Let ${G}_{\text{NUT}}$ be the Green operator on the model space, so $u^{\prime}={G}_{\text{NUT}}f$ is an $S^{1}$-invariant function. Applying Hein’s package on Poisson equations as in Corollary 2.17 with a simple scaling argument, the function $u^{\prime}$ decays like

$$|u^{\prime}|\leq CA^{-1}(A^{1/2}r+1)^{-3+\epsilon}.$$

The decay exponent $-3+\epsilon$ here comes from the quintic volume growth rate of $g_{\text{NUT}}$. Since the model space is smooth, we can bootstrap this to a weighted $C^{2,\alpha}$ estimate on $u^{\prime}$. The function $u$ is obtained by cutting off $u^{\prime}$ at a dyadic scale $r\sim A^{1/4}$. The cutoff error is controlled by the Hessian estimate. ∎