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.
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Received: 12 October 2007
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Kunstmann, P.C. H ∞-calculus for the Stokes operator on unbounded domains. Arch. Math. 91, 178–186 (2008). https://doi.org/10.1007/s00013-008-2621-0
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DOI: https://doi.org/10.1007/s00013-008-2621-0