We will prove by contradiction and suppose no such a uniform constant $C>0$ exists. That is, there exist the following sequences:

1. (1)

   a sequence of numbers $\beta_{j}\to\infty$,

2. (2)

   a sequence of gluing metrics $(\mathcal{M},g_{j})$ with weight functions $\rho_{j,\delta,\nu,\mu}^{(k+\alpha)}$ (for simplicity, we still denote by $\rho_{\delta,\nu,\mu}^{(k+\alpha)}$ because there is no ambiguity),

3. (3)

   a sequence of differential $1$-forms $\omega_{j}\in\Omega^{1}(\mathcal{M})$ such that

   (8.8)			$\displaystyle\|\omega_{j}\|_{C_{\delta,\nu,\mu}^{1,\alpha}(\mathcal{M},g_{j})}=1,$
   (8.9)			$\displaystyle\|\mathscr{D}_{g_{j}}\omega_{j}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M},g_{j})}+\|\omega_{j}\|_{C_{\delta,\nu,\mu}^{0}(\mathcal{M},g_{j})}\to 0,$

   as $j\to\infty$.

Our main goal is to prove a local version of the above weighted Schauder estimate. Precisely, it suffices to show that, there is some uniform constant $C>0$ (independent of $j$) and for every $\omega\in\Omega^{1}(\mathcal{M})$ and for every $\bm{x}_{j}\in\mathcal{M}$, there is some $r_{j}>0$ (depending on the location of $\bm{x}_{j}$) such that the following estimate holds in $B_{2r_{j}}(\bm{x}_{j})\subset(\mathcal{M},g_{j})$,

Once (8.10) is established, the contradiction immediately arises which completes the entire proof. Indeed, (8.9) implies that either $\|\rho_{\delta,\nu,\mu}^{(1)}\cdot\nabla\omega_{j}\|_{C^{0}(\mathcal{M},g_{j})}\geq\frac{1}{2}$ or $[\omega_{j}]_{C_{\delta,\nu,\mu}^{1,\alpha}(\mathcal{M},g_{j})}\geq\frac{1}{2}$. We can assume $\|\rho_{\delta,\nu,\mu}^{(1)}\cdot\nabla\omega_{j}\|_{C^{0}(\mathcal{M},g_{j})}\geq\frac{1}{2}$ because the argument for the other case is exactly the same. Hence by definition, there exists some $\bm{x}_{j}\in\mathcal{M}_{j}$ with

By (8.10), there is some $r_{j}>0$ which depends on $\bm{x}_{j}$ such that

The above estimate implies that

However, (8.13) contradicts (8.11). The proof is done.

So the main part of the proof of the proposition is to establish (8.10). In our proof, the primary strategy is to rescale the metric $g_{j}$ and the differential $1$-form $\omega$. That is, we choose some correct rescaling factors $\lambda_{j}>0$, $\kappa_{j}>0$ and define

In the previous section, we showed that, for every reference point $\bm{x}_{j}\in\mathcal{M}$, after appropriate rescaling, there is a subdomain $U_{j}$ which contains $\bm{x}_{j}$ and has uniformly bounded geometry away from at most finitely many singular points. Then the standard Schauder estimate in the rescaled spaces is available. That is, for every $\tilde{\omega}\in\Omega^{1}(\mathcal{M})$,

where $\tilde{r}_{0}>0$ is some uniform constant independent of the index $j$ such that the balls $B_{2\tilde{r}_{0}}^{\tilde{g}_{j}}(\bm{x}_{j})$ converge to a smooth space and the convergence keeps curvatures uniformly bounded. Once we obtain (8.15), we will get the weighted estimate (8.10) after an appropriate rescaling.

In the following arguments, for every fixed $\bm{x}_{j}\in\mathcal{M}$, we will choose the corresponding rescaled metrics $\tilde{g}_{j}=\lambda_{j}^{2}g_{j}$ defined in Section 7.3.

We prove (8.10) around the monopole $p_{m}\in\mathcal{P}_{m_{0}}$. Let $\lambda_{j}=\beta_{j}^{\frac{1}{2}}$ and we choose the rescaled metric $\tilde{g}_{j}\equiv\lambda_{j}^{2}g_{j}$, then

where $\tilde{g}_{\infty}$ is the standard Taub-NUT metric such that the length of the $S^{1}$-fiber at infinity equals $1$. Since the above convergence is $C^{\infty}$, the rescaled sequence $(\mathcal{M},\tilde{g}_{j},p_{m})$ has bounded geometry and thus the standard Schauder estimate holds in the geodesic ball $B_{2}^{\tilde{g}_{j}}(p_{m})$ with respect to the rescaled metric $\tilde{g}_{j}$. Precisely, there is a uniform constant such that for every $\tilde{\omega}\in\Omega^{1}(\mathcal{M})$,

Now we rescale back to the original metric $g_{j}$. First, we choose

and denote $\tilde{\omega}=\kappa_{j}\cdot\omega$. With respect to the original metric, the above Schauder estimate is equivalent to the following

where $r_{j}=\lambda_{j}^{-1}$. Therefore, by the definition of the weighted Hölder norm,

So the estimate (8.10) has been proved in Region $\I$.

We will prove (8.10) for every $\bm{x}_{j}$ in Region $\II$. As what is introduced in Section 7.3, we break down this region in $3$ cases with different rescaling geometries:

1. (a)

   There is a uniform constant $\sigma_{0}>0$ such that $2\beta_{j}^{-\frac{1}{2}}\leq d_{m}(\bm{x}_{j})\leq\frac{1}{\sigma_{0}}\cdot\beta_{j}^{-\frac{1}{2}}$.

2. (b)

   The distance to a pole $d_{m}(\bm{x}_{j})$ satisfies

   (8.21)

   $\displaystyle\frac{d_{m}(\bm{x}_{j})}{\beta_{j}^{-\frac{1}{2}}}\to\infty,\ \mbox{ and }\frac{d_{m}(\bm{x}_{j})}{\beta_{j}^{\frac{1}{2}}}\to 0.$

3. (c)

   There is some uniform constant $C_{0}>0$ such that

   (8.22)

   $$0<C_{0}\cdot\beta_{j}^{\frac{1}{2}}\leq d_{m}(\bm{x}_{j})\leq\frac{\iota_{0}^{\prime}}{4}\cdot\beta_{j}^{\frac{1}{2}}.$$

In each of the above cases, the rescaled spaces $(\mathcal{M},\tilde{g}_{j},\bm{x}_{j})$ have uniformly bounded curvatures and converge to a smooth limit space, which enables us to obtain the standard Schauder estimate in any ball of a definite radius the rescaled spaces. Specifically, let $\bm{x}_{j}$ be a fixed point in Region $\II$, then the standard Schauder estimate in $B_{1/6}^{\tilde{g}_{j}}(\bm{x}_{j})$ states that for any $\tilde{\omega}\in\Omega^{1}(\mathcal{M})$,

Now let

and denote $\omega_{j}=\tilde{\omega}_{j}/\kappa_{j}$, then rescaling to the original metrics $g_{j}$, we have

The above $r_{j}>0$ is defined as follows. If $\bm{x}_{j}$ is in Case (a) or (b), then

If $\bm{x}_{j}$ is in Case (c), then

where $d_{j}\equiv\min\limits_{1\leq m\leq m_{0}}d_{g_{j}}(p_{m},\bm{x}_{j})$. We need to show that the values $\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{y})$ for all $\bm{y}\in B_{r_{j}}(\bm{x}_{j})$ are equivalent. Indeed, by the triangle inequality, we can see that for every $\bm{y}\in B_{r_{j}}(\bm{x}_{j})$,

Therefore, by the definition of the weighted norm, the estimate (8.10) immediately follows.

If we choose $\lambda_{j}\equiv\beta_{j}^{-\frac{1}{2}}$, then the remaining arguments coincide with those in Case (c) of Region $\II$.

For fixed $\bm{x}_{j}$ in Region $\IV_{-}$, we define $\tilde{g}_{j}=\lambda_{j}^{2}g_{j}$ and

In Section 7.3, the rescaled limits were separated in the following cases:

1. (a)

   There is a constant $C_{0}>10T_{0}^{\prime}$ independent of the index $j$ such that

   (8.30)

   $$5T_{0}^{\prime}\leq d_{\tilde{g}_{j}}(p_{m},\bm{x}_{j})\equiv\lambda_{j}\cdot d_{m}(\bm{x}_{j})\leq C_{0}$$

   for each $1\leq m\leq m_{0}$.

2. (b)

   The reference points $\bm{x}_{j}$ in Region $\IV_{-}$ satisfy

   (8.31)

   $$d_{\tilde{g}_{j}}(p_{m},\bm{x}_{j})\equiv\lambda_{j}\cdot d_{m}(\bm{x}_{j})\to\infty,$$

We follow the notations in Section 7.3. Notice that, in each of the above cases, the rescaled spaces have uniformly bounded curvature and the standard Schauder estimate can be stated in the following way.

In Case (a), by assumption, we can pick some definite constant

such that for every $\tilde{\omega}\in\Omega^{1}(\mathcal{M})$,

In the above estimate, the constant $C>0$ depends only on $T_{0}^{\prime}>0$ and the flat product metric $g_{0}$ (particularly $C$ does not depend on $C_{0}$).

Now we rescale the $1$-form $\tilde{\omega}$ by choosing

and $\omega_{j}\equiv\tilde{\omega}_{j}/\kappa_{j}$. First, we rescale the above estimate to the original metric $g_{j}$ and denote

then

So the rest is to show the values of the weight function $\rho_{\delta,\nu,\mu}^{(k+\alpha)}$ are equivalent for every $\bm{y}\in B_{2r_{j}}(\bm{x}_{j})$. Indeed, denote $\zeta_{j}\equiv z(\bm{x}_{j})$, by straightforward computations, there is a uniform constant $C_{1}>0$ such that for every $\bm{y}\in B_{2r_{j}}(\bm{x}_{j})$,

Moreover, we can show that

for some uniform constant $C_{2}>0$. By the definition of the weight function in Region $\IV_{-}$,

The above estimates imply that there is a uniform constant $C_{3}>0$ such that

So the proof of (8.10) in Case (a) is done.

The proof of the estimate in Case (b) is the same.

First, we prove the weighted Schauder estimate (8.10) in Case (a). Let $\lambda_{j}\equiv(\underline{L}(\bm{x}_{j}))^{-1}$ and $\tilde{g}_{j}\equiv\lambda_{j}^{2}g_{j}$, then we have shown in Section 7.3 that

Moreover, the curvatures are uniformly bounded in the above convergence, which implies the standard Schauder estimate for every $\tilde{\omega}\in\Omega^{1}(\mathcal{M})$,

To prove the weighted estimate (8.10) in the original metrics $g_{j}$, we both rescale the metric $\tilde{g}_{j}$ and $\tilde{\omega}_{j}$ in the above estimate. First, let

and denote $\omega_{j}\equiv\tilde{\omega}_{j}/\kappa_{j}$, then

where

Now the last step is to show that all the values $\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{y})$ are equivalent for every $\bm{y}\in B_{2r_{j}}(\bm{x}_{j})$. Indeed, denote $\zeta_{j}\equiv z_{-}(\bm{x}_{j})$ and $\xi_{j}\equiv z_{-}(\bm{y})$, then straightforward computations immediately imply that

for some uniform constant $C_{0}>0$. By the definition of the weight function in Region $\V_{-}$,

Therefore,

and thus the proof of Case (a) is done.

Now we prove Case (b). We showed in Section 7.3 that the limit space $(\mathcal{M}_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty})$ is a finite rescale of $(X_{b_{-}}^{4},g_{b-},q_{-})$. Notice that, by the choice of the constant $C_{0}$ in Case (b), the geodesic ball $B_{2}^{\tilde{g}_{\infty}}(\bm{x}_{\infty})$ is contained in the end part of $\mathcal{M}_{\infty}$ which has an $S^{1}$-fibration structure. It follows that $B_{2}^{\tilde{g}_{\infty}}(\bm{x}_{\infty})$ has uniformly bounded geometry. Then for every $\tilde{\omega}\in\Omega^{1}(\mathcal{M})$, the standard Schauder estimate holds and we have

where $C>0$ is independent of the index $j$ and the constant $C_{0}$. Now we rescale the above estimate to the original metric and we also rescale $\omega$ as in Case (a), which gives

where

It is similar to Case (a) that there is some constant $C_{2}>0$ which is independent of the index $j$ and the constant $C_{0}$, such that

By definition, the weighted Schauder estimate (8.10) immediately follows.

First, we consider the case that the fixed reference point $\bm{x}_{j}$ is in Region $\VI_{-}$. We choose a ball $B_{1}(\bm{x}_{j})$ and the estimate (8.10) is the standard Schauder estimate on $B_{1}(\bm{x}_{j})$. Since the weight function in this region is uniformly bounded, the weighted Schauder estimate (8.10) is equivalent to the standard one.

Next, if $\bm{x}_{j}$ is in Region $\VI_{+}$, then

and the standard Schauder estimate states that there is a uniform constant $C>0$ such that for every $\tilde{\omega}\in\Omega^{1}(\mathcal{M})$,

So we just rescale the the $1$-form $\tilde{\omega}$ by letting $\kappa_{j}\equiv e^{\delta(2T_{-}+2T_{+})}$ and $\omega=\tilde{\omega}/\kappa_{j}$, then (8.54) is equivalent to

Up to some uniformly bounded constant, the above estimate implies the weighted Schauder estimate (8.10).

∎