  We study the spatial decay of time-periodic Navier-Stokes flow at the rate
$|x|^{-1}$ with/without wake structure in 3D exterior domains when a rigid body
moves periodically in time. In this regime the existence of time-periodic
solutions was established first in the 2006 paper by Galdi and Silvestre,
however, with little information about spatial behavior at infinity so that
uniqueness of solutions was not available. This latter issue has been addressed
by Galdi, who has recently succeeded in construction of a unique time-periodic
solution with spatial behavior mentioned above if translational and angular
velocities of the body fulfill, besides smallness and regularity, either of the
following assumptions: (i) translation or rotation is absent; (ii) both
velocities are parallel to the same constant vector. This paper shows the
existence of a unique time-periodic Navier-stokes flow in the small with values
in the weak-$L^3$ space and then deduces the desired pointwise decay of the
solution under some condition on the rigid motion of the body, that covers the
cases (i), (ii) mentioned above.
