$∥ k (f) ∥_{U} \leq B_{X} (f) .$
unlabeled · line 507
$∣ k_{j} (f) ∣ \leq \frac{C _{X} ω}{4 π ω _{c}} ∥ f ∥_{1} .$
unlabeled · line 512
$∣ k_{j} (f) ∣ \leq \frac{C _{X}}{4 π ω ^{3/2}} ∥ \partial_{t} f ∥_{1} .$

Lemma: Weighted compact-test continuity

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