Resolvent estimates in L∞ for the Stokes operator in nonsmooth domains

We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain Ω in Rd under the assumptions that Ω is C1 for d≥3 and Lipschitz for d=2. As a corollary, it follows that the Stokes operator generates a uniformly bounded analytic semigroup in the spaces of bounded solenoidal functions in Ω. The smoothness conditions on Ω are sharp. The case of exterior domains with nonsmooth boundaries is also studied. The key step in our proof involves new estimates that connect the pressure to the gradient of the velocity in the Lq average, but only on scales above certain level.