  The paper studies weak Paley-Wiener properties for group extensions by use of
Mackey's theory. The main theorem establishes sufficient conditions on the dual
action to ensure that the group has the weak Paley-Wiener property. The theorem
applies to yield the weak Paley-Wiener property for large classes of simply
connected, connected solvable Lie groups (including exponential Lie groups),
but also criteria for non-unimodular groups or motion groups.
