Minimal conditions for BMO

TL;DR

This paper establishes minimal integrability conditions using Luxemburg-type expressions related to generalized oscillations that ensure a function belongs to BMO, adaptable to various settings.

Contribution

It introduces a simple, sharp, and flexible method for characterizing BMO membership under minimal conditions across diverse mathematical contexts.

Findings

Derived minimal integrability conditions for BMO membership

Applicable to spaces of homogeneous type and non-doubling measures

Extended BMO definitions beyond classical cube bases

Abstract

We study minimal integrability conditions via Luxemburg-type expressions with respect to generalized oscillations that imply the membership of a given function $f$ to the space BMO. Our method is simple, sharp and flexible enough to be adapted to several different settings, like spaces of homogeneous type, non doubling measures on $\mathbb{R}^n$ and also BMO spaces defined over more general bases than the basis of cubes.