Note that

$$D_{\sigma}\mathfrak{F}_{k}(0,0)\dot{\sigma}=(d\dot{\sigma},*_{h}d*_{h}\dot{\sigma})+(D_{\sigma}(d\alpha_{k}|_{L})\dot{\sigma},*_{h}D_{\sigma}(d{\rm Im}\beta_{k}|_{L})\dot{\sigma})=(\mathcal{D}+V_{k})\dot{\sigma},$$

where $\mathcal{D}=d-*_{h}d*_{h}$ is the restriction of the Hodge Dirac operator $d+d^{*_{h}}$ on the space of 1-forms, and, thus, is an elliptic operator of 1-order. By the standard elliptic estimate (c.f. Proposition 1.5.2 in [23] and [20]), we have

$$\|\xi\|_{C^{1,\alpha}(L,h)}\leq C_{S}\|\mathcal{D}\xi\|_{C^{0,\alpha}(L,h)},$$

for any $\xi\in\mathfrak{B}_{1}$, and a constant $C_{S}$ independent of $k$. Hence $\mathcal{D}$ is injective. From the definition of $\mathfrak{B}_{2}$, $\mathcal{D}$ is also surjective, which implies that $\mathcal{D}$ is invertible from $\mathfrak{B}_{1}$ to $\mathfrak{B}_{2}$. Moreover,

$$\|\mathcal{D}^{-1}\|\leq C_{S}.$$

By (7) and (8),

$$\|V_{k}\|\leq C\|(d\alpha_{k},d\beta_{k})\|_{C^{1,\alpha}(Y_{2r},g)}<\frac{1}{2C_{S}},$$

for $k\gg 1$, and, thus,

$$\|\mathcal{D}^{-1}V_{k}\|<\frac{1}{2}.$$

By the standard operator’s theory (c.f. [36]), $D_{\sigma}\mathfrak{F}_{k}(0,0)=\mathcal{D}+V_{k}$ is invertible, and the inverse operator is defined by

$$D_{\sigma}\mathfrak{F}_{k}(0,0)^{-1}=(\sum_{j=0}^{\infty}(-1)^{j}(\mathcal{D}^{-1}V_{k})^{j})\mathcal{D}^{-1}.$$

We obtain

$$\|D_{\sigma}\mathfrak{F}_{k}(0,0)^{-1}\|\leq(\sum_{j=0}^{\infty}2^{-j})\|\mathcal{D}^{-1}\|\leq\overline{C},$$

for a constant $\overline{C}>0$ independent of $k$. ∎