Manuscript XVI-B@1 · XVI-B.tex · proposition · 6a0abffd

Proposition: Fixed compact smooth amplitude

Smooth root tests and their limited asymptotics · lines 1294–1348 · proof included or linked

XVI-B@1#prop:root-decay

Let $a_{μν} \in C_{c}^{\infty} (R^{4}; Sym^{2} T^{*} M)$ be independent of $λ$ . Define

Q_{a} (λ) = \int_{R^{4}} a_{μν} (y) S_{λ}^{μν} (y) d^{4} y .

The integral exists, and

Q_{a} (λ) = O (λ^{- \infty}) (λ \to + \infty)

at fixed $ε$ , $n$ , and $a$ .

Proof. The integrand is smooth and compactly supported. Since $S_{λ}$ is spatially independent, put

A (t) = \int_{R^{3}} a_{μν} (t, x) σ (t) q^{μν} d^{3} x,

B (t) = \int_{R^{3}} a_{μν} (t, x) d_{1}^{μν} (t) d^{3} x .

Both belong to $C_{c}^{\infty} (R)$ and are independent of $λ$ . Equation (eq:JDeval) gives

Q_{a} (λ) = λ^{2} \int_{R} A (t) e^{2 i λ t} d t + λ \int_{R} B (t) e^{2 i λ t} d t .

For each integer $M \geq 0$ , integration by parts yields

\int A (t) e^{2 i λ t} d t \leq (2 λ)^{- M} ∥ A^{(M)} ∥_{L^{1}},

and the analogous bound for $B$ . Given $N \geq 0$ , choose $M = N + 2$ . For $λ \geq 1$ ,

∣ Q_{a} (λ) ∣ \leq 2^{- (N + 2)} λ^{- N} (∥ A^{(N + 2)} ∥_{L^{1}} + ∥ B^{(N + 2)} ∥_{L^{1}}) .

This proves the stated rapid decay.

Displayed formulas in this passage

unlabeled · line 1300
$Q_{a} (λ) = \int_{R^{4}} a_{μν} (y) S_{λ}^{μν} (y) d^{4} y .$
unlabeled · line 1306
$Q_{a} (λ) = O (λ^{- \infty}) (λ \to + \infty)$
unlabeled · line 1316
$A (t) = \int_{R^{3}} a_{μν} (t, x) σ (t) q^{μν} d^{3} x,$
unlabeled · line 1321
$B (t) = \int_{R^{3}} a_{μν} (t, x) d_{1}^{μν} (t) d^{3} x .$
unlabeled · line 1328
$Q_{a} (λ) = λ^{2} \int_{R} A (t) e^{2 i λ t} d t + λ \int_{R} B (t) e^{2 i λ t} d t .$
unlabeled · line 1334
$\int A (t) e^{2 i λ t} d t \leq (2 λ)^{- M} ∥ A^{(M)} ∥_{L^{1}},$
unlabeled · line 1341
$∣ Q_{a} (λ) ∣ \leq 2^{- (N + 2)} λ^{- N} (∥ A^{(N + 2)} ∥_{L^{1}} + ∥ B^{(N + 2)} ∥_{L^{1}}) .$