Averaging operators on decreasing or positive functions: equivalence and optimal bounds
type: Item
instance of: scholarly article
title: Averaging operators on decreasing or positive functions: equivalence and optimal bounds
zbMATH Open document ID: 1412.26034
author: Santiago Boza
author: Javier Soria
published in: Journal of Approximation Theory
publication date: 2018-10-26T00:00:00Z
full work available at URL: http://hdl.handle.net/2117/131811
review text: The Hardy operator $S$ of a positive measurable function $f$ on $(0,\infty)$ is defined at point $x$ by the integral mean of $f(t)$ on the interval $[0,x]$, while its dual $S^*$ is the improper integral of $f(t)/t$ on $[x,\infty)$.\par The authors study optimal bounds for $S$-$I$ and $S^*$ (where $I$ is the identity operator) on the full range $1\le p\le\infty$ of $L^p$ functions for the case of decreasing, positive or general functions. There are many results, we state for example the following \begin{itemize} \item[1)] if $f$ is decreasing, then $\Vert Sf-f\Vert_p\le\frac{1}{p-1}\Vert f\Vert_p$ for $1<p<2$; \item[2)] if $2<p<\infty$ and $f$ is positive then $\Vert Sf-f\Vert_p\le\Vert f\Vert_p$; \item[3)] let $p\ge 2$ be a positive integer. Then for decreasing $f$, one has $\Vert Sf\Vert_p\le (p'/p)^{1/p}\cdot\Vert S^* f\Vert_p$, where $p'=p/(p-1)$. \end{itemize} All inequalities are sharp.
reviewed by: József Sándor
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 6966597
zbMATH Keywords: Hardy operator
zbMATH Keywords: dual operator
zbMATH Keywords: best constants
zbMATH Keywords: decreasing functions
MaRDI profile type: Publication
OpenAlex ID: W2891347206
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Wikidata QID: Q129209047
DOI: 10.1016/J.JAT.2018.09.001
description: scientific article; zbMATH DE number 6966597