Uncertainty principles like Hardy's theorem on some Lie groups

Abstract

We extend an uncertainty principle due to Cowling and Price to Euclidean spaces, Heisenberg groups and the Euclidean motion group of the plane. This uncertainty principle is a generalisation of a classical result due to Hardy. We also show that on the real line this uncertainty principle is almost equivalent to Hardy's theorem.1991 Mathematics subject classification (Amer. Math. Soc): primary 22E25, 22E30.

Variations on a theorem of Cowling and Price with applications to nilpotent Lie groups

“…The lemma shows that F = 0 whenever a t < (1 -e~4 b )/2. Since a < 2tanh2& we have coth2b < 2/a and so we can choose b\ and b 2 such that coth2fo < 4b { < 4b 2 The second claim is proved using the bound T*T < Ce~2 bH . We have With these notations we prove the following corollary of Theorem 1.4, which can be considered as the Cowling-Price theorem for the Laguerre expansion of polyradial functions.…”

“…For a > 1, y q endowed with the norm ||r||, := (tr(|r| 9 )) 1/9 is a complete subalgebra of the set of all bounded operators on L 2 (K"). In particular, for q = 2, y 2 This theorem is an uncertainty principle for the Weyl transform for the following reason. As is well known every T e S?i is of the form W(f) for some / e L 2 (C n ); that is to say…”

“…Consider functions / on K" that satisfy estimates of the form for some a, b > 0 where / is the Fourier transform of / defined by (1.1)12 S. Parui and S. Thangavelu [2] In 1983, Cowling and Price [5] obtained a generalization of Hardy's theorem. Let us define </>"(*) = e~a M \ Then we have the following result.…”

The uncertainty principle on Riemannian symmetric spaces of the noncompact type

“…These generalisations are the strong version of Morgan's theorem and the Cowling-Price theorem for R n , respectively (cf. [2]). …”

“…Remark 1.1. The two theorems above are not stated exactly in this form in [2]. However they may be derived rather easily from Beurling's theorem, namely Theorem 1.2 in [2].…”