The main part is to prove Item (1). We only prove the estimate by assuming the scale parameter $r=1$. The estimate in the general case $r\in(0,1)$ can be achieved by simple rescaling.

The proof is based on the explicit description of the $C^{k,\alpha}$-regularity scale given by Proposition 4.18. Since we have shown that, under the rescalings

the geodesic balls $B_{1/2}^{\tilde{g}}(\bm{x})$ have uniformly bounded $C^{k,\alpha}$-geometry (independent of $T$) for each $\alpha\in(0,1)$ and $k\in\{0,1\}$. So there is a uniform constant $C>0$ (independent of $T$) such that the standard Schauder estimate holds for every $u\in\mathfrak{A}$ and $\bm{x}\in\mathring{\mathcal{M}}(T_{-},T_{+})$,

Then the desired weighted Schauder estimate (4.288) will be obtained after appropriately rescaling. The argument is rather standard. In fact, the only crucial point is to verify that for every $\bm{x}\in\mathring{\mathcal{M}}(T_{-},T_{+})$, the weight function $\rho_{\delta,\nu,\mu}^{(k+\alpha)}$ is roughly a constant in the ball $B_{s(\bm{x})}(\bm{x})$ in the sense that there is a uniform constant $C>0$ such that for any $\bm{y}\in B_{s(\bm{x})}(\bm{x})$,

The verifications of the above estimate essentially follows from Corollary 4.18.1 which is the Harnack inequality for the regularity scale. As a comparison, the detailed arguments in dimension $4$ is given in Section 8 of [HSVZ18]. In the following, we only verify (4.293) in Region $\bf{I}_{1}$ and Region $\bf{I}_{2}$ as sample examples.

By Proposition 4.18, the canonical scale in this case is chosen as $\mathfrak{s}(\bm{x})=T^{\frac{1-n}{n}}$, while the rescaling factor is $\lambda(\bm{x})=T^{\frac{n-1}{n}}$ such that $(\mathcal{M}_{T},\tilde{g}_{T},\bm{x})$ is close to the Riemann product $\mathbb{C}_{TN,1}^{2}\times\mathbb{C}^{n-2}$ in the pointed $C^{2,\alpha}$-topology for any $\alpha\in(0,1)$, where $\mathbb{C}_{TN,1}^{2}$ is the Ricci-flat Taub-NUT space. Then for $k\in\{0,1\}$ and $\alpha\in(0,1)$,

Since the weight function, by definition, is constant in the geodesic ball $B_{s(\bm{x})}(\bm{x})$ for $s(\bm{x})=\frac{1}{4}\mathfrak{s}(\bm{x})$. With respect to the original metric, the standard Schauder estimate (4.292) for $u$ is equivalent to

Therefore, by the definition of the weighted Hölder space,

The proof in Region $\bf{I}_{1}$ is done.

Proposition 4.18 tells us that, in this region, $\mathfrak{s}(\bm{x})=T^{\frac{1}{n}}\cdot r(\bm{x})$ and the metric is rescaled by $\lambda(\bm{x})=\mathfrak{s}(\bm{x})^{-1}$ with

We notice that the values $\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{y})$ for all $\bm{y}\in B_{2s(\bm{x})}(\bm{x})$ are uniformly equivalent. Indeed, by Corollary 4.18.1, we can see that for every $\bm{y}\in B_{2s(\bm{x})}(\bm{x})$,

So the standard Schauder estimate (4.292) for $\tilde{u}$ is equivalent to the following estimate for $u$, is equivalent to

Therefore, by the definition of the weighted norm, the required estimate immediately follows.

For the remaining regions, the key point in the proof is in fact the same, which just requires to show that the values of the weight function at the points within the $C^{k,\alpha}$-regularity scale are uniformly equivalent. So we just skip the proof.

Now we switch to prove Item (2), which can be obtained by contradiction. Suppose there is no such a constant $C_{\mathcal{P}}>0$. Then there are a sequence of numbers $T_{j}>0$ and reference points $\bm{x}_{j}\in\mathcal{M}_{T_{j}}$ such that

but the uniform local Schauder estimate (4.288) does not hold around $\bm{x}_{j}\in\mathcal{M}_{T_{j}}$. Under the contradicting assumption (4.300), Proposition 4.18 shows that, with respect to the rescaled metrics we choose, we will obtain one of the following rescaled Gromov-Hausdorff limits depending upon the location of $\bm{x}_{j}$ in the subdivision:

1. (i)

   The Euclidean product $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$,

2. (ii)

   The cylinder $D\times\mathbb{R}$,

3. (iii)

   The Calabi space $(\mathcal{C}_{-}^{n},g_{\mathcal{C}_{-}^{n}},\bm{x}_{-})$ or $(\mathcal{C}_{+}^{n},g_{\mathcal{C}_{+}^{n}},\bm{x}_{+})$.

Moreover, away from the singularity, the convergence is $C^{k,\alpha}$ for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ by passing to the local universal cover.

First, if the convergence keeps the $C^{k,\alpha}$-geometry uniformly bounded, then the proof of the higher order estimate is just standard and routine.

Now let $\bm{x}_{j}$ stay in the regions giving the rescaled limits in (i) and (ii). Recall the discussions in Section 4.3 that, in Case (b), (c) in Region $\bf{I}_{2}$ and Case (a) in Region $\bf{I}_{3}$, singularity behavior appears in the Gromov-Hausdorff procedure. With respect to the rescaled metric $\tilde{g}_{j}=\lambda(\bm{x}_{j})^{-2}g_{j}$, the limiting geodesic ball $B_{\frac{1}{2}}^{\tilde{g}_{j}}(\bm{x}_{j})$ never contains the singularity. So it follows that every point $\bm{y}\in B_{\frac{1}{2}}^{\tilde{g}_{j}}(\bm{x}_{j})$ has a $C^{k,\alpha}$-regularity scale $r_{k,\alpha}(\bm{y})\geq\rho_{0}>0$ for all $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$. So the standard interior Schauder estimate reads as follows,

for all $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$. Rescaling back to the original metrics $g_{j}$, we obtain the desired weighted Schauder estimate for sufficiently large $j$. So the contradiction arises. This completes the proof of Item (2).

The proof of Item (3) follows from the Schauder estimate for Neumann boundary problem. As before, we only consider the case $r=1$ for simplicity. The tubular neighborhood $T_{2}(\partial\mathcal{M}_{T})$ belongs to Case (c) of Region $\bf{I}_{3}$. We only consider the left boundary $\{T=T_{-}\}$. For every $\bm{x}\in\{T=T_{-}\}$, we choose the rescaled metric $\tilde{g}=\lambda(\bm{x})^{2}\cdot g$ with

where $c_{-}>0$ is a fixed constant. The analysis in Section 4.3 tells us that, for $T\gg 1$ sufficiently large, $(\mathcal{M}_{T},\tilde{g},\bm{x})$ is Gromov-Hausdorff close to a fixed incomplete Calabi space $(\mathcal{C}_{-}^{n},g_{\mathcal{C}_{-}^{n}},\bm{x}_{\infty})$. Moreover, the tubular neighborhood $T_{2}(\partial M_{T})$ satisfies the following property: there are constants $\rho_{0}>0$ depending only the conjugate radius of $(\mathcal{C}_{-}^{n},g_{\mathcal{C}_{-}^{n}},\bm{x}_{\infty})$ such that every point $\bm{y}\in T_{2}(\partial M_{T})$ satisfies the regularity scale estimate $r_{k,\alpha}(\bm{y})\geq\rho_{0}>0$ for all $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$.

The above geometric regularity implies the following uniform boundary Schauder estimate in $B_{2}^{+}(\bm{x})$ for each $\bm{x}\in\partial\mathcal{M}_{T}$,

Here $\frac{\partial}{\partial n}$ is the exterior normal vector field, and the constant $C_{k,\alpha}>0$ depends only on $k$, $\alpha$, $\rho_{0}$. This estimate is standard in the literature (see Section 6 of [GT01] for instance). By rescaling, we obtain the desired weighted estimate.

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