Maximal Lp -Lq regularity to the Stokes problem with Navier boundary conditions

We prove in this paper some results on the complex and fractional powers of the Stokes operator with slip frictionless boundary conditions involving the stress tensor. This is fundamental and plays an important role in the associated parabolic problem and will be used to prove maximal L^{p} - L^{q} regularity results for the non-homogeneous Stokes problem.