Dyadic-BMO functions, the dyadic Gurov-Reshetnyak conditions on $[0,1]^n$ and rearrangements of functions

TL;DR

This paper introduces dyadic BMO functions on the unit cube, examines their rearrangements in BMO space, and explores functions satisfying the dyadic Gurov-Reshetnyak condition, focusing on their integrability properties.

Contribution

It defines dyadic BMO and Gurov-Reshetnyak classes on $[0,1]^n$ and analyzes the rearrangement and integrability properties of these functions.

Findings

Rearrangements of dyadic BMO functions lie in BMO((0,1])

Dyadic Gurov-Reshetnyak functions exhibit specific integrability properties

New insights into the structure of dyadic BMO and Gurov-Reshetnyak classes

Abstract

We introduce the space of dyadic bounded mean oscillation functions $f$ defined on $[0,1]^n$ and study the behavior of the nonincreasing rearrangement of $f$, as an element of the space $BMO((0,1])$. We also study the analogous class of functions that satisfy the dyadic Gurov-Reshetnyak condition and look upon their integrability properties.