Extremizers of a Fourier uncertainty principle related to averaging

We study the uncertainty principle $$\lVert\widehatμ(ξ) |ξ|^β\rVert_\infty^α \left(\int |x|^αd μ\right)^β \geq C(α,β,d){\lVertμ\rVert_{TV}^{α+β}}$$ for finite non-negative measures on $\mathbb{R}^d $. We prove that $C(α,β,d)>0$ for all $α,β>0$ 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.