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

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.