A class of monotone decreasing rearrangements
type: Item
instance of: scholarly article
title: A class of monotone decreasing rearrangements
zbMATH Open document ID: 0714.28002
DOI: 10.1016/0022-247X(90)90209-X
author: Jay B. Epperson
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: Let \(\mu\) be a Borel measure on \({\mathbb{R}}\) with no pure point support. The monotone decreasing rearrangement of a Borel measurable function f: \({\mathbb{R}}\to {\mathbb{R}}\) with respect to \(\mu\) is defined by \[ f^*(x)=\inf \{y\in {\mathbb{R}}: \mu_ f(y)\leq \mu (-\infty,x)\},\quad x\in {\mathbb{R}}, \] where \(\mu_ f(y)=\mu \{u\in {\mathbb{R}}: f(u)>y\}\). It is easy to see that \(f^*\) is a decreasing, right-continuous function on \({\mathbb{R}}\), and that f and \(f^*\) are equidistributed, i.e. \(\mu_ f(x)=\mu_{f^*}(x)\) for all \(x\in X\). The study of monotone decreasing rearrangements was initiated by Hardy and Littlewood in the process of treatment of fractional integrals. The notion also appears in various problems of probability theory One of the main goals of the paper under review is to prove the following rearrangement inequality for the derivative. Theorem. Let F be an increasing convex function on \([0,\infty)\), with \(F(0)=0\), and let \(\mu\) be an absolutely continuous measure on \({\mathbb{R}}\) with a strictly positive, symmetric decreasing, log-concave density. Then for every Lipschitz continuous function f and every Borel set \(A\subseteq {\mathbb{R}}\), \[ (1)\quad \int_{f^{-1}(A)}F(| {\mathcal D}f|)d\mu \geq \int_{f^{*^{-1}}(A)}F(| {\mathcal D}f^*|)d\mu. \] A similar inequality for the Lebesgue measure on an interval has been obtained by \textit{G. F. D. Duff} [Can. J. Math. 22, 408-430 (1970; Zbl 0194.360)]. The theorem above widens considerably the class of measures for which inequality (1) holds, involving as a subclass the very important class of Gaussian measures.
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 26A16
Mathematics Subject Classification ID: 60A10
zbMATH DE Number: 4175301
zbMATH Keywords: monotone decreasing rearrangement of a Borel measurable function
zbMATH Keywords: rearrangement inequality for the derivative
zbMATH Keywords: log-concave density
zbMATH Keywords: Lebesgue measure
zbMATH Keywords: Gaussian measures
reviewed by: S. A. Chobanjan
MaRDI profile type: Publication
cites work: Q4039801
cites work: Rearrangements of functions and convergence in orlicz spaces
cites work: Integral Inequalities for Equimeasurable Rearrangements
cites work: A General Integral Inequality for the Derivative of an Equimeasurable Rearrangement
cites work: Symétrisation dans l'espace de Gauss.
cites work: Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes
cites work: Q5565773
cites work: Logarithmic Sobolev Inequalities
cites work: Rearrangements and convexity of level sets in PDE
cites work: Q3790372
Recommended article: Approximating the decreasing rearrangement
Recommended article: A certain class of completely monotonic sequences
Recommended article: Q5748502
Recommended article: Q3974738
Recommended article: Q3483838
Recommended article: On monotonicity of some combinatorial sequences
Recommended article: Q5740501
Recommended article: Juxtaposing Catalan permutation classes with monotone ones
Recommended article: Q3325775
Recommended article: A decreasing rearrangement for functions on homogeneous trees
description: scientific article; zbMATH DE number 4175301