Three-dimensional rotating Navier-Stokes equations are considered with a constant Coriolis parameter $\Omega$ and
initial data nondecreasing at infinity. In contrast to the non-rotating case ($\Omega=0$), it is shown for the problem with rotation ($\Omega \neq 0$) that Green's function corresponding to the linear problem (Stokes + Coriolis combined operator) does not belong to $L^1({\mathbb R}^3)$.
Moreover, the corresponding integral operator is unbounded in the space $L^{\infty}_{\sigma}({\mathbb R}^3)$ of
solenoidal vector fields in ${\mathbb R}^3$ and the linear (Stokes+Coriolis) combined operator does not generate a semigroup in $L^{\infty}_{\sigma}({\mathbb R}^3)$.
Local in time unique solvability of the rotating Navier-Stokes equations is proven for initial velocity fields in the space
$L^{\infty}_{\sigma,a}({\mathbb R}^3)$ which consists
of $L^{\infty}$ solenoidal vector fields satisfying vertical averaging property such that their baroclinic component
belongs to a homogeneous Besov space
${\dot B}_{\infty,1}^0$ which is smaller than
$L^\infty$ but still contains various periodic and almost periodic functions. This restriction of initial data to $L^{\infty}_{\sigma,a}({\mathbb R}^3)$
which is a subspace of $L^{\infty}_{\sigma}({\mathbb R}^3)$
is essential for the combined linear operator (Stokes + Coriolis) to generate a semigroup. Using the rotation transformation, we also obtain local in time
solvability of the classical 3D Navier-Stokes equations
in ${\mathbb R}^3$ with initial velocity and vorticity
of the form
$\mbox{\bf{V}}(0)=\tilde{\mbox{\bf{V}}}_0(y) +
\frac{\Omega}{2} e_3 \times y$,
$\mbox{curl} \mbox{\bf{V}}(0)=\mbox{curl} \tilde{\mbox{\bf{V}}}_0(y) +
\Omega e_3$ where
$\tilde{\mbox{\bf{V}}}_0(y) \in L^{\infty}_{\sigma,a}({\mathbb R}^3)$.