Weak-type boundedness of the Hardy-Littlewood maximal operator on weighted Lorentz spaces
type: Item
instance of: scholarly article
title: Weak-type boundedness of the Hardy-Littlewood maximal operator on weighted Lorentz spaces
zbMATH Open document ID: 1357.42014
author: Elona Agora
author: Jorge Antezana
author: María J. Carro
published in: The Journal of Fourier Analysis and Applications
publication date: 2017-01-10T00:00:00Z
review text: The paper deals with the boundedness of the Hardy--Littlewood maximal operator \(M\) on the weighted Lorentz spaces \(\Lambda^p_u(w)\) and \(\Lambda^{p,\infty}_u(w)\) defined by the norms \[ \| f \|_{\Lambda_u^p(w)}^p :=\int_0^\infty pt^{p-1} W(u(\{[f|>t\})) dt, \] \[ \|f\|_{\Lambda_u^{p,\infty}(w)}^p :=\sup_{t>0} t^p W(u(\{|f|>t\})), \] where \(W(t):=\int_0^t w(s)ds\) and \(u(E):=\int_E u(x)dx\). The main result shows that a condition that was previously known to characterise the boundedness of \(M:\Lambda^p_u(w)\to \Lambda^p_u(w)\) [\textit{M. J. Carro} et al., Mem. Am. Math. Soc. 877, 128 p. (2007; Zbl 1126.42005)] is also necessary for \(M:\Lambda^p_u(w)\to \Lambda^{p,\infty}_u(w)\) when \(1<p<\infty\), and therefore these strong and weak type bounds are in fact equivalent. The proof is based on clever estimates of the super level sets of \(M\chi_E\).
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 6673272
zbMATH Keywords: weighted Lorentz spaces
zbMATH Keywords: Hardy-Littlewood maximal operator
reviewed by: Tuomas Hytönen
MaRDI profile type: Publication
full work available at URL: s00041-015-9456-4
OpenAlex ID: W2256184619
cites work: Lorentz-Shimogaki and Boyd theorems for weighted Lorentz spaces
cites work: Maximal Functions on Classical Lorentz Spaces and Hardy's Inequality with Weights for Nonincreasing Functions
cites work: Q4039801
cites work: Weighted Lorentz spaces and the Hardy operator
cites work: Recent developments in the theory of Lorentz spaces and weighted inequalities
cites work: A new characterization of the Muckenhoupt $A_p$ weights through an extension of the Lorentz-Shimogaki Theorem
cites work: Some new functional spaces
cites work: On the theory of spaces \(\Lambda\)
cites work: Weighted Norm Inequalities for the Hardy Maximal Function
cites work: Weighted norm inequalities for averaging operators of monotone functions
DOI: 10.1007/S00041-015-9456-4
description: scientific article; zbMATH DE number 6673272