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.