Resolvent Estimates and Maximal Regularity in Weighted L q -spaces of the Stokes Operator in an Infinite Cylinder

Abstract

Let $${\Omega = \Sigma \times \mathbb{R}}$$ be an infinite cylinder of $${\mathbb{R}^n}$$ , n ≥ 3, with a bounded cross-section $${\Sigma \subset \mathbb{R}^{n-1}}$$ of C 1,1-class. We study resolvent estimates and maximal regularity of the Stokes operator in $${L^{q}(\mathbb{R}; L^{r}_{\omega}(\Sigma))}$$ for 1 < q, r < ∞ and for arbitrary Muckenhoupt weights ω ∈ A r with respect to x′ ∈ Σ. The proofs 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 Σ parametrized by the phase variable of the one-dimensional partial Fourier transform.