The exact continuation of a reverse Hölder inequality and Muckenhoupt's conditions
type: Item
instance of: scholarly article
title: The exact continuation of a reverse Hölder inequality and Muckenhoupt's conditions
zbMATH Open document ID: 0807.42015
author: V. Pereyra
published in: Mathematical Notes
publication date: 1994-07-21T00:00:00Z
review text: Let \(\varphi\) be a convex downward positive function on \((0,+\infty)\) and for a measurable nonnegative function \(f\) on \([a,b]\), \[ S(\varphi,f, [a,b])= \sup_{a\leq \alpha\leq \beta\leq b}\left(\int^ b_ a \varphi(f)dx\right)\left(\varphi\left(\int^ b_ a f dx\right)\right)^{-1}, \] where \(\int^ b_ a f dx= (b- a)^{- 1}\int^ b_ a f dx\). It is proved that \[ S\bigl(\varphi, f^*,[0,b- a]\bigr)\leq S\bigl(\varphi,f, [a,b]\bigr), \] where \(f^*\) is a nonincreasing rearrangement of \(f\). Several important corollaries of this result are derived, among them the following one. Denote for \(1< q<+\infty\) by \(A_ q(M)\) the class of all functions satisfying Muckenhoupt's \(A_ q\)-condition on \([a,b]\) with constant \(M> 0\). The number \(r= r(q,M)> 1\) is found such that for \(r< p< q\) for some \(N\), \(A_ q(M)\subset A_ p(N)\) and for any \(N\), \(A_ q(M)\not\subset A_ r(N)\).
reviewed by: V. I. Burenko
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 578708
zbMATH Keywords: rearrangement
zbMATH Keywords: reverse Hölder inequality
zbMATH Keywords: Muckenhoupt's \(A_ q\)- condition
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 578708