Manuscript XVI-K@1 · XVI-K.tex · lemma · aa7553bb

Lemma: Explicit derivative defects

A coherent extension map on the finite span › Contact defects of differentiation · lines 1366–1441 · proof included or linked

XVI-K@1#lem:defect, XVI-K@1#eq:defect1, XVI-K@1#eq:defect2

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Define

D_{α} = R_{K} (2 S \partial_{α} S) - \partial_{α} R_{K} (S^{2}) .

It is supported at zero and has the form

D_{α} = ∣ γ ∣ \leq 1 \sum c_{α, γ} \partial^{γ} δ, c_{α, γ} = (- 1)^{∣ γ ∣} ⟨ q ˉ^{R}, \partial_{α} w^{(1)}_{γ} ⟩ .

Similarly,

D_{β α} = R_{K} ((\partial_{β} S) (\partial_{α} S) + S \partial_{β} \partial_{α} S) - \partial_{β} R_{K} (S \partial_{α} S)

is supported at zero, and

D_{β α} = ∣ γ ∣ \leq 2 \sum c_{β α, γ} \partial^{γ} δ, c_{β α, γ} = (- 1)^{∣ γ ∣} ⟨ R_{K} (S \partial_{α} S), \partial_{β} w^{(2)}_{γ} ⟩ .

No nonzero or zero value of these coefficients is asserted.

Proof. The two terms defining each defect agree off zero by (eq:Leibniz1) or (eq:Leibniz2). The defects are therefore point-supported. Differentiation raises scaling degree by at most one BF (Lemma 5.1), so their bounds are five and six. Only derivatives of $δ$ of orders at most one and two, respectively, can occur.

For the first defect, its product-extension term has zero moments on $w^{(1)}_{γ}$ . Distributional differentiation gives

⟨ D_{α}, w^{(1)}_{γ} ⟩ = ⟨ q ˉ^{R}, \partial_{α} w^{(1)}_{γ} ⟩ .

On the other hand,

⟨ \partial^{β} δ, w^{(1)}_{γ} ⟩ = (- 1)^{∣ β ∣} δ_{β γ} .

Thus

(- 1)^{∣ γ ∣} c_{α, γ} = ⟨ q ˉ^{R}, \partial_{α} w^{(1)}_{γ} ⟩,

which proves (eq:defect1). The same sign calculation on the order-two Taylor complement proves (eq:defect2).

Displayed formulas in this passage

XVI-K@1#eq:defect1 · line 1376
$D_{α} = ∣ γ ∣ \leq 1 \sum c_{α, γ} \partial^{γ} δ, c_{α, γ} = (- 1)^{∣ γ ∣} ⟨ q ˉ^{R}, \partial_{α} w^{(1)}_{γ} ⟩ .$
XVI-K@1#eq:defect2 · line 1396
$D_{β α} = ∣ γ ∣ \leq 2 \sum c_{β α, γ} \partial^{γ} δ, c_{β α, γ} = (- 1)^{∣ γ ∣} ⟨ R_{K} (S \partial_{α} S), \partial_{β} w^{(2)}_{γ} ⟩ .$
unlabeled · line 1369
$D_{α} = R_{K} (2 S \partial_{α} S) - \partial_{α} R_{K} (S^{2}) .$
unlabeled · line 1388
$D_{β α} = R_{K} ((\partial_{β} S) (\partial_{α} S) + S \partial_{β} \partial_{α} S) - \partial_{β} R_{K} (S \partial_{α} S)$
unlabeled · line 1422
$⟨ D_{α}, w^{(1)}_{γ} ⟩ = ⟨ q ˉ^{R}, \partial_{α} w^{(1)}_{γ} ⟩ .$
unlabeled · line 1428
$⟨ \partial^{β} δ, w^{(1)}_{γ} ⟩ = (- 1)^{∣ β ∣} δ_{β γ} .$
unlabeled · line 1433
$(- 1)^{∣ γ ∣} c_{α, γ} = ⟨ q ˉ^{R}, \partial_{α} w^{(1)}_{γ} ⟩,$

Nearby in this section

section: Contact defects of differentiation