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

Theorem: Scale-integrability route

Three sufficient verification routes › Route A: scale-integrability and Cauchy control · lines 826–858 · proof included or linked

Em01-E@2#thm:scale, Em01-E@2#eq:Dini

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Suppose for every $L ⋐ U$ ,

∣ (δ \partial_{δ} t_{δ}) (Φ) ∣ \leq C_{L} b_{L} (δ) ∥Φ ∥_{m_{L}, L}, \int_{0}^{δ_{0}} b_{L} (s) \frac{d s}{s} < \infty,

with $b_{L} \geq 0$ . Then $t_{δ} \to t_{0}$ strongly for some distribution $t_{0}$ , and

∣ (t_{δ} - t_{0}) (Φ) ∣ \leq C_{L} ∥Φ ∥_{m_{L}, L} \int_{0}^{δ} b_{L} (s) \frac{d s}{s} .

Proof. Positive-scale smoothness gives

(t_{δ_{2}} - t_{δ_{1}}) (Φ) = \int_{δ_{1}}^{δ_{2}} (s \partial_{s} t_{s}) (Φ) \frac{d s}{s} .

The estimate makes the pairings Cauchy uniformly on bounded test sets. A fixed positive-scale member plus the integrable increments supplies a finite-order local bound for the limit. It is therefore a distribution. Letting $δ_{1} ↓ 0$ gives the tail estimate.

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Em01-E@2#eq:Dini · line 829
$∣ (δ \partial_{δ} t_{δ}) (Φ) ∣ \leq C_{L} b_{L} (δ) ∥Φ ∥_{m_{L}, L}, \int_{0}^{δ_{0}} b_{L} (s) \frac{d s}{s} < \infty,$
unlabeled · line 838
$∣ (t_{δ} - t_{0}) (Φ) ∣ \leq C_{L} ∥Φ ∥_{m_{L}, L} \int_{0}^{δ} b_{L} (s) \frac{d s}{s} .$
unlabeled · line 847
$(t_{δ_{2}} - t_{δ_{1}}) (Φ) = \int_{δ_{1}}^{δ_{2}} (s \partial_{s} t_{s}) (Φ) \frac{d s}{s} .$

Theorem: Scale-integrability route

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