H ^∞-calculus for the Stokes operator on unbounded domains

Abstract.

We consider the Stokes operator A on unbounded domains $\Omega \subseteq {\mathbb{R}}^{n}$ of uniform C ^1,1-type. Recently, it has been shown by Farwig, Kozono and Sohr that – A generates an analytic semigroup in the spaces $\tilde{L}^{q}(\Omega)$, 1 < q < ∞, where $\tilde{L}^{q}(\Omega) = {L}^{q}(\Omega) \cap L^{2}(\Omega)$ for q ≥ 2 and $\tilde{L}^{q}(\Omega) = {L}^{q}(\Omega) + L^{2}(\Omega)$ for q ∈ (1, 2). Moreover, it was shown that A has maximal L ^p-regularity in these spaces for p ∈ (1,∞). In this paper we show that ɛ + A has a bounded H ^∞-calculus in $\tilde{L}^{q}(\Omega)$ for all q ∈ (1, ∞) and ɛ > 0. This allows to identify domains of fractional powers of the Stokes operator.