Lemma 4.20 (Lower bound estimate for the weight function) . [0530] 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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Lemma 4.20 (Lower bound estimate for the weight function).
For fixed constants δ > 0 \delta>0 μ , ν ∈ ℝ \mu,\nu\in\mathbb{R} α ∈ ( 0 , 1 ) \alpha\in(0,1) k ∈ ℕ k\in\mathbb{N} T ≫ 1 T\gg 1 𝐱 ∈ ℳ T \bm{x}\in\mathcal{M}_{T}
(4.278) ρ δ , ν , μ ( k + α ) ( 𝒙 ) ≥ { T ( 1 n − 1 ) ( ν + k + α ) + μ , ν + k + α ≥ 0 , T ν + k + α n + μ , ν + k + α < 0 . \displaystyle\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})\geq\begin{cases}T^{(\frac{1}{n}-1)(\nu+k+\alpha)+\mu},&\nu+k+\alpha\geq 0,\\
T^{\frac{\nu+k+\alpha}{n}+\mu},&\nu+k+\alpha<0.\end{cases}