Differential representation of measures and functions
type: Item
instance of: scholarly article
title: Differential representation of measures and functions
zbMATH Open document ID: 0909.46026
author: Juan-Miguel Gracia
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-04-08T00:00:00Z
review text: Let \({\mathcal M}\) denote the vector space of all Radon measures \(\mu\) on \(\mathbb{R}^n\) with the property that \(p_m(\mu)= \int| x|^m d|\mu|(x)<\infty\) for every \(m\geq 0\), and let \({\mathcal M}_r\) be the Banach space of those Radon measures \(\mu\) for which \(\|\mu\|_{{\mathcal M}_r}= \int(1+| x|)^{r-1}d|\mu|(x)< \infty\). A Radon measure \(\mu\) is called \(r\)-oscillating if \(\mu\in{\mathcal M}_r\) and \(\int x^\alpha d\mu(x)= 0\) for every multiindex \(\alpha= (\alpha_1,\alpha_2,\dots,\alpha_n)\) with \(|\alpha|< r\). Among other results obtained the authors show that if \(r\) is a positive integer, \(\mu\in{\mathcal M}\) and \(\mu\) is \(r\)-oscillating then \(\mu\) can be represented in the form \(\sum_{|\alpha|= r}\partial^\alpha R_\alpha(\mu)\), where \(R_\alpha:{\mathcal M}\to{\mathcal M}\) is the linear operator defined by the formula \[ \langle R_\alpha(\mu), f\rangle= (-1)^{|\alpha|} {|\alpha|\over\alpha!} \int^1_0 (1-t)^{|\alpha|- 1}\int x^\alpha f(tx)d\mu(x)dt \] valid for every continuous function \(f\) on \(\mathbb{R}^n\) with compact support.
reviewed by: Ludvík Janoš
Mathematics Subject Classification ID: 46E27
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 28C05
zbMATH DE Number: 1003958
zbMATH Keywords: differential representation of measures and functions
zbMATH Keywords: vector space of all Radon measures
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5251
OpenAlex ID: W1978210025
DOI: 10.1006/JMAA.1997.5251
description: scientific article; zbMATH DE number 1003958