Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Averaged resolvent estimate and corrected parameter ledger

Averaged resolvent estimate and corrected parameter ledger · lines 945–992

Em01-G@5#eq:finite-part, Em01-G@5#eq:parameter-seminorm

For $F \in C_{c}^{\infty} ([0, \infty))$ , put $f_{j} = F^{(j)} (0) / j!$ , and define

FP_{j} (F) = ϵ ↓ 0 lim [\int_{ϵ}^{\infty} \frac{F ( t )}{t ^{j + 1}} d t - l = 0 \sum^{j - 1} \frac{f _{l}}{j - l} ϵ^{l - j} + f_{j} lo g ϵ] .

The sum is empty for $j = 0$ .

On fixed sets $0 \leq t \leq T$ , $H \in K$ , $θ \in P$ , $∣ δ ∣ \leq δ_{0}$ , define

P_{N, r} (F) = i \leq N + 2, j \leq N + 1 ∣ γ ∣ \leq N + 2 + r, ∣ ω ∣ \leq r max sup \partial_{t}^{i} \partial_{δ}^{j} \partial_{H}^{γ} \partial_{θ}^{ω} F_{δ, H, θ} (t) .

All derivative orders are nonnegative. For a compact smooth weight $w$ , define

W_{N, r} (w) = j \leq N + 1, ∣ γ ∣ \leq N + 2 + r ∣ ω ∣ \leq r max sup ∣ \partial_{δ}^{j} \partial_{H}^{γ} \partial_{θ}^{ω} w ∣.

The additional $r$ in the $H$ -order is essential: spectator differentiation of a transformed smoothing coordinate can differentiate the original weight or amplitude in $H$ .

Use

lo g (- L + i 0) = lo g ∣ L ∣ + iπ 1_{{L > 0}}, \partial_{L} lo g (- L + i 0) = (L - i 0)^{- 1} .

Displayed formulas in this passage

Em01-G@5#eq:finite-part · line 950
$FP_{j} (F) = ϵ ↓ 0 lim [\int_{ϵ}^{\infty} \frac{F ( t )}{t ^{j + 1}} d t - l = 0 \sum^{j - 1} \frac{f _{l}}{j - l} ϵ^{l - j} + f_{j} lo g ϵ] .$
Em01-G@5#eq:parameter-seminorm · line 962
$P_{N, r} (F) = i \leq N + 2, j \leq N + 1 ∣ γ ∣ \leq N + 2 + r, ∣ ω ∣ \leq r max sup \partial_{t}^{i} \partial_{δ}^{j} \partial_{H}^{γ} \partial_{θ}^{ω} F_{δ, H, θ} (t) .$
unlabeled · line 975
$W_{N, r} (w) = j \leq N + 1, ∣ γ ∣ \leq N + 2 + r ∣ ω ∣ \leq r max sup ∣ \partial_{δ}^{j} \partial_{H}^{γ} \partial_{θ}^{ω} w ∣.$
unlabeled · line 987
$lo g (- L + i 0) = lo g ∣ L ∣ + iπ 1_{{L > 0}}, \partial_{L} lo g (- L + i 0) = (L - i 0)^{- 1} .$

Averaged resolvent estimate and corrected parameter ledger

Displayed formulas in this passage

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