Rearrangement and polarization

TL;DR

This paper introduces a new perspective on rearrangements of measurable functions, focusing on polarization, and provides structural insights into measure-preserving maps and symmetrization processes.

Contribution

It offers a novel approach to rearrangements and characterizes polarization among them, with new results on measure-preserving maps and symmetrization techniques.

Findings

New structural results on measure-preserving maps on convex bodies

Characterization of polarization as a special rearrangement

Enhanced understanding of symmetrization processes

Abstract

The paper has two main goals. The first is to take a new approach to rearrangements on certain classes of measurable real-valued functions on $\mathbb{R}^n$. Rearrangements are maps that are monotonic (up to sets of measure zero) and equimeasurable, i.e., they preserve the measure of super-level sets of functions. All the principal known symmetrization processes for functions, such as Steiner and Schwarz symmetrization, are rearrangements, and these have a multitude of applications in diverse areas of the mathematical sciences. The second goal is to understand which properties of rearrangements characterize polarization, a special rearrangement that has proved particularly useful in a number of contexts. In order to achieve this, new results are obtained on the structure of measure-preserving maps on convex bodies and of rearrangements generally.