Manuscript Em02-D@4 · Em02-D-v4.tex · lemma · fcf2fac7

Lemma: Weighted cone convolution

Weighted convolution and leading master estimates › Exact two-cone convolution · lines 779–828 · proof included or linked

Em02-D@4#lem:weighted, Em02-D@4#eq:J-unsmoothed, Em02-D@4#eq:Gamma, Em02-D@4#eq:fiber

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For $S = (T, q n)$ , $T > q > 0$ ,

J (S; w) = Θ (T - q) Γ (S; w),

where on the timelike side

Γ (S; w) = \frac{1}{8 π} \int_{0}^{1} \int_{0}^{2 π} w (X (S, θ, φ)) \frac{d φ}{2 π} d θ,

and

X^{0} X = \frac{T + q ( 2 θ - 1 )}{2}, = \frac{q + T ( 2 θ - 1 )}{2} n + T^{2} - q^{2} θ (1 - θ) (cos φ e_{1} + sin φ e_{2}) .

The convolution vanishes outside the future causal cone. There is no additional null-boundary measure. In particular, $R * R = B$ .

Proof. In the rest frame $S = (S^{2}, 0)$ , temporal and radial cone integration gives

\frac{1}{32 π ^{2}} \int_{S^{2}} w (\frac{S ^{2}}{2}, \frac{S ^{2}}{2} ω) d ω .

The cone measure $R = Θ (x^{0}) δ (x^{2}) / (2 π)$ is Lorentz invariant. Boost in direction $n$ , and put $ω_{∥} = 2 θ - 1$ . This gives the formula.

On compact sets of $S$ , retarded support bounds both propagation times, so the convolution is proper. The set of parallel angular pairs has zero product-measure mass. Therefore no extra measure is carried by the null boundary. For $w = 1$ , $Γ = 1/ (8 π)$ , proving $R * R = B$ .

Displayed formulas in this passage

Em02-D@4#eq:J-unsmoothed · line 782
$J (S; w) = Θ (T - q) Γ (S; w),$
Em02-D@4#eq:Gamma · line 787
$Γ (S; w) = \frac{1}{8 π} \int_{0}^{1} \int_{0}^{2 π} w (X (S, θ, φ)) \frac{d φ}{2 π} d θ,$
Em02-D@4#eq:fiber · line 796
$X^{0} X = \frac{T + q ( 2 θ - 1 )}{2}, = \frac{q + T ( 2 θ - 1 )}{2} n + T^{2} - q^{2} θ (1 - θ) (cos φ e_{1} + sin φ e_{2}) .$
unlabeled · line 812
$\frac{1}{32 π ^{2}} \int_{S^{2}} w (\frac{S ^{2}}{2}, \frac{S ^{2}}{2} ω) d ω .$

Lemma: Weighted cone convolution

Displayed formulas in this passage

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