Manuscript VII-C@1 · VII-C.tex · theorem · 8b2ade29

Theorem: Explicit mixed-graph divergence

The mandatory mixed triple › The logarithmic coefficient · lines 1235–1338 · proof included or linked

VII-C@1#thm:log, VII-C@1#eq:log, VII-C@1#eq:residue

For real $f, f^{'}, g$ as above,

G_{ε} (f, g, f^{'}) = C (f, g, f^{'}) lo g (ε^{- 1}) + O (1),

where

C (f, g, f^{'}) = \frac{1}{64 ( 2 π ) ^{3}} \int_{M} g (y) d^{4} y \int_{S^{2}} A_{f} (y, n) A_{f^{'}} (y, n) d Ω (n) .

The $O (1)$ bound is for the fixed three test functions and fixed mass. The coefficient of the logarithm is independent of the chosen normalized radial mollifier.

Proof. Insert Lemma lem:asymptotic into (eq:spectral) and use polar coordinates $q = r n$ . For $r \geq 1$ ,

r^{2} ω_{m} (r) = r^{3} (1 + O (r^{- 2})) .

The leading contribution is therefore

C (f, g, f^{'}) I_{ε}, I_{ε} = \int_{1}^{\infty} \frac{M _{ε} ( r ) ^{2} μ ( ε r ) ^{2}}{r} d r .

We first check that the remainder is uniformly bounded. By (eq:Ebound), the error in the product of the two square brackets in (eq:Fasymptotic) is bounded by a constant times

(r^{- 1} + ε) + (r^{- 1} + ε)^{2} .

Also, for every $N$ ,

∣ M_{ε} (r) ∣ \leq C_{N} (1 + ε r)^{- N} .

Thus the error integral is controlled by

C \int_{1}^{\infty} \frac{( 1 + ε r ) ^{- N}}{r} [r^{- 1} + ε + (r^{- 1} + ε)^{2}] d r .

This is uniformly bounded: terms containing $r^{- 1}$ are integrable without the cutoff, and

ε \int_{1}^{\infty} \frac{( 1 + ε r ) ^{- N}}{r} d r = O (ε lo g (ε^{- 1}) + ε) .

The correction from $ω_{m} (r) / r - 1 = O (r^{- 2})$ is integrable as well. The region $0 \leq r \leq 1$ is uniformly bounded by the final assertion of Lemma lem:asymptotic.

It remains to evaluate $I_{ε}$ . For $r \geq 1$ ,

r^{2} + \frac{m ^{2}}{2} - r \leq C_{m} r^{- 1},

so boundedness of $μ^{'}$ gives

∣ M_{ε} (r) - μ (ε r) ∣ \leq C_{m} ε r^{- 1} .

Consequently,

I_{ε} = \int_{1}^{\infty} \frac{μ ( ε r ) ^{4}}{r} d r + O (ε) .

Changing variables $u = ε r$ ,

\int_{1}^{\infty} \frac{μ ( ε r ) ^{4}}{r} d r = \int_{ε}^{\infty} \frac{μ ( u ) ^{4}}{u} d u = lo g (ε^{- 1}) + O (1),

because $μ (0) = 1$ , $μ (u)^{4} - 1 = O (u)$ near zero, and $μ$ decreases rapidly at infinity. This proves (eq:log) with the coefficient (eq:residue).

Displayed formulas in this passage

VII-C@1#eq:log · line 1238
$G_{ε} (f, g, f^{'}) = C (f, g, f^{'}) lo g (ε^{- 1}) + O (1),$
VII-C@1#eq:residue · line 1245
$C (f, g, f^{'}) = \frac{1}{64 ( 2 π ) ^{3}} \int_{M} g (y) d^{4} y \int_{S^{2}} A_{f} (y, n) A_{f^{'}} (y, n) d Ω (n) .$
unlabeled · line 1266
$r^{2} ω_{m} (r) = r^{3} (1 + O (r^{- 2})) .$
unlabeled · line 1270
$C (f, g, f^{'}) I_{ε}, I_{ε} = \int_{1}^{\infty} \frac{M _{ε} ( r ) ^{2} μ ( ε r ) ^{2}}{r} d r .$
unlabeled · line 1282
$(r^{- 1} + ε) + (r^{- 1} + ε)^{2} .$
unlabeled · line 1286
$∣ M_{ε} (r) ∣ \leq C_{N} (1 + ε r)^{- N} .$
unlabeled · line 1290
$C \int_{1}^{\infty} \frac{( 1 + ε r ) ^{- N}}{r} [r^{- 1} + ε + (r^{- 1} + ε)^{2}] d r .$
unlabeled · line 1299
$ε \int_{1}^{\infty} \frac{( 1 + ε r ) ^{- N}}{r} d r = O (ε lo g (ε^{- 1}) + ε) .$
unlabeled · line 1311
$r^{2} + \frac{m ^{2}}{2} - r \leq C_{m} r^{- 1},$
unlabeled · line 1317
$∣ M_{ε} (r) - μ (ε r) ∣ \leq C_{m} ε r^{- 1} .$
unlabeled · line 1321
$I_{ε} = \int_{1}^{\infty} \frac{μ ( ε r ) ^{4}}{r} d r + O (ε) .$
unlabeled · line 1327
$\int_{1}^{\infty} \frac{μ ( ε r ) ^{4}}{r} d r = \int_{ε}^{\infty} \frac{μ ( u ) ^{4}}{u} d u = lo g (ε^{- 1}) + O (1),$