A characterization of some weighted inequalities for the vector-valued weighted maximal function
type: Item
instance of: scholarly article
title: A characterization of some weighted inequalities for the vector-valued weighted maximal function
zbMATH Open document ID: 1228.42026
author: Cuilan Wu
publication date: 2011-03-02T00:00:00Z
review text: The author obtains necessary and sufficient conditions of weighted weak and strong type norm inequalities for vector-valued weighted maximal functions. More precisely, let \(f=\{f_k\}_1^\infty\) be a sequence of locally integrable functions on \(\mathbb R^n\), \(|f(x)|_r= (\sum_1^\infty| f(x)|^r)\) and \(M\omega f(x)= \sup_{x\in Q}\int_Q|f(y)|\omega(y)\,dy\). The author proves that {\parindent=6,5mm \begin{itemize}\item[(1)] There is constant \(C_{r,p}\) such that \[ u\big(\{x\in\mathbb R^n:| M\omega f(x)|_r>\lambda\}\big)\leq C_{r,p} \lambda^{-p} \int_{\mathbb R^n}|f(y)|_r^p u(y)\,dy \] if and only if \(u\in A_{p}(\omega)\). \item[(2)] There is a constant \(C_{r,p}\) such that \[ \int_{\mathbb R^n}|M_\omega f(y)|_r^p u(y)\,dy\leq C_{r,p} \int_{\mathbb R^n}|f(y)|_r^p u(y)\,dy \] if and only if \(u\in A_{p}(\omega)\). \end{itemize}} The case \(\omega(x)=1\) was proved by \textit{K. F. Andersen} and \textit{R. T. John} [Stud. Math. 69, 19--31 (1980; Zbl 0448.42016)].
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 5859392
zbMATH Keywords: \(A_p(\omega)\) weights
zbMATH Keywords: vector-valued maximal function
zbMATH Keywords: weighted norm inequalities
reviewed by: Joaquim Martin
MaRDI profile type: Publication
full work available at URL: s10114-010-8009-7
OpenAlex ID: W2397222704
cites work: Weighted Norm Inequalities for the Hardy Maximal Function
cites work: Q3154241
cites work: Q4655912
cites work: Weighted inequalities for vector-valued maximal functions and singular integrals
cites work: Q3142876
cites work: Some Maximal Inequalities
DOI: 10.1007/S10114-010-8009-7
published in: Acta Mathematica Sinica, English Series
Recommended article: A characterization of some weighted norm inequalities for maximal operators
Recommended article: Weighted inequalities for a vector-valued strong maximal function
Recommended article: Weighted and vector-valued inequalities for one-sided maximal functions
Recommended article: Q3786940
Recommended article: Q5186138
Recommended article: Q3125080
Recommended article: Sharp weighted inequalities for the vector-valued maximal function
Recommended article: On weighted norm inequalities for the maximal function
Recommended article: Weighting inequalities for one-sided vector-valued maximal operators with respect to a function
Recommended article: A characterization of some weighted norm inequalities for the fractional maximal function
description: scientific article; zbMATH DE number 5859392