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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  ≥ 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 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.

Maximal regularity in exponentially weighted Lebesgue spaces of the Stokes operator in unbounded cylinders