Manuscript X-E-ST@1 · X-E-ST.tex · proposition · 44614221

Proposition: Existence and corrected support

E4: the isolated matter-induced tree › An explicit two-point coefficient · lines 1491–1528 · proof included or linked

X-E-ST@1#prop:treesupport, X-E-ST@1#eq:treesupport

The formula (eq:t) defines a compact distribution with

supp t \subset C \times S, WF (t) \subset Z_{2} .

All its scalar external points lie in $J_{η}^{-} (K)$ . The support claim $K \times S$ need not hold for the completed tree.

Proof. Insert (eq:k1) into the affine formula using (eq:D). The coordinate composition is

\int Λ^{μ, α β} (x, z) R (z - y) d^{4} z = - a^{ν μ, α β} \partial_{x^{ν}} R_{2} (x - y)

on the relevant retarded support. The two local terms in (eq:D) give the second and third lines of (eq:t), including their signs.

The first line is supported with $x \in K$ , $y \in supp g$ , and $y ⪯ x$ . The remaining lines have $x \in supp (g F) \subset S$ , $y \in supp g$ , and $y ⪯ x$ , hence $y \in S$ . This proves the compact support in (eq:treesupport). There is no reason for $supp (g F)$ to be contained in $K$ .

Each term is a derivative of a smooth multiplier times a translation-invariant two-point distribution. The wavefront multiplier and derivative rules give the $Z_{2}$ bound Hormander.

Displayed formulas in this passage

X-E-ST@1#eq:treesupport · line 1495
$supp t \subset C \times S, WF (t) \subset Z_{2} .$
unlabeled · line 1508
$\int Λ^{μ, α β} (x, z) R (z - y) d^{4} z = - a^{ν μ, α β} \partial_{x^{ν}} R_{2} (x - y)$

Proposition: Existence and corrected support

Displayed formulas in this passage

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