Weighted a priori estimates for elliptic equations

MARÍA EUGENIA CEJAS; RICARDO DURÁN

STUDIA MATHEMATICA

POLISH ACAD SCIENCES INST MATHEMATICS

Año: 2018

0039-3223

We give a simpler proof of the a priori estimates obtained incite{DST1,DST2} for solutions of elliptic problems in weighted Sobolevnorms when the weights belong to the Muckenhoupt class $A_p$.The argument is a generalization to bounded domains of the one usedin $R^n$ to prove the continuity of singular integral operatorsin weighted norms.In the case of singular integral operators it is known thatthe $A_p$ condition is also necessary for the continuity. We do notknow whether this is also true for the a priori estimates in bounded domainsbut we are able to prove a weaker result when the operator is the Laplacianor a power of it. We prove that a necessary condition isthat the weight belongs to the local $A_p$ class.