Manuscript Em01-D@2 · Em01-D-v2.tex · proposition · d3b49151

Proposition: Direct scale-integrability

Coefficient calculation and remainder proof › An alternative certificate without a homogeneous germ · lines 1534–1565 · proof included or linked

Em01-D@2#prop:Dini

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Suppose, instead of Assumption ass:germ, that on each compact $L ⋐ U$ ,

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

Then the raw fixed-kernel family converges strongly, with

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

Proof. For $0 < δ_{1} < δ_{2}$ ,

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

The estimate gives a Cauchy pairing and the displayed tail bound. The limit is a distribution because a fixed positive-scale member plus the integrable increments has a local finite-order bound. The same estimate is uniform on bounded test sets.

Displayed formulas in this passage

unlabeled · line 1538
$∣ Ω_{Φ} (θ_{δ}) ∣ \leq C_{L} b_{L} (δ) ∥Φ ∥_{m_{L}}, \int_{0}^{δ_{0}} b_{L} (s) \frac{d s}{s} < \infty.$
unlabeled · line 1545
$∣ (t_{δ} - t_{0}) (Φ) ∣ \leq C_{L} ∥Φ ∥_{m_{L}} \int_{0}^{δ} b_{L} (s) \frac{d s}{s} .$
unlabeled · line 1555
$(t_{δ_{2}} - t_{δ_{1}}) (Φ) = \int_{δ_{1}}^{δ_{2}} Ω_{Φ} (θ_{s}) \frac{d s}{s} .$

Nearby in this section

section: An alternative certificate without a homogeneous germ