The estimate will be proved by the standard integration by parts. Since the eigenfunctions $\varphi_{k}$ satisfy

(4.126)

$$-\Delta_{h_{0}}\varphi_{k}=\Lambda_{k}\varphi_{k}$$

and $\|\varphi_{k}\|_{L^{2}(Y^{3})}=1$, we have that

	$\displaystyle\Big|\xi_{k}(z)\Big|$	$\displaystyle=\Big|\int_{Y^{3}}\xi\cdot\varphi_{k}\Big|=\Big|\int_{Y^{3}}\xi\cdot\frac{(-\Delta_{h_{0}})^{K_{0}}\varphi_{k}}{(\Lambda_{k})^{K_{0}}}\Big|$
(4.127)			$\displaystyle\leq\frac{1}{(\Lambda_{k})^{K_{0}}}\int_{Y^{3}}|\Delta_{h_{0}}^{K_{0}}\xi|\cdot|\varphi_{k}|\leq\frac{Q_{2K_{0}}\cdot\Vol_{h_{0}}(Y^{3})^{1/2}\cdot e^{\eta_{0}z}}{(\Lambda_{k})^{K_{0}}},$

where $Q_{2K_{0}}$ depends only on the asymptotic bound of $\nabla^{2K_{0}}\xi$. The proof is done.

∎