Manuscript Em01-C@3 · Em01-C-v3.tex · remark · f0953702

Remark: An optional stronger certificate

Scale identities and the direct convergence target · lines 1375–1394

If near zero the complete $Ω_{Φ}$ is represented by a continuous function $W_{Φ}$ with uniform local bounds

∣ W_{Φ} (h) - W_{Φ} (0) ∣ \leq C_{L} ∣ h ∣^{α} ∥Φ ∥_{m_{L}}, ∣ W_{Φ} (0) ∣ \leq C_{L} ∥Φ ∥_{m_{L}},

then

D_{δ} (Φ) = \int ϑ (s) [W_{Φ} (δ s) - W_{Φ} (0)] d s = O (δ^{α} ∥Φ ∥_{m_{L}}) .

Also $t_{δ} (Φ) \to W_{Φ} (0)$ , independently of the normalized compact profile. This is sufficient, not a required route.

Displayed formulas in this passage

unlabeled · line 1378
$∣ W_{Φ} (h) - W_{Φ} (0) ∣ \leq C_{L} ∣ h ∣^{α} ∥Φ ∥_{m_{L}}, ∣ W_{Φ} (0) ∣ \leq C_{L} ∥Φ ∥_{m_{L}},$
unlabeled · line 1385
$D_{δ} (Φ) = \int ϑ (s) [W_{Φ} (δ s) - W_{Φ} (0)] d s = O (δ^{α} ∥Φ ∥_{m_{L}}) .$