  Time-periodic solutions to the linearized Navier-Stokes system in the
$n$-dimensional whole-space are investigated. For time-periodic data in
$L^q$-spaces, maximal regularity and corresponding a priori estimates for the
associated time-periodic solutions are established. More specifically, a Banach
space of time-periodic vector fields is identified with the property that the
linearized Navier-Stokes operator maps this space homeomorphically onto the
$L^q$-space of time-periodic data.
