  We prove existence of a solution to the divergence equation satisfying a new
Bogovski-type estimate for the difference quotients. This enables us to give an
alternative proof of the interior regularity of the solution to the $p$-Stokes
problem, completely avoiding the pressure. Moreover, as a key preliminary
result we prove boundedness of Calder\'on-Zygmnud operators with standard
kernels in weighted Lebesgue and Orlicz spaces over a general domain.
