The key observation is that if without loss of generality $f$ has no zeroth Fourier modes, then the convolution integral

$$\nabla^{2}_{g_{a}}G_{a}*f=(\nabla^{2}_{g_{a}}G_{a}-\nabla^{2}_{g_{a}}\bar{G}_{a})*f,$$

but Lemma 3.18 says the integral kernel $\nabla^{2}_{g_{a}}G_{a}-\nabla^{2}_{g_{a}}\bar{G}_{a}$ has exponential decay, at a rate faster than the exponential decay rate of $f$ itself. Thus at any point $p$ in the region $\{\ell>A^{1/2}\}\cap\mathcal{B}^{+}_{\nu}$, the contribution to $\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f|_{p}$ from sources outside the ball $\{|p-q|_{a}^{\prime}\lesssim A^{1/2}\}$ is negligible. The contribution from sources inside the ball is treated by standard Schauder theory, and inherits the same exponential decay factor $e^{-\kappa\tilde{\ell}}$ as $f$ itself. ∎