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Journal of Lie Theory 25 (2015), No. 4, 1125--1137
Copyright Heldermann Verlag 2015

A Beurling Theorem for Exponential Solvable Lie Groups

Ahmad M. A. Alghamdi
Dept. of Mathematical Science, Faculty of Applied Science, Umm Alqura University, P. O. Box 14035, Makkah 21955, Saudi Arabia

Ali Baklouti
Dept. of Mathematics, Faculty of Sciences, Sfax University, Route de Soukra, 3000 Sfax, Tunisia

We prove in this paper an L2-version of Beurling's theorem for an arbitrary exponential solvable Lie group G with a non-trivial center, which encompasses the setting of nilpotent connected and simply connected Lie groups. When G has a trivial center, the uncertainty principle may fail to hold and an example is produced. The representation theory and a localized Plancherel formula are fundamental tools in the proof.

Keywords: Uncertainty principle, Fourier transform, Plancherel formula.

MSC: 22E25; 43A25

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