Manuscript XX-B@1 · XX-B.tex · theorem · 56d708a5

Theorem: Existence, normalization, and support

Retarded and advanced Green distributions › Reduced inverses · lines 911–953 · proof included or linked

XX-B@1#thm:greens, XX-B@1#eq:GB

There are unique continuous operators

G_{L}^{\pm} : C_{c}^{\infty} (M; V) ⟶ C^{\infty} (M; V), G_{Q}^{\pm} : C_{c}^{\infty} (M; W) ⟶ C^{\infty} (M; W)

with the left and right inverse identities and

supp (G_{P}^{\pm} f) \subset J^{\pm}_{g ˉ} (supp f), P = L, Q .

The correctly normalized reduced action inverse is

G_{B}^{\pm} = G_{L}^{\pm} A^{- 1}, B G_{B}^{\pm} f = f, G_{B}^{\pm} B f = f .

Moreover,

(G_{B}^{+})^{†} = G_{B}^{-}, (G_{Q}^{+})^{†} = G_{Q}^{-} .

Proof. Lemma lem:GH verifies global hyperbolicity. The bundles have finite rank and smooth coefficients. Theorem thm:NH and (eq:Q) verify normal hyperbolicity. No boundary or positivity assumption is needed. The Green theorem BGP (Corollary 3.4.3) therefore applies.

The inverse identities in (eq:GB) follow from $B = A L$ . The fiber map is local and invertible, so it changes neither support nor wavefront relations. Formal self-adjointness, integration by parts, and uniqueness give the adjoint identities.

Displayed formulas in this passage

XX-B@1#eq:GB · line 927
$G_{B}^{\pm} = G_{L}^{\pm} A^{- 1}, B G_{B}^{\pm} f = f, G_{B}^{\pm} B f = f .$
unlabeled · line 914
$G_{L}^{\pm} : C_{c}^{\infty} (M; V) ⟶ C^{\infty} (M; V), G_{Q}^{\pm} : C_{c}^{\infty} (M; W) ⟶ C^{\infty} (M; W)$
unlabeled · line 922
$supp (G_{P}^{\pm} f) \subset J^{\pm}_{g ˉ} (supp f), P = L, Q .$
unlabeled · line 935
$(G_{B}^{+})^{†} = G_{B}^{-}, (G_{Q}^{+})^{†} = G_{Q}^{-} .$

Theorem: Existence, normalization, and support

Displayed formulas in this passage

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