  The description of the Paley-Wiener space for compactly supported smooth
functions $C^\infty_c(G)$ on a semi-simple Lie group $G$ involves certain
intertwining conditions that are difficult to handle. In the present paper, we
make them completely explicit for $G=\mathbf{SL}(2,\mathbb{R})^d$ ($d\in
\mathbb{N}$) and $G=\mathbf{SL}(2,\mathbb{C})$. Our results are based on a
defining criterion for the Paley-Wiener space, valid for general groups of real
rank one, that we derive from Delorme's proof of the Paley-Wiener theorem. In a
forthcoming paper, we will show how these results can be used to study
solvability of invariant differential operators between sections of homogeneous
vector bundles over the corresponding symmetric spaces.
