On optimal autocorrelation inequalities on the real line

We study autocorrelation inequalities, in the spirit of Barnard and Steinerberger's work [1]. In particular, we obtain improvements on sharp constants some inequalities previously considered by these authors, also prove existence extremizers to certain specific settings. Our methods consist relating question other classical Fourier analysis, such as Hausdorff–Young inequality, employing functional analysis well measure theory tools connection a suitable dual version problem identify impose conditions extremizers.