Cases of equality in the Riesz rearrangement inequality
type: Item
instance of: scholarly article
title: Cases of equality in the Riesz rearrangement inequality
zbMATH Open document ID: 0876.26016
DOI: 10.2307/2118534
author: Almut Burchard
published in: Annals of Mathematics. Second Series
publication date: 1997-11-12T00:00:00Z
review text: The Riesz rearrangement inequality of the title is the assertion that for nonnegative measurable functions \(f\), \(g\), \(h\) in \(\mathbb{R}^n\), we have \[ \int_{\mathbb{R}^n\times\mathbb{R}^n} f(y)g(x- y)h(x)dx dy\leq \int_{\mathbb{R}^n\times\mathbb{R}^n} f^*(y)g^*(x- y)h^*(x)dx dy, \] where \(*\) denotes symmetric decreasing rearrangement in \(\mathbb{R}^n\). For \(n=1\) the inequality is due to F. Riesz in 1930, and for general \(n\) to Sobolev in 1938. The paper under review is devoted to the identification of the triples \(f\), \(g\), \(h\) for which equality holds. The author gives a complete description when \(f\), \(g\), \(h\) are characteristic functions of sets \(A\), \(B\), and \(C\) of finite positive measure: If the volume radius of one of the sets, say \(C\), is \(\geq\) the sum of the volume radii of \(A\) and \(B\), then equality holds if \(A+B\subset C\), modulo sets of measure zero. But if the volume radius of each of the sets is less than the sum of the volume radii of the other two, then \(A\), \(B\), and \(C\) must be suitable similitudes of a single ellipsoid. The former case follows fairly easily from known results related to the Brunn-Minkowski inequality, but proof of the latter case requires a good deal with inventiveness with symmetrization techniques and some new hands-on criteria for deciding when certain types of sets are ellipsoids. Via ``layer-cake'' representations [see, for example the book ``Analysis'', by \textit{E. H. Lieb} and \textit{M. Loss}, Am. Math. Soc. (1996)] and her characterization in the characteristic function case, the author deduces that for general functions \(f\), \(g\), \(h\), if at least two of the symmetric decreasing rearrangements are strictly decreasing, then, except in trivial cases, equality holds in Riesz's inequality iff there is a measure-preserving linear transformation \(T\) and constants \(a\), \(b\), \(c\) with \(c=a+b\), such that \(f= f^*\circ(T+ a)\), \(g= g^*\circ(T+ b)\), and \(h= h^*\circ(…
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 906874
zbMATH Keywords: Riesz rearrangement inequality
zbMATH Keywords: decreasing rearrangement
zbMATH Keywords: measure-preserving linear transformation
zbMATH Keywords: optimizers
zbMATH Keywords: weak-type Young inequality
zbMATH Keywords: symmetrization
reviewed by: Albert Baernstein II
MaRDI profile type: Publication
OpenAlex ID: W1987278905
description: scientific article; zbMATH DE number 906874