Weighted norm inequalities for maximal operators and Pisier's theorem on factorization through \(L^{p{\infty}}\)
type: Item
instance of: scholarly article
title: Weighted norm inequalities for maximal operators and Pisier's theorem on factorization through \(L^{p{\infty}}\)
zbMATH Open document ID: 0782.47027
DOI: 10.1007/BF01193770
published in: Integral Equations and Operator Theory
publication date: 1992-07-23T00:00:00Z
review text: Through a suite of theorems, propositions, lemmas, corollaries and remarks, the author proves a set of inequalities. Let us consider, as an example, the fractional maximal operator \(M_ \alpha\) defined by \[ M_ \alpha f(x)= \sup_ Q| Q|^{\alpha/(n-1)} \int_ Q | f| dx,\quad x\in \mathbb{R}^ n, \] where \(f\in L^ p\), \(Q\) is a cube in \(\mathbb{R}^ n\) containing \(x\) with the sides parallel to the coordinate axes, \(| Q|\) is the finite Borel measure of \(Q\) and \(0<\alpha< n/p\). In these circumstances the inequality \[ \int (M_ \alpha f)^ q d\omega\leq C\bigl(\int | f|^ p dx\bigr)^{q/p} \] holds iff \[ \sup_{x\in Q}\bigl(| Q|_ \omega/| Q|^{1-\alpha p/n}\bigr)\in L^{q/(p-q)}(\omega), \] with \(0< q< p<\infty\), \(\omega\) being an arbitrary measure.
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 45P05
zbMATH DE Number: 39670
zbMATH Keywords: inequalities
zbMATH Keywords: fractional maximal operator
author: Igor E. Verbitsky
reviewed by: I. Gottlieb
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01193770
OpenAlex ID: W2058142178
description: scientific article; zbMATH DE number 39670