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Journal of Evolution Equations

, Volume 17, Issue 1, pp 387–409 | Cite as

New criteria for the \({H^\infty}\)-calculus and the Stokes operator on bounded Lipschitz domains

  • Peer Christian Kunstmann
  • Lutz WeisEmail author
Article

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.

Keywords

Sectorial operators bounded \({H^\infty}\)-calculus Littlewood–Paley operators domains of fractional powers Stokes operator 

Mathematics Subject Classification

47 A 60 47 D 06 35 Q 30 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Institute for AnalysisKarlsruhe Institute of Technologie (KIT)KarlsruheGermany

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