Weighted a priori estimates for elliptic equations

Abstract

We give a simpler proof of the a priori estimates obtained by Durán et al. (2008, 2010) 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 Rⁿ 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.