### An ${L}^{q}\left(L\xb2\right)$-theory of the generalized Stokes resolvent system in infinite cylinders

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with $\Sigma \subset {\mathbb{R}}^{n-1}$, a bounded domain of class ${C}^{1,1}$, are obtained in the space ${L}^{q}(\mathbb{R};L\xb2\left(\Sigma \right))$, q ∈ (1,∞). As 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.