On the non-compactness of maximal operators.

Abstract

It is proved that if \(B\) is a convex quasi-density basis and \(E\) is a symmetric space on \(\mathbb{R}^n\) with respect to Lebesgue measure, then there do not exist non-orthogonal weights \(w\) and \(v\) for which the maximal operator \(M_B\) corresponding to \(B\) acts compactly from the weight space \(E_w\) to the weight space \(E_v\).