Differential properties of bases and halo problem for rearrangement-invariant spaces (Q1920859)

The author investigates connections among covering properties of differentiation bases, boundedness of the maximal operators associated to them, and the differentiation properties of bases. Roughly speaking, given a rearrangement-invariant space \(X\), he proves that a differentiation basis \(B\) differentiates \(X\) if and only if the maximal operator associated to \(B\) is bounded from \(X\) to the Marcinkiewicz space \(M(\psi_X)\), where the function \(\psi_X\) is given by means of the norm of the dilatation operator on \(X\). Using this result and the concept of the \((\phi,X)\)-covering property of \(B\) he presents necessary and sufficient conditions under which the Lorentz, Marcinkiewicz, and Orlicz spaces are differentiated by \(B\). The results are applied to solve the halo problem.