Manuscript VII-D@1 · VII-D.tex · theorem · ef67c89c

Theorem: Strong finite part with quantitative remainder

Strong finite part and dyadic certificates · lines 1810–1886 · proof included or linked

VII-D@1#thm:finitepart, VII-D@1#eq:finitepart, VII-D@1#eq:finalrate, VII-D@1#eq:subtracteddecomp, VII-D@1#eq:scalarfinite

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For every admissible fixed mollifier $ρ$ and every $μ > 0$ ,

u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r

is the strong distributional limit

T_{ε}^{ρ} - lo g (\frac{1}{μ ε}) r ⟶ u_{μ}^{ρ} in D^{'} (X) .

More precisely,

⟨ T_{ε}^{ρ} - lo g (\frac{1}{μ ε}) 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 $Rel \cap Δ_{13}$ .

Proof. The ultraviolet decomposition gives

T_{ε}^{ρ} = S + \int_{1}^{\infty} ϑ (ε r)^{4} (A (r) - r) \frac{d r}{r} + (\int_{1}^{\infty} ϑ (ε r)^{4} \frac{d r}{r}) r + E_{ε}^{ρ} .

Changing variables in the scalar radial integral gives

\int_{1}^{\infty} ϑ (ε r)^{4} \frac{d r}{r} = lo g (ε^{- 1}) + κ_{ρ} - \int_{0}^{ε} \frac{ϑ ( u ) ^{4} - 1}{u} d u = lo g (ε^{- 1}) + κ_{ρ} + O_{ρ} (ε^{2}) .

Furthermore,

∣ ϑ (ε r)^{4} - 1∣ \leq C_{ρ} min {(ε r)^{2}, 1} .

Lemma A therefore bounds the difference between the first radial integral in (eq:subtracteddecomp) and the integral in (eq:Ucommon) by

C_{K, ρ, m} p_{K, 64} (Φ) \int_{1}^{\infty} \frac{min {( ε r ) ^{2} , 1 }}{r ^{2}} d r \leq C^{'}_{K, ρ, m} ε p_{K, 64} (Φ) .

Together with (eq:Eerror) and (eq:scalarfinite), this proves (eq:finitepart)–(eq:finalrate). The compact-test estimate implies uniform convergence on every bounded subset of $D (X)$ .

Displayed formulas in this passage

VII-D@1#eq:finitepart · line 1813
$u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r$
VII-D@1#eq:finalrate · line 1826
$⟨ T_{ε}^{ρ} - lo g (\frac{1}{μ ε}) r - u_{μ}^{ρ}, Φ ⟩ \leq C_{K, ρ, m} ε (1 + ∣ lo g ε ∣) p_{K, 64} (Φ), 0 < ε \leq \frac{1}{2} .$
VII-D@1#eq:subtracteddecomp · line 1846
$T_{ε}^{ρ} = S + \int_{1}^{\infty} ϑ (ε r)^{4} (A (r) - r) \frac{d r}{r} + (\int_{1}^{\infty} ϑ (ε r)^{4} \frac{d r}{r}) r + E_{ε}^{ρ} .$
VII-D@1#eq:scalarfinite · line 1857
$\int_{1}^{\infty} ϑ (ε r)^{4} \frac{d r}{r} = lo g (ε^{- 1}) + κ_{ρ} - \int_{0}^{ε} \frac{ϑ ( u ) ^{4} - 1}{u} d u = lo g (ε^{- 1}) + κ_{ρ} + O_{ρ} (ε^{2}) .$
unlabeled · line 1820
$T_{ε}^{ρ} - lo g (\frac{1}{μ ε}) r ⟶ u_{μ}^{ρ} in D^{'} (X) .$
unlabeled · line 1869
$∣ ϑ (ε r)^{4} - 1∣ \leq C_{ρ} min {(ε r)^{2}, 1} .$
unlabeled · line 1876
$C_{K, ρ, m} p_{K, 64} (Φ) \int_{1}^{\infty} \frac{min {( ε r ) ^{2} , 1 }}{r ^{2}} d r \leq C^{'}_{K, ρ, m} ε p_{K, 64} (Φ) .$

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section: Strong finite part and dyadic certificates