Extremizers of a Fourier uncertainty principle related to averaging
Abstract
Abstract We study the uncertainty principle $$ \begin{align*} & \lVert{\widehat{\mu}(\xi) |\xi|^\beta}\rVert_\infty^{\alpha} \left(\int |x|^\alpha d \mu\right)^{\beta} \geq C_{\alpha,\beta,d}{\lVert{\mu}\rVert_{TV}^{\alpha+\beta}}\end{align*} $$ 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.
Fields of Study & Subjects
Mathematical Analysis and Transform Methods
Stability and Controllability of Differential Equations
Spectral Theory in Mathematical Physics
Publication details
- Journal / source
- International Mathematics Research Notices
- Publisher
- Oxford University Press (OUP)
- Volume
- 2026
- Issue
- 11
- Publication date
- 6 May 2026
- Language
- EN