Abstract
We show that the Stokes operator A on the Helmholtz space \({L^p_\sigma(\Omega)}\) for a bounded Lipschitz domain \({\Omega\subset\mathbb{R}^d}\), \({d \ge 3}\), has a bounded \({H^\infty}\)-calculus if \({\left|\frac{1}{p}-\frac{1}{2} \right|\le\frac{1}{2d}}\). Our proof uses a new comparison theorem for A and the Dirichlet Laplace \({-\Delta}\) on \({L^p(\Omega)^d}\), which is based on “off-diagonal” estimates of the Littlewood–Paley decompositions of A and \({-\Delta}\). This comparison theorem can be formulated for rather general sectorial operators and is well suited to extrapolate the \({H^\infty}\)-calculus from L 2(U) to the L p(U)-scale or part of it. It also gives some information on coincidence of domains of fractional powers.
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Dedicated to Jan Prüss on the occasion of his 65th birthday.
Both authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.
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Kunstmann, P.C., Weis, L. New criteria for the \({H^\infty}\)-calculus and the Stokes operator on bounded Lipschitz domains. J. Evol. Equ. 17, 387–409 (2017). https://doi.org/10.1007/s00028-016-0360-4
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DOI: https://doi.org/10.1007/s00028-016-0360-4