Uncertainty principles for integral operators

TL;DR

This paper establishes new uncertainty principles for integral operators with bounded kernels, extending classical results and applying them to Dunkl and Clifford Fourier transforms.

Contribution

It introduces novel uncertainty principles for integral operators with a Plancherel theorem, generalizing existing results and applying them to specific Fourier transforms.

Findings

Extended local uncertainty principle for integral operators.

Generalized uncertainty principle preventing simultaneous localization.

Derived a Heisenberg-type global uncertainty principle.

Abstract

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