Let $-3<\delta<0$ and $\delta+\tau<0$. Let $f$ be a $T^{2}$-invariant function on $\mathbb{C}^{3}$ supported in $\{\text{dist}(\cdot,\mathfrak{D})\gtrsim 1\}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\leq 1$. Then the second order derivatives of the Euclidean potential $\Delta_{a}^{-1}f$ satisfies

$$|\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f|_{g_{a}}\leq C\ell^{\delta}(|\vec{\mu}|_{a}+1)^{\tau}.$$

Morever if $\delta<-1$ and $\delta+\tau<-1$, then

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

The constants depend only on $k,\alpha,\delta,\tau$ and the uniform ellipticity bound on $a_{ij}$.