Proposition 4.23 (Weighted error estimate) . [0537] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
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Proposition 4.23 (Weighted error estimate).
Let Err C Y \mathrm{Err}_{CY} be the error function given by Definition 4.3 .
For fixed parameters δ > 0 \delta>0 , μ , ν ∈ ℝ \mu,\nu\in\mathbb{R} and α ∈ ( 0 , 1 ) \alpha\in(0,1) which satisfy
(4.304)
0 < δ < δ e \displaystyle 0<\delta<\delta_{e}
≡ λ 1 n ( | k − | + | k + | ) , \displaystyle\equiv\frac{\sqrt{\lambda_{1}}}{n(|k_{-}|+|k_{+}|)},
(4.305)
ν + α \displaystyle\nu+\alpha
> 0 , \displaystyle>0,
where the constants λ 1 > 0 \lambda_{1}>0 , k − > 0 k_{-}>0 and k + < 0 k_{+}<0 are given in Proposition 3.31 .
Then the weighted C 0 , α C^{0,\alpha} -estimate holds,
(4.306)
‖ Err C Y ‖ C δ , ν , μ 0 , α ( ℳ T ) = O ( T − 2 + ν + α n + μ ) . \|\mathrm{Err}_{CY}\|_{C^{0,\alpha}_{\delta,\nu,\mu}(\mathcal{M}_{T})}=O(T^{-2+\frac{\nu+\alpha}{n}+\mu}).