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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 Calderón–Zygmund operators with standard kernels in weighted Lebesgue and Orlicz spaces over a general domain.

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