Manuscript XVI-I@1 · XVI-I.tex · assumption · a7046a8d

Assumption: Matching root and polarized kernel

Classical kernel, domains, and exact matching › The classical kernel input · lines 341–384

XVI-I@1#input:kernel, XVI-I@1#eq:t-test

The classical completion is the stated XII-D insertion

C_{3, D} (f, g) [h, ϕ] = A_{1} [f, g] (u_{D} [h]; ϕ),

with the response normalization $u_{D} = - P E_{0}^{R} J_{D}$ and preparation source

J_{D} = σ_{ε} J_{3} + D_{σ_{ε}} .

Its real polarization is the one represented in revised XVI-H by a compact kernel $t_{D}$ , symmetric in its two tensor slots:

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

In the chosen component frames, its general test-function action is

⟨ t_{D}, Φ ⟩ = A, I, J ∣ α ∣ + ∣ β ∣ \leq 2 \sum (- 1)^{∣ α ∣ + ∣ β ∣} ⟨ B_{A} (y, x), c^{A}_{I J; α β} (y) (\partial_{1}^{α} \partial_{2}^{β} Φ_{I J}) (y, y, x) ⟩ .

The coefficients are smooth and may be chosen symmetric:

c^{A}_{I J; α β} = c^{A}_{J I; β α} .

The compact root kernel $B_{A} (y, x)$ is the same root kernel used in the XII-D source-root construction.

Displayed formulas in this passage

XVI-I@1#eq:t-test · line 365
$⟨ t_{D}, Φ ⟩ = A, I, J ∣ α ∣ + ∣ β ∣ \leq 2 \sum (- 1)^{∣ α ∣ + ∣ β ∣} ⟨ B_{A} (y, x), c^{A}_{I J; α β} (y) (\partial_{1}^{α} \partial_{2}^{β} Φ_{I J}) (y, y, x) ⟩ .$
unlabeled · line 344
$C_{3, D} (f, g) [h, ϕ] = A_{1} [f, g] (u_{D} [h]; ϕ),$
unlabeled · line 351
$J_{D} = σ_{ε} J_{3} + D_{σ_{ε}} .$
unlabeled · line 359
$O_{pol} (a, c; ϕ) = ⟨ t_{D}, a \otimes c \otimes ϕ ⟩ .$
unlabeled · line 379
$c^{A}_{I J; α β} = c^{A}_{J I; β α} .$