Manuscript XII-C@1 · XII-C.tex · section · dead1004

An honest classical source-pairing functional

C4: successor data without a response › An honest classical source-pairing functional · lines 1244–1276

XII-C@1#eq:source-pairing

A successor may prescribe a compact root datum: a continuous linear map

a_{f, g} : C^{\infty} (M) ⟶ C_{c}^{\infty} (M; Sym^{2} T^{*} M)

whose range is supported in a fixed compact set

K_{a} \subset J^{-} (supp f) \cap {t \geq 0} .

Here the range support and the map are given data. They are not claimed to arise from a retarded construction already performed in this paper.

For such a datum, the classical trilinear source pairing is

I_{D, a} (f, g; γ_{1}, γ_{2}, ϕ) = \int a_{f, g} [ϕ]_{μν} (y) J_{3, D}^{μν} (γ_{1}, γ_{2}) (y) d^{4} y .

It is well defined on smooth configurations: the test has compact support and the source is a local smooth polynomial.

Equation (eq:source-pairing) is an actual classical functional with an independently supplied root datum. It is not asserted to be the completed insertion of XI-E. No claim about its distributional kernel or admission is made.

Displayed formulas in this passage

XII-C@1#eq:source-pairing · line 1261
$I_{D, a} (f, g; γ_{1}, γ_{2}, ϕ) = \int a_{f, g} [ϕ]_{μν} (y) J_{3, D}^{μν} (γ_{1}, γ_{2}) (y) d^{4} y .$
unlabeled · line 1248
$a_{f, g} : C^{\infty} (M) ⟶ C_{c}^{\infty} (M; Sym^{2} T^{*} M)$
unlabeled · line 1253
$K_{a} \subset J^{-} (supp f) \cap {t \geq 0} .$