  We study resolvent estimate and maximal regularity of the Stokes operator in
$L^q$-spaces with exponential weights in the axial directions of unbounded
cylinders of ${\mathbb R}^n,n\geq 3$. For straights cylinders we obtain these
results in Lebesgue spaces with exponential weights in the axial direction and
Muckenhoupt weights in the cross-section. Next, for general cylinders with
several exits to infinity we prove that the Stokes operator in $L^q$-spaces
with exponential weight along the axial directions generates an exponentially
decaying analytic semigroup and has maximal regularity.
  The proofs for straight cylinders use an operator-valued Fourier multiplier
theorem and techniques of unconditional Schauder decompositions based on the
${\mathcal R}$-boundedness of the family of solution operators for a system in
the cross-section of the cylinder parametrized by the phase variable of the
one-dimensional partial Fourier transform. For general cylinders we use cut-off
techniques based on the result for straight cylinders and the result for the
case without exponential weight.
