Whittaker functions estimates proof

Question

I am reading the proof of the estimates of Whittaker functions from Jacquet, Hervé, Ilja Iosifovitch Piatetski-Shapiro, and Joseph Shalika. "Automorphic forms on GL (3) I." Annals of Mathematics 109.1 (1979): 169-212. I am referring to Proposition 2.2 (page 181). I don't understand a step in the proof of Lemma 2.2.1 - how from the equality $$\begin{equation} \phi \left( a_1, \dots , a_{n-1}, bc \right) = \sum \eta\left(b\right) \left( A_\eta \phi \right) \left( a_1, a_2, \dots, a_{n-1}, c \right) \end{equation}$$ for a fixed $ c $ and all $b$ with $ \left| b \right| \le 1$ they conclude that $$\begin{equation} \phi \left( a_1, \dots , a_n \right) = \sum_{j, \eta} \eta\left( a_n \right) \phi_j \left( a_n \right) \phi \left( a_1, a_2, \dots, a_{n-1}, b_j \right) \end{equation}$$ for all $ a_1, \dots, a_n \in F^{\times} $, where $ b_j \in F^{\times}$ are constants depending on $ \phi $ only, and $ \phi_j $ are Schwartz functions.

I understand why this is true for $ a_n $ with $ \left| a_n \right| \le \left| c \right| $ if one extends the set $ Y $ of finite functions such that it spans the translation spaces of its elements, but I don't understand how to deduce this equality in the case that some $ \left| a_i \right| $ $(i \ne n)$ is small and $\left| a_n \right| > \left| c \right|$.

My original thought was that the difference between the functions can be thought as a Schwartz function on $ F^n $, but I'm not sure what happens when $ a_i $ is close to zero ($i \ne n$).

Thanks

Answer 1

I found out that in Cogdell, James W., and Nadir Matringe. "The functional equation of the Jacquet-Shalika integral representation of the local exterior-square L-function." arXiv preprint arXiv:1406.1935 (2014)., [Page 9, Lemma 3.4], they prove a similar statement. In their statement they assume that functions are not only smooth, but uniformly smooth. This assumption is fine since this is the case in the Whittaker model. Using this assumption, it is not hard to show that for $ \left| a_n \right| > \left| c \right|$ such an equality holds.