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 M B constructed from an arbitrary quasidensity differential basis B is not compact if considered in a pair of weighted spaces ( X w , X v ) generated by X . For special differential bases that include convex quasidensity bases, we prove that M B is not compact in a pair of weighted spaces ( X w , X v ) 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 ∞).