Rearrangement methods in the work of Elliott Lieb

TL;DR

This paper reviews key topics in symmetric decreasing rearrangements, emphasizing Lieb's contributions to fundamental inequalities and properties relevant to analysis and mathematical physics.

Contribution

It provides a comprehensive overview of Lieb's work on rearrangement methods, including theorems and inequalities central to analysis.

Findings

Discussion of the Brascamp--Lieb--Luttinger theorem

Analysis of sharp Young and Hardy--Littlewood--Sobolev inequalities

Examination of the continuity of the rearrangement map on Sobolev spaces

Abstract

We review some topics in the theory of symmetric decreasing rearrangements with a particular focus on Lieb's fundamental contributions. Topics covered include the Brascamp--Lieb--Luttinger theorem, the sharp Young and Hardy--Littlewood--Sobolev inequalities, as well as the continuity of the rearrangement map on Sobolev spaces.