A new type of bmo space for non-doubling measures

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.