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

## TL;DR

This paper introduces solenoidal difference quotients and uses them to establish interior regularity results for the $p$-Stokes system, providing an alternative approach that avoids pressure terms and extends to weighted spaces.

## Contribution

It develops a new Bogovski-type estimate for difference quotients and applies it to prove regularity of the $p$-Stokes system without pressure dependence.

## Key findings

- Existence of divergence solutions with new estimates
- Interior regularity of $p$-Stokes solutions proven without pressure
- Boundedness of Calderón-Zygmund operators in weighted spaces

## 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.