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

Expansion of the family

Coefficient calculation and remainder proof › Expansion of the family · lines 1390–1428

Em01-D@2#eq:germexpansion, Em01-D@2#eq:explicitrem

For a sufficiently small scale, $δ supp ρ \subset B (0, r_{L})$ . Using (eq:germ) and (eq:chain),

t_{δ} (Φ) = λ \in E \sum δ^{λ} k = 0 \sum^{J_{λ}} \frac{( lo g δ ) ^{k}}{k !} H_{λ, J_{λ} - k} [Φ] (ρ) + \int ρ (h) W_{Φ} (δ h) d h .

Because $\int ρ = 1$ , the continuous contribution is

W_{Φ} (0) + \int ρ (h) (W_{Φ} (δ h) - W_{Φ} (0)) d h .

Its remainder satisfies

\int ρ (h) (W_{Φ} (δ h) - W_{Φ} (0)) d h \leq C_{L} δ^{a_{L}} ∥Φ ∥_{m_{L}} \int ∣ ρ (h) ∣∣ h ∣^{a_{L}} d h .

These identities establish (eq:mainexpansion)–(eq:mainrem). The coefficient estimates in Assumption ass:germ make every $A_{λ, k}^{ρ}$ and $T^{ρ}$ an actual distribution on $U$ .

Displayed formulas in this passage

Em01-D@2#eq:germexpansion · line 1395
$t_{δ} (Φ) = λ \in E \sum δ^{λ} k = 0 \sum^{J_{λ}} \frac{( lo g δ ) ^{k}}{k !} H_{λ, J_{λ} - k} [Φ] (ρ) + \int ρ (h) W_{Φ} (δ h) d h .$
Em01-D@2#eq:explicitrem · line 1414
$\int ρ (h) (W_{Φ} (δ h) - W_{Φ} (0)) d h \leq C_{L} δ^{a_{L}} ∥Φ ∥_{m_{L}} \int ∣ ρ (h) ∣∣ h ∣^{a_{L}} d h .$
unlabeled · line 1409
$W_{Φ} (0) + \int ρ (h) (W_{Φ} (δ h) - W_{Φ} (0)) d h .$