The following few Sections are aimed at perturbing the Kähler ansatz into a Calabi-Yau metric. This Section sets up the weighted Hölder norms and measure the volume form error

(4.30)

$$E^{(1)}=\frac{\det(W^{p\bar{q}}_{(1)})}{V_{(1)}}-1=\frac{A+Aa^{p\bar{q}}w^{p\bar{q}}+\det(w^{p\bar{q}})}{A+v}-1=\frac{\det(w^{p\bar{q}})}{A+v}.$$

There are three weight parameters:

$$\varrho=|(y_{1},y_{2},\mu)|_{a}^{\prime},\quad R=\text{dist}_{g_{a}}(\cdot,S),\quad\tilde{\ell}=\kappa_{a}\text{dist}_{g_{a}^{\prime}}(\cdot,\text{Im}(S)).$$

The parameter $\tilde{\ell}$ is useful for measuring exponential decay rates (cf. Proposition 4.13). The following definitions are parallel to Section 3.4.

Let $\delta\leq 0$. We shall define the weighted Hölder norms $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ for $S^{1}$-invariant tensor fields $T$ on $M^{-}$, by prescribing the norm on a number of overlapping regions up to uniform equivalence.

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  The region $\{R\lesssim A^{1/4}\}$ is covered by local charts $\{r\lesssim A^{1/4}\}$ introduced in Section 4.4 and 4.5, where the ansatz metric is approximated by $g_{\text{NUT}}$. Let $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ be uniformly equivalent to the norm $\left\lVert\Psi^{*}T\right\rVert_{C^{k,\alpha}_{\delta}(g_{\text{NUT}})}$ in Section 4.5. Inside $\{R\lesssim A^{-1/2}\}$ the metric ansatz is $C^{1,\alpha}$-regular, so correspondingly we should work with functions of at most $C^{2,\alpha}$-regularity and tensors of at most $C^{1,\alpha}$-regularity. Inside $\{R\gtrsim A^{-1/2}\}$ there is no restriction on regularity.

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  The region $\{\tilde{\ell}\gtrsim 1\}$ can be covered by subregions of diameter $\sim R$, where the $S^{1}$-bundle is topologically trivial. Over each subregion the metric is approximated by the periodic version of the constant solution $g_{\text{flat}}$ (cf. Section 2.2). The $x_{1},x_{2}$ variables define two periodic direction. We decompose $T$ into the part $\bar{T}$ independent of $x_{1},x_{2}$ (the ‘zeroth Fourier mode’) and the oscillatory part $T-\bar{T}$ (the ‘higher Fourier mode’), and define the weighted Hölder norm separately on the two parts:

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  On the zeroth Fourier mode, the norm $\left\lVert\bar{T}\right\rVert_{C^{k,\alpha}_{\delta,0}}$ is equivalent to

  $$A^{-3\delta/4}(\sum_{j=0}^{k}\left\lVert R^{j}\nabla^{j}\bar{T}\right\rVert_{L^{\infty}}+[R^{k}\nabla^{k}\bar{T}]_{\alpha}),$$

  where $[]_{\alpha}$ denotes the appropriately normalised Hölder seminorm.

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  On the higher Fourier modes we build in the exponential decay. Fix a parameter $0<\kappa<1$. The norm $\left\lVert T-\bar{T}\right\rVert_{C^{k,\alpha}_{\delta,0}}$ in this region is equivalent to

  $$A^{-3\delta/4}\sup_{\tilde{\ell}\gtrsim 1}e^{\kappa\tilde{\ell}}(\sum_{j=0}^{k}\left\lVert A^{j/4}\nabla^{j}(T-\bar{T})\right\rVert_{L^{\infty}}+A^{k/4}[\nabla^{k}(T-\bar{T})]_{\alpha}).$$

  An estimate in this norm is the higher order version of $|T-\bar{T}|\leq CA^{3\delta/4}e^{-\kappa\tilde{\ell}}.$

We will also need a variant weighted Hölder norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta}}$. The only difference from $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ is that in the region $\{\tilde{\ell}\gtrsim 1\}$ on the zeroth Fourier mode, $\left\lVert\bar{T}\right\rVert_{C^{k,\alpha}_{\delta}}$ is equivalent to

$$A^{-\delta/2}(\sum_{j=0}^{k}\left\lVert R^{j-\delta}\nabla^{j}\bar{T}\right\rVert_{L^{\infty}}+[R^{k-\delta}\nabla^{k}\bar{T}]_{\alpha}),$$

so an estimate in this norm is the higher order version of $|\bar{T}|=O(A^{3\delta/4}(A^{-1/4}R)^{\delta})$. We have inserted an extra decay factor $(A^{-1/4}R)^{\delta}$.