Weighted inequalities for the dyadic square function without dyadic \(A_{\infty}\)
type: Item
instance of: scholarly article
title: Weighted inequalities for the dyadic square function without dyadic \(A_{\infty}\)
zbMATH Open document ID: 0639.42016
DOI: 10.1215/S0012-7094-87-05502-5
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Let \(f^*\) and \(S(f)\) denote, respectively, the dyadic maximal and square functions of f. We prove weighted norm inequalities of the form \[ \int | f\quad *|^ 2Vdx\leq C(\tilde M)\int S^ 2(f)\tilde MVdx, \] valid for all \(f\in C_ 0^{\infty}({\mathbb{R}}^ d\)) and non-negative \(V\in L\) \(1_{loc}({\mathbb{R}}^ d\)), where \(\tilde M\) is an appropriate ``maximal function''. In general, the \(\tilde MV\)'s we consider do not have the (dyadic) Muckenhoupt \(A_{\infty}\) property. In addition, we give a necessary condition and a similar-looking sufficient condition for the two-weight inequality \(\int | f^*|^ 2Vdx\leq C\int S^ 2(f)Wdx\) to hold.
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 4042515
zbMATH Keywords: dyadic maximal function
zbMATH Keywords: square functions
zbMATH Keywords: weighted norm inequalities
zbMATH Keywords: maximal function
author: J. Michael Wilson
reviewed by: Stephen M. J. Wilson
MaRDI profile type: Publication
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cites work: The uncertainty principle
cites work: Q3864736
cites work: Q5607484
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description: scientific article; zbMATH DE number 4042515