  Consider the time-periodic flow of an incompressible viscous fluid past a
body performing a rigid motion with non-zero translational and rotational
velocity. We introduce a framework of homogeneous Sobolev spaces that renders
the resolvent problem of the associated linear problem well posed on the whole
imaginary axis. In contrast to the cases without translation or rotation, the
resolvent estimates are merely uniform under additional restrictions, and the
existence of time-periodic solutions depends on the ratio of the rotational
velocity of the body motion to the angular velocity associated with the time
period. Provided that this ratio is a rational number, time-periodic solutions
to both the linear and, under suitable smallness conditions, the nonlinear
problem can be established. If this ratio is irrational, a counterexample shows
that in a special case there is no uniform resolvent estimate and solutions to
the time-periodic linear problem do not exist.
