  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.
