Corollary 5.5.1 . [053V] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
Source coverage notes Source fidelity gap: the original arXiv HTML omits an author TeX footnote attached to equation e:def-omega, including its reference to Remark r:error-function. The retained HTML is preserved as published; the original author TeX remains available. TeX correspondence is not complete. Complete original source context · Original author HTML
Corollary 5.5.1 .
There is a dimensional constant C ( n ) > 0 C(n)>0 z ≥ 2 − 2 n n 1 n λ ¯ − 1 n z\geq 2^{-\frac{2}{n}}n^{\frac{1}{n}}\underline{\lambda}^{-\frac{1}{n}}
(5.74) C − 1 ( n ) λ k 1 4 ⋅ e − 2 λ k n ⋅ z n 2 z n − 2 4 \displaystyle\frac{C^{-1}(n)}{\lambda_{k}^{\frac{1}{4}}}\cdot\frac{e^{-2\sqrt{\frac{\lambda_{k}}{n}}\cdot z^{\frac{n}{2}}}}{z^{\frac{n-2}{4}}} ≤ 𝒟 k ( z ) ≤ C ( n ) λ k 1 4 ⋅ e − 2 λ k n ⋅ z n 2 z n − 2 4 , \displaystyle\leq\mathcal{D}_{k}(z)\leq\frac{C(n)}{\lambda_{k}^{\frac{1}{4}}}\cdot\frac{e^{-2\sqrt{\frac{\lambda_{k}}{n}}\cdot z^{\frac{n}{2}}}}{z^{\frac{n-2}{4}}},
(5.75) C − 1 ( n ) λ k 1 4 ⋅ e 2 λ k n ⋅ z n 2 z n − 2 4 \displaystyle\frac{C^{-1}(n)}{\lambda_{k}^{\frac{1}{4}}}\cdot\frac{e^{2\sqrt{\frac{\lambda_{k}}{n}}\cdot z^{\frac{n}{2}}}}{z^{\frac{n-2}{4}}} ≤ 𝒢 k ( z ) ≤ C ( n ) λ k 1 4 ⋅ e 2 λ k n ⋅ z n 2 z n − 2 4 . \displaystyle\leq\mathcal{G}_{k}(z)\leq\frac{C(n)}{\lambda_{k}^{\frac{1}{4}}}\cdot\frac{e^{2\sqrt{\frac{\lambda_{k}}{n}}\cdot z^{\frac{n}{2}}}}{z^{\frac{n-2}{4}}}.