Weighted a priori estimates for elliptic equations

TL;DR

This paper presents a simplified proof of weighted a priori estimates for elliptic equations in bounded domains, extending techniques from singular integral operators and exploring conditions on weights in the Muckenhoupt class.

Contribution

The authors provide a more straightforward proof of weighted a priori estimates and analyze necessary conditions on weights for elliptic problems in bounded domains.

Findings

Simplified proof of weighted a priori estimates in bounded domains.

Identification of local $A_p$ class as a necessary condition for weights.

Extension of techniques from singular integral operators to elliptic equations.

Abstract

We give a simpler proof of the a priori estimates obtained in the paper by Duran, Sanmartino and Toschi for solutions of elliptic problems in weighted Sobolev norms when the weights belong to the Muckenhoupt class $A_p$. The argument is a generalization to bounded domains of the one used in $\mathbb{R}^n$ to prove the continuity of singular integral operators in weighted norms. In the case of singular integral operators it is known that the $A_p$ condition is also necessary for the continuity. We do not know whether this is also true for the a priori estimates in bounded domains but we are able to prove a weaker result when the operator is the Laplacian or a power of it. We prove that a necessary condition is that the weight belongs to the local $A_p$ class.