Manuscript VII-A-ST@1 · VII-A-ST.tex · lemma · c39a67a3

Lemma: Two linear identities

First-order repair and the larger-support residual · lines 751–787 · proof included or linked

VII-A-ST@1#lem:linear, VII-A-ST@1#eq:linear, VII-A-ST@1#eq:QK-shell

For smooth $ξ, u$ ,

C [K ξ] Q_{K ξ}^{(m)} u = □_{η} ξ, = D_{ξ} P_{m} u - P_{m} X_{ξ} u .

If $ξ$ vanishes near $Σ_{0}$ , then $w [K ξ] = ξ$ . On $P_{m} ϕ_{0} = 0$ ,

Q_{K ξ}^{(m)} ϕ_{0} = - P_{m} (X_{ξ} ϕ_{0}) .

Proof. The trace of $K ξ$ is $2 \partial_{ρ} ξ^{ρ}$ ; the mixed derivative cancels in $C [K ξ]$ . The zero-data left-inverse identity then gives $w [K ξ] = ξ$ .

For the second formula,

A^{μν} (K ξ) = (\partial_{ρ} ξ^{ρ}) η^{μν} - \partial^{μ} ξ^{ν} - \partial^{ν} ξ^{μ}, a (K ξ) = \partial_{ρ} ξ^{ρ} .

Its divergence is $- □_{η} ξ^{ν}$ . Thus

Q_{K ξ}^{(m)} u = (\partial_{ρ} ξ^{ρ}) P_{m} u - (□_{η} ξ^{ν}) \partial_{ν} u - 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} u .

This is $D_{ξ} P_{m} u - P_{m} X_{ξ} u$ .

Displayed formulas in this passage

VII-A-ST@1#eq:linear · line 754
$C [K ξ] Q_{K ξ}^{(m)} u = □_{η} ξ, = D_{ξ} P_{m} u - P_{m} X_{ξ} u .$
VII-A-ST@1#eq:QK-shell · line 761
$Q_{K ξ}^{(m)} ϕ_{0} = - P_{m} (X_{ξ} ϕ_{0}) .$
unlabeled · line 773
$A^{μν} (K ξ) = (\partial_{ρ} ξ^{ρ}) η^{μν} - \partial^{μ} ξ^{ν} - \partial^{ν} ξ^{μ}, a (K ξ) = \partial_{ρ} ξ^{ρ} .$
unlabeled · line 780
$Q_{K ξ}^{(m)} u = (\partial_{ρ} ξ^{ρ}) P_{m} u - (□_{η} ξ^{ν}) \partial_{ν} u - 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} u .$

Lemma: Two linear identities

Displayed formulas in this passage

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