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

Exact two-cone convolution

Weighted convolution and leading master estimates › Exact two-cone convolution · lines 828–869

Em02-D@4#eq:G, Em02-D@4#eq:A, Em02-D@4#eq:T

Let

d (S) = S^{0} - ∣ S ∣, n (S) = S /∣ S ∣, k (S) = (1, - n (S)),

and $S = (∣ S ∣, S)$ .

At $s = q ℓ_{n}$ , define

G [w] A [w] T [w] = \frac{1}{8 π} \int_{0}^{1} w (θ s) d θ, = \frac{1}{8 π} \int_{0}^{1} [\frac{1}{2} \partial_{0} w + (θ - \frac{1}{2}) n \cdot \nabla_{x} w + \frac{q}{2} θ (1 - θ) Δ_{⊥} w] (θ s) d θ, = \frac{1}{8 π} \int_{0}^{1} θ ℓ_{n} \cdot \nabla_{x} w (θ s) d θ .

Here $Δ_{⊥} = Δ_{x} - (n \cdot \nabla_{x})^{2}$ . Formula (eq:fiber) gives

Γ (s; w) = G [w], (\partial_{S^{0}} Γ) (s; w) = A [w] .

Every propagation derivative of $Γ$ freezes $w$ .

Displayed formulas in this passage

Em02-D@4#eq:G · line 839
$G [w] A [w] T [w] = \frac{1}{8 π} \int_{0}^{1} w (θ s) d θ, = \frac{1}{8 π} \int_{0}^{1} [\frac{1}{2} \partial_{0} w + (θ - \frac{1}{2}) n \cdot \nabla_{x} w + \frac{q}{2} θ (1 - θ) Δ_{⊥} w] (θ s) d θ, = \frac{1}{8 π} \int_{0}^{1} θ ℓ_{n} \cdot \nabla_{x} w (θ s) d θ .$
Em02-D@4#eq:A · line 839
$G [w] A [w] T [w] = \frac{1}{8 π} \int_{0}^{1} w (θ s) d θ, = \frac{1}{8 π} \int_{0}^{1} [\frac{1}{2} \partial_{0} w + (θ - \frac{1}{2}) n \cdot \nabla_{x} w + \frac{q}{2} θ (1 - θ) Δ_{⊥} w] (θ s) d θ, = \frac{1}{8 π} \int_{0}^{1} θ ℓ_{n} \cdot \nabla_{x} w (θ s) d θ .$
Em02-D@4#eq:T · line 839
$G [w] A [w] T [w] = \frac{1}{8 π} \int_{0}^{1} w (θ s) d θ, = \frac{1}{8 π} \int_{0}^{1} [\frac{1}{2} \partial_{0} w + (θ - \frac{1}{2}) n \cdot \nabla_{x} w + \frac{q}{2} θ (1 - θ) Δ_{⊥} w] (θ s) d θ, = \frac{1}{8 π} \int_{0}^{1} θ ℓ_{n} \cdot \nabla_{x} w (θ s) d θ .$
unlabeled · line 831
$d (S) = S^{0} - ∣ S ∣, n (S) = S /∣ S ∣, k (S) = (1, - n (S)),$
unlabeled · line 864
$Γ (s; w) = G [w], (\partial_{S^{0}} Γ) (s; w) = A [w] .$