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 R n , n ≥ 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 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 onedimensional 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.

It is proved that the Stokes operator in L q -space on an infinite cylindrical domain of R n , n ≥ 3, with several exits to infinity generates a bounded and exponentially decaying analytic semigroup and admits a bounded H ∞ -calculus. For the resolvent estimates, the Stokes resolvent system with a prescribed divergence in an infinite straight cylinder with bounded cross-section is studied in L q (R; L r ω ( )) where 1 < q, r < ∞ and ω ∈ A r (R n−1 ) is an arbitrary Muckenhoupt weight. The proofs use cut-off techniques and the theory of Schauder decomposition of UMD spaces based on R-boundedness of operator families and on square function estimates involving Muckenhoupt weights.

No abstract

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 ≥ . 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 in nity 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-o techniques based on the result for straight cylinders and the case without exponential weight.

Abstract. Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinderAs a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.