Uncertainty principles for integral operators

Abstract

The aim of this paper is to prove new uncertainty principles for integral operators T with bounded kernel for which there is a Plancherel Theorem. The first of these results is an extension of Faris's local uncertainty principle which states that if a nonzero function f ∈ L2 (ℝd ; μ) is highly localized near a single point then T (f ) cannot be concentrated in a set of finite measure. The second result extends the Benedicks-Amrein-Berthier uncertainty principle and states that a nonzero function f ∈ L2 (ℝd ; μ) and its integral transform T (f ) cannot both have support of finite measure. From these two results we deduce a global uncertainty principle of Heisenberg type for the transformation T. We apply our results to obtain new uncertainty principles for the Dunkl and Clifford Fourier transforms.