Manuscript XX-C@1 · XX-C.tex · section · 7a8cab37

The two-Newton kernel

An intrinsic inverse-distance obstruction for two rods › The two-Newton kernel · lines 1460–1501

XX-C@1#eq:biharmonic, XX-C@1#eq:ERR

In three dimensions define

Φ (z) = - \frac{∣ z ∣}{8 π}, Δ^{2} Φ = δ, K_{ij} (z) = - \partial_{i} \partial_{j} Φ = \frac{δ _{ij} - n _{i} n _{j}}{8 π ∣ z ∣}, n = \frac{z}{∣ z ∣} .

The Fourier multiplier of $K_{ij}$ is $k_{i} k_{j} / ∣ k ∣^{4}$ . It is locally integrable at $k = 0$ . No infrared subtraction is needed for this multiplier.

The native double-rod causal distribution is

E_{R R}^{ab} (t, t^{'}; z) = K_{ij} * (C^{ia} e (t, t^{'}; \cdot) (C^{j b})^{†}) (z) .

The convolution is defined because its second factor is compactly supported in space. This prescription is the two-rod composition of the canonical kernel.

For a compact distribution $d$ , Taylor expansion of $K_{ij} (z - y)$ outside its support gives

K_{ij} * d (z) = K_{ij} (z) ⟨ d, 1 ⟩ + O (∣ z ∣^{- 2}) .

The statement holds for distributions of finite order: derivatives of the Taylor remainder on the compact support retain the inverse-square bound.

Displayed formulas in this passage

XX-C@1#eq:biharmonic · line 1463
$Φ (z) = - \frac{∣ z ∣}{8 π}, Δ^{2} Φ = δ, K_{ij} (z) = - \partial_{i} \partial_{j} Φ = \frac{δ _{ij} - n _{i} n _{j}}{8 π ∣ z ∣}, n = \frac{z}{∣ z ∣} .$
XX-C@1#eq:ERR · line 1478
$E_{R R}^{ab} (t, t^{'}; z) = K_{ij} * (C^{ia} e (t, t^{'}; \cdot) (C^{j b})^{†}) (z) .$
unlabeled · line 1493
$K_{ij} * d (z) = K_{ij} (z) ⟨ d, 1 ⟩ + O (∣ z ∣^{- 2}) .$