Rearrangement and polarization
arXiv:1912.07498 · doi:10.1016/j.aim.2020.107380
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 . 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.
44 pages. In version 2 two minor mistakes in the proofs of Lemma 6.3 and Theorem 6.6 have been corrected. Version 3 is the one accepted for publication in Advances in Mathematics