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