Uncertainty principles of Ingham and Paley–Wiener on semisimple Lie groups

Abstract

Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. Viewing these results as uncertainty principles for Fourier transforms, we prove certain analogues of these results on connected, noncompact, semisimple Lie groups with finite center. We also use these results to show unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on Riemmanian symmetric spaces of noncompact type.2010 Mathematics Subject Classification. Primary 22E30; Secondary 22E46, 43A80.

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.…”

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

“…Remark 2.4. It is easy to see that a special case of Theorem 1.4, namely when f ∈ C ∞ c (G//K), can be proved simply by using the slice projection theorem (see [6] for a more general result in this direction). However, this approach cannot be used to prove Theorem 1.4, because if an integrable K-biinvariant function f vanishes on an open set then it is not necessarily true that Af also vanishes on an open subset of a.…”

Theorems of Chernoff and Ingham for certain eigenfunction expansions

“…In the one dimensional case, this problem has been addressed by Ingham [10], Levinson [14], Paley and Wiener [15,16] and their results are in terms of non integrability of ψ(t) t −2 over [1, ∞). Recently this problem has received considerable attention and several versions have been proved in the contexts of R n , nilpotent Lie groups and compact and non-compact Riemannian symmetric spaces, see the works [2], [3], [4], [5] of Bhowmik, Pusti, Ray and Sen in various combinations of authorship. Recently, in a joint work with Bagchi and Sarkar [1] we have proved an analogue of Ingham's theorem for the operator valued Fourier transform on the Heisenberg group.…”