We will estimate the two terms in (5.204) individually, and we also divide into several cases.

First consider $j_{k}=0$ and $k=0$. In this case the solutions $u_{k}$ is given by simple integrals of $\xi_{k}$ and the conclusion is easy to see.

The second case is that $k\in\mathbb{Z}_{+}$ and $j_{k}=0$. Applying Proposition 5.5, the fundamental solutions $\mathcal{G}_{k}(z)$ and $\mathcal{D}_{k}(z)$ satisfy the uniform estimates

(5.208)		$\displaystyle\mathcal{G}_{k}(z)$	$\displaystyle\leq\frac{C}{\lambda_{k}^{\frac{1}{4}}}\cdot z^{\frac{2-n}{4}}\cdot e^{2\sqrt{\frac{\lambda_{k}}{n}}\cdot z^{\frac{n}{2}}},$
(5.209)		$\displaystyle\mathcal{D}_{k}(z)$	$\displaystyle\leq\frac{C}{\lambda_{k}^{\frac{1}{4}}}\cdot z^{\frac{2-n}{4}}\cdot e^{-2\sqrt{\frac{\lambda_{k}}{n}}\cdot z^{\frac{n}{2}}}.$

By Lemma 5.3.1, $\mathcal{W}_{k}(z)=\mathcal{W}(\mathcal{G}_{k}(z),\mathcal{D}_{k}(z))=\frac{n}{2}$. Let us denote $\tilde{\lambda}_{k}\equiv 2\sqrt{\frac{\lambda_{k}}{n}}$, then $\tilde{\lambda}_{k}\geq 2\sqrt{\frac{\lambda_{1}}{n}}=\delta_{b}$. Now the first integral term in (5.204) has the following bound,

(5.210)				$\displaystyle\frac{\mathcal{G}_{k}(z)}{\mathcal{W}_{k}(z)}\int_{z}^{\infty}\mathcal{D}_{k}(r)|\xi_{k}(r)\cdot r^{n-1}|dr$
			$\displaystyle\leq$	$\displaystyle\frac{C\cdot\mathfrak{B}_{k}}{\lambda_{k}^{\frac{1}{2}}}\cdot z^{\frac{2-n}{4}}\cdot e^{\tilde{\lambda}_{k}\cdot z^{\frac{n}{2}}}\cdot\int_{z}^{\infty}r^{\frac{3n}{4}-\frac{1}{2}}\cdot e^{(-\tilde{\lambda}_{k}+\eta_{0})\cdot r^{\frac{n}{2}}}dr.$

By assumption, $|\eta_{0}|<\frac{\delta_{b}}{2}\leq\frac{\tilde{\lambda}_{k}}{2}$, then

(5.211)		$\displaystyle\frac{\mathcal{G}_{k}(z)}{\mathcal{W}_{k}(z)}\int_{z}^{\infty}\mathcal{D}_{k}(r)|\xi_{k}(r)\cdot r^{n-1}|dr$	$\displaystyle\leq$	$\displaystyle\frac{C\cdot\mathfrak{B}_{k}}{\lambda_{k}^{\frac{1}{2}}}\cdot z^{\frac{2-n}{4}}\cdot e^{\tilde{\lambda}_{k}\cdot z^{\frac{n}{2}}}\cdot e^{(-\tilde{\lambda}_{k}+\eta^{\prime})\cdot z^{\frac{n}{2}}}$
			$\displaystyle\leq$	$\displaystyle C\cdot\mathfrak{B}_{k}\cdot e^{\eta\cdot z^{\frac{n}{2}}},$

where $\eta>\eta^{\prime}>\eta_{0}>0$. Similarly,

(5.212)

$\displaystyle\frac{\mathcal{D}_{k}(z)}{\mathcal{W}_{k}(z)}\int_{z_{0}}^{z}\mathcal{G}_{k}(r)|\xi_{k}(r)\cdot r^{n-1}|dr\leq C\cdot\mathfrak{B}_{k}\cdot e^{\eta\cdot z^{\frac{n}{2}}}.$

In the third case $j_{k}\in\mathbb{Z}_{+}$ and $Q\geq 1$, we need to apply Lemma 5.11. In fact,

(5.213)				$\displaystyle\frac{\mathcal{G}_{k}(z)}{\mathcal{W}_{k}(z)}\int_{z}^{\infty}\mathcal{D}_{k}(r)\xi_{k}(r)\cdot r^{n-1}dr$
			$\displaystyle\leq$	$\displaystyle C_{n}\cdot\frac{Q^{\frac{1}{4}}\cdot(j_{k}\cdot z^{n})^{\frac{1-2\alpha}{4}}}{\Gamma^{2}(Q+1)}\cdot\frac{e^{\widehat{G}_{k}(z)}}{\mathcal{W}_{k}(z)}\int_{z}^{\infty}e^{\widehat{F}_{k}(r)}\xi_{k}(r)\cdot r^{n-1}dr$
			$\displaystyle\leq$	$\displaystyle C_{n}\cdot\mathfrak{B}_{k}\cdot\frac{Q^{\frac{1}{4}}\cdot(j_{k}\cdot z^{n})^{\frac{1-2\alpha}{4}}}{\Gamma^{2}(Q+1)}\cdot\frac{e^{\widehat{G}_{k}(z)}}{\mathcal{W}_{k}(z)}\int_{z}^{\infty}e^{\widehat{F}_{k}(r)+\eta^{\prime}\cdot r^{\frac{n}{2}}}dr,$

where $\eta^{\prime}>\eta_{0}$. We choose any $\epsilon\in(\delta_{b}/100,\delta_{b}/10)$ and denote $\eta^{\prime\prime}\equiv\eta^{\prime}+\epsilon$, then by Lemma 5.11,

(5.214)				$\displaystyle\frac{e^{\widehat{G}_{k}(z)}}{\mathcal{W}_{k}(z)}\int_{z}^{\infty}e^{\widehat{F}_{k}(r)+\eta^{\prime}\cdot r^{\frac{n}{2}}}dr$
			$\displaystyle=$	$\displaystyle\frac{e^{\widehat{G}_{k}(z)}}{\mathcal{W}_{k}(z)}\int_{z}^{\infty}e^{\widehat{F}_{k}(r)+\eta^{\prime\prime}\cdot r^{\frac{n}{2}}}\cdot e^{-\epsilon r^{\frac{n}{2}}}dr$
			$\displaystyle\leq$	$\displaystyle\frac{e^{\widehat{F}_{k}(z)+\widehat{G}_{k}(z)+\eta^{\prime\prime}\cdot z^{\frac{n}{2}}}}{\mathcal{W}_{k}(z)}\int_{z}^{\infty}e^{-\epsilon\cdot r^{\frac{n}{2}}}dr$
			$\displaystyle\leq$	$\displaystyle C_{n}\cdot\frac{e^{\widehat{F}_{k}(z)+\widehat{G}_{k}(z)+\eta^{\prime\prime}\cdot z^{\frac{n}{2}}}}{\mathcal{W}_{k}(z)}.$

Therefore,

(5.215)

$$\frac{\mathcal{G}_{k}(z)}{\mathcal{W}_{k}(z)}\int_{z}^{\infty}\mathcal{D}_{k}(r)\xi_{k}(r)\cdot r^{n-1}dr\leq C_{n}\cdot\mathfrak{B}_{k}\cdot\frac{Q^{\frac{1}{4}}\cdot(j_{k}\cdot z^{n})^{\frac{1-2\alpha}{4}}}{\Gamma^{2}(Q+1)}\cdot\frac{e^{\widehat{F}_{k}(z)+\widehat{G}_{k}(z)+\eta^{\prime\prime}\cdot z^{\frac{n}{2}}}}{\mathcal{W}_{k}(z)}.$$

Plugging Lemma 5.10 and Proposition 5.6 into the above inequality,

(5.216)		$\displaystyle\frac{\mathcal{G}_{k}(z)}{\mathcal{W}_{k}(z)}\int_{z}^{\infty}\mathcal{D}_{k}(r)\xi_{k}(r)\cdot r^{n-1}dr$	$\displaystyle\leq$	$\displaystyle C_{n}\cdot\mathfrak{B}_{k}\cdot\frac{j_{k}^{\frac{1}{n}}\cdot e^{-Q}\cdot Q^{Q+1}}{\Gamma(Q+1)}\cdot z\cdot e^{\eta^{\prime\prime}\cdot z^{\frac{n}{2}}}.$
			$\displaystyle\leq$	$\displaystyle C_{n}\cdot\mathfrak{B}_{k}\cdot j_{k}^{\frac{1}{n}}\cdot e^{\eta\cdot z^{\frac{n}{2}}}$
			$\displaystyle\leq$	$\displaystyle C_{n}\cdot\mathfrak{B}_{k}\cdot(\Lambda_{k})^{\frac{1}{2n}}\cdot e^{\eta\cdot z^{\frac{n}{2}}}$

for any $\eta\in(\eta^{\prime\prime},\eta^{\prime\prime}+\frac{\delta_{b}}{100})$, where we used Stirling’s formula for estimating $\Gamma(Q+1)$. Similarly we get the bound for the other term of (5.204).

The fourth case is when $j_{k}\geq 1$ and $Q\leq 1$. This case is simpler and follows from Corollary 5.12.1 and the argument in the second case.

This completes the proof of the proposition.

∎