Manuscript IX-C@1 · IX-C.tex · assumption · 03ce8e6b

Assumption: VII-D calibration, as restated in IX-B-v2

Inherited data and the separation from V › Explicit analytical premise · lines 426–476

IX-C@1#ass:calibration, IX-C@1#eq:finitepart

For the prescribed radial regulator placement,

T_{ε}^{ρ} (x, y, x^{'}) = 2 K_{ε} (x, y) K_{ε} (x^{'}, y) D_{ε} (x - x^{'}) .

There are distributions $r, u_{μ}^{ρ} \in D^{'} (X)$ such that

u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r,

and, for every compact $K_{0} \subset X$ ,

⟨ T_{ε}^{ρ} - lo g ((μ ε)^{- 1}) r - u_{μ}^{ρ}, Φ ⟩ \leq C_{K_{0}, ρ, m} ε (1 + ∣ lo g ε ∣) p_{K_{0}, 64} (Φ) .

The estimate includes compacts meeting $Rel \cap Δ_{13}$ .

The residue is

r = c_{0} Ψ_{*} ν, c_{0} = \frac{1}{64 ( 2 π ) ^{3}},

where

Ψ (z, n, t, s, t^{'}) = ((t, z + t n), (s, z - s n), (t^{'}, z + t^{'} n)),

and

d ν = d^{3} z d Ω (n) d t d s d t^{'} .

Thus $Rel = supp r$ .

For the prescribed regulator family,

u_{μ}^{ρ_{a}} - u_{μ}^{ρ_{b}} = c_{ab} r, c_{ab} \in R .

The parent distributions satisfy

(u_{μ}^{ρ})^{†} = u_{μ}^{ρ}, r^{†} = r,

where dagger conjugates and exchanges $x, x^{'}$ .

Displayed formulas in this passage

IX-C@1#eq:finitepart · line 438
$⟨ T_{ε}^{ρ} - lo g ((μ ε)^{- 1}) r - u_{μ}^{ρ}, Φ ⟩ \leq C_{K_{0}, ρ, m} ε (1 + ∣ lo g ε ∣) p_{K_{0}, 64} (Φ) .$
unlabeled · line 429
$T_{ε}^{ρ} (x, y, x^{'}) = 2 K_{ε} (x, y) K_{ε} (x^{'}, y) D_{ε} (x - x^{'}) .$
unlabeled · line 434
$u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r,$
unlabeled · line 449
$r = c_{0} Ψ_{*} ν, c_{0} = \frac{1}{64 ( 2 π ) ^{3}},$
unlabeled · line 453
$Ψ (z, n, t, s, t^{'}) = ((t, z + t n), (s, z - s n), (t^{'}, z + t^{'} n)),$
unlabeled · line 461
$d ν = d^{3} z d Ω (n) d t d s d t^{'} .$
unlabeled · line 468
$u_{μ}^{ρ_{a}} - u_{μ}^{ρ_{b}} = c_{ab} r, c_{ab} \in R .$
unlabeled · line 472
$(u_{μ}^{ρ})^{†} = u_{μ}^{ρ}, r^{†} = r,$