Given a large parameter $T\gg 1$, let $\mathcal{M}_{T}$ be the neck region with an $S^{1}$-symmetric Kähler metric $\omega_{T}$ constructed in Section 4.1. Let the parameters $\mu$, $\nu$, $\alpha$, $\delta$ satisfy satisfying

(6.63)

$\displaystyle-1<\nu<0,\quad\nu+\alpha<0,\quad 0<\delta<\delta_{N},\ \mu=(1-\frac{1}{n})(\nu+2+\alpha)$

as fixed in (6.10), (6.11), (6.12) and (6.13), then there exists a uniform constant $C>0$ (independent of $T$) such that for every $u\in C^{2,\alpha}(\mathcal{M}_{T})$ satisfying the boundary condition $\frac{\partial u}{\partial n}|_{\partial\mathcal{M}_{T}}=0$, we have

(6.64)		$\displaystyle\|\nabla u\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{T})}+\|\nabla^{2}u\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{T})}\leq C\cdot\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})},$
(6.65)		$\displaystyle[u]_{C_{\delta,\nu,\mu}^{2,\alpha}(\mathcal{M}_{T})}\leq C\cdot\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})}.$