A note on decreasing rearrangement and mean oscillation on measure spaces

TL;DR

This paper establishes bounds relating the mean oscillation of a decreasing rearrangement of a function to the mean oscillation of the original function on measure spaces, with explicit dependence on the doubling constant in metric spaces.

Contribution

It provides new bounds on mean oscillation of rearranged functions in measure spaces, especially in doubling metric measure spaces, highlighting the dependence on the doubling constant.

Findings

Bounds on mean oscillation of $f^*$ in terms of $f$

Explicit dependence on doubling constant in metric spaces

Applicable to measure spaces with doubling measures

Abstract

We derive bounds on the mean oscillation of the decreasing rearrangement $f^*$ on $\mathbb{R}_+$ in terms of the mean oscillation of $f$ on a suitable measure space $X$. In the special case of a doubling metric measure space, the bound depends only on the doubling constant.