Mean oscillation bounds on rearrangements

TL;DR

This paper establishes improved bounds on the mean oscillation of rearranged functions in high dimensions, showing that the growth of constants is at most proportional to the square root of the dimension, using geometric and Calderón–Zygmund techniques.

Contribution

It provides the first dimension-dependent bounds for rearrangements of BMO functions, improving classical inequalities and introducing new geometric comparison methods.

Findings

Bound on $C_n$ grows at most as $oxed{ ext{constant} imes \sqrt{n}}$

Dimension-free Calderón–Zygmund decomposition for rectangles

First proof of BMO inequality for symmetric decreasing rearrangement

Abstract

We use geometric arguments to prove explicit bounds on the mean oscillation for two important rearrangements on $\mathbb{R}^n$. For the decreasing rearrangement $f^*$ of a rearrangeable function $f$ of bounded mean oscillation (BMO) on cubes, we improve a classical inequality of Bennett--DeVore--Sharpley, $\|f^*\|_{BMO(\mathbb{R}_+)}\leq C_n \|f\|_{BMO(\mathbb{R}^n)}$, by showing the growth of $C_n$ in the dimension $n$ is not exponential but at most of the order of $\sqrt{n}$. This is achieved by comparing cubes to a family of rectangles for which one can prove a dimension-free Calder\'{o}n--Zygmund decomposition. By comparing cubes to a family of polar rectangles, we provide a first proof that an analogous inequality holds for the symmetric decreasing rearrangement, $Sf$.