Extremizers of a Fourier uncertainty principle related to averaging

We study the uncertainty principle align* & μ(ξ) |ξ|^β_∞α (∫ |x|^α d μ)β ≥ Cα,β,dμTVα+βalign* for finite non-negative measures on Rᵈ. We prove that Cα ,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.