Suppose $(\delta,\tau)$ stays within a compact subset of the good range of weight exponents. Then the operator $\mathcal{R}^{\prime}=\nabla^{2}_{g^{(2)}}G_{g^{(2)}}$ extends to bounded linear operators between the weighted Hölder spaces

$$\mathcal{R}^{\prime}:C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})\to C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\text{Sym}^{2}),\quad\left\lVert\mathcal{R}^{\prime}\right\rVert\leq C$$

where the constant depends only on $k,\alpha$, the compact region of exponents $(\delta,\tau)$, and the scale invariant ellipticity bound (2.11). The composition with the natural projection

$$\mathcal{R}:C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})\xrightarrow{\mathcal{R}^{\prime}}C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\text{Sym}^{2})\to C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\Lambda^{1,1})$$

extends the operator $\mathcal{R}=\sqrt{-1}\partial\bar{\partial}G_{g^{(2)}},$ which takes value in closed real (1,1)-forms and is inverse to taking trace.