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 $M_\mathbf B$ constructed from an arbitrary quasidensity differential basis $\mathbf B$ is not compact if considered in a pair of weighted spaces $(X_w,X_v)$ generated by $X$. For special differential bases that includ $(X_w,X_v)$ 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 $(L^\infty,L^\infty)$.