Manuscript XV-C@1 · XV-C.tex · theorem · 5afd124d

Theorem: Complete null criterion for the specified module

Complete null tests in a finite normal-jet module › A specified local module · lines 572–639 · proof included or linked

XV-C@1#thm:jetnull, XV-C@1#eq:B, XV-C@1#eq:jetnormal, XV-C@1#eq:completekernel

For $a \in P_{r}^{jet} (T)$ , set

B_{γ} (a) = α \geq γ ∣ α ∣ \leq r \sum (- 1)^{∣ α - γ ∣} (γ α) (\partial_{q}^{α - γ} a_{α}) (x, 0) T .

Then

Rec_{T} (a) = ∣ γ ∣ \leq r \sum E_{γ} B_{γ} (a),

and

ker Rec_{T} = {a : B_{γ} (a) = 0 in D^{'} (X) for all ∣ γ ∣ \leq r} .

Thus (eq:completekernel) is a complete null-relation test for this module, not merely a list of necessary identities.

Proof. For one summand and any test $ϕ$ ,

⟨ a_{α} E_{α} T, ϕ ⟩ = (- 1)^{∣ α ∣} ⟨ T, \partial_{q}^{α} (a_{α} ϕ) (x, 0) ⟩ .

The Leibniz rule gives

\partial_{q}^{α} (a_{α} ϕ) = β \leq α \sum (β α) (\partial_{q}^{β} a_{α}) (\partial_{q}^{α - β} ϕ) .

Using (eq:insert), this becomes

a_{α} E_{α} T = β \leq α \sum (- 1)^{∣ β ∣} (β α) E_{α - β} ((\partial_{q}^{β} a_{α}) (x, 0) T) .

Collecting the coefficient of $E_{γ}$ yields (eq:B) and (eq:jetnormal). Lemma lem:extraction then proves (eq:completekernel).

Displayed formulas in this passage

XV-C@1#eq:B · line 575
$B_{γ} (a) = α \geq γ ∣ α ∣ \leq r \sum (- 1)^{∣ α - γ ∣} (γ α) (\partial_{q}^{α - γ} a_{α}) (x, 0) T .$
XV-C@1#eq:jetnormal · line 585
$Rec_{T} (a) = ∣ γ ∣ \leq r \sum E_{γ} B_{γ} (a),$
XV-C@1#eq:completekernel · line 591
$ker Rec_{T} = {a : B_{γ} (a) = 0 in D^{'} (X) for all ∣ γ ∣ \leq r} .$
unlabeled · line 606
$⟨ a_{α} E_{α} T, ϕ ⟩ = (- 1)^{∣ α ∣} ⟨ T, \partial_{q}^{α} (a_{α} ϕ) (x, 0) ⟩ .$
unlabeled · line 615
$\partial_{q}^{α} (a_{α} ϕ) = β \leq α \sum (β α) (\partial_{q}^{β} a_{α}) (\partial_{q}^{α - β} ϕ) .$
unlabeled · line 624
$a_{α} E_{α} T = β \leq α \sum (- 1)^{∣ β ∣} (β α) E_{α - β} ((\partial_{q}^{β} a_{α}) (x, 0) T) .$