Manuscript IX-A-ST@1 · IX-A-ST.tex · proposition · 6e99d1b2

Proposition: Sharp adjoint, with its surface term

The sharp adjoint and its intrinsic boundary trace · lines 556–587 · proof included or linked

IX-A-ST@1#prop:adjoint, IX-A-ST@1#eq:qa

The compact kernel of $J (a)$ is

q_{a} = L^{*} a = C^{*} (Θ (t) b) = Θ (t) C^{*} b + δ (t) B (b ∣_{Σ}) .

There is no $δ^{'} (t)$ term, and

supp q_{a} \subseteq J_{η}^{-} (supp a) \cap {t \geq 0} ⋐ U, WF (q_{a}) \subseteq N^{*} Σ ∖ 0.

Proof. The zero-data prescription gives $L h = G_{R} (Θ C h)$ on the observation region. Transpose the retarded kernel against the compact source:

J (a) [h] = ⟨ G_{A} a, Θ C h ⟩ = ⟨ C^{*} (Θ b), h ⟩ .

All relevant intersections are contained in $K_{*}$ . The product rule gives (eq:qa). The adjoint has differential order one, so it differentiates $Θ$ at most once. The advanced response to a smooth compact source is smooth; only the surface factor can contribute wavefront. The support statement follows from finite propagation.

Displayed formulas in this passage

IX-A-ST@1#eq:qa · line 559
$q_{a} = L^{*} a = C^{*} (Θ (t) b) = Θ (t) C^{*} b + δ (t) B (b ∣_{Σ}) .$
unlabeled · line 565
$supp q_{a} \subseteq J_{η}^{-} (supp a) \cap {t \geq 0} ⋐ U, WF (q_{a}) \subseteq N^{*} Σ ∖ 0.$
unlabeled · line 576
$J (a) [h] = ⟨ G_{A} a, Θ C h ⟩ = ⟨ C^{*} (Θ b), h ⟩ .$

Proposition: Sharp adjoint, with its surface term

Displayed formulas in this passage

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