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

Theorem: Derivative–spectator finite part

Observation derivatives and smooth spectators · lines 841–893 · proof included or linked

IX-B@2#thm:descendants, IX-B@2#eq:descendantlimit, IX-B@2#eq:descendantrate

Assume ass:VIID. For every fixed $a, D$ , with $d = ord D$ ,

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

More precisely, for $Φ \in D_{K} (X)$ ,

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

The estimate is uniform when $a$ and the coefficients $b_{α}$ range over sets bounded in the compact smooth seminorms through order $64 + d$ . It includes compacts meeting $Rel \cap Δ_{13}$ .

Proof. Write $R_{ε} = T_{ε}^{ρ} - lo g ((μ ε)^{- 1}) r - u_{μ}^{ρ}$ . Then

⟨ a D R_{ε}, Φ ⟩ = ⟨ R_{ε}, D^{t} (a Φ) ⟩ .

Differential operators and multiplication preserve the test support. Leibniz's rule gives

p_{K, 64} (D^{t} (a Φ)) \leq C_{K, a, D} p_{K, 64 + d} (Φ) .

All coefficient bounds used here involve at most $64 + d$ derivatives. Apply (eq:input).

Every bounded subset of $D (X)$ has a common compact support and bounded smooth seminorms. Thus the estimate proves strong convergence and its asserted uniformity. The parent input already covers the reflected-diagonal intersection; the same Leibniz transfer covers it at order $64 + d$ , without a separate intersection argument.

Displayed formulas in this passage

IX-B@2#eq:descendantlimit · line 845
$a D T_{ε}^{ρ} - lo g ((μ ε)^{- 1}) a D r ⟶ a D u_{μ}^{ρ} strongly in D^{'} (X) .$
IX-B@2#eq:descendantrate · line 853
$⟨ a D (T_{ε}^{ρ} - lo g ((μ ε)^{- 1}) r - u_{μ}^{ρ}), Φ ⟩ \leq C_{K, a, D, ρ, m} ε (1 + ∣ lo g ε ∣) p_{K, 64 + d} (Φ) .$
unlabeled · line 875
$⟨ a D R_{ε}, Φ ⟩ = ⟨ R_{ε}, D^{t} (a Φ) ⟩ .$
unlabeled · line 880
$p_{K, 64} (D^{t} (a Φ)) \leq C_{K, a, D} p_{K, 64 + d} (Φ) .$

Theorem: Derivative–spectator finite part

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