Failure of Calder\'{o}n-Zygmund estimates for degenerate elliptic PDEs with $A_p$-weights when $p > 2$

TL;DR

This paper demonstrates that weighted Calderón-Zygmund estimates fail for certain degenerate elliptic PDEs with $A_p$ weights when $p > 2$, using convex integration techniques.

Contribution

It provides the first known examples of such failures specifically for $A_p$ weights with $p > 2$, advancing understanding of elliptic PDE regularity.

Findings

Weighted Calderón-Zygmund estimates do not hold for $A_p$ weights when $p > 2$.

Convex integration techniques can construct explicit counterexamples.

The results highlight limitations of regularity theory for degenerate elliptic PDEs.

Abstract

We use convex integration techniques to provide examples of failure of weighted Calder\'{o}n-Zygmund estimates for degenerate linear elliptic PDEs when the weights are in $A_p$, $p > 2$.