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

Fabio Berra, Marilina Carena, Gladis Pradolini

TL;DR

The paper characterizes the continuity properties of Luxemburg-type maximal operators associated to a critical radius function within Orlicz and Zygmund spaces, via a Dini-type condition. It develops equivalences that connect this Dini-type control to boundedness and modular-type inequalities, including Fefferman-Stein-type results, for M_η^{ρ,σ}. It also proves weak-type modular bounds and shows boundedness of the associated Hardy-Littlewood maximal operator on L^{Φ}(w) spaces under A_p^{ρ} weights. Overall, it extends prior Luxemburg maximal operator theory to settings with critical radii, providing a robust framework for weighted continuity across generalized function spaces.

Abstract

We give a characterization of the continuity properties of a Luxemburg maximal type operator associated to a critical radius function $ρ$ 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 $ρ$ between Zygmund spaces of $L\,\log\,L$ type with $A_p$ weights.

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

TL;DR

The paper characterizes the continuity properties of Luxemburg-type maximal operators associated to a critical radius function within Orlicz and Zygmund spaces, via a Dini-type condition. It develops equivalences that connect this Dini-type control to boundedness and modular-type inequalities, including Fefferman-Stein-type results, for M_η^{ρ,σ}. It also proves weak-type modular bounds and shows boundedness of the associated Hardy-Littlewood maximal operator on L^{Φ}(w) spaces under A_p^{ρ} weights. Overall, it extends prior Luxemburg maximal operator theory to settings with critical radii, providing a robust framework for weighted continuity across generalized function spaces.

Abstract

We give a characterization of the continuity properties of a Luxemburg maximal type operator associated to a critical radius function 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 between Zygmund spaces of type with weights.

Paper Structure

This paper contains 5 sections, 14 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and definitions
  3. Proof of Theorem \ref{['teo: acotacion de M en LpLogLq']}
  4. Proof of Theorem \ref{['teo: tipo debil modular de MPhi,rho']}
  5. Proof of Theorem \ref{['teo: equivalencias']}

Key Result

Theorem 1.1

Let $\eta$ be a normalized and differentiable Young function in $\Delta_2$. Let $a,b,\phi$ and $\psi$ functions defined as in eq: definicion de phi y psi. Then the following statements are equivalent:

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 3.1
  • Proposition 3.2
  • proof : Proof of Theorem \ref{['teo: acotacion de M en LpLogLq']}
  • Proposition 4.1
  • Proposition 4.2
  • ...and 16 more