On compactness of maximal operators

Using a new approach, we show that, for any ideal space X with nonempty regular part, the maximal function operator MB constructed from an arbitrary quasidensity differential basis B is not compact if considered in a pair of weighted spaces (Xw, Xv) generated by X. For special differential bases that include convex quasidensity bases, we prove that MB is not compact in a pair of weighted spaces (Xw, Xv) generated by an arbitrary ideal space X. An example is given of a quasidensity differential basis such that the maximal function operator constructed from this basis is compact in (L∞, L∞).