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On Hardy’s uncertainty principle for solvable locally compact groups. (English) Zbl 1187.22004

Hardy’s uncertainty principle says that a non-zero integrable function \(f\) on \(\mathbb R^n\) and its Fourier transform \(\widehat f\) cannot both be very rapidly decreasing, more exactly, if \(|f(x)|\leq C\exp{(-\alpha\|x\|^2)}\) and \(\|\widehat f(y)\|\leq C\exp{(-\beta\|y\|^2)}\) for some \(\alpha>0\), \(\beta>0\) such that \(\alpha\beta>1/4\) then \(f=0\).
Analogues of this principle are established for locally compact abelian groups and for some classes of solvable Lie groups, such as diamond groups and exponential Lie groups with non-trivial center.

MSC:

22E25 Nilpotent and solvable Lie groups
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
43A80 Analysis on other specific Lie groups
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