Manuscript XV-B@2 · XV-B.tex · theorem · 4d3f12e8

Theorem: Point-puncture annihilator-preserving extension

Natural null defects and point-puncture corrections › A simultaneous correction at one point · lines 1408–1484 · proof included or linked

XV-B@2#thm:pointcorrection, XV-B@2#eq:pointD, XV-B@2#eq:allannihilators

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Let $X \subset R^{d}$ be open with $0 \in X$ , and let $T^{\circ} \in D^{'} (X ∖ {0})$ have a distributional extension $T_{0} \in D^{'} (X)$ . Define its smooth annihilator ideal

I = {f \in C^{\infty} (X) : f T^{\circ} = 0 on X ∖ {0}} .

Then there exists

D = ∣ ν ∣ \leq M \sum c_{ν} \partial^{ν} δ_{0}

for some finite $M$ such that

T = T_{0} - D

still extends $T^{\circ}$ and satisfies

f T = 0 (f \in I) .

One may take $M$ to be a distributional order bound for $T_{0}$ on a compact neighborhood of zero.

Proof. Fix such an order bound $M$ . For every $f \in I$ , the distribution $f T_{0}$ is supported at zero. If $j_{0}^{M} f = 0$ , the point case of Lemma lem:jetdefect gives $f T_{0} = 0$ .

Let

J_{0}^{M} = C^{\infty} (X) / {f : j_{0}^{M} f = 0}

be the finite-dimensional algebra of jets through order $M$ , and let $I_{M} = j_{0}^{M} (I)$ be the image ideal in that algebra. Choose $χ \in D (X)$ equal to one near zero and define

ℓ (j_{0}^{M} f) = ⟨ f T_{0}, χ ⟩, f \in I .

This is well defined: representatives with the same jet differ by an element of $I$ with vanishing $M$ -jet, whose product with $T_{0}$ is zero. The value is independent of the chosen $χ$ , since $f T_{0}$ is supported at zero.

Extend the linear functional $ℓ$ from $I_{M}$ to a linear functional $Λ$ on the finite-dimensional vector space $J_{0}^{M}$ . Define

⟨ D, ϕ ⟩ = Λ (j_{0}^{M} ϕ) .

This is a distribution supported at zero. In the jet basis $e_{ν} = [x^{ν} / ν!]$ , it has the explicit form

D = ∣ ν ∣ \leq M \sum (- 1)^{∣ ν ∣} Λ (e_{ν}) \partial^{ν} δ_{0},

which is (eq:pointD).

For $f \in I$ and $ϕ \in D (X)$ , the product $f ϕ$ is also in $I$ . Because $(f ϕ) T_{0}$ is supported at zero,

⟨ f T_{0}, ϕ ⟩ = ⟨(f ϕ) T_{0}, χ ⟩ = ℓ (j_{0}^{M} (f ϕ)) = Λ (j_{0}^{M} f \cdot j_{0}^{M} ϕ) = ⟨ f D, ϕ ⟩ .

Thus $f (T_{0} - D) = 0$ for every $f \in I$ . Subtracting $D$ changes no restriction off zero.

Displayed formulas in this passage

XV-B@2#eq:pointD · line 1418
$D = ∣ ν ∣ \leq M \sum c_{ν} \partial^{ν} δ_{0}$
XV-B@2#eq:allannihilators · line 1427
$f T = 0 (f \in I) .$
unlabeled · line 1413
$I = {f \in C^{\infty} (X) : f T^{\circ} = 0 on X ∖ {0}} .$
unlabeled · line 1423
$T = T_{0} - D$
unlabeled · line 1441
$J_{0}^{M} = C^{\infty} (X) / {f : j_{0}^{M} f = 0}$
unlabeled · line 1448
$ℓ (j_{0}^{M} f) = ⟨ f T_{0}, χ ⟩, f \in I .$
unlabeled · line 1460
$⟨ D, ϕ ⟩ = Λ (j_{0}^{M} ϕ) .$
unlabeled · line 1465
$D = ∣ ν ∣ \leq M \sum (- 1)^{∣ ν ∣} Λ (e_{ν}) \partial^{ν} δ_{0},$
unlabeled · line 1473
$⟨ f T_{0}, ϕ ⟩ = ⟨(f ϕ) T_{0}, χ ⟩ = ℓ (j_{0}^{M} (f ϕ)) = Λ (j_{0}^{M} f \cdot j_{0}^{M} ϕ) = ⟨ f D, ϕ ⟩ .$

Theorem: Point-puncture annihilator-preserving extension

Displayed formulas in this passage

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