Reverse Hölder inequalities with constant close to 1
type: Item
instance of: scholarly article
title: Reverse Hölder inequalities with constant close to 1
zbMATH Open document ID: 0746.30022
author: Ingemar Wik
published in: Ricerche di Matematica
publication date: 1992-06-27T00:00:00Z
review text: The main result of the paper is: Let \(f\) be a nonnegative function in \(L^ p(\mathbb{R}^ n)\) \((p>1)\) satisfying: \[ \int_ Q\kern-1.25em-f^ pdx\leq(1+\delta)(\int_ Q\kern-1.125em-f dx)^ p,\quad\delta>0\tag{*} \] for every cube \(Q\) with sides parallel to the axes. Then there exist constants \(c\) and \(c'\) depending on \(p\) and \(n\) such that (1) \(f\) belongs to \(L^ r_{loc}\) for every \(r<c/\sqrt\delta\), (2) for fixed \(r\) and every cube \(Q\) it results: \(\int_ Q\kern-1.25em- f^ rdx\leq(1+c'\sqrt\delta)(\int_ Q\kern-1.25em- f dx)^ r\). Results (1) and (2) give precise information on what happens to functions satisfying the reverse Hölder inequality (*) when \(\delta\) tends to zero. The nonincreasing rearrangement method is used for the proof.
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 30C70
zbMATH DE Number: 26525
zbMATH Keywords: order of integrability
zbMATH Keywords: nonincreasing rearrangement
zbMATH Keywords: Gehring class
reviewed by: Giovanni Porru
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 26525