Lemma 5.13 . [054E] 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
Lemma 5.13 .
Let K 0 ≥ 1 K_{0}\geq 1 ξ ∈ C 2 K 0 ( Y 2 n − 1 ) \xi\in C^{2K_{0}}(Y^{2n-1}) L 2 L^{2}
(5.176) ξ ( 𝒚 ) = ∑ k = 1 ∞ ξ k ⋅ φ k ( 𝒚 ) , \xi(\bm{y})=\sum\limits_{k=1}^{\infty}\xi_{k}\cdot\varphi_{k}(\bm{y}),
then for all k ∈ ℤ + k\in\mathbb{Z}_{+}
(5.177) | ξ k | ≤ C | ξ | C 2 K 0 ( Y 2 n − 1 ) ( Λ k ) K 0 , |\xi_{k}|\leq\frac{C|\xi|_{C^{2K_{0}}(Y^{2n-1})}}{(\Lambda_{k})^{K_{0}}},
where the constant C > 0 C>0 k k