The key technique is to construct a parametrix $P_{g^{(2)}}$ for the Green operator semi-explicitly, with precise control on its mapping properties.

Given a function $f$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}=1$, temporarily assumed to have sufficient decay at infinity, we will construct an approximate solution $u=P_{g^{(2)}}f$ to the Poisson equation as follows. Take a smooth cutoff function $\chi^{\prime}$

$$\chi^{\prime}=\begin{cases}1\quad&\text{dist}(\cdot,\mathfrak{D})\geq 2,\\ 0\quad&\text{dist}(\cdot,\mathfrak{D})\leq 1,\end{cases}$$

then $\chi^{\prime}f$ has norm $\left\lVert\chi^{\prime}f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\lesssim 1$ and is supported in $\{\text{dist}(\cdot,\mathfrak{D})\geq 1\}$. Applying Lemma 2.18, the function $u_{0}=\Delta_{a}^{-1}(\chi^{\prime}f)$ satisfies $\left\lVert\nabla^{2}_{g^{(2)}}u_{0}\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C$. By Lemma 2.19 we can choose $C_{2}\gg 1$ large enough independent of $f$ to ensure

$$\left\lVert\Delta_{g^{(2)}}u_{0}-f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3}\cap\{\text{dist}(\cdot,\mathfrak{D})>C_{2}/2\})}\ll 1.$$

Next we take smooth cutoff functions $\chi_{1}^{\prime},\chi_{2}^{\prime},\chi_{3}^{\prime}$ near $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$, such that

$$\chi_{1}^{\prime}=\begin{cases}1\quad&\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})\leq C_{2}/2\text{ and }|\vec{\mu}|_{a}>2C_{2},\\ 0\quad&\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})\geq C_{2}\text{ or }|\vec{\mu}|_{a}<C_{2}\end{cases}$$

and similarly with $\chi_{2}^{\prime},\chi_{3}^{\prime}$. The function

$$f_{1}=\chi_{1}^{\prime}(f-\Delta_{g^{(2)}}u_{0})=\chi_{1}^{\prime}(f-\Tr_{g^{(2)}}\nabla^{2}_{g^{(2)}}u_{0})$$

is supported in $\{\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})\leq C_{2},|\vec{\mu}|_{a}\geq C_{2}\}$ with bound $\left\lVert f_{1}\right\rVert_{C^{k,\alpha}_{\delta,\tau}}\leq C$. So we can apply Lemma 2.20 and Lemma 2.21 to find $u_{1}=\chi_{1}G_{\text{Taub}}f_{1}$ with bounds

$$\left\lVert\nabla^{2}_{g^{(2)}}u_{1}\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C,\quad\left\lVert\Delta_{{g^{(2)}}}u_{1}-f_{1}\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3}\cap\{|\vec{\mu}|_{a}>2C_{3}\})}\leq CC_{3}^{-\epsilon}\ll 1.$$

Completely analogous constructions are made near $\mathfrak{D}_{2}$ and $\mathfrak{D}_{3}$, where we obtain $u_{2},u_{3}$ with similar bounds.

Let $\chi_{4}^{\prime}$ be a smooth cutoff function

$$\chi_{4}^{\prime}=\begin{cases}1\quad&|\vec{\mu}|_{a}\leq 2C_{3},\\ 0\quad&|\vec{\mu}|_{a}\geq 4C_{3},\end{cases}$$

and define $f_{4}=\chi_{4}^{\prime}(f-\Delta_{g^{(2)}}(u_{0}+u_{1}+u_{2}+u_{3}))$, which is supported in the ball $\{|\vec{\mu}|_{a}\leq 4C_{3}\}$ and admits the bound $\left\lVert f_{4}\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C$. Then we can apply Lemma 2.22 to obtain $u_{4}=G_{g^{(2)}}f_{4}$ with bounds

$$\left\lVert\nabla^{2}_{g^{(2)}}u_{4}\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C,\quad\Delta_{{g^{(2)}}}u_{4}=f_{4}.$$

We set $P_{g^{(2)}}f=u=u_{0}+u_{1}+u_{2}+u_{3}+u_{4}$. The key point is that by construction

$$\left\lVert\nabla^{2}_{g^{(2)}}u\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C,\quad\left\lVert\Delta_{{g^{(2)}}}u-f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\ll 1,$$

namely $u$ is an approximate solution to the Poisson equation with bounds. A subtlety is that $\nabla^{2}_{g^{(2)}}u$ is fully controlled while $u$ is only controlled up to an additive constant. In any event, after extending the definition of the operator by removing the fast decay hypotheses on $f$, we have defined a bounded linear operator between weighted Hölder spaces of $T^{2}$-invariant functions and symmetric 2-tensors on $\mathbb{C}^{3}$

$$\nabla^{2}_{g^{(2)}}P_{g^{(2)}}:C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})\to C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\text{Sym}^{2}),$$

such that the operator $\Tr_{g^{(2)}}\nabla^{2}_{g^{(2)}}P_{g^{(2)}}$ is an approximation to the identity. Thus

$$\mathcal{R}^{\prime}=\nabla^{2}_{g^{(2)}}P_{g^{(2)}}(\Tr_{g^{(2)}}\nabla^{2}P_{g^{(2)}})^{-1}$$

is a bounded right inverse to $\Tr_{g^{(2)}}$. Composing with the projection to the type (1,1)-forms defines the operator

$$\mathcal{R}=\sqrt{-1}\partial\bar{\partial}P_{g^{(2)}}(\Tr_{g^{(2)}}\nabla^{2}P_{g^{(2)}})^{-1},$$

which takes value in closed (1,1)-forms and is a bounded inverse to $\Tr_{g^{(2)}}$. It is worth commenting that the same operators work for different exponents $\delta,\tau$.

It remains to relate $\mathcal{R}$ and $\mathcal{R}^{\prime}$ to the Green operator $G_{g^{(2)}}$ when $f$ has sufficient decay at infinity. The point is that for fast decay weights $\delta+\tau<-2$ and $\delta<-2$, the Hessian control $\left\lVert\nabla^{2}_{g^{(2)}}u\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq C$ together with the a priori qualitative decay $u\to 0$ at infinity, imply the quantitative bound $\left\lVert u\right\rVert_{C^{k+2,\alpha}_{\delta+2,\tau}(\mathbb{C}^{3})}\leq C.$ This enables us to extend ${P}_{g^{(2)}}$ to a bounded linear operator

$${P}_{g^{(2)}}:C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})\to C^{k+2,\alpha}_{\delta+2,\tau}(\mathbb{C}^{3})$$

and the operator

$$P_{g^{(2)}}(\Tr_{g^{(2)}}\nabla^{2}P_{g^{(2)}})^{-1}:C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})\to C^{k+2,\alpha}_{\delta+2,\tau}(\mathbb{C}^{3})$$

defines an inverse to the Laplacian $\Delta_{g^{(2)}}$. By the uniqueness of decaying solution to the Poisson equation $G_{g^{(2)}}=P_{g^{(2)}}(\Tr_{g^{(2)}}\nabla^{2}P_{g^{(2)}})^{-1}$. Hence

$$\mathcal{R}^{\prime}=\nabla^{2}_{g^{(2)}}G_{g^{(2)}},\quad\mathcal{R}=\sqrt{-1}\partial\bar{\partial}G_{g^{(2)}}$$

as required. ∎