A theorem of Levinson for Riemannian symmetric spaces of noncompact type

Abstract

A classical result of N. Levinson characterizes the existence of a nonzero integrable function vanishing on a nonempty open subset of the real line in terms of the pointwise decay of its Fourier transform. We prove an analogue of this result for Riemannian symmetric spaces of noncompact type.

Analogs of certain quasi-analiticity results on Riemannian symmetric spaces of noncompact type

“…This motivated us to have a fresh look at these results and explore the possibility of extending them beyond Euclidean spaces. Very recently we could extend the main result of [28] for Riemannian symmetric spaces X (see [5]) and in this paper our objective is to do the same with the result of Ingham [23]. A recently proved analogue of the result of Ingham on R d states the following.…”

“…We refer the reader to Theorem 4.2 for an exact analogue of Theorem 1.1. Recently we could prove a result analogous to Theorem 1.4 for Riemannian symmetric spaces X (see [5,Theorem 1.2]) where the function λ B θ(λ) was assumed to be increasing (as in [28]). It turns out that with some effort, one can construct a function θ satisfying the conditions of Theorem 1.4 such that λ θ(λ) is decreasing on an unbounded set of positive Lebesgue measure.…”

A local Levinson theorem for compact symmetric spaces

“…Most of these results deal with functions defined on the circle or on the real line. Very recently, we have extended few results of this genre in the context of higher dimenional Euclidean spaces and on certain classes of noncommutative Lie groups and the corrsponding homogeneous spaces [1,2,3]. For the d-dimensional torus T d , d ≥ 1, we have the following generalization of Theorem 1.1.…”

“…We shall now reduce the general case to the case of K-biinvariant functions by using the radialization operator S. The idea is same as we did in [2] (see step 3 of the proof of Theorem 1.2 in [2]). If possible, let f ∈ C ∞ (U/K) be a nonzero function supported on Exp(B(0, r)) and satisfies the estimate (1.5).…”

“…If f ∈ L 1 (T d ) we then define its Fourier coefficient Ff (m) by the formula It is therefore natural to explore the possibility of extending Theorem 1.1 to more general Riemannian manifolds. Recently, we have obtained an anlogue of Theorem 1.1 for Riemannian symmetric spaces of non-compact type [2]. Our aim in this paper is to do the same for Riemannian symmetric spaces of compact type which is dual of noncompact symmetric spaces.…”