The basic idea is similar to Proposition 2.18. We analyse the contribution of the source located at $q$ to the convolution integral $\nabla^{2}_{g_{a}}G_{a}*f(p)$, depending on the spatial separation between $p$ and $q$. We write $|q|_{a}^{\prime}=\varrho(q),|p|_{a}^{\prime}=\varrho(p)$.

Suppose $p$ and $q$ do not belong to the same dyadic scale, namely $|p|_{a}^{\prime}\geq A^{1/2}+2|q|_{a}^{\prime}$ or $|q|_{a}^{\prime}\geq A^{1/2}+2|p|_{a}^{\prime}$. From Lemma 3.17 we easily deduce

so the contribution from all such dyadic scales on $\text{supp}(f)$ is bounded by

where we use $\delta>-3$ to control the source $f=O(A^{\delta/4}\ell^{\delta})$ in $L^{1}$.

We are left with one dyadic scale $|q|_{a}^{\prime}\sim|p|_{a}^{\prime}\lesssim A^{1/2}e^{\nu}$. By a similar argument, the contribution from sources at $A^{1/2}\lesssim|p-q|_{a}\lesssim 2|p|_{a}^{\prime}$ is bounded by $CA^{3\delta/4}\nu.$ If $\ell(p)>\frac{1}{2}A^{1/2}$, then the contribution from sources at $|p-q|_{a}\leq\frac{1}{4}A^{1/2}$ is controlled by $CA^{3\delta/4}$ using standard Schauder theory.

If $\ell(p)\leq\frac{1}{2}A^{1/2}$ and $|p-q|_{a}^{\prime}\lesssim A^{1/2}$, then for the purpose of estimating the convolution integral we can simply replace the Green kernel $G_{a}$ by $\frac{-1}{4\pi^{2}|(\mu_{1},\mu_{2},\eta)|_{a}^{2}}$, and correspondingly for their second derivatives. The point is that at this length scale the periodicity effect is secondary, and we are essentially in the same situation as Lemma 2.18 with $-3<\delta<0$ and $\tau=0$. A careful examination of that argument there, restoring the $A$-dependence, shows that the contribution of sources inside this region towards $\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f$ is bounded by $C(A^{1/4}\ell)^{\delta}.$

Combining the above shows the claim. ∎