Weighted a priori estimates for elliptic equations
Volume 243 / 2018
Studia Mathematica 243 (2018), 13-24
MSC: Primary 42B37, 35J48; Secondary 42B20.
DOI: 10.4064/sm8704-6-2017
Published online: 8 February 2018
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 . 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.
