An Analogue of Hardy’s Theorem for Very Rapidly Decreasing Functions on Semi-simple Lie Groups

A celebrated theorem of L. Schwartz asserts that a function f on R is ‘rapidly decreasing’ (or in the ‘Schwartz class’) iff its Fourier transform is ‘rapidly decreasing’. Since this theorem is of fundamental importance in harmonic analysis, there is a whole body of literature devoted to generalizing this result to other Lie groups. (For example, see [18].) In sharp contrast to Schwartz’s theorem, is a result due to Hardy [5] which says that f and f̂ cannot both be “very rapidly decreasing”. More precisely, if |f(x)| ≤ Ae−α|x|2 and |f̂(y)| ≤ Be−β|y| 2 and αβ > 1 4 , then f ≡ 0. (See [2], pp. 155-157.) However, as far as we are aware, until very recently no systematic attempt was made to generalize Hardy’s theorem to other Lie groups. In [12], [13], and [15], this result has been generalized to the Heisenberg groups Hn, the Euclidean motion groups M(n) and for certain eigenfunction expansions. In this paper we establish an analogue of Hardy’s theorem for a class of noncompact semisimple Lie groups and all symmetric spaces of the noncompact type. Hardy’s theorem can also be viewed as a sort of ‘Uncertainty Principle’. The results in [12] and [13] are presented from this point of view. (In [1], Cowling and Price have proved an “L − L” version of Hardy’s theorem on R. The theorem of Beurling in [9] is similar in spirit to Hardy’s theorem, although far more general, and indeed Hardy’s theorem, as well as the result of Cowling and Price, can be deduced from it as special cases.)