Let $G$ be a simple algebraic group defined over $\mathbb Q$. In their Corvallis article (automorphic forms and automorphic representations), Borel and Jacquet define an automorphic form to be a smooth function on $G(\mathbb A)$ which is left-$G(\mathbb Q)$-invariant, satisfies some "finiteness" conditions, and also the following moderate growth condition:

- For every $y\in G(\mathbb A)$, the map $x\mapsto f(x\cdot y)$ on $G(\mathbb R)$ is slowly increasing.

I have a couple of questions about this condition:

Assume that $f$ appears in the discrete spectrum (but is not necessarily a cusp form). Is it true that the map $x\mapsto f(x\cdot y)$ is bounded on unipotent subgroups of $G(\mathbb R)$?

What can be said about the "growth" of the map $x\mapsto f(x\cdot y)$ for a fixed $y\in G(\mathbb A)$ when $x$ varies in $G(\mathbb Q_p)$? For example, will this map be bounded on unipotent subgroups of $G(\mathbb Q_p)$?