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Weak Type Boundedness of the Hardy Littlewood Maximal Operator on Weighted Lorentz Spaces

Elona Agora, Jorge Antezana, María J. Carro

Abstract

The main goal of this paper is to provide a complete characterization of the weak-type boundedness of the Hardy-Littlewood maximal operator, $M$, on weighted Lorentz spaces $Λ^p_u(w)$, whenever $p>1$. This solves a problem left open in \cite{crs:crs}. Moreover, with this result, we complete the program of unifying the study of the boundedness of $M$ on weighted Lebesgue spaces and classical Lorentz spaces, which was initiated in the aforementioned monograph.

Weak Type Boundedness of the Hardy Littlewood Maximal Operator on Weighted Lorentz Spaces

Abstract

The main goal of this paper is to provide a complete characterization of the weak-type boundedness of the Hardy-Littlewood maximal operator, , on weighted Lorentz spaces , whenever . This solves a problem left open in \cite{crs:crs}. Moreover, with this result, we complete the program of unifying the study of the boundedness of on weighted Lebesgue spaces and classical Lorentz spaces, which was initiated in the aforementioned monograph.
Paper Structure (2 sections, 6 theorems, 44 equations)

This paper contains 2 sections, 6 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Proof of the main result

Key Result

Theorem 1.1

For every $0<p<\infty$, is bounded if and only if there exists $q\in(0,p)$ such that, for every finite family of cubes $(Q_j)_{j=1}^J$, and every family of measurable sets $(S_j)_{j=1}^{J}$, with $S_j\subset Q_j$, for every $j$, we have that for some universal positive constant $C$ depending only on $p$ and the dimension.

Theorems & Definitions (9)

  • Theorem 1.1: crs:crs, Theorem 3.3.5
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Proposition 2.4
  • proof : Proof of Theorem \ref{['David y Goliath']}