Differentiation bases and symmetric spaces (Q1813419)

Necessary and sufficient conditions are given for a differential basis \({\mathcal B}\) of measurable subsets of a cube \(Q\subset\mathbb{R}^ n\) to differentiate every function in a rearrangement invariant space \(X\) on \(Q\), and for this basis to have some natural covering property with respect to \(X\). These conditions are expressed in terms of boundedness of the maximal operator generated by \({\mathcal B}\), considered as an operator from \(X\) to some Marcinkiewciz space.