Manuscript VII-D@1 · VII-D.tex · corollary · ef67c89c

Corollary: Summable shell certificate

Strong finite part and dyadic certificates › Shell and within-shell bounds · lines 1904–1926 · proof included or linked

VII-D@1#cor:shell, VII-D@1#eq:shellbounds

For every compact $K \subset X$ ,

∣ ⟨ E_{n}^{ρ}, Φ ⟩ ∣ 0 \leq η \leq 1 sup ∣ ⟨ J_{n, η}^{ρ}, Φ ⟩ ∣ \leq a_{n, K} p_{K, 64} (Φ), \leq b_{n, K} p_{K, 64} (Φ),

where one may take

a_{n, K} = b_{n, K} = C_{K, ρ, m} (n + 1) 2^{- n} .

In particular, $\sum_{n} a_{n, K} < \infty$ and $b_{n, K} \to 0$ .

Proof. Apply (eq:finalrate) with $μ = 1$ to the two terms in each difference. For the within-shell estimate, $2^{- (n + η)} \in [2^{- n - 1}, 2^{- n}]$ , so the bound is uniform in $η$ .

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unlabeled · line 1907
$∣ ⟨ E_{n}^{ρ}, Φ ⟩ ∣ 0 \leq η \leq 1 sup ∣ ⟨ J_{n, η}^{ρ}, Φ ⟩ ∣ \leq a_{n, K} p_{K, 64} (Φ), \leq b_{n, K} p_{K, 64} (Φ),$
VII-D@1#eq:shellbounds · line 1914
$a_{n, K} = b_{n, K} = C_{K, ρ, m} (n + 1) 2^{- n} .$

Corollary: Summable shell certificate

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