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

Strong finite part and dyadic certificates

Strong finite part and dyadic certificates · lines 1779–1809

VII-D@1#eq:kappa, VII-D@1#eq:Ucommon

Define the real profile constant

κ_{ρ} = \int_{0}^{1} \frac{ϑ ( u ) ^{4} - 1}{u} d u + \int_{1}^{\infty} \frac{ϑ ( u ) ^{4}}{u} d u .

The profile is real, even, Schwartz, and equals one at zero. Hence $ϑ (u)^{4} - 1 = O (u^{2})$ , and both integrals converge.

Define the common distribution

U_{m} = S + \int_{1}^{\infty} (A (r) - r) \frac{d r}{r} .

This integral is defined by its joint-test pairings. They are measurable in $r$ , and Lemma A gives the integrable bound

∣ ⟨ A (r) - r, Φ ⟩ ∣ \frac{1}{r} \leq C_{K, m} r^{- 2} p_{K, 64} (Φ) .

Thus (eq:Ucommon) defines a continuous distribution. Its finite radial truncations converge strongly, with tail at most $C_{K, m} R^{- 1} p_{K, 64} (Φ)$ above radius $R$ .

Displayed formulas in this passage

VII-D@1#eq:kappa · line 1782
$κ_{ρ} = \int_{0}^{1} \frac{ϑ ( u ) ^{4} - 1}{u} d u + \int_{1}^{\infty} \frac{ϑ ( u ) ^{4}}{u} d u .$
VII-D@1#eq:Ucommon · line 1793
$U_{m} = S + \int_{1}^{\infty} (A (r) - r) \frac{d r}{r} .$
unlabeled · line 1802
$∣ ⟨ A (r) - r, Φ ⟩ ∣ \frac{1}{r} \leq C_{K, m} r^{- 2} p_{K, 64} (Φ) .$

Nearby in this section

theorem: Theorem: Strong finite part with quantitative remainder