Hardy’s Uncertainty Principle on semisimple groups

Abstract

A theorem of Hardy states that, if f is a function on R such that |f (x)| ≤ C e −α|x| 2 for all x in R and |f (ξ)| ≤ C e −β|ξ| 2 for all ξ in R, where α > 0, β > 0, and αβ > 1/4, then f = 0. Sitaram and Sundari generalised this theorem to semisimple groups with one conjugacy class of Cartan subgroups and to the K-invariant case for general semisimple groups. We extend the theorem to all semisimple groups.

On Theorems of Hardy, Gelfand–Shilov and Beurling for Semisimple Groups

“…It is interesting to note that the conditions (i) and (ii) allow us to factorisef (λ, σ) as above with A α independent of λ and σ. We show that all the earlier versions of Hardy's theorem proved by Sitaram-Sundari [35], Cowling-Sitaram-Sundari [9], Sengupta [31], Narayanan-Ray [24] can be deduced from the above result. In particular we have the following result for right K-invariant functions on a semisimple group of rank one.…”

“…(In this paper we are only concerned with this principal series representations. As observed in [9], for a very rapidly decreasing function, the group Fourier transform being zero on all the principal series coming from the minimal parabolic subgroup necessarily implies that it is identically zero on the unitary dual.) If x = kak is the polar decomposition of x ∈ G then we define |x| = | log a|.…”

“…The first version of Hardy's theorem for semisimple Lie groups with one conjugacy class of Cartan subgroups was proved in Sitaram-Sundari [35]. Later this condition was removed by Cowling et al in [9]. Another proof was given by Sengupta [31].…”

Beurling's theorem for Riemannian symmetric spaces II

“…Some of the latter theorems (which follow from Beurling's) were proved independently on symmetric spaces in recent years by many authors (see [15,3,12,14,16,6], etc.). The statements of those theorems and proofs of the above implications in the case of rank 1 symmetric spaces can be found in [13].…”

$L^p$ versions of Hardy’s uncertainty principle on hyperbolic spaces

“…Hardy's uncertainty principle (see [8]) tells us, however, that they cannot both be very rapidly decreasing. Generalisations to a L p set-up have been studied and proved by, among others, Beurling ( [12]) and Cowling-Price: Analogues of Hardy's uncertainty principle and its L p versions for the Fourier transform on (semisimple) Lie groups have been the object of interest in several recent papers; see [5], [15] and the references therein. The Riemannian symmetric spaces of the non-compact type have also been studied; see [17] and [18].…”