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

Abstract

We calculate the norm of the Fourier operator from L p (X) to L q ( 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 Rényi entropies are discussed.

Fourier transform acting on the functions defined in the infinite LCA groups. A Grand Lebesgue Spaces approach

“…Remark 3.1. It is easily to verify that the source estimated of Mokshay Madiman and Peng Xu [7] are the particular cases of obtained here (3.8) and (3.17), as long as the classical Lebesgue -Riesz spaces L p are the particular cases of Grand Lebesgue Spaces.…”

“…Denote following the authors of the very interest article [7] by K(p, q) the norm of the Fourier transform in the L q (Y ) → L p (X) sense:…”

“…Mokshay Madiman and Peng Xu in the article [7] calculated the exact value of the variable K(p, q). Namely, they proved that if (p, q) ∈ Q, then…”

An Exact Upper Bound on the $L^p$ Lebesgue Constant and The $\infty$-Rényi Entropy Power Inequality for Integer Valued Random Variables

“…In particular, for Euclidean space, the norm of FT from L p (R d ) to L p (R d ) with p the Hölder dual index of p and p ∈ (1,2] on Euclidean space is the content of Hausdorff-Young inequality, and the sharp constant is proven by Beckner in [5]. For some abstract LCA groups, the norm of FT is proven by Gilbert and Rzeszotnik in [8] for the case that the group is finite, and by Madiman and Xu in [15,21] for the case that the group is infinite and discrete or compact.…”