Consider $(\mathcal{M},g_{\beta})$ with sufficiently large gluing parameter $\beta>0$. Assume that the parameters $\delta$, $\mu$ and $\nu$ satisfy

1. (1)

   $0<\delta<\frac{1}{10^{3}}\min\{\underline{\delta}_{1},\underline{\delta}_{2},\underline{\epsilon}_{1},\underline{\epsilon}_{2},\lambda_{0},\delta_{h},\delta_{q}\}$,

2. (2)

   $\mu+\nu\in(0,1)$,

where $\underline{\delta}_{1},\underline{\delta}_{2},\underline{\epsilon}_{1},\underline{\epsilon}_{2}$ are the fixed constants specified in Section 7.1, $\lambda_{0}>0$ is in Lemma 9.1, $\delta_{h}>0$ is in Theorem 5.1 and $\delta_{q}$ is in Corollary 6.5. Then for every $\alpha\in(0,1)$, there exists a uniform constant $C=C(\alpha,\delta,\mu,\nu)>0$ which is independent of $\beta$ such that for every $\omega\in\Omega^{1}(\mathcal{M})$ it holds that

(9.2)

$$\|\omega\|_{C_{\delta,\nu,\mu}^{1,\alpha}(\mathcal{M})}\leq C\cdot\|\mathscr{D}_{g_{\beta}}\omega\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}.$$