For every sufficiently large parameter $T\gg 1$, let $\mathcal{M}_{T}$ be the neck region with an $S^{1}$-invariant Kähler metric $\omega_{T}$ constructed in Section 4.1. Then the following estimates hold:

1. (1)

   (Interior estimate) Given $k\in\{0,1\}$ and $\alpha\in(0,1)$, there is some uniform constant $C_{k,\alpha}>0$ such that for any $\bm{x}\in\mathring{\mathcal{M}}(T_{-},T_{+})$, $r\in(0,1)$, $u\in C^{k+2,\alpha}(B_{s(\bm{x})}(\bm{x}))$,

   		$\displaystyle r^{k+2+\alpha}\cdot\|u\|_{C_{\delta,\nu,\mu}^{k+2,\alpha}(B_{r\cdot s(\bm{x})}(\bm{x}))}$
   (4.288)		$\displaystyle\leq$	$\displaystyle C_{k,\alpha}\Big(\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{k,\alpha}(B_{2r\cdot s(\bm{x})}(\bm{x}))}+\|u\|_{C_{\delta,\nu,\mu}^{0}(B_{2r\cdot s(\bm{x})}(\bm{x}))}\Big),$

   where $s(\bm{x})\equiv\frac{\mathfrak{s}(\bm{x})}{4}$ and $\mathfrak{s}(\bm{x})$ is the regularity scale at $\bm{x}$ given by Proposition 4.18.

2. (2)

   (Higher order estimate away from $\mathcal{P}$) There exists some large constant $C_{\mathcal{P}}>0$ such that if $\bm{x}\in\mathring{\mathcal{M}}_{T}$ satisfies

   (4.289)

   $$r(\bm{x})\geq C_{\mathcal{P}}\cdot T^{-1},$$

   then the uniform Schauder estimate (4.288) holds for all $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$.

3. (3)

   (Boundary estimate) For any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$, there exists some uniform constant $C_{k,\alpha}>0$ such that for all $\bm{x}\in\partial\mathcal{M}_{T}$, $r\in(0,1)$ and $u\in C^{k+2,\alpha}(T_{2}(\partial\mathcal{M}_{T}))$,

   		$\displaystyle r^{k+2+\alpha}\cdot\|u\|_{C_{\delta,\nu,\mu}^{k+2,\alpha}(B_{r\cdot s(\bm{x})}^{+}(\bm{x}))}$
   (4.290)		$\displaystyle\leq$	$\displaystyle C_{k,\alpha}\Big(\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{k,\alpha}(B_{2r\cdot s(\bm{x})}^{+}(\bm{x}))}+\Big\|\frac{\partial u}{\partial n}\Big\|_{C_{\delta,\nu,\mu}^{k+1,\alpha}(B_{2r\cdot s(\bm{x})}^{+}(\bm{x}))}+\|u\|_{C_{\delta,\nu,\mu}^{0}(B_{2r\cdot s(\bm{x})}^{+}(\bm{x}))}\Big),$

   where $B_{s}^{+}(\bm{x})\equiv B_{s}(\bm{x})\cap\mathcal{M}_{T}$.