Manuscript XVI-D@1 · XVI-D.tex · section · b73c6f4a

Compact-interior Green theorem

The two inherited interfaces and no-decay uniqueness › Compact-interior Green theorem · lines 482–515

XVI-D@1#eq:compactGreen, XVI-D@1#eq:compact-support

Define the real compact-interior source class

D_{int} = C_{c}^{\infty} (M_{T}^{\circ}; R) .

The complex version consists of real and imaginary parts in $D_{int}$ .

The reported XI-B compact-source theorem states, for $F \in D_{int}$ ,

(□ + μ^{2}) E_{μ}^{R} F = F, E_{μ}^{R} (□ + μ^{2}) F = F,

with zero initial data, and the corresponding advanced statement with zero final data. It also gives the causal support bounds

supp E_{μ}^{R} F supp E_{μ}^{A} F \subset J^{+} (supp F) \cap M_{T}, \subset J^{-} (supp F) \cap M_{T} .

These supports are compact in the closed slab XIB.

The support can meet a time boundary. Compactness in the closed slab is not compact support in the open slab.

Displayed formulas in this passage

XVI-D@1#eq:compactGreen · line 493
$(□ + μ^{2}) E_{μ}^{R} F = F, E_{μ}^{R} (□ + μ^{2}) F = F,$
XVI-D@1#eq:compact-support · line 501
$supp E_{μ}^{R} F supp E_{μ}^{A} F \subset J^{+} (supp F) \cap M_{T}, \subset J^{-} (supp F) \cap M_{T} .$
unlabeled · line 485
$D_{int} = C_{c}^{\infty} (M_{T}^{\circ}; R) .$