Manuscript IX-Free-2@1 · IX-Free-2.tex · section · abb2397a

The concrete degree-four relation kernel

The concrete degree-four relation kernel · lines 695–771

IX-Free-2@1#eq:P, IX-Free-2@1#eq:Prelation, IX-Free-2@1#eq:lambdaL, IX-Free-2@1#eq:lambdaR, IX-Free-2@1#eq:annihilation

For

v = i \sum α_{i} f_{i} \otimes g_{i} \in E_{L}, v (x, y) = i \sum α_{i} f_{i} (x) g_{i} (y),

use the inherited injective realization of this algebraic tensor product as smooth compact joint functions.

Let $J^{μ}_{α β} (x, z)$ be the distribution kernel of $L$ , and let $e_{00}^{α β} h_{α β} = h_{00}$ . The relation kernel is copied unchanged:

P [v] (x, z_{1}, z_{2}, z_{3}) = Sym_{(z_{i}, α_{i}, β_{i})} \partial_{μ}^{x} [v (x, z_{2}) J^{μ}_{α_{1} β_{1}} (x, z_{1}) \times δ (z_{2} - z_{3}) e_{00}^{α_{2} β_{2}} e_{00}^{α_{3} β_{3}}] .

The symmetrization is the average over the three metric slots. IX-Free establishes

p_{L} (v) = ⟨ P [v], ϕ (x) h (z_{1}) h (z_{2}) h (z_{3}) ⟩, p_{L} (v) = 0 ⟺ P [v] = 0.

For $v_{R} (y, x^{'})$ , set $v (x^{'}, y) = v_{R} (y, x^{'})$ . The right relation is

p_{R} (v_{R}) = 0 ⟺ P [v] = 0.

Full coefficient recovery uses one scalar derivative and three metric derivatives. The metric polarization factor is $3!$ . Two metric derivatives do not recover this coefficient.

The prescribed values whose descent remains to be proved are

λ^{L}_{1} (f, g, f^{'}) λ^{R}_{1} (f, g, f^{'}) = G_{13} + G_{23}, = G_{12} + G_{13}, λ^{L}_{2} (f, g, f^{'}) λ^{R}_{2} (f, g, f^{'}) = G_{13, 23}, = G_{12, 13} .

They descend precisely when

λ^{L}_{j} ((ker p_{L}) \otimes T) = 0, λ^{R}_{j} (T \otimes (ker p_{R})) = 0, j = 1, 2.

We now evaluate the kernels. No inference from injectivity of $q_{L}, q_{R}$ will be used.

Displayed formulas in this passage

IX-Free-2@1#eq:P · line 710
$P [v] (x, z_{1}, z_{2}, z_{3}) = Sym_{(z_{i}, α_{i}, β_{i})} \partial_{μ}^{x} [v (x, z_{2}) J^{μ}_{α_{1} β_{1}} (x, z_{1}) \times δ (z_{2} - z_{3}) e_{00}^{α_{2} β_{2}} e_{00}^{α_{3} β_{3}}] .$
IX-Free-2@1#eq:Prelation · line 726
$p_{L} (v) = ⟨ P [v], ϕ (x) h (z_{1}) h (z_{2}) h (z_{3}) ⟩, p_{L} (v) = 0 ⟺ P [v] = 0.$
IX-Free-2@1#eq:lambdaL · line 745
$λ^{L}_{1} (f, g, f^{'}) λ^{R}_{1} (f, g, f^{'}) = G_{13} + G_{23}, = G_{12} + G_{13}, λ^{L}_{2} (f, g, f^{'}) λ^{R}_{2} (f, g, f^{'}) = G_{13, 23}, = G_{12, 13} .$
IX-Free-2@1#eq:lambdaR · line 745
$λ^{L}_{1} (f, g, f^{'}) λ^{R}_{1} (f, g, f^{'}) = G_{13} + G_{23}, = G_{12} + G_{13}, λ^{L}_{2} (f, g, f^{'}) λ^{R}_{2} (f, g, f^{'}) = G_{13, 23}, = G_{12, 13} .$
IX-Free-2@1#eq:annihilation · line 760
$λ^{L}_{j} ((ker p_{L}) \otimes T) = 0, λ^{R}_{j} (T \otimes (ker p_{R})) = 0, j = 1, 2.$
unlabeled · line 699
$v = i \sum α_{i} f_{i} \otimes g_{i} \in E_{L}, v (x, y) = i \sum α_{i} f_{i} (x) g_{i} (y),$
unlabeled · line 736
$p_{R} (v_{R}) = 0 ⟺ P [v] = 0.$