Bounded imaginary powers and H∞-calculus of the Stokes operator in two-dimensional exterior domains

The present contribution deals with the Stokes operator A(q) on L-sigma(q) (Omega), 1 < q < infinity, where Omega is an exterior domain in R-2 of class C-2. It is proved that A(q) admits a bounded H-infinity-calculus. This implies the existence of bounded imaginary powers of Aq, which has several important applications. - So far this property was only known for exterior domains in R-n, n >= 3. - In particular, this shows that A(q) has maximal regularity on L-sigma(q)(Omega). For the proof the resolvent (lambda + A(q))(-1) has to be analyzed for |lambda| --> infinity and lambda --> 0. For large lambda this is done using an approximate resolvent based on the results of [ 3], which were obtained by applying the calculus of pseudodifferential boundary value problems. For small lambda we analyze the representation of the resolvent developed in [11] by a potential theoretical method.