On the Fourier transform of the symmetric decreasing rearrangements

TL;DR

This paper explores inequalities relating the Fourier transforms of functions and their symmetric decreasing rearrangements, providing new bounds, extending existing results, and applying findings to solutions of the free Schrödinger equation.

Contribution

It introduces new rearrangement inequalities for Fourier transforms, extends Lieb's smoothness result, and applies these to analyze solutions of the free Schrödinger equation.

Findings

Fourier transform of a function over small sets is controlled by its rearrangement.

Extension of Lieb's smoothness result for rearranged functions.

Application to solutions of the free Schrödinger equation.

Abstract

Inspired by work of Montgomery on Fourier series and Donoho-Strak in signal processing, we investigate two families of rearrangement inequalities for the Fourier transform. More precisely, we show that the $L^2$ behavior of a Fourier transform of a function over a small set is controlled by the $L^2$ behavior of the Fourier transform of its symmetric decreasing rearrangement. In the $L^1$ case, the same is true if we further assume that the function has a support of finite measure. As a byproduct, we also give a simple proof and an extension of a result of Lieb about the smoothness of a rearrangement. Finally, a straightforward application to solutions of the free Shr\"odinger equation is given.