Uncertainty principles for integral operators

Abstract

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

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• Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2)
• Infinite Determinantal Measures and The Ergodic Decomposition of Infinite Pickrell Measures II. Convergence of determinantal measures
• The Ergodic Decomposition of Infinite Pickrell Measures III. The infinite Bessel process as the limit of radial parts of finite-dimensional projections of infinite Pickrell measures
• Fourier-like multipliers and applications for integral operators
• Sharp Pitt inequality and logarithmic uncertainty principle for Dunkl transform in $L^2$
• A new construction of the Clifford-Fourier kernel
• Uncertainty principles for the Clifford-Fourier transform
• The kernel of the generalized Clifford-Fourier transform and its generating function