Assume

$$\begin{cases}\text{Either }&-2\leq\delta\leq 0,\quad\tau>-2,\\ \text{Or }&\delta\leq-2,\quad\delta+\tau>-4,\end{cases}$$

and let $0<\epsilon\ll 1$ depending on $\delta,\tau$. If $f$ is supported in the ball $\{|\vec{\mu}|_{a}<4C_{3}\}$, with bound $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq 1$ or equivalently $\left\lVert f\right\rVert_{C^{k,\alpha}_{0,0}(\mathbb{C}^{3})}\lesssim 1$, then $\left\lVert G_{g^{(2)}}f\right\rVert_{C^{k,\alpha}_{0,-2+\epsilon}(\mathbb{C}^{3})}\leq C,$ so in particular

$$\left\lVert\nabla^{2}_{g^{(2)}}G_{g^{(2)}}f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq\left\lVert\nabla^{2}_{g^{(2)}}G_{g^{(2)}}f\right\rVert_{C^{k,\alpha}_{-2,-2+\epsilon}(\mathbb{C}^{3})}\leq C.$$