  We study resolvent estimates and maximal regularity of the Stokes operator in
$L^q$-spaces with exponential weights in the axial directions of unbounded
cylinders of $\R^n,n\geq 3$. For a straight cylinder we use exponential weights
in the axial direction and Muckenhoupt weights in the cross-section. Next, for
cylinders with several exits to infinity we prove that the Stokes operator in
$L^q$-spaces with exponential weights generates an exponentially decaying
analytic semigroup and has maximal regularity.
  The proof for straight cylinders uses an operator-valued Fourier multiplier
theorem and 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 case without
exponential weight.
