  We establish an explicit Plancherel decomposition for
$\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$ where $E/F$ is a quadratic
extension of local fields of characteristic zero by making use of a local
functional equation for Asai $\gamma$-factors. We also give two applications of
this Plancherel formula: first to the global Ichino-Ikeda conjecture for
unitary groups by completing a comparison between local relative characters
that was left open by W. Zhang and secondly to the Hiraga-Ichino-Ikeda
conjecture on formal degrees in the case of unitary groups.
