Let $F$ be a number field and $S$ a finite set of places including all infinity places. Let $f\in \mathcal{A}(F_\infty)$ be a rapidly decreasing function (in the usual sense), and $K\subseteq F_{S\setminus\infty}$ be a compact open subgroup. I would like to study the bound of the series $$\sum_{x\in\mathcal{O}_F^S\cap F^\times}f(\beta^{-1}x)\mathbf{1}_K(\beta^{-1}x)$$ in terms of the norm of $\beta$, where $\beta\in \mathcal{O}_F^S\cap F^\times$ and $\mathbf{1}_K$ is the indicator function of $K$.
If $S = \infty$ consists of only infinity places, then this is basically a sum of rapidly decreasing functions over a lattice minus origin, so a trivial estimate tells, for any $N\gg0$, that $$\sum_{x\in\mathcal{O}_F\cap F^\times}f(\beta^{-1}x)\ll_{f,N} \sum_{x\in\mathcal{O}_F\cap F^\times}\max_{v\mid \infty}|\beta|^N_v|x|_v^{-N} \ll \max_{v\mid\infty}|\beta|_v^N$$ Following are my two questions.
- I wonder if there is a similar bound for an arbitrary finite set of $S$, namely $$\sum_{x\in\mathcal{O}_F^S\cap F^\times}f(\beta^{-1}x)\mathbf{1}_K(\beta^{-1}x)\ll_{f,K,N} \max_{v\in S} |\beta|_v^N.$$
My naive try is to push everything to $F_\infty$: if we incorporate the finite constraint into the sum, actually we are considering the sum $$\sum_{x\in\Lambda_\beta\setminus\{0\}}f(\beta^{-1}x)$$ where $\Lambda_\beta\leq F_\infty$ is a lattice defined by $$\Lambda_\beta = F\cap\left(F_\infty\times \beta K\times \prod\limits_{v\not\in S}\mathcal{O}_{F_v}\right).$$ But I have no idea on how to grab that $\beta$ contribution from the above expression.
- Does a bound like $$\sum_{x\in\mathcal{O}_F^S\cap F^\times}f(\beta^{-1}x)\mathbf{1}_K(\beta^{-1}x)\ll_{f,K,N}|\beta|_{F_S}^N$$ have any chance to exist?
Here $|\beta|_{F_S} = \prod\limits_{v\in S}|\beta|_{v}$. Here the notation $f\ll_?g$ means there exists a constant $C>0$ depending only on ? (in some way) such that $|f(x)|\leq C|g(x)|$ for any $x$.
Any suggestion and comment will be appreciated. Thanks!