Manuscript VII-I@1 · VII-I.tex · proposition · 907e96ab

Proposition: Exact boundary kernel

The sharp kernel and the off-shell admission obstruction · lines 530–571 · proof included or linked

VII-I@1#prop:boundary, VII-I@1#eq:boundary-kernel

The compact distribution $q_{a} = L^{*} a$ , $L = E_{0} C$ , is

q_{a}^{α β} B^{α β} (z) = θ (C^{*} b)^{α β} + δ (t) B^{α β} (b ∣_{Σ}), = - n^{(α} z^{β)} + \frac{1}{2} η^{α β} n^{μ} z_{μ}, n = d t .

There is no $δ^{'} (t)$ term. The map $B$ is injective. If $b (0, x_{0}) \neq = 0$ , every nonzero normal covector $(0, x_{0}; τ d t)$ belongs to $WF (q_{a})$ .

Proof. Transpose $E^{R}_{0} (θ C h)$ against $a$ :

ℓ_{a} [h] = ⟨ C^{*} (θ E^{A}_{0} a), h ⟩ .

The product rule gives (eq:boundary-kernel). Its support is in $J^{-} (supp a) \cap {t \geq 0}$ .

The components

B^{00} (z) = - \frac{1}{2} z_{0}, B^{0 i} (z) = \frac{1}{2} z_{i}

prove injectivity.

Locally, a nonzero component has the form $δ (t) v (x) + θ (t) r (t, x)$ , with $v (x_{0}) \neq = 0$ and smooth coefficients. After localization, the first term has normal Fourier order zero, whereas the second has order at most minus one at fixed tangential frequency. The leading nonzero term cannot cancel. Thus rapid decrease fails in both normal directions. These are actual timelike conormals, not merely an upper bound.

Displayed formulas in this passage

VII-I@1#eq:boundary-kernel · line 533
$q_{a}^{α β} B^{α β} (z) = θ (C^{*} b)^{α β} + δ (t) B^{α β} (b ∣_{Σ}), = - n^{(α} z^{β)} + \frac{1}{2} η^{α β} n^{μ} z_{μ}, n = d t .$
unlabeled · line 550
$ℓ_{a} [h] = ⟨ C^{*} (θ E^{A}_{0} a), h ⟩ .$
unlabeled · line 557
$B^{00} (z) = - \frac{1}{2} z_{0}, B^{0 i} (z) = \frac{1}{2} z_{i}$

Proposition: Exact boundary kernel

Displayed formulas in this passage

Nearby in this section