Manuscript Em02-E@1 · Em02-E-v1.tex · lemma · f3795050

Lemma: Uniform Coulomb tails

Spatial tails and ultraviolet removal are separate limits › Elementary quantitative tail bounds · lines 1386–1432 · proof included or linked

Em02-E@1#lem:tail, Em02-E@1#eq:tail-G, Em02-E@1#eq:tail-dG

Let $K \subset R^{3}$ be compact, and suppose

∣ F_{δ} (y) ∣ \leq C (1 + ∣ y ∣)^{- q}

uniformly in $δ$ . For sufficiently large $R$ ,

x \in K sup \int_{∣ y ∣ > R} ∣ G_{Δ} (x - y) F_{δ} (y) ∣ d^{3} y \leq C_{K} R^{2 - q} (q > 2) .

For a first spatial derivative of $G_{Δ}$ ,

x \in K sup \int_{∣ y ∣ > R} ∣ \partial_{i} G_{Δ} (x - y) F_{δ} (y) ∣ d^{3} y \leq C_{K} R^{1 - q} (q > 1) .

The estimates remain valid for time or spectator derivatives when their sources satisfy the same uniform bounds.

Proof. For $R$ larger than twice a ball containing $K$ ,

∣ x - y ∣ \geq ∣ y ∣/2.

The Coulomb kernel is bounded by $C ∣ y ∣^{- 1}$ , and its first derivative by $C ∣ y ∣^{- 2}$ . Spherical integration therefore reduces the tails to

C \int_{R}^{\infty} r^{1 - q} d r, C \int_{R}^{\infty} r^{- q} d r,

respectively.

Displayed formulas in this passage

unlabeled · line 1389
$∣ F_{δ} (y) ∣ \leq C (1 + ∣ y ∣)^{- q}$
Em02-E@1#eq:tail-G · line 1395
$x \in K sup \int_{∣ y ∣ > R} ∣ G_{Δ} (x - y) F_{δ} (y) ∣ d^{3} y \leq C_{K} R^{2 - q} (q > 2) .$
Em02-E@1#eq:tail-dG · line 1405
$x \in K sup \int_{∣ y ∣ > R} ∣ \partial_{i} G_{Δ} (x - y) F_{δ} (y) ∣ d^{3} y \leq C_{K} R^{1 - q} (q > 1) .$
unlabeled · line 1420
$∣ x - y ∣ \geq ∣ y ∣/2.$
unlabeled · line 1426
$C \int_{R}^{\infty} r^{1 - q} d r, C \int_{R}^{\infty} r^{- q} d r,$