Manuscript Em01-E@2 · Em01-E-v2.tex · proposition · 54f8f8c2

Proposition: Dyadic verification with interval filling

Three sufficient verification routes › Route A: scale-integrability and Cauchy control · lines 864–889 · proof included or linked

Em01-E@2#prop:dyadic

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Put $δ_{n} = 2^{- n} δ_{0}$ . Suppose for each compact $L$ there are $e_{n}, h_{n} \geq 0$ such that

∣ (t_{δ_{n + 1}} - t_{δ_{n}}) (Φ) ∣ δ_{n + 1} \leq δ \leq δ_{n} sup ∣ (t_{δ} - t_{δ_{n}}) (Φ) ∣ \leq C_{L} e_{n} ∥Φ ∥_{m_{L}, L}, \leq C_{L} h_{n} ∥Φ ∥_{m_{L}, L}, n \sum e_{n} h_{n} < \infty, ⟶ 0.

Then the full family converges strongly. On that scale interval, its error is bounded by

C_{L} (h_{n} + j \geq n \sum e_{j}) ∥Φ ∥_{m_{L}, L} .

Proof. Sum the dyadic increments. Their finite-order bounds give a distributional limit and uniform convergence on bounded test sets along the dyadic sequence. The interval estimate connects every intervening scale to that sequence.

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unlabeled · line 868
$∣ (t_{δ_{n + 1}} - t_{δ_{n}}) (Φ) ∣ δ_{n + 1} \leq δ \leq δ_{n} sup ∣ (t_{δ} - t_{δ_{n}}) (Φ) ∣ \leq C_{L} e_{n} ∥Φ ∥_{m_{L}, L}, \leq C_{L} h_{n} ∥Φ ∥_{m_{L}, L}, n \sum e_{n} h_{n} < \infty, ⟶ 0.$
unlabeled · line 879
$C_{L} (h_{n} + j \geq n \sum e_{j}) ∥Φ ∥_{m_{L}, L} .$

Proposition: Dyadic verification with interval filling

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