Manuscript X-A-conv@1 · X-A-conv.tex · lemma · c76fffa3

Lemma: Strict convolution gap

Regulator preflight and an exact positive defect · lines 1084–1122 · proof included or linked

X-A-conv@1#lem:positivegap, X-A-conv@1#eq:ceps, X-A-conv@1#eq:positiveenergy

For the fixed $f, ξ_{*}$ of (eq:fandxi), define

c_{ε} = \int_{M} f (x) (ψ (x) - S_{ε} ψ (x)) d^{4} x .

Then

c_{ε} = \frac{1}{2 N} \int_{R^{4}} ϱ_{ε} (a) \int_{R^{4}} (ψ (x) - ψ (x - a))^{2} d^{4} x d^{4} a > 0.

Proof. Translation preserves $\int ψ^{2}$ . Expanding the square in (eq:positiveenergy) and using $\int ϱ_{ε} = 1$ gives

\frac{1}{N} (\int ψ^{2} - \int ψ S_{ε} ψ),

which is (eq:ceps).

For every $a \neq = 0$ , the inner integral is strictly positive. Otherwise continuity would give $ψ (x) = ψ (x - a)$ everywhere. Iteration would make a nonzero compactly supported function invariant under arbitrarily large translations in the direction $a$ , which is impossible. The mollifier is positive on the open ball $0 < ∣ a ∣_{E} < ε$ , a set of positive measure. Thus the outer integral is strictly positive.

Displayed formulas in this passage

X-A-conv@1#eq:ceps · line 1087
$c_{ε} = \int_{M} f (x) (ψ (x) - S_{ε} ψ (x)) d^{4} x .$
X-A-conv@1#eq:positiveenergy · line 1094
$c_{ε} = \frac{1}{2 N} \int_{R^{4}} ϱ_{ε} (a) \int_{R^{4}} (ψ (x) - ψ (x - a))^{2} d^{4} x d^{4} a > 0.$
unlabeled · line 1110
$\frac{1}{N} (\int ψ^{2} - \int ψ S_{ε} ψ),$

Lemma: Strict convolution gap

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