Manuscript IX-B@2 · IX-B-v2.tex · assumption · 8ce915d6

Assumption: VII-D finite-part and parent-conversion input

Fixed data and inherited inputs › Quantified analytical input · lines 429–472

IX-B@2#ass:VIID, IX-B@2#eq:parent, IX-B@2#eq:input, IX-B@2#eq:parentconversion, IX-B@2#eq:parentscale

We use the following statements of VII-D as analytical input VIID (thm:finitepart, thm:conversion). There is a distribution $U_{m}$ , independent of $ρ$ , such that

u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r, κ_{ρ} = \int_{0}^{1} \frac{ϑ _{ρ} ( v ) ^{4} - 1}{v} d v + \int_{1}^{\infty} \frac{ϑ _{ρ} ( v ) ^{4}}{v} d v .

For every compact $K \subset X$ and every $Φ \in D_{K} (X)$ ,

⟨ T_{ε}^{ρ} - lo g ((μ ε)^{- 1}) r - u_{μ}^{ρ}, Φ ⟩ \leq C_{K, ρ, m} ε (1 + ∣ lo g ε ∣) p_{K, 64} (Φ), 0 < ε \leq \frac{1}{2} .

The estimate includes compacts meeting $Rel \cap Δ_{13}$ . For two admissible regulators,

u_{μ}^{ρ_{a}} - u_{μ}^{ρ_{b}} = c_{ab} r, c_{ab} = \int_{0}^{\infty} \frac{ϑ _{ρ_{a}} ( v ) ^{4} - ϑ _{ρ_{b}} ( v ) ^{4}}{v} d v \in R .

This is the complete parent conversion: there is no extra normal derivative, nonconstant weight, or separate term on $Rel \cap Δ_{13}$ . The inherited identities also give

c_{ab} + c_{b c} = c_{a c}, u_{μ_{2}}^{ρ} - u_{μ_{1}}^{ρ} = lo g (μ_{2} / μ_{1}) r .

Displayed formulas in this passage

IX-B@2#eq:parent · line 434
$u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r, κ_{ρ} = \int_{0}^{1} \frac{ϑ _{ρ} ( v ) ^{4} - 1}{v} d v + \int_{1}^{\infty} \frac{ϑ _{ρ} ( v ) ^{4}}{v} d v .$
IX-B@2#eq:input · line 444
$⟨ T_{ε}^{ρ} - lo g ((μ ε)^{- 1}) r - u_{μ}^{ρ}, Φ ⟩ \leq C_{K, ρ, m} ε (1 + ∣ lo g ε ∣) p_{K, 64} (Φ), 0 < ε \leq \frac{1}{2} .$
IX-B@2#eq:parentconversion · line 456
$u_{μ}^{ρ_{a}} - u_{μ}^{ρ_{b}} = c_{ab} r, c_{ab} = \int_{0}^{\infty} \frac{ϑ _{ρ_{a}} ( v ) ^{4} - ϑ _{ρ_{b}} ( v ) ^{4}}{v} d v \in R .$
IX-B@2#eq:parentscale · line 466
$c_{ab} + c_{b c} = c_{a c}, u_{μ_{2}}^{ρ} - u_{μ_{1}}^{ρ} = lo g (μ_{2} / μ_{1}) r .$

Assumption: VII-D finite-part and parent-conversion input

Displayed formulas in this passage

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