Solenoidal difference quotients and their application to the regularity theory of the $p$-Stokes system

Abstract

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.