The absolute value is estimated by Corollary 2.16:

$$|G_{g^{(2)}}f|\leq C(1+|\vec{\mu}|_{a})^{-2+\epsilon}.$$

The higher order estimate $\left\lVert\nabla^{2}_{g^{(2)}}G_{g^{(2)}}f\right\rVert_{C^{k,\alpha}_{-2,-2+\epsilon}(\mathbb{C}^{3})}\leq C$ follows by bootstrapping, which controls $C^{k,\alpha}_{\delta,\tau}$ norm for the given range of weight exponents $\delta,\tau$. ∎