Manuscript Em01-B@3 · Em01-B-v3.tex · corollary · 69189992

Corollary: A two-parameter bound

An actual estimate for the complete retained family › Complete-contraction estimate on characteristic tests · lines 1363–1387 · proof included or linked

Em01-B@3#cor:two-parameter, Em01-B@3#eq:two-parameter

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Choose coordinates $(τ, ν)$ with $N = {ν = 0}$ , $ν \in R^{8}$ , and fixed compact test supports. Set

Φ_{λ} (τ, ν) = λ^{- 8} Φ (τ, ν / λ) .

Whenever the scaled supports remain in the chart,

∣ F_{δ} (Rec A) (Φ_{λ}) ∣ \leq C p_{L, 4} (A) δ^{- q - 13} λ^{- 13} ∥Φ ∥_{5} .

Proof. The fixed coordinate transformation and its density conversion have bounded derivatives on the compact chart. The order-five seminorm of the scaled test is bounded by $C λ^{- 8 - 5} ∥Φ ∥_{5}$ . Apply (eq:moderate).

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Em01-B@3#eq:two-parameter · line 1373
$∣ F_{δ} (Rec A) (Φ_{λ}) ∣ \leq C p_{L, 4} (A) δ^{- q - 13} λ^{- 13} ∥Φ ∥_{5} .$
unlabeled · line 1368
$Φ_{λ} (τ, ν) = λ^{- 8} Φ (τ, ν / λ) .$

Corollary: A two-parameter bound

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