Equality in Hausdorff-Young for Hypergroups

TL;DR

This paper investigates the conditions under which equality holds in the Hausdorff-Young inequality for Fourier transforms on commutative hypergroups, extending classical results and exploring uncertainty principles.

Contribution

It characterizes functions attaining equality in the Hausdorff-Young inequality and analyzes the basic uncertainty principle on commutative hypergroups with non-trivial centers.

Findings

Characterization of functions achieving equality in Hausdorff-Young inequality.

Extension of Fourier transform domain to $L^p(K)$ for $1 \\leq p \\leq 2$.

Analysis of uncertainty principles on hypergroups.

Abstract

It has been shown in "On the Hausdorff-Young theorem for commutative hypergroups" by Sina Degenfeld-Schonburg, that one can extend the domain of Fourier transform of a commutative hypergroup $K$ to $L^p(K)$ for $1\leq p \leq 2$, and the Hausdorff-Young inequality holds true for these cases. In this article, we examine the structure of non-zero functions in $L^p(K)$ for which equality is attained in the Hausdorff-Young inequality, for $1<p<2$, and further provide a characterization for the basic uncertainty principle for commutative hypergroups with non-trivial centre.