Weighted a priori estimates for elliptic equations

作者：Cejas Maria E^*; Duran Ricardo G

来源：Studia Mathematica, 2018, 243(1): 13-24.

DOI：10.4064/sm8704-6-2017

• elliptic equations
• weighted a priori estimates

We give a simpler proof of the a priori estimates obtained by Duran 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 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.

• 出版日期2018