Hardy's theorem: a survey
by
Sundaram Thangavelu
Indian Statistical Institute, Bangalore, India.

A classical theorem of Hardy, proved way back in 1933, on Fourier transform pairs says that a nontrivial function f and its Fourier transform [^f] cannot have arbitrary Gaussian decay. More precisely, he proved that if f(x) = O(e^-ax2) and [^f](\xi) = O(e^-b\xi2) then f = 0 whenever ab > \frac14 and f is a constant multiple of e^-ax2 when ab = \frac14. This result is an instance of uncertainty principle which is also a characterisation of the heat kernel for the standard Laplacian.

In the recent past analogues of Hardy's theorem have been studied in various set-ups coming from noncommutative harmonic analysis. Thus there are analogues for the group Fourier transforms on semisimple and nilpotent Lie groups and also for the Helgason Fourier transforms on symmetric spaces and NA groups. In this talk we plan to survey some of the important results proved in this direction. The results are complete for SL(2, \R) , Heisenberg groups and rank one symmetric spaces but for other situations there are still open problems.

Date received: June 27, 2003