Weak Paley-Wiener property for completely solvable Lie groups (Q1306242)

Let \(G\) denote a connected, simply connected, completely solvable Lie group with unitary dual \(G^\wedge\), and \(\varphi\) a measurable, bounded and compactly supported function on \(G\). The author proves the following weak Paley-Wiener type theorem: If the group Fourier transform of \(\varphi\) vanishes on a subset of \(G^\wedge\) having positive Plancherel measure then \(\varphi=0\) almost everywhere on \(G\). -- For the case of nilpotent Lie groups the author had previously proved the theorem [J. Lie Theory 5, 165-172 (1995; Zbl 0851.43001)].