Characterization of the continuity properties of maximal operators associated to critical radius functions via Dini type conditions

Abstract

We give a characterization of the continuity properties of a Luxemburg maximal type operator associated to a critical radius function $\rho$ between Orlicz spaces. This goal is achieved by means of a Dini type condition that includes certain Young functions related to the maximal operator and the spaces involved. Our results provide not only weak Fefferman-Stein type inequalities but also a weak weighted estimate of modular type for the considered operators, which is interesting in its own right. On the other hand, we prove the boundedness of the Hardy-Littlewood maximal function associated to $\rho$ between Zygmund spaces of $L\,\log\,L$ type with $A_p$ weights.