Manuscript Em01-A@1 · Em01-A.tex · lemma · 880b67af

Lemma: Compact wave primitive

A common classical setting › The analytic Green package and its signs · lines 419–447 · proof included or linked

Em01-A@1#lem:compact-primitive

For $a \in D^{'}_{c} (M; H)$ ,

E_{+} a = 0 ⟺ a = □ r for some r \in D^{'}_{c} (M; H) .

The same equivalence holds with $E_{-}$ .

Proof. If $E_{+} a = 0$ , then

r := G_{R} a = G_{A} a .

The support inclusions give

supp r \subseteq J^{+} (supp a) \cap J^{-} (supp a) .

For compact $supp a$ , this intersection is compact in Minkowski space. Hence $r$ is compact and $□ r = a$ .

Conversely, if $a = □ r$ with $r$ compact, both Green operators give $r$ , so their difference vanishes. The $E_{-}$ statement follows from (eq:causal-signs).

Displayed formulas in this passage

unlabeled · line 422
$E_{+} a = 0 ⟺ a = □ r for some r \in D^{'}_{c} (M; H) .$
unlabeled · line 432
$r := G_{R} a = G_{A} a .$
unlabeled · line 436
$supp r \subseteq J^{+} (supp a) \cap J^{-} (supp a) .$

Lemma: Compact wave primitive

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