Manuscript VII-D@1 · VII-D.tex · section · ef67c89c

The exact distribution and intersection coordinates

The exact distribution and intersection coordinates · lines 325–369

VII-D@1#eq:strongtemplate

Write

x = (t, x), y = (s, y), x^{'} = (t^{'}, x^{'}) .

The scalar mass $m \geq 0$ and each admissible mollifier are fixed independently of $ε$ .

Our Fourier convention is

v (k) = \int e^{- ik \cdot z} v (z) d z, v (z) = \int \frac{d k}{( 2 π ) ^{d i m z}} e^{ik \cdot z} v (k) .

The dot product in this convention is the coordinate pairing, not the Lorentzian metric pairing. Radiality of $ρ$ is Euclidean.

For a compact $K \subset X$ , let

p_{K, N} (Φ) = ∣ γ ∣ \leq N max X sup ∣ \partial^{γ} Φ∣, Φ \in D_{K} (X) .

Tests are extended by zero to the ambient Euclidean space when taking Fourier transforms. Their supports are separated from the boundary of $X$ .

All estimates below can be taken with the single test order $64$ . Its size is inessential; fixing it makes independence from the cutoff and the radial parameter explicit.

Strong convergence in $D^{'} (X)$ means uniform convergence on bounded subsets of $D (X)$ . A bounded subset of $D (X)$ is contained and bounded in some fixed-support test space. Thus

∣ ⟨ U_{ε} - U, Φ ⟩ ∣ \leq C_{K} a (ε) p_{K, 64} (Φ), a (ε) ⟶ 0,

on every compact implies strong convergence.

Displayed formulas in this passage

VII-D@1#eq:strongtemplate · line 362
$∣ ⟨ U_{ε} - U, Φ ⟩ ∣ \leq C_{K} a (ε) p_{K, 64} (Φ), a (ε) ⟶ 0,$
unlabeled · line 328
$x = (t, x), y = (s, y), x^{'} = (t^{'}, x^{'}) .$
unlabeled · line 336
$v (k) = \int e^{- ik \cdot z} v (z) d z, v (z) = \int \frac{d k}{( 2 π ) ^{d i m z}} e^{ik \cdot z} v (k) .$
unlabeled · line 346
$p_{K, N} (Φ) = ∣ γ ∣ \leq N max X sup ∣ \partial^{γ} Φ∣, Φ \in D_{K} (X) .$