Given $-3<\delta<-1$ and $1\ll\nu\ll A^{3/8}$, let $f$ be an $S^{1}$-invariant function compactly supported in $M^{-}_{\nu}$ with norm $\left\lVert f\right\rVert_{C^{\alpha}_{\delta}}=1$. Then there is an $S^{1}$-invariant function $u$ approximately solving the Poisson equation:

$$\left\lVert\Delta_{g^{(2)}}u-f\right\rVert_{C^{\alpha}_{\delta}(M^{-}_{\nu})}\ll 1,$$

with the Hessian bound

$$\left\lVert\nabla^{2}_{g^{(2)}}u\right\rVert_{C^{\alpha}_{\delta}(M^{-}_{\nu})}\leq C,\quad\left\lVert du\right\rVert_{C^{1,\alpha}_{\delta+1}(M^{-}_{\nu})}\leq CA^{-1/2}.$$

The constants depend only on $\delta,\alpha,\kappa$ and the scale invariant ellipticity bound on $a_{p\bar{q}}$.