Uncertainty principles for integral operators
Volume 220 / 2014
Abstract
The aim of this paper is to prove new uncertainty principles for integral operators ${\mathcal T}$ 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 $f\in L^2({\mathbb {R}}^d,\mu )$ is highly localized near a single point then ${\mathcal T} (f)$ 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 $f\in L^2({\mathbb {R}}^d,\mu )$ and its integral transform ${\mathcal T} (f)$ cannot both have support of finite measure. From these two results we deduce a global uncertainty principle of Heisenberg type for the transformation ${\mathcal T}$. We apply our results to obtain new uncertainty principles for the Dunkl and Clifford Fourier transforms.
