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

Partial witness contraction and recovery of the density

Partial witness contraction and recovery of the density · lines 1172–1212

XVI-H@1#eq:j-pair, XVI-H@1#eq:j

Let $t_{D}^{C}$ denote the complex-linear extension of the real distribution $t_{D}$ to complex test fields. Its evaluation on complex tensor arguments and a real scalar implements the four-term extension (eq:OC).

Current XVI-G, prop:kernel-slot, supplies

I_{λ} [ϕ] = \int_{M} j_{λ} (x) ϕ (x) d^{4} x,

where

j_{λ} = - \partial_{ν} (g V_{λ}^{μν} \partial_{μ} v_{f}) - m^{2} g v_{f} b_{1} [R_{λ}] + \partial_{μ} (f X_{cand}^{μ}), v_{f} = E_{m}^{A} f .

Here

X_{cand}^{μ} (t) = - \int_{0}^{t} σ (s) V_{λ}^{0 μ} (s) d s,

and

j_{λ} \in C_{c}^{\infty} (M_{T}^{\circ}; C), supp j_{λ} \subset supp f \cup supp g .

Its extension to ambient space is smooth because the support is compact-interior.

Displayed formulas in this passage

XVI-H@1#eq:j-pair · line 1182
$I_{λ} [ϕ] = \int_{M} j_{λ} (x) ϕ (x) d^{4} x,$
XVI-H@1#eq:j · line 1187
$j_{λ} = - \partial_{ν} (g V_{λ}^{μν} \partial_{μ} v_{f}) - m^{2} g v_{f} b_{1} [R_{λ}] + \partial_{μ} (f X_{cand}^{μ}), v_{f} = E_{m}^{A} f .$
unlabeled · line 1200
$X_{cand}^{μ} (t) = - \int_{0}^{t} σ (s) V_{λ}^{0 μ} (s) d s,$
unlabeled · line 1205
$j_{λ} \in C_{c}^{\infty} (M_{T}^{\circ}; C), supp j_{λ} \subset supp f \cup supp g .$

Partial witness contraction and recovery of the density

Displayed formulas in this passage

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