On the geometric mean operator
type: Item
instance of: scholarly article
title: On the geometric mean operator
zbMATH Open document ID: 0815.47041
author: Luboš Pick
author: Bohumír Opic
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-06-25T00:00:00Z
review text: The authors give a characterization of pairs of weights \((u,v)\) such that the geometric mean operator \(Gf(x)= \exp((1/x) \int_ 0^ x \log f(t) dt)\), defined for \(f>0\) almost everywhere on \((0,\infty)\), is bounded from \(L_{p,v} (0,\infty)\) to \(L_{q,u} (0,\infty)\), where \(0<q< p\leq\infty\). Their proofs are based on the simple observation that in the case \(v\equiv 1\) and \(p>1\) the good weights for \(G\) coincide with those good for the averaging operator \(Af(x)= (1/x) \int_ 0^ x f(t)dt\). Their result applies to a certain independence on \(p\), \(q\) of weighted \(L_ p- L_ q\) inequalities involving the operator \(A\).
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47A63
zbMATH DE Number: 599926
zbMATH Keywords: geometric mean operator
zbMATH Keywords: averaging operator
zbMATH Keywords: \(L_ p- L_ q\) inequalities
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1172
OpenAlex ID: W2080670856
DOI: 10.1006/JMAA.1994.1172
description: scientific article; zbMATH DE number 599926