Manuscript XVI-H@1 · XVI-H.tex · theorem · af895076

Theorem: Compact kernel representation

Construction of the compact three-slot kernel › Definition on arbitrary test functions · lines 992–1050 · proof included or linked

XVI-H@1#thm:trep, XVI-H@1#eq:O-t

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Under Inputs input:source–input:root, (eq:t-test) defines a real compactly supported distributional kernel on ambient $M^{3}$ , symmetric under interchange of its tensor slots. For real smooth $a, c, ϕ$ ,

O_{pol} (a, c; ϕ) = ⟨ t_{D}, a \otimes c \otimes ϕ ⟩ .

The pairing is meaningful without compact support of the configuration fields.

The representation uses the root identity and does not require the response-slot formula (eq:O-A).

Proof. A derivative of a smooth test followed by evaluation at $(y_{1}, y_{2}) = (y, y)$ is smooth. For tests supported in a fixed compact set, each derivative seminorm of this evaluated function is bounded by a finite derivative seminorm of the original test, by the chain rule. Smooth multiplication by the fixed coefficients preserves these estimates. Continuity of $B$ therefore proves that (eq:t-test) is a distribution.

The definition can only detect test germs on the image of $supp (c B)$ under $(y, x) \mapsto (y, y, x)$ . That image is compact by (eq:B-support). Thus $t_{D}$ is compactly supported. Formula (eq:c-sym) and exchange of $α, β$ in the finite sum prove tensor-slot symmetry.

For a decomposable test,

(\partial_{y_{1}}^{α} \partial_{y_{2}}^{β} (a_{I} (y_{1}) c_{J} (y_{2}) ϕ (x))) (y, y, x) = (\partial^{α} a_{I}) (y) (\partial^{β} c_{J}) (y) ϕ (x) .

Consequently (eq:j-id) and (eq:B-root) give

⟨ t_{D}, a \otimes c \otimes ϕ ⟩ = ⟨ B_{A}, J_{D}^{A} (a, c) \otimes ϕ ⟩ = \int_{M} a_{f, g} [ϕ]_{A} (y) J_{D}^{A} (a, c) (y) d^{4} y .

The last expression is (eq:O-root).

Compactness supplies a cutoff-independent pairing with the smooth, possibly noncompact fields. This justifies the same calculation for all inputs in the stated real smooth domain.

Displayed formulas in this passage

XVI-H@1#eq:O-t · line 998
$O_{pol} (a, c; ϕ) = ⟨ t_{D}, a \otimes c \otimes ϕ ⟩ .$
unlabeled · line 1028
$(\partial_{y_{1}}^{α} \partial_{y_{2}}^{β} (a_{I} (y_{1}) c_{J} (y_{2}) ϕ (x))) (y, y, x) = (\partial^{α} a_{I}) (y) (\partial^{β} c_{J}) (y) ϕ (x) .$
unlabeled · line 1036
$⟨ t_{D}, a \otimes c \otimes ϕ ⟩ = ⟨ B_{A}, J_{D}^{A} (a, c) \otimes ϕ ⟩ = \int_{M} a_{f, g} [ϕ]_{A} (y) J_{D}^{A} (a, c) (y) d^{4} y .$

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section: Definition on arbitrary test functions