Let ${G}_{\text{NUT}}$ be the Green operator on the model space, so $u^{\prime}={G}_{\text{NUT}}f$ is an $S^{1}$-invariant function. Applying Hein’s package on Poisson equations as in Corollary 2.17 with a simple scaling argument, the function $u^{\prime}$ decays like

$$|u^{\prime}|\leq CA^{-1}(A^{1/2}r+1)^{-3+\epsilon}.$$

The decay exponent $-3+\epsilon$ here comes from the quintic volume growth rate of $g_{\text{NUT}}$. Since the model space is smooth, we can bootstrap this to a weighted $C^{2,\alpha}$ estimate on $u^{\prime}$. The function $u$ is obtained by cutting off $u^{\prime}$ at a dyadic scale $r\sim A^{1/4}$. The cutoff error is controlled by the Hessian estimate. ∎

Take a large collection of points $\{P_{i}\}_{i=1}^{N}$ on $S\cap\mathcal{B}^{-}_{\nu}$, such that for any point $P$ on $S\cap\mathcal{B}^{-}_{\nu}$, the number of points $P_{i}$ in the collection within $g_{a}$-distance $O(A^{-1/2})$ to $P$ is at least one but no more than $C$. Then take cutoff functions $\chi_{i}$ on $\mathcal{B}^{-}_{\nu}$ supported in $\{\text{dist}_{g_{a}}(\cdot,P_{i})\lesssim A^{-1/2}\}$ such that $\sum\chi_{i}=1$ on $\mathcal{B}^{-}_{\nu}\cap\{R\lesssim A^{-1/2}\}$. These allow us to decompose $f$ into a large number of localised contributions:

$$f=\sum_{i=1}^{N}\chi_{i}f,\quad\left\lVert\chi_{i}f\right\rVert_{C^{\alpha}_{0}}\leq C,$$

using the fact that the $C^{\alpha}_{\delta}$-norm is not sensitive to $\delta$ inside $\{R\lesssim A^{-1/2}\}$.

For each term $\chi_{i}f$ we apply Lemma 4.35 to produce an approximate local solution $u_{i}$ on $\{r_{P_{i}}\lesssim A^{1/4}\}$, with bounds prescribed in Lemma 4.35. Here $r_{P_{i}}(Q)$ is uniformly equivalent to $|P_{i}-Q|_{g_{a}}$. The candidate solution is

$$u=u_{1}+u_{2}+\ldots u_{N}.$$

By construction $u$ is supported in $\{R\lesssim A^{1/4}\}$.

We now bound $u$, focusing on the absolute estimate. Summing up the contributions

$$|u_{i}|\leq CA^{-1}(A^{1/2}r_{P_{i}}+1)^{-3+\epsilon}\left\lVert\chi_{i}f\right\rVert_{C^{\alpha}_{0}}\leq CA^{-1}(A^{1/2}r_{P_{i}}+1)^{-3+\epsilon},$$

we estimate at a point $Q$:

$$\begin{split}|u(Q)|\leq&\sum_{|P_{i}-Q|_{g_{a}}\lesssim A^{1/4}}|u_{i}|(Q)\\ \leq&CA^{-1}\sum_{|P_{i}-Q|_{g_{a}}\lesssim A^{1/4}}(A^{1/2}|P_{i}-Q|_{g_{a}}+1)^{-3+\epsilon}\\ \leq&C\int_{S\cap\{|P-Q|_{g_{a}}\lesssim A^{1/4}\}}(A^{1/2}|P-Q|_{g_{a}}+1)^{-3+\epsilon}d\mathcal{A}(P)\\ \leq&CA^{-1}(A^{1/2}R(Q)+1)^{-1+\epsilon}.\end{split}$$

The higher order version is $\left\lVert u\right\rVert_{C^{2,\alpha}_{-1-\epsilon}}\leq CA^{-1},$ so $\left\lVert u\right\rVert_{C^{2,\alpha}_{\delta}}\leq CA^{-1}$ for $\delta>-3+\epsilon$.

Next we estimate the error $\Delta_{g^{(2)}}u-f$. The error $\Delta_{g^{(2)}}u_{i}-\chi_{i}f$ has two sources: the cutoff error supported on $\{r_{P_{i}}\sim A^{1/4}\}$ from Lemma 4.35

$$\left\lVert\Delta_{g_{\text{NUT}}}u_{i}-\chi_{i}f\right\rVert_{C^{\alpha}_{-2}(g_{\text{NUT}})}\leq CA^{3(-3+\epsilon)/4},$$

and the metric deviation error $\Delta_{g_{\text{NUT}}}u_{i}-\Delta_{g^{(2)}}f$, which is controlled because by Proposition 4.19 the local diffeomorphism $\Psi$ is a $C^{1,\alpha}$-approximate isometry between $g_{\text{NUT}}$ and $g^{(2)}$, and $f$ has weighted $C^{2,\alpha}$ control. We focus on the absolute estimate:

$$|\Delta_{g^{(2)}}u_{i}-\Delta_{g_{\text{NUT}}}u_{i}|\leq CA^{-3/4}(A^{1/2}r_{P_{i}}+1)^{-3+\epsilon}(A^{1/2}R+1)^{-2}(A|\xi_{2}|+\nu),$$

so that

$$|\Delta_{g^{(2)}}u_{i}-\chi_{i}f|\leq CA^{-3/4}(A^{1/2}r_{P_{i}}+1)^{-3+\epsilon}(A^{1/2}R+1)^{-2}(A^{1/2}r_{P_{i}}+\nu).$$

Using

$$\begin{cases}\sum_{|P_{i}-Q|_{g_{a}}\lesssim A^{1/4}}(A^{1/2}r_{P_{i}}(Q)+1)^{-3+\epsilon}\leq(A^{1/2}R(Q)+1)^{-1+\epsilon},\\ \sum_{|P_{i}-Q|_{g_{a}}\lesssim A^{1/4}}(Ar_{P_{i}}(Q)+1)^{-2+\epsilon}\leq CA^{3\epsilon/4},\end{cases}$$

we sum up all contributions to deduce for $\delta>-3+\epsilon$,

$$\begin{split}&|\Delta_{g^{(2)}}u-f|\leq\sum_{|P_{i}-Q|_{g_{a}}\lesssim A^{1/4}}|\Delta_{g^{(2)}}u_{i}-\chi_{i}f|\\ \leq&CA^{-3/4}\{A^{3\epsilon/4}(A^{1/2}R+1)^{-2}+\nu(A^{1/2}R+1)^{-3+\epsilon}\}\ll(A^{1/2}R+1)^{\delta}.\end{split}$$

The Hölder version is $\left\lVert\Delta_{g^{(2)}}u-f\right\rVert_{C^{\alpha}_{\delta}}\ll 1$ as required. ∎