Manuscript I-B-VP-0@1 · I-B-VP-0-v1.tex · lemma · 8a9d0682

Lemma: Heat kernel and compact-test covariance

Prescribed geometry, bundle, and spectral state › Massless infrared finiteness · lines 283–343 · proof included or linked

I-B-VP-0@1#lem:IR, I-B-VP-0@1#eq:heat, I-B-VP-0@1#eq:heatbound

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The spatial Friedrichs heat kernel, relative to $d V$ , is

K_{β} (s; x, x^{'}) = \frac{e ^{- Δ z^{2} / (4 s)}}{4 π s} \frac{e ^{- (r^{2} + r^{'2}) / (4 s)}}{4 π α s} \times n \in Z \sum e^{i (n + β) (φ - φ^{'})} I_{ν_{n}} (\frac{r r ^{'}}{2 s}) .

On every compact regular spatial set $K$ ,

x, x^{'} \in K sup ∣ K_{β} (s; x, x^{'}) ∣ \leq C_{K} s^{- 3/2 - ν_{*}}, s \geq 1, ν_{*} = \frac{min ( β , 1 - β )}{α} .

Every compact smooth spatial section $v$ belongs to $dom (B_{β}^{- 1/4})$ , continuously in the fixed-support test topology.

Proof. The radial Friedrichs transform uses $J_{ν_{n}} (k r)$ . Weber's integral

\int_{0}^{\infty} e^{- s k^{2}} J_{ν} (k r) J_{ν} (k r^{'}) k d k = \frac{e ^{- (r^{2} + r^{'2}) / (4 s)}}{2 s} I_{ν} (\frac{r r ^{'}}{2 s})

gives (eq:heat) after the angular normalization and longitudinal heat kernel are included Watson,CheungIBG. For bounded $r, r^{'}$ and $s \geq 1$ , the Bessel power series gives

I_{ν} (\frac{r r ^{'}}{2 s}) \leq C_{K} \frac{( C _{K} / s ) ^{ν}}{Γ ( ν + 1 )} .

The orders grow linearly in $∣ n ∣$ , so their sum is bounded by a constant times $s^{- ν_{*}}$ . This proves (eq:heatbound).

The spectral identity

B_{β}^{- 1/4} v^{2} = \frac{1}{π} \int_{0}^{\infty} s^{- 1/2} ⟨ v, e^{- s B_{β}} v ⟩ d s

is finite: on $0 < s < 1$ use the contraction bound $⟨ v, e^{- s B_{β}} v ⟩ \leq ∥ v ∥^{2}$ ; on $s \geq 1$ use (eq:heatbound) and $∥ v ∥_{L^{1}}^{2}$ . The large- $s$ integrand is $O (s^{- 2 - ν_{*}})$ . These bounds also prove continuity. There is no $L^{2}$ zero eigenfunction, since zero form energy would give a parallel section, which cannot be square integrable on $N$ .

Displayed formulas in this passage

I-B-VP-0@1#eq:heat · line 286
$K_{β} (s; x, x^{'}) = \frac{e ^{- Δ z^{2} / (4 s)}}{4 π s} \frac{e ^{- (r^{2} + r^{'2}) / (4 s)}}{4 π α s} \times n \in Z \sum e^{i (n + β) (φ - φ^{'})} I_{ν_{n}} (\frac{r r ^{'}}{2 s}) .$
I-B-VP-0@1#eq:heatbound · line 298
$x, x^{'} \in K sup ∣ K_{β} (s; x, x^{'}) ∣ \leq C_{K} s^{- 3/2 - ν_{*}}, s \geq 1, ν_{*} = \frac{min ( β , 1 - β )}{α} .$
unlabeled · line 313
$\int_{0}^{\infty} e^{- s k^{2}} J_{ν} (k r) J_{ν} (k r^{'}) k d k = \frac{e ^{- (r^{2} + r^{'2}) / (4 s)}}{2 s} I_{ν} (\frac{r r ^{'}}{2 s})$
unlabeled · line 322
$I_{ν} (\frac{r r ^{'}}{2 s}) \leq C_{K} \frac{( C _{K} / s ) ^{ν}}{Γ ( ν + 1 )} .$
unlabeled · line 330
$B_{β}^{- 1/4} v^{2} = \frac{1}{π} \int_{0}^{\infty} s^{- 1/2} ⟨ v, e^{- s B_{β}} v ⟩ d s$

Lemma: Heat kernel and compact-test covariance

Displayed formulas in this passage

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