Manuscript XX-C@1 · XX-C.tex · theorem · 7a8cab37

Theorem: Sufficient one-sided covariance-tail condition

Covariance tails, coordinate pullback, and failure of unconditional infrared control › What compact testing does supply · lines 1221–1245 · proof included or linked

XX-C@1#thm:conditional, XX-C@1#eq:IR

Back to manuscript

Suppose a selected representative satisfies

C w_{f} \in C^{\infty} (J; S (R^{3})) with all seminorms in finite

for each compact test $f$ . Then

R_{x} W_{B} (x, f)

is well defined and obeys (eq:dipolebound). The estimate is uniform for bounded test families whenever the seminorm bounds in (eq:IR) are uniform on those families.

Proof. Apply Theorem thm:dipole to $b^{ia} = (\calCw_{f})^{ia}$ .

Displayed formulas in this passage

XX-C@1#eq:IR · line 1224
$C w_{f} \in C^{\infty} (J; S (R^{3})) with all seminorms in finite$
unlabeled · line 1232
$R_{x} W_{B} (x, f)$

Nearby in this section

section: What compact testing does supply
lemma: Lemma: Compact testing gives smoothness, not decay
section: What compact testing does supply