A sharp inequality for the square function
type: Item
instance of: scholarly article
title: A sharp inequality for the square function
zbMATH Open document ID: 0639.42017
DOI: 10.1215/S0012-7094-87-05542-6
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Let f *, S(f), and Mf, respectively, denote the dyadic maximal, square, and Hardy-Littlewood maximal functions of f. For Q a dyadic cube and V a non-negative function in \(L\) \(1_{loc}({\mathbb{R}}^ d\)), define \(Y(Q,V)=\int_{Q}(M(\chi_ QV)/\int_{Q}V)\int_{Q}V>0\), \(Y(Q,V)=1,\int_{Q}V=0\). Let \(\psi\) :[0,\(\infty)\mapsto [1,\infty)\) be increasing, with \(\psi (2x)\leq C_ 1\psi (x)\) for all x. Define \[ M_{\psi}V(x)=\sup_{Q\ni x,Qdyadic}\psi (\log Y(Q,V))\frac{1}{| Q|}\int_{Q}V. \] We show that if \(\sum^{\infty}_{k=0}\frac{1}{\psi (k)}\leq 1\) then (*) \(\int | f\quad *|^ 2Vdx\leq C(C_ 1,d)\int S^ 2(f)M_{\psi}Vdx\) for all \(f\in C_ 0^{\infty}({\mathbb{R}}^ d\)) and weights V. We show that (*) fails, for any \(C(C_ 1,d)<\infty\), if \(\sum 1/\psi (k)=\infty\). We obtain as a corollary a strong sufficient condition for the two-weight inequality \(\int | f\quad *|^ 2Vdx\leq \int S^ 2(f)Wdx\) to hold.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 60G42
zbMATH DE Number: 4042516
zbMATH Keywords: square function
zbMATH Keywords: weighted norm inequality
zbMATH Keywords: dyadic maximal function. Hardy-Littlewood maximal functions
author: J. Michael Wilson
reviewed by: Stephen M. J. Wilson
MaRDI profile type: Publication
cites work: Some weighted norm inequalities concerning the Schrödinger operators
cites work: The uncertainty principle
cites work: Q5607484
cites work: Weighted inequalities for the dyadic square function without dyadic \(A_{\infty}\)
full work available at URL: s0012-7094-87-05542-6
OpenAlex ID: W2053455886
description: scientific article; zbMATH DE number 4042516