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Resolvent Estimates and Maximal Regularity in Weighted Lebesgue Spaces of the Stokes Operator in Unbounded Cylinders

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.