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

TL;DR

This paper characterizes the continuity of a Luxemburg maximal operator linked to a critical radius function using Dini conditions, establishing new inequalities and boundedness results in Orlicz and Zygmund spaces.

Contribution

It introduces a Dini type condition to analyze the continuity of maximal operators associated with critical radius functions, providing new inequalities and boundedness results.

Findings

Weak Fefferman-Stein type inequalities established

Weak weighted modular estimates derived

Boundedness of Hardy-Littlewood maximal function in Zygmund spaces proven

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.