Monotone rearrangement in averaging classes

TL;DR

This paper investigates how monotone rearrangement affects function classes defined by averages on [0,1], showing it does not increase their class constants, with applications to BMO and A_2 spaces.

Contribution

It establishes that monotone rearrangement preserves or reduces class constants across a broad range of average-based function classes, including BMO and A_2.

Findings

Monotone rearrangement does not increase class constants.

The result applies to classes including BMO and A_2.

Provides a unified approach for various average-based classes.

Abstract

We consider a general collection of function classes on the interval $[0,1]$ defined in terms of certain averages and show that monotone rearrangement does not increase the class constant in each case. The formulation includes BMO and $A_2$ with a special choice of the norm and, respectively, characteristic.