The $\Delta_{a}$-harmonic function $\alpha_{2},\alpha_{3}$ are both of order $O(\frac{A^{1/4}}{|\vec{\mu}|_{a}})$. The function

$$\alpha_{1}-\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}=-\frac{1}{2\pi\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}\arctan(\frac{\sqrt{A}\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}{a_{22}\mu_{2}+a_{12}\mu_{1}})$$

is also $\Delta_{a}$-harmonic, and by the Taylor expansion of $\arctan$ is seen to be $O(\frac{A^{1/4}}{|\vec{\mu}|_{a}})$ as well. These functions are smooth on the base in the region (2.14) with regularity scale $O(|\vec{\mu}|_{a})$. The $\Delta_{a}$-harmonicity takes care of all higher order estimates. ∎