The norm of the Fourier transform on compact or discrete abelian groups

TL;DR

This paper precisely calculates the Fourier transform norm on infinite compact or discrete abelian groups, extending classical inequalities and exploring the behavior of the norm across different function spaces.

Contribution

It generalizes the Hausdorff-Young inequality to a broader class of groups and identifies where the Fourier operator's norm becomes infinite, contrasting with finite group cases.

Findings

Identified the region in (p,q)-space with infinite Fourier norm.

Extended the sharp Hausdorff-Young inequality to infinite groups.

Discussed uncertainty principles using Rényi entropies.

Abstract

We calculate the norm of the Fourier operator from $L^p(X)$ to $L^q(\hat{X})$ when $X$ is an infinite locally compact abelian group that is, furthermore, compact or discrete. This subsumes the sharp Hausdorff-Young inequality on such groups. In particular, we identify the region in $(p,q)$-space where the norm is infinite, generalizing a result of Fournier, and setting up a contrast with the case of finite abelian groups, where the norm was determined by Gilbert and Rzeszotnik. As an application, uncertainty principles on such groups expressed in terms of R\'enyi entropies are discussed.