Extremizers of a Fourier uncertainty principle related to averaging

Abstract

We study the uncertainty principle $$\Vert \hat{\mu }(\xi )|\xi {|}^{\beta }{\Vert }_{\infty }^{\alpha }{\left(\int |x{|}^{\alpha }d\mu \right)}^{\beta }\ge C(\alpha ,\beta ,d)\Vert \mu {\Vert }_{TV}^{\alpha +\beta }$$ 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.

30 pages