Extremizers of a Fourier uncertainty principle related to averaging

We study the uncertainty principle $$\parallel \widehat{\mu }(\xi )\mathrm{\mid }\xi {\mathrm{\mid }}^{\beta }{\parallel }_{\mathrm{\infty }}^{\alpha }{(\int \mathrm{\mid }x{\mathrm{\mid }}^{\alpha }d\mu )}^{\beta }\ge C(\alpha ,\beta ,d)\parallel \mu {\parallel }_{TV}^{\alpha +\beta }$$ 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.