# New criteria for the $H^\infty$-calculus and the Stokes operator on bounded Lipschitz domains

Kunstmann, Peer Christian; Weis, Lutz

##### Abstract:
We show that the Stokes operator A on the Helmholtz space Lp (Ω) for a bounded Lipschitz domain Ω ⊂ Rd, d ≥ 3, has a bounded H ∞- calculus if |1p − 1/2| ≤ 1/2d . Our proof uses a new comparison theorem A and the Dirichlet Laplace −∆ on Lp(Ω)d, which is based on “off-diagonal” estimates of the Littlewood-Paley decompositions of A and −∆. This comparison theorem can be formulated for rather general sectorial operators and is well suited to extrapolate the H ∞-calculus from L2(U ) to the Lp(U )-scale or part of it. It also gives some information on coincidence of domains of fractional powers.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht Jahr 2016 Sprache Englisch Identifikator ISSN: 2365-662X urn:nbn:de:swb:90-605607 KITopen-ID: 1000060560 Verlag KIT, Karlsruhe Umfang 25 S. Serie CRC 1173 ; 2016/26 Schlagworte sectorial operators, bounded H ∞-calculus, Littlewood-Paley operators, domains of fractional powers, Stokes operator
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