Extremizers of a Fourier uncertainty principle related to averaging

Key Points

• This research investigates the existence and characterization of extremizers for a Fourier uncertainty principle involving finite non-negative measures.
• Analyzed the uncertainty principle relating Fourier transforms of measures to their norms.
• Established conditions under which extremizers exist for various parameters.
• Characterized the asymptotic behavior and explicit forms of extremizers.
• Proved positivity of the constant C associated with the uncertainty principle for all considered parameters.
• Demonstrated that extremizers exist under specific conditions and provided an abstract characterization.
• Identified explicit extremizers for certain parameter values.