  General evolution equations in Banach spaces are investigated. Based on an
operator-valued version of de Leeuw's transference principle, time-periodic
$L^p$ estimates of maximal regularity type are established from
$\mathscr{R}$-bounds of the family of solution operators
($\mathscr{R}$-solvers) to the corresponding resolvent problems. With this
method, existence of time-periodic solutions to the Navier-Stokes equations is
shown for two configurations: in a periodically moving bounded domain and in an
exterior domain, subject to prescribed time-periodic forcing and boundary data.
