Uncertainty Principles for Integral Operators

The aim of this paper is to prove new uncertainty principles for an integral operator T with a 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 ∈ L(R, 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 ∈ L(R, 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 a new uncertainty principles for the Dunkl and Clifford Fourier transforms.