Extremizers of a Fourier uncertainty principle related to averaging

TL;DR

This paper investigates a Fourier uncertainty principle involving measures and establishes the existence, characterization, and explicit forms of extremizers, enhancing understanding of the principle's fundamental limits.

Contribution

It proves positivity and existence of extremizers for a Fourier uncertainty inequality and provides a detailed characterization and explicit descriptions for certain parameters.

Findings

Extremizers exist for all positive parameters.

Explicit extremizers are identified for specific cases.

Asymptotic behavior of extremizers is characterized.

Abstract

We study the uncertainty principle $$\lVert\widehat{\mu}(\xi) |\xi|^\beta\rVert_\infty^{\alpha} \left(\int |x|^\alpha d \mu\right)^{\beta} \geq C(\alpha,\beta,d){\lVert{\mu}\rVert_{TV}^{\alpha+\beta}}$$ for finite non-negative measures on $\mathbb{R}^d $. We prove that $C(\alpha,\beta,d)>0$ for all $\alpha,\beta>0$ and that extremizers exist. Moreover, we obtain an abstract characterization of the extremizers, which allows us to describe their asymptotic behavior and, for certain parameter values, to determine them explicitly.