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 $L^q(\hat{X})$ 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 Rényi entropies are discussed.

36 pages

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• Nazarov's uncertainty principles in higher dimension
• Some extensions of the uncertainty principle
• Forward and Reverse Entropy Power Inequalities in Convex Geometry
• Combinatorial Entropy Power Inequalities: A Preliminary Study of the Stam region
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