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Proof.
The estimate is proved by the standard integration by parts. Since the eigenfunctions φ k \varphi_{k}
(5.178) − Δ h 0 φ k = Λ k ⋅ φ k -\Delta_{h_{0}}\varphi_{k}=\Lambda_{k}\cdot\varphi_{k}
and ‖ φ k ‖ L 2 ( Y 2 n − 1 ) = 1 \|\varphi_{k}\|_{L^{2}(Y^{2n-1})}=1
| ξ k ( z ) | \displaystyle|\xi_{k}(z)| = | ∫ Y 2 n − 1 ξ ⋅ φ k | = | ∫ Y 2 n − 1 ξ ⋅ ( − Δ h 0 ) K 0 φ k ( Λ k ) K 0 | \displaystyle=\Big|\int_{Y^{2n-1}}\xi\cdot\varphi_{k}\Big|=\Big|\int_{Y^{2n-1}}\xi\cdot\frac{(-\Delta_{h_{0}})^{K_{0}}\varphi_{k}}{(\Lambda_{k})^{K_{0}}}\Big|
(5.179) ≤ 1 ( Λ k ) K 0 ∫ Y 2 n − 1 | Δ h 0 K 0 ξ | ⋅ | φ k | \displaystyle\leq\frac{1}{(\Lambda_{k})^{K_{0}}}\int_{Y^{2n-1}}|\Delta_{h_{0}}^{K_{0}}\xi|\cdot|\varphi_{k}|
(5.180) ≤ C | ξ | C 2 K 0 ( Y 2 n − 1 ) ( Λ k ) K 0 , \displaystyle\leq\frac{C|\xi|_{C^{2K_{0}}(Y^{2n-1})}}{(\Lambda_{k})^{K_{0}}},
where C > 0 C>0 Y 2 n − 1 Y^{2n-1}