# Weighted $L_{p,q}$-estimates for higher order elliptic and parabolic systems with $\mathrm{BMO}_x$ coefficients on Reifenberg flat domains

## TL;DR

This paper establishes weighted $L_{p,q}$-estimates and solvability results for higher order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains, allowing for minimal regularity assumptions.

## Contribution

It introduces new weighted estimates for systems with coefficients having small mean oscillations and no regularity in time, expanding the understanding of such systems on irregular domains.

## Key findings

- Weighted $L_{p,q}$-estimates proved for systems with irregular coefficients
- Solvability established in weighted Sobolev spaces
- Applicable to domains with minimal regularity assumptions

## Abstract

We prove weighted $L_{p,q}$-estimates for divergence type higher order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains. In particular, in the parabolic case the coefficients do not have any regularity assumptions in the time variable. As functions of the spatial variables, the leading coefficients are permitted to have small mean oscillations. The weights are in the class of Muckenhoupt weights $A_p$. We also prove the solvability of the systems in weighted Sobolev spaces.