Let $F$ be a Schwarz-Bruhat function on $\mathbb A = \mathbb A_{\mathbb Q}$, i.e. a finite sum of products of the form $\prod\limits_{p \leq \infty} f_p$, where $f_{\infty}$ is a Schwarz function on $\mathbb R$, and $f_p$ for $p < \infty$ is a locally constant and compactly supported function on $\mathbb Q_p$.
I'm trying to understand the convergence of the sum of integrals (over $\mathbb A^{\ast}/\mathbb Q^{\ast}$)
$$\int\limits_{ \substack{a \in \mathbb A^{\ast}/\mathbb Q^{\ast} \\ |a| \geq 1}} \sum\limits_{0 \neq \eta \in \mathbb Q} F(a \eta)|a| d^{\ast} a + \int\limits_{|a| \leq 1} \sum\limits_{0 \neq \eta}\hat{F}(a^{-1} \eta)d^{\ast}a $$
$$+ \hat{F}(0) \int\limits_{|a| \leq 1}(1-\chi_c(|a|^{-1}))d^{\ast}a - F(0)\int\limits_{|a| \leq 1}|a| d^{\ast}a.$$
Here $c > 1$ is a real number, $\chi_c$ is the characteristic function of $[c,\infty)$, and $\hat{F}(a) = \int\limits_{\mathbb A} F(y) \psi(ay)dy$ is the Fourier transform, where $\psi$ is a nontrivial character of $\mathbb A/\mathbb Q$. It is also a Schwarz-Bruhat function.
These appear in the geometric side of Arthur's modified trace formula for $\operatorname{GL}(2)$ (Corvallis I, Forms of GL(2) from the analytic point of view, page 236), in the proof of the convergence of the truncated kernels.
The last two integrals I understand. For example, the second to last term is
$$\hat{F}(0) \int\limits_{\mathbb A^{\ast 1}/\mathbb Q^{\ast}} \int_1^c \frac{dt}{t} d^{\ast}a = \hat{F}(0) \textrm{vol}(\mathbb A^{\ast 1}/\mathbb Q^{\ast}) \log(c).$$
How can one establish the convergence of the first two integrals? I would appreciate any hint in this direction.