In order to prove this estimate it is convenient to cut the manifold $M_{\epsilon}$ in various pieces and analyse the geometry separately in each of them. The global estimate follows by combining the “local” estimates obtained in each of these pieces.

Consider first the region $\rho_{j}\leq 2R_{0}\epsilon$ for some $j=1,\dots,8$. The rescaled metric $\epsilon^{-2}g_{\epsilon}$ is isometric to a compact region in the $D_{m_{j}}$ ALF space $(M_{j},g_{M_{j}})$.

Now, given a $1$–form $a$ on $M_{\epsilon}$, restrict $a$ to the region $\rho_{j}\leq 2R_{0}\epsilon$ and define $\tilde{a}=\epsilon^{-1-\delta}a$. The standard Schauder estimate for the elliptic operator $D$ associated with the metric $g_{M_{j}}$ is

Since $|\tilde{a}|_{\epsilon^{-2}g_{\epsilon}}=\epsilon^{-\delta}|a|_{g_{\epsilon}}$ and the norms $\nabla\tilde{a}$ and $D\tilde{a}$ are related in a similar way to those of $\nabla a$ and $Da$, the weighted Schauder estimate follows immediately.

The same argument can be applied in the region $\rho_{i}\leq 2R_{0}\epsilon$: the role of $M_{j}$ is now played by a small perturbation (cf. the beginning of Section 5.2) of the $A_{k_{i}-1}$ ALF space $N_{i}$.

Consider now the transition region $R_{0}\,\epsilon\leq\rho_{j}\leq\rho_{0}$ for some $j=1,\dots,8$. We can work on the double cover $H^{2m_{j}-4}$ and restrict to $\mathbb{Z}_{2}$–invariant forms. Scaling by $\epsilon$ as above we reduce to consider the restriction of $H^{2m_{j}-4}$ to the region $R_{0}\leq\rho\leq\frac{\rho_{0}}{\epsilon}$ in $\mathbb{R}^{3}$ endowed with a metric

Moreover, after rescaling the weight function $\rho_{\epsilon}$ coincides with the radial function $\rho$ on $\mathbb{R}^{3}$.

Fix a number $\sigma\in(0,1)$. For each point $x$ let $\pi(x)$ be its image in $\mathbb{R}^{3}$ and set $R=\sigma\rho(x)$. Up to changing $R_{0}$ and $\rho_{0}$ into $(1-\sigma)R_{0}$ and $(1+\sigma)\rho_{0}$ we can assume that $B_{R}(\pi(x))$ is contained in the annulus $R_{0}\leq\rho\leq\frac{\rho_{0}}{\epsilon}$ and that the restriction of $H^{2m_{j}-4}$ to this ball is trivial. We can then work on a “square” $B_{R}\times[-R,R]$ in the universal cover of $H^{2m_{j}-4}|_{B_{R}}$. Rescaling the metric by $R^{-2}$, applying standard Schauder estimates, rescaling back and multiplying by $R^{-\delta}$ we obtain

The case of the region $R_{0}\epsilon\leq\rho_{i}\leq\rho_{0}$ is completely analogous.

Finally, in the region where $\rho_{j},\rho_{i}\geq\tfrac{1}{2}\rho_{0}$ the weight function $\rho_{\epsilon}$ is uniformly equivalent to the constant $1$ and therefore weighted spaces coincide with standard Hölder spaces. Moreover the harmonic function $h_{\epsilon}$ is $C^{\infty}$–close to the constant $1$. The metric $g_{\epsilon}$ is therefore $C^{\infty}$–close to the metric $g_{\infty}=g_{\mathbb{T}}+\epsilon^{2}\theta^{2}$. The Schauder estimate for forms supported in this region is immediate since we can restrict to small balls in the torus $\mathbb{T}$ on which the circle bundle $P$ is trivial and then work on the universal cover, which has bounded geometry. ∎