The Stokes Operator with Power Weights Outside the Muckenhoupt Class

In this paper, we prove estimates for the Stokes resolvent problem with no-slip boundary conditions on the half space in weighted $L^p$-spaces.

The weights we consider are power weights both inside and outside the Muckenhoupt range.

Our estimates imply that the corresponding Stokes operator is the generator of a bounded analytic $C_0$-semigroup.

We furthermore show that it admits a bounded $H^\infty$-calculus of angle $0.$ This results seems to be new even within the Muckenhoupt range.