A new type of bmo space for non-doubling measures

TL;DR

This paper introduces a new bmo space for non-doubling measures that is larger than existing spaces, establishes its properties, and shows it coincides with previously known rbmo spaces.

Contribution

The paper defines a new bmo space for non-doubling measures, proves its equivalence with existing rbmo spaces, and develops new methods for analyzing these function spaces.

Findings

The new bmo space is larger than the existing rbmo space.

The four norms of the new space are shown to be equivalent.

The new space coincides with Dachun Yang's rbmo space.

Abstract

Let $\mu$ be a Radon measure on $\mathbb R^{d}$ which may be non-doubling and only satisfies $\mu(Q(x,l))\le C_{0}l^{n}$} for all $x\in \mathbb R^{d}$, $l(Q)>0$, with some fixed constants $C_{0}>0$ and $n\in (0,d]$. We introduce a new type of $bmo(\mu)$ space which looks bigger than the $rbmo(\mu)$ space of Dachun Yang (JAMS,\,2005). And its four equivalent norms are established by constructing some special types of auxiliary doubling cubes. Then we further obtain that this new $rbmo(\mu)$ space actually coincides with the $rbmo(\mu)$ space of Dachun Yang.