Monotonic rearrangements of functions with small mean oscillation

TL;DR

This paper establishes precise bounds for the monotonic rearrangement operator acting on classes like BMO and Muckenhoupt, using a geometric approach called alpha-extension to connect dyadic and continuous settings.

Contribution

It introduces a novel geometric method, alpha-extension, to derive sharp bounds for the rearrangement operator on BMO and Muckenhoupt classes.

Findings

Sharp bounds for the rearrangement operator on BMO and Muckenhoupt classes.

Connection between dyadic and continuous classes via alpha-extension.

Geometric construction simplifies the analysis of rearrangement bounds.

Abstract

We obtain sharp bounds for the monotonic rearrangement operator from "dyadic-type" classes to "continuous". In particular, for the $\mathrm{BMO}$ space and Muckenhoupt classes. The idea is to connect the problem with a simple geometric construction named $\alpha$-extension.